Patent Publication Number: US-2023142054-A1

Title: Displaying three-dimensional objects

Description:
INCORPORATION BY REFERENCE 
     The present application is a continuation of, and claims benefit under 35 USC §120 to, international applications PCT/US2021/050271 entitled “DISPLAYING THREE-DIMENSIONAL OBJECTS” and filed on Sep. 14, 2021, and PCT/US2021/050275 entitled “RECONSTRUCTING OBJECTS WITH DISPLAY ZERO ORDER LIGHT SUPPRESSION” and filed on Sep. 14, 2021, which claim priority under 35 U.S.C. §119 to U.S.S.N. 63/079,707 entitled “DISPLAYING THREE-DIMENSIONAL OBJECTS” and filed on Sep. 17, 2020, and to U.S.S.N. 63/149,964 entitled “RECONSTRUCTING OBJECTS WITH DISPLAY ZERO ORDER LIGHT SUPPRESSION” and filed on Feb. 16, 2021. The entire contents of each of the applications are incorporated by reference in its entirety herein. 
    
    
     TECHNICAL FIELD 
     This disclosure relates to three-dimensional (3D) displays, and more particularly to 3D displays with object reconstruction. 
     BACKGROUND 
     Advances in traditional two-dimensional (2D) projection and 3D rendering have led to new approaches for 3D displays, including numerous hybrid techniques that mix head and eye tracking with conventional display devices for virtual reality (VR), augmented reality (AR), and mixed reality (MR). These techniques attempt to replicate an experience of holographic imagery, combined with tracking and measurement-based calculations, to simulate stereo or in-eye light field that can be represented by an actual hologram. 
     SUMMARY 
     The present disclosure describes methods, apparatus, devices, and systems for reconstructing objects (e.g., 2D or 3D), particularly with display zero order light suppression. The present disclosure provides techniques that can efficiently suppress display zero order light (e.g., reflected, diffracted, or transmitted) from a display in a reconstructed holographic scene (or holographic content) to improve an effect of the holographic scene and accordingly a performance of a display system. As an example, when light illuminates a display for holographic reconstruction, a portion of the light is incident on and diffracted by display elements that are modulated with a hologram to form a desired holographic scene. The other portion of the light is incident on and reflected at gaps between the display elements on the display. The reflected other portion of the light can be considered as at least a part (e.g., a main order) of display zero order light that may be undesirably presented in the holographic scene. The display zero order light can also include any other unwanted light from the display, e.g., diffracted light at the gaps, reflected light from the display elements, or reflected light from a display cover on the display. Embodiments of the disclosure can suppress such display zero order light. 
     In some implementations, a hologram is configured such that a first portion of light illuminated on display elements of the display is diffracted by the display elements modulated by the hologram to have at least one characteristic different from that of display zero order light including reflected light from the display. The display zero order light can include a second portion of the light illuminated on gaps between the display elements and reflected at the gaps without modulation of the hologram. The techniques can make use of the difference between the diffracted first portion of the light and the display zero order light (e.g., the reflected second portion of the light) to cause the display zero order light to be suppressed in the holographic scene formed by the diffracted first portion of the light. The techniques can be applied individually or in a combination thereof. The techniques can be applied to any other display systems that suppress or eliminate undesired light from desired light. 
     In some examples, the display is configured to suppress higher orders of the display zero order light, e.g., by including irregular or non-uniform display elements that have different sizes. The display elements can have no periodicity, and can form a Voronoi pattern. In some examples, in the holographic scene, the display zero order light can have a much smaller power density than the diffracted first portion of the light. That is, the display zero order light is suppressed by increasing a signal to noise ratio of the holographic scene, e.g., by diverging the display zero order light without divergence of the diffracted first portion of the light, or by adjusting respective phases of the display elements within a predetermined phase range such as [0, 2π], or both. In some examples, the display zero order light is suppressed by directing the display zero order light away from the diffracted first portion of the light, e.g., by illuminating the light on the display at an incident angle and preconfiguring the hologram such that the diffracted first portion of the light still propagates around a normal axis and the display zero order light propagates at a reflected angle. The display zero order light can be redirected outside of the holographic scene formed by the diffracted first portion of the light, e.g., by adding an additional optically diffractive grating structure to further direct the display zero order light away from the holographic scene. The display zero order light can be reflected back away from the holographic scene. The display zero order light can be also absorbed before the holographic scene. 
     In the present disclosure, the terms “zero order” and “zero-order” are used interchangeably, and the terms “first order” and “first-order” are used interchangeably. 
     In the present disclosure, the terms “zero order” and “zero-order” are used interchangeably, and the terms “first order” and “first-order” are used interchangeably. 
     One aspect of the present disclosure features a method including: illuminating a display with light, a first portion of the light illuminating display elements of the display; and modulating the display elements of the display with a hologram corresponding to holographic data to i) diffract the first portion of the light to form a holographic scene corresponding to the holographic data, and ii) suppress display zero order light in the holographic scene, the display zero order light including reflected light from the display. 
     In some examples, illuminating the display with the light includes a second portion of the light illuminates gaps between adjacent display elements. The display zero order light can include at least one of: the second portion of the light reflected at the gaps of the display, the second portion of the light diffracted at the gaps of the display, reflected light from the display elements, or reflected right from a display cover covering the display. 
     The reflected light from the display forms a main order of the display zero order light, and the display can be configured to suppress one or more higher orders of the display zero order light, and where the display elements are irregular or non-uniform. In some examples, the display elements form a Voronoi pattern. 
     In some implementations, the method further includes: configuring the hologram such that the diffracted first portion of the light has at least one characteristic different from that of the display zero order light. The at least one characteristic can include at least one of: a power density; a beam divergence; a propagating direction away from the display; or a polarization state. 
     In some implementations, the display zero order light is suppressed in the holographic scene with a light suppression efficiency. The light suppression efficiency is defined as a result of one minus a ratio between an amount of the display zero light in the holographic scene with the suppression and an amount of the display zero light in the holographic scene without the suppression. In some cases, the light suppression efficiency is more than a predetermined percentage that is one of 50%, 60%, 70%, 80%, 90%, or 99%. In some cases, the light suppression efficiency is 100%. 
     In some implementations, the method further includes: for each of a plurality of primitives corresponding to an object, determining an electromagnetic (EM) field contribution to each of the display elements of the display by computing, in a global three-dimensional (3D) coordinate system, EM field propagation from the primitive to the display element; and for each of the display elements, generating a sum of the EM field contributions from the plurality of primitives to the display element. The holographic data can include the sums of the EM field contributions for the display elements of the display from the plurality of primitives of the object. The holographic scene can include a reconstructed object corresponding to the object. 
     In some implementations, the holographic data includes respective phases for the display elements of the display, and the method further includes configuring the hologram by adjusting the respective phases for the display elements to have a predetermined phase range. The predetermined phase range can be [0, 2π]. 
     In some implementations, adjusting the respective phases for the display elements includes: adjusting the respective phases according to 
     
       
         
           
             
               ∅ 
               a 
             
             = 
             A 
             
               ∅ 
               i 
             
             + 
             B 
             , 
           
         
       
     
      where ø i  represents an initial phase value of a respective phase, ø a  represents an adjusted phase value of the respective phase, and A and B are constants. 
     In some implementations, adjusting the respective phases includes: adjusting the constants A and B such that a light suppression efficiency for the holographic scene is maximized. The light suppression efficiency can be larger than 50%, 60%, 70%, 80%, 90%, or 99%. In some cases, adjusting the constants A and B includes adjusting the constants A and B by a machine vision algorithm or a machine learning algorithm. 
     In some implementations, the method further includes: diverging the diffracted first portion of the light to form the holographic scene; and diverging the display zero order light in or adjacent to the holographic scene. In some examples, diverging the diffracted first portion of the light includes guiding the diffracted first portion of the light through an optically diverging component arranged downstream the display, and diverging the display zero order light includes guiding the display zero order light through the optically diverging component. 
     In some examples, the light illuminating the display is a collimated light. The display zero order light is collimated before arriving at the optically diverging component, and the method can further include configuring the hologram such that the diffracted first portion of the light is converging before arriving at the optically diverging component. 
     In some implementations, the holographic data includes a respective phase for each of the display elements. The method can further include configuring the hologram by adding a corresponding phase to the respective phase for each of the display elements, and the corresponding phases for the display elements can be compensated by the optically diverging component such that the holographic scene corresponds to the respective phases for the display elements. The corresponding phase for each of the display elements can be expressed as: 
     
       
         
           
             ∅ 
             = 
             
               π 
               
                 λ 
                 f 
               
             
             
               
                 
                   
                     ax 
                   
                   2 
                 
                 + 
                 
                   
                     by 
                   
                   2 
                 
               
             
             , 
           
         
       
     
      where ø represents the corresponding phase for the display element, λ, represents a wavelength of the light, ƒ represents a focal length of the optically diverging component, x and y represent coordinates of the display element in a coordinate system, and a and b represent constants. 
     In some implementations, the holographic scene corresponds to a reconstruction cone with a viewing angle. The method can further include configuring the hologram by moving a configuration cone with respect to the display with respect to a global 3D coordinate system along a direction perpendicular to the display with a distance corresponding to a focal length of the optically diverging component, the configuration cone corresponding to the reconstruction cone and having an apex angle identical to the viewing angle, and generating the holographic data based on the moved configuration cone in the global 3D coordinate system. The plurality of primitives of the object can be in the moved configuration cone. 
     In some implementations, the optically diverging component is a defocusing element including at least one of a concave lens or a holographic optical element (HOE) configured to diffract the display zero order light outside of the holographic scene. 
     In some implementations, the optically diverging component is a focusing element including at least one of a convex lens or a holographic optical element (HOE) configured to diffract the display zero order light outside of the holographic scene. 
     In some implementations, the method further includes: displaying the holographic scene on a two-dimensional (2D) screen spaced away from the display along a direction perpendicular to the display. The method can further include: moving the 2D screen to obtain different slices of the holographic scene on the 2D screen. 
     In some implementations, the method further includes: guiding the light to illuminate the display. In some examples, guiding the light to illuminate the display includes: guiding the light by a beam splitter, and the diffracted first portion of the light and the display zero order light transmit through the beam splitter. 
     In some implementations, illuminating the display with the light includes: illuminating the display with the light at normal incidence. 
     In some implementations, the diffracted first portion of the light forms a reconstruction cone with a viewing angle, and illuminating the display with the light includes illuminating the display with the light at an incident angle that is larger than a half of the viewing angle. In some examples, the method further includes: configuring the hologram such that the diffracted first portion of the light forms the reconstruction cone that is same as a reconstruction cone to be formed by the diffracted first portion of the light if the light is normally incident on the display. 
     In some examples, the holographic data includes a respective phase for each of the display elements. The method can further include configuring the hologram by adding a corresponding phase to the respective phase for each of the display elements, and the corresponding phases for the display elements can be compensated by the incident angle such that the holographic scene corresponds to the respective phases for the display elements. 
     In some examples, the corresponding phase for each of the display elements can be expressed as: 
     
       
         
           
             ∅ 
             = 
             
               
                 2 
                 π 
               
               λ 
             
             
               
                 xcos 
                 θ 
                 +ycos 
                 θ 
               
             
             , 
           
         
       
     
      where ø represents the corresponding phase for the display element, λ, represents a wavelength of the light, x and y represent coordinates of the display element in a global 3D coordinate system, and θ represents an angle corresponding to the incident angle. 
     In some examples, configuring the hologram includes: moving a configuration cone with respect to the display with respect to a global 3D coordinate system, the configuration cone corresponding to the reconstruction cone and having an apex angle corresponding to the viewing angle of the reconstruction cone, and generating the holographic data based on the moved configuration cone in the global 3D coordinate system. 
     In some examples, moving the configuration cone with respect to the display in the global 3D coordinate system includes: rotating the configuration cone by a rotation angle with respect to a surface of the display with respect to the global 3D coordinate system, the rotation angle corresponding to the incident angle. 
     In some implementations, the method further includes: blocking the display zero order light from appearing in the holographic scene. A light suppression efficiency for the holographic scene can be 100%. In some examples, blocking the display zero order light includes: guiding the display zero order light towards an optically blocking component arranged downstream the display. The method can further include: guiding the diffracted first portion of the light to transmit through the optically blocking component with a transmission efficiency to form the holographic scene. The transmission efficiency can be no less than a predetermined ratio. The predetermined ratio can be 50%, 60%, 70%, 80%, 90%, or 99%. 
     In some implementations, the optically blocking component is configured to transmit a first light beam having an angle smaller than a predetermined angle and block a second light beam having an angle larger than the predetermined angle, and the predetermined angle is smaller than the incident angle and larger than the half of the viewing angle. The optically blocking component can include a plurality of microstructures or nanostructures, a metamaterial layer, or an optically anisotropic film. 
     In some implementations, the method further includes: guiding the light to illuminate the display by guiding the light through an optically diffractive component on a substrate configured to diffract the light out with the incident angle. Guiding the light to illuminate the display can include at least one of: guiding the light through a waveguide coupler to the optically diffractive component, guiding the light through a coupling prism to the optically diffractive component, or guiding the light through a wedged surface of the substrate to the optically diffractive component. 
     In some implementations, the optically diffractive component is formed on a first surface of the substrate facing to the display, and the optically blocking component is formed on a second surface of the substrate that is opposite to the first surface. 
     In some implementations, the method further includes: redirecting the display zero order light away from the holographic scene. A light suppression efficiency for the holographic scene can be 100%. 
     In some implementations, redirecting the display zero order light away from the holographic scene includes: diffracting the display zero order light away from the holographic scene by an optically redirecting component arranged downstream the display. The optically redirecting component can be configured to transmit the diffracted first portion of the light to form the holographic scene. 
     In some implementations, the optically redirecting component is configured such that the display zero order light is diffracted outside of the holographic scene in a three-dimensional (3D) space along at least one of an upward direction, a downward direction, a leftward direction, a rightward direction, or a combination thereof. 
     In some implementations, the optically redirecting component is configured to diffract a first light beam having an angle identical to a predetermined angle with a substantially larger diffraction efficiency than a second light beam having an angle different from the predetermined angle, and the predetermined angle is substantially identical to the incident angle. The optically redirecting component can include a Bragg grating. 
     In some implementations, the optically diffractive component is formed on a first surface of the substrate facing to the display, and the optically redirecting component is formed on a second surface of the substrate that is opposite to the first surface. 
     In some cases, the incident angle of the light is negative, and a diffraction angle of the display zero order light diffracted by the optically redirecting component is negative. In some cases, the incident angle of the light is positive, and a diffraction angle of the display zero order light diffracted by the optically redirecting component is positive. In some cases, the incident angle of the light is negative, and a diffraction angle of the display zero order light diffracted by the optically redirecting component is positive. In some cases, the incident angle of the light is positive, and a diffraction angle of the display zero order light diffracted by the optically redirecting component is negative. 
     In some implementations, the optically redirecting component is covered by a second substrate. The method can further include: absorbing, by an optical absorber formed on at least one of a side surface of the second substrate or a side surface of the substrate, the display zero order light redirected by the optically redirecting component and reflected by an interface between the second substrate and a surrounding medium. 
     In some implementations, the second substrate includes an anti-reflective coating on a surface of the second substrate opposite to the optically redirecting component, and the anti-reflective coating is configured to transmit the display zero order light. 
     In some implementations, the display zero order light is p polarized before arriving at the second substrate, and the optically redirecting component is configured to diffract the display zero order light to be incident at a Brewster’s angle on an interface between the second substrate and a surrounding medium, such that the display zero order light totally transmits through the second substrate. 
     In some implementations, the method further includes: converting a polarization state of the display zero order light from s polarization to p polarization before display zero order light arrives at the second substrate. In some cases, converting the polarization state of the display zero order light includes: converting the polarization state of the display zero order light by an optically polarizing device arranged upstream the optically redirecting component with respect to the display. 
     In some cases, converting the polarization state of the display zero order light includes: converting the polarization state of the display zero order light by an optically polarizing device arranged downstream the optically redirecting component with respect to the display. The optically polarizing device can include an optical retarder and an optical polarizer that are sequentially arranged downstream the optically redirecting component, and the optical retarder can be formed on a side of the second substrate opposite to the optically redirecting component, the optical polarizer being covered by a third substrate. In some examples, the optical retarder includes a broadband half-wave plate and the optical polarizer includes a linear polarizer. 
     In some implementations, the second substrate includes: a first side on top of the optically redirecting component and a second side opposite to the first side. An optically blocking component can be formed on the second side of the second substrate and configured to transmit the diffracted first portion of the light and to absorb the display zero order light diffracted by the optically redirecting component. 
     In some implementations, the optically blocking component includes an optically anisotropic transmitter configured to transmit a first light beam with an angle smaller than a predetermined angle, and absorb a second light beam with an angle larger than the predetermined angle. The predetermined angle can be larger than half of the viewing angle and smaller than a diffraction angle at which the display zero order light is diffracted by the optically redirecting component. 
     In some implementations, the optically redirecting component is configured to diffract the display zero order light to be incident with an angle larger than a critical angle on an interface between the second substrate and a surrounding medium, such that the display zero order light diffracted by the optically diffractive component is totally reflected at the interface. An optical absorber can be formed on side surfaces of the substrate and the second substrate and configured to absorb the totally reflected display zero order light. 
     In some implementations, the light includes a plurality of different colors of light, and the optically diffractive component is configured to diffract the plurality of different colors of light at the incident angle on the display. 
     In some implementations, the optical redirecting component includes a respective optically redirecting subcomponent for each of the plurality different colors of light. In some examples, the respective optically redirecting subcomponents for the plurality of different colors of light can be recorded in a same recording structure. In some examples, the respective optically directing subcomponents for the plurality of different colors of light are recorded in different corresponding recording structures. 
     In some implementations, the optical redirecting component is configured to diffract the plurality of different colors of light at different diffraction angles towards different directions in a 3D space. The optical redirecting component can be configured to diffract at least one of the plurality of different colors of light to be incident at at least one Brewster’s angle at an interface. The interface can include one of: an interface between a top substrate and a surrounding medium, or an interface between two adjacent substrates. 
     In some implementations, the optical redirecting component is configured to diffract a first color of light and a second color of light within a plane, and a third color of light orthogonal to the plane. In some implementations, the optical redirecting component includes at least two different optically redirecting subcomponents configured to diffract a same color of light of the plurality of different colors of light. The two different optically redirecting subcomponents can be sequentially arranged in the optical redirecting component. 
     In some implementations, guiding the light to illuminate the display includes: sequentially guiding the plurality of different colors of light to illuminate the display in a series of time periods. In some implementations, the optical redirecting component includes a switchable optically redirecting subcomponent configured to diffract a first color of light at a first state during a first time period and transmit a second color of light at a second state during a second time period. In some implementations, the optical redirecting component includes a switchable optically redirecting subcomponent configured to diffract a first color of light at a first state during a first time period and diffract a second color of light at a second state during a second time period. 
     In some implementations, the plurality of different colors of light includes a first color of light and a second color of light, the first color of light having a shorter wavelength than the second color of light, and in the optically redirecting component, a first optically redirecting subcomponent for the first color of light is arranged closer to the display than a second optically redirecting subcomponent for the second color of light. 
     In some implementations, fringe planes of at least two optically redirecting subcomponents for at least two different colors of light are oriented substantially differently. 
     In some implementations, the optically redirecting component includes: a first optically redirecting subcomponent configured to diffract a first color of light; a second optically redirecting subcomponent configured to diffract a second color of light; and at least one optically polarizing device arranged between the first and second optically redirecting subcomponents and configured to convert a polarization state of the first color of light such that the first color of light transmits through the second optically redirecting subcomponent. The at least one optically polarizing device can include optical retarder and an optical polarizer that are sequentially arranged downstream the first optically redirecting subcomponent. 
     In some cases, a half of the viewing angle is within a range from -10 degrees to 10 degrees or a range from -5 degrees to 5 degrees. In some cases, the incident angle is -6 degrees or 6 degrees. 
     Another aspect of the present disclosure features a method including: illuminating a display with light, a portion of the light illuminating display elements of the display; and generating a holographic scene by diffracting the portion of light, while suppressing display zero order light present in the holographic scene, where the display zero order light includes reflected light from the display. 
     In some implementations, suppressing the display zero order light present in the holographic scene includes: diverging the display zero order light. 
     In some implementations, generating a holographic scene by diffracting the portion of light includes modulating the display elements with a hologram. Suppressing the display zero order light present in the holographic scene can include adjusting a phase range of the hologram. 
     In some implementations, illuminating the display with the light includes illuminating the display with the light at an incident angle, and suppressing the display zero order light present in the holographic scene can include modulating the portion of light with a hologram configured such that the portion of the light is diffracted by the display elements at a diffraction angle different from a reflected angle at which the reflected light is reflected. In some cases, suppressing the display zero order light present in the holographic scene includes: blocking the display zero order light by an incident angle dependent material. The incident angle dependent material can include a metamaterial or an optically anisotropic material. 
     In some implementations, suppressing the display zero order light present in the holographic scene includes: redirecting the display zero order light. Redirecting the display zero order light can include diffracting the display zero order light by an optically diffractive component. The light can include different colors of light, and redirecting the display zero order light can include diffracting the different colors of light to different directions in a three-dimensional (3D) space. 
     In some implementations, suppressing the display zero order light present in the holographic scene includes: suppressing the display zero order light with a light suppression efficiency no less than a predetermined ratio. The light suppression efficiency is defined as a result of one minus a ratio between an amount of the display zero order light in the holographic scene with the suppression and an amount of the display zero order light without the suppression. The predetermined ratio can be 50%, 60%, 70%, 80%, 90%, or 100%. 
     Another feature of the present disclosure features an optical device including: an optically diffractive component and an optically blocking component. The optically diffractive component is configured to diffract light at an incident angle to illuminate a display, with a portion of the light illuminating display elements of the display, and the optically blocking component is configured to block display zero order light in a holographic scene formed by the portion of the light diffracted by the display elements, the display zero order light including reflected light from the display. 
     In some implementations, the optical device is configured to perform the method as described above. 
     In some implementations, the display is configured to be modulated with a hologram corresponding to holographic data to diffract the portion of the light to form the holographic scene, and the optically blocking component is configured to transmit the diffracted portion of the light to form the holographic scene. The diffracted portion of the light can form a reconstruction cone with a viewing angle, and the incident angle can be larger than a half of the viewing angle. 
     The optically blocking component can be configured to transmit a first light beam having an angle smaller than a predetermined angle and block a second light beam having an angle larger than the predetermined angle, and the predetermined angle can be smaller than the incident angle and larger than the half of the viewing angle. 
     In some implementations, the optically blocking component includes a metamaterial layer or an optically anisotropic film. In some implementations, the optically blocking component includes a plurality of microstructures or nanostructures. 
     In some implementations, the optical device further includes a substrate having opposite sides. The optically diffractive component and the optically blocking component can be formed on the opposite sides of the substrate. 
     Another aspect of the present disclosure features a method of fabricating the optical device as described above, including: forming the optically diffractive component on a first side of a substrate and forming the optically blocking component on a second side of the substrate opposite to the first side. 
     Another aspect of the present disclosure features an optical device including: an optically diffractive component and an optically redirecting component. The optically diffractive component is configured to diffract light at an incident angle onto a display including a plurality of display elements spaced with gaps on the display. The display is configured to diffract a portion of the light illuminating the display elements. The optically redirecting component is configured to transmit the portion of the light to form a holographic scene and to redirect display zero order light away from the holographic scene in a three-dimensional (3D) space, the display zero order light including reflected light from the display. 
     In some examples, the optically redirecting component includes a Bragg grating. 
     In some implementations, the optically diffractive component is formed on a first side of a substrate facing to the display, and the optically redirecting component is formed on a second side of the substrate that is opposite to the first side. 
     In some implementations, the optical device further includes a second substrate covering the optically redirecting component. In some implementations, the optical device further includes an optical absorber formed on at least one of a side surface of the substrate or a side surface of the second substrate, and the optical absorber is configured to absorb the display zero order light redirected by the optically redirecting component and reflected by an interface between the second substrate and a surrounding medium. 
     In some implementations, the optical device further includes: an anti-reflective coating formed on the second substrate and being opposite to the optically redirecting component, the anti-reflective coating being configured to transmit the display zero order light redirected by the optically redirecting component. 
     In some implementations, the optical device further includes: an optically polarizing device configured to convert a polarization state of the display zero order light from s polarization to p polarization before the display zero order light arrives at the second substrate, and the optically redirecting component is configured to diffract the display zero order light to be incident at a Brewster’s angle on an interface between the second substrate and a surrounding medium, such that the display zero order light totally transmits through the second substrate. The optical polarizing device can include an optical retarder and a linear polarizer that are sequentially arranged together. 
     In some implementations, the optically polarizing device is arranged upstream the optically redirecting component with respect to the display. In some implementations, the optically polarizing device is formed a side of the second substrate opposite to the optically redirecting component, the optically polarizing device being covered by a third substrate. 
     In some implementations, the optical device further includes: an optical blocking component formed on a side of the second substrate opposite to the optically redirecting component, the optical blocking component being configured to transmit the portion of the light and to absorb the display zero order light diffracted by the optically redirecting component. The optically blocking component can include an optically anisotropic transmitter. 
     In some implementations, the optically redirecting component is configured to diffract the display zero order light to be incident with an angle larger than a critical angle on an interface between the second substrate and a surrounding medium, such that the display zero order light diffracted by the optically diffractive component is totally reflected at the interface. 
     In some implementations, the light includes a plurality of different colors of light. The optically diffractive component is configured to diffract the plurality of different colors of light at the incident angle on the display, and the optical redirecting component can be configured to diffract display zero order light of the plurality of different colors of light reflected by the display at different diffraction angles towards different directions in the 3D space, the display zero order light including reflected light of the plurality of different colors of light by the display. 
     In some implementations, the optical diffractive component includes a plurality of holographic gratings for the plurality of different colors of light, and each of the plurality of holographic gratings is configured to diffract a respective color of light of the plurality of different colors of light at the incident angle on the display. 
     In some implementations, the optical redirecting component includes a plurality of redirecting holographic grating for the display zero order light of the plurality of different colors of light, and each of the plurality of redirecting holographic gratings is configured to diffract display zero order light of a respective color of light of the plurality of different colors of light at a respective diffractive angle towards a respective direction in the 3D space. 
     In some implementations, the optical redirecting component includes at least two different redirecting holographic gratings configured to diffract display zero order light of a same color of light of the plurality of different colors of light. 
     In some implementations, the optical redirecting component includes a switchable redirecting holographic grating configured to diffract a first color of light at a first state during a first time period and transmit a second color of light at a second state during a second time period. 
     In some implementations, the optical redirecting component includes a switchable redirecting holographic grating configured to diffract a first color of light at a first state during a first time period and diffract a second color of light at a second state during a second time period. 
     In some implementations, the plurality of different colors of light includes a first color of light and a second color of light, the first color of light having a shorter wavelength than the second color of light, and, in the optically redirecting component, a first redirecting holographic grating for the first color of light is arranged closer to the display than a second redirecting holographic grating for the second color of light. 
     In some implementations, fringe planes of at least two redirecting holographic gratings for at least two different colors of light are oriented substantially differently. 
     In some implementations, the optically redirecting component includes: a first redirecting holographic grating configured to diffract a first color of light; a second redirecting holographic grating configured to diffract a second color of light; and at least one optical polarizing device arranged between the first and second redirecting holographic gratings and configured to convert a polarization state of the first color of light such that the first color of light transmits through the second redirecting holographic grating. 
     In some implementations, the optical device is configured to perform the methods described above. 
     Another aspect of the present disclosure features a method of fabricating the optical device as described above, including: forming the optically diffractive component on a first side of a substrate; and forming the optically redirecting component on a second side of the substrate opposite to the first side. 
     Another aspect of the present disclosure features a system including: a display including display elements separated with gaps on the display and an optical device configured to illuminate the display with light, with a portion of the light illuminating on the display elements. The system is configured to diffract the portion of the light to form a holographic scene, while suppressing display zero order light in the holographic scene. The display zero order light can include at least one of reflected light at the gaps, diffracted light at the gaps, reflected light at the display elements, or reflected light at a display cover covering the display. 
     In some implementations, the system further includes a controller coupled to the display and configured to: modulate the display elements of the display with a hologram corresponding to holographic data to diffract the portion of the light to form the holographic scene corresponding to the holographic data. The hologram can be configured such that the display zero order light is suppressed in the holographic scene. 
     In some implementations, the system further includes a computing device configured to generate primitives of one or more objects corresponding to the holographic scene. The system can be configured to perform the methods as described above. The optical device can include one or more of the optical devices as described above. 
     In some implementations, the system further includes: an optically diverging device arranged downstream the optical device and configured to diverge the display zero order light in the holographic scene. The light illuminating the display is a collimated light. The display zero order light is collimated before arriving at the optically diverging device, and the hologram is configured such that the diffracted portion of the light is converging before arriving at the optically diverging device. The optically diverging device can includes the optically diverging component as described above. 
     In some implementations, the system further includes a two-dimensional (2D) screen arranged downstream the display. In some implementations, the optical device includes a beam splitter. In some implementations, the optical device includes a waveguide having an incoupler and an outcoupler. In some implementations, the optical device includes a lightguide including a light coupler and an optically diffractive component. The light coupler can include a coupling prism. The light coupler can also include a wedged substrate. 
     Another aspect of the present disclosure features a method of fabricating the system of as described above. 
     Another aspect of the present disclosure features an optical device including: at least two beam expanders configured to expand an input light beam in at least two dimensions to generate an output light beam by diffracting the input light beam to adjust a beam size of the input light beam in the at least two dimensions. The beam size can include a width and a height. 
     In some implementations, each of the at least two beam expanders includes a respective optically diffractive device. The input light beam can include light of a plurality of different colors, and the respective optically diffractive device can be configured to diffract the light of the plurality of different colors at respective diffracted angles that are substantially identical to each other. 
     In some examples, the respective optically diffractive device is configured such that, when the light of the different colors is incident on the respective optically diffractive device, the respective optical diffractive device separates light of individual colors of the different colors while suppressing crosstalk between the different colors. 
     In some implementations, the respective optically diffractive device includes: at least two optically diffractive components and at least one color-selective polarizer. 
     In some implementations, the respective optically diffractive device includes: at least two optically diffractive components and at least one reflective layer. The at least one reflective layer can be configured for total internal reflection of light of at least one color. 
     In some implementations, the respective optically diffractive device includes at least one of: one or more transmissive diffractive structures, or one or more reflective diffractive structures. 
     In some implementations, the at least two beam expanders include: a first one-dimensional beam expander configured to expand the input light beam in a first dimension of the at least two dimensions, to generate an intermediate light beam; and a second one-dimensional beam expander configured to expand the intermediate light beam in a second dimension of the at least two dimensions, to generate the output light beam. The intermediate light beam has a larger beam size than the input light beam in the first dimension and a same beam size as the input light beam in the second dimension, and the output light beam has a larger beam size than the intermediate light beam in the second dimension and a same beam size as the intermediate light beam in the first dimension. 
     In some implementations, the optical device is configured to couple the intermediate light beam from the first one-dimensional beam expander to the second one-dimensional beam expander using at least one of: a free-space in-air geometry, a monolithic or segmented substrate, or one or more coupling elements. 
     In some implementations, the intermediate input beam includes collinear collimated light of two or more colors, and the one or more coupling elements are configured to convert the collinear collimated light of the two or more colors to two or more independent collimated but not collinear light beams with corresponding colors of the two or more colors. 
     The present disclosure also describes methods, apparatus, devices, and systems for displaying three-dimensional (3D) objects, particularly by individually diffracting different colors of light. The present disclosure provides technology that can efficiently separate light of different colors or wavelengths to suppress (e.g., reduce or eliminate) crosstalk between the colors or wavelengths. The technology can also suppress light propagating without diffraction through an optically diffractive device and hitting at undesired angles onto a display, thereby suppressing undesired effects such as ghost images. The technology enables to reconstruct multi-color three-dimensional light fields or images with no or little crosstalk, sequentially or simultaneously. The technology enables to implement an illumination system to provide nearly normal polarized light beams of multiple different colors with relatively large incident angles. Accordingly, the technology enables to present light fields or images to viewers (e.g., observers or users) in front of a display without obstruction of an illuminator, and to reduce power loss, e.g., due to reflections, diffraction, and/or scattering. The technology also enables to implement compact optical systems for displaying three-dimensional objects. 
     The present disclosure provides technology that can overcome limitations present in known technologies. As an example, the technology disclosed herein can be implemented without the use of cumbersome wearable devices, such as “3D glasses.” As another example, the technology disclosed herein can optionally be implemented without being limited by the accuracy of tracking mechanisms, the quality of the display devices, relatively long processing times and/or relatively high computational demands, and/or by an inability to display objects to multiple viewers simultaneously. As a further example, the technology can be implemented without specialized tools and software to develop contents that extend above and beyond the tools and software used in conventional 3D content creation. Various embodiments can exhibit one or more of the foregoing advantages. For example, certain implementations of the present disclosure can produce real-time, full color, genuine 3D images that appear to be real 3D objects in the world and can be viewed without encumbrances by multiple viewers simultaneously from different points. 
     One aspect of the present disclosure features a method including: for each of a plurality of primitives corresponding to an object in a three-dimensional (3D) space, determining an electromagnetic (EM) field contribution to each of a plurality of elements of a display by computing, in a 3D coordinate system, EM field propagation from the primitive to the element; and for each of the plurality of elements, generating a sum of the EM field contributions from the plurality of primitives to the element. 
     The EM field contribution can include at least one of a phase contribution or an amplitude contribution. The primitives can include at least one of a point primitive, a line primitive, or a polygon primitive. The primitives can include a line primitive including at least one of a gradient color, a textured color, or any surface shading effect. The primitives can also include a polygon primitive including at least one of a gradient color, a textured color, or any surface shading effect. The plurality of primitives can be indexed in a particular order. 
     In some implementations, the method further includes obtaining respective primitive data for each of the plurality of primitives. The respective primitive data of each of the plurality of primitives can include respective color information of the primitive, and the determined EM field contributions for each of the elements include information corresponding to the respective color information of the primitives. The color information can include at least one of a textured color or a gradient color. The respective primitive data of each of the plurality of primitives can include texture information of the primitive. The respective primitive data of each of the plurality of primitives can include shading information on one or more surfaces of the primitive. The shading information can include a modulation on at least one of color or brightness on the one or more surfaces of the primitive. 
     In some implementations, the respective primitive data of each of the plurality of primitives includes respective coordinate information of the primitive in the 3D coordinate system. Respective coordinate information of each of the plurality of elements in the 3D coordinate system can be determined based on the respective coordinate information of the plurality of primitives in the 3D coordinate system. The respective coordinate information of each of the elements can correspond to a logical memory address for the element stored in a memory. 
     Determining the EM field contribution to each of the plurality of elements for each of the plurality of primitives can include determining, in the 3D coordinate system, at least one distance between the element and the primitive based on the respective coordinate information of the element and the respective coordinate information of the primitive. In some examples, determining the EM field contribution to each of the plurality of elements for each of the plurality of primitives includes: determining a first distance between a first primitive of the plurality of primitives and a first element of the plurality of elements based on the respective coordinate information of the first primitive and the respective coordinate information of the first element; and determining a second distance between the first primitive and a second element of the plurality of elements based on the first distance and a distance between the first element and the second element. The distance between the first element and the second element can be predetermined based on a pitch of the plurality of elements of the display. 
     In some examples, at least one of the plurality of primitives is a line primitive including first and second endpoints, and determining at least one distance between the element and the primitive includes: determining a first distance between the element and the first endpoint of the line primitive; and determining a second distance between the element and the second point of the line primitive. In some examples, at least one of the plurality of primitives is a triangle primitive including first, second, and third endpoints, and determining at least one distance between the element and the primitive includes: determining a first distance between the element and the first endpoint of the triangle primitive; determining a second distance between the element and the second point of the triangle primitive; and determining a third distance between the element and the third point of the triangle primitive. 
     In some implementations, determining the EM field contribution to each of the plurality of elements for each of the plurality of primitives includes determining the EM field contribution to the element from the primitive based on a predetermined expression for the primitive and the at least one distance. In some cases, the predetermined expression is determined by analytically calculating the EM field propagation from the primitive to the element. In some cases, the predetermined expression is determined by solving Maxwell’s equations. The Maxwell’s equations can be solved by providing a boundary condition defined at a surface of the display. The boundary condition can include a Dirichlet boundary condition or a Cauchy boundary condition. The plurality of primitives and the plurality of elements can be in the 3D space, and a surface of the display can form a portion of a boundary surface of the 3D space. In some cases, the predetermined expression includes at least one of functions including a sine function, a cosine function, or an exponential function, and determining the EM field contribution includes identifying a value of the at least one of the functions in a table stored in a memory. 
     In some implementations, determining the EM field contribution to each of the plurality of elements for each of the plurality of primitives and generating the sum of the field contributions for each of the plurality of elements includes: determining first EM field contributions from the plurality of primitives to a first element of the plurality of elements and summing the first EM field contributions for the first element; and determining second EM field contributions from the plurality of primitives to a second element of the plurality of elements and summing the second EM field contributions for the second element. Determining the first EM field contributions from the plurality of primitives to the first element can include: determining an EM field contribution from a first primitive of the plurality of primitives to the first element in parallel with determining an EM field contribution from a second primitive of the plurality of primitives to the first element. 
     In some implementations, determining the EM field contribution to each of the plurality of elements for each of the plurality of primitives includes: determining first respective EM field contributions from a first primitive of the plurality of primitives to each of the plurality of elements; and determining second respective EM field contributions from a second primitive of the plurality of primitives to each of the plurality of elements, and generating the sum of the field contributions for each of the plurality of elements can include: accumulating the EM field contributions for the element by adding the second respective EM field contribution to the first respective EM field contribution for the element. Determining the first respective EM field contributions from the first primitive to each of the plurality of elements can be performed in parallel with determining the second respective EM field contributions from the second primitive to each of the plurality of elements. 
     Determining the EM field contribution to each of the plurality of elements for each of the plurality of primitives can include: determining a first EM field contribution from a first primitive of the plurality of primitives to a first element of the plurality of elements in parallel with determining a second EM field contribution from a second primitive of the plurality of primitives to the first element. 
     In some implementations, the method further includes: for each of the plurality of elements, generating a respective control signal based on the sum of the EM field contributions from the plurality of primitives to the element, the respective control signal being for modulating at least one property of the element based on the sum of the EM field contributions from the plurality of primitives to the element. The at least one property of the element can include at least one of a refractive index, an amplitude index, a birefringence, or a retardance. The respective control signal can include an electrical signal, an optical signal, a magnetic signal, or an acoustic signal. In some cases, the method further includes: multiplying a scale factor to the sum of the field contributions for each of the elements to obtain a scaled sum of the field contributions, and the respective control signal is generated based on the scaled sum of the field contributions for the element. In some cases, the method further includes: normalizing the sum of the field contributions for each of the elements, and the respective control signal is based on the normalized sum of the field contributions for the element. The method can also include: transmitting the respective control signal to the element. 
     In some implementations, the method further includes: transmitting a control signal to an illuminator, the control signal indicating to activate the illuminator such that the illuminator emits light on the display. The control signal can be transmitted in response to determining a completion of obtaining the sum of the field contributions for each of the plurality of elements. The modulated elements of the display can cause the light to propagate in different directions to form a volumetric light field corresponding to the object in the 3D space. The volumetric light field can correspond to a solution of Maxwell’s equations with a boundary condition defined by the modulated elements of the display. The light can include a white light, and the display can be configured to diffract the white light into light with different colors. 
     In some implementations, the method further includes representing values using fixed point number representations during calculation. Each of the values can be represented as integers with an implicit scale factor. 
     In some implementations, the method further includes performing a mathematical function using fixed point number representations. The mathematical function can include at least one of sine, cosine, and arc tangent. Performing the mathematical function can include receiving an expression in a first fixed point format, and outputting a value at a second fixed point format that has a level of accuracy different from that of the first fixed point format. Performing the mathematical function can include looking up a table for calculation of the mathematical function, wherein the table includes at least one of a fully enumerated look-up table, an interpolated table, a semi-table based polynomial functions, and a semi-table based on full minimax polynomials. Performing the mathematical function can include applying a specialized range reduction for an input. Performing the mathematical function can include transforming a trigonometric calculation from a range [-π, π] into a signed 2&#39;s compliment representation in a range [-1.1]. 
     Another aspect of the present disclosure features a method that includes: obtaining respective primitive data of a plurality of primitives corresponding to an object in a three-dimensional (3D) space; calculating first respective electromagnetic (EM) field contributions from a first primitive of the plurality of primitives to each of a plurality of elements of a display; and calculating second respective EM field contributions from a second primitive of the plurality of primitives to each of the plurality of elements of the display. Calculating the first respective EM field contributions from the first primitive is at least partially in parallel with calculating the second respective EM field contributions from the second primitive. 
     In some implementations, calculating a first EM field contribution from the first primitive to a first element of the plurality of elements is in parallel with calculating a second EM field contribution from a second primitive of the plurality of primitives to the first element. The method can include calculating respective EM field contributions from each of the plurality of primitives to each of the plurality of elements. The calculation of the respective EM field contributions can be without at least one of: expanding geometry of the object into the plurality of elements; applying visibility tests before packing wavefronts; and decision making or communication between parallel calculations for different primitives. The calculation of the respective EM field contributions can be configured to cause at least one of: tuning parallel calculations for different primitives to speed, cost, size or energy optimization; reducing latency between initiating a draw and a result being ready for display; increasing accuracy using fixed point number representations; and optimizing computation speed by optimizing mathematical functions. 
     In some implementations, the method further includes representing values using fixed point number representations during calculation. Representing the values using the fixed point number representations can proceed without at least one of: denormalizing floats for gradual underflow; handling NaN results from operations including division by zero; altering floating point rounding modes; and raising floating point exceptions to an operating system. 
     In some implementations, the method further includes, for each of the plurality of elements, accumulating EM field contributions for the element by adding the second respective EM field contribution for the element to the first respective EM field contribution for the element. 
     In some implementations, the method further includes, for each of the plurality of elements, generating a respective control signal based on a sum of the EM field contributions from the plurality of primitives to the element, wherein the respective control signal is for modulating at least one property of the element based on the sum of the EM field contributions from the plurality of primitives to the element. 
     In some implementations, the method further includes scaling a first primitive adjacent to a second primitive by a predetermined factor such that a reconstruction of the first primitive does not overlap with a reconstruction of the second primitive. The predetermined factor can be determined at least partially based on a resolution of the display. The method can further include: obtaining respective primitive data for each of the plurality of primitives, wherein the respective primitive data of each of the plurality of primitives comprises respective coordinate information of the primitive in the 3D coordinate system; and determining new respective coordinate information of the first primitive based on the respective coordinate information of the first primitive and the predetermined factor. The method can further include determining an EM field contribution from the first primitive to each of the plurality of elements based on the new respective coordinate information of the first primitive. The method can further include scaling the second primitive by the predetermined factor. The first primitive and the second primitive can share a common part, wherein scaling the first primitive comprises scaling the common part of the first primitive. Scaling the first primitive can include scaling the first primitive in a predetermined direction. 
     Another aspect of the present disclosure features a method that includes: obtaining respective primitive data of a plurality of primitives corresponding to an object in a three-dimensional (3D) space; scaling a first primitive adjacent to a second primitive by a predetermined factor using the respective primitive data for the first primitive and the second primitive; and updating the respective primitive data for the first primitive based on a result of the scaling. 
     In some implementations, the respective primitive data of each of the plurality of primitives include respective coordinate information of the primitive in a 3D coordinate system, and updating the respective primitive data includes determining new respective coordinate information of the first primitive based on the respective coordinate information of the first primitive and the predetermined factor. 
     In some implementations, the predetermined factor is determined such that a reconstruction of the first primitive does not overlap with a reconstruction of the second primitive in the 3D space. 
     In some implementations, the scaling is performed such that a gap between reconstruction of the first primitive and the second primitive in the 3D space is big enough to separate the first and second primitives to minimize an overlapping effect and small enough to make the reconstruction appear seamless. 
     In some implementations, the predetermined factor is determined at least partially based on a resolution of the display or on an actual or assumed distance from the viewer to the display or to the z-depth of the primitives within the display’s 3D space. 
     In some implementations, the method further includes storing the updated primitive data for the first primitive in a buffer. 
     In some implementations, the scaling is performed during a rendering process of the object for obtaining the respective primitive data of the plurality of primitives. 
     In some implementations, the method further includes transmitting updated primitive data for the plurality of primitives to a controller, wherein the controller is configured to determining respective electromagnetic (EM) field contributions from each of the plurality of primitives to each of a plurality of elements of a display based on the updated primitive data for the plurality of primitives. 
     In some implementations, the method further includes determining an EM field contribution from the first primitive to each of a plurality of elements of a display based on the updated primitive data of the first primitive. 
     In some implementations, the method further includes scaling the second primitive by the predetermined factor. 
     In some implementations, the first primitive and the second primitive share a common part, and scaling the first primitive comprises scaling the common part of the first primitive. 
     In some implementations, scaling the first primitive includes scaling the first primitive in a predetermined direction. 
     In some implementations, scaling the first primitive includes scaling a first part of the first primitive by a first predetermined factor, and scaling a second part of the second primitive by a second predetermined factor, where the first predetermined factor is different from the second predetermined factor. 
     Another aspect of the present disclosure features a method that includes: obtaining a plurality of discrete cosine transform (DCT) weights of an image to be mapped on a specified surface of a particular primitive of a plurality of primitives corresponding to an object in a three-dimensional (3D) space; and determining a respective EM field contribution from the particular primitive to each of a plurality of elements of a display by taking into consideration of an effect of the plurality of DCT weights of the image. 
     In some implementations, the method further includes: determining a resolution for the image to be mapped on the specified surface of the particular primitive; and determining the plurality of DCT weights of the image based on the resolution. 
     In some implementations, the method further includes decoding the DCT weights of the image to obtain a respective DCT amplitude for each pixel of the image. 
     In some implementations, the method further includes storing values associated with the respective DCT amplitudes of the pixels of the image together with primitive data of the particular primitive. Determining the respective EM field contribution can include calculating the respective EM field contribution from the particular primitive to each of the plurality of elements with the values associated with the respective DCT amplitudes of the pixels of the image. 
     In some implementations, the method further includes selecting particular DCT terms to be included in the determining of the respective EM field contribution, each of the particular DCT terms having a respective DCT weight higher than a predetermined threshold. 
     Another aspect of the present disclosure features a method that includes: obtaining information of a given primitive and an occluder of the given primitive, wherein the given primitive is within a plurality of primitives corresponding to an object in a three-dimensional (3D) space; and determining one or more particular elements of a plurality of elements of a display that do not contribute to a reconstruction of the given primitive as an effect of the occluder. 
     In some implementations, the method further includes storing the information of the particular elements with the information of the given primitive and the occluder. 
     In some implementations, the determining is performed during a rendering process of the object for obtaining primitive data of the plurality of primitives. 
     In some implementations, the method further includes transmitting the stored information of the particular elements with the information of the given primitive and the occluder to a controller configured to calculate electromagnetic (EM) contributions for the plurality of primitives to the plurality of elements of the display. 
     In some implementations, the method further includes, for each one of the particular elements, generating a sum of electromagnetic (EM) field contributions from the plurality of primitives to the one of the particular elements by excluding an EM field contribution from the given primitive to the one of the particular elements. 
     In some implementations, the method further includes, for each of the plurality of elements other than the particular elements, generating a respective sum of EM field contributions from the plurality of primitives to the element. 
     In some implementations, the method further includes masking an EM field contribution of the particular elements to the given primitive. 
     In some implementations, determining the one or more particular elements includes: connecting the given primitive to endpoints of the occluder; extending the connection to the display to determine intersections between the connection and the display; and determining a particular range defined by the intersections to be the particular elements that do not contribute to the reconstruction of the given primitive at the effect of the occluder. 
     Another aspect of the present invention features a method that includes: obtaining information of a given primitive and an occluder of the given primitive, wherein the given primitive is within a plurality of primitives corresponding to an object in a three-dimensional (3D) space; and for each of a plurality of elements of a display, determining a respective part of the given primitive that does not make an electromagnetic (EM) field contribution to the element as an effect of the occluder. 
     In some implementations, the method further includes storing the information of the respective part of the given primitive with the information of the given primitive and the occluder. 
     In some implementations, the determining is performed during a rendering process of the object for obtaining primitive data of the plurality of primitives. 
     In some implementations, the method further includes transmitting the stored information of the respective part of the given information with the information of the given primitive and the occluder to a controller configured to calculate electromagnetic (EM) contributions for the plurality of primitives to the plurality of elements of the display. 
     In some implementations, the method further includes masking an EM field contribution of each of the plurality of elements to the respective part of the given primitive. 
     In some implementations, the method further includes, for each of the plurality of elements, generating a sum of EM field contributions from the plurality of primitives to the element by excluding an EM field contribution from the respective part of the given primitive to the element. Generating the sum of EM field contributions from the plurality of primitives to the element can include subtracting the EM contribution of the respective part of the given primitive to the element from the sum of EM field contributions from the plurality of primitive to the element without the effect of the occluder. Generating the sum of EM field contributions from the plurality of primitives to the element can include summing EM field contributions from one or more other parts of the given primitive to the element, the respective part and the one or more other parts forming the given primitive. 
     In some implementations, determining a respective part of the given primitive that do not make an EM field contribution to the element as an effect of the occluder includes: connecting the element to endpoints of the occluder; determining intersections between the connection and the given primitive; and determining a particular part of the given primitive that is enclosed by the intersections to be the respective part of the given primitive that does not make the EM field contribution to the element at the effect of the occluder. 
     Another aspect of the present disclosure features a method that includes obtaining respective primitive data of each of a plurality of primitives corresponding to an object in a three-dimensional (3D) space; obtaining respective geometric specular information for each of the plurality of primitives; and storing the respective geometric specular information with respective primitive data for each of the plurality of primitives. 
     In some implementations, the respective geometric specular information for each of the plurality of primitives includes a reflectivity of a surface of the primitive upon a viewing angle. 
     In some implementations, the method further includes determining a respective EM field contribution from each of the plurality of primitives to each of a plurality of elements of a display by taking into consideration of the respective geometric specular information for the primitive. 
     Another aspect of the present disclosure features a method that includes: obtaining graphic data comprising respective primitive data for a plurality of primitives corresponding to an object in a three-dimensional (3D) space; determining, for each of the plurality of primitives, an electromagnetic (EM) field contribution to each of a plurality of elements of a display by calculating, in a 3D coordinate system, an EM field propagation from the primitive to the element; generating, for each of the plurality of elements, a sum of the EM field contributions from the plurality of primitives to the element; transmitting, for each of the plurality of elements, a respective control signal to the element, the control signal being for modulating at least one property of the element based on the sum of the EM field contributions to the element; and transmitting a timing control signal to an illuminator to activate the illuminator to illuminate light on the display such that the light is caused by the modulated elements of the display to form a volumetric light field corresponding to the object. 
     Another aspect of the disclosure features a method that includes: for each of a plurality of elements of a display, altering a respective control signal with a predetermined calibration value; applying the respective altered respective control signals to the plurality of elements of the display; measuring an output of light incident on the display; and evaluating the predetermined calibration value based on the measurement of the output of the light. 
     In some implementations, the predetermined calibration value is the same for each of the plurality of elements. 
     In some implementations, the method further includes converting the respective control signals of the plurality of elements by a digital-to-analog converter (DAC), wherein altering the respective control signals for the plurality of elements includes altering digital signals of the respective control signals with the predetermined calibration value. 
     In some implementations, the predetermined value comprises a plurality of bits. 
     In some implementations, the method further includes adjusting the predetermined calibration value based on a result of the evaluation. Adjusting the predetermined calibration value can include modifying one or more values of the plurality of bits. Adjusting the predetermined calibration value can include determining a combination of values of the plurality of bits based on the predetermined calibration value and another calibration value determined from a previous evaluation. 
     In some implementations, the output of the light comprises a phase change of the light or an intensity difference between the output of the light and a background. 
     In some implementations, the respective control signal of the element is determined based on a sum of electromagnetic (EM) field contributions from a plurality of primitives corresponding to an object to the element in a 3D space. 
     Another aspect of the disclosure features a method that includes, for each of a plurality of elements of a display: obtaining a respective sum of electromagnetic (EM) field contributions from a plurality of primitives in a three-dimensional (3D) space, the plurality of primitives corresponding to an object in the 3D space; applying a respective mathematical transform to the respective sum of EM field contributions for the element to obtain a respective transformed sum of EM field contributions for the element; determining a respective control signal based on the respective transformed sum of EM field contributions for the element; and modulating a property of the element based on the determined respective control signal for the element. 
     In some implementations, the method further includes: introducing light incident on the plurality of elements of the display; measuring a first output of the light; and adjusting one or more coefficients of the respective mathematical transforms of the plurality of elements based on a result of the measurement of the first output of the light. The method can further include: changing a depth of a holographic pattern corresponding to the object in view of the display; measuring a second output of the light; and adjusting the one or more coefficients of the respective mathematical transforms based on the first and second outputs. The method can further include: changing the plurality of primitives corresponding to a first holographic pattern to a second plurality of primitives corresponding to a second holographic pattern; measuring a second output of the light; and adjusting the one or more coefficients of the respective mathematical transforms based on the first and second outputs. The first holographic pattern and the second holographic pattern can correspond to the object. The second holographic pattern can correspond to a second object different from the object related to the first holographic pattern. The first output of the light can be measured by an imaging sensor (e.g., a point sensor or a spatially integrating sensor or a three-dimensional sensor such as a light-field sensor). The imaging sensor can be configured to use a machine vision algorithm to determine what is being displayed and calculate a fitness parameter. Each of the first and second holographic patterns can include a grid of dots or other fiducial elements, wherein the fitness parameter is at least one of: how close the dots or other fiducial elements are together; how close the dots or other fiducial elements are to their intended positions colors and intensities; how well centered the dots or other fiducial elements are positioned with respect to their intended positions, and how distorted the dots or other fiducial elements are. 
     In some implementations, the mathematical transform is derived from a Zernike polynomial expression. 
     In some implementations, the mathematical transforms for the plurality of elements vary element-by-element. 
     In some implementations the method further includes: reproducing a sample set of known colors and intensities by illuminating the display; measuring an output light using a colorimeter device which can be calibrated to CIE standard observer curves; and defining the output light of the display in a color space such as a CIE color space. The method can further include: determining a deviation of values of the defined output light from known standard values; and adapting illumination into the display or the generation of output colors and intensities by the display to bring them back into alignment, e.g., conformance with standard or desired values. 
     Another aspect of the disclosure features a method that includes: determining a cell gap of a liquid crystal (LC) display based on a pitch of display elements of the LC display; and calculating a minimum value of a birefringence of an LC mixture based on the cell gap and a predetermined retardance for the LC display. 
     In some implementations, the method further includes improving a switching speed of the LC display by keeping the birefringence of the LC mixture above the minimum value. Improving the switching speed can include at least one of: increasing dielectric anisotropy of the LC mixture; and decreasing the rotational viscosity of the LC mixture. 
     In some implementations, the LC display includes a liquid crystal on silicon (LCOS or LCoS) device having a silicon backplane. 
     In some implementations, the LC display includes: a liquid crystal layer; a transparent conductive layer on top of the liquid crystal layer as a common electrode; and a backplane comprising a plurality of metal electrodes on or electrically close to the bottom of the liquid crystal layer, wherein each of the plurality of metal electrodes is isolated from each other, and the backplane is configured to control a voltage of each of the plurality of metal electrodes. 
     Another aspect of the disclosure features a display that includes: a backplane; and a plurality of display elements on the backplane, wherein at least two of the plurality of display elements have different sizes. 
     In some implementations, a larger one of the at least two display elements comprises a buffer, and a smaller one of the at least two display elements comprises no buffer. The larger display element can be connected with a first plurality of display elements by a conductive line, wherein the buffer is configured to buffer a voltage applied on the conductive line such that the voltage is only applied to a second plurality of display elements within the first plurality of display elements, a number of the second plurality of display elements being smaller a number of the first plurality of display elements. 
     In some implementations, the buffer comprises an analog circuit in a form of a transistor or a digital circuit in a form of logic gates. 
     In some implementations, a size distribution of the plurality of display elements is substantially identical to a size of a smaller one of the at least two display elements. 
     In some implementations, the display is configured to be a liquid crystal on silicon device. 
     Another aspect of the disclosure features a display that includes: a backplane; and a plurality of display elements on the backplane, wherein at least two of the plurality of display elements have different shapes. 
     In some implementations, the backplane includes a respective circuit for each of the display elements, wherein the respective circuits for the at least two display elements have shapes corresponding to the different shapes of the at least two display elements. 
     In some implementations, a size distribution of the plurality of display elements is substantially identical to a predetermined size. 
     In some implementations, the display is configured to be a liquid crystal on silicon device. 
     Another aspect of the present disclosure features a method including: obtaining graphic data including respective primitive data for a plurality of primitives corresponding to an object in a three-dimensional (3D) space; determining, for each of the plurality of primitives, an electromagnetic (EM) field contribution to each of a plurality of elements of a display by calculating, in a 3D coordinate system, an EM field propagation from the primitive to the element; generating, for each of the plurality of elements, a sum of the EM field contributions from the plurality of primitives to the element; transmitting, for each of the plurality of elements, a respective control signal to the element, the control signal being for modulating at least one property of the element based on the sum of the EM field contributions to the element; and transmitting a timing control signal to an illuminator to activate the illuminator to illuminate light on the display such that the light is caused by the modulated elements of the display to form a volumetric light field corresponding to the object. 
     Other embodiments of the aspects include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. For a system of one or more computers to be configured to perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by data processing apparatus, cause the apparatus to perform the operations or actions. 
     Another aspect of the present disclosure features a device that includes: one or more processors; and a non-transitory computer readable storage medium in communication with the one or more processors and storing instructions executable by the one or more processors and upon such execution cause the one or more processors to perform one or more of the methods disclosed herein. 
     Another aspect of the present disclosure features a non-transitory computer readable storage medium storing instructions executable by one or more processors and upon such execution cause the one or more processors to perform the method according to one or more of the methods disclosed herein. 
     Another aspect of the present disclosure features a display including a plurality of elements; and a controller coupled to the display and configured to perform one or more of the methods disclosed herein. The controller can include a plurality of computing units, each of the computing units being configured to perform operations on one or more primitives of a plurality of primitives correspond to an object in a three-dimensional (3D) space. In some implementations, the controller is locally coupled to the display, and each of the computing units is coupled to one or more respective elements of the display and configured to transmit a respective control signal to each of the one or more respective elements. The computing units can be configured to operate in parallel. 
     The controller can include at least one of an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), a programmable gate array (PGA), a central processing unit (CPU), a graphics processing unit (GPU), or standard or custom computing cells. The display can include a spatial light modulator (SLM) including a digital micro-mirror device (DMD) or a liquid crystal on silicon (LCOS or LCoS) device. The display can be configured to be phase modulated, amplitude modulated, or phase and amplitude modulated. The controller can be coupled to the display through a memory buffer. 
     In some implementations, the system includes an illuminator arranged adjacent to the display and configured to emit light on the display. The illuminator can be coupled to the controller and configured to be turned on/off based on a control signal from the controller. 
     In some cases, the illuminator is coupled to the controller through a memory buffer configured to control amplitude or brightness of one or more light emitting elements in the illuminator. The memory buffer for the illuminator can have a smaller size than a memory buffer for the display. A number of the light emitting elements in the illuminator can be smaller than a number of the elements of the display. The controller can be configured to simultaneously or sequentially activate the one or more light emitting elements of the illuminator. 
     The illuminator can be a coherent light source, a semi-coherent light source, or an incoherent light source. In some implementations, the illuminator is configured to emit a white light, and wherein the display is configured to diffract the white light into light with different colors. In some implementations, the illuminator includes two or more light emitting elements each configured to emit light with a different color. The controller can be configured to sequentially modulate the display with information associated with a first color during a first time period and modulate the display with information associated with a second color during a second, sequential time period, and the controller can be configured to control the illuminator to sequentially activate a first light emitting element to emit light with the first color during the first time period and a second light emitting element to emit light with the second color during the second time period. 
     In some implementations, the illuminator is arranged in front of a surface of the display and configured to emit the light on to the surface of the display with an incident angle within a range between 0 degree and 90 degrees, and the emitted light is diffracted from the display. In some cases, the emitted light from the illuminator includes collimated light. In some cases, the emitted light from the illuminator includes divergent light. In some cases, the emitted light from the illuminator includes convergent light. In some cases, the emitted light from the illuminator includes semi-collimated light. 
     In some implementations, the illuminator is arranged behind a rear surface of the display and configured to emit a divergent collimated, semi-collimated, or convergent light on the rear surface of the display, and the emitted light is transmitted through the display and diffracted out of the display from a front surface of the display. 
     In some implementations, the illuminator includes: a light source configured to emit the light; and a waveguide coupled to the light source and arranged adjacent to the display, the waveguide being configured to receive the emitted light from the light source and guide the emitted light to the display. In some cases, the light from the light source is coupled to the waveguide from a side cross-section of the waveguide through a light coupler. In some cases, the light source and the waveguide are integrated in a planar form and positioned on a surface of the display. The waveguide can be configured to guide the light to illuminate the display uniformly. 
     In some cases, the waveguide is positioned on or optically close to a rear surface of the display, and the light is guided to transmit into the display and transmitted and diffracted out of the display from a front surface of the display. The controller can be positioned on a rear surface of the waveguide. In some cases, the waveguide or lightguide is positioned on or optically close to a front surface of the display, and wherein the light is guided to be incident on the front surface of the display and reflected and diffracted back out through the front surface. 
     Another aspect of the present disclosure features a system including: a display including an array of elements; and an integrated circuit including an array of computing units, each of the computing units being coupled to one or more respective elements of the display and configured to: compute an electromagnetic (EM) field contribution from at least one primitive of a plurality of primitives to each of the array of elements; and generate, for each of the one or more respective elements, a respective sum of the EM field contributions from the plurality of primitives to the element. 
     Each of the computing units can be configured to: receive, from other computing units of the array of computing units, computed EM field contributions from other primitives of the plurality of primitives to each of the one or more respective elements; and generate, for each of the one or more respective elements, the respective sum of the EM field contributions by adding the received computed EM field contributions from the other primitives to the element. 
     Each of the computing units can be configured to generate, for each of the one or more respective elements, a respective control signal to modulate at least one property of the element based on the respective sum of the EM field contributions to the element. 
     In some implementations, the integrated circuit includes a respective accumulator configured to store an accumulation result of the computed EM field contribution from the plurality of primitives to each of the elements of the display. The integrated circuit can be configured to clear the accumulators at a beginning of a computation operation. In some examples, the integrated circuit includes a respective memory buffer for each of the elements, and the integrated circuit can be configured to accumulate the computed EM field contribution from the plurality of primitives to the element to obtain the respective sum of the EM field contributions as a final accumulation result in the respective accumulator and transfer the final accumulation result from the respective accumulator to the respective memory buffer for the element. 
     In some implementations, the system further includes an illuminator positioned between the integrated circuit and the display and configured to receive a control signal from the integrated circuit and illuminate light on the display based on the control signal, and the integrated circuit, the illuminator, and the display can be integrated as a single unit. 
     Another aspect of the present disclosure features a system, including: a computing device configured to generate data including respective primitive data of a plurality of primitives corresponding to an object in a three-dimensional (3D) space; and the system as disclosed herein. The system is configured to receive the graphic data from the computing device and process the graphic data for presenting the object in the 3D space. The computing device can include an application programming interface (API) configured to create the primitives with the respective primitive data by rendering a computer generated (CG) model of the object. 
     Another aspect of the present disclosure features an optical device, including: a first optically diffractive component; a second optically diffractive component; and a color-selective polarizer between the first and second optically diffractive components. When a first beam of light including a first color of light in a first polarization state is incident on the first optically diffractive component, the first optically diffractive component diffracts the first color of light in the first polarization state; when a second beam of light including a second color of light in a second polarization state is incident on the color-selective polarizer, the color-selective polarizer converts the second beam of light to a third beam of light including the second color of light in the first polarization state, the second color being different from the first color, and the second polarization state being different from the first polarization state; when the third beam of light is incident on the second optically diffractive component, the second optically diffractive component diffracts the second color of light in the first polarization state; and a diffraction efficiency with which the first optically diffractive component diffracts the second color of light in the second polarization state is substantially smaller than a diffraction efficiency with which the first optically diffractive component diffracts the first color of light in the first polarization state. 
     Another aspect of the present disclosure features an optical device including: a first optically diffractive component; a second optically diffractive component; and a color-selective polarizer between the first and second optically diffractive components. When a first color of light is incident on the first optically diffractive component at a first incident angle and in a first polarization state, the first optically diffractive component diffracts the first color of light at a first diffracted angle with a first diffraction efficiency; when a second color of light different from the first color of light is incident on the first optically diffractive component at a second incident angle in a second polarization state different from the first polarization state, the first optically diffractive component diffracts the second color of light with a diffraction efficiency that is substantially less than the first diffraction efficiency; when the second color of light in the second polarization state is incident on the color-selective polarizer, the color-selective polarizer rotates a polarization state of the second color of light from the second polarization state to the first polarization state; and when the second color of light is incident on the second optically diffractive component at the second incident angle and in the first polarization state, the second optically diffractive component diffracts the second color of light at a second diffracted angle with a second diffraction efficiency. 
     Another aspect of the present disclosure features an optical device including: a first optically diffractive component configured to: i) diffract a first color of light in a first polarization state incident at a first incident angle with a first diffraction efficiency at a first diffracted angle; and ii) diffract a second color of light in a second polarization state incident at a second incident angle with a diffraction efficiency that is substantially less than the first diffraction efficiency; a color-selective polarizer configured to rotate a polarization state of the second color of light in the second polarization state incident on the color-selective polarizer from the second polarization state to the first polarization state; and a second optically diffractive component configured to diffract the second color of light in the first polarization state incident at the second incident angle with a second diffraction efficiency at a second diffracted angle, where the color-selective polarizer is between the first and second optically diffractive components. 
     In some implementations, the second optically diffractive component is configured to diffract the first color of light in the second polarization state at the first incident angle with a diffraction efficiency substantially smaller than the second diffraction efficiency. 
     In some implementations, the first optically diffractive component, the color-selective polarizer, and the second optically diffractive component are sequentially stacked, such that the first color of light and the second color of light are incident on the first optically diffractive component before the second optically diffractive component. 
     In some implementations, the optical device further includes: a third optically diffractive component; and a second color-selective polarizer between the second and third optically diffractive components. The second color-selective polarizer is configured to: when a third color of light is incident in the second polarization state on the second color-selective polarizer, rotate a polarization state of the third color of light from the second polarization state to the first polarization state. The third optically diffractive component is configured to: when the third color of light is incident on the third optically diffractive component at a third incident angle and in the first polarization state, diffract the third color of light at a third diffracted angle with a third diffraction efficiency. 
     In some implementations, the color-selective polarizer is configured to rotate a polarization state of the first color of light from the first polarization state to the second polarization state, and the second color-selective polarizer is configured to rotate the polarization state of the second color of light from the first polarization state to the second polarization state, without rotation of the polarization state of the first color of light. 
     In some implementations, the optical device further includes: a third color-selective polarizer configured to rotate the polarization state of each of the first and second colors of light from the second polarization state to the first polarization state, without rotation of the polarization state of the third color of light. The third optically diffractive component is between the second and third color-selective polarizers. 
     In some implementations, the third optically diffractive component is configured to diffract each of the first and second colors of light incident in the second polarization state with a diffraction efficiency substantially smaller than the third diffraction efficiency. The first optically diffractive component is configured to diffract the third color of light incident in the second polarization state with a diffraction efficiency substantially smaller than the first diffraction efficiency, and the second optically diffractive component is configured to diffract each of the first and third colors of light incident in the second polarization state with a diffraction efficiency substantially smaller than the second diffraction efficiency. 
     In some implementations, the second color-selective polarizer includes a pair of a first sub-polarizer and a second sub-polarizer. The first sub-polarizer is configured to rotate the polarization state of the second color of light from the first polarization state to the second polarization state, without rotation of the polarization state of each of the first and third colors of light, and the second sub-polarizer is configured to rotate the polarization state of the third color of light from the second polarization state to the first polarization state, without rotation of the polarization state of each of the first and second colors of light. 
     In some implementations, the optical device further includes: a fourth color-selective polarizer configured to rotate a polarization state of the first color of light from the second polarization state to the first polarization state, without rotation of the polarization state of each of the second and third colors of light, where the first optically diffractive component is between the fourth color-selective polarizer and the color-selective polarizer. 
     In some implementations, each of the first, second, and third optically diffractive components includes a respective holographic grating formed in a recording medium. The recording medium can include a photosensitive polymer. The recording medium can be optically transparent. The respective holographic grating can be fixed in the recording medium. 
     In some implementations, each of the first, second, and third optically diffractive components includes a carrier film attached to a side of the recording medium. Each of the first, second, and third optically diffractive components can include a diffraction substrate attached to another side of the recording medium opposite to the carrier film. 
     In some cases, the carrier film of the first optically diffractive component is attached to a first side of the color-selective polarizer, and the diffraction substrate of the second optically diffractive component is attached to a second, opposite side of the color-selective polarizer, and the carrier film of the second optically diffractive component is attached to a first side of the second color-selective polarizer, and the diffraction substrate of the second optically diffractive component is attached to a second, opposite side of the second color-selective polarizer. 
     In some implementations, the optical device further includes a substrate, and the first optically diffractive component is between the substrate and the color-selective polarizer. In some implementations, the optical devices further includes: an anti-reflective coating on a surface of the substrate. In some implementations, the optical device includes: a front surface and a back surface, where the first color of light and the second color of light are incident on the front surface, and the optical device further includes: an anti-reflective coating on the back surface. 
     In some implementations, the optical device includes a plurality of optical components including the first optically diffractive component, the color-selective polarizer, and the second optically diffractive component, where adjacent two optical components of the plurality of components are attached together through a refractive index matching material. 
     In some implementations, each of the first and second optically diffractive components includes a respective Bragg grating formed in a recording medium, and the respective Bragg grating includes a plurality of fringe planes with a fringe tilt angle θ t  and a fringe spacing A perpendicular to the fringe planes in a volume of the recording medium. 
     In some cases, the respective Bragg grating is configured such that, when an incident angle on the recording medium is an on-Bragg angle, a respective diffracted angle θ m  is satisfied with Bragg’s equation as below: mλ = 2 n Λ sin( θ m  - θ t  ), where λ, represents a respective wavelength of a color of light in vacuum, n represents a refractive index in the recording medium, θ m , represents m th  diffraction order Bragg angle in the recording medium, and θ t , represents a fringe tilt in the recording medium. 
     In some cases, each of the first and second incident angles is substantially identical to the on-Bragg angle, and each of the first and second diffracted angles is substantially identical to first order Bragg angle. 
     In some cases, the fringe tilt angle of the respective Bragg grating is substantially identical to 45 degrees. 
     In some cases, a thickness of the recording medium is more than one order of magnitude larger than the fringe spacing. The thickness of the recording medium can be about 30 times larger than the fringe spacing. 
     In some cases, the first diffracted angle and the second diffracted angle are substantially identical to each other. 
     In some cases, each of the first and second diffracted angles is in a range from -10 degrees to 10 degrees. Each of the first and second diffracted angles can be substantially identical to 0 degrees. Each of the first and second diffracted angles can be in a range from -7 degrees to 7 degrees. Each of the first and second diffracted angles can be substantially identical to 6 degrees. 
     In some cases, each of the first and second incident angles is in a range from 70 degrees to 90 degrees. The first incident angle and the second incident angle can be substantially identical to each other. 
     In some cases, the first polarization state is s polarization, and the second polarization state is p polarization. 
     In some implementations, the first optically diffractive component is configured to diffract the second color of light incident in the second polarization state with the diffraction efficiency that is at least one order of magnitude smaller than the first diffraction efficiency. 
     In some implementations, the color-selective polarizer is configured not to rotate a polarization state of the first color of light. 
     In some implementations, the optical device further includes: a second color-selective polarizer configured to rotate a polarization state of the first color of light from the second polarization state to the first polarization state, without rotation of the polarization state of the second color of light, where the first optically diffractive component is between the second color-selective polarizer and the color-selective polarizer. 
     In some implementations, the first optically diffractive component includes a first diffractive structure, and the second optically diffractive component including a second diffractive structure, where the optical device includes a first reflective layer and a second reflective layer, where the first reflective layer is between the first and second diffractive structures, and the second diffractive structure is between the first and second reflective layers, where the first diffractive structure is configured to: i) diffract first and zero orders of the first color of light incident at the first incident angle on the first diffractive structure, the first order being diffracted at the first diffracted angle, and the zero order being transmitted at the first incident angle; and ii) transmit the second color of light incident at the second incident angle on the first diffractive structure, where the first reflective layer is configured to: i) totally reflect the first color of light incident on the first reflective layer at the first incident angle; and ii) transmit the second color of light incident on the first reflective layer at the second incident angle, where the second diffractive structure is configured to diffracts first and zero orders of the second color of light incident at the second incident angle on the second diffractive structure, the first order being diffracted at a second diffracted angle, and the zero order being transmitted at the second incident angle, and where the second reflective layer is configured to totally reflect the second color of light incident on the second reflective layer at the second incident angle. 
     Another aspect of the present disclosure features an optical device including: a first optically diffractive component including a first diffractive structure; a second optically diffractive component including a second diffractive structure; a first reflective layer; and a second reflective layer. The first reflective layer is between the first and second diffractive structures; the second diffractive structure is between the first and second reflective layers; when a first color of light is incident at a first incident angle on the first diffractive structure, the first diffraction structure diffracts first and zero orders of the first color, the first order being diffracted at a first diffracted angle, and the zero order being transmitted at the first incident angle; when a second color of light is incident at a second incident angle on the first diffractive structure, the first diffraction grating transmits the second color of light at the second incident angle; when the first color of light is incident on the first reflective layer at the first incident angle, the first reflective layer totally reflects the first color of light; when the second color of light is incident on the first reflective layer at the second incident angle, the reflective layer transmits the second color of light at the second incident angle; when the second color of light is incident at the second incident angle on the second diffractive structure, the second diffractive structure diffracts first and zero orders of the second color of light, the first order being diffracted at a second diffracted angle, and the zero order being transmitted at the second incident angle; and when the second color of light is incident on the second reflective layer at the second incident angle, the second reflective layer totally reflects the second color of light. 
     Another aspect of the present disclosure features an optical device including: a first optically diffractive component including a first diffractive structure configured to: i) diffract first and zero orders of a first color of light incident at a first incident angle on the first diffractive structure, the first order being diffracted at a first diffracted angle, and the zero order being transmitted at the first incident angle; and ii) transmit a second color of light incident at a second incident angle on the first diffractive structure; a first reflective layer configured to: i) totally reflect the first color of light incident on the first reflective layer at the first incident angle; and ii) transmit the second color of light incident on the first reflective layer at the second incident angle; a second optically diffractive component including a second diffractive structure configured to diffract first and zero orders of the second color of light incident at the second incident angle on the second diffractive structure, the first order being diffracted at a second diffracted angle, and the zero order being transmitted at the second incident angle; and a second reflective layer configured to totally reflect the second color of light incident on the second reflective layer at the second incident angle, where the first reflective layer is between the first and second diffractive structures, and the second diffractive structure is between the first and second reflective layers. 
     Another aspect of the present disclosure features an optical device including: a first optically diffractive component including a first diffractive structure configured to diffract a first color of light having a first incident angle at a first diffracted angle; a second optically diffractive component including a second diffractive structure configured to diffract a second color of light having a second incident angle at a second diffracted angle; a first reflective layer configured to totally reflect the first color of light having the first incident angle and transmit the second color of light having the second incident angle; and a second reflective layer configured to totally reflect the second color of light having the second incident angle, where the first reflective layer is between the first and second diffractive structures, and the second diffractive structure is between the first and second reflective layers. 
     In some implementations, the optical device further includes: a color-selective polarizer between the first and second diffractive structures. The first diffractive structure can be configured to: i) diffract the first color of light in a first polarization state incident at the first incident angle with a first diffraction efficiency; and ii) diffract the second color of light in a second polarization state incident at the second incident angle with a diffraction efficiency that is substantially less than the first diffraction efficiency. The color-selective polarizer can be configured to rotate a polarization state of the second color of light in the second polarization state incident on the color-selective polarizer from the second polarization state to the first polarization state. The second diffractive structure can be configured to diffract the second color of light in the first polarization state incident at the second incident angle with a second diffraction efficiency. 
     In some implementations, the optical device further includes: a side surface and an optical absorber attached to the side surface and configured to absorb totally reflected light of the first and second colors. 
     In some implementations, the first reflective layer is configured to have a refractive index smaller than that of a layer of the first optically diffractive component that is immediately adjacent to the first reflective layer, such that the first color of light having the first incident angle is totally reflected by an interface between the first reflective layer and the layer of the first optically diffractive component, without totally reflecting the second color of light having the second incident angle. 
     In some implementations, the first optically diffractive component includes a first carrier film and a first diffraction substrate attached to opposite sides of the first diffractive structure, the first carrier film being closer to the second diffractive structure than the first diffraction substrate, and the first carrier film can include the first reflective layer. 
     In some implementations, the second optically diffractive component includes a second carrier film and a second diffraction substrate attached to opposite sides of the second diffractive structure, the second diffraction substrate being closer to the first diffractive structure than the second carrier film, and the second reflective layer is attached to the second carrier film. 
     In some implementations, the optical device further includes: a third optically diffractive component including a third diffractive structure configured to diffract first and zero orders of a third color of light incident at a third incident angle on the third diffractive structure, the first order being diffracted at a third diffracted angle, and the zero order being transmitted at the third incident angle, and the second reflective layer is between the second diffractive structure and the third diffractive structure. 
     In some cases, each of the first and second reflective layers is configured to transmit the third color of light incident at the third incident angle. 
     In some implementations, the optical device further includes: a third reflective layer configured to totally reflect the third color of light incident at the third incident angle on the third reflective layer, where the third diffractive structure is between the second and third reflective layers. 
     In some implementations, the second optically diffractive components includes a second diffraction substrate and a second carrier film arranged on opposite sides of the second diffractive structure, the third optically diffractive component includes a third carrier film and a third diffraction substrate positioned on opposite sides of the third diffractive structure, and the second reflective layer is between the second and third carrier films. 
     In some implementations, each of the first and second diffractive structure includes a respective holographic grating formed in a recording medium. The recording medium can include a photosensitive polymer. The recording medium can be optically transparent. 
     In some implementations, each of the first and second optically diffractive components includes a respective Bragg grating formed in the recording medium, and the respective Bragg grating includes a plurality of fringe planes with a fringe tilt angle θ t  and a fringe spacing A perpendicular to the fringe planes in a volume of the recording medium. 
     In some implementations, the respective Bragg grating is configured such that, when an incident angle on the recording medium is an on-Bragg angle, a respective diffracted angle θ m  is satisfied with Bragg’s equation as below: 
     
       
         
           
             m 
             λ 
             = 
             2 
               
             n 
               
             Λ 
               
             sin 
             
               
                 
                   θ 
                   m 
                 
                 − 
                 
                   θ 
                   t 
                 
               
             
             , 
           
         
       
     
      where λ, represents a respective wavelength of a color of light in vacuum, n represents a refractive index in the recording medium, θ m  represents m th  diffraction order Bragg angle in the recording medium, θ t  represents the fringe tilt in the recording medium. 
     Each of the first and second incident angles can be substantially identical to a respective on-Bragg angle, and each of the first and second diffracted angles can be substantially identical to a respective first order Bragg angle. 
     In some implementations, a thickness of the recording medium is more than one order of magnitude larger than the fringe spacing. The thickness of the recording medium can be about 30 times larger than the fringe spacing. 
     In some cases, the first diffracted angle and the second diffracted angle are substantially identical to each other. In some examples, each of the first and second diffracted angles is in a range from -10 degrees to 10 degrees. In some examples, each of the first and second diffracted angles is substantially identical to 0 degrees. In some examples, each of the first and second diffracted angles is substantially identical to 6 degrees. 
     In some cases, the first incident angle is different from the second incident angle. In some cases, the first color of light has a wavelength smaller (or shorter) than the second color of light, and the first incident angle of the first color of light is larger (or longer) than the second incident angle of the second color of light. In some cases, each of the first and second incident angles is in a range from 70 degrees to 90 degrees. 
     In some implementations, the optical device includes a plurality of components including the first optically diffractive component and the second optically diffractive component, and adjacent two components of the plurality of components are attached together by an intermediate layer that includes at least one of a refractive index matching material, an OCA, a UV-cured or heat-cured optical glue, or an optical contacting material. 
     In some implementations, the second reflective layer includes the intermediate layer. 
     In some implementations, the optical device further includes a substrate having a back surface attached to a front surface of the first optically diffractive component. The substrate can include a side surface angled to the back surface and is configured to receive a plurality of different colors of light at the side surface. An angle between the side surface and the back surface of the substrate can be no less than 90 degrees. The substrate can be configured such that the plurality of different colors of light are incident on the side surface with an incident angle substantially identical to 0 degrees. In some cases, the substrate is wedged and includes a titled front surface, and an angle between the front surface and the side surface is less than 90 degrees. 
     Another aspect of the present disclosure features a system including: an illuminator configured to provide a plurality of different colors of light and any one of the optical devices described herein. The optical device is arranged adjacent to the illuminator and configured to receive the plurality of different colors of light from the illuminator and diffract the plurality of different colors of light. 
     In some implementations, the optical device is configured to diffract the plurality of different colors of light at respective diffracted angles that are substantially identical to each other. 
     In some examples, each of the respective diffracted angles is in a range of -10 degrees to 10 degrees. 
     In some implementations, the system further includes: a controller coupled to the illuminator and configured to control the illuminator to provide each of the plurality of different colors of light. 
     In some implementations, the system further includes: a display including a plurality of display elements, and the optical device is configured to diffract the plurality of colors of light to the display. 
     In some implementations, the controller is coupled to the display and configured to transmit a respective control signal to each of the plurality of display elements for modulation of at least one property of the display element. 
     In some implementations, the controller is configured to: obtain graphic data including respective primitive data for a plurality of primitives corresponding to an object in a three-dimensional space; determine, for each of the plurality of primitives, an electromagnetic (EM) field contribution to each of the plurality of display elements of the display; generate, for each of the plurality of display elements, a sum of the EM field contributions from the plurality of primitives to the display element; and generate, for each of the plurality of display elements, the respective control signal based on the sum of the EM field contributions to the display element. 
     Another aspect of the present disclosure features a system including: a display including a plurality of display elements and any one of the optical devices as described herein, and the optical device is configured to diffract a plurality of different colors of light to the display. 
     In some implementations, the optical device and the display are arranged along a direction. The optical device includes a front surface and a back surface along the direction, and the display includes a front surface and a back surface along the direction, and the front surface of the display is spaced from the back surface of the optical device. 
     In some implementations, the front surface of the display is spaced from the back surface of the optical device by a gap. At least one of the front surface of the display or the back surface of the optical device can be treated with an anti-reflection coating. 
     In some implementations, the system further includes a transparent protective layer on the back surface of the optical device. 
     In some implementations, the front surface of the display and the back surface of the optical device are attached together by an intermediate layer. The intermediate layer can be configured to have a refractive index lower than a refractive index of a layer of the optical device, such that each of the plurality of colors of light transmitted at zero order by the optical device is totally reflected at an interface between the intermediate layer and the layer of the optical device. 
     In some implementations, the system further includes a cover (e.g., a cover glass) on the front surface of the display, where the optical device is formed in the cover glass. 
     In some implementations, the optical device is configured to receive the plurality of colors of light at the front surface of the optical device. 
     In some implementations, the optical device includes a substrate in front of the optical device and is configured to receive the plurality of colors of light at a side surface of the substrate that is angled to a back surface of the substrate. 
     In some implementations, the optical device includes at least one diffractive grating supported by the substrate and configured to diffract the plurality of different colors of light towards the display. 
     In some implementations, the substrate includes a container filled with a liquid having a refractive index smaller than a recording medium of the diffractive grating. 
     In some implementations, the substrate is wedge-shaped and comprises a titled front surface. An angle between the front surface and the side surface can be less than 90 degree. 
     In some implementations, the optical device is configured to receive different portions of the plurality of different colors of light along different optical paths in the substrate and to diffract the different portions to illuminate different corresponding regions of the display. The different regions can include two or more of a lower region, an upper region, a left region, and a right region of the display. The different portions of the plurality of different colors of light can be provided by different corresponding illuminators. The optical device can be configured to receive different portions of the plurality of different colors of light from different corresponding side surfaces of the substrate. 
     In some examples, the optical device is configured to: receive a first portion of the plurality of different colors of light from a first side surface of the substrate to the back surface of the optical device and diffract the first portion to illuminate a first region of the display, and receive a second portion of the plurality of different colors of light from a second side surface of the substrate to the front surface of the optical device, reflect the second portion back to the back surface of the optical device, and diffract the second portion to illuminate a second region of the display. The first side surface and the second side surface can be a same side surface. The second portion of the plurality of different colors of light can be reflected by total internal reflection or a reflective grating in the optical device. The substrate can also include a partially reflective surface configured to separate an input light into the first portion and the second portion. 
     In some implementations, the optical device includes at least one diffractive grating arranged at the back surface of the optical device. The diffractive grating can include different sub-regions with different corresponding diffraction efficiencies. The diffractive grating can be configured to: diffract a first portion of the plurality of different colors of light incident at a first sub-region of the diffractive grating to illuminate a first region of the display and reflect a second portion of the plurality of different colors of light to the front back of the optical device that is further reflected back to the back surface of the optical device and incident at a second sub-region of the diffractive grating, and diffract the second portion to illuminate a second, different region of the display. 
     In some examples, the diffractive grating is configured such that the diffracted first portion and the diffracted second portion on the first region and the second region of the display have a substantially same optical power. The first and second regions of the display can have different reflectivities that are associated with first and second different diffraction efficiencies of the first and second sub-regions of the diffractive grating. 
     In some implementations, the diffractive grating includes a plurality of sub-regions that are tiled together. The sub-regions can be tiled along a horizontal direction. 
     In some cases, edges of the different sub-regions are configured to abut each other in an optically seamless manner. The different sub-regions can be formed by including one or more edge-defining elements in an optical path of at least one of a recording beam or an object beam during recording each sub-region in a recording medium, and the one or more edge-defining elements can include a square aperture, a rectangular aperture, or a plane-tiling aperture. 
     In some cases, two adjacent sub-regions of the diffractive grating abut with a gap. The display can include multiple tiled display devices, and the gap between the adjacent sub-regions of the diffractive grating is aligned with a gap between adj acent tiled display devices of the display. 
     In some cases, two adjacent different sub-regions have an overlap. 
     In some implementations, the diffractive grating is mechanically formed by using an embossed, nano-imprinted, or self-assembled structure. 
     In some implementations, the display has a width along a horizontal direction and a height along a vertical direction, both the horizontal direction and the vertical direction being perpendicular to the direction, and an aspect ratio between the width and the height can be larger than 16:9. 
     In some implementations, the optical device is configured to diffract a plurality of different colors of light at respective diffracted angles that are substantially identical to each other. In some examples, each of the respective diffracted angles is in a range of -10 degrees to 10 degrees. 
     In some implementations, the display is configured to diffract the diffracted colors of light back through the optical device. 
     In some implementations, an area of the optical device covers an area of the display. 
     In some implementations, the system further includes: an illuminator arranged adjacent to the optical device and configured to provide the plurality of colors of light to the optical device. The illuminator can include a plurality of light emitting elements each configured to emit a respective color of light. 
     In some implementations, centers of beams from the plurality of light emitting elements can be offset with respect to one another. The illuminator can be configured to provide a light beam with an elliptical beam profile or a rectangular beam profile. The illuminator can be configured to provide a light beam with a particular polarization orientation. The illuminator can include one or more optical components configured to independently control ellipticity and polarization orientation of each of the plurality of different colors of light. 
     In some implementations, the illuminator includes one or more optical components configured to control a uniformity of the plurality of different colors of light. The one or more optical components include apodizing optical elements or profile converters. 
     In some implementations, the system includes one or more anamorphic or cylindrical optical elements configured to increase a width of the plurality of different colors of light. 
     In some implementations, the system can further include: a prism element between the illuminator and the optical device and configured to receive the plurality of different colors of light from an input surface of the prism element; and one or more expansion gratings adjacent an exit surface of the prism element, each of the one or more expansion gratings configured to expand a beam profile of a different corresponding color of light by a factor in at least one dimension. 
     In some implementations, the system can further include: one or more reflectors downstream of the one or more expansion diffractive gratings, each of the one or more reflectors being configured to reflect a respective color of light into the optical device. A tilt angle of each of the one or more reflectors can be independently adjustable to cause a uniformity of diffraction from the optical device to the display. 
     The system can further include at least one of a color sensor or a brightness sensor configured to detect one or more optical properties of a holographic light field formed by the system, wherein the tilt angles of the one or more reflectors are adjustable based on the detected optical properties of the holographic light field. The one or more optical properties can include brightness uniformity, color uniformity, or white point. 
     In some implementations, the one or more reflectors are adjustable to correct for changes in alignment of components of the system. 
     In some implementations, an optical distance between the one or more reflectors and the optical device is configured such that each of the plurality of different colors of light is reflected by a corresponding reflector without transmission through one or more other reflectors. 
     In some implementations, the one or more reflectors are configured so that light illuminated at each of the one or more reflectors comes from a substantially different direction. 
     In some implementations, an angle between the prism element and a substrate of the optical device is adjustable to tilt a position of a holographic light field formed by the system. 
     In some implementations, the one or more expansion gratings are configured to at least partially collimate the plurality of different colors of light in one or two traverse directions. 
     In some implementations, the system further includes: a controller coupled to the illuminator and configured to control the illuminator to provide each of the plurality of colors of light. The controller can be coupled to the display and configured to transmit a respective control signal to each of the plurality of display elements for modulation of at least one property of the display element. 
     In some implementations, the controller is configured to: obtain graphic data including respective primitive data for a plurality of primitives corresponding to an object in a three-dimensional space; determine, for each of the plurality of primitives, an electromagnetic (EM) field contribution to each of the plurality of display elements of the display; generate, for each of the plurality of display elements, a sum of the EM field contributions from the plurality of primitives to the display element; and generate, for each of the plurality of display elements, the respective control signal based on the sum of the EM field contributions to the display element. 
     In some implementations, the controller is configured to: sequentially modulate the display with information associated with the plurality of colors of light in a series of time periods, and control the illuminator to sequentially emit each of the plurality of colors of light to the optical device during a respective time period of the series of time periods, such that each of the plurality of colors of light is diffracted by the optical device to the display and reflected by modulated display elements of the display to form a respective color three-dimensional light field corresponding to the object during the respective time period. 
     In some implementations, the controller is configured to modulate the display such that the respective color three-dimensional light field appears fully in front of the display, fully behind the display, or partially in front of the display and partially behind the display. 
     In some cases, the display includes a spatial light modulator (SLM) including a digital micro-mirror device (DMD) or a liquid crystal on silicon (LCOS) device. 
     In some implementations, the system further includes an optical polarizer arranged between the display and the optical device, wherein the optical polarizer is configured to change a polarization state of the plurality of different colors of light. 
     In some implementations, the optical device includes: an optical diffractive component configured to diffract light comprising the plurality of different colors of light to the display that is configured to diffract a portion of the light illuminating the display elements. 
     In some implementations, the optical device further includes: an optically redirecting component configured to transmit the portion of the light to form a holographic scene and to redirect display zero order light away from the holographic scene in a three-dimensional (3D) space, the display zero order light comprising reflected light from the display. 
     In some implementations, the optical redirecting component includes a plurality of redirecting holographic grating for the display zero order light of the plurality of different colors of light, and each of the plurality of redirecting holographic gratings is configured to diffract display zero order light of a respective color of light of the plurality of different colors of light at a respective diffractive angle towards a respective direction in the 3D space. 
     In some implementations, the optical diffractive component is configured to diffract the plurality of different colors of light to illuminate the display at an angle of about 0°, such that the optical diffractive component redirects the display zero order light reflected from the display away from the holographic scene. 
     In some implementations, a ratio between an amount of the display zero order light in the holographic scene with suppression of the optical diffractive component and the optically redirecting component and an amount of the display zero order light in the holographic scene without the suppression is less than 2%. 
     In some implementations, the optically redirecting component includes a one-dimensional suppression grating, and the holographic scene comprises a band corresponding to suppression of the display zero order light, and the system can be configured such that the band is outside of a viewing eyesight of a viewer. 
     Another aspect of the present disclosure features a system including: a display including a plurality of display elements; an optical device arranged adjacent to the display and configured to diffract light to the display; and a controller coupled to the display and configured to: obtain graphic data including respective primitive data for a plurality of primitives corresponding to an object in a three-dimensional space; determine, for each of the plurality of primitives, an electromagnetic (EM) field contribution to each of the plurality of display elements of the display by calculating, in a three-dimensional coordinate system, an EM field propagation from the primitive to the display element; generate, for each of the plurality of display elements, a sum of the EM field contributions from the plurality of primitives to the display element; and transmit, for each of the plurality of display elements, a respective control signal based on the sum of the EM field contributions to the display element for modulation of at least one property of the display element. 
     In some implementations, the optical device can include any one of the optical devices including at least one color-selective polarizer as describe herein. 
     In some implementations, the optical device includes any one of the optical devices including at least one reflective layer as described herein. 
     In some implementations, the optical device includes a holographic grating formed in a recording medium. 
     In some implementations, the optical device includes a plurality of holographic gratings formed on a recording medium, and each of the plurality of holographic gratings is configured to diffract light with a respective color having a respective incident angle to the display. 
     In some implementations, the optical device is arranged in front of the display and the display is configured to diffract the diffracted light back through the optical device to form a three-dimensional light field corresponding to the object. 
     In some implementations, the system further includes: an illuminator arranged adjacent to the optical device and configured to provide the light to the optical device. 
     In some implementations, the controller is configured to: sequentially modulate the display with information associated with a plurality of colors corresponding to a plurality of colors of light in a series of time periods, and control the illuminator to sequentially emit each of the plurality of colors of light to the optical device during a respective time period of the series of time periods, such that each of the plurality of colors of light is diffracted by the optical device to the display and reflected by modulated display elements of the display to form a respective color three-dimensional light field corresponding to the object during the respective time period. 
     Another aspect of the present disclosure features a method including: making any one of the optical devices as described herein. 
     Another aspect of the present disclosure features a method of making any one of the optical devices including at least one color-selective polarizer, including: forming the first optically diffractive component; forming the second optically diffractive component; and arranging the color-selective polarizer between the first optically diffractive component and the second optically diffractive component. 
     In some implementations, forming the first optically diffractive component includes: forming a first diffractive structure in a recording medium. 
     In some implementations, forming the first diffractive structure in the recording medium includes: recording a first holographic grating in the recording medium by illuminating a first recording object beam at a first recording object angle and a first recording reference beam at a first recording reference angle on the recording medium, where the first recording object beam and the first recording reference beam have a same wavelength and the same first polarization state. 
     In some examples, the first color of light includes a wavelength range wider than or identical to that of the first recording reference beam or the first recording object beam. In some examples, the first recording reference beam corresponds to a color different from a first color of the first color of light. 
     In some examples, the first incident angle of the first color of light is substantially identical to the first recording reference angle, and the first diffracted angle is substantially identical to the first recording object angle. 
     In some examples, the first recording reference angle is in a range from 70 degrees to 90 degrees. In some examples, the first recording reference angle is in a range from 80 degrees to 90 degrees. In some examples, the first recording object angle is in a range from -10 degrees to 10 degrees. In some examples, the first recording object angle is substantially identical to 6 degrees. In some examples, the first recording object angle is substantially identical to 0 degrees. In some examples, a sum of the first recording reference angle and the first recording object angle is substantially identical to 90 degrees. 
     In some implementations, a thickness of the recording medium is more than one order of magnitude larger than the wavelength of the first recording object beam. The thickness of the recording medium can be about 30 times larger than the wavelength of the first recording object beam. 
     In some implementations, forming the first diffractive structure in the recording medium includes: fixing the first diffractive structure in the recording medium. 
     In some implementations, the recording medium is between a carrier film and a diffraction substrate. 
     In some examples, the first diffracted angle and the second diffracted angle are substantially identical to each other. In some examples, the first incident angle and the second incident angle are substantially identical to each other. 
     In some implementations, arranging the color-selective polarizer between the first optically diffractive component and the second optically diffractive component includes: sequentially stacking the first optically diffractive component, the color-selective polarizer, and the second optically diffractive component, such that the first color of light and the second color of light are incident on the first optically diffractive component before the second optically diffractive component. 
     In some implementations, sequentially stacking the first optically diffractive component, the color-selective polarizer, and the second optically diffractive component includes: sequentially arranging the first optically diffractive component, the color-selective polarizer, and the second optically diffractive component on a substrate that is before the first optically diffractive component. 
     In some implementations, sequentially stacking the first optically diffractive component, the color-selective polarizer, and the second optically diffractive component includes: attaching the color-selective polarizer to the first optically diffractive component through a first intermediate layer; and attaching the second optically diffractive component to the color-selective polarizer through a second intermediate layer, where each of the first and second intermediate layers includes a respective refractive index matching material. 
     In some implementations, the method further includes: forming a third optically diffractive component configured to diffract a third color of light having the first polarization state and a third incident angle at a third diffracted angle with a third diffraction efficiency; and arranging a second color-selective polarizer between the second and third optically diffractive components, where the second color-selective polarizer is configured to rotate a polarization state of the third color of light from the second polarization state to the first polarization state. 
     In some implementations, the color-selective polarizer is configured to rotate a polarization state of the first color of light from the first polarization state to the second polarization state, and the second color-selective polarizer is configured to rotate the polarization state of the second color of light from the first polarization state to the second polarization state, without rotation of the polarization state of the first color of light. 
     In some implementations, the method further includes: arranging a third color-selective polarizer sequential to the third optically diffractive component such that the third optically diffractive component is between the second and third color-selective polarizers, where the third color-selective polarizer is configured to rotate the polarization state of each of the first and second colors of light from the second polarization state to the first polarization state, without rotation of the polarization state of the third color of light. 
     In some implementations, the method further includes: arranging a fourth color-selective polarizer before the first optically diffractive component such that the first optically diffractive component is between the fourth color-selective polarizer and the color-selective polarizer, where the fourth color-selective polarizer is configured to rotate a polarization state of the first color of light from the second polarization state to the first polarization state, without rotation of the polarization state of each of the second and third colors of light. 
     In some implementations, the first polarization state is s polarization, and the second polarization state is p polarization. 
     Another aspect of the present disclosure features a method of making any one of the optical devices including at least one reflective layer, including: forming the first optically diffractive component including the first diffractive structure; forming the second optically diffractive component including the second diffractive structure; arranging the first reflective layer between the first diffractive structure and the second diffractive structure, the second diffractive structure being sequential to the first diffractive structure along a direction; and arranging the second reflective layer sequential to the second diffractive structure along the direction. 
     In some implementations, the method further includes: forming an optical absorber on a side surface of the optical device, where the optical absorber is configured to absorb the totally reflected light of the first and second colors. 
     In some implementations, the first reflective layer is configured to have a refractive index smaller than that of a layer of the first optically diffractive component that is immediately adjacent to the first reflective layer, such that the first color of light having the first incident angle is totally reflected by an interface between the first reflective layer and the layer of the first optically diffractive component, without totally reflecting the second color of light having the second incident angle. 
     In some implementations, the method further includes: forming a third optically diffractive component including a third diffractive structure configured to diffract a third color of light having a third incident angle, where arranging the second reflective layer sequential to the second diffractive structure along the direction includes: arranging the second reflective layer between the second diffractive structure and the third diffractive structure along the direction. Each of the first reflective layer and the second reflective layer can be configured to transmit the third color of light having the third incident angle. 
     In some implementations, the method further includes: arranging a third reflective layer sequential to the third diffractive structure along the direction, where the third reflective layer is configured to totally reflect the third color of light having the third incident angle. 
     In some implementations, each of the first, second, and third optically diffractive components includes a respective carrier film and a respective diffraction substrate, and the first reflective layer includes a first carrier film of the first optically diffractive component. Arranging the first reflective layer between the first diffractive structure and the second diffractive structure can include: attaching a second diffraction substrate of the second optically diffractive component to the first carrier film of the first optically diffractive component by a first intermediate layer. Arranging the second reflective layer between the second diffractive structure and the third diffractive structure along the direction can include: attaching a second carrier film of the second optically diffractive component to a third carrier film of the third optically diffractive component by a second intermediate layer. The second reflective layer can include the second intermediate layer. The third reflective layer can be attached to a third diffraction substrate of the third optically diffractive component. 
     In some implementations, the method further includes: arranging the first optically diffractive component on a substrate that is before the first optically diffractive component along the direction, where the substrate includes a front surface and a back surface. 
     In some implementations, arranging the first optically diffractive component on the substrate includes: attaching a front surface of the first optically diffractive component to the back surface of the substrate through a refractive index matching material. 
     In some implementations, the substrate includes a side surface angled to the back surface of the substrate, and the substrate is configured to receive a plurality of different colors of light at the side surface. The substrate can be configured such that the plurality of different colors of light are incident on the side surface with an incident angle substantially identical to 0 degrees. 
     In some implementations, forming the first optically diffractive component including the first diffractive structure includes: forming the first diffractive structure in a recording medium. 
     In some implementations, forming the first diffractive structure in the recording medium includes: recording a first holographic grating in the recording medium by injecting a first recording object beam at a first recording object angle and a first recording reference beam at a first recording reference angle, where the first recording object beam and the first recording reference beam have a same wavelength and a same polarization state. 
     In some implementations, the first color of light includes a wavelength range wider than or identical to that of the first recording reference beam. 
     In some implementations, the first recording reference beam corresponds to a color different from a first color of the first color of light. 
     In some implementations, the first incident angle of the first color of light is substantially identical to the first recording reference angle, and the first diffracted angle is substantially identical to the first recording object angle. 
     In some examples, the first recording reference angle is in a range from 70 degrees to 90 degrees. In some examples, the first recording reference angle is in a range from 70 degrees to 80 degrees. In some examples, the first recording object angle is in a range from -10 degrees to 10 degrees. 
     In some implementations, a thickness of the recording medium is more than one order of magnitude larger than the wavelength of the first recording object beam. The thickness of the recording medium can be about 30 times larger than the wavelength of the first recording object beam. 
     In some implementations, forming the first diffractive structure in the recording medium includes: fixing the first diffractive structure in the recording medium. 
     In some implementations, the first incident angle is different from the second incident angle. In some examples, the first color of light has a wavelength smaller (or shorter) than the second color of light, and the first incident angle is larger (or longer) than the second incident angle. 
     Another aspect of the present disclosure features a method including: forming any one of the optical devices as described herein according to any one the methods as described above, and arranging the optical device and a display including a plurality of display elements, such that the optical device is configured to diffract a plurality of different colors of light to the display. 
     In some implementations, arranging the optical device and the display includes: spacing a back surface of the optical device from a front surface of the display by a gap. 
     In some implementations, the method further include: forming an anti-reflection coating on at least one of the front surface of the display or the back surface of the optical device. 
     In some implementations, arranging the optical device and the display includes: attaching a back surface of the optical device on a front surface of the display through an intermediate layer. 
     In some cases, the intermediate layer is configured to have a refractive index lower than a refractive index of a layer of the optical device, such that each of the plurality of different colors of light transmitted at zero order by the optical device is totally reflected at an interface between the intermediate layer and the layer of the optical device. 
     In some implementations, the optical device is configured to diffract the plurality of different colors of light at respective diffracted angles that are substantially identical to each other. 
     In some examples, each of the respective diffracted angles is in a range of -10 degrees to 10 degrees. 
     In some implementations, the display is configured to diffract the diffracted colors of light back through the optical device. 
     In some implementations, an area of the optical device covers an area of the display. 
     In some implementations, the optical device includes a substrate in front of the optical device and is configured to receive the plurality of different colors of light at a side surface of the substrate that is angled to a back surface of the substrate. 
     Another aspect of the present disclosure features a method including: using an optical device to convert an incoming beam including a plurality of different colors of light to individually diffracted colors of light. The optical device can be any one of the optical devices as described herein. 
     Another aspect of the present disclosure features a method including: transmitting at least one timing control signal to an illuminator to activate the illuminator to emit a plurality of different colors of light onto an optical device, such that the optical device converts the plurality of different colors of light to individually diffracted colors of light to illuminate a display including a plurality of display elements, where the optical device is any one of the optical devices as described herein; and transmitting, for each of the plurality of display elements of the display, at least one respective control signal to modulate the display element, such that the individually diffracted colors of light are reflected by the modulated display elements to form a multi-color three-dimensional light field corresponding to the respective control signals. 
     In some implementations, the method further includes: obtaining graphic data including respective primitive data for a plurality of primitives corresponding to an object in a three-dimensional space; determining, for each of the plurality of primitives, an electromagnetic (EM) field contribution to each of the plurality of display elements of the display by calculating, in a three-dimensional coordinate system, an EM field propagation from the primitive to the display element; generating, for each of the plurality of display elements, a sum of the EM field contributions from the plurality of primitives to the display element; and generating, for each of the plurality of display elements, the respective control signal based on the sum of the EM field contributions to the display element for modulation of at least one property of the display element, where the multi-color three-dimensional light field corresponds to the object. 
     In some implementations, the method includes: sequentially modulating the display with information associated with the plurality of different colors in a series of time periods, and controlling the illuminator to sequentially emit each of the plurality of different colors of light to the optical device during a respective time period of the series of time periods, such that each of the plurality of different colors of light is diffracted by the optical device to the display and reflected by the modulated display elements of the display to form a respective color three-dimensional light field corresponding to the object during the respective time period. 
     In some implementations, the plurality of different colors of light are diffracted by the optical device at a substantially same diffracted angle to the display. In some examples, the diffracted angle is within a range from -10 degrees to 10 degrees. 
     In some implementations, the illuminator and the optical device are configured such that the plurality of different colors of light are incident on the first optically diffractive component of the optical device with respective incident angles. In some examples, the respective incident angles are different from each other. In some examples, the respective incident angles are substantially identical to each other. In some examples, each of the respective incident angles is in a range from 70 degrees to 90 degrees. 
     Another aspect of the present disclosure features an optical device, including: at least two optically diffractive components and at least one color-selective polarizer, where the optical device is configured such that, when light of different colors is incident on the optical device, the optical device separates light of individual colors of the different colors while suppressing crosstalk between the different colors. 
     In some implementations, the optical device is configured such that, when the light of different colors is incident on the optical device, each of the optically diffractive components diffracts light of a respective color of the different colors. 
     In some implementations, the optical device is configured such that, in an output light beam diffracted by the optical device, a power of light of a particular color of the different colors is at least one order of magnitude higher than a power of light of one or more other colors of the different colors. 
     In some implementations, the at least one color-selective polarizer is configured to rotate a polarization state of light of at least one color of the different colors, such that light of a particular color of the different colors is incident in a first polarization state on a respective one of the optically diffractive components, while light of one or more other colors of the different colors is incident in a second polarization state different from the first polarization state on the respective one of the optically diffractive components. 
     Another aspect of the present disclosure features an optical device, including: at least two optically diffractive components and at least one reflective layer, where the optical device is configured such that, when light of different colors is incident on the optical device, the optical device separates light of individual colors of the different colors while suppressing crosstalk between the different colors, and where the at least one reflective layer is configured for total internal reflection of light of at least one of the different colors. 
     In some implementations, the optical device is configured such that an output light beam diffracted by the optical device includes only light of a particular color of the different colors without crosstalk from one or more other colors of the different colors. 
     In some implementations, the at least one reflective layer is configured to totally reflect zero order light of a particular color of the different colors transmitted by a respective one of the optically diffractive component, while transmitting one or more other colors of the different colors 
     In some implementations, the optical device is configured such that, when the light of different colors is incident on the optical device, each of the optically diffractive components diffracts light of a respective color of the different colors. 
     Another aspect of the present disclosure features a display and any one of the optical devices as described herein, where the optical device is configured to diffract a plurality of different colors of light to the display. 
     Another aspect of the present disclosure features an illuminator configured to provide a plurality of different colors of light and any one of the optical devices as described herein, where the optical device is configured to diffract the plurality of different colors of light from the illuminator. 
     Another aspect of the present disclosure features a system including: a display and an optical device including one or more transmissive diffractive structures for diffracting light to the display. 
     In some implementations, the display is a reflective display configured to diffract the light back through the optical device. In some implementations, the system further includes an illuminator configured to provide the light to the optical device, where the illuminator is arranged in a front side of the transmissive diffractive structures of the optical device. 
     In some implementations, the display is a transmissive display configured to diffract the light forwards through the optical device. In some implementations, the system further includes an illuminator configured to provide the light to the optical device, where the illuminator is arranged in a rear side of the transmissive diffractive structures of the optical device. 
     In some implementations, each of the one or more transmissive diffractive structures is configured to diffract a respective color of a plurality of different colors. 
     In some implementations, the optical device further includes one or more reflective diffractive structures, and each of the one or more transmissive diffractive structures and the one or more reflective diffractive structures is configured to diffract a respective color of a plurality of different colors. 
     Another aspect of the present disclosure features a system including: a display and an optical device including one or more reflective diffractive structures for diffracting light to the display. 
     In some implementations, the display is a reflective display configured to diffract the light back through the optical device. In some implementations, the system further includes an illuminator configured to provide the light to the optical device, where the illuminator is arranged in a rear side of the reflective diffractive structures of the optical device. 
     In some implementations, the display is a transmissive display configured to diffract the light forwards through the optical device. In some implementations, the system further includes an illuminator configured to provide the light to the optical device, where the illuminator is arranged in a front side of the reflective diffractive structures of the optical device. 
     In some implementations, each of the one or more reflective diffractive structures is configured to diffract a respective color of a plurality of different colors. 
     In some implementations, the optical device further includes one or more transmissive diffractive structures, and each of the one or more transmissive diffractive structures and the one or more reflective diffractive structures is configured to diffract a respective color of a plurality of different colors. 
     Another aspect of the present disclosure features an optical device, including: a plurality of optically diffractive components including at least one transmissive diffractive structure and at least one reflective diffractive structure, where the optical device is configured such that, when light of different colors is incident on the optical device, the optical device separates light of individual colors of the different colors while suppressing crosstalk between the different colors. 
     In some implementations, each of the transmissive diffractive structure and the reflective diffractive structure is configured to light of a respective color of the different colors. 
     In some implementations, the optical device further includes: at least one reflective layer configured for total internal reflection of light of at least one of the different colors. 
     In some implementations, the optical device further includes: at least one color-selective polarizer configured to rotate a polarization state of light of at least one color of the different colors, such that light of a particular color of the different colors is incident in a first polarization state on a respective one of the optically diffractive components, while light of one or more other colors of the different colors is incident in a second polarization state different from the first polarization state on the respective one of the optically diffractive components. 
     Another aspect of the present disclosure features a system including: a display and an optical device according to any one of the optical devices as described herein, where the optical device is configured to diffract a plurality of different colors of light to the display. 
     Another aspect of the present disclosure features a system including: an illuminator configured to provide a plurality of different colors of light and an optical device according to any one of the optical devices as described herein, where the optical device is configured to diffract the plurality of different colors of light from the illuminator. 
     In the present disclosure herein, the term “primitive” refers to a basic nondivisible element for input or output within a computing system. The element can be a geometric element or a graphical element. The term “hologram” refers to a pattern displayed by (or uploaded to) a display which contains amplitude information or phase information, or some combination thereof, regarding an object. The term “holographic reconstruction” refers to a volumetric light field (e.g., a holographic light field) from a display when illuminated. 
     The details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and associated description. Other features, aspects, and advantages of the subject matter will become apparent from the description, the drawings, and the claims. 
     It is to be understood that various aspects of implementations can be combined in different manners. As an example, features from certain methods, devices, or systems can be combined with features of other methods, devices, or systems. 
    
    
     
       DESCRIPTION OF DRAWINGS 
         FIG.  1 A  illustrates a schematic diagram of an example system including a holographic display. 
         FIG.  1 B  illustrates a schematic diagram of an example holographic display. 
         FIG.  1 C  illustrates an example system for 3D displays. 
         FIG.  2    illustrates an example configuration for electromagnetic (EM) propagation calculation. 
         FIG.  3 A  illustrates an example EM propagation for a point primitive relative to an element of a display. 
         FIG.  3 B  illustrates an example EM propagation for a line primitive relative to an element of a display. 
         FIG.  3 C  illustrates an example EM propagation for a triangle primitive relative to an element of a display. 
         FIG.  3 D  illustrates an example implementation of Maxwell holographic occlusion for a point primitive with a line primitive as an occluder. 
         FIG.  3 E  illustrates an example implementation of Maxwell holographic occlusion for a line primitive with another line primitive as an occluder. 
         FIG.  3 F  illustrates an example implementation of Maxwell holographic occlusion for a triangle primitive with a line primitive as an occluder. 
         FIG.  3 G  illustrates an example implementation of Maxwell holographic stitching. 
         FIG.  4    is a flowchart of an example process of displaying an object in 3D. 
         FIG.  5 A  illustrates an example system for 3D display including a reflective display with front illumination. 
         FIG.  5 B  illustrates another example system for 3D display including a reflective display with front illumination. 
         FIG.  5 C  illustrates another example system for 3D display including a transmissive display with back illumination. 
         FIG.  5 D  illustrates another example system for 3D display including a transmissive display with waveguide illumination. 
         FIG.  5 E  illustrates another example system for 3D display including a transmissive display with waveguide illumination. 
         FIG.  5 F  illustrates another example system for 3D display including a reflective display with waveguide illumination. 
         FIG.  5 G  illustrates another example system for 3D display including a reflective display with waveguide illumination. 
         FIG.  5 H  illustrates another example system for 3D display including a reflective display with optically diffractive illumination using a transmissive field grating based structure. 
         FIG.  5 I  illustrates another example system for 3D display including a reflective display with optically diffractive illumination using a reflective field grating based structure. 
         FIG.  5 J  illustrates another example system for 3D display including a transmissive display with optically diffractive illumination using a reflective field grating based structure. 
         FIG.  5 K  illustrates another example system for 3D display including a transmissive display with optically diffractive illumination using a transmissive field grating based structure. 
         FIG.  6 A  illustrates an example display with display elements having nonuniform shapes. 
         FIG.  6 B  illustrates an example display with display elements having different sizes. 
         FIG.  7 A  illustrates an example of recording a grating in a recording medium. 
         FIG.  7 B  illustrates an example of diffracting a replay reference beam by the grating of  FIG.  7 A . 
         FIG.  7 C  illustrates an example of recording gratings for different colors in a recording medium using different colors of light. 
         FIG.  7 D  illustrates an example of recording gratings for different colors in a recording medium using a same color of light. 
         FIG.  7 E  illustrates an example of diffracting replay reference beams of different colors by gratings for different colors. 
         FIG.  7 F  illustrates an example of crosstalk among diffracted beams of different colors. 
         FIG.  8    illustrates an example of recording a diffractive grating with a large reference angle in a recording medium. 
         FIG.  9 A  illustrates an example optical device, including diffractive gratings for two colors and corresponding color-selective polarizers, for individually diffracting the two colors of light. 
         FIG.  9 B  illustrates an example of diffracting the two colors of light by the optical device of  FIG.  9 A . 
         FIG.  10 A  illustrates an example optical device, including diffractive gratings for three colors and corresponding color-selective polarizers, for individually diffracting the three colors of light. 
         FIG.  10 B  illustrates an example of diffracting the three colors of light by the optical device of  FIG.  10 A . 
         FIG.  11    illustrates an example optical device, including diffractive gratings for two colors and corresponding reflective layers, for individually diffracting the two colors of light. 
         FIG.  12 A  illustrates an example optical device, including diffractive gratings for three colors and corresponding reflective layers, for individually diffracting the three colors of light. 
         FIG.  12 B  illustrates another example optical device including diffractive gratings for three colors and corresponding reflective layers with a wedged substrate. 
         FIG.  12 C  illustrates a further example optical device including diffractive gratings for three colors and corresponding reflective layers with a wedged input face. 
         FIGS.  13 A- 13 C  illustrate relationships between diffracted and reflected beam power with different incident angles for a blue color of light ( FIG.  13 A ), a green color of light ( FIG.  13 B ), and a red color of light ( FIG.  13 C ). 
         FIG.  14 A  is a flowchart of an example process of fabricating an optical device including holographic gratings and corresponding color-selective polarizers. 
         FIG.  14 B  is a flowchart of an example process of fabricating an optical device including holographic gratings and corresponding reflective layers. 
         FIG.  15    illustrates an example optical device including a combination of transmissive and reflective diffractive gratings. 
         FIG.  16    illustrates an example of incident light being diffracted by display elements of a display and reflected at gaps between the display elements on the display. 
         FIG.  17 A  illustrates an example of display zero order light within a holographic scene displayed on a projection screen. 
         FIG.  17 B  illustrates an example of display zero order light within a holographic scene displayed on a viewer’s eye. 
         FIG.  18    illustrates an example of suppressing display zero order light in a holographic scene displayed on a projection screen by diverging the display zero order light. 
         FIG.  19 A  illustrates an example of display zero order light in a holographic scene when the display is illuminated with light at normal incidence. 
         FIG.  19 B  illustrates an example of suppressing display zero order light in a holographic scene displayed on a projection screen by directing the display zero order light away from the holographic scene when the display is illuminated with light at an incident angle. 
         FIG.  19 C  illustrates an example of suppressing display zero order light in a holographic scene displayed on a viewer’s eye by directing the display zero order light away from the holographic scene when the display is illuminated with light at an incident angle. 
         FIG.  20 A  illustrates an example of a configuration cone and a reconstruction cone corresponding to a holographic scene with respect to a display in a 3D coordinate system. 
         FIG.  20 B  illustrates an example of adjusting the configuration cone of  FIG.  20 A  to configure a hologram corresponding to the holographic scene in the 3D coordinate system. 
         FIG.  21    illustrates an example of coupling light via a coupling prism to an optically diffractive device to illuminate a display at an incident angle for suppressing display zero order light in a holographic scene. 
         FIG.  22    illustrates an example of coupling light via a wedged substrate to an optically diffractive device to illuminate a display at an incident angle for suppressing display zero order light in a holographic scene. 
         FIG.  23 A  illustrates an example of suppressing display zero order light in a holographic scene displayed on a projection screen by absorbing the display zero order light reflected from the display with a metamaterial layer. 
         FIG.  23 B  illustrates an example of suppressing display zero order light in a holographic scene displayed on a viewer’s eye by blocking (or absorbing) the display zero order light reflected from the display with a metamaterial layer. 
         FIG.  24    illustrates a system of suppressing display zero order light in a holographic scene by redirecting the display zero order light away from the holographic scene via an optically redirecting structure. 
         FIGS.  25 A- 25 C  illustrate examples of redirecting display zero order light via optically redirecting structures to different directions in space. 
         FIGS.  26 A- 26 E  illustrate examples of redirecting display zero order light when light is input at different incident angles via optically redirecting structures to different directions in space. 
         FIG.  27 A  illustrates an example of redirecting display zero order light with p polarization to transmit at a Brewster’s angle. 
         FIGS.  27 B- 27 C  illustrate examples of redirecting display zero order light with s polarization with an optical retarder for transmission at a Brewster’s angle. 
         FIG.  28    illustrates an example of redirecting display zero order light to an anisotropic transmitter for absorbing the display zero order light. 
         FIG.  29    illustrates an example of redirecting display zero order light to totally reflect the display zero order light. 
         FIGS.  30 A- 30 B  illustrate examples of redirecting two different colors of display zero order light to different directions away from a holographic scene. 
         FIGS.  31 A- 31 B  illustrate examples of redirecting three different colors of display zero order light to different directions away from a holographic scene in a same plane. 
         FIG.  32    illustrates an example of redirecting three different colors of display zero order light to different directions away from a holographic scene in space. 
         FIG.  33    illustrates an example of redirecting three different colors of display zero order light to different directions away from a holographic scene using a switchable grating for one of the colors. 
         FIG.  34    is a flowchart of an example process of suppressing display zero order light in a holographic scene. 
         FIGS.  35 A- 35 C  illustrate an example of a system for displaying reconstructed 3D objects. 
         FIGS.  36 A- 36 C  illustrate the same views of the system of  FIGS.  35 A- 35 C , respectively, but with three colors of light propagate in the system. 
     
    
    
     Like reference numbers and designations in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
     Implementations of the present disclosure feature technologies for enabling 3D displays of complex computer-generated scenes as genuine holograms. The technologies provide a novel and deterministic solution to real time dynamic computational holography based upon Maxwell’s Equations for electromagnetic fields, which can be represented as Maxwell holography. The calculation (or computation) in Maxwell holography can be represented as Maxwell holographic calculation (or Maxwell holographic computation). In embodiments, the disclosure approaches a hologram as a Dirichlet or Cauchy boundary condition problem for a general electric field, utilizing tools including field theory, topology, analytic continuation, and/or symmetry groups, which enables to solve for holograms in real time without the limitations of legacy holographic systems. In embodiments, the technologies can be used to make phase-only, amplitude-only, or phase-and-amplitude holograms, utilizing spatial light modulators (SLMs) or any other holographic devices. 
     Implementations of the present disclosure can provide: 1) a mechanism of approximation of a hologram as an electromagnetic boundary condition, using field theory and contact geometry, instead of classic optics; 2) derivation and implementation into computer codes and application programming interfaces (APIs) of the electromagnetic boundary condition approach to computational holography, that is, implementation of the hologram calculation as a 2D analytic function on a plane of the hologram and subsequent discretization into parallel algorithms; and/or 3) implementation of a complete set of fully 3D, holographic versions of standard computer graphics primitives (e.g., point, line, triangle, and texture triangle), which can enable full compatibility with standard existing computer graphics tools and techniques. The technologies can enable devices to display general existing content that is not specifically created for holography, and simultaneously allows existing content creators to create holographic works without having to learn special techniques, or use special tools. 
     Particularly, the technologies disclosed herein can involve the use of a mathematical formulation (or expression) of light as an electromagnetic (EM) phenomenon in lieu of the mathematical formulation of classical optics that is commonly used in computational holography, e.g., the Gerchberg-Saxton (G-S) algorithm. The mathematical formulation disclosed herein is derived from Maxwell’s Equations. In embodiments, the technologies disclosed herein involve treating the displayed image as an electromagnetic field and treating a hologram as a boundary value condition that produces the electromagnetic field (e.g., a Dirichlet problem). Additionally, a desired image can be constructed using a primitive paradigm ubiquitous in computer graphics, allowing, for example, the technologies to be used to display any 3D imagery as a holographic reconstruction, e.g., a holographic light field, instead of as a projective image on a 2D screen. Compared to depth point clouds technologies that suffer from bandwidth limitations, the technologies can avoid these limitations and use any suitable types of primitives, e.g., a point primitive, a line primitive, or a polygon primitive such as a triangle primitive. Moreover, the primitives can be rendered with color information, texture information, and/or shading information. This can help achieve a recording and compression scheme for CG holographic content including holographic videos. 
     In embodiments, the technologies disclosed herein use Maxwell’s Equations to compute generated holograms as a boundary condition problem for modeling an electromagnetic field, which can remove dependency on the fast Fourier transform (FFT) and its inherent limitations, remove dependency on collimated light sources such as lasers or light emitting diodes (LEDs), and/or remove limitations of previous approaches to computational holography and nondeterministic solutions. 
     In embodiments, the technologies disclosed herein can be optimized for computational simplicity and speed through a mathematical optimization process that constrains independent inputs to a surface of the hologram, depending on parameters of computer-generated (CG) primitives needed to build the scene. This allows work to be performed in a highly parallel and highly optimal fashion in computing architectures, e.g., application specific integrated circuits (ASIC) and multicore architectures. The process of computing the hologram can be considered as a single instruction that executes on input data in a form of a computer-generated imagery (CGI) scene, and can theoretically be completed in a single clock cycle per CGI primitive. 
     In embodiments, the technologies disclosed herein treat a holographic scene as an assembly of fully 3D holographic primitive apertures which are functionally compatible with the standard primitives of conventional 3D graphics as employed in, for example, video games, movies, television, computer displays, or any other display technologies. The technologies can enable efficient implementation of these aperture primitives in hardware and software without limitations inherent in standard implementations of computational holography. Amplitude and color of the primitives can be automatically computed. Computational complexity can increase linearly with phase element number n, compared to n^2 or n*log(n) in standard computational holography. The images created are fully 3D and not an assemblage of planar images, and the technologies do not require iterative amplitude correction with unknown numbers of steps. Moreover, the generated holograms do not have “conjugate” images that take up space on the holographic device. 
     As the holographic primitives are part of a special collection of mathematical objects, they can be relatively simple and relatively fast to compute, and they can be uniquely suited to parallel, distributed computing approaches. The computability and parallelism can allow for interactive computation of large holograms to design large area holographic devices of theoretically unlimited size, which can act as holographic computer displays, phone displays, home theaters, and even holographic rooms. Moreover, the holograms can fill large areas with light, e.g., rendering large shaded areas in 3D, without limitations associated with conventional holographic computation methods which can cause elements to appear in outline instead of solid. Furthermore, the relatively simple and relatively fast computation allows for the display of real-time holograms at interactive speeds that are not constrained by n^2 computational load and by iterative amplitude correction. 
     In embodiments, the technologies can realize natural computability on modern ASIC and multicore architectures and can realize complete compatibility with modern graphics hardware, modern graphics software, and/or modern graphics tools and tool chains. For example, the technologies can implement clear and simple holographic APIs and enable high performance rendering of arbitrary CG models using conventional 3D content creation tools, e.g., 3ds Max®, SOLIDWORKS®, Maya®, or Unity, through the APIs. The APIs can enable developers or users to interact with a holographic device, e.g., a light modulator or holographic system. The holographic APIs can create computer graphics primitives as discrete holographic scene primitives, allowing for rich holographic content generation utilizing general purpose and specially designed holographic computation hardware. The creation of a mathematical and computational architecture can allow holograms to be rendered using the tools and techniques used to make conventional 3D content and software applications. The optimization of the mathematical and computational architecture can allow for performant embodiments of conventional graphics and renderings to be displayed as holographic reconstructions. 
     Algorithms in the technologies disclosed herein are relatively simple to implement in hardware. This not only allows the computational speeds needed for high quality rendering that users expect, but it also allows the algorithms to be implemented in relatively simple circuits, e.g., ASIC gate structures, as part of a holographic device. Accordingly, bandwidth issues that can plague high density displays can become irrelevant, as computation of scenes can be spread across the computing architecture built into the display device (e.g., built-in-computation) instead of having to be computed remotely and then written to each display element (or display pixel) of the display for each frame of content. It also means that the number of display elements, and thus the size of a holographic display, can be relatively unbounded by constraints that severely limit other technologies. 
     The technologies disclosed herein can enable multiple interactive technologies using structured light to be implemented relatively simply and relatively inexpensively in different applications, including, for example, solid-state light detection and ranging (LIDAR) devices, 3D printing and machining, smart illuminators, smart microdisplays, optical switching, optical tweezers, or any other applications demanding structured light. The technologies disclosed herein can be also used for optical simulations, e.g., for grating simulations. 
       FIG.  1 A  illustrates a schematic diagram of an example system  100  for 3D displays. The system 100 includes a computing device  102  and a holographic display device (or a Maxwell holographic display device)  110 . The computing device 102 is configured to prepare data for a list of primitives corresponding to an object, e.g., a 3D object, and transmit the data to the holographic display device  110  via a wired or wireless connection, e.g., USB-C connection or any other high speed serial connection. The holographic display device  110  is configured to compute electromagnetic (EM) field contributions from the list of primitives to display elements of a display (e.g., a modulator) in the holographic display device  110 , modulate the display elements with a pattern, e.g., a hologram, based on the computed EM field contributions on the display, and display upon illumination a light field corresponding to the object in 3D, e.g., a holographic reconstruction. Herein, the hologram refers to the pattern displayed on the display which contains amplitude information or phase information, or some combination thereof, regarding the object. The holographic reconstruction refers to a volumetric light field (e.g., a holographic light field) from the display when illuminated. 
     The computing device  102  can be any appropriate type of device, e.g., a desktop computer, a personal computer, a notebook, a tablet computing device, a personal digital assistant (PDA), a network appliance, a smart mobile phone, a smartwatch, an enhanced general packet radio service (EGPRS) mobile phone, a media player, a navigation device, an email device, a game console, or any appropriate combination of any two or more of these computing devices or other computing devices. 
     The computing device  102  includes an operating system (OS)  104  that can include a number of applications  106  as graphics engines. The applications  106  can process or render a scene, e.g., any arbitrary CG model using standard 3D content creation tools, e.g., 3ds Max®, SOLIDWORKS®, Maya®, or Unity. The scene can correspond to one or more real or imaginary 3D objects or a representation of objects. The applications  106  can operate in parallel to render the scene to obtain an OS graphics abstraction  101  which can be provided to a graphics processing unit (GPU)  108  for further processing. In some implementations, the OS graphics abstraction  101  is provided to the holographic display device  110  for further processing. 
     The GPU  108  can include a specialized electronic circuit designed for rapid manipulation of computer graphics and image processing. The GPU  108  can process the graphics abstraction  101  of the scene to get processed scene data  103  which can be used to obtain a list of primitives  105 , e.g., indexed in a particular order. The primitives can include at least one of a point primitive, a line primitive, or a polygon primitive. In some implementations, the GPU  108  includes a video driver configured to generate the processed scene data  103  and the list of primitives  105 . 
     In some implementations, the GPU  108  includes a conventional renderer  120 , by which the list of primitives  105  can be rendered by conventional rendering techniques, e.g., culling and clipping, into a list of items to draw on a conventional monitor  124 , e.g., a 2D display screen. The list of items can be sent via a screen buffer  122  to the conventional monitor  124 . 
     In some implementations, the GPU  108  includes a holographic renderer  130  to render the list of primitives  105  into graphic data to be displayed by the holographic display device  110 . The graphic data can include the list of primitives and corresponding primitive data. For example, the graphic data can include a hex code for each primitive. 
     In some implementations, the GPU  108  includes both the conventional renderer  120  and the holographic renderer  130 . In some implementations, the GPU  108  includes the conventional renderer  120  and the holographic display device  110  includes the holographic renderer  130 . 
     The corresponding primitive data for a primitive can also include color information (e.g., a textured color, a gradient color or both), texture information, and/or shading information. The shading information can be obtained by any customary CGI surface shading method that involves modulating color or brightness of a surface of the primitive. 
     The primitive data of a primitive can include coordinate information of the primitive in a 3D coordinate system, e.g., Cartesian coordinate system XYZ, polar coordinate system, cylindrical coordinate system, and spherical coordinate system. As discussed with further detail below, the display elements in the holographic display device  110  can also have corresponding coordinate information in the 3D coordinate system. The primitives at coordinate locations can represent a 3D object adjacent to the display elements, e.g., in front of the display elements, behind the display elements, or straddling the display elements. 
     As an example, the primitive is a shaded line, e.g., a straight line that changes smoothly from one color to another across its span. The primitive needs four elements of data to be rendered: two end points, and color information (e.g., a RGB color value) at each end point. Assume that a hex code for the line is a0, and the line stretches from a first end point (0.1, 0.1, 0.1) to a second end point (0.2, 0.2, 0.2) in the 3D coordinate system, with the color ½ Blue: RGB = (0,0,128) at the first end point and the color full Red: RGB = (255,0,0) at the second end point. The holographic renderer determines how much and what kind of data to expect for each primitive. For the line, the primitive data for the shaded line in the primitive stream can be a set of instructions as below: 
     
       
         
           
               
            
               
                              0xa0 // hex code for the shaded line 
               
               
                              0x3dcccccd // first vertex at (0.1, 0.1, 0.1) float (single) 
               
               
                              0x3dcccccd 
               
               
                              0x3dcccccd 
               
               
                              0x000080 // first vertex color is (0, 0, 128) 
               
               
                              0x3e4ccccd // second vertex at (0.2, 0.2, 0.2) float (single) 
               
               
                              0x3e4ccccd 
               
               
                              0x3e4ccccd 
               
               
                              0xff0000 // second vertex color is (255, 0, 0) 
               
            
           
         
       
     
     There are a total of 31 hex words in the primitive data for the shaded line primitive. It can be an extremely efficient way to transmit a complex scene, and the primitive data can further be compressed. Since each primitive is a deterministic Turing step, there is no need for terminators. Different from a traditional model where this line primitive is simply drawn on a 2D display screen, the primitive data for the line is transmitted to the holographic display device  110  that can compute a hologram and display a corresponding holographic reconstruction presenting a line floating in space. 
     In some implementations, the computing device 102 transmits non-primitive based data, e.g., a recorded light field video, to the holographic display device  110 . The holographic display device  110  can compute sequential holograms to display the video as sequential holographic reconstructions in space. In some implementations, the computing device  102  transmits CG holographic content simultaneously with live holographic content to the holographic display device  110 . The holographic display device  110  can also compute corresponding holograms to display the contents as corresponding holographic reconstructions. 
     As illustrated in  FIG.  1 A , the holographic display device  110  includes a controller  112  and a display  114 . The controller  112  can include a number of computing units or processing units. In some implementations, the controller  112  includes ASIC, field programmable gate array (FPGA) or GPU units, or any combination thereof. In some implementations, the controller  112  includes the holographic renderer  130  to render the list of primitives  105  into the graphic data to be computed by the computing units. In some implementations, the controller  112  receives the OS graphics abstraction  101  from the computing device  102  for further processing. The display  114  can include a number of display elements. In some implementations, the display  114  includes a spatial light modulator (SLM). The SLM can be a phase SLM, an amplitude SLM, or a phase and amplitude SLM. In some examples, the display  114  is a digital micro-mirror device (DMD) or a liquid crystal on silicon (LCOS) device. In some implementations, the holographic display device  110  includes an illuminator  116  adjacent to the display  114  and configured to emit light toward the display  114 . The illuminator  116  can include one or more coherent light sources, e.g., lasers, one or more semi-coherent light sources, e.g., LEDs (light emitting diodes) or superluminescent diodes (SLEDs), one or more incoherent light sources, or a combination of such sources. 
     Different from a conventional 3D graphics system, which takes a 3D scene and renders it on to a 2D display device, the holographic display device  110  is configured to produce a 3D output such as a holographic reconstruction  117  in a form of a light field, e.g., a 3D volume of light. In a hologram, each display element can contribute to every part of the holographic reconstruction of the scene. Hence, for the holographic display device  110 , each display element potentially needs to be modulated for every part of the scene, e.g., each primitive in the list of primitives generated by the GPU  108 , for complete holographic reproduction of the scene. In some implementations, modulation of certain elements can be omitted or simplified based on, for example, an acceptable level of accuracy in the reproduced scene or in some region of the scene. 
     In some implementations, the controller  112  is configured to compute an EM field contribution, e.g., phase, amplitude, or both, from each primitive to each display element, and generate, for each display element, a sum of the EM field contributions from the list of primitives to the display element. This can be done either by running through every primitive and accruing its contribution to a given display element, or by running through each display element for each primitive, or by a hybrid blend of these two techniques. 
     The controller  112  can compute the EM field contribution from each primitive to each display element based on a predetermined expression for the primitive. Different primitives can have corresponding expressions. In some cases, the predetermined expression is an analytic expression, as discussed with further detail below in relation to  FIGS.  3 A- 3 C . In some cases, the predetermined expression is determined by solving Maxwell’s Equations with a boundary condition defined at the display  114 . The boundary condition can include a Dirichlet boundary condition or a Cauchy boundary condition. Then, the display element can be modulated based on the sum of the EM field contributions, e.g., by modulating at least one of a refractive index, an amplitude index, a birefringence, or a retardance of the display element. 
     If values of an EM field, e.g., a solution to the Maxwell Equations, at each point on a surface that bounds the field are known, an exact, unique configuration of the EM field inside a volume bounded by a boundary surface can be determined. The list of primitives (or a holographic reconstruction of a corresponding hologram) and the display  114  define a 3D space, and a surface of the display  114  forms a portion of a boundary surface of the 3D space. By setting EM field states (e.g., phase or amplitude or phase and amplitude states) on the surface of the display  114 , for example, by illuminating light on the display surface, the boundary condition of the EM field can be determined. Due to time symmetry of the Maxwell Equations, as the display elements are modulated based on the EM field contributions from the primitives corresponding to the hologram, a volumetric light field corresponding to the hologram can be obtained as the holographic reconstruction. 
     For example, a line primitive of illumination at a specific color can be set in front of the display  114 . As discussed in further detail below with respect to  FIG.  3 B , an analytic expression for a linear aperture can be written as a function in space. Then the EM field contribution from the line primitive on a boundary surface including the display  114  can be determined. If EM field values corresponding to the computed EM field contribution are set in the display  114 , due to time-symmetry of the Maxwell Equations, the same linear aperture used in the computation can appear at a corresponding location, e.g., a coordinate position of the linear primitive in the 3D coordinate system and with the specific color. 
     In some examples, as discussed in further detail below with respect to  FIG.  3 B , suppose that there is a line of light between two points A and B in the 3D space. The light is evenly lit and has an intensity I per line distance l. At each infinitesimal dl along the line from A to B, an amount of light proportional to I*dl is emitted. The infinitesimal dl acts as a delta (point) source, and the EM field contribution from the infinitesimal dl to any point on a boundary surface around a scene corresponding to a list of primitives can be determined. Thus, for any display element of the display  114 , an analytic equation that represents the EM field contribution at the display element from the infinitesimal segment of the line can be determined. A special kind of summation/integral that marches along the line and accrues the EM field contribution of the entire line to the EM field at the display element of the display can be determined as an expression. Values corresponding to the expression can be set at the display element, e.g., by modulating the display element and illuminating the display element. Then, through time reversal and a correction constant, the line can be created in the same location defined by points A and B in the 3D space. 
     In some implementations, the controller  112  is coupled to the display  114  through a memory buffer. The control signal  112  can generate a respective control signal based on the sum of the EM field contributions to each of the display elements. The control signal is for modulating the display element based on the sum of the EM field contributions. The respective control signals are transmitted to the corresponding display elements via the memory buffer. 
     In some implementations, the controller  112  is integrated with the display  114  and locally coupled to the display  114 . As discussed with further detail in relation to  FIG.  1 B , the controller  112  can include a number of computing units each coupled to one or more respective display elements and configured to transmit a respective control signal to each of the one or more respective display elements. Each computing unit can be configured to perform computations on one or more primitives of the list of primitives. The computing units can operate in parallel. 
     In some implementations, the illuminator  116  is coupled to the controller  112  and configured to be turned on/off based on a control signal from the controller  112 . For example, the controller  112  can activate the illuminator  116  to turn on in response to the controller  112  completing the computation, e.g., all the sums of the EM field contributions for the display elements are obtained. As noted above, when the illuminator  116  emits light on the display  114 , the modulated elements of the display cause the light to propagate in different directions to form a volumetric light field corresponding to the list of primitives that correspond to the 3D object. The resulting volumetric light field corresponds to a solution of Maxwell’s equations with a boundary condition defined by the modulated elements of the display  114 . 
     In some implementations, the controller  112  is coupled to the illuminator  116  through a memory buffer. The memory buffer can be configured to control amplitude or brightness of light emitting elements in the illuminator. The memory buffer for the illuminator  116  can have a smaller size than a memory buffer for the display  114 . A number of the light emitting elements in the illuminator  116  can be smaller than a number of the display elements of the display  114 , as long as light from the light emitting elements can illuminate over substantially a total surface of the display  114 . For example, an illuminator having 64 x 64 OLEDs (organic light emitting diodes) can be used for a display having 1024 x 1024 elements. The controller  112  can be configured to simultaneously activate a number of lighting elements of the illuminator  116 . 
     In some implementations, the illuminator  116  is a monochromatic light source configured to emit a substantially monochromatic light, e.g., a red light, a green light, a yellow light, or a blue light. In some implementations, the illuminator  116  includes two or more light emitting elements, e.g., lasers or light emitting diodes (LEDs), each configured to emit light with a different color. For example, the illuminator  116  can include red, green, and blue lighting elements. To display a full-color 3D object, three or more separate holograms for colors including at least red, green, and blue, can be computed. That is, at least three EM field contributions from corresponding primitives to the display elements can be obtained. The display elements can be modulated sequentially based on the at least three EM field contributions and the illuminator  116  can be controlled to sequentially turn on the at least red, green and blue lighting elements sequentially. For example, the controller  112  can first transmit a first timing signal to turn on a blue lighting element and transmit first control signals corresponding to a blue hologram to display elements of the display  114 . After the blue hologram on the display  114  is illuminated with the blue light for a first period of time, the controller  112  can transmit a second timing signal to turn on a green lighting element and transmit second control signals corresponding to a green hologram to display elements of the display  114 . After the green hologram on the display  114  is illuminated with the green light for a second period of time, the controller  112  can transmit a third timing signal to turn on a red lighting element and transmit third control signals corresponding to a red hologram to display elements of the display  114 . After the red hologram on the display  114  is illuminated with the red light for a third period of time, the controller  112  can repeat the above steps. Depending on temporal coherence-of vision effect in an eye of a viewer, the three colors can be combined in the eye to give an appearance of full color. In some cases, the illuminator  116  is switched off during a state change of the display image (or holographic reconstruction) and switched on when a valid image (or holographic reconstruction) is presented for a period of time. This can also depend on the temporal coherence of vision to make the image (or holographic reconstruction) appear stable. 
     In some implementations, the display  114  has a resolution small enough to diffract visible light, e.g., on an order of 0.5 µm or less. The illuminator  116  can include a single, white light source and the emitted white light can be diffracted by the display  114  into different colors for holographic reconstructions. 
     As discussed in further detail below with respect to  FIGS.  5 A- 5 K , there can be different configurations for the system  100 . The display  114  can be reflective or transmissive. The display  114  can have various sizes, ranging from a small scale (e.g., 1-10 cm on a side) to a large scale (e.g., 100-1000 cm on a side). Illumination from the illuminator  116  can be from the front of the display  114  (e.g., for a reflective or transflective display) or from the rear of the display  114  (e.g., for a transmissive display). The holographic display device  110  can provide uniform illumination across the display  114 . In some implementations, an optical waveguide, as illustrated in  FIGS.  5 D- 5 G , can be used to evenly illuminate a surface of the display  114 . In some examples, the controller  112 , the illuminator 116, and the display  114  can be integrated together as a single unit. The integrated single unit can include the holographic renderer  130 , e.g., in the controller  112 . 
     In some implementations, an optically diffractive device, e.g., a field grating device or a lightguide device as illustrated in  FIGS.  5 H to  5 K , can be configured to diffract light from the illuminator  116  into the display  114 , and the display  114  can then diffract the light to a viewer’s eyes. In some examples, the light from the illuminator  116  can be incident on the optically diffractive device with a large incident angle from a side, such that the illuminator  116  does not block the viewer’s view of the display  114 . In some examples, the diffracted light from the optically diffractive device can be diffracted at a nearly normal incident angle into the display, such that the light can relatively uniformly illuminate the display and be diffracted to the viewer’s eyes with reduced (e.g., minimized) loss. 
       FIG.  1 B  illustrates a schematic diagram of an example holographic display device  150 . The holographic display device  150  can be similar to the holographic display device  110  of  FIG.  1 A . The holographic display device  150  includes a computing architecture  152  and a display  156 . The computing architecture  152  can be similar to the controller  112  of  FIG.  1 A . The computing architecture  152  can include an array of parallel computing cores  154 . A computing core can be connected to an adjacent computing core via a communication connection  159 , e.g., a USB-C connection or any other high speed serial (or parallel) connection. The connections  159  can be included in a data distribution network through which scene data  151  (e.g., scene primitives) can be distributed among the computing cores  154 . 
     The display  156  can be similar to the display  114  of  FIG.  1 A , and can include an array of display elements  160  positioned on a backplane  158 . The display elements  160  can be arranged on a front side of the backplane  158  and the computing cores  154  can be arranged on a back side of the backplane  158 . The backplane  158  can be a substrate, e.g., a wafer. The computing cores  154  can be either on the same substrate as the display  156  or bonded to the back side of the display  156 . 
     Each computing core  154  can be connected to a respective tile (or array) of display elements  160 . Each computing core  154  can be configured to perform computations on respective primitives of a number of primitives in the scene data  151  in parallel with one or more other computing cores. In some examples, the computing core  154  is configured to compute an EM field contribution from each of the respective primitives to each of the array of display elements  160  and generate a sum of EM field contributions from the number of primitives to each of the respective tiles of display elements  160 . The computing core  154  can receive, from other computing cores of the array of computing cores  154 , computed EM field contributions from other primitives of the number of primitives to each of the respective tile of display elements  160 , and generate the sum of EM field contributions based on the received computed EM field contributions. The computing core  154  can generate a control signal for each of the respective tile of display elements to modulate at least one property of each of the respective tile of display elements  160  based on the sum of EM field contributions to the display element. 
     As noted above, the computing architecture  152  can also generate a control signal to an illuminator  162 , e.g., in response to determining that the computations of the sums of the EM field contributions from the number of primitives to each of the display elements have been completed. The illuminator  162  emits an input light  153  to illuminate the modulated display elements  160  and the input light  153  is diffracted by the modulated display elements  160  to form a volumetric light field e.g., a holographic light field  155 , corresponding to the scene data  151 . 
     As illustrated in  FIG.  1 B , the tiles of display elements  160  can be interconnected into a larger display. Correspondingly, computing cores  154  can be interconnected for data communication and distribution. Note that a parameter that changes in the holographic calculations between any given two display elements is their physical locations. Thus, the task of computing the hologram can be shared between the corresponding computing cores  154  equally, and the entire display  150  can operate at the same speed as a single tile, independent of the number of tiles. 
       FIG.  1 C  illustrates an exemplary system  170  for displaying objects in a 3D space. The system  170  can include a computing device, e.g., the computing device  102  of  FIG.  1 A , and a holographic display device  172 , e.g., the holographic display  110  of  FIG.  1 A  or  150  of  FIG.  1 B . A user can use an input device, e.g., a keyboard  174  and/or a mouse  176 , to operate the system  170 . For example, the user can create a CG model for a 2D object  178  and a 3D object  180  through the computing device. The computing device or the holographic display device  172  can include a holographic renderer, e.g., the holographic renderer  130  of  FIG.  1 A , to render the CG model to generate corresponding graphic data for the 2D object  178  and the 3D object  180 . The graphic data can include respective primitive data for a list of primitives corresponding to the objects  178  and  180 . 
     The holographic display device  172  can include a controller, e.g., the controller  112  of  FIG.  1 A  or  152  of  FIG.  1 B , and a display  173 , e.g., the display  114  of  FIG.  1 A  or  156  of  FIG.  1 B . The controller can compute a respective sum of EM field contributions from the primitives to each display element of the display  173  and generate control signals for modulating each display element based on the respective sum of EM field contributions. The holographic display device  172  can further include an illuminator, e.g., the illuminator  116  of  FIG.  1 A  or the illuminator  162  of  FIG.  1 B . The controller can generate a timing control signal to activate the illuminator. When light from the illuminator illuminates a surface of the display  173 , the modulated display elements can cause the light to propagate in the 3D space to form a volumetric light field corresponding to a holographic reconstruction for the 2D views of object  178  and a holographic reconstruction for the 3D object  180 . Thus, the 2D views of object  178  and the 3D holographic reconstruction of the object  180  are displayed as respective holographic reconstructions floating in the 3D space in front of, behind, or straddling the display  173 . 
     In some implementations, the computing device transmits non-primitive based data, e.g., a recorded light field video, to the holographic display device  172 . The holographic display device  172  can compute and generate corresponding holograms, e.g., a series of sequential holograms, to display as corresponding holographic reconstructions in the 3D space. In some implementations, the computing device transmits a CG holographic content simultaneously with live holographic content to the holographic display device  172 . The holographic display device  172  can also compute and generate corresponding holograms to display the contents as corresponding holographic reconstructions in the 3D space. 
       FIG.  2    illustrates an exemplary configuration  200  for electromagnetic (EM) field calculation. A display  202 , e.g., an LCOS device, including an array of elements  204  and a list of primitives including a point primitive  206  are in a 3D space  208 . The 3D space  208  includes boundary surfaces  210 . In a 3D coordinate system XYZ, the point primitive  206  has coordinate information (x, y, z). Each display element  204  lies in a flat plane with respect to other display elements  204  and has a 2D position (u, v). The display element  204  also has a location in the 3D space. By a mathematical point transformation, the 2D position (u, v) can be transferred into six coordinates  250  in the 3D coordinate system. That is, a surface of the display  202  forms a portion of the boundary surfaces  210 . Thus, EM field contributions from the list of primitives to a display element computed by defining a boundary condition at the surface of the display  202  represent a portion of the total EM field contributions from the primitives to the display element. A scale factor, e.g., six, can be multiplied to a sum of the EM field contributions for each of the display elements to obtain a scaled sum of the field contributions, and the display element can be modulated based on the scaled sum of the field contributions. 
     Exemplary EM Field Contributions for Primitives 
     Primitives can be used for computer graphics rendering. Each type of primitive in computer graphics corresponds in the formulation of the technologies disclosed herein to a discrete mathematical function that defines a single holographic primitive for a graphical element added to a hologram. Each type of primitive can correspond to an expression for calculating an EM field contribution to a display element. A primitive can be a point primitive, a line primitive, or a polygon (e.g., a triangle) primitive. As illustrated below, an analytic expression can be derived by calculating EM field propagation from a corresponding primitive to a display element of a display. 
       FIG.  3 A  illustrates an example EM propagation from a point primitive  304  to an element  302  of a display  300 . In a 3D coordinate system XYZ, it is assumed that z coordinate is 0 across the display  300 , which means negative z values are behind the display  300  and positive z values are in front of the display  300 . The point primitive  304  has a coordinate (x, y, z), and the display element  302  has a coordinate (u, v, 0). A distance d uv  between the point primitive  304  and the display element  302  can be determined based on their coordinates. 
     The point primitive  304  can be considered as a point charge with time varying amplitude. According to electromagnetic theory, an electric field E generated by such a point charge can be expressed as: 
     
       
         
           
             
               E 
             
             ∝ 
             
               
                 exp 
                 
                   
                     
                       
                         i 
                         2 
                         π 
                         d 
                       
                       / 
                       λ 
                     
                   
                 
               
               
                 
                   d 
                   2 
                 
               
             
           
         
       
     
      where λ, represents a wavelength of an EM wave, and d represents a distance from the point charge. 
     Thus, the electric field E u,v  at the display element (u,v) can be expressed as: 
     
       
         
           
             
               
                 
                   E 
                   
                     u 
                     , 
                     v 
                   
                 
               
             
             ∝ 
             
               I 
               
                 
                   d 
                   
                     u 
                     v 
                   
                 
                 
                     
                   2 
                 
               
             
             exp 
             
               
                 
                   
                     i 
                     2 
                     π 
                     
                       d 
                       
                         u 
                         v 
                       
                     
                   
                   / 
                   λ 
                 
               
             
           
         
       
     
      where I represents a relative intensity of the holographic primitive electric field at the display element contributed from the point primitive  304 . 
     As discussed above with respect to  FIG.  2   , a surface of the display  300  forms only a portion of a boundary surface for the EM field. A scale factor δ can be applied to the electric field E u,v  to get a scaled electric field E φ  (u, v) at the display element that adjusts for the partial boundary as follows: 
     
       
         
           
             
               E 
               φ 
             
             
               
                 u 
                 , 
                 v 
               
             
             ∝ 
             
               
                 δ 
                 I 
               
               
                 
                   d 
                   
                     u 
                     v 
                   
                 
                 
                     
                   2 
                 
               
             
             exp 
             
               
                 
                   
                     i 
                     2 
                     π 
                     
                       d 
                       
                         u 
                         v 
                       
                     
                   
                   / 
                   λ 
                 
               
             
           
         
       
     
      where δ ≅ [6 + ε],0 &lt; ε ≤ 1. 
       FIG.  3 B  illustrates an example of EM propagation from a line primitive  306  to the display element  302  of the display  300  in the 3D coordinate system XYZ. As noted above, the display element  302  can have a coordinate (u, v, 0), where z = 0. The line primitive  306  has two endpoints P 0  with coordinate (x 0 , y 0 , z 0 ) and P 1  with coordinate (x 1 , y 1 , z 1 ). A distance d 0  between the endpoint P 0  and the display element can be determined based on their coordinates. Similarly, a distance d 1  between the endpoint P 1  and the display element can be determined based on their coordinates. A distance d 01  between the two endpoints P 0  and P 1  can be also determined, e.g., d 01 = d 1 -d 0 . 
     As discussed above, a line primitive can be treated as a superposition or a linear deformation, and a corresponding analytic expression for the line primitive as a linear aperture can be obtained as a distributed delta function in space. This analytic expression can be a closed expression for continuous 3D line segments as holograms. 
       FIG.  3 C  illustrates an example EM propagation from a triangle primitive  308  to the display element  302  of the display  300  in the 3D coordinate system XYZ. As noted above, the display element  302  can have a coordinate (u, v, 0), where z = 0. The triangle primitive  308  has three endpoints: P 0  (x 0 , y 0 , z 0 ), P 1  (x 1 , y 1 , z 1 ), and P 2  (x 2 , y 2 , z 2 ). Distance d 0 , d 1 , and d 2  between the display element and the endpoints P 0 , P 1 , and P 2  can be respectively determined based on their coordinates. 
     Similar to the line primitive in  FIG.  3 B , the triangle primitive can be treated as a continuous aperture in space and an analytical expression for the EM field contribution of the triangle primitive to the display element can be obtained by integration. This can be simplified to obtain an expression for efficient computation. 
     Exemplary Computations for Primitives 
     As discussed above, a controller, e.g., the controller  112  of  FIG.  1 A , can compute an EM field contribution from a primitive to a display element based on an analytical expression that can be determined as shown above. As an example, the EM field contribution for a line primitive is computed as below. 
     Each display element in a display has a physical location in space, and each display element lies in a flat plane with respect to other display elements. Assuming that the display elements and their controllers are laid out as is customary in display and memory devices, a simple mathematical point transformation can be used to transform a logical location of a given display element based on a logical memory address for the display element in a processor to an actual physical location of the display element in the space. Therefore, as the logical memory addresses of the display elements are looped over in a logical memory space of the processor, corresponding actual physical locations in the space across the surface of the display can be identified. 
     As an example, if the display has a 5 µm pitch in both x and y, each logical address increment can move 5 µm in the x direction, and when an x resolution limit of the display is reached, the next increment will move back to the initial x physical location and increment the y physical location by 5 µm. The third spatial coordinate z can be assumed to be zero across the display surface, which means that the negative z values are behind the display, and the positive z values are in front of the display. 
     To begin the line calculation, a type of scaled physical distance between the current display element and each of the two points of the line primitive can be determined to be d 0  and d 1 . As a matter of fact, d 0  and d 1  can be calculated once per primitive, as every subsequent calculation of the distances across display elements is a small perturbation of an initial value. In this way, this computation is performed in one dimension. 
     An example computation process for each primitive can include the following computation codes: 
     
       
         
           
             DD 
               
             = 
               
             f 
             
               
                 d1, 
                   
                 d0 
               
             
             , 
           
         
       
     
     
       
         
           
             iscale= 
               
             SS 
             ∗ 
             COLOR 
             ∗ 
             Alpha1, 
           
         
       
     
     
       
         
           
             C1 
               
             =-2*iscale*sin 
             
               
                 
                   
                     DD 
                   
                   / 
                   2 
                 
               
             
             * 
             sin 
             
               
                 Alpha2 
               
             
             * 
             cos 
             
               
                 Alpha3 
               
             
             , 
           
         
       
     
     
       
         
           
             C2 = -2*iscale*sin 
             
               
                 DD/2 
               
             
             * 
             sin 
             
               
                 Alpha2 
               
             
             * 
             sin 
             
               
                 Alpha4 
               
             
             , 
           
         
       
     
      where SS, Alpha1, Alpha2, Alpha3, and Alpha4 are pre-computed constants, COLOR is the RGB color value passed in with the primitive, and all values are scalar, single precision floats. Both the sine and cosine functions can be looked up in tables stored in the controller to improve computation efficiency. 
     The results in C 1  and C 2  are then accumulated for each primitive at each display element, e.g., in an accumulator for the display element, and can be normalized once at the end of the computations for the display elements. At this point, as noted above, the controller can transmit a first control signal to the display elements to modulate the display elements based on the computed results and a second control signal to an illuminator to turn on to emit light. Accordingly, a holographic reconstruction (or a holographic light field) is visible to a viewer. When illuminated, the modulated display elements can cause the light to produce a crisp, continuous color line in three dimensional space. 
     In some implementations, the computation codes include a hex code for clearing previous accumulations in the accumulator, e.g., at the beginning of the codes. The computation codes can also include a hex code for storing the accumulator results into a respective memory buffer for each display element, e.g., at the end of the codes. In some implementations, a computing device, e.g., the computing device  102  of  FIG.  1 A , transmits a number of background or static primitive hex codes to the controller at an application startup or an interval between displaying frames that does not affect a primary display frame rate. The computing device can then transmit one or more combinations of the hex codes potentially along with other foreground or dynamic primitives at a much higher rate to the controller that can form a corresponding control signal to modulate the display elements of the display. 
     The computation process can be orders of magnitude simpler and faster than the most efficient line drawing routines in conventional 2D display technology. Moreover, this computation algorithm scales linearly with the number of display elements. Thus, scaling computing units of the controller as a 2D networked processing system can keep up with computation needs of an increasing surface area of the display. 
     Exemplary Computation Implementations 
     A Maxwell holographic controller, e.g., the controller  112  of  FIG.  1 A , can compute an EM field contribution from a primitive to a display element based on an analytical expression that can be determined as shown above. The controller can be implemented in, for example, an ASIC, an FPGA or GPU, or any combination thereof. 
     In a modern GPU pipeline, a GPU takes descriptions of geometric figures as well as vertex and fragment shader programs to produce color and depth pixel outputs to one or more output image surfaces (called render targets). The process involves an explosive fan-out of information where geometry is expanded into shading fragments, followed by a visibility test to select whether work needs to be done on each of these fragments. A fragment is a record that contains all the information involved to shade that sample point, e.g., barycentric coordinates on the triangle, interpolated values like colors or texture coordinates, surface derivatives, etc. The process of creating these records then rejecting those that do not contribute to the final image is the visibility test. Fragments that pass the visibility test can be packed into work groups called wavefronts or warps that are executed in parallel by the shader engines. These produce output values that are written back to memory as pixel values, ready for display, or for use as input textures for later rendering passes. 
     In Maxwell holography, the rendering process can be greatly simplified. In Maxwell holographic calculations, every primitive can contribute to every display element. There is no need to expand geometry into pixels and no need to apply visibility tests before packing wavefronts. This can also remove the need for decision making or communication between Maxwell holographic pipelines and allow computation to become a parallel issue with a number of possible solutions each one tuned to speed, cost, size or energy optimization. The graphics pipeline is significantly shorter with fewer intermediate steps, no data copying or movement, and fewer decisions leading to lower latency between initiating a draw and the result being ready for display. This can allow Maxwell holographic rendering to create extremely low latency displays. As discussed below, this can allow Maxwell holographic calculations to increase accuracy, for example, by using fixed point numbers in the Maxwell holographic pipeline, and to optimize computation speed, for example, by optimizing mathematical functions. 
     Using Fixed Point Numbers 
     When calculating an EM contribution from each primitive at each display element (or “phasel”), intermediate calculations involve producing very large numbers. These large numbers involve special handling as they also need to retain the fractional parts during the calculation. 
     Floating point values have the disadvantage thatthey are most accurate close to the origin (zero on the number line) and lose one bit of accuracy every power-of-two when moving away from the origin. For numbers close in the range [-1,1], the accuracy of floating point values can be exquisite, but once reaching numbers in the tens of millions, e.g., reaching the point where single-precision 32-bit IEEE-754 floating point values have no fractional digits remaining, the entire significand (a.k.a mantissa) is used to represent the integer part of the value. However, it is the fractional part of large numbers that Maxwell holography is particularly interested in retaining. 
     In some cases, fixed point numbers are used in the Maxwell holographic calculations. Fixed point number representations are numbers where the decimal point does not change on a case-by-case basis. By choosing the correct numbers of bits for the integer and fractional parts of a number, the same number of fractional bits can be obtained regardless of the magnitude of the number. Fixed point numbers are represented as integers with an implicit scale factor, e.g., 14.375 can be represented as the number 3680 (0000111001100000 base-2) in a 16-bit fixed point value with 8 fractional bits. This can be also represented as an “unsigned 16.8” fixed point number, or u16.8 for short. Negative numbers can have one additional sign bit and are stored in “2s compliment” format. In such a way, the accuracy of the calculation can be greatly improved. 
     Optimization to Mathematical Functions 
     As shown above, Maxwell holographic calculations involve the use of transcendental mathematical functions, e.g., sine, cosine, arc tangent, etc. In a CPU, these functions are implemented as floating point library functions that can use specialized CPU instructions, or on a GPU as floating point units in the GPU. These functions are written to take arguments as a floating point number and the results are returned in the same floating point representation. These functions are built for the general case, to be accurate where floats are accurate, to be correctly rounded and to cope with every edge case in the floating point number representation (+/-Infinity, NaN, signed zero, and denormal floats). 
     In Maxwell holographic calculations, with the fixed point representation, there is no need to use denormal floats for gradual underflow, no need to handle NaN results from operations like division by zero, no need to alter the floating point rounding modes, and no need to raise floating point exceptions to the operating system. All of these allow simplifying (and/or optimizing) the transcendental mathematical functions, for example, as discussed below. 
     In some cases, optimizations can be made to take arguments in one fixed point format and return the value to a different level of accuracy, e.g., input s28.12 and output s15.14. This can be especially desirable when calculating the sine of large values in the 10 s of millions, the input argument can be large but the output can only need to represent the value range [-1,1], or arctangent which takes in any value but return values in the range [-π/2, π/2]. 
     In some cases, optimization can be made to freely implement the transcendental functions as fully enumerated look-up tables, as interpolated tables, as semi-table based polynomial functions, or as semi-table based full minimax polynomials, depending on the input range involved. It also allows to apply specialized range reduction methods that cope with large inputs, which the general purpose GPU pipeline calculation can skip for speed. 
     In some cases, another optimization can be transforming trigonometric calculations from the range [-π, π] into a signed 2’s compliment representation in the range [-1,1] which has the advantage of not requiring expensive modulo 2π division operations. 
     Exemplary Implementations for Occlusion 
     Occlusion is often viewed as a difficult and important topic in computer graphics, and even more so in computational holography. This is because, in at least some cases, while the occlusion problem in projective CGI is static, what is hidden and what is visible in holographic systems depend on the location, orientation, and direction of a viewer. Wave approaches of G-S holography or its derivatives have been developed to address the holographic occlusions. However, masking or blocking contributions from parts of a scene that are behind other parts of a scene can be very complicated and computationally expensive in the G-S methodology. 
     In Maxwell holography, the occlusion issue can be addressed comparatively easily, because which display elements (e.g., phasels) correspond to which primitives is completely deterministic and trivial. For example, whether or not a given display element contributes to a reconstruction of a given primitive can be determined as the calculation for the given primitive is performed. After determining that a number of display elements do not contribute to the given primitive due to occlusion, when calculating a sum of EM contributions to one of the number of display elements, the EM contribution from the given primitive is omitted from the calculation of the sum of EM contributions to the one of the number of display elements. 
     For illustration only,  FIGS.  3 D- 3 F  show a determination of display elements not contributing to a given primitive (a point in  FIG.  3 D , a line in  FIG.  3 E , and a triangle in  FIG.  3 F ) with a line primitive as an occluder. The line primitive has a starting point O 1  and an ending point O2. 
     As illustrated in  FIG.  3 D , a point primitive P0 is behind the occluder and closer to the display. By extending lines connecting O 1 -P 0  and O 2 -P 0 , a range of display elements from D 1  to D 2  in the display is determined, which do not contribute to the reconstruction of the point primitive P 0 . 
     In some examples, the coordinate information of O 1 , O 2 , and P 0  is known, e.g., stored in a “Z” buffer calculated by a GPU (e.g., the GPU  108  of  FIG.  1 A ) prior to the scene being transmitted to the Maxwell holographic controller (e.g., the controller 112 of  FIG.  1 A ). For example, in an XZ plane with y=0, the coordinate information can be O 1  (O×1, Oz1), O 2  (Ox2, Oz2), and P 0  (Px, Pz), with Oz1=Oz2=Oz. Based on the coordinate information, the coordinate information of D 1  and D 2  can be determined to be 
     
       
         
           
             Dx1 
             = 
             Px 
             + 
             ρ 
               
             
               
                 Px 
                 − 
                 Ox2 
               
             
             , 
              Dx2 
             = 
             Dx1 
             + 
             ρ 
               
             
               
                 Ox2 
                 − 
                 Ox1 
               
             
           
         
       
     
      where ρ = Pz/(Oz-P), and Dz1=Dz2=0. 
     The information of D 1  and D 2  can be stored as additional information in an “S” buffer for the Maxwell holographic controller, besides the information in a Z buffer for the point primitive P 0 . In such a way, the additional information can be used to trivially mask the contributions of specific display elements (within the range from D 1  to D 2 ) to the specific primitive P 0  in the indexed primitive list. 
       FIG.  3 E  illustrates a determination of how a specific display element contributes to a line primitive with an occluder before (or in front of) the line primitive. By connecting the specific display element D 0  to the starting point O 1  and the ending point O 2  of the occluder, two point primitives P 1  and P 2  on the line primitive are determined as the intersection points. Thus, the specific display element D 0  does not contribute to the reconstruction of the part of the line primitive from P 1  to P 2  on the line primitive. Accordingly, when calculating the sum of EM contributions to the specific display element D 0 , the EM contributions from the part P 1 -P 2  of the line primitive is not calculated. 
     This can be implemented in two ways. In the first way, the EM contributions from the part P 0 -P 1  and the part P 2 -P n  to the specific display element D 0  are summed as the EM contributions of the line primitive to the specific display element D 0 , by considering the occlusion from the occluder. In the second way, the EM contribution from the whole line primitive P 0 -P n  is calculated, together with the EM contribution from the part P 1 -P 2 , and a difference between the two calculated EM contributions can be considered as the EM contribution of the line primitive to the specific display element D 0  by considering the occlusion from the occluder. The coordinate information of P 1  and P 2  or the part P 1 -P 2  can be stored, as the part of the line primitive that does not contribute to the specific display element D 0 , in the “S” buffer of the Maxwell holographic controller, together with the information of the occluder and other information in the “Z” buffer of the GPU. 
       FIG.  3 F  illustrates a determination of how a specific display element contributes to a triangle primitive with an occluder before the triangle primitive. By connecting the specific display element D 0  to the starting point O 1  and the ending point O 2  of the occluder, four point primitives P 1 , P 2 , P 3 , and P 4  on sides of the triangle primitive are determined as the intersection points. Thus, the specific display element D 0  does not contribute to the reconstruction of the part of the triangle primitive enclosed by the points P 1 , P 2 , P 3 , P 4 , P C . Accordingly, when calculating the sum of EM contributions to the specific display element D 0 , the EM contributions from the part P 1 -P 2 -P 3 -P 4 -P C  of the triangle primitive is not calculated. That is, only the EM contributions from the first triangle formed by points P A , P 1  and P 2  and the second triangle formed by points P B , P 3 , and P 4  are summed as the EM contribution of the triangle primitive P A -P B -P C  by considering the occlusion of the occluder. The coordinate information of P 1 , P 2 , P 3 , and P 4  or the triangle primitives P A -P 1 -P 2  and P B -P 3 -P 4  can be stored, as the part of triangle primitive P A -P B -P C  that contributes to the specific display element D 0 , in the “S” buffer of the Maxwell holographic controller, together with the information of the occluder and other information in the “Z” buffer of the GPU. 
     The implementations of occlusion in Maxwell holography enables to convert the “Z” buffer in the GPU to the “S” buffer in the Maxwell holographic controller, and can mask the contributions of specific primitives (or specific parts of the primitives) in the indexed primitive list to a specific display element. This not only provides accurate, physically correct occlusion, it also saves computation time, as the primitives that do not contribute to a given display element can be ignored and computation can move on to computation for the next display element. The “S” buffer can contain additional information related to diffraction efficiency of the display. 
     The “S” buffer can also include rendering features such as Holographic specular highlights, in which a reflectivity of a surface is dependent upon the viewing angle. In traditional CGI, specular highlights are dependent only on the orientation of the rendered object, whereas in a Maxwell holographic context, the direction from which the object is viewed also plays a part. Therefore, the geometric specular information can be encoded in the “S” buffer as an additive (specular) rather than a subtractive (occlusion) contribution. In Maxwell holography, the mathematics for holographic specular highlights can be substantially the same as that for holographic occlusion. 
     Exemplary Implementations for Stitching 
     When light illuminates a display modulated with EM contributions from a list of primitives of a 3D object, the modulated display causes the light to propagate in different directions to form a volumetric light field corresponding to the primitives. The volume light field is the Maxwell holographic reconstruction. Two adjacent primitives in the 3D object, e.g., two triangle primitives, have a shared side (e.g., edge or surface). During the reconstruction, a stitching issue may raise, where the light intensity of the shared side can be doubled due to the reconstructions of the two adjacent primitives separately. This may affect the appearance of the reconstructed 3D object. 
     To address the stitching issue in Maxwell holography, as illustrated in  FIG.  3 G , the adjacent primitives can be scaled down by a predetermined factor, so that a gap can be formed between the adjacent primitives. In some cases, instead of scaling down the two adjacent primitives, only one primitive or a part of the primitive is scaled down. For example, a line of a triangle primitive can be scaled down to separate from another triangle primitive. In some cases, the scaling can include scaling different parts of a primitive with different predetermined factors. The scaling can be designed such that the gap is big enough to separate the adjacent primitives to minimize the stitching issue and small enough to make the reconstructed 3D object appear seamless. The predetermined factor can be determined based on information of the display and of the viewer, e.g., a maximum spatial resolution of the holographic light field and, in the case of a part of a primitive appearing entirely or partially behind the display, a minimum distance from the viewer to that part of the primitive. 
     In some cases, the scaling operation can be applied to primitive data of a primitive obtained from the holographic renderer, e.g., the holographic renderer  130  of  FIG.  1 A , and the scaled primitive data of the primitive is sent to the Maxwell holographic controller, e.g., the controller  112  of  FIG.  1 A . In some cases, the controller can perform the scaling operation on the primitive data obtained from the holographic renderer, before calculating EM contributions of the primitives to the display elements of the display. 
     Exemplary Implementations for Texture Mapping 
     Texture mapping is a technique developed in computer graphics. The basic idea is to take a source image and apply it as a decal to a surface in a CGI system, enabling detail to be rendered into the scene without the need for the addition of complex geometry. The texture mapping can include techniques for the creation of realistic lighting and surface effects in the CGI system, and can refer universally to the application of surface data to triangular meshes. 
     In Maxwell holography, flat shaded and also interpolated triangular meshes can be rendered in genuine 3D using the analytic relationship between arbitrary triangles in space and a phase map on a holographic device. However, to be compatible with modern rendering engines, the ability to map information on the surface of these triangles is desirable. This can present a real problem, in that the speed of the method is derived from the existence of the analytic mapping, which does not admit data-driven amplitude changes. 
     Discrete Cosine Transform (DCT) is an image compression technique and can be considered as the real-valued version of the FFT (Fast Fourier transform). DCT depends on an encode-decode process that assigns weights to cosine harmonics in a given image. The result of an encode is a set of weights equal in number to the number of pixels in the original image, and if every weight is used to reconstruct an image, there will be no loss in information. However, in many images, acceptable reconstructions can be made from a small subset of the weights, enabling large compression ratios. 
     The decode (render) process of the DCT in two dimensions involves a weighted double sum over every DCT weight and every destination pixel. This can be applied to Maxwell holography for texture mapping. In Maxwell holography, triangle rendering involves a “spiked” double integral, in phase space, to determine the phase contribution of any individual phasel to the triangle in question. The integral can be folded into a double sum which mirrors the one in the DCT reconstruction, and then re-derive the analytic triangle expression in terms of the DCT weights. This implementation of DCT technique in Maxwell holographic calculations enables to draw full, texture mapped triangles, to employ image compression to the data for the rendered texture triangles, and to take advantage of existing toolsets that automatically compress texture and image data using DCT such as JPEG. 
     In some implementations, to draw a Maxwell holographic textured triangle, a spatial resolution desired for the mapping on a specified surface is first calculated. Then a texture with the resolution is supplied, and DCT compressed with angular and origin information to correctly orient it on the triangle is obtained. Then, the triangle corners and a list of DCT weights are included in the indexed primitive list and sent to the Maxwell holographic controller. The DCT weights can be included in the EM contributions of the triangle primitive to each display element. The texture triangle can be n times slower than a flat triangle, where n is the number of (nonzero) DCT weights that are sent with the primitive. Modern techniques for “fragment shading” can be implemented in the Maxwell holographic system, with the step of the DCT encode replacing the filter step for traditional projective rendering. 
     As an example, the following expression shows the DCT weights B pq  for an image: 
     
       
         
           
             
               B 
               
                 p 
                 q 
               
             
             ≡ 
             
               σ 
               p 
             
             
               σ 
               q 
             
             
               
                 ∑ 
                 
                   m 
                   = 
                   0 
                 
                 
                   M 
                   − 
                   1 
                 
               
               
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       0 
                     
                     
                       N 
                       − 
                       1 
                     
                   
                   
                     
                       A 
                       
                         m 
                         n 
                       
                     
                     c 
                     o 
                     s 
                     
                       
                         
                           
                             π 
                             
                               
                                 2 
                                 m 
                                 + 
                                 1 
                               
                             
                             p 
                           
                           
                             2 
                             M 
                           
                         
                       
                     
                   
                 
               
             
               
             c 
             o 
             s 
             
               
                 
                   
                     π 
                     
                       
                         2 
                         n 
                         + 
                         1 
                       
                     
                     q 
                   
                   
                     2 
                     N 
                   
                 
               
             
           
         
       
     
     where  
     
       
         
           
             
               σ 
               p 
             
             = 
             
               
                 
                   
                     
                       
                         
                           
                             1 
                               
                           
                           / 
                           
                             
                               M 
                             
                           
                         
                       
                     
                     
                       
                         p 
                         = 
                         0 
                       
                     
                   
                   
                     
                       
                           
                         
                           
                             
                               2 
                               / 
                               M 
                             
                           
                         
                       
                     
                     
                       
                         e 
                         l 
                         s 
                         e 
                       
                     
                   
                 
               
             
             , 
               
               
             
               σ 
               q 
             
             = 
             
               
                 
                   
                     
                       
                         
                           
                             1 
                               
                           
                           / 
                           
                             
                               N 
                             
                           
                         
                       
                     
                     
                       
                         q 
                         = 
                         0 
                       
                     
                   
                   
                     
                       
                           
                         
                           
                             
                               2 
                               / 
                               N 
                             
                           
                         
                       
                     
                     
                       
                         e 
                         l 
                         s 
                         e 
                       
                     
                   
                 
                   
                 , 
               
             
           
         
       
     
     M and N are corners of a rectangular image, and (p, q) is a DCT term. 
     By decoding, the amplitude value A mn  can be obtained as follows: 
     
       
         
           
             
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 
                   A 
                   
                     m 
                     n 
                   
                 
                 = 
                 
                   
                     ∑ 
                     
                       p 
                       = 
                       0 
                     
                     
                       M 
                       − 
                       1 
                     
                   
                   
                     
                       
                         ∑ 
                         
                           q 
                           = 
                           0 
                         
                         
                           N 
                           − 
                           1 
                         
                       
                       
                         
                           σ 
                           p 
                         
                         
                           σ 
                           q 
                         
                         
                           A 
                           
                             m 
                             n 
                           
                           * 
                         
                       
                     
                   
                 
               
             
             
               
                 where  
                 
                   A 
                   
                     m 
                     n 
                   
                   * 
                 
                 = 
                 
                   B 
                   
                     p 
                     q 
                   
                 
                 c 
                 o 
                 s 
                 
                   
                     
                       
                         π 
                         
                           
                             2 
                             m 
                             + 
                             1 
                           
                         
                         p 
                       
                       
                         2 
                         M 
                       
                     
                   
                 
                 c 
                 o 
                 s 
                 
                   
                     
                       
                         π 
                         
                           
                             2 
                             n 
                             + 
                             1 
                           
                         
                         q 
                       
                       
                         2 
                         N 
                       
                     
                   
                 
                 . 
               
             
           
         
       
     
     When calculating the EM contribution of the textured triangle primitive to a display element (e.g., a phasel), a DCT term with a corresponding DCT weight  
     
       
         
           
             
               A 
               
                 m 
                 n 
               
               ∗ 
             
           
         
       
     
     can be included in the calculation as follows: 
     
       
         
           
             
               φ 
               
                 p 
                 q 
                   
               
             
             = 
               
             
               ∑ 
               
                 
                     
                   
                     y 
                     = 
                     0 
                   
                   Y 
                 
               
             
             
               ∑ 
               
                 
                     
                   
                     x 
                     = 
                     0 
                   
                   X 
                 
                   
                 
                   A 
                   
                     m 
                     n 
                   
                   ∗ 
                 
               
             
             T 
           
         
       
     
      where X, Y are corners of the triangle in the coordinate system, T corresponds to the EM contribution of the triangle primitive to the display element, and φ pq  is the partial contribution for non-zero term B pq  in the DCT. The number of (p,q) DCT terms can be selected by considering both the information loss in reconstruction and the information compression. 
     Exemplary Process 
       FIG.  4    is a flowchart of an exemplary process  400  of displaying an object in 3D. The process  400  can be performed by a controller for a display. The controller can be the controller  112  of  FIG.  1 A  or  152  of  FIG.  1 B . The display can the display  114  of  FIG.  1 A  or  156  of  FIG.  1 B . 
     Data including respective primitive data for primitives corresponding to an object in a 3D space is obtained ( 402 ). The data can be obtained from a computing device, e.g., the computing device  102  of  FIG.  1 A . The computing device can process a scene to generate the primitives corresponding to the object. The computing device can include a renderer to generate the primitive data for the primitives. In some implementations, the controller generates the data itself, e.g., by rendering the scene. 
     The primitives can include at least one of a point primitive, a line primitive, or a polygon primitive. The list of primitives is indexed in a particular order, e.g., by which the object can be reconstructed. The primitive data can include color information that has at least one of a textured color, a gradient color, or a constant color. For example, the line primitive can have at least one of a gradient color or a textured color, or a constant color. The polygon primitive can also have at least one of a gradient color, a textured color, or a constant color. The primitive data can also include texture information of the primitive and/or shading information on one or more surfaces of the primitive (e.g., a triangle). The shading information can include a modulation on at least one of color or brightness on the one or more surfaces of the primitive. The primitive data can also include respective coordinate information of the primitive in the 3D coordinate system. 
     The display can include a number of display elements, and the controller can include a number of computing units. Respective coordinate information of each of the display elements in the 3D coordinate system can be determined based on the respective coordinate information of the list of primitives in the 3D coordinate system. For example, a distance between the display and the object corresponding to the primitives can be predetermined. Based on the predetermined distance and the coordinate information of the primitives, the coordinate information of the display elements can be determined. The respective coordinate information of each of the display elements can correspond to a logical memory address for the element stored in a memory. In such a way, when the controller loops in a logical memory address for a display element in a logical memory space of the controller, a corresponding actual physical location for the display element in the space can be identified. 
     An EM field contribution from each of the primitives to each of the display elements is determined by calculating EM field propagation from the primitive to the element in the 3D coordinate system ( 404 ). The EM field contribution can include at least one of a phase contribution or an amplitude contribution. 
     As illustrated above with respect to  FIGS.  3 A- 3 C , at least one distance between the primitive and the display element can be determined based on the respective coordinate information of the display element and the respective coordinate information of the primitive. In some cases, for each primitive, the at least one distance can be calculated or computed just once. For example, the controller can determine a first distance between a first primitive of the primitives and a first element of the display elements based on the respective coordinate information of the first primitive and the respective coordinate information of the first element and determining a second distance between the first primitive and a second element of the elements based on the first distance and a distance between the first element and the second element. The distance between the first element and the second element can be predetermined based on a pitch of the plurality of elements of the display. 
     The controller can determine the EM field contribution to the display element from the primitive based on a predetermined expression for the primitive and the at least one distance. In some cases, as illustrated above with respect to  FIGS.  3 A- 3 C , the predetermined expression can be determined by analytically calculating the EM field propagation from the primitive to the element. In some cases, the predetermined expression is determined by solving Maxwell’s equations. Particularly, the Maxwell’s equations can be solved by providing a boundary condition defined at a surface of the display. The boundary condition can include a Dirichlet boundary condition or a Cauchy boundary condition. The primitives and the display elements are in the 3D space, and the surface of the display forms a portion of a boundary surface of the 3D space. The predetermined expression can include at least one of functions that include a sine function, a cosine function, and an exponential function. During computation, the controller can identify a value of the at least one of the functions in a table stored in a memory, which can improve a computation speed. The controller can determine the EM field contribution to each of the display elements for each of the primitives by determining a first EM field contribution from a first primitive to a display element in parallel with determining a second EM field contribution from a second primitive to the display element. 
     For each of the display elements, a sum of the EM field contributions from the list of primitives to the display element is generated ( 406 ). 
     In some implementations, the controller determines first EM field contributions from the primitives to a first display element and sums the first EM field contributions for the first element and determining second EM field contributions from the primitives to a second display element and sums the second EM field contributions for the second display element. The controller can include a number of computing units. The controller can determine an EM field contribution from a first primitive to the first element by a first computing unit in parallel with determining an EM field contribution from a second primitive to the first element by a second computing unit. 
     In some implementations, the controller determines first respective EM field contributions from a first primitive to each of the display elements and determine second respective EM field contributions from a second primitive to each of the display elements. Then the controller accumulates the EM field contributions for the display element by adding the second respective EM field contribution to the first respective EM field contribution for the display element. Particularly, the controller can determine the first respective EM field contributions from the first primitive to each of the display elements by using a first computing unit in parallel with determining the second respective EM field contributions from the second primitive to each of the display elements by using a second computing unit. 
     A first control signal is transmitted to the display, the first control signal being for modulating at least one property of each display element based on the sum of the field distributions to the display element ( 408 ). The at least one property of the element includes at least one of a refractive index, an amplitude index, a birefringence, or a retardance. 
     The controller can generate, for each of the display elements, a respective control signal based on the sum of the EM field contributions from the primitives to the element. The respective control signal is for modulating the at least one property of the element based on the sum of the EM field contributions from the primitives to the element. That is, the first control signal includes the respective control signals for the display elements. 
     In some examples, the display is controlled by electrical signals. Then the respective control signal can be an electrical signal. For example, an LCOS display includes an array of tiny electrodes whose voltage is individually controlled as element intensities. The LCOS display can be filled with a birefringent liquid crystal (LC) formulation that changes its refractive index as an applied voltage changes. Thus, the respective control signals from the controller can control the relative refractive index across the display elements and accordingly the relative phase of light passing through or reflected by the display. 
     As discussed above, the display surface forms a part of the boundary surface. The controller can multiple a scale factor to the sum of the field contributions for each of the elements to obtain a scaled sum of the field contributions, and generate the respective control signal based on the scaled sum of the field contributions for the element. In some cases, the controller can normalize the sum of the field contributions for each of the elements, e.g., among all the elements, and generate the respective control signal based on the normalized sum of the field contributions for the element. 
     A second control signal is transmitted to an illuminator as a control signal for turning on the illuminator to illuminate light on the modulated display ( 410 ). The controller can generate and transmit the second control signal in response to determining a completion of obtaining the sum of the field contributions for each of the display elements. Due to time symmetry (or conservation of energy), the modulated elements of the display can cause the light to propagate in different directions to form a volumetric light field corresponding to the object in the 3D space. The volumetric light field can correspond to a solution of Maxwell’s equations with a boundary condition defined by the modulated elements of the display. 
     In some implementations, the illuminator is coupled to the controller through a memory buffer configured to control amplitude or brightness of one or more light emitting elements in the illuminator. The memory buffer for the illuminator can have a smaller size than a memory buffer for the display. A number of the light emitting elements in the illuminator can be smaller than a number of the elements of the display. The controller can be configured to activate the one or more light emitting elements of the illuminator simultaneously. 
     In some examples, the illuminator includes two or more light emitting elements each configured to emit light with a different color. The controller can be configured to sequentially modulate the display with information associated with a first color during a first time period and modulate the display with information associated with a second color during a second, sequential time period, and to control the illuminator to sequentially turn on a first light emitting element to emit light with the first color during the first time period and a second light emitting element to emit light with the second color during the second time period. In such a way, a multi-color object can be displayed in the 3D space. 
     In some examples, the display has a resolution small enough to diffract light. The illuminator can emit a white light into the display which can diffract the white light into light with different colors to thereby display a multi-color object. 
     Exemplary Systems 
       FIGS.  5 A- 5 K  show implementations of example systems for 3D displays. Any one of the systems can correspond to, for example, the system  100  of  FIG.  1 A .  FIGS.  5 A and  5 B  show example systems having reflective displays with front illumination.  FIG.  5 C  shows an example system having a transmissive display with back illumination.  FIGS.  5 D and  5 E  show example systems having transmissive displays with waveguide illumination.  FIGS.  5 F and  5 G  show example systems having reflective displays with waveguide illumination.  FIGS.  5 H and  5 I  show example systems having reflective displays with optically diffractive illumination using a transmissive grating structure ( FIG.  5 H ) and a reflective grating structure ( FIG.  5 I ).  FIGS.  5 J and  5 K  show example systems having transmissive displays with optically diffractive illumination using a reflective grating structure ( FIG.  5 J ) and a transmissive grating structure ( FIG.  5 K ). 
       FIG.  5 A  illustrates a system  500  with a reflective display with front illumination. The system  500  includes a computer  502 , a controller  510  (e.g., an ASIC), a display  512  (e.g., an LCOS device), and an illuminator  514 . The computer  502  can be the computing device  102  of  FIG.  1 A , the controller  510  can be the controller  112  of  FIG.  1 A , the display  512  can be the display  114  of  FIG.  1 A , and the illuminator  514  can be the illuminator  116  of  FIG.  1 A . 
     As illustrated in  FIG.  5 A , the computer  502  includes an application  504  that has a renderer  503  for rendering a scene of an object. The rendered scene data is processed by a video driver  505  and then a GPU  506 . The GPU  506  can be the GPU  108  of  FIG.  1 A  and can be configured to generate a list of primitives corresponding to the scene and respective primitive data. For example, the video driver  505  can be configured to process the rendered scene data and generate a list of primitives. As noted above, the GPU  506  can include a conventional 2D renderer, e.g., the conventional 2D renderer  120  of  FIG.  1 A , to render the primitives into a list of items to draw on a 2D display  508 . The GPU  506  or the controller  510  can include a holographic renderer, e.g., the holographic renderer  130  of  FIG.  1 A , to render the list of primitives into graphic data to be displayed by the display  512 . 
     The controller  510  is configured to receive the graphic data from the computer  502 , compute EM field contributions from the list of primitives to each of elements of the display  512 , and generate a respective sum of the EM field contributions from the primitives to each of the elements. The controller  510  can generate respective control signals to each of the display elements for modulating at least one property of the display element. The controller can transmit the respective control signals to the display elements of the display  512  through a memory buffer  511  for the display  512 . 
     The controller  510  can also generate and transmit a control signal, e.g., an illumination timing signal, to activate the illuminator  514 . For example, the controller  510  can generate and transmit the control signal in response to determining that the computations of the sums of EM field contributions from the primitives to the display elements are completed. As noted above, the controller  510  can transmit the control signal to the illuminator  514  via a memory buffer. The memory buffer can be configured to control amplitude or brightness of light emitting elements in the illuminator  514  and activate the light emitting elements simultaneously or sequentially. 
     As illustrated in  FIG.  5 A , the illuminator  514  can emit a collimated light beam  516  that is incident on a front surface of the display  512  at an incident angle in a range between 0 degrees and almost ±90 degrees. The emitted light beam is diffracted from the display  512  to form a holographic light field  518 , corresponding to the object, which can be seen by a viewer. 
       FIG.  5 B  illustrates another system  520  with another reflective display  524  with front illumination. Compared to the system  500  of  FIG.  5 A , the system  520  has a larger reflective display  524 . To accommodate this, or for other packaging or aesthetic reasons, a display controller  522  is included in a housing that can be a support or enclosure for an illuminator  526 . The controller  522  is similar to the controller  510  of  FIG.  5 A  and can be configured to receive graphic data from a computer  521 , compute EM field contributions from primitives to each of display elements of the display  524 , and generate a respective sum of the EM field contributions from the primitives to each of the display elements. The controller  522  then generates respective control signals to each of the display elements for modulating at least one property of the display element and transmits the respective control signals to the display elements of the display  524  through a memory buffer  523  for the display  524 . 
     The controller  522  also transmits a control signal to the illuminator  526  to activate the illuminator  526 . The illuminator  526  emits a divergent or semi-collimated light beam  527  to cover a whole surface of the display  524 . The light beam  524  is diffracted by the modulated display  524  to form a holographic light field  528 . 
       FIG.  5 C  illustrates a system  530  with a transmissive display  534  with back illumination. The transmissive display  534 , for example, can be a large scale display. The system  530  includes a controller  532  which can be similar to the controller  510  of  FIG.  5 A . The controller  532  can be configured to receive graphic data from a computer  531 , compute EM field contributions from primitives to each of display elements of the display  534 , and generate a respective sum of the EM field contributions from the primitives to each of the display elements. The controller  532  then generates respective control signals to each of the display elements for modulating at least one property of the display element and transmits the respective control signals to the display elements of the display  534  through a memory buffer  533  for the display  534 . 
     The controller  532  also transmits a control signal to an illuminator  536  to activate the illuminator  536 . Different from the system  500  of  FIG.  5 A  and the system  520  of  FIG.  5 B , the illuminator  536  in the system  530  is positioned behind a rear surface of the display  534 . To cover a large surface of the display  534 , the illuminator  536  emits a divergent or semi-collimated light beam  535  on to the rear surface of the display  534 . The light beam  535  is transmitted through and diffracted by the modulated display  534  to form a holographic light field  538 . 
       FIG.  5 D  illustrates another system  540  with a transmissive display  544  with waveguide illumination. The system  540  also includes a controller  542  and an illuminator  546 . The controller  542  can be similar to the controller  510  of  FIG.  5 A , and can be configured to receive graphic data from a computer  541 , perform computation on the graphic data, generate and transmit control signals for modulation to the display  544  and a timing signal to activate the illuminator  546 . 
     The illuminator  546  can include a light source  545  and include or be optically attached to a waveguide  547 . Light emitted from the light source  545  can be coupled to the waveguide  547 , e.g., from a side cross-section of the waveguide. The waveguide  547  is configured to guide the light to illuminate a surface of the display  544  uniformly. The light guided by the waveguide  547  is incident on a rear surface of the display  544  and transmitted through and diffracted by the display  544  to form a holographic light field  548 . 
     Different from the system  500  of  FIG.  5 A ,  520  of  FIG.  5 B ,  530  of  FIG.  5 C , in the system  540 , the controller  542 , the display  544 , and the waveguide  547  are integrated together into a single unit  550 . In some cases, the waveguide  547  and the light source  545  can be integrated as an active waveguide illuminator in a planar form, which can further increase a degree of integration of the single unit  550 . As discussed above, the single unit  500  can be connected or tiled with other similar units  550  to form a larger holographic display device. 
       FIG.  5 E  illustrates another system  560  with another transmissive display  564  with waveguide illumination. Compared to the system  540 , the transmissive display  564  can potentially implement a display that is larger than the transmissive display  544 . For example, the transmissive display  564  can have a larger area than a controller  562 , and to accommodate this, the controller  562  can be positioned away from the display  564 . The system  560  includes an illuminator  566  that has a light source  565  and a waveguide  567 . The waveguide  567  is integrated with the display  564 , e.g., optically attached, to a rear surface of the display  564 . In some implementations, the display  564  is fabricated on a front side of a substrate and the waveguide  567  can be fabricated on a back side of the substrate. 
     The controller  562  can be similar to the controller  510  of  FIG.  1 A  and configured to receive graphic data from a computer  561 , perform computation on the graphic data, generate and transmit control signals for modulation to the display  564  through a memory buffer  563  and a timing signal to activate the light source  565 . Light emitted from the light source  565  is guided in the waveguide  567  to illuminate the rear surface of the display  564  and transmitted and diffracted through the display  564  to form a holographic light field  568 . 
       FIG.  5 F  illustrates another system  570  with a reflective display  574  with waveguide illumination. The reflective display  574 , for example, can be a large display. A waveguide  577  of an illuminator  576  is positioned on a front surface of the reflective display  574 . A controller  572 , similar to the controller  510  of  FIG.  5 A , can be configured to receive graphic data from a computer  571 , perform computation on the graphic data, generate and transmit control signals for modulation to the display  574  through a memory buffer  573  and a timing signal to activate a light source  575  of the illuminator  576 . Light coupled from a waveguide  577  of the illuminator  576  is guided to be incident on the front surface of the display  574  and diffracted by the display  574  to form a holographic light field  578 . 
       FIG.  5 G  illustrates another system  580  with a reflective display  584  with another type of waveguide illumination using a waveguide device  588 . A controller  582 , similar to the controller  510  of  FIG.  5 A , is configured to generate and transmit controls signals corresponding to holographic data (images and/or videos) for modulation of the display  584  and transmit a timing signal to activate an illuminator  586 . The illuminator  586  can provide one or more colors of light that can be collimated. The waveguide device  588  is positioned in front of the illuminator  586  and the display  584 . The waveguide device  588  can include an input coupler  588 - 1 , a waveguide  588 - 2 , and an output coupler  588 - 3 . The input coupler  588 - 1  is configured to couple the collimated light from the illuminator  586  into the waveguide  588 - 2 . The light then travels inside the waveguide  588 - 2  via total internal reflection and is incident at the end of the waveguide  588 - 2  on the output coupler  588 - 3 . The output coupler  588 - 3  is configured to couple out the light into the display  584 . The light then illuminates the display elements of the display  584  that are modulated with corresponding control signals and is diffracted by the reflective display  584  and reflected back (e.g., by a back mirror of the display  584 ) through the waveguide device  588  (e.g., the output coupler  588 - 3 ) to form a holographic light field corresponding to the holographic data in front of a viewer. 
     In some examples, light is coupled out by the output coupler  588 - 3  at an angle normal to the waveguide device  588  and/or a front surface of the reflective display  584 . In some examples, each of the input coupler  588 - 1  and the output coupler  588 - 2  can include a grating structure, e.g., a Bragg grating. The input coupler  588 - 1  and the output coupler  588 - 2  can include a similar diffraction grating with different fringe tilt angle. In some examples, the illuminator  586  provides a single color of light, and the input coupler  588 - 1  and the output coupler  588 - 2  includes a diffraction grating for the color. In some examples, the illuminator  586  provides multiple colors of light, e.g., red, green and blue light beams, and the input coupler  588 - 1  and the output coupler  588 - 2  can include a multilayer stack of three corresponding diffraction gratings (or a single layer having the three corresponding diffraction gratings) that respectively couple in or couple out the different color light beams. 
       FIG.  5 H  illustrates another system  590  with a reflective display  594  with optically diffractive illumination using an optically diffractive device  598 . The optically diffractive device  598  can be considered as a lightguide device for guiding light. The optically diffractive device  598  can be a transmissive field grating based structure that can include one or more transmissive holographic gratings. The reflective display  594  can be a reflective LCOS device. A controller  592 , similar to the controller  510  of  FIG.  5 A , can be configured to receive graphic data corresponding to one or more objects from a computer  591 , perform computation on the graphic data, and generate and transmit control signals for modulation to the display  594  through a memory buffer  593 . The controller  592  can be also coupled to an illuminator  596  and be configured to provide a timing signal to activate the illuminator  596  to provide light. The light is then diffracted by the optically diffractive device  598  to be incident on the display  594  and then diffracted by the display  594  to form a holographic light field  599  corresponding to the one or more objects. The display  594  can include a back mirror on the back of the display  594  and can reflect the light towards the viewer. The optically diffractive device  598  can be optically transparent. The illuminator  596  can be positioned below the display  594 , which can allow the illuminator  596  to be mounted or housed with other components of the system  590  and to be below an eyeline of the viewer. 
     As discussed with further details below, Bragg selectivity allows off-axis illumination light to be diffracted from the optically diffractive device  598  towards the display  594  while the returning light diffracted from the display  594  can be close to on axis and hence be off-Bragg to the gratings in the optically diffractive device  598  and hence can pass through the optically diffractive device  598  almost perfectly to the viewer without being diffracted again by the gratings in the optically diffractive device  598 . In some implementations, the light from the illuminator  596  can be incident on the optically diffractive device  598  with a large incident angle from a side of the display  594 , such that the illuminator  596  does not block the viewer’s view and is not intrusive into the holographic light field  599 . The incident angle can be a positive angle or a negative angle with respect to a normal line of the display  594 . For illustration, the incident angle is presented as a positive angle. For example, the incident angle can be in a range from 70 degrees to 90 degrees, e.g., in a range from 80 degrees to 90 degrees. In a particular example, the incident angle is 84 degrees. The diffracted light from the optically diffractive device  598  can be diffracted at close to normal incidence into the display  594 , such that the light can uniformly illuminate the display  594  and can be diffracted back near-normally through the optically diffractive device  598  to the viewer’s eyes with minimized power loss due to undesired reflections, diffractions, and/or scatterings within or at the surfaces of the optically diffractive device  598 . In some examples, the diffracted angle from the optically diffractive device  598  to the reflective display  594  can be in a range of -10° (or 10 degrees) to 10° (or 10 degrees), e.g., from -7° to 7°, or from 5° to 7°. In a particular example, the diffracted angle is 6°. In another example, the diffracted angle is 0°. 
     In some implementations, as illustrated in  FIG.  5 H , the optically diffractive device  598  is arranged in front of the reflective display  594 , e.g., along the Z direction towards the viewer. The optically diffractive device  598  can include a field grating structure  598 - 1  positioned on a substrate  598 - 2 . A back surface of the field grating structure  598 - 1  faces a front surface of the reflective display  594 , and a front surface of the field grating structure  598 - 1  is attached to the substrate  598 - 2 . The light from the illuminator  596  can be incident on the front surface of the field grating structure  598 - 1  through the substrate  598 - 2 , e.g., from a side surface of the substrate  598 - 2 . For example, the substrate  598 - 2  can have a wedged side surface, e.g., as illustrated with further details in  FIG.  12 C , such that the light at a large incident angle can have less reflection loss. 
     As discussed with further details below, if a diffraction efficiency of a diffractive structure, e.g., a holographic grating, is less than 100%, light incident at an incident angle can be diffracted by the diffractive structure into zero and first orders. Light of first order (or first order light) is diffracted by the diffractive structure at a diffracted angle towards the display to therein diffract again to reconstruct a holographic light field  599 . The first order can be also called first diffraction order. Light in the zero order (or zero order light, or undiffracted light, or the undiffracted order) is undiffracted (or undeflected) by the diffractive structure and transmitted by the diffractive structure at an angle corresponding to the incident angle. The zero order light may cause an undesired effect such as a ghost image, e.g., when the zero order light is incident upon the reflective display  598 - 1  directly or subsequent to reflection off surfaces within the optically diffractive device  598 . 
     To eliminate the undesired effect, the field grating structure  598 - 1  can be spaced from the display  594 . In some implementations, a back surface of the field grating structure  598 - 1  is spaced from a front surface of the display  594  by a gap. The gap can have any suitable distance, e.g., 1 mm. The gap can be filled with air or any lower-refractive-index material to satisfy total internal reflection (TIR) on an interface. For example, air has a refractive index (e.g., n≈1.0) which is much smaller than that of a back layer of the field grating structure  598 - 1  (e.g., n≈1.5), and hence any residual light at the incident angle (e.g., &gt; 70°) can be totally internally reflected by the back surface of the field grating structure  598 - 1  when the incident angle is larger than a critical angle (e.g., ≈41.8° for n≈1.5). That is, the residual light at the incident angle cannot reach the reflective display  594  to cause the undesired effect. In some examples, at least one of the front surface of the reflective display  594  or the back surface of the field grating structure  598 - 1  is treated with an anti-reflection coating, which can substantially reduce a part of the holographic light field reflected from the reflective display  594  back towards the reflective display  594  from the back of the field grating structure  598 - 1  which otherwise could cause further ghost images. In some examples, the back surface of the field grating structure  598 - 1  can be protected by an additional layer, e.g., a glass layer. 
     In some implementations, instead of being spaced with a gap, the back surface of the field grating structure  598 - 1  can be attached to the front surface of the reflective display  594  using an intermediate layer. The intermediate layer can be an optically clear adhesive (OCA) layer with a refractive index substantially lower than that of the back layer of the field grating structure  598 - 1 , such that total internal reflection (TIR) can occur and the residual zero order light can be totally reflected at the interface between the intermediate layer and the back layer of the field grating structure  598 - 1  back into the optically diffractive structure  598 . 
     In some implementations, the field grating structure  598 - 1  and the display  594  can be separated with a gap so that any residual light cannot reach the display  594 . The gap can be filled with any suitable transparent material, index-matching fluid, or OCA. In some implementations, the field grating structure  598 - 1  can be formed in a cover layer (e.g., a cover glass) of the display  594 . 
     In some cases, to illuminate a whole surface of the reflective display  594  by light diffracted from an active area of the field grating structure  598 - 1 , the active area of the field grating structure  598 - 1  can be no smaller than an area of the whole surface of the reflective display  594 . In some implementations, the field grating structure  598 - 1  and the reflective display  594  have a rectangular shape with a height along the X direction and a width along the Y direction. The active area of the field grating structure  598 - 1  can have a height no smaller than a height of the reflective display  594  and a width no smaller than a width of the reflective display  594 . If there is a substantial gap between the field grating structure  598 - 1  and the reflective display  594 , the field grating structure  598 - 1  and the substrate  598 - 2  can be enlarged further so that an expanding cone (or frustrum) of light from the reflective display  594 , e.g., the holographic light field  599 , can be seen through the front of the optically diffractive device  598  over an entire vertical and horizontal field of view (around the +Z axis) of the holographic light field  599 . The substrate  598 - 2  can be a little wider and higher than the field grating structure  598 - 1 . 
     As light is incident on the field grating structure  598 - 1  at a substantially off-axis angle in a dimension, e.g. the Z direction, the light can be narrower by the cosine of the incidence angle in that dimension. The light from the illuminator  596  can have a narrow rectangular shape incident into the field grating structure  598 - 1  which can then expand the light to a large rectangular shape incident into the reflective display  594 . One or more optical components, e.g., mirrors, prisms, optical slabs, and/or optical fillers, can be arranged between and within the illuminator  596 , the optically diffractive structure  598 , and the reflective display  594  to further expand the light and to filter its bandwidth. In some examples, the expanded light can have a beam area somewhat smaller than the active area of the reflective display  594 , such that the edges and surrounding area of the illuminated area of the reflective display  594  are not noticeable in reflection or scatter towards the viewer. In some examples, the expanded light can have a beam area somewhat larger than the active area of the reflective display  594 , such that the edges of the illuminated area of the reflective display  594  are fully illuminated even if the edges of the expanded light are not uniform, e.g. because of diffraction off masking edges. 
     In some implementations, the controller  592  can obtain graphic data including respective primitive data for a plurality of primitives corresponding to an object in a three-dimensional space, determine, for each of the plurality of primitives, an electromagnetic (EM) field contribution to each of a plurality of display elements of the reflective display  594 , generate, for each of the plurality of display elements, a sum of the EM field contributions from the plurality of primitives to the display element, and generate, for each of the plurality of display elements, the respective control signal based on the sum of the EM field contributions to the display element. 
     In some implementations, the illuminator  596  can include one or more color light emitting elements, e.g., red, blue, or green color lasers (or LEDs), configured to emit light of corresponding colors. The optically diffractive device  598  can be configured to diffract a plurality of different colors of light at respective diffracted angles that are substantially identical to each other. Each of the respective diffracted angles can be in a range of 0° to ±10°, e.g., substantially identical to 0°, + or - 1°, + or - 2°, + or - 3°, + or - 4°, + or - 5°, + or - 6°, + or - 7°, + or - 8°, + or - 9°, or + or - 10°. 
     In some implementations, the controller  592  is configured to sequentially modulate the display  594  with information associated with a plurality of colors of light in a series of time periods. For example, the information can include a series of color holograms or color images. The controller  592  can control the illuminator  596  to sequentially emit each of the plurality of colors of light to the optically diffractive device  598  during a respective time period of the series of time periods, such that each of the plurality of colors of light is diffracted by the optically diffractive device  598  to the reflective display  594  and diffracted by modulated display elements of the reflective display  594  to form a respective color three-dimensional holographic light field  599  corresponding to the object during the respective time period. Depending on temporal coherence-of vision effect in an eye of a viewer, the plurality of colors can be combined in the eye to give an appearance of full color. In some cases, the illuminator  596  is switched off among different light emitting elements during a state change of the display image (or holographic reconstruction) such as during black-insertion subframes between color subframes or during blanking or retrace periods of a video source or during LC rise, fall, or DC-balancing inversion transitions, or during system warm-up, or when the intended holographic light field is completely black, or during a calibration procedure, and is switched on when a valid image (or holographic reconstruction) is presented for a period of time. This can also rely on persistence of vision to make the image (or holographic reconstruction) appear stable and flicker-free. 
     If a part of the holographic light field  599  appears in front of the display  594 , as illustrated by a light field  599 - 1  in  FIG.  5 H , that part of the holographic light field  599  is a real part of the reconstructed image or holographic reconstruction (also called a real image or a real holographic reconstruction). When a viewer sees a point of light in front of the display  594 , there really is light being reflected from the display  594  to that point. If a part of the light field  599  appears to the viewer to be behind (or inside) the display  594 , as illustrated by a light field  599 - 2  in  FIG.  5 H , that part of the holographic light field  599  is a virtual part of the reconstructed image or holographic reconstruction (also called a virtual image or a virtual holographic reconstruction). When the viewer sees a point of light which appears to be behind or inside the display  594 , there is actually no light being diffracted from the display  594  to that virtual point: rather, part of the light diffracted from the display  594  appears to be originated at that virtual point. 
     The computer  591  and/or the controller  592  can be configured to adjust a computation (e.g., by equations) of the information (e.g., a two-dimensional hologram, image, or pattern) to be modulated in the display  594  to move the reconstructed holographic light field  599  back and forth along a direction (e.g., the Z direction) normal to the display  594 . The computation can be based on a holographic rendering process, e.g., as illustrated in  FIGS.  2  and  3 A- 3 G . In some cases, the holographic light field  599  can be fully in front of the display  594 . In some cases, the holographic light field  599  can appear to be all behind the display  594 . In some cases, as illustrated in  FIG.  5 H , the holographic light field can have one part in front of the display  594 , e.g., the real part  599 - 1 , and another part appearing to be behind the display, e.g., the virtual part  599 - 2 . That is, the light field  599  can appear to straddle a surface of the display  594 , which can be called image planning. 
     The optically diffractive device  598  can be implemented in different configurations. In some implementations, the optically diffractive device  598  includes a holographic grating, e.g., a Bragg grating, for a particular color, e.g., as illustrated in  FIGS.  7 A,  7 B, and  8   , and the holographic light field  599  can correspond to the particular color. In some implementations, the optically diffractive device  598  includes multiple holographic gratings for different colors in a single recording layer, e.g., as illustrated in  FIGS.  7 C,  7 D and  7 E . 
     In some implementations, the optically diffractive device  598  includes multiple holographic gratings for different colors in different recording layers, e.g., as illustrated in  FIGS.   9 A to  12 C . As illustrated in  FIG.  7 F , a grating for a particular color can diffract not only light of the particular color, but also light of other colors, which can cause crosstalk among the different colors. In some examples, as described with further details below with respect to  FIGS.  9 A to  10 B , the optically diffractive device  598  can include multiple holographic gratings with one or more color-selective polarizers to suppress (e.g., eliminate or minimize) color crosstalk. In some examples, as described with further details below with respect to  FIGS.  11  to  12 C , the optically diffractive device 598 can include multiple holographic gratings with one or more reflective layers for light of different colors incident at respective incident angles to suppress color crosstalk and zero order light. In some examples, the optically diffractive device  598  can include multiple holographic gratings with one or more color-selective polarizers, e.g., as illustrated in  FIGS.  9 A to  10 B , and one or more reflective layers, e.g., as illustrated in  FIGS.  11  to  12 C , to suppress color crosstalk and zero order diffraction. Each of the color-selective polarizers can be configured for a single color or multiple colors. Each of the reflective layers can be configured for a single color or multiple colors. 
       FIG.  5 I  illustrates another system  590 A with a reflective display  594 A with optically diffractive illumination using an optically diffractive device  598 A. The reflective display  594 A can be the same as the reflective display  594  of  FIG.  5 H . Different from the optically diffractive device  598  of the system  590  of  FIG.  5 H , the optically diffractive device  598 A of the system  590 A has a reflective field grating based structure that can include a reflective field grating structure  598 - 1 A and a substrate  598 - 2 A. The substrate  598 - 2 A can be a glass substrate. The reflective field grating structure  598 - 1 A can include one or more reflective holographic gratings for one or more different colors. The reflective field grating structure  598 - 1 A is arranged on a front surface of the substrate  598 - 2 A, e.g., along Z direction. An illuminator  596  is arranged behind the reflective field grating structure  598 - 1 A and configured to illuminate light on the reflective field grating structure  598 - 1 A at a large incident angle. The light is diffracted back (along - Z direction) to the reflective display  594 A that further diffracts the light back through the optically diffractive device  598 A to form a holographic light field  599 . 
       FIG.  5 J  illustrates another system  590 B with a transmissive display  594 B with optically diffractive illumination using an optically diffractive device  598 B. The transmissive display  594 B can be the same as the transmissive display  534  of  FIG.  5 C ,  544  of  FIG.  5 D , or  564  of  FIG.  5 E . Similar to the optically diffractive structure  598 A of  FIG.  5 I , the optically diffractive structure  598 B can be a reflective field grating based structure that can include a reflective field grating structure  598 - 1 B and a substrate  598 - 2 B. The substrate  598 - 2 B can be a glass substrate. The reflective field grating structure  598 - 1 B can include one or more reflective holographic gratings for one or more different colors. Different from the optically diffractive structure  598 A of  FIG.  5 I , the reflective field grating structure  598 - 1 B in the optically diffractive structure  598 B is arranged on a back surface of the substrate  598 - 2 B. An illuminator  596  is arranged before the reflective field grating structure  598 - 1 B and configured to illuminate light on the reflective field grating structure  598 - 1 B at a large incident angle. The light is diffracted back (along -Z direction) to the transmissive display  594 B that further diffracts the light to form a holographic light field  599 . 
       FIG.  5 K  illustrates another system  590 C with a transmissive display  594 C with optically diffractive illumination using an optically diffractive device  598 C. The transmissive display  594 C can be the same as the transmissive display  594 C of  FIG.  5 J . Similar to the optically diffractive structure  598  of  FIG.  5 H , the optically diffractive structure  598 C can be a transmissive field grating based structure that can include a transmissive field grating structure  598 - 1 C and a substrate  598 - 2 C. The substrate  598 - 2 C can be a glass substrate. The transmissive field grating structure  598 - 1 C can include one or more transmissive holographic gratings for one or more different colors. Different from the optically diffractive structure  598  of  FIG.  5 H , the transmissive field grating structure  598 - 1 C in the optically diffractive structure  598 C is arranged on a front surface of the substrate  598 - 2 C. An illuminator  596  is arranged behind the transmissive field grating structure  598 - 1 C and configured to illuminate light on the transmissive field grating structure  598 - 1 C at a large incident angle. The light is diffracted forward (along +Z direction) to the transmissive display  594 C that further diffracts the light to form a holographic light field  599 . 
     As discussed above,  FIGS.  5 H to  5 K  show different combinations of reflective/transmissive displays and reflective/transmissive field grating based optically diffractive devices. In some cases, placing an optically diffractive device on a rear side of a display can provide better protection for photopolymers if the photopolymers have not already been protected by their inherent structures or by additional glass layers. In some cases, a transmissive grating can be mechanically and optically closer to a display, and light from the transmissive grating to the display can travel a shorter distance, than from a reflective grating, which can reduce alignment, coverage, dispersion, and/or scatter issues. In some cases, transmissive gratings can have a greater wavelength tolerance and a lesser angular tolerance than reflective gratings. In some cases, transmissive grating can be less likely to mirror ambient illumination towards a viewer, e.g., ceiling lights and illuminated keyboards. In some cases, with a transmissive display, a viewer can get closer to the display, and the holographic light field may be projected closer to the display. In some cases, for a transmissive display, a glass substrate for the transmissive display can have a proven manufacturing capability up to &gt; 100″ diagonal with near-seamless tiling for cinema and architectural sizes. In some cases, reflective and transflective displays can embed a controller, e.g., Maxwell holography circuitry, behind display elements, and transmissive displays can incorporate the controller or circuitry behind inter-pixel (or inter-phasel) gaps. In some cases, reflective and transflective displays can enable light to double-pass display elements (e.g., liquid crystal material) and can have twice the refractive index change of transmissive displays that uses a single-pass through the liquid crystal material. A transflective display can represent a display with an optical layer that reflects transmitted light. 
     Exemplary Display Implementations 
     As noted above, a display in Maxwell holography can be a phase modulating device. A phase element of the display (or a display element) can be represented as a phasel. For illustration only, a liquid crystal on silicon (LCOS) device is discussed below to function as the phase modulating device. The LCOS device is a display using a liquid crystal (LC) layer on top of a silicon backplane. The LCOS device can be optimized to achieve minimum possible phasel pitch, minimum cross-talk between phasels, and/or a large available phase modulation or retardance (e.g., at least 2π). 
     A list of parameters can be controlled to optimize the performance of the LCOS device, including a birefringence of LC mixture (Δn), a cell gap (d), a dielectric anisotropy of the LC mixture (Δε), a rotational viscosity of the LC mixture (η), and the maximum applied voltage between the silicon backplane and a common electrode on top of the LC layer (V). 
     There can be a fundamental trade-off that exists between parameters of the liquid crystal material and structure. For example, a fundamental bounding parameter is the available phase modulation or retardance (Re), which can be expressed as: 
     
       
         
           
             R 
             e 
             = 
               
             4 
             π 
             ⋅ 
             Δ 
             n 
             ⋅ 
             
               d 
               / 
               λ 
             
           
         
       
     
      where λ is the wavelength of an input light. If the retardance Re needs to be at least 2π for a red light with a wavelength of about 0.633 µm, then 
     
       
         
           
             Δ 
             n 
             ⋅ 
             d 
               
               
             ⩾ 
               
             0.317 
               
             μ 
             m 
           
         
       
     
      The above expression implies that there is a direct trade-off between cell gap (d) and birefringence (Δn) of the LC mixture for any given wavelength (λ). 
     Another bounding parameter is the switching speed, or the switching time (T) it takes for the liquid crystal (LC) molecules in an LC layer to reach the desired orientation after a voltage is applied. For example, for real-time video (~ 60 Hz) using a 3-color field sequential color system, a minimum of 180 Hz modulation of the LC layer is involved, which puts an upper bound on the LC switching speed of 5.6 milliseconds (ms). Switching time (T) is related to a number of parameters including the liquid crystal mixture, the cell gap, the operating temperature, and the applied voltage. First, T is proportional to d 2 . As the cell gap d is decreased, the switching time decreases as the square. Second, the switching time is also related to the dielectric anisotropy (Δε) of the liquid crystal (LC) mixture, with a higher dielectric anisotropy resulting in a shorter switching time and a lower viscosity (which may be temperature dependent) also resulting in a shorter switching time. 
     A third bounding parameter can be the fringing field. Due to the high electron mobility of crystalline silicon, an LCOS device can be fabricated with a very small phasel size (e.g., less than 10 µm) and with submicron inter-phasel gaps. When the adjacent phasels are operated at different voltages, the LC directors near the phasel edges are distorted by the lateral component of the fringing field, which significantly degrades the electro-optic performance of the device. In addition, as the phasel gap becomes comparable to the incident light wavelength, diffraction effects can cause severe light loss. The phasel gap may need to be kept at less than or equal to a phasel pitch to keep phase noise within an acceptable level. 
     In some examples, the LCOS device is designed to have a phasel pitch of 2 µm and a cell gap of approximately 2 µm if the fringe field bounding condition is observed. According to the above expression Δn •d ≥ 0.317 µm, hence Δn needs to be equal to 0.1585 or greater, which is achievable using current liquid crystal technology. Once the minimum birefringence for a given phasel pitch is determined, the LC can be optimized for switching speed, e.g., by increasing the dielectric anisotropy and/or decreasing the rotational viscosity. 
     Nonuniform Phasels Implementations for Displays 
     In an LCOS device, a circuit chip, e.g., a complementary metal-oxide-semiconductor (CMOS) chip or equivalent, controls the voltage on reflective metal electrodes buried below the chip surface, each controlling one phasel. A common electrode for all the phasels is supplied by a transparent conductive layer made of indium tin oxide on the LCOS cover glass. The phasels can have identical sizes and same shape (e.g., square). For example, a chip can have 1024x768 (or 4096x2160) phasels, each with an independently addressable voltage. As noted above, when the inter-phasel gap becomes comparable to the incident light wavelength, diffraction effects can appear due to the periodic structure of the LCOS device, which may cause severe light loss and a strong periodic structure in the diffracted light. 
     In Maxwell holographic calculations, each phasel receives a sum of EM contributions from each primitive and is relatively independent from each other. Thus, the phasels of the LCOS device in Maxwell holography can be designed to be different from each other. For example, as illustrated in  FIG.  6 A , the LCOS device  600  can be made of a number of nonuniform (or irregular) phasels  602 . At least two phasels  602  have different shapes. The nonuniform shapes of the phasels  602  can greatly reduce or eliminate diffractive aberrations (e.g., due to the periodic structure in the diffracted light), among other effects, and thus improve image quality. Although the phasels can have nonuniform shapes, the phasels can be designed to have a size distribution with an average (e.g., about 3 µm) that satisfies a desired spatial resolution. The silicon backplane can be configured to provide a respective circuit (e.g., including a metal electrode) for each of the phasels according to the shape of the phasel. 
     In an array of phasels in an LCOS device, to select a specific phasel, a first voltage is applied to a word line connecting a row of phasels including the specific phasel and a second voltage is applied to a bit line connecting a column of phasels including the specific phasel. As each phasel has a resistance and/or a capacitance, the operational speed of the LCOS device can be limited by the switching (or rise and fall times) of these voltages. 
     As noted above, in Maxwell holography, the phasels can have different sizes. As illustrated in  FIG.  6 B , an LCOS device  650  is designed to have one or more phasels 654 having a size larger than the other phasels  652 . All of the phasels can still have a size distribution that satisfies the desired resolution. For example, 99% of the phasels have a size of 3 µm, and only 1% of the phasels have a size of 6 µm. The larger size of the phasel  654  allows to arrange at least one buffer  660  in the phasel  654  besides other circuitry same as in the phasel  652 . The buffer  660  is configured to buffer the applied voltage such that the voltage is only applied to a smaller number of phasels within a row or column of phasels. The buffer  660  can be an analog circuit, e.g., made of a transistor, or a digital circuit, e.g., made of a number of logic gates, or any combination thereof. 
     For example, as illustrated in  FIG.  6 B , a voltage is applied to a word line  651  and another voltage is applied to a bit line  653  to select a particular phasel  652   ∗ . The phasel  652   ∗  is in the same row as the larger phasel  654  including the buffer  660 . The voltage is mainly applied to the first number of phasels in the row and before the larger phasel  654  and obstructed by the buffer  660  in the larger phasel  654 . In such a way, the operational speed of the LCOS device  650  can be improved. With the larger size of the phasels  654 , other circuitry can be also arranged in the LCOS device  650  to further improve the performance of the LCOS device  650 . Although the phasels  654  and the phasels  652  in  FIG.  6 B  have square shape, the phasels can also have different shapes as illustrated in  FIG.  6 A  as long as there are one or more phasels  654  having a larger size than the other phasels  652 . 
     Exemplary Calibrations 
     The unique nature of Maxwell holography in the present disclosure allows for the protection of calibration techniques that can create a significant competitive advantage in the actual production of high quality displays. A number of calibration techniques can be implemented to be combined with the Maxwell holographic computational techniques, including:
     (i) using image sensors or light field sensors in conjunction with a Dirichlet boundary condition modulator and/or in conjunction with mechanical and software diffractive and nondiffractive calibration techniques;   (ii) software alignments and software calibrations including individual color calibrations and alignments with Dirichlet boundary condition modulators; and   (iii) embedding silicon features in the boundary condition modulators that allow for photo detection (including power and color) and/or thermometry to be built directly into the modulator that when combined with Maxwell holography creates a powerful and unique approach to simplifying manufacturing calibration processes.   

     In the following, for illustration only, three types of calibrations are implemented for phase based displays, e.g., LCOS displays. Each phase element can be represented as a phasel. 
     Phase Calibration 
     An amount of phase added to light impinging upon an LCOS phase element (or phasel) can be known directly by a voltage applied to the LCOS phasel. This is due to the birefringent liquid crystal (LC) rotating in the presence of an electric field and thus changing its index of refraction and slowing down light to alter its phase. The altered phase can depend upon electrical characteristics of the liquid crystal (LC) and the silicon device in which the LC resides. Digital signals sent to the LCOS need to be transformed into correct analog voltages to achieve high quality holographic images. Phase calibration is involved for the LCOS device to ensure that a digital signal is properly transformed into an analog signal applied to the LC such that it produces the greatest amount of phase range. This conversion is expected to result in a linear behavior. That is, as the voltage is changed by fixed increments, the phase also changes by fixed increments, regardless of the starting voltage value. 
     In some cases, an LCOS device allows a user to alter a digital-to-analog converter (DAC) such that the user has a control over the amount of analog voltage output given a digital input signal. A digital potentiometer can be applied to each input bit. For example, if there are 8 input bits, there can be 8 digital potentiometers corresponding to each input bit. The same digital inputs from the digital potentiometers can be applied to all phasels of the LCOS device. Bits set to “1” activate a voltage, and bits set to “0” do not activate the voltage. All voltages from such “1” bits are summed together to obtain the final voltage sent to each phasel. There may also be a DC voltage applied in all cases such that all “0” bits results in a baseline non-zero voltage. Thus, the phase calibration of the LCOS device can be implemented by setting values of the digital potentiometers for the LCOS device. For example, as noted above, a controller can compute EM field contributions from a list of primitives to each of phasels of a display, generate a respective sum of the EM field contributions from the primitives to each of the phasels, and generate respective control signals to each of the phasels for modulating a phase of the phasel. The same digital inputs from the digital potentiometers can be applied to adjust the respective control signals to all of the phasels of the LCOS device, which is different from a phasel-by-phasel based phase calibration. The digital inputs can be set once for a duration of an operation of the LCOS device, e.g., for displaying a hologram. 
     To determine an optimal set of phase calibration values for the digital inputs, a genetic algorithm can be applied, where there are many input values that lead to one output value, such as phase range or holographic image contrast. This output value can be reduced to one number known as the fitness. The genetic algorithm can be configured to explore different combinations of input values until it achieves an output with the highest fitness. In some cases, the algorithm can take two or more of the most fit inputs and combine a number of their constituent values together to create a new input that has characteristics of the taken inputs but is different from each of the taken inputs. In some cases, the algorithm can alter one of these constituent values to something not from either of the taken fit inputs, which is represented as a “mutation” and can add a variety to the available fit inputs. In some cases, one or more optimal values can be found by taking advantage of the knowledge gained from prior measurements with good results while trying new values so the optimal values do not be restricted to a local maximum. 
     There can be multiple ways to calculate the fitness output value. One way is to calculate the phase change of the light given a set of digital inputs applied to all the phasels on the LCOS. In this scheme, the incident light can be polarized. Upon impinging upon the LCOS, the incident light’s polarization can change depending on the rotation of the LC. The incident light can be diffracted back through another polarizer set to either the same polarization or 90 degrees different from the original polarization and then into a light detector. Therefore, when the LC rotation changes, the intensity as viewed from the light detector can change. Accordingly, the phase change of the light can be perceived indirectly through the intensity variations. Another way to calculate the phase change is to measure the intensity difference of a Maxwell holographic reconstruction from the background. This is most effective in a projective display. Measuring the intensity in such an instance may need the use of computer vision algorithms to identify the Maxwell holographic reconstruction and measure its intensity. Another way to determine the phase change is to measure or image it microscopically in an interferometric optical geometry. 
     Alignment Calibration 
     Light sources and other optical elements may not be adequately aligned within a holographic device and therefore may need to be aligned. Different liquid crystals (LC) and optically diffractive elements or diffractive optical elements can also behave differently for different wavelengths of the light sources. Moreover, especially the LC, diffractives, and light sources can change device to device and over time (aging and burn-in) and as a result of changes in the operating environment such as the operating temperature and mechanically induced deformation due to thermal or mechanical stress, giving different characteristics, e.g., object scaling, to the same input hologram when shown in a different base color or at a different time or in a different environment. Furthermore, certain hardware features can apply different optical effects to the output light, e.g., lensing, that also may need correction under these circumstances. 
     In some implementations, the problems described above can be addressed by applying mechanical translations, deformations, and rotations to one or more optical element. In some implementations, the problems described above can be addressed by applying a mathematical transform to a phase calculated for a phasel of a display. The phase is a respective sum of the EM field contributions from a list of primitives to the phasel. The mathematical transform can be derived from a mathematical expression, e.g., a Zernike polynomial, and can be varied by altering polynomial coefficients or other varying input values. The mathematical transform can vary phasel-by-phasel as well as by color. For example, there is a Zernike polynomial coefficient that corresponds to the amount of tilt to be applied to the light after it diffracts off of the display. 
     To determine these coefficients/input values, a hardware and software setup can be created where a 2D camera, a photometer, a light field camera, and/or other photometric or colorimetric instrumentation is pointed at a reflective or diffusely transmissive surface illuminated by the LCOS in the case of a projective display or pointed into the LCOS in the case of a direct-view display. One or more holographic test patterns and objects can be sent to the display and measured by the measuring instrument or instruments. 2D cameras or 3D (light field) cameras or camera arrays can use machine vision algorithms to determine what is being displayed and then calculate its fitness. For example, if a grid of dots is the test pattern, then the fitness can be determined by a statistical measure of how close they are together, how centered they are on their intended positions, how much distortion they exhibit (e.g., scale or pincushion), etc. There can be different fitness values for different performance characteristics. Depending on these values, corrections can be applied, e.g., in the form of changing coefficients to the Zernike polynomial, until the fitness reaches a predetermined satisfactory level or passes a visual or task-oriented A/B test. These test patterns can be rendered at different distances to ensure that alignment is consistent for objects at different distances, and not just at one 3D point or plane in particular. Such depth-based calibrations can involve iterative processes that involve altering the depth of the holographic test pattern or elements therein, as well as the position of the reflective or diffusely transmissive surface, and where the previous calibrations can be repeated until converging upon a solution that works at multiple depths. Finally, white dots can be displayed to show the effectiveness of the calibration. 
     Color Calibration 
     In displays, holographic or otherwise, it is important that, when any two units are rendering the same image, colors match between displays and additionally match colors defined by television (TV) and computer display standards, like the Rec.709 standard for high-definition television (HDTV) or the sRGB color space of computer monitors. Different batches of hardware components, e.g., LEDs and laser diodes, can exhibit different behaviors for the same inputs and can output different colors when perceived by the human eye. Therefore, it is important to have a color standard to which all display units can be calibrated. 
     In some implementations, an objective measurement of color specified by measurements of intensity and chromaticity can be obtained by measuring color intensity against Commission internationale de 1′éclairage (CIE) Standard Observer curves. By requesting that each display reproduces a sample set of known colors and intensities, then measuring the output light using a colorimeter device calibrated to the CIE Standard Observer curves, the color output of a device in a chosen CIE color space can be objectively defined. Any deviation of the measured values from the known good values can be used to adapt the output colors on the display to bring it back into alignment or conformance, which can be implemented using an iterative measure-adapt-measure feedback loop. Once a Maxwell holographic device produces accurate outputs for a given set of inputs, the final adaptations can be encoded as look-up tables for the illuminators that map input values to output intensities, and color matrix transformations that transform input colors to output color space values. These calibration tables can be embedded in the device itself to produce reliable objective output colors. Multiple such tables can be provided for each of a multitude of operating temperature ranges. Multiple such tables can be provided for each of a multitude of different regions of the active surface of the LCOS. Calibration values can be interpolated between tables for adjacent temperature ranges and/or adjacent surface regions. 
     Additionally, given an LCOS device with fine enough features to control diffraction with sub-wavelength accuracy, there may be no need for tri-stimulus illumination (e.g., linear mixes of red, green, and blue), and the LCOS device can be illuminated with a single wide spectrum light source and selectively tune the phasels output to produce tri-, quad-, even N-stimulus output colors which, combined with spatial dithering patterns, can reproduce a more complete spectral output of a color rather than the common tri-stimulus approximation. Given a sufficiently wide spectrum illuminator this allows Maxwell holography to produce any reflected color that lies inside the spectral focus of the human visual system or outside the spectral focus for infrared (IR) or ultraviolet (UV) structured light. 
     Exemplary Holographic Gratings 
       FIGS.  7 A- 7 F  illustrate implementations of example holographic gratings that can be included in an optically diffractive device (or a lightguide device), e.g., the optically diffractive device  598  of  FIG.  5 H ,  598 A of  FIG.  5 I ,  598 B of  FIG.  5 J , or  598 C of  FIG.  5 K .  FIGS.  7 A and  7 B  illustrate recording and replaying a holographic grating in a recording medium with a single color.  FIGS.  7 C and  7 D  illustrate recording three different color holographic gratings in a recording medium with three different colors of light ( FIG.  7 C ) and replaying them with a single color of light ( FIG.  7 D ).  FIGS.  7 E and  7 F  illustrate replaying three different color holographic gratings in a recording medium with three different colors of light, and  FIG.  7 F  illustrates color crosstalk among diffracted light of different colors. Any one of a recording reference light beam, a recording object light beam, a replaying reference light beam, and a diffracted light beam is a polarized light beam that can be s polarized or p polarized. 
       FIG.  7 A  illustrates an example of recording a holographic grating in a recording medium. The recording medium can be a photosensitive material, e.g., a photosensitive polymer or photopolymer, silver halide, or any other suitable material. The recording medium can be arranged on a substrate, e.g., a glass substrate. The substrate can be transparent or not transparent during the recording. In some implementations, the photosensitive material can be adhered to a carrier film, e.g., a TAC (cellulose triacetate) film. The photosensitive material with the carrier film can be laminated on the substrate, with the photosensitive material between the carrier film and the substrate. 
     In transmission holography, a recording reference beam and a recording object beam are incident from the same side on a same region of the recording medium with a recording reference angle θ r , and a recording object angle θ o , respectively. Each of the reference and object beams can start in air, pass through the photosensitive material, and then pass on into and through the substrate, exiting into air. The recording reference beam and the recording object beam have the same color, e.g., green color, and same polarization state, e.g., s polarized. Both of the beams can originate from a laser source with high spatial and temporal coherence so that the beams interfere strongly to form a standing pattern where the beams overlap. Within the recording medium, the pattern is recorded as a fringe pattern, e.g., a grating, including multiple parallel interference planes, as illustrated as tilted solid lines in  FIG.  7 A , at a fringe tilt angle θ t  that satisfies the following expression: 
     
       
         
           
             
               θ 
               t 
             
               
             = 
               
             
               
                 
                   
                     
                       θ 
                       o 
                     
                       
                     + 
                       
                     
                       θ 
                       r 
                     
                   
                 
               
               / 
               2 
             
           
         
       
     
      where θ t  represents the fringe tilt angle in the recording medium during recording, θ o  represents the object angle in the recording medium during recording, and θ r , represents a reference angle in the recording medium during recording. 
     A fringe spacing (or fringe period) d on a surface of the recording medium can be expressed as: 
     
       
         
           
             d 
               
             = 
               
             
               
                 
                   λ 
                   
                     record 
                   
                 
               
               / 
               
                 
                   
                     n 
                       
                     sin 
                     
                       θ 
                       
                         record 
                       
                     
                   
                 
               
             
           
         
       
     
      where λ record  represents a recording wavelength (in vacuo), n represents the refractive index of the medium surrounding the grating (e.g., air with n = 1.0), θ record  represents the inter-beam angle during recording and is identical to | θ o  - θ r  |, where θ o  represents the object incidence angle at a surface of the recording medium during recording and θ r  represents the reference incidence angle at the surface of the recording medium during recording. In some cases, the fringe spacing d has a size similar to a wavelength of a recording light, e.g., 0.5 µm. Thus, the fringe pattern can have a frequency f = 1 / d, e.g., about 2,000 fringes per mm. The thickness D of the recording medium can be more than one order of magnitude larger than the wavelength of the recording light. In some examples, the thickness of the recording medium D is about 30 times of the wavelength, e.g., about 16.0 +/- 2.0 µm. The carrier film can have a thickness larger than the recording medium, e.g., 60 µm. The substrate can have a thickness more than orders of magnitude larger than the recording medium, e.g., about 1.0 mm. 
     After the fringe pattern or grating is recorded in the recording medium, the fringe pattern can be fixed in the recording medium, e.g., for the example of a photopolymer by exposure of deep blue or ultraviolet (UV) light which can freeze the fringes in place and can also enhance the fringes’ refractive index differences. The recording medium can shrink during the fixing. The recording medium can be selected to have a low shrinkage during the fixing, e.g., less than 2% or such shrinkage can be compensated for. 
     As each beam passes through an interface between materials of different refractive indices, some portion of the beam is reflected following Fresnel’s laws, which give the percentage of power reflected at each transition. The reflection is polarization dependent. For light at a smaller incidence angle, e.g., 30°, the Fresnel reflections can be weaker. For light at a larger incident angle (e.g., 80°) and for s-polarized light, the Fresnel reflections can be stronger. When the incident angle reaches or is beyond a critical angle, total internal reflection (TIR) occurs, that is, the reflectivity is 100%. For example, from a transition from glass (n = 1.5) to air (n = 1.0), the critical angle is about 41.8°. Since the refractive index is dependent on polarization and weakly dependent on wavelength, reflected powers at large angles of incidence can become weakly wavelength dependent, and can become strongly polarization dependent. 
       FIG.  7 B  illustrates an example of diffracting a replay reference beam by the grating of  FIG.  7 A . For transmission holography, during replay the substrate is transparent. The substrate can be also an optically clear plastic, such as TAC or some other low-birefringence plastic. When the recorded grating in the recording medium is thin compared with the wavelength of the replay reference beam, e.g., the thickness of the recording medium is less than one order of magnitude larger than the replay wavelength, the grating’s diffracted angle can be described by a grating equation as below: 
     
       
         
           
             m 
               
             
               λ 
               
                 replay 
               
             
             = 
             n 
               
             d 
               
             
               
                 sin 
                 
                   θ 
                   
                     in 
                   
                 
                 − 
                 sin 
                 
                   θ 
                   
                     out 
                   
                 
               
             
           
         
       
     
      where m represents a diffraction order (integer), n represents the refractive index of the medium surrounding the grating, d represents the fringe spacing on the surface of the recording medium, θ in  represents the incident angle from the surrounding medium onto the grating, θ out  represents the output angle for the m th  order from the grating back into the surrounding medium, and λ replay  represents the replay wavelength in vacuo. 
     When the recorded grating is comparatively thick, for example, when the thickness of the recording medium is more than one order of magnitude (e.g., 30 times) larger than the replay wavelength, the grating can be called a volume grating or a Bragg grating. For volume gratings, Bragg selectivity can strongly enhance diffraction efficiency at a Bragg angle. The Bragg angle can be determined based on numerical solutions, e.g., rigorous couple-wave solutions, and/or experimentation and iteration. At off-Bragg angles, the diffraction efficiency can be substantially decreased. 
     The Bragg condition can be satisfied when an angle of incident onto the fringe planes equals the diffraction angle off of the fringe planes within the medium containing the fringe planes. The grating equation (12) can then become Bragg’s equation: 
     
       
         
           
             m 
               
             
               λ 
               
                 replay 
               
             
             = 
             2 
               
             
               n 
               
                 replay 
               
             
               
             
               Λ 
               
                 replay 
               
             
             sin 
             
               
                   
                 
                   θ 
                   m 
                 
                 − 
                 
                   θ 
                   t 
                 
               
             
           
         
       
     
      where m represents the diffraction order (or Bragg order), n replay  represents the refractive index in the medium, Λ replay  represents the fringe spacing in the recording medium, θ m  represents the m th  Bragg angle in the recording medium, θ t  represents the fringe tilt in the recording medium, and Λ replay  can be identical to d cosθ t.   
     The Bragg condition can be automatically satisfied for volume gratings recorded and replayed with the same angles and wavelengths (assuming no shrinkage during processing). For example, as illustrated in  FIG.  7 B , a volume grating is recorded and replayed with the same wavelength (e.g., green color) and reference angle (e.g., θ r ), and the grating can diffract out a first order replay beam at the angle of the recording object beam. A fraction of the incident light beam can pass through the grating as an undeflected or undiffracted zero order light beam. If the zero order light beam gets to a display such as a reflective LCOS device, the light beam can cause undesired effects, e.g., ghost images. 
     If the replay reference angle is not changed but the replay reference wavelength is changed, a diffraction efficiency η of a Bragg grating in a recording medium can be expressed as: 
     
       
         
           
             η 
               
             ∝ 
               
             2 
               
             
               D 
               
                 replay 
               
             
               
             sin 
             
               θ 
               
                 Bragg 
               
             
             
                 
               2 
             
               
             δ 
             λ 
             
               
                 cos 
                 
                   θ 
                   
                     tilt 
                     .replay 
                   
                 
               
               / 
               
                 
                   
                     
                       λ 
                       
                         Bragg 
                       
                     
                     
                         
                       2 
                     
                     cos 
                     
                       θ 
                       
                         Bragg 
                       
                     
                   
                 
               
             
           
         
       
     
      where η represents diffraction efficiency, D replay  represents a thickness of the recording medium (after shrinkage) during replay, θ Bragg  represents a replay reference angle (after shrinkage) at Bragg for an intended replay wavelength λ Bragg , δλ represents an error in a replay wavelength, that is, δλ = | λ replay - λ Bragg | , and θ tilt.replay  represents the fringe tilt in the recording medium during replay (after shrinkage). All λ, are values in vacuo. 
       FIG.  7 C  illustrates an example of recording gratings for different colors in a recording medium using different colors of light. As illustrated, three fringe patterns (or gratings) can be recorded in a single recording medium, e.g., sequentially or simultaneously. A fringe pattern corresponds to a replay color (e.g., red, green, or blue) and can be recorded with a different wavelength. The recording reference beam and the recording object beam have the same polarization state. Each beam can be s polarized. The recording reference beams for each color can be incident upon the single recording medium at the same reference beam angle θ r  (e.g., +30°). The recording object beams for each color can be incident upon the single recording medium at the same object beam angle θ o  (e.g., -20°). 
     The fringe plane tilt θ t  for each grating during recording can be the same, as θ t  is independent of wavelength, e.g., θ t , = (θ o  + θ r )/2. The fringe spacing d perpendicular to the fringe planes during recording can be different for each grating, as d depends on wavelength. In some examples, as illustrated in  FIG.  7 C , the fringe spacings are in proportion red : green : blue ≈ 123% : 100% : 89% corresponding to example wavelengths of 640 nm : 520 nm : 460 nm. 
       FIG.  7 D  illustrates an example of recording gratings for different colors in a recording medium using a same color of light. Similar to  FIG.  7 C , three fringe patterns are recorded in a single photopolymer, one fringe pattern for each replay color. Different from  FIG.  7 C , the three fringe patterns in  FIG.  7 D  can be recorded using the same wavelength, e.g., green light. To achieve this, the recording object beams for each replay color can be incident upon the single recording medium at different object beam angles, and the recording reference beams for each replay color can be incident upon the single recording photopolymer at different reference beam angles. The fringe tilt and fringe spacing in  FIG.  7 D  for a replay color can match the fringe tilt and fringe spacing for that same replay color in  FIG.  7 C . 
       FIG.  7 E  illustrates an example of diffracting replay reference beams of different colors by gratings for different colors. The gratings can be recorded as illustrated in  FIGS.  7 C or  7 D . Similar to  FIG.  7 B , for a replay color, when the recording wavelength is the same as the replay wavelength and the replay reference angle is a first Bragg angle of a grating for the replay color, the grating diffracts a first order of the replay reference beam at a diffracted angle identical to a recording object angle, and transmits a zero order of the replay reference beam at the replay reference angle. Due to Bragg selectivity, the power of the replay reference beam at the first order can be substantially larger than the power of the replay reference beam at the zero order. The three replay reference beams can have the same incident angles, e.g., 30°, and the first order diffracted beams can have the same diffracted angles, e.g., 20°. 
     Replay reference angles for each color can be neither equal to one another, nor equal to the angles for the color used during recording. For example, for green color, a grating can be recorded at 532 nm, e.g., using a high-power high-coherence green laser such as a frequency-doubled diode-pumped YaG laser, and then be replayed at 520±10 nm using a green laser diode. In some cases, the green laser having the wavelength of 532 nm can also be used to record the required fringe pattern for replay using a cheap red laser diode at 640±10 nm. For blue color, a grating can be recorded at 442 nm using a HeCd laser, and be replayed using a 460±2 nm blue laser diode. 
       FIG.  7 F  illustrates an example of crosstalk among diffracted beams of different colors. Despite Bragg selectivity, each color can also slightly diffract off the gratings recorded for each other color, which may cause crosstalk among these colors. Compare to  FIG.  7 E  providing only first order diffraction for a corresponding color,  FIG.  7 F  provides the first order diffraction of each color off each grating. 
     For example, as illustrated in  FIG.  7 F , red grating, green grating, and blue gratings for red, green, and blue colors are respectively recorded. When the red light is incident at the same reference angle 30° on the red grating, the diffracted angle of the red light at first order is 20°; but when the red light is incident at the same reference angle 30° on the green grating, the diffracted angle of the red light at first order is 32°; and when the red light is incident at the same reference angle 30° on the blue grating, the diffracted angle of the red light at first order is 42°. Thus, diffracted light can be present at unintended angles, and color crosstalk occurs. Similarly, when the green light is incident at the reference angle 30° on the green grating, the diffracted angle of the green light at first order is 20°; but when the green light is incident at the same reference angle 30° on the red grating, the diffracted angle of the green light at first order is 11°; and when the green light is incident at the same reference angle 30° on the blue grating, the diffracted angle of the green light at first order is 27°. Thus, diffracted light can be present at unintended angles, and color crosstalk occurs. Similarly, when the blue light is incident at the reference angle 30° on the blue grating, the diffracted angle of the blue light at first order is 20°; but when the blue light is incident at the same reference angle 30° on the red grating, the diffracted angle of the blue light at first order is 6°; and when the blue light is incident at the same reference angle 30° on the green grating, the diffracted angle of the blue light at first order is 14°. Thus, diffracted light can be present at unintended angles, and color crosstalk occurs. Accordingly, when a single color of light, e.g., green light, is incident on the three gratings in the recording medium, the three gratings diffract the single color of light to have a first diffracted green light at a diffracted angle of 20°, a second diffracted green light at a diffracted angle of 27°, and a third diffracted green light at a diffracted angle at 11°. The two unintended angles of each color of diffracted light can generate undesired effects. 
     In some cases, instead of recording the three different gratings for three different colors in a single recording layer, the three different gratings can instead be stored in three separated recording layers that are stacked together. Similar to  FIG.  7 F , color crosstalk can occur when three colors of light are incident at the same incident angle on any one of the gratings. Implementations of the present disclosure provide methods and devices for suppressing the color crosstalk in multiple grating stacks, as illustrated with further details in  FIGS.  9 A to  12 C . 
       FIG.  8    illustrates an example of recording a holographic grating with a large reference angle in a recording medium. Using a large replay reference beam angle can allow a thin replay system. Also, a replay output beam, that is, the diffracted angle at first order, can be normal to a display. Thus, the recording object beam can be close to normal incidence, as illustrated in  FIG.  8   . 
     For Bragg diffraction, the Fresnel reflections for p- and for s-polarized light are both low at each fringe plane, but at an angle of incidence of 45°, s polarization can be reflected orders of magnitude more strongly than p polarization. Thus, if the incidence angle of the replay reference on to the fringes in the recording medium is close to 45°, then Bragg resonance off the fringes can be highly polarization sensitive, strongly favoring s-polarization. The recording object beam can be near normal incident on the recording medium, such that the reconstructed object beam or the diffracted replay beam can be at near normal incidence on a display. As the fringe tilt in the recording medium is the average of the in-medium recording object and reference angles, to achieve, at replay, an incidence angle onto the fringes of close to 45° and hence high polarization selectivity, a recording reference angle approaching 90° in the recording medium can be used. An interbeam angle between the recording object beam and recording reference beam can be close to 90°. For example, the interbeam angle is 84° as illustrated in  FIG.  8   , and the fringe tilt of the fringe planes in the recording beam is 42°, and the incident angle of the replay reference beam onto the fringe planes is 48°, which corresponds to a polarization sensitivity of about 90: 1. 
     In some cases, to obtain a replay output (or first order) diffracted angle to be 0°, the recording object beam can be not identical to 0°, but close to 0°, which can be achieved by taking into consideration a combination of shrinkage of a recording medium during its processing and a slight wavelength difference between a recording wavelength and a replaying wavelength. For example, the recording object angle can be in a range from -10° to 10°, e.g., a range from -7° to 7°, or 5° to 7°. In some examples, the recording object angle is 0°. In some examples, the recording object angle is 6°. 
     In some implementations, to achieve large enough interbeam angles, e.g., close to 90°, during recording, a prism is applied such that each recording beam enters the prism through a prism face where its incidence angle into the prism is close to the normal of that face of the prism, and thus refraction and Fresnel losses become both negligible. The prism can be index matched to the recording medium’s cover film or substrate at an interface, such that the index mismatch is negligible at the interface, and refraction and Fresnel losses can be also negligible at the interface. 
     Exemplary Optically Diffractive Devices 
       FIGS.  9 A- 12 C  show implementations of example optically diffractive devices. Any one of the devices can correspond to, for example, the optically diffractive device  598  of  FIG.  5 H  or 598C of  FIG.  5 K . The optically diffractive devices are configured to individually diffract light with a plurality of colors to suppress (e.g., reduce or eliminate) color crosstalk among diffracted light and/or to suppress zero order undiffracted light.  FIGS.  9 A to  10 B  show example optically diffractive devices including color-selective polarizers. The color-selective polarizers can selectively change a polarization of a selected color, such that a single color of light can have s polarization to achieve high diffraction efficiency at first order while other colors of light have p polarization thus lower diffraction efficiency at the first order.  FIGS.  11  to  12 C  show example optically diffractive devices including reflective layers. The reflective layers can selectively totally reflect a single color of light of zero order while transmitting other colors of light. 
     Optically Diffractive Devices with Color-Selective Polarizers 
       FIG.  9 A  illustrates an example optically diffractive device  900  including holographic gratings for two colors and corresponding color-selective polarizers, and  FIG.  9 B  illustrates an example  950  of diffracting the two colors of light by the optically diffracted device  900  of  FIG.  9 A . For illustration, the device  900  is configured for green and blue colors of light. 
     The optically diffractive device  900  includes a first optically diffractive component  910  having a first diffractive grating (B grating)  912  for blue color of light and a second optically diffractive component  920  having a second diffractive grating (G grating)  922  for green color of light. Each of the diffractive gratings can be between a carrier film, e.g., a TAC film, and a substrate, e.g., a glass substrate. The carrier film can be after the diffractive grating and the substrate can be before the diffractive grating along the Z direction, or vice versa. As illustrated in  FIG.  9 A , the first optically diffractive component  910  includes a substrate  914  and a carrier film  916  on opposite sides of the B grating  912 , and the second optically diffractive component  920  includes a substrate  924  and a carrier film  926  on opposite sides of the G grating  922 . The optically diffractive device  900  can include a field grating substrate  902  on which the first and second optically diffractive components  910  and  920  are stacked. An anti-reflection (AR) coating  901  can be attached to or applied on a surface of the field grating substrate  902  to reduce reflection at the surface. 
     The optically diffractive device  900  can also include one or more layers of optically-clear index-matched adhesive (OCA), UV-cured or heat-cured optical glues, optical contacting, or index matching fluid to attach or stick together adjacent layers or components, e.g., the field grating substrate  902  and the BY filter  904 , the BY filter  904  and the first diffractive component  910  (or the substrate  914 ), the first diffractive component  910  (or the carrier film  916 ) and the GM filter  906 , and/or the GM filter  906  and the second diffractive components  920  (or the substrate  924 ). An order of the carrier film  914  or  924 , the substrate  916  or  926 , and the OCA layers can be determined based on their refractive indices at a wavelength of a replay light to reduce refractive index mismatch at interfaces and thus reduce Fresnel reflections at the interfaces. 
     Each of the first and second diffractive gratings can be a holographic grating (e.g., volume grating or Bragg grating) independently recorded and fixed (e.g., cured) in a recording medium, e.g., a photosensitive polymer. A thickness of the recording medium can be more than one order of magnitude larger than a recording wavelength, e.g., about 30 times. Similar to what is illustrated in  FIG.  7 A  or  FIG.  8   , a recording reference light beam incident at a recording reference angle and a recording object light beam incident at a recording object angle on the recording medium can interfere in the recording medium to form the diffractive grating. Then, similar to what is illustrated in  FIG.  7 B , a replaying reference light beam can be diffracted by the recorded diffractive grating at first order and zero order. The recording light beams and the replaying light beam can have the same s polarization state. A replaying wavelength of the replaying light beam can be substantially identical to a recording wavelength of the recording light beams. 
     In some examples, the replay incident angle can be substantially identical to the recording reference angle (or a Bragg angle), and a Bragg condition can satisfy. Light of first order (or first order light) is diffracted at a diffracted angle substantially close to the recording object angle, and light of zero order (or zero order light) is undiffracted and transmitted at the replay incident angle. Due to Bragg selectivity, the power of the first order light can be substantially higher than the power of the zero order light. The power of the zero order light (e.g., residual light or depleted light) depends on the diffraction efficiency of the diffractive grating. The higher the diffraction efficiency is, the lower the power of the zero order light is. In some examples, the recording reference angle, the recording object angle, the replay incident angle, the recording wavelength, and the replay wavelength can be configured such that the replay output angle (or diffracted angle at first order) is substantially close to 0° or normal to the grating. The diffracted angle can be in a range of -10° to 10°, e.g., in a range of -7° to 7°, 0° to 10°, or 5° to 7°. In a particular example, the diffracted angle is 6°. 
     Also, due to polarization sensitivity, the diffraction efficiency for s polarized light of a first color (e.g., blue color) incident at a replay reference angle and diffracted with first order at the diffracted angle can be substantially higher than the diffraction efficiency for p polarized light of the same color incident at the replay reference angle diffracted with first order at the diffracted angle. As illustrated in  FIG.  7 F , a second color of light (e.g., green color) incident at the same replay incident angle as the first color of light is diffracted at a diffraction angle different from the diffraction angle of the first color of light. Thus, due to both Bragg sensitivity and polarization sensitivity, the diffraction efficiency for the first color of light incident in s polarization state at the reply incident angle and diffracted with first order can be substantially higher than the diffraction efficiency for the second color of light incident in p polarization state at the same replay incident angle or at a different replay incident angle. 
     The optically diffractive device  900  can be configured to suppress crosstalk between diffracted light beams of blue and green colors. For example, when the B grating  912  is positioned in front of the G grating  922  in the device  900  along the Z direction, light is incident on the B grating  912  prior to being incident on the G grating  922 . The optically diffractive device  900  can be configured such that blue color of light is incident on the B grating  912  in s polarization state and the green color of light is incident on the B grating  912  in p polarization state and the green color of light is incident on the G grating  922  in s polarization state. In some cases, the optically diffractive device  900  can also be configured such that the residual blue color of light is incident on the G grating  922  in p polarization state. 
     In some implementations, as shown in  FIGS.  9 A and  9 B , the optically diffractive device  900  can include a color-selective polarizer  906  (also known as a color-selective retarder or filter) between the first diffractive grating  912  and the second diffractive grating  922  (or between the first diffractive component  910  and the second diffractive component  920 ). The color-selective polarizer  906  can include a GM filter configured to rotate a polarization state of green color of light by 90 degrees, e.g., from p polarization state to s polarization state, but without rotation of a polarization state of blue color of light. 
     In some implementations, as shown in  FIGS.  9 A and  9 B , the optically diffractive device  900  can include another color-selective polarizer  904  in front of the first diffractive grating  912  and the second diffractive grating  922  along the Z direction. The color-selective polarizer  904  can include a BY filter configured to rotate a polarization state of blue color of light by 90 degrees from p polarization state to s polarization state, but without rotation of a polarization state of green color of light. 
     As shown in  FIGS.  9 A and  9 B , both blue color of light  952  and green color of light  954  can be incident in p polarization state, simultaneously or sequentially, into the optically diffractive device  900 . The two colors of light can have a same incident angle 0°. When the blue color of light  952  and the green color of light  954  are first incident on the BY filter  904 , the color-selective polarizer  904  rotates the p polarization state of the blue color of light to s polarization state, without rotation of the polarization state of the green color of light, such that the blue color of light is incident on the B grating  912  in s polarization state and the green color of light is incident on the B grating  912  in p polarization state. The B grating  912  diffracts the blue color of light in s polarization state into first order blue color of light  952 ′ at a diffracted angle with a first diffraction efficiency and transmits zero order blue color of light  952 ″ at the incident angle. Due to polarization sensitivity and Bragg sensitivity, the B grating  912  diffracts the green color of light  954  in p polarization state with a diffraction efficiency substantially smaller than the first diffraction efficiency, and most of the green color of light  954  in p polarization state transmits through the B grating  912 . The color-selective polarizer  906  rotates p polarization state of the green color of light into s polarization state, without rotation of s polarization state of the blue color of light, such that the G grating  922  diffracts the green color of light in s polarization into first order green color light  954 ′ at a diffracted angle with a second diffraction efficiency and transmits zero order green color of light  954 ″ at the incident angle. Thus, the diffracted blue color of light  952 ′ and green color of light  954 ′ exit out of the optically diffracted device  900  with the same s polarization state and with the same diffracted angle, e.g., in a range from -10° to 10° or -7° to 7°, or substantially close to 0° or normal to the device  900 . 
     As shown in  FIG.  5 H , the optically diffractive device  900  can be positioned in front of a cover glass  930  of a display, e.g., the display  594  of  FIG.  5 H , along the Z direction. As discussed above in  FIG.  5 H , the optically diffractive device  900  can be attached to the cover glass  930  with an OCA layer or index-matching oil, or spaced with a gap such as an air gap. The diffracted blue color of light  952 ′ and green color of light  954 ′ can be incident in the same s polarization state and at the same incident angle (e.g., at substantially normal incidence) into the display. The display can diffract the blue color of light  952 ′ and the green color of light  954 ′ back into and through the optically diffractive device  900 . The blue color of light and green color of light diffracted from the display cannot significantly be further diffracted by the optically diffractive device  950  as they are incident on the diffractive gratings  912  and  922  at an angle far off-Bragg. 
     The display  594  can be illuminated by light polarized in a direction of the display’s alignment layer or a direction perpendicular to the display’s alignment layer. The display can be rotated in its own plane between horizontal and vertical orientations, hence which polarization is required depends on which orientation the display is in. In some implementations, the display can be illuminated with p polarized light. The blue color of light and green color of light diffracted from the optically diffractive device  900  can be incident in the same p polarization state on the display. The optically diffractive device  900  can include an additional color-selective polarizer after the G grating  922  to rotate the s polarization state of each of the blue color of light  952 ′ and the green color of light  954 ′ to p polarization state. 
     In some implementations, the blue color of light is incident in s polarization state and the green color of light is incident in p polarization state into the optically diffractive device  900 , and the optically diffractive device  900  can include no BY filter  904  before the B grating  912  to rotate the polarization state of the blue color of light. 
     In some implementations, the zero order undiffracted (or transmitted) blue color of light and/or the zero order undiffracted (or transmitted) green color of light can be totally internally reflected by one or more reflective layers arranged in the optically diffractive device  900 , as discussed with further details in  FIGS.  11  to  12 C.   
       FIG.  10 A  illustrates an example optically diffractive device  1000 , including holographic gratings for three colors and corresponding color-selective polarizers, for individually diffracting the three colors of light.  FIG.  10 B  illustrates an example of diffracting the three colors of light by the optical device of  FIG.  10 A . Compared to  FIGS.  9 A and  9 B , the optically diffractive device  1000  includes an additional diffractive component for an additional color and different color-selective polarizers for the three colors. For illustration, the device  1000  is configured for blue, red, and green colors of light. 
     As illustrated in  FIG.  10 A , the optically diffractive device  1000  can be arranged in front of a cover glass  1050  of a display, e.g., the display  594  of  FIG.  5 H , along the Z direction. The optically diffractive device  1000  includes a first diffractive component  1010 , a second diffractive component  1020 , and a third diffractive component  1030  that can be sequentially stacked together on a field grating substrate  1002  along the Z direction. An AR film  1001  can be applied to or coated on a front surface of the field grating substrate  1002  to reduce reflection of light. Each of the first, second, and third diffractive components  1010 ,  1020 ,  1030  can include a respective substrate  1014 ,  1024 ,  1034 , a respective diffractive grating  1012 ,  1022 ,  1032 , and a respective carrier film  1016 ,  1026 ,  1036 . The respective diffractive grating  1012 ,  1022 ,  1032  is between the respective substrate  1014 ,  1024 ,  1034  and the respective carrier film  1016 ,  1026 ,  1036 . In some cases, the respective substrate  1014 ,  1024 ,  1034  is in front of the respective carrier film  1016 ,  1026 ,  1036  along the Z direction. In some cases, the respective carrier film  1016 ,  1026 ,  1036  is in front of the respective substrate  1014 ,  1024 ,  1034  along the Z direction. 
     Each of the first, second, and third diffractive gratings  1012 ,  1022 , and  1032  can be configured to: diffract a single color of light in s polarization state incident at an incident angle with a diffraction efficiency substantially higher, e.g., more than one order of magnitude, two orders of magnitude, or three orders of magnitude, than a diffraction efficiency where the diffractive grating diffracts another color of light in p polarization state incident at a same or different incident angle. Each of the first, second, and third diffractive gratings  1012 ,  1022 , and  1032  can be a holographic grating, e.g., a volume grating or a Bragg grating. Each of the first, second, and third diffractive gratings  1012 ,  1022 , and  1032  can be independently recorded and fixed in a recording medium, e.g., a photosensitive polymer or a photopolymer. 
     The optically diffractive device  1000  can include multiple color-selective polarizers for the three colors of light. In some implementations, a BY filter  1004  is between a field grating substrate 1002 and the first diffractive grating  1012  of the first diffractive component  1010  and configured to rotate a polarization state of blue color of light, without rotation of a polarization state of each of red and green colors of light. A MG filter  1006  is between the first and second diffractive gratings  1012  and  1022  (or between the first and second diffractive components  1010  and  1020 ) and configured to rotate a polarization state of each of blue and red colors of light, without rotation of a polarization state of green color of light. A YB filter  1008  is between the second and third diffractive gratings  1022  and  1032  (or between the second and third diffractive components  1020  and  1030 ) and configured to rotate a polarization state of each of red and green colors of light, without rotation of a polarization state of blue color of light. An MG filter  1040  is after the third diffractive grating  1032  (or the third diffractive component  1030 ) and configured to rotate a polarization state of each of red and blue colors of light, without rotation of a polarization state of green color of light. 
     In some implementations, a color-selective polarizer is composed of two or more sub-polarizers. The sub-polarizers can be arranged in any desired order. For example, the YB filter 1008 can be composed of a RC filter  1008 - 1  and a GM filter  1008 - 2 . The RC filter  1008 - 1  can be arranged before the GM filter  1008 - 2 , or vice versa. The RC filter  1008 - 1  is configured to rotate a polarization state of red color of light, without rotation of a polarization state of each of green and blue colors of light, and the GM filter  1008 - 2  is configured to rotate a polarization state of green color of light, without rotation of a polarization state of each of red and blue colors of light. 
     Adjacent layers or components in the optically diffractive device  1000  can be attached together using one or more intermediate layers of OCA, UV-cured or heat-cured optical glues, optical contacting, or index matching fluid. As discussed in  FIG.  5 H , the optically diffractive device  1000  can be attached to the display cover glass  1050  through an intermediate layer or spaced with a gap, e.g., an air gap. 
     The optically diffractive device  1000  is configured to diffract the three colors of light (red, green, and blue) out at a same diffracted angle (e.g., substantially normal incidence) with a same polarization state (e.g., s or p) towards the display. The three colors of light can be input into the optically diffractive device  1000  at a same incident angle 0°, e.g., substantially identical to be a Bragg angle. In some cases, the three colors of light can be incident at different angles to match a Bragg angle of each color’s grating. The three colors of light can be in beams large enough to illuminate the whole region of the gratings. The three colors of light can be input into the optically diffractive device  1000  in a same polarization state (e.g., s or p). In some cases, a color of light is incident from an opposite side (e.g., at - θ°) or from the Y direction. Each color grating can be rotated to match the direction of its corresponding color replay reference light. A corresponding color-selective polarizer can be independent of the rotation of the color grating. 
       FIG.  10 B  illustrates an example  1060  of diffracting the three colors of light (blue, red, green) by the optically diffractive device  1000  of  FIG.  10 A . The three colors of light are incident into the optically diffractive device  1000  at the same incident angle θ° and in the same p polarization state. 
     As shown in  FIG.  10 B , the BY filter  1004  rotates the p polarization state of the blue color of light to s polarization state, without rotation of the p polarization state of each of the red and green colors of light. The B grating  1012  diffracts the blue color of light in the s polarization state into first order at the diffracted angle and zero order at the incident angle. The green and red colors of light incident in p polarization state at the incident angle transmit through the B grating  1012 . 
     The MG filter  1006  rotates the s polarization state of the blue color of light to p polarization state, and the p polarization state of the red color of light to s polarization state, without rotation of the p polarization state of the green color of light. The R grating  1022  diffracts the red color of light in the s polarization state into first order at the diffracted angle and zero order at the incident angle. The residual blue color of light at zero order and the green color of light incident in p polarization state at the incident angle transmit through the R grating  1022 . 
     The RC filter  1008 - 1  in the YB filter  1008  rotates the s polarization state of the red color of light to p polarization state, without rotation of the p polarization state of each of the green and blue colors of light. The GM filter  1008 - 2  of the YB filter  1008  rotates the p polarization state of the green color of light to s polarization state, without rotation of the p polarization of each of the red and blue colors of light. The residual blue color of light at zero order, the residual red color of light at zero order, and the green color of light transmit through the RC filter  1008 - 1  and the GM filter  1008 - 2 . 
     The G grating  1032  diffracts the green color of light in the s polarization state into first order at the diffracted angle and zero order at the incident angle. The residual blue color of light and the residual red color of light incident in p polarization state at the incident angle transmit through the G grating  1032 . 
     The MG filter  1040  rotates the p polarization state of each of the red and blue colors of light to s polarization state, without rotation of the s polarization state of the green color of light. The diffracted blue, red, and green colors of light in the s polarization state at the same diffracted angle propagate out of the optically diffractive device  1000 . The residual blue color of light, the residual red color of light, and the residual green color of light at zero order are also in s polarization state and at the incident angle transmit through the MG filter  1040 . 
     In some implementations, the optically diffractive device  1000  can have a larger size than the display. The residual blue, red, green colors of light at zero order can propagate at a large angle out of the device  1000  and into air. In some implementations, as discussed with further details below in  FIGS.  11  to  12 C , the optically diffractive device  1000  can include one or more reflective layers between or after diffractive gratings for total internal reflection of corresponding colors of light at zero order. 
     Exemplary Optically Diffractive Devices with Reflective Layers 
       FIGS.  11  to  12 C  show example optically diffractive devices including reflective layers. The reflective layers can selectively totally reflect a single color of light at zero order while transmitting other colors of light. Each of the optically diffractive devices includes multiple gratings each for a different color of light. Each color of light can be incident at a different replay reference angle on a corresponding grating, such that each color of light undiffracted (or transmitted) by the grating at zero order undergoes total internal reflection (TIR) from an interface subsequent to the grating which diffracts out the color of light at first order at a same diffracted angle (e.g., substantially normal), but prior to the subsequent gratings (if any) in the device. The other colors of light can transmit at the corresponding replay reference angles through the grating. 
       FIG.  11    illustrates an example optically diffractive device  1100 , including diffractive gratings for two colors and corresponding reflective layers, for individually diffracting the two colors of light. For illustration, the device  1100  is configured for green and blue colors of light. 
     The optically diffractive device  1100  includes a first diffractive component  1110  having a first diffractive grating  1112  for blue color and a second diffractive component  1120  having a second diffractive grating  1122  for green color. Each of the first and second diffractive gratings  1112 ,  1122  can be a holographic grating, e.g., a Bragg grating or a volume grating. Each of the first and second diffractive gratings  1112  and  1122  can be independently recorded and fixed in a recording medium, e.g., a photosensitive material such as a photopolymer. 
     The first diffractive component  1110  and the second diffractive component  1120  can be stacked together on a field grating substrate  1102  along a direction, e.g., the Z direction. The field grating substrate  1102  can be an optically transparent substrate, e.g., a glass substrate. The optically diffractive device  1100  can be in front of a display such as LCOS, e.g., the display  594  of  FIG.  5 H . For example, the optically diffractive device  1100  can be arranged on a cover glass  1130  of the display through an intermediately layer or spaced by a gap, e.g., an air gap. 
     Similar to the first and second diffractive components  910 ,  920  in  FIGS.  9 A and  9 B , each of the first and second diffractive components  1110  and  1120  can include a respective substrate  1114 ,  1124  and a respective carrier film  1116 ,  1126  on opposite sides of the respective diffractive grating  1112 ,  1122 . The respective diffractive grating  1112 ,  1122  is between the respective substrate  1114 ,  1124  and the respective carrier film  1116 ,  1126 . The respective substrate  1114 ,  1124  and the respective carrier film  1116 ,  1126  can be arranged in an order to reduce refractive index mismatch and thus undesired Fresnel reflection. The respective substrate  1114 ,  1124  can be a glass substrate that can have a refractive index same as or close to the refractive index of the field grating substrate  1102 . The respective carrier film  1116 ,  1126  can be a TAC film. The TAC film can have a lower refractive index than a photosensitive polymer used to record diffractive gratings  1112  and  1122 . In some examples, the respective substrate  1114 ,  1124  is arranged before the carrier film  1116 ,  1126 . 
     Adjacent layers or components in the optically diffractive device  1100  can be attached together using one or more intermediate layers of OCA, UV-cured or heat-cured optical glues, optical contacting, or index matching fluid. For example, the first diffractive component  1110  (e.g., the substrate  1114 ) can be attached to the field grating substrate  1102  through an intermediate layer  1101 , e.g., an OCA layer. The first and second diffractive components  1110  and  1120 , e.g., the carrier film  1116  and the substrate  1124 , can be attached together through another intermediate layer  1103 , e.g., an OCA layer. The optically diffractive device  1100  (e.g., the carrier film  1126 ) can be attached to the cover glass  1130  of the display through an intermediate layer  1105 , e.g., an OCA layer. 
     As shown in  FIG.  11   , each of the first and second diffractive gratings  1112 ,  1122  is configured to diffract a corresponding color of light incident at a respective incident angle into first order at a respective diffracted angle and zero order at the respective incident angle, and transmit another color of light at a different incident angle, e.g., due to Bragg selectivity. Thus, there can be no crosstalk between the different colors of light individually diffracted at corresponding diffractive gratings. Each color of light can be polarized. The polarization state of the different colors of light diffracted at first order can be the same, e.g., s or p. The respective diffracted angles for the different colors of light can be same, e.g., substantially normal. 
     The optically diffractive device  1100  can include a first reflective layer (or blocking layer) between the first grating  1112  and the second grating  1122 . The first grating  1112  is configured to diffract blue color of light incident at a first incident angle θ b , e.g., 78.4°, into first order at a diffracted angle, e.g., 0° and zero order at the first incident angle. The first reflective layer, e.g., a refractive index of the first reflective layer, is configured to totally reflect the blue color of light diffracted at the first incident angle but to transmit the green color of light incident at a second incident angle θ g , e.g., 76.5°. For example, the refractive index of the first reflective layer is lower than the refractive index of a layer immediately before the first reflective layer, e.g., the first grating  1112 . The first reflective layer can be a suitable layer between the first grating  1112  and the second grating  1122 . In some examples, the first reflective layer is the carrier film  1116 , as shown in  FIG.  11   . 
     Similarly, the optically diffractive device  1100  can include a second reflective layer after the second grating  1112  and before the display cover glass  1130 . The second grating  1112  is configured to diffract green color of light incident at the second incident angle θ g , e.g., 76.5°, into first order at a diffracted angle, e.g., 0° and zero order at the second incident angle. The second reflective layer, e.g., a refractive index of the second reflective layer, is configured to totally reflect the green color of light diffracted at the second incident angle. The second reflective layer can be a suitable layer between the second grating  1122  and the cover glass  1130 . In some examples, the second reflective layer is the intermediate layer  1105 , as shown in  FIG.  11   . 
     The totally reflected blue and green colors of light by the corresponding reflective layers are reflected back into the optically diffractive device  1100  to a side of the optically diffractive device  1100 . As illustrated in  FIG.  11   , a surface of the side can be coated with an optical absorber  1104 , e.g., a black coating, to absorb the totally reflected blue and green colors of light diffracted at zero order by the corresponding diffractive gratings. 
     The field grating substrate  1102  can be thick enough such that the replay reference light beams of different colors can enter at its edge of the field grating substrate  1102 . The field grating substrate  1102  can be also configured to fully contain the replay reference light beams such that a viewer or observer cannot insert a finger or other object into the replay reference light beams. The viewer thus cannot obstruct the replay reference light beams, which can improve laser safety as the viewer cannot get an eye (or a reflective or focusing element) into the full-power replay reference light beams. The optically diffractive device  1100  with the field grating substrate  1102  can be significantly more compact than if the replay reference light beams are incident upon the front surface of the optically diffractive device  1100  from air. 
     As the blue and green colors of light are incident at a relatively large replay reference angle (or incident angle), e.g., more than 70°, Fresnel reflection can be significant from layer interfaces (for both P and S polarization), and can rapidly increase with increasing replay reference angle. Since the optically diffractive device  1100  contains a number of interfaces between materials of different refractive indices, the Fresnel reflection losses from each such interface can add to substantially attenuate the replay output light, causing a substantially reduced replay-light power at each diffractive grating, especially the grating, e.g., the G grating  1122 , closest to the display. In some examples, a replay reference angle (or an incident angle) for a particular color of light can be selected to be just large enough to reliably undergo TIR, but not much large so that the Fresnel losses can be reduced. 
       FIGS.  13 A- 13 C  illustrate relationships between diffracted (solid lines) and reflected or blocked (dashed lines) replay reference beam powers with different incident angles for blue color of light ( FIG.  13 A ), green color of light ( FIG.  13 B ), and red color of light ( FIG.  13 C ). The diffracted replay reference beam power can be an illumination beam into a cover glass of a display, e.g., the display  594  of  FIG.  5 H , adjacent to an optically diffractive device, e.g., the optically diffractive device  598  of  FIG.  5 H . 
     As illustrated in  FIG.  13 A , for blue color of light, plot  1302  shows the diffracted replay reference beam power (or the display’s blue illumination power) as a replay reference beam angle (e.g., an incident angle in glass) is increased, and plot  1304  shows the reflected replay reference beam power from a corresponding reflective layer as the replay reference beam angle is increased. As illustrated in  FIG.  13 B , for green color of light, plot  1312  shows the diffracted replay reference beam power (or the display’s green illumination power) as a replay reference beam angle (e.g., an incident angle in glass) is increased, and plot  1314  shows the reflected replay reference beam power from a corresponding reflective layer as the replay reference beam angle is increased. As illustrated in  FIG.  13 C , for red color of light, plot  1322  shows the diffracted replay reference beam power (or the display’s red illumination power power) as a replay reference beam angle (e.g., an incident angle in glass) is increased, and plot  1324  shows the reflected replay reference beam power from a corresponding reflective layer as the replay reference beam angle is increased. 
     Replay reference angles for different colors of light can be chosen to be large enough such that for each color of light, the corresponding reflective layer can totally reflect the color of light with a reflection of 100%, while the replay reference angles can be small enough such that the Fresnel losses do not substantially eliminate the diffracted replay reference beams or the illumination in the cover glass of the display. As an example, a diffraction efficiency of each grating is 50% for blue, 60% for green, and 70% for red. A bottom layer of the optically diffractive device is parallel to the cover glass of the display. A diffracted angle of the replay object beam for each color is -6°. As shown in  FIGS.  13 A,  13 B,  13 C , the net object beam powers inside the cover glass of the display are 46.8% for blue, 33.1% for green, and 43.0% for red, when the replay reference angle is 78.4° for blue color of light at 460 nm, 76.5° for green color of light at 520 nm, and 73.5° for red color of light at 640 nm. 
       FIG.  12 A  illustrates an example optically diffractive device  1200 , including diffractive gratings for three colors and corresponding reflective layers, for individually diffracting the three colors of light. For illustration, the device  1200  is configured for blue, green and red colors of light. 
     The optically diffractive device  1200  includes a first diffractive component  1210  having a first diffractive grating  1212  for blue color, a second diffractive component  1220  having a second diffractive grating  1222  for green color, and a third diffractive component  1230  having a third diffractive grating  1232  for red color. Each of the first, second, and third diffractive gratings  1212 ,  1222 ,  1232  can be a holographic grating, e.g., a Bragg grating or a volume grating. Each of the first, second, and third diffractive gratings  1212 ,  1222 , and  1232  can be independently recorded and fixed in a recording medium, e.g., a photosensitive material such as a photopolymer. 
     The first, second, and third diffractive components  1210 ,  1220 , and  1230  can be stacked together on a field grating substrate  1202  along a direction, e.g., the Z direction. The field grating substrate  1202  can be an optically transparent substrate, e.g., a glass substrate. The optically diffractive device  1210  can be in front of a display such as LCOS, e.g., the display  594  of  FIG.  5 H . For example, the optically diffractive device  1200  can be arranged on a cover glass  1240  of the display through an intermediately layer or spaced by a gap, e.g., an air gap. 
     Similar to the first, second, and third diffractive components  1010 ,  1020 ,  1030  in  FIGS.  10 A and  10 B , each of the first, second, and third diffractive components  1210 ,  1220 ,  1230  can include a respective substrate  1214 ,  1224 ,  1234  and a respective carrier film  1216 ,  1226 ,  1236  on opposite sides of the respective diffractive grating  1212 ,  1222 ,  1232 . The respective diffractive grating  1212 ,  1222 ,  1232  is between the respective substrate  1214 ,  1224 ,  1234  and the respective carrier film  1216 ,  1226 ,  1236 . The respective substrate  1214 ,  1224 ,  1234  and the respective carrier film  1216 ,  1226 ,  1236  can be arranged in an order to reduce refractive index mismatch and thus Fresnel reflection. The respective substrate  1214 ,  1224 ,  1234  can be a glass substrate that can have a refractive index same as or close to the refractive index of the field grating substrate  1202 . The respective carrier film  1216 ,  1226 ,  1236  can be a TAC film. The TAC film can have a lower refractive index than a photosensitive polymer. In some examples, the respective substrate  1214 ,  1224  is arranged before the carrier film  1216 ,  1226 . The substrate  1234  is arranged after the carrier film  1236 . 
     Adjacent layers or components in the optically diffractive device  1100  can be attached together using one or more intermediate layers of OCA, UV-cured or heat-cured optical glues, optical contacting, or index matching fluid. For example, the first diffractive component  1210  (e.g., the substrate  1214 ) can be attached to the field grating substrate  1202  through an intermediate layer  1201 , e.g., an OCA layer. The first and second diffractive components  1210  and  1220 , e.g., the carrier film  1216  and the substrate  1224 , can be attached together through another intermediate layer  1203 , e.g., an OCA layer. The second and third diffractive components  1220  and  1230 , e.g., the carrier film  1226  and the carrier film  1236 , can be attached together through another intermediate layer  1205 , e.g., an OCA layer. The optically diffractive device  1200  (e.g., the substrate  1234 ) can be attached to the cover glass  1240  of the display through an intermediate layer  1207 , e.g., an OCA layer. 
     As shown in  FIG.  12 A , each of the first, second, and third diffractive gratings  1212 , 1222, 1232 is configured to diffract a corresponding color of light incident at a respective incident angle into first order at a respective diffracted angle and zero order at the respective incident angle, and transmit another color of light at a different incident angle, e.g., due to Bragg selectivity. Thus, there can be no or little crosstalk between the different colors of light individually diffracted at corresponding diffractive gratings. Each color of light can be polarized. The polarization state of the different colors of light diffracted at first order can be the same, e.g., s or p. The respective diffracted angles for the different colors of light can be same, e.g., substantially normal. 
     As discussed above in  FIGS.  13 A,  13 B,  13 C , different incident angles θ b , θ g , θ r , (or replay reference angles) for different colors of light (blue, green, and red) can be chosen, e.g., to be 78.4°, 76.5°, and 73.5°. The optically diffractive device  1200  can include a first reflective layer (or blocking layer) between the first grating  1212  and the second grating  1222 . The first grating  1212  is configured to diffract blue color of light incident at the first incident angle θ b  into first order at a diffracted angle, e.g., 0°, and zero order at the first incident angle. The first reflective layer, e.g., a refractive index of the first reflective layer, is configured to totally reflect the blue color of light diffracted at the first incident angle but to transmit the green color of light incident at the second incident angle θ g  and the red color of light incident at the third incident angle θ r . For example, the refractive index of the first reflective layer is lower than the refractive index of a layer immediately before the first reflective layer, e.g., the first grating  1212 . The first reflective layer can be a suitable layer between the first grating  1212  and the second grating  1222 . In some examples, the first reflective layer is the carrier film  1216 , as shown in  FIG.  12 A . Total internal reflection occurs on an interface between the first grating  1212  and the carrier film  1216 . The totally reflected blue color of light undiffracted (or transmitted) at the zero order is reflected back to the layers above the first reflective layer and can be absorbed by an optical absorber  1204  coated on a side surface of the optically diffractive device  1200 . 
     The optically diffractive device  1200  can include a second reflective layer (or blocking layer) between the second grating  1222  and the third grating  1232 . The second grating  1222  is configured to diffract the green color of light incident at the second incident angle θ g  into first order at a diffracted angle, e.g., 0°, and zero order at the second incident angle. The second reflective layer, e.g., a refractive index of the second reflective layer, is configured to totally reflect the green color of light diffracted at the second incident angle but to transmit the red color of light incident at the third incident angle θ r . For example, the refractive index of the second reflective layer is lower than the refractive index of a layer immediately before the second reflective layer. The second reflective layer can be a suitable layer between the second grating  1222  and the third grating  1232 . In some examples, the second reflective layer is the intermediate layer  1205 , as shown in  FIG.  12 A . Total internal reflection occurs on an interface between the carrier film  1226  and the intermediate layer  1205 . The totally reflected green color of light undiffracted (or transmitted) at the zero order is reflected back to the layers above the second reflective layer and can be absorbed by the optical absorber  1204 . 
     The optically diffractive device  1200  can include a third reflective layer after the third grating  1232  and before the display cover glass  1240 . The third grating  1232  is configured to diffract the red color of light incident at the third incident angle θ r , into first order at a diffracted angle, e.g., 0° and zero order at the third incident angle. The third reflective layer, e.g., a refractive index of the third reflective layer, is configured to totally reflect the red color of light diffracted at the third incident angle. The third reflective layer can be a suitable layer between the third grating  1232  and the cover glass  1240 . In some examples, the third reflective layer is the intermediate layer  1207  between the substrate  1234  and the cover glass  1240 , as shown in  FIG.  12 A . The totally reflected red color of light undiffracted (or transmitted) at the zero order is reflected back to the layers above the second reflective layer and can be absorbed by the optical absorber  1204 . 
     The field grating substrate  1202  can be thick enough such that the replay reference light beams of different colors entering at its edge of the field grating substrate  1202 . The field grating substrate  1202  can be also configured to fully contain the replay reference light beams such that a viewer or observer cannot insert a finger or other object into the replay reference light beams. The viewer thus cannot obstruct the replay reference light beams, which can improve laser safety as the viewer cannot get an eye (or a reflective or focusing element) into the full-power replay reference light beams. The optically diffractive device  1200  with the field grating substrate  1202  can be significantly more compact than if the replay reference light beams are incident upon the front surface of the optically diffractive device  1200  from air. 
     As shown in  FIG.  12 A , the field grating substrate  1202  can have a rectangular cross-section in the XZ plane. The different colors of light are incident from a side surface of the field grating substrate  1202 .  FIG.  12 B  illustrates another example optically diffractive device  1250  including a wedged field grating substrate  1252 . A wedged angle between a side surface (or an input surface for light beams)  1251  of the substrate  1252  and a top layer  1253  of the substrate  1252  can be selected, and/or the side face can be AR coated, such that an optical path taken by any light beam returning to the field grating substrate  1252  from the optically diffractive device  1250  and the display can be conveniently blocked or attenuated or directed to reduce or eliminate reflections back into the optically diffractive device  1250  and the display. The optically diffractive device  1250  can include a corresponding optical absorber  1254  coated on an opposite side surface, which can be shorter than the optical absorber  1204  of  FIG.  12 A . 
       FIG.  12 C  illustrates a further example optically diffractive device  1270  including a field grating substrate  1272  having a wedged input face  1271 . The wedged input face  1271  may be configured to reduce Fresnel losses of input light of different colors. The wedged input face  1271  may be configured such that the input light of different colors is incident on the input face  1271  at substantially normal incidence and incident on corresponding diffractive gratings at different incident angles (or replay reference angles). The wedged input face  1271  may be configured to refract input light of different colors to the desired angles of each color inside the diffractive device and from convenient directions and angles in air. For example, the wedged input face  1271  may have a wedge angle such that the in air angles cause the input beams to travel parallel to the front surface of the diffractive device or from the space behind the front surface of the diffractive device. 
     An AR coating can be formed on a front surface  1273  of the field grating substrate  1272  to reduce or eliminate the reflection of ambient light back towards a viewer. An AR coating can be also formed on a back face of the optically diffractive device  1270  closest to the display to reduce or eliminate the undesirable reflection of light reflected and/or diffracted from the display towards the viewer. 
     In some implementations, one or more layers in an optically diffractive device, e.g., the optically diffractive device  1100  of  FIG.  11   ,  1200  of  FIG.  12 A ,  1250  of  FIG.  12 B , or  1270  of  FIG.  12 C , can be slightly wedged, which can allow fine tuning of TIR and Fresnel reflection at each layer. The layers can be also configured to reduce or eliminate a visibility of Newton’s rings or interference fringes which can occur between any pair of substantially parallel surfaces within the optically diffractive device when using narrow-band light sources, e.g., laser diodes. 
     Exemplary Fabrication Processes 
       FIG.  14 A  is a flowchart of an example process  1400  of fabricating an optically diffractive device including diffractive structures and corresponding color-selective polarizers. The optically diffractive device can be the optically diffractive device  598  of  FIG.  5 H ,  598 A of  FIG.  5 I ,  598 B of  FIG.  5 J , or  598 C of  FIG.  5 K , the optically diffractive device  900  of  FIGS.  9 A and  9 B , or the optically diffractive device  1000  of  FIGS.  10 A and  10 B . 
     A first diffractive component for a first color is fabricated ( 1402 ). The first diffractive component can be the first diffractive component  910  of  FIGS.  9 A and  9 B  or  1010  of  FIGS.  10 A and  10 B . The first diffractive component includes a first diffractive structure, e.g., the B grating  912  of  FIGS.  9 A and  9 B  or the B grating  1012  of  FIGS.  10 A and  10 B , formed in a recording medium. The first diffractive structure is configured to diffract replay reference light of the first color (or the first color of light), which is incident in a first polarization state at a first incident angle on the first diffractive structure, at a first diffracted angle with a first diffraction efficiency. The first diffraction efficiency can be substantially higher than a diffraction efficiency with which the first diffractive structure diffracts the first color of light or another different color of light incident in a second polarization state different from the first polarization state at the first incident angle, e.g., due to polarization selectivity. The first polarization state can be s polarization, and the second polarization state can be p polarization. 
     The first diffractive structure can be a holographic grating, e.g., a volume grating or a Bragg grating. A thickness of the recording medium can be more than one order of magnitude larger than the wavelength of the first recording object beam, e.g., 30 times. In some examples, the first incident angle can be a Bragg angle. The first diffraction efficiency can be substantially higher than a diffraction efficiency with which the first diffractive structure diffracts the first color of light or another different color of light incident in the first or second polarization state at an incident angle different from the first incident angle, e.g., due to Bragg selectivity. 
     The recording medium can include a photosensitive material, e.g., a photosensitive polymer or photopolymer. The first diffractive structure can be formed by exposing the photosensitive material to a first recording object beam at a first recording object angle and simultaneously to a first recording reference beam at a first recording reference angle. The first recording object beam and the first recording reference beam can have a same wavelength, e.g., from a same light source, and the same first polarization state. 
     In some cases, the first color of light used for replay can include a wavelength range wider than or identical to that of the first recording reference beam or the first recording object beam. For example, the first recording reference beam and the first recording object beam can be light beams of a laser, and the first color of light for replay can be a light beam of a laser diode. In some cases, the first recording reference beam and the first recording object beam can correspond to a color different from the first color of the first color of light. For example, a green color laser light can be used to record a diffractive grating for a red color. 
     The first incident angle of the first color of light can be substantially identical to the first recording reference angle, and the first diffracted angle can be substantially identical to the first recording object angle. In some examples, the first recording reference angle is in a range from 70 degrees to 90 degrees, e.g., in a range from 80 degrees to 90 degrees. In some examples, the first recording object angle is in a range from -10 degrees to 10 degrees, e.g., -7 degrees to 7 degrees, 0 degrees or 6 degrees. In some examples, a sum of the first recording reference angle and the first recording object angle within the photosensitive material is substantially identical to 90 degrees. 
     The first diffractive structure can be fixed in the recording medium, e.g., by UV curing or heat curing. In some examples, the first diffractive component includes a carrier film, e.g., a TAC film, on the recording medium. In some examples, the first diffractive component includes a diffraction substrate, e.g., a glass substrate. The recording medium can be between a carrier film and a diffraction substrate. 
     A second diffractive component for a second color is fabricated ( 1404 ). The second diffractive component can be the second diffractive component  920  of  FIGS.  9 A and  9 B  or  1020  of  FIGS.  10 A and  10 B . The second diffractive component includes a second diffractive structure, e.g., the B grating  922  of  FIGS.  9 A and  9 B  or the R grating  1022  of  FIGS.  10 A and  10 B , formed in a second recording medium. The second diffractive structure is configured to diffract replay reference light of the second color (or the second color of light), which is incident in the first polarization state at a second incident angle on the second diffractive structure, at a second diffracted angle with a second diffraction efficiency. The second diffraction efficiency can be substantially higher than a diffraction efficiency with which the second diffractive structure diffracts the second color of light or another different color of light incident in the second polarization state at the second incident angle or an incident angle different from the second incident angle. 
     The second diffractive structure can be fabricated in a way similar to the first diffractive structure as described above. The first diffractive structure and the second diffractive structure can be independently fabricated. The second diffractive component can also include a carrier film and a diffraction substrate. 
     The first and second diffractive components can be configured such that the first diffracted angle and the second diffracted angle are substantially identical to each other, e.g., substantially normal. The first incident angle and the second incident angle can be substantially identical to each other. 
     A color-selective polarizer is arranged between the first and second optically diffractive components ( 1406 ). The color-sensitive polarizer can be the GM filter  906  of  FIGS.  9 A and  9 B , or the MG filter  1006  of  FIGS.  10 A and  10 B . The optically diffractive structure can include a field grating substrate, e.g., the substrate  902  of  FIGS.  9 A and  9 B  or the substrate  1002  of  FIGS.  10 A and  10 B . The first optically diffractive component, the color-selective polarizer, and the second optically diffractive component can be sequentially stacked on the field grating substrate, such that the first color of light and the second color of light are incident on the first optically diffractive component before the second optically diffractive component. The color-selective polarizer can be configured to rotate a polarization state of the second color of light, e.g., from the second polarization state to the first polarization state, such that the second color of light can be incident in the first polarization state on the second diffractive structure. In some cases, the color-selective polarizer can rotate a polarization state of the first color of light. In some cases, the color-selective polarizer is configured not to rotate the polarization state of the first color of light. 
     In some implementations, an additional color-selective polarizer is arranged in front of the first diffractive component. For example, the additional color-selective polarizer can be between the field grating substrate and the first diffractive component. The additional color-selective polarizer can be the BY filter 904 of  FIGS.  9 A and  9 B  or the BY filter  1004  of  FIGS.  10 A and  10 B . The additional color-selective polarizer is configured to rotate a polarization state of the first color of light, e.g., from the second polarization state to the first polarization state, such that the first color of light is incident in the first polarization state on the first diffractive structure. In some cases, the additional color-selective polarizer can rotate a polarization state of the second color of light, e.g., from the first polarization state to the second polarization state, such that the second color of light is incident in the second polarization state on the first diffractive structure. In some cases, the additional color-selective polarizer is configured not to rotate the polarization state of the second color of light, such that the second color of light is incident in the second polarization state on the first diffractive structure. 
     Adjacent components in the optically diffractive device can be attached together through an intermediate layer. The intermediate layer can be an OCA layer, a UV-cured or heat-cured optical glue, optical contacting, or an index-matching fluid. 
     In some implementations, the process  1400  can further include forming a third optically diffractive component. The third diffractive component includes a third diffractive structure, e.g., the G grating  1032  of  FIGS.  10 A and  10 B , formed in a third recording medium. The third diffractive structure is configured to diffract replay reference light of a third color (or the third color of light), which is incident in the first polarization state at a third incident angle on the third diffractive structure, at a third diffracted angle with a third diffraction efficiency. The third diffraction efficiency can be substantially higher than a diffraction efficiency with which the third diffractive structure diffracts the third color of light or another different color of light incident in the third polarization state at the second incident angle or an incident angle different from the third incident angle. 
     The third diffractive structure can be fabricated in a way similar to the first diffractive structure as described above. The first, second, and third diffractive structures can be independently fabricated. The third diffractive component can also include a carrier film and a diffraction substrate. The first, second, and third diffractive components can be configured such that the first, second, and third diffracted angles are substantially identical to each other, e.g., substantially normal. The first, second, and third incident angles can be substantially identical to each other. 
     A second color-selective polarizer can be arranged between the second and third optically diffractive components. The second color-sensitive polarizer can be YG filter of  FIGS.  10 A and  10 B . The second color-selective polarizer can be composed of two or more sub-polarizers, e.g., the RC filter  1008 - 1  and the GM filter  1008 - 2  of  FIGS.  10 A and  10 B . In some examples, the second color-selective polarizer is first attached on the third diffractive component, and then the second color-selective polarizer can be attached to the second diffractive component. In some examples, the second color-selective polarizer can be first attached to the second diffractive component, and then the third diffractive component can be attached to the second color-selective polarizer. The second color-selective polarizer can be configured to rotate a polarization state of the third color of light from the second polarization state to the first polarization state, such that the third color of light is incident in the first polarization state on the third diffractive structure. The second color-selective polarizer can be configured to rotate the polarization state of the second color of light, e.g., from the first polarization state to the second polarization state, without rotation of the polarization state of the first color of light. 
     A third color-selective polarizer can be arranged sequential to the third optically diffractive component such that the third optically diffractive component is between the second and third color-selective polarizers. The third color-selective polarizer can be the MG filter  1040  of  FIGS.  10 A and  10 B . The third color-selective polarizer is configured to rotate the polarization state of each of the first and second colors of light, e.g., from the second polarization state to the first polarization state, without rotation of the first polarization state of the third color of light, such that the diffracted first, second, and third colors of light have the same polarization state. 
       FIG.  14 B  is a flowchart of an example process  1450  of fabricating an optically diffractive device including diffractive structures and corresponding reflective layers. The optically diffractive device can be the optically diffractive device  598  of  FIG.  5 H ,  598 A of  FIG.  5 I ,  598 B of  FIG.  5 J , or  598 C of  FIG.  5 K , the optically diffractive device  1100  of  FIG.  11   , or the optically diffractive device  1200  of  FIG.  12 A ,  1250  of  FIG.  12 B , or  1270  of  FIG.  12 C . 
     A first optically diffractive component is formed ( 1452 ). The first diffractive component can be the first diffractive component  1110  of  FIG.  11   ,  1210  of  FIGS.  12 A,  12 B, or  12 C . The first diffractive component includes a first diffractive structure stored in a first recording medium. The first diffractive structure is configured to diffract a first color of light incident at a first incident angle into first order at a first diffracted angle and zero order at the first incident angle. A power of the first color of light at the first order can be substantially higher than the power of the first color of light at zero order. 
     The first diffractive structure can be a holographic grating, e.g., a volume grating or a Bragg grating. A thickness of the recording medium can be more than one order of magnitude larger than the wavelength of the first recording object beam, e.g., 30 times. In some examples, the first incident angle can be a Bragg angle. The first diffraction efficiency can be substantially higher than a diffraction efficiency with which the first diffractive structure diffracts the first color of light or another different color of light incident at an incident angle different from the first incident angle, e.g., due to Bragg selectivity. Light incident at a different incident angle can transmit through the first diffractive structure. 
     The recording medium can include a photosensitive material, e.g., a photosensitive polymer or photopolymer. The first diffractive structure can be formed similar to step  1402  of  FIG.  14 A , e.g., by exposing the photosensitive material to a first recording object beam at a first recording object angle and simultaneously to a first recording reference beam at a first recording reference angle. The first recording object beam and the first recording reference beam can have a same wavelength, e.g., from a same light source, and the same polarization state. The first incident angle of the first color of light can be substantially identical to the first recording reference angle, and the first diffracted angle can be substantially identical to the first recording object angle. In some examples, the first recording reference angle is in a range from 70 degrees to 90 degrees, e.g., in a range from 70 degrees to 80 degrees. In some examples, the first recording object angle is in a range from -10 degrees to 10 degrees, e.g., -7 degrees to 7 degrees, 0 degrees or 6 degrees. The first diffractive structure can be fixed in the recording medium, e.g., by UV curing or heat curing. In some examples, the first diffractive component includes a carrier film, e.g., a TAC film, on the recording medium. In some examples, the first diffractive component includes a diffraction substrate, e.g., a glass substrate. The recording medium can be between a carrier film and a diffraction substrate. 
     A second optically diffractive component is formed ( 1454 ). The second diffractive component can be the second diffractive component  1120  of  FIG.  11   ,  1220  of  FIGS.  12 A,  12 B, or  12 C . The second diffractive component includes a second diffractive structure stored in a second recording medium. The second diffractive structure is configured to diffract a second color of light incident at a second incident angle into first order at a second diffracted angle and zero order at the second incident angle. A power of the second color of light at the first order can be substantially higher than the power of the second color of light at the zero order. 
     The second diffractive structure can be fabricated in a way similar to the first diffractive structure in step  1452 . The first diffractive structure and the second diffractive structure can be independently fabricated. The second diffractive component can also include a carrier film and a diffraction substrate. 
     The first and second diffractive components can be configured such that the first diffracted angle and the second diffracted angle are substantially identical to each other, e.g., substantially normal. The first incident angle and the second incident angle are different from each other. The first and second incident angles can be determined, e.g., according to what is described in  FIGS.  13 A- 13 C . In some examples, the first color of light has a wavelength smaller than the second color of light, and the first incident angle is larger than the second incident angle. 
     A first reflective layer is arranged between the first and second diffractive structures (1456). The first reflective layer can be the reflective layer  1116  of  FIG.  11   , or  1216  of  FIGS.  12 A,  12 B, or  12 C . The first reflective layer is configured to totally reflect the first color of light incident at the first incident angle, such that the first color of light undiffracted (or transmitted) at the zero order can be reflected back into layers before the first reflective layer without propagating to a display behind the optically diffractive device. The first reflective layer can be configured to have a refractive index smaller than that of a layer of the first diffractive component that is immediately adjacent to the first reflective layer, such that the first color of light having the first incident angle is totally reflected by an interface between the first reflective layer and the layer of the first optically diffractive component, without totally reflecting the second color of light having the second incident angle. The first reflective layer can be any suitable layer between the first and second diffractive structures. For example, the first reflective layer can be the carrier film of the first diffractive component. 
     A second reflective layer is arranged behind the second diffractive structures ( 1458 ). The second reflective layer can be the reflective layer  1105  of  FIG.  11   , or 1205 of  FIGS.  12 A,  12 B, or  12 C . The second reflective layer is configured to totally reflect the second color of light incident at the second incident angle, such that the second color of light undiffracted (or transmitted) at the zero order can be reflected back into layers before the second reflective layer without propagating to the display behind the optically diffractive device. 
     An optical absorber can be formed on a side surface of the optically diffractive device. The optical absorber can be the optical absorber  1104  of  FIG.  11   ,  1204  of  FIGS.  12 A,  12 C , or  1254  of  FIG.  12 B . The optical absorber is configured to absorb the totally reflected light of the first and second colors. 
     In some implementations, a third optically diffractive component including a third diffractive structure is formed. The third diffractive component can be the third diffractive component  1230  of  FIGS.  12 A,  12 B, or  12 C . The third diffractive structure can be the third diffractive structure  1232  of  FIGS.  12 A,  12 B, or  12 C . The third diffractive structure is configured to diffract a third color of light incident at a third incident angle into first order at a third diffracted angle and zero order at the third incident angle. A power of the third color of light at the first order can be substantially higher than the power of the third color of light at zero order. The first, second, and third diffracted angle can be substantially identical to each other. The third incident angle can be different from the first and second incident angles. Each of the first and second reflective layers can be configured to transmit the third color of light having the third incident angle. The second reflective layer can be arranged between the second and third diffractive structures. The third diffractive structure can be fabricated in a way similar to the first diffractive structure in step  1452 . The first, second, and third diffractive structures can be independently fabricated. The third diffractive component can also include a carrier film and a diffraction substrate. 
     A third reflective layer can be arranged behind the third diffractive structure. The third reflective layer can be the third reflective layer  1207  of  FIGS.  12 A,  12 B, or  12 C . The third reflective layer is configured to totally reflect the third color of light having the third incident angle, such that the third color of light undiffracted (or transmitted) at zero order is reflected back to layers before the third reflective layer and can be absorbed by the optical absorber coated on the side surface of the optically diffractive device. 
     In some implementations, the first reflective layer includes a first carrier film of the first optically diffractive component. A second diffraction substrate of the second diffractive component is attached to the first carrier film of the first diffractive component by a first intermediate layer, e.g., an OCA layer. A second carrier film of the second diffractive component is attached to a third carrier film of the third optically diffractive component by a second intermediate layer, and the second reflective layer can include the second intermediate layer. The third reflective layer can be attached to a third diffraction substrate of the third diffractive component. 
     The process  1450  can include arranging the first diffractive component on a substrate that is before the first diffractive component. The substrate can be the field grating substrate  1102  of  FIG.  11   ,  1202  of  FIG.  12 A ,  1252  of  FIG.  12 B , or  1272  of  FIG.  12 C . The substrate can include a front surface and a back surface. A front surface of the first diffractive component can be attached to the back surface of the substrate through a refractive index matching material or an OCA layer. 
     In some examples, the substrate includes a side surface angled to the back surface of the substrate, and the substrate is configured to receive a plurality of different colors of light at the side surface. The substrate can be configured such that the plurality of different colors of light are incident on the side surface with an incident angle substantially identical to 0 degrees and incident on the back surface at respective replay reference angles. 
     Implementations of the present disclosure can provide a method of fabricating a device including an optically diffractive device and a display. The display can be the display  594  of  FIG.  5 H ,  594 A of  FIG.  5 I ,  594 B of  FIG.  5 J , or  594  of  FIG.  5 K . The optically diffractive device can be the optically diffractive device  598  of  FIG.  5 H ,  598 A of  FIG.  5 I ,  598 B of  FIG.  5 J , or  598 C of  FIG.  5 K  the optically diffractive device  900  of  FIGS.  9 A and  9 B , the optically diffractive device  1000  of  FIGS.  10 A and  10 B , the optically diffractive device  1100  of  FIG.  11   , or the optically diffractive device  1200  of  FIG.  12 A ,  1250  of  FIG.  12 B , or  1270  of  FIG.  12 C . 
     The method can include forming the optically diffractive device according to the process  1400  of  FIG.  14 A  or the process  1450  of  FIG.  14 B . In some implementations, the optically diffractive device can include one or more color-selective polarizers and one or more reflective layers for a plurality of different colors of light. The optically diffractive device can be fabricated according to a combination of the process  1400  and the process  1450 . 
     The method can further include arranging the optically diffractive device and the display, such that the optically diffractive device is configured to diffract the plurality of different colors of light to the display. 
     In some implementations, the optically diffractive device and the display can be arranged such that a back surface of the optical device is spaced from a front surface of the display by a gap, e.g., an air gap. The method can further include forming an anti-reflection coating on at least one of the front surface of the display or the back surface of the optically diffractive device. 
     In some implementations, the optically diffractive device and the display are arranged by attaching the back surface of the optically diffractive device on the front surface of the display through an intermediate layer. The intermediate layer can be configured to have a refractive index lower than a refractive index of a layer of the optically diffractive device, such that each of the plurality of different colors of light diffracted at zero order by the optically diffractive device is totally reflected at an interface between the intermediate layer and the layer of the optically diffractive device. 
     The optically diffractive device is configured to diffract the plurality of different colors of light at respective diffracted angles that are substantially identical to each other. Each of the respective diffracted angles can be in a range of -10 degrees to 10 degrees, e.g., -7 degrees to 7 degrees, 0 degrees, or 6 degrees. The display can be configured to re-diffract the diffracted colors of light back through the optically diffractive device. An area of the optically diffractive device can cover an area of the display. The optically diffractive device can include a substrate in front of the optical device that can be configured to receive the plurality of different colors of light at a side surface of the substrate that is angled to a back surface of the substrate. 
     Implementations of the present disclosure can provide a method of operating an optically diffractive device. The optically diffractive device can be the optically diffractive device  598  of  FIG.  5 H ,  598 A of  FIG.  5 I ,  598 B of  FIG.  5 J , or  598 C of  FIG.  5 K , the optically diffractive device  900  of  FIGS.  9 A and  9 B , the optically diffractive device  1000  of  FIGS.  10 A and  10 B , the optically diffractive device  1100  of  FIG.  11   , or the optically diffractive device  1200  of  FIG.  12 A ,  1250  of  FIG.  12 B , or  1270  of  FIG.  12 C . The optically diffractive device can be operated to convert an incoming beam including a plurality of different colors of light to individually diffracted colors of light, 
     Implementations of the present disclosure can provide a method of operating a system including an optically diffractive device and a display. The optically diffractive device can be the optically diffractive device  598  of  FIG.  5 H ,  598 A of  FIG.  5 I ,  598 B of  FIG.  5 J , or  598 C of  FIG.  5 K , the optically diffractive device  900  of  FIGS.  9 A and  9 B , the optically diffractive device  1000  of  FIGS.  10 A and  10 B , the optically diffractive device  1100  of  FIG.  11   , or the optically diffractive device  1200  of  FIG.  12 A ,  1250  of  FIG.  12 B , or  1270  of  FIG.  12 C . The display includes a plurality of display elements. The display can be the display  594  of  FIG.  5 H ,  594 A of  FIG.  5 I ,  594 B of  FIG.  5 J , or  594 C of  FIG.  5 K . The method can be performed by a controller, e.g., the controller  112  of  FIG.  1 A  or  592  of  FIG.  5 H . 
     The method can include: transmitting at least one timing control signal to an illuminator to activate the illuminator to emit a plurality of different colors of light onto the optically diffractive device, such that the optically diffractive device converts the plurality of different colors of light to individually diffracted colors of light to illuminate the display and transmitting, for each of the plurality of display elements of the display, at least one respective control signal to modulate the display element, such that the individually diffracted colors of light are reflected by the modulated display elements to form a multi-color three-dimensional light field corresponding to the respective control signals. 
     In some implementations, the method can further include: obtaining graphic data comprising respective primitive data for a plurality of primitives corresponding to an object in a three-dimensional space, determining, for each of the plurality of primitives, an electromagnetic (EM) field contribution to each of the plurality of display elements of the display by calculating, in a three-dimensional coordinate system, an EM field propagation from the primitive to the display element, generating, for each of the plurality of display elements, a sum of the EM field contributions from the plurality of primitives to the display element, and generating, for each of the plurality of display elements, the respective control signal based on the sum of the EM field contributions to the display element for modulation of at least one property of the display element. The multi-color three-dimensional light field corresponds to the object. 
     In some implementations, the method include: sequentially modulating the display with information associated with the plurality of different colors in a series of time periods, and controlling the illuminator to sequentially emit each of the plurality of different colors of light to the optical device during a respective time period of the series of time periods, such that each of the plurality of different colors of light is diffracted by the optical device to the display and reflected by the modulated display elements of the display to form a respective color three-dimensional light field corresponding to the object during the respective time period. 
     The plurality of different colors of light can be diffracted by the optical device at a substantially same diffracted angle to the display. The diffracted angle can be within a range from 0 degrees to 10 degrees. 
     The illuminator and the optically diffractive device can be configured such that the plurality of different colors of light are incident on the first optically diffractive component of the optically diffractive device with respective incident angles. Each of the respective incident angles is in a range from 70 degrees to 90 degrees. In some cases, the respective incident angles are different from each other. In some cases, the respective incident angles are substantially identical to each other. 
     An optically diffractive device can include a plurality of diffractive gratings for a plurality of different colors. The gratings can include a transmissive grating, a reflective grating, or a combination thereof. For example, each of the optically diffractive devices shown in  FIGS.  9 A to  12 C  includes corresponding transmissive gratings for different colors. In some implementations, an optically diffractive device can include a combination of transmissive gratings and reflective gratings that can be configured for different colors. The optically diffractive device can be configured to diffract an incoming light towards a same direction, or back to an opposite direction. 
       FIG.  15    illustrates an example optical device  1500 , including a combination of transmissive and reflective diffractive gratings for two respective colors and corresponding reflective layers, for individually diffracting the two colors of light. The optical device  1500  can include a first diffractive component  1510  having a first diffractive grating  1512  for blue color and a second diffractive component  1520  having a second diffractive grating  1522  for green color. Each of the first and second diffractive gratings  1512 ,  1522  can be a holographic grating, e.g., a Bragg grating or a volume grating. However, the first diffractive grating  1512  for the blue color is configured to be a transmissive grating that diffracts light of blue color forward with respect to the light of blue color incident on the grating  1512 , while the second diffractive grating  1522  for the green color is configured to be a reflective grating that reflects light of green color backward with respect to the light of green color incident on the grating  1522 . Each of the first and second diffractive gratings  1512  and  1522  can be independently recorded and fixed in a recording medium, e.g., a photosensitive material such as a photopolymer. 
     The first diffractive component  1510  and the second diffractive component  1520  can be stacked together on a field grating substrate  1502  along a direction, e.g., the Z direction. The field grating substrate  1502  can be an optically transparent substrate, e.g., a glass substrate. The optically diffractive device  1500  can be in front of a display such as LCOS, e.g., the display  594  of  FIG.  5 H ,  594 A of  FIG.  5 I ,  594 B of  FIG.  5 J , or  594 C of  FIG.  5 K . For example, the optically diffractive device  1500  can be arranged on a cover glass  1530  of the display through an intermediately layer or spaced by a gap, e.g., an air gap. 
     Similar to the first and second diffractive components  1110 ,  1120  in  FIG.  11   , each of the first and second diffractive components  1510  and  1520  can include a respective substrate  1514 ,  1524  and a respective carrier film  1516 ,  1526  on opposite sides of the respective diffractive grating  1512 ,  1522 . The respective diffractive grating  1512 ,  1522  is between the respective substrate  1514 ,  1524  and the respective carrier film  1516 ,  1526 . The respective substrate  1514 ,  1524  can be a glass substrate that can have a refractive index same as or close to the refractive index of the field grating substrate  1502 . The respective carrier film  1516 ,  1526  can be a TAC film. The TAC film can have a lower refractive index than a photosensitive polymer used to record diffractive gratings  1512  and  1522 . Adjacent layers or components in the optically diffractive device  1500  can be attached together using one or more intermediate layers of OCA, UV-cured or heat-cured optical glues, optical contacting, or index matching fluid. For example, the first diffractive component  1510  (e.g., the substrate  1514 ) can be attached to the field grating substrate  1502  through an intermediate layer  1501 , e.g., an OCA layer. The first and second diffractive components  1510  and  1520 , e.g., the carrier film  1516  and the substrate  1524 , can be attached together through another intermediate layer  1503 , e.g., an OCA layer. The optically diffractive device  1500  (e.g., the carrier film  1526 ) can be attached to the cover glass  1530  of the display through an intermediate layer  1505 , e.g., an OCA layer. 
     As shown in  FIG.  15   , the first diffractive grating  1512  is configured to diffract a blue color of light incident at a first incident angle θ b , e.g., 78.4°, into first order at a respective diffracted angle, e.g., normal to the display, and zero order at the respective incident angle, and transmit a green color of light at a different incident angle, e.g., due to Bragg selectivity. Thus, there can be no crosstalk between the different colors of light individually diffracted at corresponding diffractive gratings. Each color of light can be polarized. The polarization state of the different colors of light diffracted at first order can be the same, e.g., s or p. 
     The optically diffractive device  1500  can include a first reflective layer (or blocking layer) between the first grating  1512  and the second grating  1522 . The first grating  1512  is configured to diffract the blue color of light incident at the first incident angle θ b , e.g., 78.4°, into first order at a diffracted angle, e.g., 0° and zero order at the first incident angle. The first reflective layer, e.g., a refractive index of the first reflective layer, is configured to totally reflect the blue color of light diffracted at the first incident angle but to transmit the green color of light incident at a second incident angle. For example, the refractive index of the first reflective layer is lower than the refractive index of a layer immediately before the first reflective layer, e.g., the first grating  1512 . The first reflective layer can be a suitable layer between the first grating  1512  and the second grating  1522 . In some examples, the first reflective layer is the carrier film  1516 , as shown in  FIG.  15   . 
     The optically diffractive device  1500  can include a second reflective layer after the second grating  1512  and before the display cover glass  1530 . The second reflective layer can be the intermediate layer  1505  and be configured to reflect, e.g., totally, the green color of light back to the second grating  1512 . The second grating  1512  is then configured to diffract the green color of light incident at the second incident angle θ g , e.g., 76.5°, into first order at a diffracted angle, e.g., 0°, back towards the display and zero order at the second incident angle back into the optically diffractive device  1500 . 
     The totally reflected blue color of light by the reflective layer  1516  and the zero order transmitted green color of light are back into the optically diffractive device  1500  to a side of the optically diffractive device  1500 . As illustrated in  FIG.  15   , a surface of the side can be coated with an optical absorber  1504 , e.g., a black coating, to absorb the blue and green colors of light at zero order by the corresponding transmissive and reflective diffractive gratings  1512  and  1522 . 
     Each of optically diffractive devices with color-selective polarizers (e.g., as illustrated in  FIGS.  9 A to  10 B ) and optically diffractive devices with reflective layers (e.g., as illustrated in  FIGS.  11  to  12 C and  15   ) can be considered as a one-dimensional beam expander. The one-dimensional beam expander can be configured to expand an input beam with a width and a height into an output beam with either the same width and a greater height or the same height and a greater width, e.g., by diffracting the input beam at one or more diffracted angles. 
     The techniques described herein can also be used to expand an input beam into an output beam which is both wider and higher than the input beam, e.g., with a two-dimensional beam expansion. The two-dimensional beam expansion can be achieved by using a two-dimensional beam expander (or a dual beam expander) having at least two one-dimensional beam expanders in series. For example, a first one-dimensional beam expander can be configured to expand an input beam in a first dimension, either width or height, producing an intermediate beam which is wider or higher than the input beam in the first dimension. A second one-dimensional beam expander can be configured to expand the intermediate beam in a second dimension, either height or width, to produce an output beam which is higher or wider than the intermediate beam in the second dimension. Thus, the output beam can be both wider and higher than the input beam in the first dimension and the second dimension. 
     In such a two-dimensional beam expander configuration, either one or both of the one-dimensional beam expanders can use the color-selective technique, and either one or both of the one-dimensional beam expanders can use the reflective layers technique. Each one-dimensional expander can use any of the detailed embodiments herein including reflective or refractive diffractive elements or a combination of reflective and refractive diffractive elements. The one-dimensional beam expanders can be positioned in a sequential order in any suitable arrangements or configurations. 
     In some implementations, the intermediate beam between two such one-dimensional expanders can be coupled from the first one-dimensional expander into the second one-dimensional expander using a free-space in-air geometry or through a monolithic or segmented substrate made, for example, of glass or acrylic, and embodying the geometry and functionality of the substrates of both expanders. This coupling can be achieved using one or more coupling elements between the two one-dimensional expanders. The coupling elements can include a mirror, mirrors, or a mirror and a beam-splitting dichroic component, or thin-film elements of further diffractive elements. The coupling elements can take collinear collimated output light of two or more colors from the first one-dimensional expander and convert the collinear collimated output light of the two or more colors to two or more independent collimated but not collinear intermediate beams, each for one of the colors, to satisfy the color-dependent angular input requirements, if any, of the second one-dimensional expander. Similarly, the first one-dimensional expander can have as its input either collinear collimated outputs of two or more light sources (e.g., laser diodes), each with a different color, or can have as its inputs two or more independent collimated but not collinear intermediate beams, each for one color from two or more light sources. 
     Display Zero Order Light Suppression 
     A display (e.g., LCoS) includes an array of display elements (e.g., pixels or phasels). There are gaps between the display elements on the display. The gaps occupy part of an area of the display, e.g., in a range from 5% to 10%. The gaps can be considered as dead gaps because display materials (e.g., liquid crystal) at these gaps are not controlled by an input control signal and thus no holographic information can be input into these gaps. In contrast, holographic information can be input into the display elements that are controlled (or modulated) to diffract light to reconstruct a holographic scene corresponding to the holographic information. 
       FIG.  16    illustrates an example  1600  of incident light  1620  incident on a display  1610 . The display  1610  can be the display  114  of  FIG.  1 A , the display  156  of  FIG.  1 B , the display  512  of  FIG.  5 A , the display  524  of  FIG.  5 B , the display  534  of  FIG.  5 C , the display  544  of  FIG.  5 D , the display  564  of  FIG.  5 E , the display  574  of  FIG.  5 F , the display  584  of  FIG.  5 G , the display  594  of  FIG.  5 H , the display  594 A of  FIG.  5 I , the display  594 B of  FIG.  5 J , the display  594 C of  FIG.  5 K , the display  600  of  FIG.  6 A , or the display  650  of  FIG.  6 B . Other display arrangements are also possible. 
     As an example, the display  1610  can be an LCoS made of liquid crystal. The display  1610  includes an array of display elements  1612  (e.g., the display element  160  of  FIG.  1 B ) that are spaced apart by gaps  1614 . Each display element  1612  can have a square (or rectangular or any other suitable) shape that has an element width  1613 , e.g., 5 µm. The display element  1612  can also be any other suitable shape, e.g., polygon. Adjacent display elements  1612  is separated by a gap  1614  with a gap size  1615 , e.g., less than 0.5 µm. 
     The incident light  1620  can be a collimated light beam that can have a beam size larger than an entire area of the display  1610 , such that the incident light  1620  can illuminate the entire area of the display  1610 . When the incident light  1620  is incident on the display  1610  at an incident angle θ i , a first portion of the incident light  1620  (e.g., 90% to 95% of the light  1620 ) illuminates the display elements  1612  and a second portion of the incident light  1620  (e.g., 5% to 10% of the light  1620 ) illuminates the gaps  1614 . When the display elements  1612  are modulated with holographic information (e.g., a hologram corresponding to holographic data), e.g., by voltages, the first portion of the incident light  1620  can be diffracted by the modulated display elements 1612 at first order with a diffraction angle θ d  to become diffracted first order light  1622 . 
     The diffracted first order light  1622  forms a holographic light field that can be a reconstruction cone (or frustum)  1630  with a viewing angle θ a . The viewing angle θ a  is dependent on one or more characteristics of the display  1610  (e.g., the element pitch  1613 ) and one or more wavelengths of the incident light  1620 . In some examples, a half of the viewing angle θ a  is within a range from 3° to 10°, e.g., 5°. For example, for the pitch d = 3.7 µm, the viewing angle θ a  is about 7° in air for blue color of light (λ = 460 nm) and about 10° in air for red color of light at (λ, = 640 nm). Light with a larger wavelength corresponds to a larger viewing angle. 
     As the gaps  1614  of the display  1610  are not modulated by any holographic information, the display  1610  at the gaps  1614  acts like a reflective mirror. When the second portion of the incident light  1620  is incident on the gaps  1614 , the second portion of the incident light  1620  can be reflected at the gaps  1614  with a reflected angle θ r  that has an absolute value identical to that of the incident angle θ i . In the present disclosure herein, “A is identical to B” indicates that an absolute value of A is identical to that of B, and A’s direction can be either the same or different from B’s direction. The reflected second portion of the incident light  1620  can be considered as at least a part of display zero order light  1624 . If the incident angle θ 1  is less than the half of the apex angle θ a , e.g., θi = 0°, the display zero order light  1624  may undesirably appear in the reconstruction cone, which can affect an effect of the holographic scene. 
     The display zero order light can also include any other unwanted light from the display, e.g., diffracted light at the gaps, reflected light from the display elements, or reflected light from a display cover on the display. Higher orders of the display zero order light  1624  can include the diffracted light at the gaps. In some implementations, the display  1610  is configured to suppress the higher orders of the display zero order light, e.g., by including irregular or non-uniform display elements that have different sizes. The display elements can have no periodicity, and can form a Voronoi pattern, e.g., as illustrated in  FIG.  6 A . 
     In the present disclosure herein, for illustration purposes only, reflected second portion of the incident light is considered as a representative of display zero order light. 
       FIGS.  17 A- 17 B  illustrate examples  1700 ,  1750  of display zero order light within a holographic scene displayed on a projection screen ( FIG.  17 A ) and on a viewer’s eye ( FIG.  17 B ). Collimated input light  1720  is coupled by an optical device  1710  to illuminate the display  1610  at normal incidence, i.e., θ i  = 0°. The optical device  1710  can be a waveguide, a beam splitter, or an optically diffractive device. For illustration, the optical device  1710  is an optically diffractive device, e.g., the device  598  of  FIG.  5 H , that includes a grating  1714  formed on a substrate  1712 . However, as noted above, reflective optical devices may be used. 
     A first portion of the input light  1720  is incident on the display elements  1612  of the display  1610  that are modulated with holographic information, and is diffracted by the display elements  1612  to become diffracted first order light  1722 . A second portion of the input light  1720  is incident on the gaps  1614  of the display  1610 , and is reflected at the gaps  1614  to become at least a part of display zero order light  1724 . The diffracted first order light  1722  propagates in space to form a reconstruction cone with a viewing angle, e.g., 10°. As the incident angle, e.g., 0°, is less than a half of the viewing angle, e.g., 5°, the display zero order light  1724  propagating with a reflected angle identical to the incident angle, e.g., 0°, is within the reconstruction cone. 
     As illustrated in  FIG.  17 A , the diffracted first order light  1722  forms a three-dimensional holographic scene, a two-dimensional cross-section  1732  of which may be observed on a two-dimensional (2D) projection screen  1730  that is spaced away from the display  1610  along a direction perpendicular to the display  1610 . The display zero order light  1724  appears to be collimated zero order light  1734  as an undesired image (e.g., having a rectangular shape) within the holographic scene  1732 . As illustrated in  FIG.  17 B , the diffracted first order light  1722  forms a holographic scene  1762  on an eye of a viewer  1760 . The display zero order light  1724  is focused by a lens of the eye of the viewer  1760  and appears to be focused zero order light  1764  as an undesired spot within the holographic scene  1762 . 
     To improve an effect of a reconstructed holographic scene and thus a performance of a display system, it is desirable to suppress (or even eliminate) display zero order light in the reconstructed holographic scene. Implementations of the present disclosure provide multiple techniques, e.g., five techniques as described below, to suppress (or even eliminate) the display zero order light in the reconstructed holographic scene. The techniques can be applied individually or in a combination thereof. 
     The display zero order light can be suppressed in the reconstructed holographic scene with a light suppression efficiency. The light suppression efficiency is defined as one minus a ratio between an amount of the display zero order light in the holographic scene with the suppression using the technique described herein and an amount of display zero order light in the holographic scene without suppression. In some examples, the light suppression efficiency is more than a predetermined percentage, e.g., 50%, 60%, 70%, 80%, 90%, or 99%. In some examples, the light suppression efficiency is 100%. That is, all the display zero order light is eliminated in the holographic scene. 
     In a first technique referred to as “phase calibration,” phases of display elements of a display can be adjusted to have a predetermined phase range, e.g., [0, 2π]. In such a way, a signal to noise ratio (S/N) between a holographic scene formed based on the calibrated phases and display zero order light can be increased. 
     In a second technique referred to as “zero order beam divergence,” as illustrated in  FIG.  18   , a display zero order light beam is diverged by an optically defocusing device (e.g., a concave lens) to have a lower power density. In contrast, a hologram is preconfigured, such that collimated light beam incident on display elements modulated by the hologram is diffracted to become a converged light beam. The converged light beam is re-focused by the optically defocusing device to form a holographic scene with a higher power density. Thus, the display zero order light beam is diluted or suppressed in the holographic scene. 
     In a third technique referred to as “zero order light deviation,” as illustrated in  FIGS.  19 A- 19 C,  20 A- 20 B,  21 , and  22   , display zero order light is deviated away from a holographic scene. An optical device is configured to couple input light to illuminate a display at an incident angle larger than a half of a viewing angle of a reconstructed cone that forms the holographic scene. The display zero order light propagates away from the display at a reflected angle identical to the incident angle. A hologram corresponding to the holographic scene is preconfigured such that diffracted first order light propagates away from the display to form the reconstruction cone in a same way as that when the incident angle is 0°. Thus, the display zero order light is deviated from the reconstruction cone and accordingly the holographic scene. 
     In a fourth technique referred to as “zero order light blocking,” as illustrated in  FIGS.  23 A- 23 B , display zero order light is first deviated away from diffracted first order light according to the third technique and then blocked (or absorbed) by an optically blocking component, e.g., a metamaterial layer or an anisotropic optical element such as a louver film. The optically blocking component is configured to transmit a light beam having an angle smaller than a predetermined angle and block a light beam having an angle larger than the predetermined angle. The predetermined angle can be smaller than the incident angle of the input light and larger than a half of the viewing angle of the reconstruction cone. 
     In a fifth technique referred to as “zero order light redirection,” as illustrated in  FIGS.  24  to  33   , display zero order light is first deviated away from diffracted first order light according to the third technique and then redirected even further away from the diffracted first order light by an optically diffractive component, e.g., a diffractive grating. When the input light includes different colors of light simultaneously or sequentially, as illustrated in  FIGS.  30 A- 30 B,  31 A- 31 B,  32 , and  33   , the optically diffractive component can include one or more corresponding diffractive gratings that are configured to diffract the different colors of light towards different directions in a plane or in space to reduce color crosstalk among the different colors of light. 
     The above five techniques are mainly used to suppress main reflected zero order of the whole display zero order light. In a sixth technique, the display is configured to suppress higher orders of the whole display zero order light, e.g., by using irregular or nonuniform display elements having different sizes or shapes or both. The display elements can have no periodicity, and can form a Voronoi pattern or be Voronoil patterned display elements. In some implementations, the display can be the display  600  of  FIG.  6 A  or the display  650  of  FIG.  6 B . 
     In the following, the first five techniques are described with more details. 
     First technique - Phase Calibration 
     Phase calibration is a technique that can increase a contrast in a display, e.g., by pulling a direct current (DC) term of a computed hologram out, which can be implemented by a software or program instructions. Phase calibration can achieve an accuracy beyond a device calibration that may be bad or unknown. 
     In some implementations, a hologram includes respective phases for display elements of a display. As described above, the respective phase can be a computed EM contribution from one or more corresponding objects to each display element. According to the phase calibration technique, the hologram is configured by adjusting (e.g., scaling and/or shifting) the respective phases for the display elements to have a predetermined phase range, e.g., [0, 2π], to get a higher contrast in the display. 
     The respective phases can be adjusted according to an expression: 
     
       
         
           
             
               ∅ 
               a 
             
             = 
             A 
             
               ∅ 
               i 
             
             + 
             B 
           
         
       
     
      where ø i  represents an initial phase value of a respective phase, ø a  represents an adjusted phase value of the respective phase, and A and B are constants for the respective phases, A being in [0, 1] and B being in [0, 2π]. In some examples, A is the same for all display elements. In some examples, B is the same for all display elements. In some examples, A is different for different display elements. In some examples, B is different for different display elements. 
     In a perfectly calibrated and linearized display system, a pair of values (1, 0) for (A, B) works best to give the best contrast by proving the highest diffraction efficiency for the input hologram. However, due to nonlinear LC curves and inaccurate calibration of the display, the respective phases for the display elements are typically not in a range of [0, 2π], and thus the display contrast is degraded. As the input light is the same, the display zero order light will be the same. If the diffraction efficiency of the hologram is increased, the display contrast can be higher and the S/N ratio of the holographic scene can be higher. 
     According to the phase calibration technique, the display contrast can be improved by scaling and shifting the respective phases in a phase coordinate system, such that the respective phases are adjusted to have a range, e.g., exactly [0, 2π]. In some cases, the range of the adjusted respective phases can be smaller or larger than the 2π range depending on the calibration and the maximum phase shift of the working LC. Therefore, for each display, there can be a pair of (A, B) that produces the highest diffraction efficiency resulting in the highest S/N ratio. 
     The respective phases for the display elements can be adjusted by adjusting the constants A and B such that a light suppression efficiency for the holographic scene is maximized. The light suppression efficiency can be larger than a predetermined percentage, e.g., 50%, 60%, 70%, 80%, 90%, or 99%. 
     In some implementations, the constants A and B are adjusted by a machine vision algorithm or a machine learning algorithm such as an artificial intelligence (AI) algorithm. In the machine vision algorithm, a hologram is designed to create pseudo-random points focused on a transmissive diffusing screen in a plane at a specific distance from the display. Then, the hologram is computed for each of three primary colors red, green, and blue (RGB) in a way that the RGB reconstructed points are aligned perfectly on that plane. Then the algorithm is set to find a pair of values (A, B) for each color so that a display contrast is at an acceptable level. At the beginning for a pair of values (A, B), e.g., [1, 0], a camera at the specific distance takes a picture of the pattern on the screen. In the taken picture, a brightness of all the points (X) is averaged, and also one small area (Y) on a background noise is measured. The ratio of X/Y is calculated and checked if it is larger than a specific value. If not, the pair of values (A, B) will be changed and the process is automatically repeated until an acceptable pair of values (A, B) is determined. 
     Second technique - Zero Order Beam Divergence 
       FIG.  18    illustrates an example system  1800  of suppressing display zero order light in a holographic scene displayed on a projection screen  1830  by diverging the display zero order light beam. A beam splitter  1810  is positioned in front of a display  1610  and couples a collimated input light beam  1820  to illuminate the display  1610  at normal incidence. A first portion of the light beam  1820  is diffracted by display elements modulated by a hologram to become a diffracted first order light beam  1822 , and a second portion of the light beam  1820  is reflected by gaps of the display  1610  to become a display zero order light beam  1824 . An optically diverging component, e.g., a concave lens  1802 , is arranged downstream the beam splitter  1810  and before the projection screen  1830 . In some examples, the optically diverging component includes a convex lens arranged at a position further away from the projection screen  1830  than the concave lens  1802  such that a collimated light beam is first focused and then diverged towards the projection screen  1830 . 
     When the display zero order light beam  1824  comes off the display  1610 , the display zero order light beam  1824  is collimated. Thus, when the display zero order light beam  1824  transmits through the concave lens  1802 , the display zero order light beam  1824  is diverged by the concave lens  1802 , as illustrated in  FIG.  18   . Thus, a power density of the diverged display zero order light beam  1824  is decreased or diluted over the diverged beam area, compared to that of the original collimated input light beam  1820 . 
     According to the second technique, the hologram (or respective phases) modulating display elements of the display  1610  can be preconfigured such that the diffracted first order light beam 1822 is converged when coming off the display  1610 . The degree of convergence is configured to correspond to a degree of divergence of the concave lens  1802 . That is, the divergence of the concave lens is compensated by the configured convergence. Thus, when the converged diffracted first order light beam  1822  transmits through the concave lens  1802 , the diffracted first order light beam  1822  is collimated to form a reconstructed holographic scene  1832  on a projection screen  1830 , which is the same as that without the pre-configuration of the hologram and the concave lens  1802 . Thus, the reconstructed holographic scene  1832  has a power density the same as that of the collimated input light beam  1820 . In contrast, a display zero order light beam  1834  is diverged and smeared (or diluted) across the projection screen  1830  with a decreased power density. The projection screen  1830  is spaced away from the display  1610  with a specified distance, e.g., 50 cm. The display zero order light beam  1834  can be dim and appear like a background noise in the holographic scene  1832 . In such a way, a light suppression efficiency can be increased, e.g., to more than 99%, and an S/N ratio of the holographic scene  1832  can be increased. 
     In some implementations, the hologram is preconfigured by adding corresponding phases to the respective phases for the display elements of the display  1610 . The respective phases for the display elements can be the respective phases adjusted according to the first technique -phase calibration. The corresponding phase for each of the display elements is expressed as: 
     
       
         
           
             ∅ 
             = 
             
               π 
               
                 λ 
                 f 
               
             
             
               
                 
                   
                     ax 
                   
                   2 
                 
                 + 
                 
                   
                     by 
                   
                   2 
                 
               
             
           
         
       
     
      where ø represents the corresponding phase for the display element, λ, represents a wavelength of the input light  1820 , ƒ represents a focal length of the optically diverging component (e.g., the concave lens  1802 ), x and y represent coordinates of the display element in a 2D display coordinate system, and a and b represent constants. A pair of values (a, b) can be adjusted based on applications, e.g., for introducing astigmatism for people whose eyes suffer from astigmatism. If a is identical to b, e.g., a=1 and b=1, a defocusing effect of the corresponding phase is circular; if a is different from b, e.g., a=1 and b=0.5, the defocusing effect is elliptical and can match a 2:1 anamorphic focusing lens. If either a=0 or b=0, but not both, the defocusing effect can produce a line focus rather than an area focus and can match a cylindrical focusing lens. 
     In some implementations, the hologram is preconfigured by adding a virtual lens for a configuration cone when designing (or simulating) the holographic scene in a 3D software application such as Unity, e.g., the application  106  of  FIG.  1 A . The configuration cone is described with further details in  FIGS.  20 A- 20 B . The diffracted first order light beam  1822  forms a reconstruction cone with a viewing angle, and the configuration cone corresponds to the reconstruction cone and has an apex angle identical to the viewing angle. In the simulation, the configuration cone can be moved with respect to the display in a global 3D coordinate system along a direction perpendicular to the display with a distance corresponding to a focal length of the optically diverging component. The configuration cone can be moved just once for all objects in the reconstruction cone. Holographic data, e.g., primitive lists for the objects, are then generated based on the moved configuration cone in the global 3D coordinate system. 
     Third technique - Zero Order Light Deviation 
     As described above in  FIGS.  16  and  17 A- 17 B , a reconstruction cone of a holographic scene (or holographic content) has a viewing angle depending on a display and a wavelength of an input light beam. If display zero order light can be deviated outside of the reconstruction cone, the holographic scene can be observed without the display zero order light. 
       FIG.  19 A  illustrates an example system  1900  of display zero order light in a holographic scene when a display  1610  is illuminated with collimated input light  1920  at normal incidence, i.e., θi = 0°. An optical device  1910  couples the collimated input light  1920  to illuminate the display  1610  at the normal incidence. In some implementations, as illustrated in  FIG.  19 A , the optical device  1910  is a waveguide device, e.g., the waveguide device  588  of  FIG.  5 G , that includes an incoupler  1916  and an outcoupler  1914  formed on a substrate  1912 . 
     A first portion of the input light  1920  is incident on display elements of the display 1610 that are modulated with a hologram, and is diffracted by the display elements to become diffracted first order light  1922 . A second portion of the input light  1920  is incident on gaps of the display  1610 , and is reflected at the gaps to become at least a part of display zero order light  1924 . The diffracted first order light  1922  propagates in space to form a reconstruction cone with a viewing angle, e.g., 10°. As the incident angle, e.g., 0°, is less than a half of the viewing angle, e.g., 5°, the display zero order light  1924  propagating with a reflected angle identical to the incident angle, e.g., 0°, is within the reconstruction cone. As illustrated in  FIG.  19 A , the diffracted first order light  1922  forms a holographic scene  1932  on a two-dimensional (2D) projection screen  1930 . The display zero order light  1924  appears to be collimated zero order light  1934  as an undesired image within the holographic scene  1932 . 
       FIG.  19 B  illustrates an example  1950  of suppressing display zero order light in a holographic scene displayed on the projection screen  1930  by directing (or deviating) display zero order light away from the holographic scene. Different from the optical device  1910 , an optical device  1960 , including incoupler  1966  and outcoupler  1964  formed on a substrate  1962 , is configured to couple the collimated input light  1920  to illuminate the display  1610  at an incident angle θ i  larger than 0°. Due to reflection, display zero order light  1974  comes off the display  1610  at a reflected angle θ r  identical to the incident angle θ 1 . 
     According to the third technique, a hologram (or respective phases) modulating display elements of the display  1610  can be preconfigured such that diffracted first order light  1972  comes off the display  1610  at normal incidence. That is, the deviation of the incident angle is compensated by the configured hologram. Thus, the diffracted first order light beam  1972  forms a reconstruction cone that appears as a reconstructed holographic scene  1976  on the projection screen  1930 , the same as when the incident angle is at normal incidence. When the incident angle, e.g., 6°, is larger than a half of the viewing angle of the reconstruction cone, e.g., 5°, the display zero order light  1974  can be deviated or shifted away from the reconstruction cone. Accordingly, as illustrated in  FIG.  19 B , a shifted display zero order image  1978  formed by the display zero order light  1974  can be outside of the holographic scene  1976  on the projection screen  1930 . Similarly, as illustrated in  FIG.  19 C , when seen by a viewer  1990 , a display zero order spot  1994  formed by the display zero order light  1974  can be outside of a holographic scene  1992  formed by the diffracted first order light  1972  on an eye of the viewer  1990 . By configuring a direction of the incident angle, the display zero order light can be deviated up or down or to a side in space. 
     In some implementations, the hologram is preconfigured by adding corresponding phases to the respective phases for the display elements of the display  1610 . The respective phases for the display elements can be the respective phases adjusted according to the first technique -phase calibration. The corresponding phase for each of the display elements is expressed as: 
     
       
         
           
             ∅ 
             = 
             
               
                 2 
                 π 
               
               λ 
             
             
               
                 xcos 
                 θ 
                 +ycos 
                 θ 
               
             
           
         
       
     
      where ø represents the corresponding phase for the display element, λ, represents a wavelength of the input light  1920 , x and y represent coordinates of the display element in a 2D display coordinate system (or in a 3D coordinate system), and θ represents an angle corresponding to the incident angle θi, e.g., θ=θ i . 
     In some implementations, the hologram is preconfigured by adding a virtual prism for a configuration cone when designing (or simulating) the holographic scene in a 3D software application such as Unity, e.g., the application  106  of  FIG.  1 A . 
       FIG.  20 A  illustrates an example  2000  of a configuration cone  2020  and a reconstruction cone  2030  with respect to a display  2002  and an optical device  2010  in a 3D coordinate system in the 3D software application. The optical device  2010  can be a lightguide device, e.g., the optically diffractive device  598  of  FIG.  5 H , that includes a grating  2014  formed on a substrate  2012 . 
     As illustrated in  FIG.  20 A , the optical device  2010  couples input light  2040  to illuminate the display  2002  with an incident angle larger than 0°, not at normal incidence, which is identical in effect to rotating the configuration cone  2020  (together with all objects including an object  2022  within the configuration cone  2020 ) with an angle corresponding to (e.g., identical to) a reflected angle of the incident angle with respect to the 3D coordinate system. In some implementations, the configuration cone  2020  is rotated in the original 3D coordinate system. In some implementations, the original 3D coordinate system is rotated but the configuration cone  2020  is not rotated. Once the configuration cone  2020  in the 3D coordinate system is set, objects can be placed in the configuration cone  2020  without changing primitives’ vertices individually. Accordingly, the simulated reconstruction cone  2030  (with all reconstructed objects including a reconstructed object  2032 ) and display zero order light  2042  are rotated with respect to the display  2002  with the same reflected angle with respect to the 3D coordinate system. That is, the display zero order light  2042  can appear in a holographic scene when seen by a viewer. 
       FIG.  20 B  illustrates an example  2050  of adjusting the configuration cone  2020  of  FIG.  20 A  to configure a hologram corresponding to the holographic scene in the 3D coordinate system in the 3D software application. The configuration cone  2020  (together with the designed objects including the object  2022 ) can be rotated with a rotation angle with respect to a surface of the display  2002  in the 3D coordinate system. The rotation angle is corresponding to (e.g., identical to) the incident angle so that an adjusted configuration cone  2060  (with the adjusted designed objects including the adjusted object  2062 ) is at normal incidence to the display  2002 . The configuration cone  2020  can be adjusted just once for all the designed objects. Holographic data, e.g., primitive lists for the objects, are then generated based on the adjusted configuration cone  2060  in the global 3D coordinate system. The hologram is then generated based on the holographic data. 
     Accordingly, when the optical device  2010  couples the input light  2040  to illuminate the display  2002  at the incident angle, a first portion of the input light  2040  is diffracted by the display elements modulated with the preconfigured hologram. The diffracted first order light forms a reconstruction cone  2070  (with reconstructed objects including the reconstructed object  2072  of the designed object  2062 ) normal to the display  2002 . The reconstruction cone  2070  has a viewing angle θ v . In contrast, a second portion of the input light  2040  is reflected at the gaps without the modulation of the preconfigured hologram to become display zero order light  2042  that comes off the display at a reflected angle θ r  identical to the incident angle θ 1 . Thus, when the incident angle θ 1  is larger than a half of the viewing angle, i.e., θ i  &gt; θ v /2, the display zero order light  2042  is outside the reconstruction cone  2070  and accordingly the holographic scene when seen by a viewer. 
     The input light  2040  can be coupled into the optical device  2010  in any suitable way, e.g., by an incoupler such as the incoupler  1966  of  FIG.  19 B , by a prism as illustrated in  FIG.  21   , or a wedged substrate as illustrated in  FIG.  22   . 
       FIG.  21    illustrates an example  2100  of coupling collimated input light  2120  via a coupling prism  2111  to an optical device  2110  to illuminate a display  1610  at an incident angle for suppressing display zero order light in a holographic scene. The optical device  2110  includes a grating  2114  on a substrate  2112 . The coupling prism  2111  couples the input light  2120  into the substrate  2112  that guides the input light  2120  towards the grating  2114 . The grating  2114  diffracts the input light  2120  out towards the display  1610  at the incident angle. A hologram is preconfigured such that diffractive first order light  2122  comes off the display  1610  surrounding normal incidence to form a reconstruction cone, while display zero order light  2124  comes off the display  1610  at a reflected angle identical to the incident angle. When the incident angle is larger than a half of a viewing angle of the reconstruction cone, the display zero order light  2124  forms a shifted zero order spot  2134  outside of a holographic scene  2132  when seen by a viewer  2130 . 
       FIG.  22    illustrates an example system  2200  of coupling light via a wedged substrate  2212  of an optical device  2210  to illuminate a display  1610  at an incident angle for suppressing display zero order light in a holographic scene. The optical device  2210  includes a grating  2214  on the wedged substrate  2212 . The wedged substrate  2212  couples the input light  1020  into the substrate  2212  that guides the input light  2120  towards the grating  2214 . The grating  2214  diffracts the input light  2120  out towards the display  1610  at the incident angle. A hologram is preconfigured such that diffractive first order light  2222  comes off the display  1610  surrounding normal incidence to form a reconstruction cone, while display zero order light  2224  comes off the display  1610  at a reflected angle identical to the incident angle. When the incident angle is larger than a half of a viewing angle of the reconstruction cone, the display zero order light  2224  forms a shifted zero order spot  2234  outside of a holographic scene  2232  when seen by a viewer  2230 . 
     According to the third technique, the display zero order light coming off the display has a larger deviation angle than the diffracted first order light coming off the display. Thus, the display zero order light can be suppressed (or eliminated) in the holographic scene based on the angle difference, e.g., as described further in the fourth technique “zero order light blocking” and the fifth technique “zero order light redirection.” 
     Fourth technique - Zero Order Light Blocking 
       FIGS.  23 A- 23 B  illustrate example systems  2300 ,  2350  of suppressing display zero order light in a holographic scene by blocking or absorbing the display zero order light reflected from the display by an optically blocking component. The optically blocking component can be any suitable structure, e.g., an artificial structure such as a louvered layer, a metamaterial layer, a metamaterial structure, a metasurface, or any other kind of engineered microstructure or nanostructure that can exhibit the blocking property. 
     For illustration, similar to  FIG.  21   , a coupling prism  2311  couples a collimated input light  2320  into an optical device  2310  having a grating  2314  formed on a substrate  2312 . The grating  2314  is configured to diffract the input light  2320  out to illuminate a display  1610  at an incident angle, e.g., larger than a half of a viewing angle of a reconstruction cone. By applying the third technique, a hologram is preconfigured such that diffracted first order light  2322  comes off the display  1610  in a same way as that when the input light is incident on the display at normal incidence, while display zero order light  2324  propagates away from the display  1610  at a reflected angle identical to the incident angle. 
     A metamaterial layer  2316 , as an example of the optically blocking component, is formed on (e.g., deposited upon, or attached to) the substrate  2312 . As illustrated in  FIGS.  23 A- 23 B , the metamaterial layer  2316  and the grating  2314  can be formed on opposite sides of the substrate  2312 . The metamaterial layer  2316  can be made of an array of microstructures or nanostructures smaller than a wavelength of interest. By configuring a geometry of the microstructures or nanostructures individually and collectively, the metamaterial layer  2316  can be designed to interact with light in a desire manner. In the present disclosure, the metamaterial layer  2316  is configured to transmit a light beam having an angle smaller than a predetermined angle and block a light beam having an angle larger than the predetermined angle. The predetermined angle can be set to be smaller than the incident angle and larger than the half of the viewing angle of the reconstruction cone formed by the diffracted first order light  2322 . Thus, the diffracted first order light  2322  can be transmitted through the metamaterial layer  2316  with a transmission efficiency, e.g., no less than a predetermined ratio such as 50%, 60%, 70%, 80%, 90%, or 99%. In contrast, the display zero order light can be blocked or absorbed by the metamaterial layer  2316 , e.g., with a blocking efficiency of 100%. 
     A light suppression efficiency of the display zero order light in a holographic scene can be 100%. As illustrated in  FIG.  23 A , the diffracted first order light  2322  can form a holographic scene  2332  on a projection screen  2330 , without the display zero order light  2324 . As illustrated in  FIG.  23 B , when seen by a viewer  2360 , the diffracted first order light  2322  can form a holographic scene  2362  on an eye of the viewer  2360 , without the display zero order light  2324 . 
     Fifth technique - Zero Order Light Redirection 
       FIG.  24    illustrates a system  2400  of suppressing display zero order light in a holographic scene by redirecting the display zero order light away from the holographic scene via an optically redirecting structure. The optically redirecting structure can be a grating, e.g., a holographic grating such as a Bragg grating, or any other suitable redirecting structure. 
     Similar to the system  590  of  FIG.  5 H , the system  2400  includes a computer  2401  (e.g., the computer  591  of  FIG.  5 H ), a controller  2402  (e.g., the controller  592  of  FIG.  5 H ), a reflective display  2404  (e.g., the reflective display  594  of  FIG.  5 H ), and an illuminator  2406  (e.g., the illuminator  596  of  FIG.  5 H ). The system  2400  also includes an optical device  2410  that can include an optically diffractive device, e.g., the optically diffractive device  598  of  FIG.  5 H ,  598 A of  FIG.  5 I ,  598 B of  FIG.  5 J , or  598 C of  FIG.  5 K , the optically diffractive device  900  of  FIGS.  9 A and  9 B , or  1270 ,  1000  of  FIGS.  10 A and  10 B ,  1100  of  FIG.  11   ,  1200  of  FIG.  12 A ,  1250  of  FIG.  12 B  of  FIG.  12 C , or  1500  of  FIG.  15   . In some implementations, as illustrated in  FIG.  24   , the optical device  2410  includes a transmissive field grating structure  2414  as the optically diffractive device on a substrate  2412  (e.g., the substrate  598 - 2  of  FIG.  5 H ). The transmissive field grating structure  2414  can be the field grating structure  598 - 1  of  FIG.  5 H . The transmissive field grating structure  2414  can include one or more gratings for one or more different colors of light. The substrate  2412  can be a transparent glass substrate. 
     Similar to what is described above, the optical device  2410  can be arranged adjacent to a front surface of the display  2404 . In some implementations, a top surface of the optical device  2410  (e.g., a surface of the field grating structure  2414 ) is attached to the front surface of the display  2404 , e.g., through an index matching material. In some implementations, an air gap is between the top surfaces of the optical device  2410  and the display  2404 . In some implementations, a spacer, e.g., glass, is inserted in the air gap between the top surfaces of the optical device  2410  and the display  2404 . To better illustrate light propagation, the air gap is used as an example in  FIG.  24    and the following  FIGS.  26 A to  33   . 
     The controller  2402  is configured to receive graphic data corresponding to one or more objects from the computer  591  (e.g., by using a 3D software application such as Unity), perform computation on the graphic data, and generate and transmit control signals for modulation to the display  2404  through a memory buffer  2403 . The controller  2402  is also coupled to the illuminator  2406  and configured to provide a timing signal  2405  to activate the illuminator  2406  to provide input light  2420 . The input light  2420  is then diffracted by the transmissive field grating  2414  of the optical device  2410  to illuminate the display  2404 . A first portion of the input light  2420  incident on display elements of the display  2404  is diffracted by the display  2404 , and diffracted first order light  2421  forms a holographic light field  2422  towards a viewer. The holographic light field  2422  can correspond to a reconstruction cone (or frustum) that has a viewing angle. The display  2404  can include a back mirror on a back of the display  2404  and can reflect light towards the viewer. A second portion of the input light  2420  incident on gaps of the display  2404  is reflected by the display  2404 , e.g., by the back mirror, to become display zero order light  2424 . 
     As described above, the transmissive field grating  2414  can be configured to diffract the input light  2420  from the illuminator  2406  out to illuminate the display  2404  off axis at an incident angle, e.g., larger than a half of a viewing angle of the reconstruction cone (or frustum). By applying the third technique, the diffracted first order light  2421  comes off the display  2404  in the same manner as that when the input light  2420  is incident on axis at normal incidence, while the display zero order light  2424  comes off at a reflected angle that is identical to the incident angle, which is outside of the reconstruction cone. 
     As illustrated in  FIG.  24   , the system  2400  can include an optically redirecting structure  2416  configured to diffract a first light beam having an angle identical to a predetermined angle with a substantially larger diffraction efficiency at a diffraction angle than a second light beam having an angle different from the predetermined angle. The optically redirecting structure  2416  can be a holographic grating such as a Bragg grating. The diffraction angle can be substantially larger than the predetermined angle. In some implementations, the optically redirecting structure  2416  includes one or more gratings for one or more different colors of light, as illustrated further in  FIGS.  30 A- 33   . In some implementations, the optically redirecting structure  2416  is arranged downstream the optical device  2410  away from the display  2404 . In some implementations, as illustrated in  FIG.  24   , the optically redirecting structure  2416  is formed on a side of the substrate 2412 that is opposite to the transmissive field grating structure  2414 . 
     According to the fifth technique, the optically redirecting structure  2416  can be configured to have the predetermined angle identical to the reflected angle of the display zero order light  2424  or the incident angle of the input light  2420  at the display  2404 . As the display zero order light  2424  propagates at the reflected angle, the optically redirecting structure  2416  can diffract the display zero order light  2424  with a substantially larger diffraction efficiency at the diffraction angle than the diffracted first order light  2421 , while the diffracted first order light  2421  can transmit through the optically redirecting structure  2416  to form the holographic light field  2422 . In such a way, the optically redirecting structure  2416  can redirect the display zero order light  2424  further away from the holographic light field  2422 . 
       FIGS.  25 A- 25 C  illustrate examples of redirecting display zero order light via zero order redirection gratings  2500 ,  2530 ,  2550  in  FIGS.  25 A,  25 B,  25 C  to different directions in space. The zero order redirection grating  2500 ,  2530 ,  2550  can be in the optically redirection grating structure  2416  of  FIG.  24   . The redirection gratings  2500 ,  2530 ,  2550  can be fabricated according to the method illustrated in  FIG.  7 A . 
     For comparison, display zero order light  2502  is incident on the zero order redirection grating  2500 ,  2530 ,  2550  at an incident angle - 6.0°, which is a predetermined angle for the redirection grating  2500 ,  2530 ,  2550 . The redirection grating  2500 ,  2530 ,  2550  is configured to diffract the display zero order light  2502  with a high diffraction efficiency at a diffraction angle that is substantially larger than the incident angle of the display zero order light  2502 . The redirection gratings  2500 ,  2530 ,  2550  can be configured to diffract the display zero order light  2502  at different diffraction angles, for example, 60° for the grating  2500  shown in  FIG.  25 A , 56° for the grating  2530  shown in  FIG.  25 B , and -56° for the grating  2550  in  FIG.  25 C . 
       FIGS.  26 A- 26 E  illustrate examples of redirecting display zero order light when light is input at different incident angles via optically redirecting structures (e.g., zero order redirection gratings) to different directions in space. Each of the incident angles, e.g., -6° or 6° in air, is configured to be larger than a half of a viewing angle of a reconstruction cone corresponding to a holographic light field, e.g., 5° in air. 
     As illustrated in  FIG.  26 A , a system  2600  includes an optical device  2610  that can be the optical device  2410  of  FIG.  24   . The optical device  2610  includes a substrate  2612  (e.g., the substrate  2412  of  FIG.  24   ), a transmissive field grating structure  2614  (e.g., the transmissive field grating structure  2414  of  FIG.  24   ), and a zero order redirection grating structure  2616  (e.g., the zero order redirection grating structure  2416  of  FIG.  24   ). The optical device  2610  can include a cover glass  2618  on the zero order redirection grating structure  2616 . 
     Input light  2620  from the illuminator  2406  is diffracted by the transmissive field grating structure  2614  to illuminate the display  2404  with an incident angle -6° (in air). A first portion of the input light  2620  illuminating on modulated display elements of the display  2404  is diffracted to transmit through the optical device  2610  (including the zero order redirection grating structure  2616 ) to become diffracted first order light  2621  that forms a holographic light field  2622 . A second portion of the input light  2620  illuminating on gaps of the display  2404  is reflected to come off the display  2404  as display zero order light  2624 . The display zero order light  2624  is redirected by the zero order redirection grating structure  2616  at a diffraction angle substantially larger than the incident angle, e.g., -28° in glass. Due to Fresnel reflection, part of the redirected display zero order light is reflected back by an interface between the cover glass  2618  and the air to the optical device  2610 , and the reflected display zero order light, e.g., Fresnel reflection of zero order light  2625 , can be absorbed by an optical absorber  2619  formed on an edge of the optical device  2610 . The optical absorber  2619  can be similar to the optical absorber  1104  of  FIG.  11   ,  1204  of  FIGS.  12 A,  12 C , or  1254  of  FIG.  12 B . Another part of the redirected display zero order light is transmitted through the interface into the air downwards at a redirection angle of -45°, e.g., redirected zero order light  2626 , which is far away from the holographic light field  2622 . 
     As illustrated in  FIG.  26 B , a system  2630  includes an optical device  2640  that can be the optical device  2410  of  FIG.  24   . The optical device  2640  includes a substrate  2642  (e.g., the substrate  2412  of  FIG.  24   ), a transmissive field grating structure  2644  (e.g., the transmissive field grating structure  2414  of  FIG.  24   ), and a zero order redirection grating structure  2646  (e.g., the zero order redirection grating structure  2416  of  FIG.  24   ). The optical device  2640  can include a cover glass  2648  on the zero order redirection grating structure  2646 . 
     Different from the transmissive field grating structure  2614  of the optical device  2610  of  FIG.  26 A , the transmissive field grating structure  2644  of the optical device  2640  diffracts the input light  2620  from the illuminator  2406  to illuminate the display  2404  with an incident angle +6° (in air). A first portion of the input light  2620  illuminating on modulated display elements of the display  2404  is diffracted to transmit through the optical device  2640  (including the zero order redirection grating structure  2646 ) to become diffracted first order light  2631  that forms a holographic light field  2632 . A second portion of the input light  2620  illuminating on gaps of the display  2404  is reflected to come off the display  2404  as display zero order light  2634 . Different from the zero order redirection grating structure  2616  of  FIG.  26 A , the zero order redirection grating structure  2646  redirects (or diffracts) the display zero order light  2624  at a diffraction angle substantially larger than the incident angle, e.g., +28° in glass. Due to Fresnel reflection, part of the redirected display zero order light is reflected back by an interface between the cover glass  2618  and the air to the optical device  2610 , and the reflected display zero order light, e.g., Fresnel reflection of zero order light  2635 , can be absorbed by an optical absorber  2649  formed on an edge of the optical device  2640 . The optical absorber  2649  can be similar to the optical absorber  2619  of  FIG.  26 A . Another part of the redirected display zero order light is transmitted through the interface into the air upwards at a redirection angle of +45°, e.g., redirected zero order light  2636 , which is far away from the holographic light field  2622 . 
     As illustrated in  FIG.  26 C , a system  2650  includes an optical device  2660  that can be the optical device  2410  of  FIG.  24   . The optical device  2660  includes a substrate  2662  (e.g., the substrate  2412  of  FIG.  24   ), a transmissive field grating structure  2664  (e.g., the transmissive field grating structure  2414  of  FIG.  24   ), and a zero order redirection grating structure  2666  (e.g., the zero order redirection grating structure  2416  of  FIG.  24   ). The optical device  2660  can include a cover glass  2668  on the zero order redirection grating structure  2666 . 
     Same as the transmissive field grating structure  2614  of the optical device  2610  of  FIG.  26 A , the transmissive field grating structure  2664  of the optical device  2660  diffracts the input light  2620  from the illuminator  2406  to illuminate the display  2404  with an incident angle -6° (in air). A first portion of the input light  2620  illuminating on modulated display elements of the display  2404  is diffracted to transmit through the optical device  2660  (including the zero order redirection grating structure  2666 ) to become diffracted first order light  2631  that forms a holographic light field  2632 . A second portion of the input light  2620  illuminating on gaps of the display  2404  is reflected to come off the display  2404  to become at least a part of display zero order light  2654 . Different from the zero order redirection grating structure  2616  of  FIG.  26 A , the zero order redirection grating structure  2666  redirects (or diffracts) the display zero order light  2654  at a diffraction angle substantially larger than the incident angle, e.g., +28° in glass. Due to Fresnel reflection, part of the redirected display zero order light is reflected back by an interface between the cover glass  2668  and the air to the optical device  2660 , and the reflected display zero order light, e.g., Fresnel reflection of zero order light  2655 , can be absorbed by an optical absorber  2649  formed on an edge of the optical device  2640 . The optical absorber  2669  can be similar to the optical absorber  2619  of  FIG.  26 A . Another part of the redirected display zero order light is transmitted through the interface into the air upwards at a redirection angle of +45°, e.g., redirected zero order light  2656 , which is far away from the holographic light field  2622 . 
     To eliminate the effect of Fresnel reflection on the redirected display zero order light on the interface between a surface of the cover glass and the air, an anti-reflection (AR) coating can be formed on the surface of the cover glass  2668 , so that the redirected display zero order light can be transmitted with a high transmittance into the air but with little or no reflection back to the optical device. 
     As illustrated in  FIG.  26 D , a system  2670  includes an optical device  2680 . Similar to the optical device  2660  of  FIG.  26 C , the optical device  2680  is configured to diffract the input light  2620  to illuminate the display  2404  at an incident angle -6° (in air) and redirect the display zero order light  2654  into the air upwards at a redirection angle of +45°. However, different from the optical device  2660  of  FIG.  26 C , the optical device  2680  includes an AR coating layer  2682  formed on an external surface of the cover glass  2668 , such that the redirected display zero order light is substantially transmitted through the cover glass  2668  into the air at a redirection angle of +45°, e.g., redirected zero order light  2672 . In such a way, there is little or no Fresnel reflection of the redirected zero order light back into the optical device  2680 . 
       FIG.  26 E  shows another example of redirecting the display zero order light at an even larger redirection angle, e.g., +75° in air or approximately +40° in glass). As illustrated in  FIG.  26 E , a system  2690  includes an optical device  2692 . Similar to the optical device  2660  of  FIG.  26 C , the optical device  2692  is configured to diffract the input light  2620  to illuminate the display  2404  at an incident angle -6° (in air). However, different from the optical device  2660  of  FIG.  26 C , the optical device  2692  includes a zero order redirection grating structure  2694  configured to redirect the display zero order light  2654  into the air upwards at a redirection angle of +75°, e.g., redirected zero order light  2696 . Accordingly, there is larger Fresnel reflection of zero order light  2698  back into the optical device  2692 , which can be absorbed by the optical absorber  2669 . 
     When light with p polarization is incident at a Brewster’s angle at an interface between a larger refractive index medium and a smaller refractive index medium, there is no Fresnel reflection for the light with p polarization. 
       FIG.  27 A  illustrates an example system  2700  of redirecting display zero order light with p polarization to transmit into air at a Brewster’s angle. The system  2700  includes an optical device  2710  that can be the optical device  2410  of  FIG.  24   . The optical device  2710  includes a substrate  2712  (e.g., the substrate  2412  of  FIG.  24   ), a transmissive field grating structure  2714  (e.g., the transmissive field grating structure  2414  of  FIG.  24   ), and a zero order redirection grating structure  2716  (e.g., the zero order redirection grating structure  2416  of  FIG.  24   ). The optical device  2710  can include a cover glass  2718  on the zero order redirection grating structure  2716 . 
     Same as the transmissive field grating structure  2614  of the optical device  2610  of  FIG.  26 A , the transmissive field grating structure  2714  of the optical device  2710  diffracts the input light  2620  from the illuminator  2406  to illuminate the display  2404  with an incident angle -6° (in air). A first portion of the input light  2620  illuminating on modulated display elements of the display  2404  is diffracted to transmit through the optical device  2710  (including the zero order redirection grating structure  2716 ) to become diffracted first order light  2701  that forms a holographic light field  2702 . A second portion of the input light  2620  illuminating on gaps of the display  2404  is reflected to come off the display  2404  as display zero order light  2704 . The display zero order light  2704  can have p polarization state. In some cases, the input light  2620  from the illuminator  2406  has p polarization state. In some cases, the optical device  2710  includes one or more optical polarizing devices (e.g., polarizers, retarders, waveplates, or a combination thereof) configured to control a polarization state of the diffracted input light  2620  to be p polarization. In some implementations, the optical device  2710  includes an optical retarder (e.g., a broad-band half-wave retarder) followed by an optical polarizer (e.g., a linear polarizer). The optical retarder is configured to rotate each color of light from s polarization to p polarization, e.g., with corresponding efficiencies, and the optical polarizer is configured to absorb whatever percentage of each color of light has not been rotated from s polarization to p polarization. 
     Different from the zero order redirection grating structure  2616  of  FIG.  26 A , the zero order redirection grating structure  2716  redirects (or diffracts) the display zero order light  2654  with a Brewster’s angle, e.g., approximately -37° in glass, at the interface between the cover glass  2718  and the air. Thus, there is no Fresnel reflection of the redirected display zero order light back to the optical device  2710 , and almost all the redirected display zero order light is transmitted into the air at the Brewster’s angle of approximately -57°, e.g., redirected zero order light  2706 . 
       FIGS.  27 B- 27 C  illustrate examples of redirecting display zero order light with s polarization with an optically polarizing device such as an optical retarder for transmission at a Brewster’s angle. When the display zero order light comes off the display  2404  with s polarization, an optical device can include an optical retarder before an interface into air. The optical retarder can convert a polarization state of the display zero order light from s polarization state to p polarization state for transmitting at Brewster’s angle at the air interface without Fresnel reflection. 
     As illustrated in  FIG.  27 B , a system  2730  includes an optical device  2740  that can be the optical device  2410  of  FIG.  24   . The optical device  2740  includes a substrate  2742  (e.g., the substrate  2412  of  FIG.  24   ), a transmissive field grating structure  2744  (e.g., the transmissive field grating structure  2414  of  FIG.  24   ), and a zero order redirection grating structure  2746  (e.g., the zero order redirection grating structure  2416  of  FIG.  24   ). The optical device  2740  can include a cover glass  2748  on the zero order redirection grating structure  2746 . 
     Similar to the transmissive field grating structure  2714  of the optical device  2710  of  FIG.  27 A , the transmissive field grating structure  2744  of the optical device  2740  diffracts the input light  2620  from the illuminator  2406  to illuminate the display  2404  with an incident angle -6° (in air). A first portion of the input light  2620  illuminating on modulated display elements of the display  2404  is diffracted to transmit through the optical device  2740  (including the zero order redirection grating structure  2746 ) to become diffracted first order light  2731  that forms a holographic light field  2732 . A second portion of the input light  2620  illuminating on gaps of the display  2404  is reflected to come off the display  2404  as display zero order light  2734 . Different from the display zero order light  2704  in  FIG.  27 A , the display zero order light  2734  can have s polarization. In some cases, the input light  2620  from the illuminator  2406  has s polarization state. In some cases, the optical device  2740  includes one or more optically polarizing devices configured to control a polarization state of the diffracted input light  2620  to be s polarization. 
     Different from the optical device  2710  of  FIG.  27 A , the optical device  2740  includes an optical retarder  2747  that is configured to convert a polarization state of the display zero order light  2734  from s polarization to p polarization. In some examples, the polarization conversion can be achieved using a broadband half-wave retarder, which can rotate each color of light from s polarization to p polarization with differing efficiencies for each color. The half-wave retarder can be followed by a “cleanup” linear polarizer to absorb that percentage of each color of light which has not been rotated from s polarization to p polarization. In such a way, the retarder can rotate the polarization of light emerging from the optical device  2740  to another polarization more suitable for the best performance of the display  2404 , and the linear polarizer can eliminate light incident upon the display  2404  in polarizations less suitable for the best performance of the display  240 . 
     In some implementation, as illustrated in  FIG.  27 B , the optical retarder  2747  (and optionally a linear polarizer) is arranged before the zero order redirection grating structure  2746  on the substrate  2742 . Same as the zero order redirection grating structure  2716  of  FIG.  27 A , the zero order redirection grating structure  2746  redirects (or diffracts) the display zero order light  2734  with p polarization with a Brewster’s angle, e.g., approximately -37° in glass, at the interface between the cover glass  2748  and the air. Thus, there is no or negligible Fresnel reflection of the redirected display zero order light back to the optical device  2740 , and almost all the redirected display zero order light is transmitted into the air at the Brewster’s angle of approximately -57°, e.g., redirected zero order light  2736 . 
     In some implementations, as illustrated in  FIG.  27 C , in an optical device  2760  of a system  2750 , the optical retarder  2747  is arranged after the zero order redirection grating structure  2746  with respect to the substrate  2742 . The zero order redirection grating structure  2746  is arranged between the substrate  2742  and the grating cover glass  2748 . The optical retarder  2747  can be arranged between the grating cover glass  2748  and a retarder cover glass  2762 . Same as the zero order redirection grating structure  2716  of  FIG.  27 A , the zero order redirection grating structure  2746  redirects (or diffracts) the display zero order light  2734  with a Brewster’s angle, e.g., approximately -37° in glass, at the interface between the retarder cover glass  2762  and the air. Thus, there is no or negligible Fresnel reflection of the redirected display zero order light back to the optical device  2760 , and almost all the redirected display zero order light is transmitted into the air at the Brewster’s angle of approximately -57°, e.g., redirected zero order light  2752 . 
       FIG.  28    illustrates an example system  2800  of redirecting display zero order light to an anisotropic transmitter  2820  for absorbing redirected display zero order light. The anisotropic transmitter  2820  is configured to transmit a first light beam (e.g., diffracted first order light) with an angle (e.g., less than a half of a viewing angle of a reconstruction cone) smaller than a predetermined angle, and absorb a second light beam (e.g., the redirected display zero order light) with an angle (e.g., a redirection angle) larger than the predetermined angle. The predetermined angle is configured to be larger than the half of the viewing angle and smaller than the redirection angle at which the display zero order light is diffracted by an optically redirecting component. 
     The system  2800  includes an optical device  2810  that can include the optical device  2410  of  FIG.  24   . The optical device  2810  includes a substrate  2812  (e.g., the substrate  2412  of  FIG.  24   ), a transmissive field grating structure  2814  (e.g., the transmissive field grating structure  2414  of  FIG.  24   ), and a zero order redirection grating structure  2816  (e.g., the zero order redirection grating structure  2416  of  FIG.  24   ). The optical device  2810  can include a cover glass  2818  on the zero order redirection grating structure  2816 . 
     Same as the transmissive field grating structure  2414  of the optical device  2410  of  FIG.  24   , the transmissive field grating structure  2814  of the optical device  2810  diffracts the input light  2620  from the illuminator  2406  to illuminate the display  2404  with an incident angle, e.g., -6° in air. A first portion of the input light  2620  illuminating on modulated display elements of the display  2404  is diffracted to transmit through the optical device  2810  (including the zero order redirection grating structure  2816 ) to become diffracted first order light  2801  that forms a holographic light field  2802 . The incident angle is configured to be larger than a half of the viewing angle of the reconstruction cone corresponding to the holographic light field  2802 . A second portion of the input light  2620  illuminating on gaps of the display  2404  is reflected to come off the display  2404  as at least a part of display zero order light  2804 . Similar to the zero order redirection grating structure  2416  of  FIG.  24   , the zero order redirection grating structure  2816  redirects (or diffracts) the display zero order light 2804 with a redirection angle substantially larger than the incident angle, e.g., an angle corresponding to approximately 75° in air. 
     Different from the optical device  2410  of  FIG.  24   , the optical device  2810  can include the anisotropic transmitter  2820  configured to transmit the diffracted first order light  2801  and absorb the display zero order light  2804 . In some examples, the anisotropic transmitter  2820  includes a louver film configured to have a predetermined angle (or a pass angle) approximately ±30° in air or approximately ±20° in acrylic. The anisotropic transmitter  2820  substantially transmits the diffracted first order light  2801 , e.g., at approximately ±5° in air (approximately ±3° in acrylic), and absorbs the display zero order light  2804 , e.g., approximately 75° in air. The anisotropic transmitter  2820  can be index matched to the cover glass  2818 , such that there is no significant Fresnel reflection from a surface of the anisotropic transmitter  2820  back into the optical device  2810  for the display zero order light  2804  having either s-polarization or p-polarization state. Louvers in the louver film can be also index matched to the transmissive material of the louver film to eliminate Fresnel reflections off the louvers. 
     In the previous examples shown in  FIGS.  26 A- 26 E,  27 A- 27 B , and  FIG.  28   , the zero order redirection grating structures are configured to diffract display zero order light at redirection angles smaller than a critical angle for total internal reflection at an interface into air. 
       FIG.  29    illustrates an example system  2900  of redirecting display zero order light to totally reflect the display zero order light. Similar to the optical device  2610  of  FIG.  26 A , an optical device  2910  of the system  2900  includes a transmissive field grating structure  2914  formed on a substrate  2912  and configured to diffract the input light  2620  to illuminate the display  2404  at an incident angle, e.g., -6° in air and approximately -4° in glass. 
     However, different from the optical device  2610  of  FIG.  26 A , the optical device  2910  includes a zero order redirection grating structure  2916  configured to redirect display zero order light  2904  at a redirection angle, e.g., approximately +60° in glass, larger than a critical angle for total internal reflection in glass, e.g., approximately 41° for a transition from a cover glass  2918  to air at a high-to-low index interface  2919 . Thus, display zero order light  2904  is totally reflected back at the interface  2919 , and Fresnel reflection of the display zero order light  2906  can be absorbed by an optical absorber  2920  (e.g., the optical absorber  2619  of  FIG.  26 A ) formed on an edge of the optical device  2910 . In contrast, a portion of the input light  2620  illuminating on modulated display elements of the display  2404  is diffracted to transmit through the optical device  2910  (including the zero order redirection grating structure  2916 ) to become diffracted first order light  2901  that forms a holographic light field  2902 , without the display zero order light  2904 . 
     Input light illuminating a display can include multiple different colors of light, e.g., red, green, and blue. The different colors of light can be sequentially incident on the display, and corresponding different color holographic data (or holograms) can sequentially modulate display elements of the display. As described above, an optically diffractive device, e.g., the optically diffractive device  598  of  FIG.  5 H , can be configured to diffract the different colors of light to illuminate the display, and can also be configured to reduce color crosstalk among the different colors of light. For example, the optically diffractive device  598  includes multiple holographic gratings for the different colors in different recording layers, e.g., as illustrated in  FIGS.  9 A to  12 C . In some examples, as described above with respect to  FIGS.  9 A to  10 B , the optically diffractive device can include multiple holographic gratings with one or more color-selective polarizers to suppress (e.g., eliminate or minimize) color crosstalk. In some examples, as described above with respect to  FIGS.  11  to  12 C and  15   , the optically diffractive device can include multiple holographic gratings with one or more reflective layers for light of different colors incident at respective incident angles to suppress color crosstalk and zero order light. 
     Similarly, an optically redirecting device can be also configured to redirect different colors of display zero order light out of corresponding holographic scenes and can also be configured to reduce color crosstalk among the different colors of display zero order light, e.g., by redirecting the different colors of display zero order light to different directions away from the holographic scenes in plane and/or in space. In the following,  FIGS.  30 A- 30 B,  31 A- 31 B,  32 , and  33    illustrate different examples of implementations. 
       FIGS.  30 A- 30 B  illustrate examples of redirecting two different colors (e.g., blue and red) of display zero order light to different directions away from a holographic scene. 
     As illustrated in  FIG.  30 A , similar to the system  2400  of  FIG.  24   , a system  3000  includes a computer  2401  (e.g., the computer  2401  of  FIG.  24   ), a controller  3002  (e.g., the controller  2402  of  FIG.  24   ), a reflective display  3004  (e.g., the reflective display  2404  of  FIG.  24   ), and an illuminator  3006  (e.g., the illuminator  2406  of  FIG.  24   ). The system  3000  also includes an optical device  3010  that can include an optically diffractive device, e.g., the optically diffractive device  900  of  FIGS.  9 A and  9 B  or  1100  of  FIG.  11   . In some implementations, as illustrated in  FIG.  30 A , the optical device  3010  includes a transmissive field grating structure  3014  on a substrate  3012  (e.g., the substrate  2412  of  FIG.  24   ). The transmissive field grating structure  3014  can include two corresponding different gratings for the two different colors of light. 
     The controller  3002  is configured to receive graphic data corresponding to one or more objects from the computer  3001  (e.g., by using a 3D software application such as Unity), perform computation on the graphic data, generate and transmit control signals for modulation to the display  3004  through a memory buffer  3003 . The controller  3002  is also coupled to the illuminator  3006  and configured to provide a timing signal  3005  to activate the illuminator  3006  to provide input light  3020 . The input light  3020  is then diffracted by the transmissive field grating structure  3014  of the optical device  3010  to illuminate the display  3004 . A first portion of the input light  3020  incident on display elements of the display  3004  is diffracted by the display  3004 , and diffracted first order light  3021  forms a holographic light field  3022  towards a viewer. The holographic light field  3022  can correspond to a reconstruction cone (or frustum) that has a viewing angle. A second portion of the input light  3020  incident on gaps of the display  3004  is reflected by the display  3004  to become at least a part of display zero order light  3024 . 
     The transmissive field grating structure 3014 is configured to diffract the different colors of input light  3020  from the illuminator  3006  out to illuminate the display  3004  off axis at an incident angle, e.g., -6° in air or approximately -4° in glass, larger than a half of a viewing angle of the reconstruction cone (or frustum). By applying the third technique, the diffracted first order light  3021  comes off the display  3004  in the same manner as that when the input light  3020  is incident on axis at normal incidence, while the display zero order light  3024  comes off at a reflected angle that is identical to the incident angle, which is outside of the reconstruction cone. 
     As illustrated in  FIG.  30 A , the system  3000  can include an optically redirecting structure having corresponding zero order redirection gratings  3016  and  3018  for different colors (blue and red) of light. Each zero order redirection grating  3016 ,  3018  can be similar to the redirection grating  2416  of  FIG.  24   , and configured to diffract a first light beam having an angle identical to a predetermined angle with a substantially larger diffraction efficiency at a diffraction angle than a second light beam having an angle different from the predetermined angle. Each zero-order redirection grating  3016 ,  3018  can be a holographic grating such as a Bragg grating for a corresponding color of light. 
     As illustrated in  FIG.  30 A , the zero order redirection grating  3016  is configured to diffract blue color display zero order light at a reflected angle (identical to the incident angle) of at a diffraction angle of +45° in air (approximately +28° in glass), e.g., redirected blue zero order display light  3026 . The zero order redirection grating  3018  is configured to diffract red color display zero order light from approximately -6° in air (approximately -4° in glass) to approximately -45° (approximately -28° in glass), e.g., redirected red display zero order light 3028. 
     The zero order redirection gratings  3016 ,  3018  can be sequentially arranged on the substrate  3012  on an opposite side of the transmissive field grating structure  3014 . As light with a shorter wavelength tends to crosstalk more strongly off gratings intended for longer wavelengths, the zero order redirection grating  3016  for blue color of light can be arranged closer to the display than the zero order redirection grating  3018  for red color. The two zero order redirection gratings 3016, 3018 can have substantially dissimilar fringe-plane tilts, which can reduce color crosstalk. 
     In some implementations, as illustrated in  FIG.  30 A , each zero order redirection grating  3016 ,  3018  for a different color of light is recorded in a corresponding recording material, e.g., photosensitive polymer, and protected by a corresponding cover glass  3017 ,  3019 . 
     In some implementations, as illustrated in  FIG.  30 B , each zero order redirection grating  3046 ,  3048  of an optical device  3040  in a system  3030  for a different color of light is recorded in a same recording material, e.g., photosensitive polymer, and protected by a cover glass  3047 . The zero order redirection grating  3046  can be same as the zero order redirection grating  3016  and configured to diffract blue color display zero order light from approximately -6° in air (approximately -4° in glass) to approximately +45° (approximately +28° in glass), e.g., redirected blue display zero order light  3036 . The zero order redirection grating  3048  can be same as the zero order redirection grating  3018  and configured to diffract red color display zero order light from approximately -6° in air (approximately -4° in glass) to approximately -45° (approximately -28° in glass), e.g., redirected red display zero order light  3038 . 
     The optical devices  3010 ,  3040  can include optical absorbers (e.g., the optical absorber  2619  of  FIG.  26 A ) on edges of the optical devices  3010 ,  3040 , to reduce Fresnel reflection at the interface between the cover glass and the air. 
       FIGS.  31 A- 31 B  illustrate example systems  3100  and  3150  of redirecting three different colors (blue, green, red) of display zero order light to different directions away from a holographic scene in a same plane. Compared to a system for two different colors of light, e.g., as illustrated in  FIGS.  30 A or  30 B , a system for three different colors of light includes an optical diffractive structure including three different diffraction gratings for diffracting the three colors of input light to illuminate a display at a same incident angle, and an optical redirecting structure including three different zero order redirection gratings for diffracting three colors of display zero order light at different diffraction angles towards different directions. 
     As illustrated in  FIG.  31 A , similar to the system  3000  of  FIG.  30 A , a system  3100  includes a computer  3101  (e.g., the computer  3101  of  FIG.  30 A ), a controller  3102  (e.g., the controller  3002  of  FIG.  30 A ), a reflective display  3104  (e.g., the reflective display  3004  of  FIG.  30 A ), and an illuminator  3106  (e.g., the illuminator  3006  of  FIG.  30 A ). The system  3100  also includes an optical device  3110  that can include an optically diffractive device, e.g., the optically diffractive device  1000  of  FIGS.  10 A and  10 B ,  1200  of  FIG.  12 A ,  1250  of  FIG.  12 B , or  1270  of  FIG.  12 C , or  1500  of  FIG.  15   . In some implementations, as illustrated in  FIG.  31 A , the optical device  3110  includes a transmissive field grating structure  3112  on a substrate  3111 . The transmissive field grating structure  3112  can include three corresponding different gratings for the three different colors of light. 
     The controller  3102  is configured to receive graphic data corresponding to one or more objects from the computer  3101  (e.g., by using a 3D software application such as Unity), perform computation on the graphic data, generate and transmit control signals for modulation to the display  3104  through a memory buffer  3103 . The controller  3102  is also coupled to the illuminator  3106  and configured to provide a timing signal  3105  to activate the illuminator  3106  to provide input light  3120 . The input light  3120  is then diffracted by the transmissive field grating  3112  of the optical device  3110  to illuminate the display  3104 . A first portion of the input light  3120  incident on display elements of the display  3104  is diffracted by the display  3104 , and diffracted first order light  3121  forms a holographic light field  3122  towards a viewer. The holographic light field  3122  can correspond to a reconstruction cone (or frustum) that has a viewing angle. A second portion of the input light  3120  incident on gaps of the display  3104  is reflected by the display  3104  to become display zero order light  3123 . 
     The transmissive field grating  3112  is configured to diffract the different colors of input light  3120  from the illuminator  3106  out to illuminate the display  3104  off axis at an incident angle, e.g., -6° in air or approximately -4° in glass, larger than a half of a viewing angle of the reconstruction cone (or frustum). By applying the third technique, the diffracted first order light  3121  comes off the display  3104  in the same manner as that when the input light  3120  is incident on axis at normal incidence, while the display zero order light  3123  comes off at a reflected angle that is identical to the incident angle, which is outside of the reconstruction cone. 
     As illustrated in  FIG.  31 A , the system  3100  can include an optically redirecting structure having three corresponding zero order redirection gratings  3114 ,  3116 , and  3018  for the different colors (blue, green, and red) of light. Each zero order redirection grating  3114 ,  3116 ,  3118  can be similar to the redirection grating  2416  of  FIG.  24   . Each zero-order redirection grating  3114 ,  3116 ,  3118  can be a holographic grating such as a Bragg grating for a corresponding color of light. 
     The zero order redirection gratings  3114 ,  3116 ,  3118  can be sequentially arranged on the substrate  3111  on an opposite side of the transmissive field grating structure  3112 . In some implementations, as illustrated in  FIG.  31 A , each zero order redirection grating  3114 ,  3116 ,  3118  for a different color of light (blue, green, red) is recorded in a corresponding recording material, e.g., photosensitive polymer, and protected by a corresponding cover glass  3113 ,  3115 ,  3117 . As noted above, the zero order redirection gratings  3114 ,  3116 ,  3118  for the three different colors of light can be recorded in a same recording material, e.g., photosensitive polymer, and protected by a cover glass. The three zero order redirection gratings  3114 ,  3116 ,  3118  can have substantially dissimilar fringe-plane tilts, which can reduce color crosstalk. 
     As illustrated in  FIG.  31 A , the blue color zero order redirection grating  3114  is configured to diffract blue color display zero order light from approximately -6° in air (approximately -4° in glass) to approximately +45° (approximately +28° in glass), e.g., redirected blue display zero order light  3124 . The green color zero order redirection grating  3116  is configured to diffract green color display zero order light from approximately -6° in air (approximately -4° in glass) to approximately -45° (approximately -28° in glass), e.g., redirected green display zero order light  3126 . The red color zero order redirection grating  3118  is configured to diffract red color display zero order light from approximately -6° in air (approximately -4° in glass) to the Brewster’s angle approximately -57° (approximately -37° in glass), e.g., redirected red display zero order light  3128 . If the red color display zero order light has p polarization state, the red color display zero order light can be totally transmitted into the air. The optical device  3110  can include one or more optical absorbers (e.g., the optical absorber  2619  of  FIG.  26 A ) on one or more edges of the optical device  3110  to reduce Fresnel reflection of the blue and green colors of display zero order light at the interface between the cover glass and the air. 
     If all the three colors of display zero order light have p polarization state, e.g., when the input light is p polarized, an optical redirecting device can include zero order redirection gratings for the three different colors of display zero order light configured to diffract the three different colors of display zero order light into air all at the Brewster’s angle, which can reduce Fresnel reflection. One or more diffractive gratings can be used together to redirect a particular color of light. 
     As illustrated in  FIG.  31 B , an optical device  3160  of a system  3150  includes a blue color redirection grating  3164 , a pair of green color redirection grating  3166 - 1 ,  3166 - 2 , and a red color redirection grating  3168 , which are recorded in corresponding recording media and protected by corresponding cover glasses  3163 ,  3165 - 1  and  3165 - 2 , and  3167 . The blue color zero order redirection grating  3164  is configured to diffract blue color display zero order light from approximately +6° in air (approximately +4° in glass) to the Brewster’s angle of approximately -57° in air (approximately -37° in glass), e.g., redirected blue display zero order light  3154 . Green color display zero order light is first diffracted by first green color zero order redirection grating  3166 - 1  from approximately +6° in air (approximately +4° in glass) to approximately +70° (approximately +38° in glass), and then diffracted by second green color zero order redirection grating  3166 - 2  to the Brewster’s angle of approximately -57° in air (approximately -37° in glass), e.g., redirected green display zero order light  3156 . The red color zero order redirection grating 3168 is configured to diffract red color display zero order light from approximately +6° in air (approximately +4° in glass) to the Brewster’s angle of approximately +57° in air (approximately +37° in glass), e.g., redirected red display zero order light  3158 . The four zero order redirection gratings  3164 ,  3166 - 1 ,  3166 - 2 , and  3168  can have substantially dissimilar fringe-plane tilts, which can reduce color crosstalk. 
     To reduce color crosstalk among different colors of display zero order light, an optical redirecting device can be configured to redirect the different colors of display zero order light towards different directions in a sample plane, as illustrated in  FIGS.  30 A- 30 B and  31 A- 31 B . The optical redirecting device can also be configured to redirect the different colors of display zero order light towards different planes in space, as illustrated in  FIG.  32    below. 
       FIG.  32    illustrates an example system  3200  including an optical device  3210  of redirecting three different colors (e.g., blue, green, and red) of display zero order light to different directions away from corresponding holographic scenes in space. 
     Similar to the optical device  3110  of  FIG.  31 A , the optical device  3210  includes a transmissive field grating structure  3212  that is same as the transmissive field grating structure  3112  of  FIG.  31 A  and configured to diffract each color input light to illuminate the display  3104  off axis at an incident angle, e.g., -6° in air or approximately -4° in glass, larger than a half of a viewing angle of the reconstruction cone (or frustum). By applying the third technique, the diffracted first order light comes off the display  3104  in the same manner as that when the input light is incident on axis at normal incidence. As noted above, light with a larger wavelength corresponds to a larger viewing angle. As illustrated in  FIG.  32   , blue color diffracted first order light forms a blue color holographic light field  3220 , green color diffracted first order light forms a green color holographic light field  3222 , and red color diffracted first order light forms a red color holographic light field  3224 . 
     Similar to the optical device  3110  of  FIG.  31 A , the optical device  3210  includes blue, green, red color redirection gratings  3214 ,  3216 ,  3218  recorded in different recording media and sequentially arranged on an opposite side of a substrate  3211  with respect to the transmissive field grating structure  3212 . The blue, green, red color redirection gratings  3214 ,  3216 ,  3218  are protected by corresponding blue, green, red cover glasses  3213 ,  3215 ,  3217 . However, different from the redirection gratings  3114 ,  3116 ,  3118  of  FIG.  31 A , the redirection gratings  3214 ,  3216 ,  3218  redirect corresponding colors of display zero order light into different planes. 
     For example, as illustrated in  FIG.  32   , the blue color redirection grating  3214  diffracts the blue color display zero order light from approximately -6° in air (approximately -4° in glass) to an upwards Brewster’s angle of approximately +57° in air (approximately +37° in glass), e.g., upwards redirected blue color zero order light  3230 . The red color redirection grating  3218  redirects the red color display zero order light from approximately -6° in air (approximately -4° in glass) to a downwards Brewster’s angle of approximately -57° in air (approximately -37° in glass), e.g., downwards redirected red color zero order light  3234 . The green color redirection grating  3216  redirects the green color display zero order light from approximately -6° in air (approximately -4° in glass) to a rightwards Brewster’s angle (approximately +57° in air, approximately +37° in glass), e.g., rightwards redirected green color zero order light  3232 , which is orthogonal to the plane of the upwards redirected blue color zero order light  3230  and downwards redirected red color zero order light  3234 . Note that the blue and red color redirection gratings  3214 ,  3218  have different fringe-plane tilts and/or orientations than the green color redirection grating  3216 , which can suppress color crosstalk. 
       FIG.  33    illustrates another example system  3300  of redirecting three different colors of display zero order light to different directions away from a holographic scene using at least one switchable grating for at least one corresponding color display zero order light. 
     Similar to the optical device  3110  of  FIG.  31 A , an optical device  3310  in the system 3300 includes blue, green, red color redirection gratings  3314 ,  3316 ,  3318  sequentially arranged on an opposite side of the substrate  3111  with respect to the transmissive field grating structure  3112 . The blue, green, red color redirection gratings  3314 ,  3316 ,  3318  are protected by corresponding blue, green, red cover glasses  3313 ,  3315 ,  3317 . Similar to the blue and red color redirection gratings  3114 ,  3118  of  FIG.  31 A , the blue and red color redirection gratings  3314 ,  3318  are permanently recorded in corresponding recording media. 
     However, different from the green color redirection grating  3116  of  FIG.  31 A  that is permanently recorded in the corresponding recording medium, the green color redirection grating  3316  is recorded in a switchable recording material, e.g., an electrically switchable Holographic Polymer Dispersed Liquid Crystal (HPDLC) material, and configured to be switchable between different states. For example, the green color redirection grating  3316  can be switched to a first state during first intervals of a field-sequential color (FSC) illumination sequence when only green color of light is present. During the first green-only intervals, the switchable green color redirection grating  3316  in the first state diffracts green color display zero order light from approximately -6° in air (approximately -4° in glass) to a downwards angle of approximately -45° in air (approximately -28° in glass), e.g., redirected green color display zero order light  3338 . 
     During other intervals of the FSC color illumination sequence, when only red or blue color of light is present, the switchable green color redirection grating  3316  is switched to a second state in which the switchable green color redirection grating does not diffract red or blue color of light. As illustrated in  FIG.  32   , the blue color redirection grating  3314  diffracts the blue color display zero order light from approximately -6° in air (approximately -4° in glass) to an upwards angle of approximately +45° in air (approximately +28° in glass), e.g., upwards redirected blue color zero order light  3336 . The red color redirection grating  3318  redirects the red color display zero order light from approximately -6° in air (approximately -4° in glass) to a downwards angle of approximately -45° in air (approximately -28° in glass), e.g., downwards redirected red color zero order light  3340 . Although the redirected red color zero order light  3340  has the same direction as the redirected green color zero order light  3338 , the switchable green color redirection grating  3316  is switched between the first state during all, part, or parts of the first intervals for redirecting the green color of light and the second state during all, part, or parts of the other intervals for transmitting the red or blue color of light, which can suppress color crosstalk. 
     In some implementations, two or more separate switchable gratings can be used for two or more corresponding colors, with fewer or no permanently-recorded gratings, which may further suppress color crosstalk. In some implementations, binary (on/off) switchable gratings can be replaced by switchable gratings in which a first switched state diffracts a first color, and a second switched state diffracts a second color, which can enable the use of fewer or no permanently recorded gratings. 
       FIG.  34    is a flowchart of an example process  3400  of suppressing display zero order light in a holographic scene. The process  3400  can be implemented in a system for reconstructing 2D or 3D objects. The system can be any suitable system, e.g., the system  500  of  FIG.  5 A ,  520  of  FIG.  5 B ,  530  of  FIG.  5 C ,  540  of  FIG.  5 D ,  560  of  FIG.  5 E ,  570  of  FIG.  5 F ,  580  of  FIG.  5 G ,  590  of  FIG.  5 H ,  590 A of  FIG.  5 I ,  590 B of  FIG.  5 J ,  590 C of  FIG.  5 K ,  1800  of  FIG.  18   ,  1950  of  FIG.  19 B  ,1980 of  FIG.  19 C ,  2100  of  FIG.  21   ,  2200  of  FIG.  22   ,  2300  of  FIG.  23 A ,  2350  of  FIG.  23 B ,  2400  of  FIG.  24   ,  2600  of  FIG.  26 A ,  2630  of  FIG.  26 B ,  2650  of  FIG.  26 C ,  2670  of  FIG.  26 D ,  2690  of  FIG.  26 E ,  2700  of  FIG.  27 A ,  2730  of  FIG.  27 B ,  2750  of  FIG.  27 C ,  2800  of  FIG.  28   ,  2900  of  FIG.  29   ,  3000  of  FIG.  30 A ,  3030  of  FIG.  30 B ,  3100  of  FIG.  31 A ,  3150  of  FIG.  31 B ,  3200  of  FIG.  32   , or  3300  of  FIG.  33   . 
     At  3402 , a display is illuminated with light. A first portion of the light illuminates display elements of the display. In some cases, a second portion of the light illuminates gaps between adjacent display elements. The display can be the display  1610  of  FIG.  16   , the display elements can be the display elements  1612  of  FIG.  16   , and the gaps can be the gaps  1614  of  FIG.  16   . 
     At  3404 , the display elements of the display are modulated with a hologram corresponding to holographic data to diffract the first portion of the light to form a holographic scene corresponding to the holographic data and to suppress display zero order light in the holographic scene. The display zero order light can include reflected light from the display, e.g., the second portion of the light reflected at the gaps. The reflected light from the display can be a main order of the display zero order light. The display zero order light can also include any unwanted or undesirable light, e.g., diffracted light at the gaps, reflected light at surfaces of the display elements, and reflected light at a surface of a display cover covering the display. The holographic scene corresponds to a reconstruction cone (or frustum) with a viewing angle. The hologram is configured such that the display zero order light is suppressed in the holographic scene. The hologram can be configured such that the diffracted first portion of the light has at least one characteristic different from that of the display zero order light. The at least one characteristic can include at least one of a power density (e.g., as illustrated in  FIG.  18   ), a beam divergence (e.g., as illustrated in  FIG.  18   ), a propagating direction away from the display (e.g., as illustrated in  FIGS.  19 B,  19 C,  20 B, and  21 - 33   ), or a polarization state. 
     The display zero order light is suppressed in the holographic scene with a light suppression efficiency. The light suppression efficiency can be defined as a result of one minus a ratio between an amount of the display zero order light in the holographic scene using the suppression and an amount of the display zero order light in the holographic scene without any suppression. In some examples, the light suppression efficiency is more than a predetermined percentage that is one of 50%, 60%, 70%, 80%, 90%, or 99%. In some examples, the light suppression efficiency is 100%. 
     In some implementations, the process  3400  further includes: for each of a plurality of primitives corresponding to an object, determining an electromagnetic (EM) field contribution to each of the display elements of the display by computing, in a global three-dimensional (3D) coordinate system, EM field propagation from the primitive to the display element, and for each of the display elements, generating a sum of the EM field contributions from the plurality of primitives to the display element. The holographic data can include the sums of the EM field contributions for the display elements of the display from the plurality of primitives of the object. When the display is phase modulated, the holographic data can include respective phases for the display elements of the display. The holographic scene can include a reconstructed object corresponding to the object. The holographic data can include information of two or more objects. 
     In some implementations, as discussed above with respect to the first technique, “phase calibration,” the hologram can be configured by adjusting the respective phases for the display elements to have a predetermined phase range, e.g., [0, 2 π ]. In some implementations, the respective phases can be adjusted according to the expression (15) below: 
     
       
         
           
             
               ∅ 
               a 
             
             = 
             A 
             
               ∅ 
               i 
             
             + 
             B 
             , 
           
         
       
     
      where Ø i  represents an initial phase value of a respective phase, Ø a  represents an adjusted phase value of the respective phase, and A and B are constants for the respective phases. The constants A and B can be adjusted such that the light suppression efficiency for the holographic scene is maximized or larger than a predetermined threshold, e.g., 50%, 60%, 70%, 80%, 90%, or 99%. In some implementations, the constants A and B are adjusted according to a machine vision algorithm or a machine learning algorithm. 
     In some implementations, as discussed above with respect to the second technique, “zero order beam divergence,” an optically diverging component is arranged downstream the display. The optically diverging component can be a defocusing element including a concave lens. e.g., the concave lens  1802  of  FIG.  18   . The optically diverging component can be a focusing element including a convex lens. The diffracted first portion of the light is guided through the optically diverging component to form the holographic scene, while the display zero order light is diverged in the holographic scene. The light illuminating the display can be collimated, and the display zero order light can be collimated before arriving at the optically diverging component, and the hologram is configured such that the diffracted first portion of the light is converging before arriving at the optically diverging component. The optically diverging component can be a focusing element including a cylindrical lens. The optically diverging component can be a lenslet array including concave, convex, or cylindrical lenses, or a combination thereof. The optically diverging component can be one or more Holographic Optical Elements (HOEs), either added to the optical device, or incorporated within one or more of the other diffractive layers of the optical device. The one or more HOEs can be configured to converge, diverge or linearly focus light, or to impose a more complicated transfer function on the optically diverging component such as directing the display zero order light to a region or regions outside the reconstruction cone of the holographic scene. The region can include an annular or peripheral region or parts of an annular or peripheral region. The light illuminating the display can be collimated, and the hologram can be configured such that the diffracted first portion of the light is shaped with a shaping effect before arriving at the optically diverging component such that the effect of the optically diverging component on the first portion of the light compensates the shaping effect. 
     In some examples, the hologram is configured by adding a virtual lens, e.g., by adding a corresponding phase to the respective phase for each of the display elements, and the corresponding phases for the display elements are compensated by the optically diverging component such that the holographic scene corresponds to the respective phases for the display elements. The corresponding phase for each of the display elements can be expressed by the expression (16) below: 
     
       
         
           
             ∅ 
             = 
             
               π 
               
                 λ 
                 f 
               
             
             
               
                 
                   
                     ax 
                   
                   2 
                 
                 + 
                 
                   
                     by 
                   
                   2 
                 
               
             
             , 
           
         
       
     
      where Ø represents the corresponding phase for the display element, λ represents a wavelength of the light, f represents a focal length of the optically diverging component, x and y represent coordinates of the display element in a coordinate system, and a and b represent constants. 
     In some examples, the hologram is configured in a 3D software application, e.g., Unity, by moving a configuration cone with respect to the display with respect to a global 3D coordinate system along a direction perpendicular to the display with a distance corresponding to a focal length of the optically diverging component. The configuration cone corresponds to the reconstruction cone and has an apex angle identical to the viewing angle. The software application can generate primitives for objects based on the moved configuration cone in the global 3D coordinate system. 
     The process  3400  can include displaying the holographic scene on a two-dimensional (2D) screen, e.g., the projection screen  1830  of  FIG.  18   , spaced away from the display along the direction perpendicular to the display. The 2D screen can be moved along the direction to obtain different slices of the holographic scene on the 2D screen. 
     The process  3400  can further include guiding the light to illuminate the display. In some examples, the light is guided by a beam splitter, e.g., the beam splitter  1810  of  FIG.  18   , to illuminate the display, and the diffracted first portion of the light and the display zero order light transmit through the beam splitter. 
     In some implementations, the display is illuminated with the light at normal incidence, e.g., as illustrated in  FIGS.  18  or  19 A . In some implementations, the display is illuminated with the light at an incident angle that can be larger than a half of the viewing angle, as illustrated in  FIGS.  19 B or  19 C . 
     In some implementations, as discussed above with respect to the third technique, “zero order light deviation,” the hologram is configured such that the diffracted first portion of the light forms the reconstruction cone that is the same as a reconstruction cone to be formed by the diffracted first portion of the light if the light is normally incident on the display, while the reflected second portion of the light comes off the display at a reflected angle identical to the incident angle, as illustrated in  FIGS.  19 B or  19 C . 
     In some examples, the hologram is configured by adding a virtual prism, e.g., by adding a corresponding phase to the respective phase for each of the display elements, and the corresponding phases for the display elements are compensated by the incident angle such that the holographic scene corresponds to the respective phases for the display elements. The corresponding phase for each of the display elements can be expressed by the expression (17) below: 
     
       
         
           
             ∅ 
             = 
             
               
                 2 
                 π 
               
               λ 
             
             
               
                 xcos 
                 θ 
                 +ycos 
                 θ 
               
             
             , 
           
         
       
     
      where ø represents the corresponding phase for the display element, λ represents a wavelength of the light, x and y represent coordinates of the display element in the global 3D coordinate system, and θ represents an angle corresponding to the incident angle. 
     In some examples, the hologram is configured by moving the configuration cone with respect to the display with respect to the global 3D coordinate system, e.g., as illustrated in  FIG.  20 B , by rotating the configuration cone by a rotation angle with respect to a surface of the display with respect to the global 3D coordinate system, the rotation angle corresponding to the incident angle. 
     In some implementations, as discussed above with respect to the fourth technique, “zero order light blocking,” the display zero order light is blocked to appear in the holographic scene. The light suppression efficiency for the holographic scene can be 100%. 
     In some examples, an optically blocking component is arranged downstream the display. The optically blocking component can include a plurality of microstructures or nanostructures. The optically blocking component can include a metamaterial layer, e.g., the metamaterial layer  2316  of  FIGS.  23 A- 23 B , or a louver film, e.g., the anisotropic transmitter of  FIG.  28   . The optically blocking component is configured to transmit a first light beam having an angle smaller than a predetermined angle and block a second light beam having an angle larger than the predetermined angle, and the predetermined angle is smaller than the incident angle and larger than the half of the viewing angle. Thus, as illustrated in  FIGS.  23 A,  23 B , the display zero order light is blocked by the optically blocking component, and the diffracted first portion of the light transmits through the optically blocking component with a transmission efficiency to form the holographic scene. The transmission efficiency is no less than a predetermined ratio, e.g., 50%, 60%, 70%, 80%, 90%, or 99%. 
     In some implementations, the process  3400  further includes: guiding the light to illuminate the display by guiding the light through an optically diffractive component on a substrate configured to diffract the light out with the incident angle. The optically diffractive component can the outcoupler  1914  of  FIG.  19 A ,  1964  of  FIGS.  19 B or  19 C , or the transmissive field grating structure  2414  of  FIG.  24   . In some examples, the light is guided through a waveguide coupler, e.g., the incoupler  1916  of  FIG.  19 A , or  1966  of  FIGS.  19 B or  19 C , to the optically diffractive component. In some examples, the light is guided through a coupling prism, e.g., the coupling prism  2111  of  FIG.  21    or  2311  of  FIGS.  23 A or  23 B , to the optically diffractive component. In some examples, the light is guided through a wedged surface of the substrate to the optically diffractive component, e.g., as illustrated in  FIG.  22   . 
     As illustrated in  FIGS.  23 A or  23 B , the optically diffractive component is formed on a first surface of the substrate facing to the display, and the optically blocking component is formed on a second surface of the substrate that is opposite to the first surface. 
     In some implementations, as discussed above with respect to the fifth technique, “zero order light redirection,” an optically redirecting component is arranged downstream the display and configured to transmit the diffracted first portion of the light to form the holographic scene and redirect the display zero order light away from the holographic scene. The optically redirecting component can be the zero order redirection grating structure  2416  of  FIG.  24   ,  2616  of  FIG.  26 A ,  2646  of  FIG.  26 B ,  2666  of  FIGS.  26 C or  26 D ,  2694  of  FIG.  26 E ,  2716  of  FIG.  27 A ,  2746  of  FIGS.  27 B or  27 C ,  2816  of  FIG.  28   ,  2916  of  FIG.  29   ,  3016  and  3018  of  FIG.  30 A ,  3046  and  3048  of  FIG.  30 B ,  3114 ,  3116 , and  3118  of  FIG.  31 A ,  3164 ,  3166 - 1 ,  3166 - 2 , and  3168  of  FIG.  31 B , or  3214 ,  3216 , and  3218  of  FIG.  32   , or  3314 ,  3316 , and  3318  of  FIG.  33   . 
     The optically redirecting component can be configured to diffract a first light beam having an angle identical to a predetermined angle with a substantially larger diffraction efficiency than a second light beam having an angle different from the predetermined angle, and the predetermined angle is substantially identical to the incident angle. The optically redirecting component can include one or more holographic gratings such as Bragg gratings. 
     In some implementations, the optically diffractive component is formed on a first surface of the substrate facing towards the display, and the optically redirecting component is formed on a second surface of the substrate that is opposite to the first surface, e.g., as illustrated in  FIGS.  24  to  33   . 
     The optically redirecting component is configured such that the display zero order light is diffracted outside of the holographic scene in a three-dimensional (3D) space along at least one of an upward direction, a downward direction, a leftward direction, a rightward direction, or a combination thereof. The light suppression efficiency for the holographic scene can be 100%. In some examples, as illustrated in  FIG.  26 A , the incident angle of the light is negative, e.g., -6° in air, and a diffraction angle of the display zero order light diffracted by the optically redirecting component is negative, e.g., -45° in air. In some examples, as illustrated in  FIG.  26 B , the incident angle of the light is positive, e.g., +6° in air, and a diffraction angle of the display zero order light diffracted by the optically redirecting component is positive, e.g., +45° in air. In some examples, as illustrated in  FIGS.  26 C or  26 D , the incident angle of the light is negative, e.g., -6° in air, and a diffraction angle of the display zero order light diffracted by the optically redirecting component is positive, e.g., +45° in air. In some examples, the incident angle of the light is positive, +6° in air, and a diffraction angle of the display zero order light diffracted by the optically redirecting component is negative, e.g., -45° in air. 
     The optically redirecting component can be covered by a second substrate, e.g., the cover glass  2618  of  FIG.  26 A . The optically redirecting component can be configured to redirect the display zero order light to an optical absorber, e.g., the optical absorber  2619  of  FIG.  26 A  or  2649  of  FIG.  26 B , formed on at least one of a side surface of the second substrate or a side surface of the substrate. The second substrate can include an anti-reflective (AR) coating, e.g., the AR coating  2682  of  FIG.  26 D , on a surface of the second substrate opposite to the optically redirecting component. The anti-reflective coating is configured to transmit the display zero order light to prevent Fresnel reflection of the display zero order light. An anti-reflective coating can also be configured to reduce or eliminate reflections of ambient light from the viewer and the environment reflected off the viewer-facing front surface of the second substrate, e.g., the AR coating  2682  of  FIG.  26 D . The final AR coating can be designed such that it does not interfere with those of the five techniques as described herein which depend upon the properties of the final transition into air on the viewer’s side. Preventing Fresnel reflection from the front surface prevents the viewer seeing themselves and room lights mirrored by the front surface. Deeper surfaces within the optical device involve only comparatively small refractive index changes, and hence minimal Fresnel reflection of the observer and room light back towards the viewer, or the surfaces can also be AR coated, or, as in the case of the rear-reflector of the display, the surfaces are behind multiple absorptive layers, such as linear polarizers, through which ambient illumination can make a double pass and be hence attenuated, an effect which can be enhanced by adding to or incorporating within the device a layer of a material with a cumulative optical density in the range 0.2 to 1.0. 
     In some implementations, the display zero order light is p polarized before arriving at the second substrate. As illustrated in  FIG.  27 A , the optically redirecting component can be configured to diffract the display zero order light to be incident at a Brewster’s angle on an interface between the second substrate and a surrounding medium, e.g., air, such that the display zero order light totally transmits through the second substrate. 
     In some implementations, the display zero order light is s polarized before arriving at the second substrate. The process  3400  can further include: converting a polarization state of the display zero order light from s polarization to p polarization. In some examples, converting the polarization state of the display zero order light is by an optical retarder (e.g., the optical retarder  2747  of  FIG.  27 B ) (and optionally a linear polarizer) arranged upstream the optically redirecting component with respect to the display. In some examples, converting the polarization state of the display zero order light is by an optical retarder (e.g., the optical retarder  2747  of  FIG.  27 C ) (and optionally a linear polarizer) arranged downstream the optically redirecting component with respect to the display. The optical retarder can be formed on a side of the second substrate opposite to the optically redirecting component, and the optical retarder can be covered by a third substrate (e.g., the retarder cover glass  2762  of  FIG.  27 C ). 
     In some implementations, as illustrated in  FIG.  28   , an optically blocking component is formed on a side of the second substrate opposing to the optically redirecting component. The optically blocking component is configured to transmit the diffracted first portion of the light and to absorb the display zero order light diffracted by the optically redirecting component. In some examples, the optically blocking component includes an anisotropic transmitter (e.g., the anisotropic transmitter  2820  of  FIG.  28   ) configured to transmit a first light beam with an angle smaller than a predetermined angle, and absorb a second light beam with an angle larger than the predetermined angle. The predetermined angle is larger than half of the viewing angle and smaller than a diffraction angle at which the display zero order light is diffracted by the optically redirecting component. 
     In some implementations, as illustrated in  FIG.  29   , the optically redirecting component is configured to diffract the display zero order light to be incident with an angle larger than a critical angle on an interface between the second substrate and a surrounding medium, such that the display zero order light diffracted by the optically diffractive component is totally reflected at the interface. An optical absorber, e.g., the optical absorber  2920  of  FIG.  29   , can be formed on side surfaces of the substrate and the second substrate and configured to absorb the totally reflected display zero order light. 
     In some implementations, as illustrated in  FIGS.  30 A to  33   , the light includes a plurality of different colors of light, and the optically diffractive component is configured to diffract the plurality of different colors of light at the incident angle on the display. The optical redirecting component comprises a respective optically redirecting subcomponent for each of the plurality of different colors of light. 
     In some implementations, as illustrated in  FIG.  30 B , the respective optically redirecting subcomponents for the plurality of different colors of light are recorded in a same recording structure, or in recording structure which are adjacent and separated only by a thin optical indexing, contacting, or adhesive layer. In some implementations, as illustrated in  FIGS.  30 A,  31 A,  31 B,  32 ,  33   , the respective optically directing subcomponents for the plurality of different colors of light are recorded in different corresponding recording structures which may be separated by cover glasses. 
     The optical redirecting component can be configured to diffract the plurality of different colors of light at different diffraction angles towards different directions in a 3D space. In some examples, as illustrated in  FIGS.  31 A- 31 B , the optical redirecting component is configured to diffract at least one of the plurality of different colors of light to be incident at at least one Brewster’s angle at an interface. The interface can include one of an interface between a top substrate and a surrounding medium or an interface between two adjacent substrates. 
     In some implementations, as illustrated in  FIG.  32   , the optical redirecting component is configured to diffract a first color of light (e.g., blue) and a second color of light (e.g., red) within a plane, and a third color of light (e.g., green) orthogonal to the plane. 
     In some implementations, as illustrated in  FIG.  31 B , the optical redirecting component includes at least two different optically redirecting subcomponents (e.g., the redirection gratings  3166 - 1 ,  3166 - 2  of  FIG.  31 B ) configured to diffract a same color of light of the plurality of different colors of light. The two different optically redirecting subcomponents can be sequentially arranged in the optical redirecting component. 
     Guiding the light to illuminate the display can include sequentially guiding the plurality of different colors of light to illuminate the display in a series of time periods. In some implementations, as illustrated in  FIG.  33   , the optical redirecting component can include a switchable optically redirecting subcomponent (e.g., the switchable green redirection grating  3316  of  FIG.  33   ) configured to diffract a first color of light at a first state during all, part, or parts of a first time period and transmit a second color of light at a second state during all, part, or parts of a second time period. 
     In some implementations, the switchable optically redirecting subcomponent is configured to diffract a first color of light at a first state during all, part, or parts of a first time period and diffract a second color of light at a second state during all, part, or parts of a second time period. 
     The plurality of different colors of light can include a first color of light and a second color of light, the first color of light having a shorter wavelength than the second color of light. In the optically redirecting component, a first optically redirecting subcomponent for the first color of light can be arranged closer to the display than a second optically redirecting subcomponent for the second color of light, as illustrated in  FIGS.  30 A to  33   . 
     In some implementations, fringe planes of at least two optically redirecting subcomponents for at least two different colors of light are oriented substantially differently. 
     In some implementations, the optically redirecting component includes: a first optically redirecting component configured to diffract a first color of light, a second optically redirecting component configured to diffract a second color of light, and at least one optical retarder (and optionally a linear polarizer) arranged between the first and second optically redirecting subcomponent and configured to convert a polarization state of the first color of light such that the first color of light transmits through the second optically redirecting component. 
     The reflected second portion of the light has a reflected angle identical to the incident angle and propagates outside of the holographic scene. In some examples, a half of the viewing angle is within a range from -10 degrees to 10 degrees or a range from -5 degrees to 5 degrees. In some examples, the incident angle is -6 degrees or 6 degrees. 
     In some implementations, the optical redirecting component is configured to allow the display zero order light to pass through unchanged, and redirect the diffracted first portion of the light to form a holographic scene corresponding to a cone or frustum having a predetermined angle, which is away from the display zero order light. 
     In some implementations, the optical redirecting component is configured to redirect the display zero order light towards a first direction and redirect the diffracted first portion of the light towards a second direction away from the first direction. For example, the diffracted first portion of the light can be redirected to be normal to a wedged surface of a substrate, and the display zero order light can be redirected to hit the wedged surface beyond a critical angle and hence undergo total-internal-reflection (TIR) back into the substrate. 
     Additional Aspects of Displaying Reconstructed Three-dimensional Objects 
     Implementations of the present disclosure provide a display system for displaying reconstructed three-dimensional (3D) objects in a holographic light field, e.g., the holographic light field  518  of  FIG.  5 A ,  528  of  FIG.  5 B ,  538  of  FIG.  5 C ,  548  of  FIG.  5 D ,  568  of  FIG.  5 E ,  578  of  FIG.  5 F ,  599 - 1  or  599 - 2  of  FIGS.  5 H,  5 I,  5 J, or  5 K ,  2422  of  FIG.  24   ,  2622  of  FIG.  26 A ,  2632  of  FIG.  26 B ,  2652  of  FIGS.  26 C,  26 D or  26 E ,  2702  of  FIG.  27 A ,  2732  of  FIGS.  27 B or  27 C ,  2802  of  FIG.  28   ,  2902  of  FIG.  29   ,  3022  of  FIGS.  30 A or  30 B ,  3122  of  FIGS.  31 A or  31 B or  33    , or  3220 ,  3222 ,  3224  of  FIG.  32   . Techniques described herein can improve one or more characteristics (e.g., size or zero order suppression) of the holographic light field to thereby improve a performance of the display system, e.g., by using larger reflective displays, using larger gratings, and/or controlling input light. For illustration purpose only, the techniques are discussed with reference to the system  3100  in  FIG.  31 A . 
     First Exemplary Method - Using Larger Reflective Displays 
     One method to increase a size of the holographic light field  3122  of  FIG.  31 A  is to build the same optical geometry using a larger reflective display  3104  and a proportionately larger substrate  3111  with unchanged beam angles. 
     As the linear extent of the reflective display  3104  increases, the front-area of the substrate  3111  increases as a square of the increase in the linear extent of the reflective display  3104 . If the beam angles and beam distributions remain unchanged, then the thickness of the substrate  3111  increases as the increase in the linear extent of the reflective display  3104 . As a result, a volume of the substrate  3111  can increase as a cube of the increase in the linear extent of the reflective display  3104 . For example, doubling the width of the reflective display  3104 , while maintaining the same width-to-height aspect ratio of the reflective display  3104  and a proportional thickness of the substrate  3111 , quadruples the front-area of the substrate  3111  and increases the volume of the substrate  3111  by a factor of eight. Eventually the large thickness and the high cost of the substrate  3111  may become undesirable, e.g., because it may be desirable that the substrate  3111  maintains an optical-grade clarity, substantially free from significant inclusions, absorption, scatter, birefringence, and/or other visible optical defects or imperfections. 
     The weight of the substrate  3111  also may become undesirable. For example, the substrate  3111  may have a thickness of approximately 20% of the height of the reflective display  3104 . As an example, for a 686 mm (27″) diagonal reflective display  3104  with a 16:9 aspect-ratio (typical dimensions for a computer monitor), the substrate  3111  may have dimensions of 598 mm x 336 mm x 68 mm or greater. If such a substrate  3111  were made from a solid block of acrylic with a density of 1.17 to 1.20 g/cm 3 , the weight of substrate  3111  could be at least 16 kg (35 pounds). For a similar 1,650 mm (65″) diagonal reflective display  3104  with the 16:9 aspect-ration, the substrate  3111  can be at least 165 mm thick and weigh at least 225 kg (495 pounds), which can be challenging to ship, install, and move. Mounting and support structures for such a block of acrylic may also be large and heavy. 
     Further, if all or part of the holographic light field  3122  is projected into a viewing space in front of the final cover glass  3113 , then it may be desirable for the holographic light field  3122  to be positioned proportionately further in front of the front cover glass  3113  (e.g., more than 165 mm in front of the reflective display  3104  with a 1,650 mm diagonal). This could reduce its field of view and resolution. If a lesser, zero, or negative z-axis translation is applied, the holographic light field  3122  may appear deeper behind the front surface of the front cover glass  3113 . 
     To address the above issues, the substrate  3111  can be made thinner, which may reduce its mas, cost, and cause the substrate to have lesser constraints on its z-position and field of view. 
     In some embodiments, the substrate  3111  can be made of a material with a lower density and/or with a refractive index permitting more extreme angles and beam-angle changes for the beams entering, within, and exiting the substrate  3111 . For example, a liquid-filled substrate  3111  can be used with a liquid, e.g., water or oil, with a refractive index that can be smaller (e.g., 17% to 20% smaller) than a refractive index of acrylic. The liquid can be enclosed in a tank, which may help resolve certain potential shipping and installation issues because the tank can be transported empty and then filled in situ. 
     In certain embodiments, the angle of the input light  3120  as refracted into the substrate  3111  can be increased for one or more wavelengths of the input light  3120 . This can allow for the use of a relatively thin substrate  3111  for the input light  3120 , e.g., to illuminate a same area of the reflective display  3104 . In some cases, it may be desirable to choose the angle(s) to achieve a particular diffraction efficiency and/or to meet desired critical-angle properties. 
     In some embodiments, the substrate  3111  can be wedged, e.g., similar to the substrate  1252  of  FIG.  12 B  or the substrate  1272  of  FIG.  12 C , such that incident angles of the input light  3120  on the field grating  3112  can be relatively large. 
     In certain embodiments, two or more illuminators can be used to illuminate different regions of the reflective display  3104 , e.g., respectively from upper and lower directions. For example, a first illuminator  3106  providing first input light  3120  into a first edge-face of the substrate  3111  (e.g., a lower edge-face of substrate  3111 ) can be used to illuminate only a first region (e.g., a lower half) of the reflective display  3104 . A second illuminator (which can be similar to the first illuminator  3106 ) providing second input light (which can be similar to the first input light  3120 ) into a second edge-face of the substrate  3111  (e.g., an upper edge-face of the substrate  3111 ) can be used to illuminate only a second region (e.g., an upper half) of the reflective display  3104 . Such an arrangement can allow the reflective display  3104  to be fully illuminated while allowing the substrate  3111  to be relatively thin (e.g., allowing the thickness of the substrate  3111  to be halved). Optionally, a third, fourth, or greater number of input lights, each entering through a different corresponding edge-face of the substrate  3111  (e.g., left and right edge-faces of the substrate  3111 ), can be used to illuminate, respectively, regions (e.g., a left region and a right region, respectively) of the reflective display  3104 . 
     In some embodiments, input light can illuminate different regions of the reflective display  3104  along different optical paths. For example, a first illuminator  3106 , providing first input light  3120  into an edge-face of the substrate  3111  (e.g., a lower edge-face of the substrate  3111 ) and directly illuminating the transmissive field grating  3112 , can be used in combination with a second illuminator, providing second input light into an edge-face of substrate  3111  (which may be the same edge-face as used by the first input light) but with the second input light being initially directed forwards towards the redirection grating  3114  and subsequently being reflected back towards the transmissive field grating  3112  such that the first input light illuminates a first region (e.g., an upper half) of the reflective display  3104  and the second input light illuminates a second adjacent region (e.g., a lower half) of the reflective display  3104 . Such reflection of the second input light may be achieved by using total internal reflection (TIR) or a reflective grating at a surface of or prior to the redirection grating  3114  (e.g., by an interface between the substrate  3111  and the redirection grating  3114 ). Alternatively, a partially reflective surface (e.g., a 50:50 or gradient or patterned beamsplitter) can be incorporated into the substrate  3111  to split a single input light  3120  within the substrate  3111  into two beams, including a first beam proceeding directly to the transmissive field grating  3112  with a reduced optical power and a second beam initially proceeding away from the transmissive field grating  3112 , also with reduced optical power, and subsequently being directed back towards the transmissive field grating  3112 , e.g., by TIR or a reflective grating at a surface of or prior to the redirection grating  3114 . 
     In certain embodiments, the diffraction efficiency of the transmissive field grating  3112  may be patterned such that, when the input light  3120  first encounters a sub-region of the transmissive field grating  3112 , only a chosen percentage of the input light  3120  is diffracted out towards the reflective display  3104 , while all or part of the remainder of the input light  3120  is reflected back into the substrate  3111 . The reflected input light  3120  in the substrate  3111  is further reflected by TIR off, for example, the front surface of the substrate  3111  back towards a second sub-region of the transmissive field grating  3112  which couples out a second portion towards the reflective display  3104  with a diffraction efficiency adjusted such that two such regions of the transmissive field grating  3112  illuminate two corresponding sub-regions of the reflective display  3104  with a substantially similar optical power. The above process can be extended to three or more such sub-regions of the transmissive field grating  3112  and accordingly three or more corresponding sub-regions of the reflective display  3104 . 
     In some embodiments, light not initially diffracted to a reflective display is recycled to illuminate the reflective display. For example, the diffraction efficiency of the transmissive field grating  3112  can be patterned or chosen such that, when the input light  3120  first encounters a first sub-region of the transmissive field grating  3112 , only a chosen percentage of such input light  3120  is diffracted out towards the reflective display  3104 , while all or part of the remainder of the input light  3120  is reflected back into the substrate  3111 . The reflected input light  3120  can eventually make its way (e.g., by TIR within substrate  3111  or via a direct path) to a reflective element attached to or subsequent to an edge face of the substrate  3111  (e.g., a mirror or a reflective grating in place of the absorber  1203  of  FIG.  12 B ) which reflects it back through the substrate  3111  to reilluminate (directly, or after further TIR or diffractive redirections) the first sub-region of the transmissive field grating  3112  or a second sub-region of the transmissive field grating  3112  where the sub-region of the transmissive field grating  3112  diffracts it out towards the reflective display  3104 . 
     In some embodiments, each of sub-regions of the reflective display  3104  is made of an individual display device (e.g., LCoS) or any other reflective display device, and the reflective display  3104  is formed by a tiled array of smaller display devices. This can allow differences in diffraction efficiency and hence in device illumination for each sub-region of the transmissive field grating  3112  to be compensated for by operating such smaller display devices with different reflectivities. 
     In certain embodimnets, a relatievly high aspect ratio of the width to the height of the reflective display is used to increase the size of the holographic light field. Becuase the thickness of the substrate  3111  generally depends on illuminated height of the reflective display  3104  but not on the illuminated width of the reflective display  3104 , the thickness of the substrate  3111  does not have to be increased if the aspect ratio of the reflective display  3104  is increased such that its width is increased without necessarily a corresponding increase in its height. For example, rather than the 16:9 aspect ratio of width:height, an aspect ratio of 20:9 may be used. Increasing the aspect ratio of the reflective display  3104  in this manner can increase the size of a holographic light field, because the viewer typically has two eyes in a predominantly horizontal arrangement, affording stereopsis. 
     In some cases, when multiple viewers observe the holographic light field display at the same time, the viewers are likely to be positioned side-by-side (rather than one looking over the head of the other), so the wider field of view afforded by a high-aspect ratio can be suitable for group viewing. Further, empirically it has been observed that most viewers of holographic light fields, e.g., casual viewers, are more likely to move their heads from side to side rather than up and down, so again a higher aspect ratio with a wider width can be implemented to increase the performance of the system. 
     In some cases, a useful and pleasing holographic light field display may have a very high aspect ratio (a strip or slit display). A wider aspect ratio can be achieved with a comparatively thin substrate  3111 , e.g., if gratings  3112 ,  3114 ,  3116 , and  3118  are tiled in the horizontal direction. 
     In general, irrespective of the aspect ratio of the reflective display  3104  (and hence of the substrate  3111  and the gratings  3112 ,  3114 ,  3116 , and  3118 ), it is desirable for the width of input light  3110  to be sufficient to illuminate the width of the reflective display  3104  (and the width of the substrate  3111  and the gratings  3112 ,  3114 ,  3116 , and  3118 ). For low aspect ratios of the reflective dispaly  3104 , the input light  3120  can have a mildly extended rectangular profile or cross-section (or even a square profile or cross-section), which can be implemented by masking or otherwise truncating a sufficiently large circular or elliptical beam profile from the illuminator  3106 . 
     Second Exemplary Method - Using Larger Gratings 
     If the reflective display  3104  and the substrate  3111  are enlarged, then the transmissive field grating  3112  and the display zero-order redirecting gratings  3114 ,  3116 , and  3118  can also be enlarged to match. 
     In some embodiments, the transmissive field grating  3112  can be split into two or more regions, each utilizing an input light entering substrate  3111  through a different edge face of the substrate  3111  as noted above. 
     In certain embodimetns, larger gratings  3112 ,  3114 ,  3116 , and  3118  can be produced by enlarging corresponding optical elements and recording materials of their respective production systems. 
     In some embodiments, larger gratings  3112 ,  3114 ,  3116 , and  3118  can be produced by tiled optical-recording, in which sub-regions of each of the gratings can be recorded in sequence using smaller optical elements and full-sized recording materials in a step-and-repeat process. This can allow fore the use of smaller optical components, which are often relatively inexpensive. Additionally or alternatively, this can allow for the use of lower recording powers (e.g., rather than increasing recording exposure durations), which can allow for the use of relatively inexpensive recording laser sources, and/or a relative large range of laser technologies, wavelengths, and vendors available to provide such sources. Such tiled-gratings also may be used to provide multiple regions for enlarging the transmissive field grating  3112  using multiple input lights. 
     Edges of the tiled sub-regions of gratings can abut each other with a slight gap between the sub-regions of the gratings. Optionally, the sub-regions can join seamlessly, or the sub-regions can overlap slightly or substantially. Combinations of such approaches are possible. In some cases, slight gaps can be invisible or may have low visibility to the viewer. For example, when the holographic light field  3122  occupies optical distances from the viewer which do not include the optical distance of the grating from the viewer, the gaps may be out-of-focus when the viewer’s eyes are focused on the holographic light field  3122 . In certain cases, slight overlaps may have little or no visibility to the viewer. Substantial overlaps, e.g., a 50% overlap, between two sub-regions of the gratings may be implemented to smooth and/or reduce the visibility of the tiling and/or to improve the net uniformity of the overlapped gratings. 
     In some cases, to reduce the visibility of such slight gaps or overlaps between the tiled sub-regions of gratings, the sub-regions of gratings can be aligned with gaps between smaller display devices forming the reflective display  3104  as a tiled array of smaller display devices. 
     In some cases, effectively seamless gratings, with neither a significant gap nor a significant overlap, can be implemented by including one or more edge-defining elements, e.g., a square, a rectangular, or otherwise a plane-tiling aperture, in the optics of the recording reference and/or object beams when recording the gratings for a sub-region, and projecting or re-imaging the edge or edges so formed such that the edges are substantially in a sharp focus within the recording material during the recording of the grating or gratings. Sharply well defined edges can also be achieved, for example, using reflective or transmissive phase masks in the optics of the recording reference and/or object beams when recording the gratings for a sub-region. 
     In some embodiments, larger gratings  3112 ,  3114 ,  3116 , and  3118  can be produced using mechanical rather than optical means, e.g., embossed, nano-imprinted, or self-assembled structures, and such mechanically produced gratings can also be tiled in one or more dimensions, e.g., by the use of roller embossing in a roll-to-roll system. 
     Third Exemplary Method - Controlling Input Light 
     As noted above, as the aspect ratio of reflective display  3104  is increased, a more extended rectangular profile for the input light  3110  can become desirable, and a more elliptical beam profile from the illuminator  3106  can also become desirable. Because many laser-diodes produce elliptical beams, in some cases, the desired beam profile from the illuminator  3106  can be implemented by rotating the ellipticity of laser diode sources within the illuminator  3106 , e.g., by mechanically or optically rotating the laser diode sources within the illuminator  3106 . 
     Because many laser diodes emit substantially polarized light, and because certain other components of the optical device  3110  may perform better for a particular polarization orientation (e.g., may require a particular polarization orientation), it may be desirable to rotate the ellipticity and polarization orientation of light sources within the illuminator  3106  independently, e.g., by using a broad-wavelength-band half-wave retarder to rotate the polarization of all of the input light  3120 , or by using individual narrow-wavelength-band half-wave retarders to rotate the polarization of each color of input light  3120 , separately. Because the profile or cross-section of the input light  3120  may be quite extensive in both width and height, low cost half-wave plates such as polymer waveplates or liquid-crystal waveplates may be more suitable than high cost half-wave plates fabricated form for example quartz. 
     In some embodiments, the uniformity of the input light  3120  can be improved by using apodizing optical elements or profile converters, e.g., arrangements of optical elements like lenses or holographic optical elements (HOEs) or integrating rods to effect, for example, Gaussian to top-hat and/or circular to rectangular profile conversion, or by using polarization recycling elements. 
     In certain embodiments, anamorphic optics can be implemented. The aspect ratio of the reflective display  3104  can be increased to such an extent that the a desired degree of anamophicity of the input light  3120  may exceed a threshold degree which can conveniently be provided by cost effective light sources in the illuminator  3160  without masking off and hence wasting an unacceptable proportion of the light source power. In such cases, the width of the input light  3120  can be further increased by the use of anamorphic optics, e.g., anamorphic lenses or cylindrical lenses, or HOEs performing as anamorphic or cylindrical lenses or mirrors. 
     Exemplary System 
       FIGS.  35 A-C  illustrate an example system  3500  for displaying reconstructed 3D objects.  FIGS.  36 A-C  show the same views of the system  3500  as  FIGS.  35 A-C , respectively, but with three colors of light (e.g., red, green, blue) propagate through the system  3500 . 
     A rectangular section of substantially-coaxial elliptical beams  3501  (as illustrated in  FIG.  36 A ) from an illuminator  3501 S (e.g., made of three laser diodes for three different colors such as red, green, and blue) is reflected off a mirror  3502  and then refracted into a first face  3503  of a prism element  3504 . The beams  3501  have a width defined between an upper beam and a lower beam, as illustrated in  FIG.  36 A . The different colors of light beams refracted into the prism element  3504  can be stacked together along a first direction (e.g., as illustrated in  FIG.  36 A ) and spaced from (or overlapping with) one another along a second direction (e.g., as illustrated in  FIG.  36 B ). A second surface  3505  of the prism element  3504  reflects the beams to a third surface  3506  of the prism element  3504  on which one or more transmissive expansion gratings  3507  are optically stacked (generally, one grating per color). Each expansion grating is illuminated by its corresponding color at a relatively high angle of incidence within the prism element  3504 , for example 68°, and is configured to diffract a portion of its illuminating light out towards a series of reflectors  3508 . In effect, the gratings  3507  expand the original rectangular section of light beams  3501  from the laser diodes by a substantial factor (e.g., a factor of approximately 6) in one dimension (e.g., in width as illustrated in  FIG.  36 A ). Light beams reflected by the third surface  3506  and/or the expansion gratings  3507  and/or a cover layer applied to the expansion gratings  3507 , back into the prism element  3504  can be absorbed by an absorptive layer  3504 A applied to a surface of the prism element  3504  (e.g., as illustrated in  FIG.  36 A ). 
     Because the light incident upon the gratings  3507  is incident at a high angle, the depth of prism element  3504  (e.g., the length of its face  3505 , part of which at least is reflective) can be comparatively small. The incidence angle can exceed criticality if the light is incident from air (refractive index ∼1.0) upon the gratings  3507  at such a large angle, causing all of the incident light to reflect away from the gratings. In the system  3500 , the light is incident from the prism element  3504  that can be made of, for example, glass or acrylic with a high refractive index (e.g., ∼1.5), and thus, the incident angle does not exceed the critical angle. 
     In some embodiments, the reflectors  3508  can include three dichroic reflectors, one per color, or two dichroics and a mirror for one color, or one dichroic reflector for two colors and a mirror for one color, that are arranged in the beam (all three colors)  3509  diffracted out by expansion gratings  3507 , to reflect each color into a cover plate  3510  attached to a shaped substrate  3511 . Each color of light is incident on the cover plate  3510  at a different angle and over a different region of the cover plate  3510 , and is refracted into the cover plate  3510  (and thereafter into the shaped substrate  3511 ) at such angles that the colors of light subsequently are reflected off, for example, a low-index layer formed on the front face  3512  of the shaped substrate  3511 , then diffracted out of three stacked field gratings (one per color)  3513  attached to the back face  3514  of the shaped substrate  3511 . All three colors of light are incident on an array of reflective display devices  3515  at substantially the same angle for each color and with each color illuminating substantially the entirety of the reflective area formed by one or more reflective display devices  3515 . The refletive display devices reflect and diffract each color back through the field gratings  3513 , through the shaped substrate  3511 , and into a stack of three stacked display (e.g., LCoS) Zero-order Suppression (LZOS) gratings  3516  (one per color) (elsewhere herein referred to as redirection gratings, e.g., redirection gratings  3114 ,  3116 , and  3118  of  FIG.  31 A ) attached to the front face  3512  of the sub strate  3511 . 
     A proportion of each color incident on the reflective display devices  3515  is reflected into a display zero-order beam  3521 , and a proportion of each color which is incident upon each display device (e.g., LCoS) is diffracted by each display device into a corresponding holographic light field  3522 , e.g., the holographic light field  3220 ,  3222 ,  3224  of  FIG.  32   , which may be seen by a viewer. As discussed elsewhere herein, the display zero-order suppression gratings (or redirection gratings)  3516  are angle-selective transmission gratings which substantially diffract light incident upon them at the display zero-order angle but substantially transmit light incident upon them at greater or lesser angles, separating the reflected display zero-order light from the diffracted holographic light field. The rejected display zero-order light  3523  may exit the front of the redirection gratings at a substantial angle as shown in  FIG.  36 B , or may be reflected back into the shaped substrate  3511  by TIR or by reflection gratings as described elsewhere herein. 
     In some embodiments, the tilt angle of the reflective elements  3508  can be adjusted to achieve greater uniformity of diffraction from the transmissive field gratings  3513  (e.g., by causing the transmissive field gratings  3513  to be illuminated at or close to their replay Bragg angles), and/or to achieve greater brightness of diffraction from the transmissive field gratings  3513  (e.g., by causing the transmissive field gratings  3513  to be illuminated at or close to their replay Bragg angles). Such adjustments can be made substantially independently for each color by adjusting the tilt angle of a respective one of the reflective elements  3508 . 
     In some embodiments, the adjustments can be made as a one-off adjustment during manufacture or assembly. Optionally, the adjustments can be made by the user or installer in the field. In certain embodiments, the adjustments can be performed automatically, for example as part of a feedback loop utilizing color and/or brightness sensors to detect and optimize optical properties of the holographic light field, e.g., brightness, uniformity, color uniformity, or white-point. In some cases, the tilt angles of the reflective elements  3508  orthogonal to the tilt angles shown in  FIG.  35 B  are adjusted to optimize the performance of the display system  3500 . These approaches can be combined as appropriate. 
     In some cases, tilt adjustments of the reflective elements  3508  can be used to correct for changes or errors in alignment of the components of the display system caused by factors, e.g., manufacturing and assembly tolerances, shipping, storage, and in-use vibration and shock, thermal expansion and contraction, aging of the gratings, laser-diodes or other wavelength-dependent components, and wavelength shifts of the laser-diodes due to aging, operating temperature, operating duty cycle, and/or part-to-part variations. 
     In some cases, substantially larger or substantially smaller tilt adjustments of the reflective elements  3508  can be used to maintain alignment even if the angle between the expansion prism  3504  and the shaped substrate  3511  is changed substantially from 90° (as shown in  FIG.  35 B ) for example by tilting or rotating the shaped substrate  3511  backwards or forwards to tilt the holographic light field respectively upwards or downwards. 
     To achieve relatively uniform illumination on the reflective display  3515 , the centers of the beams from the laser diodes can be offset, which can also maintain color uniformity in the holographic light field. Small differences in the path travelled by each color to and from the display devices  3515  (in general, primarily due to chromatic dispersion of the beams), for example at their entry into prism element  3504 , can otherwise slightly misalign the concentrations of the three colors. This can also be corrected for by adjusting the diffraction efficiency of the reflective display devices  3515  in a spatially variant manner (e.g., in one or two dimensions). Such adjustment can be made on-the-fly as the diffraction efficiency is a function of computer generated holograms (CGHs), or by utilizing elements before or after the display devices  3515  with constant or adjustable spatially varying transmissivities or absorbances (e.g., in one or two dimensions). 
     In some cases, input light  3517  (e.g., as illustrated in  FIG.  36 B ) into the substrate  3511  can be p-polarized at the edge surface of the substrate  3511  where the input light  3517  enters the substrate  3511  (or the cover glass  3510  if used) to reduce Fresnel losses at the surface, or the surface can be tilted or anti-reflection coated to reduce such Fresnel losses. A broad-wavelength-band halfwave retarder affixed to the surface or subsequent to the surface can convert such p-polarization to s-polarization if s-polarization light is the required or desired polarization for the transmissive field grating  3513 . 
     In some cases, a broad-wavelength-band retarder positioned between the transmissive field grating  3513  and the reflective display devices  3515  can be used to further adjust the polarization of illumination light upon the reflective display device  3515  to provide the required or desired or optimal polarization state for the reflective display devices  3515 . Such a retarder can be affixed to the exit face of the field grating  3513 , or to the outer surface of the reflective display devices  3515 , or to both, and can be a halfwave plate to provide p-polarization or s-polarization or can be a quarterwave plate to provide circular polarization or can have a retardance of another value, which can also vary spatially and/or temporally and/or by wavelength, to provide optimal polarization at every point on the reflective display devices  3515  for each color. In so far as such a waveplate provides a polarization state, for the reflected holographic light field from the reflective display devices  3515 , which may be not the desired or optimal polarization state for subsequent polarization-dependent elements, e.g., redirection gratings  3516 . In some cases, one or more further waveplates can be provided prior to such an element or elements with fixed or with spatially or temporally or chromatically varying retardances to further adjust the polarization to satisfy the element or elements. 
     In some cases, an optical distance between the substrate  3511  and the coupling reflective elements  3508  can be proportionately large to allow the three colors of light to be separated further at their reflections of the reflective elements  3508  so that each color can be reflected by a corresponding reflective element without having to be transmitted through one or two other reflective elements, or even made so large that the three colors of light separate enough to be reflected using three mirrors with no transmissions through other reflective elements. 
     In certain embodiments, the coupling reflective elements  3508  can be positioned and tilted such that the illumination of each of the reflective elements  3508  comes from a substantially different direction rather than from substantially optically-coaxial laser beams. This may allow the illuminator  3501 S to be split into two or three separate illuminators each providing one or two of the three illumination colors, which can be cheaper and/or more efficient than using optics within the illuminator  3501 S to combine the light from three laser diodes into a combined white input light which provide input light  3501 . 
     In some embodiments, the shaped substrate  3511  can be formed monolithically, e.g., by computer numerical control (CNC) machined from a larger block of material, can be formed by optically bonding or indexing two or more simpler (and hence more manufacturable) shapes, or can be formed by additive or subtractive manufacturing techniques. 
     In certain embodiments, the reflective display  3515  (or an array of reflective display devices  3515 ) with a greater vertical extent can be illuminated by increasing the height of the input light  3517 , which is subject to the input light  3517  actually entering the cover glass  3510  (which may be omitted) at the tip of the shaped substrate  3511  that forms a first lower cutoff for display illumination, and subject to the input light  3517  missing a corner  3518  of the shaped substrate  3511  that forms an upper cutoff and a second lower cutoff for display illumination. 
     In some embodiments, the illumination of the reflective display  3515  is at an angle of approximately 6°, which can be changed to approximately 0° because the transmissive field grating  3513  can also act as a zero-order suppression element, similar to the redirection gratings 3516. In such embodiments, the field grating  3513  can reflect rather than transmit, entrapping specularly-reflected zero-order light from the reflective display  3515  within the shaped substrate  3511 , where TIR can guide it up and out of the top of the shaped substrate  3511  or into an absorber  3524  formed thereupon. Using the field grating  3513  at or near 0° in combination with the redirection gratings  10016  can reduce residual display zero-order to a very high degree, e.g., less than 2% residual display zero-order light or even &lt; 1 %. 
     In certain embodiments, when one-dimensional suppression gratings are used, the display zero-order suppression appears as a dark band across the reflective display  3515 , not a point, with the zero-order of each illumination color just visible as a point of that color within this dark band. If the viewer is more likely to look into the reflective display  3515  from above the normal to the reflective display  3515 , as is commonly the case for a desk or table display, then the system can be configured to arrange the band to be above (but, in angular-space, close to) the holographic light field, where it is less likely to be noticed or objectionable, rather than below or on either side of the holographic light field. Similarly, if the viewer is more likely to look into the display from below the normal to the reflective display  3515 , then the system can be configured to arrange the band to be below the holographic light field. If most viewers look into the display using two eyes distributed predominantly horizontally, then the band can be arranged at up or below, instead of left or right, of the holographic light field. 
     In some embodiments in which the illuminator  3501 S derives from light sources with spectral bandwidths on an order of a few nm or a few tens of nm, diffraction in the expansion gratings  3505  and the field gratings  3507  can spectrally disperse the illumination light incident upon the reflective display  3515 . The illumination light can then exhibit spectral diversity (from the spectral bandwidths of the laser diodes) and spatial diversity (from the dispersion of light from the laser diodes by these gratings, and, to a lesser extent, from the source size of the laser diodes). These multiple orthogonal degrees of diversity can cause significant reduction in visible laser speckle in the holographic light field, compared to those provided just by the spectral and spatial diversity of the laser diodes themselves. 
     In some embodiments, expansion gratings  3505  can be formed with an optical power such that the expanson gratings  3505  can fully or partially collimate the input light  3501  in one or two transverse directions, reducing or eliminating the need for laser-diode collimation in the illuminator  3501 S. 
     The incidence angles of the input light  3517  upon the cover plate  3510  may be selected such that two or more such incidence angles are substantially equal, and in this case the number of reflective elements  3508  may be reduced since a single such reflective element may suffice to reflect two or more colors. Further, the final reflective element in  3508  may be provided as a reflective coating upon a surface of, or within the substrate of, the previous reflective element, which substrate may be wedged to provide a different reflection angle for this final reflector. 
     Implementations of the subject matter and the functional operations described in this specification can be implemented in digital electronic circuitry, in tangibly embodied computer software or firmware, in computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. Implementations of the subject matter described in this specification can be implemented as one or more computer programs, such as, one or more modules of computer program instructions encoded on a tangible, non-transitory computer-storage medium for execution by, or to control the operation of, data processing apparatus. Alternatively or in addition, the program instructions can be encoded on an artificially generated propagated signal, such as, a machine-generated electrical, optical, or electromagnetic signal that is generated to encode information for transmission to suitable receiver apparatus for execution by a data processing apparatus. The computer-storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, or a combination of one or more of them. 
     The terms “data processing apparatus,” “computer,” or “electronic computer device” (or equivalent as understood by one of ordinary skill in the art) refer to data processing hardware and encompass all kinds of apparatus, devices, and machines for processing data, including by way of example, a programmable processor, a computer, or multiple processors or computers. The apparatus can also be or further include special purpose logic circuitry, for example, a central processing unit (CPU), an FPGA (field programmable gate array), or an ASIC (application-specific integrated circuit). In some implementations, the data processing apparatus and special purpose logic circuitry may be hardware-based and software-based. The apparatus can optionally include code that creates an execution environment for computer programs, for example, code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them. The present specification contemplates the use of data processing apparatuses with or without conventional operating systems. 
     A computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a computing environment. A computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data, for example, one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, for example, files that store one or more modules, sub-programs, or portions of code. A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a communication network. While portions of the programs illustrated in the various figures are shown as individual modules that implement the various features and functionality through various objects, methods, or other processes, the programs may instead include a number of sub-modules, third-party services, components, libraries, and such, as appropriate. Conversely, the features and functionality of various components can be combined into single components as appropriate. 
     The processes and logic flows described in this specification can be performed by one or more programmable computers executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, such as, a CPU, a GPU, an FPGA, or an ASIC. 
     Computers suitable for the execution of a computer program can be based on general or special purpose microprocessors, both, or any other kind of CPU. Generally, a CPU will receive instructions and data from a read-only memory (ROM) or a random access memory (RAM) or both. The main elements of a computer are a CPU for performing or executing instructions and one or more memory devices for storing instructions and data. Generally, a computer will also include, or be operatively coupled to, receive data from or transfer data to, or both, one or more mass storage devices for storing data, for example, magnetic, magneto-optical disks, or optical disks. However, a computer need not have such devices. Moreover, a computer can be embedded in another device, for example, a mobile telephone, a personal digital assistant (PDA), a mobile audio or video player, a game console, a global positioning system (GPS) receiver, or a portable storage device, for example, a universal serial bus (USB) flash drive, to name just a few. 
     Computer readable media (transitory or non-transitory, as appropriate) suitable for storing computer program instructions and data include all forms of non-volatile memory, media and memory devices, including by way of example semiconductor memory devices, for example, erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), and flash memory devices; magnetic disks, for example, internal hard disks or removable disks; magneto-optical disks; and CD-ROM, DVD- R, DVD-RAM, and DVD-ROM disks. The memory may store various objects or data, including caches, look-up-tables, classes, frameworks, applications, backup data, jobs, web pages, web page templates, database tables, repositories storing business and dynamic information, and any other appropriate information including any parameters, variables, algorithms, instructions, rules, constraints, or references thereto. Additionally, the memory may include any other appropriate data, such as logs, policies, security or access data, reporting files, as well as others. The processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry. 
     To provide for interaction with a user, implementations of the subject matter described in this specification can be implemented on a computer having a display device, for example, a cathode ray tube (CRT), liquid crystal display (LCD), light emitting diode (LED), holographic or light field display, or plasma monitor, for displaying information to the user and a keyboard and a pointing device, for example, a mouse, trackball, or trackpad by which the user can provide input to the computer. Input may also be provided to the computer using a touchscreen, such as a tablet computer surface with pressure sensitivity, a multi-touch screen using capacitive or electric sensing, or other type of touchscreen. Other kinds of devices can be used to provide for interaction with a user as well; for example, feedback provided to the user can be any form of sensory feedback, for example, visual feedback, auditory feedback, or tactile feedback; and input from the user can be received in any form, including acoustic, speech, or tactile input. In addition, a computer can interact with a user by sending documents to and receiving documents from a device that is used by the user; for example, by sending web pages to a web browser on a user’s client device in response to requests received from the web browser. 
     The term “graphical user interface,” or “GUI,” may be used in the singular or the plural to describe one or more graphical user interfaces and each of the displays of a particular graphical user interface. Therefore, a GUI may represent any graphical user interface, including but not limited to, a web browser, a touch screen, or a command line interface (CLI) that processes information and efficiently presents the information results to the user. In general, a GUI may include multiple user interface (UI) elements, some or all associated with a web browser, such as interactive fields, pull-down lists, and buttons operable by the business suite user. These and other UI elements may be related to or represent the functions of the web browser. 
     Implementations of the subject matter described in this specification can be implemented in a computing system that includes a back-end component, for example, as a data server, or that includes a middleware component, for example, an application server, or that includes a front-end component, for example, a client computer having a graphical user interface or a web browser through which a user can interact with an implementation of the subject matter described in this specification, or any combination of one or more such back-end, middleware, or front-end components. The components of the system can be interconnected by any form or medium of wireline or wireless digital data communication, for example, a communication network. Examples of communication networks include a local area network (LAN), a radio access network (RAN), a metropolitan area network (MAN), a wide area network (WAN), worldwide interoperability for microwave access (WIMAX), a wireless local area network (WLAN) using, for example, 902.11 a/b/g/n and 902.20, all or a portion of the Internet, and any other communication system or systems at one or more locations. The network may communicate with, for example, internet protocol (IP) packets, frame relay frames, asynchronous transfer mode (ATM) cells, voice, video, data, or other suitable information between network addresses. 
     The computing system can include clients and servers. A client and server are generally remote from each other and typically interact through a communication network. The relationship of client and server arises by virtue of computer programs running on the respective computers and having a client-server relationship to each other. 
     In some implementations, any or all of the components of the computing system, both hardware and software, may interface with each other or the interface using an application programming interface (API) or a service layer. The API may include specifications for routines, data structures, and object classes. The API may be either computer language-independent or -dependent and refer to a complete interface, a single function, or even a set of APIs. The service layer provides software services to the computing system. The functionality of the various components of the computing system may be accessible for all service consumers via this service layer. Software services provide reusable, defined business functionalities through a defined interface. For example, the interface may be software written in any suitable language providing data in any suitable format. The API and service layer may be an integral or a stand-alone component in relation to other components of the computing system. Moreover, any or all parts of the service layer may be implemented as child or sub-modules of another software module, enterprise application, or hardware module without departing from the scope of this specification. 
     While this specification contains many specific implementation details, these should not be construed as limitations on the scope of any invention or on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations of particular inventions. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub-combination. Moreover, although features may be described as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination. 
     Particular implementations of the subject matter have been described. Other implementations, alterations, and permutations of the described implementations are within the scope of the following claims as will be apparent to those skilled in the art. While operations are depicted in the drawings or claims in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed (some operations may be considered optional), to achieve desirable results. In certain circumstances, multitasking or parallel processing may be advantageous and performed as deemed appropriate. 
     For the sake of brevity, conventional techniques for construction, use, and/or the like of holographic gratings, LCOS devices, and other optical structures and systems may not be described in detail herein. Furthermore, the connecting lines shown in various figures contained herein are intended to represent exemplary functional relationships, signal or optical paths, and/or physical couplings between various elements. It should be noted that many alternative or additional functional relationships, signal or optical paths, or physical connections may be present in an exemplary holographic grating, LCOS, or other optical structure or system, and/or component thereof. 
     The detailed description of various exemplary embodiments herein makes reference to the accompanying drawings and pictures, which show various exemplary embodiments by way of illustration. While these various exemplary embodiments are described in sufficient detail to enable those skilled in the art to practice the disclosure, it should be understood that other exemplary embodiments may be realized and that logical, optical, and mechanical changes may be made without departing from the spirit and scope of the disclosure. Thus, the detailed description herein is presented for purposes of illustration only and not of limitation. For example, the steps recited in any of the method or process descriptions may be executed in any suitable order and are not limited to the order presented unless explicitly so stated. Moreover, any of the functions or steps may be outsourced to or performed by one or more third parties. Modifications, additions, or omissions may be made to the systems, apparatuses, and methods described herein without departing from the scope of the disclosure. For example, the components of the systems and apparatuses may be integrated or separated. Moreover, the operations of the systems and apparatuses disclosed herein may be performed by more, fewer, or other components and the methods described may include more, fewer, or other steps. 
     As used in this document, “each” refers to each member of a set or each member of a subset of a set. Furthermore, any reference to singular includes plural exemplary embodiments, and any reference to more than one component may include a singular exemplary embodiment. Although specific advantages have been enumerated herein, various exemplary embodiments may include some, none, or all of the enumerated advantages. 
     Benefits, other advantages, and solutions to problems have been described herein with regard to specific exemplary embodiments. However, the benefits, advantages, solutions to problems, and any elements that may cause any benefit, advantage, or solution to occur or become more pronounced are not to be construed as critical, required, or essential features or elements of the disclosure. The scope of the disclosure is accordingly limited by nothing other than the appended claims, in which reference to an element in the singular is not intended to mean “one and only one” unless explicitly so stated, but rather “one or more.” Moreover, where a phrase similar to ‘at least one of A, B, and C’ or ‘at least one of A, B, or C’ is used in the claims or specification, it is intended that the phrase be interpreted to mean that A alone may be present in an exemplary embodiment, B alone may be present in an exemplary embodiment, C alone may be present in an exemplary embodiment, or that any combination of the elements A, B and C may be present in a single exemplary embodiment; for example, A and B, A and C, B and C, or A and B and C. 
     Accordingly, the earlier provided description of example implementations does not define or constrain this specification. Other changes, substitutions, and alterations are also possible without departing from the spirit and scope of this specification.