Patent Publication Number: US-2022236561-A1

Title: Display device and system

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     The present application is a continuation of U.S. patent application Ser. No. 17/186,752, filed Feb. 26, 2021, which claims the benefit of priority of United Kingdom Patent Application no. 2004067.1, filed Mar. 20, 2020. The entire disclosure contents of these applications are herewith incorporated by reference into the present application in their entirety. 
    
    
     FIELD 
     The present disclosure relates to a projector. More specifically, the present disclosure relates to a holographic display device, a method of holographic projection and holographic projection system. Some embodiments relate to a head-up display. Some embodiments relate to an improved pupil expander and method for forming a viewing window of a display device. 
     BACKGROUND AND INTRODUCTION 
     Light scattered from an object contains both amplitude and phase information. This amplitude and phase information can be captured on, for example, a photosensitive plate by well-known interference techniques to form a holographic recording, or “hologram”, comprising interference fringes. The hologram may be reconstructed by illumination with suitable light to form a two-dimensional or three-dimensional holographic reconstruction, or replay image, representative of the original object. 
     Computer-generated holography may numerically simulate the interference process. A computer-generated hologram may be calculated by a technique based on a mathematical transformation such as a Fresnel or Fourier transform. These types of holograms may be referred to as Fresnel/Fourier transform holograms or simply Fresnel/Fourier holograms. A Fourier hologram may be considered a Fourier domain/plane representation of the object or a frequency domain/plane representation of the object. A computer-generated hologram may also be calculated by coherent ray tracing or a point cloud technique, for example. 
     A computer-generated hologram may be encoded on a spatial light modulator arranged to modulate the amplitude and/or phase of incident light. Light modulation may be achieved using electrically-addressable liquid crystals, optically-addressable liquid crystals or micro-mirrors, for example. 
     A spatial light modulator typically comprises a plurality of individually-addressable pixels which may also be referred to as cells or elements. The light modulation scheme may be binary, multilevel or continuous. Alternatively, the device may be continuous (i.e. is not comprised of pixels) and light modulation may therefore be continuous across the device. The spatial light modulator may be reflective meaning that modulated light is output in reflection. The spatial light modulator may equally be transmissive meaning that modulated light is output in transmission. 
     A holographic display device, such as a holographic projector, may be provided using the system described herein. Such projectors have found application in head-up displays, “HUD”, and head-mounted displays, “HMD”, including near-eye devices, for example. 
     A moving diffuser may be used to improve image quality in devices which use coherent light such as holographic projectors. 
     SUMMARY 
     Aspects of the present disclosure are defined in the appended independent claims. 
     There is disclosed herein a display device having a viewing window. The viewing window is disposed on a viewing plane. The display device comprises a picture generating unit, a first waveguide pupil expander and a second waveguide pupil expander. The picture generating unit is arranged to display a picture (or image) on a display plane. The picture is a holographic reconstruction formed from a hologram of the picture (or image). The first waveguide pupil expander comprises a first input port. The first input port is arranged to receive light of the picture (or image). The first waveguide pupil expander is arranged to expand a first exit pupil thereof in a first dimension. The second waveguide pupil expander comprises a second input port. The second input port is arranged to receive light of the picture. The second waveguide pupil expander is arranged to expand a second exit pupil thereof in the first dimension. The first dimension corresponds to a dimension of the viewing window. 
     There is disclosed herein a display device having a viewing window. The viewing window is disposed on a viewing plane. The display device comprises a picture generating unit, a first waveguide pupil expander and a second waveguide pupil expander. The picture generating unit is arranged to display a picture (or image) on a display plane. The picture is a holographic reconstruction formed from a hologram of the picture (or image). The first waveguide pupil expander comprises a first input port. The first input port is arranged to receive light of the picture (or image). The first waveguide pupil expander is arranged to replicate the received light of the picture (or image) in order to form a first (expanded) exit pupil of the first waveguide expander (e.g. expand the viewing window in a first sub-area of the viewing plane). The second waveguide pupil expander comprises a second input port. The second input port is also arranged to receive light of the picture (or image). The second waveguide pupil expander is arranged to replicate the received light of the picture (or image) in order to form a second (expanded) exit pupil of the second waveguide pupil expander (e.g. expand the viewing window in a second sub-area of the viewing plane). The first (expanded) exit pupil/sub-area may correspond to one of the left and right eyes of the viewer at the viewing plane and the second (expanded) exit pupil/sub-area may correspond to the other of the left and right eyes of the viewer at the viewing plane. 
     In a display device having a viewing window on a viewing plane, a waveguide pupil expander is arranged to expand the exit pupil/viewing window in a first dimension by a series of internal reflections along the length of the waveguide. In particular, the waveguide receives input light from a picture generating unit at one end thereof and, by a series of internal reflections from opposed surfaces thereof, forms a plurality of replicas along its length by partial transmission of the received light from one of the opposed surfaces. The input light should enter the waveguide at an angle that gives rise to the series of internal reflections. For example, in order to form a plurality of generally evenly spaced replicas with rays that are parallel to each other, the light should be incident on, and reflect from, each opposed surface at the substantially the same angle (i.e. each “bounce” of the light rays should be at substantially the same angle). This may be achieved by orientating the waveguide in a second dimension orthogonal to the first dimension, the second dimension parallel to the direction of propagation of the light of the replicas. Thus, the waveguide is orientated at an angle (e.g. of less than 90°) relative to the viewing plane and so occupies a “height” in the second dimension, as described in more detail below. As a result of this angled orientation, the waveguide consumes a volume within the display device that is significantly larger than the volume of the waveguide itself. 
     The first and second waveguide pupil expanders form a pupil expander arranged to expand the viewing window of the display device in the first dimension. In embodiments, each of the first and second waveguide pupil expanders comprises a pair of parallel reflective surfaces arranged to guide received light from an input port to an exit pupil by a series of internal reflections. A first reflective surface of the pair of parallel reflective surfaces is partially transmissive and partially reflective of the received light, and a second reflective surface of the pair of parallel reflective surfaces is fully reflective of the received light. 
     The first exit pupil of the first waveguide pupil expander is formed at the first reflective surface thereof, and the second exit pupil of the second waveguide pupil expander is formed at the second reflective surface thereof. It may be said that the first reflective surface of each of the first and second waveguide pupil expanders acts to expand the exit pupil/viewing window by transmitting a plurality of replicas along its length (corresponding to the first direction). 
     In embodiments, the first input port of the first waveguide pupil expander is arranged to receive an input beam, comprising light of the picture, from the picture generating unit, and the second input port of the second waveguide pupil expander is arranged to receive an input beam, comprising light of the picture, from the picture generating unit. 
     In embodiments, the output light of the replicas may be in the same direction (i.e. along the same optical/projection axis) as the input light. Thus, the optical/projection axis of the input and output light is orthogonal to the first dimension of the viewing plane. 
     In embodiments, the first waveguide pupil expander is arranged at a first angle less than 90° with respect to the first dimension/viewing window/input beam. The second waveguide pupil expander is arranged at a second angle less than 90° with respect to the first dimension/viewing window/input beam. The angled orientation of each of the first and second waveguide pupil expanders advantageously provides substantially uniform pupil expansion, by the formation of evenly spaced replicas along the length thereof, whilst minimising the overall volume occupied by the pupil expander of the display device. 
     Each of the first and second waveguide pupil expanders replicates the received light of the picture (or image) in order to expand the exit pupil thereof. In particular, each of the first and second waveguide pupil expanders forms a plurality of replicas of the received light of the picture (or image). Each waveguide pupil expander may form n replicas of the received light of the picture (or image). The total number of replicas formed by the first and second waveguide pupil expander, collectively, is therefore 2n. The length of each first/second waveguide pupil expander is half the length of a single waveguide pupil expander arranged to form 2n replicas of the exit pupil by itself. By using first and second waveguide pupil expanders of shorter length (e.g. arranged in an appropriate composite configuration, each with an appropriate angled orientation) the size of the composite waveguide pupil expander structure in the second dimension (e.g. height or light propagation direction) may be reduced, and thus the volume required thereby. In some embodiments, the first waveguide pupil expander is configured to provide a first exit pupil for a first eye of a viewer (disposed on the viewing plane) and the second waveguide pupil expander is configured to provide a second exit pupil for a second eye of the viewer. 
     In some embodiments, the first waveguide pupil expander and second waveguide pupil expander are arranged in a symmetrical configuration. For example, respective end surfaces of the first waveguide pupil expander and the second waveguide pupil expander may be abutted to form a V-shaped configuration. The first input port may be adjacent the second input port. Thus, the first input port and the second input port may receive a common input beam from the picture generating unit. The first waveguide pupil expander may be arranged to replicate the received light of the picture in order to expand the first exit pupil of the first waveguide pupil expander in a first direction. The second waveguide pupil expander may be arranged to replicate the received light of the picture in order to expand the second exit pupil of the second waveguide pupil expander in a second direction. The first direction may be opposite to the second direction. 
     In some embodiments, the first direction and second direction are parallel to the viewing plane. It may be said that the viewing plane comprises the first direction and second direction. The viewing window is an area within which a viewer may view the picture (or image) at the viewing plane. Accordingly, the first exit pupil and the second exit pupil are expanded in the respective first and second directions along a first dimension of the viewing plane. 
     In other embodiments, the first and second waveguide pupil expanders are arranged in a different configuration, which may be symmetrical or asymmetrical. For example, the first and second waveguide pupil expanders may be spatially separated and/or spatially offset. In this case, the input ports of the first and second waveguide pupil expanders are also spatially separated, and separate input light beams from picture generating unit may be directed thereto by appropriate optical components, such as a beam splitter. 
     In embodiments, each waveguide pupil expander comprises a pair of parallel reflective surfaces. Each waveguide pupil expander is arranged to guide the received light of the picture (or image) from the input port to the exit pupil by a series of internal reflections. A first reflective surface of the pair of parallel reflective surfaces is partially transmissive and partially reflective of the received light and a second reflective surface of the pair of parallel reflective surfaces is fully reflective of the received light. The reflectivity of the second reflective surface may be graded along the length of the waveguide pupil expander. 
     In some embodiments, the reflectivity of the first reflective surface of at least one of the first and second waveguide pupil expanders is graded along the length thereof. For example, the reflectivity of the first or second waveguide pupil expander may be graded so that the first reflective surface outputs each replica with substantially the same intensity. 
     In some embodiments, each wavelength pupil expander may comprise a slab waveguide. The slab waveguide may be formed of an optically transparent solid material comprising a pair of parallel surfaces. In particular, a first surface may have a first fully reflective coating and a second surface, opposite to the first surface, may have a second partially reflective/partially transmissive coating. In other embodiments, each pupil expander comprises a pair of opposed mirrors spatially separated by an air gap. One mirror of the pair of mirrors is fully reflective and the other mirror of the pair of mirrors is partially reflective/partially transmissive. 
     In some embodiments, the first waveguide pupil expander and second waveguide pupil expander are adjoined such that the first input surface of the first waveguide pupil expander and second input surface of the second waveguide pupil expander form a continuous surface that faces the display plane. 
     Each of the first and second waveguides may be configured with first and second end surfaces at one end thereof, wherein the first end surface comprises the respective input port. The input port provides an optically transparent aperture in the first end surface for light to enter the waveguide. The first waveguide pupil expander and second waveguide pupil expander may be arranged in a configuration having abutting second end surfaces. In particular, a second end surface of the first waveguide pupil expander may be abutted against a second end surface of the second waveguide pupil expander. 
     In some embodiments, the display plane may comprise a display screen such as a diffuse screen. 
     In embodiments, each waveguide pupil expander comprises two reflective surfaces. One reflective surface—e.g. the first reflective surface—is partially transmissive and partially reflective. The partial transmissivity leads to the formation of replicas of the light rays received by the waveguide. The other reflective surface—e.g. the second reflective surface—of the waveguide is a perfect or near-perfect reflector—e.g. a mirror. Light is received at an input port of the waveguide. The waveguide may be substantially slab-shaped, wherein the separation of the two reflective surfaces is less than the dimensions of the two reflective surfaces. A light transmissive material between the two reflective surfaces may be air or a transparent solid material such as glass. Each reflective surface is substantially planar. The waveguide is configured such that the viewer should look at the large surface of one of the reflective surfaces—e.g. the partially reflective surface. This surface may be referred to as the viewing surface. The input port may be on the same side of the slab as the viewer or it may be on the opposite side. The input port may comprise a break or gap in the reflective component—e.g. reflective layer or coating—of the fully reflective surface or the partially reflective surface. The input port may alternatively begin at the end of the reflective component. 
     The viewing window may be an eye-box or an eye-motion-box. The waveguide enlarges the viewing window due to the generation of extra rays by division of amplitude of the incident wavefront. In accordance with this disclosure, the incident light (that is, the light received by the waveguide) is spatially-modulated light. The spatially-modulated light may be encoded with a hologram of the picture or encoded with the picture itself. In either case, the incident light may be collimated. 
     The waveguide replicates the rays of the incident wavefront in order to expand the viewing window/pupil. Each successive replica is formed by one more bounce of the light rays within the waveguide. In some embodiments, the reflectivity/transmissivity of the first reflective surface/viewing surface is graded—that is, changes with distance—in order to compensate for the decrease in light intensity along the waveguide owing to partial escape of the light at each bounce. The reflectivity of the graded coating therefore decreases with distance from the input port. 
     In some embodiments, the light of the picture (or image) comprises spatially-modulated light that is formed on an intermediate plane between the picture generating unit and first and second waveguide expanders/viewing window. The picture formed at the intermediate plane may be formed in free-space or may be formed on a screen such as a diffuser. In these embodiments, the amplitude of the light is spatially-modulated in accordance with the displayed picture. The picture may be formed on the intermediate plane or screen by holographic projection. That is, a spatial light modulator may display a hologram that is illuminated and forms a picture on the intermediate plane or screen (e.g. diffuser). In embodiments in which the picture (i.e. holographic reconstruction) is formed on a screen, it may be preferred to consider the screen to be part of the picture generating unit. In this case, it may be said that the screen is disposed between the hologram plane and the viewing plane. It may be said that the screen forms a display plane of the picture generating unit. 
     In other embodiments, the light of the picture (or image) comprises spatially-modulated light that is encoded with a hologram. In these embodiments, a hologram is displayed by a spatial light modulator and light spatially-modulated in accordance with the hologram pattern may be used to reconstruct the picture (or image). It is sometimes said that, in this configuration, the lens of the viewer&#39;s eye performs the hologram-to-image (e.g. frequency domain to spatial domain) transformation. The holographic reconstruction may therefore be formed on the viewer&#39;s retina. Alternatively, or additionally, the holographic reconstruction may be formed in free space between the hologram plane and viewing plane. In these other embodiments, a lens may be included between the display/hologram and viewer in order to relay the holographic reconstruction. The hologram may be a Fourier hologram. Embodiments in which the viewer looks directly at the hologram (e.g. spatial light modulator displaying the hologram) may be referred to as direct view. Direct view embodiments are characterised by the absence of a screen (e.g. diffuser) between the viewer and hologram. Direct view embodiments may be advantageous because the problems associated with using a screen/diffuser are not encountered. 
     There is further provided a method of expanding a viewing window on a viewing plane of a display device. The method displays a picture (or image) on a display plane. The picture is a holographic reconstruction formed from a hologram of the picture (or image). The method receives light of the picture (or image) at an input port by a first waveguide expander. The method expands, by the first waveguide pupil expander, a first exit pupil thereof in a first dimension. The method further receives light of the picture (or image) an input port of a second waveguide expander. The method expands, by the second waveguide pupil expander, a second exit pupil thereof in the first dimension. The first dimension corresponds to a dimension of the viewing window. For example, the first dimension may correspond to a width of the viewing window. 
     In some embodiments, the method may expand the first exit pupil in a first sub-area of the viewing window corresponding to a first eye of a viewer (at the viewing plane) and the method may expand the second exit pupil in a second sub-area of the viewing window corresponding to a second eye of the viewer. 
     The term “hologram” is used to refer to the recording which contains amplitude information or phase information, or some combination thereof, regarding the object. The term “holographic reconstruction” is used to refer to the optical reconstruction of the object which is formed by illuminating the hologram. The system disclosed herein is described as a “holographic projector” because the holographic reconstruction is a real image and spatially-separated from the hologram. The term “replay field” is used to refer to the 2D area within which the holographic reconstruction is formed and fully focused. If the hologram is displayed on a spatial light modulator comprising pixels, the replay field will be repeated in the form of a plurality diffracted orders wherein each diffracted order is a replica of the zeroth-order replay field. The zeroth-order replay field generally corresponds to the preferred or primary replay field because it is the brightest replay field. Unless explicitly stated otherwise, the term “replay field” should be taken as referring to the zeroth-order replay field. The term “replay plane” is used to refer to the plane in space containing all the replay fields. The terms “image”, “replay image” and “image region” refer to areas of the replay field illuminated by light of the holographic reconstruction. In some embodiments, the “image” may comprise discrete spots which may be referred to as “image spots” or, for convenience only, “image pixels”. 
     The terms “encoding”, “writing” or “addressing” are used to describe the process of providing the plurality of pixels of the SLM with a respective plurality of control values which respectively determine the modulation level of each pixel. It may be said that the pixels of the SLM are configured to “display” a light modulation distribution in response to receiving the plurality of control values. Thus, the SLM may be said to “display” a hologram and the hologram may be considered an array of light modulation values or levels. 
     It has been found that a holographic reconstruction of acceptable quality can be formed from a “hologram” containing only phase information related to the Fourier transform of the original object. Such a holographic recording may be referred to as a phase-only hologram. Embodiments relate to a phase-only hologram but the present disclosure is equally applicable to amplitude-only holography. 
     The present disclosure is also equally applicable to forming a holographic reconstruction using amplitude and phase information related to the Fourier transform of the original object. In some embodiments, this is achieved by complex modulation using a so-called fully complex hologram which contains both amplitude and phase information related to the original object. Such a hologram may be referred to as a fully-complex hologram because the value (grey level) assigned to each pixel of the hologram has an amplitude and phase component. The value (grey level) assigned to each pixel may be represented as a complex number having both amplitude and phase components. In some embodiments, a fully-complex computer-generated hologram is calculated. 
     Reference may be made to the phase value, phase component, phase information or, simply, phase of pixels of the computer-generated hologram or the spatial light modulator as shorthand for “phase-delay”. That is, any phase value described is, in fact, a number (e.g. in the range 0 to 2π) which represents the amount of phase retardation provided by that pixel. For example, a pixel of the spatial light modulator described as having a phase value of π/2 will retard the phase of received light by π/2 radians. In some embodiments, each pixel of the spatial light modulator is operable in one of a plurality of possible modulation values (e.g. phase delay values). The term “grey level” may be used to refer to the plurality of available modulation levels. For example, the term “grey level” may be used for convenience to refer to the plurality of available phase levels in a phase-only modulator even though different phase levels do not provide different shades of grey. The term “grey level” may also be used for convenience to refer to the plurality of available complex modulation levels in a complex modulator. 
     The hologram therefore comprises an array of grey levels—that is, an array of light modulation values such as an array of phase-delay values or complex modulation values. The hologram is also considered a diffractive pattern because it is a pattern that causes diffraction when displayed on a spatial light modulator and illuminated with light having a wavelength comparable to, generally less than, the pixel pitch of the spatial light modulator. Reference is made herein to combining the hologram with other diffractive patterns such as diffractive patterns functioning as a lens or grating. For example, a diffractive pattern functioning as a grating may be combined with a hologram to translate the replay field on the replay plane or a diffractive pattern functioning as a lens may be combined with a hologram to focus the holographic reconstruction on a replay plane in the near field. 
     Although different embodiments and groups of embodiments may be disclosed separately in the detailed description which follows, any feature of any embodiment or group of embodiments may be combined with any other feature or combination of features of any embodiment or group of embodiments. That is, all possible combinations and permutations of features disclosed in the present disclosure are envisaged. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Specific embodiments are described by way of example only with reference to the following figures: 
         FIG. 1  is a schematic showing a reflective SLM producing a holographic reconstruction on a screen; 
         FIG. 2A  illustrates a first iteration of an example Gerchberg-Saxton type algorithm; 
         FIG. 2B  illustrates the second and subsequent iterations of the example Gerchberg-Saxton type algorithm; 
         FIG. 2C  illustrates alternative second and subsequent iterations of the example Gerchberg-Saxton type algorithm; 
         FIG. 3  is a schematic of a reflective LCOS SLM; 
         FIG. 4  is a schematic of an example pupil expander comprising a waveguide formed by a pair of mirrors; 
         FIG. 5  shows an example pupil expander comprising a slab waveguide formed by a slab of glass having reflective opposed major surfaces; 
         FIG. 6  shows a holographic display device in an indirect view configuration comprising a pupil expander in accordance with embodiments; 
         FIGS. 7A-C  show the geometry of the pupil expander in accordance with embodiments; 
         FIG. 8  shows a holographic display device in a direct view configuration comprising a pupil expander in accordance with embodiments; and 
         FIG. 9  shows a pupil expander in accordance with embodiments. 
     
    
    
     The same reference numbers will be used throughout the drawings to refer to the same or like parts. 
     DETAILED DESCRIPTION 
     The present invention is not restricted to the embodiments described in the following but extends to the full scope of the appended claims. That is, the present invention may be embodied in different forms and should not be construed as limited to the described embodiments, which are set out for the purpose of illustration. 
     Terms of a singular form may include plural forms unless specified otherwise. 
     A structure described as being formed at an upper portion/lower portion of another structure or on/under the other structure should be construed as including a case where the structures contact each other and, moreover, a case where a third structure is disposed there between. 
     In describing a time relationship—for example, when the temporal order of events is described as “after”, “subsequent”, “next”, “before” or such like—the present disclosure should be taken to include continuous and non-continuous events unless otherwise specified. For example, the description should be taken to include a case which is not continuous unless wording such as “just”, “immediate” or “direct” is used. 
     Although the terms “first”, “second”, etc. may be used herein to describe various elements, these elements are not to be limited by these terms. These terms are only used to distinguish one element from another. For example, a first element could be termed a second element, and, similarly, a second element could be termed a first element, without departing from the scope of the appended claims. 
     Features of different embodiments may be partially or overall coupled to or combined with each other, and may be variously inter-operated with each other. Some embodiments may be carried out independently from each other, or may be carried out together in co-dependent relationship. 
     Optical Configuration 
       FIG. 1  shows an embodiment in which a computer-generated hologram is encoded on a single spatial light modulator. The computer-generated hologram is a Fourier transform of the object for reconstruction. It may therefore be said that the hologram is a Fourier domain or frequency domain or spectral domain representation of the object. In this embodiment, the spatial light modulator is a reflective liquid crystal on silicon, “LCOS”, device. The hologram is encoded on the spatial light modulator and a holographic reconstruction is formed at a replay field, for example, a light receiving surface such as a screen or diffuser. 
     A light source  110 , for example a laser or laser diode, is disposed to illuminate the SLM  140  via a collimating lens  111 . The collimating lens causes a generally planar wavefront of light to be incident on the SLM. In  FIG. 1 , the direction of the wavefront is off-normal (e.g. two or three degrees away from being truly orthogonal to the plane of the transparent layer). However, in other embodiments, the generally planar wavefront is provided at normal incidence and a beam splitter arrangement is used to separate the input and output optical paths. In the embodiment shown in  FIG. 1 , the arrangement is such that light from the light source is reflected off a mirrored rear surface of the SLM and interacts with a light-modulating layer to form an exit wavefront  112 . The exit wavefront  112  is applied to optics including a Fourier transform lens  120 , having its focus at a screen  125 . More specifically, the Fourier transform lens  120  receives a beam of modulated light from the SLM  140  and performs a frequency-space transformation to produce a holographic reconstruction at the screen  125 . 
     Notably, in this type of holography, each pixel of the hologram contributes to the whole reconstruction. There is not a one-to-one correlation between specific points (or image pixels) on the replay field and specific light-modulating elements (or hologram pixels). In other words, modulated light exiting the light-modulating layer is distributed across the replay field. 
     In these embodiments, the position of the holographic reconstruction in space is determined by the dioptric (focusing) power of the Fourier transform lens. In the embodiment shown in  FIG. 1 , the Fourier transform lens is a physical lens. That is, the Fourier transform lens is an optical Fourier transform lens and the Fourier transform is performed optically. Any lens can act as a Fourier transform lens but the performance of the lens will limit the accuracy of the Fourier transform it performs. The skilled person understands how to use a lens to perform an optical Fourier transform. 
     Hologram Calculation 
     In some embodiments, the computer-generated hologram is a Fourier transform hologram, or simply a Fourier hologram or Fourier-based hologram, in which an image is reconstructed in the far field by utilising the Fourier transforming properties of a positive lens. The Fourier hologram is calculated by Fourier transforming the desired light field in the replay plane back to the lens plane. Computer-generated Fourier holograms may be calculated using Fourier transforms. 
     A Fourier transform hologram may be calculated using an algorithm such as the Gerchberg-Saxton algorithm. Furthermore, the Gerchberg-Saxton algorithm may be used to calculate a hologram in the Fourier domain (i.e. a Fourier transform hologram) from amplitude-only information in the spatial domain (such as a photograph). The phase information related to the object is effectively “retrieved” from the amplitude-only information in the spatial domain. In some embodiments, a computer-generated hologram is calculated from amplitude-only information using the Gerchberg-Saxton algorithm or a variation thereof. 
     The Gerchberg Saxton algorithm considers the situation when intensity cross-sections of a light beam, I A (x, y) and I B (x, y), in the planes A and B respectively, are known and I A (x, y) and I B (x, y) are related by a single Fourier transform. With the given intensity cross-sections, an approximation to the phase distribution in the planes A and B, ψ A (x, y) and ψ B (x, y) respectively, is found. The Gerchberg-Saxton algorithm finds solutions to this problem by following an iterative process. More specifically, the Gerchberg-Saxton algorithm iteratively applies spatial and spectral constraints while repeatedly transferring a data set (amplitude and phase), representative of I A (x, y) and I B (x, y), between the spatial domain and the Fourier (spectral or frequency) domain. The corresponding computer-generated hologram in the spectral domain is obtained through at least one iteration of the algorithm. The algorithm is convergent and arranged to produce a hologram representing an input image. The hologram may be an amplitude-only hologram, a phase-only hologram or a fully complex hologram. 
     In some embodiments, a phase-only hologram is calculated using an algorithm based on the Gerchberg-Saxton algorithm such as described in British patent 2,498,170 or 2,501,112 which are hereby incorporated in their entirety by reference. However, embodiments disclosed herein describe calculating a phase-only hologram by way of example only. In these embodiments, the Gerchberg-Saxton algorithm retrieves the phase information ψ [u, v] of the Fourier transform of the data set which gives rise to a known amplitude information T[x, y], wherein the amplitude information T[x, y] is representative of a target image (e.g. a photograph). Since the magnitude and phase are intrinsically combined in the Fourier transform, the transformed magnitude and phase contain useful information about the accuracy of the calculated data set. Thus, the algorithm may be used iteratively with feedback on both the amplitude and the phase information. However, in these embodiments, only the phase information ψ[u, v] is used as the hologram to form a holographic representative of the target image at an image plane. The hologram is a data set (e.g. 2D array) of phase values. 
     In other embodiments, an algorithm based on the Gerchberg-Saxton algorithm is used to calculate a fully-complex hologram. A fully-complex hologram is a hologram having a magnitude component and a phase component. The hologram is a data set (e.g. 2D array) comprising an array of complex data values wherein each complex data value comprises a magnitude component and a phase component. 
     In some embodiments, the algorithm processes complex data and the Fourier transforms are complex Fourier transforms. Complex data may be considered as comprising (i) a real component and an imaginary component or (ii) a magnitude component and a phase component. In some embodiments, the two components of the complex data are processed differently at various stages of the algorithm. 
       FIG. 2A  illustrates the first iteration of an algorithm in accordance with some embodiments for calculating a phase-only hologram. The input to the algorithm is an input image  210  comprising a 2D array of pixels or data values, wherein each pixel or data value is a magnitude, or amplitude, value. That is, each pixel or data value of the input image  210  does not have a phase component. The input image  210  may therefore be considered a magnitude-only or amplitude-only or intensity-only distribution. An example of such an input image  210  is a photograph or one frame of video comprising a temporal sequence of frames. The first iteration of the algorithm starts with a data forming step  202 A comprising assigning a random phase value to each pixel of the input image, using a random phase distribution (or random phase seed)  230 , to form a starting complex data set wherein each data element of the set comprising magnitude and phase. It may be said that the starting complex data set is representative of the input image in the spatial domain. 
     First processing block  250  receives the starting complex data set and performs a complex Fourier transform to form a Fourier transformed complex data set. Second processing block  253  receives the Fourier transformed complex data set and outputs a hologram  280 A. In some embodiments, the hologram  280 A is a phase-only hologram. In these embodiments, second processing block  253  quantises each phase value and sets each amplitude value to unity in order to form hologram  280 A. Each phase value is quantised in accordance with the phase-levels which may be represented on the pixels of the spatial light modulator which will be used to “display” the phase-only hologram. For example, if each pixel of the spatial light modulator provides 256 different phase levels, each phase value of the hologram is quantised into one phase level of the 256 possible phase levels. Hologram  280 A is a phase-only Fourier hologram which is representative of an input image. In other embodiments, the hologram  280 A is a fully complex hologram comprising an array of complex data values (each including an amplitude component and a phase component) derived from the received Fourier transformed complex data set. In some embodiments, second processing block  253  constrains each complex data value to one of a plurality of allowable complex modulation levels to form hologram  280 A. The step of constraining may include setting each complex data value to the nearest allowable complex modulation level in the complex plane. It may be said that hologram  280 A is representative of the input image in the spectral or Fourier or frequency domain. In some embodiments, the algorithm stops at this point. 
     However, in other embodiments, the algorithm continues as represented by the dotted arrow in  FIG. 2A . In other words, the steps which follow the dotted arrow in  FIG. 2A  are optional (i.e. not essential to all embodiments). 
     Third processing block  256  receives the modified complex data set from the second processing block  253  and performs an inverse Fourier transform to form an inverse Fourier transformed complex data set. It may be said that the inverse Fourier transformed complex data set is representative of the input image in the spatial domain. 
     Fourth processing block  259  receives the inverse Fourier transformed complex data set and extracts the distribution of magnitude values  211 A and the distribution of phase values  213 A. Optionally, the fourth processing block  259  assesses the distribution of magnitude values  211 A. Specifically, the fourth processing block  259  may compare the distribution of magnitude values  211 A of the inverse Fourier transformed complex data set with the input image  510  which is itself, of course, a distribution of magnitude values. If the difference between the distribution of magnitude values  211 A and the input image  210  is sufficiently small, the fourth processing block  259  may determine that the hologram  280 A is acceptable. That is, if the difference between the distribution of magnitude values  211 A and the input image  210  is sufficiently small, the fourth processing block  259  may determine that the hologram  280 A is a sufficiently-accurate representative of the input image  210 . In some embodiments, the distribution of phase values  213 A of the inverse Fourier transformed complex data set is ignored for the purpose of the comparison. It will be appreciated that any number of different methods for comparing the distribution of magnitude values  211 A and the input image  210  may be employed and the present disclosure is not limited to any particular method. In some embodiments, a mean square difference is calculated and if the mean square difference is less than a threshold value, the hologram  280 A is deemed acceptable. If the fourth processing block  259  determines that the hologram  280 A is not acceptable, a further iteration of the algorithm may be performed. However, this comparison step is not essential and in other embodiments, the number of iterations of the algorithm performed is predetermined or preset or user-defined. 
       FIG. 2B  represents a second iteration of the algorithm and any further iterations of the algorithm. The distribution of phase values  213 A of the preceding iteration is fed-back through the processing blocks of the algorithm. The distribution of magnitude values  211 A is rejected in favour of the distribution of magnitude values of the input image  210 . In the first iteration, the data forming step  202 A formed the first complex data set by combining distribution of magnitude values of the input image  210  with a random phase distribution  230 . However, in the second and subsequent iterations, the data forming step  202 B comprises forming a complex data set by combining (i) the distribution of phase values  213 A from the previous iteration of the algorithm with (ii) the distribution of magnitude values of the input image  210 . 
     The complex data set formed by the data forming step  202 B of  FIG. 2B  is then processed in the same way described with reference to  FIG. 2A  to form second iteration hologram  280 B. The explanation of the process is not therefore repeated here. The algorithm may stop when the second iteration hologram  280 B has been calculated. However, any number of further iterations of the algorithm may be performed. It will be understood that the third processing block  256  is only required if the fourth processing block  259  is required or a further iteration is required. The output hologram  280 B generally gets better with each iteration. However, in practice, a point is usually reached at which no measurable improvement is observed or the positive benefit of performing a further iteration is out-weighted by the negative effect of additional processing time. Hence, the algorithm is described as iterative and convergent. 
       FIG. 2C  represents an alternative embodiment of the second and subsequent iterations. The distribution of phase values  213 A of the preceding iteration is fed-back through the processing blocks of the algorithm. The distribution of magnitude values  211 A is rejected in favour of an alternative distribution of magnitude values. In this alternative embodiment, the alternative distribution of magnitude values is derived from the distribution of magnitude values  211  of the previous iteration. Specifically, processing block  258  subtracts the distribution of magnitude values of the input image  210  from the distribution of magnitude values  211  of the previous iteration, scales that difference by a gain factor α and subtracts the scaled difference from the input image  210 . This is expressed mathematically by the following equations, wherein the subscript text and numbers indicate the iteration number: 
     
       
         
           
             
               
                 
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     where:
 
F′ is the inverse Fourier transform;
 
F is the forward Fourier transform;
 
R[x, y] is the complex data set output by the third processing block  256 ;
 
T[x, y] is the input or target image;
 
∠ is the phase component;
 
ψ is the phase-only hologram  280 B;
 
η is the new distribution of magnitude values  211 B; and
 
α is the gain factor.
 
     The gain factor α may be fixed or variable. In some embodiments, the gain factor α is determined based on the size and rate of the incoming target image data. In some embodiments, the gain factor α is dependent on the iteration number. In some embodiments, the gain factor α is solely function of the iteration number. 
     The embodiment of  FIG. 2C  is the same as that of  FIG. 2A  and  FIG. 2B  in all other respects. It may be said that the phase-only hologram ψ(u, v) comprises a phase distribution in the frequency or Fourier domain. 
     In some embodiments, the Fourier transform is performed using the spatial light modulator. Specifically, the hologram data is combined with second data providing optical power. That is, the data written to the spatial light modulation comprises hologram data representing the object and lens data representative of a lens. When displayed on a spatial light modulator and illuminated with light, the lens data emulates a physical lens—that is, it brings light to a focus in the same way as the corresponding physical optic. The lens data therefore provides optical, or focusing, power. In these embodiments, the physical Fourier transform lens  120  of  FIG. 1  may be omitted. It is known how to calculate data representative of a lens. The data representative of a lens may be referred to as a software lens. For example, a phase-only lens may be formed by calculating the phase delay caused by each point of the lens owing to its refractive index and spatially-variant optical path length. For example, the optical path length at the centre of a convex lens is greater than the optical path length at the edges of the lens. An amplitude-only lens may be formed by a Fresnel zone plate. It is also known in the art of computer-generated holography how to combine data representative of a lens with a hologram so that a Fourier transform of the hologram can be performed without the need for a physical Fourier lens. In some embodiments, lensing data is combined with the hologram by simple addition such as simple vector addition. In some embodiments, a physical lens is used in conjunction with a software lens to perform the Fourier transform. Alternatively, in other embodiments, the Fourier transform lens is omitted altogether such that the holographic reconstruction takes place in the far-field. In further embodiments, the hologram may be combined in the same way with grating data—that is, data arranged to perform the function of a grating such as image steering. Again, it is known in the field how to calculate such data. For example, a phase-only grating may be formed by modelling the phase delay caused by each point on the surface of a blazed grating. An amplitude-only grating may be simply superimposed with an amplitude-only hologram to provide angular steering of the holographic reconstruction. The second data providing lensing and/or steering may be referred to as a light processing function or light processing pattern to distinguish from the hologram data which may be referred to as an image forming function or image forming pattern. 
     In some embodiments, the Fourier transform is performed jointly by a physical Fourier transform lens and a software lens. That is, some optical power which contributes to the Fourier transform is provided by a software lens and the rest of the optical power which contributes to the Fourier transform is provided by a physical optic or optics. 
     In some embodiments, there is provided a real-time engine arranged to receive image data and calculate holograms in real-time using the algorithm. In some embodiments, the image data is a video comprising a sequence of image frames. In other embodiments, the holograms are pre-calculated, stored in computer memory and recalled as needed for display on a SLM. That is, in some embodiments, there is provided a repository of predetermined holograms. 
     Embodiments relate to Fourier holography and Gerchberg-Saxton type algorithms by way of example only. The present disclosure is equally applicable to Fresnel holography and Fresnel holograms which may be calculated by a similar method. The present disclosure is also applicable to holograms calculated by other techniques such as those based on point cloud methods. 
     Light Modulation 
     A spatial light modulator may be used to display the diffractive pattern including the computer-generated hologram. If the hologram is a phase-only hologram, a spatial light modulator which modulates phase is required. If the hologram is a fully-complex hologram, a spatial light modulator which modulates phase and amplitude may be used or a first spatial light modulator which modulates phase and a second spatial light modulator which modulates amplitude may be used. 
     In some embodiments, the light-modulating elements (i.e. the pixels) of the spatial light modulator are cells containing liquid crystal. That is, in some embodiments, the spatial light modulator is a liquid crystal device in which the optically-active component is the liquid crystal. Each liquid crystal cell is configured to selectively-provide a plurality of light modulation levels. That is, each liquid crystal cell is configured at any one time to operate at one light modulation level selected from a plurality of possible light modulation levels. Each liquid crystal cell is dynamically-reconfigurable to a different light modulation level from the plurality of light modulation levels. In some embodiments, the spatial light modulator is a reflective liquid crystal on silicon (LCOS) spatial light modulator but the present disclosure is not restricted to this type of spatial light modulator. 
     A LCOS device provides a dense array of light modulating elements, or pixels, within a small aperture (e.g. a few centimetres in width). The pixels are typically approximately 10 microns or less which results in a diffraction angle of a few degrees meaning that the optical system can be compact. It is easier to adequately illuminate the small aperture of a LCOS SLM than it is the larger aperture of other liquid crystal devices. An LCOS device is typically reflective which means that the circuitry which drives the pixels of a LCOS SLM can be buried under the reflective surface. The results in a higher aperture ratio. In other words, the pixels are closely packed meaning there is very little dead space between the pixels. This is advantageous because it reduces the optical noise in the replay field. A LCOS SLM uses a silicon backplane which has the advantage that the pixels are optically flat. This is particularly important for a phase modulating device. 
     A suitable LCOS SLM is described below, by way of example only, with reference to  FIG. 3 . An LCOS device is formed using a single crystal silicon substrate  302 . It has a 2D array of square planar aluminium electrodes  301 , spaced apart by a gap  301   a , arranged on the upper surface of the substrate. Each of the electrodes  301  can be addressed via circuitry  302   a  buried in the substrate  302 . Each of the electrodes forms a respective planar mirror. An alignment layer  303  is disposed on the array of electrodes, and a liquid crystal layer  304  is disposed on the alignment layer  303 . A second alignment layer  305  is disposed on the planar transparent layer  306 , e.g. of glass. A single transparent electrode  307  e.g. of ITO is disposed between the transparent layer  306  and the second alignment layer  305 . 
     Each of the square electrodes  301  defines, together with the overlying region of the transparent electrode  307  and the intervening liquid crystal material, a controllable phase-modulating element  308 , often referred to as a pixel. The effective pixel area, or fill factor, is the percentage of the total pixel which is optically active, taking into account the space between pixels  301   a . By control of the voltage applied to each electrode  301  with respect to the transparent electrode  307 , the properties of the liquid crystal material of the respective phase modulating element may be varied, thereby to provide a variable delay to light incident thereon. The effect is to provide phase-only modulation to the wavefront, i.e. no amplitude effect occurs. 
     The described LCOS SLM outputs spatially modulated light in reflection. Reflective LCOS SLMs have the advantage that the signal lines, gate lines and transistors are below the mirrored surface, which results in high fill factors (typically greater than 90%) and high resolutions. Another advantage of using a reflective LCOS spatial light modulator is that the liquid crystal layer can be half the thickness than would be necessary if a transmissive device were used. This greatly improves the switching speed of the liquid crystal (a key advantage for the projection of moving video images). However, the teachings of the present disclosure may equally be implemented using a transmissive LCOS SLM. 
     Waveguide Pupil/Viewing Window Expander 
     In holographic projection systems such as head-up displays (HUDs) it is desirable to expand the exit pupil corresponding to the eye box region or viewing window of the HUD. In particular, the viewer needs to be able to move his or her head around and be able to see the complete image from any position within a limited area at the eye box/viewing distance. This is known as the eye motion box (EMB) or viewing window. Thus, a pupil expander may be employed to enlarge the EMB or viewing window. Typically, the pupil expander enlarges the EMB by generating extra rays by division of the amplitude of the incident wavefront. 
       FIG. 4  schematically illustrates an example pupil expander comprising a waveguide formed by a pair of mirrors. The general principle of a waveguide is known in the art and not described in detail herein. A waveguide guides light within a layer between a pair of parallel reflective surfaces by internal reflection. A pupil expander is formed from a waveguide comprising a first partially reflective surface (e.g. a partially reflective mirror having varying reflectivity with distance) and a second fully reflective surface (e.g. a mirror having substantially 100% reflectivity). The first reflective surface is opposite (i.e. faces) the second reflective surface. In the waveguide of  FIG. 4 , first partially reflective mirror  420  comprises a reflective coating the reflectivity of which decreases along the length of the waveguide, and second fully reflective mirror  410  comprises a reflective coating having substantially full reflectivity. The first and second mirrors are spatially-separated to form an air gap within which light propagates by internal reflection. 
     As shown in  FIG. 4 , an input light beam  402  (which may comprise light of a picture/image or spatially modulated light encoded with a hologram as described below) comprising input light rays enters the waveguide through an opening between first and second mirrors  410 , 420  at one end thereof. The waveguide is arranged to guide the received light along its length to a viewing window. The viewing window is an area or volume within which a viewer may view an image as described herein. The input light beam  402  is bounced along the waveguide due to internal reflections by first partially reflective mirror  420  and second fully reflective mirror  410 . Example rays are illustrated in  FIG. 4 . 
     Due to the partial reflectivity of first mirror  420 , a proportion of light is transmitted by first mirror  420  to provide a plurality of output light rays  404   a - f  (herein called “replicas” because they replicate the input light rays) along the length of the waveguide. For ease of illustration, refraction of the output light rays  404   a - f  at the waveguide-air interface is not illustrated in  FIG. 4 . The replicas are evenly spaced along the length of the waveguide, since the light rays are incident on, and reflect from, the respective first and second mirrors  420 ,  410  at substantially the same angle at each bounce. It may be said that replicas are output uniformly along the length of the waveguide. Thus, first reflective surface  420  forms a viewing surface. It is said that the pupil (or viewing window) is expanded by the replicas formed by the waveguide. In particular, by forming a plurality of replicas  404   a - f  along the length of the waveguide, the viewing window is increased in size. Since the replicas are evenly spaced, it may be said that the waveguide uniformly expands the exit pupil (or viewing window). Each replica  404   a - f  corresponds to a proportion of the amplitude (intensity or brightness) of the input light beam  402 . It is desirable that the grading provides a decrease in reflectivity (or conversely an increase in transmissivity) of the first reflective surface  420  along the length of the waveguide such that each replica  404   a - f  has substantially the same amplitude or intensity. Thus, a viewer&#39;s eyes  430 L and  430 R located within an eye box (i.e. pupil or viewing window) at a viewing distance from the first mirror  420  are able to see the image at any position within an expanded viewing window and with substantially the same brightness, as illustrated by arrows  440  denoting the viewing range. 
     The waveguide shown in  FIG. 4  expands the viewing window in one dimension (illustrated as the x-axis/direction)—corresponding to the lengthwise direction along which the light beam propagates within the waveguide or the width of the viewing window from the perspective of the viewer—as shown by arrows  440 . As the skilled person will appreciate, it is possible to expand the viewing window in two dimensions, when required, by using two orthogonal waveguides. This leads to a viewing range in two dimensions—i.e. a viewing plane located at the viewing distance from the waveguide. 
       FIG. 5  shows an example pupil expander comprising a waveguide  500  arranged in a known configuration. The pupil expander of  FIG. 5  is used a comparative example to embodiments described below with reference to  FIG. 9 . In contrast to the waveguide of  FIG. 4 , the waveguide  500  comprises a block of optically transparent solid material. Such a waveguide is sometimes known as a “slab waveguide”. Thus, the layer within which light propagates by internal reflection in the waveguide  500  of  FIG. 5  is a transparent solid material. For example, the layer may be glass or Perspex. The waveguide may therefore be a glass or Perspex block or slab. Waveguide  500  comprises first and second reflective surfaces  503   a ,  503   b  formed by a pair of parallel opposed (facing) side surfaces of the slab or block, coated with reflective coatings. The first reflective surface  503   a  of the waveguide  500  may be coated with a coating comprising a large number of thin films (e.g. 25 or more thin films) in order to provide the necessary graded partial reflectivity. In particular, as described above, such thin films or similar coatings may provide decreasing reflectivity, and thus increasing transmissivity, with propagation distance such that the brightness (ray intensity) of each replica R 0  to R 7  is substantially constant. The amplitude of the propagating light beam reduces with propagation distance along the length of the waveguide  500  due to output of the replicas R 0  to R 7  and losses due to internal reflection. Thus, the grading of the first reflective surface  503   a  is designed to take into account the drop in intensity of the propagating light beam with propagation distance, whilst ensuring that each replica R 0  to R 7  has substantially the same intensity so that the image seen has uniform brightness throughout the viewing window (i.e. at all viewing positions). The second reflective surface  503   b  may be coated with a fully reflective coating or may be highly polished or otherwise treated to provide 100% reflectivity. 
     An input light beam  510  (which may comprise light of a picture/image or spatially modulated light encoded with a hologram as described below) comprising input light rays enters the waveguide  500  through input surface  501  which forms an input port. In the illustrated arrangement, the input surface  501  is provided on one or a pair of side surfaces  501 ,  502  formed adjacent one end of waveguide  500 . The input port comprises an optically transparent aperture for the input beam  510 . The waveguide  500  is arranged to guide light received from the input port to a viewing window. In the illustrated arrangement, the input light beam  510  is incident on first partially reflective surface  503   a , which (partially) transmits light rays of a first replica R 0 . The light rays reflected by first partially reflective surface  503   a  propagate along the length of the waveguide  500  due to internal reflection by first partially reflective surface  503   a  and second fully reflective surface  503   b . Example rays are illustrated in  FIG. 5 . 
     As in the example pupil expander of  FIG. 4 , a plurality of replicas R 0  to R 7  are transmitted by first partially reflective surface  503   a  along the length of the waveguide  500  so as to expand the exit pupil or viewing window in one dimension (illustrated as the x-axis). For ease of illustration, refraction of the output light rays of replicas R 0  to F 7  at the waveguide-air interface is not illustrated in  FIG. 5 . The replicas are evenly spaced along the length of the waveguide, since the light rays are incident on, and reflect from, the respective first and second reflective surfaces  503   a ,  503   b  at substantially the same angle at each bounce. To achieve this, waveguide  500  is orientated at an angle relative to the plane of the viewing window and light rays of replicas R 0  to R 7  are transmitted substantially parallel to each other. Thus, the waveguide pupil expander  500  has an overall height  520  in a second dimension (illustrated as the y-axis/direction), corresponding to the direction of light rays of replicas R 0  to R 7 . In the illustrated example, the first partially reflective surface  503   a  comprises a graded coating, as described above, so that each replica R 0  to R 7  has substantially the same amplitude/intensity. Thus, when a viewer&#39;s eyes  530 L and  530 R are located within an eye box (i.e. pupil or viewing window) at a viewing distance, the viewer is able to see the image at any position within an expanded viewing window, as illustrated by arrows  540  denoting the viewing range. In the illustrated arrangement, the viewing window is at a fixed position in the y-direction, on an (x, z) viewing plane. 
     As shown in  FIG. 5 , the waveguide  500  is orientated along its length at an acute angle α relative to the viewing plane. In the illustrated arrangement, the input beam  510  and the light rays of replicas R 0  to R 7  propagate along an optical axis in the second dimension (illustrated as the y-axis/direction), which is orthogonal to the viewing plane. Thus, it also may be said that the waveguide  500  is orientated along its length at an acute angle (90°−α) relative to the second dimension/direction of input beam  510  and/or replicas R 0  to R 7 . The length of waveguide  500  is chosen according to the required amount of expansion of the viewing window in the first direction (e.g. to form a defined number of replicas). As the skilled person will appreciate, the length and orientation (angle) of the waveguide  500  are determined according to application requirements, such as the size of the eye-box, the size of the picture (or image) and the like. Thus, the angled orientation of waveguide  500  leads to an increased height  520  of the pupil expander in the second dimension (the direction of propagation of replicas R 0  to R 7 —illustrated as the y-direction), and thus increases the total volume required to accommodate the waveguide  500  in an imaging system. Thus, the package height is relatively bulky. 
     Example implementations of the present disclosure comprise holographic display devices and methods that use a pupil expander comprising a pair of waveguides. The waveguides may be configured to reduce the effective height of the pupil expander. This may result in a less bulky imaging system. 
     First Embodiments—Indirect View Holographic Display Device 
     It should be understood that various changes and modifications to the presently preferred embodiments described herein will be apparent to those skilled in the art. Such changes and modifications can be made without departing from the spirit and scope of the present invention and without diminishing its intended advantages. It is therefore intended that such changes and modifications be covered by the appended claims. 
       FIG. 6  shows a holographic display device comprising a waveguide forming a waveguide pupil expander in accordance with first example embodiments of the present disclosure. 
     Holographic display device comprises a picture generating unit arranged to form a first picture (also called “first image”) and a second picture (also called “second image”). A first single colour channel (also called “first display channel”) is arranged to form the first picture and comprises a first light source  610 , a first collimating lens  612  and a first dichroic mirror  614 . First dichroic mirror  614  is arranged to reflect light of a first wavelength along a common optical path so as to illuminate a spatial light modulator (SLM)  640 . The first wavelength of light corresponds to the first display channel of a first colour (e.g. red). A second single colour channel (also called “second display channel”) is arranged to form the second picture and comprises a second light source  620 , a second collimating lens  622  and a second mirror  624 . Second mirror  624  is arranged to reflect light of a second wavelength along the common optical path so as to illuminate the SLM  640 . The second wavelength of light corresponds to the second single colour channel of a second colour (e.g. green). As described below, in other embodiments, the picture generating unit may comprises a third single colour/display channel (equivalent to the first and second channels) arranged to form a third picture, wherein the third colour channel corresponds to a wavelength of light of a third colour (e.g. blue). In the illustrated embodiment, SLM  640  comprises a single array of light modulating pixels (e.g. LCOS) that is illuminated by light of both the first and second wavelengths. In other embodiments, SLM  640  may comprise separate arrays of light modulating pixels that are illuminated by light of the respective first and second wavelengths. 
     Holographic display device further comprises a holographic controller  602  arranged to control the picture generating unit, specifically the light output by picture generating unit as described herein. First spatially modulated light of the first colour corresponding to the first picture is output by SLM  640  to form a first single colour image (e.g. red image) on a light receiving surface  670 , such as a screen or diffuser. A first single colour computer-generated hologram is calculated by a holographic controller  602  and encoded on SLM  640 , for example by a display driver  642 . The SLM  640  displays the first hologram and is illuminated by light of the first colour from the first colour/display channel to form a first holographic reconstruction on the light receiving surface  670  which is positioned at the replay plane. Similarly, second spatially modulated light of the second colour corresponding to the second picture is output by SLM  640  to form a second single colour image (e.g. green image) on the light receiving surface  670 . A second single colour computer-generated hologram is encoded on SLM  640  by holographic controller  602 . The SLM  640  displays the second hologram and is illuminated by light of the second colour from the second colour/display channel to form a second holographic reconstruction on the light receiving surface at the replay plane. In the illustrated arrangement, a beam splitter cube  630  is arranged to separate input light to SLM  640  and spatially modulated light output by SLM  640 . A Fourier lens  650  and mirror  660  are provided in the optical path of the output spatially modulated light to light receiving surface  670 . It may be said that a first/second picture is formed on the light receiving surface  670 . The first/second pictures are first/second holographic reconstructions of the respective first/second holograms. Thus, a composite colour picture may be formed on light receiving surface  670  combining the first and second pictures. A projection lens  680  is arranged to project the first and second pictures formed on the light receiving surface  672  to an input port of a pupil expander in the form of a waveguide  690 . A viewer  608  may view a magnified image of the pictures from the expanded eye box—the “viewing window”-formed by waveguide  690  owing to optical power of projection lens  680 . Waveguide  690  comprises an optically transparent medium separated by first and second reflective mirrors/surfaces as described herein. Thus, holographic display device has an “indirect view” configuration—that is the viewer  608  does not directly view the holographic reconstruction, but rather views pictures formed on light receiving surface  670 . 
     Waveguide Geometry 
       FIGS. 7A-C  show the geometry of the waveguide pupil expander in accordance with embodiments. In particular,  FIGS. 7B and 7C  show the positioning, and propagation along waveguide, of the chief rays of an input image beam  702  (as a described above) and the output of replicas visible to viewer  730 . Waveguide comprises first partially reflective surface  720  and second fully reflective surface  710  as described above. 
     The geometry shown in  FIG. 7C  may be represented by the following equations. 
     
       
         
           
             
               
                 
                   
                     
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     Second Embodiments—Direct View Holographic Display Device 
       FIG. 8  shows a holographic display device comprising waveguide pupil expander in accordance with second example embodiments of the present disclosure. 
     The holographic display device illustrated in  FIG. 8  is similar to the holographic display device of  FIG. 6 . In particular, holographic display device comprises a picture generating unit arranged to form a first picture (or first image) and a second picture (or second image). A first single colour/display channel (e.g. red colour channel) comprises a first light source  810 , a first collimating lens  812  and a first dichroic mirror  814  arranged to illuminate SLM  840  with light of the first wavelength. A second single colour/display channel (e.g. green colour channel) comprises a second light source  820 , a second collimating lens  822  and a second mirror  824  arranged to illuminate the SLM  840  with light of the second wavelength. The first display channel is arranged to form the first image (e.g. red image). A first single colour computer-generated hologram is encoded on SLM  840  by a holographic controller  802 . The SLM  840  displays the first hologram and is illuminated by light from the first colour channel to form a first holographic reconstruction at a replay plane. Similarly, the second display channel is arranged to form the second image (e.g. green image). A second single colour computer-generated hologram is encoded on SLM  840  by holographic controller  802 . The SLM  840  displays the second hologram and is illuminated by light from the second colour channel to form a second holographic reconstruction at a replay plane. 
     The holographic display device further comprises a beam splitter cube  830 , arranged to separate input light to and output light from SLM  840 . However, in contrast  FIG. 6 , the holographic display device is a direct view system. In the illustrated arrangement, an optional lens  850  is positioned in the optical path of the spatially modulated light output by SLM  840 , which determines the position of the replay plane along the optical path. A viewer  808  may directly view the spatially modulated light from the spatial light modulator. In some embodiments, as described above, the lens of the viewer&#39;s eye forms a holographic reconstruction on the retina of the eye. In these embodiments, it may be said that the viewer receives spatially modulated light encoded with the hologram. In other embodiments, the viewer receives light of the picture or light encoded with the picture. The picture may be formed at an intermediate plane in free space. Waveguide  890  comprises an optically transparent medium separated by first and second reflective surfaces as described above. Thus, holographic display device has an “direct view” configuration—that is the viewer looks directly at the display device (i.e. spatial light modulator) and no light receiving surface is used. 
     It should be noted that the holographic display devices of the embodiments illustrated in  FIGS. 6 and 8  comprise picture generating units with two colour channels to form a colour picture (or image) by way of example only. The skilled person will appreciate that arrangements comprising a single colour channel to form a single colour picture (or image) or three or more colour channels to form full colour (polychromatic) images are equally possible. 
     Pupil Expander-Dual-Waveguide Configuration for 1D Pupil Expansion 
     The above described first and second embodiments of a holographic display device comprise a picture generating unit and a pupil expander that expands the exit pupil/viewing window in one dimension. In accordance with the present disclosure, the pupil expander comprises a first waveguide pupil expander and a second waveguide pupil expander. The first and second waveguide pupil expanders are arranged in a composite configuration to reduce the volume occupied or required compared to a conventional single waveguide pupil expander that expands the exit pupil/viewing window in one dimension, for example as shown in  FIG. 5 . 
       FIG. 9  shows an example pupil expander in accordance with embodiments of the present disclosure. In particular, the pupil expander is a “dual waveguide” arrangement comprising a first waveguide pupil expander  900  and a second waveguide pupil expander  900 ′. Each waveguide  900 ,  900 ′ is approximately half the length of a single waveguide pupil expander for providing the desired amount of exit pupil expansion (e.g. number of replicas or distance). The first waveguide pupil expander  900  and the second waveguide pupil expander  900 ′ are arranged symmetrically in a V-shaped configuration. In particular, first (proximal) ends of first and second waveguides  900 ,  900 ′ are adjacent each of other and second (distal) ends of first and second waveguides  900 ,  900 ′ are remote from each other. In addition, first and second waveguides  900 ,  900 ′ are orientated symmetrically about the first proximal ends thereof. Thus, the first waveguide  900  is arranged at an (acute/non-orthogonal) angle θ relative to the plane of the viewing window in a first direction, and the second waveguide  900 ′ is arranged at substantially the same (acute/non-orthogonal) angle θ to the plane of the viewing window in a second direction, opposite to the first direction, as shown in  FIG. 9 . 
     The first waveguide  900  comprises a first input surface  901  adjacent a first end thereof. In the illustrated arrangement, the first waveguide pupil expander  900  comprises a slab waveguide comprising an optically transparent solid material. Thus, first input surface  901  may be formed by grinding and/or polishing a side surface of the slab waveguide (e.g. the side surface opposite to the partially reflective surface forming the viewing surface) adjacent first end surface  902 . First input surface  901  forms a first input port that is arranged to receive an input beam  910  comprising light of the picture (or image). The first waveguide  900  is arranged to guide the received light from the first input surface  901  in a first direction (towards the second (distal) end thereof) by a series of internal reflections, as described above with reference to  FIGS. 4 and 5 . In particular, the partially reflective surface of first waveguide pupil expander  900  is arranged to transmit replicas R 0  to R 3  of the received light of the picture (or image) along the length thereof in order to expand the exit pupil of the first waveguide pupil expander  900  in the first direction. It may be said that the first waveguide pupil expander  900  is arranged to expand the exit pupil (or viewing window) in the first direction of a first dimension (illustrated by the x-axis) of viewing plane  940  (e.g. width of viewing plane). The second waveguide  900 ′ comprises a second input surface  901 ′ at a first end thereof. Second input surface  901 ′ may be provided in the same way as first input surface  901  as described above. Second input surface  901 ′ of second waveguide  900 ′ is adjacent the first input surface  901  of first waveguide  900 . Thus, second input surface  901 ′ forms a second input port that is also arranged to receive the input beam  910  comprising light of the picture (or image). The second waveguide  900 ′ is arranged to guide the received light from the second input surface  901 ′ in a second direction (towards a second end thereof), which is opposite the first direction, by a series of internal reflections, as described above with reference to  FIGS. 4 and 5 . In particular, the partially reflective surface of second waveguide pupil expander  900 ′ is arranged to transmit replicas R 0 ′ to R 3 ′ of the received light of the picture (or image) in order to expand the exit pupil of the second waveguide pupil expander  900 ′ in the second direction. It may be said that the second waveguide pupil expander  900 ′ is arranged to expand the exit pupil (or viewing window) in the second direction of the first dimension (illustrated by the x-axis) of viewing plane  940 . 
     Similar to the single waveguide pupil expander of  FIG. 5 , the dual waveguide pupil expander of  FIG. 9  is arranged to maintain the direction of the optical/projection axis. Thus, the output light rays of the replicas R 0  to R 3  and R 0 ′ to R 3 ′ are in the same direction as the input light beam  910 . Accordingly, the optical/projection axis of the input and output light is in a second dimension (illustrated as the y-axis/direction) orthogonal to the first dimension of the viewing plane  940 . 
     In the illustrated arrangement, first end surface  902  of first waveguide pupil expander  900  is in abutment with first end surface  902 ′ of second waveguide pupil expander  900 ′. Thus, first input surface  901  of first waveguide pupil expander  900  adjoins second input surface  901 ′ of second waveguide pupil expander  900 ′ so as to form a continuous surface for receiving the input beam  910 . Accordingly, when implemented in a display device, the continuous surface formed by first and second input surfaces  901 ,  901 ′ is configured to face the display plane of a picture generating unit so as to receive the input beam  910 . 
     As shown in  FIG. 9 , first waveguide  900  is orientated at an angle relative to the plane of the viewing window and light rays of replicas R 0  to R 3  are evenly spaced and transmitted parallel to each other. Second waveguide  900 ′ is arranged symmetrically to first waveguide  900 , and so is orientated at the acute angle relative to the plane of the viewing window, but in the opposite direction. Thus, each waveguide pupil expander  900 ,  900 ′ has an overall height  920  in the second dimension (illustrated by y-axis), corresponding to the direction of light rays of replicas R 0  to R 3  and R 0 ′ to R 3 ′, respectively. For the avoidance of doubt, the orientation of the first and second waveguides  900 ,  900 ′ refers to the orientation of the length (or longitudinal axis) of the respective waveguide. 
     As previously mentioned, in the illustrated arrangement, each of the first and second waveguide pupil expanders  900 ,  900 ′ is approximately half the length of a single waveguide pupil expander for providing the same amount of pupil expansion (e.g. same number of replicas), as shown by the comparative example waveguide pupil expander  500  of  FIG. 5 . In particular, in the comparative example of  FIG. 5 , the pupil expander is arranged to form eight replicas by means of a single waveguide  500  with a first length and orientated at angle α. In the embodiments of  FIG. 9 , the pupil expander is also arranged to form eight replicas by means of a pair of waveguides  900 , 900 ′, each with a second length and orientated at angle β. Thus, in the case α=β, the second length of each of the first and second pupil expanders  900 ,  900 ′ may be half the first length of the single waveguide  500  to form four replicas each (R 0  to R 3  and R 0 ′ to R 3 ′). Thus, in accordance with the present disclosure, the overall height  920  of the composite (dual waveguide) pupil expander comprising the first and second waveguide pupil expanders  900 ,  900 ′ is reduced compared to a single waveguide pupil expander that provides the same amount of pupil expansion, such as the waveguide pupil expander  500  of  FIG. 5 . Thus, the volume occupied by the pupil expander is reduced. 
     Referring again to  FIG. 9 , it can be seen that first waveguide pupil expander  900  is arranged to form replicas R 0  to R 3 , which expand the exit pupil thereof in the first dimension (illustrated by the x-axis) in a first sub-area of the viewing window corresponding to the viewer&#39;s right eye  930 R. Similarly, second waveguide pupil expander  900 ′ is arranged to form replicas R 0 ′ to R 3 ′, which expand the exit pupil thereof in the first dimension (illustrated by the x-axis) in a second sub-area of the viewing window corresponding to the viewer&#39;s left eye  930 L. Thus the viewer is able to see the complete image with the right eye  930 R from any position in the first-sub-area as shown by associated arrows, and/or the left eye  930 L from any position within the second sub-area as shown be associated arrows, where the first and second sub-areas are on the viewing plane  940  at the viewing distance. Thus, the viewing window at the viewing plane is formed by both the first and second sub-areas extending in the first dimension (illustrated by the x-axis). It may be said that the composite (dual waveguide) pupil expander of  FIG. 9  forms a dual monocular system, in contrast to the single pupil expander of  FIG. 5  which forms a binocular system. In particular, in the composite arrangement of  FIG. 9 , each of the first and second waveguides forms replicas that are seen (exclusively) by only one eye of the viewer—as in a monocular system. 
     As the skilled person will appreciate, the arrangement shown in  FIG. 9  is just one example of possible forms and/or configurations of the first and second waveguide pupil expanders in accordance with the present disclosure. The first and second waveguides may be arranged in a complementary configuration (e.g. symmetrical about ends thereof), in which each waveguide extends lengthwise along the (expanded) first dimension of the viewing window. The first and second waveguides may have substantially the same length in order to provide a similar amount of pupil expansion. Equally, the first and second waveguides may have different lengths. In either case, the length of each of the first and second waveguides is reduced compared to the length of a comparable single waveguide providing the same amount of pupil expansion in one dimension (i.e. forming the same number of replicas). 
     Some embodiments may comprise first and second waveguide pupil expanders formed by pairs of mirrors, as in the example waveguide of  FIG. 4 . In such embodiments, the first ends of the first and second waveguides may be spatially separated to provide respective input ports for the input beam. Alternatively, the input ports may be formed in the second ends of the first and second waveguides. In such arrangements, suitable optics may be required to direct light from the display plane of the picture generating unit into the respective input ports of the first and second waveguides. 
     In further embodiments, first and second waveguide pupil expanders formed by respective slab waveguides may be arranged in different configurations to provide the required pupil expansion. For example, the input ports of the first and second waveguide pupil expanders may be located at opposite ends thereof. In such arrangements, each waveguide will be orientated at the required angle to provide internal reflection and light propagation through the waveguide. 
     Head-Up Display 
     Commercially-viable displays for the automotive industry are proposed by the applications using dynamic spatial light modulators including phase-only spatial light modulators. The main advantages of phase-only modulation are light efficiency and fault tolerance. 
     The physical extent of the replay field is limited by the maximum diffraction angle, θm, of the SLM. Associated with each point in the replay field is a pencil of rays, which diverges with angle, ϕ, and of which the chief ray also diverges. In Fourier holographic replay, the angle, ϕ, gives the field of view (FOV) of the reconstruction. A small FOV is a correlate of a large replay field size. The angle of the chief rays is determined by the curvature of the replay field. A projection optic (e.g. lens or powered mirror) is used to view a virtual image of the replay field at a large distance (e.g. 2 m, 5 m, 15 m, etc). The combiner is typically the windscreen of the car. If the chief ray from each ray pencil is projected back, they will meet at a distance from the lens, u, determined by the curvature of the replay field surface. The eye pupil placed at a distance, v, from the projection lens, where v is given by the lens equation, will view the complete image. The FOV is now the total field subtended at the eye pupil. 
     In addition to viewing the complete image, it is essential that the driver be able to move his/her head around within a limited area at the designed distance, v. In the automotive industry, this is known as the eye motion box (EMB)—but referred to more generically herein as the viewing window. In order to be able to keep looking at the complete image as the eye moves around the EMB, the waveguide pupil expander is employed. The pupil expander enlarges the EMB due to the generation of extra rays by division of amplitude of the incident wavefront. The extra rays augment the visual field directly below the eye. A further consequence is that a larger image size can be supported and, hence, a larger FOV. 
     The applicant has fabricated slab waveguides based on glass, and slabs with an air cavity to function as the pupil expander. The uniformity of the expansion can be tailored using either graded reflectors or tailored diffraction gratings on the surface of the slab or in the bulk. Alternative technologies may be used for the pupil expander. For example, microlens arrays that result in a virtual image location which is close behind the expander. The main virtue of a HUD including the display device of the present disclosure is the reduction of eye focus correction between the virtual image and the road viewed by the driver. 
     Additional Features 
     Embodiments refer to an electrically-activated LCOS spatial light modulator by way of example only. The teachings of the present disclosure may equally be implemented on any spatial light modulator capable of displaying a computer-generated hologram in accordance with the present disclosure such as any electrically-activated SLMs, optically-activated SLM, digital micromirror device or microelectromechanical device, for example. 
     In some embodiments, the light source is a laser such as a laser diode. In some embodiments, the detector is a photodetector such as a photodiode. In some embodiments, the light receiving surface is a diffuser surface or screen such as a diffuser. The holographic projection system of the present disclosure may be used to provide an improved head-up display (HUD) or head-mounted display. In some embodiments, there is provided a vehicle comprising the holographic projection system installed in the vehicle to provide a HUD. The vehicle may be an automotive vehicle such as a car, truck, van, lorry, motorcycle, train, airplane, boat, or ship. 
     The quality of the holographic reconstruction may be affected by the so-called zero order problem which is a consequence of the diffractive nature of using a pixelated spatial light modulator. Such zero-order light can be regarded as “noise” and includes for example specularly reflected light, and other unwanted light from the SLM. 
     In the example of Fourier holography, this “noise” is focused at the focal point of the Fourier lens leading to a bright spot at the centre of the holographic reconstruction. The zero order light may be simply blocked out however this would mean replacing the bright spot with a dark spot. Some embodiments include an angularly selective filter to remove only the collimated rays of the zero order. Embodiments also include the method of managing the zero-order described in European patent 2,030,072, which is hereby incorporated in its entirety by reference. 
     In embodiments, only the primary replay field is utilised and system comprises physical blocks, such as baffles, arranged to restrict the propagation of the higher order replay fields through the system. 
     In the above described embodiments, the holographic reconstruction is a composite colour image. In some embodiments, an approach known as spatially-separated colours, “SSC”, is used to provide colour holographic reconstruction. In other embodiments, an approach known as frame sequential colour, “FSC”, is used. 
     The method of SSC uses three spatially-separated arrays of light-modulating pixels for the three single-colour holograms. An advantage of the SSC method is that the image can be very bright because all three holographic reconstructions may be formed at the same time. However, if due to space limitations, the three spatially-separated arrays of light-modulating pixels are provided on a common SLM, the quality of each single-colour image is sub-optimal because only a subset of the available light-modulating pixels is used for each colour. Accordingly, a relatively low-resolution colour image is provided. 
     The method of FSC can use all pixels of a common spatial light modulator to display the three single-colour holograms in sequence. The single-colour reconstructions are cycled (e.g. red, green, blue, red, green, blue, etc.) fast enough such that a human viewer perceives a polychromatic image from integration of the three single-colour images. An advantage of FSC is that the whole SLM is used for each colour. This means that the quality of the three colour images produced is optimal because all pixels of the SLM are used for each of the colour images. However, a disadvantage of the FSC method is that the brightness of the composite colour image is lower than with the SSC method—by a factor of about 3-because each single-colour illumination event can only occur for one third of the frame time. This drawback could potentially be addressed by overdriving the lasers, or by using more powerful lasers, but this requires more power resulting in higher costs and an increase in the size of the system. 
     The methods and processes described herein may be embodied on a computer-readable medium. The term “computer-readable medium” includes a medium arranged to store data temporarily or permanently such as random-access memory (RAM), read-only memory (ROM), buffer memory, flash memory, and cache memory. The term “computer-readable medium” shall also be taken to include any medium, or combination of multiple media, that is capable of storing instructions for execution by a machine such that the instructions, when executed by one or more processors, cause the machine to perform any one or more of the methodologies described herein, in whole or in part. 
     The term “computer-readable medium” also encompasses cloud-based storage systems. The term “computer-readable medium” includes, but is not limited to, one or more tangible and non-transitory data repositories (e.g., data volumes) in the example form of a solid-state memory chip, an optical disc, a magnetic disc, or any suitable combination thereof. In some example embodiments, the instructions for execution may be communicated by a carrier medium. Examples of such a carrier medium include a transient medium (e.g., a propagating signal that communicates instructions). 
     It will be apparent to those skilled in the art that various modifications and variations can be made without departing from the scope of the appended claims. The present disclosure covers all modifications and variations within the scope of the appended claims and their equivalents.