Patent Publication Number: US-2019179063-A1

Title: Wide field-of-view holographic skew mirrors

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation-in-part of International Application No. PCT/US16/48499, filed Aug. 24, 2016, and entitled “Skew Mirrors, Methods of Use, and Methods of Manufacture,” which is a continuation-in-part of U.S. application Ser. No. 15/174,938, filed Jun. 6, 2016, and entitled “Skew Mirrors, Methods of Use, and Methods of Manufacture,” which in turn claims the priority benefit, under 35 U.S.C. § 119, of U.S. Application No. 62/318,917, filed Apr. 6, 2016, and entitled “Skew Mirrors, Methods of Use, and Methods of Manufacture,” and of U.S. Application No. 62/209,290, filed Aug. 24, 2015, and entitled “Multiwavelength Diffraction Grating Mirrors, Methods of Use, and Methods of Manufacture.” This application also claims the priority benefit, under 35 U.S.C. § 119, of U.S. Application No. 62/435,676, filed Dec. 16, 2016, and entitled “Wide Field of View Skew Mirror,” and of U.S. Application No. 62/407,994, filed Oct. 13, 2016, and entitled “TIGER Prisms and Methods of Use.” Each of these applications is incorporated by reference herein. 
    
    
     BACKGROUND 
     A holographic skew mirror is a holographic optical element that reflects incident light about a reflective axis that need not be perpendicular to the surface upon which the incident light impinges. In other words, a holographic skew mirror&#39;s reflective axis does not have to be parallel to or coincident with the surface normal of the holographic optical element. The angle between the reflective axis and the surface normal is referred to as the reflective axis angle and can be selected based on the desired application of the holographic skew mirror. 
     The term “reflection” and similar terms are used in this disclosure in some cases where “diffraction” might ordinarily be considered an appropriate term. This use of “reflection” is consistent with mirror-like properties exhibited by skew mirrors and helps avoid potentially confusing terminology. For example, where a grating or skew mirror is said to be structured to “reflect” incident light, a conventional artisan might prefer to say the grating structure is structured to “diffract” incident light, since grating structures are generally thought to act on light by diffraction. However, such use of the term “diffract” would result in expressions such as “incident light is diffracted about substantially constant reflective axes,” which could be confusing. 
     Accordingly, where incident light is said to be “reflected” by a grating structure, persons of ordinary skill in art, given the benefit of this disclosure, will recognize that the grating structure is in fact “reflecting” the light by a diffractive mechanism. Such use of “reflect” is not without precedent in optics, as conventional dielectric mirrors are generally said to “reflect” light despite the predominant role diffraction plays in such reflection. Artisans of ordinary skill thus recognize that most “reflection” includes characteristics of diffraction, and “reflection” by a skew mirror or components thereof also includes diffraction. 
     SUMMARY 
     Embodiments of the present technology include holographic optical elements, including but not limited to holographic skew mirrors, holographic input/output couplers, and other holographic optical reflective devices. One example is an optical reflective device that includes a grating structure residing in a grating medium. This grating structure is structured to principally reflect incident light as reflected light, where the incident and reflected light both include a first wavelength. The incident light of the first wavelength and the reflected light of the first wavelength form an angle bisected by a reflective axis, which varies by less than 1 degree where the incident light is incident upon the grating medium at a range of internal angles of incidence spanning at least 15 degrees. In addition, the reflective axis differs from a surface normal of the grating medium by at least 2.0 degrees. 
     In some implementations of this optical reflective device, the reflective axis varies by less than 1 degree where the incident light is incident upon the grating medium at a range of internal angles of incidence spanning at least 30 degrees. Likewise, the grating structure may include one or more holograms having a grating frequency (|K G |) spanning a range of at least 2.00×10 5  radians per meter. 
     In some cases, the incident and reflected light both include a second wavelength that differs from the first wavelength by at least about 50 nm (e.g., the first wavelength may be 50 nm, 60 nm, 70 nm, 80 nm, 90 nm, 100 nm, or more greater than the second wavelength). And in some of these cases, the incident and reflected light includes a third wavelength that differs from each of the first wavelength and the second wavelength by at least about 50 nm (e.g., the first wavelength may be 50-100 nm greater than the second wavelength, which in turn may be 50-100 nm greater than the third wavelength). For example, the first wavelength may reside in the red region of the electromagnetic spectrum, the second wavelength may reside in the green region, and the third wavelength may reside in the blue region. 
     The grating structure in the optical reflective device may include one or more holograms having a grating frequency (|K G |) spanning a range of at least 1.68×10 6  radians per meter, at least 5.01×10 6  radians per meter, or at least 1.24×10 7  radians per meter. For instance, the grating structure may include one or more holograms having grating frequencies (|K G |) spanning a range of greater than 5.10×10 5  radians per meter and less than 3.15×10 7  radians per meter. 
     In some examples, the grating structure includes at least nine holograms. The mean adjacent |ΔK G | of these holograms may reside in a range between 5.0×10 3  rad/m and 1.0×10 7  rad/m. 
     The optical reflective device may be configured or structured to act as an output coupler, with the incident light is incident upon the grating structure from inside the optical reflective device and the reflected light exiting the optical reflected device. 
     The optical reflective device may further comprise at least one substrate adjacent to the grating medium. For example, the optical reflective device may include two substrates with the grating medium disposed between the two substrates. In these cases, the grating medium may comprise a photopolymer medium at least 100 μm thick, and the two substrates may transmit at least 60% of the incident light and at least 60% of the reflected light. The refractive indices of the grating medium and the two substrates may be within about 0.1 of each other. 
     Other embodiments of the present technology include a method of using an optical reflective device. This method comprises illuminating a grating structure residing in a grating medium with incident light at a first wavelength. This incident light reflects off the grating structure to produce reflected light at the first wavelength. The incident and reflected light form an angle bisected by a reflective axis tilted by at least about 2.0 degrees with a surface normal of the grating medium. This reflective axis varies by less than 1 degree where the incident light is incident upon the grating structure within the grating medium at a range of internal angles of incidence spanning at least 15 degrees. In some cases, the reflective axis varies by less than 1 degree where the incident light is incident upon the grating medium at a range of internal angles of incidence spanning at least 30 degrees. 
     In examples of this method, illuminating the grating structure comprises coupling the incident light into the grating medium, e.g., via a holographic input coupler, prism, or edge coupling, and totally internally reflecting the incident light within the grating medium. Put differently, the grating medium may guide the incident light at least partway to the grating structure. 
     As mentioned above, the incident and reflected light may comprise a second wavelength different from the first wavelength by at least about 50 nm. The incident and reflected light may also comprise a third wavelength different from each of the first wavelength and the second wavelength by at least about 50 nm. 
     Examples of this method may also include coupling the reflected light out of the grating medium at an angle of about 25 degrees with respect to the surface normal of the grating medium. The grating medium may couple this reflected light towards a human eye in optical communication with the grating medium such that the reflected beam at least partially illuminates the human eye. In these cases, illuminating the grating structure may comprise illuminating the grating structure with an image such that a reflected image appears visible to the human eye. 
     Another example of the present technology includes a method of imaging. This method comprises disposing a grating medium containing a grating structure in optical communication with a human eye. This grating medium has a proximate surface defining a surface normal. A visible image is coupled into the grating medium and guided to the grating structure via at least one total internal reflection within the grating medium. The grating structure reflects the visible image about an axis of reflection forming an angle of at least about 2 degrees with respect to the surface normal. The visible image is coupled out of the grating medium via the proximate surface towards the human eye. This visible image spans a field of view in free space of at least about 30 degrees. 
     Yet another example of the present technology includes a method of writing holographic gratings in a photosensitive medium. This method includes disposing the photosensitive medium between an oblique face of a first prism and an oblique face of a second prism. A first beam is coupled into the photosensitive medium via a first surface of the photosensitive medium and the oblique face of the first prism. This first beam forms a first angle with respect to a surface normal of the first surface. A second beam is coupled into the photosensitive medium via a second surface of the photosensitive medium and the oblique face of the second prism. This second beam forms a second angle with respect to a surface normal of the second surface with a magnitude substantially equal to that of the first angle. In some case, this method also includes interfering a third beam and a fourth beam in the photosensitive medium to form a second holographic grating within the photosensitive medium. 
     Still another example of the present technology includes a device with a holographic optical element having at least one grating structured to reflect visible light over a field of view of at least about 50° in a first dimension. The field of view is measured outside the holographic optical element and is substantially centered on a surface normal of the holographic optical element. And the grating has a grating vector oriented at an angle of at least about 15° to about 45° with respect to the surface normal. 
     In some implementations, the holographic optical element includes a single grating structured to reflect visible light at wavelengths in a range from about 400 nm to about 700 nm over the field of view. In other implementations, the holographic optical element includes a plurality of gratings, each structured to reflect incident light at one wavelength of the visible light at a different angle within the field of view. In these implementations, the device may also include at least one light source, in optical communication with the holographic optical element, to illuminate the plurality of gratings with the visible light. 
     The field of view may be at least about 30° in a second dimension orthogonal to the first dimension. In addition, the angle formed by the axis of reflection and the surface normal may be about 20° to about 40°. And the holographic optical element may be substantially free of photo-initiators sensitive to visible light. 
     Another instance of the present technology includes a method of reflecting light. This method comprises illuminating at least one grating in a holographic optical element with visible light. The grating reflects at least a portion of the light over a field of view of at least about 50°. This field of view is centered on an axis of reflection that forms an angle of at least about 15° to about 45° with respect to a surface normal of the holographic optical element. 
     Still another instance of the present technology includes method of making a holographic optical element (and the resulting holographic optical element). This method comprises interfering a first beam and a second beam inside a holographic recording medium to form a first grating. The holographic recording medium has a planar surface. The first grating is structured to reflect incident light at a first visible wavelength over a field of view of at least about 50°. This field of view is centered on an axis of reflection that forms an angle of at least about 15° to about 45° with respect to a surface normal of the planar surface of the holographic optical element. In some cases, interfering the first beam and the second beam comprises coupling the first beam into the holographic recording medium via an oblique surface of a first prism and coupling the second beam into the holographic recording medium via an oblique surface of a second prism. 
     Yet another instance of the present technology includes device comprising a holographic optical element having a plurality of reflection gratings. Each reflection grating in the plurality of reflection gratings has a grating vector K G  forming an angle of about 15° to about 45° with a surface normal of the holographic optical element and a grating frequency (K G S) of at least 2.00×10 5  radians per meter. 
     All combinations of the foregoing concepts and additional concepts discussed in greater detail below (provided such concepts are not mutually inconsistent) are part of the inventive subject matter disclosed herein. In particular, all combinations of claimed subject matter appearing at the end of this disclosure are part of the inventive subject matter disclosed herein. The terminology used herein that also may appear in any disclosure incorporated by reference should be accorded a meaning most consistent with the particular concepts disclosed herein. 
    
    
     
       BRIEF DESCRIPTIONS OF THE DRAWINGS 
       The skilled artisan will understand that the drawings primarily are for illustrative purposes and are not intended to limit the scope of the inventive subject matter described herein. The drawings are not necessarily to scale; in some instances, various aspects of the inventive subject matter disclosed herein may be shown exaggerated or enlarged in the drawings to facilitate an understanding of different features. In the drawings, like reference characters generally refer to like features (e.g., functionally similar and/or structurally similar elements). 
         FIG. 1  shows a holographic skew mirror with a relatively narrow field of view. 
         FIGS. 2A and 2B  show a k-space representation of the holographic skew mirror shown in  FIG. 1  without and with an incident beam, respectively. 
         FIGS. 3A and 3B  show an in-plane holographic recording system suitable for making holographic skew mirrors. 
         FIGS. 4A and 4B  show k-space representations of making a holographic skew mirror using the in-plane holographic recording geometry of  FIGS. 3A and 3B , respectively. 
         FIGS. 5A and 5C  show real-space perspective views of an out-of-plane holographic skew mirror writing geometry. 
         FIGS. 5B and 5D  show k-space representations of the real-space views shown in  FIGS. 5A and 5C , respectively. 
         FIG. 6  is a plot that shows the angular recording bands achievable with both in-plane and out-of-plane holographic skew mirror recording geometries. 
         FIGS. 7A-7C  show different views of a holographic recording medium sandwiched between Total Internal Grazing-Extension Rotation (TIGER) prisms for writing holographic skew mirrors with wide fields-of-view. 
         FIG. 7D  shows a perspective view of the TIGER prisms used in the holographic recording geometry of  FIGS. 7A-7C . 
         FIGS. 8A-8C  shows a holographic recording system with TIGER prisms and the out-of-plane holographic recording geometry shown in  FIGS. 7A-7C . 
         FIG. 9  illustrates angle-correction using pairs of wedges. 
         FIG. 10  shows a plan view of a holographic skew input/output coupler with a 60° diagonal field of view (53.4° horizontal field-of-view, 31.6° horizontal field-of-view, and 16:9 aspect ratio) made using an out-of-plane holographic recording system. 
         FIG. 11  shows a k-space representation of the recording beams for the first and 228 th  gratings in the holographic skew mirror of  FIG. 10 . 
         FIG. 12  is a plot of skew mirror internal angle recording bands for a 53.4° field-of-view holographic skew mirror output coupler. 
         FIG. 13  illustrates an experimentally realized holographic skew output coupler with a 53.4° horizontal field of view and a 31.6° vertical field of view coupled to a waveguide. 
         FIG. 14  is a mosaic of modulation transfer function (MTF) plots of the holographic skew mirror of  FIG. 13 . 
         FIG. 15  shows a head-mounted display with a wide field-of-view holographic skew mirror. 
     
    
    
     DETAILED DESCRIPTION 
     Holographic Skew Mirrors 
       FIG. 1  shows a real-space representation of a holographic skew mirror  100 . This holographic skew mirror  100  includes a grating structure  120  recorded in a holographic grating medium  110 , such as Tapestry® holographic photopolymer media from Akonia Holographics LLC of Longmont, Colo., or Bayfol® HX200 light-sensitive, self-developing photopolymer film from Covestro AG of Leverkusen, Germany. The grating structure  120  may comprise many discrete holographic gratings, each of which reflects light over a narrow range of angles and/or wavelengths. 
     In this case, the grating structure  120  includes many holographic gratings that define both a skew axis  121  and a reflective axis. The grating vector for each holographic grating is parallel to or coincident with the skew axis  121 , which forms a skew angle θ with respect to a surface normal  111  of the holographic grating medium  110 . As mentioned briefly above, each holographic grating reflects light at a particular wavelength or range of wavelengths over a particular range of internal angles of incidence, which are incidence angles on the grating structure  120  as measured within the holographic grating medium  110 . The axis about which each holographic grating reflects incident light is called the reflective axis. 
     Each holographic grating&#39;s reflective axis may vary slightly from the skew axis  121  with wavelength, e.g., by about less than 0.1°, less than 0.01°, less than by 0.001°, etc. In light of this very slight variation, a skew axis/reflective axis may be called a skew axis when referring to making a skew mirror (for example when describing recording a hologram in a skew mirror grating medium), and as a reflective axis when referring to light reflective properties of a skew mirror. 
     A mean skew angle for a hologram (including a mean skew angle for a collection of holograms) can be substantially identical to a reflective axis angle, meaning the skew angle or mean skew angle is within 1.0°, 0.1°, 0.05°, 0.02°, 0.0167° (1 arcmin), or less of the reflective axis angle. Given the benefit of the present disclosure, persons skilled in the art will recognize that the skew angle and reflective axis angle can be theoretically identical. However, due to limits in system precision and accuracy, shrinkage of recording medium that occurs during recording holograms, and other sources of error, the skew angle or mean skew angle as measured or estimated based on recording beam angles may not perfectly match the reflective axis angle as measured by incidence angles and reflection angles of light reflected by a skew mirror. This variation occurs at the single hologram level and is inversely proportional to the hologram&#39;s thickness. Nevertheless, a skew angle determined based on recording beam angles can be within 1.0°, 0.1°, 0.05°, 0.02°, 0.0167°, or less of the reflective axis angle determined based on angles of incident light and its reflection, even where medium shrinkage and system imperfections contribute to errors in estimating skew angle and reflective axis angle. 
     In  FIG. 1 , an incident beam  101 ′ of visible light impinges on a surface  112  of the holographic grating medium  110  at an angle θ i ′ with respect to the skew axis  121 . This beam  101 ′ may be a monochromatic, polychromatic, or broadband visible beam of light. The holographic grating medium  110  has a higher refractive index than the surrounding air, so the incident beam  101 ′ refracts at the surface  112  to form a refracted incident beam  101 . The refracted incident beam  101  illuminates the volume hologram  120  at an angle θ i  with respect to the skew axis  121 . The angle θ i  is also called the internal angle of incidence because it is the angle of incidence on the volume hologram  120  measured inside the holographic grating medium  110 . 
     The volume hologram  120  reflects at least a portion of the refracted incident beam  101  at an angle θ r  with respect to the skew axis  121 . The angle θ r  is also called the internal angle of reflection and is equal to the internal angle of incidence θ i  as shown in  FIG. 1 . Put differently, the skew axis  121  bisects an angle equal to twice the internal angle of incidence θ i . 
     The reflected portion of the refracted incident beam  101  is called the principal reflected beam  103 . The principal reflected beam  103  impinges on the surface  112  of the holographic grating medium  110 . It refracts at this boundary to form a refracted principal reflected beam  103 ′ at an angle θ r ′ with respect to the skew axis  121 . The holographic skew mirror&#39;s field-of-view, as measured in free space outside the holographic grating medium  110 , is determined by the range of external angles of reflection θ r ′. 
     k-Space Representation of a Holographic Skew Mirror 
       FIGS. 2A and 2B  show a k-space representation of the holographic skew mirror  100  shown in  FIG. 1  with and without the incident beam  101  and principal reflected beam  103 , respectively. As readily appreciated by those of skill in the art, this k-space representation includes multiple concentric circles, each of which is a two-dimensional projection of a k-sphere representing the optical propagation vectors, or wave vectors, for light at a particular wavelength in the holographic medium. The length of the wave vector can be represented as: 
     
       
         
           
             
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     where n is the refractive index and λ is the wavelength. 
     In normally dispersive media, including the holographic grating medium  110 , the wave vectors (and hence the k-sphere radii) are longer for shorter wavelengths. Thus, the innermost circle  290  represents wave vectors for red light in the holographic grating medium  110 , the second innermost circle  291  represents wave vectors for green light in the holographic grating medium  110 , the second outermost circle  292  represents wave vectors for blue light in the holographic grating medium  110 , and the outermost circle  293  represents wave vectors at the recording wavelength in the holographic grating medium  110 . 
       FIGS. 2A and 2B  also show the volume hologram  120 , which appears in k-space as a line segment-like distribution whose distribution of grating vectors K G  is parallel to the reflective/skew axis  121 .  FIG. 2B  also shows the wave vectors of the incident refracted beam  101  and the principal reflected beam  103  with respect to the volume hologram&#39;s grating vector. In k-space, the wave vector of the principal reflected beam  103  is the vector sum of the volume hologram&#39;s grating vector and the wave vector of the incident refracted beam  101 . 
     An In-Plane Holographic Skew Mirror Recording System 
       FIGS. 3A and 3B  illustrate a skew mirror recording system  300  that uses in-plane recording prisms  330   a  and  330   b  (collectively, in-plane recording prisms  330 ) to couple light into a holographic recording medium  310 , which is disposed between a pair of transparent substrates (not shown). The recording medium  310  and substrates are sandwiched between the in-plane recording prisms  330  so that a signal beam  331   a  and a reference beam  331   b , also called recording beams  331 , can be introduced into the holographic recording medium  310  at angles that would produce total internal reflection (TIR) at the substrate-air boundary were the in-plane recording prisms  330  not present. The in-plane recording prisms are typically index-matched to the substrates, and an index-matching fluid may be applied at a boundary between the prisms  330  and substrates (not shown) to reduce reflection and refraction at the prism/substrate boundaries. In practice, index-matching may mean that the refractive indices of the holographic recording medium  310 , substrates, and prisms  330  are within about 0.1 or less. 
     Mirrors  350   a  and  350   b  (collectively, mirrors  350 ) reflect the recording beams  331   a  and  331   b , respectively, into the holographic recording medium  310  via the prisms  330   a  and  330   b , respectively. Each mirror  350   a ,  350   b  is oriented to direct the corresponding recording beam  331   a ,  331   b  such that it illuminates a base of the corresponding prism  330   a ,  330   b . The recording beams  331  may refract at the air/base interface, then propagate into the holographic medium  310 , where they interfere to produce a (reflection) grating recorded by the holographic recording medium  310 . The holographic recording medium  310  and prisms  330  are translated back and forth along the z G  axis with respect to the mirrors  350  using a translation stage (not shown) and the mirrors  350  are rotated to record the series of gratings that make up the skew mirror as shown in  FIG. 3B . 
       FIGS. 3A and 3B  also illustrate global, or recorder, coordinates (x G , y G , z G ) for the case of in-plane prisms. The origin of the global coordinates shown in  FIGS. 3A and 3B  is defined to be in the center of the output coupler in the center of the recording layer of the holographic recording medium  310 . The Global Angle for recording, θ G , is defined as the angle of the recording beam  331   a  with respect to the x G  axis within the holographic recording medium  310 /prism  330   a . Note that the nominal angle of the other recording beam  331   b  is 180°−θ G  (not marked), so that recorded grating vectors are aligned substantially with the x G  axis. The angle between the recordings beams  331 , or interbeam angle, within the holographic recording medium  310 /prism  330   a  is marked as α. The global skew angle is the angle between the x G  and z axes and is marked as ϕ G . 
     Given the benefit of this disclosure, one skilled in the art will ascertain that standard coordinates (Cartesian coordinates in the frame of reference of the holographic recording medium  310 ) may be converted to global coordinates for the case of in-plane prisms by 
     
       
         
           
             
               
                 
                   
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     Conversion from global to standard coordinates may also easily be derived. 
     In the global coordinate frame, the surface normal of the holographic recording medium  310  forms an angle ϕ G  (the global skew angle) with respect to the z G  axis. In other words, the angle between the holographic recording medium  310  and the z G  axis sets the skew axis of the holographic skew mirror. This skew axis can be changed, e.g., by rotating the holographic recording medium  310  and prisms  330  with respect to the recording beams  331  using an appropriate combination of stages and mounts. 
     For more information on holographic skew mirrors and methods of making and using holographic skew mirrors, see U.S. application Ser. No. 15/174,938, filed Jun. 6, 2016, and entitled “Skew Mirrors, Methods of Use, and Methods of Manufacture,” which is incorporated herein by reference in its entirety. 
     Constraints on Field-of-View with In-Plane Recording 
     Unfortunately, in-plane recording systems generally cannot be used to make holographic skew mirrors with wide fields of view. This is due to constraints on the geometry and wavelengths of the beams used to record the volume holographic gratings for the holographic skew mirror. These constraints include the skew angle, the grating frequency, which determines the angle of reflection, and the difference between the recording beam wavelength, which is usually in the deep blue region of the electromagnetic spectrum (e.g., 400 nm to 430 nm), and the reading beam wavelength, which is usually in the visible region of the electromagnetic spectrum. 
     As mentioned briefly above, a holographic grating&#39;s grating frequency, which can be expressed as the magnitude of the grating wave vector, |K G |, determines its angle of reflection: the smaller the grating frequency, the larger the angle of reflection. For a skew mirror, increasing the range of grating vector magnitudes in the grating structure is necessary to increase or widen the field of view. But the holographic recording medium&#39;s refractive index and the skew angle both limit the range of recording angles that can be accessed via in-plane prisms like those shown in  FIGS. 3A and 3B . For certain combinations of skew angles and angles between the recording beams, one or both of the recording beams may become parallel to the holographic recording medium surface, making it difficult if not impossible to interfere the recording beams within the holographic recording medium. 
     To see how these geometrical constraints limit the holographic skew mirror&#39;s field of view, consider the in-plane recording system  300  shown in  FIGS. 3A and 3B . As shown in  FIG. 3B , one recording beam  331   b ′ makes a grazing angle that is steeper (higher) with respect to surface of the recording medium  310  than that made by the other recording beam  331   a ′. Increasing this grazing angle reduces the spatial frequency (magnitude) of the holographic grating recorded by the holographic recording medium, which in turn increases the field of view of the skew mirror. Unfortunately, increasing the grazing angle may degrade the recording quality since it magnifies the effects of aberrations and refractive index mismatch between the prisms  330  and the holographic recording medium  310 . 
     Furthermore, Snell&#39;s Law may limit the maximum grazing angle (the exact limit depends on the recording wavelength, the refractive indices of the recording medium and surrounding media, and the skew angle). Above this limit, the recording beam  331   b ′ may reflect off the holographic recording medium  310  instead of coupling into it. The upper bound on the grazing angle may limit the ability to record lower frequency holograms, which may limit the skew mirror&#39;s field of view for some colors, particularly for large skew angles. 
       FIGS. 4A and 4B  show k-space representations of the in-plane recording geometries in  FIGS. 3A and 3B , respectively. In  FIG. 4A , the recording beams  331   a  and  331   b  are incident on the holographic recording medium such that their wave vectors form an interbeam angle α and record a holographic grating with a grating vector K G  that is parallel to the skew axis  421 . In  FIG. 4B , the recording beams  331   a ′ and  331   b ′ are incident on the holographic recording medium such that their wave vectors form an interbeam angle α′ and record a holographic grating with a grating vector K G ′ that is also parallel to the skew axis  421 . 
     The size of the grating vector determines the internal incidence angle(s) over which the corresponding holographic grating reflects incident light. Holographic gratings with smaller gratings reflect light at larger internal incidence angles as measured from the skew axis, and holographic gratings with larger gratings reflect light at smaller internal incidence angles as measured from the skew axis. The largest possible grating vector is recorded when the recording beams&#39; wave vectors are antiparallel and aligned with the skew axis  421 . The corresponding holographic grating retro-reflects light incident on the grating medium along the skew axis  421  (“normal incidence” for a holographic skew mirror). 
     As the angle between the recording beam  331   b ′ and the x axis shrinks, the interbeam angle α also shrinks, reducing the size of the grating vector K G  and increasing the possible field of view. Eventually, though, the angle between the recording beam  331   b ′ and the x axis becomes so small that the recording beam  331   b ′ becomes parallel to the surface of the holographic recording medium  310  instead of refracting into the holographic recording medium  310 . In other words, the limit occurs when the grating vector of the recording beam  331   b ′ is aligned with k x , i.e., when the recording beam  331   b ′ is parallel to the surface of the holographic recording medium  310 . At this point, the recording beam  331   b ′ no longer interferes with the other recording beam  331   a ′ within the holographic recording medium  310  to record a reflection grating. This limits the smallest size of the grating vector and hence the field of view. Although rotating the skew axis may compensate for this effect, it also limits the range of permissible skew angle/field of view combinations. 
     In sum,  FIGS. 3A, 3B, 4A, and 4B  illustrate the tradeoff between permissible skew angle and permissible field of view in the in-plane recording geometry: generally, it is possible to have a large skew angle or a large field of view with in-plane recording, but not both. 
     The smallest accessible angular difference between recording beams depends in part on the wavelengths of the recording beams and the reading beams and dispersion of the holographic recording medium. Most holographic recording media is optimized to record gratings at deep blue wavelengths, e.g., 405 nm, and to be insensitive to visible light at longer wavelengths. But it is difficult to impossible to interfere deep blue beams within a holographic recording medium at angular differences small enough to generate reflective gratings at spatial frequencies low enough to produce a wide field-of-view at visible wavelengths in an in-plane recording system. 
     Increasing the recording beam wavelength would alleviate this problem, but would also require a holographic recording medium sensitive to longer-wavelength light. But increasing the holographic recording medium&#39;s sensitivity to longer-wavelength light would make the holographic recording medium more susceptible to incomplete bleaching at visible wavelengths. This is because holographic recording media with photo-initiators sensitive to visible light may polymerize when exposed to visible light and is therefore less suitable for making holographic optical elements that operate at visible wavelengths. Moreover, photo-initiators sensitive to visible light can cause undesirable visible light absorbance in the grating medium. This would make the holographic recording medium less suitable for use in a skew mirror that reflects light at visible wavelengths. 
     Out-of-Plane Recording for Making Wide Field-of-View Holographic Skew Mirrors 
     As explained above, geometry limits the range of accessible interbeam angles (and hence the maximum field of view) for writing holographic skew mirrors with in-plane recording systems. But the inventors have recognized that it is possible to access smaller interbeam angles by tilting the surface normal out of the plane accessed by an in-plane recording system. In other words, rotating the medium 90° about the skew axis relieves the constraints shown in  FIGS. 3A, 3B, 4A and 4B . 
       FIGS. 5A-5D  show how rotating recording beams  531   a  and  531   b  (collectively, recording beams  531 ) about the y G  axis, which is in the x-z plane of the holographic recording medium  310 , makes it possible to record shorter holographic gratings than is possible with an in-plane recording geometry.  FIGS. 5A and 5C  show real-space views from different perspectives of the recording beams  531  incident on a holographic recording medium  510  that is tilted about the z G  axis of the recording system&#39;s coordinate frame (i.e., they axis of the holographic recording medium  510 ). The recording beams  531  are rotated within the x G -z G  plane to record grating vectors that also lie in the x G -z G  plane. 
       FIGS. 5B and 5D  are k-space representations of the real-space views shown in  FIGS. 5A and 5C , respectively. As shown in both  FIGS. 5B and 5D , the wave vectors of the recording beams  531  lie within the x G -z G  plane, which forms an off-axis slice of a k-sphere  591  representing the recording beams&#39; momentum within the holographic recording medium  510 . As in the in-plane recording geometry, changing the interbeam angle between the recording beams  531  changes the length of the grating vector. The longest grating vector (maximum |K G |) is written when the recording beams  531  are counter-propagating along the x G  axis. and the smallest grating vector (minimum |K G |) occurs when the recording beams  531 ′ are co-propagating along the z G  axis (y axis). This is the grazing condition for both recording beams  531 ′ as shown in  FIG. 5A . 
     Those of skill in the art will appreciate that  FIGS. 5A-5C  show just one of the many possible orientations of recording beams and skew angles. The skew angle and each recording beam can be adjusted as desired, within the constraints imposed by the writing wavelength and the holographic recording medium&#39;s refractive index, to record holographic gratings at a wide variety of spatial frequencies. The exact number and spatial frequencies of holographic gratings depend on the desired field of view of the holographic skew mirror, among other things. 
     In-Plane Versus Out-of-Plane Recording 
       FIG. 6  is a plot illustrating the capabilities of out-of-plane vs. in-plane recording prisms for a specific recording geometry. The plot indicates the spatial grating frequencies for a waveguided, head-mounted display (HMD) skew mirror output coupler with skew axis ϕ=−30.25° supporting a horizontal field of view of 53.4°. The horizontal axis is grating frequency (in rad/m), and the vertical axis shows the Bragg-matching angle with respect to the grating vector/skew axis. The five curves illustrate the Bragg-matching angle for five different indicated wavelengths: curve  690  is for the wavelength at which the holograms are recorded (405 nm); curve  691  is for 463 nm (blue);  692  is for 522 nm (green); curve  693  is for 622 nm (red); and curve  694  is for 860 nm. The horizontal lines at 47.75° and 12.75° demark the range of spatial grating frequencies required for the red, green, and blue wavelengths, coded by color. 
     With in-plane recording prisms, the reference beam grazing condition occurs where the writing wavelength curve crosses 59.75° (=90°−ϕ), as indicated by the solid arrow. This indicates that the in-plane recording system cannot record gratings that fall to the left of the solid arrow. Gratings close to the right side of the solid arrow may suffer degradation since the reference beam is incident on the index-matched boundary at a very shallow angle. 
     With out-of-plane recording, however, the grazing condition does not occur until the writing wavelength curve crosses 90° at the left edge of the plot. The lowest writing angle, indicated by the open arrow, makes angles of about 22° with respect to the internal boundaries for both the reference and signal beams, an angle which is easily practicable. 
     TIGER Prisms for Out-of-Plane Holographic Skew Mirror Recording Systems 
       FIGS. 7A-7C  show different perspectives of an out-of-plane holographic recording system  700  that can record holographic skew mirrors with wide fields of view. In this holographic recording system  700 , a holographic recording medium  710  is disposed between a pair of total internal grazing-extension rotation (TIGER) prisms  730   a  and  730   b  (collectively, TIGER prisms  730 ). The holographic recording medium  710  may also be sandwiched between a pair of transparent substrates (not shown) with index-matching fluid disposed on the surfaces of the transparent substrates contacting the prisms  730 . These substrates may transmit 60%, 70%, 80%, 90%, or more of light at visible wavelengths. The TIGER prisms  730  enable introducing recording beams into the holographic recording medium  710  at angles that are inaccessible using in-plane recording geometries because of total internal reflection (TIR) and grazing angles constraints. 
       FIGS. 7A-7C  (and  FIGS. 5A-5D ) also show the symmetry of the recording beams  731   a  and  731   b  ( 531   a  and  531   b  in  FIGS. 5A-5D ). More specifically, these figures show that the magnitudes angles between the recording beams and the surface normals of the holographic recording medium  710  are substantially equal. Put differently, if recording beam  731   a  forms a first angle (e.g., 32°) with respect to the holographic recording medium&#39;s surface normal, then recording beam  731   b  and the holographic recording medium&#39;s surface normal form an angle with the same magnitude (e.g., −32°). (Here, the recording beams  731  are incident on surfaces of the holographic recording medium  710  that are parallel and therefore have coincident/parallel surface normals.) This holds as the recording beams  731  are rotated with respect to the holographic recording medium  710  as shown in  FIGS. 5A-5D . 
     As shown in  FIG. 7D , each TIGER prism  730  has a prism primary face  732   a ,  732   b  (collectively, primary faces  732 ) that is oblique to the prism base. In this example, the primary face  732  of each TIGER prism  730  is a hexagonal face residing parallel to and immediately proximate the holographic recording medium  710  where the prism  730  and the holographic recording medium  710  reside in a skew mirror recording system. The obliqueness of the primary faces  730  makes it possible to access out-of-plane recording beam angles and skew angles like those illustrated in  FIGS. 5A-5D . 
     The TIGER prisms  730  can be visualized by imagining a cutting a glass cuboid, or rectangular prism, into two sections. The cut in the cuboid is along a plane that joins a diagonal line connecting adjacent sides of one of the cuboid&#39;s faces (prism faces  734   a  and  734   b ) with another diagonal line connecting the other two sides of the opposite face of the cuboid (prism faces  736   a  and  736   b ). The resulting sections of the cuboid form a matched pair of TIGER prisms  730 . 
     In practice, TIGER prisms can be of any suitable shape, so long as they have oblique faces or facets angled to enable access of out-of-plane recording angles. For instance, a TIGER prism may be formed as a section of any suitable polyhedra, including parallelepipeds and regular right (geometrical prisms). Likewise, the face/facet may oriented or angled as desired and does not necessarily have to result in a symmetric division of a polyhedron. The face (and the holographic recording medium) could also be curved, e.g., to form at least a portion of a spherical, cylindrical, or conical surface. Other surfaces are also possible, including arbitrarily curved or warped surfaces. 
     The TIGER prisms&#39; oblique primary faces  732  and other faces (e.g., faces  734  and  736 ) can be used to define the two different coordinate systems shown in  FIGS. 7A-7C  show. As in the in-plane recording system shown in  FIGS. 3A and 3B , axes x G , y G , and z G  are Cartesian coordinates in the frame of reference of the TIGER prisms  730 . And axes x, y, and z are Cartesian coordinates (aka standard coordinates) in the frame of reference of the holographic recording medium  710 , with the z axis extending normal to the surface of the holographic recording medium  710 . The axes x, y, and z are the real-space equivalents of the k-space axes k x , k y , and k z  shown in  FIGS. 5A-5D . 
       FIG. 7C  illustrates how the standard coordinate axes are aligned in the prisms of a skew mirror recorder for the case of Tiger prisms. More specifically, it shows a view of the TIGER prisms  730  along the z G  axis with the holographic recording medium  710  sandwiched between them at an angle equal to ϕ. Note that in the illustrated geometry, ϕ has a negative sign (e.g., ϕ=−30.25°). Because the holographic recording medium  710  is tilted with respect to the y G  axis, the conversion from standard coordinates to global coordinates is different for the TIGER prism recording system  700  than for the in-plane recording system  300  shown in  FIGS. 3A and 3B . Given the benefit of this disclosure, one skilled in the art will ascertain that standard coordinates may be converted to global coordinates for the case of TIGER prisms by equation (2): 
     
       
         
           
             
               
                 
                   
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     In contrast to the “in-plane” configuration of  FIGS. 3A and 3B , which imposes a worst-case angle on the recording beam  331   b  grazing angle, the TIGER prisms  730  allow the recording medium  710  to be rotated about the x G  axis, “splitting the difference” between the recording beams  731   a  and  731   b  grazing angles. Both the TIGER prism configuration  700  and the in-plane configuration  300  can record grating vectors aligned with the x G  axis, and hence result in equivalent written skew mirrors. However, the TIGER prism configuration  700  can also access smaller recording angles and thus can record gratings of lower spatial frequency than the in-plane configuration. 
       FIGS. 8A-8C  show a TIGER prism-based skew mirror recorder  800  that uses TIGER prisms  730  to implement the recording geometry  700  shown in  FIGS. 7A-7C  for making wide-field-of-view holographic skew mirrors. It includes mirrors  850   a  and  850   b  (collectively, mirrors  850 ) that directed the recording beams  731   a  and  731   b , respectively, to the holographic recording medium  710 , which is mounted between the TIGER prisms  730  in a mount  860 . The TIGER prism-based skew mirror recorder  800  also includes stages to adjust the angular and translational alignment of the recording beams  731  with respect to the holographic recording medium  730 . These stages may include three goniometers  870   a - 870   c  (collectively, goniometers  870 ), a vertical translation stage  880 , a rotation stage  872   a  and  872   b  (collectively, rotation stages  872 ) for each mirror  850 , and a horizontal translation stage (not shown) for moving the mounted holographic recording medium  710  and TIGER prisms  730  back and forth. 
     In the case of in-plane recording prisms, refraction correction and other adjustments are typically performed by rotating the mirrors  350  and translating the holographic recording medium. With TIGER prisms, however, there may be desired out-of-plane angle adjustments that cannot be made by rotating the mirrors  850  or translating the holographic recording medium. This is why the TIGER prism skew recorder  800  may be equipped with other actuators, such as the goniometers  870  and vertical stage  880 , to perform out-of-plane angle adjustments. 
     The first goniometer  870   a  is situated on top of the first rotating stage  872   a , beneath rotating mirror  850   a , to allow rotation of the mirror  850   a  about an axis substantially aligned to the horizontal mid-line of the mirror surface. Actuation of the first goniometer  870   a  allows recording beam  731   a  to be reflected up or down by up to several degrees out of the x G -z G  plane. Similarly, the second goniometer  870   b  is situated so as to allow recording beam  731   b  to also be reflected up or down independently by mirror  850   b . The third goniometer  870   c  similarly allows a mirror (not labeled) upstream of rotating mirror  850   a  to tip up and down so that, in combination, the first and third goniometers  870   a ,  870   c  can produce any desired beam  731   a  height and vertical angle combination (within mechanical limits). The vertical stage  880  can raise or lower the height (y G  coordinate) of the entire prism package  860 , including the recording medium  710 . 
     An additional method for performing the correction would be to add an additional goniometer (not shown) for adjusting the path of recording beam  731   b  to produce any desired height and vertical angle combination in much the same manner as the first and third goniometers  870   a ,  870   c  produce the desired height and vertical angle combination. 
       FIG. 9  illustrates another method for achieving this refraction correction uses a pair of optical wedges in rotation mounts which can be aligned with respect to each other to achieve angles within a cone of twice the magnitude that one wedge achieves. 
     Given the benefit of this disclosure, one skilled in the art will recognize that this configuration will allow small arbitrary vertical angle components to be introduced into each recording beam, while also maintaining overlap between the beams and the recording medium. For instance, one may set a desired vertical angle component of reference beam  731   b  using goniometer  870   b , and then set the vertical stage height so that the reference beam  731   b  passes through the desired recording region. Then, one may set goniometers  870   a  and  870   c  so that the signal beam  731   a  is introduced at a desired vertical angle at a height matching that set by the vertical stage. Typically, only a few degrees of vertical angle range and a couple centimeters of vertical motion will be sufficient to implement desired refraction correction and other adjustments. 
     Selecting Interbeam and Skew Angles for Out-of-Plane Holographic Skew Mirrors 
     The out-of-plane skew mirror recorder  800  shown in  FIGS. 8A-8C  can be used to make wide field-of-view holographic skew mirror by recording one or more volume holograms within the volume of the holographic recording medium. The interbeam and skew angles chosen to record these holograms depend on the desired field of view and operating wavelength range of the holographic skew mirror. 
     In some cases, the out-of-plane skew mirror recorder  800  may be used to record many discrete gratings, each of which principally reflects light at one or more wavelengths over a different, narrow range of incident angles. If these ranges of incident angles overlap or are close to each other, the gratings will reflect light over a wide range of incident angles to provide a wide field of view. Alternatively, the holographic skew mirror may include a holographic grating that is written by continuously recording the interference between a pair of recording beams as the interbeam angle is varied. This continuously recorded grating reflects light over wide range of incident angles to provide a wide field of view. Other combinations of grating are also possible, e.g., to produce a holographic skew mirror that reflects light over certain ranges of angles of incidence but not others, or certain ranges of wavelengths but no others. 
     In at least one example, the vector form of Snell&#39;s law may be used to calculate the direction of a recording beam upon refraction at an internal boundary between a recording medium and a prism. As understood by those skilled in the art of holography, the vector form of Snell&#39;s law describes what happens to light rays that impinge on an optical boundary, such as a prism surface, at an angle that includes non-zero components in more than one coordinate axis. The vector form of Snell&#39;s law gives the resulting refraction at such a surface as: 
     
       
         
           
             
               
                 
                   
                     
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     where {circumflex over (N)} is the unit normal vector of the optical boundary, ŝ 1  and ŝ 2  are the normalized incident and refracted ray direction vectors, and n 1  and n 2  are the refractive indices of the first and second materials. For TIGER prisms, such a refraction typically includes non-zero components in more than one axis of the global coordinate system. 
     In some embodiments, refractive index mismatches between optical elements such as a recording medium and a recording prism are corrected using Snell&#39;s law. For instance, an internal ray direction vector ŝ S,int  might be desired for a signal beam within recording medium  710  ( FIGS. 7A-7D and 8A-8C ) during a recording exposure. In order to determine the external angle that must be applied to produce ŝ S,int  at internal angle θ G  within medium  710 . To this purpose, Snell&#39;s law may be applied at the internal boundary between recording prism  730   a  and recording medium  710  to determine ŝ S,prism , the ray direction vector within prism  730   a  (note that even a small index mismatch may produce significant refraction). Then, Snell&#39;s law may be may be again applied at the external surface of recording prism  730   b  to determine the external ray direction vector ŝ S,ext  from ŝ S,prism . External ray direction vector ŝ S,ext  thus directly determines θ G , which may be set by rotating mirror  850   b . Similarly, the angle for rotating mirror  850   a  may be determined from the desired reference ray direction vector ŝ R,int  by tracing through the internal and external surfaces of recording prism  730   a.    
     In some embodiments, adjustments to recording angles may be performed for reasons other than refraction correction. Examples of other adjustments include dispersion compensation, media shrinkage pre-compensation, and empirical adjustments to improve the modulation transfer function (MTF) or color plane separation. For instance, these adjustments may be made to compensate for instrument error, shrinkage, index mismatch, etc. Such error can be determined empirically by writing a complete test skew mirror with (imperfect) holograms and ascertaining the holograms&#39; imperfections by measuring the test skew mirror&#39;s angular dispersion as a function of wavelength. These measurements can be used to adjust the design angles. Once the design angles have been adjusted, it is possible to record holographic skew mirrors that are virtually free of imperfections. 
     A Wide Field-of-View Holographic Skew Mirror 
     In practice, an out-of-plane skew mirror recorder can create a holographic skew mirror with a wide field of view by recording one or more (e.g., tens, hundreds, or thousands) of holographic gratings within a holographic recording medium. Writing a discrete set of gratings over a series of exposures as opposed to a single grating in a single continuous exposure with angularly scanning beams offers several advantages. First, it reduces the need to suppress or compensate vibration during exposure. Second, compared to a continuous grating, discrete gratings preserve the change in refractive index, Δn, at the expense of spectrally subsampling incoming light (the discrete gratings apply a reflective comb function to light illuminating skew mirror). Third, picking the grating spacing to match the light source&#39;s spectrum makes the device more efficient. 
       FIG. 10  shows a wide field-of-view holographic input/output coupler  1000  made using an out-of-plane writing geometry. The holographic skew input/output coupler  1000  comprises a holographic grating structure  1020  recorded in a holographic grating medium  1010 , which may be about 100 microns thick or thicker. The grating structure  1020  comprises 228 gratings, each of which is recorded with a different interbeam recording beam angle and hence has a different grating frequency (|K G |) at the recording wavelength. These gratings are oriented about a skew axis  1021  that forms a skew angle of about ϕ=−30.25° with respect to a surface normal  1011  of the holographic grating medium  1010 . In practice, other skew angles are possible, e.g., skew angles greater than 2.0°, 5.0°, 10.0°, 15.0°, 30.0°, 45.0°, 60.0°, and so on. The skew angles may range from about 15.0° to about 45.0° (e.g., about 20.0° to about 40.0°, about 25.0° to about 35.0°, about 27.5° to about 32.5°, and so on). 
     Together, the gratings cause the holographic skew mirror  1000  to reflect internally incident light about a substantially constant reflective axis  1021  at incident angles ranging from 13.1° to 47.6°, for a range of incidence angles θ RAI =34.5°, as measured from the reflective axis  1021 . This corresponds to a field of view as measured in air outside the holographic grating medium of about θ FOV =54.3°. Rays propagating within shaded region  1001 , which spans a range of angles measuring from 13.1° to 47.6° with respect to the skew axis  1021  (an angular range of about 34.5°) illuminate the grating structure  1020 . The grating structure  1020  principally reflects this light into shaded region  1003 , which spans the same angular range (−13.1° to −47.6°) on the other side of the skew axis  1021 . The principally reflected light with the third shaded region  1003  refracts at the surface  1020  into a fourth shaded region  1003 ′ spanning a horizontal field of view of about 54.3°. 
     Referring to  FIG. 10 , this means that incident light following ray  1091  at an angle of 47.6° measured from the skew axis  1021  reflects off the grating structure  1020  along ray  1093  that is symmetric with ray  1091  about the skew axis  1021 . This principally reflected light  1093  refracts at the surface  1012  along ray  1093 ′. Similarly, incident light following ray  1081 ′ totally internally reflects off the surface  1012  of the holographic skew mirror  1000  along ray  1081  at an angle of 13.1° measured from the skew axis  1021 . The grating structure  1020  reflects this incident light into along ray  1083  that is symmetric with ray  1081  about the skew axis  1021 . And this principally reflected light  1083  refracts at the surface  1012  along ray  1083 ′. 
       FIG. 11  shows a k-space representation of the grating vectors for a first grating K G1  and a 228 th  grating K G228  along with the wave vectors for the recording beams used to record the gratings. The grating and wave vectors are shown with respect to k-sphere  1191  for the holographic grating medium  1020 , which has a refractive index of about 1.5471 at a recording wavelength of 405 nm. When projected into a plane, the tips of the grating and wave vectors lie on an ellipse. First and second recording beam wave vectors for the first grating, R 1   1  and R 2   1 , respectively, form angles in the holographic grating medium of 32.0° and 148.0°, respectively, with respect to the reflective axis  1021  to produce a first grating with a grating frequency of about 4.1×10 7  rad/m. The wave vectors for the 228 th  grating, R 1   228  and R 2   228 , form angles in the holographic grating medium of 64.1° and 115.9°, respectively, with respect to the skew axis  1021  to produce a 228 th  grating with a grating frequency of about 2.1×10 7  rad/m. Each grating vector is angled at −30.25° relative to the surface normal  1011 . 
     This grating vectors in the grating structure  1020  of  FIG. 10  span a range of grating frequencies that extends about 2.0×10 7  rad/m. Other grating frequencies and ranges of grating frequencies are also possible; in practice, range of grating frequencies, or difference between the maximum and minimum grating frequency, may be about 2.00×10 5  radians per meter to about 3.15×10 7  radians per meter (e.g., about 2.00×10 5  radians per meter, 1.68×10 6  radians per meter, 5.01×10 6  radians per meter, 1.24×10 7  radians per meter, 3.15×10 7  radians per meter, or any other value or sub-range). 
     The gratings may be spaced uniformly or non-uniformly in k-space. For about 228 uniformly spaced gratings with grating frequencies of between about 2.1×10 7  rad/m and about 4.1×10 7  rad/m, the difference in grating frequency between adjacent grating vectors is about 8.68×10 4  rad/m. Other spacings are also possible, including spacings within a range of about 5.0×10 3  rad/m and 1.0×10 7  rad/m. Non-uniform spacing is also possible, e.g., if the holographic skew mirror should reflect light at some wavelengths or angles but not others. For instance, the grating frequencies may be selected based on the spectrum of the incident light and/or the range of expected incidence angles for increased efficiency. 
     Because each grating has a different grating frequency, each grating reflects light from a different incident angle to a different principally reflected angle. The range of possible angles of incidence depends on the range of grating frequencies and determines the field of view. For example, each grating may reflect light at one wavelength (e.g., 460 nm, 518 nm, or 618 nm), two wavelengths (e.g., 460 nm and 518 nm or 518 nm and 618 nm), three wavelengths (e.g., 460 nm, 518 nm, and 618 nm), or more. Gratings may reflect light at visible wavelengths, near infrared (NIR) wavelengths, near ultraviolet wavelengths, or combinations thereof. This enables the skew mirror to reflect narrowband light (e.g., light from a laser), broadband light (e.g., light from an organic light-emitting diode (OLED)), and even natural light (e.g., sunlight). 
     For an input/output coupler, the skew axis may be selected to couple light into or out of the grating medium at angles near the surface normal, e.g., as shown in  FIG. 10 . In these cases, the skew angle may be based on the critical angle for total internal reflection, which is about 40.81° at a boundary between air and the grating medium (e.g., n=1.53 at visible wavelengths). 
     Table 1 lists recording beam angles and grating frequencies for each of the 228 uniformly spaced gratings. First recording beam angle θ R1  and second recording beam angle θ R2  are relative to a skew axis having a skew angle of −30.25 degrees relative to surface normal of the recording medium. Accordingly, grating vectors listed in Table 1 are oriented at −30.25 degrees relative to surface normal of the recording medium, which is referred to as a grating medium after all 228 gratings are recorded. θ R1  and θ R2  are analogous to θ GR1  and θ GR2  respectively, as illustrated in  FIG. 7B , except that θ R1  and θ R2  are measured within medium  710 , no prism  730 . 
     Collectively, the 228 gratings in Table 1 are structured to reflect incident light at 460 nm, 518 nm, and 618 nm about a substantially constant reflective axis, at angles of incidence ranging from 13.1-47.6 degrees (a range of 34.5 degrees) relative to the reflective axis. The reflective axis has a reflective axis angle of −30.25 degrees relative to surface normal. The gratings can be grouped into three overlapping subsets, each of which is structured to reflect incident light of a specified wavelength about the reflective axis, at a range of incidence angles. 
     Subset 1, comprising gratings 1 through 146, is structured to reflect incident light (which can be referred to as a probe beam) at or near 460 nm (e.g., over 20-40 nm band centered at 460 nm) about the substantially constant reflective axis, at angles of incidence ranging from 13.1 to 47.7 degrees relative to the reflective axis. Gratings 1 through 228 (i.e., all of the gratings in Table 1), collectively can reflect incident light at 460 nm about the substantially constant reflective axis, at angles of incidence ranging from 13.1 to 59.8 degrees (a range of 46.7 degrees) relative to the reflective axis. The range from 13.1 to 47.6 degrees is of interest for the skew mirror structured to reflect blue, green, and red light at common incidence angles relative to the substantially constant reflective axis. 
     Subset 2, comprising gratings 53 through 182, is structured to reflect incident light at 518 nm about the substantially constant reflective axis, at angles of incidence ranging from 12.8 to 47.7 degrees relative to the reflective axis. Collectively, gratings 43 through 228 can reflect incident light at 518 nm about the substantially constant reflective axis, at angles of incidence ranging from 3.1 to 55.6 degrees (a range of 52.5 degrees) relative to the reflective axis. The range from 13.1 to 47.6 is of interest for purposes of the present discussion. 
     Subset 3, comprising gratings 120 through 228, is structured to reflect incident light at 618 nm about the substantially constant reflective axis, at angles of incidence ranging from 12.5 to 47.6 degrees relative to the reflective axis. Gratings 112 through 228 are structured to reflect incident light at 618 nm about the substantially constant reflective axis, at angles of incidence ranging from 3.0 to 47.6 degrees (a range of 44.6 degrees) relative to the reflective axis. The range from 13.1 to 47.6 is of interest for purposes of the present discussion. 
     At least gratings 198 through 228 cannot be recorded using in-plane recording such as illustrated in  FIGS. 3A and 3B  because the in-plane recording geometry results in impermissible recording beam angles of 90 degrees and greater relative to surface normal of the recording medium. Practically speaking, even gratings 115 through 198, although theoretically possible, would likely problematic using in-plane architecture because recording beam angles approach a grazing condition (i.e., approach 90 degrees). Out-of-plane recording using TIGER prisms, such as illustrated in  FIGS. 7A-7C , enables writing all gratings in Table 1. 
     Selecting Recording Beam Angles and Exposure Times 
     Computer code can be used to compute the writing beam angles and exposure times for making a holographic skew mirror with an out-of-plane writing geometry. The following snippet of computer code calculates the horizontal and vertical fields of view from the diagonal field of view. As mentioned above, this holographic skew mirror  1000  has a 60° diagonal field of view (measured outside the holographic recording medium  1020 ). When the variable g.aspect=9/16, it also has an aspect ratio of 16:9, which is common for many displays: 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 g.dFoV = 60; 
                 % diagonal angle 
               
               
                   
                 g.dia = 2 * tand( g.dFoV/2); 
                 % diagonal size @ dist=1.0 
               
            
           
           
               
               
            
               
                   
                 g.width = g.dia * cos( atan( g.aspect)); 
               
               
                   
                 g.height = g.dia * sin( atan( g.aspect)); 
               
               
                   
                 g.vFoV = 2 * atand( g.height/2); 
               
               
                   
                 g.hFoV = 2 * atand( g.width/2); 
               
               
                   
                   
               
            
           
         
       
     
     A 60° diagonal field of view and a 16:9 aspect ratio correspond to a 53.4° horizontal field of view and a 31.6° vertical (Bragg degenerate) field of view (again, as measured outside the holographic recording medium). (The choice of orientation is arbitrary and could be reversed, i.e., the horizontal field of view could be 31.6° and the vertical field of view could be 53.4°.) For a holographic recording medium with a refractive index of about 1.5 (e.g., n=1.53), the range of horizontal incident angles on the holographic gratings measured inside the medium is about 35° (e.g., 34.17°). 
       FIG. 12  illustrates a set of curves, produced by a different computer code, illustrating the holograms (holographic gratings) for each color band. These curves represent the skew mirror internal angle wavelength bands  1201   a - 1201   e  for a holographic output coupler with a 53.4° horizontal field-of-view. The left bands  1201   a  represent holograms that reflect red light. The middle bands  1201   c  represent holograms that are used for all three color bands. The middle-left bands  1201   b  represent holograms that are shared for the green and red band. The middle-right bands  1201   d  represent holograms shared for the blue and green bands. And the right bands  1201   e  represent holograms that reflect blue light. 
     The code also produced a table of recording parameters, shown below in Table 1. The parameters were selected to support the 53.4° horizontal field-of-view for an output coupler at the red-green-blue (RGB) color bands centered at 620 nm, 520 nm, and 460 nm, respectively. 
     The 228 rows of Table 1 correspond to 228 exposures for programming the skew mirror using the out-of-plane writing geometry and system shown in  FIGS. 7A-7D and 8A-8C . The first column, Global Angle, indicates the angle θ G  of signal beam  731   a  ( FIG. 7A ) with respect to the x G  axis inside the medium, which is set by rotating mirror  850   a  ( FIG. 8A ). Rotating mirror  850   b  is set to deliver reference beam  731   b  at an angle of 180°−θ G  inside the medium. Column  3 , Adjustment Angle, is used to set the goniometers  870  to produce the indicated out-of-plane angular components for both beams  731  inside the medium. The adjustments are of equal magnitude but opposite sign for the two beams  731 , so that the signal beam is delivered with an upward angle and the reference beam is delivered with a downward angle of the same magnitude or vice versa. A linear stage and the vertical stage  880  are set to center the recording medium  710  at the intersection of the recording beams  731 . Then a shutter is opened to expose the recording medium  710  for the time indicated in column  2 . All exposures in Table 1 are so recorded, and then the recording medium  710  is removed from the skew mirror recorder and post-cured with an incoherent UV LED source immediately after exposure. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 RECORDING PARAMETERS FOR A SKEW MIRROR EMBODIMENT 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 θ R1  First 
                   
                   
                   
                   
               
               
                   
                 Recording Beam 
                 θ R2  Second 
                 Interbeam 
                 |K G | Grating 
               
               
                 Grating 
                 Angle 
                 Recording Beam 
                 Angle (α) 
                 Frequency 
                 Exposure Time 
               
               
                 Number 
                 (degrees)* 
                 Angle (degrees)* 
                 (degrees) 
                 (rad/m)** 
                 (seconds) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 1 
                 32.00 
                 148.00 
                 116.00 
                 4.0709E+07 
                 9.9473E−01 
               
               
                 2 
                 32.20 
                 147.80 
                 115.61 
                 4.0622E+07 
                 7.4488E−01 
               
               
                 3 
                 32.39 
                 147.61 
                 115.22 
                 4.0535E+07 
                 7.4502E−01 
               
               
                 4 
                 32.58 
                 147.42 
                 114.83 
                 4.0448E+07 
                 7.4517E−01 
               
               
                 5 
                 32.78 
                 147.22 
                 114.45 
                 4.0361E+07 
                 7.4532E−01 
               
               
                 6 
                 32.97 
                 147.03 
                 114.07 
                 4.0275E+07 
                 7.4547E−01 
               
               
                 7 
                 33.16 
                 146.84 
                 113.69 
                 4.0188E+07 
                 7.4562E−01 
               
               
                 8 
                 33.35 
                 146.65 
                 113.31 
                 4.0101E+07 
                 7.4577E−01 
               
               
                 9 
                 33.53 
                 146.47 
                 112.93 
                 4.0014E+07 
                 7.4592E−01 
               
               
                 10 
                 33.72 
                 146.28 
                 112.56 
                 3.9927E+07 
                 7.4607E−01 
               
               
                 11 
                 33.91 
                 146.09 
                 112.19 
                 3.9840E+07 
                 7.4622E−01 
               
               
                 12 
                 34.09 
                 145.91 
                 111.82 
                 3.9754E+07 
                 7.4636E−01 
               
               
                 13 
                 34.28 
                 145.72 
                 111.45 
                 3.9667E+07 
                 7.4651E−01 
               
               
                 14 
                 34.46 
                 145.54 
                 111.08 
                 3.9580E+07 
                 7.4666E−01 
               
               
                 15 
                 34.64 
                 145.36 
                 110.71 
                 3.9493E+07 
                 7.4681E−01 
               
               
                 16 
                 34.82 
                 145.18 
                 110.35 
                 3.9406E+07 
                 7.4696E−01 
               
               
                 17 
                 35.01 
                 144.99 
                 109.99 
                 3.9319E+07 
                 7.4711E−01 
               
               
                 18 
                 35.19 
                 144.81 
                 109.63 
                 3.9232E+07 
                 7.4726E−01 
               
               
                 19 
                 35.37 
                 144.63 
                 109.27 
                 3.9146E+07 
                 7.4741E−01 
               
               
                 20 
                 35.54 
                 144.46 
                 108.91 
                 3.9059E+07 
                 7.4756E−01 
               
               
                 21 
                 35.72 
                 144.28 
                 108.56 
                 3.8972E+07 
                 7.4770E−01 
               
               
                 22 
                 35.90 
                 144.10 
                 108.20 
                 3.8885E+07 
                 7.4785E−01 
               
               
                 23 
                 36.08 
                 143.92 
                 107.85 
                 3.8798E+07 
                 7.4800E−01 
               
               
                 24 
                 36.25 
                 143.75 
                 107.50 
                 3.8711E+07 
                 7.4815E−01 
               
               
                 25 
                 36.43 
                 143.57 
                 107.15 
                 3.8625E+07 
                 7.4830E−01 
               
               
                 26 
                 36.60 
                 143.40 
                 106.80 
                 3.8538E+07 
                 7.4845E−01 
               
               
                 27 
                 36.77 
                 143.23 
                 106.45 
                 3.8451E+07 
                 7.4860E−01 
               
               
                 28 
                 36.95 
                 143.05 
                 106.11 
                 3.8364E+07 
                 7.4875E−01 
               
               
                 29 
                 37.12 
                 142.88 
                 105.76 
                 3.8277E+07 
                 7.4890E−01 
               
               
                 30 
                 37.29 
                 142.71 
                 105.42 
                 3.8190E+07 
                 7.4904E−01 
               
               
                 31 
                 37.46 
                 142.54 
                 105.08 
                 3.8104E+07 
                 7.4919E−01 
               
               
                 32 
                 37.63 
                 142.37 
                 104.74 
                 3.8017E+07 
                 7.4934E−01 
               
               
                 33 
                 37.80 
                 142.20 
                 104.40 
                 3.7930E+07 
                 7.4949E−01 
               
               
                 34 
                 37.97 
                 142.03 
                 104.06 
                 3.7843E+07 
                 7.4964E−01 
               
               
                 35 
                 38.14 
                 141.86 
                 103.72 
                 3.7756E+07 
                 7.4979E−01 
               
               
                 36 
                 38.31 
                 141.69 
                 103.39 
                 3.7669E+07 
                 7.4994E−01 
               
               
                 37 
                 38.47 
                 141.53 
                 103.06 
                 3.7582E+07 
                 7.5009E−01 
               
               
                 38 
                 38.64 
                 141.36 
                 102.72 
                 3.7496E+07 
                 7.5024E−01 
               
               
                 39 
                 38.80 
                 141.20 
                 102.39 
                 3.7409E+07 
                 7.5039E−01 
               
               
                 40 
                 38.97 
                 141.03 
                 102.06 
                 3.7322E+07 
                 7.5053E−01 
               
               
                 41 
                 39.13 
                 140.87 
                 101.73 
                 3.7235E+07 
                 7.5068E−01 
               
               
                 42 
                 39.30 
                 140.70 
                 101.40 
                 3.7148E+07 
                 7.5083E−01 
               
               
                 43 
                 39.46 
                 140.54 
                 101.08 
                 3.7061E+07 
                 7.5098E−01 
               
               
                 44 
                 39.62 
                 140.38 
                 100.75 
                 3.6975E+07 
                 7.5113E−01 
               
               
                 45 
                 39.79 
                 140.21 
                 100.43 
                 3.6888E+07 
                 7.5128E−01 
               
               
                 46 
                 39.95 
                 140.05 
                 100.10 
                 3.6801E+07 
                 7.5143E−01 
               
               
                 47 
                 40.11 
                 139.89 
                 99.78 
                 3.6714E+07 
                 7.5158E−01 
               
               
                 48 
                 40.27 
                 139.73 
                 99.46 
                 3.6627E+07 
                 7.5173E−01 
               
               
                 49 
                 40.43 
                 139.57 
                 99.14 
                 3.6540E+07 
                 7.5187E−01 
               
               
                 50 
                 40.59 
                 139.41 
                 98.82 
                 3.6454E+07 
                 7.5202E−01 
               
               
                 51 
                 40.75 
                 139.25 
                 98.50 
                 3.6367E+07 
                 7.5217E−01 
               
               
                 52 
                 40.91 
                 139.09 
                 98.19 
                 3.6280E+07 
                 7.5232E−01 
               
               
                 53 
                 41.06 
                 138.94 
                 97.87 
                 3.6193E+07 
                 7.5247E−01 
               
               
                 54 
                 41.22 
                 138.78 
                 97.55 
                 3.6106E+07 
                 7.5262E−01 
               
               
                 55 
                 41.38 
                 138.62 
                 97.24 
                 3.6019E+07 
                 7.6572E−01 
               
               
                 56 
                 41.54 
                 138.46 
                 96.93 
                 3.5932E+07 
                 7.6587E−01 
               
               
                 57 
                 41.69 
                 138.31 
                 96.62 
                 3.5846E+07 
                 7.6602E−01 
               
               
                 58 
                 41.85 
                 138.15 
                 96.30 
                 3.5759E+07 
                 7.6617E−01 
               
               
                 59 
                 42.00 
                 138.00 
                 95.99 
                 3.5672E+07 
                 7.6632E−01 
               
               
                 60 
                 42.16 
                 137.84 
                 95.68 
                 3.5585E+07 
                 7.6647E−01 
               
               
                 61 
                 42.31 
                 137.69 
                 95.38 
                 3.5498E+07 
                 7.6662E−01 
               
               
                 62 
                 42.47 
                 137.53 
                 95.07 
                 3.5411E+07 
                 7.6677E−01 
               
               
                 63 
                 42.62 
                 137.38 
                 94.76 
                 3.5325E+07 
                 7.6691E−01 
               
               
                 64 
                 42.77 
                 137.23 
                 94.46 
                 3.5238E+07 
                 7.6706E−01 
               
               
                 65 
                 42.92 
                 137.08 
                 94.15 
                 3.5151E+07 
                 7.6721E−01 
               
               
                 66 
                 43.08 
                 136.92 
                 93.85 
                 3.5064E+07 
                 7.6736E−01 
               
               
                 67 
                 43.23 
                 136.77 
                 93.54 
                 3.4977E+07 
                 7.6751E−01 
               
               
                 68 
                 43.38 
                 136.62 
                 93.24 
                 3.4890E+07 
                 7.6766E−01 
               
               
                 69 
                 43.53 
                 136.47 
                 92.94 
                 3.4804E+07 
                 7.6781E−01 
               
               
                 70 
                 43.68 
                 136.32 
                 92.64 
                 3.4717E+07 
                 7.6796E−01 
               
               
                 71 
                 43.83 
                 136.17 
                 92.34 
                 3.4630E+07 
                 7.6811E−01 
               
               
                 72 
                 43.98 
                 136.02 
                 92.04 
                 3.4543E+07 
                 7.6826E−01 
               
               
                 73 
                 44.13 
                 135.87 
                 91.74 
                 3.4456E+07 
                 7.6840E−01 
               
               
                 74 
                 44.28 
                 135.72 
                 91.45 
                 3.4369E+07 
                 7.6855E−01 
               
               
                 75 
                 44.43 
                 135.57 
                 91.15 
                 3.4282E+07 
                 7.6870E−01 
               
               
                 76 
                 44.57 
                 135.43 
                 90.85 
                 3.4196E+07 
                 7.6885E−01 
               
               
                 77 
                 44.72 
                 135.28 
                 90.56 
                 3.4109E+07 
                 7.6900E−01 
               
               
                 78 
                 44.87 
                 135.13 
                 90.26 
                 3.4022E+07 
                 7.6915E−01 
               
               
                 79 
                 45.01 
                 134.99 
                 89.97 
                 3.3935E+07 
                 7.6930E−01 
               
               
                 80 
                 45.16 
                 134.84 
                 89.68 
                 3.3848E+07 
                 7.6945E−01 
               
               
                 81 
                 45.31 
                 134.69 
                 89.39 
                 3.3761E+07 
                 7.6960E−01 
               
               
                 82 
                 45.45 
                 134.55 
                 89.10 
                 3.3675E+07 
                 7.6974E−01 
               
               
                 83 
                 45.60 
                 134.40 
                 88.80 
                 3.3588E+07 
                 7.6989E−01 
               
               
                 84 
                 45.74 
                 134.26 
                 88.51 
                 3.3501E+07 
                 7.7004E−01 
               
               
                 85 
                 45.89 
                 134.11 
                 88.23 
                 3.3414E+07 
                 7.7019E−01 
               
               
                 86 
                 46.03 
                 133.97 
                 87.94 
                 3.3327E+07 
                 7.7034E−01 
               
               
                 87 
                 46.18 
                 133.82 
                 87.65 
                 3.3240E+07 
                 7.7049E−01 
               
               
                 88 
                 46.32 
                 133.68 
                 87.36 
                 3.3154E+07 
                 7.7064E−01 
               
               
                 89 
                 46.46 
                 133.54 
                 87.08 
                 3.3067E+07 
                 7.7079E−01 
               
               
                 90 
                 46.60 
                 133.40 
                 86.79 
                 3.2980E+07 
                 7.7094E−01 
               
               
                 91 
                 46.75 
                 133.25 
                 86.51 
                 3.2893E+07 
                 7.7108E−01 
               
               
                 92 
                 46.89 
                 133.11 
                 86.22 
                 3.2806E+07 
                 7.7123E−01 
               
               
                 93 
                 47.03 
                 132.97 
                 85.94 
                 3.2719E+07 
                 7.7138E−01 
               
               
                 94 
                 47.17 
                 132.83 
                 85.65 
                 3.2632E+07 
                 7.7153E−01 
               
               
                 95 
                 47.31 
                 132.69 
                 85.37 
                 3.2546E+07 
                 7.7168E−01 
               
               
                 96 
                 47.45 
                 132.55 
                 85.09 
                 3.2459E+07 
                 7.7183E−01 
               
               
                 97 
                 47.60 
                 132.40 
                 84.81 
                 3.2372E+07 
                 7.7198E−01 
               
               
                 98 
                 47.74 
                 132.26 
                 84.53 
                 3.2285E+07 
                 7.7213E−01 
               
               
                 99 
                 47.88 
                 132.12 
                 84.25 
                 3.2198E+07 
                 7.7228E−01 
               
               
                 100 
                 48.01 
                 131.99 
                 83.97 
                 3.2111E+07 
                 7.7242E−01 
               
               
                 101 
                 48.15 
                 131.85 
                 83.69 
                 3.2025E+07 
                 7.7257E−01 
               
               
                 102 
                 48.29 
                 131.71 
                 83.41 
                 3.1938E+07 
                 7.7272E−01 
               
               
                 103 
                 48.43 
                 131.57 
                 83.14 
                 3.1851E+07 
                 7.7287E−01 
               
               
                 104 
                 48.57 
                 131.43 
                 82.86 
                 3.1764E+07 
                 7.7302E−01 
               
               
                 105 
                 48.71 
                 131.29 
                 82.58 
                 3.1677E+07 
                 7.7317E−01 
               
               
                 106 
                 48.85 
                 131.15 
                 82.31 
                 3.1590E+07 
                 7.7332E−01 
               
               
                 107 
                 48.98 
                 131.02 
                 82.03 
                 3.1504E+07 
                 7.7347E−01 
               
               
                 108 
                 49.12 
                 130.88 
                 81.76 
                 3.1417E+07 
                 7.7362E−01 
               
               
                 109 
                 49.26 
                 130.74 
                 81.48 
                 3.1330E+07 
                 7.7376E−01 
               
               
                 110 
                 49.39 
                 130.61 
                 81.21 
                 3.1243E+07 
                 7.7391E−01 
               
               
                 111 
                 49.53 
                 130.47 
                 80.94 
                 3.1156E+07 
                 7.7406E−01 
               
               
                 112 
                 49.67 
                 130.33 
                 80.67 
                 3.1069E+07 
                 7.7421E−01 
               
               
                 113 
                 49.80 
                 130.20 
                 80.39 
                 3.0982E+07 
                 7.7436E−01 
               
               
                 114 
                 49.94 
                 130.06 
                 80.12 
                 3.0896E+07 
                 7.7451E−01 
               
               
                 115 
                 50.07 
                 129.93 
                 79.85 
                 3.0809E+07 
                 7.7466E−01 
               
               
                 116 
                 50.21 
                 129.79 
                 79.58 
                 3.0722E+07 
                 7.7481E−01 
               
               
                 117 
                 50.34 
                 129.66 
                 79.31 
                 3.0635E+07 
                 7.7496E−01 
               
               
                 118 
                 50.48 
                 129.52 
                 79.04 
                 3.0548E+07 
                 7.7511E−01 
               
               
                 119 
                 50.61 
                 129.39 
                 78.78 
                 3.0461E+07 
                 7.7525E−01 
               
               
                 120 
                 50.75 
                 129.25 
                 78.51 
                 3.0375E+07 
                 7.7540E−01 
               
               
                 121 
                 50.88 
                 129.12 
                 78.24 
                 3.0288E+07 
                 7.7555E−01 
               
               
                 122 
                 51.01 
                 128.99 
                 77.97 
                 3.0201E+07 
                 7.8434E−01 
               
               
                 123 
                 51.15 
                 128.85 
                 77.71 
                 3.0114E+07 
                 7.8449E−01 
               
               
                 124 
                 51.28 
                 128.72 
                 77.44 
                 3.0027E+07 
                 7.8464E−01 
               
               
                 125 
                 51.41 
                 128.59 
                 77.18 
                 2.9940E+07 
                 7.8478E−01 
               
               
                 126 
                 51.54 
                 128.46 
                 76.91 
                 2.9854E+07 
                 7.8493E−01 
               
               
                 127 
                 51.68 
                 128.32 
                 76.65 
                 2.9767E+07 
                 7.8508E−01 
               
               
                 128 
                 51.81 
                 128.19 
                 76.38 
                 2.9680E+07 
                 7.8523E−01 
               
               
                 129 
                 51.94 
                 128.06 
                 76.12 
                 2.9593E+07 
                 7.8538E−01 
               
               
                 130 
                 52.07 
                 127.93 
                 75.86 
                 2.9506E+07 
                 7.8553E−01 
               
               
                 131 
                 52.20 
                 127.80 
                 75.59 
                 2.9419E+07 
                 7.8568E−01 
               
               
                 132 
                 52.33 
                 127.67 
                 75.33 
                 2.9332E+07 
                 7.8583E−01 
               
               
                 133 
                 52.47 
                 127.53 
                 75.07 
                 2.9246E+07 
                 7.8598E−01 
               
               
                 134 
                 52.60 
                 127.40 
                 74.81 
                 2.9159E+07 
                 7.8612E−01 
               
               
                 135 
                 52.73 
                 127.27 
                 74.55 
                 2.9072E+07 
                 7.8627E−01 
               
               
                 136 
                 52.86 
                 127.14 
                 74.29 
                 2.8985E+07 
                 7.8642E−01 
               
               
                 137 
                 52.99 
                 127.01 
                 74.03 
                 2.8898E+07 
                 7.8657E−01 
               
               
                 138 
                 53.12 
                 126.88 
                 73.77 
                 2.8811E+07 
                 7.8672E−01 
               
               
                 139 
                 53.25 
                 126.75 
                 73.51 
                 2.8725E+07 
                 7.8687E−01 
               
               
                 140 
                 53.37 
                 126.63 
                 73.25 
                 2.8638E+07 
                 7.8702E−01 
               
               
                 141 
                 53.50 
                 126.50 
                 72.99 
                 2.8551E+07 
                 7.8717E−01 
               
               
                 142 
                 53.63 
                 126.37 
                 72.73 
                 2.8464E+07 
                 7.8732E−01 
               
               
                 143 
                 53.76 
                 126.24 
                 72.48 
                 2.8377E+07 
                 7.8747E−01 
               
               
                 144 
                 53.89 
                 126.11 
                 72.22 
                 2.8290E+07 
                 7.8761E−01 
               
               
                 145 
                 54.02 
                 125.98 
                 71.96 
                 2.8204E+07 
                 7.8776E−01 
               
               
                 146 
                 54.15 
                 125.85 
                 71.71 
                 2.8117E+07 
                 7.8791E−01 
               
               
                 147 
                 54.27 
                 125.73 
                 71.45 
                 2.8030E+07 
                 7.8806E−01 
               
               
                 148 
                 54.40 
                 125.60 
                 71.20 
                 2.7943E+07 
                 7.8821E−01 
               
               
                 149 
                 54.53 
                 125.47 
                 70.94 
                 2.7856E+07 
                 7.8836E−01 
               
               
                 150 
                 54.66 
                 125.34 
                 70.69 
                 2.7769E+07 
                 7.8851E−01 
               
               
                 151 
                 54.78 
                 125.22 
                 70.43 
                 2.7682E+07 
                 7.8866E−01 
               
               
                 152 
                 54.91 
                 125.09 
                 70.18 
                 2.7596E+07 
                 7.8881E−01 
               
               
                 153 
                 55.04 
                 124.96 
                 69.93 
                 2.7509E+07 
                 7.8895E−01 
               
               
                 154 
                 55.16 
                 124.84 
                 69.67 
                 2.7422E+07 
                 7.8910E−01 
               
               
                 155 
                 55.29 
                 124.71 
                 69.42 
                 2.7335E+07 
                 7.8925E−01 
               
               
                 156 
                 55.41 
                 124.59 
                 69.17 
                 2.7248E+07 
                 7.8940E−01 
               
               
                 157 
                 55.54 
                 124.46 
                 68.92 
                 2.7161E+07 
                 7.8955E−01 
               
               
                 158 
                 55.67 
                 124.33 
                 68.67 
                 2.7075E+07 
                 7.8970E−01 
               
               
                 159 
                 55.79 
                 124.21 
                 68.42 
                 2.6988E+07 
                 7.8985E−01 
               
               
                 160 
                 55.92 
                 124.08 
                 68.17 
                 2.6901E+07 
                 7.9000E−01 
               
               
                 161 
                 56.04 
                 123.96 
                 67.92 
                 2.6814E+07 
                 7.9015E−01 
               
               
                 162 
                 56.17 
                 123.83 
                 67.67 
                 2.6727E+07 
                 7.9029E−01 
               
               
                 163 
                 56.29 
                 123.71 
                 67.42 
                 2.6640E+07 
                 7.9044E−01 
               
               
                 164 
                 56.42 
                 123.58 
                 67.17 
                 2.6554E+07 
                 7.9059E−01 
               
               
                 165 
                 56.54 
                 123.46 
                 66.92 
                 2.6467E+07 
                 7.9074E−01 
               
               
                 166 
                 56.66 
                 123.34 
                 66.67 
                 2.6380E+07 
                 7.9089E−01 
               
               
                 167 
                 56.79 
                 123.21 
                 66.42 
                 2.6293E+07 
                 7.9104E−01 
               
               
                 168 
                 56.91 
                 123.09 
                 66.18 
                 2.6206E+07 
                 7.9119E−01 
               
               
                 169 
                 57.04 
                 122.96 
                 65.93 
                 2.6119E+07 
                 7.9134E−01 
               
               
                 170 
                 57.16 
                 122.84 
                 65.68 
                 2.6032E+07 
                 7.9149E−01 
               
               
                 171 
                 57.28 
                 122.72 
                 65.43 
                 2.5946E+07 
                 7.9163E−01 
               
               
                 172 
                 57.41 
                 122.59 
                 65.19 
                 2.5859E+07 
                 7.9178E−01 
               
               
                 173 
                 57.53 
                 122.47 
                 64.94 
                 2.5772E+07 
                 7.9193E−01 
               
               
                 174 
                 57.65 
                 122.35 
                 64.70 
                 2.5685E+07 
                 7.9208E−01 
               
               
                 175 
                 57.77 
                 122.23 
                 64.45 
                 2.5598E+07 
                 7.9223E−01 
               
               
                 176 
                 57.90 
                 122.10 
                 64.21 
                 2.5511E+07 
                 7.9238E−01 
               
               
                 177 
                 58.02 
                 121.98 
                 63.96 
                 2.5425E+07 
                 7.9253E−01 
               
               
                 178 
                 58.14 
                 121.86 
                 63.72 
                 2.5338E+07 
                 7.9268E−01 
               
               
                 179 
                 58.26 
                 121.74 
                 63.47 
                 2.5251E+07 
                 7.9283E−01 
               
               
                 180 
                 58.38 
                 121.62 
                 63.23 
                 2.5164E+07 
                 7.9297E−01 
               
               
                 181 
                 58.51 
                 121.49 
                 62.99 
                 2.5077E+07 
                 7.9312E−01 
               
               
                 182 
                 58.63 
                 121.37 
                 62.74 
                 2.4990E+07 
                 7.9327E−01 
               
               
                 183 
                 58.75 
                 121.25 
                 62.50 
                 2.4904E+07 
                 7.9342E−01 
               
               
                 184 
                 58.87 
                 121.13 
                 62.26 
                 2.4817E+07 
                 7.9357E−01 
               
               
                 185 
                 58.99 
                 121.01 
                 62.02 
                 2.4730E+07 
                 7.9372E−01 
               
               
                 186 
                 59.11 
                 120.89 
                 61.78 
                 2.4643E+07 
                 7.9387E−01 
               
               
                 187 
                 59.23 
                 120.77 
                 61.53 
                 2.4556E+07 
                 7.9402E−01 
               
               
                 188 
                 59.35 
                 120.65 
                 61.29 
                 2.4469E+07 
                 7.9417E−01 
               
               
                 189 
                 59.47 
                 120.53 
                 61.05 
                 2.4382E+07 
                 7.9432E−01 
               
               
                 190 
                 59.59 
                 120.41 
                 60.81 
                 2.4296E+07 
                 7.9446E−01 
               
               
                 191 
                 59.71 
                 120.29 
                 60.57 
                 2.4209E+07 
                 7.9461E−01 
               
               
                 192 
                 59.83 
                 120.17 
                 60.33 
                 2.4122E+07 
                 7.9476E−01 
               
               
                 193 
                 59.95 
                 120.05 
                 60.09 
                 2.4035E+07 
                 7.9491E−01 
               
               
                 194 
                 60.07 
                 119.93 
                 59.85 
                 2.3948E+07 
                 7.9506E−01 
               
               
                 195 
                 60.19 
                 119.81 
                 59.61 
                 2.3861E+07 
                 7.9521E−01 
               
               
                 196 
                 60.31 
                 119.69 
                 59.37 
                 2.3775E+07 
                 7.9536E−01 
               
               
                 197 
                 60.43 
                 119.57 
                 59.14 
                 2.3688E+07 
                 7.9551E−01 
               
               
                 198 
                 60.55 
                 119.45 
                 58.90 
                 2.3601E+07 
                 7.9566E−01 
               
               
                 199 
                 60.67 
                 119.33 
                 58.66 
                 2.3514E+07 
                 7.9580E−01 
               
               
                 200 
                 60.79 
                 119.21 
                 58.42 
                 2.3427E+07 
                 7.9595E−01 
               
               
                 201 
                 60.91 
                 119.09 
                 58.19 
                 2.3340E+07 
                 7.9610E−01 
               
               
                 202 
                 61.03 
                 118.97 
                 57.95 
                 2.3254E+07 
                 7.9625E−01 
               
               
                 203 
                 61.14 
                 118.86 
                 57.71 
                 2.3167E+07 
                 7.9640E−01 
               
               
                 204 
                 61.26 
                 118.74 
                 57.47 
                 2.3080E+07 
                 7.9655E−01 
               
               
                 205 
                 61.38 
                 118.62 
                 57.24 
                 2.2993E+07 
                 7.9670E−01 
               
               
                 206 
                 61.50 
                 118.50 
                 57.00 
                 2.2906E+07 
                 7.9685E−01 
               
               
                 207 
                 61.62 
                 118.38 
                 56.77 
                 2.2819E+07 
                 7.9700E−01 
               
               
                 208 
                 61.73 
                 118.27 
                 56.53 
                 2.2732E+07 
                 7.9714E−01 
               
               
                 209 
                 61.85 
                 118.15 
                 56.30 
                 2.2646E+07 
                 7.9729E−01 
               
               
                 210 
                 61.97 
                 118.03 
                 56.06 
                 2.2559E+07 
                 7.9744E−01 
               
               
                 211 
                 62.09 
                 117.91 
                 55.83 
                 2.2472E+07 
                 7.9759E−01 
               
               
                 212 
                 62.20 
                 117.80 
                 55.59 
                 2.2385E+07 
                 7.9774E−01 
               
               
                 213 
                 62.32 
                 117.68 
                 55.36 
                 2.2298E+07 
                 7.9789E−01 
               
               
                 214 
                 62.44 
                 117.56 
                 55.12 
                 2.2211E+07 
                 7.9804E−01 
               
               
                 215 
                 62.56 
                 117.44 
                 54.89 
                 2.2125E+07 
                 7.9819E−01 
               
               
                 216 
                 62.67 
                 117.33 
                 54.66 
                 2.2038E+07 
                 7.9834E−01 
               
               
                 217 
                 62.79 
                 117.21 
                 54.42 
                 2.1951E+07 
                 7.9848E−01 
               
               
                 218 
                 62.91 
                 117.09 
                 54.19 
                 2.1864E+07 
                 7.9863E−01 
               
               
                 219 
                 63.02 
                 116.98 
                 53.96 
                 2.1777E+07 
                 7.9878E−01 
               
               
                 220 
                 63.14 
                 116.86 
                 53.72 
                 2.1690E+07 
                 7.9893E−01 
               
               
                 221 
                 63.25 
                 116.75 
                 53.49 
                 2.1604E+07 
                 7.9908E−01 
               
               
                 222 
                 63.37 
                 116.63 
                 53.26 
                 2.1517E+07 
                 7.9923E−01 
               
               
                 223 
                 63.49 
                 116.51 
                 53.03 
                 2.1430E+07 
                 7.9938E−01 
               
               
                 224 
                 63.60 
                 116.40 
                 52.80 
                 2.1343E+07 
                 7.9953E−01 
               
               
                 225 
                 63.72 
                 116.28 
                 52.57 
                 2.1256E+07 
                 7.9968E−01 
               
               
                 226 
                 63.83 
                 116.17 
                 52.33 
                 2.1169E+07 
                 7.9983E−01 
               
               
                 227 
                 63.95 
                 116.05 
                 52.10 
                 2.1082E+07 
                 7.9997E−01 
               
               
                 228 
                 64.06 
                 115.94 
                 51.87 
                 2.0996E+07 
                 8.0012E−01 
               
               
                   
               
               
                 *beam angles are relative to the skew axis, which is oriented −30.25° relative to surface normal 
               
               
                 **gratings have grating vector angles oriented −30.25° relative to surface normal 
               
            
           
         
       
     
     Experimental Demonstration 
       FIG. 13  shows a slab waveguide  1350  with a holographic skew mirror output coupler  1300  (e.g., like the output coupler  1000  shown in  FIG. 10 ) that was fabricated according to the parameters shown in Table 1. The holographic skew output coupler  1300  had a 53.4° horizontal field-of-view and a 31.6° vertical (Bragg degenerate) field-of-view. The skew mirror output coupler was programmed into the recording media according to the parameters of Table 1. An optically flat waveguide package was fabricated using two 1″×2″ 500 μm thick Eagle x G  glass substrates  1354  with a 500 μm recording layer  1310  of Akonia formulation AK291 photopolymer medium. These substrates transmit about 90% of incident visible light in both directions. A TIGER prism skew recorder delivered collimated signal and reference beams approximately 40 mm in diameter at an optical power of approximately 2 mW/cm 2  for each beam. Each beam was apodized by a rectangular aperture measuring 25×21 mm (width by height). 
     The resulting waveguide  1350  and output coupler  1300  were tested to verify their properties. A coupling prism was affixed to left (x&lt;0) end of the waveguide  1350  using an optical adhesive, and an image  1301  was projected into the waveguide through the coupling prism using an off-the-shelf picoprojector. This image was guided within the recording layer via total internal reflection at the substrate boundaries to the gratings within the output coupler  1300 . These gratings reflect the image out of the coupler  1300  (e.g., towards an eye) about an axis of reflection forming an angle of about −30.25 degrees with respect to the surface normal. The output image  1303 ′ was visually inspected to approximately confirm the 53.4° horizontal field-of-view (the picoprojector had only a ˜30° field of view, so it was manually rotated to examine both ends of the waveguide range). 
     A modulation transfer function (MTF) test was performed across the field of view to verify projected image quality.  FIG. 14  depicts a mosaic of nine plots of the MTF measured across the field of view, where the position of the plot within the figure corresponds to the position in the field of view (i.e., the top left plot corresponds to the top left of the field, the center corresponds to the center, etc.). The horizontal axis of each plot in  FIG. 14  is spatial frequency (cycles/degree) and the vertical axis is contrast ratio (CR). The darker lines correspond to vertical MTF and lighter lines to horizontal MTF. A large portion of the degradation is due to the projector lens, demonstrated by the low CR of the vertical MTF, which is not deleteriously effected by the output coupler. 
     Skew Mirror-Based Head-Mounted Display 
       FIG. 15  shows a head-mounted display  1500  with wide field of view skew mirror-based couplers for projecting images to an eye  1599  of a viewer. An image source  1502 , such as a microdisplay illuminated by one or more lasers or light-emitting diodes (LEDs), disposed in or along an eyewear temple  1504  emits image light  1501  at one or more colors (e.g., red, green, and blue light) in a direction substantially parallel to the eyewear temple  1504 . A skew input coupler  1510 , which comprises a grating structure recorded in a grating medium sandwiched between a pair of transparent substrates  1512 , couples the light into a slab waveguide  1520 . (A prism or edge coupling can also be used to couple the light  1501  from the image source  1502  into the slab waveguide  1520 .) The slab waveguide  1520  guides this light  1511  to a skew output coupler  1530  like the one shown in  FIG. 10 . 
     This skew output coupler  1530  comprises another grating structure recorded in more grating medium sandwiched between the transparent substrates  1512 . The skew output coupler  1530  couples this light  1531  out towards to the viewer over a wide field of view, e.g., a field of view spanning about 50 degrees horizontally and about 30 degrees vertically as perceived by the viewer. This causes the viewer to perceive an image with a wide field of view. As shown in  FIG. 15 , the skew input coupler  1510  has a skew angle of about +30.25 degrees and the skew output coupler  1530  has a skew angle of about −30.25 degrees (e.g., like, the skew input/output coupler shown in  FIG. 10 ). 
     CONCLUSION 
     While various inventive embodiments have been described and illustrated herein, those of ordinary skill in the art will readily envision a variety of other means and/or structures for performing the function and/or obtaining the results and/or one or more of the advantages described herein, and each of such variations and/or modifications is deemed to be within the scope of the inventive embodiments described herein. More generally, those skilled in the art will readily appreciate that all parameters, dimensions, materials, and configurations described herein are meant to be exemplary and that the actual parameters, dimensions, materials, and/or configurations will depend upon the specific application or applications for which the inventive teachings is/are used. Those skilled in the art will recognize, or be able to ascertain using no more than routine experimentation, many equivalents to the specific inventive embodiments described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described and claimed. Inventive embodiments of the present disclosure are directed to each individual feature, system, article, material, kit, and/or method described herein. In addition, any combination of two or more such features, systems, articles, materials, kits, and/or methods, if such features, systems, articles, materials, kits, and/or methods are not mutually inconsistent, is included within the inventive scope of the present disclosure. 
     The above-described embodiments can be implemented in any of numerous ways. For example, embodiments of designing and making the technology disclosed herein may be implemented using hardware, software or a combination thereof. When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers. 
     The various methods or processes (e.g., of designing and making the technology disclosed above) outlined herein may be coded as software that is executable on one or more processors that employ any one of a variety of operating systems or platforms. Additionally, such software may be written using any of a number of suitable programming languages and/or programming or scripting tools, and also may be compiled as executable machine language code or intermediate code that is executed on a framework or virtual machine. 
     In this respect, various inventive concepts may be embodied as a computer readable storage medium (or multiple computer readable storage media) (e.g., a computer memory, one or more floppy discs, compact discs, optical discs, magnetic tapes, flash memories, circuit configurations in Field Programmable Gate Arrays or other semiconductor devices, or other non-transitory medium or tangible computer storage medium) encoded with one or more programs that, when executed on one or more computers or other processors, perform methods that implement the various embodiments of the invention discussed above. The computer readable medium or media can be transportable, such that the program or programs stored thereon can be loaded onto one or more different computers or other processors to implement various aspects of the present invention as discussed above. 
     The terms “program” or “software” or “code” are used herein in a generic sense to refer to any type of computer code or set of computer-executable instructions that can be employed to program a computer or other processor to implement various aspects of embodiments as discussed above. Additionally, it should be appreciated that according to one aspect, one or more computer programs that when executed perform methods of the present invention need not reside on a single computer or processor, but may be distributed in a modular fashion amongst a number of different computers or processors to implement various aspects of the present invention. 
     Computer-executable instructions may be in many forms, such as program modules, executed by one or more computers or other devices. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Typically the functionality of the program modules may be combined or distributed as desired in various embodiments. 
     Also, data structures may be stored in computer-readable media in any suitable form. For simplicity of illustration, data structures may be shown to have fields that are related through location in the data structure. Such relationships may likewise be achieved by assigning storage for the fields with locations in a computer-readable medium that convey relationship between the fields. However, any suitable mechanism may be used to establish a relationship between information in fields of a data structure, including through the use of pointers, tags or other mechanisms that establish relationship between data elements. 
     Also, various inventive concepts may be embodied as one or more methods, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments. 
     All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and/or ordinary meanings of the defined terms. 
     The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.” 
     The phrase “and/or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with “and/or” should be construed in the same fashion, i.e., “one or more” of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and/or” clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to “A and/or B”, when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc. 
     As used herein in the specification and in the claims, “or” should be understood to have the same meaning as “and/or” as defined above. For example, when separating items in a list, “or” or “and/or” shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only terms clearly indicated to the contrary, such as “only one of” or “exactly one of,” or, when used in the claims, “consisting of,” will refer to the inclusion of exactly one element of a number or list of elements. In general, the term “or” as used herein shall only be interpreted as indicating exclusive alternatives (i.e. “one or the other but not both”) when preceded by terms of exclusivity, such as “either,” “one of,” “only one of,” or “exactly one of” “Consisting essentially of,” when used in the claims, shall have its ordinary meaning as used in the field of patent law. 
     As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, “at least one of A and B” (or, equivalently, “at least one of A or B,” or, equivalently “at least one of A and/or B”) can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including more than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc. 
     In the claims, as well as in the specification above, all transitional phrases such as “comprising,” “including,” “carrying,” “having,” “containing,” “involving,” “holding,” “composed of,” and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases “consisting of” and “consisting essentially of” shall be closed or semi-closed transitional phrases, respectively, as set forth in the United States Patent Office Manual of Patent Examining Procedures, Section 2111.03.