Akonia Holographics Patent | Skew Mirrors, Methods Of Use, And Methods Of Manufacture

Patent: Skew Mirrors, Methods Of Use, And Methods Of Manufacture

Publication Number: 20170242176

Publication Date: 20170824

Applicants: Akonia Holographics

Abstract

An optical reflective device referred to as a skew mirror, having a reflective axis that need not be constrained to surface normal, is described. Examples of skew mirrors are configured to reflect light about substantially constant reflective axes across a relatively wide range of wavelengths. In some examples, a skew mirror has substantially constant reflective axes across a relatively wide range of angles of incidence. Exemplary methods for making and using skew mirrors are also disclosed. Skew mirrors include a grating structure, which in some examples comprises a hologram.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

[0001] This application claims priority from co-pending U.S. application Ser. No. 15/174,938, filed 6 Jun. 2016 and titled “SKEW MIRRORS, METHODS OF USE, AND METHODS OF MANUFACTURE,” 62/209,290, filed 24 Aug. 2015 and titled “MULTIWAVELENGTH DIFFRACTION GRATING MIRRORS, METHODS OF USE, AND METHODS OF MANUFACTURE,” and 62/318,917, filed 6 Apr. 2016 and titled “SKEW MIRRORS, METHODS OF USE, AND METHODS OF MANUFACTURE.” The above applications are incorporated herein by reference, in their entireties.

BACKGROUND

[0002] Conventional dielectric mirrors are produced by coating a surface (typically glass) with layers of materials that differ from each other in their electric permittivity. The layers of materials are typically arranged so that Fresnel reflections from layer boundaries reinforce constructively, producing large net reflectivity. Broadband dielectric mirrors can be designed by ensuring that this condition obtains over a relatively broad specified range of wavelengths and incidence angles. However, because the layers are deposited on a surface, the reflective axis of a dielectric mirror is necessarily coincident with surface normal, i.e. the reflective axis is perpendicular to the mirror surface. Because of this constraint on the reflective axis, a dielectric mirror is entirely inadequate for some purposes. Moreover, glass dielectric mirrors tend to be relatively heavy, making them suboptimal or inappropriate for applications requiring a relatively lightweight reflective component.

[0003] Conversely, conventional grating structures can reflect light about a reflective axis that differs from surface normal of the medium in which the grating structure resides. However, for a given angle of incidence, angles of reflection for conventional grating structures typically co-vary with wavelength of incident light. Thus, using a conventional grating structure to reflect light avoids the constraint inherent in dielectric mirrors that reflective axes must coincide with surface normal. However, where a constant reflective axis is required, a conventional grating structure is typically limited to a single wavelength or very narrow range of wavelengths for a given angle of incidence. Similarly, a conventional grating structure is limited to a single angle of incidence or very narrow range of incidence angles in order to reflect light of a specified wavelength about a constant reflective axis. Accordingly, a conventional grating structure does not have a constant reflective axis over any significant range of wavelengths or angles of incident light.

[0004] Accordingly, requirements for a relatively simple device that reflects light about a reflective axis not constrained to surface normal, and whose angle of reflection for a given angle of incidence is substantially constant at multiple wavelengths, are not met by currently available reflective devices comprising either reflective grating structures or dielectric mirrors. A need therefore exists for such a reflective device, and such need may be acute in head mounted display devices.

BRIEF DESCRIPTION OF THE DRAWINGS

[0005] 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).

[0006] FIG. 1A is a cross-section view of a hologram recorded in a grating medium.

[0007] FIG. 1B is a cross-section view of a k-space representation of a single sinusoidal hologram.

[0008] FIG. 2A is a cross-section view of a k-space representation of a single sinusoidal hologram.

[0009] FIG. 2B cross-section view of a k-space representation of a single sinusoidal hologram.

[0010] FIG. 3 is a cross-section real view illustrating reflective properties of a skew mirror in real space, according to an embodiment.

[0011] FIG. 4A is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

[0012] FIG. 4B is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

[0013] FIG. 5A is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

[0014] FIG. 5B is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

[0015] FIG. 6A is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

[0016] FIG. 6B is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

[0017] FIG. 6C is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

[0018] FIG. 6D is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

[0019] FIG. 7A is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

[0020] FIG. 7B is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

[0021] FIG. 8A is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

[0022] FIG. 8B is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

[0023] FIG. 8C is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

[0024] FIG. 9A is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

[0025] FIG. 9B is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

[0026] FIG. 10A is a cross-section view of a k-space representation of a skew mirror according to an embodiment.

[0027] FIG. 10B is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

[0028] FIG. 11A is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

[0029] FIG. 11B is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

[0030] FIG. 12A is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

[0031] FIG. 12B is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.

[0032] FIG. 13 is a cross-section view of a system for making a skew mirror, according to an embodiment.

[0033] FIG. 14 is a cross-section view illustrating a method of making a skew mirror, according to an embodiment.

[0034] FIG. 15 is a plan view illustrating reflective properties of a skew mirror according to an embodiment.

[0035] FIG. 16A is a cross-section view illustrating a system for making a skew mirror, according to an embodiment.

[0036] FIG. 16B is a cross-section view illustrating a system for making a skew mirror, according to an embodiment.

DETAILED DESCRIPTION

[0037] Embodiments of the present invention include a reflective device comprising a grating medium within which resides a volume hologram or other grating structure. The grating medium, by virtue of the grating structure residing therein, has physical properties that allow it to diffract light about an axis, referred to as a reflective axis, wherein angle of diffraction (henceforth referred to as angle of reflection) varies by less than 1.degree. for multiple wavelengths of light incident upon the grating medium at a given angle of incidence. In some embodiments, the above phenomenon is observed for multiple angles of incidence.

[0038] Similarly, embodiments typically have substantially constant reflective axes (i.e., reflective axes have reflective axis angles that vary by less than 1.0 degree) across a range of incidence angles for incident light of a given wavelength, and this phenomenon may be observed with incident light at various wavelengths. In some embodiments,* the reflective axes remain substantially constant for every combination of a set of multiple incidence angles and a set of multiple wavelengths*

[0039] In some embodiments, the grating structure includes a hologram generated by interference between multiple light beams referred to as recording beams. Typically, but not necessarily, the grating structure includes multiple holograms. The multiple holograms may be recorded using recording beams incident upon the grating medium at angles that vary among the multiple holograms (i.e. angle multiplexed) , and/or using recording beams whose wavelengths vary among the multiple holograms (i.e. wavelength multiplexed). In some embodiments, the grating structure includes a hologram recorded using two recording beams whose angles of incidence upon the grating medium vary while the hologram is being recorded, and/or whose wavelengths vary while the hologram is being recorded. Embodiments further include a device wherein the reflective axis differs from surface normal of the grating medium by at least 1.0 degree; or at least by 2.0 degrees; or at least by 4.0 degrees; or at least by 9.0 degrees.

* k-Space Formalism for Holography*

[0040] The k-space formalism is a method for analyzing holographic recording and diffraction. In k-space, propagating optical waves and holograms are represented by three dimensional Fourier transforms of their distributions in real space. For example, an infinite collimated monochromatic reference beam can be represented in real space and k-space by equation (1),

E r ( r ) = A r exp ( i k r r ) .fwdarw. E r ( k ) = A r .delta. ( k - k r ) , ( 1 ) ##EQU00001##

where E,() is the optical scalar field distribution at all ={x,y,z} 3D spatial vector locations, and its transform E, () is the optical scalar field distribution at all ={k.sub.x, k.sub.y, k.sub.z} 3D spatial frequency vectors. A.sub.r is the scalar complex amplitude of the field; and is the wave vector, whose length indicates the spatial frequency of the light waves, and whose direction indicates the direction of propagation. In some embodiments, all beams are composed of light of the same wavelength, so all optical wave vectors must have the same length, i.e., ||=k.sub.n. Thus, all optical propagation vectors must lie on a sphere of radius k.sub.n=2 .pi. n.sub.0/.lamda., where n.sub.0 is the average refractive index of the hologram (“bulk index”), and .lamda. is the vacuum wavelength of the light. This construct is known as the k-sphere. In other embodiments, light of multiple wavelengths may be decomposed into a superposition of wave vectors of differing lengths, lying on different k-spheres.

[0041] Another important k-space distribution is that of the holograms themselves. Volume phase holograms usually consist of spatial variations of the index of refraction within a grating medium. The index of refraction spatial variations, typically denoted .DELTA.n(), can be referred to as index modulation patterns, the k-space distributions of which are typically denoted .DELTA.n(). The index modulation pattern created by interference between a first recording beam and a second recording beam is typically proportional to the spatial intensity of the recording interference pattern, as shown in equation (2),

.DELTA.n().varies.|E.sub.1()+E.sub.2()|.sup.2=|E.sub.1()|.sup.2+|E.sub.2- ()|.sup.2+E.sub.1*()E.sub.2()+E.sub.1()E.sub.2*(), (2)

where E.sub.1() is the spatial distribution of the signal first recording beam field and E.sub.2 () is the spatial distribution of the second recording beam field. The unary operator * denotes complex conjugation. The final term in equation (2), E.sub.1()E.sub.2*(), maps the incident second recording beam into the diffracted first recording beam. Thus we can write equation (3),

E 1 ( r ) E 2 * ( r ) .fwdarw. E 1 ( k ) E 2 ( k ) , ( 3 ) ##EQU00002##

[0042] where is the 3D cross correlation operator. This is to say, the product of one optical field and the complex conjugate of another in the spatial domain becomes a cross correlation of their respective Fourier transforms in the frequency domain.

[0043] FIG. 1A illustrates a real space representation of recording a hologram 105 in a grating medium 110 using a second recording beam 115 and a first recording beam 114. The grating medium typically includes a recording layer configured to record interference patterns as holograms. FIG. 1A omits grating medium components other than the recording layer, such as an additional layer that might serve as a substrate or protective layer for the recording layer. The second recording beam 115 and first recording beam 114 are counter-propagating. Each of the second recording beam 115 and first recording beam 114 are typically plane wave beams having the same wavelength as each other, and the first recording beam 114 typically contains no encoded information that is not also present in the second recording beam. Thus the first and second recording beams, which can be referred to as signal and reference beams, are typically substantially identical to each other except for angles at which they are incident upon the recording medium 110.

[0044] FIG. 1B illustrates a k-space representation of the first and second recording beams, and the hologram. The hologram illustrated in FIGS. 1A and 1B is a simple Bragg reflection hologram generated with the counter-propagating first recording beam 114 and second recording beam 115, and recorded in recording medium 110. FIG. 1A shows the second recording beam 115 and the first recording beam 114 impinging on opposite sides of the grating medium 110. Optical scalar field distributions at ={x,y,z} 3D spatial vector locations for each of the second recording beam 115 and the first recording beam 114 can be represented as E.sub.2() and E.sub.1(), respectively. The recording beams 114, 115 form planar interference fringes, which are recorded as a hologram 105 within the grating medium 110. The hologram 105 comprises a sinusoidal refractive index modulation pattern, and can be represented as .DELTA.n(). In a counter-propagating configuration, the recorded planar interference fringes have a spacing exactly half that of the (internal) wavelength of the light used to record the hologram.

[0045] FIG. 1B shows a k-space representation of the situation illustrated in real space by FIG. 1A. The recording beams are represented in FIG. 1B by point-like k-space distributions lying on opposite sides of the recording k-sphere 170. As illustrated in FIG. 1B, the second recording beam has a k-space distribution 162, and the first recording beam has a k-space distribution 163. The second recording beam k-space distribution 162 can be represented as E.sub.2() and the first recording beam k-space distribution 163 can be represented as E.sub.1(). Each of the second recording beam k-space distribution 162 and the first recording beam k-space distribution 163 are “point-like.” Second recording beam wave vector 164 and first recording beam wave vector 165, are shown extending from the origin to the second recording beam k-space distribution 162 and first recording beam k-space distribution 163, respectively. The second recording beam wave vector 164 can be represented as E.sub.2() and the first recording beam wave vector 165 can be represented as E.sub.1(). The hologram itself is represented in FIG. 1B by two conjugate sideband k-space distributions 168, each of which can be represented as .DELTA.n() and referred to as a .DELTA.n() k-space distribution. The two .DELTA.n() k-space distributions 168 have a small, finite size, but are “point-like” in the sense that they are typically several orders of magnitude smaller than their distance to the origin, or other features of FIG. 1B. For instance, if the thickness of grating medium 110 is 200 .mu.m with refractive index 1.5 and the recording beams have a wavelength of 532 nm, then distributions 168 each resemble a sinc function along the k.sub.z dimension with size 3.14.times.10.sup.4 radians per meter (rad/m) null-to-null. However, their distance from the origin is 3.56.times.10.sup.7 rad/m, which is more than 1000 times as large. Unless specified otherwise, all recited wavelengths refer to vacuum wavelengths.

[0046] Typically, the hologram constitutes a refractive index distribution that is real-valued in real space. Locations of the two .DELTA.n({right arrow over (k)}) k-space distributions 168 of the hologram may be determined mathematically from the cross-correlation operations E.sub.2({right arrow over (k)})E.sub.1({right arrow over (k)}) and E.sub.1({right arrow over (k)})E.sub.2({right arrow over (k)}), respectively, or geometrically from vector differences .sub.G+={right arrow over (k)}.sub.1-{right arrow over (k)}.sub.2 and .sub.G-={right arrow over (k .sub.2)}-{right arrow over (k)}.sub.1, where .sub.G+ and .sub.G are grating vectors from the respective hologram .DELTA.n({right arrow over (k)}) k-space distributions to the origin (not shown individually). A grating vector 169, which can be represented as .sub.G, comprising both .sub.G+ and .sub.G- grating vectors, is shown in FIG. 1B as double headed arrow 169 extending between the second recording beam k-space distribution 162 and the first recording beam k-space distribution 163. Note that by convention, wave vectors are represented by a lowercase “k,” and grating vectors by uppercase “K.”

[0047] Once recorded, the hologram may be illuminated by a probe beam to produce a diffracted beam. For purposes of the present disclosure, the diffracted beam can be considered a reflection of the probe beam, which can be referred to as an incident light beam. The probe beam and its reflected beam are angularly bisected by a reflective axis (i.e. the angle of incidence of the probe beam relative to the reflective axis has the same magnitude as the angle of reflection of the reflected beam relative to the reflective axis). The diffraction process can be represented by a set of mathematical and geometric operations in k-space similar to those of the recording process. In the weak diffraction limit, the diffracted light distribution of the diffracted beam is given by equation (4),

E d ( k ) .varies. .DELTA. n ( k ) * E p ( k ) | | k | = k n , ( 4 ) ##EQU00003##

where E.sub.d({right arrow over (k)}) and E.sub.p({right arrow over (k)}) are k-space distributions of the diffracted beam and the probe beam, respectively; and “*” is the 3D convolution operator. The notation “

| | k | = k n ##EQU00004##

” indicates that the preceding expression is evaluated only where |{right arrow over (k)}|={right arrow over (k)}.sub.n, i.e., where the result lies on the k-sphere. The convolution .DELTA.n({right arrow over (k)})*E.sub.p({right arrow over (k)}) represents a polarization density distribution, and is proportional to the macroscopic sum of the inhomogeneous electric dipole moments of the grating medium induced by the probe beam, E.sub.p().

[0048] Typically, when the probe beam resembles one of the recording beams used for recording, the effect of the convolution is to reverse the cross correlation during recording, and the diffracted beam will substantially resemble the other recording beam used to record the hologram. When the probe beam has a different k-space distribution than the recording beams used for recording, the hologram may produce a diffracted beam that is substantially different than the beams used to record the hologram. Note also that while the recording beams are typically mutually coherent, the probe beam (and diffracted beam) is not so constrained. A multiwavelength probe beam may be analyzed as a superposition of single-wavelength beams, each obeying Equation (4) with a different k-sphere radius.

[0049] FIGS. 2A and 2B illustrate cases of Bragg-matched and Bragg-mismatched reconstructions, respectively, generated by illuminating the hologram depicted in FIGS. 1A and 1B. In both the Bragg-matched and Bragg-mismatched cases, the hologram is illuminated with a probe beam having a shorter wavelength than the recording beams used to record the hologram. The shorter wavelength corresponds to a longer wave vector. Accordingly, a probe k-sphere 172 has a greater radius than that of the recording k-sphere 170. oth the probe k-sphere 172 and the recording k-sphere 170 are indicated in FIGS. 2A and 2.

[0050] FIG. 2A shows a case where the probe beam is designed to produce a diffracted beam k-space distribution 175 (represented as E.sub.d()) that is point-like and lies on the probe beam k-sphere 172. The diffracted beam k-space distribution 175 is produced according to the convolution of Equation (4). The probe beam has a k-space distribution 176 (represented as E.sub.p()) that is also point-like. In this case, the probe beam is said to be “Bragg-matched” to the hologram, and the hologram may produce significant diffraction, even though the probe beam wavelength differs from the wavelength of the recording beams used to record the hologram. As shown in FIG. 2A, the convolution operation may also be represented geometrically by the vector sum .sub.d=.sub.p+{right arrow over (K)}.sub.G+, where .sub.d represents a diffracted beam wave vector 177, .sub.p represents a probe beam wave vector 178, and {right arrow over (K)}.sub.G+ represents a sideband grating vector 179.

[0051] FIG. 2A shows a k-space representation of a mirror-like diffraction (which can be referred to as a reflection) of the probe beam by the hologram, where the probe beam angle of incidence with respect to the k.sub.z axis is equal to the diffracted beam angle of reflection with respect to the k.sub.z axis. FIG. 2B shows a k-space representation of a Bragg-mismatched case, wherein a k-space polarization density distribution 180, which can be represented as .DELTA.n()* E.sub.p(), does not lie on the probe k-sphere 172, and thus no significant diffraction of the probe beam occurs. This non-diffracted k-space distribution 180 in the Bragg-mismatched case illustrated in FIG. 2B is somewhat analogous to the diffracted beam k-space distribution 175 in the Bragg-matched case illustrated in FIG. 2A, but k-space distribution 180 should not be referred to as a diffracted beam k-space distribution because no significant diffraction of the probe beam occurs.

[0052] Comparing the Bragg-matched and Bragg-mismatched cases, it is evident that the hologram will only produce mirror-like diffraction over a very small range of input angles for a given probe wavelength, if at all. Those skilled in the art will recognize that this range may be somewhat extended by over-modulating the hologram, or by using a very thin recording layer; but that these steps may still not lead to mirror-like behavior over a larger range of wavelengths and angles. These steps may also lead to undesired chromatic dispersion.

* Skew Mirror Embodiment in k-Space*

[0053] FIGS. 1A, 1B, 2A, and 2B represent a reflection hologram constituted by a single sinusoidal grating. As illustrated, this hologram exhibits mirror-like reflectivity in a narrow band of wavelengths and incidence angles. The specific properties of such a hologram may be determined by application of the well-known coupled wave theory of Kogelnik. Conversely, embodiments of the present invention exhibit novel mirror-like reflectivity across relatively broad ranges of wavelengths and angles by creating a more complex grating structure comprising multiple gratings.

[0054] FIG. 3 shows a geometry illustrating the Bragg selectivity of a single sinusoidal grating. Grating medium 310 contains a single sinusoidal grating of thickness d which reflects incident light 324 of a single wavelength, .lamda..sub.0, as principal reflected light 327. At the Bragg-matched condition, incident light 324 impinges at angle .theta..sub.i, and reflects as reflected light 327 at angle .theta..sub.r, both angles measured with respect to the z axis. Incident light 324 and reflected light 327 also define a reflective axis 338, about which the angular magnitudes of incidence .theta..sub.i’ and reflection .theta..sub.r’ are equal. Reflective axis 338 is thus an angular bisector of incident light 324 and reflected light 327.

[0055] As is known to those skilled in the art, the sinusoidal grating of FIG. 3 will exhibit both angular and wavelength Bragg selectivity. If incident light 324 impinges at non-Bragg-matched angle .theta..sub.i+.DELTA..theta..sub.i, the diffraction efficiency may be diminished compared to the Bragg-matched diffraction efficiency. The selectivity of a sinusoidal grating may be characterized by its angular Bragg selectivity, .DELTA..theta..sub.B, given by equation (5):

.DELTA..theta. B = .lamda. cos .theta. r n 0 d sin ( .theta. i - .theta. r ) . ( 5 ) ##EQU00005##

Those skilled in the art will recognize that in a weakly-diffracting sinusoidal grating, the angle .theta..sub.i+.DELTA..theta..sub.B represents the first null in the angular diffraction efficiency plot. The quantity .DELTA..theta..sub.B can thus be said to represent the angular width of the sinusoidal grating in that diffraction can be greatly diminished when the angle of incidence deviates from the Bragg-matched angle .theta..sub.i by more than several times .DELTA..theta..sub.B. Similarly, for a weakly-diffracting sinusoidal grating, the skilled artisan would expect a reflective axis to vary considerably for monochromatic incident light whose angle of incidence varies by more than several times .DELTA..theta..sub.B.

[0056] Conversely, skew mirrors according to present disclosure exhibit relatively stable diffraction and substantially constant reflective axes for incident light whose angle of incidence varies by many times .DELTA..theta..sub.B. Some skew mirror embodiments exhibit substantially constant reflective axes across a range of incident light angles of incidence of 20.times..DELTA..theta..sub.B. In embodiments, reflective axis angles across a range of incident light angles of incidence of 20.times..DELTA..theta..sub.B change by less than 0.250 degree; or by less than 0.10 degree; or by less than 0.025 degree.

[0057] Similarly, a sinusoidal grating may be characterized by its wavelength Bragg selectivity, .DELTA..lamda..sub.B, given by equation (6):

.DELTA..lamda. B = .lamda. 0 2 cos .theta. r 2 n 0 2 d sin 2 ( .theta. i - .theta. r ) . ( 6 ) ##EQU00006##

Those skilled in the art will recognize that in a weakly-diffracting sinusoidal grating, the wavelength .lamda..sub.0+.DELTA..lamda..sub.B represents the first null in the wavelength diffraction efficiency plot. The quantity .DELTA..lamda..sub.B can thus be said to represent the wavelength width of the sinusoidal grating in that no significant diffraction will occur when the incident wavelength deviates from the Bragg-matched wavelength .lamda..sub.0 by more than several times .DELTA..lamda..sub.B. Those skilled in the art will also recognize that equations (5) and (6) apply to changes in angle and wavelength only, respectively, and that changing both angle and wavelength simultaneously may result in another Bragg-matched condition.

[0058] A grating may also be characterized by its diffracted angle response. For a sinusoidal grating, the diffracted angle response may be expressed by equation (7):

.DELTA..theta..sub.r cos .theta..sub.r=-.DELTA..theta..sub.i cos .theta..sub.i. (7)

The diffracted angle response expresses the change in the angle of reflection, .DELTA..theta..sub.r, in response to small changes in the angle of incidence, .DELTA..theta..sub.i. In contrast, a true mirror has an angle response expressed by equation (8):

.DELTA..theta..sub.r=-.DELTA..theta..sub.i. (8)

A device that has a diffracted angle response substantially characterized by equation (7) may be said to exhibit grating-like reflective behavior, whereas a device that has a diffracted angle response substantially characterized by equation (8) may be said to exhibit mirror-like reflective behavior. A device exhibiting grating-like reflective behavior will necessarily also exhibit a reflective axis that changes with angle of incidence, unless that reflective axis is normal to the device surface, in which case cos .theta..sub.r=cos .theta..sub.i. Accordingly, requirements for a relatively simple device that reflects light about a reflective axis not constrained to surface normal, and whose angle of reflection for angles of incidence spanning multiples of its angular Bragg selectivity is constant at wavelengths spanning multiples of its wavelength Bragg selectivity, may not be met by a single sinusoidal grating.

[0059] FIG. 3 illustrates a device geometry in a reflective configuration. Those skilled in the art will recognize that the preceding analysis also applies to device geometries in transmissive configurations and to device geometries in which one or both beams are guided by total internal reflection within the device.

[0060] FIGS. 4A and 4B illustrate operation of a skew mirror in k-space according to an embodiment. FIG. 4A shows two .DELTA.n() k-space distributions 488 for a hologram recorded in a grating medium and configured to produce multiwavelength mirror-like diffraction according to an embodiment. A red k-sphere 490, green k-sphere 492, and blue k-sphere 493 in FIGS. 4A and 4B indicate k-spheres corresponding to wavelengths of light residing in the red, green, and blue regions of the visible spectrum, respectively.

[0061] Instead of two .DELTA.n() k-space distributions constituting a single sinusoidal grating (and which therefore can be characterized as “point-like”), the .DELTA.n() k-space distributions 488 shown in FIG. 4A are situated along a substantially straight line in k-space, and thus can be characterized as “line segment-like”. In some embodiments, line segment-like .DELTA.n() k-space distributions comprise continuously-modulated sub-segments of a substantially straight line in k-space. In some embodiments, line segment-like .DELTA.n() k-space distributions substantially consist of point-like distributions situated along a substantially straight line in k-space. The line segment-like .DELTA.n() k-space distributions 488 are situated symmetrically about the origin, and thus may be realized as conjugate sidebands of a real-valued refractive index distribution in real space (represented as .DELTA.n()). In some embodiments, the modulation may include absorptive and/or emissive components, and thus may not exhibit conjugate symmetry in k-space. The complex amplitude of the distribution may be uniform, or it may vary in amplitude and/or phase while still exhibiting substantially multiwavelength mirror-like diffraction according to embodiments of the present invention. In an embodiment, the line segment-like .DELTA.n() k-space distributions are situated substantially along the k.sub.z axis, which, by convention, is the thickness direction of a grating medium.

[0062] FIG. 4B illustrates a multiwavelength mirror-like reflective property of the hologram. Illumination of the hologram by a collimated probe beam with point-like k-space distribution 476 (represented as E.sub.p()) results in a k-space polarization density distribution 480 (represented as .DELTA.n()*E.sub.p()) according to Equation (4). Because the probe beam k-space distribution 476 is point-like, polarization density distribution 480 resembles a simple translation of .DELTA.n() k-space distribution 488 from the origin to the tip of probe beam wave vector 478 (.sub.p). Then, also according to Equation (4), only the part of the k-space polarization density distribution 480 (.DELTA.n()*E.sub.p()) intersecting the k-sphere 492 of the probe beam k-space distribution 476 (E.sub.p()) contributes to diffraction. This produces the diffracted beam k-space distribution 475, (E.sub.d()), constituting the diffracted beam. Because .DELTA.n() k-space distribution 488 resembles a line segment parallel to the k.sub..tau. axis, it is evident that the magnitude of the angle of reflection 482 (.theta..sub.r,) is substantially equal to the magnitude of the angle of incidence 481 (.theta..sub.i,) so that the hologram exhibits mirror-like behavior. Furthermore, it is also evident that this property typically holds for any incidence angle and wavelength that produces any diffraction at all, and for any superposition of probe beams producing diffraction. A k-space polarization distribution .DELTA.n()*E.sub.p() will intersect the probe k-sphere at a single point with mirror-symmetry about the k.sub.x axis (or about the k.sub.x, k.sub.y plane in the 3D case). Thus, the hologram of FIG. 4A is configured to exhibit mirror-like behavior at a relatively broad range of wavelengths and angles, and thus constitutes a broadband holographic mirror.

[0063] Embodiments typically, but not necessarily, exhibit a gap in .DELTA.n() k-space distribution 488 near the origin, as shown in FIG. 4A. The presence of the gap can limit performance at very high .DELTA..theta. (i.e., grazing angles of both incidence and reflection).

[0064] According to an embodiment, a skew mirror .DELTA.n() k-space distribution may be rotated to an arbitrary angle with respect to the k.sub.x, k.sub.y, and k.sub.z axes. In some embodiments, the .DELTA.n() k-space distribution is not perpendicular to the relevant reflecting surface in real space. In other words, the reflective axis of a skew mirror embodiment is not constrained to coincident with surface normal.

[0065] FIGS. 5A and 5B illustrate a skew mirror in k-space. FIGS. 5A and 5B are identical to FIGS. 4A and 4B, respectively, excepting that all distributions and vectors have been rotated by approximately 45.degree. about the origin. Following the discussion of FIG. 4B, it is evident that the skew mirror of FIG. 5B also produces mirror-like diffraction for all probe beam wavelengths and angles that produce diffraction. The diffraction is mirror-like with respect to the reflective axis 461 defined by the line segment-like .DELTA.n() k-space distribution 488, i.e., the angle of incidence 481 magnitude with respect to the reflective axis 461 is equal to the angle of reflection 482 magnitude with respect to the reflective axis 461. FIG. 5B illustrates one such case.

[0066] FIG. 6A illustrates the operation of a skew mirror in real space. Skew mirror 610 is characterized by reflective axis 638 at angle -13.degree. measured with respect to the z axis, which is normal to the skew mirror surface 612. Skew mirror 610 is illuminated with incident light 624 with internal incidence angle -26.degree. measured with respect to the z axis. Principal reflected light 627 is reflected with internal reflection angle 180.degree. measured with respect to the z axis.

[0067] FIG. 6B illustrates the skew mirror 610 of FIG. 6A in k-space. Line segment-like .DELTA.n() k-space distribution 688 passes through the origin, and has an angle of -13.degree. with respect to the z axis, equal to that of reflective axis 638. Recording k-sphere 670 is the k-sphere corresponding to the writing wavelength of 405 nm. A red k-sphere 690, green k-sphere 692, and blue k-sphere 693 in FIGS. 6B and 6D indicate k-spheres corresponding to wavelengths of light residing in the red, green, and blue regions of the visible spectrum, respectively.

[0068] FIG. 6C illustrates a highly magnified portion of FIG. 6B showing the left intersection between recording k-sphere 670 and line segment-like .DELTA.n() k-space distribution 688 according to an embodiment. In this view, line segment-like .DELTA.n() k-space distribution 688 can be seen to be include multiple discrete holograms. Each of the multiple discreet holograms 605 is represented by a horizontal line demarking the first null-to-first null spacing of the hologram in the k.sub.z direction. In some embodiments, the spacing of the discrete holograms may be higher or lower than illustrated in 6C. In some embodiments, the spacing may be low enough to create gaps in line segment-like .DELTA.n() k-space distribution 688. In some embodiments with gaps, the use of broadband illumination may substantially mask any effect of the gaps upon the reflected light. In some embodiments, this approach may result in a net diffraction efficiency increase. In other embodiments, the spacing of the discrete holograms may be so dense as to approximate or be equivalent to a continuous distribution.

[0069] FIG. 6D illustrates the reflection of blue incident light by the skew mirror of FIG. 6A in k-space. Incident light having a probe beam wave vector 678 impinges with an internal incidence angle of -26.degree. measured with respect to the z axis. The tip of probe beam wave vector 678 lies on blue k-sphere 693, indicating the position of point-like probe beam k-space distribution 676 (E.sub.p()). Polarization density distribution 680 is given by the convolution .DELTA.n()*E.sub.p(), which resembles line segment-like .DELTA.n() k-space distribution 688 (seen in FIG. 6C) translated to the tip of probe beam wave vector 678. Principal reflected light having diffracted beam wave vector 677 is determined from equation (4) by evaluating polarization density distribution 680 at blue k-sphere 693. Principal reflected light having diffracted beam wave vector 677 is reflected with internal propagation angle 180.degree. measured with respect to the z axis.

[0070] Persons skilled in the art will recognize that the term probe beam, typically used here when describing skew mirror properties in k-space, is analogous to the term incident light, which is typically used here when describing skew mirror reflective properties in real space. Similarly, the term diffracted beam, typically used here when describing skew mirror properties in k-space, is analogous to the term principal reflected light, typically used here when describing skew mirror properties in real space. Thus when describing reflective properties of a skew mirror in real space, it is typical to state that incident light is reflected by a hologram (or other grating structure) as principal reflected light, though to state that a probe beam is diffracted by the hologram to produce a diffracted beam says essentially the same thing. Similarly, when describing reflective properties of a skew mirror in k-space, it is typical to state that a probe beam is diffracted by a hologram (or other grating structure) to produce a diffracted beam, though to state that incident light is reflected by the grating structure to produce principal reflected light has the same meaning in the context of embodiments of the present invention.

[0071] As shown in FIG. 6D, probe beam wave vector 678 and diffracted beam wave vector 677 necessarily form the legs of a substantially isosceles triangle with line segment-like polarization density distribution 680 as the base. The equal angles of this triangle are necessarily congruent with the angle of incidence, 608, and angle of reflection 609, both measured with respect to reflective axis 638. Thus, skew mirror 610 reflects light in a substantially mirror-like manner about reflective axis 638.

[0072] The isosceles triangle construction of FIG. 6D obtains whenever .DELTA.n() k-space distribution 688 substantially resembles a segment of a line passing through the origin, as shown in FIG. 6C. Polarization density distribution 680 hence substantially resembles the straight base of an isosceles triangle, leading to mirror-like reflection about reflective axis 638 for any incident internal wave vectors of any length that diffracts. In some embodiments, dispersion of the grating medium may cause internal wave vectors of the same direction but differing lengths to refract in different directions in an external medium according to Snell’s law. Similarly, dispersion may cause external wave vectors of the same direction and differing lengths to refract in different directions in the internal grating medium. Accordingly, if it is desired to minimize the effects of dispersion in a skew mirror, it may be desirable to impart a curve to line segment-like .DELTA.n() k-space distribution 688, or to otherwise deviate from a line that passes through the origin. Such an approach may reduce net angular dispersion in reflections involving external refraction according to some metric. Since the dispersion of useful grating media is typically quite low, the deviation from a straight line passing through the origin may be small.

[0073] FIG. 7A illustrates the reflection of green incident light by the skew mirror of FIG. 6A in k-space. Incident light with wave vector 778A impinges with internal propagation angle -35.degree. measured with respect to the z axis. Principal reflected light with wave vector 777A is reflected with internal propagation angle -171.degree. measured with respect to the z axis. The magnitudes of angle of incidence 708A and angle of reflection 709A are both substantially equal to 22 degrees measured with respect to reflective axis 638, thus constituting a mirror-like reflection about reflective axis 638. Polarization density distribution 780A is also illustrated in FIG. 7A.

[0074] FIG. 7B illustrates the reflection of red incident light by the skew mirror of FIG. 10A in k-space. Incident light having probe beam wave vector 778 impinges with internal propagation angle -35.degree. measured with respect to the z axis. Principal reflected light having diffracted beam wave vector 777 is reflected with internal propagation angle -171.degree. measured with respect to the z axis. The magnitudes of angle of incidence 708 and angle of reflection 709 are both substantially equal to 22.degree. measured with respect to reflective axis 638, thus constituting a mirror-like reflection about reflective axis 638. Polarization density distribution 780 is also illustrated in FIG. 7B.

[0075] FIGS. 7A and 7B show the reflection of green and red light at the same angles of incidence and reflection, illustrating the achromatic reflection property of the skew mirror. Those skilled in the art will recognize that the geometrical constructions of FIGS. 6A-D and 7A-B will produce mirror-like reflection at all angle/wavelength combinations that produce reflection, including angles and wavelengths not specifically illustrated.

Skew Mirror Optical Properties

[0076] Embodiments of a skew mirror effect a mirror-like reflection with respect to internal propagation angles, external angles must be determined using Snell’s law at the relevant boundaries. Because of this, a skew mirror may introduce aberrations, dispersion, and/or field distortion to external wavefronts. In some embodiments, aberrations, dispersion, and/or field distortions may be mitigated by the use of compensating optics. In some embodiments, the compensating optics may include another skew mirror in a symmetric relationship.

[0077] A relatively thin skew mirror may introduce lowered angular resolution in the reflected beam in proportion to the beam’s projection onto the thin axis. In some cases it may be advantageous to increase the thickness of the recording layer in order to mitigate this effect.

Skew Mirror Reflectivity

[0078] Embodiments of a skew mirror may be either fully or partially reflective. Embodiments of a skew mirror may require relatively high dynamic range recording medium to achieve high reflectivity over a relatively wide wavelength bandwidth and angle range. In an embodiment, a skew mirror with an angular range spanning 105.degree. at 405 nm down to 20.degree. at 650 nm may require 183 individual holograms in a 200 .mu.m recording layer. This configuration has a reflectivity of approximately 7.5% using a state-of-the-art photosensitive recording medium with a maximum refractive index modulation of 0.03. In some embodiments, increasing recording medium thickness may not lead to increased reflectivity since diffractive selectivity also increases with thickness.

Skew Mirror Applications

[0079] The preceding exposition pertains to internal wavelengths and propagation angles, although in one case a slab-like hologram with thickness in the z direction was described. Many other configurations are possible within the scope of the invention. Without implying limitation, a few exemplary embodiments are illustrated here.

[0080] FIG. 8A illustrates an embodiment referred to as a skew window comprising grating structure 805 in a grating medium, and including a reflective axis 861 about which incident light is symmetrically refracted. The skew window is a transmissive analog of the skew mirror. FIG. 8B shows a skew coupler embodiment, which uses a skew mirror to couple external light into or out of a waveguide 894. Transmissive skew couplers are also possible. FIG. 8C shows a skew prism embodiment, which may fold an optical path and/or invert an image.

[0081] FIG. 9A illustrates a pupil relay embodiment formed by a slab waveguide 994 with two skew couplers, each of which comprises a grating medium 910 having a reflective axis 961 that differs from surface normal of the grating medium. Since this device is configured to relay input rays to output rays with a uniform 1:1 mapping, it can transmit an image at infinity through the waveguide 994 to the eye or other sensor. Such a configuration may be useful for head mounted displays (HMDs), among other applications. In the reverse direction, it may relay an image of the eye, possibly for the purposes of eye tracking. FIG. 9B shows a skew mirror 900 used as a concentrator/diffuser, which can transform a large dim beam into a bright small one, and/or vice-versa.

[0082] FIGS. 10A and 10B illustrate an angle filter embodiment of a skew mirror. In FIG. 10A, a .DELTA.n() k-space 1088 distribution is indicated with a higher low frequency cut-off (i.e., larger center gap) compared to the distribution illustrated in FIG. 8A. As a consequence, the skew mirror will reflect only the low .theta. (i.e., near normal incidence) angular components of narrow band incident beam E.sub.inc, into reflected beam E.sub.r, while transmitting high .theta. angular components in E.sub.t. One skilled in the art will readily discern that an arbitrary circularly-symmetric transfer function may be so realized by modulating the amplitude and/or phase of the line segment-like .DELTA.n() distribution according to an embodiment of the invention. Angular filtering may also be accomplished with skew mirrors, and in configurations involving multiple skew mirrors recorded in one or more media. These configurations may not be constrained to be circularly-symmetric, and may achieve some level of achromatic operation.

A First Embodiment Skew Mirror

[0083] Inventive aspects of a first embodiment skew mirror include the mirror being configured to reflect incident light having a wavelength of 532 nm and incident light having a wavelength of 513 nm about reflective axes that collectively have a mean reflective axis angle of +13.73 degrees relative to surface normal. In a further inventive aspect, the mean reflective axis angle (+13.759 degrees) for 532 nm light incident upon the skew mirror at internal angles of incidence ranging from -4.660 to +1.933 degrees differs by only 0.066 degree from the mean reflective axis angle (+13.693 degrees) for 513 nm light incident upon the skew mirror at the same angles of incidence as the 532 nm incident light. The reflective axes are thus substantially constant for the 532 nm to 513 nm wavelength range, a condition that obtains for internal angles of incidence (relative to surface normal) from -4.660 degrees to +1.993 degrees.

[0084] The first embodiment skew mirror 1100 is illustrated in FIGS. 11A and 11B. The first embodiment skew mirror 1100 comprises a grating structure 1105 (shown by diagonal hatch lines in FIGS. 11A and 11B) residing in a grating medium 1110. For purposes of clarity, the diagonal hatch lines are omitted in a region within the grating medium 1110 proximate figure elements indicating light, axes, and angles. However, persons skilled in the art will recognize that the grating structure 1105 typically occupies the region described above. The grating structure 1105 of the first embodiment includes multiple holograms that at least partially spatially overlap with each other in the grating medium 1110.

[0085] The multiple holograms are recorded into the grating medium internal volume and thus extend below the grating medium surface 1112. Accordingly, they are sometimes referred to as volume holograms. The multiple holograms of the first embodiment comprise forty eight (48) volume holograms, recorded with recording beams having a wavelength of 405 nm. Each of the 48 volume holograms typically at least partially spatially overlaps all others of the 48 volume holograms in the grating medium 1110. In some embodiments, each of the multiple holograms at least partially spatially overlaps at least one, but not all, of the other of the multiple holograms. Recording the 48 holograms of the first embodiment skew mirror is described below in a first method of making a skew mirror. In some embodiments, the grating structure includes between 1 and 48 holograms; or between 4 and 25 holograms; or at least 5 holograms; or at least 9 holograms; or at least 11 holograms; or at least 24 holograms.

[0086] The first embodiment grating medium 1110 is a proprietary photosensitive polymeric optical recording medium, designated AK174-200, available from Akonia Holographics, LLC (Longmont, Colo.). The AK174-200 recording medium of the first embodiment is approximately 200 .mu.m thick, has an M/# of approximately 18, and a refractive index of approximately 1.50 for 405 nm light. Optical recording mediums such as the AK174-200 medium are a type of grating medium in which grating structures can be recorded by optical means. Grating mediums are typically, but not necessarily, at least 70 .mu.m thick to approximately 1.2 mm thick. The AK174-200 medium typically undergoes relatively little shrinkage (usually about 0.1% to 0.2%) as a result of recording volume holograms. Variations of grating mediums include, but are not limited to, photorefractive crystals, dichromated gelatin, photo-thermo-refractive glass, and film containing dispersed silver halide particles.

[0087] Variations of the first embodiment skew mirror 1100 may include an additional layer such as a glass cover or glass substrate (not shown in FIGS. 11A and 11B). The additional layer may serve to protect the grating medium from contamination, moisture, oxygen, reactive chemical species, damage, and the like. The additional layer is typically refractive index matched to the grating medium 1110. ecause the refractive index for the additional layer is usually very close to the refractive index of the grating medium, refraction of light at the interface of the additional layer and the grating medium can sometimes be ignored. For the first embodiment, refractive indices for both the additional layer and the grating medium are approximately 1.5 for light having a wavelength of 405 nm. For clarity, the additional layer is not shown in FIGS. 11A and 11B.

[0088] As shown in FIG. 11A, the grating structure 1105 of the first embodiment has the physical property of being configured to reflect a first incident light 1124A, 1124B, about a first reflective axis 1138 (shown in broken line). The first incident light has a first wavelength of 532 nm and is incident upon the grating medium 1110 at a specific site 1117. The first reflective axis 1138 differs from surface normal 1122 of the grating medium by a first reflective axis angle 1135 of +13.759 degrees (internal, relative to surface normal), where the first incident light has an first internal angle of incidence 1125A, 1125B relative to surface normal, from -4.660 degrees (shown as first incident light 1124A) to +1.933 degrees (shown as first incident light 1124B), resulting in a range of 6.593 degrees. The first internal angles of incidence for the first incident light include one hundred (100) different internal angles spaced at angle intervals of about 0.067 degrees, from -4.660 degrees to +1.933 degrees, as shown in Table 1. In some variations of the first embodiment skew mirror, the first internal angles of incidence for the first incident light include ten (10) different internal angles spaced at angle intervals of about 0.67 degrees, from -4.660 degrees to +1.933 degrees. Throughout this specification and appended claims, identified angles and angle values refer to internal angles relative to surface normal, unless clearly indicated otherwise.

[0089] As shown FIG. 11A, first incident light 1124A, having a first internal angle of incidence of 1125A of -4.660 degrees relative to surface normal, is reflected by the grating structure 1105 as first reflected light 1127A, having a first internal angle of reflection 1126A of +32.267 degrees relative to surface normal. First incident light 1124, having a first internal angle of incidence 1125B relative to surface normal of +1.933 degrees, is reflected as first reflected light 1127B having a first internal angle of reflection 1126B of +25.668 degrees. First reflected light 1127A, 1127B has the first wavelength, i.e. in the first embodiment the first reflected light has a wavelength of 532 nm. First incident light angles, first reflected light angles, and first reflective axis angles for the first embodiment skew mirror are shown in Table 1.

TABLE-US-00001 TABLE 1 ANGLES OF FIRST INCIDENT LIGHT, FIRST REFLECTED LIGHT, AND FIRST REFLECTIVE AXIS, FOR A FIRST EMBODIMENT SKEW MIRROR; WAVELENGTH = 532 nm; AK174-200 RECORDING MEDIUM; N = 100 Angle of Angle Of Angle Of Angle of Reflection First First Incidence Reflection Incidence of First Internal First Internal of First of First of First Reflected First Angle of Reflective Angle of Incident Reflected Incident Light Reflective Reflection Axis Angle Incidence Light Light Light (external, Axis Angle (relative (internal, (relative (external, (external, (external, relative (external, to surface relative to to surface relative to relative to relative to to surface relative to normal, surface normal, reflective reflective surface normal, surface in normal, in in axis, in axis, in normal, in in normal, in degrees) degrees) degrees) degrees) degrees) degrees) degrees) degrees) 25.668 13.800 1.933 -11.867 11.867 2.900 40.521 21.711 25.680 13.773 1.866 -11.907 11.907 2.800 40.542 21.671 25.691 13.746 1.800 -11.946 11.946 2.701 40.563 21.632 25.814 13.774 1.733 -12.041 12.041 2.600 40.782 21.691 25.938 13.803 1.667 -12.136 12.136 2.501 41.003 21.752 26.005 13.802 1.600 -12.202 12.202 2.400 41.122 21.761 25.904 13.719 1.533 -12.185 12.185 2.300 40.942 21.621 25.971 13.719 1.466 -12.252 12.252 2.200 41.062 21.631 26.094 13.747 1.400 -12.347 12.347 2.101 41.283 21.692 26.216 13.775 1.333 -12.442 12.442 2.000 41.502 21.751 26.339 13.803 1.267 -12.536 12.536 1.901 41.723 21.812 26.350 13.775 1.200 -12.575 12.575 1.800 41.742 21.771 26.472 13.803 1.134 -12.669 12.669 1.701 41.963 21.832 26.538 13.802 1.067 -12.736 12.736 1.600 42.082 21.841 26.660 13.830 1.001 -12.830 12.830 1.501 42.303 21.902 26.780 13.857 0.933 -12.924 12.924 1.399 42.521 21.960 26.738 13.802 0.867 -12.935 12.935 1.301 42.443 21.872 26.803 13.801 0.800 -13.001 13.001 1.200 42.561 21.881 26.923 13.829 0.734 -13.095 13.095 1.101 42.781 21.941 26.989 13.828 0.667 -13.161 13.161 1.000 42.901 21.951 26.946 13.773 0.601 -13.173 13.173 0.901 42.822 21.862 27.066 13.800 0.533 -13.266 13.266 0.800 43.041 21.921 26.913 13.690 0.467 -13.223 13.223 0.701 42.762 21.732 27.088 13.744 0.400 -13.344 13.344 0.600 43.081 21.841 27.263 13.798 0.334 -13.464 13.464 0.501 43.402 21.952 27.436 13.852 0.267 -13.585 13.585 0.400 43.721 22.061 27.230 13.715 0.201 -13.515 13.515 0.301 43.342 21.822 27.241 13.687 0.133 -13.554 13.554 0.200 43.361 21.781 27.416 13.742 0.067 -13.674 13.674 0.101 43.683 21.892 27.589 13.794 0.000 -13.794 13.794 0.000 44.002 22.001 27.600 13.766 -0.067 -13.833 13.833 -0.100 44.022 21.961 27.664 13.766 -0.133 -13.899 13.899 -0.200 44.142 21.971 27.837 13.818 -0.200 -14.018 14.018 -0.300 44.462 22.081 27.955 13.844 -0.267 -14.111 14.111 -0.400 44.682 22.141 28.074 13.870 -0.333 -14.203 14.203 -0.499 44.903 22.202 28.030 13.815 -0.401 -14.215 14.215 -0.601 44.822 22.111 28.042 13.788 -0.467 -14.254 14.254 -0.700 44.844 22.072 28.106 13.786 -0.533 -14.320 14.320 -0.800 44.964 22.082 28.224 13.812 -0.600 -14.412 14.412 -0.900 45.184 22.142 28.288 13.811 -0.667 -14.477 14.477 -1.000 45.304 22.152 28.298 13.783 -0.733 -14.516 14.516 -1.100 45.324 22.112 28.362 13.781 -0.800 -14.581 14.581 -1.200 45.444 22.122 28.427 13.781 -0.866 -14.646 14.646 -1.299 45.566 22.134 28.437 13.752 -0.933 -14.685 14.685 -1.400 45.585 22.093 28.607 13.804 -0.999 -14.803 14.803 -1.499 45.906 22.204 28.670 13.802 -1.067 -14.868 14.868 -1.600 46.026 22.213 28.734 13.800 -1.133 -14.933 14.933 -1.700 46.146 22.223 28.797 13.798 -1.200 -14.998 14.998 -1.800 46.266 22.233 28.808 13.771 -1.266 -15.037 15.037 -1.899 46.287 22.194 28.923 13.795 -1.333 -15.128 15.128 -2.000 46.506 22.253 28.829 13.715 -1.399 -15.114 15.114 -2.099 46.327 22.114 28.996 13.765 -1.466 -15.231 15.231 -2.200 46.646 22.223 29.007 13.737 -1.532 -15.270 15.270 -2.299 46.667 22.184 29.069 13.735 -1.600 -15.335 15.335 -2.400 46.786 22.193 29.028 13.681 -1.666 -15.347 15.347 -2.499 46.707 22.104 29.142 13.705 -1.733 -15.438 15.438 -2.600 46.926 22.163 29.309 13.755 -1.799 -15.554 15.554 -2.699 47.247 22.274 29.475 13.804 -1.866 -15.670 15.670 -2.800 47.566 22.383 29.330 13.699 -1.932 -15.631 15.631 -2.899 47.287 22.194 29.392 13.696 -1.999 -15.696 15.696 -3.000 47.406 22.203 29.558 13.746 -2.065 -15.812 15.812 -3.099 47.727 22.314 29.670 13.769 -2.133 -15.902 15.902 -3.200 47.946 22.373 29.630 13.716 -2.199 -15.914 15.914 -3.299 47.867 22.284 29.640 13.687 -2.266 -15.953 15.953 -3.400 47.886 22.243 29.752 13.710 -2.333 -16.043 16.043 -3.500 48.106 22.303 29.916 13.759 -2.399 -16.158 16.158 -3.600 48.426 22.413 29.825 13.680 -2.465 -16.145 16.145 -3.699 48.247 22.274 29.988 13.728 -2.532 -16.260 16.260 -3.800 48.566 22.383 30.151 13.776 -2.598 -16.374 16.374 -3.899 48.887 22.494 30.160 13.747 -2.665 -16.413 16.413 -4.000 48.906 22.453 30.170 13.719 -2.732 -16.451 16.451 -4.100 48.926 22.413 30.332 13.767 -2.799 -16.565 16.565 -4.200 49.246 22.523 30.394 13.765 -2.865 -16.629 16.629 -4.299 49.368 22.535 30.302 13.685 -2.932 -16.617 16.617 -4.400 49.187 22.394 30.363 13.683 -2.998 -16.681 16.681 -4.499 49.308 22.405 30.474 13.704 -3.065 -16.769 16.769 -4.600 49.527 22.464 30.634 13.752 -3.131 -16.883 16.883 -4.699 49.848 22.575 30.694 13.748 -3.198 -16.946 16.946 -4.800 49.967 22.584 30.654 13.695 -3.264 -16.959 16.959 -4.899 49.888 22.495 30.814 13.741 -3.331 -17.072 17.072 -5.000 50.208 22.604 30.874 13.738 -3.397 -17.135 17.135 -5.099 50.329 22.615 30.834 13.685 -3.464 -17.149 17.149 -5.200 50.248 22.524 30.894 13.682 -3.530 -17.212 17.212 -5.299 50.369 22.535 31.051 13.727 -3.597 -17.324 17.324 -5.400 50.688 22.644 31.160 13.749 -3.663 -17.411 17.411 -5.499 50.909 22.705 31.169 13.720 -3.730 -17.450 17.450 -5.600 50.928 22.664 31.180 13.692 -3.796 -17.488 17.488 -5.699 50.949 22.625 31.336 13.736 -3.863 -17.599 17.599 -5.800 51.268 22.734 31.443 13.757 -3.929 -17.686 17.686 -5.899 51.488 22.795 31.549 13.777 -3.996 -17.772 17.772 -6.000 51.706 22.853 31.704 13.821 -4.062 -17.883 17.883 -6.099 52.027 22.964 31.713 13.792 -4.129 -17.921 17.921 -6.200 52.046 22.923 31.723 13.764 -4.195 -17.959 17.959 -6.299 52.067 22.884 31.636 13.687 -4.262 -17.949 17.949 -6.400 51.886 22.743 31.695 13.684 -4.327 -18.011 18.011 -6.499 52.007 22.754 31.848 13.727 -4.395 -18.121 18.121 -6.600 52.326 22.863 31.858 13.699 -4.460 -18.159 18.159 -6.699 52.347 22.824 31.963 13.718 -4.527 -18.245 18.245 -6.800 52.566 22.883 32.116 13.762 -4.593 -18.355 18.355 -6.899 52.888 22.995 32.267 13.804 -4.660 -18.464 18.464 -7.000 53.207 23.104 Mean = 13.759 Mean = 22.234 Std. 0.047 Dev. =

[0090] Incident light and its reflection are bisected by the reflective axis such that the internal angle of incidence of the incident light relative to the reflective axis has the same magnitude as the internal angle of reflection of the reflected light relative to the reflective axis. Thus it can be said that the incident light and its reflection exhibit bilateral symmetry about the reflective axis.

[0091] As shown in FIG. 11B, the grating structure 1105 of the first embodiment is further configured to reflect second incident light 1130A, 1130B about a second reflective axis 1139. The second incident light has a second wavelength of 513 nm and is incident upon the grating medium 1110 at the specific site 1117. The specific site 1117 includes an area of the grating medium surface 1112 upon which both the first and second incident light shine. The second reflective axis 1139 differs from surface normal 1122 of the grating medium by a second reflective axis angle 1136 of +13.693 degrees (internal) relative to surface normal, where the second incident light has a second internal angle of incidence, relative to surface normal, from -4.660 degrees to +1.933 degrees. The second internal angle of incidence includes one hundred (100) different internal angles spaced at angle intervals of approximately 0.067 degrees, from -4.660 degrees to +1.933 degrees. In some variations of the first embodiment skew mirror, the second internal angles of incidence for the second incident light include ten (10) different internal angles spaced at angle intervals of about 0.67 degrees, from -4.660 degrees to +1.933 degrees.

[0092] As shown in FIG. 11B, second incident light 1130A, having a second internal angle of incidence 1128A of -4.660 degrees relative to surface normal, is reflected by the grating structure 1105 as second reflected light 1133A, having a second internal angle of reflection 1133A of +32.075 degrees relative to surface normal. Second incident light 1130B, having a second internal angle of incidence 1128B relative to surface normal of +1.933 degrees, is reflected as second reflected light 1133B having a second internal angle of reflection 1129B of +25.273 degrees. Second reflected light 1133A, 1133B has the second wavelength, i.e. in the first embodiment the second reflected light has a wavelength of 513 nm. Second incident light angles, second reflected light angles, and second reflective axis angles for the first embodiment skew mirror, are shown in Table 2.

TABLE-US-00002 TABLE 2 ANGLES OF SECOND INCIDENT LIGHT, SECOND REFLECTED LIGHT, AND SECOND REFLECTIVE AXIS, FOR A FIRST EMBODIMENT SKEW MIRROR; WAVELENGTH = 513 nm; AK174-200 RECORDING MEDIUM; N = 100 Angle of Angle Of Angle Of Angle of Reflection Second Second Incidence Reflection Incidence of Second Internal Second Internal of Second of Second of Second Reflected Second Angle of Reflective Angle of Incident Reflected Incident Light Reflective Reflection Axis Angle Incidence Light Light Light (external, Axis Angle (relative (internal, (relative (external, (external, (external, relative (external, to surface relative to to surface relative to relative to relative to to surface relative to normal, surface normal, reflective reflective surface normal, surface in normal, in in axis, in axis, in normal, in in normal, in degrees) degrees) degrees) degrees) degrees) degrees) degrees) degrees) 25.273 13.603 1.933 -11.670 11.670 2.900 39.821 21.361 25.341 13.604 1.866 -11.737 11.737 2.800 39.942 21.371 25.466 13.633 1.800 -11.833 11.833 2.701 40.163 21.432 25.645 13.689 1.733 -11.956 11.956 2.600 40.481 21.541 25.769 13.718 1.667 -12.051 12.051 2.501 40.702 21.602 25.780 13.690 1.600 -12.090 12.090 2.400 40.721 21.561 25.959 13.746 1.533 -12.213 12.213 2.300 41.041 21.671 25.915 13.691 1.466 -12.224 12.224 2.200 40.961 21.581 25.982 13.691 1.400 -12.291 12.291 2.100 41.081 21.591 26.160 13.746 1.333 -12.413 12.413 2.000 41.400 21.700 26.171 13.719 1.267 -12.452 12.452 1.900 41.420 21.660 26.181 13.691 1.200 -12.491 12.491 1.800 41.439 21.620 26.249 13.691 1.134 -12.557 12.557 1.701 41.560 21.631 26.259 13.663 1.067 -12.596 12.596 1.600 41.579 21.590 26.438 13.719 1.001 -12.718 12.718 1.501 41.900 21.701 26.448 13.691 0.933 -12.757 12.757 1.400 41.919 21.660 26.515 13.691 0.867 -12.824 12.824 1.301 42.040 21.671 26.636 13.718 0.800 -12.918 12.918 1.200 42.259 21.730 26.592 13.663 0.734 -12.929 12.929 1.101 42.180 21.641 26.769 13.718 0.667 -13.051 13.051 1.000 42.500 21.750 26.780 13.690 0.601 -13.090 13.090 0.901 42.520 21.711 26.845 13.689 0.533 -13.156 13.156 0.800 42.639 21.720 26.912 13.690 0.467 -13.222 13.222 0.701 42.760 21.731 26.977 13.689 0.400 -13.289 13.289 0.600 42.879 21.740 26.989 13.661 0.334 -13.327 13.327 0.501 42.900 21.701 27.108 13.687 0.266 -13.421 13.421 0.399 43.118 21.759 27.229 13.715 0.201 -13.514 13.514 0.301 43.340 21.821 27.240 13.686 0.133 -13.553 13.553 0.200 43.359 21.780 27.360 13.714 0.067 -13.646 13.646 0.101 43.580 21.841 27.425 13.713 0.000 -13.713 13.713 0.000 43.700 21.850 27.490 13.712 -0.066 -13.778 13.778 -0.099 43.820 21.861 27.555 13.711 -0.133 -13.844 13.844 -0.200 43.939 21.870 27.565 13.683 -0.200 -13.883 13.883 -0.300 43.959 21.830 27.630 13.682 -0.267 -13.949 13.949 -0.400 44.079 21.840 27.750 13.709 -0.333 -14.041 14.041 -0.499 44.300 21.901 27.760 13.680 -0.400 -14.080 14.080 -0.600 44.319 21.860 27.825 13.680 -0.466 -14.146 14.146 -0.699 44.440 21.871 27.889 13.678 -0.533 -14.211 14.211 -0.800 44.559 21.880 28.007 13.703 -0.600 -14.303 14.303 -0.900 44.778 21.939 28.017 13.675 -0.667 -14.342 14.342 -1.000 44.798 21.899 28.135 13.701 -0.733 -14.434 14.434 -1.100 45.018 21.959 28.253 13.726 -0.800 -14.526 14.526 -1.200 45.238 22.019 28.264 13.699 -0.866 -14.565 14.565 -1.299 45.259 21.980 28.274 13.670 -0.933 -14.604 14.604 -1.400 45.278 21.939 28.338 13.669 -0.999 -14.669 14.669 -1.499 45.399 21.950 28.455 13.694 -1.067 -14.761 14.761 -1.600 45.619 22.010 28.572 13.719 -1.133 -14.852 14.852 -1.700 45.839 22.070 28.635 13.718 -1.200 -14.917 14.917 -1.800 45.959 22.080 28.646 13.690 -1.267 -14.956 14.956 -1.900 45.979 22.040 28.709 13.688 -1.333 -15.021 15.021 -2.000 46.099 22.050 28.720 13.660 -1.399 -15.060 15.060 -2.099 46.120 22.011 28.835 13.684 -1.466 -15.151 15.151 -2.200 46.339 22.070 28.899 13.683 -1.532 -15.216 15.216 -2.299 46.460 22.081 29.013 13.707 -1.600 -15.307 15.307 -2.400 46.679 22.140 29.024 13.679 -1.666 -15.345 15.345 -2.499 46.700 22.101 29.087 13.677 -1.733 -15.410 15.410 -2.600 46.819 22.110 29.150 13.675 -1.799 -15.474 15.474 -2.699 46.940 22.121 29.264 13.699 -1.866 -15.565 15.565 -2.800 47.159 22.180 29.326 13.697 -1.932 -15.629 15.629 -2.899 47.280 22.191 29.388 13.694 -1.999 -15.694 15.694 -3.000 47.399 22.200 29.502 13.718 -2.065 -15.784 15.784 -3.099 47.620 22.261 29.667 13.767 -2.133 -15.900 15.900 -3.200 47.939 22.370 29.678 13.739 -2.199 -15.938 15.938 -3.299 47.960 22.331 29.790 13.762 -2.266 -16.028 16.028 -3.400 48.180 22.390 29.647 13.657 -2.333 -15.990 15.990 -3.500 47.900 22.200 29.760 13.680 -2.399 -16.079 16.079 -3.600 48.120 22.260 29.822 13.678 -2.465 -16.143 16.143 -3.699 48.241 22.271 29.882 13.675 -2.532 -16.207 16.207 -3.800 48.360 22.280 29.944 13.672 -2.599 -16.271 16.271 -3.900 48.480 22.290 30.056 13.695 -2.665 -16.361 16.361 -4.000 48.700 22.350 30.066 13.667 -2.732 -16.399 16.399 -4.100 48.721 22.311 30.229 13.715 -2.799 -16.514 16.514 -4.200 49.041 22.421 30.290 13.713 -2.865 -16.577 16.577 -4.299 49.162 22.432 30.349 13.709 -2.932 -16.641 16.641 -4.400 49.280 22.440 30.360 13.681 -2.998 -16.679 16.679 -4.499 49.301 22.401 30.420 13.677 -3.065 -16.742 16.742 -4.600 49.420 22.410 30.531 13.700 -3.131 -16.831 16.831 -4.699 49.641 22.471 30.590 13.696 -3.198 -16.894 16.894 -4.800 49.760 22.480 30.651 13.694 -3.264 -16.957 16.957 -4.899 49.881 22.491 30.710 13.690 -3.331 -17.021 17.021 -5.000 50.000 22.500 30.820 13.712 -3.397 -17.109 17.109 -5.099 50.221 22.561 30.830 13.683 -3.464 -17.147 17.147 -5.200 50.240 22.520 30.939 13.705 -3.530 -17.235 17.235 -5.299 50.461 22.581 30.949 13.676 -3.597 -17.273 17.273 -5.400 50.480 22.540 31.009 13.673 -3.663 -17.336 17.336 -5.499 50.602 22.552 31.068 13.669 -3.730 -17.399 17.399 -5.600 50.721 22.561 31.225 13.714 -3.797 -17.511 17.511 -5.700 51.041 22.671 31.284 13.710 -3.863 -17.573 17.573 -5.800 51.161 22.681 31.293 13.682 -3.929 -17.611 17.611 -5.900 51.181 22.641 31.352 13.678 -3.996 -17.674 17.674 -6.000 51.302 22.651 31.460 13.699 -4.062 -17.761 17.761 -6.099 51.522 22.712 31.517 13.694 -4.129 -17.823 17.823 -6.200 51.641 22.721 31.528 13.667 -4.195 -17.861 17.861 -6.299 51.662 22.682 31.682 13.710 -4.262 -17.972 17.972 -6.400 51.981 22.791 31.692 13.682 -4.327 -18.010 18.010 -6.499 52.002 22.752 31.798 13.701 -4.395 -18.096 18.096 -6.600 52.221 22.811 31.904 13.722 -4.460 -18.182 18.182 -6.699 52.442 22.872 31.913 13.693 -4.527 -18.220 18.220 -6.800 52.461 22.831 31.970 13.689 -4.593 -18.282 18.282 -6.899 52.582 22.842 32.075 13.707 -4.660 -18.368 18.368 -7.000 52.801 22.901 Mean = 13.693 Mean = 22.110 Std. 0.025 Dev. =

[0093] The first wavelength (.lamda..sub.1=532 nm) differs from the second wavelength (.lamda..sub.2=513 nm) by 19 nm, which can be represented by a value referred to as a wave fraction (WF), defined as WF=|.lamda..sub.1-.lamda..sub.2|/[(.lamda..sub.1+.lamda..sub.2)/2]. Thus where the multiple wavelengths include a first wavelength of 532 nm and a second wavelength of 513 nm, WF=0.036. Similarly, where the multiple wavelengths consist of a continuous spectrum from 390 nm or less to at least 700 nm, WF.gtoreq.0.57. Embodiments include, but are not limited to, variations in which WF.gtoreq.0.005; WF.gtoreq.0.010; WF.gtoreq.0.030; WF.gtoreq.0.10; WF.gtoreq.0.250; WF.gtoreq.1.0; or WF.gtoreq.2.0. The wave fraction (WF) defined by a first (.lamda..sub.1) and a second (.lamda..sub.2) wavelength in the range may, but does not necessarily, includes a continuous spectrum of wavelengths between .lamda..sub.1 and .lamda..sub.2.

[0094] The second reflective axis angle 1136 differs from the first reflective axis angle 1135 by 0.066 degree. Accordingly, the second reflective axis is substantially coincident with the first reflective axis, meaning that the second reflective axis angle 1136 differs from first reflective axis angle 1135 by 1.0 degree or less. Such small difference between reflecting axis angles across a range of wavelengths (in this case, across a WF of 0.039) means that the grating structure acts as a nondispersive mirror. For some applications, the difference between reflective axis angles should be 0.250 degree or less for WF=0.030. Similarly, for some other applications, the difference between reflective axis angles should equal 0.10 degree or less for WF=0.030.

[0095] Relative to the first reflective axis, internal angles of incidence of the first incident light range from -11.867 degrees to -18.464 degrees. Relative to the second reflective axis, internal angles of incidence of the second incident light range from -11.670 degrees to -18.368 degrees. Thus it can be said that each of the first incident light and second incident light is offset from the first reflective axis by at least 11.670 degrees. In embodiments, incident light may be offset from its reflective axis by an internal angle of at least 1.0 degree; by at least 2.0 degrees; by at least 5.0 degrees; or by at least 9.0 degrees. A skew mirror or other reflective device configured to reflect incident light that is offset from the incident light’s reflective axis can be advantageous in some applications. For example, in a head mounted display it may be advantageous to reflect an image toward a user’s eye, but not to retroreflect the image back toward its source. Such reflection toward a user’s eye typically requires that incident light be offset from its reflective axis by an internal angle of at least 5.0 degrees, and more typically by at least 9.0 degrees. Similarly, a device utilizing total internal reflection typically requires that incident light be offset from its reflective axis.

[0096] First embodiment external angles relative to surface normal for incident light and its reflection are also illustrated in FIGS. 11A and 11B. As seen in FIG. 11A, external angles relative to surface normal for first incident light 1124A, 1124B ranges from first incident light external angle 1113A of -7.000 degrees to first incident light external angle 1113B of +2.900 degrees. As seen in FIG. 11B, external angles relative to surface normal for second incident light 1130A, 1130B ranges from second incident light external angle 1115A of -7.000 to second incident light external angle 1115B of +2.900 degrees. First reflected light external angles 1114A, 1114B and second reflected light external angles 1116A, 1116B are also illustrated in FIGS. 11A and 11B, respectively. External angles are measured with the skew mirror residing in air, with refraction occurring at the skew mirror/air boundary. Angles of incidence and angles of reflection, and reflective axis angles are tabulated in Tables 1 and 2.

[0097] The physical properties of the first embodiment enable it to reflect light having other wavelengths, and to reflect light incident upon the grating medium at other angles, about substantially constant reflective axes. For example, the first embodiment grating structure’s reflective properties enable it to reflect light having a wavelength of 520.4 nm about reflective axes having a mean reflective axis angle of +13.726 degrees, where the reflective axis angles vary by 0.10 degree or less for angles of incidence ranging from -6.862 degrees to +13.726 degrees and all angles in between (a range of 20.588 degrees). In another example of its reflective properties, the first embodiment is configured to reflect incident light about reflective axes (having a mean reflective axis angle of)+13.726.degree., where the reflective axis angles vary by 0.20 degree or less for wavelengths at 503 nm and 537 nm (a range of 34 nm, WF=0.065, including a continuous spectrum of wavelengths between 503 nm and 537 nm), where the angle of incidence (internal, relative to surface normal) is -1.174 degrees.

[0098] For clarity, light in FIGS. 11A and 11B is illustrated as being reflected at a point residing proximate a center of the grating structure 1105. However, persons skilled in the art recognize that light is typically reflected throughout the grating structure rather than at a specific point.

[0099] In some embodiments, the first incident light and the second incident light have wavelengths other than 532 and 513, respectively. Similarly, embodiments include first and second reflective axes that may be coincident with surface normal, or may differ from surface normal.

A Second Embodiment Skew Mirror

[0100] Inventive aspects of a second embodiment skew mirror include the mirror being configured to reflect incident light having a wavelength of 532 nm and incident light having a wavelength of 513 nm about reflective axes that collectively have a mean reflective axis angle of +14.62 degrees relative to surface normal. In a further inventive aspect, the mean reflective axis angle (+14.618 degrees) for 532 nm light incident upon the skew mirror at internal angles of incidence ranging from -9.281 to -2.665 degrees differs by less than 0.001 degree from the mean reflective axis angle (+14.617 degrees) for 513 nm light incident upon the skew mirror at the same angles of incidence as the 532 nm incident light. The reflective axes are thus substantially constant for the 532 nm to 513 nm wavelength range, a condition that obtains for internal angles of incidence (relative to surface normal) from -9.281 degrees to -2.665 degrees.

[0101] A second embodiment skew mirror 1200 is illustrated in FIGS. 12A and 12B. The second embodiment skew mirror 1200 comprises a grating structure 1205 (shown by diagonal hatch lines in FIGS. 12A and 12B) residing in a grating medium 1210. For purposes of clarity, the diagonal hatch lines are omitted in a region within the grating medium 1210 proximate figure elements indicating light, axes, and angles. However, persons skilled in the art will recognize that the grating structure 1205 typically occupies the region described above. The grating structure 1205 of the second embodiment includes multiple holograms that at least partially overlap with each other in the grating medium 1210. The multiple holograms of the second embodiment comprise forty nine (49) volume holograms, recorded with recording beams having a wavelength of 405 nm. The 49 volume holograms overlap each other in the grating medium 1210, and are recorded in a manner similar to the first embodiment skew mirror, except that recording beam internal angles of incidence are adjusted to account for media shrinkage. Recording the 49 holograms of the second embodiment skew mirror is described below in a second method of making a skew mirror.

[0102] The second embodiment grating medium 1210 is a proprietary photosensitive polymeric optical recording medium, designated AK233-200, available from Akonia Holographics, LLC (Longmont, Colo.). The AK233-200 recording medium of the second embodiment is approximately 200 .mu.m thick, has an M/# of approximately 24, and a refractive index of approximately 1.50 for light having a wavelength of 405 nm. The AK233-200 medium typically shrinks about 0.50% as a result of recording volume holograms.

[0103] Variations of the second embodiment skew mirror 1200 may include an additional layer such as a glass cover or glass substrate (not shown in FIGS. 12A and 12B). The additional layer is typically refractive index matched to the grating medium, and a thin film of index matching fluid may reside between the grating medium 1210 and the additional layer.

[0104] As shown in FIG. 12A, the grating structure 1205 of the second embodiment has the physical property of being configured to reflect a first incident light 1224A, 1224B, about a first reflective axis 1238 (shown in broken line). The first incident light has a first wavelength of 532 nm and is incident upon the grating medium 1210 at a specific site 1217. The first reflective axis 1238 differs from surface normal 1222 of the grating medium by a first reflective axis angle 1235 of +14.618 degrees (internal) relative to surface normal, where the first incident light has a first internal angle of incidence 1225A, 1225B, relative to surface normal, residing between -9.281 degrees to -2.665 degrees, inclusive (a range of 6.616 degrees). The first internal angle of incidence includes one hundred one (101) different internal angles spaced at angle intervals of approximately 0.066 degrees, from -9.281 degrees to -2.665 degrees. In some variations of the second embodiment skew mirror, the first internal angles of incidence for the first incident light include ten (10) different internal angles spaced at angle intervals of about 0.66 degrees, from -9.281 degrees to -2.665 degrees.

[0105] As shown FIG. 12A, first incident light 1224A, having a first internal angle of incidence 1225A of -9.281 degrees relative to surface normal, is reflected by the grating structure 1205 as first reflected light 1227A, having a first internal angle of reflectance 1226A of +38.610 degrees relative to surface normal. First incident light 1224B, having a first internal angle of incidence 1225B relative to surface normal of -2.665 degrees, is reflected as first reflected light 1227B having a first internal angle of reflectance 1226B of +31.836 degrees. First reflected light 1224A, 1224B has the first wavelength, i.e. in the second embodiment the first reflected light has a wavelength of 532 nm. First incident light angles, first reflected light angles, and first reflective axis angles, for the second embodiment skew mirror are shown in Table 3.

TABLE-US-00003 TABLE 3 ANGLES OF FIRST INCIDENT LIGHT, FIRST REFLECTED LIGHT, AND FIRST REFLECTIVE AXIS, FOR A SECOND EMBODIMENT SKEW MIRROR; WAVELENGTH = 532 nm; AK233-200 RECORDING MEDIUM; N = 101 Angle of Angle Of Angle Of Angle of Reflection First First Incidence Reflection Incidence of First Internal First Internal of First of First of First Reflected First Angle of Reflective Angle of Incident Reflected Incident Light Reflective Reflection Axis Angle Incidence Light Light Light (external, Axis Angle (relative (internal, (relative (external, (external, (external, relative (external, to surface relative to to surface relative to relative to relative to to surface relative to normal, surface normal, reflective reflective surface normal, surface in normal, in in axis, in axis, in normal, in in normal, in degrees) degrees) degrees) degrees) degrees) degrees) degrees) degrees) 31.836 14.585 -2.665 -17.250 17.250 -4.000 52.300 24.150 31.941 14.604 -2.732 -17.336 17.336 -4.100 52.520 24.210 31.998 14.600 -2.799 -17.398 17.398 -4.200 52.640 24.220 32.103 14.619 -2.865 -17.484 17.484 -4.299 52.861 24.281 32.160 14.614 -2.932 -17.546 17.546 -4.400 52.980 24.290 32.217 14.610 -2.998 -17.607 17.607 -4.499 53.101 24.301 32.321 14.628 -3.065 -17.693 17.693 -4.600 53.320 24.360 32.378 14.623 -3.131 -17.754 17.754 -4.699 53.441 24.371 32.433 14.618 -3.198 -17.816 17.816 -4.800 53.560 24.380 32.490 14.613 -3.264 -17.877 17.877 -4.899 53.681 24.391 32.546 14.607 -3.331 -17.938 17.938 -5.000 53.800 24.400 32.602 14.603 -3.397 -18.000 18.000 -5.099 53.921 24.411 32.704 14.620 -3.464 -18.084 18.084 -5.200 54.140 24.470 32.760 14.615 -3.530 -18.145 18.145 -5.299 54.261 24.481 32.815 14.609 -3.597 -18.206 18.206 -5.400 54.380 24.490 32.871 14.604 -3.664 -18.267 18.267 -5.500 54.500 24.500 32.926 14.598 -3.730 -18.328 18.328 -5.600 54.620 24.510 33.027 14.616 -3.796 -18.412 18.412 -5.699 54.841 24.571 33.082 14.609 -3.863 -18.472 18.472 -5.800 54.960 24.580 33.137 14.604 -3.929 -18.533 18.533 -5.899 55.081 24.591 33.191 14.598 -3.996 -18.594 18.594 -6.000 55.200 24.600 33.291 14.615 -4.062 -18.677 18.677 -6.099 55.421 24.661 33.345 14.608 -4.129 -18.737 18.737 -6.200 55.540 24.670 33.400 14.603 -4.195 -18.797 18.797 -6.299 55.661 24.681 33.498 14.618 -4.262 -18.880 18.880 -6.400 55.880 24.740 33.552 14.612 -4.327 -18.940 18.940 -6.499 56.001 24.751 33.605 14.605 -4.395 -19.000 19.000 -6.600 56.120 24.760 33.659 14.600 -4.460 -19.060 19.060 -6.699 56.241 24.771 33.757 14.615 -4.527 -19.142 19.142 -6.800 56.460 24.830 33.810 14.608 -4.593 -19.201 19.201 -6.899 56.580 24.841 33.862 14.601 -4.660 -19.261 19.261 -7.000 56.699 24.850 33.916 14.595 -4.726 -19.321 19.321 -7.099 56.820 24.861 34.012 14.609 -4.793 -19.402 19.402 -7.200 57.039 24.920 34.064 14.603 -4.859 -19.462 19.462 -7.299 57.160 24.931 34.116 14.595 -4.926 -19.521 19.521 -7.400 57.279 24.940 34.169 14.588 -4.992 -19.580 19.580 -7.500 57.399 24.950 34.264 14.603 -5.058 -19.661 19.661 -7.600 57.619 25.010 34.316 14.596 -5.124 -19.720 19.720 -7.699 57.740 25.021 34.368 14.588 -5.191 -19.779 19.779 -7.800 57.860 25.030 34.462 14.602 -5.257 -19.860 19.860 -7.900 58.080 25.090 34.513 14.595 -5.324 -19.918 19.918 -8.000 58.199 25.100 34.606 14.608 -5.390 -19.998 19.998 -8.100 58.419 25.160 34.699 14.622 -5.456 -20.078 20.078 -8.200 58.639 25.220 34.750 14.614 -5.522 -20.136 20.136 -8.299 58.760 25.231 34.842 14.626 -5.589 -20.216 20.216 -8.401 58.978 25.289 34.893 14.619 -5.655 -20.274 20.274 -8.500 59.100 25.300 34.943 14.611 -5.721 -20.332 20.332 -8.600 59.220 25.310 35.035 14.624 -5.787 -20.411 20.411 -8.699 59.441 25.371 35.084 14.615 -5.854 -20.469 20.469 -8.800 59.560 25.380 35.134 14.607 -5.919 -20.527 20.527 -8.899 59.681 25.391 35.224 14.619 -5.986 -20.605 20.605 -9.000 59.900 25.450 35.273 14.611 -6.052 -20.662 20.662 -9.099 60.021 25.461 35.321 14.601 -6.119 -20.720 20.720 -9.200 60.140 25.470 35.411 14.613 -6.184 -20.798 20.798 -9.299 60.361 25.531 35.459 14.604 -6.251 -20.855 20.855 -9.400 60.479 25.540 35.548 14.616 -6.316 -20.932 20.932 -9.499 60.700 25.601 35.595 14.606 -6.383 -20.989 20.989 -9.600 60.819 25.610 35.683 14.617 -6.449 -21.066 21.066 -9.699 61.040 25.671 35.731 14.608 -6.516 -21.123 21.123 -9.800 61.159 25.680 35.817 14.618 -6.582 -21.200 21.200 -9.900 61.379 25.740 35.865 14.608 -6.648 -21.256 21.256 -10.000 61.499 25.750 35.951 14.618 -6.714 -21.332 21.332 -10.100 61.719 25.810 35.997 14.609 -6.780 -21.389 21.389 -10.200 61.839 25.820 36.083 14.619 -6.845 -21.464 21.464 -10.299 62.060 25.881 36.168 14.628 -6.912 -21.540 21.540 -10.400 62.279 25.940 36.214 14.618 -6.977 -21.596 21.596 -10.499 62.400 25.951 36.298 14.627 -7.044 -21.671 21.671 -10.600 62.619 26.010 36.343 14.617 -7.110 -21.726 21.726 -10.699 62.739 26.020 36.426 14.625 -7.176 -21.801 21.801 -10.800 62.958 26.079 36.471 14.615 -7.242 -21.856 21.856 -10.899 63.079 26.090 36.553 14.623 -7.308 -21.931 21.931 -11.000 63.298 26.149 36.635 14.631 -7.373 -22.004 22.004 -11.099 63.519 26.210 36.679 14.620 -7.440 -22.060 22.060 -11.200 63.638 26.219 36.761 14.628 -7.505 -22.133 22.133 -11.299 63.859 26.280 36.804 14.616 -7.572 -22.188 22.188 -11.400 63.978 26.289 36.885 14.624 -7.637 -22.261 22.261 -11.499 64.199 26.350 36.964 14.630 -7.704 -22.334 22.334 -11.600 64.418 26.409 37.007 14.619 -7.769 -22.388 22.388 -11.699 64.539 26.420 37.086 14.625 -7.836 -22.461 22.461 -11.800 64.758 26.479 37.164 14.631 -7.901 -22.533 22.533 -11.900 64.978 26.539 37.241 14.637 -7.967 -22.604 22.604 -12.000 65.198 26.599 37.284 14.625 -8.033 -22.658 22.658 -12.100 65.318 26.609 37.360 14.630 -8.099 -22.729 22.729 -12.200 65.537 26.669 37.436 14.636 -8.165 -22.800 22.800 -12.300 65.757 26.729 37.512 14.640 -8.231 -22.871 22.871 -12.400 65.977 26.789 37.553 14.629 -8.296 -22.924 22.924 -12.499 66.098 26.800 37.627 14.633 -8.362 -22.995 22.995 -12.600 66.317 26.859 37.702 14.637 -8.427 -23.064 23.064 -12.699 66.538 26.920 37.774 14.640 -8.494 -23.134 23.134 -12.800 66.756 26.978 37.848 14.645 -8.559 -23.203 23.203 -12.899 66.978 27.040 37.920 14.648 -8.625 -23.273 23.273 -13.000 67.197 27.099 37.960 14.635 -8.690 -23.325 23.325 -13.099 67.318 27.110 38.031 14.637 -8.756 -23.394 23.394 -13.200 67.537 27.169 38.102 14.640 -8.822 -23.462 23.462 -13.300 67.757 27.229 38.172 14.642 -8.888 -23.530 23.530 -13.400 67.977 27.289 38.242 14.644 -8.953 -23.597 23.597 -13.499 68.197 27.349 38.310 14.645 -9.019 -23.664 23.664 -13.600 68.415 27.408 38.379 14.647 -9.084 -23.731 23.731 -13.699 68.636 27.469 38.446 14.648 -9.150 -23.798 23.798 -13.800 68.855 27.528 38.514 14.649 -9.215 -23.864 23.864 -13.899 69.076 27.589 38.610 14.664 -9.281 -23.946 23.946 -14.000 69.395 27.698 Mean = 14.618 Mean = 25.594 Std. 0.016 Dev. =

[0106] As shown in FIG. 12B, the grating structure 1205 of the second embodiment is further configured to reflect second incident light 1230A, 1230B about a second reflective axis 1239. The second incident light has a second wavelength of 513 nm, and the second wavelength therefore differs from the first wavelength by 19 nm, or a wave fraction (WF) of 0.036. The second incident light is incident upon the grating medium 1210 at the specific site 1217. The specific site 1217 of the second embodiment includes an area of the grating medium surface 1212 upon which both the first and second incident light shine. The second reflective axis 1239 differs from surface normal 1222 of the grating medium by a second reflective axis angle 1236 of +14.617 degrees (internal) relative to surface normal, where the second incident light has a second internal angle of incidence 1228A, 1228B relative to surface normal, spanning a range of -9.281 degrees to -2.665 degrees. The second internal angle of incidence of the second incident light includes one hundred one (101) different internal angles spaced at angle intervals of approximately 0.066 degrees, from -9.281 degrees to -2.665 degrees. In some variations of the second embodiment skew mirror, the second internal angles of incidence for the second incident light include ten (10) different internal angles spaced at angle intervals of about 0.66 degrees, from -9.281 degrees to -2.665 degrees.

[0107] As shown in FIG. 12B, second incident light 1230A, having a second internal angle incidence 1228A of -9.281 degrees relative to surface normal, is reflected by the grating structure 1205 as second reflected light 1233A, having a second internal angle of reflectance 1229A of +38.598 degrees relative to surface normal. Second incident light 1230B, having a second internal angle of incidence 1228B relative to surface normal of -2.655 degrees, is reflected as second reflected light 1233B having a second internal angle of reflectance 1229B of +31.836 degrees. Second reflected light 1233A, 1233B has the second wavelength, i.e. in the second embodiment the second reflected light has a wavelength of 513 nm. Second incident light angles, second reflected light angles, and second reflective axis angles for the second embodiment skew mirror 1200 are shown in Table 4.

TABLE-US-00004 TABLE 4 ANGLES OF SECOND INCIDENT LIGHT, SECOND REFLECTED LIGHT, AND SECOND REFLECTIVE AXIS, FOR A SECOND EMBODIMENT SKEW MIRROR; WAVELENGTH = 513 nm; AK233-200 RECORDING MEDIUM; N = 101 Angle of Angle Of Angle Of Angle of Reflection Second Second Incidence Reflection Incidence of Second Internal Second Internal of Second of Second of Second Reflected Second Angle of Reflective Angle of Incident Reflected Incident Light Reflective Reflection Axis Angle Incidence Light Light Light (external, Axis Angle (relative (internal, (relative (external, (external, (external, relative (external, to surface relative to to surface relative to relative to relative to to surface relative to normal, surface normal, reflective reflective surface normal, surface in normal, in in axis, in axis, in normal, in in normal, in degrees) degrees) degrees) degrees) degrees) degrees) degrees) degrees) 31.836 14.585 -2.665 -17.250 17.250 -4.000 52.300 24.150 31.941 14.604 -2.732 -17.336 17.336 -4.100 52.520 24.210 32.022 14.612 -2.799 -17.410 17.410 -4.200 52.690 24.245 32.080 14.608 -2.865 -17.472 17.472 -4.299 52.811 24.256 32.160 14.614 -2.932 -17.546 17.546 -4.400 52.980 24.290 32.240 14.621 -2.998 -17.619 17.619 -4.499 53.150 24.326 32.297 14.616 -3.065 -17.681 17.681 -4.600 53.270 24.335 32.378 14.623 -3.131 -17.754 17.754 -4.699 53.441 24.371 32.434 14.618 -3.198 -17.816 17.816 -4.800 53.561 24.381 32.514 14.625 -3.264 -17.889 17.889 -4.899 53.732 24.417 32.570 14.619 -3.331 -17.950 17.950 -5.000 53.851 24.426 32.626 14.615 -3.397 -18.011 18.011 -5.099 53.972 24.437 32.705 14.620 -3.464 -18.084 18.084 -5.200 54.141 24.471 32.737 14.604 -3.530 -18.134 18.134 -5.299 54.212 24.457 32.816 14.610 -3.597 -18.207 18.207 -5.400 54.382 24.491 32.872 14.605 -3.663 -18.267 18.267 -5.500 54.503 24.502 32.950 14.610 -3.730 -18.340 18.340 -5.600 54.672 24.536 33.006 14.605 -3.796 -18.401 18.401 -5.699 54.794 24.548 33.060 14.598 -3.863 -18.461 18.461 -5.800 54.912 24.556 33.137 14.604 -3.929 -18.533 18.533 -5.899 55.082 24.592 33.215 14.609 -3.996 -18.605 18.605 -6.000 55.252 24.626 33.292 14.615 -4.062 -18.677 18.677 -6.099 55.423 24.662 33.346 14.608 -4.129 -18.737 18.737 -6.200 55.541 24.671 33.423 14.614 -4.195 -18.809 18.809 -6.299 55.713 24.707 33.477 14.608 -4.262 -18.869 18.869 -6.400 55.833 24.717 33.554 14.613 -4.327 -18.941 18.941 -6.499 56.004 24.753 33.607 14.606 -4.395 -19.001 19.001 -6.600 56.123 24.762 33.683 14.611 -4.460 -19.072 19.072 -6.699 56.294 24.798 33.758 14.615 -4.527 -19.143 19.143 -6.800 56.463 24.832 33.812 14.609 -4.593 -19.202 19.202 -6.899 56.584 24.843 33.886 14.613 -4.660 -19.273 19.273 -7.000 56.752 24.876 33.939 14.607 -4.726 -19.333 19.333 -7.099 56.874 24.888 33.992 14.599 -4.793 -19.392 19.392 -7.200 56.994 24.897 34.067 14.604 -4.859 -19.463 19.463 -7.299 57.165 24.933 34.141 14.608 -4.926 -19.533 19.533 -7.400 57.335 24.968 34.192 14.600 -4.992 -19.592 19.592 -7.500 57.454 24.977 34.266 14.604 -5.058 -19.662 19.662 -7.600 57.624 25.012 34.318 14.597 -5.124 -19.721 19.721 -7.699 57.745 25.023 34.391 14.600 -5.191 -19.791 19.791 -7.800 57.915 25.058 34.443 14.593 -5.257 -19.850 19.850 -7.900 58.036 25.068 34.258 14.467 -5.324 -19.791 19.791 -8.000 57.606 24.803 34.418 14.514 -5.390 -19.904 19.904 -8.100 57.977 24.939 34.576 14.560 -5.456 -20.016 20.016 -8.200 58.348 25.074 34.733 14.606 -5.522 -20.127 20.127 -8.299 58.719 25.210 34.846 14.629 -5.589 -20.217 20.217 -8.401 58.988 25.294 34.897 14.621 -5.654 -20.276 20.276 -8.500 59.109 25.305 34.967 14.623 -5.721 -20.344 20.344 -8.600 59.279 25.340 35.018 14.615 -5.787 -20.402 20.402 -8.699 59.400 25.351 35.108 14.627 -5.854 -20.481 20.481 -8.800 59.618 25.409 35.137 14.609 -5.919 -20.528 20.528 -8.899 59.690 25.396 35.207 14.610 -5.986 -20.596 20.596 -9.000 59.859 25.430 35.277 14.612 -6.052 -20.664 20.664 -9.099 60.030 25.466 35.345 14.613 -6.119 -20.732 20.732 -9.200 60.198 25.499 35.414 14.615 -6.184 -20.799 20.799 -9.299 60.368 25.535 35.482 14.615 -6.251 -20.866 20.866 -9.400 60.536 25.568 35.551 14.617 -6.316 -20.934 20.934 -9.499 60.708 25.605 35.618 14.617 -6.383 -21.001 21.001 -9.600 60.876 25.638 35.666 14.608 -6.449 -21.058 21.058 -9.699 60.996 25.649 35.753 14.619 -6.516 -21.134 21.134 -9.800 61.216 25.708 35.820 14.619 -6.582 -21.201 21.201 -9.900 61.385 25.743 35.887 14.619 -6.648 -21.267 21.267 -10.000 61.555 25.778 35.954 14.620 -6.713 -21.334 21.334 -10.100 61.727 25.814 36.020 14.620 -6.780 -21.400 21.400 -10.200 61.897 25.849 36.067 14.611 -6.845 -21.456 21.456 -10.299 62.017 25.859 36.170 14.629 -6.912 -21.541 21.541 -10.400 62.286 25.943 36.217 14.620 -6.977 -21.597 21.597 -10.499 62.407 25.954 36.282 14.619 -7.044 -21.663 21.663 -10.600 62.577 25.989 36.365 14.628 -7.110 -21.737 21.737 -10.699 62.798 26.050 36.429 14.627 -7.176 -21.803 21.803 -10.800 62.967 26.084 36.475 14.617 -7.242 -21.858 21.858 -10.899 63.089 26.095 36.557 14.625 -7.308 -21.933 21.933 -11.000 63.309 26.155 36.621 14.624 -7.373 -21.997 21.997 -11.099 63.480 26.191 36.665 14.612 -7.440 -22.053 22.053 -11.200 63.599 26.200 36.746 14.620 -7.505 -22.126 22.126 -11.299 63.819 26.260 36.826 14.627 -7.572 -22.199 22.199 -11.400 64.037 26.319 36.888 14.626 -7.637 -22.263 22.263 -11.499 64.209 26.355 36.950 14.623 -7.704 -22.327 22.327 -11.600 64.379 26.390 37.029 14.630 -7.769 -22.399 22.399 -11.699 64.600 26.451 37.107 14.636 -7.836 -22.472 22.472 -11.800 64.819 26.510 37.185 14.642 -7.901 -22.543 22.543 -11.900 65.039 26.570 37.228 14.630 -7.967 -22.598 22.598 -12.000 65.159 26.580 37.305 14.636 -8.033 -22.669 22.669 -12.100 65.380 26.640 37.364 14.633 -8.099 -22.731 22.731 -12.200 65.549 26.675 37.440 14.638 -8.165 -22.803 22.803 -12.300 65.770 26.735 37.499 14.634 -8.231 -22.865 22.865 -12.400 65.940 26.770 37.557 14.631 -8.296 -22.926 22.926 -12.499 66.111 26.806 37.632 14.635 -8.362 -22.997 22.997 -12.600 66.330 26.865 37.706 14.639 -8.427 -23.067 23.067 -12.699 66.551 26.926 37.779 14.643 -8.494 -23.136 23.136 -12.800 66.770 26.985 37.852 14.647 -8.559 -23.206 23.206 -12.899 66.991 27.046 37.908 14.641 -8.625 -23.266 23.266 -13.000 67.159 27.080 37.980 14.645 -8.690 -23.335 23.335 -13.099 67.380 27.141 38.051 14.647 -8.756 -23.404 23.404 -13.200 67.599 27.200 38.121 14.650 -8.822 -23.472 23.472 -13.300 67.819 27.260 38.176 14.644 -8.888 -23.532 23.532 -13.400 67.989 27.295 38.245 14.646 -8.953 -23.599 23.599 -13.499 68.208 27.355 38.314 14.647 -9.019 -23.666 23.666 -13.600 68.427 27.414 38.398 14.657 -9.084 -23.741 23.741 -13.699 68.697 27.499 38.465 14.657 -9.150 -23.808 23.808 -13.800 68.916 27.558 38.517 14.651 -9.215 -23.866 23.866 -13.899 69.087 27.594 38.598 14.658 -9.281 -23.940 23.940 -14.000 69.355 27.678 Mean = 14.617 Mean = 25.593 Std. 0.025 Dev. =

[0108] For clarity, light in FIGS. 12A and 12B is illustrated as being reflected at a point residing proximate a center of the grating structure 1205. However, persons skilled in the art recognize that light is typically reflected throughout the grating structure rather than at a specific point.

[0109] In the second embodiment, the second reflective axis angle differs from the first reflective axis angle by approximately 0.0005 degree across WF=0.036. This very low level of change can approach the level of precision of instrumentation used to measure reflection angles. Accordingly, for the purposes of the present invention, the second reflective axis can be said to not differ from the first reflective axis. For some applications, the difference between reflective axis angles should be 0.025 degree or less. For some other applications, the difference between reflective axis angles should be 0.010 degree or less across WF.gtoreq.0.036. The second embodiment skew mirror meets these requirements. A Student’s t-test (two-tailed) indicates no difference between the first reflective axis angle and the second reflective axis angle (N=101 per group; P=0.873). Moreover, a difference of 0.001 degree or less challenges the precision of instrumentation used to measure skew mirror reflection angles. Accordingly, for purposes of the present invention, where a second reflective axis differs from a first reflective axis by 0.001 degree or less, the second reflective axis can be said to not differ from the first reflective axis.……
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