# Patent

## Title
Skew mirrors, methods of use, and methods of manufacture

## 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.

## Background
CROSS-REFERENCE TO RELATED PATENT APPLICATIONS
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
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.
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.
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
The skilled artisan will understand that the drawings primarily are for illustrative purposes and are not intended to limit the scope of the inventive subject matter described herein. The drawings are not necessarily to scale; in some instances, various aspects of the inventive subject matter disclosed herein may be shown exaggerated or enlarged in the drawings to facilitate an understanding of different features. In the drawings, like reference characters generally refer to like features (e.g., functionally similar and/or structurally similar elements).
FIG. 1A is a cross-section view of a hologram recorded in a grating medium.
FIG. 1B is a cross-section view of a k-space representation of a single sinusoidal hologram.
FIG. 2A is a cross-section view of a k-space representation of a single sinusoidal hologram.
FIG. 2B cross-section view of a k-space representation of a single sinusoidal hologram.
FIG. 3 is a cross-section real view illustrating reflective properties of a skew mirror in real space, according to an embodiment.
FIG. 4A is a cross-section view of a k-space representation of a skew mirror according to an embodiment.
FIG. 4B is a cross-section view of a k-space representation of a skew mirror according to an embodiment.
FIG. 5A is a cross-section view of a k-space representation of a skew mirror according to an embodiment.
FIG. 5B is a cross-section view of a k-space representation of a skew mirror according to an embodiment.
FIG. 6A is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.
FIG. 6B is a cross-section view of a k-space representation of a skew mirror according to an embodiment.
FIG. 6C is a cross-section view of a k-space representation of a skew mirror according to an embodiment.
FIG. 6D is a cross-section view of a k-space representation of a skew mirror according to an embodiment.
FIG. 7A is a cross-section view of a k-space representation of a skew mirror according to an embodiment.
FIG. 7B is a cross-section view of a k-space representation of a skew mirror according to an embodiment.
FIG. 8A is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.
FIG. 8B is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.
FIG. 8C is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.
FIG. 9A is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.
FIG. 9B is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.
FIG. 10A is a cross-section view of a k-space representation of a skew mirror according to an embodiment.
FIG. 10B is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.
FIG. 11A is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.
FIG. 11B is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.
FIG. 12A is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.
FIG. 12B is a cross-section view illustrating reflective properties of a skew mirror according to an embodiment.
FIG. 13 is a cross-section view of a system for making a skew mirror, according to an embodiment.
FIG. 14 is a cross-section view illustrating a method of making a skew mirror, according to an embodiment.
FIG. 15 is a plan view illustrating reflective properties of a skew mirror according to an embodiment.
FIG. 16A is a cross-section view illustrating a system for making a skew mirror, according to an embodiment.
FIG. 16B is a cross-section view illustrating a system for making a skew mirror, according to an embodiment.



DETAILED DESCRIPTION
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° 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.
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
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
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
                          ⇀
                        
                      
                    
                    )
                  
                
              
              ⁢
              
                →
                F
              
              ⁢
              
                
                  E
                  r
                
                ⁡
                
                  (
                  
                    k
                    ⇀
                  
                  )
                
              
            
            =
            
              
                A
                r
              
              ⁢
              
                δ
                ⁡
                
                  (
                  
                    
                      k
                      ⇀
                    
                    -
                    
                      
                        k
                        ⇀
                      
                      r
                    
                  
                  )
                
              
            
          
        
        ,
      
    
    
      
        (
        1
        )
      
    
  




where Er() is the optical scalar field distribution at all ={x,y,z} 3D spatial vector locations, and its transform Er() is the optical scalar field distribution at all ={kx, ky, kz} 3D spatial frequency vectors. Ar 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., ||=kn. Thus, all optical propagation vectors must lie on a sphere of radius kn=2 π n0/λ, where n0 is the average refractive index of the hologram (“bulk index”), and λ 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.

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 Δn(), can be referred to as index modulation patterns, the k-space distributions of which are typically denoted Δn(). 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),

Δn()∝|E1()+E2()|2=|E1()|2+|E2()|2+E1*()E2()+E1()E2*(),  (2)

where E1() is the spatial distribution of the signal first recording beam field and E2 () is the spatial distribution of the second recording beam field. The unary operator * denotes complex conjugation. The final term in equation (2), E1()E2*(), maps the incident second recording beam into the diffracted first recording beam. Thus we can write equation (3),





  
    
      
        
          
            
              
                E
                1
              
              ⁡
              
                (
                
                  r
                  ⇀
                
                )
              
            
            ⁢
            
              
                E
                2
                *
              
              ⁡
              
                (
                
                  r
                  ⇀
                
                )
              
            
          
          ⁢
          
            →
            F
          
          ⁢
          
            
              
                E
                1
              
              ⁡
              
                (
                
                  k
                  ⇀
                
                )
              
            
            ⊗
            
              
                E
                2
              
              ⁡
              
                (
                
                  k
                  ⇀
                
                )
              
            
          
        
        ,
      
    
    
      
        (
        3
        )
      
    
  




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.
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.
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 E2() and E1(), 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 Δn(). 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.
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 E2() and the first recording beam k-space distribution 163 can be represented as E1(). 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 E2() and the first recording beam wave vector 165 can be represented as E1(). The hologram itself is represented in FIG. 1B by two conjugate sideband k-space distributions 168, each of which can be represented as Δn() and referred to as a Δn() k-space distribution. The two Δn() 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 μm 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 kz dimension with size 3.14×104 radians per meter (rad/m) null-to-null. However, their distance from the origin is 3.56×107 rad/m, which is more than 1000 times as large. Unless specified otherwise, all recited wavelengths refer to vacuum wavelengths.
Typically, the hologram constitutes a refractive index distribution that is real-valued in real space. Locations of the two Δn({right arrow over (k)}) k-space distributions 168 of the hologram may be determined mathematically from the cross-correlation operations E2({right arrow over (k)})E1({right arrow over (k)}) and E1({right arrow over (k)})E2({right arrow over (k)}), respectively, or geometrically from vector differences G+={right arrow over (k)}1−{right arrow over (k)}2 and G−={right arrow over (k)}2−{right arrow over (k)}1, where G+ and G are grating vectors from the respective hologram Δn({right arrow over (k)}) k-space distributions to the origin (not shown individually). A grating vector 169, which can be represented as G, comprising both G+ and 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.”
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
                  ⇀
                
                )
              
            
            ∝
            
              Δ
              ⁢
              
                
              
              ⁢
              
                n
                ⁡
                
                  (
                  
                    k
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                  )
                
              
              *
              
                
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                  p
                
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                  (
                  
                    k
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                  )
                
              
            
          
          ⁢
          
            |
            
              
                |
                
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                |
              
              =
              
                k
                n
              
            
          
        
        ,
      
    
    
      
        (
        4
        )
      
    
  




where Ed({right arrow over (k)}) and Ep({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
      
    
  
  ”




indicates that the preceding expression is evaluated only where |{right arrow over (k)}|={right arrow over (k)}n, i.e., where the result lies on the k-sphere. The convolution Δn({right arrow over (k)})*Ep({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, Ep().

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.
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. Both the probe k-sphere 172 and the recording k-sphere 170 are indicated in FIGS. 2A and 2.
FIG. 2A shows a case where the probe beam is designed to produce a diffracted beam k-space distribution 175 (represented as Ed()) 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 Ep()) 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 d=p+{right arrow over (K)}G+, where d represents a diffracted beam wave vector 177, p represents a probe beam wave vector 178, and {right arrow over (K)}G+ represents a sideband grating vector 179.
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 kz axis is equal to the diffracted beam angle of reflection with respect to the kz 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 Δn()*Ep(), 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.
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
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.
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, λ0, as principal reflected light 327. At the Bragg-matched condition, incident light 324 impinges at angle θi, and reflects as reflected light 327 at angle θr, 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 θi′ and reflection θr′ are equal. Reflective axis 338 is thus an angular bisector of incident light 324 and reflected light 327.
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 θi+Δθi, 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, ΔθB, given by equation (5):




  
    
      
        
          Δθ
          B
        
        =
        
          
            
              λ
              ⁢
              
                
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              
                θ
                r
              
            
            
              
                n
                0
              
              ⁢
              d
              ⁢
              
                
              
              ⁢
              
                sin
                ⁡
                
                  (
                  
                    
                      θ
                      i
                    
                    -
                    
                      θ
                      r
                    
                  
                  )
                
              
            
          
          .
        
      
    
    
      
        (
        5
        )
      
    
  




Those skilled in the art will recognize that in a weakly-diffracting sinusoidal grating, the angle θi+ΔθB represents the first null in the angular diffraction efficiency plot. The quantity ΔθB 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 θi by more than several times ΔθB. 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 ΔθB.

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 ΔθB. Some skew mirror embodiments exhibit substantially constant reflective axes across a range of incident light angles of incidence of 20×ΔθB. In embodiments, reflective axis angles across a range of incident light angles of incidence of 20×ΔθB change by less than 0.250 degree; or by less than 0.10 degree; or by less than 0.025 degree.
Similarly, a sinusoidal grating may be characterized by its wavelength Bragg selectivity, ΔλB, given by equation (6):




  
    
      
        
          Δλ
          B
        
        =
        
          
            
              
                λ
                0
                2
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              
                θ
                r
              
            
            
              2
              ⁢
              
                n
                0
                2
              
              ⁢
              d
              ⁢
              
                
              
              ⁢
              
                
                  sin
                  2
                
                ⁡
                
                  (
                  
                    
                      θ
                      i
                    
                    -
                    
                      θ
                      r
                    
                  
                  )
                
              
            
          
          .
        
      
    
    
      
        (
        6
        )
      
    
  




Those skilled in the art will recognize that in a weakly-diffracting sinusoidal grating, the wavelength λ0+ΔλB represents the first null in the wavelength diffraction efficiency plot. The quantity ΔλB 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 λ0 by more than several times ΔλB. 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.

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):

Δθr cos θr=−Δθi cos θi.  (7)

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

Δθr=−Δθi.  (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 θr=cos θi. 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.

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.
FIGS. 4A and 4B illustrate operation of a skew mirror in k-space according to an embodiment. FIG. 4A shows two Δn() 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.
Instead of two Δn() k-space distributions constituting a single sinusoidal grating (and which therefore can be characterized as “point-like”), the Δn() 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 Δn() k-space distributions comprise continuously-modulated sub-segments of a substantially straight line in k-space. In some embodiments, line segment-like Δn() k-space distributions substantially consist of point-like distributions situated along a substantially straight line in k-space. The line segment-like Δn() 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 Δn()). 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 Δn() k-space distributions are situated substantially along the kz axis, which, by convention, is the thickness direction of a grating medium.
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 Ep()) results in a k-space polarization density distribution 480 (represented as Δn()*Ep()) according to Equation (4). Because the probe beam k-space distribution 476 is point-like, polarization density distribution 480 resembles a simple translation of Δn() k-space distribution 488 from the origin to the tip of probe beam wave vector 478 (p). Then, also according to Equation (4), only the part of the k-space polarization density distribution 480 (Δn()*Ep()) intersecting the k-sphere 492 of the probe beam k-space distribution 476 (Ep()) contributes to diffraction. This produces the diffracted beam k-space distribution 475, (Ed()), constituting the diffracted beam. Because Δn() k-space distribution 488 resembles a line segment parallel to the kτ axis, it is evident that the magnitude of the angle of reflection 482 (θr,) is substantially equal to the magnitude of the angle of incidence 481 (θi,) 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 Δn()*Ep() will intersect the probe k-sphere at a single point with mirror-symmetry about the kx axis (or about the kx, ky 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.
Embodiments typically, but not necessarily, exhibit a gap in Δn() k-space distribution 488 near the origin, as shown in FIG. 4A. The presence of the gap can limit performance at very high Δθ (i.e., grazing angles of both incidence and reflection).
According to an embodiment, a skew mirror Δn() k-space distribution may be rotated to an arbitrary angle with respect to the kx, ky, and kz axes. In some embodiments, the Δn() 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.
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° 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 Δn() 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.
FIG. 6A illustrates the operation of a skew mirror in real space. Skew mirror 610 is characterized by reflective axis 638 at angle −13° 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° measured with respect to the z axis. Principal reflected light 627 is reflected with internal reflection angle 180° measured with respect to the z axis.
FIG. 6B illustrates the skew mirror 610 of FIG. 6A in k-space. Line segment-like Δn() k-space distribution 688 passes through the origin, and has an angle of −13° 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.
FIG. 6C illustrates a highly magnified portion of FIG. 6B showing the left intersection between recording k-sphere 670 and line segment-like Δn() k-space distribution 688 according to an embodiment. In this view, line segment-like Δn() 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 kz 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 Δn() 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.
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° 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 (Ep()). Polarization density distribution 680 is given by the convolution Δn()*Ep(), which resembles line segment-like Δn() 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° measured with respect to the z axis.
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.
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.
The isosceles triangle construction of FIG. 6D obtains whenever Δn() 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 Δn() 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.
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° measured with respect to the z axis. Principal reflected light with wave vector 777A is reflected with internal propagation angle −171° 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.
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° measured with respect to the z axis. Principal reflected light having diffracted beam wave vector 777 is reflected with internal propagation angle −171° 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° 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.
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
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.
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
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° at 405 nm down to 20° at 650 nm may require 183 individual holograms in a 200 μm 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
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.
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.
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.
FIGS. 10A and 10B illustrate an angle filter embodiment of a skew mirror. In FIG. 10A, a Δn() 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 θ (i.e., near normal incidence) angular components of narrow band incident beam Einc, into reflected beam Er, while transmitting high θ angular components in Et. 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 Δn() 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
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.
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.
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.
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 μm 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 μm 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.
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. Because 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.
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.
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 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. =









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.
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.
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 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. =









The first wavelength (λ1=532 nm) differs from the second wavelength (λ2=513 nm) by 19 nm, which can be represented by a value referred to as a wave fraction (WF), defined as WF=|λ1−λ2|/[(λ1+λ2)/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≧0.57. Embodiments include, but are not limited to, variations in which WF≧0.005; WF≧0.010; WF≧0.030; WF≧0.10; WF≧0.250; WF≧1.0; or WF≧2.0. The wave fraction (WF) defined by a first (λ1) and a second (λ2) wavelength in the range may, but does not necessarily, includes a continuous spectrum of wavelengths between λ1 and λ2.
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.
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.
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.
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°, 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.
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.
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
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.
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.
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 μm 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.
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.
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.
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 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. =









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.
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 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. =









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.
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≧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.
For the second embodiment skew mirror, angles of incidence of the first incident light vary from −17.250 degrees to −23.946 degrees relative to the first reflective axis. Angles of incidence of the second incident light relative to the second reflective axis vary from −17.250 degrees to −23.940 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 17.20 degrees. For the second embodiment skew mirror, angles if incidence and angles of reflection relative to reflective axis, for incident light and its reflection, respectively, are tabulated in Tables 3 and 4.
Second embodiment external angles relative to surface normal for incident light and its reflection are also illustrated in FIGS. 12A and 12B. As seen in FIG. 12A, external angles relative to surface normal for first incident light 1224A, 1224B ranges from first incident light external angle 1213A of −14.000 degrees to first incident light external angle 1213B of −4.000 degrees. As seen in FIG. 12A, external angles relative to surface normal for second incident light 1230A, 1230B ranges from second incident light external angle 1215A of −14.000 to second incident light external angle 1215B of −4.000 degrees. First reflected light external angles 1214A, 1214B and second reflected light external angles 1216A, 1216B are also illustrated in FIGS. 12A and 12B, respectively.
Persons skilled in the art will recognize that incident light and its reflection can typically be reversed, such that what was previously an angle of reflection becomes and angle of incidence, and vice versa. However, for purposes of the present disclosure, recitation or description of a range of incidence angles refers to incident light oriented to one side or the other of a reflective axis, but not both, or in the case of retroreflected incident light, an incidence angle of zero (0) relative to the reflective axis. Accordingly, a range of incidence angles does not include angles that are both positive and negative with respect to the reflective axes. As illustrated and described here, incidence angles relative to their respective reflective axes are negative (i.e. in a clockwise direction). However, this convention is used for convenience and simplicity and is not meant to teach, suggest, or imply that a skew mirror can only reflect incident light residing to one side of a reflective axis.
A Third Embodiment Skew Mirror
A third embodiment skew mirror comprises a grating structure residing in a grating medium, wherein the grating structure comprises twenty one (21) volume holograms that overlap each other in the grating medium.
The third embodiment grating medium is a commercial photosensitive polymeric optical recording medium, designated BAYFOL® HX TP photopolymer film, available from Covestro AG (formerly Bayer MaterialScience AG) (Leverkusen, Germany) The BAYFOL® HX TP recording medium of the third embodiment is approximately 70 μm thick, and typically shrinks about 1.0% as a result of recording volume holograms. Accordingly, shrinkage compensation is typically employed when recording volume holograms in the third embodiment grating medium. Shrinkage compensation is described below in the method of making the third embodiment skew mirror.
Variations of the third embodiment skew mirror may include an additional layer such as a glass cover or glass substrate. The additional layer is typically refractive index matched to the grating medium, and a thin film of index matching fluid may reside between the third embodiment grating medium and the additional layer.
The grating structure of the third embodiment has the physical property of being configured to reflect a first incident light about a first reflective axis. The first incident light has a first wavelength of 532 nm and is incident upon the grating medium at a specific site. The first reflective axis differs from surface normal of the grating medium by a first reflective axis angle of +9.419 degrees (internal) relative to surface normal, where the first incident light has an internal angle, relative to surface normal, residing between −6.251 degrees and +0.334 degrees, inclusive (a range of 6.585 degrees). The internal angle of the first incident light includes multiple angles spanning a range of approximately 6.59 degrees, the multiple angles including one hundred (100) different internal angles spaced at angle intervals of approximately 0.067 degrees, from −6.251 degrees to +0.334 degrees.
Third embodiment first incident light having an internal angle of −6.251 degrees relative to surface normal, is reflected by the grating structure as first reflected light having an internal angle of +25.027 degrees relative to surface normal. First incident light having an internal angle relative to surface normal of +0.334 degrees is reflected as first reflected light having an internal angle of +18.487 degrees. First reflected light has the first wavelength, i.e. in the third embodiment the first reflected light has a wavelength of 532 nm.
The grating structure of the third embodiment is further configured to reflect second incident light about a second reflective axis. The second incident light has a second wavelength of 513 nm, and the second wavelength therefor 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 at the specific site. The second reflective axis differs from surface normal of the grating medium by a second reflective axis angle of +9.400 degrees (internal) relative to surface normal, where the second incident light has in internal angle, relative to surface normal, spanning a range from −6.251 degrees to +0.334 degrees. The internal angle of the second incident light includes one hundred (100) different internal angles spaced at angle intervals of approximately 0.067 degrees, from −6.251 degrees to +0.334 degrees.
Third embodiment second incident light, having an internal angle of −6 . . . 251 degrees relative to surface normal, is reflected by the grating structure as second reflected light, having an internal angle of +24.967 degrees relative to surface normal. Second incident light having an internal angle relative to surface normal of +0.334 degrees is reflected as second reflected light having an internal angle of +18.425 degrees. Second reflected light has the second wavelength, i.e. in the third embodiment the second reflected light has a wavelength of 513 nm. The second reflective axis of the third embodiment is substantially coincident with the first reflective axis.
Tables 5 summarizes reflective properties of first, second, and third embodiment skew mirrors.







TABLE 5







DIFFERENCE BETWEEN REFLECTIVE AXIS


ANGLES AT λ = 532 nm AND λ = 513 nm











FIRST EMBODIMENT
SECOND
THIRD EMBODIMENT



SKEW MIRROR
EMBODIMENT SKEW
SKEW MIRROR



(AK174-200 recording
MIRROR (AK233-200
(BAYFOL ® HX



medium)
recording medium)
recording medium)



N = 100
N = 101
N = 100



measurements
measurements
measurements













Mean reflective axis
13.693°
14.617°
9.400°


INTERNAL angle at λ =





532 nm*





Mean reflective axis
13.759°
14.618°
9.419°


INTERNAL angle at λ =





513 nm*





Difference between
0.066°
0.0005°
0.018°


reflective axis INTERNAL





angle at λ = 532 nm and





at λ = 513 nm**





Incident Light INTERNAL
−4.660° to +1.933°
−9.281° to −2.665°
−6.251° to +0.334°


Angles***
(range = 6.593°)
(range = 6.616°)
(range = 6.585°)


Mean reflective axis
22.234°
25.594°
14.720°


EXTERNAL angle at λ =





532 nm*





Mean reflective axis
22.110°
25.593°
14.690°


EXTERNAL angle at λ =





513 nm*





Difference between
0.124°
0.0005°
0.030°


reflective axis EXTERNAL





angle at λ = 532 nm and





at λ = 513 nm**





Incident Light EXTERNAL
−7.000° to 2.900°
−14.000° to −4.000°
−9.400° to +0.501°


Angles***





*mean angles are relative to surface normal, and are the means of N measurements at N incident light angles of incidence; both incident and reflected light have the specified wavelength (λ).


**differences between mean reflective axis angles at λ = 532 nm and at λ = 513 nm are absolute values and thus excludes negative numbers.


***incident light angles of incidence, relative to surface normal.






The range of angles of incidence across which a reflective axis remains constant can by expressed in terms of ΔθB. As shown in Table 6 below, the reflective axis angle for the first embodiment skew mirror varies by less than 0.015 degree for incident light having a range of angles of incidence of ≧20×ΔθB, at multiple wavelengths that differ from each other by WF≧0.036. For the second embodiment skew mirror the reflective axis angle varies by less than 0.020 degree for incident light having a range of angles of incidence ≧20×ΔθB, at multiple wavelengths that differ from each other by WF≧0.036.







TABLE 6







CHANGE IN REFLECTIVE AXIS ANGLES ACROSS


AN INCIDENCE ANGLE RANGE OF APPROXIMATELY 20 × ΔθB











Skew Mirror

Difference In reflective
Incident Light



Embodiment
λ*
Axis Angles**
Angle Range***
ΔθB†





FIRST
532 nm
0.012°
−3.198° to +0.400°
0.177°


EMBODIMENT


(3.598° = 20.3 × ΔθB)



SKEW MIRROR






(AK174-200
513 nm
0.012°
−3.198° to +0.400°
0.171°


recording medium)


(3.598° = 21.0 × ΔθB)



SECOND
532 nm
0.019°
−7.242° to −4.726°
0.126°


EMBODIMENT


(2.516° = 20.0 × ΔθB)



SKEW MIRROR






(AK233-200
513 nm
0.016°
−7.242° to −4.726°
0.122°


recording medium)


(2.516° = 20.6 × ΔθB)





*wavelength of both incident and reflected light.


**difference in reflective axis angles (internal, relative to surface normal) for incident light having a change in angle of incidence of approximately 20 × ΔθB.


***range of incident light angles of incidence (internal, relative to surface normal) approximately equal to 20 × ΔθB, for which the Difference In Reflective Axis Angles is reported in this table.


†ΔθB is calculated for an incident light angle of incidence at the midpoint of the Incident Light Angle Range reported in this table.






Methods of Making a Skew Mirror

An exemplary system 1350 for making a skew mirror is illustrated in FIG. 13. The exemplary system 1350 includes a grating medium 1310 disposed between a first mirror 1352A and a second mirror 1352B. The first and second mirrors are arranged to direct a first recording beam 1354 and a second recording beam 1355 such that the recording beams intersect and interfere with each other to form an interference pattern that is recorded as a hologram 1305 in the grating medium 1310. The hologram 1305 is an example of a grating structure.
The recording beams may be referred to as a reference beam and a signal beam according to a convention sometimes used by persons skilled in the holographic arts. However, each of the first and second recording beams are typically monochromatic collimated plane wave beams that are identical to each other (except for angles at which they are incident upon the grating medium). Moreover, the so-called signal beam typically includes no data encoded therein that is not also present in the so-called reference beam. Thus designation of one recording beam as a signal beam and the other recording beam as a reference beam can be arbitrary, with the designation of “signal” and “reference” serving to distinguish between the two recording beams, rather than to indicate that the one recording beam includes encoded data not present in the other recording beam.
In some embodiments the recording beams may have widths that differ from each other, or they may be the same. The recording beams may each have the same intensity as each other, or intensity can differ among the beams. The grating medium 1310 is typically secured in place between a first prism 1359A and second prism 1359B using a fluid index matched to both the prisms and the grating medium. A skew axis 1361 resides at a skew angle 1364 relative to surface normal 1322. The first and second recording beams 1354, 1355 reside at a first recording beam internal angle 1356 and a second recording beam internal angle 1357, respectively, relative surface normal 1322. Beam difference angle (α) 1358 is an angle of the first and second recording beams 1354, 1355, relative to each other. In embodiments, a resides in a range from 0 to 180 degrees. The skew angle 1364 for each hologram can be calculated according to equation (9),

θskew=(θR1+θR2−180°)/2  (9)

where: θskew is the skew angle, i.e., the internal angle of the skew axis relative to surface normal;

θR1 is the first recording beam internal angle relative to surface normal; and
θR2 is the second recording beam internal angle relative to surface normal.
As can be seen in FIG. 13, the first and second recording beams 1354, 1355 are symmetrical about the skew axis 1361 such that the first recording beam internal angle relative to the skew axis 1366 plus the second recording beam internal angle relative to the skew axis 1367 is equal to 180 degrees. The internal angles of the first and second recording beams relative to the skew axis 1366, 1367 are readily calculated from the first and second recording beam internal angles 1356, 1357, respectively, and the skew angle 1364.
Each of the first and second recording beams are typically collimated plane wave beams originating from a laser light source. The plane wave beams may be illustrated using multiple light ray depictions for each recording beam. For clarity however, in FIG. 13 the first and second recording beams are illustrated using a single light ray depiction for each recording beam.
Refraction at air/prism boundaries, for example where the first recording beam 1354 intersects an air/prism boundary of the first prism 1359A and where the second recording beam 1355 intersects an air/prism boundary of the second prism 1359, is shown figuratively rather than strictly quantitatively in FIG. 13. Because the prisms are typically index matched to the grating medium 1310, refraction at the prism/grating medium boundary can usually be ignored. In embodiments, the grating medium and prisms each have an index of refraction of approximately 1.50.
A skew angle for a hologram (including a mean skew angle for a collection of holograms) can be substantially identical to a reflective axis angle, meaning the skew angle or mean skew angle is within 1.0 degree of the reflective axis angle. Persons skilled in the art will recognize that the skew angle and reflective axis angle can be theoretically identical. However, due to limits in system precision and accuracy, shrinkage of recording medium that occurs during recording holograms, and other sources of error, the skew angle or mean skew angle as measured or estimated based on recording beam angles may not perfectly match the reflective axis angle as measured by incidence angles and reflection angles of light reflected by a skew mirror. Nevertheless, a skew angle determined based on recording beam angles can be within 1.0 degree of the reflective axis angle determined based on angles of incident light and its reflection, even where medium shrinkage and system imperfections contribute to errors in estimating skew angle and reflective axis angle. A skew axis/reflective axis is generally called a skew axis when referring to making a skew mirror (for example when describing recording a hologram in a skew mirror grating medium), and as a reflective axis when referring to light reflective properties of a skew mirror.
Angles at which the first and second recording beams 1354, 1355 are incident upon the grating medium are adjusted by rotating the first and second beam mirrors, 1352A, 1352B, respectively. Rotation of the beam mirrors, indicated by rotation arrows 1353, not only adjusts incidence angles, but would also change where in the grating medium 1310 the recording beams interfere with each other. Accordingly, when beam mirrors are rotated in order to adjust incidence angles, the grating medium 1310 and prisms 1359A, 1359B are moved translationally in order to record new holograms at approximately the same location in the grating medium as previously recorded holograms. Translation of the grating medium 1310 is indicated by translation arrow 1360.
In a variation of the exemplary system 1350, a variable wavelength laser is used to vary the wavelength of the first and second recording beams. Incidence angles of the first and second recording beams may be, but are not necessarily, held constant while the wavelength of the first and second recording beams is changed.
A First Method of Making a Skew Mirror
A first method of making a skew mirror is illustrated in FIG. 14. The skew mirror of the first method is the first embodiment skew mirror 1100, which is also illustrated in FIGS. 11A and 11B, and whose reflective properties are described above. The first method typically utilizes a system for making a skew mirror such as the exemplary system 1350 illustrated in FIG. 13 and described above. For clarity however, in FIG. 14 first and second prisms are omitted, and recording beams are illustrated without showing refraction at air/grating medium boundaries or air/prism boundaries. However, persons skilled in the art will recognize that refraction typically occurs at an air/prism boundary (or air/grating medium boundary, where index matched prisms are not used), and should be accounted for when designing a system or method to achieve the internal angles described.
A first recording beam 1154 and a second recording beam 1155 are directed at the first embodiment grating medium 1110, where the recording beams interfere with each other to create an interference pattern, which is recorded as a volume hologram in the grating medium 1110. The recording beams are typically created by splitting a 405 nm light beam from an external cavity, tunable diode laser into two separate beams. The light beam is split using a polarizing beam splitter, and a half wave plate is used to alter polarity of one of the two separate beams from p-polarized to s-polarized, such that both of the two separate beams are s-polarized. One of the s-polarized beams becomes the first recording beam 1154 and the other of the s-polarized beams becomes the second recording beam 1155. Each of the first and second recording beams is a collimated, plane wave beam having a wavelength of 405 nm.
The first embodiment skew mirror benefits from having reflective properties that allow it to reflect light at a substantially different wavelength, and in particular a considerably longer wavelength, than the recording beam wavelength. The AK174-200 grating medium, in which first embodiment holograms are recorded with 405 nm wavelength recording beams, absorbs 405 nm light at approximately 0.07 absorbance units for the 200 μm thick medium. Conversely, the AK174-200 grating medium has negligible absorbance for visible wavelengths of light greater than 425 nm (conservatively estimated at less than 0.002 absorbance units per 200 μm; the negligible absorbance is typically indistinguishable from zero). Thus the AK174-200 grating medium absorbs recording beam light (at 405 nm) at least 35 times more strongly than green light (for example, in a range of 503 nm to 537 nm) the first embodiment skew mirror is configured to reflect.
The grating structure 1105 of the first embodiment skew mirror 1100 is created by recording 48 volume holograms in the grating medium 1110. Each of the 48 holograms is recorded at its own unique first recording beam internal angle 1156 and its own unique second recording beam internal angle 1157. The first recording beam internal angle 1156 is an internal angle of the first recording beam 1154 relative to surface normal 1122 of the grating medium 1110 and the second recording beam internal angle 1157 is an internal angle of the second recording beam 1155 relative to surface normal 1122. Beam difference angle (α) 1158 is an angle of the first and second recording beams 1154, 1155, relative to each other.
Each of the first and second recording beams for the first embodiment skew mirror has irradiance of approximately 3 mW/cm2. Typically, the first of the 48 holograms is recorded with an energy dose of 35 mJ/cm2, and the dose is increased by about 1.5% for each subsequent hologram. The total energy dose for recording all 48 holograms is typically about 2.5 J/cm2. Irradiance and energy doses described here are merely exemplary. Other embodiments of skew mirrors and methods of making skew mirrors may use different levels of irradiance and energy dose.
A first hologram is recorded using a first recording beam internal angle 1156 of +53.218 degrees and a second recording beam internal angle 1157 of +154.234 degrees, resulting in a beam difference angle (α) 1158 of 101.016 degrees. The skew axis 1161 for each of the 48 holograms has a skew angle 1164 of +13.726 degrees relative to surface normal 1122, and the mean skew angle for the 48 holograms is therefore also +13.726 degrees. The skew angle for each hologram is calculated according to equation (9) above. For each subsequent hologram of the grating structure, the first and second recording beam internal angles 1156, 1157 are typically changed by amounts that are approximately equal in magnitude to each other but having opposite signs, which maintains symmetry of the first and second recording beams about the skew axis.
For example, for a second hologram, the first recording beam internal angle is changed by +0.091 degree and the second recording beam internal angle is adjusted by −0.091 degree, such that the first recording beam internal angle 1156 becomes +53.309 degrees and the second recording beam internal angle +154.143 degrees; α=100.834 degrees. The magnitudes of changes in recording beam internal angles from one hologram to the next hologram vary slightly across the 48 volume holograms (i.e., the change in recording beam internal angles from one hologram to the next varies), from 0.091 degree for changes in recording beam internal angles from the first hologram to the second hologram, to 0.084 degree for changes in recording beam internal angles from the 47th hologram to the 48th hologram. However, for each change of first and second recording beam internal angles, the magnitude of change is the same and the sign is opposite for each of the first and second beam angles. The first and second recording beam internal angles 1156, 1157 for the last (48th) hologram of the first embodiment grating structure 1105 are +57.332 and +150.120 degrees, respectively, and α=92.788 degrees. In some embodiments, the magnitude of change of the first recording beam internal angle may differ very slightly from the magnitude of change of the second recording beam internal angle, which may compensate for system imprecision, for Snell effects, for dispersion, or for shrinkage of the grating medium that results from recording the holograms.
Position of the grating medium 1110 is adjusted (as indicated by translation arrow 1160) between recording one hologram and the next hologram such that at least part of each of the 48 holograms is recorded in a common space in the grating medium shared with at least part of all others of the 48 holograms. Accordingly, each of the 48 holograms at least partially spatially overlaps all others of the 48 holograms in the grating medium.
The first recording beam internal angle 1156 ranges from +53.218 to +57.332 degrees (a range of 4.114 degrees) and the second recording beam internal angle 1157 ranges from +154.234 to +150.120 degrees (a range of 4.114 degrees). As can be seen in FIG. 14, for each hologram of the first method, the first and second recording beams 1154, 1155 are symmetrical about the skew axis 1161 such that the internal angle of the first recording beam relative to the skew axis 1166 (+38.492 degrees for the first hologram) plus the internal angle of the second recording beam relative to the skew axis 1167 (+141.508 degrees for the first hologram) is equal to 180 degrees (38.492°+141.508°=180°). The internal angles of the first and second recording beams relative to the skew axis 1166, 1167 are readily calculated from the first and second recording beam internal angles 1156, 1157, respectively, and the skew angle 1164. First and second recording beam internal angles (which are defined as internal angles relative to surface normal of the grating medium) and internal angles relative to the skew axis of the first and second recording beams are listed in Table 7. After recording the 48 volume holograms, the AK174-200 recording medium is light cured by a process familiar to persons skilled in the art. [need more here]
Beam difference angle α can be used to calculate KG according to equations (10) and (11).




  
    
      
        
          
            
              K
              G
            
            ≡
          
          |
          
            
              K
              ⇀
            
            G
          
          |
        
        =
        
          
            2
            ⁢
            
              
                k
                2
              
              ⁡
              
                (
                
                  1
                  -
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    α
                  
                
                )
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  
  
    
      
        
          
            k
            ≡
          
          |
          
            
              k
              ⇀
            
            1
          
          |
        
        =
        
          
            |
            
              
                k
                ⇀
              
              2
            
            |
          
          =
          
            
              2
              ⁢
              π
              ⁢
              
                
              
              ⁢
              n
            
            λ
          
        
      
    
    
      
        (
        11
        )
      
    
  




where: G is a grating vector in k-space for a hologram, comprising both G+ and G−;

    
    
        {right arrow over (k)}1 and {right arrow over (k)}2 are wave vectors in k-space for first and second recording beams, respectively, used for recording the hologram;
        α is the beam difference angle as described above;
        λ is the vacuum wavelength of light of the first and second recording beams; and
        n is the mean refractive index of the hologram (referred to as the “bulk index,” for which the refractive index of the grating medium in which the hologram is recorded is used as an approximation).
    
    


A more detailed explanation of G and {right arrow over (k)} is found above in a section of this specification titled k-Space Formalism for Holography.
For the AK174-200 grating medium used for the first embodiment skew mirror, n is estimated at 1.50 for light at 405 nm. The wavelength of the first and second recording beams used to record holograms for making the first embodiment skew mirror is 405 nm=405×10−9 m. Accordingly, k=2.327×107 radians per meter (rad/m) for the first and second recording beams in AK174-200 grating medium. KG for the holograms of the first embodiment skew mirror ranges from 3.592×107 rad/m for the first hologram to 3.370×107 rad/m for the 48th hologram.
The absolute value of the difference in grating vectors |ΔKG| between any two holograms can be a useful metric for describing hologram “spacing” (i.e. how close to each other in k-space are grating vectors for the any two holograms). |ΔKG| for each first embodiment hologram and its adjacent hologram(s) is relatively constant, with a mean value for all 48 holograms of 4.715×104 rad/m and a coefficient of variation of 0.11%. The adjacent hologram(s) for each hologram consist of a hologram or holograms having the next highest or the next lowest KG compared to KG for the each hologram. |ΔKG| for each first embodiment hologram and its adjacent hologram(s) reside in a range between 4.70×104 and less than 4.73×104 rad/m. |ΔKG| between the first and 48th holograms is 2.22×106 rad/m.
In skew mirror embodiments, |ΔKG| between a hologram and an adjacent hologram (which can be referred to as adjacent |ΔKG|) has a mean value for multiple holograms that typically, but not necessarily, resides in a range between 5.0×103 and 1.0×107 rad/m, more typically in a range between 1.0×104 and 5×106 rad/m, and more typically still in a range between 1.0×104 and 1.0×106. In some embodiments, mean adjacent |ΔKG| for multiple holograms resides in a range between 8.0×104 and 5.0×106 rad/m, and may reside in a range between 1.0×105 and 1.0×106 rad/m.
In some embodiments, mean adjacent |ΔKG| for multiple holograms strongly influences skew mirror performance Relatively small mean adjacent |ΔKG| for a set of holograms can correspond to relatively high skew mirror image fidelity. However, where adjacent |ΔKG| for a set of holograms is relatively small, the total number of holograms in the set is larger in order to span a given |ΔKG| range for the hologram set. Moreover, given that recording capacity for grating mediums is typically limited by dynamic range (usually expressed as M/#), recording more holograms in a set usually means that each hologram in the set is weaker (i.e., is recorded more faintly in the medium). Accordingly, tension exists between having relatively small adjacent |ΔKG| for a hologram set (which requires more holograms, other things being equal), and having a larger adjacent |ΔKG| for the set, which enables recording fewer, but stronger holograms. Fewer, stronger holograms typically results in stronger reflectance by a skew mirror. Furthermore, the use of relatively broadband illumination sources (e.g., LEDs instead of lasers) may reduce image fidelity loss in a skew mirror with a larger mean adjacent |ΔKG|. In some embodiments, a sweet spot exists where mean adjacent |ΔKG| for multiple holograms resides in the range between 5.0×103 rad/m and 1.0×107 rad/m. Embodiments of skew mirrors where mean adjacent |ΔKG| resides in the sweet spot typically exhibit a desirable balance of image fidelity and reflectance.
Values for a, KG, and |ΔKG| for each of the 48 holograms of the first embodiment skew mirror can be found in Table 7.







TABLE 7







RECORDING BEAM ANGLES FOR A FIRST METHOD OF MAKING A SKEW


MIRROR; SKEW ANGLE = 13.726° RELATIVE TO SURFACE NORMAL


















Internal
Internal



|ΔKG|



First
Second
Angle of
Angle of
Magnitude


Between



Recording
Recording
First
Second
of Angle


Hologram



Beam
Beam
Recording
Recording
Difference


and Its



Angle
Angle
Beam
Beam
From


Preceding















(internal, relative to
Relative To
Relative To
Previous


Adjacent



surface normal, in
Skew Axis
Skew Axis
Hologram
α
KG
Hologram


#
degrees)
(degrees)
(degrees)
(degrees)
(degrees)
(×107 rad/m)
(×104 rad/m)


















1
53.218
154.234
39.492
140.508

101.016
3.592



2
53.309
154.143
39.583
140.417
0.091
100.833
3.587
4.709


3
53.400
154.052
39.674
140.326
0.091
100.652
3.582
4.704


4
53.491
153.961
39.765
140.235
0.091
100.470
3.578
4.713


5
53.581
153.871
39.855
140.145
0.091
100.289
3.573
4.708


6
53.672
153.780
39.946
140.054
0.090
100.109
3.568
4.709


7
53.762
153.690
40.036
139.964
0.090
99.928
3.563
4.704


8
53.852
153.600
40.126
139.874
0.090
99.748
3.559
4.713


9
53.942
153.510
40.216
139.784
0.090
99.568
3.554
4.707


10
54.031
153.421
40.305
139.695
0.090
99.389
3.549
4.709


11
54.121
153.331
40.395
139.605
0.090
99.210
3.545
4.710


12
54.210
153.242
40.484
139.516
0.089
99.031
3.540
4.712


13
54.300
153.152
40.574
139.426
0.089
98.853
3.535
4.713


14
54.389
153.063
40.663
139.337
0.089
98.674
3.530
4.707


15
54.478
152.974
40.752
139.248
0.089
98.496
3.526
4.715


16
54.567
152.885
40.841
139.159
0.089
98.319
3.521
4.710


17
54.655
152.797
40.929
139.071
0.089
98.142
3.516
4.711


18
54.744
152.708
41.018
138.982
0.089
97.965
3.512
4.719


19
54.832
152.620
41.106
138.894
0.088
97.788
3.507
4.712


20
54.920
152.532
41.194
138.806
0.088
97.612
3.502
4.713


21
55.008
152.444
41.282
138.718
0.088
97.436
3.497
4.714


22
55.096
152.356
41.370
138.630
0.088
97.260
3.493
4.715


23
55.184
152.268
41.458
138.542
0.088
97.085
3.488
4.708


24
55.271
152.181
41.545
138.455
0.088
96.909
3.483
4.716


25
55.359
152.093
41.633
138.367
0.087
96.734
3.479
4.717


26
55.446
152.006
41.720
138.280
0.087
96.560
3.474
4.717


27
55.533
151.919
41.807
138.193
0.087
96.386
3.469
4.710


28
55.620
151.832
41.894
138.106
0.087
96.211
3.464
4.718


29
55.707
151.745
41.981
138.019
0.087
96.038
3.460
4.718


30
55.794
151.658
42.068
137.932
0.087
95.864
3.455
4.718


31
55.881
151.571
42.155
137.845
0.087
95.691
3.450
4.718


32
55.967
151.485
42.241
137.759
0.086
95.518
3.446
4.711


33
56.053
151.399
42.327
137.673
0.086
95.346
3.441
4.719


34
56.139
151.313
42.413
137.587
0.086
95.173
3.436
4.719


35
56.225
151.227
42.499
137.501
0.086
95.001
3.431
4.718


36
56.311
151.141
42.585
137.415
0.086
94.829
3.427
4.718


37
56.397
151.055
42.671
137.329
0.086
94.658
3.422
4.718


38
56.483
150.969
42.757
137.243
0.086
94.486
3.417
4.718


39
56.568
150.884
42.842
137.158
0.086
94.315
3.413
4.725


40
56.654
150.798
42.928
137.072
0.085
94.145
3.408
4.717


41
56.739
150.713
43.013
136.987
0.085
93.974
3.403
4.717


42
56.824
150.628
43.098
136.902
0.085
93.804
3.398
4.724


43
56.909
150.543
43.183
136.817
0.085
93.634
3.394
4.723


44
56.994
150.458
43.268
136.732
0.085
93.464
3.389
4.715


45
57.079
150.373
43.353
136.647
0.085
93.295
3.384
4.722


46
57.163
150.289
43.437
136.563
0.085
93.126
3.380
4.721


47
57.248
150.204
43.522
136.478
0.085
92.956
3.375
4.728


48
57.332
150.120
43.606
136.394
0.084
92.788
3.370
4.719









MEAN
4.715









STDEV
0.0054









% CV
0.11









In a variation of the first method of making a skew mirror, a hologram is created by continuously and synchronously adjusting the first and second recording beam internal angles while maintaining the symmetry of the first and second recording beams about the skew axis as described above. Accordingly, a single hologram is recorded while the first recording beam is scanned from a first recording beam internal angle of +53.218 degrees to a first recording beam angle of +57.332 degrees. Simultaneously, the second recording beam is scanned from a second recording beam internal angle of +154.234 degrees to +150.120 degrees. Accordingly, a varies from 101.016 degrees to 92.788 degrees and KG varies from 3.592×107 rad/m to 3.370×107 rad/m while the single hologram is being recorded. Position of the grating medium is adjusted while the single hologram is being recorded such that the single hologram is recorded in a relatively compact space in the grating medium, rather than being smeared across a relatively broad space as the recording beam angles change. The single hologram thus exhibits reflective properties very similar to the 48 discrete holograms recorded with 48 sets of unique first recording beam and second recording beam internal angles, and the total energy dose for recording the single hologram is typically about the same (2.5 J/cm2) as for the 48 holograms.
A Second Method of Making a Skew Mirror
A second method of making a skew mirror is described below. The skew mirror made by the second method is the second embodiment skew mirror 1200, which is also illustrated in FIGS. 12A and 12B, and whose reflective properties are described above.
The second method is identical to the first method except that first and second recording beam internal angles are different than with the first method, which gives the second embodiment skew mirror reflective properties that differ from those of the first embodiment. In the present example, the second method is performed using a grating medium (AK233-200) that differs from that of the first method (AK174-200). Like the first embodiment, the second embodiment skew mirror benefits from having reflective properties that allow it to reflect light at a substantially different wavelength, and in particular a considerably longer wavelength, than the recording beam wavelength.
The grating structure 1205 of the second embodiment skew mirror 1200 is created by recording 49 volume holograms in the grating medium 1210. Each of the 49 holograms of the second method is recorded at its own unique first recording beam internal angle and its own unique second recording beam internal angle. The first recording beam internal angle is an internal angle of the first recording beam relative to surface normal of the grating medium and the second recording beam internal angle is an internal angle of the second recording beam relative to surface normal. Each of the first and second recording beams for the first embodiment skew mirror has irradiance of approximately 3 mW/cm2. Typically, the first of the 49 holograms is recorded with an energy dose of 35 mJ/cm2, and the dose is increased by about 1.5% for each subsequent hologram. The total dose for recording all 49 holograms is typically about 2.5 J/cm2.
According to the second method, a first hologram is recorded using a first recording beam internal angle of +55.913 degrees and a second recording beam internal angle of +153.323 degrees; a therefore is 97.410 degrees. The skew axis for each of the 49 holograms has a skew angle of +14.618 degrees relative to surface normal. The skew angle for each hologram is calculated according to equation (9) above. For each subsequent hologram of the grating structure, the first and second recording beam internal angles are typically changed by amounts that are approximately equal in magnitude to each other, but having opposite signs, which maintains symmetry of the first and second recording beams about the skew axis.
For example, for recording a second hologram according to the second method, the first recording beam internal angle is changed by +0.095 degree and the second recording beam internal angle is adjusted by −0.095 degree, such that the first recording beam internal angle becomes +56.008 degrees and the second recording beam internal angle +153.228 degrees; α=97.220 degrees. The magnitudes of changes in recording beam internal angles from one hologram to the next hologram typically vary slightly across the 49 volume holograms (i.e. the change in change in recording beam internal angles from one hologram to the next varies), from a magnitude of 0.095 degree for changes in recording beam internal angles from the first hologram to the second hologram, to a magnitude of 0.087 degree for changes in recording beam internal angles from the 48th hologram to the 49th hologram. However, the magnitude of change is the same for each of the first and second recording beam internal angles, and the sign of the change is opposite for each of the first and second recording beam internal angles. The first and second recording beam internal angles for the last (49th) hologram of the second embodiment grating structure are +60.252 and +148.984 degrees, respectively, and α=88.732. In some embodiments, the magnitude of change of the first recording beam internal angle may differ very slightly from the magnitude of change of the second recording beam internal angle in order to compensate for factors such as system imprecision, Snell effects, dispersion, or shrinkage of the grating medium that results from recording the holograms.
The position of the grating medium is adjusted between recording one hologram and the next such that at least part of each of the 49 holograms is recorded in a common space shared with at least part of all others of the 49 holograms. Accordingly, each of the 49 holograms at least partially spatially overlaps all others of the 49 holograms in the grating medium.
Thus according to the second method first recording beam internal angles range from +55.913 to +60.252 degrees (a range of 4.339 degrees) and the second recording beam internal angles range from +153.323 to +148.984 degrees (a range of 4.339 degrees). As with the first method, for each hologram of the second method the first and second recording beams are symmetrical about the skew axis such that the internal angle of the first recording beam relative to the skew axis (+41.295 degrees for the first hologram) plus the internal angle of the second recording beam relative to the skew axis (+138.705 for the first hologram)=180 degrees (41.295°+138.705°=180°). The internal angles of the first and second recording beams relative to the skew axis are readily calculated from the first and second recording beam internal angles relative to surface normal, respectively, and the skew angle. For the second method of making a skew mirror, first and second recording beam internal angles (which are defined as internal angles relative to surface normal of the grating medium) and internal angles relative to the skew axis for the first and second recording beams are listed in Table 8. After recording the 49 volume holograms, the AK233-200 recording medium is light cured by a process familiar to persons skilled in the art. For example, in some embodiments light curing comprises exposure to near-ultraviolet uniform coherent light from a light emitting diode, until substantially all photoinitiator, photoactive monomer, of other photoactive chemistry has been consumed.







TABLE 8







RECORDING BEAM ANGLES FOR A SECOND METHOD OF MAKING A


SKEW MIRROR; SKEW ANGLE = 14.618° RELATIVE TO SURFACE NORMAL


















Internal
Internal







First
Second
Angle of
Angle of
Magnitude



Recording
Recording
First
Second
of Angle



Beam
Beam
Recording
Recording
Difference



Angle
Angle
Beam
Beam
From















(internal, relative to
Relative To
Relative To
Previous






surface normal, in
Skew Axis
Skew Axis
Hologram
α
KG
ΔKG


#
degrees)
(degrees)
(degrees)
(degrees)
(degrees)
(×107 rad/m)
(×104 rad/m)


















1
55.913
153.323
41.295
138.705

97.410
3.497



2
56.008
153.228
41.390
138.610
0.095
97.220
3.492
5.098


3
56.102
153.134
41.484
138.516
0.094
97.032
3.487
5.053


4
56.196
153.040
41.578
138.422
0.094
96.844
3.482
5.063


5
56.290
152.946
41.672
138.328
0.094
96.656
3.477
5.072


6
56.384
152.852
41.766
138.234
0.094
96.468
3.471
5.081


7
56.477
152.759
41.859
138.141
0.093
96.282
3.466
5.037


8
56.571
152.665
41.953
138.047
0.094
96.094
3.461
5.100


9
56.664
152.572
42.046
137.954
0.093
95.908
3.456
5.055


10
56.757
152.479
42.139
137.861
0.093
95.722
3.451
5.064


11
56.849
152.387
42.231
137.769
0.092
95.538
3.446
5.019


12
56.942
152.294
42.324
137.676
0.093
95.352
3.441
5.082


13
57.034
152.202
42.416
137.584
0.092
95.168
3.436
5.036


14
57.127
152.109
42.509
137.491
0.093
94.982
3.431
5.100


15
57.219
152.017
42.601
137.399
0.092
94.798
3.426
5.054


16
57.311
151.925
42.693
137.307
0.092
94.614
3.421
5.063


17
57.402
151.834
42.784
137.216
0.091
94.432
3.416
5.017


18
57.494
151.742
42.876
137.124
0.092
94.248
3.411
5.081


19
57.585
151.651
42.967
137.033
0.091
94.066
3.406
5.034


20
57.676
151.560
43.058
136.942
0.091
93.884
3.401
5.043


21
57.767
151.469
43.149
136.851
0.091
93.702
3.396
5.051


22
57.858
151.378
43.240
136.760
0.091
93.520
3.391
5.060


23
57.949
151.287
43.331
136.669
0.091
93.338
3.385
5.068


24
58.040
151.196
43.422
136.578
0.091
93.156
3.380
5.077


25
58.130
151.106
43.512
136.488
0.090
92.976
3.375
5.029


26
58.220
151.016
43.602
136.398
0.090
92.796
3.370
5.038


27
58.310
150.926
43.692
136.308
0.090
92.616
3.365
5.046


28
58.400
150.836
43.782
136.218
0.090
92.436
3.360
5.054


29
58.490
150.746
43.872
136.128
0.090
92.256
3.355
5.063


30
58.579
150.657
43.961
136.039
0.089
92.078
3.350
5.015


31
58.669
150.567
44.051
135.949
0.090
91.898
3.345
5.079


32
58.758
150.478
44.140
135.860
0.089
91.720
3.340
5.031


33
58.847
150.389
44.229
135.771
0.089
91.542
3.335
5.039


34
58.936
150.300
44.318
135.682
0.089
91.364
3.330
5.047


35
59.025
150.211
44.407
135.593
0.089
91.186
3.325
5.055


36
59.113
150.123
44.495
135.505
0.088
91.010
3.320
5.006


37
59.202
150.034
44.584
135.416
0.089
90.832
3.315
5.071


38
59.290
149.946
44.672
135.328
0.088
90.656
3.310
5.022


39
59.378
149.858
44.760
135.240
0.088
90.480
3.305
5.030


40
59.466
149.770
44.848
135.152
0.088
90.304
3.300
5.037


41
59.554
149.682
44.936
135.064
0.088
90.128
3.295
5.045


42
59.642
149.594
45.024
134.976
0.088
89.952
3.290
5.053


43
59.730
149.506
45.112
134.888
0.088
89.776
3.285
5.061


44
59.817
149.419
45.199
134.801
0.087
89.602
3.280
5.011


45
59.904
149.332
45.286
134.714
0.087
89.428
3.275
5.018


46
59.991
149.245
45.373
134.627
0.087
89.254
3.270
5.026


47
60.078
149.158
45.460
134.540
0.087
89.080
3.264
5.033


48
60.165
149.071
45.547
134.453
0.087
88.906
3.259
5.041


49
60.252
148.984
45.634
134.366
0.087
88.732
3.254
5.048









MEAN
5.050









STDEV
0.0235









% CV
0.47









For the AK233-200 grating medium used for the second embodiment skew mirror, n is estimated at 1.50 for light at 405 nm. The wavelength of the first and second recording beams used to record holograms for making the second embodiment skew mirror is 405 nm=405×10−9 m. Accordingly, k=2.327×107 rad/m for the first and second recording beams in AK233-200 grating medium. KG for the holograms of the second embodiment skew mirror ranges from 3.497×107 rad/m for the first hologram to 3.254×107 rad/m for the 49th hologram.
Adjacent |ΔKG| for the second embodiment holograms is relatively constant, with a mean value for all 49 holograms of 5.050×104 rad/m and a coefficient of variation of 0.47%. Adjacent |ΔKG| for each second embodiment hologram resides in a range between 5.01×104 to 5.10×104 rad/m. |ΔKG| between the first and 49th holograms is 2.42×106 rad/m.
In a variation of the second method of making a skew mirror, a hologram is created by continuously and synchronously adjusting the first and second recording beam internal angles while maintaining the symmetry of the first and second recording beams about the skew axis as described above. Accordingly, a single hologram is recorded while the first recording beam is scanned from a first recording beam internal angle of +55.913 degrees to a first recording beam angle of +60.252 degrees. Simultaneously, the second recording beam is scanned from a second recording beam internal angle of +153.323 degrees to +148.984 degrees. The single hologram is thus equivalent to the 49 discrete holograms recorded with 49 sets of unique first recording beam and second recording beam internal angles. The total energy dose for recording the single hologram is typically 2.5 J/cm2 for the single hologram.
A Third Method of Making a Skew Mirror
A third method of making a skew mirror is described below. Like the first method, the third method typically utilizes a system for making a skew mirror such as the exemplary system 1350 illustrated in FIG. 13 and described above.
According to the third method, a grating structure is created by recording three sets of holograms in a grating medium. The first hologram set includes 21 holograms, the second hologram set includes 19 holograms, and the third hologram set includes 16 holograms, for a total of 56 holograms. In some embodiments, each of the first, second, and third holograms sets includes at least 6 holograms, or at least 9 holograms. Each of the multiple holograms in the first hologram set typically at least partially spatially overlaps at least one other of the multiple holograms in the first hologram set, and at least one of the multiple holograms in the first hologram set may at least partially spatially overlap at least one of the holograms in the second hologram set. In some embodiments, each of the multiple holograms in the first hologram set at least partially spatially overlaps all others of the multiple holograms in the first hologram set.
Similarly, each of the multiple holograms in the second hologram set typically at least partially spatially overlaps at least one other of the multiple holograms in the second hologram set, and at least one of the multiple holograms in the second hologram set may at least partially spatially overlap at least one of the holograms in the first hologram set or the third hologram set. In some embodiments, each of the multiple holograms in the second hologram set at least partially spatially overlaps all others of the multiple holograms in the second hologram set.
Similarly, each of the multiple holograms in the third hologram set typically at least partially spatially overlaps at least one other of the multiple holograms in the third hologram set, and at least one of the multiple holograms in the third hologram set may at least partially spatially overlap at least one of the holograms in the second hologram set. In some embodiments, each of the multiple holograms in the third hologram set at least partially spatially overlaps all others of the multiple holograms in the third hologram set. In some embodiments, all holograms of the first, second, and third hologram sets at least partially spatially overlap with each other.
Each of the 56 total holograms is recorded using first and second recording beams, each of which is incident upon the grating medium at its own unique first recording beam internal angle and its own unique second recording beam internal angle. In some embodiments, not every first and second recording beam internal angle is unique. For example, in some embodiments multiple holograms having the same recording beam internal angles as each other may be written in in locations in the skew mirror that differ from each other. The first recording beam internal angle is an internal angle of the first recording beam relative to surface normal of the grating medium and the second recording beam internal angle is an internal angle of the second recording beam relative to surface normal. Each of the first and second recording beams for the first embodiment skew mirror is a monochromatic collimated light beam having irradiance of approximately 3 mW/cm2. Typically, the first of the 56 holograms is recorded with an energy dose of 35 mJ/cm2, and the dose is increased by about 0.9% for each subsequent hologram. The total dose for recording all 56 holograms is typically about 2.5 J/cm2.
The first hologram set of the third method includes a first hologram recorded using a first recording beam internal angle of +43.519 degrees and a second recording beam internal angle of +163.882 degrees, resulting in a beam difference angle (α) of 120.363 degrees. The first and second recording beams of the first hologram are symmetrical about a skew axis having a skew angle of 13.700 degrees. For each subsequent hologram of first hologram set, the first and second recording beam internal angles are typically changed by amounts that are approximately equal in magnitude to each other, but having opposite signs. For example, for recording a second hologram of the first hologram set, the first recording beam internal angle is changed by +0.351 degree and the second recording beam internal angle is adjusted by −0.355 degree, such that the first recording beam internal angle becomes +43.870 degrees and the second recording beam internal angle +163.527 degrees (α=119.657 degrees). The first and second recording beams of the second hologram are symmetrical about a skew axis having a skew angle of 13.699 degrees. The magnitudes of changes in recording beam internal angles from one hologram to the next hologram typically vary slightly across the 21 volume holograms of the first hologram set (i.e. the change in change in recording beam internal angles from one hologram to the next varies), from a magnitude of approximately 0.353 degree for changes in recording beam internal angles from the first hologram to the second hologram, to a magnitude of approximately 0.299 degree for changes in recording beam internal angles from the 20th hologram to the 21st hologram. However, the magnitude of change is approximately the same for each of the first and second recording beam internal angles, and the sign of the change is opposite for each of the first and second recording beam internal angles. The first and second recording beam internal angles for the last (21′′) hologram of the first hologram set are +49.960 and +157.379 degrees, respectively, and α=107.419 degrees. The first and second recording beams of the 21st hologram are symmetrical about a skew axis having a skew angle of 13.670 degrees.
According to the third method, the first recording beam internal angle of the first hologram set ranges from +43.519 to +49.960 degrees (a range of 6.441 degrees) and the second recording beam internal angle of the first hologram set ranges from +163.882 to +157.379 degrees (a range of 6.503 degrees). For each hologram of the first hologram set, the first recording beam and its respective second recording beam are symmetrical about a skew axis. Thus the internal angle of the first recording beam relative to a skew axis (+29.819 degrees for the first hologram) plus the internal angle of the second recording beam relative to the skew axis (+150.182 degrees for the first hologram) is equal to 180.0 degrees (29.818°+150.182=180.0°). The internal angles of the first and second recording beams relative to the skew axis are readily calculated from the first and second recording beam internal angles respectively, and the skew angle. The mean skew angle for all holograms of the first hologram set is 13.685 degrees and all skew angles of the first set are within 0.015 degree of the mean. For the first hologram set of the third method of making a skew mirror, first and second recording beam internal angles relative to surface normal of the grating medium and internal angles relative to the skew axis are listed in Table 9.
For many skew mirror applications, all skew angles for a set of holograms are within 2.0 degrees of the mean skew angle for all holograms in the set, in order to achieve adequate reflective performance. In some skew mirror applications, all skew angles for a set of holograms are within 1.0 degree of the hologram set mean skew angle in order to achieve adequate reflective performance. For more demanding applications, all skew angles of a set of holograms are within 0.5 degree of the hologram set mean skew angle in order to achieve adequate reflective performance. For still more demanding applications, all skew angles of a set of holograms are within 0.10 degree of the hologram set mean skew angle in order to achieve adequate reflective performance. For especially demanding applications, all skew angles in a set of holograms are within 0.01 degree of the hologram set mean skew angle.
KG for the holograms of the first hologram set ranges from 4.140×107 rad/m for the first hologram to 3.846×107 rad/m for the 21st hologram, based on n=1.538 for 405 nm light and AK283 photosensitive grating medium, resulting in k=2.386×107 rad/m. The third method can be, but is not necessarily, practiced using the AK283 grating medium having a thickness of 500 μm. Adjacent |ΔKG| for each hologram of the first hologram set is 1.469×105 rad/m. |ΔKG| between the first and 21st holograms is 2.939×106 rad/m. Values for α, KG, and |ΔKG| for each of the 21 holograms of the first hologram set of the third method can be found in Table 9.
The second hologram set of the third method includes a first hologram recorded using a first recording beam internal angle of +53.704 degrees and a second recording beam internal angle of +153.696 degrees, resulting in a =99.992 degrees. The first and second recording beams of the first hologram are symmetrical about a skew axis having a skew angle of 13.700 degrees. For recording a second hologram of the second hologram set, the first recording beam internal angle is changed by +0.272 degree and the second recording beam internal angle is adjusted by −0.275 degree, such that the first recording beam internal angle becomes +53.976 degrees and the second recording beam internal angle +153.421 degrees (α=99.445 degrees). The first and second recording beams of the second hologram are symmetrical about a skew axis having a skew angle of 13.699 degrees. The magnitudes of changes in recording beam internal angles from one hologram to the next hologram typically vary slightly across the 19 volume holograms of the second hologram set (i.e. the change in change in recording beam internal angles from one hologram to the next varies), from a magnitude of approximately 0.274 degree for changes in recording beam internal angles from the first hologram to the second hologram, to a magnitude of 0.252 degree for changes in recording beam internal angles from the 18th hologram to the 19th hologram. However, the magnitude of change is approximately the same for each of the first and second recording beam internal angles, and the sign of the change is opposite for each of the first and second recording beam internal angles. The first and second recording beam internal angles for the last (19th) hologram of the second hologram set are +58.393 and +148.957 degrees, respectively, and α=90.564 degrees. The first and second recording beams of the 19th hologram are symmetrical about a skew axis having a skew angle of 13.675 degrees.
KG for the holograms of the second hologram set ranges from 3.655×107 for the first hologram to 3.391×107 for the 19th hologram (n=1.538; k=2.386×107). Adjacent |ΔKG| for each hologram of the second hologram set is 1.469×105. |ΔKG| between the first and 19th holograms is 2.645×106. Values for α, KG, and |ΔKG| for each of the 19 holograms of the second hologram set of the third method can be found in Table 9.
According to the third method, the first recording beam internal angle of the second hologram set ranges from +53.704 to +58.393 degrees (a range of 4.689 degrees) and the second recording beam internal angle of the second hologram set ranges from +153.696 to +148.597 degrees (a range of 4.736 degrees). For each hologram of the second hologram set, the first recording beam and its respective second recording beam are symmetrical about a skew axis. Thus the internal angle of the first recording beam relative to a skew axis (+40.004 degrees for the first hologram) plus the internal angle of the second recording beam relative to the skew axis (+139.996 degrees for the first hologram) is equal to 180.0 degrees (40.004°+139.996=180.0°). The internal angles of the first and second recording beams relative to the skew axis are readily calculated from the first and second recording beam internal angles respectively, and the skew angle. The mean skew angle for all holograms of the second hologram set is 13.688 degrees and all skew angles of the first set are within 0.013 degree of the mean. For the second hologram set of the third method of making a skew mirror, first and second recording beam internal angles relative to surface normal of the grating medium and internal angles relative to the skew axis are listed in Table 9.
The third hologram set of the third method includes a first hologram recorded using a first recording beam internal angle of +63.696 degrees and a second recording beam internal angle of +143.704 degrees, resulting in a =80.008 degrees. The first and second recording beams of the first hologram are symmetrical about a skew axis having a skew angle of 13.700 degrees. For recording a second hologram of the third hologram set, the first recording beam internal angle is changed by +0.229 degree and the second recording beam internal angle is adjusted by −0.231 degree, such that the first recording beam internal angle becomes +63.925 degrees and the second recording beam internal angle +143.473 degrees (α=79.548 degrees). The first and second recording beams of the first hologram are symmetrical about a skew axis having a skew angle of 13.699 degrees. The magnitudes of changes in recording beam internal angles from one hologram to the next hologram typically vary slightly across the 16 volume holograms of the third hologram set (i.e. the change in change in recording beam internal angles from one hologram to the next varies), from a magnitude of approximately 0.230 degree for changes in recording beam internal angles from the first hologram to the second hologram, to a magnitude of approximately 0.219 degree for changes in recording beam internal angles from the 18th hologram to the 19th hologram. However, the magnitude of change is approximately the same for each of the first and second recording beam internal angles, and the sign of the change is opposite for each of the first and second recording beam internal angles. The first and second recording beam internal angles for the last (16th) hologram of the third hologram set are +67.051 and +140.313 degrees, respectively, and a =73.262 degrees. The first and second recording beams of the 16th hologram are symmetrical about a skew axis having a skew angle of 13.682 degrees.
KG for the holograms of the third hologram set ranges from 3.068×107 for the first hologram to 2.847×107 for the 16th hologram (n=1.538; k=2.386×107). Adjacent |ΔKG| for each hologram of the third hologram set is 1.469×105. |ΔKG| between the first and 16th holograms is 2.204×106. Values for a, KG, and |ΔKG| for each of the 16 holograms of the third hologram set of the third method can be found in Table 9.
According to the third method, the first recording beam internal angle of the third hologram set ranges from +63.696 to +67.051 degrees (a range of 3.355 degrees) and the second recording beam internal angle of the third hologram set ranges from +143.704 to +140.313 degrees (a range of 3.391 degrees). For each hologram of the third hologram set, the first recording beam and its respective second recording beam are symmetrical about a skew axis. Thus the internal angle of the first recording beam relative to a skew axis (+49.996 degrees for the first hologram) plus the internal angle of the second recording beam relative to the skew axis (+130.004 degrees for the first hologram) is equal to 180.0 degrees (49.996°+130.004=180.0°). The internal angles of the first and second recording beams relative to the skew axis are readily calculated from the first and second recording beam internal angles respectively, and the skew angle. The mean skew angle for all holograms of the third hologram set is 13.691 degrees and all skew angles of the first set are within 0.009 degree of the mean. For the third hologram set of the third method of making a skew mirror, first and second recording beam internal angles relative to surface normal of the grating medium and internal angles relative to the skew axis are listed in Table 9.







TABLE 9







RECORDING BEAM ANGLES AND RELATED DATA


FOR A THIRD METHOD OF MAKING A SKEW MIRROR


















Skew
Internal
Internal








Angle
Angle of
Angle of



First
Second
(internal,
First
Second



Recording
Recording
in
Recording
Recording
Magnitude



Beam
Beam
degrees,
Beam
Beam
of Angle



Angle
Angle
relative
Relative
Relative
Difference















(internal, in degrees,
to
To Skew
To Skew
From





relative to surface
surface
Axis
Axis
Previous
α
KG


#
normal)
normal)
(degrees)
(degrees)
Hologram
(degrees)
(×107 rad/m)










First Set of Holograms















1
43.519
163.882
13.700
29.819
150.182

120.363
4.140


2
43.870
163.527
13.699
30.171
149.829
0.354
119.657
4.126


3
44.218
163.177
13.697
30.521
149.479
0.351
118.959
4.111


4
44.562
162.830
13.696
30.866
149.134
0.347
118.268
4.096


5
44.903
162.486
13.695
31.208
148.792
0.344
117.583
4.082


6
45.240
162.146
13.693
31.547
148.453
0.340
116.906
4.067


7
45.574
161.809
13.692
31.883
148.117
0.337
116.235
4.052


8
45.905
161.475
13.690
32.215
147.785
0.334
115.570
4.037


9
46.233
161.144
13.689
32.545
147.455
0.331
114.911
4.023


10
46.558
160.816
13.687
32.871
147.129
0.328
114.258
4.008


11
46.880
160.491
13.686
33.195
146.805
0.325
113.611
3.993


12
47.200
160.169
13.684
33.515
146.485
0.322
112.969
3.979


13
47.516
159.849
13.683
33.834
146.166
0.320
112.333
3.964


14
47.830
159.532
13.681
34.149
145.851
0.317
111.702
3.949


15
48.142
159.218
13.680
34.462
145.538
0.314
111.076
3.935


16
48.451
158.905
13.678
34.773
145.227
0.312
110.455
3.920


17
48.757
158.596
13.676
35.081
144.919
0.310
109.838
3.905


18
49.061
158.288
13.675
35.387
144.613
0.307
109.227
3.891


19
49.363
157.983
13.673
35.690
144.310
0.305
108.620
3.876


20
49.663
157.680
13.672
35.991
144.009
0.303
108.017
3.861


21
49.960
157.379
13.670
36.290
143.710
0.301
107.419
3.846









MEAN SKEW ANGLE =
13.685








Second Set of Holograms















1
53.704
153.696
13.700
40.004
139.996

99.992
3.655


2
53.976
153.421
13.699
40.278
139.722
0.275
99.445
3.641


3
54.247
153.148
13.698
40.550
139.450
0.273
98.900
3.626


4
54.517
152.876
13.696
40.820
139.180
0.272
98.359
3.611


5
54.785
152.605
13.695
41.090
138.910
0.270
97.821
3.597


6
55.051
152.336
13.694
41.357
138.643
0.269
97.286
3.582


7
55.316
152.069
13.692
41.624
138.377
0.268
96.753
3.567


8
55.579
151.803
13.691
41.888
138.112
0.266
96.223
3.553


9
55.842
151.538
13.690
42.152
137.848
0.265
95.696
3.538


10
56.102
151.274
13.688
42.414
137.586
0.264
95.172
3.523


11
56.362
151.012
13.687
42.675
137.325
0.262
94.650
3.509


12
56.620
150.751
13.685
42.935
137.065
0.261
94.131
3.494


13
56.877
150.491
13.684
43.193
136.807
0.260
93.614
3.479


14
57.133
150.232
13.683
43.450
136.550
0.259
93.100
3.464


15
57.387
149.975
13.681
43.706
136.294
0.257
92.588
3.450


16
57.640
149.719
13.680
43.961
136.039
0.256
92.079
3.435


17
57.892
149.464
13.678
44.214
135.786
0.255
91.571
3.420


18
58.143
149.210
13.677
44.467
135.533
0.254
91.067
3.406


19
58.393
148.957
13.675
44.718
135.282
0.253
90.564
3.391









MEAN SKEW ANGLE =
13.688








Third Set of Holograms















1
63.696
143.704
13.700
49.996
130.004

80.008
3.068


2
63.925
143.473
13.699
50.226
129.774
0.231
79.548
3.053


3
64.153
143.243
13.698
50.455
129.545
0.230
79.090
3.038


4
64.380
143.013
13.697
50.683
129.317
0.230
78.633
3.024


5
64.607
142.785
13.696
50.911
129.089
0.229
78.178
3.009


6
64.833
142.556
13.694
51.138
128.862
0.228
77.724
2.994


7
65.058
142.329
13.693
51.364
128.636
0.227
77.272
2.980


8
65.282
142.102
13.692
51.590
128.410
0.227
76.821
2.965


9
65.506
141.877
13.691
51.815
128.186
0.226
76.371
2.950


10
65.728
141.651
13.690
52.039
127.961
0.225
75.923
2.935


11
65.951
141.427
13.689
52.262
127.738
0.225
75.476
2.921


12
66.172
141.203
13.687
52.485
127.515
0.224
75.031
2.906


13
66.393
140.979
13.686
52.707
127.293
0.223
74.586
2.891


14
66.613
140.757
13.685
52.928
127.072
0.223
74.144
2.877


15
66.832
140.534
13.683
53.149
126.851
0.222
73.702
2.862


16
67.051
140.313
13.682
53.369
126.631
0.222
73.262
2.847









MEAN SKEW ANGLE =
13.691









A Multicolor Skew Mirror Embodiment

A skew mirror produced by the third method of making a skew mirror can be referred to as a multicolor skew mirror because its grating medium is configured to reflect blue, green, and red light about substantially constant reflective axes. The first hologram set is configured to reflect incident light residing in a blue region of the visible spectrum about substantially constant first reflective axes that differ by at least 2.0 degrees from surface normal of the grating medium. For purposes of the present disclosure, incident light in the blue region of the visible spectrum has a wavelength in the range of 405 nm to 492 nm. The first hologram set is more specifically configured to reflect blue incident light having a wavelength of 463 nm about substantially constant first reflective axes having a mean reflective axis angle of +13.685 degrees, where (i) the blue incident light has internal angles of incidence (relative to surface normal) that range from +8.615 degrees to −8.606 degrees, and (ii) the internal angles of incidence include at least 21 different incidence angles, each of which is separated from all others of the at least 21 different incidence angles by 0.52 degrees or more. In some embodiments, the internal angles of incidence of the blue incident light include at least 4 different incidence angles, each of which is separated from all others of the at least 4 difference incidence angles by 1.0 degrees or more.
The incident light is reflected at an internal angle of reflection (relative to surface normal) ranging from +18.785 degrees to +35.946 degrees, respectively, and the reflected light has the same wavelength as the incident light. Persons skilled in the art recognize that the incident light and its reflection are interchangeable such that where the 463 nm incident light has internal angles of incidence that range from +18.785 degrees to +35.946 degrees, it is reflected about the substantially constant reflective axes at an internal angles of reflection ranging from +8.615 degrees to −8.606 degrees, respectively.
The second hologram set is configured to reflect incident light residing in a green region of the visible spectrum about substantially constant second reflective axes that differ by at least 2.0 degrees from surface normal of the grating medium. For purposes of the present disclosure, incident light in the green region of the visible spectrum has a wavelength in the range of 493 nm to 577 nm. The second hologram set is more specifically configured to reflect green incident light having a wavelength of 522 nm about substantially constant second reflective axes having a mean reflective axis angle of +13.688 degrees, where (i) the green incident light has internal angles of incidence (relative to surface normal) that range from +7.813 degrees to −8.993 degrees, and (ii) the internal angles of incidence include at least 19 different incidence angles, each of which is separated from all others of the at least 19 different incidence angles by 0.60 degrees or more. In some embodiments, the internal angles of incidence of the green incident light include at least 4 different incidence angles, each of which is separated from all others of the at least 4 difference incidence angles by 1.2 degrees or more.
The green incident light is reflected at internal angles of reflection ranging from +19.587 degrees to +36.342 degrees, respectively, and the reflected light has the same wavelength as the incident light. Persons skilled in the art recognize that the incident light and its reflection are interchangeable such that where the 522 nm incident light has an internal angle of incidence that ranges from +19.587 degrees to +36.342 degrees, it is reflected about the substantially constant reflective axes at an internal angle of reflection ranging from +7.813 degrees to −8.993 degrees, respectively.
The third hologram set is configured to reflect incident light residing in a red region of the visible spectrum about substantially constant third reflective axes that differ by at least 2.0 degrees from surface normal of the grating medium. For purposes of the present disclosure, incident light in the red region of the visible spectrum has a wavelength in the range of 610 nm to 780 nm. The third hologram set is more specifically configured to reflect red incident light having a wavelength of 622 nm about substantially constant third reflective axes having a mean reflective axis angle of +13.691 degrees, where (i) the red incident light has internal angles of incidence (relative to surface normal) that range from +10.370 degrees to −8.391 degrees, and (ii) the internal angles of incidence include at least 16 different incidence angles, each of which is separated from others of the at least 16 different incidence angles by 0.74 degrees or more. In some embodiments, the internal angles of incidence of the red incident light include at least 4 different incidence angles, each of which is separated from all others of the at least 4 difference incidence angles by 1.5 degrees or more.
The red incident light is reflected at internal angles of reflection ranging from +17.030 degrees to +35.791 degrees, respectively, and the reflected light has the same wavelength as the incident light. Persons skilled in the art recognize that the red incident light and its reflection are interchangeable such that where the 622 nm incident light has an internal angle of incidence that ranges from +17.030 degrees to +35.791 degrees, it is reflected about the substantially constant reflective axis at internal angles of reflection ranging from +10.370 degrees to −8.391 degrees, respectively.
As described above, the first hologram set is configured to reflect blue incident light having a wavelength of 463 nm about reflective axes that are substantially constant, having a mean reflective axis angle of +13.7 degrees, where the 463 nm light is incident upon the grating medium at multiple internal angles ranging from −8.6 degrees to +8.6 degrees relative to surface normal. The second hologram set is configured to reflect green incident light having a wavelength of 522 nm about reflective axes that are substantially constant, having a mean reflective axis angle of +13.7 degrees, where the 522 nm light is incident upon the grating medium at multiple internal angles of incidence ranging from −9.0 degrees to +7.8 degrees relative to surface normal. The third hologram set is configured to reflect red incident light having a wavelength of 622 nm about reflective axes that are substantially constant, having a mean reflective axis angle of +13.7 degrees, where the 622 nm light is incident upon the grating medium at multiple internal angles ranging from −8.4 degrees to +10.4 degrees relative to surface normal.
Thus the multicolor skew mirror's reflective properties enable it to reflect blue, green, and red incident light about substantially constant reflective axes having a mean reflective axis angle of 13.7 degrees, where the blue, green, and red incident light is incident upon the mirror at internal angles of incidence ranging from −8.4 degrees to +7.8 degrees (a range of 16.2 degrees) relative to surface normal. In embodiments, a skew mirror's reflective properties enable it to reflect blue, green, and red incident light about substantially constant reflective axes, where the blue, green, and red incident light is incident upon the grating medium at multiple internal angles of incidence that span a range of at least 4.0 degrees, or at least 8.0 degrees.
A Multiwavelength Method of Making a Skew Mirror
In a multiwavelength method of making a skew mirror, six volume holograms are recorded in AK233-200 grating medium, with each of the six holograms being recorded using its own unique first and second recording beam internal angles of incidence. In addition, for each of the six volume holograms, wavelengths of the first and second recording beams are adjusted continuously and synchronously from 403 nm to 408 nm, using a variable wavelength laser. Wavelengths of the first and second recording beams are kept equal to each other while recording each of the six volume holograms. Total energy dose delivered in recording the six volume holograms according to the multiwavelength method is typically, but not necessarily, 2.5 J/cm2 for first and second recording beam internal angles of incidence for the multiwavelength method of making a skew mirror are provided below in Table 10. A skew mirror made by the multiwavelength method has the same reflective characteristics of the second embodiment skew mirror described above.







TABLE 10







RECORDING BEAM ANGLES FOR A


MULTIWAVELENGTH METHOD


OF MAKING A SKEW MIRROR










First Recording Beam
Second Recording Beam


HOLOGRAM
Angle of Incidence*
Angle of Incidence*





1
56.235°
153.001°


2
57.033°
152.203°


3
57.813°
151.423°


4
58.568°
150.668°


5
59.303°
149.933°


6
60.018°
149.218°





*internal, relative to grating medium surface normal






Other Skew Mirror Embodiments
Embodiments of a skew mirror can be created in a grating medium comprising a volumetric dielectric medium, such as a photosensitive recording medium. Skew mirror embodiments may be formed by constraining a spatial dielectric modulation spectrum as described herein. In an embodiment, dielectric modulation is accomplished holographically by recording an interference pattern of two or more coherent light beams in a photosensitive recording medium. In other embodiments, dielectric modulation can be accomplished by other means.
FIG. 15 illustrates another skew mirror embodiment, a “narcissist's mirror” includes several skew mirrors 1500 whose reflective axes 1561 intersect. A narcissist can sit at the point of convergence and see several images of them self.
Skew Mirror Fabrication
Skew mirrors may be recorded holographically according to an embodiment. Skew mirrors may be recorded holographically or fabricated by with non-holographic means according to embodiments.
Holographic Recording
FIGS. 16A and 16B illustrate additional methods for recording skew mirrors. In FIG. 16A, substantially collimated recording beams are used to illuminate a grating medium to create a desired Δn() distribution. In one embodiment, illustrated in FIG. 16A, a recording beam pair consisting of a first recording beam 1654A and a second recording beam 1655A at wavelength λilluminate the grating medium 1610 in order to record a first point-like subset of the desired line segment-like Δn() distribution, e.g., the highest spatial frequency components (the outer tips of Δn()). The angles of incidence θ1 and θ2 of a recording apparatus are then adjusted to produce another set of recording beams consisting of another first recording beam 1654 and another second recording beam 1655, which are also at wavelength A. The other first and second recording beams 1654, 1655 illuminate the medium to record a second point-like subset of the desired line segment-like Δn() distribution. This process is repeated using yet another set of recording beams consisting of yet another first recording beam 1654C and yet another second recording beam 1655C etc . . . , until an entire desired line segment-like Δn() distribution has been recorded.
In some embodiments, this recording may be made in one continuous exposure wherein θr and θs are adjusted continuously and synchronously in order to produce the desired distribution. In other embodiments, separate, discreet exposures where θr and θs are fixed during exposure and changed only between exposures are used. Still other embodiments may combine these methods. In some embodiments, the components of Δn() may be written in an arbitrary order. In some embodiments, intensity may be varied across one or both beams in order to control the spatial diffraction efficiency profile. In some embodiments, a phase control element (e.g., a mirror mounted on a piezo-electric actuator) may be inserted into one or both beam paths in order to control the phase of each exposure. In some embodiments, more than one skew mirror or broadband skew mirror might be recorded into the same medium.
In the case of discreet exposures, the number and angular density of exposures is sufficient to produce a smooth, continuous line segment-like Δn() distribution. One skilled in the art will readily calculate the angular selectivity of each hologram produced by a discreet exposure using Kogelnik's theory. In one embodiment, exposures are made at angular increments corresponding to a function of this angular selectivity, e.g., at the angular spacing of the full-width-quarter-maximum (FWQM) of the diffraction efficiency peaks. In other embodiments, the angular exposure density might be finer than this in order to assure a smooth final distribution.
The number of FWQM peaks necessary to span the line segment-like Δn() distribution may be regarded as an equivalent number of holograms, M, required to form the distribution. Accordingly, the maximum possible diffraction efficiency of the resulting skew mirror may be estimated by η=(M/M/#)z where η is the diffraction efficiency, and M/# is a material parameter characterizing the dynamic range of the recording medium. One skilled in the art will readily determine how to refine this estimate according to the geometry of each individual exposure or the overlap of neighboring exposures.
FIG. 16B illustrates an embodiment where a first prism 1659A and a second prism 1659B are incorporated to produce internal beam angles that are not otherwise accessible due to refraction at the grating medium 1610 surface. This method is typically used, for example, to fabricate the skew coupler of FIG. 12B. One skilled in the art will readily perceive how to modify the configurations of FIGS. 13A and 13B to achieve a desired distribution.
In some embodiments, a single recording wavelength λ may be chosen to write the entire line segment-like Δn() distribution. For example, in an embodiment it is possible to write a skew mirror that operates across all visible wavelengths using only a 405 nm laser source. This has an advantage of requiring sufficient recording medium sensitivity at only a single wavelength, as well as an advantage of simplicity. In some embodiments, more than one recording wavelength is used. In still other cases, a continuously-variable wavelength source is used. In one such embodiment, the recording angles Or and a are held constant, and the recording wavelength is instead changed in order to produce the entire line segment-like Δn() distribution, or a subset thereof.
Other Fabrication Methods
Other methods for producing a skew mirror fall within the scope of the present invention. In one embodiment, for example, a very thick dielectric layer structure is built up using conventional optical coating means. The structure is designed to produce broadband reflectivity within sub-layers, typically by repetition of a conventional broadband reflective coating design. The thick structure is then ground and polished to produce a surface at an oblique angle to the coating layers. The resulting structure typically exhibits mirror-like behavior with respect to a reflective axis substantially defined by the normal of the coating layers rather than the polished surface, and thus constitutes a skew mirror. In some embodiments, atomically-precise manufacturing methods enable fabrication of skew mirrors by composing dielectric structures atom-by-atom without regard to external surfaces.
Non-Flat Mirrors
Skew mirrors may be said to be non-flat in two senses: 1) When the physical shape of the recording medium is not flat; and 2) when the holographic fringes are not planar.
Non-Slab-Like Mirrors
Embodiments of mirrors according to the present invention, including examples of skew mirrors, broadband mirrors, and holographic mirrors, include holograms recorded in medium that is not slab-like in shape. In an example, in an embodiment, a recording layer is cast with a uniform thickness, but on a curved surface. In another example, a non-uniform recording layer (e.g., wedge-shaped) is utilized. In still another example, an arbitrary shape (e.g., spherical) is molded. In these non-slab-like mirror cases, whether the designation “skew mirror” is appropriate depends on the geometry of the relevant surface(s). Non-slab-like holographic mirrors typically exhibit broadband mirror-like properties.
Mirrors with Non-Planar Holographic Fringes
In some embodiments, it is desirable to introduce optical power or other deliberate aberrations into a reflection. This can be accomplished with an embodiment of a skew mirror by locally varying the direction of the reflective axis, for example so that a plane-wave incident beam is reflected to form a spherical-wave reflected beam, as occurs with a conventional parabolic mirror. Such a skew mirror can be fabricated, for instance, by using one converging and one diverging beam in the fabrication method of FIG. 13 and by recording while changing the wavelength instead of the angle of incidence. Such a mirror can also be fabricated by polishing dielectric layers deposited on a non-flat surface, or by using advanced atomically-precise manufacturing methods.
Other Fabrication Embodiments
Some holographic recording system embodiments incorporates mirrors, lenses and prisms to direct first and second recording beams into the grating medium in such a way that translation of the grating medium is not required to record multiple holograms at varying recording beam internal angles, at approximately the same location in the grating medium.
In some embodiments a prism in addition to the coupling prism may be used to fabricate the skew mirror. In some embodiments a variety of coupling prisms and flat pieces of glass may be used. In some embodiments multiple beams, Er_N and Es_N, at multiple wavelengths, λN, may be used. In some embodiments multiple wavelengths may be used to fabricate multiple discrete line segment-like Δn() distributions. In some embodiments multiple wavelengths may be used to fabricate a line segment-like Δn() distribution that may be continuous or may include closely spaced sections. In some embodiments the incident angle of the signal and/or reference beam may be adjusted to compensate for shrinkage of the sample material. In some embodiments the sample may be rotated to compensate for shrinkage of the sample material. In some embodiments the wavelength may be changed to compensate for shrinkage of the sample material.
Alternative Embodiments and Variations
The various embodiments and variations thereof, illustrated in the accompanying Figures and/or described above, are merely exemplary and are not meant to limit the scope of the invention. It is to be appreciated that numerous other variations of the invention have been contemplated, as would be obvious to one of ordinary skill in the art, given the benefit of this disclosure. All variations of the invention that read upon appended claims are intended and contemplated to be within the scope of the invention.
Terminology
The terms and phrases as indicated in quotation marks (“ ”) in this section are intended to have the meaning ascribed to them in this Terminology section applied to them throughout this document, including in the claims, unless clearly indicated otherwise in context. Further, as applicable, the stated definitions are to apply, regardless of the word or phrase's case, to the singular and plural variations of the defined word or phrase.
References in the specification to “one embodiment,” “an embodiment,” “another embodiment,” “a preferred embodiment,” “an alternative embodiment,” “one variation,” “a variation,” and similar phrases mean that a particular feature, structure, or characteristic described in connection with the embodiment or variation, is included in at least an embodiment or variation of the invention. The phrase “in one embodiment,” “in one variation” or similar phrases, as used in various places in the specification, are not necessarily meant to refer to the same embodiment or the same variation.
The term “approximately,” as used in this specification and appended claims, refers to plus or minus 10% of the value given.
The term “about,” as used in this specification and appended claims, refers to plus or minus 20% of the value given.
The term “generally,” as used in this specification and appended claims, mean mostly, or for the most part.
The term “principally,” as used in this specification and appended claims with respect to reflected light, refers to light reflected by a grating structure. Light that is principally reflected at a recited angle includes more light than is reflected at any other angle (excluding surface reflections). Light that is principally reflected about a recited reflective axis includes more reflected light than is reflected about any other reflective axis (excluding surface reflections). Light reflected by a device surface is not included when considering principally reflected light.
The term “reflective axis,” as used in this specification and appended claims, refers to an axis that bisects an angle of an incident light ray relative to its reflected light ray. The incident light ray, reflective axis, and reflected light ray all reside in one common plane, which can be referred to as a plane of incidence. The plane of incidence for a skew mirror need not be coincident with surface normal, although it can be. The magnitude of an angle of incidence of the incident light ray relative to the reflective axis is equal to the magnitude of an angle of reflection of the reflected light ray relative to the reflective axis. For purposes of the foregoing definition of “reflective axis,” and angles are internal angles. For conventional dielectric and metal mirrors, the reflective axis is coincident with surface normal, i.e. the reflective axis is perpendicular to the mirror surface, as is the plane of incidence. Conversely, embodiments of skew mirrors according to the present invention may have a reflective axis that differs from surface normal, or may have a reflective axis that is coincident with surface normal. Whether or not a skew mirror's reflective axis is coincident with surface normal is independent of whether or not the skew mirror's plane of incidence is coincident with surface normal. Angles of incidence and angles of reflection are usually, but not necessarily, determined empirically, with multiple measurements (generally three or more) typically used to generate a mean value.
The term “reflection” and similar terms are used in this disclosure in some cases where “diffraction” might ordinarily be considered an appropriate term. This use of “reflection” is consistent with mirror-like properties exhibited by skew mirrors and helps avoid potentially confusing terminology. For example, where a grating structure is said to be configured to “reflect” incident light, a conventional artisan might prefer to say the grating structure is configured to “diffract” incident light, since grating structures are generally thought to act on light by diffraction. However, such use of the term “diffract” would result in expressions such as “incident light is diffracted about substantially constant reflective axes,” which could be confusing. Accordingly, where incident light is said to be “reflected” by a grating structure, persons of ordinary skill in art, given the benefit of this disclosure, will recognize that the grating structure is in fact “reflecting” the light by a diffractive mechanism. Such use of “reflect” is not without precedent in optics, as conventional dielectric mirrors are generally said to “reflect” light despite the predominant role diffraction plays in such reflection. Artisans of ordinary skill thus recognize that most “reflection” includes characteristics of diffraction, and “reflection” by a skew mirror or components thereof also includes diffraction.
The terms “angle interval” and “angle intervals,” as used in this specification and appended claims, refer to angular spacing between multiple light beams incident upon a skew mirror within a recited range of angles of incidence.
The terms “hologram” and “holographic grating,” as used in this specification and appended claims, refer to a recording of an interference pattern generated by interference between multiple intersecting light beams. A hologram or holographic grating is an example of a grating structure.
While various inventive embodiments have been described and illustrated herein, those of ordinary skill in the art will readily envision a variety of other means and/or structures for performing the function and/or obtaining the results and/or one or more of the advantages described herein, and each of such variations and/or modifications is deemed to be within the scope of the inventive embodiments described herein. More generally, those skilled in the art will readily appreciate that all parameters, dimensions, materials, and configurations described herein are meant to be exemplary and that the actual parameters, dimensions, materials, and/or configurations will depend upon the specific application or applications for which the inventive teachings is/are used. Those skilled in the art will recognize, or be able to ascertain using no more than routine experimentation, many equivalents to the specific inventive embodiments described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described and claimed. Inventive embodiments of the present disclosure are directed to each individual feature, system, article, material, kit, and/or method described herein. In addition, any combination of two or more such features, systems, articles, materials, kits, and/or methods, if such features, systems, articles, materials, kits, and/or methods are not mutually inconsistent, is included within the inventive scope of the present disclosure.
The above-described embodiments can be implemented in any of numerous ways. For example, embodiments of designing and making the technology disclosed herein may be implemented using hardware, software or a combination thereof. When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers.
Further, it should be appreciated that a computer may be embodied in any of a number of forms, such as a rack-mounted computer, a desktop computer, a laptop computer, or a tablet computer. Additionally, a computer may be embedded in a device not generally regarded as a computer but with suitable processing capabilities, including a Personal Digital Assistant (PDA), a smart phone or any other suitable portable or fixed electronic device.
Also, a computer may have one or more input and output devices. These devices can be used, among other things, to present a user interface. Examples of output devices that can be used to provide a user interface include printers or display screens for visual presentation of output and speakers or other sound generating devices for audible presentation of output. Examples of input devices that can be used for a user interface include keyboards, and pointing devices, such as mice, touch pads, and digitizing tablets. As another example, a computer may receive input information through speech recognition or in other audible format.
Such computers may be interconnected by one or more networks in any suitable form, including a local area network or a wide area network, such as an enterprise network, and intelligent network (IN) or the Internet. Such networks may be based on any suitable technology and may operate according to any suitable protocol and may include wireless networks, wired networks or fiber optic networks.
The various methods or processes (e.g., of designing and making the coupling structures and diffractive optical elements disclosed above) outlined herein may be coded as software that is executable on one or more processors that employ any one of a variety of operating systems or platforms. Additionally, such software may be written using any of a number of suitable programming languages and/or programming or scripting tools, and also may be compiled as executable machine language code or intermediate code that is executed on a framework or virtual machine.
In this respect, various inventive concepts may be embodied as a computer readable storage medium (or multiple computer readable storage media) (e.g., a computer memory, one or more floppy discs, compact discs, optical discs, magnetic tapes, flash memories, circuit configurations in Field Programmable Gate Arrays or other semiconductor devices, or other non-transitory medium or tangible computer storage medium) encoded with one or more programs that, when executed on one or more computers or other processors, perform methods that implement the various embodiments of the invention discussed above. The computer readable medium or media can be transportable, such that the program or programs stored thereon can be loaded onto one or more different computers or other processors to implement various aspects of the present invention as discussed above.
The terms “program” or “software” are used herein in a generic sense to refer to any type of computer code or set of computer-executable instructions that can be employed to program a computer or other processor to implement various aspects of embodiments as discussed above. Additionally, it should be appreciated that according to one aspect, one or more computer programs that when executed perform methods of the present invention need not reside on a single computer or processor, but may be distributed in a modular fashion amongst a number of different computers or processors to implement various aspects of the present invention.
Computer-executable instructions may be in many forms, such as program modules, executed by one or more computers or other devices. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Typically the functionality of the program modules may be combined or distributed as desired in various embodiments.
Also, data structures may be stored in computer-readable media in any suitable form. For simplicity of illustration, data structures may be shown to have fields that are related through location in the data structure. Such relationships may likewise be achieved by assigning storage for the fields with locations in a computer-readable medium that convey relationship between the fields. However, any suitable mechanism may be used to establish a relationship between information in fields of a data structure, including through the use of pointers, tags or other mechanisms that establish relationship between data elements.
Also, various inventive concepts may be embodied as one or more methods, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.
All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and/or ordinary meanings of the defined terms.
The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.”
The phrase “and/or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with “and/or” should be construed in the same fashion, i.e., “one or more” of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and/or” clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to “A and/or B”, when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc.
As used herein in the specification and in the claims, “or” should be understood to have the same meaning as “and/or” as defined above. For example, when separating items in a list, “or” or “and/or” shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only terms clearly indicated to the contrary, such as “only one of” or “exactly one of,” or, when used in the claims, “consisting of,” will refer to the inclusion of exactly one element of a number or list of elements. In general, the term “or” as used herein shall only be interpreted as indicating exclusive alternatives (i.e. “one or the other but not both”) when preceded by terms of exclusivity, such as “either,” “one of,” “only one of,” or “exactly one of.” “Consisting essentially of,” when used in the claims, shall have its ordinary meaning as used in the field of patent law.
As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, “at least one of A and B” (or, equivalently, “at least one of A or B,” or, equivalently “at least one of A and/or B”) can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including more than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc.
In the claims, as well as in the specification above, all transitional phrases such as “comprising,” “including,” “carrying,” “having,” “containing,” “involving,” “holding,” “composed of,” and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases “consisting of” and “consisting essentially of” shall be closed or semi-closed transitional phrases, respectively, as set forth in the United States Patent Office Manual of Patent Examining Procedures, Section 221.03.

## Claims

1. A skew mirror comprising:
a grating structure residing in a grating medium, wherein:
the grating structure is configured to reflect first incident light, the first incident light being incident upon the grating medium at a specific site and having a first wavelength and a first internal angle of incidence relative to grating medium surface normal;
the first incident light is principally reflected by the grating medium as first reflected light, the first reflected light having the first wavelength and a first internal angle of reflection relative to the surface normal;
the first incident light and the first reflected light are bisected by a first reflective axis having a first reflective axis angle relative to the surface normal;
the grating structure is further configured to reflect second incident light, the second incident light being incident on the grating medium at the specific site and having a second wavelength and a second internal angle of incidence relative to the surface normal;
the second incident light is principally reflected by the grating medium as second reflected light, the second reflected light having the second wavelength and a second internal angle of reflection relative to the surface normal;
the second incident light and the second reflected light are bisected by a second reflective axis having a second reflective axis angle relative to the surface normal;
the first internal angle of incidence is the same as the second internal angle of incidence;
the first reflective axis angle differs from the surface normal by at least 2.0 degrees;
the first wavelength differs from the second wavelength by a wave fraction of at least 0.030; and
the first reflective axis angle differs from the second reflective axis angle by 0.25 degree or less.

2. The skew mirror of claim 1, wherein the first reflective axis angle differs from the second reflective axis angle by 0.025 degree or less.

3. The skew mirror of claim 2, wherein the first wavelength differs from the second wavelength by a wave fraction of at least 0.036.

4. The skew mirror of claim 1, wherein the first incident light is offset from the first reflective axis by at least 1.0 degree.

5. The skew mirror of claim 4, wherein:
the grating structure comprises at least 9 volume holograms;
each of the at least 9 volume holograms at least partially spatially overlaps at least one other of the at least 9 volume holograms; and
the grating medium is at least 200 μm thick.

6. The skew mirror of claim 1, wherein:
the grating structure comprises multiple volume holograms;
each of the multiple volume holograms spatially overlaps at least one other of the multiple volume holograms; and
the grating medium is at least 70 μm thick.

7. The skew mirror of claim 6, wherein:
the multiple volume holograms includes at least 9 holograms; and
each of the multiple volume holograms at least partially spatially overlaps all others of the multiple volume holograms.

8. The skew mirror of claim 7, wherein adjacent |ΔKG| for the at least 9 holograms has a mean value that resides between 1.0×104 and 1.0×106 radians per meter (rad/m).

9. A method of making the skew mirror of claim 1, the method comprising:
creating the grating structure by recording multiple volume holograms in the grating medium, wherein:
each of the multiple volume holograms is recorded using a first recording beam and a second recording beam, each of the first and second recording beams including a collimated, plane wave beam, and the first recording beam having the same wavelength as the second recording beam;
each of the multiple volume holograms is recorded with the first recording beam being incident upon the grating medium at a unique first recording beam internal angle relative to the surface normal and the second recording beam being incident upon the grating medium at a unique second recording beam internal angle relative to the surface normal;
each of the multiple volume holograms is recorded with the first recording beam and the second recording beam being symmetrical about a skew axis such that a first recording beam internal angle relative to the skew axis plus a second recording beam angle relative to the skew axis is equal to 180 degrees;
each of the multiple volume holograms at least partially spatially overlaps at least one other of the multiple holograms;
the skew axes of the multiple volume holograms have substantially constant skew angles relative to the surface normal; and
the skew axes of the multiple volume holograms have a mean skew angle that is substantially identical to both the first reflective axis angle and the second reflective axis angle.

10. The method of claim 9, wherein each of the multiple volume holograms at least partially spatially overlaps all others of the multiple volume holograms.

11. A method of making the skew mirror of claim 1, the method comprising:
creating the grating structure by recording a volume hologram in the grating medium, wherein:
the volume hologram is recorded using a first recording beam and a second recording beam, each of the first and second recording beams including a collimated, plane wave beam, and the first recording beam having the same wavelength as the second recording beam;
the volume hologram is recorded with the first recording beam being incident upon the grating medium at a first recording beam internal angle relative to the surface normal and the second recording beam being incident upon the grating medium at a second recording beam internal angle relative to the surface normal;
the first recoding beam and the second recording beam are symmetrical about a skew axis such that an internal angle of the first recording beam relative to the skew axis plus an internal angle of the second recording beam relative to the skew axis is equal to 180 degrees; and
the skew axis has a skew axis angle relative to the surface normal that is substantially identical to both the first reflective axis angle and the second reflective axis angle;

changing the first recording beam internal angle relative to the surface normal while recording the volume hologram; and
changing the second recording beam internal angle relative to the surface normal while recording the volume hologram, wherein the symmetry of the first recording beam and the second recording beam about the skew axis is maintained during said changing the first and second recording beam internal angles.

12. A method of making the skew mirror of claim 1, the method comprising:
creating the grating structure by recording a volume hologram in the grating medium, wherein:
the volume hologram is recorded using a first recording beam and a second recording beam, each of the first and second recording beams including a collimated, plane wave beam, and the first recording beam having the same wavelength as the second recording beam;
the volume hologram is recorded with the first recording beam being incident upon the grating medium at a first recording beam internal angle relative to the surface normal and the second recording beam being incident upon the grating medium at a second recording beam internal angle relative to the surface normal;
the first recoding beam and the second recording beam are symmetrical about a skew axis such that an internal angle of the first recording beam relative to the skew axis plus an internal angle of the second recording beam relative to the skew axis is equal to 180 degrees; and
the skew axis has a skew axis angle relative to the surface normal that is substantially identical to both the first reflective axis angle and the second reflective axis angle;

changing the first recording beam wavelength while recording the volume hologram; and
changing the second recording beam wavelength while recording the volume hologram, wherein the first recording beam wavelength remains the same as the second recording beam wavelength during said changing of the first and second recording beam wavelengths.

13. A method of making a skew mirror comprising:
creating a grating structure in a grating medium by recording multiple volume holograms in the grating medium, wherein:
each of the multiple volume holograms is recorded using a first recording beam and a second recording beam, each of the first and second recording beams including a collimated, plane wave beam, and the first recording beam having a same wavelength as the second recording beam;
each of the multiple volume holograms is recorded with the first recording beam being incident upon the grating medium at a unique first recording beam internal angle relative to a surface normal of the grating medium and the second recording beam being incident upon the grating medium at a unique second recording beam internal angle relative to the surface normal;
each of the multiple volume holograms is recorded with the first recording beam and the second recording beam being symmetrical about a skew axis such that a first recording beam internal angle relative to the skew axis plus a second recording beam angle relative to the skew axis is equal to 180 degrees, the skew axis having a skew axis angle relative to surface normal;
each of the multiple volume holograms at least partially spatially overlaps at least one other of the multiple holograms; and
the skew axis has a skew axis angle relative to the surface normal having a magnitude of at least 2.0 degrees.

14. The method of claim 13, wherein each of the multiple volume holograms at least partially spatially overlaps all others of the multiple volume holograms.

15. The method of claim 14, wherein:
the multiple volume holograms includes at least 9 holograms;
all of the unique first recording beam internal angles collectively span a range of at least 4.0 degrees; and
all of the unique second recording beam internal angles collectively span a range of at least 4.0 degrees.

16. The method of claim 13, wherein:
adjacent |ΔKG| for the at least 9 holograms has a mean value that resides between 1.0×104 and 1.0×106 rad/m.

17. The method of claim 13, wherein:
the grating structure is configured to reflect first incident light, the first incident light being incident upon the grating medium at a specific site and having a first wavelength and a first internal angle of incidence relative to grating medium surface normal;
the first incident light is principally reflected by the grating medium as first reflected light, the first reflected light having the first wavelength and a first internal angle of reflection relative to the surface normal;
the first incident light and the first reflected light are bisected by a first reflective axis having a first reflective axis angle relative to the surface normal;
the grating structure is further configured to reflect second incident light, the second incident light being incident on the grating medium at the specific site and having a second wavelength and a second internal angle of incidence relative to the surface normal;
the second incident light is principally reflected by the grating medium as second reflected light, the second reflected light having the second wavelength and a second internal angle of reflection relative to the surface normal;
the second incident light and the second reflected light are bisected by a second reflective axis having a second reflective axis angle relative to the surface normal;
the first internal angle of incidence is the same as the second internal angle of incidence;
both the first reflective axis angle and the second reflective axis angle are substantially identical to the skew axis angle; and
the first wavelength differs from the second wavelength by a wave fraction of at least 0.010.

18. The method of claim 17, wherein:
the first wavelength differs from the second wavelength by a wave fraction of at least 0.030.

19. A skew mirror comprising:
a grating structure residing in a grating medium, wherein:
the grating structure is configured to reflect first incident light, the first incident light being incident upon the grating medium at a specific site and having a first internal angle of incidence relative to grating medium surface normal;
the first incident light is principally reflected by the grating medium as first reflected light, the first reflected light having a first internal angle of reflection relative to the surface normal;
the first incident light and the first reflected light are bisected by a first reflective axis having a first reflective axis angle relative to the surface normal;
the grating structure is further configured to reflect second incident light, the second incident light being incident on the grating medium at the specific site and having a second internal angle of incidence relative to the surface normal;
the second incident light is principally reflected by the grating medium as second reflected light, the second reflected light having a second internal angle of reflection relative to the surface normal;
the second incident light and the second reflected light are bisected by a second reflective axis having a second reflective axis angle relative to the surface normal;
the first incident light, the first reflected light, the second incident light, and the second reflected light have the same wavelength as each other;
the first internal angle of incidence differs from the second internal angle of incidence by 20 times ΔθB;
the first reflective axis angle differs from the surface normal by at least 2.0 degrees; and
the first reflective axis angle differs from the second reflective axis angle by 0.25 degree or less.

20. The skew mirror of claim 19, wherein:
the first reflective axis angle differs from the second reflective axis angle by 0.10 degree or less.

21. The skew mirror of claim 20, wherein:
the first reflective axis angle differs from the second reflective axis angle by 0.025 degree or less.

22. The skew mirror of claim 21, wherein:
each of the first incident light and the second incident light are offset from the first reflective axis by at least 5.0 degrees.

23. The skew mirror of claim 19, wherein:
each of the first incident light and the second incident light are offset from the first reflective axis by at least 5.0 degrees.

24. The skew mirror of claim 19, wherein:
each of the first incident light and the second incident light are offset from the first reflective axis by at least 9.0 degrees.

25. A skew mirror comprising:
a grating structure residing in a grating medium, wherein:
the grating structure includes a first hologram set, a second hologram set, and a third hologram set, each of which comprises multiple volume holograms;
the first hologram set is configured to reflect blue incident light about a substantially constant first reflective axis having a first mean reflective axis angle relative to surface normal;
the second hologram set is configured to reflect green incident light about a second reflective axis having a mean second reflective axis angle relative to surface normal;

the third hologram set is configured to reflect red incident light about a third reflective axis having a third mean reflective axis angle relative to surface normal;
each of the first, second, and third mean reflective axes angles (i) differs from surface normal of the grating medium by at least 2.0 degrees, and (ii) is within 1.0 degree of others of the first, second, and third mean reflective axis angles;
within each of the first, second, and third hologram sets, each of the multiple volume holograms at least partially spatially overlaps at least one other of the multiple volume holograms.

26. The skew mirror of claim 25, wherein the grating medium is at least 200 μm thick.

27. The skew mirror of claim 25, wherein the grating medium is at least 500 μm thick.

28. The skew mirror of claim 25, wherein:
each of the blue, green, and red incident light is incident upon the grating medium at multiple internal angles of incidence that span a range of at least 4.0 degrees.

29. The skew mirror of claim 28, wherein;
within each of the first, second, and third hologram sets, each of the multiple holograms at least partially spatially overlaps all others of the multiple volume holograms.

30. The skew mirror of claim 28, wherein:
each of the blue, green, and red incident light is incident upon the grating medium at multiple internal angles of incidence that span a range of at least 8.0 degrees.

31. The skew mirror of claim 30, wherein adjacent |ΔKG| for each of the first, second, and third hologram sets has a mean value that resides between 1.0×104 and 1.0×106 rad/m.

32. The skew mirror of claim 31, wherein the adjacent |ΔKG| for each of the first, second, and third hologram sets has a mean value greater than 8.0×104 rad/m.

33. The skew mirror of claim 32, wherein each of the first, second, and third hologram sets includes at least 5 holograms.

34. A method of using a skew mirror comprising:
projecting light at a skew mirror, wherein:
the skew mirror comprises a grating medium within which resides a grating structure;
the grating medium is at least 70 μm thick;
the light includes first incident light, the first incident light being incident upon the grating medium at a specific site and having a first wavelength and a first internal angle of incidence relative to grating medium surface normal;
the first incident light is principally reflected by the grating medium as first reflected light, the first reflected light having the first wavelength and a first internal angle of reflection relative to the surface normal;
the first incident light and the first reflected light are bisected by a first reflective axis having a first reflective axis angle relative to the surface normal;
the light further includes second incident light, the second incident light being incident on the grating medium at the specific site and having a second wavelength and a second internal angle of incidence relative to the surface normal;
the second incident light is principally reflected by the grating medium as second reflected light, the second reflected light having the second wavelength and a second internal angle of reflection relative to the surface normal;
the second incident light and the second reflected light are bisected by a second reflective axis having a second reflective axis angle relative to the surface normal;
the first internal angle of incidence is the same as the second internal angle of incidence;
the first reflective axis angle differs from the surface normal by at least 2.0 degrees;
the first wavelength differs from the second wavelength by a wave fraction of at least 0.030; and
the first reflective axis angle differs from the second reflective axis angle by 0.10 degree or less.

35. The method of claim 34, wherein the first reflective axis angle differs from the surface normal by at least 4.0 degrees.

36. The method of claim 35, wherein the first incident light is offset from the first reflective axis by at least 1.0 degree.

37. The method of claim 36, wherein: the multiple volume holograms includes at least 9 holograms; and each of the multiple volume holograms at least partially spatially overlaps all others of the multiple volume holograms.

38. The method of claim 34, wherein: the grating medium is at least 200 μm thick; the grating structure comprises multiple volume holograms; and each of the multiple volume holograms spatially overlaps at least one other of the multiple volume holograms.

39. The method of claim 34, wherein the grating structure comprises at least 9 volume holograms; each of the at least 9 volume holograms spatially overlaps at least one other of the at least 9 volume holograms; and adjacent |ΔKG| for the at least 9 holograms has a mean value that resides between 1.0×104 and 1.0×106 radians per meter (rad/m).