Abstract

A comprehensive treatment is presented for the diffraction efficiencies of transmission holographic elements and cascade lenses when subject to broad spectral and field angle detunings. Experimental measurements are made in support of our theory on holographic optical elements fabricated in bleached silver-halide emulsions and in dichromated gelatin. The theory of holographic grating diffraction efficiency is studied through two approaches. A numerical treatment based on the theory of thin grating decomposition is implemented and shown to be in close agreement with other theories. Additionally, a more approximate approach is pursued in which the volume grating is treated as a phased array of scatterers. The latter approach leads to closed-form formulas in addition to a simple physical picture of volume effects. It is found that three-element cascades can exhibit spectral and field angle bandwidths essentially as broad as two-element cascades and that these bandwidths are in excess of 2300 Å and 7° respectively.

© 1985 Optical Society of America

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References

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  1. E. N. Leith, J. Upatnieks, “Reconstructed Wavefronts and Communication Theory,” J. Opt. Soc. Am. 52, 1123 (1962).
    [CrossRef]
  2. R. H. Katyl, “Compensating Optical Systems. 1: Broadband Holographic Reconstruction,” Appl. Opt. 11, 1241 (1972).
    [CrossRef] [PubMed]
  3. J. N. Latta, “Analysis of Multiple Hologram Optical Elements with Low Dispersion and Low Aberrations,” Appl. Opt. 11, 1686 (1972).
    [CrossRef] [PubMed]
  4. B. J. Chang, “Doubly Modulated On-Axis Thick Hologram Optical Elements,” J. Opt. Soc. Am. 67, 1435(A) (1977).
  5. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, N. Massey, “Holographic Data Storage in Three-Dimensional Media,” Appl. Opt. 5, 1303 (1966).
    [CrossRef] [PubMed]
  6. C. B. Burckhardt, “Diffraction of a Plane Wave at a Sinusoidally Stratified Dielectric Grating,” J. Opt. Soc. Am. 56, 1502 (1966).
    [CrossRef]
  7. N. George, J. W. Matthews, “Holographic Diffraction Gratings,” Appl. Phys. Lett. 9, 212 (1966).
    [CrossRef]
  8. E. J. Saccocio, “Application of the Dynamical Theory of X-Ray Diffraction to Holography,” J. Appl. Phys. 38, 3994 (1967).
    [CrossRef]
  9. H. Kogelnik, “Reconstructing Response and Efficiency of Hologram Gratings,” Proceedings, Symposium on Modern Optics (Polytechnic Institute of Brooklyn, 1967), p. 605.
  10. H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909 (1969).
  11. M. Chang, N. George, “Holographic Dielectric Grating: Theory and Practice,” Appl. Opt. 9, 713 (1970).
    [CrossRef] [PubMed]
  12. F. G. Kaspar, “Diffraction by Thick, Periodically Stratified Gratings with Complex Dielectric Constant,” J. Opt. Soc. Am. 63, 37 (1973).
    [CrossRef]
  13. S. F. Su, T. K. Gaylord, “Calculation of Arbitrary-Order Diffraction Efficiencies of Thick Gratings with Arbitrary Grating Shape,” J. Opt. Soc. Am. 65, 59 (1975).
    [CrossRef]
  14. R. Alferness, “Analysis of Optical Propagation in Thick Holographgic Gratings,” Appl. Phys. 7, 29 (1975).
    [CrossRef]
  15. R. Alferness, “Analysis of Propagation at the Second-Order Bragg Angle of a Thick Holographic Grating,” J. Opt. Soc. Am. 66, 353 (1976).
    [CrossRef]
  16. R. Alferness, “Optical Propagation in Holographic Gratings,” Ph.D. Thesis, U. Michigan, Ann Arbor (1976).
  17. D. L. Jaggard, C. Elachi, “Floquet and Coupled-Waves Analysis of Higher-Order Bragg Coupling in a Periodic Medium,” J. Opt. Soc. Am. 66, 674 (1976).
    [CrossRef]
  18. R. S. Chu, J. A. Kong, “Modal Theory of Spatially Periodic Media,” Trans. IEEE Microwave Theory Tech. MTT-25, 18 (1977).
  19. J. A. Kong, “Second-Order Coupled Mode Equations for Spatially Periodic Media,” J. Opt. Soc. Am. 67, 825 (1977).
    [CrossRef]
  20. R. Magnusson, T. K. Gaylord, “Analysis of Multiwave Diffraction of Thick Gratings,” J. Opt. Soc. Am. 67, 1165 (1977).
    [CrossRef]
  21. R. Magnusson, T. K. Gaylord, “Diffraction Regimes of Transmission Gratings,” J. Opt. Soc. Am. 68, 809 (1978).
    [CrossRef]
  22. M. G. Moharam, T. K. Gaylord, “Rigorous Coupled-Wave Analysis of Planar-Grating Diffraction,” J. Opt. Soc. Am. 71, 811 (1981).
    [CrossRef]
  23. M. G. Moharam, T. K. Gaylord, “Rigorous Coupled-Wave Analysis of Grating Diffraction—E-Mode Polarization and Losses,” J. Opt. Soc. Am. 73, 451 (1983).
    [CrossRef]
  24. M. G. Moharam, T. K. Gaylord, “Three-Dimensional Vector Coupled-Wave Analysis of Planar-Grating Diffraction,” J. Opt. Soc. Am. 73, 1105 (1983).
    [CrossRef]
  25. R. Alferness, “Equivalence of the Thin-Grating Decomposition and Coupled-Wave Analysis of Thick Holographic Gratings,” Opt. Commun. 15, 209 (1975).
    [CrossRef]
  26. R. Magnusson, T. K. Gaylord, “Equivalence of Multiwave Coupled-Wave Theory and Modal Theory for Periodic-Media Diffraction,” J. Opt. Soc. Am. 68, 1777 (1978).
    [CrossRef]
  27. R. Magnusson, T. K. Gaylord, “Diffraction Efficiencies of Thin Phase Gratings with Arbitrary Grating Shape,” J. Opt. Soc. Am. 68, 806 (1978).
    [CrossRef]
  28. T. Stone, N. George, “Bandwidth of Holographic Optical Elements,” Opt. Lett. 7, 445 (1982).
    [CrossRef] [PubMed]
  29. W. R. Klein, C. B. Tipnis, E. A. Hiedemann, “Experimental Study of Fraunhofer Light Diffraction by Ultrasonic Beams of Moderately High Frequency at Oblique Incidence,” J. Acoust. Soc. Am. 38, 229 (1965).
    [CrossRef]
  30. R. C. Enger, S. K. Case, “High-Frequency Holographic Transmission Gratings in Photoresist,” J. Opt. Soc. Am. 73, 1113 (1983).
    [CrossRef]
  31. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Eq. (MO-27), p. 973.
  32. T. G. Georgekutty, J. G. Duthie, “Fabrication of Holographic Optical Elements for Achromatic Matched Filtering System,” Technical Report RR-82-2, U.S. Army Missile Command (Redstone Arsenal, 1982).
  33. J. Upatnieks, J. G. Duthie, P. R. Ashley, “Matched Filtering With Achromatic Optical Correlators,” Technical Report RR-82-5, U.S. Army Missile Command (Redstone Arsenal, 1982).
  34. N. George, G. M. Morris, “Matched Filtering in White Light Illumination,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 11 (1983).
  35. C. D. Leonard, B. D. Guenther, “A Cookbook for Dichromated Gelatin Holograms,” Technical Report T-79-17, U.S. Army Missile Research and Development Command (Redstone Arsenal, 1979).
  36. C. E. K. Mees, T. H. James, The Theory of the Photographic Process (Macmillan, New York, 1966).
  37. C. R. Berry, “Turbidity of Monodisperse Suspensions of AgBr,” J. Opt. Soc. Am. 52, 888 (1962).
    [CrossRef]
  38. D. H. Napper, R. H. Ottewill, “Studies on the Light Scattering of Silver Bromide Particles,” J. Photogr. Sci. 11, 84 (1963).

1983 (4)

1982 (1)

1981 (1)

1978 (3)

1977 (4)

J. A. Kong, “Second-Order Coupled Mode Equations for Spatially Periodic Media,” J. Opt. Soc. Am. 67, 825 (1977).
[CrossRef]

R. Magnusson, T. K. Gaylord, “Analysis of Multiwave Diffraction of Thick Gratings,” J. Opt. Soc. Am. 67, 1165 (1977).
[CrossRef]

R. S. Chu, J. A. Kong, “Modal Theory of Spatially Periodic Media,” Trans. IEEE Microwave Theory Tech. MTT-25, 18 (1977).

B. J. Chang, “Doubly Modulated On-Axis Thick Hologram Optical Elements,” J. Opt. Soc. Am. 67, 1435(A) (1977).

1976 (2)

1975 (3)

R. Alferness, “Analysis of Optical Propagation in Thick Holographgic Gratings,” Appl. Phys. 7, 29 (1975).
[CrossRef]

R. Alferness, “Equivalence of the Thin-Grating Decomposition and Coupled-Wave Analysis of Thick Holographic Gratings,” Opt. Commun. 15, 209 (1975).
[CrossRef]

S. F. Su, T. K. Gaylord, “Calculation of Arbitrary-Order Diffraction Efficiencies of Thick Gratings with Arbitrary Grating Shape,” J. Opt. Soc. Am. 65, 59 (1975).
[CrossRef]

1973 (1)

1972 (2)

1970 (1)

1969 (1)

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909 (1969).

1967 (1)

E. J. Saccocio, “Application of the Dynamical Theory of X-Ray Diffraction to Holography,” J. Appl. Phys. 38, 3994 (1967).
[CrossRef]

1966 (3)

1965 (1)

W. R. Klein, C. B. Tipnis, E. A. Hiedemann, “Experimental Study of Fraunhofer Light Diffraction by Ultrasonic Beams of Moderately High Frequency at Oblique Incidence,” J. Acoust. Soc. Am. 38, 229 (1965).
[CrossRef]

1963 (1)

D. H. Napper, R. H. Ottewill, “Studies on the Light Scattering of Silver Bromide Particles,” J. Photogr. Sci. 11, 84 (1963).

1962 (2)

Alferness, R.

R. Alferness, “Analysis of Propagation at the Second-Order Bragg Angle of a Thick Holographic Grating,” J. Opt. Soc. Am. 66, 353 (1976).
[CrossRef]

R. Alferness, “Equivalence of the Thin-Grating Decomposition and Coupled-Wave Analysis of Thick Holographic Gratings,” Opt. Commun. 15, 209 (1975).
[CrossRef]

R. Alferness, “Analysis of Optical Propagation in Thick Holographgic Gratings,” Appl. Phys. 7, 29 (1975).
[CrossRef]

R. Alferness, “Optical Propagation in Holographic Gratings,” Ph.D. Thesis, U. Michigan, Ann Arbor (1976).

Ashley, P. R.

J. Upatnieks, J. G. Duthie, P. R. Ashley, “Matched Filtering With Achromatic Optical Correlators,” Technical Report RR-82-5, U.S. Army Missile Command (Redstone Arsenal, 1982).

Berry, C. R.

Burckhardt, C. B.

Case, S. K.

Chang, B. J.

B. J. Chang, “Doubly Modulated On-Axis Thick Hologram Optical Elements,” J. Opt. Soc. Am. 67, 1435(A) (1977).

Chang, M.

Chu, R. S.

R. S. Chu, J. A. Kong, “Modal Theory of Spatially Periodic Media,” Trans. IEEE Microwave Theory Tech. MTT-25, 18 (1977).

Duthie, J. G.

T. G. Georgekutty, J. G. Duthie, “Fabrication of Holographic Optical Elements for Achromatic Matched Filtering System,” Technical Report RR-82-2, U.S. Army Missile Command (Redstone Arsenal, 1982).

J. Upatnieks, J. G. Duthie, P. R. Ashley, “Matched Filtering With Achromatic Optical Correlators,” Technical Report RR-82-5, U.S. Army Missile Command (Redstone Arsenal, 1982).

Elachi, C.

Enger, R. C.

Gaylord, T. K.

George, N.

N. George, G. M. Morris, “Matched Filtering in White Light Illumination,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 11 (1983).

T. Stone, N. George, “Bandwidth of Holographic Optical Elements,” Opt. Lett. 7, 445 (1982).
[CrossRef] [PubMed]

M. Chang, N. George, “Holographic Dielectric Grating: Theory and Practice,” Appl. Opt. 9, 713 (1970).
[CrossRef] [PubMed]

N. George, J. W. Matthews, “Holographic Diffraction Gratings,” Appl. Phys. Lett. 9, 212 (1966).
[CrossRef]

Georgekutty, T. G.

T. G. Georgekutty, J. G. Duthie, “Fabrication of Holographic Optical Elements for Achromatic Matched Filtering System,” Technical Report RR-82-2, U.S. Army Missile Command (Redstone Arsenal, 1982).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Eq. (MO-27), p. 973.

Guenther, B. D.

C. D. Leonard, B. D. Guenther, “A Cookbook for Dichromated Gelatin Holograms,” Technical Report T-79-17, U.S. Army Missile Research and Development Command (Redstone Arsenal, 1979).

Hiedemann, E. A.

W. R. Klein, C. B. Tipnis, E. A. Hiedemann, “Experimental Study of Fraunhofer Light Diffraction by Ultrasonic Beams of Moderately High Frequency at Oblique Incidence,” J. Acoust. Soc. Am. 38, 229 (1965).
[CrossRef]

Jaggard, D. L.

James, T. H.

C. E. K. Mees, T. H. James, The Theory of the Photographic Process (Macmillan, New York, 1966).

Kaspar, F. G.

Katyl, R. H.

Klein, W. R.

W. R. Klein, C. B. Tipnis, E. A. Hiedemann, “Experimental Study of Fraunhofer Light Diffraction by Ultrasonic Beams of Moderately High Frequency at Oblique Incidence,” J. Acoust. Soc. Am. 38, 229 (1965).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909 (1969).

H. Kogelnik, “Reconstructing Response and Efficiency of Hologram Gratings,” Proceedings, Symposium on Modern Optics (Polytechnic Institute of Brooklyn, 1967), p. 605.

Kong, J. A.

J. A. Kong, “Second-Order Coupled Mode Equations for Spatially Periodic Media,” J. Opt. Soc. Am. 67, 825 (1977).
[CrossRef]

R. S. Chu, J. A. Kong, “Modal Theory of Spatially Periodic Media,” Trans. IEEE Microwave Theory Tech. MTT-25, 18 (1977).

Kozma, A.

Latta, J. N.

Leith, E. N.

Leonard, C. D.

C. D. Leonard, B. D. Guenther, “A Cookbook for Dichromated Gelatin Holograms,” Technical Report T-79-17, U.S. Army Missile Research and Development Command (Redstone Arsenal, 1979).

Magnusson, R.

Marks, J.

Massey, N.

Matthews, J. W.

N. George, J. W. Matthews, “Holographic Diffraction Gratings,” Appl. Phys. Lett. 9, 212 (1966).
[CrossRef]

Mees, C. E. K.

C. E. K. Mees, T. H. James, The Theory of the Photographic Process (Macmillan, New York, 1966).

Moharam, M. G.

Morris, G. M.

N. George, G. M. Morris, “Matched Filtering in White Light Illumination,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 11 (1983).

Napper, D. H.

D. H. Napper, R. H. Ottewill, “Studies on the Light Scattering of Silver Bromide Particles,” J. Photogr. Sci. 11, 84 (1963).

Ottewill, R. H.

D. H. Napper, R. H. Ottewill, “Studies on the Light Scattering of Silver Bromide Particles,” J. Photogr. Sci. 11, 84 (1963).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Eq. (MO-27), p. 973.

Saccocio, E. J.

E. J. Saccocio, “Application of the Dynamical Theory of X-Ray Diffraction to Holography,” J. Appl. Phys. 38, 3994 (1967).
[CrossRef]

Stone, T.

Su, S. F.

Tipnis, C. B.

W. R. Klein, C. B. Tipnis, E. A. Hiedemann, “Experimental Study of Fraunhofer Light Diffraction by Ultrasonic Beams of Moderately High Frequency at Oblique Incidence,” J. Acoust. Soc. Am. 38, 229 (1965).
[CrossRef]

Upatnieks, J.

Appl. Opt. (4)

Appl. Phys. (1)

R. Alferness, “Analysis of Optical Propagation in Thick Holographgic Gratings,” Appl. Phys. 7, 29 (1975).
[CrossRef]

Appl. Phys. Lett. (1)

N. George, J. W. Matthews, “Holographic Diffraction Gratings,” Appl. Phys. Lett. 9, 212 (1966).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909 (1969).

J. Acoust. Soc. Am. (1)

W. R. Klein, C. B. Tipnis, E. A. Hiedemann, “Experimental Study of Fraunhofer Light Diffraction by Ultrasonic Beams of Moderately High Frequency at Oblique Incidence,” J. Acoust. Soc. Am. 38, 229 (1965).
[CrossRef]

J. Appl. Phys. (1)

E. J. Saccocio, “Application of the Dynamical Theory of X-Ray Diffraction to Holography,” J. Appl. Phys. 38, 3994 (1967).
[CrossRef]

J. Opt. Soc. Am. (17)

B. J. Chang, “Doubly Modulated On-Axis Thick Hologram Optical Elements,” J. Opt. Soc. Am. 67, 1435(A) (1977).

C. B. Burckhardt, “Diffraction of a Plane Wave at a Sinusoidally Stratified Dielectric Grating,” J. Opt. Soc. Am. 56, 1502 (1966).
[CrossRef]

F. G. Kaspar, “Diffraction by Thick, Periodically Stratified Gratings with Complex Dielectric Constant,” J. Opt. Soc. Am. 63, 37 (1973).
[CrossRef]

S. F. Su, T. K. Gaylord, “Calculation of Arbitrary-Order Diffraction Efficiencies of Thick Gratings with Arbitrary Grating Shape,” J. Opt. Soc. Am. 65, 59 (1975).
[CrossRef]

R. Alferness, “Analysis of Propagation at the Second-Order Bragg Angle of a Thick Holographic Grating,” J. Opt. Soc. Am. 66, 353 (1976).
[CrossRef]

D. L. Jaggard, C. Elachi, “Floquet and Coupled-Waves Analysis of Higher-Order Bragg Coupling in a Periodic Medium,” J. Opt. Soc. Am. 66, 674 (1976).
[CrossRef]

J. A. Kong, “Second-Order Coupled Mode Equations for Spatially Periodic Media,” J. Opt. Soc. Am. 67, 825 (1977).
[CrossRef]

R. Magnusson, T. K. Gaylord, “Analysis of Multiwave Diffraction of Thick Gratings,” J. Opt. Soc. Am. 67, 1165 (1977).
[CrossRef]

R. Magnusson, T. K. Gaylord, “Diffraction Regimes of Transmission Gratings,” J. Opt. Soc. Am. 68, 809 (1978).
[CrossRef]

R. Magnusson, T. K. Gaylord, “Equivalence of Multiwave Coupled-Wave Theory and Modal Theory for Periodic-Media Diffraction,” J. Opt. Soc. Am. 68, 1777 (1978).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous Coupled-Wave Analysis of Planar-Grating Diffraction,” J. Opt. Soc. Am. 71, 811 (1981).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous Coupled-Wave Analysis of Grating Diffraction—E-Mode Polarization and Losses,” J. Opt. Soc. Am. 73, 451 (1983).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Three-Dimensional Vector Coupled-Wave Analysis of Planar-Grating Diffraction,” J. Opt. Soc. Am. 73, 1105 (1983).
[CrossRef]

R. C. Enger, S. K. Case, “High-Frequency Holographic Transmission Gratings in Photoresist,” J. Opt. Soc. Am. 73, 1113 (1983).
[CrossRef]

R. Magnusson, T. K. Gaylord, “Diffraction Efficiencies of Thin Phase Gratings with Arbitrary Grating Shape,” J. Opt. Soc. Am. 68, 806 (1978).
[CrossRef]

E. N. Leith, J. Upatnieks, “Reconstructed Wavefronts and Communication Theory,” J. Opt. Soc. Am. 52, 1123 (1962).
[CrossRef]

C. R. Berry, “Turbidity of Monodisperse Suspensions of AgBr,” J. Opt. Soc. Am. 52, 888 (1962).
[CrossRef]

J. Photogr. Sci. (1)

D. H. Napper, R. H. Ottewill, “Studies on the Light Scattering of Silver Bromide Particles,” J. Photogr. Sci. 11, 84 (1963).

Opt. Commun. (1)

R. Alferness, “Equivalence of the Thin-Grating Decomposition and Coupled-Wave Analysis of Thick Holographic Gratings,” Opt. Commun. 15, 209 (1975).
[CrossRef]

Opt. Lett. (1)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

N. George, G. M. Morris, “Matched Filtering in White Light Illumination,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 11 (1983).

Trans. IEEE Microwave Theory Tech. (1)

R. S. Chu, J. A. Kong, “Modal Theory of Spatially Periodic Media,” Trans. IEEE Microwave Theory Tech. MTT-25, 18 (1977).

Other (7)

R. Alferness, “Optical Propagation in Holographic Gratings,” Ph.D. Thesis, U. Michigan, Ann Arbor (1976).

C. D. Leonard, B. D. Guenther, “A Cookbook for Dichromated Gelatin Holograms,” Technical Report T-79-17, U.S. Army Missile Research and Development Command (Redstone Arsenal, 1979).

C. E. K. Mees, T. H. James, The Theory of the Photographic Process (Macmillan, New York, 1966).

H. Kogelnik, “Reconstructing Response and Efficiency of Hologram Gratings,” Proceedings, Symposium on Modern Optics (Polytechnic Institute of Brooklyn, 1967), p. 605.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Eq. (MO-27), p. 973.

T. G. Georgekutty, J. G. Duthie, “Fabrication of Holographic Optical Elements for Achromatic Matched Filtering System,” Technical Report RR-82-2, U.S. Army Missile Command (Redstone Arsenal, 1982).

J. Upatnieks, J. G. Duthie, P. R. Ashley, “Matched Filtering With Achromatic Optical Correlators,” Technical Report RR-82-5, U.S. Army Missile Command (Redstone Arsenal, 1982).

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Figures (16)

Fig. 1
Fig. 1

Volume grating coordinate system. The refractive index varies sinusoidally in the z ˆ direction with planes of constant index inclined at an angle ψ with respect to the y axis.

Fig. 2
Fig. 2

Thin grating decomposition approach to volume grating analysis. The thick element is treated as the cascade of many thin grating slabs, g1gN, which couple energy from each of the incident orders into each of the diffracted orders.

Fig. 3
Fig. 3

First-order diffraction efficiency for a 30° bias phase grating is shown vs wavelength (with no angular detuning) for a range of element thicknesses. As the thickness decreases, the peak efficiency drops, and the sine curve shape loses symmetry.

Fig. 4
Fig. 4

Field angle dependence of first-order diffraction efficiency (with no wavelength detuning) for a 30° bias phase grating and a range of element thicknesses.

Fig. 5
Fig. 5

Diffraction efficiency vs phase modulation as calculated using modified thin grating decomposition (solid curve) compared with published diffraction efficiencies calculated using the multiwave coupled wave approach of Magnusson and Gaylord.20

Fig. 6
Fig. 6

Polar representation of the +1-order grating factor (solid line) and array factor (dashed line) for the case of Bragg resonance (no wavelength or angular detunings) in a 30° bias grating of 7.5-μm thickness.

Fig. 7
Fig. 7

Polar representation of the grating factor (solid line) and array factor (dashed line) for the same grating represented in Fig. 6 but with a wavelength detuning Δλ = +1906 Å. The peak angle of both factors has increased from the original Γ = 30°, but the grating factor has swept at a faster rate resulting in a net detuning of the two factors. In this case, the peak of their products is 0.5, as shown in the inset.

Fig. 8
Fig. 8

Spectral bandwidth curves for the 30° bias holographic grating of 7.5-, 15-, and 30-μm thicknesses. The solid curves represent efficiencies calculated using the thin grating decomposition approach, while the dashed curves are the values of the array factor.

Fig. 9
Fig. 9

Field angle bandwidth curves for the 30° bias holographic phase grating of 7.5-, 15-, and 30-μm thicknesses. As with the spectral bandwidths, the thin grating decomposition values (solid curves) are slightly narrower than the array factor dependencies (dashed curves).

Fig. 10
Fig. 10

Off-axis holographic optical element configurations with bias angle θb: (a) off-axis element; (b) close cascade with Venetian blind insert; (c) separated Δ cascade.

Fig. 11
Fig. 11

Theoretical spectral bandwidth of (a) single grating, (b) two-element cascade, (c) triangle cascade, and (d) dual-vertex triangle cascade. All composite elements are the canonical 30° bias gratings of 7.5-μm thickness with the exception of the vertex element in (c), which has twice the spatial frequency. The spectral bandwidth of configuration (c) is identical to that of configuration (b) regardless of vertex element thickness.

Fig. 12
Fig. 12

Bragg effects in the triangle configuration. The Bragg resonant condition is illustrated by the solid line G. Rays with wavelength detuning are represented by dashed lines R and B.

Fig. 13
Fig. 13

Spectral bandwidth of a two-element cascade with Venetian blind zero-order block insert. Experimentally measured bandwidth (solid line) is from two bleached 6-μm thick elements cladded around 3-M Light Control Film. The dashed curve represents the normalized bandwidth of a theoretical model which ignores scattering.

Fig. 14
Fig. 14

Experimentally measured angular bandwidth of several 3-M Light Control Film materials with varied illumination conditions and detection geometries. Some light is diffracted by the slats in the Light Control Film, as is evidenced by the higher diffuse transmission values than those measured specularly.

Fig. 15
Fig. 15

Spectral bandwidth and short-wavelength loss effects of three-element triangle cascades formed with 6- and 14-μm base elements in bleached silver-halide emulsions.

Fig. 16
Fig. 16

Spectral bandwidth of a triangle cascade formed with 9-μm thick dichromated gelatin base elements and the same bleached vertex element used in the configurations of Fig. 15. The only major scatter/absorption losses are now due to the bleached vertex element, as is evidenced on the blue side of the curve.

Equations (40)

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Q = 2 π λ L Λ 2 cos Ω ;
β a = L / ( Λ cos Ω ) β p = λ / Λ .
β = β a β p = ( L / cos Ω Λ ) ( λ Λ ) .
n ( r ) = n 0 + n 1 cos [ 2 π ( z cos ψ y sin ψ ) / Λ ] ,
n ( r ) = n 0 + n 1 cos [ 2 π f ¯ ( z y tan ψ ) ] ,
n ( z ) = n 0 + n 1 cos [ 2 π f ¯ ( z y n tan ψ ) ] ,
Γ ( z ) = sin 1 { n 0 sin Γ n 0 + n 1 cos [ 2 π Λ z ( z y n tan ψ ) ] } .
E x ( x , y n , z ) = exp ( i k · r ) = exp ( i 2 π λ 0 n 0 z sin Γ ) ,
E x ( x , y n + Δ y , z ) = exp ( i k · r ) = exp [ i 2 π λ 0 n ( z ) ( Δ y cos Γ + z sin Γ ) ] ,
E x ( x , y n + Δ y , z ) = E x ( x , y n , z ) T s g .
T s g = exp ( i 2 π λ 0 { n ( z ) Δ y cos Γ + z [ n ( z ) sin Γ n 0 sin Γ ] } ) .
T s g = exp [ i 2 π λ 0 n ( z ) Δ y cos Γ ] .
T s g = exp ( i 2 π λ 0 n ( z ) Δ y { 1 [ n 0 n ( z ) sin Γ ] 2 } 1 / 2 ) .
T s g exp ( i 2 π λ 0 Δ y n ( z ) cos Γ × { 1 + 2 n 1 n 0 tan 2 Γ cos [ 2 π f ¯ ( z y n tan ψ ) ] } 1 / 2 )
exp ( i 2 π λ 0 Δ y { n 0 cos Γ + n 1 cos [ 2 π f ¯ ( z y n tan ψ ) ] cos Γ } ) ,
T s g = exp ( i 2 π λ 0 n 0 Δ y cos Γ ) × q = i q J q ( 2 π n 1 Δ y λ 0 cos Γ ) exp [ i 2 π q f ¯ ( z y n tan ψ ) ] ,
E x n ( y n + Δ y , z ) = E x n 1 ( y n , z ) T s g ,
E x ( y 1 , z ) 0 = A 0 0 exp [ i 2 π λ 0 n 0 ( z sin Γ 0 0 + y 1 cos Γ 0 0 ) ] ,
E x 1 ( y 1 + Δ y , z ) = A 0 0 exp [ i 2 π λ 0 n 0 ( y 1 + Δ y ) cos Γ 0 0 ] × q = i q J q ( 2 π n 1 Δ y λ 0 cos Γ 0 0 ) × exp [ i 2 π ( f 0 + q f ¯ ) z i 2 π q f ¯ y 1 tan ψ ] .
f q = f 0 + q f ¯ ;
Γ q = sin 1 ( λ f q ) .
E x n 1 ( y n , z ) = q = S R A q n 1 exp [ i 2 π λ 0 n 0 ( z sin Γ q + y n cos Γ q ) ] ,
A n = C n A n 1 ;
A n 1 = [ A S n 1 A 0 n 1 A R n 1 ] and A n = [ A S n A 0 n A R n ] .
E x n 1 ( y n , z ) = q = S R A q n 1 exp [ i 2 π n 0 λ 0 ( z sin Γ q + y n cos Γ q ) ] ,
E x n ( y n + Δ y , z ) = q = S R A q n exp [ i 2 π n 0 λ 0 ( z sin Γ q + y n cos Γ q ) ] .
F = A p n 1 exp { i 2 π [ f p z + y n n 0 cos ( Γ p ) / λ 0 ] + i 2 π n 0 Δ y cos ( Γ p ) / λ 0 } × q = i q J q ( 2 π n 1 Δ y λ 0 cos Γ p ) exp [ i 2 π q f ¯ ( z y n tan ψ ) ] .
c p + q , p n = i q J q ( 2 π n 1 Δ y λ 0 cos Γ p ) exp { i 2 π [ n 0 Δ y cos ( Γ p ) / λ 0 q f ¯ y n tan ψ ] } .
A r n = p = S R c r p n A p n 1 , f o r r = S , , 0 , R .
A N = C N C N 1 C n C 2 C 1 A 0 ,
N ( r 1 ) = N 0 + N 1 cos { 2 π Λ z [ z ( y tan ψ ) ] } ,
E x scat = C N 0 θ 0 T 0 + C N 1 2 ( θ + 1 T + 1 + θ 1 T 1 ) ,
θ q = sinc { L 1 λ γ 1 } sinc { L 2 λ ( γ 3 sin Ω q λ Λ z ) } ,
T q = sinc { L λ ( γ 2 cos Ω + q λ Λ z tan ψ ) } ,
C = E 0 L 1 L 2 L k 2 ( 2 1 ) R 3 r 0 ( 2 + 2 1 ) ( 1 sin 2 θ cos 2 ϕ ) exp [ i ( k r 0 ω t ) ] .
E x ± 1 scat = C N 1 2 θ ± 1 T ± 1 ;
γ 3 sin Ω q λ Λ z = 0
sin Γ G sin Ω = + q λ Λ z
γ 2 cos Ω + q λ Λ z tan ψ = 0
cos Γ A cos Ω = q λ Λ z tan ψ .

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