Abstract

We derive the response of a volume grating to arbitrary small deformations, using a perturbative approach. This result is of interest for two applications: (a) when a deformation is undesirable and one seeks to minimize the diffracted field’s sensitivity to it and (b) when the deformation itself is the quantity of interest and the diffracted field is used as a probe into the deformed volume where the hologram was originally recorded. We show that our result is consistent with previous derivations motivated by the phenomenon of shrinkage in photopolymer holographic materials. We also present the analysis of the grating’s response to deformation due to a point indenter and present experimental results consistent with theory.

© 2005 Optical Society of America

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References

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  1. W. S. Colburn, K. A. Haines, “Volume hologram formation in photopolymer materials,” Appl. Opt. 10, 1636–1641 (1971).
    [CrossRef] [PubMed]
  2. B. L. Booth, “Photopolymer material for holography,” Appl. Opt. 14, 593–601 (1975).
    [CrossRef] [PubMed]
  3. D. H. R. Vilkomerson, D. Bostwick, “Some effects of emulsion shrinkage on a hologram’s image space,” Appl. Opt. 6, 1270–1272 (1967).
    [CrossRef] [PubMed]
  4. P. Hariharan, Optical Holography: Principles, Techniques, and Applications (Cambridge U. Press, 1984).
  5. N. Chen, “Aberrations of volume holographic grating,” Opt. Lett. 10, 472–474 (1985).
    [CrossRef] [PubMed]
  6. D. Gabor, “Microscopy by reconstructed wavefronts,” Proc. R. Soc. London, Ser. A 197, 454–487 (1949).
    [CrossRef]
  7. D. Gabor, “Microscopy by reconstructed wavefronts II,” Proc. Phys. Soc. London, Sect. B 64, 449–469 (1949).
    [CrossRef]
  8. X. Yi, P. Yeh, C. Gu, S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE 87, 1912–1930 (1999).
    [CrossRef]
  9. R. M. Shelby, D. A. Waldman, R. T. Ingwall, “Distortions in pixel-matched holographic data storage due to lateral dimensional change of photopolymer storage media,” Opt. Lett. 25, 713–715 (2000).
    [CrossRef]
  10. D. Psaltis, D. Brady, G. Xiang-Guang, S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
    [CrossRef] [PubMed]
  11. C. Zhao, J. Liu, Z. Fu, R. T. Chen, “Shrinkage-corrected volume holograms based on photopolymeric phase media for surface-normal optical interconnects,” Appl. Phys. Lett. 71, 1464–1466 (1997).
    [CrossRef]
  12. A. Sinha, W. Sun, T. Shi, G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. 43, 1533–1551 (2004).
    [CrossRef] [PubMed]
  13. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, N. Massey, “Holographic data storage in three-dimensional media,” Appl. Opt. 5, 1303–1311 (1966).
    [CrossRef] [PubMed]
  14. A. A. Friesem, J. L. Walker, “Experimental investigations of some anomalies in photographic plates,” Appl. Opt. 8, 1504–1506 (1969).
    [CrossRef] [PubMed]
  15. P. Hariharan, C. M. Chidley, “Rehalogenating bleaches for photographic phase holograms: the influence of halide type and concentration on diffraction efficiency and scattering,” Appl. Opt. 26, 3895–3898 (1987).
    [CrossRef] [PubMed]
  16. P. Hariharan, C. M. Chidley, “Rehalogenating bleaches for photographic phase holograms: II. Spatial frequency effects,” Appl. Opt. 27, 3852–3854 (1988).
    [CrossRef] [PubMed]
  17. J. T. Gallo, C. M. Verber, “Model for the effects of material shrinkage on volume holograms,” Appl. Opt. 33, 6797–6804 (1994).
    [CrossRef] [PubMed]
  18. D. A. Waldman, H.-Y. S. Li, M. G. Horner, “Volume shrinkage in slant fringe gratings of a cationic ring-opening holographic recording material,” J. Imaging Sci. Technol. 41, 497–514 (1997).
  19. C. M. Vest, Holographic Interferometry (Wiley, 1979).
  20. P. K. Rastogi, Holographic Interferometry (Springer-Verlag, 1994).
    [CrossRef]
  21. V. P. Shchepinov, V. S. Pisarev, S. A. Novikov, V. V. Balalov, I. N. Odintsev, M. M. Bondarenko, Strain and Stress Analysis by Holographic and Speckle Interferometry (Wiley, 1996).
  22. P. S. Theocaris, Moire Fringes in Strain Analysis (Pergamon, 1969).
  23. A. J. Durelli, V. J. Parks, Moire Analysis of Strain (Prentice-Hall, 1970).
  24. M. A. Kronrod, L. P. Yaroslavski, N. S. Merzlyakov, “Computer synthesis of transparency holograms,” Sov. Phys. Tech. Phys. 17, 329–332 (1972).
  25. U. Schnars, W. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
    [CrossRef] [PubMed]
  26. Y. Pu, H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197 (2000).
    [CrossRef]
  27. J. Zhang, B. Tao, J. Katz, “Turbulent flow measurements in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–389 (1997).
    [CrossRef]
  28. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  29. H. Coufal, D. Psaltis, and G. Sincerbox, eds., Holographic Data Storage (Springer, 2000).
    [CrossRef]
  30. C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Hermann/Wiley-Interscience, 1977).
  31. K. L. Johnson, Contact Mechanics (Cambridge U. Press, 1985).
    [CrossRef]

2004

2000

1999

X. Yi, P. Yeh, C. Gu, S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE 87, 1912–1930 (1999).
[CrossRef]

1997

J. Zhang, B. Tao, J. Katz, “Turbulent flow measurements in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–389 (1997).
[CrossRef]

C. Zhao, J. Liu, Z. Fu, R. T. Chen, “Shrinkage-corrected volume holograms based on photopolymeric phase media for surface-normal optical interconnects,” Appl. Phys. Lett. 71, 1464–1466 (1997).
[CrossRef]

D. A. Waldman, H.-Y. S. Li, M. G. Horner, “Volume shrinkage in slant fringe gratings of a cationic ring-opening holographic recording material,” J. Imaging Sci. Technol. 41, 497–514 (1997).

1994

1990

D. Psaltis, D. Brady, G. Xiang-Guang, S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

1988

1987

1985

1975

1972

M. A. Kronrod, L. P. Yaroslavski, N. S. Merzlyakov, “Computer synthesis of transparency holograms,” Sov. Phys. Tech. Phys. 17, 329–332 (1972).

1971

1969

1967

1966

1949

D. Gabor, “Microscopy by reconstructed wavefronts,” Proc. R. Soc. London, Ser. A 197, 454–487 (1949).
[CrossRef]

D. Gabor, “Microscopy by reconstructed wavefronts II,” Proc. Phys. Soc. London, Sect. B 64, 449–469 (1949).
[CrossRef]

Balalov, V. V.

V. P. Shchepinov, V. S. Pisarev, S. A. Novikov, V. V. Balalov, I. N. Odintsev, M. M. Bondarenko, Strain and Stress Analysis by Holographic and Speckle Interferometry (Wiley, 1996).

Barbastathis, G.

Bondarenko, M. M.

V. P. Shchepinov, V. S. Pisarev, S. A. Novikov, V. V. Balalov, I. N. Odintsev, M. M. Bondarenko, Strain and Stress Analysis by Holographic and Speckle Interferometry (Wiley, 1996).

Booth, B. L.

Bostwick, D.

Brady, D.

D. Psaltis, D. Brady, G. Xiang-Guang, S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

Campbell, S.

X. Yi, P. Yeh, C. Gu, S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE 87, 1912–1930 (1999).
[CrossRef]

Chen, N.

Chen, R. T.

C. Zhao, J. Liu, Z. Fu, R. T. Chen, “Shrinkage-corrected volume holograms based on photopolymeric phase media for surface-normal optical interconnects,” Appl. Phys. Lett. 71, 1464–1466 (1997).
[CrossRef]

Chidley, C. M.

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Hermann/Wiley-Interscience, 1977).

Colburn, W. S.

Diu, B.

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Hermann/Wiley-Interscience, 1977).

Durelli, A. J.

A. J. Durelli, V. J. Parks, Moire Analysis of Strain (Prentice-Hall, 1970).

Friesem, A. A.

Fu, Z.

C. Zhao, J. Liu, Z. Fu, R. T. Chen, “Shrinkage-corrected volume holograms based on photopolymeric phase media for surface-normal optical interconnects,” Appl. Phys. Lett. 71, 1464–1466 (1997).
[CrossRef]

Gabor, D.

D. Gabor, “Microscopy by reconstructed wavefronts II,” Proc. Phys. Soc. London, Sect. B 64, 449–469 (1949).
[CrossRef]

D. Gabor, “Microscopy by reconstructed wavefronts,” Proc. R. Soc. London, Ser. A 197, 454–487 (1949).
[CrossRef]

Gallo, J. T.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gu, C.

X. Yi, P. Yeh, C. Gu, S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE 87, 1912–1930 (1999).
[CrossRef]

Haines, K. A.

Hariharan, P.

Horner, M. G.

D. A. Waldman, H.-Y. S. Li, M. G. Horner, “Volume shrinkage in slant fringe gratings of a cationic ring-opening holographic recording material,” J. Imaging Sci. Technol. 41, 497–514 (1997).

Ingwall, R. T.

Johnson, K. L.

K. L. Johnson, Contact Mechanics (Cambridge U. Press, 1985).
[CrossRef]

Juptner, W.

Katz, J.

J. Zhang, B. Tao, J. Katz, “Turbulent flow measurements in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–389 (1997).
[CrossRef]

Kozma, A.

Kronrod, M. A.

M. A. Kronrod, L. P. Yaroslavski, N. S. Merzlyakov, “Computer synthesis of transparency holograms,” Sov. Phys. Tech. Phys. 17, 329–332 (1972).

Laloë, F.

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Hermann/Wiley-Interscience, 1977).

Leith, E. N.

Li, H.-Y. S.

D. A. Waldman, H.-Y. S. Li, M. G. Horner, “Volume shrinkage in slant fringe gratings of a cationic ring-opening holographic recording material,” J. Imaging Sci. Technol. 41, 497–514 (1997).

Lin, S.

D. Psaltis, D. Brady, G. Xiang-Guang, S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

Liu, J.

C. Zhao, J. Liu, Z. Fu, R. T. Chen, “Shrinkage-corrected volume holograms based on photopolymeric phase media for surface-normal optical interconnects,” Appl. Phys. Lett. 71, 1464–1466 (1997).
[CrossRef]

Marks, J.

Massey, N.

Meng, H.

Y. Pu, H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197 (2000).
[CrossRef]

Merzlyakov, N. S.

M. A. Kronrod, L. P. Yaroslavski, N. S. Merzlyakov, “Computer synthesis of transparency holograms,” Sov. Phys. Tech. Phys. 17, 329–332 (1972).

Novikov, S. A.

V. P. Shchepinov, V. S. Pisarev, S. A. Novikov, V. V. Balalov, I. N. Odintsev, M. M. Bondarenko, Strain and Stress Analysis by Holographic and Speckle Interferometry (Wiley, 1996).

Odintsev, I. N.

V. P. Shchepinov, V. S. Pisarev, S. A. Novikov, V. V. Balalov, I. N. Odintsev, M. M. Bondarenko, Strain and Stress Analysis by Holographic and Speckle Interferometry (Wiley, 1996).

Parks, V. J.

A. J. Durelli, V. J. Parks, Moire Analysis of Strain (Prentice-Hall, 1970).

Pisarev, V. S.

V. P. Shchepinov, V. S. Pisarev, S. A. Novikov, V. V. Balalov, I. N. Odintsev, M. M. Bondarenko, Strain and Stress Analysis by Holographic and Speckle Interferometry (Wiley, 1996).

Psaltis, D.

D. Psaltis, D. Brady, G. Xiang-Guang, S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

Pu, Y.

Y. Pu, H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197 (2000).
[CrossRef]

Rastogi, P. K.

P. K. Rastogi, Holographic Interferometry (Springer-Verlag, 1994).
[CrossRef]

Schnars, U.

Shchepinov, V. P.

V. P. Shchepinov, V. S. Pisarev, S. A. Novikov, V. V. Balalov, I. N. Odintsev, M. M. Bondarenko, Strain and Stress Analysis by Holographic and Speckle Interferometry (Wiley, 1996).

Shelby, R. M.

Shi, T.

Sinha, A.

Sun, W.

Tao, B.

J. Zhang, B. Tao, J. Katz, “Turbulent flow measurements in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–389 (1997).
[CrossRef]

Theocaris, P. S.

P. S. Theocaris, Moire Fringes in Strain Analysis (Pergamon, 1969).

Upatnieks, J.

Verber, C. M.

Vest, C. M.

C. M. Vest, Holographic Interferometry (Wiley, 1979).

Vilkomerson, D. H. R.

Waldman, D. A.

R. M. Shelby, D. A. Waldman, R. T. Ingwall, “Distortions in pixel-matched holographic data storage due to lateral dimensional change of photopolymer storage media,” Opt. Lett. 25, 713–715 (2000).
[CrossRef]

D. A. Waldman, H.-Y. S. Li, M. G. Horner, “Volume shrinkage in slant fringe gratings of a cationic ring-opening holographic recording material,” J. Imaging Sci. Technol. 41, 497–514 (1997).

Walker, J. L.

Xiang-Guang, G.

D. Psaltis, D. Brady, G. Xiang-Guang, S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

Yaroslavski, L. P.

M. A. Kronrod, L. P. Yaroslavski, N. S. Merzlyakov, “Computer synthesis of transparency holograms,” Sov. Phys. Tech. Phys. 17, 329–332 (1972).

Yeh, P.

X. Yi, P. Yeh, C. Gu, S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE 87, 1912–1930 (1999).
[CrossRef]

Yi, X.

X. Yi, P. Yeh, C. Gu, S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE 87, 1912–1930 (1999).
[CrossRef]

Zhang, J.

J. Zhang, B. Tao, J. Katz, “Turbulent flow measurements in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–389 (1997).
[CrossRef]

Zhao, C.

C. Zhao, J. Liu, Z. Fu, R. T. Chen, “Shrinkage-corrected volume holograms based on photopolymeric phase media for surface-normal optical interconnects,” Appl. Phys. Lett. 71, 1464–1466 (1997).
[CrossRef]

Appl. Opt.

E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, N. Massey, “Holographic data storage in three-dimensional media,” Appl. Opt. 5, 1303–1311 (1966).
[CrossRef] [PubMed]

B. L. Booth, “Photopolymer material for holography,” Appl. Opt. 14, 593–601 (1975).
[CrossRef] [PubMed]

P. Hariharan, C. M. Chidley, “Rehalogenating bleaches for photographic phase holograms: the influence of halide type and concentration on diffraction efficiency and scattering,” Appl. Opt. 26, 3895–3898 (1987).
[CrossRef] [PubMed]

P. Hariharan, C. M. Chidley, “Rehalogenating bleaches for photographic phase holograms: II. Spatial frequency effects,” Appl. Opt. 27, 3852–3854 (1988).
[CrossRef] [PubMed]

U. Schnars, W. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
[CrossRef] [PubMed]

J. T. Gallo, C. M. Verber, “Model for the effects of material shrinkage on volume holograms,” Appl. Opt. 33, 6797–6804 (1994).
[CrossRef] [PubMed]

W. S. Colburn, K. A. Haines, “Volume hologram formation in photopolymer materials,” Appl. Opt. 10, 1636–1641 (1971).
[CrossRef] [PubMed]

A. Sinha, W. Sun, T. Shi, G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. 43, 1533–1551 (2004).
[CrossRef] [PubMed]

A. A. Friesem, J. L. Walker, “Experimental investigations of some anomalies in photographic plates,” Appl. Opt. 8, 1504–1506 (1969).
[CrossRef] [PubMed]

D. H. R. Vilkomerson, D. Bostwick, “Some effects of emulsion shrinkage on a hologram’s image space,” Appl. Opt. 6, 1270–1272 (1967).
[CrossRef] [PubMed]

Appl. Phys. Lett.

C. Zhao, J. Liu, Z. Fu, R. T. Chen, “Shrinkage-corrected volume holograms based on photopolymeric phase media for surface-normal optical interconnects,” Appl. Phys. Lett. 71, 1464–1466 (1997).
[CrossRef]

Exp. Fluids

Y. Pu, H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197 (2000).
[CrossRef]

J. Zhang, B. Tao, J. Katz, “Turbulent flow measurements in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–389 (1997).
[CrossRef]

J. Imaging Sci. Technol.

D. A. Waldman, H.-Y. S. Li, M. G. Horner, “Volume shrinkage in slant fringe gratings of a cationic ring-opening holographic recording material,” J. Imaging Sci. Technol. 41, 497–514 (1997).

Nature

D. Psaltis, D. Brady, G. Xiang-Guang, S. Lin, “Holography in artificial neural networks,” Nature 343, 325–330 (1990).
[CrossRef] [PubMed]

Opt. Lett.

Proc. IEEE

X. Yi, P. Yeh, C. Gu, S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE 87, 1912–1930 (1999).
[CrossRef]

Proc. Phys. Soc. London, Sect. B

D. Gabor, “Microscopy by reconstructed wavefronts II,” Proc. Phys. Soc. London, Sect. B 64, 449–469 (1949).
[CrossRef]

Proc. R. Soc. London, Ser. A

D. Gabor, “Microscopy by reconstructed wavefronts,” Proc. R. Soc. London, Ser. A 197, 454–487 (1949).
[CrossRef]

Sov. Phys. Tech. Phys.

M. A. Kronrod, L. P. Yaroslavski, N. S. Merzlyakov, “Computer synthesis of transparency holograms,” Sov. Phys. Tech. Phys. 17, 329–332 (1972).

Other

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

H. Coufal, D. Psaltis, and G. Sincerbox, eds., Holographic Data Storage (Springer, 2000).
[CrossRef]

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Hermann/Wiley-Interscience, 1977).

K. L. Johnson, Contact Mechanics (Cambridge U. Press, 1985).
[CrossRef]

P. Hariharan, Optical Holography: Principles, Techniques, and Applications (Cambridge U. Press, 1984).

C. M. Vest, Holographic Interferometry (Wiley, 1979).

P. K. Rastogi, Holographic Interferometry (Springer-Verlag, 1994).
[CrossRef]

V. P. Shchepinov, V. S. Pisarev, S. A. Novikov, V. V. Balalov, I. N. Odintsev, M. M. Bondarenko, Strain and Stress Analysis by Holographic and Speckle Interferometry (Wiley, 1996).

P. S. Theocaris, Moire Fringes in Strain Analysis (Pergamon, 1969).

A. J. Durelli, V. J. Parks, Moire Analysis of Strain (Prentice-Hall, 1970).

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Figures (14)

Fig. 1
Fig. 1

Deformation of holograms.

Fig. 2
Fig. 2

Fourier geometry with plane-wave reference and plane-wave signal.

Fig. 3
Fig. 3

K-sphere explanation of condition (19).

Fig. 4
Fig. 4

Locus of maximum Bragg mismatch.

Fig. 5
Fig. 5

Illumination of the deformation when a point load is exerted on the half-space.

Fig. 6
Fig. 6

Experiment geometry when a point load is exerted on a transmission hologram.

Fig. 7
Fig. 7

Simulated and experimental results when a point load is exerted on a transmission hologram. Parameters are the wavelength λ = 488 nm , the angle of reference beam θ f = 7.5 ° , the angle of signal beam θ S = 20 ° , the thickness of the hologram L = 2 mm , the estimated force P = 700 N , and the focal length of the Fourier lens F = 400 mm . The intensities before and after deformation were each normalized by their own maximum. The maximum intensity after deformation is 19.65% of the maximum intensity before deformation.

Fig. 8
Fig. 8

Fringe patterns of a transmission hologram due to point load; the parameters are the same as those for Fig. 7.

Fig. 9
Fig. 9

K-sphere explanation for the twin peaks.

Fig. 10
Fig. 10

Experiment geometry when a point load is exerted on a reflection hologram.

Fig. 11
Fig. 11

Simulated and experimental results when a point load is exerted on a reflection hologram. Parameters are the wavelength λ = 632 nm , the angle of reference beam θ f = 172 ° , the angle of signal beam θ S = 8 ° , the thickness of the hologram L = 1.5 mm , the estimated force P = 19 N , and the focal length of the Fourier lens F = 400 mm . The intensities before and after deformation were each normalized by their own maximum. The maximum intensity after deformation is 34.55% of the maximum intensity before deformation.

Fig. 12
Fig. 12

Fringe patterns of a reflection hologram due to point load; the parameters are the same as those for Fig. 11.

Fig. 13
Fig. 13

K-sphere explanation for the “triplet peak.”

Fig. 14
Fig. 14

Calculation of the locus of maximum Bragg mismatch.

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

Δ ϵ ( x , y , z ) = Δ ϵ ( f x ( x , y , z ) , f y ( x , y , z ) , f z ( x , y , z ) ) ,
Δ ϵ ( r ) = Δ ϵ ( f ( r ) ) .
Δ ϵ ( f ( r ) ) Δ ϵ ( r ) + j = 1 N { [ f ( r ) r ] } j j ! Δ ϵ ( r ) .
Δ ϵ ( r ) = ϵ 1 exp ( i K g r ) ,
j Δ ϵ ( r ) = ( i K g ) j Δ ϵ ( r ) ,
i K g .
Δ ϵ ( f ( r ) ) Δ ϵ ( r ) + j = 1 N [ Δ f ( r ) i K g ] j j ! Δ ϵ ( r ) ,
Δ f ( r ) = ( Δ f x ( x , y , z ) Δ f y ( x , y , z ) Δ f z ( x , y , z ) ) = [ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] ( x y z ) = A r ,
Δ ϵ ( f ( r ) ) Δ ϵ ( r ) + j = 1 N ( K g A i r ) j j ! Δ ϵ ( r ) .
K g j Δ ϵ ( r ) = ( i r ) j Δ ϵ ( r ) ,
i r K g .
Δ ϵ ( f ( r ) ) Δ ϵ ( r ) + j = 1 N ( K g A K g ) j j ! Δ ϵ ( r ) .
E d ( r ) = E p ( r ) Δ ϵ ( r ) V ( r ) exp ( i 2 π λ x x + y y F ) exp [ i 2 π λ ( 1 x 2 + y 2 2 F 2 ) z ] d 3 r ,
V ( r ) = { 1 inside volume hologram 0 outside volume hologram } ,
E ̃ d ( r ) = E p ( r ) Δ ϵ ( r ) V ̃ ( r ) exp ( i 2 π λ x x + y y F ) × exp [ i 2 π λ ( 1 x 2 + y 2 2 F 2 ) z ] d 3 r .
E ̃ d ( r ) E d ( r ) + E p ( r ) j = 1 N ( K g A K g ) j j ! Δ ϵ ( r ) × V ( r ) exp ( i 2 π λ x x + y y F ) exp [ i 2 π λ ( 1 x 2 + y 2 2 F 2 ) z ] d 3 r .
E ̃ d ( r ) E d ( r ) + j = 1 N ( K g A K g ) j j ! E d ( r ) .
E ̃ d ( r ) E d ( r ) + K g A K g E d ( r ) ,
( A T K g ) K g .
sin θ δ = k sin θ S k δ ,
E d ( r ) = sinc { L 2 π [ K g z + k p z k 2 ( K g x + k p x ) 2 ( K g y + k p y ) 2 ] } exp { i [ ( K g x + k p x ) x + ( K g y + k p y ) y + k 2 ( K g x + k p x ) 2 ( K g y + k p y ) 2 z ] } ,
K g E d ( r ) ( K g x + k p x ) x ̂ + ( K g y + k p y ) y ̂ + k 2 ( K g x + k p x ) 2 ( K g y + k p y ) 2 z ̂ ,
L = ( 1 s ) L ,
A = [ 0 0 0 0 0 0 0 0 s ] .
E ̃ d ( r ) E d ( r ) + j = 1 N ( K g z s K g z ) j j ! E d ( r ) = sinc { L 2 π [ ( 1 + s ) K g z + k g z k 2 ( K g x + k p x ) 2 ( K g y + k p y ) 2 ] } exp { i [ ( K g x + k p x ) x + ( K g y + k p y ) y + k 2 ( K g x + k p x ) 2 ( K g y + k p y ) 2 z ] } ,
u x = P 4 π G [ x z ρ 3 ( 1 2 ν ) x ρ ( ρ + z ) ] ,
u y = P 4 π G [ y z ρ 3 ( 1 2 ν ) y ρ ( ρ + z ) ] ,
u z = P 4 π G [ z 2 ρ 3 + 2 ( 1 ν ) ρ ] ,
Δ ϵ ( r ) = a ( r ) Δ ϵ ( f ( r ) ) ,
E ̃ d ( r ) E ̂ d ( r ) + E p ( r ) a ( r ) j = 1 N ( K g A K g ) j j ! Δ ϵ ( r ) × V ( r ) exp ( i 2 π λ x x + y y F ) exp [ i 2 π λ ( 1 x 2 + y 2 2 F 2 ) z ] d 3 r ,
E ̂ d ( r ) = E p ( r ) a ( r ) Δ ϵ ( r ) V ( r ) exp ( i 2 π λ x x + y y F ) exp [ i 2 π λ ( 1 x 2 + y 2 2 F 2 ) z ] d 3 r ,
E ̃ d ( r ) E ̂ d ( r ) + j = 1 N ( K g A K g ) j j ! E ̂ d ( r ) .
Δ ϵ ( r ) = m , n , l ϵ m n l exp [ i 2 π ( m x L x + n y L y + l z L z ) ] = m , n , l ϵ m n l exp [ i K g ( m n l ) r ] ,
Δ ϵ ( f ( r ) ) Δ ϵ ( r ) + m , n , l j = 1 N [ K g ( m n l ) A K g ( m n l ) ] j j ! Δ ϵ m n l ( r ) ,
Δ ϵ m n l ( r ) = ϵ m n l exp ( i K g ( m n l ) r ) .
E ̃ d ( r ) E d ( r ) + m , n , l E p ( r ) j = 1 N [ K g ( m n l ) A K g ( m n l ) ] j j ! Δ ϵ m n l ( r ) V ( r ) exp ( i 2 π λ x x + y y F ) exp [ i 2 π λ ( 1 x 2 + y 2 2 F 2 ) z ] d 3 r .
E ̃ d ( r ) E d ( r ) + m , n , l j = 1 N [ K g ( m n l ) A K g ( m n l ) ] j j ! E d ( m n l ) ( r ) ,
E d ( m n l ) ( r ) = E p ( r ) Δ ϵ m n l ( r ) V ( r ) exp ( i 2 π λ x x + y y F ) exp [ i 2 π λ ( 1 x 2 + y 2 2 F 2 ) z ] d 3 r ,
E d ( r ) = m , n , l E d ( m n l ) ( r ) .
z 2 + x 2 = k 2 ,
( z k cos θ S ) 2 + ( x k sin θ S ) 2 = δ 2 .
Δ K g ( x ) = k cos θ S + δ 2 ( x k sin θ S ) 2 k 2 x 2 .
Δ K g ( x ) x = x k sin θ S δ 2 ( x k sin θ S ) 2 + x k 2 x 2 = 0 .
x max = k 2 sin θ S k δ ,
sin θ δ = x max k sin θ S δ = ± k sin θ S k δ .
lim δ 0 sin θ δ = ± sin θ S .

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