Abstract

The diffraction properties of fixed volume gratings, including the effect of additional photorefractive energy coupling between the incident wave and the diffracted wave, are considered. Both transmission gratings and reflection gratings are treated. Analytical solutions for phase gratings are derived, and numerical solutions for absorption gratings are also obtained. The results are presented and discussed.

© 1990 Optical Society of America

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References

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  1. See, for example, G. C. Valley and P. Yeh, eds., Feature on Photorefractive Materials, Effects, and Devices, J. Opt. Soc. Am. B 5, 1681–1821 (1988). Many papers on photorefracts are included.
    [Crossref]
  2. See, for example, P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
    [Crossref]
  3. See, for example, P. Yeh, A. E. Chiou, J. Hong, P. Beckwith, T. Chang, and M. Khoshnevisan, “Photorefractive nonlinear optics and optical computing,” Opt. Eng. 28, 328–343 (1989).
    [Crossref]
  4. P. J. van Heerden, “Theory of optical information storage in solids,” Appl. Opt. 2, 393–400 (1963).
    [Crossref]
  5. A. E. T. Chiou and P. Yeh, “Parallel image subtraction using a phase-conjugate Michelson interferometer,” Opt. Lett. 11, 306–308 (1986).
    [Crossref] [PubMed]
  6. T. Y. Chang, P. H. Beckwith, and P. Yeh, “Real-time optical image subtraction using dynamic holographic interference in photorefractive media,” Opt. Lett. 13, 586–588 (1988).
    [Crossref] [PubMed]
  7. P. Yeh, A. E. T. Chiou, and J. Hong, “Optical interconnections using photorefractive dynamic holograms,” Appl. Opt. 27, 2093–2096 (1988).
    [Crossref] [PubMed]
  8. A. Chiou and P. Yeh, “Energy efficiency of optical interconnections using photorefractive holograms,” Appl. Opt. 29, 1111–1117 (1990).
    [Crossref] [PubMed]
  9. H. Lee, X. G. Gu, and D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
    [Crossref]
  10. D. Psaltis, X. G. Gu, and D. Brady, “Holographic implementations of neural networks,” in An Introduction to Neural and Electronic Networks, S. F. Zornetzer, J. L. Davis, and C. Lau, eds. (Academic, New York, 1990), pp. 339–348.
  11. D. Psaltis, D. Brady, X. G. Gu, and S. Lin, “Holography in artificial neural networks,” Nature (London) 343, 325–330 (1990).
    [Crossref]
  12. D. L. Staebler and J. J. Amodei, “Thermally fixed holograms in LiNbO3,” Ferroelectrics 3, 107–113 (1972).
    [Crossref]
  13. F. Micheron and G. Bismuth, “Electrical control of fixation and erasure of holographic patterns in ferroelectric materials,” Appl. Phys. Lett. 20, 79–81 (1972).
    [Crossref]
  14. F. Micheron, C. Mayeux, and J. C. Trotier, “Electrical control in photoferroelectric materials for optical storage,” Appl. Opt. 13, 784–787 (1974).
    [Crossref] [PubMed]
  15. D. von der Linde, A. M. Glass, and K. F. Rodgers, “High-sensitivity optical recording in KTN by two-photon absorption,” Appl. Phys. Lett. 26, 22–24 (1975).
    [Crossref]
  16. A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, and J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
    [Crossref]
  17. J. Hong, S. Campbell, and P. Yeh, “Optical pattern classifier with perceptron learning,” Appl. Opt. 29, 3019–3025 (1990).
    [Crossref] [PubMed]
  18. J. Hong, S. Campbell, and P. Yeh, “Optical learning machine for pattern classification,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), paper WJ3.
  19. D. Psaltis, D. Brady, and K. Wagner, “Adaptive optical networks using photorefractive crystals,” Appl. Opt. 27, 1752–1759 (1988).
    [Crossref]
  20. E. G. Paek, J. Wullert, and J. S. Patel, “Holographic implementation of a learning-machine based on a multicategory perceptron algorithm,” Opt. Lett. 14, 1303–1305 (1989).
    [Crossref] [PubMed]
  21. D. L. Staebler and J. J. Amodei, Coupled-wave analysis of holographic storage in LiNbO3,” J. Appl. Phys. 43, 1042–1049 (1972).
    [Crossref]
  22. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2945 (1969).
    [Crossref]
  23. R. Saxena, F. Vachss, I. McMichael, and P. Yeh, “Diffraction properties of multiple-beam photorefractive gratings,” J. Opt. Soc. Am. B 7, 1210–1215 (1990).
    [Crossref]
  24. T. Chang and P. Yeh, “Dark rings from photorefractive conical diffraction in a BaTiO3crystal,” Proc. Soc. Photo-Opt. Instrum. Eng. 739, 109–116 (1987).
  25. N. A. Vainos, S. L. Clapham, and R. W. Eason, “Multiplexed permanent and real-time holographic recording in photorefractive BSO,” Appl. Opt. 28, 4381–4385 (1989).
    [Crossref] [PubMed]
  26. P. Yeh and M. Khoshnevisan, “Nonlinear-optical Bragg scattering in Kerr media,” J. Opt. Soc. Am. B 4, 1954–1957 (1987). We believe that there may be a slight error in this reference (typesetting error in the asymptotic form of the diffraction efficiency for b ≫ 1).
    [Crossref]
  27. F. Vachss, I. McMichael, M. Khoshnevisan, and P. Yeh, “Enhanced acousto-optic diffraction in electrostrictive media,” J. Opt. Soc. Am. B 7, 859–867 (1990).
    [Crossref]

1990 (5)

1989 (6)

A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, and J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
[Crossref]

N. A. Vainos, S. L. Clapham, and R. W. Eason, “Multiplexed permanent and real-time holographic recording in photorefractive BSO,” Appl. Opt. 28, 4381–4385 (1989).
[Crossref] [PubMed]

E. G. Paek, J. Wullert, and J. S. Patel, “Holographic implementation of a learning-machine based on a multicategory perceptron algorithm,” Opt. Lett. 14, 1303–1305 (1989).
[Crossref] [PubMed]

H. Lee, X. G. Gu, and D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[Crossref]

See, for example, P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[Crossref]

See, for example, P. Yeh, A. E. Chiou, J. Hong, P. Beckwith, T. Chang, and M. Khoshnevisan, “Photorefractive nonlinear optics and optical computing,” Opt. Eng. 28, 328–343 (1989).
[Crossref]

1988 (4)

1987 (2)

1986 (1)

1975 (1)

D. von der Linde, A. M. Glass, and K. F. Rodgers, “High-sensitivity optical recording in KTN by two-photon absorption,” Appl. Phys. Lett. 26, 22–24 (1975).
[Crossref]

1974 (1)

1972 (3)

D. L. Staebler and J. J. Amodei, “Thermally fixed holograms in LiNbO3,” Ferroelectrics 3, 107–113 (1972).
[Crossref]

F. Micheron and G. Bismuth, “Electrical control of fixation and erasure of holographic patterns in ferroelectric materials,” Appl. Phys. Lett. 20, 79–81 (1972).
[Crossref]

D. L. Staebler and J. J. Amodei, Coupled-wave analysis of holographic storage in LiNbO3,” J. Appl. Phys. 43, 1042–1049 (1972).
[Crossref]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2945 (1969).
[Crossref]

1963 (1)

Amodei, J. J.

D. L. Staebler and J. J. Amodei, “Thermally fixed holograms in LiNbO3,” Ferroelectrics 3, 107–113 (1972).
[Crossref]

D. L. Staebler and J. J. Amodei, Coupled-wave analysis of holographic storage in LiNbO3,” J. Appl. Phys. 43, 1042–1049 (1972).
[Crossref]

Beckwith, P.

See, for example, P. Yeh, A. E. Chiou, J. Hong, P. Beckwith, T. Chang, and M. Khoshnevisan, “Photorefractive nonlinear optics and optical computing,” Opt. Eng. 28, 328–343 (1989).
[Crossref]

Beckwith, P. H.

Bismuth, G.

F. Micheron and G. Bismuth, “Electrical control of fixation and erasure of holographic patterns in ferroelectric materials,” Appl. Phys. Lett. 20, 79–81 (1972).
[Crossref]

Brady, D.

D. Psaltis, D. Brady, X. G. Gu, and S. Lin, “Holography in artificial neural networks,” Nature (London) 343, 325–330 (1990).
[Crossref]

D. Psaltis, D. Brady, and K. Wagner, “Adaptive optical networks using photorefractive crystals,” Appl. Opt. 27, 1752–1759 (1988).
[Crossref]

D. Psaltis, X. G. Gu, and D. Brady, “Holographic implementations of neural networks,” in An Introduction to Neural and Electronic Networks, S. F. Zornetzer, J. L. Davis, and C. Lau, eds. (Academic, New York, 1990), pp. 339–348.

Campbell, S.

J. Hong, S. Campbell, and P. Yeh, “Optical pattern classifier with perceptron learning,” Appl. Opt. 29, 3019–3025 (1990).
[Crossref] [PubMed]

J. Hong, S. Campbell, and P. Yeh, “Optical learning machine for pattern classification,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), paper WJ3.

Chang, T.

See, for example, P. Yeh, A. E. Chiou, J. Hong, P. Beckwith, T. Chang, and M. Khoshnevisan, “Photorefractive nonlinear optics and optical computing,” Opt. Eng. 28, 328–343 (1989).
[Crossref]

T. Chang and P. Yeh, “Dark rings from photorefractive conical diffraction in a BaTiO3crystal,” Proc. Soc. Photo-Opt. Instrum. Eng. 739, 109–116 (1987).

Chang, T. Y.

Chiou, A.

Chiou, A. E.

See, for example, P. Yeh, A. E. Chiou, J. Hong, P. Beckwith, T. Chang, and M. Khoshnevisan, “Photorefractive nonlinear optics and optical computing,” Opt. Eng. 28, 328–343 (1989).
[Crossref]

Chiou, A. E. T.

Clapham, S. L.

Delboulbe, A.

A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, and J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
[Crossref]

Eason, R. W.

Fromont, C.

A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, and J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
[Crossref]

Glass, A. M.

D. von der Linde, A. M. Glass, and K. F. Rodgers, “High-sensitivity optical recording in KTN by two-photon absorption,” Appl. Phys. Lett. 26, 22–24 (1975).
[Crossref]

Gu, X. G.

D. Psaltis, D. Brady, X. G. Gu, and S. Lin, “Holography in artificial neural networks,” Nature (London) 343, 325–330 (1990).
[Crossref]

H. Lee, X. G. Gu, and D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[Crossref]

D. Psaltis, X. G. Gu, and D. Brady, “Holographic implementations of neural networks,” in An Introduction to Neural and Electronic Networks, S. F. Zornetzer, J. L. Davis, and C. Lau, eds. (Academic, New York, 1990), pp. 339–348.

Herriau, J. P.

A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, and J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
[Crossref]

Hong, J.

J. Hong, S. Campbell, and P. Yeh, “Optical pattern classifier with perceptron learning,” Appl. Opt. 29, 3019–3025 (1990).
[Crossref] [PubMed]

See, for example, P. Yeh, A. E. Chiou, J. Hong, P. Beckwith, T. Chang, and M. Khoshnevisan, “Photorefractive nonlinear optics and optical computing,” Opt. Eng. 28, 328–343 (1989).
[Crossref]

P. Yeh, A. E. T. Chiou, and J. Hong, “Optical interconnections using photorefractive dynamic holograms,” Appl. Opt. 27, 2093–2096 (1988).
[Crossref] [PubMed]

J. Hong, S. Campbell, and P. Yeh, “Optical learning machine for pattern classification,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), paper WJ3.

Huignard, J. P.

A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, and J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
[Crossref]

Khoshnevisan, M.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2945 (1969).
[Crossref]

Lee, H.

H. Lee, X. G. Gu, and D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[Crossref]

Lin, S.

D. Psaltis, D. Brady, X. G. Gu, and S. Lin, “Holography in artificial neural networks,” Nature (London) 343, 325–330 (1990).
[Crossref]

Mallick, S.

A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, and J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
[Crossref]

Mayeux, C.

McMichael, I.

Micheron, F.

F. Micheron, C. Mayeux, and J. C. Trotier, “Electrical control in photoferroelectric materials for optical storage,” Appl. Opt. 13, 784–787 (1974).
[Crossref] [PubMed]

F. Micheron and G. Bismuth, “Electrical control of fixation and erasure of holographic patterns in ferroelectric materials,” Appl. Phys. Lett. 20, 79–81 (1972).
[Crossref]

Paek, E. G.

Patel, J. S.

Psaltis, D.

D. Psaltis, D. Brady, X. G. Gu, and S. Lin, “Holography in artificial neural networks,” Nature (London) 343, 325–330 (1990).
[Crossref]

H. Lee, X. G. Gu, and D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[Crossref]

D. Psaltis, D. Brady, and K. Wagner, “Adaptive optical networks using photorefractive crystals,” Appl. Opt. 27, 1752–1759 (1988).
[Crossref]

D. Psaltis, X. G. Gu, and D. Brady, “Holographic implementations of neural networks,” in An Introduction to Neural and Electronic Networks, S. F. Zornetzer, J. L. Davis, and C. Lau, eds. (Academic, New York, 1990), pp. 339–348.

Rodgers, K. F.

D. von der Linde, A. M. Glass, and K. F. Rodgers, “High-sensitivity optical recording in KTN by two-photon absorption,” Appl. Phys. Lett. 26, 22–24 (1975).
[Crossref]

Saxena, R.

Staebler, D. L.

D. L. Staebler and J. J. Amodei, Coupled-wave analysis of holographic storage in LiNbO3,” J. Appl. Phys. 43, 1042–1049 (1972).
[Crossref]

D. L. Staebler and J. J. Amodei, “Thermally fixed holograms in LiNbO3,” Ferroelectrics 3, 107–113 (1972).
[Crossref]

Trotier, J. C.

Vachss, F.

Vainos, N. A.

van Heerden, P. J.

von der Linde, D.

D. von der Linde, A. M. Glass, and K. F. Rodgers, “High-sensitivity optical recording in KTN by two-photon absorption,” Appl. Phys. Lett. 26, 22–24 (1975).
[Crossref]

Wagner, K.

Wullert, J.

Yeh, P.

A. Chiou and P. Yeh, “Energy efficiency of optical interconnections using photorefractive holograms,” Appl. Opt. 29, 1111–1117 (1990).
[Crossref] [PubMed]

F. Vachss, I. McMichael, M. Khoshnevisan, and P. Yeh, “Enhanced acousto-optic diffraction in electrostrictive media,” J. Opt. Soc. Am. B 7, 859–867 (1990).
[Crossref]

R. Saxena, F. Vachss, I. McMichael, and P. Yeh, “Diffraction properties of multiple-beam photorefractive gratings,” J. Opt. Soc. Am. B 7, 1210–1215 (1990).
[Crossref]

J. Hong, S. Campbell, and P. Yeh, “Optical pattern classifier with perceptron learning,” Appl. Opt. 29, 3019–3025 (1990).
[Crossref] [PubMed]

See, for example, P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[Crossref]

See, for example, P. Yeh, A. E. Chiou, J. Hong, P. Beckwith, T. Chang, and M. Khoshnevisan, “Photorefractive nonlinear optics and optical computing,” Opt. Eng. 28, 328–343 (1989).
[Crossref]

T. Y. Chang, P. H. Beckwith, and P. Yeh, “Real-time optical image subtraction using dynamic holographic interference in photorefractive media,” Opt. Lett. 13, 586–588 (1988).
[Crossref] [PubMed]

P. Yeh, A. E. T. Chiou, and J. Hong, “Optical interconnections using photorefractive dynamic holograms,” Appl. Opt. 27, 2093–2096 (1988).
[Crossref] [PubMed]

T. Chang and P. Yeh, “Dark rings from photorefractive conical diffraction in a BaTiO3crystal,” Proc. Soc. Photo-Opt. Instrum. Eng. 739, 109–116 (1987).

P. Yeh and M. Khoshnevisan, “Nonlinear-optical Bragg scattering in Kerr media,” J. Opt. Soc. Am. B 4, 1954–1957 (1987). We believe that there may be a slight error in this reference (typesetting error in the asymptotic form of the diffraction efficiency for b ≫ 1).
[Crossref]

A. E. T. Chiou and P. Yeh, “Parallel image subtraction using a phase-conjugate Michelson interferometer,” Opt. Lett. 11, 306–308 (1986).
[Crossref] [PubMed]

J. Hong, S. Campbell, and P. Yeh, “Optical learning machine for pattern classification,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), paper WJ3.

Appl. Opt. (7)

Appl. Phys. Lett. (3)

D. von der Linde, A. M. Glass, and K. F. Rodgers, “High-sensitivity optical recording in KTN by two-photon absorption,” Appl. Phys. Lett. 26, 22–24 (1975).
[Crossref]

A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, and J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
[Crossref]

F. Micheron and G. Bismuth, “Electrical control of fixation and erasure of holographic patterns in ferroelectric materials,” Appl. Phys. Lett. 20, 79–81 (1972).
[Crossref]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2945 (1969).
[Crossref]

Ferroelectrics (1)

D. L. Staebler and J. J. Amodei, “Thermally fixed holograms in LiNbO3,” Ferroelectrics 3, 107–113 (1972).
[Crossref]

IEEE J. Quantum Electron. (1)

See, for example, P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[Crossref]

J. Appl. Phys. (2)

H. Lee, X. G. Gu, and D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[Crossref]

D. L. Staebler and J. J. Amodei, Coupled-wave analysis of holographic storage in LiNbO3,” J. Appl. Phys. 43, 1042–1049 (1972).
[Crossref]

J. Opt. Soc. Am. B (4)

Nature (London) (1)

D. Psaltis, D. Brady, X. G. Gu, and S. Lin, “Holography in artificial neural networks,” Nature (London) 343, 325–330 (1990).
[Crossref]

Opt. Eng. (1)

See, for example, P. Yeh, A. E. Chiou, J. Hong, P. Beckwith, T. Chang, and M. Khoshnevisan, “Photorefractive nonlinear optics and optical computing,” Opt. Eng. 28, 328–343 (1989).
[Crossref]

Opt. Lett. (3)

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

T. Chang and P. Yeh, “Dark rings from photorefractive conical diffraction in a BaTiO3crystal,” Proc. Soc. Photo-Opt. Instrum. Eng. 739, 109–116 (1987).

Other (2)

D. Psaltis, X. G. Gu, and D. Brady, “Holographic implementations of neural networks,” in An Introduction to Neural and Electronic Networks, S. F. Zornetzer, J. L. Davis, and C. Lau, eds. (Academic, New York, 1990), pp. 339–348.

J. Hong, S. Campbell, and P. Yeh, “Optical learning machine for pattern classification,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), paper WJ3.

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Figures (12)

Fig. 1
Fig. 1

Schematic drawings of incident and scattered beams in the case of (a) transmission gratings and (b) reflection gratings.

Fig. 2
Fig. 2

Diffraction efficiency as a function of b for transmission phase gratings with different values of κL. Solid curve, κL = π/6; dotted–dashed curve, κL = π/2; dashed curve, κL = π.

Fig. 3
Fig. 3

Grating amplitudes as functions of z for b equal to 10, 1, 0, −1, and −10 in the case of transmission phase gratings.

Fig. 4
Fig. 4

Intensities of the incident and diffracted beams as functions of (α/4)z for transmission absorption gratings (κ = 0) with various values of b. Solid curves, I1; dotted curves, I2. (a) b = 1, (b) b = 0, (c) b = −1.

Fig. 5
Fig. 5

Diffraction efficiency as a function of b for transmission absorption gratings (κ = 0) with various values of αL. Solid curve, αL = 0.2π; dotted curve, αL = 0.4π; dashed curve, αL = 0.8π.

Fig. 6
Fig. 6

Intensities of the incident and diffracted beams as functions of z for fixed gratings with both phase and absorption variations [κ = cos(π/30) = 0.9945, α = 4 sin(π/30) = 0.4181] and various values of b. Solid curves, I1; dotted curves, I2. (a) b = 1, (b) b = 0.5, (c) b = 0, (d) b = −0.5, (e) b = −1. The units of κ, α, and z are chosen arbitrarily, since κz and αz are dimensionless.

Fig. 7
Fig. 7

Diffraction efficiency as a function of b for fixed gratings with both phase and absorption variations [κ = cos(π/30) = 0.9945, α = 4 sin(π/30) = 0.4181] and various values of L. Solid curve, L = π/6; Dotted–dashed curve, L = π/2; dashed curve, L = π. The units of κ, α, and L are chosen arbitrarily, since κz and αz are dimensionless.

Fig. 8
Fig. 8

Intensities of incident and diffracted beams as functions of z for reflection phase gratings (κ = 1, α = 0). (a) b = 1, (b) b = 0, (c) b = −1.

Fig. 9
Fig. 9

(a) Diffraction efficiency for reflection phase gratings as a function of κL for b equal to 0, ±1, and ±2. (b) Diffraction efficiency as a function of b for reflection phase gratings with various values of κL. Dotted curve, κL = ⅙; dotted–dashed curve, κL = ¼ dashed curve, κL = ½.

Fig. 10
Fig. 10

Grating amplitudes as functions of z for b equal to 10, 1, 0, −1, and −10, in the case of reflection phase gratings.

Fig. 11
Fig. 11

Intensities of incident and diffracted beams as functions of z for reflection absorption gratings (κ = 0, α = 4). (a) b = 1, (b) b = 0, (c) b = −1.

Fig. 12
Fig. 12

Diffraction efficiency as a function of b for reflection absorption gratings (κ = 0) with various values of αL. Dotted curve, αL = ⅔; dotted–dashed curve, αL = 1; dashed curve, αL = 2.

Equations (49)

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

E j = A j exp [ i ( ω t k j · r ) ] ,
k 2 = k 1 ± K .
n = n 0 + n p cos K · r i n a ( 1 + cos K · r ) + [ n 1 2 exp ( i ϕ ) A 1 * A 2 I 0 exp ( i K · r ) + c . c . ] ,
I 0 = | A 1 | 2 + | A 2 | 2 .
| d 2 d z 2 A j | | β j d d z A j |
| k j | 2 = ( ω 2 / c 2 ) n 0 2 ,
2 i β 1 d d z A 1 = ω 2 c 2 [ n 0 n 1 e i ϕ | A 2 | 2 I 0 A 1 + ( n 0 n p i n 0 n a ) A 2 2 i n 0 n a A 1 ] ,
2 i β 2 d d z A 2 = ω 2 c 2 [ n 0 n 1 e i ϕ | A 1 | 2 I 0 A 2 + ( n 0 n p i n 0 n a ) A 1 2 i n 0 n a A 2 ] ,
β 1 = β 2 = ( 2 π / λ ) n 0 cos θ .
d d z A 1 = Γ 2 | A 2 | 2 I 0 A 1 i ( κ i α 4 ) A 2 α 2 A 1 ,
d d z A 2 = Γ * 2 | A 1 | 2 I 0 A 2 i ( κ i α 4 ) A 1 α 2 A 2 ,
Γ = i 2 π n 1 λ cos θ e i ϕ ,
κ = π n p λ cos θ .
α = 4 π n a λ cos θ .
A 1 = I 1 exp ( i ψ 1 ) ,
A 2 = I 2 exp ( i ψ 2 ) ,
d d z I 1 = γ I 1 I 2 I 1 + I 2 2 κ sin Δ ψ I 1 I 2 ,
d d z I 2 = γ I 1 I 2 I 1 + I 2 + 2 κ sin Δ ψ I 1 I 2
d d z ψ 1 = β I 2 I 1 + I 2 + κ cos Δ ψ I 2 / I 1 ,
d d z ψ 2 = β I 1 I 1 + I 2 + κ cos Δ ψ I 1 / I 2 ,
Δ ψ = ψ 2 ψ 1
γ = 2 π n 1 λ cos θ sin ϕ ,
β = π n 1 λ cos θ cos ϕ .
Δ ψ = π / 2
Δ ψ = π / 2 .
d d z I 1 = γ I 1 I 2 I 1 + I 2 2 κ I 1 I 2 ,
d d z I 2 = γ I 1 I 2 I 1 + I 2 + 2 κ I 1 I 2 .
I 1 ( z ) = I 0 cos 2 u ,
I 2 ( z ) = I 0 sin 2 u ,
tan u = tan [ κ z ( 1 b 2 ) 1 / 2 ] ( 1 b 2 ) 1 / 2 b tan [ κ z ( 1 b 2 ) 1 / 2 ] ,
b = γ / ( 4 κ ) .
η = I 2 ( L ) I 0 = sin 2 u ( z = L ) .
η = 1 1 4 b 2 [ 1 4 b 2 exp ( 2 κ L b ) ] 2 = 1 1 4 b 2 [ 1 8 b 2 exp ( 2 κ L b ) ] .
η = 1 4 b 2 [ 1 + 1 4 b 2 2 exp ( 2 κ L | b | ) ] .
b = γ [ ( 4 κ ) 2 + α 2 ] 1 / 2 .
β 1 = β 2 = ( 2 π / λ ) n 0 cos θ .
d d z A 1 = Γ 2 | A 2 | 2 I 0 A 1 i ( κ i α 4 ) A 2 α 2 A 1 ,
d d z A 2 = Γ * 2 | A 1 | 2 I 0 A 2 + i ( κ i α 4 ) A 1 + α 2 A 2 .
d d z I 1 = γ I 1 I 2 I 1 + I 2 2 κ I 1 I 2 ,
d d z I 2 = γ I 1 I 2 I 1 + I 2 2 κ I 1 I 2 .
I 1 ( z ) = C cosh 2 u .
I 2 ( z ) = C sinh 2 u ,
e 2 u [ cosh ( 2 u ) + b sinh ( 2 u ) ] b = exp [ 2 κ ( 1 b 2 ) ( z L ) ] ,
u = 2 κ ( L z ) 1 4 + 1 4 e 4 u ,
u = 2 κ ( L z ) 1 4 + 1 4 e 4 u .
η = I 2 ( 0 ) I 1 ( 0 ) = tanh 2 u ( z = 0 ) .
u = 2 κ L 1 4 + 1 4 e 4 u .
η = 1 2 b exp ( 2 κ L b ) .
η = 1 4 b 2 [ 1 2 exp ( 2 κ L | b | ) ] .

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