Abstract

We study detuning effects that modify the Bragg condition in photorefractive crystals and analyze the influence of these effects on the retrieval of information that is stored in such crystals. We examine and integrate several mechanisms that induce Bragg detuning, including the effects of temperature changes, applied electric fields, and polarization changes, taking into account the anisotropic nature of photorefractive crystals. Transfer functions based on these detuning effects are defined and computed to examine and quantify the signal quality that is obtained during the readout of image-bearing holograms. The different recording configurations and geometries that are typically used in photorefractive applications are considered and compared. Finally, we study more specifically the consequences of temperature-induced detuning in the context of a holographic data storage system that uses thermal fixing. We show theoretically and experimentally that distortions (in this case limited field of view) occur on readout, and we propose solutions to improve the quality of the retrieved images.

© 1996 Optical Society of America

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  1. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  2. V. G. Sidorovich, V. V. Shkunov, “Spectral selectivity of 3-D holograms,” Opt. Spectrosc. 44, 586–589 (1978).
  3. H.-C. Külich, “Transfer function for image formation of objects reconstructed from volume holograms with different wavelengths,” Appl. Opt. 31, 2461–2477 (1992).
    [CrossRef] [PubMed]
  4. H.-C. Külich, “Reconstructing volume holograms without image field losses,” Appl. Opt. 30, 2850–2857 (1991).
    [CrossRef] [PubMed]
  5. M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and nonlinear image processing in electrooptic Bi12SiO20crystals,” Opt. Commun. 31, 301–305 (1979).
    [CrossRef]
  6. M. P. Petrov, S. I. Stepanov, A. A. Kamshilin, “Light diffraction from volume holograms in electrooptic birefringent crystals,” Opt. Commun. 29, 44–48 (1979).
    [CrossRef]
  7. J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749–752 (1994).
    [CrossRef] [PubMed]
  8. G. W. Burr, F. H. Mok, D. Psaltis, “Storage of 10,000 holograms in LiNbO3:Fe,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 9.
  9. K. Curtis, C. Gu, D. Psaltis, “Cross talk in wavelength-multiplexed holographic memories,” Opt. Lett. 18, 1001–1003 (1993).
    [CrossRef] [PubMed]
  10. S. I. Stepanov, A. A. Kamshilin, M. P. Petrov, “Electrically controlled optical diffraction by volume holograms in electrooptic crystals,” Pis’ma Zh. Tekh. Fiz 3, 89–93 (1977);English translation: Sov. Tech. Phys. Lett. 3, 36–38 (1977).
  11. R. De Vré, L. Hesselink, “Diffraction analysis of layered structures of photorefractive gratings,” J. Opt. Soc. Am. A 13, 285–295 (1996).
    [CrossRef]
  12. J. P. Wilde, L. Hesselink, “Electric-field-controlled diffraction in photorefractive strontium barium niobate,” Opt. Lett. 17, 853–855 (1992).
    [CrossRef] [PubMed]
  13. N. V. Kukhtarev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Pis’ma Zh. Tekh. Fiz 2, 1114 (1976);English translation: Sov. Tech. Phys. Lett. 2, 438–440 (1976).
  14. J. T. Sheridan, “A comparison of diffraction theories for off-Bragg replay,” J. Modern Opt. 39, 1709–1718 (1992).
    [CrossRef]
  15. P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
    [CrossRef]
  16. R. De Vré, M. Jeganathan, J. P. Wilde, L. Hesselink, “Effect of applied fields on the Bragg condition and the diffraction efficiency in photorefractive crystals,” Opt. Lett. 19, 910–912 (1994).
    [CrossRef] [PubMed]
  17. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  18. K. R. Sangameswara, Thermal Expansion of Crystals (Pergamon, Oxford, 1979).
  19. S. Ducharme, J. Feinberg, R. R. Neurgaonkar, “Electrooptic and piezoelectric measurements in photorefractive barium titanate and strontium barium niobate,” IEEE J. Quantum Electron. 23, 2116–2121 (1987).
    [CrossRef]
  20. J. J. Amodei, D. L. Staebler, “Holographic pattern fixing in electro-optic crystals,” Appl. Phys. Lett. 18, 540–542 (1971).
    [CrossRef]
  21. D. L. Staebler, J. J. Amodei, “Thermally fixed holograms in LiNbO3,” Ferroelectrics 3, 107–113 (1972).
    [CrossRef]
  22. M. Carrascosa, F. Agulló-López, “Theoretical modeling of the fixing and developing of holographic gratings in LiNbO3,” J. Opt. Soc. Am. B 7, 2317–2322 (1990).
    [CrossRef]
  23. A. Yariv, S. Orlov, G. Rakuljic, V. Leyva, “Holographic fixing, readout, and storage dynamics in photorefractive materials,” Opt. Lett. 20, 1334–1336 (1995).
    [CrossRef] [PubMed]
  24. R. Müller, M. T. Santos, L. Arizmendi, J. M. Cabrera, “A narrow-band interference filter with photorefractive LiNbO3,”J. Phys. D: Appl. Phys. 27, 241–246 (1994).
    [CrossRef]
  25. J. F. Heanue, K. Gürkan, M. C. Bashaw, R. De Vré, L. Hesselink, “Thermal fixing in digital holographic data storage,” presented at OSA Annual Meeting, Portland, Oregon, 1995.
  26. M. C. Bashaw, J. F. Heanue, A. Aharoni, J. F. Walkup, L. Hesselink, “Cross-talk considerations for angular and phase-encoded multiplexing in volume holography,” J. Opt. Soc. Am. B 11, 1820–1836 (1994).
    [CrossRef]
  27. D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, “Multiple storage and erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
    [CrossRef]
  28. J. P. Wilde, R. De Vré, M. Jeganathan, L. Hesselink, “Electric field control of image-bearing volume holograms stored in photorefractive media,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 98–99.
  29. Yu. L. Korzinin, V. I. Sukhanov, “Diffraction of light by 3-D holograms with a continuous spectrum of spatial frequencies. System of equations for coupled waves,” Opt. Spectrosc. 56, 467–469 (1984).
  30. Yu. L. Korzinin, V. I. Sukhanov, “Diffraction of light by 3-D holograms with a continuous spectrum of spatial frequencies,” Opt. Spectrosc. 56, 572–574 (1984).
  31. Yu. L. Korzinin, V. I. Sukhanov, “Diffraction efficiency of a 3-D hologram of a diffuse object,” Opt. Spectrosc. 58, 86–88 (1985).
  32. Properties of lithium niobate. INSPEC (information service), EMIS Datareviews Series 5, (1989).
  33. J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital information,” in Conference on Lasers and Electro-Optics, Volume 15 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 27.
  34. G. A. Rakuljic, V. Leyva, A. Yariv, “Optical data storage by using orthogonal wavelength-multiplexed volume holograms,” Opt. Lett. 17, 1471–1473 (1992).
    [CrossRef] [PubMed]
  35. G. A. Rakuljic, A. Yariv, R. R. Neurgaonkar, “Photorefractive properties of undoped, cerium-doped, and iron-doped single-crystal Sr0.6Ba0.4Nb2O6,” Opt. Eng. 25, 1212–1216 (1986).
    [CrossRef]
  36. R. Müller, J. V. Alvarez-Bravo, L. Arizmendi, J. M. Cabrera, “Tuning of photorefractive interference filters in LiNbO3,”J. Phys. D: Appl. Phys. 27, 1628–1632 (1994).
    [CrossRef]
  37. G. A. Rakuljic, V. Leyva, “Volume holographic narrow-band optical filter,” Opt. Lett. 18, 459–461 (1993).
    [CrossRef] [PubMed]
  38. J. F. Heanue, “Volume holographic storage of digital data implemented in photorefractive media,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1995).

1996 (1)

1995 (1)

1994 (5)

M. C. Bashaw, J. F. Heanue, A. Aharoni, J. F. Walkup, L. Hesselink, “Cross-talk considerations for angular and phase-encoded multiplexing in volume holography,” J. Opt. Soc. Am. B 11, 1820–1836 (1994).
[CrossRef]

R. Müller, M. T. Santos, L. Arizmendi, J. M. Cabrera, “A narrow-band interference filter with photorefractive LiNbO3,”J. Phys. D: Appl. Phys. 27, 241–246 (1994).
[CrossRef]

R. Müller, J. V. Alvarez-Bravo, L. Arizmendi, J. M. Cabrera, “Tuning of photorefractive interference filters in LiNbO3,”J. Phys. D: Appl. Phys. 27, 1628–1632 (1994).
[CrossRef]

R. De Vré, M. Jeganathan, J. P. Wilde, L. Hesselink, “Effect of applied fields on the Bragg condition and the diffraction efficiency in photorefractive crystals,” Opt. Lett. 19, 910–912 (1994).
[CrossRef] [PubMed]

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749–752 (1994).
[CrossRef] [PubMed]

1993 (2)

1992 (4)

1991 (1)

1990 (1)

1989 (2)

Properties of lithium niobate. INSPEC (information service), EMIS Datareviews Series 5, (1989).

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

1987 (1)

S. Ducharme, J. Feinberg, R. R. Neurgaonkar, “Electrooptic and piezoelectric measurements in photorefractive barium titanate and strontium barium niobate,” IEEE J. Quantum Electron. 23, 2116–2121 (1987).
[CrossRef]

1986 (1)

G. A. Rakuljic, A. Yariv, R. R. Neurgaonkar, “Photorefractive properties of undoped, cerium-doped, and iron-doped single-crystal Sr0.6Ba0.4Nb2O6,” Opt. Eng. 25, 1212–1216 (1986).
[CrossRef]

1985 (1)

Yu. L. Korzinin, V. I. Sukhanov, “Diffraction efficiency of a 3-D hologram of a diffuse object,” Opt. Spectrosc. 58, 86–88 (1985).

1984 (2)

Yu. L. Korzinin, V. I. Sukhanov, “Diffraction of light by 3-D holograms with a continuous spectrum of spatial frequencies. System of equations for coupled waves,” Opt. Spectrosc. 56, 467–469 (1984).

Yu. L. Korzinin, V. I. Sukhanov, “Diffraction of light by 3-D holograms with a continuous spectrum of spatial frequencies,” Opt. Spectrosc. 56, 572–574 (1984).

1979 (2)

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and nonlinear image processing in electrooptic Bi12SiO20crystals,” Opt. Commun. 31, 301–305 (1979).
[CrossRef]

M. P. Petrov, S. I. Stepanov, A. A. Kamshilin, “Light diffraction from volume holograms in electrooptic birefringent crystals,” Opt. Commun. 29, 44–48 (1979).
[CrossRef]

1978 (1)

V. G. Sidorovich, V. V. Shkunov, “Spectral selectivity of 3-D holograms,” Opt. Spectrosc. 44, 586–589 (1978).

1977 (1)

S. I. Stepanov, A. A. Kamshilin, M. P. Petrov, “Electrically controlled optical diffraction by volume holograms in electrooptic crystals,” Pis’ma Zh. Tekh. Fiz 3, 89–93 (1977);English translation: Sov. Tech. Phys. Lett. 3, 36–38 (1977).

1976 (1)

N. V. Kukhtarev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Pis’ma Zh. Tekh. Fiz 2, 1114 (1976);English translation: Sov. Tech. Phys. Lett. 2, 438–440 (1976).

1975 (1)

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, “Multiple storage and erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

1972 (1)

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

1971 (1)

J. J. Amodei, D. L. Staebler, “Holographic pattern fixing in electro-optic crystals,” Appl. Phys. Lett. 18, 540–542 (1971).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Agulló-López, F.

Aharoni, A.

Alvarez-Bravo, J. V.

R. Müller, J. V. Alvarez-Bravo, L. Arizmendi, J. M. Cabrera, “Tuning of photorefractive interference filters in LiNbO3,”J. Phys. D: Appl. Phys. 27, 1628–1632 (1994).
[CrossRef]

Amodei, J. J.

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, “Multiple storage and erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

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

J. J. Amodei, D. L. Staebler, “Holographic pattern fixing in electro-optic crystals,” Appl. Phys. Lett. 18, 540–542 (1971).
[CrossRef]

Arizmendi, L.

R. Müller, M. T. Santos, L. Arizmendi, J. M. Cabrera, “A narrow-band interference filter with photorefractive LiNbO3,”J. Phys. D: Appl. Phys. 27, 241–246 (1994).
[CrossRef]

R. Müller, J. V. Alvarez-Bravo, L. Arizmendi, J. M. Cabrera, “Tuning of photorefractive interference filters in LiNbO3,”J. Phys. D: Appl. Phys. 27, 1628–1632 (1994).
[CrossRef]

Bashaw, M. C.

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749–752 (1994).
[CrossRef] [PubMed]

M. C. Bashaw, J. F. Heanue, A. Aharoni, J. F. Walkup, L. Hesselink, “Cross-talk considerations for angular and phase-encoded multiplexing in volume holography,” J. Opt. Soc. Am. B 11, 1820–1836 (1994).
[CrossRef]

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital information,” in Conference on Lasers and Electro-Optics, Volume 15 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 27.

J. F. Heanue, K. Gürkan, M. C. Bashaw, R. De Vré, L. Hesselink, “Thermal fixing in digital holographic data storage,” presented at OSA Annual Meeting, Portland, Oregon, 1995.

Burke, W. J.

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, “Multiple storage and erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

Burr, G. W.

G. W. Burr, F. H. Mok, D. Psaltis, “Storage of 10,000 holograms in LiNbO3:Fe,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 9.

Cabrera, J. M.

R. Müller, M. T. Santos, L. Arizmendi, J. M. Cabrera, “A narrow-band interference filter with photorefractive LiNbO3,”J. Phys. D: Appl. Phys. 27, 241–246 (1994).
[CrossRef]

R. Müller, J. V. Alvarez-Bravo, L. Arizmendi, J. M. Cabrera, “Tuning of photorefractive interference filters in LiNbO3,”J. Phys. D: Appl. Phys. 27, 1628–1632 (1994).
[CrossRef]

Carrascosa, M.

Curtis, K.

De Vré, R.

R. De Vré, L. Hesselink, “Diffraction analysis of layered structures of photorefractive gratings,” J. Opt. Soc. Am. A 13, 285–295 (1996).
[CrossRef]

R. De Vré, M. Jeganathan, J. P. Wilde, L. Hesselink, “Effect of applied fields on the Bragg condition and the diffraction efficiency in photorefractive crystals,” Opt. Lett. 19, 910–912 (1994).
[CrossRef] [PubMed]

J. F. Heanue, K. Gürkan, M. C. Bashaw, R. De Vré, L. Hesselink, “Thermal fixing in digital holographic data storage,” presented at OSA Annual Meeting, Portland, Oregon, 1995.

J. P. Wilde, R. De Vré, M. Jeganathan, L. Hesselink, “Electric field control of image-bearing volume holograms stored in photorefractive media,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 98–99.

Ducharme, S.

S. Ducharme, J. Feinberg, R. R. Neurgaonkar, “Electrooptic and piezoelectric measurements in photorefractive barium titanate and strontium barium niobate,” IEEE J. Quantum Electron. 23, 2116–2121 (1987).
[CrossRef]

Feinberg, J.

S. Ducharme, J. Feinberg, R. R. Neurgaonkar, “Electrooptic and piezoelectric measurements in photorefractive barium titanate and strontium barium niobate,” IEEE J. Quantum Electron. 23, 2116–2121 (1987).
[CrossRef]

Gu, C.

Gürkan, K.

J. F. Heanue, K. Gürkan, M. C. Bashaw, R. De Vré, L. Hesselink, “Thermal fixing in digital holographic data storage,” presented at OSA Annual Meeting, Portland, Oregon, 1995.

Heanue, J. F.

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749–752 (1994).
[CrossRef] [PubMed]

M. C. Bashaw, J. F. Heanue, A. Aharoni, J. F. Walkup, L. Hesselink, “Cross-talk considerations for angular and phase-encoded multiplexing in volume holography,” J. Opt. Soc. Am. B 11, 1820–1836 (1994).
[CrossRef]

J. F. Heanue, K. Gürkan, M. C. Bashaw, R. De Vré, L. Hesselink, “Thermal fixing in digital holographic data storage,” presented at OSA Annual Meeting, Portland, Oregon, 1995.

J. F. Heanue, “Volume holographic storage of digital data implemented in photorefractive media,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1995).

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital information,” in Conference on Lasers and Electro-Optics, Volume 15 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 27.

Hesselink, L.

R. De Vré, L. Hesselink, “Diffraction analysis of layered structures of photorefractive gratings,” J. Opt. Soc. Am. A 13, 285–295 (1996).
[CrossRef]

R. De Vré, M. Jeganathan, J. P. Wilde, L. Hesselink, “Effect of applied fields on the Bragg condition and the diffraction efficiency in photorefractive crystals,” Opt. Lett. 19, 910–912 (1994).
[CrossRef] [PubMed]

M. C. Bashaw, J. F. Heanue, A. Aharoni, J. F. Walkup, L. Hesselink, “Cross-talk considerations for angular and phase-encoded multiplexing in volume holography,” J. Opt. Soc. Am. B 11, 1820–1836 (1994).
[CrossRef]

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749–752 (1994).
[CrossRef] [PubMed]

J. P. Wilde, L. Hesselink, “Electric-field-controlled diffraction in photorefractive strontium barium niobate,” Opt. Lett. 17, 853–855 (1992).
[CrossRef] [PubMed]

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital information,” in Conference on Lasers and Electro-Optics, Volume 15 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 27.

J. F. Heanue, K. Gürkan, M. C. Bashaw, R. De Vré, L. Hesselink, “Thermal fixing in digital holographic data storage,” presented at OSA Annual Meeting, Portland, Oregon, 1995.

J. P. Wilde, R. De Vré, M. Jeganathan, L. Hesselink, “Electric field control of image-bearing volume holograms stored in photorefractive media,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 98–99.

Jeganathan, M.

R. De Vré, M. Jeganathan, J. P. Wilde, L. Hesselink, “Effect of applied fields on the Bragg condition and the diffraction efficiency in photorefractive crystals,” Opt. Lett. 19, 910–912 (1994).
[CrossRef] [PubMed]

J. P. Wilde, R. De Vré, M. Jeganathan, L. Hesselink, “Electric field control of image-bearing volume holograms stored in photorefractive media,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 98–99.

Kamshilin, A. A.

M. P. Petrov, S. I. Stepanov, A. A. Kamshilin, “Light diffraction from volume holograms in electrooptic birefringent crystals,” Opt. Commun. 29, 44–48 (1979).
[CrossRef]

S. I. Stepanov, A. A. Kamshilin, M. P. Petrov, “Electrically controlled optical diffraction by volume holograms in electrooptic crystals,” Pis’ma Zh. Tekh. Fiz 3, 89–93 (1977);English translation: Sov. Tech. Phys. Lett. 3, 36–38 (1977).

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Korzinin, Yu. L.

Yu. L. Korzinin, V. I. Sukhanov, “Diffraction efficiency of a 3-D hologram of a diffuse object,” Opt. Spectrosc. 58, 86–88 (1985).

Yu. L. Korzinin, V. I. Sukhanov, “Diffraction of light by 3-D holograms with a continuous spectrum of spatial frequencies. System of equations for coupled waves,” Opt. Spectrosc. 56, 467–469 (1984).

Yu. L. Korzinin, V. I. Sukhanov, “Diffraction of light by 3-D holograms with a continuous spectrum of spatial frequencies,” Opt. Spectrosc. 56, 572–574 (1984).

Kukhtarev, N. V.

N. V. Kukhtarev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Pis’ma Zh. Tekh. Fiz 2, 1114 (1976);English translation: Sov. Tech. Phys. Lett. 2, 438–440 (1976).

Külich, H.-C.

Kulikov, V. V.

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and nonlinear image processing in electrooptic Bi12SiO20crystals,” Opt. Commun. 31, 301–305 (1979).
[CrossRef]

Leyva, V.

Miridonov, S. V.

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and nonlinear image processing in electrooptic Bi12SiO20crystals,” Opt. Commun. 31, 301–305 (1979).
[CrossRef]

Mok, F. H.

G. W. Burr, F. H. Mok, D. Psaltis, “Storage of 10,000 holograms in LiNbO3:Fe,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 9.

Müller, R.

R. Müller, M. T. Santos, L. Arizmendi, J. M. Cabrera, “A narrow-band interference filter with photorefractive LiNbO3,”J. Phys. D: Appl. Phys. 27, 241–246 (1994).
[CrossRef]

R. Müller, J. V. Alvarez-Bravo, L. Arizmendi, J. M. Cabrera, “Tuning of photorefractive interference filters in LiNbO3,”J. Phys. D: Appl. Phys. 27, 1628–1632 (1994).
[CrossRef]

Neurgaonkar, R. R.

S. Ducharme, J. Feinberg, R. R. Neurgaonkar, “Electrooptic and piezoelectric measurements in photorefractive barium titanate and strontium barium niobate,” IEEE J. Quantum Electron. 23, 2116–2121 (1987).
[CrossRef]

G. A. Rakuljic, A. Yariv, R. R. Neurgaonkar, “Photorefractive properties of undoped, cerium-doped, and iron-doped single-crystal Sr0.6Ba0.4Nb2O6,” Opt. Eng. 25, 1212–1216 (1986).
[CrossRef]

Orlov, S.

Petrov, M. P.

M. P. Petrov, S. I. Stepanov, A. A. Kamshilin, “Light diffraction from volume holograms in electrooptic birefringent crystals,” Opt. Commun. 29, 44–48 (1979).
[CrossRef]

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and nonlinear image processing in electrooptic Bi12SiO20crystals,” Opt. Commun. 31, 301–305 (1979).
[CrossRef]

S. I. Stepanov, A. A. Kamshilin, M. P. Petrov, “Electrically controlled optical diffraction by volume holograms in electrooptic crystals,” Pis’ma Zh. Tekh. Fiz 3, 89–93 (1977);English translation: Sov. Tech. Phys. Lett. 3, 36–38 (1977).

Phillips, W.

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, “Multiple storage and erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

Psaltis, D.

K. Curtis, C. Gu, D. Psaltis, “Cross talk in wavelength-multiplexed holographic memories,” Opt. Lett. 18, 1001–1003 (1993).
[CrossRef] [PubMed]

G. W. Burr, F. H. Mok, D. Psaltis, “Storage of 10,000 holograms in LiNbO3:Fe,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 9.

Rakuljic, G.

Rakuljic, G. A.

Sangameswara, K. R.

K. R. Sangameswara, Thermal Expansion of Crystals (Pergamon, Oxford, 1979).

Santos, M. T.

R. Müller, M. T. Santos, L. Arizmendi, J. M. Cabrera, “A narrow-band interference filter with photorefractive LiNbO3,”J. Phys. D: Appl. Phys. 27, 241–246 (1994).
[CrossRef]

Sheridan, J. T.

J. T. Sheridan, “A comparison of diffraction theories for off-Bragg replay,” J. Modern Opt. 39, 1709–1718 (1992).
[CrossRef]

Shkunov, V. V.

V. G. Sidorovich, V. V. Shkunov, “Spectral selectivity of 3-D holograms,” Opt. Spectrosc. 44, 586–589 (1978).

Sidorovich, V. G.

V. G. Sidorovich, V. V. Shkunov, “Spectral selectivity of 3-D holograms,” Opt. Spectrosc. 44, 586–589 (1978).

Staebler, D. L.

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, “Multiple storage and erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

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

J. J. Amodei, D. L. Staebler, “Holographic pattern fixing in electro-optic crystals,” Appl. Phys. Lett. 18, 540–542 (1971).
[CrossRef]

Stepanov, S. I.

M. P. Petrov, S. I. Stepanov, A. A. Kamshilin, “Light diffraction from volume holograms in electrooptic birefringent crystals,” Opt. Commun. 29, 44–48 (1979).
[CrossRef]

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and nonlinear image processing in electrooptic Bi12SiO20crystals,” Opt. Commun. 31, 301–305 (1979).
[CrossRef]

S. I. Stepanov, A. A. Kamshilin, M. P. Petrov, “Electrically controlled optical diffraction by volume holograms in electrooptic crystals,” Pis’ma Zh. Tekh. Fiz 3, 89–93 (1977);English translation: Sov. Tech. Phys. Lett. 3, 36–38 (1977).

Sukhanov, V. I.

Yu. L. Korzinin, V. I. Sukhanov, “Diffraction efficiency of a 3-D hologram of a diffuse object,” Opt. Spectrosc. 58, 86–88 (1985).

Yu. L. Korzinin, V. I. Sukhanov, “Diffraction of light by 3-D holograms with a continuous spectrum of spatial frequencies,” Opt. Spectrosc. 56, 572–574 (1984).

Yu. L. Korzinin, V. I. Sukhanov, “Diffraction of light by 3-D holograms with a continuous spectrum of spatial frequencies. System of equations for coupled waves,” Opt. Spectrosc. 56, 467–469 (1984).

Walkup, J. F.

Wilde, J. P.

R. De Vré, M. Jeganathan, J. P. Wilde, L. Hesselink, “Effect of applied fields on the Bragg condition and the diffraction efficiency in photorefractive crystals,” Opt. Lett. 19, 910–912 (1994).
[CrossRef] [PubMed]

J. P. Wilde, L. Hesselink, “Electric-field-controlled diffraction in photorefractive strontium barium niobate,” Opt. Lett. 17, 853–855 (1992).
[CrossRef] [PubMed]

J. P. Wilde, R. De Vré, M. Jeganathan, L. Hesselink, “Electric field control of image-bearing volume holograms stored in photorefractive media,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 98–99.

Yariv, A.

A. Yariv, S. Orlov, G. Rakuljic, V. Leyva, “Holographic fixing, readout, and storage dynamics in photorefractive materials,” Opt. Lett. 20, 1334–1336 (1995).
[CrossRef] [PubMed]

G. A. Rakuljic, V. Leyva, A. Yariv, “Optical data storage by using orthogonal wavelength-multiplexed volume holograms,” Opt. Lett. 17, 1471–1473 (1992).
[CrossRef] [PubMed]

G. A. Rakuljic, A. Yariv, R. R. Neurgaonkar, “Photorefractive properties of undoped, cerium-doped, and iron-doped single-crystal Sr0.6Ba0.4Nb2O6,” Opt. Eng. 25, 1212–1216 (1986).
[CrossRef]

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

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

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Appl. Opt. (2)

Appl. Phys. Lett. (2)

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, “Multiple storage and erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

J. J. Amodei, D. L. Staebler, “Holographic pattern fixing in electro-optic crystals,” Appl. Phys. Lett. 18, 540–542 (1971).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Ferroelectrics (1)

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

IEEE J. Quantum Electron. (2)

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

S. Ducharme, J. Feinberg, R. R. Neurgaonkar, “Electrooptic and piezoelectric measurements in photorefractive barium titanate and strontium barium niobate,” IEEE J. Quantum Electron. 23, 2116–2121 (1987).
[CrossRef]

J. Modern Opt. (1)

J. T. Sheridan, “A comparison of diffraction theories for off-Bragg replay,” J. Modern Opt. 39, 1709–1718 (1992).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (2)

J. Phys. D: Appl. Phys. (2)

R. Müller, M. T. Santos, L. Arizmendi, J. M. Cabrera, “A narrow-band interference filter with photorefractive LiNbO3,”J. Phys. D: Appl. Phys. 27, 241–246 (1994).
[CrossRef]

R. Müller, J. V. Alvarez-Bravo, L. Arizmendi, J. M. Cabrera, “Tuning of photorefractive interference filters in LiNbO3,”J. Phys. D: Appl. Phys. 27, 1628–1632 (1994).
[CrossRef]

Opt. Commun. (2)

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and nonlinear image processing in electrooptic Bi12SiO20crystals,” Opt. Commun. 31, 301–305 (1979).
[CrossRef]

M. P. Petrov, S. I. Stepanov, A. A. Kamshilin, “Light diffraction from volume holograms in electrooptic birefringent crystals,” Opt. Commun. 29, 44–48 (1979).
[CrossRef]

Opt. Eng. (1)

G. A. Rakuljic, A. Yariv, R. R. Neurgaonkar, “Photorefractive properties of undoped, cerium-doped, and iron-doped single-crystal Sr0.6Ba0.4Nb2O6,” Opt. Eng. 25, 1212–1216 (1986).
[CrossRef]

Opt. Lett. (6)

Opt. Spectrosc. (4)

V. G. Sidorovich, V. V. Shkunov, “Spectral selectivity of 3-D holograms,” Opt. Spectrosc. 44, 586–589 (1978).

Yu. L. Korzinin, V. I. Sukhanov, “Diffraction of light by 3-D holograms with a continuous spectrum of spatial frequencies. System of equations for coupled waves,” Opt. Spectrosc. 56, 467–469 (1984).

Yu. L. Korzinin, V. I. Sukhanov, “Diffraction of light by 3-D holograms with a continuous spectrum of spatial frequencies,” Opt. Spectrosc. 56, 572–574 (1984).

Yu. L. Korzinin, V. I. Sukhanov, “Diffraction efficiency of a 3-D hologram of a diffuse object,” Opt. Spectrosc. 58, 86–88 (1985).

Pis’ma Zh. Tekh. Fiz (2)

N. V. Kukhtarev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Pis’ma Zh. Tekh. Fiz 2, 1114 (1976);English translation: Sov. Tech. Phys. Lett. 2, 438–440 (1976).

S. I. Stepanov, A. A. Kamshilin, M. P. Petrov, “Electrically controlled optical diffraction by volume holograms in electrooptic crystals,” Pis’ma Zh. Tekh. Fiz 3, 89–93 (1977);English translation: Sov. Tech. Phys. Lett. 3, 36–38 (1977).

Properties of lithium niobate. INSPEC (information service) (1)

Properties of lithium niobate. INSPEC (information service), EMIS Datareviews Series 5, (1989).

Science (1)

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749–752 (1994).
[CrossRef] [PubMed]

Other (7)

G. W. Burr, F. H. Mok, D. Psaltis, “Storage of 10,000 holograms in LiNbO3:Fe,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 9.

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital information,” in Conference on Lasers and Electro-Optics, Volume 15 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 27.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

K. R. Sangameswara, Thermal Expansion of Crystals (Pergamon, Oxford, 1979).

J. P. Wilde, R. De Vré, M. Jeganathan, L. Hesselink, “Electric field control of image-bearing volume holograms stored in photorefractive media,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 98–99.

J. F. Heanue, “Volume holographic storage of digital data implemented in photorefractive media,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1995).

J. F. Heanue, K. Gürkan, M. C. Bashaw, R. De Vré, L. Hesselink, “Thermal fixing in digital holographic data storage,” presented at OSA Annual Meeting, Portland, Oregon, 1995.

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Figures (16)

Fig. 1
Fig. 1

Space representation of the different writing and readout wave vectors illustrating the relations K = kskr and kσ = kρ + K.

Fig. 2
Fig. 2

Recording configurations: T = transmission, P = perpendicular, and R = reflection geometry.

Fig. 3
Fig. 3

Crystal orientation with respect to the propagation axis. β = 0° for the reflection geometry, β = 45° for the perpendicular geometry, and β = 90° for the transmission geometry.

Fig. 4
Fig. 4

k-space representation of an image-bearing hologram when a change of Bragg condition occurs during readout. On readout, in general only one angular component of the image (θs0) can be exactly Bragg matched.

Fig. 5
Fig. 5

Transfer function in the perpendicular geometry with use of LiNbO3 for a ΔT = −150 °C. The solid curve represents the case in which only the angle of the readout beam is adjusted [case (i)], the dashed curve represents the case in which only the wavelength of the readout beam in adjusted [case (ii)], and the dotted line represents the case in which both the angle and the wavelength are simultaneously adjusted [case (iii)].

Fig. 6
Fig. 6

Field of view (FOV) (expressed in degrees inside the crystal) as a function of temperature in the perpendicular geometry with use of LiNbO3. Solid curve, case (i); dashed curve, case (ii).

Fig. 7
Fig. 7

(a) Angular detuning (degrees outside the crystal) and (b) wavelength detuning as a function of temperature. These are the detunings necessary to Bragg match the dc component of the image during readout. Solid curve, case (i); dashed curve, case (ii); dotted line, case (iii).

Fig. 8
Fig. 8

Relative angular shift of the diffracted beam [ Δ θ ^ σ ( θ s ) - Δ θ ^ σ ( θ 0 0 )] (given in degrees outside the crystal) as a function of θs for ΔT = −150 °C. Solid curve, case (i) [Δ θ ^ σ (θs0) = 0.05°]; dashed curve, case (ii) [Δ θ ^ σ (θs0) = −0.075°]; dotted curve, case (iii) [Δ θ ^ σ (θs0) = −0.17°].

Fig. 9
Fig. 9

Relative detuning changes (%) as a function of the angle of the reference beam θr (in the perpendicular geometry and for a ΔT = −150 °C). If angular multiplexing is used, each reference beam angle corresponds to an image number. Solid curve, case (i); dashed curve, case (ii); dotted line, case (iii).

Fig. 10
Fig. 10

Field of view (expressed in degrees inside the crystal) as a function of θr in the three recording geometries for a ΔT = −150 °C. The shaded areas represent the forbidden regions resulting from the crystal boundary conditions and Snell’s law. (a) Change in angle only (angular multiplexing), (b) change in wavelength only (wavelength multiplexing).

Fig. 11
Fig. 11

Field of view (expressed in degrees inside the crystal) as a function of the number of SLM pixels that are fully reconstructed for three different focal lengths f.

Fig. 12
Fig. 12

Transfer functions in the reflection geometry with use of SBN:75 and extraordinary polarization for two different applied fields ΔE0, 5 kV/cm and 10 kV/cm.

Fig. 13
Fig. 13

Circular aperture recorded in LiNbO3 with the perpendicular geometry: (a) thermally fixed image with the low-high-low process, (b)–(d) fixed images with the high-low process, taken at three different readout angles (ΔT = −115 °C).

Fig. 14
Fig. 14

Transfer function corresponding to the experimental images of Fig. 13T = −115 °C). The experimental field of view is compared with the theoretical field of view computed given an interaction length l = 1 cm (solid curve) and l = 0.55 cm (dashed curve).

Fig. 15
Fig. 15

Angular selectivity: theoretical curve corresponding to l = 1 cm, unfixed grating, fixed grating (low-high-low), and fixed grating (high-low). The interaction length of the fixed grating can be estimated to be l = 0.55 cm.

Fig. 16
Fig. 16

Strains and deformations along the principal axis of the crystal (Λ = 2π/|K|, Λd = 2π/|Kd|).

Tables (1)

Tables Icon

Table 1 Parameter for the Photorefractive Crystals LiNbO3 and Ce-SBN:75

Equations (67)

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Δ n ( r ) = Δ n p exp [ i ( K · r + ϕ p ) ] ,
k σ = k ρ + K ,
cos θ ρ d a ρ d z = - i π Δ n p λ 0 exp [ i ( 2 ξ z l + ϕ p ) ] a σ , cos θ σ d a σ d z = - i π Δ n p λ 0 exp [ - i ( 2 ξ z l + ϕ p ) ] a ρ ,
ξ = l / 2 ( k ρ z - k σ z + K z ) ,
η = ν 2 sin 2 ξ ξ 2 ,
ν = π Δ n p l λ 0 cos θ σ .
ξ Δ n = l 2 ( k ρ z - k σ z - k r z + k s z ) .
ξ Δ n = π n ¯ l λ 0 cos θ s [ ( Δ n ρ n ¯ - Δ λ 0 λ 0 ) cos ( θ r - θ s ) - ( Δ n σ n ¯ - Δ λ 0 λ 0 ) - Δ θ ρ sin ( θ r - θ s ) ] .
sin θ ^ ρ = n ρ sin θ ρ .
Δ θ ρ = Δ θ ^ ρ n ¯ cos θ ^ r cos θ r - Δ n ρ n ¯ tan θ r ,
cos θ ^ ρ = n ρ cos θ ρ ,
Δ θ ρ = Δ θ ^ ρ n ¯ sin θ ^ r sin θ r + Δ n ρ n ¯ cot θ r .
1 n e 2 ( θ ) = sin 2 ( θ - β ) n e 2 + cos 2 ( θ - β ) n o 2 ,
Δ n o = - 1 2 r 13 n o 3 Δ E 0 + n o T Δ T , Δ n e = - 1 2 r 33 n e 3 Δ E 0 + n e T Δ T ,
1 [ n e ( θ ) + Δ n e ( θ ) ] 2 = sin 2 ( θ - β ) ( n e + Δ n e ) 2 + cos 2 ( θ - β ) ( n o + Δ n o ) 2 .
Δ n e ( θ ) = sin 2 ( θ - β ) Δ n e + cos 2 ( θ - β ) Δ n o .
ξ Δ Λ = l / 2 [ k ρ z + K z d - k 2 - ( k ρ + K d ) 2 ] ,
ξ Δ Λ = π l n ¯ λ 0 cos θ s { [ cos ( θ r - θ s ) - 1 ] [ Δ Λ c Λ c cos 2 ( θ r - β ) + Δ Λ a Λ a sin 2 ( θ r - β ) ] - sin ( θ r - θ s ) × ( Δ Λ a Λ a - Δ Λ c Λ c ) sin ( θ r - β ) cos ( θ r - β ) } .
i j = α i j Δ T ,
c = 33 = Δ Λ c Λ c = α c Δ T a = 11 = Δ Λ a Λ a = α a Δ T ,
i j = j d i j k E 0 k ,
c = Δ Λ c Λ c = d 33 Δ E 0 a = Δ Λ a Λ a = d 13 Δ E 0 ,
ξ = ξ Δ n + ξ Δ Λ .
ξ = π l n ¯ λ 0 cos θ s { Δ n ρ - Δ n σ n ¯ + [ c cos 2 ( θ r - β ) + a sin 2 ( θ r - β ) + Δ n ρ n ¯ - Δ λ 0 λ 0 ] [ cos ( θ r - θ s ) - 1 ] - [ Δ θ ρ ( Δ θ ^ ρ , Δ n ρ ) + ( a - c ) sin ( θ r - β ) × cos ( θ r - β ) ] sin ( θ r - θ s ) } .
ξ = π l n ¯ λ 0 cos θ s { Δ n e - Δ n o n ¯ sin ( θ r - θ s ) sin ( θ r + θ s - 2 β ) + [ ( c + Δ n o n ¯ ) cos 2 ( θ r - β ) + ( a + Δ n e n ¯ ) × sin 2 ( θ r - β ) - Δ λ 0 λ 0 ] [ cos ( θ r - θ s ) - 1 ] - [ Δ θ ρ ( Δ θ ^ ρ , Δ n e , Δ n o ) + ( a - c ) sin ( θ r - β ) × cos ( θ r - β ) ] sin ( θ r - θ s ) } .
H ( θ s ) = exp [ - i ξ 0 ( θ s ) ] sin ξ 0 ( θ s ) ξ 0 ( θ s ) ,
ξ 0 ( θ s ) = - π l n ¯ Δ T λ 0 ( 1 n ¯ n o T + α c + α a 2 ) θ s ,
Δ θ ^ ρ = n ¯ ( 1 n ¯ n o T + α c ) Δ T ,
ξ 0 ( θ s ) = - π l n ¯ Δ T λ 0 ( α a - α c 2 ) θ s ,
Δ λ 0 λ 0 = ( 1 n ¯ n o T + α c ) Δ T ,
ξ 0 ( θ s ) = 0
Δ λ 0 λ 0 = ( 1 n ¯ n o T + α c + α a 2 ) Δ T ,
Δ θ ^ ρ = n ¯ ( α c - α a 2 ) Δ T .
ξ 0 ( θ s ) = π l ( Δ n o - Δ n e ) λ 0 θ s 2 .
Δ λ 0 λ 0 = ( 1 n n o T + α c ) Δ T + ( d 33 - ½ r 13 n ¯ 2 ) Δ E 0 , Δ θ ^ ρ = 0.
ξ Δ n = l 2 ( k ρ z - k σ z - k r z + k s z ) .
k ρ z = 2 π λ 0 + Δ λ 0 n ρ cos θ ρ , k ρ = 2 π λ 0 + Δ λ 0 n ρ sin θ ρ , k σ z = 2 π λ 0 + Δ λ 0 n σ cos θ σ , k σ = 2 π λ 0 + Δ λ 0 n σ sin θ σ , k r z = 2 π λ 0 n r cos θ r , k r = 2 π λ 0 n r sin θ r , k s z = 2 π λ 0 n s cos θ s , k s = 2 π λ 0 n s sin θ s ,
θ ρ = θ r + Δ θ ρ θ σ = θ s + Δ θ σ ,
n σ λ 0 + Δ λ 0 sin θ σ = n ρ λ 0 + Δ λ 0 sin θ ρ + n s λ 0 sin θ s - n r λ 0 sin θ r ,
n σ ( sin θ s + Δ θ σ cos θ s ) = n ρ ( sin θ r + Δ θ ρ cos θ r ) + ( n s sin θ s - n r sin θ r ) × ( 1 + Δ λ 0 λ 0 )
n σ = n σ ( θ σ ) = n σ ( θ s + Δ θ σ ) = n σ ( θ s ) .
Δ n ρ = n ρ - n r , Δ n σ = n σ - n s .
Δ θ σ = 1 cos θ s [ ( Δ n ρ n ¯ - Δ λ 0 λ 0 ) sin θ r - ( Δ n σ n ¯ - Δ λ 0 λ 0 ) sin θ s + Δ θ ρ cos θ r ] .
ξ Δ n = π n ¯ l λ 0 cos θ s [ ( Δ n ρ n ¯ - Δ λ 0 λ 0 ) cos ( θ r - θ s ) - ( Δ n σ n ¯ - Δ λ 0 λ 0 ) - Δ θ ρ sin ( θ r - θ s ) ] .
ξ Δ Λ = l / 2 [ k ρ z + K z d - k 2 - ( k ρ + K d ) 2 ] ,
k ρ = k r , k 2 = k r 2 , K d = K + Δ K ,
ξ Δ Λ = l / 2 [ k s z + Δ K z - k 2 - ( k s + Δ K ) 2 ] ,
ξ Δ Λ = l / 2 ( Δ K z + tan θ s Δ K ) .
K c = K · e ^ c = 2 π Λ c Λ a 2 + Λ c 2 , K a = K · e ^ a = 2 π Λ a Λ a 2 + Λ c 2 .
Δ K c K c = K a 2 - K c 2 K 2 Δ Λ c Λ c - 2 K a 2 K 2 Δ Λ a Λ a Δ K a K a = K c 2 - K a 2 K 2 Δ Λ a Λ a - 2 K c 2 K 2 Δ Λ c Λ c .
K c = 2 π n ¯ λ 0 [ cos ( θ s - β ) - cos ( θ r - β ) ] , K a = 2 π n ¯ λ 0 [ sin ( θ s - β ) - sin ( θ r - β ) ] .
Δ K z = Δ K c cos β - Δ K a sin β , Δ K = Δ K c sin β + Δ K a cos β .
ξ Δ Λ = l 2 cos θ s { cos ( θ s - β ) K c K 2 [ ( K a 2 - K c 2 ) Δ Λ c Λ c - 2 K a 2 Δ Λ a Λ a ] + sin ( θ s - β ) K a K 2 [ ( K c 2 - K a 2 ) Δ Λ a Λ a - 2 K c 2 Δ Λ c Λ c ] } ,
ξ Δ Λ = π l n ¯ λ 0 cos θ s { [ cos ( θ r - θ s ) - 1 ] Δ Λ c Λ c × cos 2 ( θ r - β ) + Δ Λ a Λ a sin 2 ( θ r - β ) ] - sin ( θ r - β ) ( Δ Λ a Λ a - Δ Λ c Λ c ) × sin ( θ r - β ) cos ( θ r - β ) } .
L = Δ Λ c Λ c cos 2 ( θ r - β ) + Δ Λ a Λ a sin 2 ( θ r - β ) + Δ n ρ n ¯ ,
T = ( Δ Λ a Λ a - Δ Λ c Λ c ) sin ( θ r - β ) cos ( θ r - β ) - Δ n ρ n ¯ tan θ r             R , T = ( Δ Λ a Λ a - Δ Λ c Λ c ) sin ( θ r - β ) cos ( θ r - β ) + Δ n ρ n ¯ cot θ r             P ,
θ ρ = cos θ ^ r n ¯ cos θ r Δ θ ^ ρ R , T = sin θ ^ r n ¯ sin θ r Δ θ ^ ρ P .
N ( θ s ) = Δ n ρ - Δ n σ n ¯ .
ξ ( θ s ) = π l n ¯ λ 0 cos θ s { N ( θ s ) + ( L - Δ λ 0 λ 0 ) [ cos ( θ r - θ s ) - 1 ] - ( T + θ ρ ) sin ( θ r - θ s ) } .
H ( θ s ) = exp [ - i ξ 0 ( θ s ) ] sin ξ 0 ( θ s ) ξ 0 ( θ s ) ,
ξ 0 ( θ s ) = π l n ¯ λ 0 cos θ s sin ( θ r - θ s 0 ) { N ( θ s ) sin ( θ r - θ s 0 ) - N ( θ s 0 ) sin ( θ r - θ s ) + L [ sin ( θ r - θ s ) + sin ( θ s - θ s 0 ) + sin ( θ s 0 - θ r ) ] } ,
θ ρ = N ( θ s 0 ) + L [ cos ( θ r - θ s 0 ) - 1 ] - T sin ( θ r - θ s 0 ) sin ( θ r - θ s 0 ) ;
ξ 0 ( θ s ) = π l n ¯ λ 0 cos θ s [ cos ( θ r - θ s 0 ) - 1 ] { N ( θ s ) × [ cos ( θ r - θ s 0 ) - 1 ] - N ( θ s 0 ) [ cos ( θ r - θ s ) - 1 ] + T [ sin ( θ r 0 - θ s ) + sin ( θ s - θ s 0 ) + sin ( θ s 0 - θ r ) ] } ,
Δ λ 0 λ 0 = N ( θ s 0 ) + L [ cos ( θ r - θ s 0 ) - 1 ] - T sin ( θ r - θ s 0 ) cos ( θ r - θ s 0 ) - 1 ;
ξ 0 ( θ s ) = π l n ¯ λ 0 cos θ s sin ( θ r - θ s 0 ) { N ( θ s ) sin ( θ r - θ s 0 ) - N ( θ s 0 ) sin ( θ r - θ s ) + sin ( θ r - θ s ) + sin ( θ s - θ s 0 ) + sin ( θ s 0 - θ r ) cos ( θ r - θ s 0 ) - 1 × [ sin ( θ r - θ s 0 ) d N d θ s | o + cos ( θ r - θ s 0 ) N ( θ s 0 ) ] } ,
- Δ λ 0 λ 0 + L = sin ( θ r - θ s 0 ) d N d θ s | o + cos ( θ r - θ s 0 ) N ( θ s 0 ) cos ( θ r - θ s 0 ) - 1 ,
θ ρ + T = d N d θ s | o + 1 + cos ( θ r - θ s 0 ) sin ( θ r - θ s 0 ) N ( θ s 0 ) .

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