Abstract

We investigate photorefractive media for which quasi-stabilized ionic gratings can be used to prolong readout lifetime. We use coupled-transport-mode theory to describe the coevolution of photorefractive gratings that arise from free-electron transport and ionic transport. We evaluate in detail the differences between low-temperature and high-temperature recording for typical conditions required by multiplex holography. We provide general normalized examples for simple diffusion transport and specific examples for photovoltaic LiNbO3. We introduce a common formalism to compare widely varying results present in the literature and to guide the materials and system development processes.

© 1997 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref]
  3. H. Kurz, “Lithium niobate as a material for holographic information storage,” Philips Tech. Rev. 37, 109 (1977).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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  23. Quasi-stabilized ionic gratings are frequently called fixed gratings in the literature. Because they eventually decay, ionic gratings are not truly fixed but can be considered to be fixed over a time scale of interest.
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    [Crossref]
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    [Crossref]
  29. H. Vormann, G. Weber, S. Kapphan, and E. Krätzig, “Hydrogen as origin of thermal fixing in LiNbO3:Fe,” Solid State Commun. 40, 543 (1981).
    [Crossref]
  30. D. Kirillov and J. Feinberg, “Fixable complementary gratings in photorefractive BaTiO3,” Opt. Lett. 16, 1520 (1991).
    [Crossref] [PubMed]
  31. L. Arizmendi, “Thermal fixing of holographic gratings in Bi12SiO20,” J. Appl. Phys. 65, 423 (1989).
    [Crossref]
  32. G. Montemezzani and P. Günter, “Thermal hologram fixing in pure and doped KNbO3 crystals,” J. Opt. Soc. Am. B 7, 2323 (1990).
    [Crossref]
  33. S. W. McCahon, D. Rytz, G. C. Valley, M. B. Klein, and B. A. Wechsler, “Hologram fixing in Bi12TiO20 using heating and an ac electric field,” Appl. Opt. 28, 1967 (1989).
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    [Crossref]
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    [Crossref] [PubMed]
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  38. S. I. Stepanov and G. S. Trofimov, “Holographic storage mechanisms in photorefractive crystals with a complex impurity level structure,” Zh. Tekh. Fiz. 55, 559 (1985) [Sov. Phys. Tech. Phys. 30, 331 (1985)].
  39. M. C. Bashaw, T.-P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube, “Theory of complementary holograms arising from electron-hole transport in photorefractive media,” J. Opt. Soc. Am. B 7, 2329 (1990).
    [Crossref]
  40. N. V. Kukhtarev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Pis’ma Zh. Tekh. Fiz. 2, 1114 (1976) [Sov. Tech. Phys. Lett. 2, 438 (1976)].
  41. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
    [Crossref]
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  43. R. A. Mullen, “Photorefractive measurements of physical parameters,” in Photorefractive Materials and Their Applications I, P. Günter and J.-P. Huignard, eds. (Springer-Verlag, Berlin, 1988), p. 167.
  44. H. W. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909 (1969).
    [Crossref]
  45. These values are found in or calculated from Ref. 7 and references therein.
  46. M. C. Bashaw, T.-P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube, “Introduction, revelation, and evolution of complementary holograms in photorefractive bismuth silicon oxide,” Phys. Rev. B 42, 5641 (1990).
    [Crossref]
  47. M. C. Bashaw, M. Jeganathan, and L. Hesselink, “Theory of two-species electron-hole transport in photorefractive media,” J. Opt. Soc. Am. B 11, 1743 (1994).
    [Crossref]
  48. S. Orlov, D. Psaltis, and R. R. Neurgaonkar, “Dynamic electronic compensation of fixed gratings in photorefractive media,” Appl. Phys. Lett. 63, 2466 (1993).
    [Crossref]
  49. K. Blotekjaer, “Limitations on holographic storage capacity on photochormic and photorefractive media,” Appl. Opt. 18, 57 (1979).
    [Crossref]
  50. E. S. Maniloff and K. M. Johnson, “Maximized photorefractive holographic storage,” J. Appl. Phys. 70, 4702 (1991).
    [Crossref]
  51. E. S. Maniloff and K. M. Johnson, “Incremental recording for photorefractive hologram multiplexing: comment,” Opt. Lett. 17, 961 (1992).
    [Crossref]
  52. Y. Taketomi, J. E. Ford, H. Sasaki, J. Ma, Y. Fainman, and S. H. Lee, “Incremental recording for photorefractive hologram multiplexing,” Opt. Lett. 16, 1774 (1991).
    [Crossref] [PubMed]
  53. Y. Taketomi, J. E. Ford, H. Sasaki, J. Ma, Y. Fainman, and S. H. Lee, “Incremental recording for photorefractive hologram multiplexing,” Opt. Lett. 17, 962 (1992).
    [Crossref]
  54. M. Jeganathan and M. C. Bashaw, “Evolution and propagation of grating envelopes during erasure in bulk photorefractive media,” J. Opt. Soc. Am. B 12, 1370 (1995).
    [Crossref]
  55. M. Segev, Y. Ophir, and B. Fischer, “Multi two-wave mixing, the fanning process, and its bleaching in photorefractive media,” Opt. Commun. 77, 265 (1990).
    [Crossref]
  56. R. De Vré, J. F. Heanue, K. Gürkan, and L. Hesselink, “Transfer functions based on Bragg detuning effects for image-bearing holograms recorded in photorefractive crystals,” J. Opt. Soc. Am. A 13, 1331 (1996).
    [Crossref]
  57. Properties of Lithium Niobate, Vol. 5 of EMIS DataReview Series (Inspec, London, 1989).
  58. M. C. Bashaw, J. F. Heanue, A. Aharoni, J. F. Walkup, and L. Hesselink, “Crosstalk considerations for angular and phase-encoded multiplexing in volume holography,” J. Opt. Soc. Am. B 11, 1820 (1994).
    [Crossref]
  59. W. Meyer, P. Würfel, R. Munser, and G. Müller-Vogt, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171 (1979).
    [Crossref]
  60. S. Orlov and A. Yariv, “Long-lifetime hologram fixing and ionic conductivity in photorefractive lithium niobate,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest SeriesOptical Society of America, Washington, D.C., 1996), paper CTuA3.
  61. A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
    [Crossref]
  62. G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, R4457 (1994).
    [Crossref] [PubMed]
  63. M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095 (1995).
    [Crossref] [PubMed]

1996 (3)

1995 (3)

1994 (5)

1993 (3)

1992 (7)

1991 (6)

Y. Taketomi, J. E. Ford, H. Sasaki, J. Ma, Y. Fainman, and S. H. Lee, “Incremental recording for photorefractive hologram multiplexing,” Opt. Lett. 16, 1774 (1991).
[Crossref] [PubMed]

E. S. Maniloff and K. M. Johnson, “Maximized photorefractive holographic storage,” J. Appl. Phys. 70, 4702 (1991).
[Crossref]

C. Gu, J. Hong, H.-Y. Li, D. Psaltis, and P. Yeh, “Dynamics of grating formation in photovoltaic media,” J. Appl. Phys. 69, 1167 (1991).
[Crossref]

K. Buse, L. Holtmann, and E. Krätzig, “Activation of BaTiO3 for infrared holographic recording,” Opt. Commun. 85, 183 (1991).
[Crossref]

L. Arizmendi, P. D. Townsend, M. Carrascosa, J. Baquedano, and J. M. Cabrera, “Photorefractive fixing and related thermal effects in LiNbO3,” J. Phys. Condens. Matter 3, 5399 (1991).
[Crossref]

D. Kirillov and J. Feinberg, “Fixable complementary gratings in photorefractive BaTiO3,” Opt. Lett. 16, 1520 (1991).
[Crossref] [PubMed]

1990 (5)

G. Montemezzani and P. Günter, “Thermal hologram fixing in pure and doped KNbO3 crystals,” J. Opt. Soc. Am. B 7, 2323 (1990).
[Crossref]

M. C. Bashaw, T.-P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube, “Theory of complementary holograms arising from electron-hole transport in photorefractive media,” J. Opt. Soc. Am. B 7, 2329 (1990).
[Crossref]

M. Carrascosa and F. Agulló-López, “Theoretical modeling of the fixing and developing of holographic gratings in LiNbO3,” J. Opt. Soc. Am. B 7, 2317 (1990).
[Crossref]

M. C. Bashaw, T.-P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube, “Introduction, revelation, and evolution of complementary holograms in photorefractive bismuth silicon oxide,” Phys. Rev. B 42, 5641 (1990).
[Crossref]

M. Segev, Y. Ophir, and B. Fischer, “Multi two-wave mixing, the fanning process, and its bleaching in photorefractive media,” Opt. Commun. 77, 265 (1990).
[Crossref]

1989 (2)

1985 (1)

S. I. Stepanov and G. S. Trofimov, “Holographic storage mechanisms in photorefractive crystals with a complex impurity level structure,” Zh. Tekh. Fiz. 55, 559 (1985) [Sov. Phys. Tech. Phys. 30, 331 (1985)].

1981 (1)

H. Vormann, G. Weber, S. Kapphan, and E. Krätzig, “Hydrogen as origin of thermal fixing in LiNbO3:Fe,” Solid State Commun. 40, 543 (1981).
[Crossref]

1979 (5)

V. V. Kulikov and S. I. Stepanov, “Mechanisms of holographic recording and thermal fixing in photorefractive LiNbO3:Fe,” Fiz. Tverd. Tela (Leningrad) 21, 3204 (1979) [Sov. Phys. Solid State 21, 1849 (1979)].

E. K. Gulanyan, I. R. Dorosh, V. D. Iskin, A. L. Mikaelyan, and M. A. Maiorchuk, “Nondestructive readout of holograms in iron-doped lithium niobate crystals,” Kvantovaya Elektron. (Moscow) 62097 (1979) [Sov. J. Quantum Electron. 9, 647 (1979)].
[Crossref]

W. Meyer, P. Würfel, R. Munser, and G. Müller-Vogt, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171 (1979).
[Crossref]

K. Blotekjaer, “Limitations on holographic storage capacity on photochormic and photorefractive media,” Appl. Opt. 18, 57 (1979).
[Crossref]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
[Crossref]

1978 (1)

D. L. Staebler, W. J. Burke, W. Phillips, and G. A. Alphonse, “Volume phase holographic storage in ferroelectric crystals,” Opt. Eng. 17, 313 (1978).

1977 (2)

H. Kurz, “Photorefractive recording dynamics and multiple storage of volume holograms in photorefractive LiNbO3,” Opt. Acta 24, 463 (1977).
[Crossref]

H. Kurz, “Lithium niobate as a material for holographic information storage,” Philips Tech. Rev. 37, 109 (1977).

1976 (2)

D. von der Linde, A. M. Glass, and K. F. Rodgers, “Optical storage using refractive index changes induced by two-step excitation,” J. Appl. Phys. 47, 217 (1976).
[Crossref]

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

1975 (2)

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

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 (1975).
[Crossref]

1974 (2)

L. d’Auria, J.-P. Huignard, C. Slezak, and E. Spitz, “Experimental holographic read–write memory using 3-D storage,” Appl. Opt. 13, 808 (1974).
[Crossref]

D. L. Staebler and W. Phillips, “Hologram storage in photochromic LiNbO3,” Appl. Phys. Lett. 24, 268 (1974).
[Crossref]

1973 (1)

V. I. Bobrinev, Z. G. Vasil’eva, E. K. Gulanyan, and A. L. Mikaelyan, “Multiple rerecording and fixation of holograms in lithium niobate crystals doped with iron,” Zh. Eksp. Teor. Fiz. Pis’ma Red. 18, 267 (1973) [Sov. Phys. JETP Lett. 18, 159 (1973)].

1972 (1)

1971 (1)

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

1969 (1)

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

1966 (1)

A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
[Crossref]

Agulló-López, F.

Aharoni, A.

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

A. Aharoni, M. Jeganathan, M. C. Bashaw, and L. Hesselink, “Prolonged readout of photorefractive holograms by replay at a longer wavelength,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CTuJ4.

Alphonse, G. A.

D. L. Staebler, W. J. Burke, W. Phillips, and G. A. Alphonse, “Volume phase holographic storage in ferroelectric crystals,” Opt. Eng. 17, 313 (1978).

Amodei, J. J.

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

J. J. Amodei, W. Phillips, and D. L. Staebler, “Improved electrooptic materials and fixing techniques for holographic recording,” Appl. Opt. 11, 390 (1972).
[Crossref] [PubMed]

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

Arizmendi, L.

L. Arizmendi, P. D. Townsend, M. Carrascosa, J. Baquedano, and J. M. Cabrera, “Photorefractive fixing and related thermal effects in LiNbO3,” J. Phys. Condens. Matter 3, 5399 (1991).
[Crossref]

L. Arizmendi, “Thermal fixing of holographic gratings in Bi12SiO20,” J. Appl. Phys. 65, 423 (1989).
[Crossref]

Ashkin, A.

A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
[Crossref]

Bai, Y. S.

Ballman, A. A.

A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
[Crossref]

Bann, S.

Baquedano, J.

L. Arizmendi, P. D. Townsend, M. Carrascosa, J. Baquedano, and J. M. Cabrera, “Photorefractive fixing and related thermal effects in LiNbO3,” J. Phys. Condens. Matter 3, 5399 (1991).
[Crossref]

Barker, R. C.

M. C. Bashaw, T.-P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube, “Theory of complementary holograms arising from electron-hole transport in photorefractive media,” J. Opt. Soc. Am. B 7, 2329 (1990).
[Crossref]

M. C. Bashaw, T.-P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube, “Introduction, revelation, and evolution of complementary holograms in photorefractive bismuth silicon oxide,” Phys. Rev. B 42, 5641 (1990).
[Crossref]

Bashaw, M. C.

J. F. Heanue, M. C. Bashaw, A. J. Daiber, R. Snyder, and L. Hesselink, “Digital holographic storage system incorporating thermal fixing in lithium niobate,” Opt. Lett. 21, 1615 (1996).
[Crossref] [PubMed]

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095 (1995).
[Crossref] [PubMed]

M. Jeganathan and M. C. Bashaw, “Evolution and propagation of grating envelopes during erasure in bulk photorefractive media,” J. Opt. Soc. Am. B 12, 1370 (1995).
[Crossref]

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

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, R4457 (1994).
[Crossref] [PubMed]

M. C. Bashaw, M. Jeganathan, and L. Hesselink, “Theory of two-species electron-hole transport in photorefractive media,” J. Opt. Soc. Am. B 11, 1743 (1994).
[Crossref]

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

M. C. Bashaw, T.-P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube, “Introduction, revelation, and evolution of complementary holograms in photorefractive bismuth silicon oxide,” Phys. Rev. B 42, 5641 (1990).
[Crossref]

M. C. Bashaw, T.-P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube, “Theory of complementary holograms arising from electron-hole transport in photorefractive media,” J. Opt. Soc. Am. B 7, 2329 (1990).
[Crossref]

A. Aharoni, M. Jeganathan, M. C. Bashaw, and L. Hesselink, “Prolonged readout of photorefractive holograms by replay at a longer wavelength,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CTuJ4.

Blotekjaer, K.

Bobrinev, V. I.

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Boj, S.

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A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
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D. L. Staebler, W. J. Burke, W. Phillips, and G. A. Alphonse, “Volume phase holographic storage in ferroelectric crystals,” Opt. Eng. 17, 313 (1978).

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K. Buse, L. Holtmann, and E. Krätzig, “Activation of BaTiO3 for infrared holographic recording,” Opt. Commun. 85, 183 (1991).
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L. Arizmendi, P. D. Townsend, M. Carrascosa, J. Baquedano, and J. M. Cabrera, “Photorefractive fixing and related thermal effects in LiNbO3,” J. Phys. Condens. Matter 3, 5399 (1991).
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M. C. Bashaw, T.-P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube, “Introduction, revelation, and evolution of complementary holograms in photorefractive bismuth silicon oxide,” Phys. Rev. B 42, 5641 (1990).
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M. C. Bashaw, T.-P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube, “Theory of complementary holograms arising from electron-hole transport in photorefractive media,” J. Opt. Soc. Am. B 7, 2329 (1990).
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A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
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Feinberg, J.

Fejer, M. M.

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095 (1995).
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G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, R4457 (1994).
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M. Segev, Y. Ophir, and B. Fischer, “Multi two-wave mixing, the fanning process, and its bleaching in photorefractive media,” Opt. Commun. 77, 265 (1990).
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D. von der Linde, A. M. Glass, and K. F. Rodgers, “Optical storage using refractive index changes induced by two-step excitation,” J. Appl. Phys. 47, 217 (1976).
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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 (1975).
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C. Gu, J. Hong, H.-Y. Li, D. Psaltis, and P. Yeh, “Dynamics of grating formation in photovoltaic media,” J. Appl. Phys. 69, 1167 (1991).
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Günter, P.

Gürkan, K.

Heanue, J. F.

Hesselink, L.

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K. Buse, L. Holtmann, and E. Krätzig, “Activation of BaTiO3 for infrared holographic recording,” Opt. Commun. 85, 183 (1991).
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C. Gu, J. Hong, H.-Y. Li, D. Psaltis, and P. Yeh, “Dynamics of grating formation in photovoltaic media,” J. Appl. Phys. 69, 1167 (1991).
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Iskin, V. D.

E. K. Gulanyan, I. R. Dorosh, V. D. Iskin, A. L. Mikaelyan, and M. A. Maiorchuk, “Nondestructive readout of holograms in iron-doped lithium niobate crystals,” Kvantovaya Elektron. (Moscow) 62097 (1979) [Sov. J. Quantum Electron. 9, 647 (1979)].
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Külich, H.-C.

Kulikov, V. V.

V. V. Kulikov and S. I. Stepanov, “Mechanisms of holographic recording and thermal fixing in photorefractive LiNbO3:Fe,” Fiz. Tverd. Tela (Leningrad) 21, 3204 (1979) [Sov. Phys. Solid State 21, 1849 (1979)].

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A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
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A. Yariv, S. Orlov, G. Rakuljic, and V. Leyva, “Hologram fixing, readout, and storage dynamics in photorefractive materials,” Opt. Lett. 20, 1334 (1995).
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A. Yariv, S. Orlov, G. Rakuljic, and V. Leyva, “Holographic fixing, readout, and storage dynamics in photorefractive materials,” in Conference on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1995), paper TPC24.

Li, H.-Y.

C. Gu, J. Hong, H.-Y. Li, D. Psaltis, and P. Yeh, “Dynamics of grating formation in photovoltaic media,” J. Appl. Phys. 69, 1167 (1991).
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Li, H.-Y. S.

Ma, J.

Ma, T.-P.

M. C. Bashaw, T.-P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube, “Theory of complementary holograms arising from electron-hole transport in photorefractive media,” J. Opt. Soc. Am. B 7, 2329 (1990).
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M. C. Bashaw, T.-P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube, “Introduction, revelation, and evolution of complementary holograms in photorefractive bismuth silicon oxide,” Phys. Rev. B 42, 5641 (1990).
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E. K. Gulanyan, I. R. Dorosh, V. D. Iskin, A. L. Mikaelyan, and M. A. Maiorchuk, “Nondestructive readout of holograms in iron-doped lithium niobate crystals,” Kvantovaya Elektron. (Moscow) 62097 (1979) [Sov. J. Quantum Electron. 9, 647 (1979)].
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E. S. Maniloff and K. M. Johnson, “Incremental recording for photorefractive hologram multiplexing: comment,” Opt. Lett. 17, 961 (1992).
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E. S. Maniloff and K. M. Johnson, “Maximized photorefractive holographic storage,” J. Appl. Phys. 70, 4702 (1991).
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N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
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Meyer, W.

W. Meyer, P. Würfel, R. Munser, and G. Müller-Vogt, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171 (1979).
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E. K. Gulanyan, I. R. Dorosh, V. D. Iskin, A. L. Mikaelyan, and M. A. Maiorchuk, “Nondestructive readout of holograms in iron-doped lithium niobate crystals,” Kvantovaya Elektron. (Moscow) 62097 (1979) [Sov. J. Quantum Electron. 9, 647 (1979)].
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V. I. Bobrinev, Z. G. Vasil’eva, E. K. Gulanyan, and A. L. Mikaelyan, “Multiple rerecording and fixation of holograms in lithium niobate crystals doped with iron,” Zh. Eksp. Teor. Fiz. Pis’ma Red. 18, 267 (1973) [Sov. Phys. JETP Lett. 18, 159 (1973)].

Mok, F. H.

Montemezzani, G.

Mroczkowski, S.

M. C. Bashaw, T.-P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube, “Theory of complementary holograms arising from electron-hole transport in photorefractive media,” J. Opt. Soc. Am. B 7, 2329 (1990).
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M. C. Bashaw, T.-P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube, “Introduction, revelation, and evolution of complementary holograms in photorefractive bismuth silicon oxide,” Phys. Rev. B 42, 5641 (1990).
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Müller-Vogt, G.

W. Meyer, P. Würfel, R. Munser, and G. Müller-Vogt, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171 (1979).
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W. Meyer, P. Würfel, R. Munser, and G. Müller-Vogt, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171 (1979).
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A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
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Y. S. Bai, R. R. Neurgaonkar, and R. Kachru, “Resonant two-photon photorefractive gating in praeseodymium-doped strontium barium niobate with cw lasers,” Opt. Lett. 21, 567 (1996).
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N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
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Ophir, Y.

M. Segev, Y. Ophir, and B. Fischer, “Multi two-wave mixing, the fanning process, and its bleaching in photorefractive media,” Opt. Commun. 77, 265 (1990).
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A. Yariv, S. Orlov, G. Rakuljic, and V. Leyva, “Hologram fixing, readout, and storage dynamics in photorefractive materials,” Opt. Lett. 20, 1334 (1995).
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S. Orlov, D. Psaltis, and R. R. Neurgaonkar, “Dynamic electronic compensation of fixed gratings in photorefractive media,” Appl. Phys. Lett. 63, 2466 (1993).
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A. Yariv, S. Orlov, G. Rakuljic, and V. Leyva, “Holographic fixing, readout, and storage dynamics in photorefractive materials,” in Conference on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1995), paper TPC24.

S. Orlov and A. Yariv, “Long-lifetime hologram fixing and ionic conductivity in photorefractive lithium niobate,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest SeriesOptical Society of America, Washington, D.C., 1996), paper CTuA3.

Pauliat, G.

Phillips, W.

D. L. Staebler, W. J. Burke, W. Phillips, and G. A. Alphonse, “Volume phase holographic storage in ferroelectric crystals,” Opt. Eng. 17, 313 (1978).

D. L. Staebler, W. J. Burke, W. Phillips, and J. J. Amodei, “Multiple storage and erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182 (1975).
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D. L. Staebler and W. Phillips, “Hologram storage in photochromic LiNbO3,” Appl. Phys. Lett. 24, 268 (1974).
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J. J. Amodei, W. Phillips, and D. L. Staebler, “Improved electrooptic materials and fixing techniques for holographic recording,” Appl. Opt. 11, 390 (1972).
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D. Psaltis, F. H. Mok, and H.-Y. S. Li, “Nonvolatile storage in photorefractive crystals,” Opt. Lett. 19, 210 (1994).
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S. Orlov, D. Psaltis, and R. R. Neurgaonkar, “Dynamic electronic compensation of fixed gratings in photorefractive media,” Appl. Phys. Lett. 63, 2466 (1993).
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C. Gu, J. Hong, H.-Y. Li, D. Psaltis, and P. Yeh, “Dynamics of grating formation in photovoltaic media,” J. Appl. Phys. 69, 1167 (1991).
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Rajbenbach, H.

Rakuljic, G.

A. Yariv, S. Orlov, G. Rakuljic, and V. Leyva, “Hologram fixing, readout, and storage dynamics in photorefractive materials,” Opt. Lett. 20, 1334 (1995).
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A. Yariv, S. Orlov, G. Rakuljic, and V. Leyva, “Holographic fixing, readout, and storage dynamics in photorefractive materials,” in Conference on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1995), paper TPC24.

Rakuljic, G. A.

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D. von der Linde, A. M. Glass, and K. F. Rodgers, “Optical storage using refractive index changes induced by two-step excitation,” J. Appl. Phys. 47, 217 (1976).
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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 (1975).
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Rytz, D.

Sasaki, H.

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M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095 (1995).
[Crossref] [PubMed]

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, R4457 (1994).
[Crossref] [PubMed]

M. Segev, Y. Ophir, and B. Fischer, “Multi two-wave mixing, the fanning process, and its bleaching in photorefractive media,” Opt. Commun. 77, 265 (1990).
[Crossref]

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Smith, R. G.

A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
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Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
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Staebler, D. L.

D. L. Staebler, W. J. Burke, W. Phillips, and G. A. Alphonse, “Volume phase holographic storage in ferroelectric crystals,” Opt. Eng. 17, 313 (1978).

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

D. L. Staebler and W. Phillips, “Hologram storage in photochromic LiNbO3,” Appl. Phys. Lett. 24, 268 (1974).
[Crossref]

J. J. Amodei, W. Phillips, and D. L. Staebler, “Improved electrooptic materials and fixing techniques for holographic recording,” Appl. Opt. 11, 390 (1972).
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Taketomi, Y.

Taya, M.

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095 (1995).
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L. Arizmendi, P. D. Townsend, M. Carrascosa, J. Baquedano, and J. M. Cabrera, “Photorefractive fixing and related thermal effects in LiNbO3,” J. Phys. Condens. Matter 3, 5399 (1991).
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S. I. Stepanov and G. S. Trofimov, “Holographic storage mechanisms in photorefractive crystals with a complex impurity level structure,” Zh. Tekh. Fiz. 55, 559 (1985) [Sov. Phys. Tech. Phys. 30, 331 (1985)].

Valley, G. C.

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095 (1995).
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G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, R4457 (1994).
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Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949 (1979).
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D. von der Linde, A. M. Glass, and K. F. Rodgers, “Optical storage using refractive index changes induced by two-step excitation,” J. Appl. Phys. 47, 217 (1976).
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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 (1975).
[Crossref]

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H. Vormann, G. Weber, S. Kapphan, and E. Krätzig, “Hydrogen as origin of thermal fixing in LiNbO3:Fe,” Solid State Commun. 40, 543 (1981).
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Weber, G.

H. Vormann, G. Weber, S. Kapphan, and E. Krätzig, “Hydrogen as origin of thermal fixing in LiNbO3:Fe,” Solid State Commun. 40, 543 (1981).
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W. Meyer, P. Würfel, R. Munser, and G. Müller-Vogt, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171 (1979).
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Yariv, A.

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

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, R4457 (1994).
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[Crossref]

A. Yariv, S. Orlov, G. Rakuljic, and V. Leyva, “Holographic fixing, readout, and storage dynamics in photorefractive materials,” in Conference on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1995), paper TPC24.

S. Orlov and A. Yariv, “Long-lifetime hologram fixing and ionic conductivity in photorefractive lithium niobate,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest SeriesOptical Society of America, Washington, D.C., 1996), paper CTuA3.

Yeh, P.

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

Opt. Acta (1)

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[Crossref]

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[Crossref]

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[Crossref]

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Phys. Rev. B (1)

M. C. Bashaw, T.-P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube, “Introduction, revelation, and evolution of complementary holograms in photorefractive bismuth silicon oxide,” Phys. Rev. B 42, 5641 (1990).
[Crossref]

Phys. Status Solidi A (1)

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[Crossref]

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Science (1)

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[Crossref] [PubMed]

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H. Vormann, G. Weber, S. Kapphan, and E. Krätzig, “Hydrogen as origin of thermal fixing in LiNbO3:Fe,” Solid State Commun. 40, 543 (1981).
[Crossref]

Zh. Eksp. Teor. Fiz. Pis’ma Red. (1)

V. I. Bobrinev, Z. G. Vasil’eva, E. K. Gulanyan, and A. L. Mikaelyan, “Multiple rerecording and fixation of holograms in lithium niobate crystals doped with iron,” Zh. Eksp. Teor. Fiz. Pis’ma Red. 18, 267 (1973) [Sov. Phys. JETP Lett. 18, 159 (1973)].

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Other (8)

A. Yariv, S. Orlov, G. Rakuljic, and V. Leyva, “Holographic fixing, readout, and storage dynamics in photorefractive materials,” in Conference on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1995), paper TPC24.

R. A. Mullen, “Photorefractive measurements of physical parameters,” in Photorefractive Materials and Their Applications I, P. Günter and J.-P. Huignard, eds. (Springer-Verlag, Berlin, 1988), p. 167.

These values are found in or calculated from Ref. 7 and references therein.

S. Orlov and A. Yariv, “Long-lifetime hologram fixing and ionic conductivity in photorefractive lithium niobate,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest SeriesOptical Society of America, Washington, D.C., 1996), paper CTuA3.

Properties of Lithium Niobate, Vol. 5 of EMIS DataReview Series (Inspec, London, 1989).

J. P. Wilde, “Spectroscopic characterization of photorefractive materials for holographic storage applications,” in Fluorescence Detection IV, E. R. Menzel, ed., Proc. SPIE2705, 82 (1996).
[Crossref]

A. Aharoni, M. Jeganathan, M. C. Bashaw, and L. Hesselink, “Prolonged readout of photorefractive holograms by replay at a longer wavelength,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CTuJ4.

Quasi-stabilized ionic gratings are frequently called fixed gratings in the literature. Because they eventually decay, ionic gratings are not truly fixed but can be considered to be fixed over a time scale of interest.

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Figures (14)

Fig. 1
Fig. 1

Relative index perturbation δn as a function of relative grating wave number k/kD for simple diffusion transport.

Fig. 2
Fig. 2

Index perturbation normalized to modulation depth, δn/m, as a function of grating frequency for typical parameters of LiNbO3 displaying a strong photovoltaic effect. Here the concentration of reduced impurities, N, is chosen such that the Debye period of 0.17 µm occurs for the minimum N for several impurity concentrations. The dashed curve indicates the index perturbation that would result in the absence of the photovoltaic effect. The parameters for the other data are 1, N=1018 cm-3, N=6.0×1016; 2, N=1017 cm-3, N=6.0 ×1016; 3, N=6.0×1015 cm-3, N=6.0×1016.

Fig. 3
Fig. 3

Index perturbation normalized to modulation depth, δn/m, as a function of grating frequency for typical parameters of LiNbO3 displaying a strong photovoltaic effect. Here, the impurities are half-reduced, so N=N. The dashed curve indicates the index perturbation that would result in the absence of the photovoltaic effect for a Debye period of 0.17 µm. The parameters for the other data are 1, N=1019 cm-3; 2, N=1018 cm-3; and 3, N=2.4×1017 cm-3.

Fig. 4
Fig. 4

Relative concentration of trapped charge, normalized to modulation depth, as a function of relative grating wave number k/kD for simple diffusion transport when mobile ions are present. Shown for several relative concentrations of ions, M/NR, are the first-order coefficients for (a) the electronic transport system, N1/mNR, and (b) the ionic transport system, M1/mNR.

Fig. 5
Fig. 5

Relative index perturbation δn as a function of relative grating wave number k/kD for simple diffusion transport when mobile ions are present, shown for several relative concentrations of ions, M/NR.

Fig. 6
Fig. 6

Relative concentration of trapped charge, normalized to modulation depth, as a function of grating frequency, for mobile-ion transport in a sample of LiNbO3 displaying a strong photovoltaic effect. Shown are the first-order coefficients for (a) the electronic transport system, N1/m, and (b) the ionic transport system, M1/m. The concentration of impurities is N=2.4 ×1017 cm-3, and the oxidation and reduction fractions are even, N=N. The dashed curve indicates diffusion only. The concentrations of mobile ions are 1, M=1019 cm-3; 2, M=1018 cm-3; 3, M=6.0×1016 cm-3; and 4, M=1016 cm-3.

Fig. 7
Fig. 7

Index perturbation normalized to modulation depth, δn/m, as a function of grating frequency, for typical parameters of LiNbO3 displaying a strong photovoltaic effect. The concentration of impurities is N=2.4×1017 cm-3, and the oxidation and reduction fractions are even, N=N. The dashed curve indicates diffusion only. The concentrations of mobile ions are 1, M=1019 cm-3; 2, M=1018 cm-3; 3, M=6.0×1016 cm-3; and 4, M=1016 cm-3.

Fig. 8
Fig. 8

Response rates as a function of grating frequency, for typical parameters of LiNbO3 displaying a strong photovoltaic effect, at a temperature of 423 K. The concentration of impurities is N=2.4×1017 cm-3, and the oxidation and reduction fractions are even, N=N. Curve 0 represents the response rate for the electronic system, for which Γ+=Γe, and does not change appreciably for ion concentrations of interest. The remaining solid curves represent rates for coupled mobile-ion transport Γ-, and the dashed curves represent rates corresponding to ion transport in isolation Γi. The concentrations of mobile ions are 1, M=1019 cm-3; 2, M=1018 cm-3; 3, M=6.0 ×1016 cm-3; and 4, M=1016 cm-3.

Fig. 9
Fig. 9

(a) Relative index perturbation, δn/m, for a quasi-stabilized grating established by postcompensation, as a function of relative grating wave number k/kD for simple diffusion transport with mobile ions, for several relative concentrations of ions, M/NR. The dashed curve represents unstabilized electron-only transport at room temperature. (b) Relative index perturbation, δn/m, for a grating quasi-stabilized by simultaneous compensation, as a function of relative grating wave number k/kD for simple diffusion transport with mobile ions, for the same set of relative concentrations of ions. The dashed curve represents unstabilized electron-only transport at the recording temperature. (c) Ratio of quasi-stabilized gratings comparing simultaneous compensation and postcompensation for simple diffusion-only transport.

Fig. 10
Fig. 10

(a) Index perturbation normalized to modulation depth, δn/m, for a grating quasi-stabilized by postcompensation, as a function of grating frequency, for typical parameters of LiNbO3 displaying a strong photovoltaic effect, with mobile ions, for several concentrations of ions. The concentration of impurities is N=2.4×1017 cm-3, and the oxidation and reduction fractions are even, N=N. The dashed curve represents unstabilized electron-only transport at room temperature. The concentrations of mobile ions are 1, M=1019 cm-3; 3, M=6.0 ×1016 cm-3; and 4, M=1016 cm-3. (b) Index perturbation normalized to modulation depth, δn/m, for a grating quasi-stabilized by simultaneous compensation, as a function of grating frequency, for typical parameters of LiNbO3 displaying a strong photovoltaic effect, with mobile ions, for the same set of relative concentrations of ions. The dashed curve represents unstabilized electron-only transport at the elevated temperature. The concentrations of mobile ions are the same. (c) Ratio of quasi-stabilized gratings comparing simultaneous compensation and postcompensation, for typical parameters of LiNbO3 displaying a strong photovoltaic effect.

Fig. 11
Fig. 11

Response rates as a function of grating frequency, for typical parameters of LiNbO3 displaying a strong photovoltaic effect, at a temperature of 300 K. The concentration of impurities is N=2.4×1017 cm-3, and the oxidation and reduction fractions are even, N=N. Curve 0 represents the response rate for the electronic system, for which Γ+=Γe, and does not change appreciably for ion concentrations of interest. The remaining solid curves represent rates for coupled mobile-ion transport Γ-, and the dashed curves represent rates corresponding to ion transport in isolation Γi. The concentrations of mobile ions are 1, M=1019 cm-3; 2, M=1018 cm-3; 3, M=6.0 ×1016 cm-3; and 4, M=1016 cm-3.

Fig. 12
Fig. 12

Decay of quasi-stabilized gratings with typical parameters of LiNbO3 at 300 K for several ion concentrations, shown for (a) the concentration of impurities N=2.4×1017 cm-3, even oxidation, and even reduction fractions N=N and (b) N=6.0×1019 cm-3 and N=6.0×1017 cm-3.

Fig. 13
Fig. 13

Index perturbation normalized to modulation depth, δn/m, as a function of oxidation ratio for several impurity concentrations for typical parameters of LiNbO3 displaying (a) a strong photovoltaic effect and (b) no appreciable photovoltaic effect.

Fig. 14
Fig. 14

Index perturbation normalized to modulation depth, δn/m, as a function of oxidation ratio, when mobile ions are present, for different concentrations of ions. The dashed curves represent unstabilized electron-only transport at the recording temperature. Shown are (a) postcompensation and (b) simultaneous compensation.

Tables (1)

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Table 1 Parameters for LiNbO3

Equations (83)

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Nt=(sI+β)N-γnN,
je=eμenE+kBTμen+κsIN,
nt=(sI+β)N-γnN+·jee,
·E=ρ/(0),
ρ=e(N-N0-n).
 E·dr=V,
me=m sI0sI0+β.
E(x, t)=E0+Re{E10[1-exp(-Γet)]exp(ikx)}.
E(x, t)=E0+Re[E10 exp(-Γet)exp(ikx)],
Γe=ΓReΓdie+ΓIeΓDe-iΓIeΓEe-iΓIeΓPeΓRe+ΓDe-iΓEe,
Γe=Γdie 1+k2/kDe2-ik/kEe-ik/kp1+k2/qDe2-ik/qEe.
qDe2kDe2=qEekEe=0γμeNN0.
E10=-ime ED-i(E0+Ep)1+ED/Eq-i[E0+(N/N)Ep]/Eq,
E10=-ime kVT-i(E0+Eph)1+k2/kDe2-ik/kEe-ik/kP.
ζ(k)=1/(1+k2/kDe2-ik/kEe-ik/kP),
1kE+1kP=1koc,
δn(x)=-12n3rE(x),
η=πn3rE10L2λ cos θ2,
ji=eμiME-kBTμiM,
Mt=-·jie,
ρ=e(M-M0)+ρf.
μi=(Di0/kBT)exp[-Ei/(kBT)],
ρf=Re[ρf1 exp(ikx)],
ρs=Re[ζi(k)ρf1 exp(ikx)]
ζi(k)=1/(1+k2/kDi2).
Γi=Γdii(1+k2/kDi2),
E10=-iai(k) ρf1k0,
ai(k)=1-ζi(k)=k2/kDi21+k2/kDi2,
ai(K)=EDEqi+ED.
ρ=e(N-N0+M-M0-n).
dN1dt=meκζeΓe(ED-iEP)-ΓeN1-ζeΓeM1,
dM1dt=-ζiΓiN1-ΓiM1,
E1=-iN1/κ-iM1/κ.
N10=καm kVT-iEph(1+kDi2/k2)-1+k2/kDe2-iαk/kp,
M10=-ζiN1,0=-καm 11+k2/kDi2×kVT-iEph(1+kDi2/k2)-1+k2/kDe2-iαk/kp,
E10=-iaiN1,0/κ=-iαm 11+kDi2/k2×kVT-iEph(1+kDi2/k2)-1+k2/kDe2-iαk/kp,
ddtN1M1=ΓeζiΓiζeΓeΓiN1M1.
(Γ-Γe)(Γ-Γi)=ζeζiΓeΓi,
N1M1=N10M10+C+A+ exp(-Γ+t)+C-A- exp(-Γ-t).
A±=A±NA±M=Γi-ζeΓe-Γ±Γe-ζiΓi-Γ±.
C+=-N10A-M+M10A-NA+NA-M-A-NA+M,
C-=-N10A+M+M10A+NA-NA+M-A+NA-M.
Γ+=Γe,
Γ-=(1-ζeζi)Γi,
A+=10,
A-=ζe-1.
Γ+=Γi,
Γ-=(1-ζeζi)Γe,
A+=01,
A-=1-ζi.
Γ-=(k2/kDr2)Γi,
A-=1-1.
Γ-=(k2/kDr2)Γe,
N10=2/3αmNR,
M10=-1/3αmM,
E10=-1/3αmkDeVT.
ρf=Re[ρf1 exp(ikx)],
ρ=e(N-N0-n)+ρf.
ρscreen=Re ζe(k)ρf1 exp(ikx),
ζe(k)=1/(1+k2/kDe2-ik/kEe-ik/kP).
E10=-iae(k) ρf1k0,
ae(k)=k2/kDe2-ik/kEe-ik/kP1+k2/kDe2-ik/kEe-ik/kP.
ae(k)=k2/kDe2-ik/koc1+k2/kDe2-ik/koc,
E10pc=-iκae(TL)ζi(TH)ρe(TL),
E10pc=-iαm k2/kDe2-ik/kE-ik/kP1+k2/kDe2-ik/kE-ik/kP11+k2/kDi2×kVT-iE0-iEph1+k2/kDe2-ik/kE-ik/kP,
E10pc=-iαmVTkDe2kDi2k-3.
E10pc=-iαmVTkDe-2k3.
E10pc=αmEpkP-1k,
E10sc=-iκae(TL)ρi(TH)=-iκae(TL)ζi(TH)ρe(TH),
E10sc=-iαm k2/[kDe(TL)]2-ik/kE-ik/kP1+k2/[kDe(TL)]2-ik/kE-ik/kP×11+k2/kDi2×kVT-iEph(1+kDi2/k2)-1+k2/[kDe(TH)]2-ik/kP.
E10sc=-iαmVT[kDe(TH)]2kDi2k-3.
E10sc=-iαmVTkDr2[kDe(TL)]-2k,
E10pc=αmEp.
δn1scδn1pc=1+k2/[kDe(TL)]2-ik/kE-iαk/kP(1+kDi2/k2)-1+k2/[kDe(TH)]2-ik/kE-iαk/kP.
δn1=-12n3rE1rev(Γ/I)J,
δn1=-i2αmn3r(Γdie/I)aeζi(k)×kVT-i(E0-Eph)1+k2rDe2-ikrEJ.
δn1=-i2αmn3r(Γdii/I)ae×kVT-iEph(1+kDi2/k2)-1+k2/kDe2-ik/kPJ.
δn1=δn1rev exp(-Γt),
δn1=δn1revNexp(-Γt),
Γ=t112ln ηiηf-ln N.
Δθ=1n0n0T+αcΔT,
Δλ/λ0=1n0n0T+αaΔT.
δθ=2Ln0ΔTλ01n0n0T+αa+αc2-1,

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