Abstract

Multiwave mixing in Bi12SiO20 causes the formation of multiple grating because of the photorefractive effect. These primary gratings interact and cause the formation of additional gratings at frequencies that are combinations of the primary grating frequencies. The nonlinear combinations of gratings stem from the nonlinear response of the space-charge field to the incident optical interference pattern through the generation of electrons into the conduction band. We investigate drift-dominated recording. The effects of nonlinear combinations of gratings are investigated quantitatively in a three-wave mixing configuration, in which one reference beam and two closely situated object beams induce three primary gratings. Using a sinusoidal phase modulation on one of the object beams provides absolute control of the primary grating strengths. Specifically, two of the primary gratings may be completely erased, and the nonlinear effect may be obtained as the relative change in the diffraction efficiency from the remaining grating. For drift-dominated recording an expression for the total space-charge field is derived, including multiple spatial frequencies. The derivation is based on the band-transport model. A numerical model is presented in which the relative change in diffraction efficiency is calculated from the corresponding change in the grating strengths. The grating strengths are found from the corresponding frequency components of the total space-charge field. The model is valid in the limit of low diffraction efficiencies and small coupling constants. The investigation is carried out with different values of the intensity ratio between the reference beam and the sum of the two object beams, the applied field to the crystal, and the separation angle between the two object beams. It is shown that relative changes of 200% in the diffraction efficiency occur. Thus the magnitude of the additional grating strengths may be substantial and even comparable with those of the primary grating. Comparing the predictions of the numerical model with the experimental data shows excellent agreement.

© 1995 Optical Society of America

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  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, 72–74 (1966).
    [CrossRef]
  2. F. S. Chen, "Optically induced change of refractive indices in LiNbO3 and LiTaO3," J. Appl. Phys. 40, 3389–3396 (1969).
    [CrossRef]
  3. J. W. Goodman, F. J. Leonberger, S.-Y. Kung, and R. A. Athale, "Optical interconnections for VLSI systems," Proc. IEEE 72, 850–866 (1984).
    [CrossRef]
  4. P. B. Berra, A. Ghafoor, M. Guizani, S. J. Marcinkowski, and P. A. Mitkas, "Optics and supercomputing," Proc. IEEE 77, 1798–1815 (1989).
    [CrossRef]
  5. S. I. Stepanov, "Applications of photorefractive crystals," Rep. Prog. Phys. 57, 36–116 (1994).
    [CrossRef]
  6. F. Vachss and L. Hesselink, "Nonlinear photorefractive response at high modulation depths," J. Opt. Soc. Am. A 5, 690–701 (1988).
    [CrossRef]
  7. L. B. Au and L. Solymar, "Higher harmonic gratings in photorefractive materials at large modulation with moving fringes," J. Opt. Soc. Am. A 7, 1554–1561 (1990).
    [CrossRef]
  8. J. P. Huignard and B. Ledu, "Collinear Bragg diffraction in photorefractive Bi12SiO20," Opt. Lett. 7, 310–312 (1982).
    [CrossRef] [PubMed]
  9. S. Fries, S. Bauschulte, E. Krátzig, K. Ringhofer, and Y. Yacoby, "Spatial frequency mixing in lithium niobate," Opt. Commun. 84, 251–257 (1991).
    [CrossRef]
  10. P. Buchhave, P. E. Andersen, P. M. Petersen, and M. V. Vasnetsov, "Nonlinear grating interaction in photorefractive Bi12SiO20," Appl. Phys. Lett. 66, 792–794 (1995).
    [CrossRef]
  11. P. E. Andersen, P. Buchhave, P. M. Petersen, and M. Vasnetsov, "Nonlinear combinations of gratings in Bi12SiO20: theory and experiments," J. Opt. Soc. Am. B 12, 1422–1433 (1995).
    [CrossRef]
  12. L. B. Au and L. Solymar, "Higher diffraction orders in photorefractive materials," IEEE J. Quantum Electron. 24, 162–168 (1988).
    [CrossRef]
  13. A. Marrakchi, J. P. Huignard, and P. Gülnter, "Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals," Appl. Phys. 24, 131–138 (1981).
    [CrossRef]
  14. P. E. Andersen, P. M. Petersen, and P. Buchhave, "Crosstalk in dynamic optical interconnects in photorefractive crystals," Appl. Phys. Lett. 65, 271–273 (1994).
    [CrossRef]
  15. 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, 949–960 (1979).
    [CrossRef]
  16. H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909–2947 (1969).
  17. P. M. Johansen, "Enhanced four-wave mixing in photorefractive BSO produced by temporal phase shifts," J. Phys. D 22, 247–253 (1989).
    [CrossRef]
  18. A. Marrakchi, W. M. Hubbard, S. F. Habiby, and J. S. Patel, "Dynamic holographic interconnects with analog weights in photorefractive crystals," Opt. Eng. 29, 215–225 (1990).
    [CrossRef]
  19. A. Marrakchi, R. V. Johnson, and A. R. Tanguay, Jr., "Polarization properties of photorefractive diffraction in electrooptic and optically active sillenite crystals (Bragg regime)," J. Opt. Soc. Am. B 3, 321–336 (1986).
    [CrossRef]
  20. P. Refrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, "Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments," J. Appl. Phys. 58, 45–57 (1985).
    [CrossRef]
  21. C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. Lee, "An analytical solution for large modulation effects in photorefractive two-wave couplings," Opt. Commun. 105, 353–358 (1994).
    [CrossRef]
  22. B. Edvold, P. E. Andersen, P. Buchhave, and P. M. Petersen, "Polarization properties of a photorefractive Bi12SiO20 crystal and their application in an optical correlator," IEEE J. Quantum Electron. 30, 1075–1089 (1994).
    [CrossRef]
  23. P. D. Foote and T. J. Hall, "Influence of optical activity on two beam coupling constants in photorefractive Bi12SiO20," Opt. Commun. 57, 201–206 (1986).
    [CrossRef]

1995 (2)

P. Buchhave, P. E. Andersen, P. M. Petersen, and M. V. Vasnetsov, "Nonlinear grating interaction in photorefractive Bi12SiO20," Appl. Phys. Lett. 66, 792–794 (1995).
[CrossRef]

P. E. Andersen, P. Buchhave, P. M. Petersen, and M. Vasnetsov, "Nonlinear combinations of gratings in Bi12SiO20: theory and experiments," J. Opt. Soc. Am. B 12, 1422–1433 (1995).
[CrossRef]

1994 (4)

P. E. Andersen, P. M. Petersen, and P. Buchhave, "Crosstalk in dynamic optical interconnects in photorefractive crystals," Appl. Phys. Lett. 65, 271–273 (1994).
[CrossRef]

S. I. Stepanov, "Applications of photorefractive crystals," Rep. Prog. Phys. 57, 36–116 (1994).
[CrossRef]

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. Lee, "An analytical solution for large modulation effects in photorefractive two-wave couplings," Opt. Commun. 105, 353–358 (1994).
[CrossRef]

B. Edvold, P. E. Andersen, P. Buchhave, and P. M. Petersen, "Polarization properties of a photorefractive Bi12SiO20 crystal and their application in an optical correlator," IEEE J. Quantum Electron. 30, 1075–1089 (1994).
[CrossRef]

1991 (1)

S. Fries, S. Bauschulte, E. Krátzig, K. Ringhofer, and Y. Yacoby, "Spatial frequency mixing in lithium niobate," Opt. Commun. 84, 251–257 (1991).
[CrossRef]

1990 (2)

L. B. Au and L. Solymar, "Higher harmonic gratings in photorefractive materials at large modulation with moving fringes," J. Opt. Soc. Am. A 7, 1554–1561 (1990).
[CrossRef]

A. Marrakchi, W. M. Hubbard, S. F. Habiby, and J. S. Patel, "Dynamic holographic interconnects with analog weights in photorefractive crystals," Opt. Eng. 29, 215–225 (1990).
[CrossRef]

1989 (2)

P. B. Berra, A. Ghafoor, M. Guizani, S. J. Marcinkowski, and P. A. Mitkas, "Optics and supercomputing," Proc. IEEE 77, 1798–1815 (1989).
[CrossRef]

P. M. Johansen, "Enhanced four-wave mixing in photorefractive BSO produced by temporal phase shifts," J. Phys. D 22, 247–253 (1989).
[CrossRef]

1988 (2)

L. B. Au and L. Solymar, "Higher diffraction orders in photorefractive materials," IEEE J. Quantum Electron. 24, 162–168 (1988).
[CrossRef]

F. Vachss and L. Hesselink, "Nonlinear photorefractive response at high modulation depths," J. Opt. Soc. Am. A 5, 690–701 (1988).
[CrossRef]

1986 (2)

1985 (1)

P. Refrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, "Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments," J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

1984 (1)

J. W. Goodman, F. J. Leonberger, S.-Y. Kung, and R. A. Athale, "Optical interconnections for VLSI systems," Proc. IEEE 72, 850–866 (1984).
[CrossRef]

1982 (1)

1981 (1)

A. Marrakchi, J. P. Huignard, and P. Gülnter, "Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals," Appl. Phys. 24, 131–138 (1981).
[CrossRef]

1979 (1)

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, 949–960 (1979).
[CrossRef]

1969 (2)

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

F. S. Chen, "Optically induced change of refractive indices in LiNbO3 and LiTaO3," J. Appl. Phys. 40, 3389–3396 (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, 72–74 (1966).
[CrossRef]

Andersen, P. E.

P. E. Andersen, P. Buchhave, P. M. Petersen, and M. Vasnetsov, "Nonlinear combinations of gratings in Bi12SiO20: theory and experiments," J. Opt. Soc. Am. B 12, 1422–1433 (1995).
[CrossRef]

P. Buchhave, P. E. Andersen, P. M. Petersen, and M. V. Vasnetsov, "Nonlinear grating interaction in photorefractive Bi12SiO20," Appl. Phys. Lett. 66, 792–794 (1995).
[CrossRef]

P. E. Andersen, P. M. Petersen, and P. Buchhave, "Crosstalk in dynamic optical interconnects in photorefractive crystals," Appl. Phys. Lett. 65, 271–273 (1994).
[CrossRef]

B. Edvold, P. E. Andersen, P. Buchhave, and P. M. Petersen, "Polarization properties of a photorefractive Bi12SiO20 crystal and their application in an optical correlator," IEEE J. Quantum Electron. 30, 1075–1089 (1994).
[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, 72–74 (1966).
[CrossRef]

Athale, R. A.

J. W. Goodman, F. J. Leonberger, S.-Y. Kung, and R. A. Athale, "Optical interconnections for VLSI systems," Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Au, L. B.

L. B. Au and L. Solymar, "Higher harmonic gratings in photorefractive materials at large modulation with moving fringes," J. Opt. Soc. Am. A 7, 1554–1561 (1990).
[CrossRef]

L. B. Au and L. Solymar, "Higher diffraction orders in photorefractive materials," IEEE J. Quantum Electron. 24, 162–168 (1988).
[CrossRef]

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, 72–74 (1966).
[CrossRef]

Bauschulte, S.

S. Fries, S. Bauschulte, E. Krátzig, K. Ringhofer, and Y. Yacoby, "Spatial frequency mixing in lithium niobate," Opt. Commun. 84, 251–257 (1991).
[CrossRef]

Berra, P. B.

P. B. Berra, A. Ghafoor, M. Guizani, S. J. Marcinkowski, and P. A. Mitkas, "Optics and supercomputing," Proc. IEEE 77, 1798–1815 (1989).
[CrossRef]

Boyd, G. D.

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, 72–74 (1966).
[CrossRef]

Buchhave, P.

P. E. Andersen, P. Buchhave, P. M. Petersen, and M. Vasnetsov, "Nonlinear combinations of gratings in Bi12SiO20: theory and experiments," J. Opt. Soc. Am. B 12, 1422–1433 (1995).
[CrossRef]

P. Buchhave, P. E. Andersen, P. M. Petersen, and M. V. Vasnetsov, "Nonlinear grating interaction in photorefractive Bi12SiO20," Appl. Phys. Lett. 66, 792–794 (1995).
[CrossRef]

P. E. Andersen, P. M. Petersen, and P. Buchhave, "Crosstalk in dynamic optical interconnects in photorefractive crystals," Appl. Phys. Lett. 65, 271–273 (1994).
[CrossRef]

B. Edvold, P. E. Andersen, P. Buchhave, and P. M. Petersen, "Polarization properties of a photorefractive Bi12SiO20 crystal and their application in an optical correlator," IEEE J. Quantum Electron. 30, 1075–1089 (1994).
[CrossRef]

Chen, F. S.

F. S. Chen, "Optically induced change of refractive indices in LiNbO3 and LiTaO3," J. Appl. Phys. 40, 3389–3396 (1969).
[CrossRef]

Dziedzic, J. M.

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, 72–74 (1966).
[CrossRef]

Edvold, B.

B. Edvold, P. E. Andersen, P. Buchhave, and P. M. Petersen, "Polarization properties of a photorefractive Bi12SiO20 crystal and their application in an optical correlator," IEEE J. Quantum Electron. 30, 1075–1089 (1994).
[CrossRef]

Foote, P. D.

P. D. Foote and T. J. Hall, "Influence of optical activity on two beam coupling constants in photorefractive Bi12SiO20," Opt. Commun. 57, 201–206 (1986).
[CrossRef]

Fries, S.

S. Fries, S. Bauschulte, E. Krátzig, K. Ringhofer, and Y. Yacoby, "Spatial frequency mixing in lithium niobate," Opt. Commun. 84, 251–257 (1991).
[CrossRef]

Ghafoor, A.

P. B. Berra, A. Ghafoor, M. Guizani, S. J. Marcinkowski, and P. A. Mitkas, "Optics and supercomputing," Proc. IEEE 77, 1798–1815 (1989).
[CrossRef]

Goodman, J. W.

J. W. Goodman, F. J. Leonberger, S.-Y. Kung, and R. A. Athale, "Optical interconnections for VLSI systems," Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Guizani, M.

P. B. Berra, A. Ghafoor, M. Guizani, S. J. Marcinkowski, and P. A. Mitkas, "Optics and supercomputing," Proc. IEEE 77, 1798–1815 (1989).
[CrossRef]

Gülnter, P.

A. Marrakchi, J. P. Huignard, and P. Gülnter, "Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals," Appl. Phys. 24, 131–138 (1981).
[CrossRef]

Habiby, S. F.

A. Marrakchi, W. M. Hubbard, S. F. Habiby, and J. S. Patel, "Dynamic holographic interconnects with analog weights in photorefractive crystals," Opt. Eng. 29, 215–225 (1990).
[CrossRef]

Hall, T. J.

P. D. Foote and T. J. Hall, "Influence of optical activity on two beam coupling constants in photorefractive Bi12SiO20," Opt. Commun. 57, 201–206 (1986).
[CrossRef]

Hesselink, L.

Hubbard, W. M.

A. Marrakchi, W. M. Hubbard, S. F. Habiby, and J. S. Patel, "Dynamic holographic interconnects with analog weights in photorefractive crystals," Opt. Eng. 29, 215–225 (1990).
[CrossRef]

Huignard, J. P.

P. Refrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, "Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments," J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

J. P. Huignard and B. Ledu, "Collinear Bragg diffraction in photorefractive Bi12SiO20," Opt. Lett. 7, 310–312 (1982).
[CrossRef] [PubMed]

A. Marrakchi, J. P. Huignard, and P. Gülnter, "Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals," Appl. Phys. 24, 131–138 (1981).
[CrossRef]

Jeong, J. S.

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. Lee, "An analytical solution for large modulation effects in photorefractive two-wave couplings," Opt. Commun. 105, 353–358 (1994).
[CrossRef]

Johansen, P. M.

P. M. Johansen, "Enhanced four-wave mixing in photorefractive BSO produced by temporal phase shifts," J. Phys. D 22, 247–253 (1989).
[CrossRef]

Johnson, R. V.

Kogelnik, H.

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

Krátzig, E.

S. Fries, S. Bauschulte, E. Krátzig, K. Ringhofer, and Y. Yacoby, "Spatial frequency mixing in lithium niobate," Opt. Commun. 84, 251–257 (1991).
[CrossRef]

Kukhtarev, N. V.

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, 949–960 (1979).
[CrossRef]

Kung, S.-Y.

J. W. Goodman, F. J. Leonberger, S.-Y. Kung, and R. A. Athale, "Optical interconnections for VLSI systems," Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Kwak, C. H.

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. Lee, "An analytical solution for large modulation effects in photorefractive two-wave couplings," Opt. Commun. 105, 353–358 (1994).
[CrossRef]

Ledu, B.

Lee, E.

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. Lee, "An analytical solution for large modulation effects in photorefractive two-wave couplings," Opt. Commun. 105, 353–358 (1994).
[CrossRef]

Leonberger, F. J.

J. W. Goodman, F. J. Leonberger, S.-Y. Kung, and R. A. Athale, "Optical interconnections for VLSI systems," Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Levinstein, J. J.

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, 72–74 (1966).
[CrossRef]

Marcinkowski, S. J.

P. B. Berra, A. Ghafoor, M. Guizani, S. J. Marcinkowski, and P. A. Mitkas, "Optics and supercomputing," Proc. IEEE 77, 1798–1815 (1989).
[CrossRef]

Markov, V. B.

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, 949–960 (1979).
[CrossRef]

Marrakchi, A.

A. Marrakchi, W. M. Hubbard, S. F. Habiby, and J. S. Patel, "Dynamic holographic interconnects with analog weights in photorefractive crystals," Opt. Eng. 29, 215–225 (1990).
[CrossRef]

A. Marrakchi, R. V. Johnson, and A. R. Tanguay, Jr., "Polarization properties of photorefractive diffraction in electrooptic and optically active sillenite crystals (Bragg regime)," J. Opt. Soc. Am. B 3, 321–336 (1986).
[CrossRef]

A. Marrakchi, J. P. Huignard, and P. Gülnter, "Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals," Appl. Phys. 24, 131–138 (1981).
[CrossRef]

Mitkas, P. A.

P. B. Berra, A. Ghafoor, M. Guizani, S. J. Marcinkowski, and P. A. Mitkas, "Optics and supercomputing," Proc. IEEE 77, 1798–1815 (1989).
[CrossRef]

Nassau, K.

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, 72–74 (1966).
[CrossRef]

Odulov, S. G.

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, 949–960 (1979).
[CrossRef]

Park, S. Y.

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. Lee, "An analytical solution for large modulation effects in photorefractive two-wave couplings," Opt. Commun. 105, 353–358 (1994).
[CrossRef]

Patel, J. S.

A. Marrakchi, W. M. Hubbard, S. F. Habiby, and J. S. Patel, "Dynamic holographic interconnects with analog weights in photorefractive crystals," Opt. Eng. 29, 215–225 (1990).
[CrossRef]

Petersen, P. M.

P. Buchhave, P. E. Andersen, P. M. Petersen, and M. V. Vasnetsov, "Nonlinear grating interaction in photorefractive Bi12SiO20," Appl. Phys. Lett. 66, 792–794 (1995).
[CrossRef]

P. E. Andersen, P. Buchhave, P. M. Petersen, and M. Vasnetsov, "Nonlinear combinations of gratings in Bi12SiO20: theory and experiments," J. Opt. Soc. Am. B 12, 1422–1433 (1995).
[CrossRef]

P. E. Andersen, P. M. Petersen, and P. Buchhave, "Crosstalk in dynamic optical interconnects in photorefractive crystals," Appl. Phys. Lett. 65, 271–273 (1994).
[CrossRef]

B. Edvold, P. E. Andersen, P. Buchhave, and P. M. Petersen, "Polarization properties of a photorefractive Bi12SiO20 crystal and their application in an optical correlator," IEEE J. Quantum Electron. 30, 1075–1089 (1994).
[CrossRef]

Rajbenbach, H.

P. Refrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, "Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments," J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Refrégier, P.

P. Refrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, "Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments," J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Ringhofer, K.

S. Fries, S. Bauschulte, E. Krátzig, K. Ringhofer, and Y. Yacoby, "Spatial frequency mixing in lithium niobate," Opt. Commun. 84, 251–257 (1991).
[CrossRef]

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, 72–74 (1966).
[CrossRef]

Solymar, L.

L. B. Au and L. Solymar, "Higher harmonic gratings in photorefractive materials at large modulation with moving fringes," J. Opt. Soc. Am. A 7, 1554–1561 (1990).
[CrossRef]

L. B. Au and L. Solymar, "Higher diffraction orders in photorefractive materials," IEEE J. Quantum Electron. 24, 162–168 (1988).
[CrossRef]

P. Refrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, "Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments," J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

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, 949–960 (1979).
[CrossRef]

Stepanov, S. I.

S. I. Stepanov, "Applications of photorefractive crystals," Rep. Prog. Phys. 57, 36–116 (1994).
[CrossRef]

Suh, H. H.

C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. Lee, "An analytical solution for large modulation effects in photorefractive two-wave couplings," Opt. Commun. 105, 353–358 (1994).
[CrossRef]

Tanguay, A. R.

Vachss, F.

Vasnetsov, M.

Vasnetsov, M. V.

P. Buchhave, P. E. Andersen, P. M. Petersen, and M. V. Vasnetsov, "Nonlinear grating interaction in photorefractive Bi12SiO20," Appl. Phys. Lett. 66, 792–794 (1995).
[CrossRef]

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, 949–960 (1979).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Three-wave mixing configuration.

Fig. 2
Fig. 2

Experimental setup: M’s, M2, mirrors; MPZ, piezo mirror; B1, B2, beam splitters.

Fig. 3
Fig. 3

Diffraction pattern in the output plane. The separation angle is Δθ = 1.9 mrad, and the intensity ratio is β = 9.9. Readout is performed with the He–Ne laser. The applied field is 5.0 kV/cm.

Fig. 4
Fig. 4

Diffraction pattern in the output plane. The separation angle is Δθ = 1.9 mrad, and the intensity ratio is β = 0.014. Readout is performed with the He–Ne laser. The applied field is 5.0 kV/cm.

Fig. 5
Fig. 5

Nonlinear effect Δθ as a function of β for series 1 and 2. The circles correspond to the experimental values and the solid curve to the numerical computations. The separation angle is 2.6 mrad. (a) The applied field is 5.0 kV/cm and the reduction factor (1 − δ) is 0.78. (b) The applied field is 3.3 kV/cm and the reduction factor (1 − δ) is 0.85. Other parameters appear in Table 1.

Fig. 6
Fig. 6

Nonlinear effect Δη as a function of β for series 3 and 4. The circles correspond to the experimental values and the solid curve to the numerical computations. The separation angle is 4.5 mrad. (a) The applied field is 5.0 kV/cm and the reduction factor (1 − δ) is 0.78. (b) The applied field is 3.3 kV/cm and the reduction factor (1 − δ) is 0.85. Other parameters appear in Table 1.

Fig. 7
Fig. 7

Nonlinear effect Δη as a function of β with the applied field as a parameter. The circles correspond to experimental values with EA = 5.0 kV/cm, the squares to EA = 3.3 kV/cm for a separation angle of 2.6 mrad. The triangles correspond experimental values with EA = 0 kV/cm and a separation angle 2.1 mrad. The solid curves correspond to the fitted curves.

Fig. 8
Fig. 8

Nonlinear effect Δη as a function of κ with β = 0.03. The separation angle is 2.6 mrad, and the reduction factor (1 − δ) is 0.77, corresponding to an applied field of 5 kV/cm (other parameters appear in Table 1).

Fig. 9
Fig. 9

Nonlinear effect Δη as a function of Δθ with β = 0.03. The object beam intensity ratio is κ = 1.0, and the reduction factor (1 − δ) is 0.77, corresponding to an applied field of 5 kV/cm (other parameters appear in Table 1).

Tables (1)

Tables Icon

Table 1 Parameters from the Three-Wave Mixing Experiments

Equations (13)

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I T ( x ) = I 0 [ 1 + m 1 R cos ( K 1 R x ) + m 2 R cos ( K 2 R x ) + m 12 cos ( K 12 x ) ] ,
I T = I 0 [ 1 + m 1 R cos ( K 1 R x ) + m 2 R J 0 ( A 0 ) cos ( K 2 R x ) + m 12 J 0 ( A 0 ) cos ( K 12 x ) ] ,
E sc ( x ) = J e μ n ( x ) = J e μ n 0 1 1 + M 1 R cos ( K 1 R x ) + J 0 ( A 0 ) M 2 R cos ( K 2 R x ) + J 0 ( A 0 ) M 12 cos ( K 12 x ) ,
J d e μ n 0 0 d d x 1 + M 1 R cos ( K 1 R x ) + J 0 ( A 0 ) M 2 R cos ( K 2 R x ) + J 0 ( A 0 ) M 12 cos ( K 12 x ) = V A d ,
η i [ G ( K i R ) ] 2 ,
G ( K i R ) M i R F ( K i R ) { E sc } = M i R E E sc ( K i R ) .
E A = E d 0 d 1 1 + M 1 R cos ( K 1 R x ) + J 0 ( A 0 ) M 2 R cos ( K 2 R x ) + J 0 ( A 0 ) M 12 cos ( K 12 x ) d x ,
E sc ( x ) = N h N exp ( i N K 12 x ) ,
h N ( x ) = E K 12 2 π 0 2 π / K 12 exp ( - i N K 12 x ) 1 + M 1 R cos ( K 1 R x ) + J 0 ( A 0 ) M 2 R cos ( K 2 R x ) + J 0 ( A 0 ) M 12 cos ( K 12 x ) d x .
h N ( x ) = E K 12 2 π 0 2 π / K 12 cos ( N K 12 x ) 1 + M 1 R cos ( K 1 R x ) + J 0 ( A 0 ) M 2 R cos ( K 2 R x ) + J 0 ( A 0 ) M 12 cos ( K 12 x ) d x .
h n ( x ) = E K 12 2 π 0 2 π / K 12 cos [ ( n K 12 + K 1 R ) x ] 1 + M 1 R cos ( K 1 R x ) + J 0 ( A 0 ) M 2 R cos ( K 2 R x ) + J 0 ( A 0 ) M 12 cos ( K 12 x ) d x = E E sc ( K i R ) .
Δ η = η 1 , a - η 1 , b η 1 , b [ G 1 , a ( K 1 R ) ] 2 - [ G 1 , b ( K 1 R ) ] 2 [ G 1 , b ( K 1 R ) ] 2 ,
M i R = 2 ( I i I 1 + I 2 ) 1 / 2 β 1 + β ( 1 - δ ) , M 12 = 2 I 1 I 2 ( I 1 + I 2 ) ( 1 + β ) ( 1 - δ ) ,

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