Abstract

The steady-state beam-coupling gain during two-wave mixing in photorefractive materials has been analyzed in the strong nonlinear regime (high modulation depths). Numerical simulations have been carried out for B12SiO20, for which detailed information on photorefractive parameters is already available. First the amplitude and the phase mismatch (with regard to the light) of the recorded index grating and consequently the coupling gain coefficient were obtained under an applied field E0=5 kV/cm. Then the evolution of the intensities of the two interfering beams during propagation was determined for several light-modulation depths. The energy exchange was considerably enhanced with regard to the linear regime. The light and index fringe profiles at large modulation m were also obtained. The bending for both kinds of fringe differed appreciably from that previously calculated with a linear approach to the material equations.

[Optical Society of America ]

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References

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  1. T. Tschudi , A. Herden , J. Goltz , H. Klumb , F. Laeri , and J. Albers , Image amplification by two- and four-wave mixing in BaTiO 3 crystals , IEEE J. Quantum Electron. IEJQA7 QE-22 , 1493 ( 1986
    [CrossRef]
  2. S. I. Stepanov , Applications of photorefractive crystals , Rep. Prog. Phys. RPPHAG 57 , 39 ( 1994
    [CrossRef]
  3. B. I. Sturman , Interaction of two light waves in a crystal caused by photoelectron diffusion and drift , Sov. Phys. Tech. Phys. SPTPA3 23 , 589 ( 1978
  4. N. V. Kukhtarev , V. B. Markov , S. G. Odulov , M. S. Soskin , and V. L. Vinetskii , Holographic storage in electrooptic crystals , Ferroelectrics FEROA8 22 , 949 ( 1979
    [CrossRef]
  5. J. M. Heaton , P. A. Mills , E. G. S. Paige , L. Solymar , and T. Wilson , Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials , Opt. Acta OPACAT 31 , 885 ( 1984
    [CrossRef]
  6. J. Goltz , C. Denz , and T. Tschudi , Dynamics of hologram readout in photorefractive crystals for broken Bragg-condition , Opt. Commun. OPCOB8 68 , 228 ( 1988
    [CrossRef]
  7. R. de Vre , M. Jeganathan , J. P. Wilde , and L. Hesselink , Effect of applied fields on the Bragg condition and the diffraction efficiency in photorefractive crystals , Opt. Lett. OPLEDP 19 , 910 ( 1994
    [CrossRef] [PubMed]
  8. S. Tao , Z. H. Song , and D. R. Selviah , Bragg shift of holographic gratings in photorefractive Fe:LiNbO 3 crystals , Opt. Commun. OPCOB8 108 , 144 ( 1994
    [CrossRef]
  9. K. Buse , S. Kamper , J. Frejlich , R. Pankrath , and K. H. Ringhofer , Tilting of holograms in photorefractive Sr 0.61 Ba 0.39 Nb 2 O 6 crystals by self-diffraction , Opt. Lett. OPLEDP 20 , 2249 ( 1995
    [CrossRef]
  10. A. A. Freschi , P. M. Garcia , Y. Rasnik , J. Frejlich , and K. Buse , Interaction of two light waves in a crystal caused by photoelectron diffusion and drift , Opt. Lett. OPLEDP 21 , 152 ( 1996
    [CrossRef] [PubMed]
  11. J. G. Murillo , L. F. Magan a , M. Carrascosa , and F. Agullo -Lo pez , Effects of light modulation on grating phase-shifts in photorefractive recording , Opt. Commun. OPCOB8 139 , 81 ( 1997
    [CrossRef]
  12. Ch. H. Kwak , S. Yeon Park , J. S. Jeong , H. H. Suh , and E.-H. Lee , An analytical solution for large modulation effects in photorefractive two-wave couplings , Opt. Commun. OPCOB8 105 , 353 ( 1994
    [CrossRef]
  13. Ch. H. Kwak , S. Y. Park , and E. Lee , Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal , Opt. Commun. OPCOB8 115 , 315 ( 1995
    [CrossRef]
  14. M. R. Belic , D. Timotijevic , M. Petrovic , and M. V. Jaric , Exact solution to photorefractive two-wave mixing with arbitrary modulation depth , Opt. Commun. OPCOB8 123 , 201 ( 1996
    [CrossRef]
  15. Zh. Zhou , X. Sun , Y. Li , Y. Jiang , H. Zhao , K. Xu , and Q. Wan , Dynamic solutions of the photorefractive two-wave coupling at large modulation depths , Opt. Commun. OPCOB8 132 , 128 ( 1997
    [CrossRef]
  16. P. Refre gier , L. Solymar , H. Rajbenbach , and J. P. Huignard , Two-beam coupling in photorefractive Bi 12 SiO 20 crystals with moving grating: theory and experiments , J. Appl. Phys. JAPIAU 58 , 45 ( 1985
    [CrossRef]
  17. L. B. Au and L. Solymar , Space-charge field in photorefractive materials at large modulation , Opt. Lett. OPLEDP 13 , 660 ( 1988
    [CrossRef] [PubMed]
  18. G. A. Brost , Photorefractive grating formations at large modulation with alternating electric fields , J. Opt. Soc. Am. B JOBPDE 9 , 1454 ( 1992
    [CrossRef]
  19. G. A. Brost , Numerical analysis of photorefractive grating formation dynamics at large modulation in BSO , Opt. Commun. OPCOB8 96 , 113 ( 1993
    [CrossRef]
  20. J. G. Murillo , L. F. Magan a , M. Carrascosa , and F. Agullo -Lo pez , Temporal evolution of the physical response during photorefractive grating formation and erasure for BSO , J. Appl. Phys. JAPIAU 78 , 5686 ( 1995
    [CrossRef]
  21. J. V. Alvarez-Bravo , M. Carrascosa , and L. Arizmendi , Experimental effects of light intensity modulation on the recording and erasure of holographic gratings in BSO crystals , Opt. Commun. OPCOB8 103 , 22 ( 1993
    [CrossRef]
  22. Y. H. Lee and R. W. Hellwarth , Spatial harmonics of photorefractive gratings in a barium titanate crystal , J. Appl. Phys. JAPIAU 71 , 916 ( 1992
    [CrossRef]
  23. E. Serrano , M. Carrascosa , and F. Agullo -Lo pez , Nonperturbative analytical solution for steady-state photorefractive recording , Opt. Lett. OPLEDP 20 , 1910 ( 1995
    [CrossRef] [PubMed]
  24. E. Serrano and M. Carrascosa y F. Agullo -Lo pez , Analytical and numerical study of the photorefractive kinetics at high modulation depth , J. Opt. Soc. Am. B JOBPDE 13 , 2587 ( 1996
    [CrossRef]

Carrascosa y F. Agullo-Lopez, M

Lee, E

Ch. H. Kwak , S. Y. Park , and E. Lee , Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal , Opt. Commun. OPCOB8 115 , 315 ( 1995
[CrossRef]

Mills, P. A

J. M. Heaton , P. A. Mills , E. G. S. Paige , L. Solymar , and T. Wilson , Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials , Opt. Acta OPACAT 31 , 885 ( 1984
[CrossRef]

Murillo, J. G

J. G. Murillo , L. F. Magan a , M. Carrascosa , and F. Agullo -Lo pez , Effects of light modulation on grating phase-shifts in photorefractive recording , Opt. Commun. OPCOB8 139 , 81 ( 1997
[CrossRef]

Park, S. Yeon

Ch. H. Kwak , S. Yeon Park , J. S. Jeong , H. H. Suh , and E.-H. Lee , An analytical solution for large modulation effects in photorefractive two-wave couplings , Opt. Commun. OPCOB8 105 , 353 ( 1994
[CrossRef]

Rasnik, Y

Selviah, D. R

S. Tao , Z. H. Song , and D. R. Selviah , Bragg shift of holographic gratings in photorefractive Fe:LiNbO 3 crystals , Opt. Commun. OPCOB8 108 , 144 ( 1994
[CrossRef]

Song, Z. H

S. Tao , Z. H. Song , and D. R. Selviah , Bragg shift of holographic gratings in photorefractive Fe:LiNbO 3 crystals , Opt. Commun. OPCOB8 108 , 144 ( 1994
[CrossRef]

Tao, S

S. Tao , Z. H. Song , and D. R. Selviah , Bragg shift of holographic gratings in photorefractive Fe:LiNbO 3 crystals , Opt. Commun. OPCOB8 108 , 144 ( 1994
[CrossRef]

Wan, Q

Zh. Zhou , X. Sun , Y. Li , Y. Jiang , H. Zhao , K. Xu , and Q. Wan , Dynamic solutions of the photorefractive two-wave coupling at large modulation depths , Opt. Commun. OPCOB8 132 , 128 ( 1997
[CrossRef]

Zhou, Zh

Zh. Zhou , X. Sun , Y. Li , Y. Jiang , H. Zhao , K. Xu , and Q. Wan , Dynamic solutions of the photorefractive two-wave coupling at large modulation depths , Opt. Commun. OPCOB8 132 , 128 ( 1997
[CrossRef]

Other

T. Tschudi , A. Herden , J. Goltz , H. Klumb , F. Laeri , and J. Albers , Image amplification by two- and four-wave mixing in BaTiO 3 crystals , IEEE J. Quantum Electron. IEJQA7 QE-22 , 1493 ( 1986
[CrossRef]

S. I. Stepanov , Applications of photorefractive crystals , Rep. Prog. Phys. RPPHAG 57 , 39 ( 1994
[CrossRef]

B. I. Sturman , Interaction of two light waves in a crystal caused by photoelectron diffusion and drift , Sov. Phys. Tech. Phys. SPTPA3 23 , 589 ( 1978

N. V. Kukhtarev , V. B. Markov , S. G. Odulov , M. S. Soskin , and V. L. Vinetskii , Holographic storage in electrooptic crystals , Ferroelectrics FEROA8 22 , 949 ( 1979
[CrossRef]

J. M. Heaton , P. A. Mills , E. G. S. Paige , L. Solymar , and T. Wilson , Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials , Opt. Acta OPACAT 31 , 885 ( 1984
[CrossRef]

J. Goltz , C. Denz , and T. Tschudi , Dynamics of hologram readout in photorefractive crystals for broken Bragg-condition , Opt. Commun. OPCOB8 68 , 228 ( 1988
[CrossRef]

R. de Vre , M. Jeganathan , J. P. Wilde , and L. Hesselink , Effect of applied fields on the Bragg condition and the diffraction efficiency in photorefractive crystals , Opt. Lett. OPLEDP 19 , 910 ( 1994
[CrossRef] [PubMed]

S. Tao , Z. H. Song , and D. R. Selviah , Bragg shift of holographic gratings in photorefractive Fe:LiNbO 3 crystals , Opt. Commun. OPCOB8 108 , 144 ( 1994
[CrossRef]

K. Buse , S. Kamper , J. Frejlich , R. Pankrath , and K. H. Ringhofer , Tilting of holograms in photorefractive Sr 0.61 Ba 0.39 Nb 2 O 6 crystals by self-diffraction , Opt. Lett. OPLEDP 20 , 2249 ( 1995
[CrossRef]

A. A. Freschi , P. M. Garcia , Y. Rasnik , J. Frejlich , and K. Buse , Interaction of two light waves in a crystal caused by photoelectron diffusion and drift , Opt. Lett. OPLEDP 21 , 152 ( 1996
[CrossRef] [PubMed]

J. G. Murillo , L. F. Magan a , M. Carrascosa , and F. Agullo -Lo pez , Effects of light modulation on grating phase-shifts in photorefractive recording , Opt. Commun. OPCOB8 139 , 81 ( 1997
[CrossRef]

Ch. H. Kwak , S. Yeon Park , J. S. Jeong , H. H. Suh , and E.-H. Lee , An analytical solution for large modulation effects in photorefractive two-wave couplings , Opt. Commun. OPCOB8 105 , 353 ( 1994
[CrossRef]

Ch. H. Kwak , S. Y. Park , and E. Lee , Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal , Opt. Commun. OPCOB8 115 , 315 ( 1995
[CrossRef]

M. R. Belic , D. Timotijevic , M. Petrovic , and M. V. Jaric , Exact solution to photorefractive two-wave mixing with arbitrary modulation depth , Opt. Commun. OPCOB8 123 , 201 ( 1996
[CrossRef]

Zh. Zhou , X. Sun , Y. Li , Y. Jiang , H. Zhao , K. Xu , and Q. Wan , Dynamic solutions of the photorefractive two-wave coupling at large modulation depths , Opt. Commun. OPCOB8 132 , 128 ( 1997
[CrossRef]

P. Refre gier , L. Solymar , H. Rajbenbach , and J. P. Huignard , Two-beam coupling in photorefractive Bi 12 SiO 20 crystals with moving grating: theory and experiments , J. Appl. Phys. JAPIAU 58 , 45 ( 1985
[CrossRef]

L. B. Au and L. Solymar , Space-charge field in photorefractive materials at large modulation , Opt. Lett. OPLEDP 13 , 660 ( 1988
[CrossRef] [PubMed]

G. A. Brost , Photorefractive grating formations at large modulation with alternating electric fields , J. Opt. Soc. Am. B JOBPDE 9 , 1454 ( 1992
[CrossRef]

G. A. Brost , Numerical analysis of photorefractive grating formation dynamics at large modulation in BSO , Opt. Commun. OPCOB8 96 , 113 ( 1993
[CrossRef]

J. G. Murillo , L. F. Magan a , M. Carrascosa , and F. Agullo -Lo pez , Temporal evolution of the physical response during photorefractive grating formation and erasure for BSO , J. Appl. Phys. JAPIAU 78 , 5686 ( 1995
[CrossRef]

J. V. Alvarez-Bravo , M. Carrascosa , and L. Arizmendi , Experimental effects of light intensity modulation on the recording and erasure of holographic gratings in BSO crystals , Opt. Commun. OPCOB8 103 , 22 ( 1993
[CrossRef]

Y. H. Lee and R. W. Hellwarth , Spatial harmonics of photorefractive gratings in a barium titanate crystal , J. Appl. Phys. JAPIAU 71 , 916 ( 1992
[CrossRef]

E. Serrano , M. Carrascosa , and F. Agullo -Lo pez , Nonperturbative analytical solution for steady-state photorefractive recording , Opt. Lett. OPLEDP 20 , 1910 ( 1995
[CrossRef] [PubMed]

E. Serrano and M. Carrascosa y F. Agullo -Lo pez , Analytical and numerical study of the photorefractive kinetics at high modulation depth , J. Opt. Soc. Am. B JOBPDE 13 , 2587 ( 1996
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of the experimental configuration.

Fig. 2
Fig. 2

Dependence of Γ sin φ on m as obtained from our numerical calculations (solid curve) corresponding to Λ=1 μm and E0=5 kV/cm. The dashed curve, derived from the phenomenological law f(m)/m=1+1.75 m4, is also included for comparison.

Fig. 3
Fig. 3

Evolution of the steady-state coupling coefficient Γ sin φ with penetration depth Z in the sample for various values of initial modulation depth m0 (Λ=1 μm and E0=5 kV/cm).

Fig. 4
Fig. 4

Evolution of steady-state coupling coefficient Γ with penetration depth Z in the sample for various values of initial modulation depth m0 and zero applied electric field (sin φ=1).

Fig. 5
Fig. 5

Evolution of the intensity of the two beams as they penetrate the sample for m0=0.9 E0=5 kV/cm (solid curves). The analytical solutions obtained by Heaton et al.7 (dashed curves) are included for comparison.

Fig. 6
Fig. 6

Variation of light modulation m along crystal thickness Z for various initial (Z=0) light-modulation values m0. Solid curves, full numerical approach; dashed curves, analytical linear solution.

Fig. 7
Fig. 7

Evolution of the intensity of the two beams as they penetrate the sample. Solid curves, full numerical solution as in Fig. 5; dashed curves, solution obtained by Heaton et al.,7 but with the nonlinear value associated with the initial modulation m0 used for coupling coefficient Γ sin φ.

Fig. 8
Fig. 8

Phase Δψ=ψ(x, y)-ψ(x, 0) (in rad) of the light pattern obtained for various values of the initial light modulation m0. Dashed curves, analytical solution obtained by Heaton et al.7; solid curves, the full numerical solution.

Fig. 9
Fig. 9

Phases (in rad) of the light (dashed curve) and index (solid curve) fringes for m0=0.9.

Tables (1)

Tables Icon

Table 1 Numerical Parameters for the Simulation

Equations (15)

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dA˜+dz+iκ*A˜-=0,
dA˜-dz+iκA˜+=0,
κ=n˜1ω2c cos θ,
n˜1=n3rEs A˜+ A˜-|A0|2 exp(iφ)
dI+(z)dz+Γ sin φ I+ I-I0=0,
dI-(z)dz-Γ sin φ I+ I-I0=0,
Γ=2πn3rEsλ cos θ.
dψ+dz+Γ cos φ I-2I0=0,
dψ-dz+Γ cos φ I+2I0=0.
dψdz-Γ cos φ I1+-I1-2I0=0,
I+(Z)=I01+r0 exp(ΓZ sin φ),
I-(Z)=r0I0 exp(ΓZ sin φ)1+r0 exp(ΓZ sin φ),
ψ(x, Z)=ψ(x, 0)+(Γ/2)Z cos φ,
x=-Γ cos φ2K Z+x0.
ψ(x, Z)=ψ(x, 0)+12 cot φ ln(1+r0)2 exp(ΓZ sin φ)[1+r0 exp(ΓZ sin φ)]2.

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