Abstract

We numerically studied the spatial fidelity of coherent image amplification by two-wave mixing configurations in photorefractive BaTiO3:Ce using a three-dimensional analysis. The results are given for the case when the input one-dimensional rectangular amplitude of the image-bearing extraordinary beam is finite in the plane of incidence and when the amplitude is finite in the orthogonal plane. The fidelity and the gain versus the angle between the propagating direction of the image-bearing beam and the crystal c axis, the pump-to-image intensity ratio, and the input beamwidth, are analyzed.

© 1996 Optical Society of America

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  1. A. Marrackchi, J. P. Huignard, P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131–138 (1981).
  2. H. Rajbenbach, J. P. Huignard, B. Loiseaux, “Spatial frequency dependence of the energy transfer in two-wave mixing experiments with BSO crystals,” Opt. Commun. 48, 247–252 (1983).
  3. F. Laeri, T. Tshuidi, J. Albers, “Coherent CW image amplifier and oscillator using two-wave interaction in BaTiO3 crystals,” Opt. Commun. 47, 387–390 (1983).
  4. Y. Fainman, E. Klancnik, S. H. Lee, “Optimal coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. 25, 228–234 (1986).
  5. M. Segev, Y. Ophir, B. Fischer, “Nonlinear multi two-wave mixing, the fanning process and its bleaching in photorefractive media,” Opt. Commun. 77, 265–274 (1990).
  6. P. Xie, Y. H. Hong, J. H. Dai, Y. Zhu, H. J. Zhang, “Theoretical and experimental studies of fanning effects in photorefractive crystals,” J. Appl. Phys. 74, 813–818 (1993).
  7. J. Feinberg, D. Heiman, A. R. Tanguay, R. W”. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).
  8. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soski, V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).
  9. M. Cronin-Golumb, B. Fischer, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
  10. G. C. Valley, “Evolution of phase-conjugate waves in stimulated photorefractive backscattering,” J. Opt. Soc. Am. B 9, 1440–1448 (1992).
  11. Y. Zhu, P. Bernasconi, M. Zgonik, P. Günter, “Low frequency electro-optic coefficient measurements in Ce:BaTiO3 crystals,” J. Synth. Cryst. 23, 242 (1994), in Chinese.

1994

Y. Zhu, P. Bernasconi, M. Zgonik, P. Günter, “Low frequency electro-optic coefficient measurements in Ce:BaTiO3 crystals,” J. Synth. Cryst. 23, 242 (1994), in Chinese.

1993

P. Xie, Y. H. Hong, J. H. Dai, Y. Zhu, H. J. Zhang, “Theoretical and experimental studies of fanning effects in photorefractive crystals,” J. Appl. Phys. 74, 813–818 (1993).

1992

1990

M. Segev, Y. Ophir, B. Fischer, “Nonlinear multi two-wave mixing, the fanning process and its bleaching in photorefractive media,” Opt. Commun. 77, 265–274 (1990).

1986

Y. Fainman, E. Klancnik, S. H. Lee, “Optimal coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. 25, 228–234 (1986).

1984

M. Cronin-Golumb, B. Fischer, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).

1983

H. Rajbenbach, J. P. Huignard, B. Loiseaux, “Spatial frequency dependence of the energy transfer in two-wave mixing experiments with BSO crystals,” Opt. Commun. 48, 247–252 (1983).

F. Laeri, T. Tshuidi, J. Albers, “Coherent CW image amplifier and oscillator using two-wave interaction in BaTiO3 crystals,” Opt. Commun. 47, 387–390 (1983).

1981

A. Marrackchi, J. P. Huignard, P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131–138 (1981).

1980

J. Feinberg, D. Heiman, A. R. Tanguay, R. W”. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).

1979

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soski, V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).

Albers, J.

F. Laeri, T. Tshuidi, J. Albers, “Coherent CW image amplifier and oscillator using two-wave interaction in BaTiO3 crystals,” Opt. Commun. 47, 387–390 (1983).

Bernasconi, P.

Y. Zhu, P. Bernasconi, M. Zgonik, P. Günter, “Low frequency electro-optic coefficient measurements in Ce:BaTiO3 crystals,” J. Synth. Cryst. 23, 242 (1994), in Chinese.

Cronin-Golumb, M.

M. Cronin-Golumb, B. Fischer, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).

Dai, J. H.

P. Xie, Y. H. Hong, J. H. Dai, Y. Zhu, H. J. Zhang, “Theoretical and experimental studies of fanning effects in photorefractive crystals,” J. Appl. Phys. 74, 813–818 (1993).

Fainman, Y.

Y. Fainman, E. Klancnik, S. H. Lee, “Optimal coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. 25, 228–234 (1986).

Feinberg, J.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W”. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).

Fischer, B.

M. Segev, Y. Ophir, B. Fischer, “Nonlinear multi two-wave mixing, the fanning process and its bleaching in photorefractive media,” Opt. Commun. 77, 265–274 (1990).

M. Cronin-Golumb, B. Fischer, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).

Günter, P.

Y. Zhu, P. Bernasconi, M. Zgonik, P. Günter, “Low frequency electro-optic coefficient measurements in Ce:BaTiO3 crystals,” J. Synth. Cryst. 23, 242 (1994), in Chinese.

A. Marrackchi, J. P. Huignard, P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131–138 (1981).

Heiman, D.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W”. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).

Hellwarth, R. W”.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W”. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).

Hong, Y. H.

P. Xie, Y. H. Hong, J. H. Dai, Y. Zhu, H. J. Zhang, “Theoretical and experimental studies of fanning effects in photorefractive crystals,” J. Appl. Phys. 74, 813–818 (1993).

Huignard, J. P.

H. Rajbenbach, J. P. Huignard, B. Loiseaux, “Spatial frequency dependence of the energy transfer in two-wave mixing experiments with BSO crystals,” Opt. Commun. 48, 247–252 (1983).

A. Marrackchi, J. P. Huignard, P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131–138 (1981).

Klancnik, E.

Y. Fainman, E. Klancnik, S. H. Lee, “Optimal coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. 25, 228–234 (1986).

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soski, V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).

Laeri, F.

F. Laeri, T. Tshuidi, J. Albers, “Coherent CW image amplifier and oscillator using two-wave interaction in BaTiO3 crystals,” Opt. Commun. 47, 387–390 (1983).

Lee, S. H.

Y. Fainman, E. Klancnik, S. H. Lee, “Optimal coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. 25, 228–234 (1986).

Loiseaux, B.

H. Rajbenbach, J. P. Huignard, B. Loiseaux, “Spatial frequency dependence of the energy transfer in two-wave mixing experiments with BSO crystals,” Opt. Commun. 48, 247–252 (1983).

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soski, V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).

Marrackchi, A.

A. Marrackchi, J. P. Huignard, P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131–138 (1981).

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soski, V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).

Ophir, Y.

M. Segev, Y. Ophir, B. Fischer, “Nonlinear multi two-wave mixing, the fanning process and its bleaching in photorefractive media,” Opt. Commun. 77, 265–274 (1990).

Rajbenbach, H.

H. Rajbenbach, J. P. Huignard, B. Loiseaux, “Spatial frequency dependence of the energy transfer in two-wave mixing experiments with BSO crystals,” Opt. Commun. 48, 247–252 (1983).

Segev, M.

M. Segev, Y. Ophir, B. Fischer, “Nonlinear multi two-wave mixing, the fanning process and its bleaching in photorefractive media,” Opt. Commun. 77, 265–274 (1990).

Soski, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soski, V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).

Tanguay, A. R.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W”. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).

Tshuidi, T.

F. Laeri, T. Tshuidi, J. Albers, “Coherent CW image amplifier and oscillator using two-wave interaction in BaTiO3 crystals,” Opt. Commun. 47, 387–390 (1983).

Valley, G. C.

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soski, V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).

White, J. O.

M. Cronin-Golumb, B. Fischer, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).

Xie, P.

P. Xie, Y. H. Hong, J. H. Dai, Y. Zhu, H. J. Zhang, “Theoretical and experimental studies of fanning effects in photorefractive crystals,” J. Appl. Phys. 74, 813–818 (1993).

Yariv, A.

M. Cronin-Golumb, B. Fischer, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).

Zgonik, M.

Y. Zhu, P. Bernasconi, M. Zgonik, P. Günter, “Low frequency electro-optic coefficient measurements in Ce:BaTiO3 crystals,” J. Synth. Cryst. 23, 242 (1994), in Chinese.

Zhang, H. J.

P. Xie, Y. H. Hong, J. H. Dai, Y. Zhu, H. J. Zhang, “Theoretical and experimental studies of fanning effects in photorefractive crystals,” J. Appl. Phys. 74, 813–818 (1993).

Zhu, Y.

Y. Zhu, P. Bernasconi, M. Zgonik, P. Günter, “Low frequency electro-optic coefficient measurements in Ce:BaTiO3 crystals,” J. Synth. Cryst. 23, 242 (1994), in Chinese.

P. Xie, Y. H. Hong, J. H. Dai, Y. Zhu, H. J. Zhang, “Theoretical and experimental studies of fanning effects in photorefractive crystals,” J. Appl. Phys. 74, 813–818 (1993).

Appl. Phys.

A. Marrackchi, J. P. Huignard, P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20 crystals,” Appl. Phys. 24, 131–138 (1981).

Ferroelectrics

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soski, V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).

IEEE J. Quantum Electron.

M. Cronin-Golumb, B. Fischer, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).

J. Appl. Phys.

P. Xie, Y. H. Hong, J. H. Dai, Y. Zhu, H. J. Zhang, “Theoretical and experimental studies of fanning effects in photorefractive crystals,” J. Appl. Phys. 74, 813–818 (1993).

J. Feinberg, D. Heiman, A. R. Tanguay, R. W”. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).

J. Opt. Soc. Am. B

J. Synth. Cryst.

Y. Zhu, P. Bernasconi, M. Zgonik, P. Günter, “Low frequency electro-optic coefficient measurements in Ce:BaTiO3 crystals,” J. Synth. Cryst. 23, 242 (1994), in Chinese.

Opt. Commun.

H. Rajbenbach, J. P. Huignard, B. Loiseaux, “Spatial frequency dependence of the energy transfer in two-wave mixing experiments with BSO crystals,” Opt. Commun. 48, 247–252 (1983).

F. Laeri, T. Tshuidi, J. Albers, “Coherent CW image amplifier and oscillator using two-wave interaction in BaTiO3 crystals,” Opt. Commun. 47, 387–390 (1983).

M. Segev, Y. Ophir, B. Fischer, “Nonlinear multi two-wave mixing, the fanning process and its bleaching in photorefractive media,” Opt. Commun. 77, 265–274 (1990).

Opt. Eng.

Y. Fainman, E. Klancnik, S. H. Lee, “Optimal coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. 25, 228–234 (1986).

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Figures (6)

Fig. 1
Fig. 1

Schematic of image amplification by two-wave mixing in photorefractive BaTiO3.

Fig. 2
Fig. 2

Intensity distributions of the image beam for α = 10° and β = 40°. The left column corresponds to distributions in real transversal space and the right column to those in spatial frequency space: a and b, input beam; c and d, the output distributions for case A; e and f, for case B. The input intensity ratio of the pump to the zero-order frequency component of the image beam is r = I p /|f(0, 0)|2 = 1 × 104.

Fig. 3
Fig. 3

Output intensity distributions of the image beam (a and b) for case A and (c and d) for case B. The left column corresponds to distributions in real transversal space and the right column to those in spatial frequency space; α = 10° and β = 50°.

Fig. 4
Fig. 4

Fidelity, a, and gain, b, of the image beam versus β at α = 10° for case A. The dashed curves correspond to an input intensity ratio of r = 0.25 × 104, the solid curves represent r = 1.0 × 104, and the dotted curves represent r = ∞.

Fig. 5
Fig. 5

Fidelity, a, and gain, b, of the image beam versus β at α = 10° for case B. The dashed curves represent r = 0.25 × 104, the solid curves correspond to r = 1.0 × 104, and the dotted curves represent r = ∞.

Fig. 6
Fig. 6

Fidelity of the image beam as a function of d for case A, a, and case B, b; α = 10° and β = 40°. The dashed curves correspond to r = 0.25 × 104, the solid curves represent r = 1.0 × 104, and the dotted curves represent r = ∞.

Equations (26)

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E i ( x , y , z , t ) = A i ( x , y , z ) exp [ i ( k z - ω t ) ] + c . c . ,
E p ( x , z , t ) A p ( z ) exp [ i ( k x x + k z z - ω t ) ] + c . c . ,
A i ( x , y , z ) = f ( q x , q y , z ) exp { i [ q x x + q y y + ( β q - k ) z ] } ,
e ^ i = cos θ 1 x ^ - sin θ 1 z ^ ,
e ^ i = cos θ 1 x ^ - sin θ 1 z ^ ,
K = ( cos θ 2 sin θ 1 - cos θ 2 sin θ 1 ) x ^ + ( sin θ 2 - sin θ 2 ) y ^ + ( cos θ 2 cos θ 1 - cos θ 2 cos θ 1 ) z ^ .
r eff = e ^ i * · ɛ ω · ( R · k ^ ) · ɛ ω · e ^ i / ( n o 3 n e ) ,
r eff ( θ 1 , θ 2 , θ 1 , θ 2 ) = { n o 4 r 13 [ sin β ( cos θ 2 sin θ 1 - cos θ 2 sin θ 1 ) + cos β ( cos θ 2 cos θ 1 - cos θ 2 cos θ 1 ) ] × cos ( β - θ 1 ) cos ( β - θ 1 ) + n o 2 n e 2 r 42 [ cos β ( cos θ 2 sin θ 1 - cos θ 2 sin θ 1 ) - sin β ( cos θ 2 cos θ 1 - cos θ 2 cos θ 1 ) ] sin ( 2 β - θ 1 - θ 1 ) + n e 4 r 33 [ sin β ( cos θ 2 sin θ 1 - cos θ 2 sin θ 1 ) + cos β ( cos θ 2 cos θ 1 - cos θ 2 cos θ 1 ) ] × sin ( β - θ 1 ) sin ( β - θ 1 ) } / ( n o 3 n e K ) .
E s c = k B T q K 1 + K 2 / K 0 2 e ^ i · e ^ i * ,
K 0 = ( N q 2 ɛ ɛ 0 k B T ) 1 / 2 .
ɛ = ɛ a sin 2 χ c + ɛ c cos 2 χ c ,
cos χ c = [ sin β ( cos θ 2 sin θ 1 - cos θ 2 sin θ 1 ) + cos β ( cos θ 2 cos θ 1 - cos θ 2 cos θ 1 ) ] / K .
γ ( θ 1 , θ 2 , θ 1 , θ 2 ) = ω n o 3 2 c r eff ( θ 1 , θ 2 , θ 1 , θ 2 ) E sc .
e ^ p = cos α x ^ + sin α z ^ ,
e ^ i = cos θ 1 x ^ - sin θ 1 z ^ ,
K = ( sin θ 1 cos θ 2 + sin α ) x ^ + sin θ 2 y ^ + ( cos θ 1 cos θ 2 - cos α ) z ^ .
r p eff ( α , θ 1 , θ 2 ) = { - n o 4 r 13 [ cos ( α + β ) - cos θ 2 × cos ( β - θ 1 ) ] cos ( α + β ) cos ( β - θ 1 ) + n o 2 n e 2 r 42 [ sin ( α + β ) - cos θ 2 × sin ( β - θ 1 ) ] sin ( α + 2 β - θ 1 ) - n e 4 r 33 [ cos ( α + β ) - cos θ 1 × cos ( β - θ 1 ) ] sin ( α + β ) × sin ( β - θ 1 ) } / ( n o 3 n e { 2 [ 1 - cos θ 2 × cos ( α + θ 1 ) ] } 1 / 2 ) .
χ p c = sin ( α + θ 1 ) sin ( β - θ 1 ) + [ cos θ 3 - cos ( α + θ 1 ) ] cos ( β - θ 1 ) { 2 [ 1 - cos θ 3 cos ( α + θ 1 ) ] } 1 / 2 .
γ p ( α , θ 1 , θ 2 ) = ω n o 3 2 c r p eff ( α , θ 1 , θ 2 ) E p sc = γ ( - α , 0 , θ 1 , θ 2 ) .
cos θ 1 cos θ 2 d f ( θ 1 , θ 2 , z ) d z = [ γ p ( α , θ 1 , θ 2 ) f ( θ 1 , θ 2 , z ) A p * A p - θ 1 θ 2 γ ( θ 1 , θ 2 , θ 1 , θ 2 ) f ( θ 1 , θ 2 , z ) f * ( θ 1 , θ 2 , z ) f ( θ 1 , θ 2 , z ) ] / I 0 , cos α d A p d z = - θ 1 θ 2 γ p ( α , θ 1 , θ 2 ) A p f * ( θ 1 , θ 2 , z ) f ( θ 1 , θ 2 , z ) ] / I 0 , I 0 = A p 2 + θ 1 θ 2 f ( θ 1 , θ 2 , z ) 2 ,
F I = θ 1 θ 2 f * ( θ 1 , θ 2 , 0 ) f ( θ 1 , θ 2 , L ) + c . c . 2 [ θ 1 θ 2 f ( θ 1 , θ 2 , 0 ) 2 θ 1 θ 2 f ( θ 1 , θ 2 , L ) 2 ] 1 / 2 .
GAIN = θ 1 θ 2 f ( θ 1 , θ 2 , L ) 2 θ 1 θ 2 f ( θ 1 , θ 2 , 0 ) 2 .
F I = A i * ( x , y , 0 ) A i ( x , y , L ) d x d y + c . c . 2 [ ( A i ( x , y , 0 ) 2 d x d y ) ( A i ( x , y , L ) 2 d x d y ) ] 1 / 2 ,
A i ( x , y , L ) = θ 1 θ 2 f ( θ 1 , θ 2 , L ) exp [ i q x ( θ 1 , θ 2 ) x + i q y ( θ 1 , θ 2 ) y ] .
A i ( x , y , 0 ) = { const . - d / 2 < x d / 2 0 x - d / 2 or x > d / 2             y ( - , + ) ,
A i ( x , y , 0 ) = { const . - d / 2 < y d / 2 0 y - d / 2 or y > d / 2             x ( - , + ) .

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