Abstract

We have analytically solved the problem of N mutually incoherent pairs of beams in photorefractive media, each pair of which shares a common grating. The results are applied to study simultaneous read–write of dynamic photorefractive holograms with beams of comparable intensity. The diffraction efficiency is shown to be a nonlinear function of the read-beam intensity and is nonreciprocal with respect to readout from the two input ports. A complete energy transfer between the two write beams occurs in a finite thickness of the photorefractive crystal, in contrast to the infinite thickness required in the standard two-beam coupling case.

© 1990 Optical Society of America

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References

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1989 (3)

1986 (2)

B. Fischer, J. O. White, M. Cronin-Golomb, A. Yariv, Opt. Lett. 11, 239 (1986).
[Crossref]

A. Bledowski, W. Krolikowski, A. Kujawski, IEEE J. Quantum Electron. QE-22, 1547 (1986); N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, G. Tomberger, Appl. Phys. B 41, 259 (1986).
[Crossref]

1985 (2)

M. R. Belić, Phys. Rev. A 31, 3169 (1985); M. R. Belić, M. Lax, Opt. Commun. 56, 197 (1985).
[Crossref]

Ph. Refreiger, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

1984 (3)

1983 (2)

1982 (2)

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, Ferroelectrics 22, 961 (1979).
[Crossref]

1975 (1)

D. W. Vahey, J. Appl. Phys. 46, 3510 (1975).
[Crossref]

1972 (1)

D. L. Staebler, J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
[Crossref]

1969 (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[Crossref]

Amodei, J. J.

D. L. Staebler, J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
[Crossref]

Belic, M. R.

M. R. Belić, Phys. Rev. A 31, 3169 (1985); M. R. Belić, M. Lax, Opt. Commun. 56, 197 (1985).
[Crossref]

Bledowski, A.

A. Bledowski, W. Krolikowski, A. Kujawski, IEEE J. Quantum Electron. QE-22, 1547 (1986); N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, G. Tomberger, Appl. Phys. B 41, 259 (1986).
[Crossref]

Cronin-Golomb, M.

Fischer, B.

Goodman, J.

Heaton, J. M.

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymar, T. Wilson, Opt. Acta 31, 885 (1984).
[Crossref]

Hesselink, L.

Huignard, J. P.

Ph. Refreiger, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

Ja, Y. H.

Y. H. Ja, Opt. Quantum Electron. 15, 539 (1983).
[Crossref]

Y. H. Ja, Opt. Quantum Electron. 14, 547 (1982).
[Crossref]

Khyzniak, A.

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[Crossref]

Kondilenko, V.

Krolikowski, W.

A. Bledowski, W. Krolikowski, A. Kujawski, IEEE J. Quantum Electron. QE-22, 1547 (1986); N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, G. Tomberger, Appl. Phys. B 41, 259 (1986).
[Crossref]

Kucherov, Y.

Kujawski, A.

A. Bledowski, W. Krolikowski, A. Kujawski, IEEE J. Quantum Electron. QE-22, 1547 (1986); N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, G. Tomberger, Appl. Phys. B 41, 259 (1986).
[Crossref]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, Ferroelectrics 22, 961 (1979).
[Crossref]

Lesnik, S.

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, Ferroelectrics 22, 961 (1979).
[Crossref]

McRuer, R.

Mills, P. A.

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymar, T. Wilson, Opt. Acta 31, 885 (1984).
[Crossref]

Odulov, S.

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, Ferroelectrics 22, 961 (1979).
[Crossref]

Paige, E. G. S.

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymar, T. Wilson, Opt. Acta 31, 885 (1984).
[Crossref]

Rajbenbach, H.

Ph. Refreiger, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

Refreiger, Ph.

Ph. Refreiger, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

Roosen, G.

Sincerbox, G. T.

Solymar, L.

Ph. Refreiger, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymar, T. Wilson, Opt. Acta 31, 885 (1984).
[Crossref]

Soskin, M.

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, Ferroelectrics 22, 961 (1979).
[Crossref]

Staebler, D. L.

D. L. Staebler, J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
[Crossref]

Vahey, D. W.

D. W. Vahey, J. Appl. Phys. 46, 3510 (1975).
[Crossref]

Valley, G. C.

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, Ferroelectrics 22, 961 (1979).
[Crossref]

White, J. O.

Wilde, J.

Wilson, T.

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymar, T. Wilson, Opt. Acta 31, 885 (1984).
[Crossref]

Yariv, A.

Yeh, P.

P. Yeh, IEEE J. Quantum Electron 25, 484 (1989).
[Crossref]

P. Yeh, Appl. Opt. 28, 1961 (1989).
[Crossref] [PubMed]

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
[Crossref]

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, Ferroelectrics 22, 961 (1979).
[Crossref]

IEEE J. Quantum Electron (1)

P. Yeh, IEEE J. Quantum Electron 25, 484 (1989).
[Crossref]

IEEE J. Quantum Electron. (1)

A. Bledowski, W. Krolikowski, A. Kujawski, IEEE J. Quantum Electron. QE-22, 1547 (1986); N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, G. Tomberger, Appl. Phys. B 41, 259 (1986).
[Crossref]

J. Appl. Phys. (3)

Ph. Refreiger, L. Solymar, H. Rajbenbach, J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

D. L. Staebler, J. J. Amodei, J. Appl. Phys. 43, 1042 (1972).
[Crossref]

D. W. Vahey, J. Appl. Phys. 46, 3510 (1975).
[Crossref]

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

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

Opt. Acta (1)

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymar, T. Wilson, Opt. Acta 31, 885 (1984).
[Crossref]

Opt. Lett. (3)

Opt. Quantum Electron. (2)

Y. H. Ja, Opt. Quantum Electron. 14, 547 (1982).
[Crossref]

Y. H. Ja, Opt. Quantum Electron. 15, 539 (1983).
[Crossref]

Phys. Rev. A (1)

M. R. Belić, Phys. Rev. A 31, 3169 (1985); M. R. Belić, M. Lax, Opt. Commun. 56, 197 (1985).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Schematic representation of (a) two-wave mixing and (b) four-wave mixing in photorefractive media.

Fig. 2
Fig. 2

(a) Schematic representation of multibeam coupling in photorefractive media, (b) Vector diagram showing the Bragg condition.

Fig. 3
Fig. 3

Diffraction efficiency η as a function of read-beam intensity I A 2 ( 0 ) for various coupling strengths ΓL. The normalized, equal intensities of the write beams are unity: I A 1 ( 0 ) = I B 1 ( 0 ) = 1.

Fig. 4
Fig. 4

Intensities I(x) of the four beams as a function of the depth x within the medium. The absorption coefficient α is taken to be zero, the coupling coefficient Γ is equal to 5 cm−1, and the interaction length L is 1 cm. The normalized, equal intensities of the write beams (shown as solid curves) are unity: I A 1 ( 0 ) = I B 1 ( 0 ) = 1 and I B 2 ( 0 ) = 0. The read beam I A 2 ( 0 ) has the following intensity corresponding to each figure: (a) 0.01, (b) 1, (c) 10, (d) 100.

Fig. 5
Fig. 5

Diffraction efficiency η as a function of read-beam intensity I A 2 ( 0 ) for various negative coupling strengths ΓL. The normalized, equal intensities of the write beams are unity: I A 1 ( 0 ) = I B 1 ( 0 ) = 1.

Fig. 6
Fig. 6

Diffraction efficiency η as a function of read-beam intensity I A 2 ( 0 ) in the presence of absorption (dashed curves, 2αL = 1). The normalized, equal intensities of the write beams are unity: I A 1 ( 0 ) = I B 1 ( 0 ) = 1.

Fig. 7
Fig. 7

Diffraction efficiency η as a function of coupling strength ΓL for various read-beam intensities I A 2 ( 0 ). The normalized, equal intensities of the write beams are unity: I A 1 ( 0 ) = I B 1 ( 0 ) = 1, and absorptive losses are assumed to be negligible (α = 0).

Equations (21)

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E A n ( r , t ) = 1 2 A n ( x ) exp [ i ( k A n · r ω n t ) + c . c ., ]
I ( r ) = 1 2 n { | A n | 2 + | B n | 2 + [ A n B n * exp ( i K n · r ) + c . c . ] } ,
n = n b + Δ n 2 I 0 [ e i ϕ exp ( i K · r ) n A n B n * + c . c . ] ,
I 0 n ( | A n | 2 + | B n | 2 )
d A n d x = γ GB n α A n , d B n d x = γ * G * A n α B n ,
γ i ω Δ n e i ϕ 2 c cos θ ,
G n A n B n * / I 0 .
d a n d x = Γ g b n , d b n d x = Γ g * a n ,
d | g | 2 d x = 2 Γ | g | 2 f , d f d x = 4 Γ | g | 2 .
f ( x ) = C tanh ( C Γ x D ) ,
C 2 f 2 ( 0 ) + 4 | g ( 0 ) | 2 , D 1 2 ln C + f ( 0 ) C f ( 0 )
| g ( x ) | = C 2 sech ( C Γ x D ) .
d g n d x = Γ g f n , d f n d x = 2 Γ g * g n + c . c .
d r n d x = Γ | g | f n , d s n d x = 0 , d f n d x = 4 Γ | g | r n .
d i a n d x = 2 Γ | g | r n , d i b n d x = 2 Γ | g | r n ,
i a n ( x ) = i a n ( 0 ) i a n ( 0 ) i b n ( 0 ) 2 × [ 1 + tanh ( C Γ x D ) tanh D sech ( C Γ x D ) sech D ] [ i i n ( 0 ) i b n ( 0 ) ] 1 / 2 cos [ ϕ g n ( 0 ) ϕ g ( 0 ) ] × [ tanh ( C Γ x D ) sech D + sech ( C Γ x D ) tanh D ] ,
I A 1 ( x ) = I A 1 ( 0 ) e 2 α x [ 1 tanh D sech ( C Γ x D ) sech D tanh ( C Γ x D ) ] , I A 2 ( x ) = 1 2 I A 2 ( 0 ) e 2 α x [ 1 + sech D sech ( C Γ x D ) tanh D tanh ( C Γ x D ) ] ,
η = e 2 α L 2 [ 1 + tanh D tanh ( C Γ L D ) sech D sech ( C Γ L D ) ] .
I A 1 ( L ) I A 1 ( 0 ) ( 1 sech D ) , I A 2 ( L ) 1 2 I A 2 ( 0 ) ( 1 tanh D ) , η 1 2 ( 1 + tanh D ) .
η = e 2 α L 2 [ 1 tanh D tanh ( C Γ L + D ) sech D sech ( C Γ L + D ) ] ,
I A 1 ( L ) I A 1 ( 0 ) e 2 α L [ 1 tanh ( Γ L ) ] , I B 1 ( L ) I A 1 ( 0 ) e 2 α L [ 1 + tanh ( Γ L ) ] , η e 2 α L 2 [ 1 sech ( Γ L ) ] .

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