Abstract

The photorefractive coupling of two polarized electromagnetic waves in cubic crystals is considered. Exact solutions for the cross-polarization coupling are obtained. Both codirectional and contradirectional coupling are considered. The results are presented and discussed.

© 1987 Optical Society of America

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References

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  1. See, for example, P. Yeh, Appl. Opt. 26, 602 (1987), and references therein.
    [Crossref] [PubMed]
  2. M. B. Klein, Opt. Lett. 9, 350 (1984).
    [Crossref] [PubMed]
  3. G. Albanese, J. Kumar, and W. H. Steier, Opt. Lett. 11, 560 (1986).
    [Crossref]
  4. J. Kumar, G. Albanese, W. H. Steier, and M. Ziari, Opt. Lett. 12, 120 (1987).
    [Crossref] [PubMed]
  5. See, for example, A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  6. B. Fischer, J. O. White, M. Cronin-Golomb, and A. Yariv, Opt. Lett. 11, 239 (1986).
    [Crossref]
  7. M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, Opt. Lett. 7, 313 (1982).
    [Crossref] [PubMed]
  8. P. Yeh, Opt. Commun. 45, 323 (1983).
    [Crossref]

1987 (2)

1986 (2)

1984 (1)

1983 (1)

P. Yeh, Opt. Commun. 45, 323 (1983).
[Crossref]

1982 (1)

Albanese, G.

Cronin-Golomb, M.

Fischer, B.

Klein, M. B.

Kumar, J.

Steier, W. H.

White, J. O.

Yariv, A.

Yeh, P.

See, for example, P. Yeh, Appl. Opt. 26, 602 (1987), and references therein.
[Crossref] [PubMed]

P. Yeh, Opt. Commun. 45, 323 (1983).
[Crossref]

See, for example, A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Ziari, M.

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Figures (3)

Fig. 1
Fig. 1

Schematic drawing of photorefractive two-beam coupling in a cubic crystal.

Fig. 2
Fig. 2

(a) A configuration for codirectional cross polarization coupling in a cubic crystal of point group symmetry 4 ¯3m. (b) A configuration for contradirectional cross-polarization coupling in the same class of crystal.

Fig. 3
Fig. 3

Intensities of the four waves are plotted as functions of distance for various interaction situations. (a) Both incident beams are s polarized [i.e., (Ap(0) = Bp(0) = 0, c2/c1 = 0.1]. (b) The pump beam is linearly polarized at an azimuth angle of 30 deg relative to the s direction, and the signal beam is s polarized; c2/c1 = 0.01.

Equations (42)

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E = ( s A s + p 1 A p ) exp ( - i k 1 · r ) + ( s B s + p 2 B p ) exp ( - i k 2 · r ) ,
E * E = A s * A s + A p * A p + B s * B s + B p * B p + [ ( A s B s * + A p B p * p 1 · p 2 ) exp ( i K · r ) + c . c . ] ,
( Δ ) i j = - 0 n 4 r i j k E k ,
Δ = - 0 1 [ ( A s B s * + A p B p * cos θ ) × exp ( i K · r + ϕ ) + c . c . ] / I 0 ,
I 0 = A s * A s + A p * A p + B s * B s + B p * B p .
1 = n 4 r 41 [ 0 E z E y E z 0 E x E y E x 0 ] .
d d z A s = i 2 β 1 e i ϕ ( Γ s s B s + Γ s p 2 B p ) ( A s B s * + A p B p * cos θ ) / I 0 , d d z B s = i 2 β 2 e - i ϕ ( Γ s s A s + Γ s p 1 A p ) ( A s * B s + A p * B p cos θ ) / I 0 , d d z A p = i 2 β 1 e i ϕ ( Γ p 1 s B s + Γ p 1 p 2 B p ) ( A s B s * + A p B p * cos θ ) / I 0 , d d z B p = i 2 β 2 e - i ϕ ( Γ p 2 s A s + Γ p 2 p 1 A p ) ( A s * B s + A p * B p cos θ ) / I 0 ,
Γ = ω 2 μ 0 1
Γ i j = i Γ j ,             i , j = s , p 1 , p 2
1 = 1 2 n 4 r 41 [ 0 0 1 0 0 1 1 1 0 ] E sc ,
Γ ss = Γ p 1 p 2 = 0 , Γ s p 1 = Γ p 1 s = Γ s p 2 = Γ p 2 s = ( 2 π / λ ) 2 n 4 r 41 E sc cos ( θ / 2 ) ,
β 1 = β 2 = ( 2 π / λ ) n cos ( θ / 2 ) .
d d z A s = - γ B p ( A s B s * + A p B p * cos θ ) / I 0 , d d z B s = γ A p ( A s * B s + A p * B p cos θ ) / I 0 , d d z A p = - γ B s ( A s B s * + A p B p * cos θ ) / I 0 , d d z B p = γ A s ( A s * B s + A p * B p cos θ ) / I 0 ,
γ = ½ ( 2 π / λ ) n 3 r 41 E sc .
A s A s * + B p B p * = c 1 ,
A p A p * + B s B s * = c 2 ,
A s A p * + B s * B p = c 3 ,
A s B s - A p B p = c 4 .
d d z g = - γ ( g 2 c 2 cos θ + g σ * - c 1 ) ,
d d z f = γ ( f 2 c 1 cos θ + f σ - c 2 ) ,
f = A p / A s ,             g = B p / B s ,
σ = c 3 - c 3 * cos θ .
f = A p / A s = [ - σ + q tanh ( - q γ z / 2 + C ) ] / ( 2 c 1 cos θ ) , g = B p / B s = [ - σ * + q * tanh ( q * γ z / 2 + C ) ] / ( 2 c 2 cos θ ) ,
q 2 = 4 c 1 c 2 + σ 2 ,
A p 2 = f 2 c 1 - f g 2 c 2 1 - f g 2 , B p 2 = g 2 c 2 - f g 2 c 1 1 - f g 2 , A s 2 = A p 2 f 2 = c 1 - g 2 c 2 1 - f g 2 , B s 2 = B p 2 g 2 = c 2 - f 2 c 1 1 - f g 2 .
A p ( 0 ) = B p ( 0 ) = 0.
f = - q tanh ( q γ z / 2 ) / 2 c 1 cos θ , g = q tanh ( q γ z / 2 ) / 2 c 2 cos θ ,
A s 2 = c 1 1 1 + tanh 2 ( q γ z / 2 ) , A p 2 = c 2 tanh 2 ( q γ z / 2 ) 1 + tanh 2 ( q γ z / 2 ) , B s 2 = c 2 1 1 + tanh 2 ( q γ z / 2 ) , B p 2 = c 1 tanh 2 ( q γ z / 2 ) 1 + tanh 2 ( q γ z / 2 ) ,
d d z B s = γ a B p + γ b B s , d d z B p = γ c B p + γ d B s ,
a = A p 2 cos θ / I 0 , b = A p A s * / I 0 , c = A s A p * cos θ / I 0 , d = A s 2 / I 0 .
B s ( z ) = { [ b B s ( 0 ) + a B p ( 0 ) ] exp [ ( b + c ) γ z ] + [ c B s ( 0 ) - a B p ( 0 ) ] } / ( b + c ) , B p ( z ) = { c [ b B s ( 0 ) + a B p ( 0 ) ] exp [ ( b + c ) γ z ] - b [ c B s ( 0 ) - a B p ( 0 ) ] } / [ a ( b + c ) ] ,
B s ( z ) = B s ( 0 ) { b exp [ ( b + c ) γ z ] + c } / ( b + c ) , B p ( z ) = B s ( 0 ) b c { exp [ ( b + c ) γ z ] - 1 } / [ a ( b + c ) ] .
B s ( z ) = B s ( 0 ) + B s ( 0 ) A p A s * I 0 γ z , B p ( z ) = B s ( 0 ) A s 2 I 0 γ z .
1 = n 4 r 41 [ 0 1 0 1 0 0 0 0 0 ] E sc ,
Γ s s = Γ p 1 p 2 = 0 , Γ s p 1 = Γ p 1 s = Γ s p 2 = Γ p 2 s = n 4 r 41 E s c cos ( θ / 2 ) ,
β 1 = - β 2 = - ( 2 π / λ ) n cos ( θ / 2 ) .
d d z A s = γ B p ( A s B s * + A p B p * cos θ ) / I 0 , d d z B s = γ A p ( A s * B s + A p * B p cos θ ) / I 0 , d d z A p = γ B s ( A s B s * + A p B p * cos θ ) / I 0 , d d z B p = γ A S ( A s * B s + A p * B p cos θ ) / I 0 ,
γ = ½ ( 2 π / λ ) n 3 r 41 E sc .
A s * A s - B p * B p = c 1 ,
A p * A p - B s * B s = c 2 .
A s A p * - B s B p * = c 3 ,
A s B s - A p B p = c 4 .

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