Abstract

We consider the backward optical wave stimulated by a multimode, monochromatic, incident optical wave in a waveguide filled with a transparent nonlinear medium, when the incident wave is negligibly perturbed by the nonlinear processes. We derive the conditions on guide length, area, mode number, and Stokes shift in order that a given high percentage of the power in the backscattered field be the “phase conjugate” of the incident field, i.e., be proportional to its complex conjugate in the entrance plane of the waveguide.

© 1978 Optical Society of America

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  1. B. Y. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faisullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’stam-Brillouin scattering,” Pisma Zh. Eksp. Teor. Fiz. 15, 160–164 (1972) [JETP Lett. 15, 109–113 (1972)].
  2. O. Y. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faisullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror,’” Pisma Zh. Eksp. Teor. Fiz. 16, 617–621 (1972) [JETP Lett. 16, 435–438 (1972)].
  3. V. G. Sidorovich, “Theory of the ‘Brillouin mirror,’” Zh. Tekh. Fiz. 46, 2168–2174 (1976) [Sov. Phys. Tech. Phsy. 21, 1270–1274 (1976)].
  4. B. Ya. Zel’dovich and V. V. Shkunov, “Wavefront reproduction in stimulated Raman scattering,” Kvantovaya Elektron. (Moscow) 4, 1090–1098 (1977) [Sov. J. Quantum Electron. 7, 610–615 (1977)].
  5. B. Ya. Zel’dovich, N. A. Mel’nikov, N. F. Pilipetskii, and V. V. Ragul’skii, “Observation of wave-front inversion in stimulated Raman scattering of light,” Pisma Zh. Eksp. Fiz. 25, 41–44 (1977) [JETP Lett. 25, 36–38 (1977)].
  6. V. Wang and C. R. Giuliano, “Correction of phase aberrations via stimulated Brillouin scattering,” Opt. Lett. 2, 4–6 (1978).
    [Crossref] [PubMed]
  7. The degeneracies referred to here are those arising from symmetry, such as occur for similar right- and left-circularly polarized modes in a cylindrical waveguide. That Δk cannot approach zero (to within less than L−1) for any other cases is because otherwise um2+ui2 cannot come within S−1of un2+uj2 without, at the same time, there being a drastic reduction in the x-y integral in (9). This may be appreciated by studying the example of Eq. (50) for un2. Here, without the aforementioned degeneracies occuring, one sees that ΔkL≳ L/Sk, and L/Sk is never much less than unity when guiding occurs.
  8. A waveguide which imitates well the interaction of focused unguided waves may be imagined as follows. Construct that complete orthonormal set of free-space Guassian-beam modes whose parameters are such that the smallest number N of modes need be superposed to give a good representation of the (focused) incident beam. Then let the waveguide axis coincide with the z axis of these modes and let it barely encompass the beam waist over the length where the waist size does not change appreciably. This length is generally of order Sλ −1N−1/2. Calculate the backscattered wave using the mode decompositions in Eqs. (5) and (8)

1978 (1)

1977 (2)

B. Ya. Zel’dovich and V. V. Shkunov, “Wavefront reproduction in stimulated Raman scattering,” Kvantovaya Elektron. (Moscow) 4, 1090–1098 (1977) [Sov. J. Quantum Electron. 7, 610–615 (1977)].

B. Ya. Zel’dovich, N. A. Mel’nikov, N. F. Pilipetskii, and V. V. Ragul’skii, “Observation of wave-front inversion in stimulated Raman scattering of light,” Pisma Zh. Eksp. Fiz. 25, 41–44 (1977) [JETP Lett. 25, 36–38 (1977)].

1976 (1)

V. G. Sidorovich, “Theory of the ‘Brillouin mirror,’” Zh. Tekh. Fiz. 46, 2168–2174 (1976) [Sov. Phys. Tech. Phsy. 21, 1270–1274 (1976)].

1972 (2)

B. Y. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faisullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’stam-Brillouin scattering,” Pisma Zh. Eksp. Teor. Fiz. 15, 160–164 (1972) [JETP Lett. 15, 109–113 (1972)].

O. Y. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faisullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror,’” Pisma Zh. Eksp. Teor. Fiz. 16, 617–621 (1972) [JETP Lett. 16, 435–438 (1972)].

Faisullov, F. S.

B. Y. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faisullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’stam-Brillouin scattering,” Pisma Zh. Eksp. Teor. Fiz. 15, 160–164 (1972) [JETP Lett. 15, 109–113 (1972)].

O. Y. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faisullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror,’” Pisma Zh. Eksp. Teor. Fiz. 16, 617–621 (1972) [JETP Lett. 16, 435–438 (1972)].

Giuliano, C. R.

Mel’nikov, N. A.

B. Ya. Zel’dovich, N. A. Mel’nikov, N. F. Pilipetskii, and V. V. Ragul’skii, “Observation of wave-front inversion in stimulated Raman scattering of light,” Pisma Zh. Eksp. Fiz. 25, 41–44 (1977) [JETP Lett. 25, 36–38 (1977)].

Nosach, O. Y.

O. Y. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faisullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror,’” Pisma Zh. Eksp. Teor. Fiz. 16, 617–621 (1972) [JETP Lett. 16, 435–438 (1972)].

Pilipetskii, N. F.

B. Ya. Zel’dovich, N. A. Mel’nikov, N. F. Pilipetskii, and V. V. Ragul’skii, “Observation of wave-front inversion in stimulated Raman scattering of light,” Pisma Zh. Eksp. Fiz. 25, 41–44 (1977) [JETP Lett. 25, 36–38 (1977)].

Popovichev, V. I.

O. Y. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faisullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror,’” Pisma Zh. Eksp. Teor. Fiz. 16, 617–621 (1972) [JETP Lett. 16, 435–438 (1972)].

B. Y. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faisullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’stam-Brillouin scattering,” Pisma Zh. Eksp. Teor. Fiz. 15, 160–164 (1972) [JETP Lett. 15, 109–113 (1972)].

Ragul’skii, V. V.

B. Ya. Zel’dovich, N. A. Mel’nikov, N. F. Pilipetskii, and V. V. Ragul’skii, “Observation of wave-front inversion in stimulated Raman scattering of light,” Pisma Zh. Eksp. Fiz. 25, 41–44 (1977) [JETP Lett. 25, 36–38 (1977)].

B. Y. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faisullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’stam-Brillouin scattering,” Pisma Zh. Eksp. Teor. Fiz. 15, 160–164 (1972) [JETP Lett. 15, 109–113 (1972)].

O. Y. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faisullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror,’” Pisma Zh. Eksp. Teor. Fiz. 16, 617–621 (1972) [JETP Lett. 16, 435–438 (1972)].

Shkunov, V. V.

B. Ya. Zel’dovich and V. V. Shkunov, “Wavefront reproduction in stimulated Raman scattering,” Kvantovaya Elektron. (Moscow) 4, 1090–1098 (1977) [Sov. J. Quantum Electron. 7, 610–615 (1977)].

Sidorovich, V. G.

V. G. Sidorovich, “Theory of the ‘Brillouin mirror,’” Zh. Tekh. Fiz. 46, 2168–2174 (1976) [Sov. Phys. Tech. Phsy. 21, 1270–1274 (1976)].

Wang, V.

Zel’dovich, B. Y.

B. Y. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faisullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’stam-Brillouin scattering,” Pisma Zh. Eksp. Teor. Fiz. 15, 160–164 (1972) [JETP Lett. 15, 109–113 (1972)].

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, N. A. Mel’nikov, N. F. Pilipetskii, and V. V. Ragul’skii, “Observation of wave-front inversion in stimulated Raman scattering of light,” Pisma Zh. Eksp. Fiz. 25, 41–44 (1977) [JETP Lett. 25, 36–38 (1977)].

B. Ya. Zel’dovich and V. V. Shkunov, “Wavefront reproduction in stimulated Raman scattering,” Kvantovaya Elektron. (Moscow) 4, 1090–1098 (1977) [Sov. J. Quantum Electron. 7, 610–615 (1977)].

Kvantovaya Elektron. (Moscow) (1)

B. Ya. Zel’dovich and V. V. Shkunov, “Wavefront reproduction in stimulated Raman scattering,” Kvantovaya Elektron. (Moscow) 4, 1090–1098 (1977) [Sov. J. Quantum Electron. 7, 610–615 (1977)].

Opt. Lett. (1)

Pisma Zh. Eksp. Fiz. (1)

B. Ya. Zel’dovich, N. A. Mel’nikov, N. F. Pilipetskii, and V. V. Ragul’skii, “Observation of wave-front inversion in stimulated Raman scattering of light,” Pisma Zh. Eksp. Fiz. 25, 41–44 (1977) [JETP Lett. 25, 36–38 (1977)].

Pisma Zh. Eksp. Teor. Fiz. (2)

B. Y. Zel’dovich, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faisullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’stam-Brillouin scattering,” Pisma Zh. Eksp. Teor. Fiz. 15, 160–164 (1972) [JETP Lett. 15, 109–113 (1972)].

O. Y. Nosach, V. I. Popovichev, V. V. Ragul’skii, and F. S. Faisullov, “Cancellation of phase distortions in an amplifying medium with a ‘Brillouin mirror,’” Pisma Zh. Eksp. Teor. Fiz. 16, 617–621 (1972) [JETP Lett. 16, 435–438 (1972)].

Zh. Tekh. Fiz. (1)

V. G. Sidorovich, “Theory of the ‘Brillouin mirror,’” Zh. Tekh. Fiz. 46, 2168–2174 (1976) [Sov. Phys. Tech. Phsy. 21, 1270–1274 (1976)].

Other (2)

The degeneracies referred to here are those arising from symmetry, such as occur for similar right- and left-circularly polarized modes in a cylindrical waveguide. That Δk cannot approach zero (to within less than L−1) for any other cases is because otherwise um2+ui2 cannot come within S−1of un2+uj2 without, at the same time, there being a drastic reduction in the x-y integral in (9). This may be appreciated by studying the example of Eq. (50) for un2. Here, without the aforementioned degeneracies occuring, one sees that ΔkL≳ L/Sk, and L/Sk is never much less than unity when guiding occurs.

A waveguide which imitates well the interaction of focused unguided waves may be imagined as follows. Construct that complete orthonormal set of free-space Guassian-beam modes whose parameters are such that the smallest number N of modes need be superposed to give a good representation of the (focused) incident beam. Then let the waveguide axis coincide with the z axis of these modes and let it barely encompass the beam waist over the length where the waist size does not change appreciably. This length is generally of order Sλ −1N−1/2. Calculate the backscattered wave using the mode decompositions in Eqs. (5) and (8)

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Equations (58)

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ω = ν - ω B ,
ω B 2 n v s ν / c
θ < Q - 1 / 2 ,
P ω nl = - i G E ν E ν * · E ω ,
E ν = m A m ê m ( x , y ) e i k m ν z .
d x d y ê m * · ê n = δ m n .
k m ν 2 + u m 2 = n ν 2 ν 2 / c 2 ,
E ω = n B n ê n ( x , y ) e - i k n ω z - γ z / 2
γ k B m = 4 π ω 2 c - 2 G i j n d x d y ê m * · ê i ê j * · ê n × K m i j n A i A j * B n ,
K m i j n 0 L e i Δ k z d z / L
K m i j n = x - 1 sin x ,
n ( H m n + V m n + U m n ) B n = Y B m ,
H m n α ¯ δ m n + β ¯ a m * a n
V m n = l ( α n l - α ¯ ) a l * a l δ n m + ( β m n - β ¯ ) a m * a n .
a n = A n ( m A m * A m ) - 1 / 2 .
α m n d x d y ê m * · ê n 2
β m n d x d y ( ê m * · ê n ) 2 K m n m n .
α ¯ m n α m n a m 2 a n 2
U m n i j a i a j * d x d y ê m * · ê i ê j * · ê n K m i j n ,
N L > S k .
Y p w = 2 / S .
γ ( cm - 1 ) = ½ Y ( cm - 2 ) g ( cm / MW ) P ( MW ) ,
G ( 0 ) = α ¯ + β ¯
G ( ν ) = α ¯ .
α ¯ β ¯
G 0 2 G ν .
a n = N - 1 / 2 e - i ϕ n .
n x , n y = 0 , 1 , 2 , , M - 1
α n l = β n l = [ 2 + δ ( n x , l x ) ] [ 2 + δ ( n y , l y ) ] / 4 S ,
B n ( l x , l y ) = M - 1 exp [ i ϕ n + 2 π i ( n x l x + n y l y ) / M ] ,
l x , l y = 0 , 1 , , M - 1
Y 00 = α ¯ + β ¯ .
Y 0 ν = α ¯ + ( 1 + 2 M ) / 4 S M 2 .
Y μ ν = α ¯ + 1 / 4 S M 2 .
α ¯ = β ¯ = ( 1 + M - 1 + M - 2 / 4 ) / S .
Y 00 - Y 0 ν = ( 1 - ½ N - 1 / 2 ) / S
B 0 n = a n * + c n .
c n = ν 0 b n ( ν ) b l ( ν ) * V l m a m * / ( G ( 0 ) - G ( ν ) )
a m V m n a n * = 0 ,
c n = V n m a m * / β ¯ ,
f = r / ( 1 - r ) ,
r = a l 2 θ l m a m 2 θ m n a n 2 / β ¯ 2 ,
Y 0 = G ( 0 ) + r β ¯
a n 2 = g x ( l x ) g y ( l y ) .
g + g + ,             g - g - ,             g + g - ,             or 0
r = 4 [ u ± 2 2 / u ± 4 - 1 ] ,
u ± 1 + ½ g ± / ( N + g + + N - g - )
w ± g ± N ± / ( N + g + + N - g - ) .
0 N + M ,             0 N - M ,             0 g - / g + 1.
r max ~ 0.0669 at g - ~ 0.20 g + .
r max ~ 0.1 N + - 2
u n 2 = π 2 ( n x 2 + n y 2 ) / S
Δ k = q ( m x 2 - n x 2 + m y 2 - n y 2 ) ,
q = π Δ λ / 4 S
Δ λ ( c / 2 π ) ( 1 / n ω ω - 1 / n ν ν ) .
4 S β m n = 0 L d z [ 2 e i q z ( m x 2 - n x 2 ) + δ ( m x , n x ) ] × [ 2 e i q z ( m y 2 - n y 2 ) + δ ( m y , n y ) ] / L ,
Δ r = 2 45 ( π Δ λ M 2 L 4 S ) 2 ( 1 + O { q 2 L 2 } + O { M - 1 } ) ,
L 6 r 0 1 / 2 S / N Δ λ