Abstract

A quantitative description of the enhancement gain profile of phase conjugation by four-wave mixing in plasma, based on stimulated Brillouin scattering, is presented. The Landau damping mechanism is taken into account, and general geometrical configurations are discussed.

© 1989 Optical Society of America

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References

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  1. D. G. Steel, J. F. Lam, “Degenerate four-wave mixing in plasmas,” Opt. Lett. 4, 363–365 (1979).
    [CrossRef] [PubMed]
  2. J. F. Federici, D. K. Mansfield, “Degenerate four-wave mixing and phase conjugation in a collisional plasma,” J. Opt. Soc. Am. B 3, 1588–1595 (1986).
    [CrossRef]
  3. F. F. Chen, Introduction to Plasma Physics and Controlled Fusion (Plenum, New York, 1984).
    [CrossRef]
  4. A. M. Scott, “Efficient phase conjugation by Brillouin enhanced four-wave mixing,” Opt. Commun. 45, 127–132 (1983).
    [CrossRef]
  5. A. M. Scott, M. S. Hazell, “High-efficiency scattering in transient Brillouin-enhanced four-wave mixing,” IEEE J. Quantum Electron. QE-22, 1248–1257(1986).
    [CrossRef]
  6. M. D. Skeldon, P. Narum, R. W. Boyd, “Non-frequency-shifted, high-fidelity phase conjugation with aberrated pump waves by Brillouin-enhanced four-wave mixing,” Opt. Lett. 12, 343–345 (1987).
    [CrossRef] [PubMed]
  7. A. Postan, Y. Ben-Aryeh, “Enhancement of phase conjugation in plasma,” J. Opt. Soc. Am. B 5, 1379–1385 (1988).
    [CrossRef]
  8. S. Ichimaru, Basic Principles of Plasma Physics, a Statistical Approach (Benjamin, Reading, Mass., 1973).
  9. I. Nebenzahl, A. Ron, N. Rostoker, “Reflected phase-conjugate wave in a plasma,” Phys. Rev. Lett. 60, 1030–1032 (1988).
    [CrossRef] [PubMed]
  10. H. C. Praddaude, D. W. Schudder, B. Lax, “Coherent four-wave scattering in plasmas—application to plasma diagnostics,” Appl. Phys. Lett. 35, 766–768 (1979).
    [CrossRef]
  11. Q. Zhong, “Doubly degenerate four-wave mixing in plasma,” Chin. Phys. 2, 141–145 (1982).
  12. A. Yariv, D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977).
    [CrossRef] [PubMed]
  13. R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).
  14. D. M. Bloom, G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977).
    [CrossRef]
  15. Y. Ben-Aryeh, A. Postan, “Enhancement of phase conjugation by stimulated Brillouin scattering in flowing and in non-flowing gases,” Opt. Commun. 66, 47–51 (1988).
    [CrossRef]

1988 (3)

I. Nebenzahl, A. Ron, N. Rostoker, “Reflected phase-conjugate wave in a plasma,” Phys. Rev. Lett. 60, 1030–1032 (1988).
[CrossRef] [PubMed]

Y. Ben-Aryeh, A. Postan, “Enhancement of phase conjugation by stimulated Brillouin scattering in flowing and in non-flowing gases,” Opt. Commun. 66, 47–51 (1988).
[CrossRef]

A. Postan, Y. Ben-Aryeh, “Enhancement of phase conjugation in plasma,” J. Opt. Soc. Am. B 5, 1379–1385 (1988).
[CrossRef]

1987 (1)

1986 (2)

J. F. Federici, D. K. Mansfield, “Degenerate four-wave mixing and phase conjugation in a collisional plasma,” J. Opt. Soc. Am. B 3, 1588–1595 (1986).
[CrossRef]

A. M. Scott, M. S. Hazell, “High-efficiency scattering in transient Brillouin-enhanced four-wave mixing,” IEEE J. Quantum Electron. QE-22, 1248–1257(1986).
[CrossRef]

1983 (1)

A. M. Scott, “Efficient phase conjugation by Brillouin enhanced four-wave mixing,” Opt. Commun. 45, 127–132 (1983).
[CrossRef]

1982 (1)

Q. Zhong, “Doubly degenerate four-wave mixing in plasma,” Chin. Phys. 2, 141–145 (1982).

1979 (2)

H. C. Praddaude, D. W. Schudder, B. Lax, “Coherent four-wave scattering in plasmas—application to plasma diagnostics,” Appl. Phys. Lett. 35, 766–768 (1979).
[CrossRef]

D. G. Steel, J. F. Lam, “Degenerate four-wave mixing in plasmas,” Opt. Lett. 4, 363–365 (1979).
[CrossRef] [PubMed]

1977 (2)

A. Yariv, D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977).
[CrossRef] [PubMed]

D. M. Bloom, G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977).
[CrossRef]

Ben-Aryeh, Y.

A. Postan, Y. Ben-Aryeh, “Enhancement of phase conjugation in plasma,” J. Opt. Soc. Am. B 5, 1379–1385 (1988).
[CrossRef]

Y. Ben-Aryeh, A. Postan, “Enhancement of phase conjugation by stimulated Brillouin scattering in flowing and in non-flowing gases,” Opt. Commun. 66, 47–51 (1988).
[CrossRef]

Bjorklund, G. C.

D. M. Bloom, G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977).
[CrossRef]

Bloom, D. M.

D. M. Bloom, G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977).
[CrossRef]

Boyd, R. W.

Chen, F. F.

F. F. Chen, Introduction to Plasma Physics and Controlled Fusion (Plenum, New York, 1984).
[CrossRef]

Federici, J. F.

Hazell, M. S.

A. M. Scott, M. S. Hazell, “High-efficiency scattering in transient Brillouin-enhanced four-wave mixing,” IEEE J. Quantum Electron. QE-22, 1248–1257(1986).
[CrossRef]

Ichimaru, S.

S. Ichimaru, Basic Principles of Plasma Physics, a Statistical Approach (Benjamin, Reading, Mass., 1973).

Lam, J. F.

Lax, B.

H. C. Praddaude, D. W. Schudder, B. Lax, “Coherent four-wave scattering in plasmas—application to plasma diagnostics,” Appl. Phys. Lett. 35, 766–768 (1979).
[CrossRef]

Mansfield, D. K.

Narum, P.

Nebenzahl, I.

I. Nebenzahl, A. Ron, N. Rostoker, “Reflected phase-conjugate wave in a plasma,” Phys. Rev. Lett. 60, 1030–1032 (1988).
[CrossRef] [PubMed]

Pepper, D. M.

Postan, A.

Y. Ben-Aryeh, A. Postan, “Enhancement of phase conjugation by stimulated Brillouin scattering in flowing and in non-flowing gases,” Opt. Commun. 66, 47–51 (1988).
[CrossRef]

A. Postan, Y. Ben-Aryeh, “Enhancement of phase conjugation in plasma,” J. Opt. Soc. Am. B 5, 1379–1385 (1988).
[CrossRef]

Praddaude, H. C.

H. C. Praddaude, D. W. Schudder, B. Lax, “Coherent four-wave scattering in plasmas—application to plasma diagnostics,” Appl. Phys. Lett. 35, 766–768 (1979).
[CrossRef]

Ron, A.

I. Nebenzahl, A. Ron, N. Rostoker, “Reflected phase-conjugate wave in a plasma,” Phys. Rev. Lett. 60, 1030–1032 (1988).
[CrossRef] [PubMed]

Rostoker, N.

I. Nebenzahl, A. Ron, N. Rostoker, “Reflected phase-conjugate wave in a plasma,” Phys. Rev. Lett. 60, 1030–1032 (1988).
[CrossRef] [PubMed]

Schudder, D. W.

H. C. Praddaude, D. W. Schudder, B. Lax, “Coherent four-wave scattering in plasmas—application to plasma diagnostics,” Appl. Phys. Lett. 35, 766–768 (1979).
[CrossRef]

Scott, A. M.

A. M. Scott, M. S. Hazell, “High-efficiency scattering in transient Brillouin-enhanced four-wave mixing,” IEEE J. Quantum Electron. QE-22, 1248–1257(1986).
[CrossRef]

A. M. Scott, “Efficient phase conjugation by Brillouin enhanced four-wave mixing,” Opt. Commun. 45, 127–132 (1983).
[CrossRef]

Skeldon, M. D.

Steel, D. G.

Yariv, A.

Zhong, Q.

Q. Zhong, “Doubly degenerate four-wave mixing in plasma,” Chin. Phys. 2, 141–145 (1982).

Appl. Phys. Lett. (2)

H. C. Praddaude, D. W. Schudder, B. Lax, “Coherent four-wave scattering in plasmas—application to plasma diagnostics,” Appl. Phys. Lett. 35, 766–768 (1979).
[CrossRef]

D. M. Bloom, G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave mixing,” Appl. Phys. Lett. 31, 592–594 (1977).
[CrossRef]

Chin. Phys. (1)

Q. Zhong, “Doubly degenerate four-wave mixing in plasma,” Chin. Phys. 2, 141–145 (1982).

IEEE J. Quantum Electron. (1)

A. M. Scott, M. S. Hazell, “High-efficiency scattering in transient Brillouin-enhanced four-wave mixing,” IEEE J. Quantum Electron. QE-22, 1248–1257(1986).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (2)

A. M. Scott, “Efficient phase conjugation by Brillouin enhanced four-wave mixing,” Opt. Commun. 45, 127–132 (1983).
[CrossRef]

Y. Ben-Aryeh, A. Postan, “Enhancement of phase conjugation by stimulated Brillouin scattering in flowing and in non-flowing gases,” Opt. Commun. 66, 47–51 (1988).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. Lett. (1)

I. Nebenzahl, A. Ron, N. Rostoker, “Reflected phase-conjugate wave in a plasma,” Phys. Rev. Lett. 60, 1030–1032 (1988).
[CrossRef] [PubMed]

Other (3)

R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).

F. F. Chen, Introduction to Plasma Physics and Controlled Fusion (Plenum, New York, 1984).
[CrossRef]

S. Ichimaru, Basic Principles of Plasma Physics, a Statistical Approach (Benjamin, Reading, Mass., 1973).

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Figures (2)

Fig. 1
Fig. 1

Enhancement parameter |κ| (in reciprocal meters) as a function of the frequency detuning, Ω1, according to Eq. (26) for the symmetrical case in which Δω′ = Δω″, the angle between k1 and k3 is θ = 2π/3, n0 = 1020 m−3, λ0 = 10.6 μm, Te = 2 eV, and the laser pump intensity is 106 W/cm2.

Fig. 2
Fig. 2

Reflectivity R as a function of M1 [the Mach number that represents the frequency detuning; see Eq. (15)] for the symmetrical case (Δω′ = Δω″). The conditions are as specified for Fig. 1 except that n0 = 1023 m−3, the laser pump intensity is 108 W/cm2, and θ is arbitrary.

Equations (32)

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ω 1 = ω - Δ ω , ω 2 = ω + Δ ω , ω 3 = ω - Δ ω , ω 4 = ω + Δ ω .
Δ ω = ω Δ k k cos θ ,             Δ ω = ω Δ k k cos θ ,
E = 1 2 j = 1 4 ( E j + c . c . ) ,             E j = A j exp [ - i ω j t + i ( k j - Δ k ) · r ] ,
n 0 β 0 2 E 2 = - n 0 β 0 j = 1 4 q j 2 F j exp ( - i Ω ˜ j t + i q j · r ) ,
β 0 = Z e 2 4 m e m i ω 2
q 1 = k 4 - k 1 ,             q 2 = 2 k 1 ,             q 3 = 2 k 4 ,             q 4 = k 1 - k 3 ,
Ω ˜ 1 = Δ ω + Δ ω ,             Ω ˜ 2 = - 2 Δ ω , Ω ˜ 3 = - 2 Δ ω ,             Ω ˜ 4 = Δ ω - Δ ω ,
F 1 = A 1 * A 4 + A 2 A 3 * ,             F 2 = A 1 A 2 * , F 3 = A 3 * A 4 ,             F 4 = A 1 A 3 * + A 2 * A 4 .
2 n t 2 - s 2 2 n = n 0 β 0 2 E 2 ,
n = n 0 + Δ n = n 0 + [ j = 1 4 n j 0 exp ( - i Ω ˜ j t + i q j · r ) + c . c . ] .
( Ω ˜ j 2 - s 2 q j 2 ) n j 0 = β 0 n 0 q j 2 F j ,             j = 1 , 2 , 3 , 4.
Ω ˜ j = Ω j ( 1 - i Γ L ) ,
Γ L = ( π 8 m e m i ) 1 / 2 .
[ Ω j 2 ( 1 - 2 i Γ L ) - s 2 q j 2 ] n j 0 = β 0 n 0 q j 2 F j .
n j 0 = - n 0 β j s 2 F j ,
β j = β 0 1 - M j 2 + 2 i M j 2 Γ L ,             M j = Ω j q j s .
2 E - η 2 c 2 2 E t 2 = 0 ,
η 2 = η 0 2 - Δ n n c .
A 1 ξ 1 = i γ β 2 ( A 2 * A 2 ) A 1 , A 2 ξ 2 = i γ β 2 * ( A 1 * A 1 ) A 2 ,
A 3 ξ 3 = i γ [ ( β 4 * A 1 * A 1 + β 1 * A 2 * A 2 ) A 3 + ( β 1 * + β 4 * ) A 1 A 2 A 4 * ] , A 4 ξ 4 = i γ [ ( β 1 A 1 * A 1 + β 4 A 2 * A 2 ) A 4 + ( β 1 + β 4 ) A 1 A 2 A 3 * ] ,
γ = k n 0 2 η 0 2 n c s 2
κ = γ ( β 1 + β 4 ) A 1 A 2
A 3 = A ˜ 3 exp [ i γ ( β 1 * + β 4 * ) I ξ 3 ] , A 4 = A ˜ 4 exp [ i γ ( β 1 + β 4 ) I ξ 4 ] ,
d A ˜ 3 d ξ 3 = i κ A ˜ 4 * ,             d A ˜ 4 d ξ 4 = i κ A ˜ 3 * ,
A ˜ 4 ( 0 ) = i κ κ 0 tan ( κ 0 L ) A ˜ 3 * ( 0 ) ,
R = r 2 = A 4 ( 0 ) 2 A 3 ( 0 ) 2 = tan 2 ( κ 0 L ) = sin 2 ( κ 0 R L ) + sinh 2 ( κ 0 I L ) cos 2 ( κ 0 R L ) + sinh 2 ( κ 0 I L ) ,
κ = k n 0 2 η 0 2 n c s 2 A 1 A 2 β 0 × ( | 1 1 - M 1 2 + 2 i Γ L M 1 2 + 1 1 - M 4 2 + 2 i Γ L M 4 2 | ) .
κ max = k n 0 2 η 0 2 n c s 2 A 1 A 2 β 0 1 2 Γ L .
L 0 = π 2 κ 0 R ,
R 0 = 1 + 1 sinh 2 ( π κ 0 I 2 κ 0 R ) .
κ 0 R = k n 0 I β 0 2 η 0 2 n c s 2 [ 1 - M 1 2 ( 1 - M 1 2 ) 2 + 4 Γ L 2 M 1 4 ) + 1 ]
κ 0 I κ 0 R = - 2 Γ L M 1 2 ( 1 - M 1 2 ) ( 2 - M 1 2 ) + 4 Γ L 2 M 1 4 .

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