Abstract

We investigate phase conjugation of a laser beam aberrated by a thin random-phase screen when the wave front is incompletely sampled by the phase conjugate mirror (PCM). We derive expressions for the average intensity after a double pass of the phase screen both when the PCM is aperture limited and when it is limited by its number of spatial modes.

© 1996 Optical Society of America

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References

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  1. D. M. Pepper, “Nonlinear optical phase conjugation,” in Laser Handbook 4, M. L. Stitch, M. Bass, eds. (North-Holland, Amsterdam, 1985), pp. 333–486.
  2. B. Ya. Zel’dovich, N. F. Pilepetsky, V. V. Shkunov, Principles of Phase Conjugation, Vol. 42 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1985).
  3. M. Gower, D. Proch, eds., Optical Phase Conjugation (Springer-Verlag, Berlin, 1994).
  4. M. Gower, “The physics of phase conjugate mirrors,” Prog. Quantum Electron. 9, 101–147 (1985).
    [CrossRef]
  5. H. Bruesselbach, D. C. Jones, D. A. Rockwell, R. C. Lind, “Real-time atmospheric compensation by stimulated Brillouin scattering phase conjugation,” J. Opt. Soc. Am. B 12, 1434–1447 (1995).
    [CrossRef]
  6. K. D. Ridley, A. M. Scott, “Brillouin-induced four-wave mixing,” Chap. 3 of Ref. 3.
  7. Yu. A. Kravtsov, A. I. Saichev, “Properties of coherent waves reflected in a turbulent medium,” J. Opt. Soc. Am. A 2, 2100–2105 (1985).
    [CrossRef]
  8. C. Gu, P. Yeh, “Partial phase conjugation, fidelity, and reciprocity,” Opt. Commun. 107, 353–357 (1994).
    [CrossRef]
  9. E. Jakeman, “Enhanced backscattering through a deep random phase screen,” J. Opt. Soc. Am. A 5, 1638–1648 (1988).
    [CrossRef]
  10. E. Jakeman, “Active imaging through a random phase screen,” J. Phys. D 24, 227–232 (1991).
    [CrossRef]
  11. I. B. Esipov, V. V. Zosimov, “Observation of partial phase conjugation in the case of reflection in a randomly in-homogeneous medium,” Opt. Spectrosc. (USSR) 60, 234–236 (1986).
  12. B. Ya. Zel’dovich, V. V. Shkunov, “Wavefront reproduction in stimulated Raman scattering,” Sov. J. Quantum Electron. 7, 610–615 (1977).
    [CrossRef]
  13. G. J. Crofts, R. P. M. Green, M. J. Damzen, “Investigation of multipass geometries for efficient degenerate four-wave mixing in Nd:YAG,” Opt. Lett. 17, 920–922 (1992).
    [CrossRef] [PubMed]
  14. K. D. Ridley, E. Jakeman, “Incomplete phase conjugation through a random-phase screen. II. Numerical simulations,” submitted to J. Opt. Soc. Am. A.
  15. See, for example, N. G. Basov, V. I. Efimkov, I. G. Zubarev, A. V. Kotov, A. B. Mironov, S. I. Mikhailov, M. G. Smirnov, “Influence of certain radiation parameters on wavefront reversal of a pump wave in a Brillouin mirror,” Sov. J. Quantum Electron. 9, 455–458 (1979).
    [CrossRef]
  16. E. Jakeman, “The physical optics of enhanced backscatter-ing,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, Amsterdam, 1990), pp. 111–123.
  17. See, for example, E. Jakeman, J. G. McWhirter, “Correlation function dependence of the scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
    [CrossRef]

1995 (1)

1994 (1)

C. Gu, P. Yeh, “Partial phase conjugation, fidelity, and reciprocity,” Opt. Commun. 107, 353–357 (1994).
[CrossRef]

1992 (1)

1991 (1)

E. Jakeman, “Active imaging through a random phase screen,” J. Phys. D 24, 227–232 (1991).
[CrossRef]

1988 (1)

1986 (1)

I. B. Esipov, V. V. Zosimov, “Observation of partial phase conjugation in the case of reflection in a randomly in-homogeneous medium,” Opt. Spectrosc. (USSR) 60, 234–236 (1986).

1985 (2)

1979 (1)

See, for example, N. G. Basov, V. I. Efimkov, I. G. Zubarev, A. V. Kotov, A. B. Mironov, S. I. Mikhailov, M. G. Smirnov, “Influence of certain radiation parameters on wavefront reversal of a pump wave in a Brillouin mirror,” Sov. J. Quantum Electron. 9, 455–458 (1979).
[CrossRef]

1977 (2)

See, for example, E. Jakeman, J. G. McWhirter, “Correlation function dependence of the scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

B. Ya. Zel’dovich, V. V. Shkunov, “Wavefront reproduction in stimulated Raman scattering,” Sov. J. Quantum Electron. 7, 610–615 (1977).
[CrossRef]

Basov, N. G.

See, for example, N. G. Basov, V. I. Efimkov, I. G. Zubarev, A. V. Kotov, A. B. Mironov, S. I. Mikhailov, M. G. Smirnov, “Influence of certain radiation parameters on wavefront reversal of a pump wave in a Brillouin mirror,” Sov. J. Quantum Electron. 9, 455–458 (1979).
[CrossRef]

Bruesselbach, H.

Crofts, G. J.

Damzen, M. J.

Efimkov, V. I.

See, for example, N. G. Basov, V. I. Efimkov, I. G. Zubarev, A. V. Kotov, A. B. Mironov, S. I. Mikhailov, M. G. Smirnov, “Influence of certain radiation parameters on wavefront reversal of a pump wave in a Brillouin mirror,” Sov. J. Quantum Electron. 9, 455–458 (1979).
[CrossRef]

Esipov, I. B.

I. B. Esipov, V. V. Zosimov, “Observation of partial phase conjugation in the case of reflection in a randomly in-homogeneous medium,” Opt. Spectrosc. (USSR) 60, 234–236 (1986).

Gower, M.

M. Gower, “The physics of phase conjugate mirrors,” Prog. Quantum Electron. 9, 101–147 (1985).
[CrossRef]

Green, R. P. M.

Gu, C.

C. Gu, P. Yeh, “Partial phase conjugation, fidelity, and reciprocity,” Opt. Commun. 107, 353–357 (1994).
[CrossRef]

Jakeman, E.

E. Jakeman, “Active imaging through a random phase screen,” J. Phys. D 24, 227–232 (1991).
[CrossRef]

E. Jakeman, “Enhanced backscattering through a deep random phase screen,” J. Opt. Soc. Am. A 5, 1638–1648 (1988).
[CrossRef]

See, for example, E. Jakeman, J. G. McWhirter, “Correlation function dependence of the scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

E. Jakeman, “The physical optics of enhanced backscatter-ing,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, Amsterdam, 1990), pp. 111–123.

K. D. Ridley, E. Jakeman, “Incomplete phase conjugation through a random-phase screen. II. Numerical simulations,” submitted to J. Opt. Soc. Am. A.

Jones, D. C.

Kotov, A. V.

See, for example, N. G. Basov, V. I. Efimkov, I. G. Zubarev, A. V. Kotov, A. B. Mironov, S. I. Mikhailov, M. G. Smirnov, “Influence of certain radiation parameters on wavefront reversal of a pump wave in a Brillouin mirror,” Sov. J. Quantum Electron. 9, 455–458 (1979).
[CrossRef]

Kravtsov, Yu. A.

Lind, R. C.

McWhirter, J. G.

See, for example, E. Jakeman, J. G. McWhirter, “Correlation function dependence of the scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

Mikhailov, S. I.

See, for example, N. G. Basov, V. I. Efimkov, I. G. Zubarev, A. V. Kotov, A. B. Mironov, S. I. Mikhailov, M. G. Smirnov, “Influence of certain radiation parameters on wavefront reversal of a pump wave in a Brillouin mirror,” Sov. J. Quantum Electron. 9, 455–458 (1979).
[CrossRef]

Mironov, A. B.

See, for example, N. G. Basov, V. I. Efimkov, I. G. Zubarev, A. V. Kotov, A. B. Mironov, S. I. Mikhailov, M. G. Smirnov, “Influence of certain radiation parameters on wavefront reversal of a pump wave in a Brillouin mirror,” Sov. J. Quantum Electron. 9, 455–458 (1979).
[CrossRef]

Pepper, D. M.

D. M. Pepper, “Nonlinear optical phase conjugation,” in Laser Handbook 4, M. L. Stitch, M. Bass, eds. (North-Holland, Amsterdam, 1985), pp. 333–486.

Pilepetsky, N. F.

B. Ya. Zel’dovich, N. F. Pilepetsky, V. V. Shkunov, Principles of Phase Conjugation, Vol. 42 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1985).

Ridley, K. D.

K. D. Ridley, E. Jakeman, “Incomplete phase conjugation through a random-phase screen. II. Numerical simulations,” submitted to J. Opt. Soc. Am. A.

K. D. Ridley, A. M. Scott, “Brillouin-induced four-wave mixing,” Chap. 3 of Ref. 3.

Rockwell, D. A.

Saichev, A. I.

Scott, A. M.

K. D. Ridley, A. M. Scott, “Brillouin-induced four-wave mixing,” Chap. 3 of Ref. 3.

Shkunov, V. V.

B. Ya. Zel’dovich, V. V. Shkunov, “Wavefront reproduction in stimulated Raman scattering,” Sov. J. Quantum Electron. 7, 610–615 (1977).
[CrossRef]

B. Ya. Zel’dovich, N. F. Pilepetsky, V. V. Shkunov, Principles of Phase Conjugation, Vol. 42 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1985).

Smirnov, M. G.

See, for example, N. G. Basov, V. I. Efimkov, I. G. Zubarev, A. V. Kotov, A. B. Mironov, S. I. Mikhailov, M. G. Smirnov, “Influence of certain radiation parameters on wavefront reversal of a pump wave in a Brillouin mirror,” Sov. J. Quantum Electron. 9, 455–458 (1979).
[CrossRef]

Yeh, P.

C. Gu, P. Yeh, “Partial phase conjugation, fidelity, and reciprocity,” Opt. Commun. 107, 353–357 (1994).
[CrossRef]

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, V. V. Shkunov, “Wavefront reproduction in stimulated Raman scattering,” Sov. J. Quantum Electron. 7, 610–615 (1977).
[CrossRef]

B. Ya. Zel’dovich, N. F. Pilepetsky, V. V. Shkunov, Principles of Phase Conjugation, Vol. 42 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1985).

Zosimov, V. V.

I. B. Esipov, V. V. Zosimov, “Observation of partial phase conjugation in the case of reflection in a randomly in-homogeneous medium,” Opt. Spectrosc. (USSR) 60, 234–236 (1986).

Zubarev, I. G.

See, for example, N. G. Basov, V. I. Efimkov, I. G. Zubarev, A. V. Kotov, A. B. Mironov, S. I. Mikhailov, M. G. Smirnov, “Influence of certain radiation parameters on wavefront reversal of a pump wave in a Brillouin mirror,” Sov. J. Quantum Electron. 9, 455–458 (1979).
[CrossRef]

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

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

J. Phys. A (1)

See, for example, E. Jakeman, J. G. McWhirter, “Correlation function dependence of the scintillation behind a deep random phase screen,” J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

J. Phys. D (1)

E. Jakeman, “Active imaging through a random phase screen,” J. Phys. D 24, 227–232 (1991).
[CrossRef]

Opt. Commun. (1)

C. Gu, P. Yeh, “Partial phase conjugation, fidelity, and reciprocity,” Opt. Commun. 107, 353–357 (1994).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (USSR) (1)

I. B. Esipov, V. V. Zosimov, “Observation of partial phase conjugation in the case of reflection in a randomly in-homogeneous medium,” Opt. Spectrosc. (USSR) 60, 234–236 (1986).

Prog. Quantum Electron. (1)

M. Gower, “The physics of phase conjugate mirrors,” Prog. Quantum Electron. 9, 101–147 (1985).
[CrossRef]

Sov. J. Quantum Electron. (2)

See, for example, N. G. Basov, V. I. Efimkov, I. G. Zubarev, A. V. Kotov, A. B. Mironov, S. I. Mikhailov, M. G. Smirnov, “Influence of certain radiation parameters on wavefront reversal of a pump wave in a Brillouin mirror,” Sov. J. Quantum Electron. 9, 455–458 (1979).
[CrossRef]

B. Ya. Zel’dovich, V. V. Shkunov, “Wavefront reproduction in stimulated Raman scattering,” Sov. J. Quantum Electron. 7, 610–615 (1977).
[CrossRef]

Other (6)

K. D. Ridley, A. M. Scott, “Brillouin-induced four-wave mixing,” Chap. 3 of Ref. 3.

D. M. Pepper, “Nonlinear optical phase conjugation,” in Laser Handbook 4, M. L. Stitch, M. Bass, eds. (North-Holland, Amsterdam, 1985), pp. 333–486.

B. Ya. Zel’dovich, N. F. Pilepetsky, V. V. Shkunov, Principles of Phase Conjugation, Vol. 42 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1985).

M. Gower, D. Proch, eds., Optical Phase Conjugation (Springer-Verlag, Berlin, 1994).

E. Jakeman, “The physical optics of enhanced backscatter-ing,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, Amsterdam, 1990), pp. 111–123.

K. D. Ridley, E. Jakeman, “Incomplete phase conjugation through a random-phase screen. II. Numerical simulations,” submitted to J. Opt. Soc. Am. A.

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Figures (3)

Fig. 1
Fig. 1

Geometry for incomplete phase conjugation through a thin phase screen. An incident field, which will usually be a high-quality beam, passes through a phase screen, after which part of it is conjugated by a PCM of limited aperture. The beam reflected from the PCM goes back through the phase screen, and the intensity profile is observed in a plane a distance R from the phase screen.

Fig. 2
Fig. 2

Geometry in which the phase conjugation takes place in a cylindrical region of aperture W and length L. Systems having different aperture sizes can be compared by assuming that a suitable imaging telescope of magnification M can be used to match the incident-beam size to the aperture size.

Fig. 3
Fig. 3

Plots of three different two-dimensional phase correlation functions: (a) correct function for a deeply modulated Gaussian phase screen, (b) approximation used in Eq. (5) with P = 1, (c) approximation used for the more accurate calculation in Appendix A.

Equations (35)

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E 3 ( r 3 ) = i k 2 π d d 2 r a E 2 ( r a ) × exp [ i ϕ ( r a ) + i k 2 d | r a r 3 | 2 ] .
E 1 ( r ) = i k 2 π R ( k 2 π d ) 2 d 2 r 3 d 2 r a d 2 r a E 2 * ( r a ) × exp { i [ ϕ ( r a ) ϕ ( r a ) ] | r 3 | 2 W 2 + i k 2 R | r a r | 2 + i k 2 d ( | r a r 3 | 2 | r a r 3 | 2 ) } ,
I 1 ( r ) = ( k 2 R ) 2 ( k W 2 π d ) 4 × d 2 r b d 2 r b d 2 r a d 2 r a E 2 * ( r a ) E 2 ( r b ) × B ( r a , r a , r b , r b ) exp [ i k 2 R ( | r a r | 2 | r b r | 2 ) + i k 2 d ( | r a | 2 | r a | 2 | r b | 2 + | r b | 2 ) k 2 W 2 4 d 2 ( | r a r a | 2 + | r b r b | 2 ) ] ,
B = exp { i [ ϕ ( r a ) ϕ ( r a ) + ϕ ( r b ) ϕ ( r b ) ] } .
B = exp { i [ ϕ ( r a ) ϕ ( r a ) ] } exp { i [ ϕ ( r b ) ϕ ( r b ) ] } + P exp { i [ ϕ ( r a ) ϕ ( r b ) ] } exp { i [ ϕ ( r a ) ϕ ( r b ) ] } .
ρ ( r ) = ϕ ( 0 ) ϕ ( r ) / ϕ 0 2 = exp ( | r | 2 / ξ 2 ) ,
exp { i [ ϕ ( 0 ) ϕ ( r ) ] } = exp ( m o 2 k 2 4 | r | 2 ) .
B = exp [ m o 2 k 2 4 ( | r a r a | 2 + | r b r b | 2 ) ] + P exp [ m o 2 k 2 4 ( | r a r b | 2 + | r a r b | 2 ) ] .
E 2 ( r a ) = σ 4 π R 2 exp ( i k 2 R | r a r s | 2 ) ,
I 1 ( r ) = σ W 2 8 ( R + d ) 2 ( 1 2 π ( k W R + d ) 2 × exp [ k 2 W 2 2 ( R + d ) 2 ( 1 + m o 2 d 2 W 2 ) | r r s | 2 ] + P 2 π [ m o 2 R 2 + ( R + d k W ) 2 ] × exp { | r r s | 2 2 [ m o 2 R 2 + ( R + d k W ) 2 ] } ) .
σ W 2 8 ( R + d ) 2 ( 1 1 + m o 2 d 2 W 2 + P ) .
P = 1 1 + W 2 m o 2 d 2 .
[ 1 + ( k m o R W R + d ) 2 ] ( 1 + W 2 m o 2 d 2 ) .
f ( r a r a ) = exp { i [ ϕ ( r a ) ϕ ( r a ) ] } .
E 1 ( r ) = i k W 2 4 ( R + d ) 2 σ π f ( r s r 1 + R / d ) × exp [ k 2 W 2 | r s r | 2 4 ( R + d ) 2 + i k ( | r s | 2 | r | 2 ) 2 ( R + d ) ] .
E 2 ( r a ) = A 0 exp ( k | r a | 2 2 z 0 ) ,
I c ( θ ) R 2 = | A 0 | 2 k 2 W 4 z 0 2 [ exp ( θ 2 / θ a 2 ) 4 d 2 + ( k W 2 + k m o 2 d 2 + 2 z 0 ) 2 + P exp ( θ 2 / θ b 2 ) 4 ( d 2 + z 0 2 ) + k 2 W 4 + 4 k m o 2 d 2 z 0 + 2 k 3 W 4 m o 2 z 0 + 2 k W 2 ( k m o 2 d 2 + 2 z 0 ) ( 1 + k m o 2 z 0 ) ] ,
θ a 2 = 2 k 2 ( m o 2 d 2 + W 2 ) + 1 k z 0 + 2 ( d 2 + z 0 2 ) m o 2 d 2 + W 2 [ 1 + 2 k z 0 k 2 ( m o 2 d 2 + W 2 ) ] ,
θ b 2 = 4 ( d 2 + z 0 2 ) + k 2 W 4 + 4 k m o 2 d 2 z 0 + 2 k 3 W 4 m o 2 z 0 + 2 k W 2 ( k m o 2 d 2 + 2 z 0 ) ( 1 + k m o 2 z 0 ) k 2 W 2 ( 2 d 2 + 2 z 0 2 + 2 k m o 2 d 2 z 0 + k W 2 z 0 ) .
P = 1 ( 1 + W 2 m o 2 d 2 ) [ 1 + k W 2 z 0 2 ( d 2 + z 0 2 ) + k m o 2 d 2 z 0 ] .
Coherent fraction = k W 2 z 0 ( 2 d 2 + 2 z 0 2 + 2 k m o 2 d 2 z 0 + k W 2 z 0 ) ( k m o 2 d 2 z 0 + k W 2 z 0 ) ( k m o 2 d 2 z 0 + k W 2 z 0 + 2 d 2 + 2 z 0 2 ) .
k 2 ( m o 2 d 2 + W 2 ) k z 0 2 ( d 2 + z 0 2 ) m o 2 d 2 + W 2 .
W 2 + m o 2 d 2 2 d 2 k z 0 + 2 ω 0 2 ω 0 2 2 .
Coherent fraction = 2 + 1 π 2 N + k z 0 m o 2 π 2 N ( 1 + k z 0 m o 2 2 π 2 N ) ( k z 0 m o 2 2 π 2 N + 2 + 1 π 2 N ) .
E c ( r , z ) = γ E * ( r , z ) + E * ( r , z ) ,
F = | E E c d 2 r | 2 | E | 2 d 2 r | E c | 2 d 2 r = | γ | 2 | E | 2 d 2 r | E c | 2 d 2 r .
F = | | E | 2 a ( r 3 ) d 2 r 3 | 2 | E | 2 d 2 r 3 | E | 2 | a ( r 3 ) | 2 d 2 r 3 .
| | E | 2 a ( r 3 ) d 2 r 3 | 2 | E | 2 d 2 r 3 | E | 2 | a ( r 3 ) | 2 d 2 r 3 .
k W 2 z 0 ( 2 d 2 + 2 z 0 2 + 2 k m o 2 d 2 z 0 + k W 2 z 0 ) ( d 2 + z 0 2 + k m o 2 d 2 z 0 + k W 2 z 0 ) 2 .
| | E | 2 a ( r 3 ) d 2 r 3 | 2 | E | 2 d 2 r 3 | E | 2 | a ( r 3 ) | 2 d 2 r 3
E c = α exp ( i φ ) ,
Reduction in fidelity = ( A I d A ) 2 A A I d A ,
I c ( r ) = σ 4 π [ k 2 W 2 4 π R d ( R + d ) ] 2 d 2 r 1 d 2 r 2 × exp [ i k R r 1 ( r 2 + r r s ) + i k 2 d r 1 r 2 k 2 W 2 2 d 2 | r 2 | 2 ] × exp { i [ ϕ ( 0 ) ϕ ( r 2 ) + ϕ ( r 1 + r 2 ) ϕ ( r 1 ) ] } .
B = exp { ϕ 0 2 [ 2 2 ρ ( x 1 ) 2 ρ ( x 2 ) + ρ ( x 1 + x 2 ) + ρ ( x 1 x 2 ) ] } .
k ξ 2 ϕ 0 ( 1 R + 1 d ) 1 ,

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