Abstract

The problem of adaptive laser beam projection onto an extended object (target) having a randomly rough surface in an optically inhomogeneous medium (atmosphere) is analyzed. Outgoing beam precompensation is considered through conjugation of either the target-return wave phase or the complex field. It is shown that in the presence of “frozen” turbulence, both phase-conjugate (PC) and field-conjugate (FC) precompensation can result in a superfocusing effect, which suggests the possibility of achieving a brighter target hit spot in volume turbulence than in vacuum. This superfocusing effect is significantly more distinct for FC precompensation. In the quasi-stationary case (slowly moving turbulence or target), PC and FC beam control lead to enhanced intensity fluctuations at the target surface associated with intermittent formation and disintegration of bright target hit spots that sporadically attach to the extended target surface. This intensity fluctuation level exceeds intensity fluctuations in the absence of beam control and is higher for FC precompensation. In the nonstationary case, both PC and FC lead to an increase of beam width and centroid wander at the extended target surface compared with conventional projection of a collimated or focused beam.

© 2007 Optical Society of America

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  7. F. Schonfeld, "Instability in saturated full-field compensation for thermal blooming," J. Opt. Soc. Am. B 9, 1794-1799 (1992).
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  11. F. Yu. Kanev and V. P. Lukin, "Amplitude phase beam control with the help of a two-mirror adaptive system," Atmosph. Opt. 4, 878-881 (1991).
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    [CrossRef]
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    [CrossRef]
  18. M. C. Rytov, Yu A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics IV: Wave Propagation through Random Media (Springer, 1989).
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  22. In the case of a point-source target, the pupil-plane phase screen and an outgoing beam with spatially uniform intensity distribution (flat-top beam), phase and field conjugation are equivalent.
  23. This implies that measurement of phase φ(r,t) is performed over a time τph (sensor's photo-receiver integration time) shorter than the characteristic times τs and τat for target surface roughness and propagation medium refractive index realization update (τph<min{τs,τat}).
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    [CrossRef]
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    [CrossRef]
  28. The time delay ΔtC includes time required for wavefront phase measurement and the outgoing beam phase control.
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    [CrossRef]
  30. M. A. Vorontsov, "Phase conjugation method for thermal blooming compensation of light beams," Sov. J. Quantum Electron. 9, 1221-1233 (1979).
    [CrossRef]
  31. V. V. Kolosov and S. I. Sysoev, "Analysis of some algorithms for minimizing angular divergence of partially coherent optical radiation," Atmosph. Opt. 3, 70-76 (1990).
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  34. For a nearly diffraction-limited hit-spot size, the speckle size (radius) asp≃a0 for a Gaussian beam. The speckle size is 1.22π times smaller for a flat-top beam.
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    [CrossRef]
  38. V. V. Kolosov and S. I. Sysoev, "Minimization of angular characteristics of partially coherent optical radiation," Atmosph. Opt. 2, 297-301 (1989).
  39. V. V. Dudorov, M. A. Vorontsov, and V. V. Kolosov, "Speckle-field propagation in 'frozen' turbulence: brightness function approach," J. Opt. Soc. Am. A 23, 1924-1936 (2006).
    [CrossRef]
  40. M. I. Charnotskii, V. A. Myakinin, and V. Zavorotnyy, "Observation of superresolution in nonisoplanatic imaging through turbulence," J. Opt. Soc. Am. A 7, 1345-1350 (1990).
    [CrossRef]
  41. M. I. Charnotskii, "Imaging in turbulence beyond the diffraction limit," Proc. SPIE 2534, 289-293 (1995).
    [CrossRef]
  42. C. C. Yang and M. A. Plonus, "Superresolution effects in weak turbulence," Appl. Opt. 32, 7528-7531 (1993).
    [CrossRef] [PubMed]
  43. M. A. Vorontsov and G. W. Carhart, "Anisoplanatic imaging through turbulent media: image recovery by local information fusion from a set of short-exposure images," J. Opt. Soc. Am. A 18, 1312-1324 (2001).
    [CrossRef]
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    [CrossRef]
  45. T. R. O'Meara, "The multi-dither principle in adaptive optics," J. Opt. Soc. Am. 67, 306-315 (1977).
    [CrossRef]
  46. M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, and D. G. Voelz, "Image quality criteria for an adaptive imaging system based on statistical analysis of the speckle field," J. Opt. Soc. Am. A 13, 1456-1466 (1996).
    [CrossRef]
  47. J. W. Goodman, Statistical Optics (Wiley, 1985).
  48. T. Weyrauch and M. A. Vorontsov, "Dynamic wave-front distortion compensation with 134-control-channel sub-millisecond adaptive system," Opt. Lett. 27, 751-753 (2002).
    [CrossRef]
  49. E. Polak, "An historical survey of computational methods in optimal control," SIAM Rev. 15, 553-584 (1973).
    [CrossRef]
  50. J. C. Dunn, "Convergence rates for conditional gradient sequences generated by implicit length rules," SIAM J. Control Optim. 18, 473-487 (1980).
    [CrossRef]

2006 (1)

2005 (2)

B. Hatfield and S. Enguehard, Adaptive Optics (Perseus, 2005).

M. A. Vorontsov and V. V. Kolosov, "Target-in-the-loop beam control: basic considerations for analysis and wavefront sensing," J. Opt. Soc. Am. A 21, 126-141 (2005).
[CrossRef]

2004 (1)

Phase Conjugate Laser Optics, A.Brignon and J.P.Huignard, eds., Wiley Series in Lasers and Applications (Wiley, 2004).

2002 (3)

2001 (1)

2000 (1)

1999 (1)

S. Haykin, Neural Networks: A Comprehensive Foundation2nd ed. (Prentice Hall, 1999).

1998 (3)

1996 (1)

1995 (1)

M. I. Charnotskii, "Imaging in turbulence beyond the diffraction limit," Proc. SPIE 2534, 289-293 (1995).
[CrossRef]

1993 (2)

1992 (1)

1991 (2)

F. Yu. Kanev and V. P. Lukin, "Amplitude phase beam control with the help of a two-mirror adaptive system," Atmosph. Opt. 4, 878-881 (1991).

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).

1990 (2)

V. V. Kolosov and S. I. Sysoev, "Analysis of some algorithms for minimizing angular divergence of partially coherent optical radiation," Atmosph. Opt. 3, 70-76 (1990).

M. I. Charnotskii, V. A. Myakinin, and V. Zavorotnyy, "Observation of superresolution in nonisoplanatic imaging through turbulence," J. Opt. Soc. Am. A 7, 1345-1350 (1990).
[CrossRef]

1989 (2)

V. V. Kolosov and S. I. Sysoev, "Minimization of angular characteristics of partially coherent optical radiation," Atmosph. Opt. 2, 297-301 (1989).

M. C. Rytov, Yu A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics IV: Wave Propagation through Random Media (Springer, 1989).

1987 (1)

A. A. Vasil'ev, M. A. Vorontsov, I. A. Kudryashov, and V. I. Shmalhauzen, "Adaptive focusing of radiation on a diffusely scattering reflector under nonlinear refraction conditions," Sov. J. Quantum Electron. 17, 1106-1115 (1987).
[CrossRef]

1985 (4)

N. V. Vysotina, N. N. Rozanov, V. E. Semenov, and V. A. Smirnov, "Amplitude-phase adaptive control over optically inhomogeneous paths with deformable mirrors," Izv. Vyssh. Uchebn. Zaved. Fiz. 11, 42-50 (1985).

B. Ya. Zeldovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer, 1985).

M. A. Vorontsov and V. I. Shmal'hauzen, Principles of Adaptive Optics, (Nauka, Moscow, 1985).

J. W. Goodman, Statistical Optics (Wiley, 1985).

1984 (1)

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz'minskii, and V. I. Shmalhauzen, "Speckle effects in adaptive optical systems," Sov. J. Quantum Electron. 14, 761-766 (1984).
[CrossRef]

1983 (1)

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

1981 (1)

V. V. Kolosov and A. V. Kuzikovskii, "Phase compensation for refractive distortions of partially coherent beam," Sov. J. Quantum Electron. 11, 301-303 (1981).
[CrossRef]

1980 (3)

M. A. Vorontsov and V. I. Shmal'hauzen, "Interference criteria for the light focusing problem," Sov. J. Quantum Electron. 10, 285-289 (1980).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

J. C. Dunn, "Convergence rates for conditional gradient sequences generated by implicit length rules," SIAM J. Control Optim. 18, 473-487 (1980).
[CrossRef]

1979 (2)

M. A. Vorontsov, "Phase conjugation method for thermal blooming compensation of light beams," Sov. J. Quantum Electron. 9, 1221-1233 (1979).
[CrossRef]

F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, 1979).

1978 (1)

J. W. Hardy, "Active optics: a new technology for the control of light," Proc. IEEE 66, 651-697 (1978).
[CrossRef]

1977 (3)

1976 (1)

1974 (1)

1973 (1)

E. Polak, "An historical survey of computational methods in optimal control," SIAM Rev. 15, 553-584 (1973).
[CrossRef]

1971 (1)

1965 (1)

Barchers, J. D.

Bass, F. G.

F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, 1979).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Brandewie, R. A.

Buffington, A.

Carhart, G. W.

Charnotskii, M. I.

Davis, W. C.

Dudorov, V. V.

Dunn, J. C.

J. C. Dunn, "Convergence rates for conditional gradient sequences generated by implicit length rules," SIAM J. Control Optim. 18, 473-487 (1980).
[CrossRef]

Enguehard, S.

B. Hatfield and S. Enguehard, Adaptive Optics (Perseus, 2005).

Fried, D. L.

Fuks, I. M.

F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, 1979).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).

J. W. Hardy, "Active optics: a new technology for the control of light," Proc. IEEE 66, 651-697 (1978).
[CrossRef]

Hatfield, B.

B. Hatfield and S. Enguehard, Adaptive Optics (Perseus, 2005).

Hayes, C. L.

Haykin, S.

S. Haykin, Neural Networks: A Comprehensive Foundation2nd ed. (Prentice Hall, 1999).

Kanev, F. Yu.

F. Yu. Kanev and V. P. Lukin, "Amplitude phase beam control with the help of a two-mirror adaptive system," Atmosph. Opt. 4, 878-881 (1991).

Karnaukhov, V. N.

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz'minskii, and V. I. Shmalhauzen, "Speckle effects in adaptive optical systems," Sov. J. Quantum Electron. 14, 761-766 (1984).
[CrossRef]

Koivunen, A. C.

Kokorovski, S. A.

Kokorowski, S. A.

Kolosov, V. V.

V. V. Dudorov, M. A. Vorontsov, and V. V. Kolosov, "Speckle-field propagation in 'frozen' turbulence: brightness function approach," J. Opt. Soc. Am. A 23, 1924-1936 (2006).
[CrossRef]

M. A. Vorontsov and V. V. Kolosov, "Target-in-the-loop beam control: basic considerations for analysis and wavefront sensing," J. Opt. Soc. Am. A 21, 126-141 (2005).
[CrossRef]

V. V. Kolosov and S. I. Sysoev, "Analysis of some algorithms for minimizing angular divergence of partially coherent optical radiation," Atmosph. Opt. 3, 70-76 (1990).

V. V. Kolosov and S. I. Sysoev, "Minimization of angular characteristics of partially coherent optical radiation," Atmosph. Opt. 2, 297-301 (1989).

V. V. Kolosov and A. V. Kuzikovskii, "Phase compensation for refractive distortions of partially coherent beam," Sov. J. Quantum Electron. 11, 301-303 (1981).
[CrossRef]

Kravtsov, Y. A.

Kravtsov, Yu A.

M. C. Rytov, Yu A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics IV: Wave Propagation through Random Media (Springer, 1989).

Kudryashov, I. A.

A. A. Vasil'ev, M. A. Vorontsov, I. A. Kudryashov, and V. I. Shmalhauzen, "Adaptive focusing of radiation on a diffusely scattering reflector under nonlinear refraction conditions," Sov. J. Quantum Electron. 17, 1106-1115 (1987).
[CrossRef]

Kuzikovskii, A. V.

V. V. Kolosov and A. V. Kuzikovskii, "Phase compensation for refractive distortions of partially coherent beam," Sov. J. Quantum Electron. 11, 301-303 (1981).
[CrossRef]

Kuz'minskii, A. L.

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz'minskii, and V. I. Shmalhauzen, "Speckle effects in adaptive optical systems," Sov. J. Quantum Electron. 14, 761-766 (1984).
[CrossRef]

Lee, D. J.

Lukin, V. P.

F. Yu. Kanev and V. P. Lukin, "Amplitude phase beam control with the help of a two-mirror adaptive system," Atmosph. Opt. 4, 878-881 (1991).

Mevers, G. E.

Miller, R. A.

Myakinin, V. A.

O'Meara, T. R.

Pearson, J. E.

Pedinoff, M. E.

Pilipetsky, N. F.

B. Ya. Zeldovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer, 1985).

Plonus, M. A.

Polak, E.

E. Polak, "An historical survey of computational methods in optimal control," SIAM Rev. 15, 553-584 (1973).
[CrossRef]

Pruidze, D. V.

Ricklin, J. C.

Roggemann, M. C.

Rozanov, N. N.

N. V. Vysotina, N. N. Rozanov, V. E. Semenov, and V. A. Smirnov, "Amplitude-phase adaptive control over optically inhomogeneous paths with deformable mirrors," Izv. Vyssh. Uchebn. Zaved. Fiz. 11, 42-50 (1985).

Rytov, M. C.

M. C. Rytov, Yu A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics IV: Wave Propagation through Random Media (Springer, 1989).

Schonfeld, F.

Semenov, V. E.

N. V. Vysotina, N. N. Rozanov, V. E. Semenov, and V. A. Smirnov, "Amplitude-phase adaptive control over optically inhomogeneous paths with deformable mirrors," Izv. Vyssh. Uchebn. Zaved. Fiz. 11, 42-50 (1985).

Shapiro, J. H.

Shkunov, V. V.

B. Ya. Zeldovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer, 1985).

Shmalhauzen, V. I.

A. A. Vasil'ev, M. A. Vorontsov, I. A. Kudryashov, and V. I. Shmalhauzen, "Adaptive focusing of radiation on a diffusely scattering reflector under nonlinear refraction conditions," Sov. J. Quantum Electron. 17, 1106-1115 (1987).
[CrossRef]

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz'minskii, and V. I. Shmalhauzen, "Speckle effects in adaptive optical systems," Sov. J. Quantum Electron. 14, 761-766 (1984).
[CrossRef]

Shmal'hauzen, V. I.

M. A. Vorontsov and V. I. Shmal'hauzen, Principles of Adaptive Optics, (Nauka, Moscow, 1985).

M. A. Vorontsov and V. I. Shmal'hauzen, "Interference criteria for the light focusing problem," Sov. J. Quantum Electron. 10, 285-289 (1980).
[CrossRef]

Smirnov, V. A.

N. V. Vysotina, N. N. Rozanov, V. E. Semenov, and V. A. Smirnov, "Amplitude-phase adaptive control over optically inhomogeneous paths with deformable mirrors," Izv. Vyssh. Uchebn. Zaved. Fiz. 11, 42-50 (1985).

Sysoev, S. I.

V. V. Kolosov and S. I. Sysoev, "Analysis of some algorithms for minimizing angular divergence of partially coherent optical radiation," Atmosph. Opt. 3, 70-76 (1990).

V. V. Kolosov and S. I. Sysoev, "Minimization of angular characteristics of partially coherent optical radiation," Atmosph. Opt. 2, 297-301 (1989).

Tatarskii, V. I.

M. C. Rytov, Yu A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics IV: Wave Propagation through Random Media (Springer, 1989).

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).

Vasil'ev, A. A.

A. A. Vasil'ev, M. A. Vorontsov, I. A. Kudryashov, and V. I. Shmalhauzen, "Adaptive focusing of radiation on a diffusely scattering reflector under nonlinear refraction conditions," Sov. J. Quantum Electron. 17, 1106-1115 (1987).
[CrossRef]

Voelz, D. G.

Vorontsov, M. A.

V. V. Dudorov, M. A. Vorontsov, and V. V. Kolosov, "Speckle-field propagation in 'frozen' turbulence: brightness function approach," J. Opt. Soc. Am. A 23, 1924-1936 (2006).
[CrossRef]

M. A. Vorontsov and V. V. Kolosov, "Target-in-the-loop beam control: basic considerations for analysis and wavefront sensing," J. Opt. Soc. Am. A 21, 126-141 (2005).
[CrossRef]

T. Weyrauch and M. A. Vorontsov, "Dynamic wave-front distortion compensation with 134-control-channel sub-millisecond adaptive system," Opt. Lett. 27, 751-753 (2002).
[CrossRef]

M. A. Vorontsov and G. W. Carhart, "Adaptive phase distortion correction in strong speckle-modulation conditions," Opt. Lett. 27, 2155-2157 (2002).
[CrossRef]

M. A. Vorontsov and G. W. Carhart, "Anisoplanatic imaging through turbulent media: image recovery by local information fusion from a set of short-exposure images," J. Opt. Soc. Am. A 18, 1312-1324 (2001).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, and D. G. Voelz, "Image quality criteria for an adaptive imaging system based on statistical analysis of the speckle field," J. Opt. Soc. Am. A 13, 1456-1466 (1996).
[CrossRef]

A. A. Vasil'ev, M. A. Vorontsov, I. A. Kudryashov, and V. I. Shmalhauzen, "Adaptive focusing of radiation on a diffusely scattering reflector under nonlinear refraction conditions," Sov. J. Quantum Electron. 17, 1106-1115 (1987).
[CrossRef]

M. A. Vorontsov and V. I. Shmal'hauzen, Principles of Adaptive Optics, (Nauka, Moscow, 1985).

M. A. Vorontsov, V. N. Karnaukhov, A. L. Kuz'minskii, and V. I. Shmalhauzen, "Speckle effects in adaptive optical systems," Sov. J. Quantum Electron. 14, 761-766 (1984).
[CrossRef]

M. A. Vorontsov and V. I. Shmal'hauzen, "Interference criteria for the light focusing problem," Sov. J. Quantum Electron. 10, 285-289 (1980).
[CrossRef]

M. A. Vorontsov, "Phase conjugation method for thermal blooming compensation of light beams," Sov. J. Quantum Electron. 9, 1221-1233 (1979).
[CrossRef]

Vysotina, N. V.

N. V. Vysotina, N. N. Rozanov, V. E. Semenov, and V. A. Smirnov, "Amplitude-phase adaptive control over optically inhomogeneous paths with deformable mirrors," Izv. Vyssh. Uchebn. Zaved. Fiz. 11, 42-50 (1985).

Weyrauch, T.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Yang, C. C.

Zavorotnyy, V.

Zeldovich, B. Ya.

B. Ya. Zeldovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer, 1985).

Appl. Opt. (3)

Atmosph. Opt. (3)

V. V. Kolosov and S. I. Sysoev, "Minimization of angular characteristics of partially coherent optical radiation," Atmosph. Opt. 2, 297-301 (1989).

F. Yu. Kanev and V. P. Lukin, "Amplitude phase beam control with the help of a two-mirror adaptive system," Atmosph. Opt. 4, 878-881 (1991).

V. V. Kolosov and S. I. Sysoev, "Analysis of some algorithms for minimizing angular divergence of partially coherent optical radiation," Atmosph. Opt. 3, 70-76 (1990).

Izv. Vyssh. Uchebn. Zaved. Fiz. (1)

N. V. Vysotina, N. N. Rozanov, V. E. Semenov, and V. A. Smirnov, "Amplitude-phase adaptive control over optically inhomogeneous paths with deformable mirrors," Izv. Vyssh. Uchebn. Zaved. Fiz. 11, 42-50 (1985).

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (8)

M. C. Roggemann and A. C. Koivunen, "Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction," J. Opt. Soc. Am. A 17, 53-62 (2000).
[CrossRef]

M. A. Vorontsov and G. W. Carhart, "Anisoplanatic imaging through turbulent media: image recovery by local information fusion from a set of short-exposure images," J. Opt. Soc. Am. A 18, 1312-1324 (2001).
[CrossRef]

J. D. Barchers, "Closed-loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations," J. Opt. Soc. Am. A 19, 926-945 (2002).
[CrossRef]

M. A. Vorontsov and V. V. Kolosov, "Target-in-the-loop beam control: basic considerations for analysis and wavefront sensing," J. Opt. Soc. Am. A 21, 126-141 (2005).
[CrossRef]

D. L. Fried, "Branch point problem in adaptive optics," J. Opt. Soc. Am. A 15, 2759-2768 (1998).
[CrossRef]

M. I. Charnotskii, V. A. Myakinin, and V. Zavorotnyy, "Observation of superresolution in nonisoplanatic imaging through turbulence," J. Opt. Soc. Am. A 7, 1345-1350 (1990).
[CrossRef]

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Other (16)

The time delay ΔtC includes time required for wavefront phase measurement and the outgoing beam phase control.

S. Haykin, Neural Networks: A Comprehensive Foundation2nd ed. (Prentice Hall, 1999).

For a nearly diffraction-limited hit-spot size, the speckle size (radius) asp≃a0 for a Gaussian beam. The speckle size is 1.22π times smaller for a flat-top beam.

F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, 1979).

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).

M. C. Rytov, Yu A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics IV: Wave Propagation through Random Media (Springer, 1989).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

In the case of a point-source target, the pupil-plane phase screen and an outgoing beam with spatially uniform intensity distribution (flat-top beam), phase and field conjugation are equivalent.

This implies that measurement of phase φ(r,t) is performed over a time τph (sensor's photo-receiver integration time) shorter than the characteristic times τs and τat for target surface roughness and propagation medium refractive index realization update (τph<min{τs,τat}).

M. A. Vorontsov and V. I. Shmal'hauzen, Principles of Adaptive Optics, (Nauka, Moscow, 1985).

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

B. Ya. Zeldovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer, 1985).

Phase Conjugate Laser Optics, A.Brignon and J.P.Huignard, eds., Wiley Series in Lasers and Applications (Wiley, 2004).

B. Hatfield and S. Enguehard, Adaptive Optics (Perseus, 2005).

J. W. Goodman, Statistical Optics (Wiley, 1985).

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Figures (13)

Fig. 1
Fig. 1

Schematic diagram of PC precompensation of the outgoing beam for a pupil-plane phase distorting layer and point-source (a), (b) and extended (c), (d) targets. In (a) and (c) target illumination is shown prior to PC compensation, and in (b) and (d) after a single PC compensation. The phase conjugation in (b) results in compensation of the phase aberration φ a t ( r , t ) and formation of the convergvent undistorted wavefront φ q ( r , t ) . Conjugation of the phase φ ( r , t ) in (d) results in the aberrated outgoing wave phase φ s ( r , t ) .

Fig. 2
Fig. 2

Efficiency of PC and FC precompensation of an (a) initially collimated and (b) focused outgoing flat-top beam in vacuum for two different target surface roughness realizations (A and B): (a) normalized dependence of the target-plane metric J ̂ J J dif [ J ̂ is J ̂ 2 J 2 J 2 dif , or I ̂ max I max I max dif , or J ̂ int J int J int dif ] on the PC iteration number n; (b) metric J ̂ 2 evolution curves for PC and FC precompensation. The gray-scale images are: (c) target-plane intensity distributions for the collimated (center) and optimally focused (inset) flat-top beams; (d), (f) return speckle-field intensity and (e), (g) phase for a (d), (e) collimated and (f), (g) focused beam at n = 0 . The transceiver aperture of radius a 0 is indicated by a dashed circle. The propagation distance L = 0.05 k a 0 2 .

Fig. 3
Fig. 3

Metric J 2 evolution curves averaged over a set of 100 different surface roughness realizations for PC and FC precompensation (curves with vertical bars) for stationary and nonstationary targets (in vacuum): (a) initially collimated and (b) focused beam. The length of the error bars corresponds to the standard deviation of metric J 2 fluctuations. The propagation distance L = 0.05 k a 0 2 .

Fig. 4
Fig. 4

PC precompensation of a flat-top outgoing beam for a target with stationary Lambertian surface in vacuum: target plane (top row) and return-field (middle row) intensities, and the return-field phase (bottom row) for iteration numbers n = 1 (left column), n = 5 (central column), and n = 50 (right column). The surface roughness realization corresponds to case A in Fig. 2a.

Fig. 5
Fig. 5

Laser beam attachment to a moving Lambertian target surface during PC precompensation in vacuum. The length of the vertical line corresponds to the transceiver aperture radius a 0 . The dashed horizontal line shows the initial position of the bright-spot centroid at n = 24 .

Fig. 6
Fig. 6

Laser beam projection onto a quasi-stationary moving Lambertian surface in a vacuum using (a) PC and (b) FC precompensation of the outgoing flat-top collimated beam. The propagation distance is L = 0.05 k a 0 2 .

Fig. 7
Fig. 7

PC precompensation in a layered phase-distorting medium: target-plane metric J 2 averaged over atmospheric turbulence and surface roughness realizations versus iteration number n for (a) an initially focused beam with R 0 = L , and (b) initially collimated beams. Atmospheric turbulence is modeled using a set of 10 Kolmogorov phase-screens characterized by the ratio D r 0 , where D = 2 a 0 is the flat-top beam diameter. The propagation distance is L = 0.05 z d ( z d = k a 0 2 ) . Computer simulations are performed using a 512 × 512 numerical grid. The metric standard deviation is shown in (a) by vertical bars; error bars are shown for D r 0 = 0 in the positive direction only and for D r 0 = 8 in the negative direction only.

Fig. 8
Fig. 8

FC precompensation of an initially focused beam in a layered phase-distorting medium. Averaged beam quality metric J 2 versus iteration number n: (a) for 0 D r 0 16 and L = 0.05 z d ; (b) for L = 0.05 z d and D r 0 16 (dashed curves) and for D r 0 = 16 and L 0.05 z d (solid curves). The dotted line corresponds to the diffraction-limited value. Numerical simulation parameters are the same as in Fig. 7. The curve L = 0.05 z d and D r 0 = 16 is repeated in (a) and (b) for convenience. The metric standard deviation is shown in (a) by vertical bars; error bars are shown for D r 0 = 16 in the positive direction only and for D r 0 = 0 in the negative direction only.

Fig. 9
Fig. 9

Superfocusing effect in a volume turbulence. Top panel, instantaneous beam quality metric I ̂ max I max I max dif (solid curves) and J ̂ 2 J 2 J 2 dif (dashed curves) evolution curves selected from 600 FC (curves 1 and 2) and PC (curves 3) precompensation trials for different values of D r 0 and propagation distances L z d ( z d = k a 0 2 ) . Curve parameters are (1) D r 0 = 16 and L z d = 0.1 , (2) D r 0 = 40 and L z d = 0.05 , and (3) D r 0 = 8 and L z d = 0.05 . Numerical simulation parameters are the same as in Figs. 8, 7. Gray-scale images (a) through (f) are the intensity distribution patterns I ( r , z ) along the propagation path of length L = 0.05 z d for two different “lucky” FC trials with D r 0 = 16 : (a)–(c) images correspond to the first and (d)–(f) images to the second trial: z L = 0.1 for (a) and (d), z L = 0.7 for (b) and (e), and z = L for (c) and (f). The achieved metric values are J ̂ 2 = 2.6 , I ̂ max = 3.1 for (c), and J ̂ 2 = 0.92 , I ̂ max = 1.47 for (f). The insets in (a)–(c) are the intensity patterns for propagation in vacuum (optimally focused, flat-top beam of radius a 0 ).

Fig. 10
Fig. 10

Probability distribution functions (a)–(c) p I ( I ̂ max ) and (d) p J ( J ̂ 2 ) for the target plane metrics I ̂ max = I max I max dif and J ̂ 2 = J 2 J 2 dif obtained using N = 1200 FC precompensation trials for the extended target. The propagation distance is L = 0.05 z d .

Fig. 11
Fig. 11

Schematic illustration of the FC-induced waveguide formation between the transceiver and an extended target surface: wave propagation geometry (a) at the beginning of FC pre-compensation and (b) after FC process convergence. In (a) target hit spots A and C result in speckle structure with speckles B, D, and E inside the transceiver aperture. The FC-induced waveguide in (b) is formed between hit spot A and speckle B.

Fig. 12
Fig. 12

Temporal dependence of the normalized target-plane peak intensity I ̂ max = I max I max dif and hit-spot centroid coordinates x ̂ c = x c a 0 , y ̂ c = y c a 0 for (a), (b), (d) PC and (c) FC precompensation in a layered phase-distorting medium with ten equidistant Kolmogorov phase screens. Distorting layers ( D r 0 = 16 ) are moving in the x direction with velocity v ( z ) = v 0 = const. for (a) and (c), and with velocity v ( z ) = z v 0 L (tracking regime) for (b) and (d). The flat-top laser beam of radius a 0 propagates a distance L = 0.05 k a 0 2 to an extended stationary target with a Lambertian surface roughness in (a)–(c), and with a flat mirror-like surface in (d). The outgoing beam phase prior to PC/FC precompensation corresponds to an optimally focused beam in vacuum. The dashed curve in (d) corresponds to a focused beam (no control). For all cases Δ t ap Δ t C = 512 .

Fig. 13
Fig. 13

Temporal dependence of the normalized target-plane peak intensity I ̂ max = I max I max dif for FC precompensation in a layered phase-distorting medium moving along the x axis with velocity v ( z ) = z v 0 L (tracking regime) for Δ t ap Δ t C = (a) 512, (b) 256, (c) 64, (d) 32 (solid curve) and 16 (dashed curve). Propagation parameters are the same as in Fig. 12. The horizontal dashed lines in (a)–(c) indicate the diffraction-limited peak intensity level.

Equations (23)

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u opt ( r , t ) = φ at ( r , t ) φ q ( r ) ,
2 i k A ( r , z , t ) z = 2 A ( r , z , t ) + 2 k 2 n 1 ( r , z , t ) A ( r , z , t ) ,
2 i k ψ ( r , z , t ) z = 2 ψ ( r , z , t ) + 2 k 2 n 1 ( r , z , t ) ψ ( r , z , t ) ,
ψ ( r , z = L , t ) = T ( r , t ) A ( r , z = L , t ) ,
A ( r , z = 0 , t n + 1 ) = P ( r ) A 0 ( r ) exp [ i arg ψ ( r , z = 0 , t n ) ]
A ( r , z = 0 , t n + 1 ) = α P ( r ) ψ * ( r , z = 0 , t n )
J int = T ( r ) A 2 ( r , z = L ) d 2 r .
J int = T ( r ) A 2 ( r , L ) d 2 r = A ( r , L ) ψ ( r , L ) d 2 r = A ( r , 0 ) ψ ( r , 0 ) d 2 r
J int = P ( r ) A 0 ( r ) ψ ( r , 0 ) exp { i [ u ( r ) + φ ( r ) ] } d 2 r .
J 2 = I T 2 ( r ) d 2 r ,
J int ( n ) = I T ( r , t n ) exp [ i φ ( r , t n ) + φ ψ ( r , t n ) ] d 2 r ,
u ( r , t ) = j = 1 N a j ( t ) S j ( r ) , φ ( r , t ) = j = 1 N b j ( t ) S j ( r ) ,
J int = J int ( a 1 , , a N ; b 1 , , b N ) .
J ̂ int [ u ( r ) ] = T ( r ) A 2 ( r , z = L ) d 2 r .
Δ J [ u ] δ J [ u ] = 2 Re T ( r ) A ( r , z = L ) δ A ( r , z = L ) d 2 r 2 Re ( ψ δ A ) z = L d 2 r ,
δ J = 2 Re ( ψ δ A ) z = L d 2 r = 2 Re 0 L d z z [ ψ ( r , z ) δ A ( r , z ) ] d 2 r + 2 Re ( ψ δ A ) z = 0 d 2 r ,
δ J = 2 Re P ( r ) A 0 ( r ) ψ ( r , 0 ) exp [ i φ ( r ) ] { exp [ i u ( r ) + i h ( r ) ] exp [ i u ( r ) ] } d 2 r ,
max ν δ J [ ν ] = 2 max ν Re P ( r ) A 0 ( r ) ψ ( n ) ( r , 0 ) exp i φ ( n ) ( r ) [ exp i v ( r ) exp i u ( n ) ( r ) ] d 2 r ,
max v P ( r ) A 0 ( r ) ψ ( n ) ( r , 0 ) cos [ v ( r ) + φ ( n ) ( r ) ] d 2 r .
v ( r ) = u ( n + 1 ) ( r ) = φ ( n ) ( r ) + 2 m π .
2 i k [ ψ A z + A ψ z ] = ψ 2 A A 2 ψ .
2 i k ( A ψ ) z d 2 r = [ ψ 2 A A 2 ψ ] d 2 r .
z A ( r , z ) ψ ( r , z ) d 2 r = 0 .

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