Abstract

Adaptive optical systems for laser beam projection onto an extended target embedded in an optically inhomogeneous medium are considered. A new adaptive optics wavefront control technique—speckle-average (SA) phase conjugation—is introduced. In this technique mitigation of speckle effects related to laser beam scattering off the rough target surface is achieved by measuring the SA wavefront slopes of the target return wave using a conventional Shack–Hartmann wavefront sensor. For statistically representative speckle averaging we consider the generation of an incoherent light source, referred to here as a Collett–Wolf beacon, directly on the target surface using a rapid steering (scanning) auxiliary laser beam. Our numerical simulations and experiment show that control of the outgoing beam phase using SA phase conjugation can lead to efficient compensation of turbulence effects and results in an increase of the projected laser beam power density on a remote extended target. The impact of both target anisoplanatism and the Collett–Wolf beacon size on adaptive system performance is studied.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. 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).
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    [CrossRef]
  22. R. R. Parenti and R. J. Sasiela, “Laser-guide-star systems for astronomical applications,” J. Opt. Soc. Am. A 11, 288-309(1994).
    [CrossRef]
  23. D. L. Fried and J. F. Belsher, “Analysis of fundamental limits of artificial-guide-star adaptive-optics system performance for astronomical imaging,” J. Opt. Soc. Am. A. 11, 277-287 (1994).
    [CrossRef]
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    [CrossRef]
  28. G. Rousset, “Wave-front sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge University Press, 1999), pp. 91-130.
    [CrossRef]
  29. We assume here that the obtained vector-function α(r,t) represents a path-independent vector field (at least at compact areas of receiver aperture).
  30. M. C. Rytov, Yu A. Kravtsov, and V. I. Tatarskii, eds., Principles of Statistical Radiophysics 4, Wave Propagation Through Random Media (Springer-Verlag, 1989).
  31. F. G. Bass and I. M. Fuks, eds., Wave Scattering from Statistically Rough Surfaces (Pergamon, 1979).
  32. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).
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    [CrossRef]
  35. J. C. Dainty, ed., Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, 1984).
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    [CrossRef]
  37. M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).
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    [CrossRef]
  39. B. M. Welsh and C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69-80 (1991).
    [CrossRef]

2007 (2)

M. A. Vorontsov, V. V. Kolosov, and A. Kohnle, “Adaptive laser beam projection on an extended target: phase- and field-conjugate precompensation,” J. Opt. Soc. Am. A 24, 1975-1993 (2007).
[CrossRef]

P. Piatrou and M. Roggemann, “Beaconless stochastic parallel gradient descent laser beam control: numerical experiments,” Appl. Opt. 46, 6831-6842 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (1)

2002 (2)

1998 (1)

1996 (1)

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]

1994 (3)

N. C. Mehta and C. W. Allen, “Dynamic compensation of atmospheric turbulence with far-field optimization,” J. Opt. Soc. Am. A 11, 434-443 (1994).
[CrossRef]

R. R. Parenti and R. J. Sasiela, “Laser-guide-star systems for astronomical applications,” J. Opt. Soc. Am. A 11, 288-309(1994).
[CrossRef]

D. L. Fried and J. F. Belsher, “Analysis of fundamental limits of artificial-guide-star adaptive-optics system performance for astronomical imaging,” J. Opt. Soc. Am. A. 11, 277-287 (1994).
[CrossRef]

1991 (1)

1985 (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. Vysshikh Uchebnykh Zavedenii, Fizika 11, 42-50 (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)

F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149-154(1983).
[CrossRef]

1982 (1)

1978 (1)

1977 (2)

1976 (1)

1974 (1)

1971 (1)

J. H. Shapiro, “Reciprocity of the turbulent atmosphere,” J. Opt. Soc. Am . 61, 492-495 (1971).
[CrossRef]

1965 (1)

Allen, C. W.

Barchers, J. D.

Belsher, J. F.

D. L. Fried and J. F. Belsher, “Analysis of fundamental limits of artificial-guide-star adaptive-optics system performance for astronomical imaging,” J. Opt. Soc. Am. A. 11, 277-287 (1994).
[CrossRef]

Buffington, A.

Carhart, G. W.

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, 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]

Collett, E.

Dudorov, V. V.

Fried, D. L.

Gardner, C. S.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

Gori, F.

F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149-154(1983).
[CrossRef]

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]

Kohnle, A.

M. A. Vorontsov, V. V. Kolosov, and A. Kohnle, “Adaptive laser beam projection on an extended target: phase- and field-conjugate precompensation,” J. Opt. Soc. Am. A 24, 1975-1993 (2007).
[CrossRef]

Kokorowski, S. A.

Kolosov, V.

Kolosov, V. V.

M. A. Vorontsov, V. V. Kolosov, and A. Kohnle, “Adaptive laser beam projection on an extended target: phase- and field-conjugate precompensation,” J. Opt. Soc. Am. A 24, 1975-1993 (2007).
[CrossRef]

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]

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]

Mehta, N. C.

Miller, R. A.

O'Meara, T. R.

Parenti, R. R.

R. R. Parenti and R. J. Sasiela, “Laser-guide-star systems for astronomical applications,” J. Opt. Soc. Am. A 11, 288-309(1994).
[CrossRef]

Pearson, J. E.

Pedinoff, M. E.

Piatrou, P.

Pruidze, D. V.

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]

Ricklin, J. C.

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]

Roggemann, M.

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. Vysshikh Uchebnykh Zavedenii, Fizika 11, 42-50 (1985).

Sasiela, R. J.

R. R. Parenti and R. J. Sasiela, “Laser-guide-star systems for astronomical applications,” J. Opt. Soc. Am. A 11, 288-309(1994).
[CrossRef]

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. Vysshikh Uchebnykh Zavedenii, Fizika 11, 42-50 (1985).

Shapiro, J. H.

J. H. Shapiro, “Reciprocity of the turbulent atmosphere,” J. Opt. Soc. Am . 61, 492-495 (1971).
[CrossRef]

Shmalhauzen, V. I.

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]

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. Vysshikh Uchebnykh Zavedenii, Fizika 11, 42-50 (1985).

Tatarskii, I.

I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Voelz, D. G.

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]

Vorontsov, M. A.

M. A. Vorontsov, V. V. Kolosov, and A. Kohnle, “Adaptive laser beam projection on an extended target: phase- and field-conjugate precompensation,” J. Opt. Soc. Am. A 24, 1975-1993 (2007).
[CrossRef]

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. Kolosov, “Target-in-the-loop beam control: basic considerations for analysis and wave-front sensing,” J. Opt. Soc. Am. A 22, 126-141 (2005).
[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, 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]

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]

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. Vysshikh Uchebnykh Zavedenii, Fizika 11, 42-50 (1985).

Welsh, B. M.

Wolf, E.

Appl. Opt. (1)

Izv. Vysshikh Uchebnykh Zavedenii, Fizika (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. Vysshikh Uchebnykh Zavedenii, Fizika 11, 42-50 (1985).

J. Opt. Soc. Am (1)

J. H. Shapiro, “Reciprocity of the turbulent atmosphere,” J. Opt. Soc. Am . 61, 492-495 (1971).
[CrossRef]

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (7)

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

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]

D. L. Fried and J. F. Belsher, “Analysis of fundamental limits of artificial-guide-star adaptive-optics system performance for astronomical imaging,” J. Opt. Soc. Am. A. 11, 277-287 (1994).
[CrossRef]

Opt. Commun. (1)

F. Gori, “Mode propagation of the field generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149-154(1983).
[CrossRef]

Opt. Lett. (2)

Sov. J. Quantum Electron. (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]

Other (16)

L. Mandel and E. Wolf, eds., Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

G. Rousset, “Wave-front sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge University Press, 1999), pp. 91-130.
[CrossRef]

We assume here that the obtained vector-function α(r,t) represents a path-independent vector field (at least at compact areas of receiver aperture).

M. C. Rytov, Yu A. Kravtsov, and V. I. Tatarskii, eds., Principles of Statistical Radiophysics 4, Wave Propagation Through Random Media (Springer-Verlag, 1989).

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

N. Ageorges and C. Dainty, eds., Laser Guide Star Adaptive Optics for Astronomy (Kluwer Academic, 2000).

J. C. Dainty, ed., Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, 1984).

M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).

M. A. Vorontsov and V. I. Shmalhauzen, eds., Principles of Adaptive Optics (Nauka, 1985).

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

M. A. Vorontsov, A. V. Koriabin, and V. I. Shmalhauzen, eds., Controlling Optical Systems (Nauka, 1988).

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

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

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

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Figures (8)

Fig. 1
Fig. 1

Stationary beacons (top row) and their impact on the characteristics of the target return wave: the instantaneous speckle-field intensity | ψ 0 ( r ) | 2 (second row) and phase φ ( r ) = arg [ ψ 0 ( r ) ] (third row) for the coherent beacons and the SA phase Φ ( r ) for the Collett–Wolf beacons (bottom row). Intensity patterns for the beacons are (from left to right) the diffraction-limited (Airy) beacon, square beacons with b sq = 2 b dif and b sq = 4 b dif , and a cigar beacon with b cg = 4 b dif . The propagation conditions (the set of Kolmogorov phase screens with D / r 0 = 5 and the distance to the target L = 0.05 ka 0 2 ) used in computations for all beacons are identical. In phase patterns (third and forth row) are shown with the removed parabolic phase component φ q ( r ) = k r 2 / ( 2 L ) .

Fig. 2
Fig. 2

Target-plane intensity obtained with conjugation of the instantaneous phase (PC control) in the top row and SA phase (SA PC control) in the bottom row for the Airy beacon (first column), square beacon with b sq = 2 b dif (second column) and with b sq = 4 b dif (third column), and for the cigar beacon with b cg = 4 b dif (fourth column). The corresponding patterns of beacon intensity, phase, and SA phase are shown in Fig. 1. The diffraction-limited intensity pattern is shown in the inset. The values of the target-plane metric J 2 and St are given below the corresponding intensity patterns. The propagation parameters are the same as in Fig. 1.

Fig. 3
Fig. 3

Speckle-average phase conjugation using the Collett–Wolf beacon created by a small amplitude steering of the outgoing beam. The Collett–Wolf beacon intensity (brightness) pattern in (a) is obtained for the beam steering along the vertical line of length b cg = 4 b dif . This beacon is used for calculation of the SA phase Φ ( r ) in (b). The target-plane intensity I T ( r ) in (c) is obtained using conjugation of phase Φ ( r ) . The propagation conditions (phase screens and distance L) are identical in Fig. 1.

Fig. 4
Fig. 4

Impact of anisoplanatism on the outgoing beam precompensation efficiency using a square Collett–Wolf for SA PC (lines with dots) and an unresolved coherent Gaussian beacon for PC (solid lines) control for D / r 0 = 5 (a), (b) and D / r 0 = 8 (c), (d). Atmospheric-average metrics St at in (a), (c) and J 2 in (b), (d) are shown as functions of the beacon size b sq for the SA PC precompensation and displacement l (distance between the location of the unresolved beacon and the outgoing beam aim point at the target plane) for the PC control. The horizontal lines correspond to the outgoing beam that is focused on the target. The threshold beacon size b th , the isoplanatic distance l is that is defined by the e 1 fall-off of the Strehl ratio, and l is ^ = 0.57 r 0 are normalized on the diffraction-limited beam radius b dif (Airy radius) for the flat-top beam of radius a 0 in vacuum. The propagation conditions (phase screens and distance L) are identical in Fig. 1.

Fig. 5
Fig. 5

Efficiency of phase control based on SA PC control for laser beam projection onto an extended target with rapidly moving Lambertian surface. Dependence of the atmospheric-average target-plane metric J 2 at on SA PC iteration number n for different D / r 0 : solid lines—initial plane phase u ( r , 0 ) = 0 ; dotted lines—focused beam with u ( r , 0 ) = u q ( r ) = k r 2 / ( 2 L ) . The propagation distance is L = 0.2 ka 0 2 , where a 0 is the outgoing flat-top beam radius. Phase distortions are modeled by N = 20 equidistant Kolmogorov phase screens. Gray-scale images correspond to target-plane intensity distributions: (a) and (b) prior to compensation for D / r 0 = 6.0 ; (a) with u ( r , 0 ) = 0 and (b) with u ( r , 0 ) = u q ( r ) ; (c) and (d) SA PC for n = 5 , (c) for D / r 0 = 6.0 and (d) for D / r 0 = 0.2 .

Fig. 6
Fig. 6

Schematic of the bench-top adaptive optical system for laser beam projection on an extended target based on SA PC feedback control. Insets are image of the target (a), geometry of the deformable mirror (DM) electrodes (b), target-plane intensity of the projected beam without (c) and without (d) adaptive compensation of a random phase aberrations, the long-exposure intensity distribution of the beacon beam with one (e) and two-dimensional (f) scanning, and the projected beam (bright spot in the middle) inside the square Collett–Wolf beacon created by two-dimensional scanning of the beacon laser beam (g). The images in (c)—(g) correspond to a 110 × 110 μm area.

Fig. 7
Fig. 7

Impact of phase aberration on the return speckle-field intensity (a) and instantaneous phase (b) for the projected beam, and on the SA phase (c), (d) obtained for the beacon beam with one- (c) and two-dimensional (d) scanning.

Fig. 8
Fig. 8

Dependence of the averaged Strehl ratio St on the iteration number n obtained in the phase distortion compensation experiments with PC and SA PC feedback control of wavefront phase for the laser beam projected onto an extended stationary target in Fig. 6a. Phase distortions are created by applying random voltages to the deformable mirror (DM) electrodes. Averaging is performed using a set of 20 different phase aberration patterns. Two-dimensional scanning of the beacon was used to generate the square Collett–Wolf beacon shown in Fig. 6g. The length of vertical bars indicates the standard deviation in the Strehl ratio.

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

τ c τ ph < τ AO < τ at .
τ s τ ph < τ AO < τ at .
ψ ( r , z = 0 , t ) ψ 0 ( r , t ) = I 0 1 / 2 ( r , t ) exp [ i φ ( r , t ) ] ,
{ α j ( t ) } { k 1 φ j ( t ) } = { r j c ( t ) } / F , ( j = 1. , ... , N ) ,
α ( r , t ) = 1 k φ ( r , t ) .
α ( r , t ) = S ( r , t ) / I 0 ( r , t ) ,
S ( r , t ) = 1 2 i k [ ψ 0 * ( r , t ) ψ 0 ( r , t ) ψ 0 ( r , t ) ψ 0 * ( r , t ) ]
α ( r , t ) s = S ( r , t ) s I 0 ( r , t ) s = ψ 0 * ( r , t ) ψ 0 ( r , t ) s ψ 0 ( r , t ) ψ 0 * ( r , t ) s 2 i k ψ 0 ( r , t ) ψ 0 * ( r , t ) s .
Γ 0 ( r 1 , r 2 , t ) ψ 0 ( r 1 , t ) ψ 0 * ( r 2 , t ) s ,
Γ 0 ( ρ , R , t ) ψ 0 ( R + ρ / 2 , t ) ψ 0 * ( R ρ / 2 , t ) s .
α ( R , t ) s S ( R , t ) s / I 0 ( R , t ) s = ( i k ) 1 ρ ln Γ 0 ( ρ 0 , R ) ,
B 0 ( θ , R , t ) = 1 ( 2 π ) 2 Γ 0 ( ρ , R , t ) exp ( i k θ ρ ) d 2 ρ ,
α ( r , t ) s = θ B 0 ( θ , r , t ) d 2 θ B 0 ( θ , r , t ) d 2 θ .
α ( r , t ) s = Φ ( r , t ) / k .
S ( r , t ) s I 0 ( r , t ) s = 1 k Φ ( r , t ) .
u ( r , t ) = Φ ( r , t ) ,
Φ ( r , t ) = φ at ( r , t ) + φ q ( r ) + φ tilt ( r , t ) .
r c ( t ) = r I T ( r , t ) d 2 r I T ( r , t ) d 2 r .
u ( r , t ) = Φ ( r , t ) = u opt ( r , t ) φ tilt ( r , t ) ,
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 ) ,
A ( r , z = 0 , t ) = A 0 ( r ) exp [ i u ( r , t ) ] ,
ψ ( r , z = L , t ) = T ( r , t ) A ( r , z = L , t ) ,
d R ( z , t ) d z = θ ( z , t ) , d θ ( z , t ) d z = R n 1 ( R , z , t ) .
B L ( θ , R , t 0 ) = c I T ( R , t 0 ) ,
I T Airy ( r ) = I T 0 [ 2 J 1 ( kra 0 / L ) kra 0 / L ] 2 ,
I T L ( r , t 0 ) = l 1 L I T ( r , r c , t 0 ) d l ( r c ) ,
B L ( θ , R , t 0 ) = c I T L ( R , t 0 ) .
I T L ( r , t 0 ) = l 1 L I T ( r r c , t 0 ) d l ( r c ) .
I T ( r ) = I T sq ( r ) I T 0 exp [ ( x / b sq ) 16 ( y / b sq ) 16 ] ,
I T ( r ) = I T cg ( r ) I T Airy ( x ) exp [ ( y / b cg ) 16 ] ,
ψ ( r , z = L ) = I T 1 / 2 ( r ) exp [ i ξ ( r ) ] ,
J 2 = I T 2 ( r ) d 2 r / I T Airy ( r ) d 2 r .
u ( r ) = u ( m ) ( r ) = u q ( r ) + u tilt ( m ) ( r ) , m = 0 , , M .
u ( r ) = φ ref ( r ) + u tilt ( r ) ,
A ( r , z = 0 , t n + 1 ) = A 0 ( r ) exp [ i Φ ( r , t n ) ] ,

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