Abstract

Wave-front sensing and deformable mirror control algorithms in adaptive optics systems are designed on the premise that a continuous phase function exists in the telescope pupil that can be conjugated with a deformable mirror for the purpose of projecting a laser beam. However, recent studies of coherent wave propagation through turbulence have shown that under conditions where scintillation is not negligible, a truly continuous phase function does not in general exist as a result of the presence of branch points in the complex optical field. Because of branch points and the associated branch cuts, least-squares wave-front reconstruction paradigms can have large errors. We study the improvement that can be obtained by implementing wave-front reconstructors that can sense the presence of branch points and reconstruct a discontinuous phase function in the context of a laser beam projection system. This study was conducted by fitting a finite-degree-of-freedom deformable mirror to branch-point and least-squares reconstructions of the phase of the beacon field, propagating the corrected field to the beacon plane, and evaluating performance in the beacon plane. We find that the value of implementing branch-point reconstructors with a finite-degree-of-freedom deformable mirror is significant for optical paths that cause saturated log-amplitude fluctuations.

© 2000 Optical Society of America

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References

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1998 (2)

1994 (1)

1993 (1)

1992 (1)

1990 (1)

1988 (1)

1977 (1)

1976 (1)

1969 (1)

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote probing,” Proc. IEEE 57, 375–406 (1969).
[CrossRef]

1966 (1)

Beland, R. R.

R. R. Beland, “Propagation through atmospheric optical turbulence,” in IR/EO Handbook, F. G. Smith, ed. (SPIE Press, Bellingham, Wash., 1993), Vol. 2, pp. 157–232.

Boeke, B. R.

Carlson, L.

T. Goldring, L. Carlson, “Analysis and implementation of non-Kolmogorov phase screens appropriate to structured environments,” in Nonlinear Optical Beam Manipulation and High Energy Beam Propagation through the Atmosphere, R. A. Fisher, L. E. Wilson, eds., Proc. SPIE1060, 244–264 (1989).
[CrossRef]

Cleis, R. A.

Ellerbroek, B. L.

Flatte, S. M.

Fried, D. L.

Fugate, R. Q.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Ghiglia, D. C.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley Interscience, New York, 1998).

Goldring, T.

T. Goldring, L. Carlson, “Analysis and implementation of non-Kolmogorov phase screens appropriate to structured environments,” in Nonlinear Optical Beam Manipulation and High Energy Beam Propagation through the Atmosphere, R. A. Fisher, L. E. Wilson, eds., Proc. SPIE1060, 244–264 (1989).
[CrossRef]

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1993).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

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

Harp, J. C.

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote probing,” Proc. IEEE 57, 375–406 (1969).
[CrossRef]

Higgins, C. H.

Jelonek, M. P.

Knox, K. T.

Lange, W. J.

Lee, D. J.

Lee, R. W.

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote probing,” Proc. IEEE 57, 375–406 (1969).
[CrossRef]

Lukin, V. P.

V. P. Lukin, Atmospheric Adaptive Optics (SPIE Press, Bellingham, Wash., 1995).

Martin, J. M.

McGlamery, B. L.

Moroney, J. F.

Oliker, M. D.

Pritt, M. D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley Interscience, New York, 1998).

Roggemann, M. C.

Ruane, R. E.

Sindle, D. W.

Slavin, A. C.

Spinhirne, J. M.

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulence Atmosphere on Wave Propagation, (National Technical Information Service, Springfield, Va., 1971).

Vaughn, J. L.

Wang, G. Y.

Welsh, B. M.

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Wild, W. J.

Winker, D. M.

Woods, R. E.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1993).

Wynia, J. M.

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Proc. IEEE (1)

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote probing,” Proc. IEEE 57, 375–406 (1969).
[CrossRef]

Other (10)

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

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

V. P. Lukin, Atmospheric Adaptive Optics (SPIE Press, Bellingham, Wash., 1995).

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley Interscience, New York, 1998).

V. I. Tatarskii, The Effects of the Turbulence Atmosphere on Wave Propagation, (National Technical Information Service, Springfield, Va., 1971).

R. R. Beland, “Propagation through atmospheric optical turbulence,” in IR/EO Handbook, F. G. Smith, ed. (SPIE Press, Bellingham, Wash., 1993), Vol. 2, pp. 157–232.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1993).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

T. Goldring, L. Carlson, “Analysis and implementation of non-Kolmogorov phase screens appropriate to structured environments,” in Nonlinear Optical Beam Manipulation and High Energy Beam Propagation through the Atmosphere, R. A. Fisher, L. E. Wilson, eds., Proc. SPIE1060, 244–264 (1989).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Geometry for laser beam projection system.

Fig. 2
Fig. 2

Example of ϕLMSE calculation: (a) gray-scale rendering of continuous input phase, (b) gray-scale rendering of the wrapped phase shown in (a), (c) x-axis slices of ϕL and ϕLMSE for this case.

Fig. 3
Fig. 3

Example of branch-point reconstruction by the Goldstein algorithm for a laser beam propagation problem: (a) wrapped incident phase ψI, (b) unwrapped phase ϕI output by the Goldstein algorithm with positive and negative branch points indicated by + and ○, respectively, (c) rewrapped phase given by arg{exp[jϕI(i, j)]}.

Fig. 4
Fig. 4

Theoretical Rytov variance and simulated σχ2 as a function of Cn2.

Fig. 5
Fig. 5

Radial averaged far-field intensity patterns for the case Cn2=3×10-17.

Fig. 6
Fig. 6

Strehl ratio results for branch-point and least-squares reconstruction and for no compensation: (a) Strehl ratio versus Cn2 and (b) Strehl ratio versus Rytov variance σχ,R2.

Equations (20)

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ψ(x)=tan-1Im[U(x)]Re[U(x)],
Δϕ¯=Pϕ¯,
ϕ¯LMSE=(PTP)-1PTΔϕ¯.
σχ,R2=0.56k7/60Ldz Cn2(z)zL5/6(L-z)5/6,
σχ,R2=0.124k7/6L11/6Cn2.
σχ,layer2=0.56k7/6ΔLCn2n=110nΔLL(L-nΔL)5/6,
UR(x)=UA(x)exp[jψR(x)],
tl(x)=exp-jk2L(|x|2),
Δx(i, j)=arg(exp{j[ψI(i+1, j)-ψI(i, j)]}),
Δy(i, j)=arg(exp{j[ψI(i, j+1)-ψI(i, j)]}),
Σ(i, j)=-Δx(i, j)-Δy(i+1, j)+Δx(i, j+1)+Δy(i, j).
ek(x)=Λx-xkD,
pk=i, jW(i, j)ϕLMSE,BP(i, j)ek(i, j),
rkl=dx W(x)ek(x)el(x).
c¯=R-1p¯,
ϕLMSE,BPDM(i, j)=kckek(i, j),
N2λz(Δx)2.
Φψ(f)=0.00969k2zlCn2[f2+(1/L0)2]11/6,
r0=2.1(1.46k2zlCn2)-3/5,
SR=Actualon-axisintensityOn-axisintensitywithnoatmosphere.

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