Abstract

An approach for compensation of turbulence-induced amplitude and phase distortions is described. Two deformable mirrors are placed optically conjugate to the collecting aperture and to a finite range from this aperture. Two control algorithms are presented. The first is a sequential generalized projection algorithm (SGPA) that is similar to the Gerchberg–Saxton phase retrieval algorithm. The second is a parallel generalized projection algorithm (PGPA) that introduces constraints that minimize the number of branch points in the control commands for the deformable mirrors. These approaches are compared with the approach of placing the second deformable mirror conjugate to the far field of the collecting aperture and using the Gerchberg–Saxton algorithm to determine the optimal mirror commands. Simulation results show that placing the second deformable mirror at a finite range can achieve near-unity Strehl ratio regardless of the strength of the scintillation induced by propagation through extended paths, while the maximum Strehl ratio of the far-field approach drops off with increasing scintillation. The feasibility of the solutions is evaluated by counting the branch points contained in the deformable mirror commands. There are large numbers of branch points contained in the control commands that are generated by the Gerchberg–Saxton SGPA-based algorithms, irrespective of where the second deformable mirror is located. However, the control commands generated by the PGPA with branch point constraints achieves excellent Strehl ratio and minimizes the number of branch points.

© 2001 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
  7. M. C. Roggemann, S. Deng, “Scintillation compensation for laser beam projection using segmented deformable mirrors,” in Propagation and Imaging through the Atmosphere III , M. C. Roggemann, Luc R. Bissonnette, eds., Proc. SPIE3763, 29–40 (1999).
    [Crossref]
  8. B. M. Welsh, M. C. Roggemann, G. L. Wilson, “Phase retrieval-based algorithms for far-field beam steering and shaping,” in Propagation and Imaging through the Atmosphere III , M. C. Roggemann, Luc R. Bissonnette, eds., Proc. SPIE3763, 11–22 (1999).
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  16. R. J. Cook, “Fundamental notions in the theory of adaptive optics,” (unpublished lecture notes, available from Brent Ellerbroek, Gemini Observatory, 670 N. A’ohaku Place, Hilo, Hawaii 96720).
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  18. H. Stark, Y. Yang, Vector Space Projections (Wiley, New York, 1998).
  19. J. D. Barchers, B. L. Ellerbroek, “Improved compensation of turbulence induced amplitude and phase distortions by means of multiple near field phase adjustments,” (Starfire Optical Range, Kirtland Air Force Base, N.M., 2000).
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1999 (2)

1998 (2)

1997 (1)

1995 (2)

1994 (2)

1992 (1)

1984 (1)

1974 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane images,” Optik 35, 225–246 (1972).

Baharev, Y.

Barchers, J. D.

J. D. Barchers, B. L. Ellerbroek, “Improved compensation of turbulence induced amplitude and phase distortions by means of multiple near field phase adjustments,” (Starfire Optical Range, Kirtland Air Force Base, N.M., 2000).

Chen, N.-X.

Conan, J.-M.

T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation and Imaging through the Atmosphere III , M. C. Roggemann, Luc R. Bissonnette, eds., Proc. SPIE3763, 125–133 (1999).
[Crossref]

Cong, W.-X.

Cook, R. J.

R. J. Cook, “Fundamental notions in the theory of adaptive optics,” (unpublished lecture notes, available from Brent Ellerbroek, Gemini Observatory, 670 N. A’ohaku Place, Hilo, Hawaii 96720).

Deng, S.

M. C. Roggemann, S. Deng, “Scintillation compensation for laser beam projection using segmented deformable mirrors,” in Propagation and Imaging through the Atmosphere III , M. C. Roggemann, Luc R. Bissonnette, eds., Proc. SPIE3763, 29–40 (1999).
[Crossref]

Ellerbroek, B. L.

B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence com-pensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[Crossref]

J. D. Barchers, B. L. Ellerbroek, “Improved compensation of turbulence induced amplitude and phase distortions by means of multiple near field phase adjustments,” (Starfire Optical Range, Kirtland Air Force Base, N.M., 2000).

Fried, D.

Fried, D. L.

Fusco, T.

T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation and Imaging through the Atmosphere III , M. C. Roggemann, Luc R. Bissonnette, eds., Proc. SPIE3763, 125–133 (1999).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane images,” Optik 35, 225–246 (1972).

Gonsalves, R. A.

Goodman, J. W.

J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1968).

Gu, B.-Y.

Gurkan, D.

Johnston, D. C.

Kotzer, T.

Lee, D. J.

Levi, A.

Michau, V.

T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation and Imaging through the Atmosphere III , M. C. Roggemann, Luc R. Bissonnette, eds., Proc. SPIE3763, 125–133 (1999).
[Crossref]

Mugnier, L. M.

T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation and Imaging through the Atmosphere III , M. C. Roggemann, Luc R. Bissonnette, eds., Proc. SPIE3763, 125–133 (1999).
[Crossref]

Roggemann, M. C.

M. C. Roggemann, D. J. Lee, “A two deformable mirror concept for correcting scintillation effects in laser beam projection through the turbulent atmosphere,” Appl. Opt. 37, 4577–4485 (1998).
[Crossref]

M. C. Roggemann, S. Deng, “Scintillation compensation for laser beam projection using segmented deformable mirrors,” in Propagation and Imaging through the Atmosphere III , M. C. Roggemann, Luc R. Bissonnette, eds., Proc. SPIE3763, 29–40 (1999).
[Crossref]

B. M. Welsh, M. C. Roggemann, G. L. Wilson, “Phase retrieval-based algorithms for far-field beam steering and shaping,” in Propagation and Imaging through the Atmosphere III , M. C. Roggemann, Luc R. Bissonnette, eds., Proc. SPIE3763, 11–22 (1999).
[Crossref]

Rosen, J.

Rousset, G.

T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation and Imaging through the Atmosphere III , M. C. Roggemann, Luc R. Bissonnette, eds., Proc. SPIE3763, 125–133 (1999).
[Crossref]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane images,” Optik 35, 225–246 (1972).

Shamir, J.

Stark, H.

Vaughn, J. L.

Welsh, B. M.

D. C. Johnston, B. M. Welsh, “Analysis of multiconjugate adaptive optics,” J. Opt. Soc. Am. A 11, 394–408 (1994).
[Crossref]

B. M. Welsh, M. C. Roggemann, G. L. Wilson, “Phase retrieval-based algorithms for far-field beam steering and shaping,” in Propagation and Imaging through the Atmosphere III , M. C. Roggemann, Luc R. Bissonnette, eds., Proc. SPIE3763, 11–22 (1999).
[Crossref]

Wilson, G. L.

B. M. Welsh, M. C. Roggemann, G. L. Wilson, “Phase retrieval-based algorithms for far-field beam steering and shaping,” in Propagation and Imaging through the Atmosphere III , M. C. Roggemann, Luc R. Bissonnette, eds., Proc. SPIE3763, 11–22 (1999).
[Crossref]

Yang, Y.

Yura, H. T.

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Opt. Lett. (1)

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane images,” Optik 35, 225–246 (1972).

Other (7)

M. C. Roggemann, S. Deng, “Scintillation compensation for laser beam projection using segmented deformable mirrors,” in Propagation and Imaging through the Atmosphere III , M. C. Roggemann, Luc R. Bissonnette, eds., Proc. SPIE3763, 29–40 (1999).
[Crossref]

B. M. Welsh, M. C. Roggemann, G. L. Wilson, “Phase retrieval-based algorithms for far-field beam steering and shaping,” in Propagation and Imaging through the Atmosphere III , M. C. Roggemann, Luc R. Bissonnette, eds., Proc. SPIE3763, 11–22 (1999).
[Crossref]

T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation and Imaging through the Atmosphere III , M. C. Roggemann, Luc R. Bissonnette, eds., Proc. SPIE3763, 125–133 (1999).
[Crossref]

H. Stark, Y. Yang, Vector Space Projections (Wiley, New York, 1998).

J. D. Barchers, B. L. Ellerbroek, “Improved compensation of turbulence induced amplitude and phase distortions by means of multiple near field phase adjustments,” (Starfire Optical Range, Kirtland Air Force Base, N.M., 2000).

J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1968).

R. J. Cook, “Fundamental notions in the theory of adaptive optics,” (unpublished lecture notes, available from Brent Ellerbroek, Gemini Observatory, 670 N. A’ohaku Place, Hilo, Hawaii 96720).

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Figures (7)

Fig. 1
Fig. 1

Approach for two-DM AO system. The field at the aperture has both large amplitude and phase distortions owing to propagation along an extended path. The first DM is conjugate to the pupil while the second DM is conjugate to a finite range z.

Fig. 2
Fig. 2

Spatial distribution of branch points in the control command for the second DM generated by the PGPA algorithm with w¯=[0.30.60.10.0]. The number of branch points as a function of aperture radius (in pixels) are shown. Note the sharp increase in the number of branch points beyond the pupil radius of 32 pixels. There are 63 branch points at a radius of 32 pixels and 370 branch points in the full control command ϕ2.

Fig. 3
Fig. 3

Ensemble average (128 independent realizations) performance results for a two-layer toy problem: (a) Strehl; (b) total number of branch points in the control commands as a function of aperture radius. Results are shown for various MCAO control approaches as well as perfect and least-squares phase-only compensation. Control by the SGPA results in many branch points but excellent performance. Control by the PGPA minimizes the number of branch points but reduces performance.

Fig. 4
Fig. 4

Distance optimization for the imaging problem at a zenith angle of 70 deg. The imaging Strehl and the number of branch points in the DM’s are shown. Up to a point, increasing the distance also increases the number of branch points in DM-2. Beyond this point, which varies with Ψ, the number of branch points falls off, but the Strehl also falls off. Strehl ratios above 0.90 can be achieved in the region where the number of branch points increases with the range of the second DM.

Fig. 5
Fig. 5

Distance optimization for the energy projection problem at a zenith angle of 70°. Solid curve, energy projection Strehl; dashed curves, number of branch points in the DM’s. Note that, unlike in the imaging application, the number of branch points in the second DM does not decrease with decreasing range in the short-distance limit. Instead, the number of branch points in the second DM decreases only with increasing range.

Fig. 6
Fig. 6

Ensemble average performance of various control approaches for imaging through HV 5/7 turbulence at zenith angles from 60° to 80°. (a) Strehl (b) number of branch points for the various approaches. Strehl drops off as a function of zenith angle for single-DM phase-only compensation and for the far-field MCAO system. The near-field MCAO system achieves near-unity Strehl at the expense of a large number of branch points. The near-field MCAO system controlled by the PGPA has a reduced number of branch points but also reduced performance.

Fig. 7
Fig. 7

Ensemble average performance of various control approaches for energy projection through HV 5/7 turbulence at zenith angles from 60° to 80°: (a) Strehl (b) number of branch points for the various approaches. The phase-only compensation Strehl, and particularly the least-squares phase-only compensation Strehl ratio, drops off rapidly with increasing zenith angle. The Strehl for the far-field MCAO system is roughly constant with zenith angle, but the far-field MCAO system has a great number of branch points. With a propagation distance of 3 km between the two DM’s, the near-field MCAO system controlled by the SGPA achieves near-unity Strehl at the expense of a large number of branch points. When controlled by the PGPA, the same system has good Strehl with a significantly reduced number of branch points. With a longer propagation distance to the second DM and controlled by the SGPA, the MCAO system has good Strehl and a reasonable number of branch points in the control commands.

Tables (1)

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Table 1 Performance Results for a Two-Layer Toy Problem with and without a Finite Aperture

Equations (40)

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Uz(r¯)=Tz[U0(r¯)]=F -1{F [U0(r¯)]exp(iπλzκ¯2)}.
Uz(r¯)=Ul*(r¯)exp[-iϕ2(r¯)],
U0(r¯)=Tz[Uz(r¯)]exp[iϕ1(r¯)],
IdrU¯b(r¯)U0(r¯)2,
I=drU¯b(r¯)exp[iϕ1(r¯)]Tz{Ul*(r¯)exp[-iϕ2(r¯)]}2drU¯b(r¯)Ub*(r¯)dr¯Ul(r¯)Ul*(r¯).
IUb(r¯)exp[iϕ1(r¯)],Tz*{Ul(r¯)exp[iϕ2(r¯)]}2
exp[iϕ1(r¯)], Ub*(r¯)Tz*{Ul(r¯)exp[iϕ2(r¯)]}2,
argmaxϕ1(r¯) I[ϕ1(r¯)]=arg(Ub*(r¯)Tz*{Ul(r¯)exp[iϕ2(r¯)]}).
drf¯(r¯)g*(r¯)=dκ¯F (f )(κ¯)[F (g)(κ¯)]*
I=dr¯Ul(r¯)exp[iϕ2(r¯)]Tz*{Ub*(r¯)exp[-iϕ1(r¯)]}2drU¯b(r¯)Ub*(r¯)dr¯Ul(r¯)Ul*(r¯)
argmaxϕ2(r¯) I[ϕ2(r¯)]=arg(Ul*(r¯)Tz{Ub(r¯)exp[iϕ1(r¯)]}).
C1={U1(r¯)L2 : |exp[i arg Ul*(r¯)]×Tz[Ub(r¯)U1(r¯)]|=|Ul(r¯)|}.
C2={U1(r¯)L2 : |U1(r¯)|=1
for|r¯|R,U1(r¯)=0elsewhere}.
P1[U1(r¯)]=Ub-1Tz*{Ul(r¯)exp[iϕ2(r¯)]},
P2[U1(r¯)]=exp[i arg U1(r¯)]
for|r¯|R,0elsewhere.
U1k+1(r¯)=P2{P1[U1k(r¯)]}
D=Ub[P1(U1k)-U1k]+|Ub|[P2U1k)-U1k].
U1k+1(r¯)=i=1NwiPi[U1k(r¯)],
D=i=1NwiPi[U1k]-U1k221/2,
U1k+1(r¯)=[w1|Ub|2+1-w1]-1w1|Ub|2P1[U1k(r¯)]+i=2NwiPi[U1k(r¯)],
D=w1Ub[P1(U1k)-U1k]22+i=2NwiPi[U1k]-U1k221/2.
C˜3=U(r¯)L2:PV [ arg U(r¯)]=0forallclosedcontours.
C3={U(r¯)l2 : PPV [Gϕ(r¯)]=0},
P3[U0]=argminUC3U-U022.
LS [ϕ]=(GTG)-1GTPV [Gϕ].
minUC3 J=(U-U0)*(U-U0).
J=n=1NA2(n)+A02(n)-2A(n)A0(n)×cos[ϕ0(n)-ϕ(n)].
J/A=2A-2A0cos(ϕ0-ϕ)=0.
A=A0cos(ϕ0-ϕ).
J=n=1NA02(n)sin2[ϕ0(n)-ϕ(n)]=A0sin(ϕ0-ϕ)22.
A=max[A0cos(ϕ0-ϕ), ].
min J=ϕ0-ϕ22
f=PPV [Gϕ]=0.
ϕ(k+1)=LS{ϕ(k)+α[ϕ0-ϕ(k)]}.
C3={U1(r¯)l2 : PPV [Gϕ1(r¯)]=0}
P3[U1(r¯)]=max{A1cos[ϕ1(r¯)-ϕ^1(r¯)], }×exp[-iϕ^1(r¯)],
C4=[U1(r¯)l2 : PPV(G arg{Ul*(r¯)Tz[Ub(r¯)U1(r¯)]})]=0
P4[U1(r¯)]=exp[-iϕb(r¯)]Tz*(Ul(r¯)exp[iϕ^2(r¯)]),

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