Abstract

A modification of the parallel generalized projection algorithm is presented that allows for the use of projections in a weighted norm. Convergence properties of the modified algorithm, denoted the weighted parallel generalized projection algorithm, are developed. The weighted parallel generalized projection algorithm is applied to the control of two finite-resolution deformable mirrors to compensate for both the amplitude and the phase fluctuations that result from propagation through a turbulent medium. Numerical results are shown that indicate that a two-deformable-mirror system can provide improved performance over that of a single-deformable-mirror system.

© 2002 Optical Society of America

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  1. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane images,” Optik (Stuttgart) 35, 225–246 (1972).
  2. H. Stark, Y. Yang, Vector Space Projections (Wiley, New York, 1998).
  3. A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  4. H. Stark, Y. Yang, D. Gurkan, “Factors affecting convergence in the design of diffractive optics by iterative vector-space methods,” J. Opt. Soc. Am. A 16, 149–159 (1999).
    [CrossRef]
  5. T. Kotzer, J. Rosen, J. Shamir, “Application of serial and parallel projection methods to correlation filter design,” Appl. Opt. 34, 3883–3895 (1995).
    [CrossRef] [PubMed]
  6. J. D. Barchers, B. L. Ellerbroek, “Improved compensation of turbulence induced amplitude and phase distortions by means of multiple near field phase adjustments,” J. Opt. Soc. Am. A 18, 399–411 (2001).
    [CrossRef]
  7. 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, 4577–4585 (1998).
    [CrossRef]
  8. 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, L. R. Bissonnette, eds., Proc. SPIE3763, 29–40 (1999).
    [CrossRef]
  9. 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, L. R. Bissonnette, eds., Proc. SPIE3763, 11–22 (1999).
    [CrossRef]
  10. R. A. Gonsalves, “Compensation of scintillation with a phase-only adaptive optic,” Opt. Lett. 22, 588–590 (1997).
    [CrossRef] [PubMed]
  11. J. D. Barchers, “Evaluation of the impact of finite-resolution effects on scintillation compensation using two deformable mirrors,” J. Opt. Soc. Am. A 18, 3098–3109 (2001).
    [CrossRef]
  12. J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1968).
  13. H. T. Yura, “Physical model for strong optical-amplitude fluctuations in a turbulent medium,” J. Opt. Soc. Am. 64, 59–67 (1974).
    [CrossRef]
  14. R. J. Cook, “Fundamental notions in the theory of adaptive optics” (unpublished lecture notes, 1975); contact: Brent L. Ellerbroek, Gemini Observatory, 670 North A’ohaku Place, Hilo, Hawaii 96720.
  15. D. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
    [CrossRef]
  16. B. L. Ellerbroek, (personal communication, 2000). (Contact Jeffrey D. Barchers, AFRL/DES, 3550 Aberdeen Avenue SE, Kirtland Air Force Base, N.M. 87117).
  17. D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).
  18. G. A. Tyler, “Reconstruction and assessment of the least-squares and slope discrepancy components of the phase,” J. Opt. Soc. Am. A 17, 1828–1839 (2000).
    [CrossRef]
  19. M. Athans, Lecture Notes for Multivariable Control Systems (6.245) (Massachusetts Institute of Technology School of Engineering, Cambridge, Mass., 1996).
  20. M. A. Dahleh, I. J. Diaz-Bobillo, Control of Uncertain Systems (Prentice-Hall, Englewood Cliffs, N.J., 1995).
  21. S. J. Orfanidis, Optimum Signal Processing (Macmillan, New York, 1985).
  22. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

2001 (2)

2000 (1)

1999 (1)

1998 (3)

1997 (1)

1995 (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 (Stuttgart) 35, 225–246 (1972).

Athans, M.

M. Athans, Lecture Notes for Multivariable Control Systems (6.245) (Massachusetts Institute of Technology School of Engineering, Cambridge, Mass., 1996).

Barchers, J. D.

Cook, R. J.

R. J. Cook, “Fundamental notions in the theory of adaptive optics” (unpublished lecture notes, 1975); contact: Brent L. Ellerbroek, Gemini Observatory, 670 North A’ohaku Place, Hilo, Hawaii 96720.

Dahleh, M. A.

M. A. Dahleh, I. J. Diaz-Bobillo, Control of Uncertain Systems (Prentice-Hall, Englewood Cliffs, N.J., 1995).

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, L. R. Bissonnette, eds., Proc. SPIE3763, 29–40 (1999).
[CrossRef]

Diaz-Bobillo, I. J.

M. A. Dahleh, I. J. Diaz-Bobillo, Control of Uncertain Systems (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Ellerbroek, B. L.

J. D. Barchers, B. L. Ellerbroek, “Improved compensation of turbulence induced amplitude and phase distortions by means of multiple near field phase adjustments,” J. Opt. Soc. Am. A 18, 399–411 (2001).
[CrossRef]

B. L. Ellerbroek, (personal communication, 2000). (Contact Jeffrey D. Barchers, AFRL/DES, 3550 Aberdeen Avenue SE, Kirtland Air Force Base, N.M. 87117).

Fried, D.

Fried, D. L.

D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).

Gerchberg, R. W.

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

Gonsalves, R. A.

Goodman, J. W.

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

Gurkan, D.

Kotzer, T.

Lee, D. J.

Levi, A.

Orfanidis, S. J.

S. J. Orfanidis, Optimum Signal Processing (Macmillan, New York, 1985).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

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, 4577–4585 (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, L. 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, L. R. Bissonnette, eds., Proc. SPIE3763, 11–22 (1999).
[CrossRef]

Rosen, J.

Saxton, W. O.

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

Shamir, J.

Stark, H.

Tyler, G. A.

Welsh, B. M.

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, L. 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, L. R. Bissonnette, eds., Proc. SPIE3763, 11–22 (1999).
[CrossRef]

Yang, Y.

Yura, H. T.

Appl. Opt. (2)

Atmos. Oceanic Opt. (1)

D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).

J. Opt. Soc. Am. (1)

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

Opt. Lett. (1)

Optik (Stuttgart) (1)

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

Other (10)

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

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, L. 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, L. R. Bissonnette, eds., Proc. SPIE3763, 11–22 (1999).
[CrossRef]

M. Athans, Lecture Notes for Multivariable Control Systems (6.245) (Massachusetts Institute of Technology School of Engineering, Cambridge, Mass., 1996).

M. A. Dahleh, I. J. Diaz-Bobillo, Control of Uncertain Systems (Prentice-Hall, Englewood Cliffs, N.J., 1995).

S. J. Orfanidis, Optimum Signal Processing (Macmillan, New York, 1985).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

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

R. J. Cook, “Fundamental notions in the theory of adaptive optics” (unpublished lecture notes, 1975); contact: Brent L. Ellerbroek, Gemini Observatory, 670 North A’ohaku Place, Hilo, Hawaii 96720.

B. L. Ellerbroek, (personal communication, 2000). (Contact Jeffrey D. Barchers, AFRL/DES, 3550 Aberdeen Avenue SE, Kirtland Air Force Base, N.M. 87117).

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Figures (4)

Fig. 1
Fig. 1

Visualization of the action of the WPGPA with an inequality constraint enforced to ensure convergence.

Fig. 2
Fig. 2

Approach for a two-DM AO system. The field at the aperture has both large amplitude and phase distortions that are due 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. 3
Fig. 3

Optimization of the control weights for (a) the α-WPGPAμ and (b) the α-WPGPAλ. Application of weight to the branch-point constraint (w3) leads to reduced performance. Increased weight on the near-field amplitude constraint (w1) improves performance.

Fig. 4
Fig. 4

Comparison of the performance of a two-DM system with the performance of a single-DM system as a function of the Rytov number for a fixed value of d/r0=1/4. The α-WPGPA (diamonds) and the α-WSGPA (solid curve) exhibit nearly identical performance (in fact, the curves appear superimposed over each other) and show an improvement over a single DM controlled by measurements from a self-referencing interferometer (circles). The conventional system in use today (Hartmann sensor in the Fried geometry controlled by a least-squares reconstructor, LS FH) is shown for reference (squares). Note the poor performance for large values of the Rytov number.

Equations (51)

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P˜i(x)=argminAiBiP˜i(x)C˜iBiP˜i(x)-Cix2,
|X|2=i=1Nwixi22,
C˜={X|X=(x1, x2,, xN)H,xiC˜i  i=1, 2,, N},
D˜={X|X=(x1, x2,, xN)H,x1=x2==xN}.
A̲X=(A1(x1), A2(x2),, AN(xN)),
B̲X=(B1(x1), B2(x2),, BN(xN)),
C̲X=(C1(x1), C2(x2),, CN(xN)),
P˜C˜(X)=argminA̲C˜B̲C˜PC˜(X)C˜|B̲C˜P˜C˜(X)-C̲C˜X|
=(P˜1(x1), P˜1(x2),, P˜N(xN)),
P˜D˜(X)=argminA̲D˜B̲D˜PD˜(X)D˜|B̲D˜P˜D˜(X)-C̲D˜X|.
J(Xk)=|B̲C˜P˜C˜(Xk)-C̲C˜Xk|+|B̲D˜P˜D˜(Xk)-C̲D˜Xk|.
J(Xk)=|B̲C˜P˜C˜(Xk)-C̲C˜Xk|.
J(Xk+1)=|B̲C˜P˜C˜P˜D˜P˜C˜(Xk)-C̲C˜PD˜PC˜(Xk)||B̲C˜P˜C˜(Xk)-C̲C˜PD˜PC˜(Xk)|.
|B̲C˜P˜C˜(Xk)-C̲C˜P˜D˜P˜C˜(Xk)|
|B̲C˜P˜C˜(Xk)-C̲C˜(Xk)|.
P¯D˜P˜C˜(Xk)=argminA̲D˜B̲D˜P¯D˜P˜C˜(Xk)D˜,|B̲C˜P˜C˜(Xk)-C̲C˜P¯D˜P˜C˜(Xk)|R×|B̲D˜P¯D˜P˜C˜(Xk)-B̲D˜P˜C˜(Xk)|,
P¯D˜P˜C˜Xk=argminμ0L(y, μ, p),
L(y, μ, p)=i=1NwiBD˜,i[y-P˜C˜,i(xk,i)]22+μi=1NwiBC˜,iP˜C˜,i(xk, i)-CC˜,iy||22-R2+p2.
y=i=1NwiBD˜,i*BD˜,i-1i=1NwiBD˜,i*BD˜,iP˜C˜,i(xk,i).
y=i=1Nwi(BD˜,i*BD˜,i+μCC˜,i*CC˜,i)-1×i=1Nwi(BD˜,i*BD˜,i+μCC˜,i*BC˜,i)P˜C˜,i(xk,i),
T˜D˜P˜C˜(Xk)=Xk+λD˜[P˜D˜P˜C˜(Xk)-Xk],
Tz()=F-1{F()exp(iπλzκ¯2)}.
S=|dr¯1 Ub(r¯1)exp[iϕ1(r¯1)]Tz{Ul*(r¯2)exp[-iϕ2(r¯2)]}|2|dr¯1 Ub(r¯1)Ub*(r¯1)dr¯2 Ul(r¯2)Ul*(r¯2)|.
Tz,α()=F-1F()expiπλzκ¯2-sgn(α)πα22κ¯2,
 
Pz,α()=expiπλzκ¯2-sgn(α)πα22κ¯2().
C1,α={U1(r¯1)l2: |Tz,-α[Ub(r¯1)U1(r¯1)]exp[i arg Ul*(r¯2)]|=|Ul(r¯2)|}.
C˜1, α={U(r¯2)l2: |U(r¯2)|=|Ul(r¯2)|}.
P˜C˜,1(U)=argmin|A1B1P˜C˜,1(U)|C˜1, αB1P˜C˜,1(U)-C1U2,
P˜C˜,1[(U)(r¯1)]=Ub-1(r¯1)T-z, α{Ul(r¯2)exp[iϕ2(r¯2)]},
 
ϕ2(r¯2)=arg {Ul*(r¯2)Tz, α[Ub(r¯1)U1(r¯1)]}.
C˜2={U1(r¯1)l2: |U1(r¯1)|=1for|r¯1|RandU1(r¯1)=0elsewhere
P˜C˜,2[U1(r¯1)]=exp[i arg U1(r¯1)]for|r¯1|Rand0elsewhere.
C˜3={U1(r¯1)l2: PPV[Gϕ1(r¯1)]=0},
P˜C˜,3[U1(r¯1)]=max{|U1|cos[ϕ1(r¯1)-ϕ^1(r¯1)], }exp[-iϕ^1(r¯1)],
C˜={X|X=(x1, x2, x3)H,x1C˜1, α, x2C˜2, x3C˜3},
D˜={X|X=(x1 , x2, x3)H,x1=x2=x3},
A̲C˜X=[F-1[Pz,0x1]exp[i arg Ul*(r¯2)],Ub-1x2,Ub-1x3],
B̲C˜X=[F[Ubx1]P0,-α,Ubx2,Ubx3],
C̲C˜X=[F[Ubx1]P0, α,Ubx2,Ubx3],
A̲D˜X=[Ub-1x1,Ub-1x2,Ub-1x3],
B̲D˜X=[Ubx1,Ubx2,Ubx3],
C̲D˜X=[Ubx1,Ubx2,Ubx3].
U1,k+1=i=13wiP˜C˜,i(U1,k).
U1,k+1=(1+μ)Ub-1F-1[I+μw1P0, α2+μ(1-w1)]-1FUbi=13wiP˜C˜,i[U1,k],
LS(ϕ)=(GTG)-1GTPV(Gϕ),
ϕ˘1=LS(ϕ1)+arg{exp(iϕ1)exp[-iLS(ϕ1)]},
ϕ˘2=LS(ϕ2)+arg{exp(iϕ2)exp[-iLS(ϕ2)]}.
U^b(r¯1)=D(r¯1)dr¯Ub(r¯)D(r¯1)dr¯|Ub(r¯)|,
Δϕb(r¯1)=D(r¯1)dr¯|Ub(r¯)|2ϕb(r¯)D(r¯1)dr¯|Ub(r¯)|2.

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