Abstract

The impact of finite-resolution deformable mirrors and wave-front sensors is evaluated as it applies to full-wave conjugation using two deformable mirrors. The first deformable mirror is fixed conjugate to the pupil, while the second deformable mirror is at a finite range. The control algorithm to determine the mirror commands for the two deformable mirrors is based on a modification of the sequential generalized projection algorithm. The modification of the algorithm allows the incorporation of Gaussian spatial filters into the optimization process to limit the spatial-frequency content applied to the two deformable mirrors. Simulation results are presented for imaging and energy projection scenarios that establish that the optimal spatial filter waist to be applied is equal to the subaperture side length in strong turbulence. The effect of varying the subaperture side length is examined, and it is found that to effect a significant degree of scintillation compensation, the subapertures, and corresponding spacing between actuators, must be much smaller than the coherence length of the input field.

© 2001 Optical Society of America

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References

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  1. 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]
  2. 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).
  3. 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 Atmo-sphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE3763, 11–22 (1999).
    [CrossRef]
  4. 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]
  5. D. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
    [CrossRef]
  6. D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
    [CrossRef] [PubMed]
  7. 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]
  8. 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]
  9. 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. Bissonette, eds., Proc. SPIE3763, 29–40 (1999).
    [CrossRef]
  10. J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1968).
  11. H. T. Yura, “Physical model for strong optical-amplitude fluctuations in a turbulent medium,” J. Opt. Soc. Am. 64, 59–67 (1974).
    [CrossRef]
  12. R. J. Cook, “Fundamental notions in the theory of adaptive optics” (unpublished lecture notes, 1975), available from Brent L. Ellerbroek, Gemini Observatory, 670 North A’ohaku Place, Hilo, Hawaii 96720.
  13. H. Stark, Y. Yang, Vector Space Projections (Wiley, New York, 1998).
  14. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–379 (1977).
    [CrossRef]
  15. S. F. Clifford, G. R. Ochs, R. S. Lawrence, “Saturation of optical scintillation by strong turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
    [CrossRef]
  16. D. Malacara, Optical Shop Testing (Wiley, New York, 1992).

2001 (1)

1998 (2)

1995 (1)

1992 (1)

1984 (1)

1977 (1)

1974 (2)

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).

Barchers, J. D.

Clifford, S. F.

Cook, R. J.

R. J. Cook, “Fundamental notions in the theory of adaptive optics” (unpublished lecture notes, 1975), available from Brent L. Ellerbroek, Gemini Observatory, 670 North 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, L. R. Bissonette, eds., Proc. SPIE3763, 29–40 (1999).
[CrossRef]

Ellerbroek, B. L.

Ellerbroek, Brent L.

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

Fried, D.

Fried, D. L.

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).

Goodman, J. W.

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

Hudgin, R. H.

Kotzer, T.

Lawrence, R. S.

Lee, D. J.

Levi, A.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, New York, 1992).

Ochs, G. R.

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]

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 Atmo-sphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE3763, 11–22 (1999).
[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. Bissonette, eds., Proc. SPIE3763, 29–40 (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.

Vaughn, J. L.

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 Atmo-sphere 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 Atmo-sphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE3763, 11–22 (1999).
[CrossRef]

Yang, Y.

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

Yura, H. T.

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

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 (6)

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 Atmo-sphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE3763, 11–22 (1999).
[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. Bissonette, eds., Proc. SPIE3763, 29–40 (1999).
[CrossRef]

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), available from Brent L. Ellerbroek, Gemini Observatory, 670 North A’ohaku Place, Hilo, Hawaii 96720.

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

D. Malacara, Optical Shop Testing (Wiley, New York, 1992).

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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 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. WFS denotes the wave-front sensor.

Fig. 2
Fig. 2

Optical implementation of the propagation and filtering operator. A field that is conjugate to some distance z0 is brought to focus. A quadratic field lens and a filter or an amplifier are applied so that after propagation through focus and recollimation of the beam, the resultant field is conjugate to a distance z0+z and is a spatially filtered or amplified version of the input field.

Fig. 3
Fig. 3

Optical computer implementation of the WSGPA for control of two DMs. The algorithm is similar to the Gerchberg–Saxton phase retrieval algorithm. After propagation and spatial filtering of the beacon field corrected by the first DM, the command for the second DM is generated. Then a new command for the first DM is generated by propagation and spatial filtering of the laser beam profile corrected by the second DM.

Fig. 4
Fig. 4

Measured log-amplitude variance of the fields used in this paper as a function of the Rytov number. With one exception (a constant-turbulence profile, with d/r0=1), agreement with the Rytov theory is excellent for small values of the Rytov number, a slight amplification is noted in the intermediate regime, and saturation occurs for large values of the Rytov number. This result is consistent with expectation.

Fig. 5
Fig. 5

Plots (a) and (b) present the performance of the α-WSGPA for energy projection (EP) and imaging, respectively, as a function of α/d. While the Strehl ratio available to the algorithm, S, increases with decreasing α, the actual Strehl ratio, S, has a clear maximum at α=d. This result is emphasized for strong turbulence (Ψ=80°). Plots (c) and (d) present the number of branch points in the control commands for energy projection and imaging, respectively, as a function of α/d. The increase in number of branch points as α decreases is consistent with intuition that suggests that branch cuts applied to continuous facesheet DMs can lead to a degradation in performance.

Fig. 6
Fig. 6

(a) Energy projection and (b) imaging Strehl ratios as a function of the Rytov number for a constant distribution of turbulence for d/r0=1/4, 1/2, and 1 for MCAO and single-DM control. The performance improvement achieved by an MCAO system is greatest for d/r0=1/4.

Fig. 7
Fig. 7

(a) Energy projection and (b) imaging Strehl ratios as a function of sec Ψ for HV 5/7 turbulence and 64, 32, and 16 subapertures across the 1-m aperture. The maximum performance improvement is noted for the smallest subapertures. Plots (c) and (d) present the number of branch points in the control commands for energy projection and imaging, respectively. The number of branch points in the first DM follows that for single-DM compensation, indicating that few additional branch points in the first DM are required to accomplish scintillation compensation. Note, however, that there is still a nontrivial number of branch points in the second DM. PO denotes phase-only control.

Tables (2)

Tables Icon

Table 1 Turbulence Profile Used for Approximation of HV 5/7 Turbulence

Tables Icon

Table 2 Range to Second DM Selected for Evaluationa

Equations (35)

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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)]}2dr¯1 Ub(r¯1)Ub*(r¯1)dr¯2 Ul(r¯2)Ul*(r¯2).
C1={U1(r¯1)l2 : |Tz[Ub(r¯1)U1(r¯1)]|=|Ul(r¯2)|}.
C2={U1(r¯1)l2 : |U1(r¯1)|=1for|r¯1|RandU1(r¯1)=0elsewhere}.
P1[U1(r¯1)]=Ub-1(r¯1)Tz*{Ul(r¯2)exp[iϕ2(r¯2)]},
P2[U1(r¯1)]=exp[i arg U1(r¯1)]for|r¯1|Rand0elsewhere.
U1k+1(r¯1)=P2{P1[U1k(r¯1)]}.
J=Ub[P1(U1k)-U1k]2+Ub[P2(U1k)-U1k]2,
P˜i(U)=arg minAiBi[P˜i(U)]C˜iBiP˜i(U)-CiU2,
C˜i={Ul2 : |U|=Zi},
P˜i(U)=Bi-1Ai-1Zi exp(iθ),
θ=arg(Ai-*CiU).
U1k+1(r¯1)=P˜2{P˜1[U1k(r¯1)]},
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˜1,α(U)=arg min|A1B1[P˜1,α(U)]|C˜1,αB1P˜1,α(U)-C1U2,
A1(·)=F-1[Pz,0(·)]exp[i arg Ul*(r¯2)],
B1(·)=F[Ub(·)]P0,-α,
C1(·)=F[Ub(·)]P0,α.
P˜1,α[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)]}.
ϕ1(r¯1)=arg(Ub*(r¯1)T-z,α{Ul(r¯2)exp[iϕ2(r¯2)]}),
ϕ2(r¯2)=arg(Ul*(r¯2)Tz,α{Ub(r¯1)exp[iϕ1(r¯1)]}).
Ub(r¯1)exp[iϕ1(r¯1)], T-z,α{Ul(r¯2)exp[iϕ2(r¯2)]}.
LS(ϕ)=(GTG)-1GTPV(Gϕ).
ϕˆ1=LS(ϕ1)+arg{exp[iϕ1]exp[-iLS(ϕ1)]}.
Uˆb(r¯1)=D(r¯1)drU¯b(r¯)D(r¯1)dr¯|Ub(r¯)|,
x=arg min|ABx|=ZBx-Cy22,
x=B-1A-1Z exp(iθ).
J=A-1Z exp(iθ)-Cy22.
Jθ=i[exp(iθ)ZATA*TZexp(iθ)exp(iθ)ZA*A1Zexp(iθ)+exp(iθ)ZA*Cyexp(iθ)ZATC*Ty*],
Jθ=i[exp(iθ)ZA*Cyexp(iθ)ZATC*Ty*].
θ=arg(A-*Cy).

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