Abstract

A class of adaptive-optics problems is described in which phase distortions caused by atmospheric turbulence are corrected by adaptive wave-front reconstruction with a deformable mirror, i.e., the control loop that drives the mirror adapts in real time to time-varying atmospheric conditions, as opposed to the linear time-invariant control loops used in conventional adaptive optics. The basic problem is posed as an adaptive disturbance-rejection problem with many channels. The solution given is an adaptive feedforward control loop built around a multichannel adaptive lattice filter. Simulation results are presented for a 1-m telescope with both one-layer and two-layer atmospheric turbulence profiles. These results demonstrate the significant improvement in imaging resolution produced by the adaptive control loop compared with a classical linear time-invariant control loop.

© 2000 Optical Society of America

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References

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  4. F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
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    [CrossRef]
  15. M. Lloyd-Hart, P. McGuire, “Spatio-temporal prediction for adaptive optics wave-front reconstructors,” in Adaptive Optics (European Southern Observatory, Garching, Germany, 1996), Vol. 54, pp. 95–101.
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    [CrossRef]
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  19. E. Gendron, P. Léna, “Astronomical adaptive optics II. Experimental results of an optimized modal control,” Astron. Astrophys. Suppl. Ser. 111, 153–167 (1995).
  20. C. Dessenne, P.-Y. Madec, G. Rousset, “Optimization of a predictive controller for closed-loop adaptive optics,” Appl. Opt. 37, 4623–4633 (1998).
    [CrossRef]
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1998 (1)

1997 (2)

1995 (3)

S.-B. Jiang, J. S. Gibson, “An unwindowed multichannel lattice filter with orthogonal channels,” IEEE Trans. Signal Process. 43, 2831–2842 (1995).
[CrossRef]

T. E. Bell, “Electronics and the stars,” IEEE Spectrum 32(8), 16–24 (1995).
[CrossRef]

E. Gendron, P. Léna, “Astronomical adaptive optics II. Experimental results of an optimized modal control,” Astron. Astrophys. Suppl. Ser. 111, 153–167 (1995).

1994 (5)

1991 (1)

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

1990 (1)

G. Rousset, J. C. Fontanella, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

1980 (1)

1977 (2)

Aitken, G. J. M.

G. J. M. Aitken, D. McGaughey, “Predictability of atmospherically distorted wave fronts,” in Adaptive Optics (European Southern Observatory, Garching, Germany, 1996), Vol. 54, p. 89.

M. B. Jorgenson, G. J. M. Aitken, “Neural network prediction of turbulence induced wavefront degradations with applications to adaptive optics,” in Adaptive and Learning Systems, F. A. Sadjadi, ed., Proc. SPIE1706, 113–121 (1992).
[CrossRef]

Bell, T. E.

T. E. Bell, “Electronics and the stars,” IEEE Spectrum 32(8), 16–24 (1995).
[CrossRef]

Boeke, B. R.

Cleis, R. A.

Dessenne, C.

Ellerbroek, B. L.

Fontanella, J. C.

G. Rousset, J. C. Fontanella, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Fried, D. L.

Fugate, R. Q.

Gendron, E.

E. Gendron, P. Léna, “Astronomical adaptive optics II. Experimental results of an optimized modal control,” Astron. Astrophys. Suppl. Ser. 111, 153–167 (1995).

E. Gendron, P. Léna, “Astronomical adaptive optics I. Modal control optimization,” Astron. Astrophys. 291, 337–347 (1994).

Gibson, J. S.

S.-B. Jiang, J. S. Gibson, “An unwindowed multichannel lattice filter with orthogonal channels,” IEEE Trans. Signal Process. 43, 2831–2842 (1995).
[CrossRef]

Graves, J. E.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Hardy, J. W.

J. W. Hardy, “Adaptive optics,” Sci. Am. 270(6) , 60–65 (1994).
[CrossRef]

Herrmann, J.

Higgins, C. H.

Hudgin, R. H.

Jelonek, M. P.

Jiang, S.-B.

S.-B. Jiang, J. S. Gibson, “An unwindowed multichannel lattice filter with orthogonal channels,” IEEE Trans. Signal Process. 43, 2831–2842 (1995).
[CrossRef]

Jorgenson, M. B.

M. B. Jorgenson, G. J. M. Aitken, “Neural network prediction of turbulence induced wavefront degradations with applications to adaptive optics,” in Adaptive and Learning Systems, F. A. Sadjadi, ed., Proc. SPIE1706, 113–121 (1992).
[CrossRef]

Lange, W. J.

Léna, P.

E. Gendron, P. Léna, “Astronomical adaptive optics II. Experimental results of an optimized modal control,” Astron. Astrophys. Suppl. Ser. 111, 153–167 (1995).

E. Gendron, P. Léna, “Astronomical adaptive optics I. Modal control optimization,” Astron. Astrophys. 291, 337–347 (1994).

Ling, F.

J. G. Proakis, C. M. Rader, F. Ling, C. L. Nikias, Advanced Digital Signal Processing (Macmillan, New York, 1992).

Lloyd-Hart, M.

M. Lloyd-Hart, P. McGuire, “Spatio-temporal prediction for adaptive optics wave-front reconstructors,” in Adaptive Optics (European Southern Observatory, Garching, Germany, 1996), Vol. 54, pp. 95–101.

Madec, P.-Y.

Manolakis, D. G.

J. G. Proakis, D. G. Manolakis, Digital Signal Processing, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1996).

McGaughey, D.

G. J. M. Aitken, D. McGaughey, “Predictability of atmospherically distorted wave fronts,” in Adaptive Optics (European Southern Observatory, Garching, Germany, 1996), Vol. 54, p. 89.

McGuire, P.

M. Lloyd-Hart, P. McGuire, “Spatio-temporal prediction for adaptive optics wave-front reconstructors,” in Adaptive Optics (European Southern Observatory, Garching, Germany, 1996), Vol. 54, pp. 95–101.

Moroney, J. F.

Nikias, C. L.

J. G. Proakis, C. M. Rader, F. Ling, C. L. Nikias, Advanced Digital Signal Processing (Macmillan, New York, 1992).

Northcott, M.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Oliker, M. D.

Pitsianis, N. P.

Plemmons, R. J.

Proakis, J. G.

J. G. Proakis, D. G. Manolakis, Digital Signal Processing, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1996).

J. G. Proakis, C. M. Rader, F. Ling, C. L. Nikias, Advanced Digital Signal Processing (Macmillan, New York, 1992).

Rader, C. M.

J. G. Proakis, C. M. Rader, F. Ling, C. L. Nikias, Advanced Digital Signal Processing (Macmillan, New York, 1992).

Rhoadarmer, T. A.

B. L. Ellerbroek, T. A. Rhoadarmer, “Optimizing the performance of closed-loop adaptive-optics control systems on the basis of experimentally measured performance data,” J. Opt. Soc. Am. A 14, 1975–1987 (1997).
[CrossRef]

B. L. Ellerbroek, T. A. Rhoadarmer, “Real-time adaptive optimization of wave-front reconstruction algorithms for closed-loop adaptive-optical systems,” in Adaptive Optical Systems Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1174–1183 (1998).
[CrossRef]

Roddier, F.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Rousset, G.

Ruane, R. E.

Slavin, A. C.

Spinhirne, J. M.

Swindle, D. W.

Van Loan, C.

Wild, W. J.

Winker, D. M.

Wynia, J. M.

Appl. Opt. (1)

Astron. Astrophys. (2)

G. Rousset, J. C. Fontanella, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

E. Gendron, P. Léna, “Astronomical adaptive optics I. Modal control optimization,” Astron. Astrophys. 291, 337–347 (1994).

Astron. Astrophys. Suppl. Ser. (1)

E. Gendron, P. Léna, “Astronomical adaptive optics II. Experimental results of an optimized modal control,” Astron. Astrophys. Suppl. Ser. 111, 153–167 (1995).

IEEE Spectrum (1)

T. E. Bell, “Electronics and the stars,” IEEE Spectrum 32(8), 16–24 (1995).
[CrossRef]

IEEE Trans. Signal Process. (1)

S.-B. Jiang, J. S. Gibson, “An unwindowed multichannel lattice filter with orthogonal channels,” IEEE Trans. Signal Process. 43, 2831–2842 (1995).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Lett. (1)

Publ. Astron. Soc. Pac. (1)

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Sci. Am. (1)

J. W. Hardy, “Adaptive optics,” Sci. Am. 270(6) , 60–65 (1994).
[CrossRef]

Other (6)

G. J. M. Aitken, D. McGaughey, “Predictability of atmospherically distorted wave fronts,” in Adaptive Optics (European Southern Observatory, Garching, Germany, 1996), Vol. 54, p. 89.

M. B. Jorgenson, G. J. M. Aitken, “Neural network prediction of turbulence induced wavefront degradations with applications to adaptive optics,” in Adaptive and Learning Systems, F. A. Sadjadi, ed., Proc. SPIE1706, 113–121 (1992).
[CrossRef]

M. Lloyd-Hart, P. McGuire, “Spatio-temporal prediction for adaptive optics wave-front reconstructors,” in Adaptive Optics (European Southern Observatory, Garching, Germany, 1996), Vol. 54, pp. 95–101.

B. L. Ellerbroek, T. A. Rhoadarmer, “Real-time adaptive optimization of wave-front reconstruction algorithms for closed-loop adaptive-optical systems,” in Adaptive Optical Systems Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1174–1183 (1998).
[CrossRef]

J. G. Proakis, D. G. Manolakis, Digital Signal Processing, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1996).

J. G. Proakis, C. M. Rader, F. Ling, C. L. Nikias, Advanced Digital Signal Processing (Macmillan, New York, 1992).

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Figures (14)

Fig. 1
Fig. 1

Block diagram for closed-loop AO: The adaptive lattice filter chooses L(z) to minimize the variance of ê = E 2 s + D 2 z -1 v. The notation z c -1 indicates a one-step computational delay; the z s -1 block represents a one-step delay in reading out s.

Fig. 2
Fig. 2

Equivalent block diagram for closed-loop AO when Ĝ = G.

Fig. 3
Fig. 3

Actuator locations on the DM.

Fig. 4
Fig. 4

Eigenvalues λ i of the weighting matrix W.

Fig. 5
Fig. 5

Representative eigenvectors of the weighting matrix W.

Fig. 6
Fig. 6

Performance as measured with ρ(t) [defined in Eq. (4)]: (a) No control, i.e., an open loop [ρ(t) = ‖ϕ(t)‖w]. A LTI feedback loop only with (b) N v = 57, (c) N v = 69, (d) N v = 76.

Fig. 7
Fig. 7

Performance as measured with ρ(t) with LTI feedback augmented by an adaptive loop with E 2 = E 2Opt, N L = 8, and N r = 69: (a) γ y = γ G = 0, (b) γ y = 0 and γ G = 0.05, (c) γ y = 0.05 and γ G = 0, (d) γ y = 0.05 and γ G = 0.05.

Fig. 8
Fig. 8

Strehl ratios for PSF’s averaged over 100-point intervals: (a) E 2Opt minimizes J E 2 (1:8000), (b) E 2(3001:3500) minimizes J E 2 (3001:3500), and E 2(7001:7500) minimizes J E 2 (7001:7500).

Fig. 9
Fig. 9

PSF’s averaged over the 100-point interval (5901:6000): (a) open loop, i.e., no control, (b) feedback loop only, i.e., without adaptive loop, (c) adaptive loop and feedback loop. The grid size is 37 × 37.

Fig. 10
Fig. 10

Images from the PSF’s averaged over the 100-point interval (5901:6000): (a) open loop, i.e., no control, (b) feedback loop only, i.e., without adaptive loop, (c) adaptive loop and feedback loop. The grid size is 37 × 37.

Fig. 11
Fig. 11

Strehl ratios for PSF’s averaged over 100-point intervals: (a) one-layer model and (b) two-layer model. The open circles on the curves plotted in (b) indicate the time for which the PSF’s and the images are shown in Figs. 910 and 1314.

Fig. 12
Fig. 12

Comparison of control loops with N v = N r = 69 and N v = N r = 57 (VR). Shown are the Strehl ratios for PSF’s averaged over 100-point intervals: (a) γ y = γ G = 0 and (b) γ y = γ G = 0.05. In each case with the adaptive loop the optimal value of E 2 is used.

Fig. 13
Fig. 13

PSF’s averaged over the interval (5901:6000) for an adaptive loop and a feedback loop: (a) E 2 = E 2Opt and γ y = γ G = 0.05, (b) E 2 = E 0, (c) E 2 = E 2Opt and γ y = γ G = 0. The grid size is 37 × 37.

Fig. 14
Fig. 14

Images from the PSF’s that were averaged over the interval 5901:6000) for an adaptive loop and a feedback loop: (a) E 2 = E 2Opt and γ y = γ G = 0.05, (b) E 2 = E 0, (c) E 2 = E 2Opt and γ y = γ G = 0. The grid size is 37 × 37.

Tables (5)

Tables Icon

Table 1 Definitions of Parameters for Vector Signals, Matrices, and Filters

Tables Icon

Table 2 Atmosphere Modelsa

Tables Icon

Table 3 Values of the Performance Index J(6001, 8000)a

Tables Icon

Table 4 Values of the Normalized Performance Index J(6001, 8000) for LTI Lattice Filters (VR)

Tables Icon

Table 5 Values of J E 2 (t 1, t 2) and J(t 1, t 2) for E 2 Chosen to Optimize J E 2 (t 1, t 2) over Different Intervalsa

Equations (36)

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

 ϕibi- ci+ηibi= ibi,
J=limt2-t1 Jt1, t2,
Jt1, t2=1t2-t1t=t1t2 ρ2t1/2,
ρt=TtWt1/2,
v0t=VTWV-1VTWψt.
c=Vv,
VTV=I,  VTWV=Λ,
V#=Λ-1VTW
V#V=I.
ρ2t=ψTtWψt-v0TtΛv0t+eTtΛet,
v0t=V#ψt,
et=v0t-vt=V#t.
c=V˜v˜.
V=V˜U,
V˜TWV˜U=V˜TV˜UΛ,  UTV˜TV˜U=I.
E0Gˆ=I.
D1=E1Gˆ,
D2=E2Gˆ-I,
zeˆ=e+E2y-V#ϕ+V#-E2Hη+E2Gˆ-Gv.
JE2t1, t2=t=t1t2 E2yt-V#ϕtΛ2t=t1t2 V#ϕtΛ21/2,
F0z=1,
F1z=K1zz-1,  K1=0.5.
Teuz=-K1z-11-z-1+K1z-2,
Teˆuz=z-1Teuz.
F0z=K0b0za0z,  F1z=K1b1za1z,
Teˆuz=-z-1a0zK1b1za0za1z+z-2K0K1b0zb1z.
r˜=E1s+z-1v=z-1E1y-Hη+z-1D1-E1Gv,
Jeˆ=limt2-t11t2-t1t=t1t2 eˆTteˆt1/2.
Lz=i=1NL Aiz-i,
y=y0+γyw0,
G=HV,  Gˆ=G+γGΔHV,
Nv=69,  Nr=69,  VR,
Nv=69,  Nr=57,  VR,
Nv=57,  Nr=57,  VR,
Nv=57,  Nr=57,  RV.
E0=GˆTGˆ-1GˆT=pseudoinverse of Gˆ,

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