Abstract

An optimal filter algorithm for adaptive optics provides a powerful method for phase correction for propagation through the Earth’s turbulent atmosphere involving anisoplanatism. In the new algorithm the outward phase correction is the sum of the product of a weighting function (the optimal filter) and all the wave-front measurements at the pupil, greatly improving the Strehl ratio. Two simplified cases are presented for illustration: (1) a collimated beam traversing a layer of uniform isotropic turbulence (angle anisoplanatism) and (2) focus anisoplanatism. It compares favorably with tomographic techniques. The technique can be extended to the case of thick, strong turbulence in the far field of a subaperture of an adaptive optics system.

© 2003 Optical Society of America

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References

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  1. R. Fugate, D. L. Fried, G. A. Amear, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
    [CrossRef]
  2. D. W. Hanson, “Reduction of anisoplanatic errors,” RADC-TR-81-122 (U.S. Air Force Rome Air Development Center, Griffiss Air Force Base, N.Y., 1981).
  3. M. R. Whiteley, B. M. Welsh, M. C. Roggemann, “Optimal modal wave-front compensation for anisoplanatism in adaptive optics,” J. Opt. Soc. Am. A 15, 2097–2106 (1998).
    [CrossRef]
  4. D. P. Looze, M. Kasper, S. Hippler, “Optimal compensation and implementation for adaptive optics systems,” in Proceedings of the 38th Conference on Decision and Control (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1715–1720.
  5. A. Tokovinin, W. Viard, “Limiting precision of tomographic phase estimation,” J. Opt. Soc. Am. A 18, 873–882 (2001).
    [CrossRef]
  6. D. Korff, M. G. Miller, G. L. Dryden, G. W. Sutton, “Optimal filtering techniques for adaptive optics,” Rep. AMP 641 (Avco-Everett Research Laboratory, Everett, Mass., 1982), available from M. Miller, Physical Sciences, Inc., 20 New England Business Center, Andover, Mass. 01810.
  7. D. Korff, “Corrections for focus anisoplanaticism,” Rep. AERLP 664A (Avco-Everett Research Laboratory, Everett, Mass., 1982), available from M. Miller, Physical Sciences, Inc., 20 New England Business Center, Andover, Mass. 01810.
  8. G. W. Sutton, “A new method of phase correction for anisoplanatism,” presented at the Eleventh Annual American Institute of Aeronautics and Astronautics and Missle Defense Agency Technology Conference, Monterey, Calif., 29 July–2 Aug. 2002.
  9. This typically entails use of a powerful, repetitively pulsed laser focused at the sodium layer at ∼85-km altitude, with a wavelength tuned to the sodium resonance line.
  10. G. W. Sutton, “Effect of turbulent fluctuations in a optically active fluid medium,” AIAA J. 7, 1737–1743 (1969).
    [CrossRef]

2001 (1)

1998 (1)

1991 (1)

R. Fugate, D. L. Fried, G. A. Amear, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

1969 (1)

G. W. Sutton, “Effect of turbulent fluctuations in a optically active fluid medium,” AIAA J. 7, 1737–1743 (1969).
[CrossRef]

Amear, G. A.

R. Fugate, D. L. Fried, G. A. Amear, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Boeke, B. R.

R. Fugate, D. L. Fried, G. A. Amear, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Browne, S. L.

R. Fugate, D. L. Fried, G. A. Amear, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Dryden, G. L.

D. Korff, M. G. Miller, G. L. Dryden, G. W. Sutton, “Optimal filtering techniques for adaptive optics,” Rep. AMP 641 (Avco-Everett Research Laboratory, Everett, Mass., 1982), available from M. Miller, Physical Sciences, Inc., 20 New England Business Center, Andover, Mass. 01810.

Fried, D. L.

R. Fugate, D. L. Fried, G. A. Amear, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Fugate, R.

R. Fugate, D. L. Fried, G. A. Amear, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Hanson, D. W.

D. W. Hanson, “Reduction of anisoplanatic errors,” RADC-TR-81-122 (U.S. Air Force Rome Air Development Center, Griffiss Air Force Base, N.Y., 1981).

Hippler, S.

D. P. Looze, M. Kasper, S. Hippler, “Optimal compensation and implementation for adaptive optics systems,” in Proceedings of the 38th Conference on Decision and Control (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1715–1720.

Kasper, M.

D. P. Looze, M. Kasper, S. Hippler, “Optimal compensation and implementation for adaptive optics systems,” in Proceedings of the 38th Conference on Decision and Control (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1715–1720.

Korff, D.

D. Korff, M. G. Miller, G. L. Dryden, G. W. Sutton, “Optimal filtering techniques for adaptive optics,” Rep. AMP 641 (Avco-Everett Research Laboratory, Everett, Mass., 1982), available from M. Miller, Physical Sciences, Inc., 20 New England Business Center, Andover, Mass. 01810.

D. Korff, “Corrections for focus anisoplanaticism,” Rep. AERLP 664A (Avco-Everett Research Laboratory, Everett, Mass., 1982), available from M. Miller, Physical Sciences, Inc., 20 New England Business Center, Andover, Mass. 01810.

Looze, D. P.

D. P. Looze, M. Kasper, S. Hippler, “Optimal compensation and implementation for adaptive optics systems,” in Proceedings of the 38th Conference on Decision and Control (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1715–1720.

Miller, M.

D. Korff, “Corrections for focus anisoplanaticism,” Rep. AERLP 664A (Avco-Everett Research Laboratory, Everett, Mass., 1982), available from M. Miller, Physical Sciences, Inc., 20 New England Business Center, Andover, Mass. 01810.

D. Korff, M. G. Miller, G. L. Dryden, G. W. Sutton, “Optimal filtering techniques for adaptive optics,” Rep. AMP 641 (Avco-Everett Research Laboratory, Everett, Mass., 1982), available from M. Miller, Physical Sciences, Inc., 20 New England Business Center, Andover, Mass. 01810.

Miller, M. G.

D. Korff, M. G. Miller, G. L. Dryden, G. W. Sutton, “Optimal filtering techniques for adaptive optics,” Rep. AMP 641 (Avco-Everett Research Laboratory, Everett, Mass., 1982), available from M. Miller, Physical Sciences, Inc., 20 New England Business Center, Andover, Mass. 01810.

Roberts, P. H.

R. Fugate, D. L. Fried, G. A. Amear, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Roggemann, M. C.

Ruane, R. E.

R. Fugate, D. L. Fried, G. A. Amear, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Sutton, G. W.

G. W. Sutton, “Effect of turbulent fluctuations in a optically active fluid medium,” AIAA J. 7, 1737–1743 (1969).
[CrossRef]

D. Korff, M. G. Miller, G. L. Dryden, G. W. Sutton, “Optimal filtering techniques for adaptive optics,” Rep. AMP 641 (Avco-Everett Research Laboratory, Everett, Mass., 1982), available from M. Miller, Physical Sciences, Inc., 20 New England Business Center, Andover, Mass. 01810.

G. W. Sutton, “A new method of phase correction for anisoplanatism,” presented at the Eleventh Annual American Institute of Aeronautics and Astronautics and Missle Defense Agency Technology Conference, Monterey, Calif., 29 July–2 Aug. 2002.

Tokovinin, A.

Tyler, G. A.

R. Fugate, D. L. Fried, G. A. Amear, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Viard, W.

Welsh, B. M.

Whiteley, M. R.

Wopat, L. M.

R. Fugate, D. L. Fried, G. A. Amear, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

AIAA J. (1)

G. W. Sutton, “Effect of turbulent fluctuations in a optically active fluid medium,” AIAA J. 7, 1737–1743 (1969).
[CrossRef]

J. Opt. Soc. Am. A (2)

Nature (London) (1)

R. Fugate, D. L. Fried, G. A. Amear, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Other (6)

D. W. Hanson, “Reduction of anisoplanatic errors,” RADC-TR-81-122 (U.S. Air Force Rome Air Development Center, Griffiss Air Force Base, N.Y., 1981).

D. P. Looze, M. Kasper, S. Hippler, “Optimal compensation and implementation for adaptive optics systems,” in Proceedings of the 38th Conference on Decision and Control (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1715–1720.

D. Korff, M. G. Miller, G. L. Dryden, G. W. Sutton, “Optimal filtering techniques for adaptive optics,” Rep. AMP 641 (Avco-Everett Research Laboratory, Everett, Mass., 1982), available from M. Miller, Physical Sciences, Inc., 20 New England Business Center, Andover, Mass. 01810.

D. Korff, “Corrections for focus anisoplanaticism,” Rep. AERLP 664A (Avco-Everett Research Laboratory, Everett, Mass., 1982), available from M. Miller, Physical Sciences, Inc., 20 New England Business Center, Andover, Mass. 01810.

G. W. Sutton, “A new method of phase correction for anisoplanatism,” presented at the Eleventh Annual American Institute of Aeronautics and Astronautics and Missle Defense Agency Technology Conference, Monterey, Calif., 29 July–2 Aug. 2002.

This typically entails use of a powerful, repetitively pulsed laser focused at the sodium layer at ∼85-km altitude, with a wavelength tuned to the sodium resonance line.

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Figures (10)

Fig. 1
Fig. 1

Nomenclature for parallel rays A and B.

Fig. 2
Fig. 2

Parallel rays from a beacon source at an angle α to the imaging direction.

Fig. 3
Fig. 3

Improvement in residual phase variance by use of the KMDS algorithm versus the simple phase algorithm for angular misalignment between source and object.

Fig. 4
Fig. 4

Schematic for a nearby source for the wave-front sensor.

Fig. 5
Fig. 5

Phase variances for phase conjugation and the KMDS optimal filter.

Fig. 6
Fig. 6

Comparison of phase variance with phase conjugation versus optimal filter.

Fig. 7
Fig. 7

Comparison of the KMDS algorithm for adaptive optics with that of Tolkvinin and Viard5 for a natural guide star tomographic optimal filter.

Fig. 8
Fig. 8

Illustration of the problem of adaptive optics for a moving object.

Fig. 9
Fig. 9

Schematic of adaptive optics for a thick turbulent region in which the turbulence is in the far field of the adaptive optics subaperture.

Fig. 10
Fig. 10

Diagram for correlation function.

Equations (36)

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

Cij=k,l FijklMkl,
Cr=exp-r2/Λ2,
ϕA=k 0L ΔnyAdyA,ϕB=k 0L ΔnyBdyB.
ϕAϕB=k20L ΔnyAdyA0L ΔnyBdyB.
ϕAϕB=k20LdyB-yBL-yB ΔnyB+ηΔnyBdη.
ϕAϕB=k2Δn2 0LdyB-yBL-yB Crdη,
r2=η2+ξ2,
ϕAϕB=k2Δn2πΛL exp-ξ2/Λ2.
ϕc=Gx00x0 ϕBxdx,
ϕe,KMDS2ϕ2=1-ψ,
limαL/Λ0ϕe,KMDS2ϕ2α2L26Λ2.
ϕe,SPC2ϕ2=2-πΛαLerfαLΛ,
limαL/Λ0ϕe,SPC2ϕ2=23αLΛ2,
ϕe2¯=1AA ϕe2dA=1α020α0 ϕe2dα2.
ϕe2KMDSϕ2¯=1+1β020β021-exp-xx×dx-2 πβ0erfβ0+2 1β02×1-exp-β02,
ϕe2pcϕ2¯=2-2 πβ0erfβ0+2 1β021-exp-β02,
Ipeak,focal plane=I0D4 exp-ϕe216λ2F2,
Ipeak, focal plane=I0D4 exp-Kϕe2D2L2/42Λ216λ2F2,
Irel,pc=D4 exp-D2,Irel,KMDS=D4 exp-D2/4.
ϕe2T-Vϕ2¯=1-π4θL/Λerf2θLΛ
ϕe,KMDS=ϕA-ϕc=ϕA-Gx00x0 ϕBxdx.
ϕe,KMDS2=ϕA2-2ϕAGx00x0 ϕBxdxIII+G2x020x0 ϕBxdx 0x0 ϕBxdxIII.
ϕA2=ϕ2=k2Δn2πΛL.
ϕA00x0 ϕBxdx=0x0 ϕA0ϕBxdx.
ϕA0ϕBx=k2 ΔnBdsΔnAdy=k2  ΔnBΔnAdsdy=k2Δn2  Crdsdy.
r2=η2+ξ2-2ηξ cos α;η=y1-y;ξ=s1-s,
ϕAϕB=k20L Δn2dy exp-α2η2/Λ2×-exp-ξ-η1-12 α22/Λ2dξ.
ϕAϕB=-k2Δn2πΛ y1y1-Ldη exp-α2η2/Λ2=-k2Δn2πΛ01y1-Ldη exp-α2η2/Λ2-01y1dη exp-α2η2/Λ2=k2Δn2πΛ01L-y1dη exp-α2η2/Λ2+01y1dη exp-α2η2/Λ2=k2Δn2πΛ22αerfαL-y1Λ+erfαy1Λ.
ϕAϕB=k2Δn2πΛ22αerfx0-xΛ+erfxΛ,
1x00x0 ϕAϕBdx=k2Δn2πΛ22α0x0erfx0-xΛdx+0x0erfxΛdx.
1x00x0 ϕAϕBdx=ϕ2πΛ2α2L2αLΛerfαLΛ+exp-α2L2/Λ2-1π,
1x020x00x0 ϕBxϕBxdxdx
ϕ2πΛ21-α2/2α2L2αLΛerfαL/Λ+1πexp-α2L2/Λ2-1.
ϕe,KMDS2=ϕ21-2Gψ+G2ψ1-α22,
ψ=πΛ2α2L2αLΛerfαL/Λ+1πexp-α2L2/Λ2-1.
ϕe,KMDS2ϕ2=1-1-α22ψ.

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