Abstract

We examine how modal aberration measurements degraded by turbulence-induced anisoplanatism may be used to optimally conjugate atmospheric phase aberrations. By examining the form of the aperture-averaged mean square residual phase error, we show that atmospheric compensation is suboptimal when the measured coefficients from off-axis or finite-altitude guide stars are applied directly. The optimal compensation is obtained only when conjugate phase coefficients are estimated, given the guide-star measurements and knowledge of the spatial correlation of the on-axis and measured phase coefficients, by use of a minimum-mean-square-error (MMSE) estimator. The form of this estimator is outlined, thus motivating the need to quantify the spatial cross correlation of the Zernike coefficients of the phase aberrations. With a knowledge of the modal cross correlation, we show that wave-front compensation performance can be enhanced by use of the MMSE estimator over use of the beacon measurements directly for all orders of correction. For high-order off-axis natural-guide-star correction, equivalent imaging performance is obtained at a beacon offset 10% larger than when beacon measurements are used directly. For high-order laser-guide-star correction, equivalent imaging performance is obtained at laser-guide-star altitudes 20% lower when the MMSE estimator is employed.

© 1998 Optical Society of America

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    [CrossRef]
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  36. S. E. Troxel, B. M. Welsh, M. C. Roggemann, “Off-axis optical transfer function calculations in an adaptive-optics system by means of a diffraction calculation for weak index fluctuations,” J. Opt. Soc. Am. A 11, 2100–2111 (1994).
    [CrossRef]
  37. J. D. H. Pilkington, “Artificial guide stars for adaptive imaging,” Nature 330, 116 (1987).
    [CrossRef]
  38. M. S. Belen’kii, “Fundamental limitation in adaptive optics: how to eliminate it? A full aperture tilt measurement technique with a laser guide star,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 321–323 (1994).
    [CrossRef]
  39. M. S. Belen’kii, “Full aperture tilt measurement technique with a laser guide star,” in Atmospheric Propagations and Remote Sensing IV, J. C. Dainty, ed., Proc. SPIE2471, 289–300 (1995).
    [CrossRef]
  40. M. S. Belen’kii, “Tilt angular correlation and tilt sensing techniques with a laser guide star,” in Optics in Atmospheric Propagation, Adaptive Systems, and Lidar Techniques for Remote Sensing, A. D. Devir, A. Kohnle, C. Werner, eds., Proc. SPIE2956, 206–217 (1996).
    [CrossRef]
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    [CrossRef]

1998 (1)

1997 (1)

1996 (1)

1995 (2)

1994 (9)

G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” J. Opt. Soc. Am. A 11, 358–367 (1994).
[CrossRef]

S. E. Troxel, B. M. Welsh, M. C. Roggemann, “Off-axis optical transfer function calculations in an adaptive-optics system by means of a diffraction calculation for weak index fluctuations,” J. Opt. Soc. Am. A 11, 2100–2111 (1994).
[CrossRef]

J. Stone, P. H. Hu, S. P. Mills, S. Ma, “Anisoplanatic effects in finite-aperture optical systems,” J. Opt. Soc. Am. A 11, 347–357 (1994).
[CrossRef]

D. G. Sandler, S. Stahl, J. R. P. Angel, M. Lloyd-Hart, D. McCarthy, “Adaptive optics for diffraction-limited infrared imaging with 8-m telescopes,” J. Opt. Soc. Am. A 11, 925–945 (1994).
[CrossRef]

D. L. Fried, J. F. Belsher, “Analysis of fundamental limits to artificial-guide-star adaptive-optics-system performance for astronomical imaging,” J. Opt. Soc. Am. A 11, 277–287 (1994).
[CrossRef]

G. A. Tyler, “Merging: a new method for tomography through random media,” J. Opt. Soc. Am. A 11, 409–424 (1994).
[CrossRef]

R. J. Sasiela, “Wave-front correction by one or more synthetic beacons,” J. Opt. Soc. Am. A 11, 379–393 (1994).
[CrossRef]

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

G. A. Tyler, “Wave-front compensation for imaging with off-axis guide stars,” J. Opt. Soc. Am. A 11, 339–346 (1994).
[CrossRef]

1992 (1)

M. A. Von Bokern, R. N. Paschall, B. M. Welsh, “Modal control for and adaptive optics system using LQG compensation,” Comput. Elect. Eng. 18, 421–433 (1992).

1991 (4)

B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[CrossRef]

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature 353, 141–143 (1991).
[CrossRef]

R. Q. Fugate, D. L. Fried, G. A. Ameer, 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 353, 144–146 (1991).
[CrossRef]

D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A 8, 1568–1573 (1991).
[CrossRef]

1990 (2)

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

1989 (1)

1987 (2)

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature 328, 229–231 (1987).
[CrossRef]

J. D. H. Pilkington, “Artificial guide stars for adaptive imaging,” Nature 330, 116 (1987).
[CrossRef]

1985 (1)

R. Foy, A. Labeyrie, “Feasibility of adaptive telescopes with laser probe,” Astron. Astrophys. 152, 129–131 (1985).

1982 (1)

1979 (1)

1978 (1)

1976 (1)

1975 (1)

1966 (1)

1965 (1)

Ameer, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, 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 353, 144–146 (1991).
[CrossRef]

Angel, J. R. P.

Barclay, H. T.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature 353, 141–143 (1991).
[CrossRef]

Belen’kii, M. S.

M. S. Belen’kii, “Fundamental limitation in adaptive optics: how to eliminate it? A full aperture tilt measurement technique with a laser guide star,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 321–323 (1994).
[CrossRef]

M. S. Belen’kii, “Full aperture tilt measurement technique with a laser guide star,” in Atmospheric Propagations and Remote Sensing IV, J. C. Dainty, ed., Proc. SPIE2471, 289–300 (1995).
[CrossRef]

M. S. Belen’kii, “Tilt angular correlation and tilt sensing techniques with a laser guide star,” in Optics in Atmospheric Propagation, Adaptive Systems, and Lidar Techniques for Remote Sensing, A. D. Devir, A. Kohnle, C. Werner, eds., Proc. SPIE2956, 206–217 (1996).
[CrossRef]

Belsher, J. F.

Boeke, B. R.

R. Q. Fugate, D. L. Fried, G. A. Ameer, 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 353, 144–146 (1991).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Browne, S. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, 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 353, 144–146 (1991).
[CrossRef]

Craig, A. T.

R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics, 5th ed. (Macmillan, New York, 1995).

Dai, G.

Dryden, G.

Ellerbroek, B. L.

Esposito, S.

Foy, R.

R. Foy, A. Labeyrie, “Feasibility of adaptive telescopes with laser probe,” Astron. Astrophys. 152, 129–131 (1985).

Fried, D. L.

Fugate, R. Q.

R. Q. Fugate, D. L. Fried, G. A. Ameer, 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 353, 144–146 (1991).
[CrossRef]

Gardner, C. S.

B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[CrossRef]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature 328, 229–231 (1987).
[CrossRef]

Hogg, R. V.

R. V. Hogg, A. T. Craig, Introduction to Mathematical Statistics, 5th ed. (Macmillan, New York, 1995).

Hu, P. H.

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Upper Saddle River, N.J., 1993).

Korff, D.

Labeyrie, A.

R. Foy, A. Labeyrie, “Feasibility of adaptive telescopes with laser probe,” Astron. Astrophys. 152, 129–131 (1985).

Leavitt, R. P.

Lloyd-Hart, M.

Ma, S.

Markey, J. K.

McCarthy, D.

Mills, S. P.

Molodij, G.

Murphy, D. V.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature 353, 141–143 (1991).
[CrossRef]

Noll, R. J.

Page, D. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature 353, 141–143 (1991).
[CrossRef]

Paschall, R. N.

M. A. Von Bokern, R. N. Paschall, B. M. Welsh, “Modal control for and adaptive optics system using LQG compensation,” Comput. Elect. Eng. 18, 421–433 (1992).

Pilkington, J. D. H.

J. D. H. Pilkington, “Artificial guide stars for adaptive imaging,” Nature 330, 116 (1987).
[CrossRef]

Primmerman, C. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature 353, 141–143 (1991).
[CrossRef]

Ragazzoni, R.

Riccardi, A.

Roberts, P. H.

R. Q. Fugate, D. L. Fried, G. A. Ameer, 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 353, 144–146 (1991).
[CrossRef]

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1981), Vol. XIX, pp. 283–376.

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Roggemann, M. C.

Rousset, G.

Ruane, R. E.

R. Q. Fugate, D. L. Fried, G. A. Ameer, 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 353, 144–146 (1991).
[CrossRef]

Sandler, D. G.

Sasiela, R. J.

Stahl, S.

Stanley, T.

Stone, J.

Takato, N.

Thompson, L. A.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature 328, 229–231 (1987).
[CrossRef]

Troxel, S. E.

Tyler, G. A.

Valley, G. C.

Von Bokern, M. A.

M. A. Von Bokern, R. N. Paschall, B. M. Welsh, “Modal control for and adaptive optics system using LQG compensation,” Comput. Elect. Eng. 18, 421–433 (1992).

Wandzura, S. M.

Wang, J. Y.

Welsh, B.

M. C. Roggemann, B. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Welsh, B. M.

Whiteley, M. R.

Winker, D. M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Wopat, L. M.

R. Q. Fugate, D. L. Fried, G. A. Ameer, 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 353, 144–146 (1991).
[CrossRef]

Yamaguchi, I.

Zollars, B. G.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature 353, 141–143 (1991).
[CrossRef]

Astron. Astrophys. (1)

R. Foy, A. Labeyrie, “Feasibility of adaptive telescopes with laser probe,” Astron. Astrophys. 152, 129–131 (1985).

Comput. Elect. Eng. (1)

M. A. Von Bokern, R. N. Paschall, B. M. Welsh, “Modal control for and adaptive optics system using LQG compensation,” Comput. Elect. Eng. 18, 421–433 (1992).

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (17)

G. Dai, “Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen–Loève functions,” J. Opt. Soc. Am. A 12, 2182–2193 (1995).
[CrossRef]

J. Stone, P. H. Hu, S. P. Mills, S. Ma, “Anisoplanatic effects in finite-aperture optical systems,” J. Opt. Soc. Am. A 11, 347–357 (1994).
[CrossRef]

N. Takato, I. Yamaguchi, “Spatial correlation of Zernike phase-expansion coefficients for atmospheric turbulence with finite outer scale,” J. Opt. Soc. Am. A 12, 958–963 (1995).
[CrossRef]

G. Molodij, G. Rousset, “Angular correlation of Zernike polynomials for a laser guide star in adaptive optics,” J. Opt. Soc. Am. A 14, 1949–1966 (1997).
[CrossRef]

G. A. Tyler, “Merging: a new method for tomography through random media,” J. Opt. Soc. Am. A 11, 409–424 (1994).
[CrossRef]

R. J. Sasiela, “Wave-front correction by one or more synthetic beacons,” J. Opt. Soc. Am. A 11, 379–393 (1994).
[CrossRef]

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

G. A. Tyler, “Wave-front compensation for imaging with off-axis guide stars,” J. Opt. Soc. Am. A 11, 339–346 (1994).
[CrossRef]

B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[CrossRef]

S. E. Troxel, B. M. Welsh, M. C. Roggemann, “Off-axis optical transfer function calculations in an adaptive-optics system by means of a diffraction calculation for weak index fluctuations,” J. Opt. Soc. Am. A 11, 2100–2111 (1994).
[CrossRef]

M. R. Whiteley, M. C. Roggemann, B. M. Welsh, “Temporal properties of the Zernike expansion coefficients of turbulence-induced phase aberrations for aperture and source motion,” J. Opt. Soc. Am. A 15, 993–1005 (1998).
[CrossRef]

D. L. Fried, J. F. Belsher, “Analysis of fundamental limits to artificial-guide-star adaptive-optics-system performance for astronomical imaging,” J. Opt. Soc. Am. A 11, 277–287 (1994).
[CrossRef]

S. Esposito, A. Riccardi, R. Ragazzoni, “Focus anisoplanatism effects on tip–tilt compensation for adaptive optics with use of a sodium laser beacon as a tracking reference,” J. Opt. Soc. Am. A 13, 1916–1923 (1996).
[CrossRef]

P. H. Hu, J. Stone, T. Stanley, “Application of Zernike polynomials to atmospheric propagation problems,” J. Opt. Soc. Am. A 6, 1595–1608 (1989).
[CrossRef]

D. G. Sandler, S. Stahl, J. R. P. Angel, M. Lloyd-Hart, D. McCarthy, “Adaptive optics for diffraction-limited infrared imaging with 8-m telescopes,” J. Opt. Soc. Am. A 11, 925–945 (1994).
[CrossRef]

D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A 8, 1568–1573 (1991).
[CrossRef]

G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” J. Opt. Soc. Am. A 11, 358–367 (1994).
[CrossRef]

Nature (4)

J. D. H. Pilkington, “Artificial guide stars for adaptive imaging,” Nature 330, 116 (1987).
[CrossRef]

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature 328, 229–231 (1987).
[CrossRef]

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature 353, 141–143 (1991).
[CrossRef]

R. Q. Fugate, D. L. Fried, G. A. Ameer, 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 353, 144–146 (1991).
[CrossRef]

Opt. Eng. (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Proc. IEEE (1)

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Other (9)

D. L. Fried, “Varieties of isoplanatism,” in Imaging Through the Atmosphere, J. C. Wyant, ed., Proc. SPIE75, 20–29 (1976).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Upper Saddle River, N.J., 1993).

M. C. Roggemann, B. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).

M. S. Belen’kii, “Fundamental limitation in adaptive optics: how to eliminate it? A full aperture tilt measurement technique with a laser guide star,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 321–323 (1994).
[CrossRef]

M. S. Belen’kii, “Full aperture tilt measurement technique with a laser guide star,” in Atmospheric Propagations and Remote Sensing IV, J. C. Dainty, ed., Proc. SPIE2471, 289–300 (1995).
[CrossRef]

M. S. Belen’kii, “Tilt angular correlation and tilt sensing techniques with a laser guide star,” in Optics in Atmospheric Propagation, Adaptive Systems, and Lidar Techniques for Remote Sensing, A. D. Devir, A. Kohnle, C. Werner, eds., Proc. SPIE2956, 206–217 (1996).
[CrossRef]

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F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1981), Vol. XIX, pp. 283–376.

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Figures (5)

Fig. 1
Fig. 1

Analysis geometry used for computing the cross correlation of Zernike coefficients from spatially separated sources.

Fig. 2
Fig. 2

(a) Normalized aperture-averaged mean square residual phase error, N2(D/r0)-5/3, for the NGS beacon offset θb/θ0 obtained with the MMSE estimator (solid curve) and obtained with beacon measurements directly (dashed curve). Horizontal lines indicate levels at which N2=1 rad2 for the corresponding value of D/r0. (b) Corresponding Strehl-ratio performance of an imaging system with relative aperture D/r0=5.

Fig. 3
Fig. 3

(a) Normalized aperture-averaged mean square residual phase error, N2(D/r0)-5/3, for the NGS beacon offset θb/θ0 obtained with the MMSE estimator (solid curve) and obtained with beacon measurements directly (dashed curve). Horizontal lines indicate levels at which N2=1 rad2 for the corresponding value of D/r0. (b) Corresponding Strehl-ratio performance of an imaging system with relative aperture D/r0=10.

Fig. 4
Fig. 4

(a) Normalized aperture-averaged mean square residual phase error, N2(D/r0)-5/3, for an on-axis LGS at altitude 10–30 km and a tilt-compensation NGS offset by θb/θ0=5 obtained with the MMSE estimator (solid curve) and obtained with beacon measurements directly (dashed curve). Horizontal lines indicate levels at which N2=1 rad2 for the corresponding values of D/r0. (b) Corresponding Strehl-ratio performance of an imaging system with relative aperture D/r0=5.

Fig. 5
Fig. 5

(a) Normalized aperture-averaged mean square residual phase error, N2(D/r0)-5/3, for an on-axis LGS at altitude 10–30 km and a tilt-compensation NGS offset by θb/θ0=5 obtained with the MMSE estimator (solid curve) and obtained with beacon measurements directly (dashed curve). Horizontal lines indicate levels at which N2=1 rad2 for the corresponding values of D/r0. (b) Corresponding Strehl-ratio performance of an imaging system with relative aperture D/r0=10.

Equations (41)

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2=dρW(ρ)E{[ϕo(Rρ)-ϕ˜o(Rρ)]2},
W(ρ)=1πif |ρ|10otherwise
Strehlratioexp(-2).
ϕ˜o(Rρ)=i=2Na˜oiZi(ρ),
N2=dρW(ρ)E{ϕo2(Rρ)}-i=2NE{aoi2}+i=2NE{(aoi-a˜oi)2},
aoidρW(ρ)ϕo(Rρ)Zi(ρ).
lima˜oiaoi N2=dρW(ρ)E{ϕo2(Rρ)}-i=2NE{aoi2}.
abidρW(ρ)ϕb(Rρ)Zi(ρ).
y=ao2ao3ao4aoN.
yˆ=aˆo2aˆo3aˆo4aˆoN.
x=ab2ab3ab4abN.
yˆ=E{y|x},
yˆ=CyxCxx-1x,
Cyx=E{yxT},
Cxx=E{xxT}.
i=2NE{(aoi-aˆoi)2}=Tr[Cyy-CyxCxx-1Cxy],
i=2NE{(aoi-abi)2}=Tr[Cyy+Cxx-2Cyx].
[Cyx]mn=E{ao(m+1)ab(n+1)}=Bao(m+1)ab(n+1)(ra, ro, rb),
[Cxx]mn=E{ab(m+1)ab(n+1)}=Bab(m+1)ab(n+1)(ra, rb, rb).
sl=Avl(rsv-ra)-Aul(rsu-ra),
Aul=zl-(ra·zˆ)(rsu-ra)·zˆ,
Avl=zl-(ra·zˆ)(rsv-ra)·zˆ.
Aul=0,
Avl=zl-(ra·zˆ)(rsv-ra)·zˆ,
sl=Avl(rsv-ra)-[zl-(ra·zˆ)]×(tan θu cos ϕuxˆ+tan θu sin ϕuyˆ+zˆ),
Aul=zl-(ra·zˆ)(rsu-ra)·zˆ,
Avl=0,
sl=[zl-(ra·zˆ)](tan θv cos ϕvxˆ+tan θv sin ϕvyˆ+zˆ)-Aul(rsu-ra),
Aul=0,
Avl=0,
sl=[zl-(ra·zˆ)][(tan θv cos ϕv-tan θu cos ϕu)xˆ+(tan θv sin ϕv-tan θu sin ϕu)yˆ].
Bauiavj(ra, rsu, rsv)=Dr05/33 Γ83 2-5/36.882.91Flwl[(1-Aul)(1-Avl)]-1×G0 dxx(x2+x02)-11/6J(mi+mj)2slDxJ(ni+1)[(1-Aul)x]J(nj+1)[(1-Avl)x]+H0 dxx(x2+x02)-11/6J|mi-mj|2slDxJ(ni+1)[(1-Aul)x]J(nj+1)[(1-Avl)x],
F[(ni+1)(nj+1)]1/2(-1)12(ni+nj)2[1-12(δmi0+δmj0)](-1)mj,
G(-1)32(mi+mj) cos(mi+mj)θsl+π4{(1-δmi0)[(-1)i-1]+(1-δmj0)[(-1)j-1]},
H(-1)32|mi-mj| cos(mi-mj)θsl+π4{(1-δmi0)[(-1)i-1]-(1-δmj0)[(-1)j-1]},
σθ2=0.182(λ/D)2(D/r0)5/3.
wl=ΔzlCn2(zl)dzCn2(z),
θ0=58.1×10-3λ6/5dzCn2(z)z5/3-3/5,
dρW(ρ)E{ϕo2(Rρ)}=Dr05/33Γ(8/3)2-5/36.882.91×310x0-5/3-20 dxx(x2+x02)-11/6J12(x),
MMSE:N2=dρW(ρ)E{ϕo2(Rρ)}-TR[CyxCxx-1Cxy].
N2=dρW(ρ)E{ϕo2(Rρ)}-TR[2Cyx-Cxx].

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