Abstract

Power series formulas are developed to efficiently compute the covariance of the integrated turbulence-induced phase distortions along a pair of ray paths through the atmosphere from two points in a telescope aperture to a pair of sources at finite or infinite range. These covariances may be used to evaluate and optimize the predicted performance of adaptive optical (AO) systems. The power series formulas apply when one or both of the phase distortions is temporally filtered by the closed-loop impulse response function of the AO control loop, thereby allowing the effects of a finite servo bandwidth to be included in AO system performance modeling without the introduction of additional numerical integrations. Results are presented for the Kolomogorov turbulence spectrum with an infinite outer scale, as well as for the case of a finite outer scale with the von Kármán turbulence spectrum. Amplitude scintillation effects are neglected. The Taylor, or frozen flow, hypothesis is used to model the temporal behavior of the turbulence, by using a fixed windspeed profile w(z) and a random wind direction profile θw(z) for which the mean values of the quantities cos[kθw(z)] and sin[kθw(z)] can be computed. The resulting formulas for the covariances are weighted integrals of the refractive-index structure constant Cn2(z), where the weighting functions are power series in from one to three indices depending on the choices made regarding the atmospheric turbulence spectrum and the direction of the wind. The integral with respect to z may also be evaluated analytically, provided that (i) the Cn2(z) profile is a sum of terms of the form zp exp(-cz) and (ii) the phase distortion profiles are not temporally filtered.

© 1999 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

1997 (1)

1996 (1)

1994 (2)

F. Dalaudier, C. Sidi, M. Crochet, J. Vernin, “Direct evidence of ‘sheets’ in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (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]

1991 (1)

1984 (1)

1983 (1)

Allen, J. G.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Ameer, G. A.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Brown, J. M.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Christou, J. C.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

D. J. Lee, B. L. Ellerbroek, J. C. Christou, “First results with the Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–75 (1998).

Crochet, M.

F. Dalaudier, C. Sidi, M. Crochet, J. Vernin, “Direct evidence of ‘sheets’ in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

Dalaudier, F.

F. Dalaudier, C. Sidi, M. Crochet, J. Vernin, “Direct evidence of ‘sheets’ in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

Duncan, T. S.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Eager, R. J.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Ealey, M. A.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Ellerbroek, B. L.

B. L. Ellerbroek, “Including outer scale effects in zonal adaptive optics calculations,” Appl. Opt. 36, 9456–9467 (1997).
[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]

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

V. P. Pauca, B. L. Ellerbroek, N. P. Pitsianis, R. J. Plemmons, X. Sun, “Performance modeling of adaptive-optics imaging systems using fast Hankel transforms,” in Advanced Signal Processing Algorithms, Architectures, and Implementations VII, F. Luk, ed., Proc. SPIE3461, 339–347 (1998).
[CrossRef]

D. J. Lee, B. L. Ellerbroek, J. C. Christou, “First results with the Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–75 (1998).

Fugate, R. Q.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Gardner, C. S.

Hirsch, M. W.

M. W. Hirsch, S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, New York, 1974).

Jones, G. W.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Kuhns, R. M.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Lee, D. J.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

D. J. Lee, B. L. Ellerbroek, J. C. Christou, “First results with the Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–75 (1998).

Link, D. J.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Lowrey, W. H.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Oliker, M. D.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Pauca, V. P.

V. P. Pauca, B. L. Ellerbroek, N. P. Pitsianis, R. J. Plemmons, X. Sun, “Performance modeling of adaptive-optics imaging systems using fast Hankel transforms,” in Advanced Signal Processing Algorithms, Architectures, and Implementations VII, F. Luk, ed., Proc. SPIE3461, 339–347 (1998).
[CrossRef]

Pitsianis, N. P.

V. P. Pauca, B. L. Ellerbroek, N. P. Pitsianis, R. J. Plemmons, X. Sun, “Performance modeling of adaptive-optics imaging systems using fast Hankel transforms,” in Advanced Signal Processing Algorithms, Architectures, and Implementations VII, F. Luk, ed., Proc. SPIE3461, 339–347 (1998).
[CrossRef]

Plemmons, R. J.

V. P. Pauca, B. L. Ellerbroek, N. P. Pitsianis, R. J. Plemmons, X. Sun, “Performance modeling of adaptive-optics imaging systems using fast Hankel transforms,” in Advanced Signal Processing Algorithms, Architectures, and Implementations VII, F. Luk, ed., Proc. SPIE3461, 339–347 (1998).
[CrossRef]

Ruane, R. E.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Rudin, W.

W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1994).

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, Berlin, 1994).

Sidi, C.

F. Dalaudier, C. Sidi, M. Crochet, J. Vernin, “Direct evidence of ‘sheets’ in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

Smale, S.

M. W. Hirsch, S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, New York, 1974).

Spinhirne, J. M.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Sun, X.

V. P. Pauca, B. L. Ellerbroek, N. P. Pitsianis, R. J. Plemmons, X. Sun, “Performance modeling of adaptive-optics imaging systems using fast Hankel transforms,” in Advanced Signal Processing Algorithms, Architectures, and Implementations VII, F. Luk, ed., Proc. SPIE3461, 339–347 (1998).
[CrossRef]

Swindle, D. W.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

Tyler, G. A.

Vernin, J.

F. Dalaudier, C. Sidi, M. Crochet, J. Vernin, “Direct evidence of ‘sheets’ in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

Voas, J. R.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Wallner, E. P.

Welsh, B. M.

Wild, W. J.

W. J. Wild, “Predictive optimal estimators for adaptive optics systems,” Opt. Lett. 21, 1433–1435 (1996).
[CrossRef] [PubMed]

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Wilson, K. B.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Wynia, J. L.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

Appl. Opt. (1)

J. Atmos. Sci. (1)

F. Dalaudier, C. Sidi, M. Crochet, J. Vernin, “Direct evidence of ‘sheets’ in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Lett. (1)

Other (12)

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. R. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–03 (1998).

D. J. Lee, B. L. Ellerbroek, J. C. Christou, “First results with the Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical System Technologies, D. Bonaccini, ed., Proc. SPIE3353, 3353–75 (1998).

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, Berlin, 1994).

W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1994).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

V. P. Pauca, B. L. Ellerbroek, N. P. Pitsianis, R. J. Plemmons, X. Sun, “Performance modeling of adaptive-optics imaging systems using fast Hankel transforms,” in Advanced Signal Processing Algorithms, Architectures, and Implementations VII, F. Luk, ed., Proc. SPIE3461, 339–347 (1998).
[CrossRef]

M. W. Hirsch, S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, New York, 1974).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), Eq. 9.1.79, p. 363.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), Eq. 6.1.1, p. 255.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), Eq. 6.1.18, p. 256.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), Eq. 6.1.15, p. 256.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), Eq. 6.1.22, p. 256.

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Figures (4)

Fig. 1
Fig. 1

Coordinate system for ray paths and atmospheric profiles. Here r is a point in the telescope aperture plane, and the z axis of the coordinate system is the optical axis of the telescope. The ray path from a point source to the point r is denoted by the curve [p(r, z), z], where z is the range from the aperture and p(r, z) is the offset of the ray path from the optical axis at this range. The curve [p(r, z), z] is the ray path from a second point source to a second aperture point r. The same coordinate system is also used for the atmospheric turbulence and wind velocity profiles.

Fig. 2
Fig. 2

I3 computation errors for the power series and asymptotic series evaluation formulas. These results are for the parameters L0=, w=5m/s, k=0, p=0, and c=100π.

Fig. 3
Fig. 3

I3 computation errors with an infinite outer scale. In each subfigure the top three curves plot the magnitude of the function I3 for the three values of the windspeed parameter w and the bottom three curves plot the minimum of the numerical errors in evaluating I3 with the power series and asymptotic series evaluation formulas. Parts (a), (b), and (c) correspond to the parameter values k=0, p=0, k=0, p=2, and k=10, p=0, respectively. c=100π for all three subfigures.

Fig. 4
Fig. 4

I3 computation error with a 10-m outer scale. These plots are similar to those in Fig. 2, with analogous parameters values except for the 10-m outer scale.

Tables (1)

Tables Icon

Table 1 Parameter Values for Numerical Results

Equations (88)

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C=ϕ(r, t)ϕ(r, t),
C=0.144r05/30dzCn2(z)-10ZdzCn2(z)×I0(L0, p(r, z)-p(r, z), (t-t)w(z)).
I0(L0, d, τw)=0 κdκ(κ2+L0-2)11/6J0(2πκ|d+τw|)w,
φ(r, t)=0dτh(τ)ϕ(r, t-τ),
φ(r, t)=0dτh(τ)ϕ(r, t-τ),
h(τ)=i=1naiτpi exp(-ciτ),
h1(τ)=2πf exp(-2πfτ),
h2(τ)=4πf exp(-2πfτ)cos(2πfτ),
I1=I1(p, c, L0, d, w)=0dττp exp(-cτ)I0(L0, d, τw)=0dττp exp(-cτ)0 κdκ(κ2+L0-2)11/6×J0(2πκ|d+τw|)w,
φ(r, t)ϕ(r, t)=i=1nai0dττpi exp(-ciτ)×ϕ(r, t-τ)ϕ(r, t).
0dττpi exp(-ciτ)ϕ(r, t-τ)ϕ(r, t)=0.144r05/3dzCn2(z)-1×0ZdzCn2(z)0dττpi exp(-ciτ)×I0(L0, p(r, z)-p(r, z), τw(z))=0.144r05/3dzCn2(z)-1×0ZdzCn2(z)I1(pi, ci, L0, p(r, z)-p(r, z), w(z)).
φ(r, t)φ(r, t)=i=1nj=1naiaj00dτdττpi(τ)pj×exp(-ciτ-cjτ)×ϕ(r, t-τ)ϕ(r, t-τ).
τ+=12(τ+τ),
τ-=τ-τ,
I2=00dτdττp(τ)q exp(-λτ-μτ)×ϕ(r, t-τ)ϕ(r, t-τ)=-dτ-(1/2)|τ-|dτ+(τ++12τ-)p(τ+-12τ-)q×exp[-(λ+μ)τ+-12(λ-μ)τ-]×ϕ(r, t-τ+-12τ-)ϕ(r, t-τ++12τ-).
I2=n=0ppn12nm=0qqm-12m×-dτ-τ-n+m exp[-12(λ-μ)τ-]×ϕ(r, t-τ-)ϕ(r, t)×(1/2)|τ-|dτ+τ+p+q-n-m exp[-(λ+μ)τ+].
I2=n=0ppn12nm=0qqm-12m×l=0p+q-n-m12p+q-n-m-l(λ+μ)-l-1×(p+q-n-m)!(p+q-n-m-l)!×-dτ|τ|p+q-n-m-lτn+m×exp{-12[(λ-μ)τ+(λ+μ)|τ|]}×ϕ(r, t-τ)ϕ(r, t).
I2=12p+q(λ+μ)n=0ppnm=0qqm×l=0p+q-n-m2λ+μl ×(p+q-n-m)!(p+q-n-m-l)!(-1)m0dττp+q-l exp(-λτ)×ϕ(r, t)ϕ(r, t-τ)+(-1)n0dττp+q-l exp(-μτ)×ϕ(r, t)ϕ(r, t-τ).
Cn2(z)=i=1nzpi exp(-ciz),
p(r, z)=r+zθ(r),
p(r, z)=r+zθ(r),
>I1=I1(p, c, L0, d, w)=0dττp exp(-cτ)0 κdκ(κ2+L0-2)11/6×J0(2πκ|d+τw|)w.
J0(2πκ|d+τw|)=J0(2πκd)J0(2πτκw)+2k=1 cos[k(θd-θw)]×Jk(2πκd)Jk(2πτκw).
cos[k(θd-θw)]Jk(2πτκw)w=Jk(2πτκw)cos[k(θd-θ¯w)]cos[k(θ¯w-θw)]w=Jk(2πτκw)cos[k(θd-θ¯w)][sin(kδ)/(kδ)].
I3=I3(d, w, k, p, c, L0)=0dττp exp(-cτ)0 κdκ(κ2+L0-2)11/6×Jk(2πκd)Jk(2πτκw),
I1=I3(d, w, 0, p, c, L0)+2k=1 cos[k(θd-θw¯)]×sin kδkδI3(d, w, k, p, c, L0).
τcτ,
κκ/L0.
I3(d, w, k, p, c, L0)=L05/3cp+10dττp exp(-τ)I42πdL0, 2πwτcL0, k,
I4(a, b, k)=0 κdκ(1+κ2)11/6Jk(aκ)Jk(bκ).
I4(a, b, k)=18Γ(11/6)(2πi)2t0-it0+idts0-is0+ids×Γ1+s2+t2, 56-s2-t2, k2-s2, k2-t21+k2+s2, 1+k2+t2×a2sb2t,
Γa1,, anb1,, bm=Γ(a1)Γ(an)Γ(b1)Γ(bm).
ss/2,
tt/2,
u=πdL02,
v=2πwcL02,
I3=I3(d, w, k, p, c, L0)=L05/3c-p-12Γ(11/6)(2πi)2t0-it0+idts0-is0+ids×Γ1+s+t, 56-s-t, k2-s, k2-t1+k2+s, 1+k2+tu2v4t×0dττp+2t exp(-τ).
0dττp+2t exp(-τ)=Γ(1+p+2t)=2p+2tπΓ(12+p2+t)Γ(1+p2+t).
I3=L05/3c-p-12p-1πΓ(11/6)(2πi)2t0-it0+idts0-is0+ids×Γ1+s+t, 56-s-t, k2-s, k2-t, 12+p2+t, 1+p2+t1+k2+s, 1+k2+tusvt,
s0=t0=-1/6+,
k=2l+j,
I3=L05/3c-p-12p-1πΓ(11/6)(2πi)n=0 (-1)nunn!×uk/2(n+k)!t0-it0+idtΓ1+k2+n+t, 56-k2-n-t, k2-t, 12+p2+t, 1+p2+t1+k2+tvt+u5/6t0-it0+idtΓ116+n,-56+k2-n+t, k2-t, 12+p2+t, 1+p2+t116+k2+n-t, 1+k2+tvut,
 12πit0-it0+idtΓ1+k2+n+t, 56-k2-n-t, k2-t, 12+p2+t, 1+p2+t1+k2+tvt=R1+R2for|v|<1R3+R4+R5for|v|>1,
R1=m=0 (-1)mm!vk/2+m(k+m)!×Γ[1+k+n+m, 56-k-n-m, 12+k2+p2+m, 1+k2+p2+m],
R2=m=l+n (-1)mm!v5/6-k/2-n+m×Γ116+m,-56+k+n-m, 43-k2+p2-n+m, 116-k2+p2-n+m116-n+m,
R3=-m=0n+l-1 (-1)mm!v5/6-k/2-n+m×Γ116+m,-56+k+n-m,43-k2+p2-n+m, 116-k2+p2-n+m116-n+m,
 R4=m=0 (-1)mm!v-1/2-p/2-m×Γ12+k2-p2+n-m, 43-k2+p2-n+m, 12+k2+p2+m, 12-m12+k2-p2-m,
R5=m=0 (-1)mm!v-1-p/2-m×Γk2-p2+n-m, 116-k2+p2-n+m, 1+k2+p2+m,-12-mk2-p2-m.
 12πit0-it0+idtΓ116+n,-56+k2-n+t, k2-t, 12+p2+t, 1+p2+t116+k2+n-t, 1+k2+t(v/u)t=m=m1 (-1)mm!(v/u)5/6-k/2+n-mΓ116+n,-56+k-n+m, 43-k2+p2+n-m, 116-k2+p2+n-m1+k+m,116+n-m+m=0 (-1)mm!(v/u)-1/2-p/2-mΓ116+n,-43+k2-p2-n-m, 12+k2+p2+m, 12-m73+k2+p2+n+m, 12+k2-p2-m+m=0 (-1)mm!(v/u)-1-p/2-mΓ116+n,-116+k2-p2-n-m, 1+k2+p2+m,-12-m176+k2+p2+n+m, k2-p2-m,
m1=max(0, 1-l+n).
m2=max(0, 1-k+n-ma).
12πit0-it0+idtΓ116+n,-56+k2-n+t, k2-t, 12+p2+t, 1+p2+t116+k2+n-t, 1+k2+t(v/u)tm=0ma (-1)mm!(v/u)k/2+mΓ116+n,-56+k-n+m, 12+k2+p2+m, 1+k2+p2+m1+k+m, 116+n-m-m=m2n-l (-1)mm!(v/u)5/6-k/2+n-mΓ116+n,-56+k-n+m, 43-k2+p2+n-m, 116-k2+p2+n-m1+k+m, 116+n-m.
Γ(x+1)=xΓ(x).
(a)n=Γ(a+n)Γ(a).
(a)0=1,
(a)n=(a+n-1)(a)n-1,
(a)-k=(-1)k(1-a)k.
a1,,anb1,,bmk=(a1)k(an)k(b1)k(bm)k.
I3(d, w, k, p, c, L0)=2p-1L05/3πΓ(11/6)cp+1×[S1(k, p, u, v)+S2(k, p, u, v)],
S1(k, p, u, v)=T1+T2for|v|<1T3+T4+T5for|v|>1,
T1=(-1)kΓ56, 16, 12+k2+p2,1+k2+p21+k, 16+kuk/2vk/2n=0 un(1, 16+k)n×m=0vm1+k+n, 12+k2+p2, 1+k2+p21, 1+k, 16+k+nm,
T2=(-1)lΓ-56+k-l, 43-j2+p2, 116-j2+p2k+1, l+1uk/2v5/6-j/2n=0un116+l1, 1+k, 1+ln×m=0vm116+l+n, 43-j2+p2, 116-j2+p21+l+n, 116-k+l, 116+lm,
T3=-Γ-56+k, 43-k2+p2, 116-k2+p2k+1(u/v)k/2v5/6n=0(u/v)n-56,-56+k1, 1+k,-13+k2-p2,-56+k2-p2n×m=0l-1+nvm116, 43-k2+p2-n,116-k2+p2-n1, 116-k-n, 116-nm,
T4=Γ12, 12+k2+p2, 43-k2+p2k+1uk/2v-1/2-p/2×n=0v-n43-k2+p2,12+k2+p21, 12nm=0um12+k2-p2-n1, 1+k,-13+k2-p2-nm
T5=Γ-12, 1+k2+p2, 116-k2+p2k+1uk/2v-1-p/2×n=0v-n116-k2+p2, 1+k2+p232, 1nm=0umk2-p2-n1, 1+k,-56+k2-p2-nm,
S2(k,p,u,v)=(1)l1Γ[16+kl,1312+p2,56j2+p2k+1,l](u/v)k/2v1/6+l×n=0l1vn(1l,16+kl23+j2p2,16+j2p2)n×m=0l1(u/v)m(16+kl+n,16l+n1,k+1,23+j2p2,56j2+p2+n)m(1)lΓ[116+l,16+kl,13j2+p2,56j2+p22+k,1+l,56+l]u1+k/2v1/6j/2×n=0un(116+l2,k+2,l+1)nm=0(u/v)m(16+kl,16l2+n,2+k+n,23+j2p2,16+j2p2)m+Γ[116,43+k2p2,12+k2+p2,1273+k2+p2,12+k2p2]u5/6(u/v)1/2+p/2n=0un(1161,73+k2+p2,73k2+p2)n×m=0(u/v)m(12+k2+p2,12k2+p21,12,73+k2+p2+n,73k2+p2+n)m +Γ116,-116+k2-p2, 1+k2+p2,-12176+k2+p2, k2-p2u5/6(u/v)1+p/2 n=0un1161, 176+k2+p2, 176-k2+p2n×m=0(u/v)m1+k2+p2, 1-k2+p21, 32176+k2+p2+n, 176-k2+p2+nm.
S2(k, p, u, v)Γ-56+k, 12+k2+p2, 1+k2+p2k+1u5/6(v/u)k/2n=0 un(1, 116-k)n×m=0ma(n)(v/u)m-56+k-n,-56-n, 12+k2+p2, 1+k2+p21, k+1m-(-1)lΓ-56+k-l, 43-j2+p2, 116-j2+p2k+1, l+1uk/2v5/6-j/2×n=0un116+l1, k+1, l+1nm=0m3(n)(v/u)m-n,-n-k, 43-j2+p2, 116-j2+p2116+l, 116-k+lm,
m3(n)=min[n, k-l-1+ma(n)].
I3(d, w, k, p, c, L0)=L05/3cp+1I3dL0, wcL0, k, p, 1, 1.
I3(d, w, k, p, c, )=limL0L05/3cp+1I3dL0, wcL0, k, p, 1, 1.
u=πdr02=L0r02u,
v=2πwcr02=L0r02v,
u/v=u/v=(dc/2w)2,
I3=I3(d, w, k, p, c, )=2p-1r05/3πΓ(11/6)cp+1[S1(k, p, u, v)+S2(k, p, u, v)],
S1=T1+T2,
T1=(L0/r0)5/3Γ[56, 12+p2, 1+p2]fork=00otherwise,
T2=Γ-56,43-j2+p2,116k+1(u/v)k/2(v)5/6fork10otherwise.
S2=T3+T5+T6fork1T4+T5+T6otherwise,
 T3=-Γ116, 16+k, 13-j2+p2, 56-j2+p22+k, 56(u/v)k/2+1(v)5/6×m=0(u/v)m16+k, 162, 2+k, 23+j2-p2, 16+j2-p2m,
T4=(-1)l-1Γ16+k-l, 13-j2+p2,56-j2+p2k+1,l1-l, 16+k-l23+j2-p2, 16+j2-p2l-1(u/v)k/2(v)5/6×m=0(u/v)m-56+k,-561, 1+k,-13+j2-p2+l,-56+j2-p2+lm,
T5=Γ116,-43+k2-p2, 12+k2+p2, 1273+k2+p2, 12+k2-p2(u/v)p/2+1/2(u)5/6m=0(u/v)m12+k2+p2, 12-k2+p21, 12, 73+k2+p2, 73-k2+p2m,
T6=Γ116,-116+k2-p2,1+k2+p2,-12176+k2+p2, k2-p2(u/v)p/2+1(u)5/6m=0(u/v)m1+k2+p2, 1-k2+p21, 32, 176+k2+p2, 176-k2+p2m.
S2T7+T8,
T7=Γ-56+k, 12+k2+p2, 1+k2+p2k+1×(v/u)k/2(u)5/6m=0ma(v/u)m×-56+k,-56, 12+k2+p2, 1+k2+p21, k+1m,
T8=-Γ-56+k, 43-j2+p2, 116-j2+p2k+1×(u/v)k/2(v)5/6fork1andk-1+ma00otherwise.
[φ(r, t)-φ(r, t)]2=[φ(r, t)]2+[φ(r, t)]2-2φ(r, t)φ(r, t),
ϕ(r, t)ϕ(r, t)=0.144r05/30dzCn2(z)-1i=1n0Zdzzpi exp(-ciz)×0 κdκ(κ2+L0-2)11/6×J0(2πκ|r-r+z[θ(r)-θ(r)]|)=0.144r05/30dzCn2(z)-1×i=1n0Zdzzpi exp(-ciz)×I0(L0, r-r, z[θ(r)-θ(r)]|).
0Zdzzpi exp(-ciz)×I0(L0,r-r, z[θ(r)-θ(r)])=0dzzpi exp(-ciz)×I0(L0, r-r, z[θ(r)-θ(r)])-Zdzzpi exp(-ciz)×I0(L0, r-r, z[θ(r)-θ(r)])=I1[pi,ci,L0,r-r,θ(r)-θ(r)]-0dz(z+Z)pi exp[-ci(z+Z)]I0(L0, r-r,(z+Z)[θ(r)-θ(r)])=I1(pi, ci, L0, r-r, θ(r)-θ(r))-exp(-ciZ)j=0pipijZpi-jI1(j, ci, L0, r-r+Z[θ(r)-θ(r)],θ(r)-θ(r)),

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