Abstract

Spatial correlation functions of phase-expansion coefficients are derived for phase fluctuations of a wave that has propagated through a layer of atmospheric turbulence. First, a general expression is derived giving the correlation of the coefficients of phase-expansion functions orthogonal over an arbitrary circularly symmetric weighting function for an isotropic turbulence spectrum. Second, the correlations are evaluated analytically for a Gaussian weight and numerically for a uniform circular weight. Finally, some of the correlations for a layer are integrated over the Hufnagel 1974 structure constant model to obtain results applicable to ground-to-space propagation.

© 1979 Optical Society of America

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  1. J. H. Shapiro, "Point-ahead limitation on reciprocal tracking," J. Opt. Soc. Am. 65, 65–68 (1975).
  2. R. K. Buchheim, R. Pringle, and H. W. Schaefgen, "Lead-angle effects in turbulence—compensating reciprocity tracking systems," Applied Optics 17, 165–166 (1978).
  3. D. P. Greenwood, "Tracking turbulence-included tilt errors with shared and adjacent apertures, J Opt. Soc. Am. 67, 282–290 (1977).
  4. D. L. Fried, "Differential angle of arrival: Theory, evaluation, and measurement feasibility, Radio Science 10, 71–76 (1975).
  5. R. E. Wagner, "Post-Processing of Imagery from Active Optics—Some Pitfalls," Proc. of the SPIE/SPSE, Imaging through the Atmosphere 75, 136 (1976).
  6. D. L. Fried, "Varieties of Isoplanatism," Proc. of the SPIE/SPSE, Imaging through the Atmosphere 75, 20 (1976).
  7. R. S. Lawrence and J. W. Strohbehn, "A Survey of Clear-Air Propagation Effects Relevant to Optical Communications," Proc. IEEE 58, 1523–1545 (1970).
  8. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic, New York, 1965) pp. 694, 933, 1058.
  9. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975) pp. 464–466, 767–772.
  10. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Govt. Printing Office, Washington, D.C., 1965) p. 793.
  11. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207–211 (1976).
  12. R. E. Hufnagel, "Variations of Atmospheric Turbulence," paper presented at Topical Meeting on Optical Propagation through Turbulence, Wal-1, July 9–11, 1974, Univ. of Colorado, Boulder, Co. (unpublished).
  13. J. S. Meditch, Stochastic Optimal Linear Estimation and Control, (McGraw-Hill, New York, 1969)

1978 (1)

R. K. Buchheim, R. Pringle, and H. W. Schaefgen, "Lead-angle effects in turbulence—compensating reciprocity tracking systems," Applied Optics 17, 165–166 (1978).

1977 (1)

D. P. Greenwood, "Tracking turbulence-included tilt errors with shared and adjacent apertures, J Opt. Soc. Am. 67, 282–290 (1977).

1976 (1)

1975 (2)

D. L. Fried, "Differential angle of arrival: Theory, evaluation, and measurement feasibility, Radio Science 10, 71–76 (1975).

J. H. Shapiro, "Point-ahead limitation on reciprocal tracking," J. Opt. Soc. Am. 65, 65–68 (1975).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Govt. Printing Office, Washington, D.C., 1965) p. 793.

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975) pp. 464–466, 767–772.

Buchheim, R. K.

R. K. Buchheim, R. Pringle, and H. W. Schaefgen, "Lead-angle effects in turbulence—compensating reciprocity tracking systems," Applied Optics 17, 165–166 (1978).

Fried, D. L.

D. L. Fried, "Differential angle of arrival: Theory, evaluation, and measurement feasibility, Radio Science 10, 71–76 (1975).

D. L. Fried, "Varieties of Isoplanatism," Proc. of the SPIE/SPSE, Imaging through the Atmosphere 75, 20 (1976).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic, New York, 1965) pp. 694, 933, 1058.

Greenwood, D. P.

D. P. Greenwood, "Tracking turbulence-included tilt errors with shared and adjacent apertures, J Opt. Soc. Am. 67, 282–290 (1977).

Hufnagel, R. E.

R. E. Hufnagel, "Variations of Atmospheric Turbulence," paper presented at Topical Meeting on Optical Propagation through Turbulence, Wal-1, July 9–11, 1974, Univ. of Colorado, Boulder, Co. (unpublished).

Lawrence, R. S.

R. S. Lawrence and J. W. Strohbehn, "A Survey of Clear-Air Propagation Effects Relevant to Optical Communications," Proc. IEEE 58, 1523–1545 (1970).

Meditch, J. S.

J. S. Meditch, Stochastic Optimal Linear Estimation and Control, (McGraw-Hill, New York, 1969)

Noll, R. J.

Pringle, R.

R. K. Buchheim, R. Pringle, and H. W. Schaefgen, "Lead-angle effects in turbulence—compensating reciprocity tracking systems," Applied Optics 17, 165–166 (1978).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic, New York, 1965) pp. 694, 933, 1058.

Schaefgen, H. W.

R. K. Buchheim, R. Pringle, and H. W. Schaefgen, "Lead-angle effects in turbulence—compensating reciprocity tracking systems," Applied Optics 17, 165–166 (1978).

Shapiro, J. H.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Govt. Printing Office, Washington, D.C., 1965) p. 793.

Strohbehn, J. W.

R. S. Lawrence and J. W. Strohbehn, "A Survey of Clear-Air Propagation Effects Relevant to Optical Communications," Proc. IEEE 58, 1523–1545 (1970).

Wagner, R. E.

R. E. Wagner, "Post-Processing of Imagery from Active Optics—Some Pitfalls," Proc. of the SPIE/SPSE, Imaging through the Atmosphere 75, 136 (1976).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975) pp. 464–466, 767–772.

Applied Optics (1)

R. K. Buchheim, R. Pringle, and H. W. Schaefgen, "Lead-angle effects in turbulence—compensating reciprocity tracking systems," Applied Optics 17, 165–166 (1978).

J Opt. Soc. Am. (1)

D. P. Greenwood, "Tracking turbulence-included tilt errors with shared and adjacent apertures, J Opt. Soc. Am. 67, 282–290 (1977).

J. Opt. Soc. Am. (2)

Radio Science (1)

D. L. Fried, "Differential angle of arrival: Theory, evaluation, and measurement feasibility, Radio Science 10, 71–76 (1975).

Other (8)

R. E. Wagner, "Post-Processing of Imagery from Active Optics—Some Pitfalls," Proc. of the SPIE/SPSE, Imaging through the Atmosphere 75, 136 (1976).

D. L. Fried, "Varieties of Isoplanatism," Proc. of the SPIE/SPSE, Imaging through the Atmosphere 75, 20 (1976).

R. S. Lawrence and J. W. Strohbehn, "A Survey of Clear-Air Propagation Effects Relevant to Optical Communications," Proc. IEEE 58, 1523–1545 (1970).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic, New York, 1965) pp. 694, 933, 1058.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975) pp. 464–466, 767–772.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Govt. Printing Office, Washington, D.C., 1965) p. 793.

R. E. Hufnagel, "Variations of Atmospheric Turbulence," paper presented at Topical Meeting on Optical Propagation through Turbulence, Wal-1, July 9–11, 1974, Univ. of Colorado, Boulder, Co. (unpublished).

J. S. Meditch, Stochastic Optimal Linear Estimation and Control, (McGraw-Hill, New York, 1969)

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