Abstract

A general expression of the spatial correlation functions of quantities related to the phase fluctuations of a wave that have propagated through the atmospheric turbulence are derived. A generalization of the method to integrand containing the product of an arbitrary number of hypergeometric functions is presented. The formalism is able to give the coefficients of phase-expansion functions orthogonal over an arbitrary circularly symmetric weighting function for an isotropic turbulence spectrum, as well as to describe the effect of the finite outer and inner scales of the turbulence and to describe the spherical propagation or to derive the effects of the analytical operators acting on the phase such as the derivatives of any order. The derivation of the generalized integrals with multiparameters is based on the Mellin transforms integration method.

© 2011 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium(Dover, 1961).
  2. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposure,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  3. R. F. Lutomirski and H. Yura, “Wave structure function and mutual coherence function of an optical wave in a turbulent atmosphere,” J. Opt. Soc. Am. A 61, 482–487 (1971).
    [CrossRef]
  4. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
    [CrossRef]
  5. G. C. Valley, “Long and short term Strehl ratio for turbulence with finite inner and outer scales,” Appl. Opt. 18, 984–987(1979).
    [CrossRef] [PubMed]
  6. A. Consortini, L. Ronchi, and E. Moroder, “Role of the outer scale of turbulence in atmospheric degradation of optical images,” J. Opt. Soc. Am. 63, 1246–1248 (1973).
    [CrossRef]
  7. 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]
  8. M. Bertolotti, M. Carnevale, A. Consortini, and L. G. Ronchi, “Optical propagation—problems and trends,” Opt. Acta 26, 507–529 (1979).
    [CrossRef]
  9. R. G. Frehlich and G. R. Ochs, “Effects of saturation on the optical scintillometer,” Appl. Opt. 29, 548–553, (1990).
    [CrossRef] [PubMed]
  10. J. H. Churnside and S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987).
    [CrossRef]
  11. J. Borgnino, F. Martin, and A. Ziad, “Effect of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91, 267–279 (1992).
    [CrossRef]
  12. G. C. Valley and S. M. Wandzura, “Spatial correlation of phase expension coefficients for propagation through atmospheric turbulence,” J. Opt. Soc. Am. 69, 712–717 (1979).
    [CrossRef]
  13. F. Chassat, “Calcul du domaine d’isoplanetisme d’un systeme d’optique adaptative fonctionnant a travers la turbulence atmospherique,” J. Opt. 20, 13–23 (1989).
    [CrossRef]
  14. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  15. S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J.W.Strohbehn, ed. (Springer, 1978), pp. 9–43.
  16. G. Molodij and G. Rousset, “Angular correlation of Zernike polynomial coefficients using laser guide star in adaptive optics,” J. Opt. Soc. Am. A 14, 1949–1966 (1997).
    [CrossRef]
  17. R. J. Sasiela and J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of the outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1993).
    [CrossRef]
  18. R. E. Hufnagel, Optical Propagation Through Turbulence, OSA Technical Digest Series, (Optical Society of America, 1974), pp. WA1-1–WA1-4.
  19. D. J. Shelton and R. J. Sasiela, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2618 (1993).
    [CrossRef]
  20. G. A. Tyler, “Analysis of propagation through turbulence: evaluation of an integral involving the product of three Bessel fonctions,” J. Opt. Soc. Am. A 7, 1218–1223 (1990).
    [CrossRef]
  21. F. Roddier, “The effect of atmospheric turbulence in optical astromomy,” in Progress in Optics, E.Wolf, ed., (North Holland, 1981), Vol.  19, pp. 283–376.
    [CrossRef]
  22. A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large reynolds,” Dokl. Akad. Nauk SSSR 30, 301–305 (1941).
  23. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225(1988).
    [CrossRef] [PubMed]
  24. G. Molodij and J. Rayrole, “Performance analysis for T.H.E.M.I.S(*) image stabilizer optical system. II. Anisoplanatism limitations (*) Telescope Heliographique pour l’Etude DU Magnetisme et des Instabilites de l’atmosphere Solaire,” Astron. Astrophys. Suppl. 128, 229–244 (1998).
    [CrossRef]
  25. G. Rousset, “Adaptative optics,” in Adaptive Optics for Astronomy, D.Alloin and J.M.Mariotti, ed., NATO Advanced Study Institute Series (Kluwer, 1994), pp. 115–137.
  26. E. Gendron and P. Lena, “Astronomical adaptive optics. I: Modal control optimization,” Astron. Astrophys. 291, 337–347(1994).
  27. R. Dautray and J. L. Lions, Transformations de Mellin Analyse Mathematique et Calcul Numerique (Masson, 1987), Vol.  3.
  28. S. Colombo, Les Transformations de Mellin et de Hankel (CNRS, 1959).
  29. J. M. Conan, P. Y. Madec, and G. Rousset, “Wavefront temporal spectra in high resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
    [CrossRef]
  30. F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. I,” Astron. Astrophys. Suppl. 108, 173–180(1994).
  31. A. Agabi, J. Borgnino, F. Martin, A. Tokovinin, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. II,” Astron. Astrophys. Suppl. Ser. 109, 557–562(1995).
  32. J. Primot, G. Rousset, J. Fontanella, “Deconvolution from wave-front sensing—a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608(1990).
    [CrossRef]
  33. G. Molodij, J. Rayrole, P. Y. Madec, and F. Colson, “Performance analysis for T.H.E.M.I.S(*) image stabilizer optical system. I,” Astron. Astrophys. Suppl. Ser. 118, 169–179 (1996).
    [CrossRef]
  34. R. J. Sasiela, “A unified approch to electromagnetic wave propagation in turbulence and evaluation of multiparameter integrals,” Tech. Rep. 807 (MIT Lincoln Laboratory, 1988).

1998

G. Molodij and J. Rayrole, “Performance analysis for T.H.E.M.I.S(*) image stabilizer optical system. II. Anisoplanatism limitations (*) Telescope Heliographique pour l’Etude DU Magnetisme et des Instabilites de l’atmosphere Solaire,” Astron. Astrophys. Suppl. 128, 229–244 (1998).
[CrossRef]

1997

1996

G. Molodij, J. Rayrole, P. Y. Madec, and F. Colson, “Performance analysis for T.H.E.M.I.S(*) image stabilizer optical system. I,” Astron. Astrophys. Suppl. Ser. 118, 169–179 (1996).
[CrossRef]

1995

A. Agabi, J. Borgnino, F. Martin, A. Tokovinin, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. II,” Astron. Astrophys. Suppl. Ser. 109, 557–562(1995).

J. M. Conan, P. Y. Madec, and G. Rousset, “Wavefront temporal spectra in high resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
[CrossRef]

1994

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

F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. I,” Astron. Astrophys. Suppl. 108, 173–180(1994).

1993

D. J. Shelton and R. J. Sasiela, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2618 (1993).
[CrossRef]

R. J. Sasiela and J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of the outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1993).
[CrossRef]

1992

J. Borgnino, F. Martin, and A. Ziad, “Effect of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91, 267–279 (1992).
[CrossRef]

1991

1990

1989

F. Chassat, “Calcul du domaine d’isoplanetisme d’un systeme d’optique adaptative fonctionnant a travers la turbulence atmospherique,” J. Opt. 20, 13–23 (1989).
[CrossRef]

1988

1987

1979

1976

1975

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

1973

1971

R. F. Lutomirski and H. Yura, “Wave structure function and mutual coherence function of an optical wave in a turbulent atmosphere,” J. Opt. Soc. Am. A 61, 482–487 (1971).
[CrossRef]

1966

1941

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large reynolds,” Dokl. Akad. Nauk SSSR 30, 301–305 (1941).

Agabi, A.

A. Agabi, J. Borgnino, F. Martin, A. Tokovinin, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. II,” Astron. Astrophys. Suppl. Ser. 109, 557–562(1995).

F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. I,” Astron. Astrophys. Suppl. 108, 173–180(1994).

Bertolotti, M.

M. Bertolotti, M. Carnevale, A. Consortini, and L. G. Ronchi, “Optical propagation—problems and trends,” Opt. Acta 26, 507–529 (1979).
[CrossRef]

Borgnino, J.

A. Agabi, J. Borgnino, F. Martin, A. Tokovinin, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. II,” Astron. Astrophys. Suppl. Ser. 109, 557–562(1995).

F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. I,” Astron. Astrophys. Suppl. 108, 173–180(1994).

J. Borgnino, F. Martin, and A. Ziad, “Effect of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91, 267–279 (1992).
[CrossRef]

Carnevale, M.

M. Bertolotti, M. Carnevale, A. Consortini, and L. G. Ronchi, “Optical propagation—problems and trends,” Opt. Acta 26, 507–529 (1979).
[CrossRef]

Chassat, F.

F. Chassat, “Calcul du domaine d’isoplanetisme d’un systeme d’optique adaptative fonctionnant a travers la turbulence atmospherique,” J. Opt. 20, 13–23 (1989).
[CrossRef]

Churnside, J. H.

Clifford, S. F.

J. H. Churnside and S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987).
[CrossRef]

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J.W.Strohbehn, ed. (Springer, 1978), pp. 9–43.

Colombo, S.

S. Colombo, Les Transformations de Mellin et de Hankel (CNRS, 1959).

Colson, F.

G. Molodij, J. Rayrole, P. Y. Madec, and F. Colson, “Performance analysis for T.H.E.M.I.S(*) image stabilizer optical system. I,” Astron. Astrophys. Suppl. Ser. 118, 169–179 (1996).
[CrossRef]

Conan, J. M.

Consortini, A.

M. Bertolotti, M. Carnevale, A. Consortini, and L. G. Ronchi, “Optical propagation—problems and trends,” Opt. Acta 26, 507–529 (1979).
[CrossRef]

A. Consortini, L. Ronchi, and E. Moroder, “Role of the outer scale of turbulence in atmospheric degradation of optical images,” J. Opt. Soc. Am. 63, 1246–1248 (1973).
[CrossRef]

Dautray, R.

R. Dautray and J. L. Lions, Transformations de Mellin Analyse Mathematique et Calcul Numerique (Masson, 1987), Vol.  3.

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Fontanella, J.

Frehlich, R. G.

Fried, D. L.

Gendron, E.

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

Hufnagel, R. E.

R. E. Hufnagel, Optical Propagation Through Turbulence, OSA Technical Digest Series, (Optical Society of America, 1974), pp. WA1-1–WA1-4.

Kolmogorov, A. N.

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large reynolds,” Dokl. Akad. Nauk SSSR 30, 301–305 (1941).

Lena, P.

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

Lions, J. L.

R. Dautray and J. L. Lions, Transformations de Mellin Analyse Mathematique et Calcul Numerique (Masson, 1987), Vol.  3.

Lutomirski, R. F.

R. F. Lutomirski and H. Yura, “Wave structure function and mutual coherence function of an optical wave in a turbulent atmosphere,” J. Opt. Soc. Am. A 61, 482–487 (1971).
[CrossRef]

Madec, P. Y.

G. Molodij, J. Rayrole, P. Y. Madec, and F. Colson, “Performance analysis for T.H.E.M.I.S(*) image stabilizer optical system. I,” Astron. Astrophys. Suppl. Ser. 118, 169–179 (1996).
[CrossRef]

J. M. Conan, P. Y. Madec, and G. Rousset, “Wavefront temporal spectra in high resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
[CrossRef]

Martin, F.

A. Agabi, J. Borgnino, F. Martin, A. Tokovinin, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. II,” Astron. Astrophys. Suppl. Ser. 109, 557–562(1995).

F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. I,” Astron. Astrophys. Suppl. 108, 173–180(1994).

J. Borgnino, F. Martin, and A. Ziad, “Effect of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91, 267–279 (1992).
[CrossRef]

Molodij, G.

G. Molodij and J. Rayrole, “Performance analysis for T.H.E.M.I.S(*) image stabilizer optical system. II. Anisoplanatism limitations (*) Telescope Heliographique pour l’Etude DU Magnetisme et des Instabilites de l’atmosphere Solaire,” Astron. Astrophys. Suppl. 128, 229–244 (1998).
[CrossRef]

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

G. Molodij, J. Rayrole, P. Y. Madec, and F. Colson, “Performance analysis for T.H.E.M.I.S(*) image stabilizer optical system. I,” Astron. Astrophys. Suppl. Ser. 118, 169–179 (1996).
[CrossRef]

Moroder, E.

Noll, R. J.

Ochs, G. R.

Primot, J.

Rayrole, J.

G. Molodij and J. Rayrole, “Performance analysis for T.H.E.M.I.S(*) image stabilizer optical system. II. Anisoplanatism limitations (*) Telescope Heliographique pour l’Etude DU Magnetisme et des Instabilites de l’atmosphere Solaire,” Astron. Astrophys. Suppl. 128, 229–244 (1998).
[CrossRef]

G. Molodij, J. Rayrole, P. Y. Madec, and F. Colson, “Performance analysis for T.H.E.M.I.S(*) image stabilizer optical system. I,” Astron. Astrophys. Suppl. Ser. 118, 169–179 (1996).
[CrossRef]

Roddier, F.

F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225(1988).
[CrossRef] [PubMed]

F. Roddier, “The effect of atmospheric turbulence in optical astromomy,” in Progress in Optics, E.Wolf, ed., (North Holland, 1981), Vol.  19, pp. 283–376.
[CrossRef]

Ronchi, L.

Ronchi, L. G.

M. Bertolotti, M. Carnevale, A. Consortini, and L. G. Ronchi, “Optical propagation—problems and trends,” Opt. Acta 26, 507–529 (1979).
[CrossRef]

Rousset, G.

Sasiela, R. J.

R. J. Sasiela and J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of the outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1993).
[CrossRef]

D. J. Shelton and R. J. Sasiela, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2618 (1993).
[CrossRef]

R. J. Sasiela, “A unified approch to electromagnetic wave propagation in turbulence and evaluation of multiparameter integrals,” Tech. Rep. 807 (MIT Lincoln Laboratory, 1988).

Shelton, D. J.

D. J. Shelton and R. J. Sasiela, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2618 (1993).
[CrossRef]

Shelton, J. D.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium(Dover, 1961).

Tokovinin, A.

A. Agabi, J. Borgnino, F. Martin, A. Tokovinin, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. II,” Astron. Astrophys. Suppl. Ser. 109, 557–562(1995).

F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. I,” Astron. Astrophys. Suppl. 108, 173–180(1994).

Tyler, G. A.

Valley, G. C.

Wandzura, S. M.

Winker, D. M.

Yura, H.

R. F. Lutomirski and H. Yura, “Wave structure function and mutual coherence function of an optical wave in a turbulent atmosphere,” J. Opt. Soc. Am. A 61, 482–487 (1971).
[CrossRef]

Ziad, A.

A. Agabi, J. Borgnino, F. Martin, A. Tokovinin, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. II,” Astron. Astrophys. Suppl. Ser. 109, 557–562(1995).

F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. I,” Astron. Astrophys. Suppl. 108, 173–180(1994).

J. Borgnino, F. Martin, and A. Ziad, “Effect of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91, 267–279 (1992).
[CrossRef]

Appl. Opt.

Astron. Astrophys.

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

Astron. Astrophys. Suppl.

G. Molodij and J. Rayrole, “Performance analysis for T.H.E.M.I.S(*) image stabilizer optical system. II. Anisoplanatism limitations (*) Telescope Heliographique pour l’Etude DU Magnetisme et des Instabilites de l’atmosphere Solaire,” Astron. Astrophys. Suppl. 128, 229–244 (1998).
[CrossRef]

F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. I,” Astron. Astrophys. Suppl. 108, 173–180(1994).

Astron. Astrophys. Suppl. Ser.

A. Agabi, J. Borgnino, F. Martin, A. Tokovinin, and A. Ziad, “A grating scale monitor for atmospheric turbulence measurements. II,” Astron. Astrophys. Suppl. Ser. 109, 557–562(1995).

G. Molodij, J. Rayrole, P. Y. Madec, and F. Colson, “Performance analysis for T.H.E.M.I.S(*) image stabilizer optical system. I,” Astron. Astrophys. Suppl. Ser. 118, 169–179 (1996).
[CrossRef]

Dokl. Akad. Nauk SSSR

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large reynolds,” Dokl. Akad. Nauk SSSR 30, 301–305 (1941).

J. Math. Phys.

D. J. Shelton and R. J. Sasiela, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2618 (1993).
[CrossRef]

J. Opt.

F. Chassat, “Calcul du domaine d’isoplanetisme d’un systeme d’optique adaptative fonctionnant a travers la turbulence atmospherique,” J. Opt. 20, 13–23 (1989).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Primot, G. Rousset, J. Fontanella, “Deconvolution from wave-front sensing—a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608(1990).
[CrossRef]

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

J. H. Churnside and S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987).
[CrossRef]

G. A. Tyler, “Analysis of propagation through turbulence: evaluation of an integral involving the product of three Bessel fonctions,” J. Opt. Soc. Am. A 7, 1218–1223 (1990).
[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]

R. J. Sasiela and J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of the outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1993).
[CrossRef]

J. M. Conan, P. Y. Madec, and G. Rousset, “Wavefront temporal spectra in high resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
[CrossRef]

R. F. Lutomirski and H. Yura, “Wave structure function and mutual coherence function of an optical wave in a turbulent atmosphere,” J. Opt. Soc. Am. A 61, 482–487 (1971).
[CrossRef]

Opt. Acta

M. Bertolotti, M. Carnevale, A. Consortini, and L. G. Ronchi, “Optical propagation—problems and trends,” Opt. Acta 26, 507–529 (1979).
[CrossRef]

Opt. Commun.

J. Borgnino, F. Martin, and A. Ziad, “Effect of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91, 267–279 (1992).
[CrossRef]

Proc. IEEE

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Other

V. I. Tatarski, Wave Propagation in a Turbulent Medium(Dover, 1961).

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J.W.Strohbehn, ed. (Springer, 1978), pp. 9–43.

R. E. Hufnagel, Optical Propagation Through Turbulence, OSA Technical Digest Series, (Optical Society of America, 1974), pp. WA1-1–WA1-4.

F. Roddier, “The effect of atmospheric turbulence in optical astromomy,” in Progress in Optics, E.Wolf, ed., (North Holland, 1981), Vol.  19, pp. 283–376.
[CrossRef]

R. J. Sasiela, “A unified approch to electromagnetic wave propagation in turbulence and evaluation of multiparameter integrals,” Tech. Rep. 807 (MIT Lincoln Laboratory, 1988).

G. Rousset, “Adaptative optics,” in Adaptive Optics for Astronomy, D.Alloin and J.M.Mariotti, ed., NATO Advanced Study Institute Series (Kluwer, 1994), pp. 115–137.

R. Dautray and J. L. Lions, Transformations de Mellin Analyse Mathematique et Calcul Numerique (Masson, 1987), Vol.  3.

S. Colombo, Les Transformations de Mellin et de Hankel (CNRS, 1959).

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Figures (3)

Fig. 1
Fig. 1

Geometry of the propagation problem.

Fig. 2
Fig. 2

Optimal field of view α cor (plain lines) and optimal image quality related to the residual phase variance σ J 2 (dashed lines) versus the number of compensated modes J of the AO for three different telescope apertures (1, 2, and 4 m class telescope indicated in blue, green, and red, respectively). Vertical axis on the left shows both the residual phase variance expressed in rad 2 and the optimum field of view (arcsec) in log units. Right vertical axis indicates the Strehl ratio related to the residual phase variance. The conventional diffraction-limit aberration level is set at a Strehl value of 0.8 ( σ 2 = 0.2 rad 2 ).

Fig. 3
Fig. 3

Effect of the outer scale on the normalized correlations of the Zernike polynomials n = 1 , 2, and 4, the angle- of-arrivals, and the differentials of angle-of-arrivals versus the angular separation α, between the object and the reference target, normalized by the telescope radius R. The different value of the outer scale are indicated by plain line curves for L o = , diamond dot curves for L o = 5 D , and square dot curves for L o = D , respectively. The derivation use the Hufnagel profile model [18] ( r o = 10 cm at λ = 0.5 μm ).

Tables (1)

Tables Icon

Table 1 Parity Rules

Equations (69)

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

g i = d 2 ρ W ( ρ ) φ i ( R i ρ ) M ( ρ ) ,
W ( ρ ) = { 1 π : if     | ρ | 1 0 : else where .
g 1 g 2 * ( α ) = d 2 ρ 1 W ( ρ 1 ) φ 1 ( R 1 ρ 1 ) M ( ρ 1 ) d 2 ρ 2 W ( ρ 2 ) φ 2 * ( R 2 ρ 2 ) M * ( ρ 2 ) .
W ( ρ ) M ( ρ ) = d 2 κ M ˜ ( κ ) exp [ 2 i π κ ρ ] .
g 1 g 2 * ( α ) = d 2 κ 1 d 2 κ 2 M 1 ˜ ( κ 1 ) M 2 * ˜ ( κ 2 ) d 2 ρ 1 d 2 ρ 2 exp [ 2 i π ( κ 2 ρ 2 κ 1 ρ 1 ) ] φ 1 ( R 1 ρ 1 ) φ 2 * ( R 2 ρ 2 ) .
B φ [ R ( ρ 1 ρ 2 ) , α ] = layer s l B φ l [ α h l i + R 1 ( h l ) ρ 1 R 2 ( h l ) ρ 2 ] ,
Let ζ ( h l ) = R 2 ( h l ) R 1 ( h l ) , η ( h l ) = ρ 1 ζ ( h l ) ρ 2 , and ρ = ρ 2 .
g 1 g 2 * ( α ) = layer s l d 2 κ 1 d 2 κ 2 M l 1 ˜ ( κ 1 ) M l 2 * ˜ ( κ 2 ) exp [ 2 i π ( κ 2 ρ 2 κ 1 ρ 1 ) ] d 2 η ( h l ) d 2 ρ exp [ 2 i π ρ ( κ 2 κ 1 ζ ( h l ) ) ] exp [ 2 i π κ 1 η ( h l ) ] B φ l [ α h l i + R 1 ( h l ) η ( h l ) ] .
δ [ κ 2 ζ ( h l ) κ 1 ] = d 2 ρ exp [ 2 i π ρ ( κ 2 ζ ( h l ) κ 1 ) ] ,
g 1 g 2 * ( α ) = layer s l d 2 κ 1 M l 1 ˜ ( κ 1 ) M l 2 * ˜ [ ζ ( h l ) κ 1 ] exp [ 2 i π κ 1 ( ρ 1 ζ ( h l ) ρ 2 ) ] × d 2 η ( h l ) exp [ 2 i π κ 1 η ( h l ) ] B φ l [ α h l i + R 1 ( h l ) η ( h l ) ] .
W φ ( | K | ) = k 2 cos Ω 0.033 ( 2 π ) 2 3 ( K 2 + K 0 2 ) 11 6 exp [ ( K / k 0 ) 2 ] C n 2 ( h ) d h ,
g 1 g 2 * ( α ) = 0.033 ( 2 π ) 2 / 3 k 2 cos Ω 0 L atm d h C n 2 ( h ) R 1 2 ( h ) d 2 κ M ˜ ( κ ) M ˜ * [ ζ ( h ) κ ] × [ ( κ ) 2 + ( 2 π L 0 ) 2 ] 11 6 exp [ ( | κ | 2 π l 0 ) 2 ] exp [ 2 i π α h κ i R 1 ( h ) ] exp [ 2 i π κ ( ρ 1 ζ ( h ) ρ 2 ) ] .
M ˜ ( κ ) = 2 J 1 ( 2 R π κ ) 2 R π κ .
M ( R ρ ) = δ φ ( R ρ ) δ x , and M ˜ ( κ ) = 2 i π κ x .
G ( R ρ ) = W ( ρ ) δ φ ( R ρ ) δ x .
M ( κ , ϕ ) = 2 i cos ( ϕ ) J 1 ( 2 π κ ) ,
M ( κ , ϕ ) = 2 i sin ( ϕ ) J 1 ( 2 π κ ) .
G ( R ρ ) = W ( ρ ) ( δ 2 δ x 2 φ ( R ρ ) + δ 2 δ y 2 φ ( R ρ ) ) ,
G ( R ρ ) = W ( ρ ) Z j ( ρ ) φ ( R ρ ) ,
Z j ( ρ , θ ) = n + 1 { R n m ( ρ ) 2 cos ( m θ ) : j     even and m 0 R n m ( ρ ) 2 sin ( m θ ) : j     odd and m 0 R n 0 ( ρ ) : m = 0 ,
with , R n m ( ρ ) = s = 0 n m 2 ( 1 ) s ( n s ) ! s ! [ n + m 2 s ] ! [ n m 2 s ] ! ρ n 2 s .
Q j ( κ , ϕ ) = n + 1 J n + 1 ( 2 π κ ) π κ { ( 1 ) n m 2 i m 2 cos ( m ϕ ) : j     even and m 0 ( 1 ) n m 2 i m 2 sin ( m ϕ ) : j     odd and m 0 ( 1 ) n 2 : m = 0 ,
G ( R ρ ) = W ( ρ ) [ δ ( ρ ρ 2 ) δ ( ρ ρ 1 ) ] M ( R ρ ) φ ( R ρ ) ,
G ( R ρ ) = [ δ ( ρ ρ 2 ) δ ( ρ + ρ 2 ) ] W ( ρ ) φ ( R ρ ) .
G ( R ρ ) = [ δ ( ρ ρ 2 ) δ ( ρ + ρ 2 ) ] W ( ρ ) δ φ ( R ρ ) δ x .
M ˜ ( α ) = exp [ ( 2 i π α h κ · i ) ] .
σ 2 ( α ) = j = J + 1 ( a j ) 2 + 2 j = 2 J [ ( a j ) 2 a j ( obs ) a j ( ref ) ( α ) ] ,
( a j ) 2 a j ( obs ) a j ( ref ) ( α ) > 0.
C 1 / 2 ( α , n , D ) = 1 2 = 0 d h C n 2 ( h ) 0 d K K 14 / 3 J n + 1 2 ( K ) J 0 ( α h K R ) 0 d h C n 2 ( h ) 0 d K K 14 / 3 J n + 1 2 ( K ) .
C t e . 0 d K K 14 / 3 J n + 1 2 ( K ) 0 d h J 0 ( α h K R ) .
F m ( s ) = 0 x s 1 f ( x ) d x ,
0 d h J 0 ( α h K R ) = ( α K R ) 1 .
0 d K K 14 / 3 J n + 1 2 ( K ) [ 1 D α K ] = 0.
J ν ( K ) 1 ν ! 1 2 ν K ν ,
J ν ( K ) 2 π K cos [ K ( 2 ν + 1 ) π 4 ] ,
[ 1 ( n + 1 ) ! 1 2 n + 1 K c n + 1 ] 2 = 1 2 π K c
( n + 1 ) ! ( n + 1 ) n + 1 2 π ( n + 1 ) exp [ ( n + 1 ) ]
0 d K K 14 / 3 δ ( K K ( n ) ) [ 1 D α cor K ] = 0 ,
α cor = D K ( n ) D n + 1 .
α cor D J max ,
σ J 2 = 0.2944 J 3 2 ( D r 0 ) 5 3 or σ J 2 ( D 2 J ) 5 6 .
C δ φ δ x , y ( α ) = 1.947 ( D r 0 ) 5 / 3 0 L d h C n 2 ( h ) 0 d κ κ η J 1 2 ( κ ) [ J 0 ( α h κ R ) ± J 2 ( α h κ R ) ] [ 1 + ( R L 0 κ ) 2 ] 11 / 6 ,
V α ¯ = 1.947 ( D r o ) 5 / 3 0 d κ κ η J 1 2 ( κ ) [ 1 + ( R L 0 κ ) 2 ] 11 / 6 .
V n = 3.895 ( n + 1 ) ( D r 0 ) 5 / 3 0 d κ κ 14 / 3 J n + 1 2 ( κ ) [ 1 + ( R L 0 κ ) 2 ] 11 / 6 .
V 1 = 0.45 ( D r 0 ) 5 / 3 { 1 0.77 ( D L 0 ) 1 / 3 + 0.09 ( D L 0 ) 2 0.05 ( D L 0 ) 7 / 3 } .
V 2 = 0.023 ( D r 0 ) 5 / 3 { 1 0.39 ( D L 0 ) 2 + 0.26 ( D L 0 ) 7 / 3 } .
V α ¯ = 1.68 ( D r 0 ) 5 / 3 { 1 0.82 ( D L 0 ) 1 / 3 + 0.14 ( D L 0 ) 2 0.09 ( D L 0 ) 7 / 3 } .
V d α ¯ = 1.28 ( D r 0 ) 5 / 3 { 1 + 0.603 ( D L 0 ) 2 0.46 ( D L 0 ) 7 / 3 } .
G ( R ρ ) = W ( ρ ) Z j ( ρ ) φ ( R ρ ) ,
φ i ( R i ρ ) = j = 2 a j i Z j ( ρ ) ,
a j i = d 2 ρ W ( ρ ) φ i ( R i ρ ) Z j ( ρ ) ,
a j 1 1 a j 2 2 ( α ) = 0.033 ( 2 π ) 2 3 k 2 R 2 cos Ω 0 L atm d h C n 2 ( h ) d 2 κ Q j 1 ( κ ) Q j 2 * [ ζ ( h ) κ ] ( κ R ) 11 3 exp [ 2 i π α h κ i ] .
a j 1 1 a j 2 2 ( α ) = 3.895 ( D r 0 ) 5 / 3 0 L atm d h C n 2 ( h ) ζ 1 ( h ) I ( α h R 1 ( h ) , ζ ( h ) ) 0 L atm d h C n 2 ( h ) ( R 1 ( h ) R 1 ( 0 ) ) 5 / 3 ,
r 0 = [ 0.033 ( 2 π ) 2 / 3 k 2 0.023 cos γ 0 L atm d h C n 2 ( h ) ( R 1 ( h ) R 1 ( 0 ) ) 5 / 3 ] 3 / 5 ,
I ( x , y ) = n 1 + 1 n 2 + 1 0 d κ κ 14 / 3 J n 1 + 1 ( 2 π κ ) J n 2 + 1 ( 2 π y κ ) b ( x ) ,
b ( x ) = ( 1 ) ( n 1 + n 2 m 1 m 2 ) 2 0 2 π d ϕ { ( i ) m 1 2 cos 2 ( m 1 ϕ ) ( i ) m 1 2 sin 2 ( m 1 ϕ ) 1 } { ( i ) m 2 2 cos 2 ( m 2 ϕ ) ( i ) m 2 2 sin 2 ( m 2 ϕ ) 1 } × [ cos ( 2 π x κ cos ϕ ) i sin ( 2 π x κ cos ϕ ) ] .
I ( x , y ) = ( 1 ) n 1 + n 2 m 1 m 2 2 ( n 1 + 1 ) ( n 2 + 1 ) [ K 1 , 2 + 0 d κ κ 14 / 3 J n 1 + 1 ( 2 π κ ) J n 2 + 1 ( 2 π y κ ) × J m 1 + m 2 ( 2 π x ) + K 1 , 2 0 d κ κ 14 / 3 J n 1 + 1 ( 2 π κ ) J n 2 + 1 ( 2 π y κ ) J | m 1 m 2 | ( 2 π x ) ] ,
I ( x , y , η ) = 0 d κ κ η [ 1 + ( κ y ) 2 ] γ J μ 2 ( κ ) J ν ( κ x ) ,
I ( x , y , η ) = 1 ( 2 i π ) 2 i + i i + i Γ [ t , γ + t , t s + 1 2 ( η + 1 ) + μ , t + s + η 2 , s + ν 2 γ , μ + t + s + 1 2 ( η + 1 ) , t + s + 1 2 ( η + 1 ) , 1 s + ν 2 ] × 1 2 π ( x 2 ) 2 s y 2 t d s d t ,
Γ [ x 1 , x 2 , . , x n y 1 , y 2 , . , y m ] = Γ ( x 1 ) Γ ( x 2 ) . Γ ( x n ) Γ ( y 1 ) Γ ( y 2 ) . Γ ( y m ) .
Γ ( s ) = O d κ exp ( κ ) κ s 1 = n = 0 ( 1 ) n n ! 1 s + n + 1 d κ exp ( κ ) κ s 1 ,
h ( κ ) = d κ κ s Π i = 1 A Γ [ a i + α i s ] Π j = 1 B Γ [ b j β j s ] Π k = 1 C Γ [ c k + γ k s ] Π m = 1 D Γ [ d m δ m s ] ,
Δ = i = 1 A α i + m = 1 D δ m j = 1 B β j k = 1 C γ k .
I 1 ( x , y ) = 1 2 π p = 0 q = 0 ( 1 ) p + q p ! q ! ( x 2 ) 2 q 2 p 2 μ + η 1 y 2 q Γ [ γ + q , p + μ + 1 2 , q + p + μ + 1 2 ( ν η + 1 ) γ , q + 2 μ + 1 , q + μ + 1 , p + ν + 1 ] ,
I 2 ( x , y ) = 1 2 π p = 0 q = 0 ( 1 ) p + q p ! q ! ( x 2 ) 2 q + 2 p + η y 2 q Γ [ γ + q , p + μ + 1 2 , q p + 1 2 ( ν η ) γ , q + p + 1 + 1 2 ( ν + η ) , μ p + 1 2 , p + 1 2 ] ,
I 3 ( x , y ) = 1 2 π p = 0 q = 0 ( 1 ) p + q p ! q ! ( x 2 ) 2 p + ν y 2 q Γ [ γ + q , p + μ q + 1 2 ( ν η + 1 ) , q p + 1 2 ( ν η ) γ , μ + q p + 1 2 ( ν η + 1 ) , q p + 1 2 ( ν η + 1 ) , p + ν + 1 ]
I 8 ( x , y ) = 1 2 π p = 0 q = 0 ( 1 ) p + q p ! q ! Γ [ μ q p + 1 2 ( ν η 1 ) , p + μ + q + γ + 1 2 ( ν η + 1 ) , q + 1 2 + μ γ , 2 μ + q + 1 , μ + q + 1 , p + ν + 1 ] ( x 2 ) 2 p + ν y 2 q 2 p ν + η 1 2 μ .
I ( x ) = 1 2 π i c i c + i G m ( t ) H m ( η t + 1 ) x t + η 1 d t ,
I ( x ) = 1 2 π p = 0 ( 1 ) p p ! { ( L 0 R ) 2 p Γ [ p + μ + 1 2 ( 1 η ) , p + η 2 , p + γ γ , μ + p + η 2 + 1 2 , p + η 2 + 1 2 ] + ( L 0 R ) 2 p 2 μ + η 1 Γ [ p + μ + 1 2 , p μ + 1 2 ( η 1 ) , p + μ + γ + 1 2 ( 1 η ) γ , 2 μ + p + 1 , μ + p + 1 ] } ,

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