Abstract

On the basis of the generalized diffraction integral formula for misaligned optical systems in the spatial domain, an analytical propagation expression for the elements of the cross-spectral density matrix of a random electromagnetic beam passing through a misaligned optical system in turbulent atmosphere is derived. Some analyses are illustrated by numerical examples relating to changes in the state of polarization of an electromagnetic Gaussian Schell-model beam propagating through such an optical system. It is shown that the misalignment has a significant influence on the intensity profile and the state of polarization of the beam, but the influence becomes smaller for the beam propagating in strong turbulent atmosphere. The method in this paper can be applied for sources that are either isotropic or anisotropic. It is shown that the isotropic sources and the anisotropic sources have different polarization properties on beam propagation.

© 2008 Optical Society of America

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References

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  1. M. S. Belenkii, A. I. Kon, and V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287-290 (1977).
    [CrossRef]
  2. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297-1304 (1979).
    [CrossRef]
  3. Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source beam,” Radio Sci. 18, 551-556 (1983).
    [CrossRef]
  4. J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671-684 (1990).
    [CrossRef]
  5. J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355-1363 (1991).
    [CrossRef]
  6. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794-1802 (2002).
    [CrossRef]
  7. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598 (2002).
    [CrossRef]
  8. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094-1102 (2003).
    [CrossRef]
  9. G. P. Berman and A. A. Chumak, “Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere,” Phys. Rev. A 74, 013805 (2006).
    [CrossRef]
  10. X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere,” Opt. Commun. 275, 292-300 (2007).
    [CrossRef]
  11. Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9, 1123-1131 (2007).
    [CrossRef]
  12. H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891-2901 (2007).
    [CrossRef]
  13. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
    [CrossRef]
  14. Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
    [CrossRef]
  15. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909-16915 (2007).
    [CrossRef] [PubMed]
  16. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  17. E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078-1080 (2003).
    [CrossRef] [PubMed]
  18. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
    [CrossRef]
  19. O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 15, 353-364 (2005).
    [CrossRef]
  20. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
    [CrossRef]
  21. H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91-97 (2007).
    [CrossRef]
  22. X. Chu, “Propagation of a Cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15, 17613-17618 (2007).
    [CrossRef] [PubMed]
  23. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931-1948 (1987).
    [CrossRef]
  24. H. T. Yura and S. G. Hanson, “Second-order statistics for wave propagation through complex optical systems,” J. Opt. Soc. Am. A 6, 564-575 (1989).
    [CrossRef]
  25. D. Zhao and S. Wang, “Effect of misalignment on optical fractional Fourier transforming systems,” Opt. Commun. 198, 281-286 (2001).
    [CrossRef]
  26. X. Du and D. Zhao, “Changes in the polarization and coherence of a random electromagnetic beam propagating through a misaligned optical system,” J. Opt. Soc. Am. A 25, 773-779 (2008).
    [CrossRef]
  27. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).
  28. X. Du and D. Zhao, “Propagation of random electromagnetic beams through axially nonsymmetrical optical systems,” Opt. Commun. 281, 2711-2715 (2008).
    [CrossRef]
  29. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35-43 (2005).
    [CrossRef]
  30. D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067-3070 (2008).

2008

X. Du and D. Zhao, “Propagation of random electromagnetic beams through axially nonsymmetrical optical systems,” Opt. Commun. 281, 2711-2715 (2008).
[CrossRef]

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067-3070 (2008).

X. Du and D. Zhao, “Changes in the polarization and coherence of a random electromagnetic beam propagating through a misaligned optical system,” J. Opt. Soc. Am. A 25, 773-779 (2008).
[CrossRef]

2007

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891-2901 (2007).
[CrossRef]

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909-16915 (2007).
[CrossRef] [PubMed]

X. Chu, “Propagation of a Cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express 15, 17613-17618 (2007).
[CrossRef] [PubMed]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91-97 (2007).
[CrossRef]

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere,” Opt. Commun. 275, 292-300 (2007).
[CrossRef]

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9, 1123-1131 (2007).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

2006

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

G. P. Berman and A. A. Chumak, “Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere,” Phys. Rev. A 74, 013805 (2006).
[CrossRef]

2005

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35-43 (2005).
[CrossRef]

2004

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[CrossRef]

2003

2002

2001

D. Zhao and S. Wang, “Effect of misalignment on optical fractional Fourier transforming systems,” Opt. Commun. 198, 281-286 (2001).
[CrossRef]

1991

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

1990

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

1989

1987

1983

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source beam,” Radio Sci. 18, 551-556 (1983).
[CrossRef]

1979

1977

M. S. Belenkii, A. I. Kon, and V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287-290 (1977).
[CrossRef]

Baykal, Y.

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91-97 (2007).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891-2901 (2007).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source beam,” Radio Sci. 18, 551-556 (1983).
[CrossRef]

Belenkii, M. S.

M. S. Belenkii, A. I. Kon, and V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287-290 (1977).
[CrossRef]

Berman, G. P.

G. P. Berman and A. A. Chumak, “Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere,” Phys. Rev. A 74, 013805 (2006).
[CrossRef]

Boardman, A. D.

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

Cai, Y.

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91-97 (2007).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891-2901 (2007).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

Chen, Z.

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9, 1123-1131 (2007).
[CrossRef]

Chu, X.

Chumak, A. A.

G. P. Berman and A. A. Chumak, “Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere,” Phys. Rev. A 74, 013805 (2006).
[CrossRef]

Davidson, F. M.

Dogariu, A.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094-1102 (2003).
[CrossRef]

Du, X.

Eyyuboglu, H. T.

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91-97 (2007).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891-2901 (2007).
[CrossRef]

Gbur, G.

Hanson, S. G.

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

Ji, X.

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere,” Opt. Commun. 275, 292-300 (2007).
[CrossRef]

Kon, A. I.

M. S. Belenkii, A. I. Kon, and V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287-290 (1977).
[CrossRef]

Korotkova, O.

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909-16915 (2007).
[CrossRef] [PubMed]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35-43 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

Lin, Q.

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Lü, B.

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere,” Opt. Commun. 275, 292-300 (2007).
[CrossRef]

Mironov, V. L.

M. S. Belenkii, A. I. Kon, and V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287-290 (1977).
[CrossRef]

Plonus, M. A.

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source beam,” Radio Sci. 18, 551-556 (1983).
[CrossRef]

S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297-1304 (1979).
[CrossRef]

Pu, J.

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9, 1123-1131 (2007).
[CrossRef]

Ricklin, J. C.

Salem, M.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

Shirai, T.

Wang, S.

D. Zhao and S. Wang, “Effect of misalignment on optical fractional Fourier transforming systems,” Opt. Commun. 198, 281-286 (2001).
[CrossRef]

Wang, S. C. H.

Wang, S. J.

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source beam,” Radio Sci. 18, 551-556 (1983).
[CrossRef]

Wolf, E.

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067-3070 (2008).

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35-43 (2005).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[CrossRef]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078-1080 (2003).
[CrossRef] [PubMed]

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094-1102 (2003).
[CrossRef]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598 (2002).
[CrossRef]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

Wu, J.

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

Yura, H. T.

Zhang, E.

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere,” Opt. Commun. 275, 292-300 (2007).
[CrossRef]

Zhao, D.

X. Du and D. Zhao, “Propagation of random electromagnetic beams through axially nonsymmetrical optical systems,” Opt. Commun. 281, 2711-2715 (2008).
[CrossRef]

X. Du and D. Zhao, “Changes in the polarization and coherence of a random electromagnetic beam propagating through a misaligned optical system,” J. Opt. Soc. Am. A 25, 773-779 (2008).
[CrossRef]

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067-3070 (2008).

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909-16915 (2007).
[CrossRef] [PubMed]

D. Zhao and S. Wang, “Effect of misalignment on optical fractional Fourier transforming systems,” Opt. Commun. 198, 281-286 (2001).
[CrossRef]

Appl. Phys. B: Lasers Opt.

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91-97 (2007).
[CrossRef]

Appl. Phys. Lett.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

J. Mod. Opt.

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9, 1123-1131 (2007).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

D. Zhao and S. Wang, “Effect of misalignment on optical fractional Fourier transforming systems,” Opt. Commun. 198, 281-286 (2001).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[CrossRef]

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere,” Opt. Commun. 275, 292-300 (2007).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

X. Du and D. Zhao, “Propagation of random electromagnetic beams through axially nonsymmetrical optical systems,” Opt. Commun. 281, 2711-2715 (2008).
[CrossRef]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35-43 (2005).
[CrossRef]

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067-3070 (2008).

Opt. Express

Opt. Lett.

Phys. Lett. A

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Phys. Rev. A

G. P. Berman and A. A. Chumak, “Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere,” Phys. Rev. A 74, 013805 (2006).
[CrossRef]

Radio Sci.

Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source beam,” Radio Sci. 18, 551-556 (1983).
[CrossRef]

Sov. J. Quantum Electron.

M. S. Belenkii, A. I. Kon, and V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287-290 (1977).
[CrossRef]

Waves Random Complex Media

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

Waves Random Media

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

Other

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

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Figures (5)

Fig. 1
Fig. 1

Optical transforming system with a misaligned thin lens in atmosphere.

Fig. 2
Fig. 2

Matrix for the misaligned optical transforming system.

Fig. 3
Fig. 3

Changes in the state of polarization on the z axis as a function of misalignment parameter ε x . (a) Degree of polarization, (b) orientation angle of the polarization ellipse, (c) degree of ellipticity. l = 3000 m , f 1 = 30 m . The source is assumed to be a Gaussian Schell-model source with λ = 632.8 nm , A x = 2 , A y = 1 , B x y = 0.2 exp ( i π 3 ) , σ x = 40 mm , σ y = 40 mm , δ x x = 7 mm , δ y y = 6 mm , and δ x y = 8 mm . The unit of C n 2 is m 2 3 .

Fig. 4
Fig. 4

Intensity distributions on the focal plane. The source has the same values as Fig. 3, and the misaligned parameter of the lens is ε x = 1 mm .

Fig. 5
Fig. 5

State of polarization on the focal plane for an isotropic or anisotropic source. (a) Degree of polarization, (b) orientation angle of the polarization ellipse, (c) degree of ellipticity. The source is assumed to be Gaussian Schell-model source with λ = 632.8 nm , A x = 2 , A y = 1 , B x y = 0.2 exp ( i π 3 ) , δ x x = 7 mm , δ y y = 6 mm , and δ x y = 8 mm . σ x and σ y are assumed to have different values, which are illustrated in the legends, C n 2 = 10 14 m 2 3 , ε x = 1 mm .

Equations (37)

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

( a b α T ε β T ε c d γ T ε δ T ε 0 0 1 0 0 0 0 1 ) = ( a 3 b 3 0 0 c 3 d 3 0 0 0 0 1 0 0 0 0 1 ) ( a 2 b 2 α ε β ε c 2 d 2 γ ε δ ε 0 0 1 0 0 0 0 1 ) ( a 1 b 1 0 0 c 1 d 1 0 0 0 0 1 0 0 0 0 1 ) ,
α = 1 a 2 , β = l b 2 , γ = c 2 , δ = ± 1 d 2 .
E i ( ρ , z , ω ) = i λ [ Det ( B ) ] 1 2 E i ( 0 ) ( ρ , ω ) exp [ i k 2 ( ρ T B 1 A ρ 2 ρ T B 1 ρ + ρ T D B 1 ρ ) ] exp [ i k 2 ( ρ T B 1 e f + ρ T B 1 g h ) ] exp [ ψ ( ρ , ρ , z , ω ) ] d ρ , ( i = x , y ) ,
A = [ a 0 0 a ] , B = [ b 0 0 b ] , C = [ c 0 0 c ] ,
D = [ d 0 0 d ] .
e = 2 ( α T ε x + β T ε x ) ,
f = 2 ( α T ε y + β T ε y ) ,
g = 2 ( b γ T d α T ) ε x + 2 ( b δ T d β T ) ε x ,
h = 2 ( b γ T d α T ) ε y + 2 ( b δ T d β T ) ε y ,
W ( r 1 , r 2 , ω ) [ W i j ( r 1 , r 2 , ω ) ] = [ E i * ( r 1 , ω ) E j ( r 2 , ω ) ] ,
( i = x , y ; j = x , y ) ,
W i j ( ρ 12 , z , ω ) = 1 λ 2 [ Det ( B ¯ ) ] 1 2 W i j ( 0 ) ( ρ 12 , ω ) exp [ i k 2 ( ρ 12 T B ¯ 1 A ¯ ρ 12 2 ρ 12 T B ¯ 1 ρ 12 + ρ 12 T D ¯ B ¯ 1 ρ 12 ) ] exp [ i k 2 ( ρ 12 T B ¯ 1 e ¯ f + ρ 12 T B ¯ 1 g ¯ h ) ] exp [ ψ * ( ρ 1 , ρ 1 , z , ω ) + ψ ( ρ 2 , ρ 2 , z , ω ) ] m d 4 ρ 12 ,
A ¯ = [ A 0 0 A ] , B ¯ = [ B 0 0 B ] , C ¯ = [ C 0 0 C ] ,
D ¯ = [ D 0 0 D ] ,
e ¯ f = [ e f e f ] , g ¯ h = [ g h g h ] .
exp [ ψ * ( ρ 1 , ρ 1 , z , ω ) + ψ ( ρ 2 , ρ 2 , z , ω ) ] m = exp [ ( 1 2 ) D ψ ( ρ d , ρ d ) ] exp [ ( 1 ρ 0 2 ) ( ρ d 2 + ρ d ρ d + ρ d 2 ) ] ,
ρ 0 = b σ 0 = b [ 1.46 k 2 C n 2 0 L b 5 3 ( z ) d z ] 3 5 ,
W i j ( ρ 12 , z , ω ) = 1 λ 2 [ Det ( B ¯ ) ] 1 2 W i j ( 0 ) ( ρ 12 , ω ) exp [ i k 2 ( ρ 12 T B ¯ 1 A ¯ ρ 12 2 ρ 12 T B ¯ 1 ρ 12 + ρ 12 T D ¯ B ¯ 1 ρ 12 ) ] exp [ i k 2 ( ρ 12 T B ¯ 1 e ¯ f + ρ 12 T B ¯ 1 g ¯ h ) ] exp [ i k 2 ( ρ 12 T P ¯ ρ 12 + ρ 12 T P ¯ ρ 12 + ρ 12 T P ¯ ρ 12 ) ] d 4 ρ 12 ,
P ¯ = 2 i k ρ 0 2 [ I I I I ] ,
W i j ( 0 ) ( ρ 1 , ρ 2 , ω ) = S i ( 0 ) ( ρ 1 , ω ) S i ( 0 ) ( ρ 2 , ω ) μ i j ( 0 ) ( ρ 2 ρ 1 , ω ) ,
( i = x , y , j = x , y ) ,
W i j ( 0 ) ( ρ 1 , ρ 2 , ω ) = A i A j B i j exp ( ρ 1 2 4 σ i 2 ) exp ( ρ 2 2 4 σ j 2 ) exp ( ρ 2 ρ 1 2 2 δ i j 2 ) ,
W i j ( 0 ) ( ρ 12 , ω ) = A i A j B i j exp ( i k 2 ρ 12 T M i j 1 ρ 12 ) ,
M i j 1 = [ i 2 k σ i 2 i k δ i j 2 0 i k δ i j 2 0 0 i 2 k σ i 2 i k δ i j 2 0 i k δ i j 2 i k δ i j 2 0 i 2 k σ j 2 i k δ i j 2 0 0 i k δ i j 2 0 i 2 k σ j 2 i k δ i j 2 ] .
W i j ( ρ 12 , z , ω ) = A i A j B i j [ Det ( A ¯ + B ¯ M i j 1 + B ¯ P ¯ ) ] 1 2 exp { i k 2 ρ 12 T [ D ¯ B ¯ 1 + P ¯ ( B ¯ 1 1 2 P ¯ ) T ( B ¯ 1 A ¯ + M i j 1 + P ¯ ) 1 ( B ¯ 1 1 2 P ¯ ) ] ρ 12 } exp { i k 2 ρ 12 T [ B ¯ 1 g ¯ h + ( B ¯ 1 1 2 P ¯ ) T ( B ¯ 1 A ¯ + M i j 1 + P ¯ ) 1 B ¯ 1 e ¯ f ] } exp [ i k 8 e ¯ f T B ¯ 1 T ( B ¯ 1 A ¯ + M i j 1 + P ¯ ) 1 B ¯ 1 e ¯ f ] .
P ( ρ 12 , z , ω ) = 1 4 Det W ( ρ 12 , z , ω ) [ Tr W ( ρ 12 , z , ω ) ] 2 ,
θ 0 ( ρ 12 , z , ω ) = 1 2 arctan ( 2 Re [ W x y ( ρ 12 , z , ω ) ] W x x ( ρ 12 , z , ω ) W y y ( ρ 12 , z , ω ) ) ,
ε = A min ( ρ 12 , z , ω ) A max ( ρ 12 , z , ω ) ,
A max 2 ( ρ 12 , z , ω ) = ( [ W x x ( ρ 12 , z , ω ) W y y ( ρ 12 , z , ω ) ] 2 + 4 W x y ( ρ 12 , z , ω ) 2 + [ W x x ( ρ 12 , z , ω ) W y y ( ρ 12 , z , ω ) ] 2 + 4 [ Re W x y ( ρ 12 , z , ω ) ] 2 ) 2 ,
A min 2 ( ρ 12 , z , ω ) = ( [ W x x ( ρ 12 , z , ω ) W y y ( ρ 12 , z , ω ) ] 2 + 4 W x y ( ρ 12 , z , ω ) 2 [ W x x ( ρ 12 , z , ω ) W y y ( ρ 12 , z , ω ) ] 2 + 4 [ Re W x y ( ρ 12 , z , ω ) ] 2 ) 2 .
( a b α T ε β T ε c d γ T ε δ T ε 0 0 1 0 0 0 0 1 ) = ( 1 f 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ( 1 0 0 0 1 f 1 1 ε f 1 0 0 0 1 0 0 0 0 1 ) ( 1 l 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) = ( 0 f 1 ε 0 1 f 1 1 l f 1 ε f 1 0 0 0 1 0 0 0 0 1 ) .
A = ( 0 0 0 0 ) , B = ( f 1 0 0 f 1 ) , C = ( 1 f 1 0 0 1 f 1 ) ,
D = ( 1 l f 1 0 0 1 l f 1 ) ,
α T = 1 , β T = 0 , γ T = 1 f 1 , δ T = 0 ,
e = 2 ε x , f = 0 , g = 2 l f 1 ε x , h = 0 .
b ( z ) = { f 1 0 < z < l ( l + f 1 z ) l < z < l + f 1 } .
ρ 0 = [ 0.1825 C n 2 k 2 ( 3 f 1 + 8 l ) ] 3 5 .

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