## Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence |

Optics Express, Vol. 21, Issue 22, pp. 27246-27259 (2013)

http://dx.doi.org/10.1364/OE.21.027246

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### Abstract

A recently introduced class of scalar cosine-Gaussian Schell-Model [CGSM] beams is generalized to electromagnetic theory. The realizability conditions and the beam conditions on the source parameters are derived. Analytical formulas for the cross-spectral density matrix elements of the electromagnetic cosine-Gaussian Schell-model [EM CGSM] beams propagating in isotropic random medium are derived. It is found that the EM CGSM beams possess single-ring or double-ring intensity profiles, depending of source parameters. As two examples, the statistical characteristics of the EM CGSM beams propagating in free space and non-Kolmogorov turbulent atmosphere are studied numerically. The effects of the fractal constant of the atmospheric spectrum and the refractive-index structure constant on such characteristics are analyzed in detail.

© 2013 Optical Society of America

## 1. Introduction

17. G. Gbur and T. D. Visser, “Can spatial coherence effects produce a local minimum of intensity at focus,” Opt. Lett. **28**(18), 1627–1629 (2003). [CrossRef] [PubMed]

18. Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A **372**(25), 4654–4660 (2008). [CrossRef]

19. M. Alavinejad, G. Taherabadi, N. Hadilou, and B. Ghafary, “Changes in the coherence properties of partially coherent dark hollow beam propagating through atmospheric turbulence,” Opt. Commun. **288**, 1–6 (2013). [CrossRef]

20. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. **38**(14), 2578–2580 (2013). [CrossRef] [PubMed]

20. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. **38**(14), 2578–2580 (2013). [CrossRef] [PubMed]

21. Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express **21**(15), 17512–17519 (2013). [CrossRef] [PubMed]

*Electromagnetic cosine-Gaussian Schell-model*(EM CGSM) beams, in which all the correlations are prescribed with the help of the scalar CGSM distributions. We stress that unlike the existing partially coherent cosine Gaussian beams [22

22. H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. **278**(1), 17–22 (2007). [CrossRef]

23. G. Zhou and X. Chu, “Propagation of a partially coherent cosine-Gaussian beam through an ABCD optical system in turbulent atmosphere,” Opt. Express **17**(13), 10529–10534 (2009). [CrossRef] [PubMed]

20. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. **38**(14), 2578–2580 (2013). [CrossRef] [PubMed]

21. Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express **21**(15), 17512–17519 (2013). [CrossRef] [PubMed]

24. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. **283**(7), 1229–1235 (2010). [CrossRef]

29. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. **47**(2), 026003 (2008). [CrossRef]

## 2. Electromagnetic cosine-Gaussian Schell-model source

10. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. **11**(8), 085706 (2009). [CrossRef]

15. Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A **29**(10), 2154–2158 (2012). [CrossRef] [PubMed]

**38**(14), 2578–2580 (2013). [CrossRef] [PubMed]

21. Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express **21**(15), 17512–17519 (2013). [CrossRef] [PubMed]

*Electromagnetic cosine-Gaussian Schell-model*(EM CGSM) sources.

*p*(

_{αβ}*v*) must be non-negative definite [10

10. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. **11**(8), 085706 (2009). [CrossRef]

*p*(

_{αβ}*v*) ≥ 0 andfor any

*p*(

_{αβ}*v*) is surely nonnegative, and substituting it into Eq. (9) implies that it is satisfied if

*n*and summarize the results in Table 1. One can see that the minimum values of

*z*axis, the spectral density in Eq. (13) must be negligible except when unit vector

*z*axis. Since

8. O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. **29**(11), 1173–1175 (2004). [CrossRef] [PubMed]

## 3. EM CGSM beam propagating in free space and in linear random medium

29. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. **47**(2), 026003 (2008). [CrossRef]

*к*), in which the slope 11/3 of the conventional van Karman spectrum is generalized to an arbitrary parameter

*α*, i.e.where

*S*, the spectral degree of coherence

*μ*, and the spectral degree of polarization

*P*in the turbulent atmosphere are calculated by the expressions [25] where Det and Tr stand for the determinant and the trace of the matrix. Further, the state of polarization of the polarized portion of the beam may be described in terms of the parameters of the polarization ellipse specified by the orientation angle

*φ*and the degree of ellipticity [12

12. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. **246**(1–3), 35–43 (2005). [CrossRef]

## 4. Examples: EM CGSM beams in free space

*z*from the source plane on propagation in free space with

*n*and on the r.m.s. correlation widths we plot in Fig. 2 its evolution in the transverse beam cross-sections, at several distances

*z*from the source plane, for

*n*= 0, 1, 2 and for

**38**(14), 2578–2580 (2013). [CrossRef] [PubMed]

**21**(15), 17512–17519 (2013). [CrossRef] [PubMed]

*n*. However, for

*n*= 0 does not lead to substantial deviation from Gaussian profile, starting from

*n*= 1 single or double ring profiles are generated. For values of

*n*> 2 (not shown) still only two rings are generated with maxima occurring at larger radial positions for larger values of

*n*.

*n*substantial quantitative changes in the degree of coherence start to occur at distances much shorter (< 1 m) than those for the spectral density (>10 m).

## 5. Examples: EM CGSM beams in atmospheric turbulence

6. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A **29**(10), 2159–2164 (2012). [CrossRef] [PubMed]

28. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express **18**(10), 10650–10658 (2010). [CrossRef] [PubMed]

## 6. Concluding remarks

30. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. **7**(5), 232–237 (2005). [CrossRef]

30. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. **7**(5), 232–237 (2005). [CrossRef]

## Acknowledgments

## References and links

1. | L. Mandel and E. Wolf, |

2. | F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J |

3. | F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J |

4. | H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. |

5. | S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. |

6. | O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A |

7. | Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. |

8. | O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. |

9. | H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. |

10. | F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. |

11. | F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. |

12. | O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. |

13. | 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 |

14. | Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. |

15. | Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A |

16. | Z. Mei, Z. Tong, and O. Korotkova, “Electromagnetic non-uniformly correlated beams in turbulent atmosphere,” Opt. Express |

17. | G. Gbur and T. D. Visser, “Can spatial coherence effects produce a local minimum of intensity at focus,” Opt. Lett. |

18. | Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A |

19. | M. Alavinejad, G. Taherabadi, N. Hadilou, and B. Ghafary, “Changes in the coherence properties of partially coherent dark hollow beam propagating through atmospheric turbulence,” Opt. Commun. |

20. | Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. |

21. | Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express |

22. | H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. |

23. | G. Zhou and X. Chu, “Propagation of a partially coherent cosine-Gaussian beam through an ABCD optical system in turbulent atmosphere,” Opt. Express |

24. | A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. |

25. | E. Wolf, |

26. | X. Du and D. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express |

27. | I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE |

28. | E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express |

29. | I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. |

30. | T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. |

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(030.1640) Coherence and statistical optics : Coherence

(260.5430) Physical optics : Polarization

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: September 6, 2013

Revised Manuscript: October 23, 2013

Manuscript Accepted: October 25, 2013

Published: November 1, 2013

**Citation**

Zhangrong Mei and Olga Korotkova, "Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence," Opt. Express **21**, 27246-27259 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-27246

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### References

- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
- F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun.64(4), 311–316 (1987). [CrossRef]
- F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett.33(16), 1857–1859 (2008). [CrossRef] [PubMed]
- H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett.36(20), 4104–4106 (2011). [CrossRef] [PubMed]
- S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett.37(14), 2970–2972 (2012). [CrossRef] [PubMed]
- O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A29(10), 2159–2164 (2012). [CrossRef] [PubMed]
- Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett.38(2), 91–93 (2013). [CrossRef] [PubMed]
- O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett.29(11), 1173–1175 (2004). [CrossRef] [PubMed]
- H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun.249(4–6), 379–385 (2005). [CrossRef]
- F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt.11(8), 085706 (2009). [CrossRef]
- F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt.3(1), 1–9 (2001). [CrossRef]
- O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun.246(1–3), 35–43 (2005). [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. Express15(25), 16909–16915 (2007). [CrossRef] [PubMed]
- Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt.15(2), 025705 (2013). [CrossRef]
- Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A29(10), 2154–2158 (2012). [CrossRef] [PubMed]
- Z. Mei, Z. Tong, and O. Korotkova, “Electromagnetic non-uniformly correlated beams in turbulent atmosphere,” Opt. Express20(24), 26458–26463 (2012). [CrossRef] [PubMed]
- G. Gbur and T. D. Visser, “Can spatial coherence effects produce a local minimum of intensity at focus,” Opt. Lett.28(18), 1627–1629 (2003). [CrossRef] [PubMed]
- Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A372(25), 4654–4660 (2008). [CrossRef]
- M. Alavinejad, G. Taherabadi, N. Hadilou, and B. Ghafary, “Changes in the coherence properties of partially coherent dark hollow beam propagating through atmospheric turbulence,” Opt. Commun.288, 1–6 (2013). [CrossRef]
- Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett.38(14), 2578–2580 (2013). [CrossRef] [PubMed]
- Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express21(15), 17512–17519 (2013). [CrossRef] [PubMed]
- H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun.278(1), 17–22 (2007). [CrossRef]
- G. Zhou and X. Chu, “Propagation of a partially coherent cosine-Gaussian beam through an ABCD optical system in turbulent atmosphere,” Opt. Express17(13), 10529–10534 (2009). [CrossRef] [PubMed]
- A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun.283(7), 1229–1235 (2010). [CrossRef]
- E. Wolf, Introduction to the Theories of Coherence and Polarization of Light (Cambridge University, 2007).
- X. Du and D. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express17(6), 4257–4262 (2009). [CrossRef] [PubMed]
- I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE6551(65510E), 65510E (2007). [CrossRef]
- E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express18(10), 10650–10658 (2010). [CrossRef] [PubMed]
- I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng.47(2), 026003 (2008). [CrossRef]
- T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt.7(5), 232–237 (2005). [CrossRef]

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