## Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere

Optics Express, Vol. 16, Issue 22, pp. 18437-18442 (2008)

http://dx.doi.org/10.1364/OE.16.018437

Acrobat PDF (271 KB)

### Abstract

Analytical formulas for the elements of the 2×2 cross-spectral density matrix of a kind of stochastic electromagnetic array beam propagating through the turbulent atmosphere are derived with the help of vector integration. Two types of superposition (i.e. the correlated superposition and the uncorrelated superposition) are considered. The changes in the spectral density and in the spectral degree of polarization of such an array beam generated by isotropic or anisotropic electromagnetic Gaussian Schell-model sources on propagation are determined by the use of the analytical formulas. It is shown by numerical calculations that for the array beam composed by isotropic Gaussian-Schell model sources, the spectral degree of polarization in the sufficiently far field returns to the value of the array source; for the array beam composed by anisotropic sources, the spectral degree of polarization in the far field approaches a fixed value that is different from the source.

© 2008 Optical Society of America

## 1. Introduction

1. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. **11**, 1399–1406 (1972). [CrossRef] [PubMed]

6. 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]

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

8. T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. **28**, 610–612 (2003). [CrossRef] [PubMed]

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 **15**, 16909–16915 (2007). [CrossRef] [PubMed]

## 2. Theory

*M*×

*N*stochastic electromagnetic Gaussian Schell-model sources, positioned at the plane of

*z*=0 (see Fig. 1). We assume that

*M*and

*N*are odd numbers, the extension to the even numbers is straightforward.

### 2.1. Uncorrelated superposition

*x*

_{0}and

*y*

_{0}are the separation distances of the beamlets. The coefficients

*A*,

_{i}*A*,

_{j}*B*and the variances

_{ij}*σ*,

_{i}*σ*,

_{j}*σ*are independent of position but may depend on frequency [9

_{ij}9. 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**′

^{-1}

*is a 4×4 matrix of the following form:*

_{ij}**I**is a 2×2 unitary matrix. Then the elements of the cross-spectral density matrix of the laser array source in the plane

*z*=0 can be expressed as:

3. 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]

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 **15**, 16909–16915 (2007). [CrossRef] [PubMed]

1. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. **11**, 1399–1406 (1972). [CrossRef] [PubMed]

3. 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]

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 **15**, 16909–16915 (2007). [CrossRef] [PubMed]

**ρ**′

*=*

_{d}**ρ**′

_{1}-

**ρ**′

_{2},

**ρ**

*=*

_{d}**ρ**

_{1}

**ρ**

_{2}°

*ρ*

_{0}(0.545

*C*

^{2}

_{n}k^{2}

*z*)

^{-3/5}is the coherence length of a spherical wave propagating through the turbulent medium and

*C*

^{2}

*is the structure parameter of the refractive index. We have employed a quadratic approximation [1*

_{n}1. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. **11**, 1399–1406 (1972). [CrossRef] [PubMed]

*z*>0 as follows [13

**15**, 16909–16915 (2007). [CrossRef] [PubMed]

**ρ**̄′

^{T}

_{12}=(

**ρ**′

^{T}

_{1},

**ρ**′

^{T}

_{2})=(

*x*′

_{1},

*y*′

_{1},

*x*′

_{2},

*y*′

_{2}) and

**ρ**̄′

^{T}

_{12}=(

**ρ**′

^{T}

_{1},

**ρ**′

^{T}

_{2})=(

*x*′

_{1},

*y*′

_{1},

*x*′

_{2},

*y*′

_{2}) are fourdimensional vectors, T stands for the matrix transposition, Det is the determinant, and

*z*>0 turn out to be

### 2.2 Correlated superposition

*z*>0:

**M**

^{-1}

_{2ij}has the same form with Eq. (10). With the choice of

**ρ**

_{1}=

**ρ**

_{2}=

**ρ**, one can obtain the elements of the cross-spectral density matrix

*W*(

_{ij}**ρ**,

*z*,

*ω*) at any two coincident points in the half-space

*z*>0. The spectral density and the spectral degree of polarization of the array beam at the point (

**ρ**,

*z*) with

**ρ**̄

^{T}

_{12}=(

**ρ**

^{T},

**ρ**

^{T}) are given by [7

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

**15**, 16909–16915 (2007). [CrossRef] [PubMed]

## 3. Numerical calculations and analyses

*z*-axis (

**ρ**̄

_{12}=0) of an electromagnetic Gaussian Schell-model array beam passing through the turbulent atmosphere. One can find that the spectral degree of polarization has the same value in the source plane or in the sufficiently far field for the two types of superposition, while it has different values on propagation. In Fig. 3(a),

*σ*and

_{i}*σ*are assumed to be equal, the spectral degree of polarization returns to its value in the source plane after the beam propagates a sufficiently long distance. This phenomenon can be considered as a consequence of the fact that the polarization components of the source field have different spatial coherence properties, i.e., the coherence-induced polarization change. In Fig. 3(b), we show the spectral degree of polarization of an array beam generated by the beamlets of anisotropic source, i.e., the source with

_{j}*σ*≠

_{i}*σ*. It can be seen that the spectral degree of polarization always changes on propagation even though each beamlet satisfies

_{j}*δ*=

_{xx}*δ*=

_{yy}*δ*, which can be considered as the anisotropic-induced polarization change. In Fig. 3(c), the source of each beamlet is anisotropic and has different coherent properties of the different components of the electromagnetic field. It can be seen that the spectral degree of polarization in the sufficiently far field approaches a fixed value that is different from the source plane. This phenomenon can be considered as the combinations of the coherence-induced polarization change and the anisotropic-induced polarization change.

_{xy}## 4. Conclusions

## Acknowledgments

## References and links

1. | H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. |

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. |

3. | 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 |

4. | G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A |

5. | G. Gbur and O. Korotkova, “Angular spectrum representation for the propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence,” J. Opt. Soc. Am. A |

6. | T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A |

7. | E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A |

8. | T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. |

9. | 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. |

10. | 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 |

11. | M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves Random Media |

12. | H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B |

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. | X. Ji, E. Zhang, and B. Lü, “Superimposed partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. B |

15. | H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B |

16. | 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. |

17. | B. Li and B. Lü, “Characterization of off-axis superposition of partially coherent beams,” J. Opt. A |

18. | E. Wolf, |

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(030.1640) Coherence and statistical optics : Coherence

(260.5430) Physical optics : Polarization

**ToC Category:**

Atmospheric and oceanic optics

**History**

Original Manuscript: July 25, 2008

Revised Manuscript: September 9, 2008

Manuscript Accepted: October 16, 2008

Published: October 24, 2008

**Citation**

Yingbin Zhu, Daomu Zhao, and Xinyue Du, "Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere," Opt. Express **16**, 18437-18442 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-18437

Sort: Year | Journal | Reset

### References

- H. T. Yura, "Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium," Appl. Opt. 11, 1399-1406 (1972). [CrossRef] [PubMed]
- 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]
- 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]
- G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002). [CrossRef]
- G. Gbur and O. Korotkova, "Angular spectrum representation for the propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence," J. Opt. Soc. Am. A 24, 745-752 (2007). [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]
- E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003). [CrossRef]
- T. Shirai, A. Dogariu, and E. Wolf, "Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence," Opt. Lett. 28, 610-612 (2003). [CrossRef] [PubMed]
- 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]
- 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]
- H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (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. Ji, E. Zhang, and B. Lü, "Superimposed partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. B 25, 825-833 (2008). [CrossRef]
- H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Scintillations of laser array beams," Appl. Phys. B 91, 265-271 (2008). [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]
- B. Li and B. Lü, "Characterization of off-axis superposition of partially coherent beams," J. Opt. A 5, 303-307 (2003). [CrossRef]
- E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.