## Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere

Optics Express, Vol. 16, Issue 20, pp. 15834-15846 (2008)

http://dx.doi.org/10.1364/OE.16.015834

Acrobat PDF (1309 KB)

### Abstract

Propagation of stochastic electromagnetic beams through paraxial ABCD optical systems operating through turbulent atmosphere is investigated with the help of the ABCD matrices and the generalized Huygens-Fresnel integral. In particular, the analytic formula is derived for the cross-spectral density matrix of an electromagnetic Gaussian Schell-model (EGSM) beam. We applied our analysis for the ABCD system with a single lens located on the propagation path, representing, in a particular case, the unfolded double-pass propagation scenario of active laser radar. Through a number of numerical examples we investigated the effect of local turbulence strength and lens’ parameters on spectral, coherence and polarization properties of the EGSM beam.

© 2008 Optical Society of America

## 1. Introduction

9. Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express **15**, 15480–15492 (2007). [CrossRef] [PubMed]

10. E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. **25**, 293–296 (1978). [CrossRef]

11. F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. **34**, 301–305 (1978). [CrossRef]

12. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun **41**, 383–387 (1982). [CrossRef]

19. Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. **31**, 685–687 (2006) [CrossRef] [PubMed]

28. 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]

33. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. **52**, 1611–1618 (2005). [CrossRef]

38. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. **281**, 2342–2348 (2008). [CrossRef]

28. 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]

33. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. **52**, 1611–1618 (2005). [CrossRef]

38. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. **281**, 2342–2348 (2008). [CrossRef]

38. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. **281**, 2342–2348 (2008). [CrossRef]

54. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am **A 4**, 1931 (1987). [CrossRef]

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

40. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, *Laser Beam Scintillation with Applications* (*SPIE Press, Washington*, 2001). [CrossRef]

## 2. Theory

54. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am **A 4**, 1931 (1987). [CrossRef]

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

*E*(

**r**

_{1},0) and

*E*(

_{ρ1},

*l*) are the electric fields of the laser beam in the source plane (

*z*=0) and the output plane (

*z*=

*l*), respectively.

**r**

^{T}_{1}=(

*x*

_{1}

*y*

_{1}) and

**ρ**

^{T}_{1}=(

*ρ*

_{1x}

*ρ*

_{1y}) with

**r**

_{1}and

**ρ**

_{1}being the position vectors in the source plane and output planes,

**Ψ**(

**r**

_{1},

**ρ**

_{1}) is the Rytov perturbation being the random part of the complex phase of the beam induced by atmospheric fluctuations,

*k*=2

*π*/

*λ*is the wave number,

*λ*is the wavelength of light. Here we note that

**A**,

**B**,

**C**and

**D**are the 2×2 sub-matrices of the astigmatic optical system [58, 59], satisfying the following Luneburg relations that describe the symplecticity of an astigmatic optical system [60]

**r**

_{1},

**r**

_{2}in the source plane by

*E*(

**r**

_{1}),

*E*(

**r**

_{2}) and the optical fields at the two arbitrary points

**ρ**

_{1},

**ρ**

_{2}in the output plane by

*E*(

**ρ**

_{1}),

*E*(

**ρ**

_{2}), respectively, we may write the expressions for the cross-spectral density in the source and output planes as:

5. 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]

42. 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]

43. J. C. Leader, “Atmospheric propagation of partially coherent radiation”, J. Opt. Soc. Am. **68**, 175–185 (1978). [CrossRef]

54. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am **A 4**, 1931 (1987). [CrossRef]

55. H. T. Yura and S. G. Hanson, “Second-order statistics for wave propagation through complex optical systems,” J. Opt. Soc. Am **A 6**, 564 (1989). [CrossRef]

_{0}being the coherence length of a spherical wave propagating in the turbulent medium given by the expression [54

**A 4**, 1931 (1987). [CrossRef]

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

*z*) is the sub-matrix for back-propagpation from output plane to propagation distance z [54

**A 4**, 1931 (1987). [CrossRef]

**15**, 17613–17618 (2007). [CrossRef] [PubMed]

*C*

^{2}

_{n}is the structure constant of turbulent atmosphere. Here, following [42

42. 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]

**15**, 17613–17618 (2007). [CrossRef] [PubMed]

*d*

**r̃**=

*d*

**r**

_{1}

*d*

**r**

_{2},

**r̃**

*=(*

^{T}**r**

^{T}_{1}

**r**

^{T}_{2}),

ρ ˜

*=(*

^{T}**ρ**

^{T}_{1}

**ρ**

^{T}_{2}) and

**I**being a 2×2 unit matrix. In the absence of turbulence (

*ρ*

_{0}→∞, i.e.,

*C*

^{2}

*=0),*

_{n}**P̃**=0, Eq. (7) reduces to the generalized Collins formula for treating propagation of a partially coherent beam through a general astigmatic ABCD optical system in free space [16

16. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. **27**, 216–218 (2002). [CrossRef]

**r**

_{1},

**r**

_{2},0) specified at any two points with position vectors

**r**

_{1}and

**r**

_{2}in the source plane with elements [21–25]

*A*,

_{α}*B*=|

_{αβ}*B*|exp(

_{αβ}*iϕ*)=

_{αβ}*B*

^{*}

*,*

_{αβ}*σ*and

_{α}*β*are independent of position but, in general, depend on the frequency. In Eq. (9) and everywhere else in this paper we have omitted the dependence on the oscillation frequency for conciseness. The nine real parameters

_{αδ}*A*,

_{x}*A*,

_{y}*σ*,

_{x}*σ*,

_{y}*|B*|,

_{xy}*ϕ*,

_{xy}*δ*,

_{xx}*δ*and

_{yy}*δ*entering the general model are shown to satisfy several intrinsic constraints and obey some simplifying assumptions (e.g. the phase difference between the x- an y-components of the field is removable, i.e.

_{xy}*ϕ*=0[29

_{αβ}29. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. **7**, 232–237 (2005). [CrossRef]

30. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. **249**, 379–385 (2005). [CrossRef]

16. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. **27**, 216–218 (2002). [CrossRef]

27. Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure App. Opt. **5**, 453–459 (2003). [CrossRef]

*k*=2

*π*/

*λ*is the wave number,

*λ*is the wavelength,

**r̃**

*=(*

^{T}**r**

^{T}_{1}

**r**

^{T}_{2}), and the 4×4 tensor has the form

*C*

^{2}

*=0, and hence, ρ*

_{n}_{0}→∞)

**P̃**=0. Equation (12) then reduces to the propagation formula for an EGSM beam passing through a general astigmatic ABCD optical system in free space [31

31. D. Ge, Y. Cai, and Q. Lin, “Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems,” Chin. Phys. **14**, 128–132 (2005). [CrossRef]

16. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. **27**, 216–218 (2002). [CrossRef]

31. D. Ge, Y. Cai, and Q. Lin, “Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems,” Chin. Phys. **14**, 128–132 (2005). [CrossRef]

**M**

^{-1}

_{0αβ}can be expressed as [35

35. H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. **32**, 2215–2217 (2007). [CrossRef] [PubMed]

*σ*

^{2}

_{a}*σ*

^{2}

*and*

_{β}*δ*

^{2}

*all are 2.2 matrices with transpose symmetry [15, 16*

_{αβ}**27**, 216–218 (2002). [CrossRef]

*ρ*

_{0}=(0 0.545

*C*

^{2}

_{n}*k*

^{2}

*l*)

^{-3/5}. Equation (16) agrees well with existing propagation formula for a scalar partially coherent GSM beam for atmospheric propagation [48

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

61. 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]

## 3. Focusing properties of an EGSM beam in a turbulent atmosphere

*focusing properties*) of an EGSM beam on propagation in a turbulent atmosphere by applying the formulae derived in section 2.

*z*≤

*l*

_{1}, the transformation matrix for back-propagation from output plane to plane located at distance z from the source is given by

*l*

_{1}<

*z*≤

*l*

_{1}+

*f*, the transformation matrix for back-propagation from output plane to plane located at distance z from the source is given by

**ρ**

_{1}and

**ρ**

_{2}is defined by the formula

*A*=

_{x}*A*=0.707,

_{y}*B*=

_{xy}*B*=0.2,

_{yx}*σ*=

_{x}*σ*=1

_{y}*mm*,

*λ*=590

*nm*,

*f*=50

*m*and

*l*

_{1}=4.95

*km*. The polarization properties are uniform across the source plane with P(

**r**

_{1},0)=0.2.

*C*

^{2}

*and by the source correlation coefficients: with increase in*

_{n}*C*

^{2}

*the distribution becomes shorter and flatter, with increase in source correlations it becomes higher and narrower. The later statement is valid in free space as well.*

_{n}_{1}-y

_{2}=0) of an EGSM beam versus the spatial difference vectors x

_{1}-x

_{2}and y

_{1}-y

_{2}at the geometrical focal plane for different values of the structure constant of turbulent atmosphere and the source correlation coefficients. One can see from these figures that the spectral degree of coherence is of Gaussian profile. The width of the Gaussian profile decreases as the value of the structure constant increases, which means the atmospheric turbulence degrades the coherence of the EGSM beam. Similar phenomenon is known for laser (coherent) Gaussian beams [62

62. T. L. Ho, “Coherence degradation of Gaussian beams in a turbulent atmosphere,” J. Opt. Soc. Am **60**, 667–673(1970). [CrossRef]

## 4. Summary

## Appendix A. Derivation of propagation Eq. (12)

## Acknowledgments

## References and links

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

2. | D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am |

3. | Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. |

4. | A. Belendez, L. Carretero, and A. Fimia, “The use of partially coherent light to reduce the efficiency of silve-halide noise gratings,” Opt. Commun. |

5. | 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. |

6. | Y. Cai and S. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. |

7. | Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. |

8. | T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. |

9. | Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express |

10. | E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. |

11. | F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. |

12. | A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun |

13. | Q. S. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. |

14. | E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A. |

15. | R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. |

16. | Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. |

17. | Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. |

18. | F. Wang, Y. Cai, and S. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express |

19. | Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. |

20. | C. Brosseau, Fundamentals of polarized light-a statistical approach (Wiley, New York, 1998). |

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

22. | F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. |

23. | 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. |

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

25. | E. Wolf, |

26. | G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. |

27. | Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure App. Opt. |

28. | 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. |

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

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

31. | D. Ge, Y. Cai, and Q. Lin, “Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems,” Chin. Phys. |

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

33. | H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. |

34. | O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media |

35. | H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. |

36. | W. Gao and O. Korotkova, “Changes in the state of polarization of a random electromagnetic beam propagating through tissue”, Opt. Comm. |

37. | F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. |

38. | O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. |

39. | A. Ishimaru, |

40. | L. C. Andrews, R. L. Phillips, and C. Y. Hopen, |

41. | V. A. Banakh and V. L. Mironov, “Phase approximation of the Huygens - Kirchhoff method in problems of space-limited optical-beam propagation in turbulent atmosphere,” Opt. Lett. |

42. | S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. |

43. | J. C. Leader, “Atmospheric propagation of partially coherent radiation”, J. Opt. Soc. Am. |

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

45. | H. T. Eyyuboğlu and Y. Baykal, “Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Express |

46. | Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. |

47. | Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express |

48. | Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. |

49. | Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. |

50. | R. J. Noriega-Manez and J. C. Gutierrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express |

51. | H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. |

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

53. | A. Gerrard and J. M. Burch, |

54. | H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am |

55. | H. T. Yura and S. G. Hanson, “Second-order statistics for wave propagation through complex optical systems,” J. Opt. Soc. Am |

56. | X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express |

57. | V. A. Banakh and L. V. Mironov, |

58. | J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in |

59. | Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik |

60. | R. K. Luneburg, |

61. | 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 |

62. | T. L. Ho, “Coherence degradation of Gaussian beams in a turbulent atmosphere,” J. Opt. Soc. Am |

**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: May 29, 2008

Revised Manuscript: August 30, 2008

Manuscript Accepted: September 12, 2008

Published: September 22, 2008

**Citation**

Yangjian Cai, Olga Korotkova, Halil T. Eyyuboglu, and Yahya Baykal, "Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere," Opt. Express **16**, 15834-15846 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15834

Sort: Year | Journal | Reset

### References

- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
- D. Kermisch, "Partially coherent image processing by laser scanning," J. Opt. Soc. Am 65, 887-891 (1975). [CrossRef]
- Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression," Phys. Rev. Lett. 53, 1057-1060 (1984). [CrossRef]
- A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silve-halide noise gratings," Opt. Commun. 98, 236-240 (1993). [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]
- Y. Cai and S. Zhu, "Ghost interference with partially coherent radiation," Opt. Lett. 29, 2716-2718 (2004). [CrossRef] [PubMed]
- Y. Cai and S. Zhu, "Ghost imaging with incoherent and partially coherent light radiation," Phys. Rev. E 71, 056607 (2005).
- T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004). [CrossRef] [PubMed]
- Y. Cai and U. Peschel, "Second-harmonic generation by an astigmatic partially coherent beam," Opt. Express 15, 15480-15492 (2007). [CrossRef] [PubMed]
- E. Wolf and E. Collett, "Partially coherent sources which produce same far-field intensity distribution as a laser," Opt. Commun. 25, 293-296 (1978). [CrossRef]
- F. Gori, "Collet-Wolf sources and multimode lasers," Opt. Commun. 34, 301-305 (1978). [CrossRef]
- A. T. Friberg and R. J. Sudol, "Propagation parameters of Gaussian Schell-model beams," Opt. Commun 41, 383-387 (1982). [CrossRef]
- Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988). [CrossRef]
- E. Tervonen, A. T. Friberg, and J. Turunen, "Gaussian Schell-model beams generated with synthetic acousto-optic holograms," J. Opt. Soc. Am. A. 9, 796-803 (1992). [CrossRef]
- R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants," Phys. Rev. A 31, 2419-2434 (1985).
- Q. Lin, Y. Cai, "Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams," Opt. Lett. 27, 216-218 (2002). [CrossRef]
- Y. Cai, Q. Lin, and S. Zhu, "Coincidence fractional Fourier transform with entangled photon pairs and incoherent light," Appl. Phy. Lett. 86, 021112 (2005). [CrossRef]
- F. Wang, Y. Cai, and S. He, "Experimental observation of coincidence fractional Fourier transform with a partially coherent beam," Opt. Express 14, 6999-7004 (2006). [CrossRef] [PubMed]
- Y. Cai and L. Hu, "Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system," Opt. Lett. 31, 685-687 (2006) [CrossRef] [PubMed]
- C. Brosseau, Fundamentals of polarized light-a statistical approach (Wiley, New York, 1998).
- E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
- F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998). [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, 1-9 (2001). [CrossRef]
- O. Korotkova, M. Salem, and E. Wolf, "Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source," Opt. Lett. 29, 1173-1175 (2004). [CrossRef] [PubMed]
- E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).
- G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002). [CrossRef]
- Y. Cai, D. Ge, and Q. Lin, "Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams," J. Opt. A: Pure App. Opt. 5, 453-459 (2003). [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]
- T. Shirai, O. Korotkova, and E. Wolf, "A method of generating electromagnetic Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 7, 232-237 (2005). [CrossRef]
- H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 249, 379-385 (2005). [CrossRef]
- D. Ge, Y. Cai and Q. Lin, "Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems," Chin. Phys. 14, 128-132 (2005). [CrossRef]
- O Korotkova, E Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation", Opt. Commun. 246, 35-43 (2005). [CrossRef]
- H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005). [CrossRef]
- O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005). [CrossRef]
- H. Wang, X. Wang, A. Zeng, and K. Yang, "Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation," Opt. Lett. 32, 2215-2217 (2007). [CrossRef] [PubMed]
- W. Gao and O. Korotkova, "Changes in the state of polarization of a random electromagnetic beam propagating through tissue," Opt. Comm. 270, 474-478 (2007). [CrossRef]
- F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, "Realizability condition for electromagnetic Schell-model sources," J. Opt. Soc. Am. A 25, 1016-1021 (2008). [CrossRef]
- O. Korotkova, "Scintillation index of a stochastic electromagnetic beam propagating in random media," Opt. Commun. 281, 2342-2348 (2008). [CrossRef]
- A. Ishimaru, Wave propagation and scattering in random media, (Academic Press, New York, 1978) Vol. 2.
- L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Washington, 2001). [CrossRef]
- V. A. Banakh and V. L. Mironov, "Phase approximation of the Huygens - Kirchhoff method in problems of space-limited optical-beam propagation in turbulent atmosphere," Opt. Lett. 4, 259-261 (1979). [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. Leader, "Atmospheric propagation of partially coherent radiation", J. Opt. Soc. Am. 68, 175-185 (1978). [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]
- H. T. Eyyuboglu and Y. Baykal, "Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere," Opt. Express 12, 4659-4674 (2004). [CrossRef] [PubMed]
- 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]
- Y. Cai and S. He, "Propagation of various dark hollow beams in a turbulent atmosphere," Opt. Express 14, 1353-1367 (2006). [CrossRef] [PubMed]
- 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, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, "Scintillation index of elliptical Gaussian beam in turbulent atmosphere," Opt. Lett. 32, 2405-2407 (2007). [CrossRef] [PubMed]
- R. J. Noriega-Manez and J. C. Gutierrez-Vega, "Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere," Opt. Express 15, 16328-16341 (2007). [CrossRef] [PubMed]
- 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]
- 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).
- A. Gerrard and J. M. Burch, Introduction to matrix methods in optics (John Wiley and Sons, 1975).
- H. T. Yura and S. G. Hanson, "Optical beam wave propagation through complex optical systems," J. Opt. Soc. Am A 4, 1931 (1987). [CrossRef]
- H. T. Yura and S. G. Hanson, "Second-order statistics for wave propagation through complex optical systems," J. Opt. Soc. Am A 6, 564 (1989). [CrossRef]
- X. Chu, "Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere," Opt. Express 15, 17613-17618 (2007). [CrossRef] [PubMed]
- V. A. Banakh and L. V. Mironov, LIDAR in a turbulent atmosphere (Artech House, Dedham 1987).
- J. A. Arnaud, "Hamiltonian theory of beam mode propagation," in Progress in Optics, Vol. XI, E. Wolf, ed. (North-Holland, 1973), pp. 247-304.
- Q. Lin, S. Wang, J. Alda, E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik 85, 67 (1990).
- R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1964).
- 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]
- T. L. Ho, "Coherence degradation of Gaussian beams in a turbulent atmosphere," J. Opt. Soc. Am 60, 667-673(1970). [CrossRef]

## 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.