## Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal |

Optics Express, Vol. 19, Issue 14, pp. 13312-13325 (2011)

http://dx.doi.org/10.1364/OE.19.013312

Acrobat PDF (1402 KB)

### Abstract

Nonparaxial propagation theory of coherent beams in a uniaxial crystal is extended to the partially coherent case. An analytical formula for the

© 2011 OSA

## 1. Introduction

14. C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. **36**(12), 2251–2253 (2011). [CrossRef] [PubMed]

15. E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. **25**(3), 293–296 (1978). [CrossRef]

17. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. **41**(6), 383–387 (1982). [CrossRef]

18. F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A **24**(7), 1937–1944 (2007). [CrossRef] [PubMed]

19. E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A **9**(5), 796–803 (1992). [CrossRef]

20. R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A **15**(9), 2373–2382 (1998). [CrossRef]

7. G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. **36**(10), 1939–1941 (2011). [CrossRef] [PubMed]

16. F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. **34**(3), 301–305 (1990). [CrossRef]

17. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. **41**(6), 383–387 (1982). [CrossRef]

20. R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A **15**(9), 2373–2382 (1998). [CrossRef]

30. F. Wang and Y. Cai, “Second-order statistics of a twisted gaussian Schell-model beam in turbulent atmosphere,” Opt. Express **18**(24), 24661–24672 (2010). [CrossRef] [PubMed]

31. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optic,” Phys. Rev. A **11**(4), 1365–1370 (1975). [CrossRef]

38. D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. B **26**(11), 2044–2049 (2009). [CrossRef]

39. L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B , doi:. [CrossRef]

45. X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B **102**(1), 205–213 (2011). [CrossRef]

46. D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxially crystals orthogonal to the optical axis,” Eur. Phys. J. D **54**(1), 95–101 (2009). [CrossRef]

47. D. Liu and Z. Zhou, “Propagation of partially coherent flat-topped beams in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A **26**(4), 924–930 (2009). [CrossRef] [PubMed]

48. Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B , doi:. [CrossRef]

50. J. Stamnes and G. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. **66**(8), 780–788 (1976). [CrossRef]

56. B. Tang, “Hermite-cosine-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A **26**(12), 2480–2487 (2009). [CrossRef] [PubMed]

## 2. Nonparaxial theory of a partially coherent beam propagating in a uniaxial crystal orthogonal to the optical axis

*x*-direction, is incident on a uniaxial crystal at the plane z = 0. The optical axis of the crystal coincides with the

*x*-axis, and the dielectric tensor of the crystal can be expressed as

*ω*in linear media is described by the following equation [49]:

*ε*is the relative dielectric tensor, and

**E**is the complex amplitude of the electric field. According to [52

52. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A **20**(11), 2163–2171 (2003). [CrossRef] [PubMed]

52. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A **20**(11), 2163–2171 (2003). [CrossRef] [PubMed]

52. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A **20**(11), 2163–2171 (2003). [CrossRef] [PubMed]

42. K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A **21**(10), 1924–1932 (2004). [CrossRef] [PubMed]

43. K. Duan and B. Lü, “Wigner-distribution-function matrix and its application to partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. B **22**(8), 1585–1593 (2005). [CrossRef]

57. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **66**(1), 016615 (2002). [CrossRef] [PubMed]

*x*,

*y*and z directions, respectively.

## 3. Statistical properties of a nonparaxial GSM beam propagating in a uniaxial crystal orthogonal to the optical axis

*x*-direction, is incident on a uniaxial crystal at the plane z = 0. The CSD matrix of the initial linearly polarized GSM beam at z = 0 is expressed as [1]

*x*,

*y*and

*z*components of the field, respectively. For a paraxial GSM beam in a uniaxial crystal, its intensity distribution is expressed as

57. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **66**(1), 016615 (2002). [CrossRef] [PubMed]

58. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A **312**(5-6), 263–267 (2003). [CrossRef]

15. E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. **25**(3), 293–296 (1978). [CrossRef]

19. E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A **9**(5), 796–803 (1992). [CrossRef]

4. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. **89**(4), 041117 (2006). [CrossRef]

11. Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express **15**(23), 15480–15492 (2007). [CrossRef] [PubMed]

## 4. Summary

## Acknowledgments

## References and links

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

2. | L. C. Andrews and R. L. Phillips, |

3. | Y. Baykal, “Average transmittance in turbulence for partially coherent sources,” Opt. Commun. |

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

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. | O. Korotkova, L. C. Andrews, and R. L. Phillips, “LIDAR model for a rough-surface target: method of partial coherence,” Proc. SPIE |

7. | G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. |

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

9. | M. von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head mounted retinal projection displays,” Opt. Eng. |

10. | Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

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

12. | 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. |

13. | C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express |

14. | C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. |

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

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

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

18. | F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A |

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

20. | R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A |

21. | D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. |

22. | S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

23. | Q. Lin and Y. Cai, “Tensor |

24. | Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. |

25. | Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A |

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

27. | 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. |

28. | F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A |

29. | G. Wu, Y. Cai, and J. Chen, “Shaping the beam profile of a partially coherent beam by a phase aperture,” Opt. Commun. |

30. | F. Wang and Y. Cai, “Second-order statistics of a twisted gaussian Schell-model beam in turbulent atmosphere,” Opt. Express |

31. | M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optic,” Phys. Rev. A |

32. | H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. |

33. | A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A |

34. | G. P. Agrawal and D. N. Pattanayak, “Gaussain beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. |

35. | S. R. S. Seshadri, “Virtual source for the Bessel-Gaussian beam,” Opt. Lett. |

36. | K. Duan and B. Lü, “Nonparaxial analysis of far-field properties of Gaussian beams diffracted at a circular aperture,” Opt. Express |

37. | G. Zhou, “The analytical vectorial structure of a nonparaxial Gaussian beam close to the source,” Opt. Express |

38. | D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. B |

39. | L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B , doi:. [CrossRef] |

40. | K. Duan and B. Lü, “Partially coherent nonparaxial beams,” Opt. Lett. |

41. | Y. Zhang and B. Lü, “Propagation of the Wigner distribution function for partially coherent nonparaxial beams,” Opt. Lett. |

42. | K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A |

43. | K. Duan and B. Lü, “Wigner-distribution-function matrix and its application to partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. B |

44. | F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a partially coherent field,” J. Opt. Soc. Am. A |

45. | X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B |

46. | D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxially crystals orthogonal to the optical axis,” Eur. Phys. J. D |

47. | D. Liu and Z. Zhou, “Propagation of partially coherent flat-topped beams in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A |

48. | Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B , doi:. [CrossRef] |

49. | M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999). |

50. | J. Stamnes and G. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. |

51. | J. J. Stamnes and V. Dhayalan, “Transmission of a twodimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A |

52. | A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A |

53. | B. Lü and S. Luo, “Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. |

54. | D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. |

55. | D. Liu and Z. Zhou, “Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. |

56. | B. Tang, “Hermite-cosine-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A |

57. | T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

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

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(260.5430) Physical optics : Polarization

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: May 20, 2011

Revised Manuscript: June 14, 2011

Manuscript Accepted: June 15, 2011

Published: June 24, 2011

**Citation**

Lina Zhang and Yangjian Cai, "Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal," Opt. Express **19**, 13312-13325 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13312

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### References

- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
- L. C. Andrews and R. L. Phillips, Laser Beam Propagation in the Turbulent Atmosphere, 2nd ed. (SPIE, 2005).
- Y. Baykal, “Average transmittance in turbulence for partially coherent sources,” Opt. Commun. 231(1-6), 129–136 (2004). [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(4), 041117 (2006). [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(3), 745–752 (2007). [CrossRef] [PubMed]
- O. Korotkova, L. C. Andrews, and R. L. Phillips, “LIDAR model for a rough-surface target: method of partial coherence,” Proc. SPIE 5237, 49–60 (2004). [CrossRef]
- G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. 36(10), 1939–1941 (2011). [CrossRef] [PubMed]
- D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am. 65(8), 887–891 (1975). [CrossRef]
- M. von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head mounted retinal projection displays,” Opt. Eng. 43(7), 1552–1560 (2004). [CrossRef]
- Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005). [CrossRef] [PubMed]
- Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007). [CrossRef] [PubMed]
- 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(11), 1057–1060 (1984). [CrossRef]
- C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009). [CrossRef] [PubMed]
- C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011). [CrossRef] [PubMed]
- E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978). [CrossRef]
- F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1990). [CrossRef]
- A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982). [CrossRef]
- F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007). [CrossRef] [PubMed]
- E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992). [CrossRef]
- R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15(9), 2373–2382 (1998). [CrossRef]
- D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994). [CrossRef]
- S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 (2001). [CrossRef] [PubMed]
- Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002). [CrossRef] [PubMed]
- Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27(19), 1672–1674 (2002). [CrossRef] [PubMed]
- Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19(10), 2036–2042 (2002). [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(6), 685–687 (2006). [CrossRef] [PubMed]
- 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(1), 157–167 (2007). [CrossRef]
- F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25(8), 2001–2010 (2008). [CrossRef] [PubMed]
- G. Wu, Y. Cai, and J. Chen, “Shaping the beam profile of a partially coherent beam by a phase aperture,” Opt. Commun. 284(18), 4129–4135 (2011). [CrossRef]
- F. Wang and Y. Cai, “Second-order statistics of a twisted gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010). [CrossRef] [PubMed]
- M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optic,” Phys. Rev. A 11(4), 1365–1370 (1975). [CrossRef]
- H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998). [CrossRef]
- A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9(5), 765–774 (1992). [CrossRef]
- G. P. Agrawal and D. N. Pattanayak, “Gaussain beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69(4), 575–578 (1979). [CrossRef]
- S. R. S. Seshadri, “Virtual source for the Bessel-Gaussian beam,” Opt. Lett. 27, 988–1000 (2002).
- K. Duan and B. Lü, “Nonparaxial analysis of far-field properties of Gaussian beams diffracted at a circular aperture,” Opt. Express 11(13), 1474–1480 (2003). [CrossRef] [PubMed]
- G. Zhou, “The analytical vectorial structure of a nonparaxial Gaussian beam close to the source,” Opt. Express 16(6), 3504–3514 (2008). [CrossRef] [PubMed]
- D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. B 26(11), 2044–2049 (2009). [CrossRef]
- L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B , doi:. [CrossRef]
- K. Duan and B. Lü, “Partially coherent nonparaxial beams,” Opt. Lett. 29(8), 800–802 (2004). [CrossRef] [PubMed]
- Y. Zhang and B. Lü, “Propagation of the Wigner distribution function for partially coherent nonparaxial beams,” Opt. Lett. 29(23), 2710–2712 (2004). [CrossRef] [PubMed]
- K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A 21(10), 1924–1932 (2004). [CrossRef] [PubMed]
- K. Duan and B. Lü, “Wigner-distribution-function matrix and its application to partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. B 22(8), 1585–1593 (2005). [CrossRef]
- F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a partially coherent field,” J. Opt. Soc. Am. A 27(5), 1120–1126 (2010). [CrossRef] [PubMed]
- X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102(1), 205–213 (2011). [CrossRef]
- D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxially crystals orthogonal to the optical axis,” Eur. Phys. J. D 54(1), 95–101 (2009). [CrossRef]
- D. Liu and Z. Zhou, “Propagation of partially coherent flat-topped beams in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26(4), 924–930 (2009). [CrossRef] [PubMed]
- Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B , doi:. [CrossRef]
- M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).
- J. Stamnes and G. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. 66(8), 780–788 (1976). [CrossRef]
- J. J. Stamnes and V. Dhayalan, “Transmission of a twodimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18(7), 1662–1669 (2001). [CrossRef]
- A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20(11), 2163–2171 (2003). [CrossRef] [PubMed]
- B. Lü and S. Luo, “Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36(1), 51–56 (2004). [CrossRef]
- D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008). [CrossRef]
- D. Liu and Z. Zhou, “Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 10(9), 095005 (2008). [CrossRef]
- B. Tang, “Hermite-cosine-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26(12), 2480–2487 (2009). [CrossRef] [PubMed]
- T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002). [CrossRef] [PubMed]
- E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]

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