## Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam |

Optics Express, Vol. 19, Issue 9, pp. 8700-8714 (2011)

http://dx.doi.org/10.1364/OE.19.008700

Acrobat PDF (1459 KB)

### Abstract

Analytical formula for the cross-spectral density matrix of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam truncated by a circular phase aperture propagating in free space is derived with the help of a tensor method, which provides a reliable and fast way for studying the propagation and transformation of a truncated EGSM beam. Statistics properties, such as the spectral intensity, the degree of coherence, the degree of polarization and the polarization ellipse of a truncated EGSM beam in free space are studied numerically. The propagation factor of a truncated EGSM beam is also analyzed. Our numerical results show that we can modulate the spectral intensity, the polarization, the coherence and the propagation factor of an EGSM beam by a circular phase aperture. It is found that the phase aperture can be used to shape the beam profile of an EGSM beam and generate electromagnetic partially coherent dark hollow or flat-topped beam, which is useful in some applications, such as optical trapping, material processing, free-space optical communications.

© 2011 OSA

## 1. Introduction

20. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express **18**(12), 12587–12598 (2010). [CrossRef] [PubMed]

2. 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]

4. 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]

20. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express **18**(12), 12587–12598 (2010). [CrossRef] [PubMed]

15. 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**(4-6), 225–230 (2004). [CrossRef]

23. O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B **94**(4), 681–690 (2009). [CrossRef]

24. C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express **17**(24), 21472–21487 (2009). [CrossRef] [PubMed]

25. Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. **283**(20), 3838–3845 (2010). [CrossRef]

26. M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. **33**(19), 2266–2268 (2008). [CrossRef] [PubMed]

29. S. Zhu and Y. Cai, “M^{2}-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express **18**(26), 27567–27581 (2010). [CrossRef]

30. M. Yao, Y. Cai, O. Korotkova, Q. Lin, and Z. Wang, “Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings,” Opt. Express **18**(21), 22503–22514 (2010). [CrossRef] [PubMed]

31. C. Ding, L. Pan, and B. Lu, “Changes in the state of polarization of apertured stochastic electromagnetic modified Bessel-Gauss beams in free-space propagation,” Appl. Phys. B **99**(1-2), 307–315 (2010). [CrossRef]

32. Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express **19**(7), 5979–5992 (2011). [CrossRef] [PubMed]

36. N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. **7**(3), 216–220 (2000). [CrossRef]

37. Y. Cai and S. He, “Partially coherent flattened Gaussian beam and its paraxial propagation properties,” J. Opt. Soc. Am. A **23**(10), 2623–2628 (2006). [CrossRef]

52. V. A. Zenkin, V. R. Kushnir, and L. V. Tarasov, “Phase aperture as means for selecting the lowest transverse mode,” Sov. J. Quantum Electron. **11**(2), 243–245 (1981). [CrossRef]

37. Y. Cai and S. He, “Partially coherent flattened Gaussian beam and its paraxial propagation properties,” J. Opt. Soc. Am. A **23**(10), 2623–2628 (2006). [CrossRef]

41. X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. **43**(3), 577–585 (2011). [CrossRef]

42. C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. **33**(12), 1389–1391 (2008). [CrossRef] [PubMed]

43. F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. **33**(16), 1795–1797 (2008). [CrossRef] [PubMed]

44. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M^{2}-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express **17**(20), 17344–17356 (2009). [CrossRef] [PubMed]

50. X. Chu, Z. Li, and Y. Wu, “Relay propagation of partially coherent flattened Gaussian beam in turbulent atmosphere,” Opt. Commun. **282**(23), 4486–4489 (2009). [CrossRef]

51. K. Ait-Ameur, “Effects of a phase aperture on the fundamental mode of a hard-apertured cavity,” J. Mod. Opt. **49**(7), 1157–1168 (2002). [CrossRef]

52. V. A. Zenkin, V. R. Kushnir, and L. V. Tarasov, “Phase aperture as means for selecting the lowest transverse mode,” Sov. J. Quantum Electron. **11**(2), 243–245 (1981). [CrossRef]

49. X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A **24**(11), 3554–3563 (2007). [CrossRef]

53. Z. Jiang, Q. Lu, and Z. Liu, “Relative phase shifts of apertured Gaussian beams and transformation of a Gaussian beam through a phase aperture,” Appl. Opt. **36**(4), 772–778 (1997). [CrossRef] [PubMed]

54. G. Zhou, “Far-field structure property of a Gaussian beam diffracted by a phase aperture,” Opt. Commun. **284**(1), 8–14 (2011). [CrossRef]

## 2. Statistics properties of an EGSM beam truncated by a circular phase aperture in free space

*ϕ*and radius is

*a*propagating in free space. Within the validity of the paraxial approximation, the propagation of a partially coherent beam truncated by a circular phase aperture in free space can be studied with the help of the following generalized Huygens-Fresnel integral [55

55. 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]

*λ*being the wavelength,

56. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. **83**(5), 1752–1756 (1988). [CrossRef]

57. 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]

56. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. **83**(5), 1752–1756 (1988). [CrossRef]

*M*= 10 assures a very good description of the diffracted beam [56

56. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. **83**(5), 1752–1756 (1988). [CrossRef]

57. 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]

**83**(5), 1752–1756 (1988). [CrossRef]

57. 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]

58. 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]

**I**is a

4. 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]

26. M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. **33**(19), 2266–2268 (2008). [CrossRef] [PubMed]

*T*stands for vector transposition and

*α*direction,

*x*component of the field, of the

*y*component of the field and of the mutual correlation function of

*x*and

*y*field components, respectively. In Eqs. (9) and (10) as well as in all the formulas below the explicit dependence of the parameters

**ρ**are defined by the expressions [1]andThe degree of coherence of the EGSM beam at a pair of transverse points

*m*for different values of the phase delay

*ϕ*and correlation coefficients

59. 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]

*ϕ*with

*ϕ*of the circular phase aperture. To learn about the behavior of the on-axis polarization ellipse of the truncated EGSM beam on propagation. We calculate in Fig. 7 the polarization ellipse of the truncated EGSM beam at several propagation distances in free space with

*m*for different values of the phase delay

*ϕ*in free space with

14. 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]

## 3. Propagation factor of an EGSM beam truncated by a circular phase aperture

61. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A **16**(1), 106–112 (1999). [CrossRef]

20. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express **18**(12), 12587–12598 (2010). [CrossRef] [PubMed]

29. S. Zhu and Y. Cai, “M^{2}-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express **18**(26), 27567–27581 (2010). [CrossRef]

29. S. Zhu and Y. Cai, “M^{2}-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express **18**(26), 27567–27581 (2010). [CrossRef]

*J*is given by:

*W*. The mean values

*W*in Eqs. (24)–(30) with

*ϕ*with

*a*with

## 4. Conclusion

## Acknowledgments

## References and links

1. | E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007). |

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

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

5. | T. Saastamoinen, J. Turunen, J. Tervo, T. Setälä, and A. T. Friberg, “Electromagnetic coherence theory of laser resonator modes,” J. Opt. Soc. Am. A |

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

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

8. | X. Ji, E. Zhang, and B. Lu, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere,” Opt. Commun. |

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

10. | B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett. |

11. | L. Pan, M. Sun, C. Ding, Z. Zhao, and B. Lü, “Effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic beams passing through an astigmatic optical system,” Opt. Express |

12. | L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett. |

13. | Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B |

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

15. | 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. |

16. | 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. |

17. | 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 |

18. | Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. |

19. | H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. |

20. | S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express |

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

22. | Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express |

23. | O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B |

24. | C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express |

25. | Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. |

26. | M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. |

27. | O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “State of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A |

28. | M. Yao, Y. Cai, and O. Korotkova, “Spectral shift of a stochastic electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Commun. |

29. | S. Zhu and Y. Cai, “M |

30. | M. Yao, Y. Cai, O. Korotkova, Q. Lin, and Z. Wang, “Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings,” Opt. Express |

31. | C. Ding, L. Pan, and B. Lu, “Changes in the state of polarization of apertured stochastic electromagnetic modified Bessel-Gauss beams in free-space propagation,” Appl. Phys. B |

32. | Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express |

33. | J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in |

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

35. | 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. |

36. | N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. |

37. | Y. Cai and S. He, “Partially coherent flattened Gaussian beam and its paraxial propagation properties,” J. Opt. Soc. Am. A |

38. | F. Wang, Y. Cai, and X. Ma, “Circular partially coherent flattened Gaussian beam,” Opt. Lasers Eng. |

39. | Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation,” J. Opt. Soc. Am. B |

40. | X. Lu and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A |

41. | X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. |

42. | C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. |

43. | F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. |

44. | Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M |

45. | Y. Qiu, Z. Chen, and L. Liu, “Partially coherent dark hollow beams propagating through real ABCD optical systems in a turbulent atmosphere,” J. Mod. Opt. |

46. | M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. |

47. | F. Cheng and Y. Cai, “Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. |

48. | Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express |

49. | X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A |

50. | X. Chu, Z. Li, and Y. Wu, “Relay propagation of partially coherent flattened Gaussian beam in turbulent atmosphere,” Opt. Commun. |

51. | K. Ait-Ameur, “Effects of a phase aperture on the fundamental mode of a hard-apertured cavity,” J. Mod. Opt. |

52. | V. A. Zenkin, V. R. Kushnir, and L. V. Tarasov, “Phase aperture as means for selecting the lowest transverse mode,” Sov. J. Quantum Electron. |

53. | Z. Jiang, Q. Lu, and Z. Liu, “Relative phase shifts of apertured Gaussian beams and transformation of a Gaussian beam through a phase aperture,” Appl. Opt. |

54. | G. Zhou, “Far-field structure property of a Gaussian beam diffracted by a phase aperture,” Opt. Commun. |

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

56. | J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. |

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

58. | 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 |

59. | 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 |

60. | A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE |

61. | M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A |

**OCIS Codes**

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

(050.1220) Diffraction and gratings : Apertures

(140.3300) Lasers and laser optics : Laser beam shaping

(260.5430) Physical optics : Polarization

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: February 28, 2011

Revised Manuscript: April 13, 2011

Manuscript Accepted: April 14, 2011

Published: April 19, 2011

**Citation**

Gaofeng Wu and Yangjian Cai, "Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam," Opt. Express **19**, 8700-8714 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8700

Sort: Year | Journal | Reset

### References

- E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).
- 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]
- E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]
- 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]
- T. Saastamoinen, J. Turunen, J. Tervo, T. Setälä, and A. T. Friberg, “Electromagnetic coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 22(1), 103–108 (2005). [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]
- H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007). [CrossRef] [PubMed]
- X. Ji, E. Zhang, and B. Lu, “Changes in the spectrum and polarization of polychromatic partially coherent electromagnetic beams in the turbulent atmosphere,” Opt. Commun. 275(2), 292–300 (2007). [CrossRef]
- F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schellmodel sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008). [CrossRef]
- B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett. 33(20), 2410–2412 (2008). [CrossRef] [PubMed]
- L. Pan, M. Sun, C. Ding, Z. Zhao, and B. Lü, “Effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic beams passing through an astigmatic optical system,” Opt. Express 17(9), 7310–7321 (2009). [CrossRef] [PubMed]
- L. Pan, Z. Zhao, C. Ding, and B. Lu, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett. 95(18), 181112 (2009). [CrossRef]
- Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009). [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]
- 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(4-6), 225–230 (2004). [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(11), 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(3), 353–364 (2005). [CrossRef]
- Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9(12), 1123–1130 (2007). [CrossRef]
- H. Wang, X. Wang, A. Zeng, and W. Lu, “Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 276(2), 218–221 (2007). [CrossRef]
- S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010). [CrossRef] [PubMed]
- O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
- Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008). [CrossRef] [PubMed]
- O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009). [CrossRef]
- C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009). [CrossRef] [PubMed]
- Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (2010). [CrossRef]
- M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008). [CrossRef] [PubMed]
- O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “State of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008). [CrossRef]
- M. Yao, Y. Cai, and O. Korotkova, “Spectral shift of a stochastic electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Commun. 283(22), 4505–4511 (2010). [CrossRef]
- S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18(26), 27567–27581 (2010). [CrossRef]
- M. Yao, Y. Cai, O. Korotkova, Q. Lin, and Z. Wang, “Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings,” Opt. Express 18(21), 22503–22514 (2010). [CrossRef] [PubMed]
- C. Ding, L. Pan, and B. Lu, “Changes in the state of polarization of apertured stochastic electromagnetic modified Bessel-Gauss beams in free-space propagation,” Appl. Phys. B 99(1-2), 307–315 (2010). [CrossRef]
- Y. Dong, Y. Cai, C. Zhao, and M. Yao, “Statistics properties of a cylindrical vector partially coherent beam,” Opt. Express 19(7), 5979–5992 (2011). [CrossRef] [PubMed]
- J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 44, E. Wolf, ed., North-Holland, Amsterdam, 2003, pp.119–204.
- Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006). [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]
- N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7(3), 216–220 (2000). [CrossRef]
- Y. Cai and S. He, “Partially coherent flattened Gaussian beam and its paraxial propagation properties,” J. Opt. Soc. Am. A 23(10), 2623–2628 (2006). [CrossRef]
- F. Wang, Y. Cai, and X. Ma, “Circular partially coherent flattened Gaussian beam,” Opt. Lasers Eng. 49(4), 481–489 (2011). [CrossRef]
- Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation,” J. Opt. Soc. Am. B 23(7), 1398–1407 (2006). [CrossRef]
- X. Lu and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369(1-2), 157–166 (2007). [CrossRef]
- X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011). [CrossRef]
- C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008). [CrossRef] [PubMed]
- F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33(16), 1795–1797 (2008). [CrossRef] [PubMed]
- Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009). [CrossRef] [PubMed]
- Y. Qiu, Z. Chen, and L. Liu, “Partially coherent dark hollow beams propagating through real ABCD optical systems in a turbulent atmosphere,” J. Mod. Opt. 57(8), 662–669 (2010). [CrossRef]
- M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008). [CrossRef]
- F. Cheng and Y. Cai, “Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 284(1), 30–37 (2011). [CrossRef]
- Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008). [CrossRef] [PubMed]
- X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24(11), 3554–3563 (2007). [CrossRef]
- X. Chu, Z. Li, and Y. Wu, “Relay propagation of partially coherent flattened Gaussian beam in turbulent atmosphere,” Opt. Commun. 282(23), 4486–4489 (2009). [CrossRef]
- K. Ait-Ameur, “Effects of a phase aperture on the fundamental mode of a hard-apertured cavity,” J. Mod. Opt. 49(7), 1157–1168 (2002). [CrossRef]
- V. A. Zenkin, V. R. Kushnir, and L. V. Tarasov, “Phase aperture as means for selecting the lowest transverse mode,” Sov. J. Quantum Electron. 11(2), 243–245 (1981). [CrossRef]
- Z. Jiang, Q. Lu, and Z. Liu, “Relative phase shifts of apertured Gaussian beams and transformation of a Gaussian beam through a phase aperture,” Appl. Opt. 36(4), 772–778 (1997). [CrossRef] [PubMed]
- G. Zhou, “Far-field structure property of a Gaussian beam diffracted by a phase aperture,” Opt. Commun. 284(1), 8–14 (2011). [CrossRef]
- 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]
- J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988). [CrossRef]
- 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]
- 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]
- 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]
- A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2C14 (1990).
- M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999). [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.