Abstract

We derive the analytical expressions for the second-order moments of an electromagnetic Gaussian Schell-model (EGSM) beam propagating in a uniaxial crystal. With the help of the derived formulas, we study the evolution properties of the propagation factor, the effective radius of curvature and the Rayleigh range of an EGSM beam in a uniaxial crystal. It is found that the evolution properties of an EGSM beam in a uniaxial crystal are much different from its evolution properties in free space and are closely determined by the initial beam parameters and the parameters of the uniaxial crystal. The uniaxial crystal provides one way for modulating the properties of an EGSM beam.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  2. C. Brosseau, Fundamentals of Polarized Light: A Statistical Approach (Wiley, 1998).
  3. D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
    [CrossRef]
  4. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
    [CrossRef]
  5. G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17, 2019–2023 (2000).
    [CrossRef]
  6. F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert–Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
    [CrossRef]
  7. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
    [CrossRef]
  8. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
    [CrossRef]
  9. 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]
  10. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  11. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
    [CrossRef]
  12. 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]
  13. 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]
  14. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A 7, 232–237 (2005).
    [CrossRef]
  15. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
    [CrossRef]
  16. T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
    [CrossRef]
  17. Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian–Schell-model beams,” J. Opt. A 5, 453–459 (2003).
    [CrossRef]
  18. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
    [CrossRef]
  19. S. Zhu, L. Liu, Y. Chen, and Y. Cai, “State of polarization and propagation factor of a stochastic electromagnetic beam in a gradient-index fiber,” J. Opt. Soc. Am. A 30, 2306–2313 (2013).
    [CrossRef]
  20. O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
    [CrossRef]
  21. 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]
  22. 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, 103–108 (2005).
    [CrossRef]
  23. E. Wolf, “Coherence and polarization properties of electromagnetic laser modes,” Opt. Commun. 265, 60–62 (2006).
    [CrossRef]
  24. 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, 2266–2268 (2008).
    [CrossRef]
  25. 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, 2710–2720 (2008).
    [CrossRef]
  26. Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96, 499–507 (2009).
    [CrossRef]
  27. E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett. 32, 3400–3401 (2007).
    [CrossRef]
  28. M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33, 1180–1182 (2008).
    [CrossRef]
  29. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18, 12587–12598 (2010).
    [CrossRef]
  30. S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18, 27567–27581 (2010).
    [CrossRef]
  31. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
    [CrossRef]
  32. O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 681–690 (2009).
    [CrossRef]
  33. Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16, 15834–15846 (2008).
    [CrossRef]
  34. C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17, 21472–21487 (2009).
    [CrossRef]
  35. Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283, 3838–3845 (2010).
    [CrossRef]
  36. L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284, 1111–1117 (2011).
    [CrossRef]
  37. 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, 22503–22514 (2010).
    [CrossRef]
  38. G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express 19, 8700–8714 (2011).
    [CrossRef]
  39. F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36, 2722–2724 (2011).
    [CrossRef]
  40. C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108, 891–895 (2012).
    [CrossRef]
  41. M. Salem and G. P. Agrawal, “Coupling of stochastic electromagnetic beams into optical fibers,” Opt. Lett. 34, 2829–2831 (2009).
    [CrossRef]
  42. M. Salem and G. P. Agrawal, “Effects of coherence and polarization on the coupling of stochastic electromagnetic beams into optical fibers,” J. Opt. Soc. Am. A 26, 2452–2458 (2009).
    [CrossRef]
  43. M. Salem and G. P. Agrawal, “Effects of coherence and polarization on the coupling of stochastic electromagnetic beams into optical fibers: errata,” J. Opt. Soc. Am. A 28, 307 (2011).
    [CrossRef]
  44. J. J. Stamnes and V. Dhayalan, “Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18, 1662–1669 (2001).
    [CrossRef]
  45. 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, 2163–2171 (2003).
    [CrossRef]
  46. A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79–92 (2004).
    [CrossRef]
  47. B. Tang, “Hermite–cosine–Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26, 2480–2487 (2009).
    [CrossRef]
  48. A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
    [CrossRef]
  49. L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express 19, 13312–13325 (2011).
    [CrossRef]
  50. L. Zhang and Y. Cai, “Evolution properties of a twisted Gaussian Schell-model beam in a uniaxial crystal,” J. Mod. Opt. 58, 1224–1232 (2011).
    [CrossRef]
  51. G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20, 2196–2205 (2012).
    [CrossRef]
  52. C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz–Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57, 375–384 (2010).
    [CrossRef]
  53. D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
    [CrossRef]
  54. 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, 95–101 (2009).
    [CrossRef]
  55. D. Liu and Z. Zhou, “Generalized Stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A 11, 065710 (2009).
    [CrossRef]
  56. X. Du and D. Zhao, “Propagation of uniformly polarized stochastic electromagnetic beams in uniaxial crystals,” J. Electromagn. Waves Appl. 24, 971–981 (2010).
    [CrossRef]
  57. 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, 17344–17356 (2009).
    [CrossRef]
  58. F. Wang and Y. Cai, “Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18, 24661–24672 (2010).
    [CrossRef]
  59. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
    [CrossRef]
  60. R. Martinez-Herrero and P. M. Mejias, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18, 1669–1671 (1993).
    [CrossRef]
  61. R. Martinez-Herrero, P. M. Mejias, and M. Arias, “Parametric characterization of coherent, lowest-order Gaussian beams propagating through hard-edged apertures,” Opt. Lett. 20, 124–126 (1995).
    [CrossRef]
  62. A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
  63. F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
    [CrossRef]
  64. 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, 106–112 (1999).
    [CrossRef]
  65. X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
    [CrossRef]
  66. H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010).
    [CrossRef]
  67. S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
    [CrossRef]
  68. G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2011).

2013 (3)

S. Zhu, L. Liu, Y. Chen, and Y. Cai, “State of polarization and propagation factor of a stochastic electromagnetic beam in a gradient-index fiber,” J. Opt. Soc. Am. A 30, 2306–2313 (2013).
[CrossRef]

D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[CrossRef]

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

2012 (2)

G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20, 2196–2205 (2012).
[CrossRef]

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108, 891–895 (2012).
[CrossRef]

2011 (7)

2010 (9)

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, 22503–22514 (2010).
[CrossRef]

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

S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18, 27567–27581 (2010).
[CrossRef]

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010).
[CrossRef]

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

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz–Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57, 375–384 (2010).
[CrossRef]

X. Du and D. Zhao, “Propagation of uniformly polarized stochastic electromagnetic beams in uniaxial crystals,” J. Electromagn. Waves Appl. 24, 971–981 (2010).
[CrossRef]

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

2009 (9)

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

M. Salem and G. P. Agrawal, “Coupling of stochastic electromagnetic beams into optical fibers,” Opt. Lett. 34, 2829–2831 (2009).
[CrossRef]

M. Salem and G. P. Agrawal, “Effects of coherence and polarization on the coupling of stochastic electromagnetic beams into optical fibers,” J. Opt. Soc. Am. A 26, 2452–2458 (2009).
[CrossRef]

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, 17344–17356 (2009).
[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, 95–101 (2009).
[CrossRef]

D. Liu and Z. Zhou, “Generalized Stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A 11, 065710 (2009).
[CrossRef]

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96, 499–507 (2009).
[CrossRef]

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 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, 21472–21487 (2009).
[CrossRef]

2008 (6)

2007 (1)

2006 (1)

E. Wolf, “Coherence and polarization properties of electromagnetic laser modes,” Opt. Commun. 265, 60–62 (2006).
[CrossRef]

2005 (6)

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, 103–108 (2005).
[CrossRef]

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

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
[CrossRef]

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

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

2004 (3)

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]

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]

A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79–92 (2004).
[CrossRef]

2003 (3)

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, 2163–2171 (2003).
[CrossRef]

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

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

2002 (3)

T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

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]

A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
[CrossRef]

2001 (2)

J. J. Stamnes and V. Dhayalan, “Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18, 1662–1669 (2001).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

2000 (2)

1999 (1)

1998 (1)

1995 (1)

1994 (1)

1993 (1)

1992 (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

1991 (1)

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

1990 (1)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).

Agrawal, G. P.

Arias, M.

Baykal, Y.

Borghi, R.

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Approach (Wiley, 1998).

Cai, Y.

S. Zhu, L. Liu, Y. Chen, and Y. Cai, “State of polarization and propagation factor of a stochastic electromagnetic beam in a gradient-index fiber,” J. Opt. Soc. Am. A 30, 2306–2313 (2013).
[CrossRef]

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108, 891–895 (2012).
[CrossRef]

F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36, 2722–2724 (2011).
[CrossRef]

L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express 19, 13312–13325 (2011).
[CrossRef]

L. Zhang and Y. Cai, “Evolution properties of a twisted Gaussian Schell-model beam in a uniaxial crystal,” J. Mod. Opt. 58, 1224–1232 (2011).
[CrossRef]

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284, 1111–1117 (2011).
[CrossRef]

G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express 19, 8700–8714 (2011).
[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, 22503–22514 (2010).
[CrossRef]

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

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

S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18, 27567–27581 (2010).
[CrossRef]

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz–Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57, 375–384 (2010).
[CrossRef]

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

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, 17344–17356 (2009).
[CrossRef]

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

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96, 499–507 (2009).
[CrossRef]

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

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16, 15834–15846 (2008).
[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, 2266–2268 (2008).
[CrossRef]

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, 2710–2720 (2008).
[CrossRef]

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

Chen, C.

D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[CrossRef]

Chen, R.

Chen, Y.

Chu, X.

Ciattoni, A.

Cincotti, G.

Deng, D.

D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[CrossRef]

Dhayalan, V.

Dogariu, A.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

Dong, Y.

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108, 891–895 (2012).
[CrossRef]

Du, S.

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Du, X.

X. Du and D. Zhao, “Propagation of uniformly polarized stochastic electromagnetic beams in uniaxial crystals,” J. Electromagn. Waves Appl. 24, 971–981 (2010).
[CrossRef]

Ellis, J.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

Eyyuboglu, H. T.

Friberg, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Friberg, A. T.

Gbur, G.

G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2011).

Ge, D.

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

Gori, F.

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]

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]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert–Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
[CrossRef]

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, 106–112 (1999).
[CrossRef]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
[CrossRef]

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

James, D. F. V.

Ji, X.

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010).
[CrossRef]

Kaivola, M.

T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Korotkova, O.

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108, 891–895 (2012).
[CrossRef]

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284, 1111–1117 (2011).
[CrossRef]

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283, 3838–3845 (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, 22503–22514 (2010).
[CrossRef]

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

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96, 499–507 (2009).
[CrossRef]

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 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, 21472–21487 (2009).
[CrossRef]

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, 17344–17356 (2009).
[CrossRef]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16, 15834–15846 (2008).
[CrossRef]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
[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, 2266–2268 (2008).
[CrossRef]

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, 2710–2720 (2008).
[CrossRef]

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

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
[CrossRef]

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

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]

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]

Li, H.

D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[CrossRef]

Liang, C.

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Lin, Q.

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, 22503–22514 (2010).
[CrossRef]

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

Liu, D.

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, 95–101 (2009).
[CrossRef]

D. Liu and Z. Zhou, “Generalized Stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A 11, 065710 (2009).
[CrossRef]

Liu, L.

Liu, X.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Martinez-Herrero, R.

Mejias, P. M.

Mondello, A.

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]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

Palma, C.

Piquero, G.

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]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert–Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
[CrossRef]

Ponomarenko, S.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

Qu, J.

Ramírez-Sánchez, V.

Romanini, P.

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]

Roychowdhury, H.

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

Saastamoinen, T.

Salem, M.

Santarsiero, M.

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]

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]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert–Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
[CrossRef]

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, 106–112 (1999).
[CrossRef]

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

Setälä, T.

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, 103–108 (2005).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Shirai, T.

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

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

Sona, A.

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

Stamnes, J. J.

Tang, B.

Tervo, J.

Tong, Z.

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

Turunen, J.

Vahimaa, P.

Wang, F.

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108, 891–895 (2012).
[CrossRef]

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284, 1111–1117 (2011).
[CrossRef]

F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36, 2722–2724 (2011).
[CrossRef]

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

Wang, Z.

Watson, E.

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

Weber, H.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

Wolf, E.

G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2011).

M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33, 1180–1182 (2008).
[CrossRef]

E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett. 32, 3400–3401 (2007).
[CrossRef]

E. Wolf, “Coherence and polarization properties of electromagnetic laser modes,” Opt. Commun. 265, 60–62 (2006).
[CrossRef]

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
[CrossRef]

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

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

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]

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

G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17, 2019–2023 (2000).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Wu, G.

Yao, M.

Yuan, Y.

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

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, 17344–17356 (2009).
[CrossRef]

Zhang, L.

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284, 1111–1117 (2011).
[CrossRef]

L. Zhang and Y. Cai, “Evolution properties of a twisted Gaussian Schell-model beam in a uniaxial crystal,” J. Mod. Opt. 58, 1224–1232 (2011).
[CrossRef]

L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express 19, 13312–13325 (2011).
[CrossRef]

Zhao, C.

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108, 891–895 (2012).
[CrossRef]

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz–Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57, 375–384 (2010).
[CrossRef]

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

Zhao, D.

X. Du and D. Zhao, “Propagation of uniformly polarized stochastic electromagnetic beams in uniaxial crystals,” J. Electromagn. Waves Appl. 24, 971–981 (2010).
[CrossRef]

Zhao, X.

D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[CrossRef]

Zhou, G.

Zhou, Z.

D. Liu and Z. Zhou, “Generalized Stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A 11, 065710 (2009).
[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, 95–101 (2009).
[CrossRef]

Zhu, S.

Appl. Phys. B (5)

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96, 499–507 (2009).
[CrossRef]

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

C. Zhao, Y. Dong, G. Wu, F. Wang, Y. Cai, and O. Korotkova, “Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber,” Appl. Phys. B 108, 891–895 (2012).
[CrossRef]

D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010).
[CrossRef]

Eur. Phys. J. D (1)

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, 95–101 (2009).
[CrossRef]

J. Electromagn. Waves Appl. (1)

X. Du and D. Zhao, “Propagation of uniformly polarized stochastic electromagnetic beams in uniaxial crystals,” J. Electromagn. Waves Appl. 24, 971–981 (2010).
[CrossRef]

J. Mod. Opt. (3)

L. Zhang and Y. Cai, “Evolution properties of a twisted Gaussian Schell-model beam in a uniaxial crystal,” J. Mod. Opt. 58, 1224–1232 (2011).
[CrossRef]

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz–Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57, 375–384 (2010).
[CrossRef]

G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2011).

J. Opt. A (4)

D. Liu and Z. Zhou, “Generalized Stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A 11, 065710 (2009).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

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

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

J. Opt. Soc. Am. A (13)

S. Zhu, L. Liu, Y. Chen, and Y. Cai, “State of polarization and propagation factor of a stochastic electromagnetic beam in a gradient-index fiber,” J. Opt. Soc. Am. A 30, 2306–2313 (2013).
[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]

G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17, 2019–2023 (2000).
[CrossRef]

D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
[CrossRef]

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, 2710–2720 (2008).
[CrossRef]

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, 103–108 (2005).
[CrossRef]

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

A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
[CrossRef]

M. Salem and G. P. Agrawal, “Effects of coherence and polarization on the coupling of stochastic electromagnetic beams into optical fibers,” J. Opt. Soc. Am. A 26, 2452–2458 (2009).
[CrossRef]

M. Salem and G. P. Agrawal, “Effects of coherence and polarization on the coupling of stochastic electromagnetic beams into optical fibers: errata,” J. Opt. Soc. Am. A 28, 307 (2011).
[CrossRef]

J. J. Stamnes and V. Dhayalan, “Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18, 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, 2163–2171 (2003).
[CrossRef]

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, 106–112 (1999).
[CrossRef]

Opt. Commun. (11)

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79–92 (2004).
[CrossRef]

E. Wolf, “Coherence and polarization properties of electromagnetic laser modes,” Opt. Commun. 265, 60–62 (2006).
[CrossRef]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (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, 225–230 (2004).
[CrossRef]

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

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

L. Zhang, F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Commun. 284, 1111–1117 (2011).
[CrossRef]

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]

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

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

Opt. Express (11)

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, 22503–22514 (2010).
[CrossRef]

G. Wu and Y. Cai, “Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam,” Opt. Express 19, 8700–8714 (2011).
[CrossRef]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16, 15834–15846 (2008).
[CrossRef]

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

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

S. Zhu and Y. Cai, “M2-factor of a stochastic electromagnetic beam in a Gaussian cavity,” Opt. Express 18, 27567–27581 (2010).
[CrossRef]

L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express 19, 13312–13325 (2011).
[CrossRef]

G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20, 2196–2205 (2012).
[CrossRef]

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, 17344–17356 (2009).
[CrossRef]

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

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
[CrossRef]

Opt. Laser Technol. (1)

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Opt. Lett. (11)

R. Martinez-Herrero and P. M. Mejias, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18, 1669–1671 (1993).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, and M. Arias, “Parametric characterization of coherent, lowest-order Gaussian beams propagating through hard-edged apertures,” Opt. Lett. 20, 124–126 (1995).
[CrossRef]

M. Salem and G. P. Agrawal, “Coupling of stochastic electromagnetic beams into optical fibers,” Opt. Lett. 34, 2829–2831 (2009).
[CrossRef]

E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett. 32, 3400–3401 (2007).
[CrossRef]

M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33, 1180–1182 (2008).
[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, 2266–2268 (2008).
[CrossRef]

F. Wang, G. Wu, X. Liu, S. Zhu, and Y. Cai, “Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source,” Opt. Lett. 36, 2722–2724 (2011).
[CrossRef]

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
[CrossRef]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert–Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
[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]

Opt. Quantum Electron. (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

Phys. Lett. A (1)

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

Phys. Rev. E (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Proc. SPIE (1)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).

Other (3)

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

C. Brosseau, Fundamentals of Polarized Light: A Statistical Approach (Wiley, 1998).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

Geometry of the propagation of a beam in a uniaxial crystal orthogonal to the optical axis.

Fig. 2.
Fig. 2.

Propagation factors of an EGSM beam versus the ratio of extraordinary index to ordinary refractive index ne/n0 for different values of the rms widths of correlation functions δxx and δyy.

Fig. 3.
Fig. 3.

Propagation factors of an EGSM beam versus the ratio of extraordinary index to ordinary refractive index ne/n0 for different values of the rms widths of the spectral densities σx and σy.

Fig. 4.
Fig. 4.

Propagation factors of an EGSM beam versus the ratio of extraordinary index to ordinary refractive index ne/n0 for different values of Ax/Ay.

Fig. 5.
Fig. 5.

Effective radius of curvature of an EGSM beam along the x or y direction versus the propagation distance z for different values of the ratio of extraordinary index to ordinary refractive index ne/n0.

Fig. 6.
Fig. 6.

Rayleigh ranges of an EGSM beam along the x and y directions versus the ratio of extraordinary index to ordinary refractive index ne/n0 for different values of Ax/Ay, δxx, δyy, σx, and σy.

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

W(ρ1,ρ2,0)=[Wxx(ρ1,ρ2,0)Wxy(ρ1,ρ2,0)Wyx(ρ1,ρ2,0)Wyy(ρ1,ρ2,0)],
Wαβ(ρ1,ρ2,0)=Eα*(ρ1,0)Eβ(ρ2,0),(α,β=x,y),
Wαβ(ρx1,ρy1,ρx2,ρy2,0)=AαAβBαβexp[ρx12+ρx22+ρy12+ρy224σα2(ρx1ρx2)2+(ρy1ρy2)22δαβ2],
ε=(ne2000no2000no2),
Ex(x,y,z)=kno2iπzexp(iknez)Ex(ρx,ρy,0)×exp{k2izne[no2(xρx)2+ne2(yρy)2]}dρxdρy,
Ey(x,y,z)=kno2iπzexp(iknoz)Ey(ρx,ρy,0)×exp{kno2iz[(xρx)2+(yρy)2]}dρxdρy,
Wxx(x1,y1,x2,y2,z)=k2no24π2z2Wxx(ρx1,ρy1,ρx2,ρy2,0)exp{k2izne[no2(x1ρx1)2+ne2(y1ρy1)2]}dρx1dρy1×exp{k2izne[no2(x2ρx2)2+ne2(y2ρy2)2]}dρx2dρy2,
Wxy(x1,y1,x2,y2,z)=k2no24π2z2exp[ikz(none)]Wxy(ρx1,ρy1,ρx2,ρy2,0)exp{k2izne[no2(x1ρx1)2+ne2(y1ρy1)2]}dρx1dρy1×exp{kno2iz[(x2ρx2)2+(y2ρy2)2]}dρx2dρy2,
Wyx(x1,y1,x2,y2,z)=Wxy*(x2,y2,x1,y1,z),
Wyy(x1,y1,x2,y2,z)=k2no24π2z2Wyy(ρx1,ρy1,ρx2,ρy2,0)exp{kno2iz[(x1ρx1)2+(y1ρy1)2]}dρx1dρy1×exp{kno2iz[(x2ρx2)2+(y2ρy2)2]}dρx2dρy2.
Wxx(x1,y1,x2,y2,z)=Ax2k2no24z2(axx3axx41/4δxx4)(axx1axx21/4δxx4)exp[kno2x122iznek2no4x124axx1z2ne2k2no4x124axx1z2ne2(4axx1axx2δxx41)]×exp[kney122izk2ne2y124axx3z2k2ne2y124axx3z2(4axx3axx4δxx41)]×exp[k2no4δxx2z2ne2(4axx1axx2δxx41)x1x2+k2ne2δxx2z2(4axx3axx4δxx41)y1y2]×exp[kno22iznex22k2no4δxx4axx1z2ne2(4axx1axx2δxx41)x22]exp[kne2izy22k2ne2δxx4axx3z2(4axx3axx4δxx41)y22],
Wxy(x1,y1,x2,y2,z)=AxAyBxyk2no2exp[ikz(none)]4z2(axy2axy51/4δxx4)(axy4axy51/4δxx4)exp[(kno22iznek2no44axy2z2ne2)x12kno2izx22]×exp[(kne22iznek2ne24axy4z2)y12kno2izy22]exp[axy2k2δxy4z2(4axy5axy2δxy41)(no22axy2δxy2nex1+nox2)2]×exp[axy4k2δxy4z2(4axy5axy4δxy41)(ne2axy4δxy2y1+noy2)2],
Wyy(x1,y1,x2,y2,z)=Ay2k2no2δyy4z2(4ayy5ayy6δyy41)exp{[(kno2izk2no24ayy5z2)(x12+y12)+kno2iz(x22+y22)]}×exp{ayy5k2no2δyy4z2(4ayy5ayy6δyy41)[(x12ayy5δyy2x2)2+(y12ayy5δyy2y2)2]},
aαβ1=14σα2+12δαβ2+kno22izne,aαβ2=14σα2+12δαβ2kno22izne,aαβ3=14σα2+12δαβ2+kne2iz,aαβ4=14σα2+12δαβ2kne2iz,aαβ5=14σα2+12δαβ2+kno2iz,aαβ6=14σα2+12δαβ2kno2iz.
Wtr(x,y,qx,qy,z)=Wxx(x,y,qx,qy,z)+Wyy(x,y,qx,qy,z),
x=x1+x22,y=y1+y22,qx=x1x2,qy=y1y2.
htr(x,y,θx,θy,z)=(k2π)2Wtr(x,y,qx,qy,z)×exp(ikθxqxikθyqy)dqxdqy,
htr(x,y,θx,θy,z)=Ax2k2π(1/2σx2+2/δxx2)exp[12σx2(ρxzneθx/no2)212σx2(ρyzθy/ne)2]exp[k2(θx2+θy2)1/2σx2+2/δxx2]+Ay2k2π(1/2σy2+2/δyy2)exp[k2(θx2+θy2)1/2σy2+2/δyy2]×exp[12σy2(ρxzθx/no)212σy2(ρyzθy/no)2].
x2=Ax2σx2π(4σx2+δxx2)(z2ne22σx2no4+2k2σx2δxx24σx2+δxx2)Qk2δxx2+Ay2σy2π(4σy2+δyy2)(z22σy2no2+2k2σy2δyy24σy2+δyy2)Qk2δyy2,
y2=Ax2σx2π(4σx2+δxx2)(z22σx2ne2+2k2σx2δxx24σx2+δxx2)Qk2δxx2+Ay2σy2π(4σy2+δyy2)(z22σy2no2+2k2σy2δyy24σy2+δyy2)Qk2δyy2,
θx2=Ax2πne22Qk2no4(1+4σx2/δxx2)+Ay2π2Qk2no2(1+4σy2/δyy2),
θy2=Ax2π2Qk2ne2(1+4σx2/δxx2)+Ay2π2Qk2no2(1+4σyy2/δyy2),
xθx=Ax2πzne22Qk2no2(1+4σx2/δxx2)+Ay2πz2Qk2no2(1+4σy2/δyy2),
yθy=Ax2πz2Qk2ne2(1+4σx2/δxx2)+Ay2πz2Qk2no2(1+4σy2/δyy2),
xθy=0,yθx=0,xy=0,θxθy=0,
14σx2+1δxx22π2λ2,14σy2+1δxx22π2λ2.
Mx2(z)=2kno(x2θx2xθx2)1/2,
My2(z)=2kne(ρy2θy2ρyθy2)1/2.
Mx2(z)=[ne2no2(1+4σx2δxx2)Ax4σx4+Ax2Ay2σy4(Ax2σx2+Ay2σy2)2+(1+4σy2δyy2)Ay4σy4+Ax2Ay2σx4(Ax2σx2+Ay2σy2)2]1/2,
My2(z)=[(1+4σx2δxx2)(Ax4σx4+Ax2Ay2σy4)(Ax2σx2+Ay2σy2)2+ne2n02(1+4σy2δyy2)(Ay4σy4+Ax2Ay2σx4)(Ax2σx2+Ay2σy2)2]1/2.
Mx2(z)=My2(z)=[(1+4σx2δxx2)Ax4σx4+Ax2Ay2σy4(Ax2σx2+Ay2σy2)2+(1+4σy2δyy2)Ay4σy4+Ax2Ay2σx4(Ax2σx2+Ay2σy2)2]1/2.
Mx2(z)=My2(z)=(1+4σα2/δαα2)1/2,(α=xory).
Rx(z)=x2/xθx,Ry(z)=y2/yθy.
Rx(z)=Ax2σx2(4σx2+δxx2)(z2ne22σx2no4+2k2σx2δxx24σx2+δxx2)Ax2zδxx2ne22no2(1+4σx2/δxx2)+Ay2zδxx22no2(1+4σy2/δyy2)+Ay2σy2(4σy2+δyy2)(z22σy2no2+2k2σy2δyy24σy2+δyy2)Ax2zδyy2ne22no2(1+4σx2/δxx2)+Ay2zδyy22no2(1+4σy2/δyy2),
Ry(z)=Ax2σx2(4σx2+δxx2)(z22σx2ne2+2k2σx2δxx24σx2+δxx2)Ax2δxx2z2ne2(1+4σx2/δxx2)+Ay2δxx2z2no2(1+4σy2/δyy2)+Ay2σy2(4σy2+δyy2)(z22σy2no2+2k2σy2δyy24σy2+δyy2)Ax2δyy2z2ne2(1+4σx2/δxx2)+Ay2δyy2z2no2(1+4σy2/δyy2).
x2z=zRx2x2z=0=0,
y2z=zRy2y2z=0=0.
zRx=2kσ2no2δxxδyy×(Ax2+Ay2)Ax2δyy2ne2(4σx2+δxx2)+Ay2δxx2no2(4σy2+δyy2),
zRy=2kσ2noneδxxδyy×(Ax2+Ay2)Ax2δyy2no2(4σx2+δxx2)+Ay2δxx2ne2(4σy2+δyy2).

Metrics