Abstract

With the help of a tensor method, an explicit expression for the M2-factor of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam in a Gaussian cavity is derived. Evolution properties of the M 2-factor of an EGSM beam in a Gaussian cavity are studied numerically in detail. It is found that the behavior of the M2-factor of an EGSM beam in a Gaussian cavity is determined by the statistical properties of the source beam and the parameters of the cavity. Thermal lens effect induced changes of the M 2-factor of an EGSM beam in a Gaussian cavity is also investigated. Our results will be useful in many applications, such as free-space optical communications, laser radar system, optical trapping and optical imaging, where stochastic electromagnetic beams are required.

© 2010 OSA

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    [CrossRef] [PubMed]
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2010 (6)

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

G. Zhou, “Generalized beam propagation factors of truncated partially coherent cosine-Gaussian and cosh-Gaussian beams,” Opt. Laser Technol. 42(3), 489–496 (2010).
[CrossRef]

S. Yin, B. Zhang, and Y. Dan, “Effects of the deformation of reflection volume Bragg gratings on the M2-factor of super-Gaussian laser beams,” Opt. Commun. 283(7), 1418–1423 (2010).
[CrossRef]

Y. Yuan, Y. Cai, C. Zhao, H. T. Eyyuboğlu, and Y. Baykal, “Propagation factors of laser array beams in turbulent atmosphere,” J. Mod. Opt. 57(8), 621–631 (2010).
[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]

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]

2009 (3)

2008 (5)

2006 (2)

E. Wolf, “Coherence and polarization properties of electromagnetic laser modes,” Opt. Commun. 265(1), 60–62 (2006).
[CrossRef]

T. Xu and S. Wang, “Propagation of Ince-Gaussian beams in a thermal lens medium,” Opt. Commun. 265(1), 1–5 (2006).
[CrossRef]

2005 (4)

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]

D. Deng, “Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture,” Phys. Lett. A 341(1-4), 352–356 (2005).
[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(1), 103–108 (2005).
[CrossRef]

D. Deng, X. Fu, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution and M2 factor of hollow Gaussian beams,” Appl. Opt. 44(33), 7187–7190 (2005).
[CrossRef] [PubMed]

2004 (2)

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]

D. Deng, “Propagation properties of off-axis cosh Gaussian beam combinations through a first-order optical system,” Phys. Lett. A 333(5-6), 485–494 (2004).
[CrossRef]

2003 (4)

D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite-Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. 5(5), 489–494 (2003).
[CrossRef]

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

J. Tervo, T. Setala, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003).
[CrossRef] [PubMed]

X. Chu, B. Zhang, and Q. Wen, “Generalized M2 factor of a partially coherent beam propagating through a circular hard-edged aperture,” Appl. Opt. 42(21), 4280–4284 (2003).
[CrossRef] [PubMed]

2002 (3)

2001 (1)

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]

1999 (3)

1998 (3)

J. F. Urchueguía, G. J. de Valcarcel, and E. Roldan, “Laser instabilities in a Gaussian cavity mode with Gaussian pump profile,” J. Opt. Soc. Am. B 15(5), 1512–1520 (1998).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[CrossRef]

C. Palma, G. Cardone, and G. Cincotti, “Spectral changes in Gaussian-cavity lasers,” IEEE J. Quantum Electron. 34(7), 1082–1088 (1998).
[CrossRef]

1997 (1)

1996 (1)

P. De Santis, A. Mascello, C. Palma, and M. R. Perrone, “Coherence growth of laser radiation in Gaussian cavities,” IEEE J. Quantum Electron. 32(5), 802–812 (1996).
[CrossRef]

1995 (1)

1994 (3)

J. J. Chang, “Time-resolved beam-quality characterization of copper-vapor lasers with unstable resonators,” Appl. Opt. 33(12), 2255–2265 (1994).
[CrossRef] [PubMed]

G. N. Lawrence, “Proposed international standard for laser-beam quality falls short,” Laser Focus World 109–114 (1994).

G. Cincotti, P. DeSantis, G. Guattari, and C. Palma, “Propagation of partially coherent beams in a periodic sequence of lenses and Gaussian apertures,” Pure Appl. Opt. 3(4), 561–571 (1994).
[CrossRef]

1993 (3)

P. A. Belanger, “Beam quality factor of the LP01 mode of the step-index fiber,” Opt. Eng. 32(9), 2107–2109 (1993).
[CrossRef]

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993).
[CrossRef]

R. Martinez-Herrero and P. M. Mejias, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18(19), 1669–1671 (1993).
[CrossRef] [PubMed]

1991 (1)

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).
[CrossRef]

1985 (1)

S. Wang, “Matrix methods in treating decentred optical systems,” Opt. Quantum Electron. 17(1), 1–14 (1985).
[CrossRef]

1984 (2)

1982 (1)

1980 (1)

F. Gori, “Propagation of the mutual intensity through a periodic structure,” Atti Fond. Giorgio Ronchi 35, 434–447 (1980).

1975 (1)

1974 (1)

J. R. Whinnery, “Laser measurement of optical absorption in liquids,” Acc. Chem. Res. 7(7), 225–231 (1974).
[CrossRef]

1973 (1)

1966 (1)

1965 (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36(1), 3–8 (1965).
[CrossRef]

1963 (1)

E. Wolf, “Spatial coherence of resonant modes in a maser interferometer,” Phys. Lett. 3(4), 166–168 (1963).
[CrossRef]

1961 (2)

A. G. Fox and T. Li, “Resonate modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

G. D. Boyd and J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

Agarwal, G. S.

Arias, M.

Baykal, Y.

Belanger, P. A.

P. A. Belanger, “Beam quality factor of the LP01 mode of the step-index fiber,” Opt. Eng. 32(9), 2107–2109 (1993).
[CrossRef]

Borghi, R.

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]

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]

Boyd, G. D.

G. D. Boyd and J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

Cai, Y.

Y. Yuan, Y. Cai, C. Zhao, H. T. Eyyuboğlu, and Y. Baykal, “Propagation factors of laser array beams in turbulent atmosphere,” J. Mod. Opt. 57(8), 621–631 (2010).
[CrossRef]

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

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]

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (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(20), 17344–17356 (2009).
[CrossRef] [PubMed]

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]

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]

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, 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]

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]

S. Zhu, F. Zhou, Y. Cai, and L. Zhang, “Thermal lens effect induced changes of polarization, coherence and spectrum of a stochastic electromagnetic beam in a Gaussian cavity,” Appl. Phys. B DOI: .

Cardone, G.

C. Palma, G. Cardone, and G. Cincotti, “Spectral changes in Gaussian-cavity lasers,” IEEE J. Quantum Electron. 34(7), 1082–1088 (1998).
[CrossRef]

Carter, C. A.

Casperson, L. W.

Chang, J. J.

Chen, X.

D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite-Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. 5(5), 489–494 (2003).
[CrossRef]

Chu, X.

Cincotti, G.

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]

C. Palma, G. Cardone, and G. Cincotti, “Spectral changes in Gaussian-cavity lasers,” IEEE J. Quantum Electron. 34(7), 1082–1088 (1998).
[CrossRef]

G. Cincotti, P. DeSantis, G. Guattari, and C. Palma, “Propagation of partially coherent beams in a periodic sequence of lenses and Gaussian apertures,” Pure Appl. Opt. 3(4), 561–571 (1994).
[CrossRef]

Dan, Y.

S. Yin, B. Zhang, and Y. Dan, “Effects of the deformation of reflection volume Bragg gratings on the M2-factor of super-Gaussian laser beams,” Opt. Commun. 283(7), 1418–1423 (2010).
[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]

De Santis, P.

P. De Santis, A. Mascello, C. Palma, and M. R. Perrone, “Coherence growth of laser radiation in Gaussian cavities,” IEEE J. Quantum Electron. 32(5), 802–812 (1996).
[CrossRef]

de Valcarcel, G. J.

Deng, D.

D. Deng, X. Fu, C. Wei, J. Shao, and Z. Fan, “Far-field intensity distribution and M2 factor of hollow Gaussian beams,” Appl. Opt. 44(33), 7187–7190 (2005).
[CrossRef] [PubMed]

D. Deng, “Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture,” Phys. Lett. A 341(1-4), 352–356 (2005).
[CrossRef]

D. Deng, “Propagation properties of off-axis cosh Gaussian beam combinations through a first-order optical system,” Phys. Lett. A 333(5-6), 485–494 (2004).
[CrossRef]

D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite-Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. 5(5), 489–494 (2003).
[CrossRef]

DeSantis, P.

G. Cincotti, P. DeSantis, G. Guattari, and C. Palma, “Propagation of partially coherent beams in a periodic sequence of lenses and Gaussian apertures,” Pure Appl. Opt. 3(4), 561–571 (1994).
[CrossRef]

Eyyuboglu, H. T.

Fan, Z.

Fox, A. G.

A. G. Fox and T. Li, “Resonate modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Friberg, A.

Friberg, A. T.

Fu, X.

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36(1), 3–8 (1965).
[CrossRef]

G. D. Boyd and J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

Gori, F.

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]

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]

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).
[CrossRef]

F. Gori, “Propagation of the mutual intensity through a periodic structure,” Atti Fond. Giorgio Ronchi 35, 434–447 (1980).

Guattari, G.

G. Cincotti, P. DeSantis, G. Guattari, and C. Palma, “Propagation of partially coherent beams in a periodic sequence of lenses and Gaussian apertures,” Pure Appl. Opt. 3(4), 561–571 (1994).
[CrossRef]

Guo, H.

D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite-Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. 5(5), 489–494 (2003).
[CrossRef]

Harris, J. M.

Hilgevoord, J.

J. Hilgevoord, “The standard deviation is not an adequate measure of quantum uncertainty,” Am. J. Phys. 70(10), 983 (2002).
[CrossRef]

Hu, C.

Knight, L. V.

Kogelnik, H.

Kong, H.

D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite-Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. 5(5), 489–494 (2003).
[CrossRef]

Korotkova, O.

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]

Z. Tong, Y. Cai, and O. Korotkova, “Ghost imaging with electromagnetic stochastic beams,” Opt. Commun. 283(20), 3838–3845 (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(20), 17344–17356 (2009).
[CrossRef] [PubMed]

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (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]

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]

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

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]

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]

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]

Lawrence, G. N.

G. N. Lawrence, “Proposed international standard for laser-beam quality falls short,” Laser Focus World 109–114 (1994).

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36(1), 3–8 (1965).
[CrossRef]

Li, Q.

Li, T.

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966).
[CrossRef] [PubMed]

A. G. Fox and T. Li, “Resonate modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Lin, Q.

Lü, B.

Lunnam, S. D.

Ma, H.

Martinez-Herrero, R.

Martínez-Herrero, R.

Mascello, A.

A. Mascello, M. R. Perrone, and C. Palma, “Coherence evolution of laser beams in cavities with variable-reflectivity mirrors,” J. Opt. Soc. Am. A 14(8), 1890–1901 (1997).
[CrossRef]

P. De Santis, A. Mascello, C. Palma, and M. R. Perrone, “Coherence growth of laser radiation in Gaussian cavities,” IEEE J. Quantum Electron. 32(5), 802–812 (1996).
[CrossRef]

Mejias, P. M.

Mejías, P. M.

Mondello, A.

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]

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36(1), 3–8 (1965).
[CrossRef]

Neuenschwander, B.

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11(2-3), 245–254 (1999).
[CrossRef]

Palma, C.

C. Palma, G. Cardone, and G. Cincotti, “Spectral changes in Gaussian-cavity lasers,” IEEE J. Quantum Electron. 34(7), 1082–1088 (1998).
[CrossRef]

A. Mascello, M. R. Perrone, and C. Palma, “Coherence evolution of laser beams in cavities with variable-reflectivity mirrors,” J. Opt. Soc. Am. A 14(8), 1890–1901 (1997).
[CrossRef]

P. De Santis, A. Mascello, C. Palma, and M. R. Perrone, “Coherence growth of laser radiation in Gaussian cavities,” IEEE J. Quantum Electron. 32(5), 802–812 (1996).
[CrossRef]

G. Cincotti, P. DeSantis, G. Guattari, and C. Palma, “Propagation of partially coherent beams in a periodic sequence of lenses and Gaussian apertures,” Pure Appl. Opt. 3(4), 561–571 (1994).
[CrossRef]

Perrone, M. R.

A. Mascello, M. R. Perrone, and C. Palma, “Coherence evolution of laser beams in cavities with variable-reflectivity mirrors,” J. Opt. Soc. Am. A 14(8), 1890–1901 (1997).
[CrossRef]

P. De Santis, A. Mascello, C. Palma, and M. R. Perrone, “Coherence growth of laser radiation in Gaussian cavities,” IEEE J. Quantum Electron. 32(5), 802–812 (1996).
[CrossRef]

Piquero, G.

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]

Porto, S. P. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36(1), 3–8 (1965).
[CrossRef]

Qu, J.

Roldan, E.

Saastamoinen, T.

Saghafi, S.

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[CrossRef]

Salem, M.

Santarsiero, M.

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]

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]

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).
[CrossRef]

Setala, T.

Setälä, T.

Shao, J.

Sheldon, S. J.

Sheppard, C. J. R.

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[CrossRef]

Shirai, T.

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]

Siegman, A. E.

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993).
[CrossRef]

Simon, R.

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]

Sona, A.

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).
[CrossRef]

Tervo, J.

Thorne, J. M.

Tong, Z.

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

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009).
[CrossRef]

Townsend, S. W.

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993).
[CrossRef]

Turunen, J.

Urchueguía, J. F.

Vahimaa, P.

Wang, F.

Wang, S.

T. Xu and S. Wang, “Propagation of Ince-Gaussian beams in a thermal lens medium,” Opt. Commun. 265(1), 1–5 (2006).
[CrossRef]

S. Wang, “Matrix methods in treating decentred optical systems,” Opt. Quantum Electron. 17(1), 1–14 (1985).
[CrossRef]

Weber, H. P.

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11(2-3), 245–254 (1999).
[CrossRef]

Weber, R.

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11(2-3), 245–254 (1999).
[CrossRef]

Wei, C.

Wen, Q.

Whinnery, J. R.

J. R. Whinnery, “Laser measurement of optical absorption in liquids,” Acc. Chem. Res. 7(7), 225–231 (1974).
[CrossRef]

C. Hu and J. R. Whinnery, “New thermooptical measurement method and a comparison with other methods,” Appl. Opt. 12(1), 72–79 (1973).
[CrossRef] [PubMed]

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36(1), 3–8 (1965).
[CrossRef]

Wolf, E.

E. Wolf, “Coherence and polarization properties of electromagnetic laser modes,” Opt. Commun. 265(1), 60–62 (2006).
[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]

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]

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

E. Wolf and G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1(5), 541–546 (1984).
[CrossRef]

E. Wolf, “Spatial coherence of resonant modes in a maser interferometer,” Phys. Lett. 3(4), 166–168 (1963).
[CrossRef]

Xu, T.

T. Xu and S. Wang, “Propagation of Ince-Gaussian beams in a thermal lens medium,” Opt. Commun. 265(1), 1–5 (2006).
[CrossRef]

Yao, M.

Yin, S.

S. Yin, B. Zhang, and Y. Dan, “Effects of the deformation of reflection volume Bragg gratings on the M2-factor of super-Gaussian laser beams,” Opt. Commun. 283(7), 1418–1423 (2010).
[CrossRef]

Yuan, Y.

Y. Yuan, Y. Cai, C. Zhao, H. T. Eyyuboğlu, and Y. Baykal, “Propagation factors of laser array beams in turbulent atmosphere,” J. Mod. Opt. 57(8), 621–631 (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(20), 17344–17356 (2009).
[CrossRef] [PubMed]

Zhang, B.

Zhang, L.

S. Zhu, F. Zhou, Y. Cai, and L. Zhang, “Thermal lens effect induced changes of polarization, coherence and spectrum of a stochastic electromagnetic beam in a Gaussian cavity,” Appl. Phys. B DOI: .

Zhao, C.

Y. Yuan, Y. Cai, C. Zhao, H. T. Eyyuboğlu, and Y. Baykal, “Propagation factors of laser array beams in turbulent atmosphere,” J. Mod. Opt. 57(8), 621–631 (2010).
[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]

Zhou, F.

S. Zhu, F. Zhou, Y. Cai, and L. Zhang, “Thermal lens effect induced changes of polarization, coherence and spectrum of a stochastic electromagnetic beam in a Gaussian cavity,” Appl. Phys. B DOI: .

Zhou, G.

G. Zhou, “Generalized beam propagation factors of truncated partially coherent cosine-Gaussian and cosh-Gaussian beams,” Opt. Laser Technol. 42(3), 489–496 (2010).
[CrossRef]

Zhu, S.

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]

S. Zhu, F. Zhou, Y. Cai, and L. Zhang, “Thermal lens effect induced changes of polarization, coherence and spectrum of a stochastic electromagnetic beam in a Gaussian cavity,” Appl. Phys. B DOI: .

Acc. Chem. Res. (1)

J. R. Whinnery, “Laser measurement of optical absorption in liquids,” Acc. Chem. Res. 7(7), 225–231 (1974).
[CrossRef]

Am. J. Phys. (1)

J. Hilgevoord, “The standard deviation is not an adequate measure of quantum uncertainty,” Am. J. Phys. 70(10), 983 (2002).
[CrossRef]

Appl. Opt. (8)

Appl. Phys. B (2)

Z. Tong, O. Korotkova, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Correlation properties of random electromagnetic beams in laser resonators,” Appl. Phys. B 97(4), 849–857 (2009).
[CrossRef]

S. Zhu, F. Zhou, Y. Cai, and L. Zhang, “Thermal lens effect induced changes of polarization, coherence and spectrum of a stochastic electromagnetic beam in a Gaussian cavity,” Appl. Phys. B DOI: .

Atti Fond. Giorgio Ronchi (1)

F. Gori, “Propagation of the mutual intensity through a periodic structure,” Atti Fond. Giorgio Ronchi 35, 434–447 (1980).

Bell Syst. Tech. J. (2)

A. G. Fox and T. Li, “Resonate modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

G. D. Boyd and J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

IEEE J. Quantum Electron. (3)

P. De Santis, A. Mascello, C. Palma, and M. R. Perrone, “Coherence growth of laser radiation in Gaussian cavities,” IEEE J. Quantum Electron. 32(5), 802–812 (1996).
[CrossRef]

C. Palma, G. Cardone, and G. Cincotti, “Spectral changes in Gaussian-cavity lasers,” IEEE J. Quantum Electron. 34(7), 1082–1088 (1998).
[CrossRef]

A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993).
[CrossRef]

J. Appl. Phys. (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36(1), 3–8 (1965).
[CrossRef]

J. Mod. Opt. (1)

Y. Yuan, Y. Cai, C. Zhao, H. T. Eyyuboğlu, and Y. Baykal, “Propagation factors of laser array beams in turbulent atmosphere,” J. Mod. Opt. 57(8), 621–631 (2010).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (3)

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]

D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite-Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. 5(5), 489–494 (2003).
[CrossRef]

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

J. Opt. Soc. Am. A (6)

J. Opt. Soc. Am. B (1)

Laser Focus World (1)

G. N. Lawrence, “Proposed international standard for laser-beam quality falls short,” Laser Focus World 109–114 (1994).

Opt. Commun. (7)

F. Gori, M. Santarsiero, and A. Sona, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82(3-4), 197–203 (1991).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998).
[CrossRef]

S. Yin, B. Zhang, and Y. Dan, “Effects of the deformation of reflection volume Bragg gratings on the M2-factor of super-Gaussian laser beams,” Opt. Commun. 283(7), 1418–1423 (2010).
[CrossRef]

T. Xu and S. Wang, “Propagation of Ince-Gaussian beams in a thermal lens medium,” Opt. Commun. 265(1), 1–5 (2006).
[CrossRef]

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

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

E. Wolf, “Coherence and polarization properties of electromagnetic laser modes,” Opt. Commun. 265(1), 60–62 (2006).
[CrossRef]

Opt. Eng. (1)

P. A. Belanger, “Beam quality factor of the LP01 mode of the step-index fiber,” Opt. Eng. 32(9), 2107–2109 (1993).
[CrossRef]

Opt. Express (7)

Opt. Laser Technol. (1)

G. Zhou, “Generalized beam propagation factors of truncated partially coherent cosine-Gaussian and cosh-Gaussian beams,” Opt. Laser Technol. 42(3), 489–496 (2010).
[CrossRef]

Opt. Lett. (6)

Opt. Mater. (1)

R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater. 11(2-3), 245–254 (1999).
[CrossRef]

Opt. Quantum Electron. (1)

S. Wang, “Matrix methods in treating decentred optical systems,” Opt. Quantum Electron. 17(1), 1–14 (1985).
[CrossRef]

Phys. Lett. (1)

E. Wolf, “Spatial coherence of resonant modes in a maser interferometer,” Phys. Lett. 3(4), 166–168 (1963).
[CrossRef]

Phys. Lett. A (3)

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

D. Deng, “Generalized M2-factor of hollow Gaussian beams through a hard-edge circular aperture,” Phys. Lett. A 341(1-4), 352–356 (2005).
[CrossRef]

D. Deng, “Propagation properties of off-axis cosh Gaussian beam combinations through a first-order optical system,” Phys. Lett. A 333(5-6), 485–494 (2004).
[CrossRef]

Pure Appl. Opt. (1)

G. Cincotti, P. DeSantis, G. Guattari, and C. Palma, “Propagation of partially coherent beams in a periodic sequence of lenses and Gaussian apertures,” Pure Appl. Opt. 3(4), 561–571 (1994).
[CrossRef]

Other (4)

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2C14 (1990).

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

N. Hodgson, and H. Weber, Optical resonators-Fundamentals, advanced concepts and applications (London Springer-Verlag, 1997).

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

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of a Gaussian cavity and its equivalent (unfolded) version

Fig. 2
Fig. 2

M 2 -factor of an EGSM beam in a Gaussian cavity versus N for different values of the cavity parameter g and the source correlation coefficients δ x x and δ y y

Fig. 3
Fig. 3

M 2 -factor of an EGSM beam in a Gaussian cavity versus N for different values of the cavity parameter g and the degree of polarization P 0 of the source beam

Fig. 4
Fig. 4

M 2 -factor of an EGSM beam in a Gaussian cavity versus N for different values of the mirror spot size of the cavity ε

Fig. 5
Fig. 5

Schematic diagram of a Gaussian cavity containing thermal lens medium and its equivalent (unfolded) version

Fig. 6
Fig. 6

M 2 -factor of an EGSM beam in a Gaussian cavity for different values of the length of the thermal lens medium and the source correlation coefficients δ x x and δ y y

Fig. 7
Fig. 7

M 2 -factor of an EGSM beam in a Gaussian cavity for different values of the thermal lens medium coefficientγ and the source correlation coefficients δ x x and δ y y

Fig. 8
Fig. 8

M 2 -factor of an EGSM beam in a Gaussian cavity for different values of the length of the thermal lens medium and the degree of polarization P 0 of the source beam

Fig. 9
Fig. 9

M 2 -factor of an EGSM beam in a Gaussian cavity for different values of the thermal lens medium coefficient γ and the degree of polarization P 0 of the source beam

Fig. 10
Fig. 10

M 2 -factor of an EGSM beam in a Gaussian cavity for different values of the length of the thermal lens medium and the mirror spot size of the cavity ε

Fig. 11
Fig. 11

M 2 -factor of an EGSM beam in a Gaussian cavity for different values of the thermal lens medium coefficientγ and the mirror spot size of the cavity ε

Equations (33)

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

W α β ( r ˜ ) = A α A β B α β exp [ i k 2 r ˜ T M 0 α β 1 r ˜ ] , ( α = x , y ; β = x , y ) ,
M 0 α β 1 = ( 1 i k ( 1 2 σ a 2 + 1 δ α β 2 ) I i k δ α β 2 I i k δ α β 2 I 1 i k ( 1 2 σ β 2 + 1 δ α β 2 ) I ) ,
W α β ( ρ ˜ ) = A α A β B α β [ Det ( A ¯ + B ¯ M 0 α β 1 ) ] 1 / 2 exp [ i k 2 ρ ˜ T M 1 α β 1 ρ ˜ ] , ( α = x , y ; β = x , y ) ,
M 1 α β 1 = ( C ¯ + D ¯ M 0 α β 1 ) ( A ¯ + B ¯ M 0 α β 1 ) 1 ,
A ¯ = ( A 0 I 0 I A * ) , B ¯ = ( B 0 I 0 I B * ) , C ¯ = ( C 0 I 0 I C * ) , D ¯ = ( D 0 I 0 I D * ) ,
( A B C D ) = ( A 1 B 1 C 1 D 1 ) N , ( A 1 B 1 C 1 D 1 ) = ( I L I ( 2 R i λ π ε 2 ) I ( 1 2 L R i λ L π ε 2 ) I ) ,
W tr ( ρ ˜ ) = Tr W ( ρ ˜ ) = W x x ( ρ ˜ ) + W y y ( ρ ˜ ) .
M x 2 = 4 π Δ x Δ p x , M y 2 = 4 π Δ y Δ p y ,
Δ x = 1 P ( ρ x ρ x ¯ ) 2 W ( ρ x , ρ y , ρ x , ρ y ) d ρ x d ρ y ,
Δ y = 1 P ( ρ y ρ y ¯ ) 2 W ( ρ x , ρ y , ρ x , ρ y ) d ρ x d ρ y ,
Δ p x = 1 P ( p x p x ¯ ) 2 W ˜ ( p x , p y , p x , p y ) d p x d p y ,
Δ p y = 1 P ( p y p y ¯ ) 2 W ˜ ( p x , p y , p x , p y ) d p x d p y ,
P = W ( ρ x , ρ y , ρ x , ρ y ) d ρ x d ρ y = W ˜ ( p x , p y , p x , p y ) d p x d p y ,
M 1 α β 1 = ( Q α β 11 Q α β 12 Q α β 13 Q α β 14 Q α β 21 Q α β 22 Q α β 23 Q α β 24 Q α β 31 Q α β 32 Q α β 33 Q α β 34 Q α β 41 Q α β 42 Q α β 43 Q α β 44 ) ,
W tr ( ρ x 1 , ρ y 1 , ρ x 2 , ρ y 2 ) = G x x exp { i ω 2 c [ ρ x 1 2 Q x x 11 + ρ x 2 2 Q x x 33 + ρ y 1 2 Q x x 22 + ρ y 2 2 Q x x 44 + ( Q x x 13 + Q x x 31 ) ρ x 1 ρ x 2 + ( Q x x 24 + Q x x 42 ) ρ y 1 ρ y 2 ] } + G y y exp { i ω 2 c [ ρ x 1 2 Q y y 11 + ρ x 2 2 Q y y 33 + ρ y 1 2 Q y y 22 + ρ y 2 2 Q y y 44 + ( Q y y 13 + Q y y 31 ) ρ x 1 ρ x 2 + ( Q y y 24 + Q y y 42 ) ρ y 1 ρ y 2 ] } ,
G x x = A x [ Det ( A ¯ + B ¯ M 0 x x 1 ) ] 1 / 2 , G y y = A y [ Det ( A ¯ + B ¯ M 0 y y 1 ) ] 1 / 2 .
W tr ( p x , p y , p x , p y , z ) = 16 π 2 G x x k 2 [ ( Q x x 13 + Q x x 31 ) 2 4 Q x x 11 Q x x 33 ] exp [ 8 π 2 i ( Q x x 33 + Q x x 11 + Q x x 13 + Q x x 31 ) k [ ( Q x x 13 + Q x x 31 ) 2 4 Q x x 11 Q x x 33 ] ( p x 2 + p y 2 ) ] + 16 π 2 G y y k 2 [ ( Q y y 13 + Q y y 31 ) 2 4 Q y y 11 Q y y 33 ] exp [ 8 π 2 i ( Q y y 33 + Q y y 11 + Q y y 13 + Q y y 31 ) k [ ( Q y y 13 + Q y y 31 ) 2 4 Q y y 11 Q y y 33 ] ( p x 2 + p y 2 ) ] .
Δ x 2 = Δ y 2 = 2 π k 2 P [ G x x ( Q x x 11 + Q x x 13 + Q x x 31 + Q x x 33 ) 2 + G y y ( Q y y 11 + Q y y 13 + Q y y 31 + Q y y 33 ) 2 ] ,
Δ p y 2 = Δ p x 2 = 1 8 π P [ G x x [ ( Q x x 13 + Q x x 31 ) 2 4 Q x x 11 Q x x 33 ] ( Q x x 11 + Q x x 13 + Q x x 31 + Q x x 33 ) 2 + G y y [ ( Q y y 13 + Q y y 31 ) 2 4 Q y y 11 Q y y 33 ] ( Q y y 11 + Q y y 13 + Q y y 31 + Q y y 33 ) 2 ] ,
P = 2 π i k [ G x x ( Q x x 11 + Q x x 13 + Q x x 31 + Q x x 33 ) + G y y ( Q y y 11 + Q y y 13 + Q y y 31 + Q y y 33 ) ] .
M x 2 = M y 2 = M 2 = Q N x x Q N y y G x x Q N y y + G y y Q N x x ( G x x Q N x x 2 + G y y Q N y y 2 ) ( G x x Q M x x Q N x x 2 + G y y Q M y y Q N y y 2 ) ,
M x 2 = M y 2 = M 2 = 4 Q α α 11 Q α α 33 ( Q α α 13 + Q α α 31 ) 2 ( Q α α 11 + Q α α 13 + Q α α 31 + Q α α 33 ) 2 , ( α = x , y )
exp [ a x 2 + b y 2 ] d x d y = π a b , x 2 exp [ a x 2 + b y 2 + d x y ] d x d y = 4 π b ( 4 a b d 2 ) 3 / 2 .
M 2 = { ( A x 2 σ x 4 + A y 2 σ y 4 ) [ 4 A x 2 δ y y 2 σ x 2 + δ x x 2 ( A x 2 δ y y 2 + A y 2 δ y y 2 + 4 A y 2 σ y 2 ) ] δ x x 2 δ y y 2 ( A x 2 σ x 2 + A y 2 σ y 2 ) 2 } 1 / 2 ,
M 2 ( z ) = ( 1 + 4 σ α 2 δ α α 2 ) 1 / 2 , ( α = x , y )
W ( r 1 , r 2 ) = ( W x x ( r 1 , r 2 ) 0 0 W y y ( r 1 , r 2 ) ) .
P 0 ( r ) = 1 4 Det W ( r , r ) [ Tr W ( r , r ) ] 2 .
( A t B t C t D t ) = ( cos ( β l ) I sin ( β l ) / ( β n 0 ) I β n 0 sin ( β l ) I cos ( β l ) I ) = ( ( 1 + γ l 2 ) I l / n 0 I 2 γ n 0 l I ( 1 + γ l 2 ) I ) ,
( A B C D ) = ( A 1 B 1 C 1 D 1 ) N ,
A 1 = ( 1 + 2 n 0 γ l 1 l 2 + γ l 2 2 ) I ,
B 1 = [ l 2 n 0 + 2 n 0 γ l 1 2 l 2 + 2 l 1 ( 1 + γ l 2 2 ) ] I ,
C 1 = [ 2 n 0 γ ( 1 + ( 2 R i λ π ε 2 ) l 1 ) l 2 + ( 2 R i λ π ε 2 ) ( 1 + γ l 2 2 ) ] I ,
D 1 = [ 1 + ( 2 R i λ π ε 2 ) l 2 n 0 + 2 n 0 γ l 1 2 l 2 ( 2 R i λ π ε 2 ) + γ l 2 2 + 4 l 1 R 2 i λ l 1 π ε 2 + 2 n 0 γ l 2 l 1 + l 1 ( 4 R 2 i λ π ε 2 ) γ l 2 2 ] I .

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