Abstract

The scattering of a stochastic electromagnetic plane-wave pulse on a deterministic spherical medium is investigated. An analytical formula for the degree of polarization (DOP) of the scattered field in the far zone is derived. Letting pulse duration T0, our formula can be applied to study the scattering of a stationary stochastic electromagnetic light wave. Numerical results show that the DOP of the far zone field is closely determined by the size of the spherical medium when the incident field is a stochastic electromagnetic plane-wave pulse. This is much different from the case when the incident field is a stationary stochastic electromagnetic light wave, where the DOP of the far zone field is independent of the size of the medium. One may obtain the information of the spherical medium by measuring the scattering-induced changes in the DOP of a stochastic electromagnetic plane-wave pulse.

© 2012 Optical Society of America

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2011 (3)

2010 (6)

2009 (9)

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef]

B. Kanseri, S. Rath, and H. C. Kandpal, “Determination of thebeam coherence-polarization matrix of a random electromagnetic beam,” IEEE J. Quantum Electron. 45, 1163–1167 (2009).
[CrossRef]

L. Z. Pan, Z. G. Zhao, C. L. Ding, and B. D. Lü, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett. 95, 181112 (2009).
[CrossRef]

C. L. Ding, L. Z. Pan, and B. D. Lü, “Characterization of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” New J. Phys. 11, 083001 (2009).
[CrossRef]

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. 11, 073004 (2009).
[CrossRef]

B. Kanseri, S. Rath, and H. C. Kandpal, “Direct determination of the generalized Stokes parameters from the usual Stokes parameters,” Opt. Lett. 34, 719–721 (2009).
[CrossRef]

L. Z. Pan, M. L. Sun, C. L. Ding, Z. G. Zhao, and B. D. Lü, “Effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic beams passing through an astigmatic optical system,” Opt. Express 17, 7310–7321 (2009).
[CrossRef]

T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett. 34, 3394–3396 (2009).
[CrossRef]

C. L. Ding, B. D. Lü, and L. Z. Pan, “Phase singularities and spectral changes of spectrally partially coherent higher-order Bessel-Gauss pulsed beams,” J. Opt. Soc. Am. A 26, 2654–2661 (2009).
[CrossRef]

2008 (4)

2007 (7)

2006 (2)

2005 (4)

H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536–1545 (2005).
[CrossRef]

J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett. 30, 2973–2975 (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]

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

2004 (2)

2003 (4)

E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078–1080 (2003).
[CrossRef]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003).
[CrossRef]

Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219, 65–70 (2003).
[CrossRef]

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[CrossRef]

2002 (3)

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (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]

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

2001 (1)

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

1998 (1)

1994 (2)

1989 (2)

E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989).
[CrossRef]

J. T. Foley and E. Wolf, “Frequency shifts of spectral lines generated by scattering from space-time fluctuations,” Phys. Rev. A 40, 588 (1989).
[CrossRef]

Agrawal, G. P.

Andres, P.

Andrès, P.

Baykal, Y.

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688–690 (2006).
[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. Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics7th ed. (Cambridge University, 1999).

Cada, M.

Cai, Y. J.

Chen, H.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[CrossRef]

Chen, Y. R.

Y. Xin, Y. R. Chen, Q. Zhao, and M. C. Zhou, “Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media,” Opt. Commun. 278, 247–252 (2007).
[CrossRef]

Ding, C. L.

Ding, K. H.

L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley-Interscience, 2000).

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]

Du, X. Y.

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.

Fischer, D. G.

T. Van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant Intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef]

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef]

D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11, 1128–1135 (1994).
[CrossRef]

Foley, J. T.

E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989).
[CrossRef]

J. T. Foley and E. Wolf, “Frequency shifts of spectral lines generated by scattering from space-time fluctuations,” Phys. Rev. A 40, 588 (1989).
[CrossRef]

Friberg, A. T.

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. 11, 073004 (2009).
[CrossRef]

T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett. 34, 3394–3396 (2009).
[CrossRef]

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77, 043811 (2008).
[CrossRef]

V. Torres-Company, G. Mínguez-Vega, J. Lancis, and A. T. Friberg, “Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper,” Opt. Lett. 32, 1608–1610 (2007).
[CrossRef]

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photon. 1, 228–231 (2007).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

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

Gao, W. R.

Gbur, G.

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688–690 (2006).
[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. Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

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

E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989).
[CrossRef]

Huang, W. H.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

Jacks, H. C.

James, D. F. V.

Kaivola, M.

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. 11, 073004 (2009).
[CrossRef]

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photon. 1, 228–231 (2007).
[CrossRef]

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

Kandpal, H. C.

B. Kanseri, S. Rath, and H. C. Kandpal, “Determination of thebeam coherence-polarization matrix of a random electromagnetic beam,” IEEE J. Quantum Electron. 45, 1163–1167 (2009).
[CrossRef]

B. Kanseri, S. Rath, and H. C. Kandpal, “Direct determination of the generalized Stokes parameters from the usual Stokes parameters,” Opt. Lett. 34, 719–721 (2009).
[CrossRef]

Kanseri, B.

B. Kanseri, S. Rath, and H. C. Kandpal, “Direct determination of the generalized Stokes parameters from the usual Stokes parameters,” Opt. Lett. 34, 719–721 (2009).
[CrossRef]

B. Kanseri, S. Rath, and H. C. Kandpal, “Determination of thebeam coherence-polarization matrix of a random electromagnetic beam,” IEEE J. Quantum Electron. 45, 1163–1167 (2009).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley-Interscience, 2000).

Korotkova, O.

H. C. Jacks and O. Korotkova, “Intensity-intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A 28, 1139–1144 (2011).
[CrossRef]

C. L. Ding, Y. J. Cai, O. Korotkova, Y. T. Zhang, and L. Z. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36, 517–519 (2011).
[CrossRef]

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

Z. S. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
[CrossRef]

M. Yao, Y. J. Cai, H. T. Eyyuboglu, 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]

S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78, 063815(2008).
[CrossRef]

D. M. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32, 3483–3485 (2007).
[CrossRef]

O. Korotkova and G. Gbur, “Angular spectrum representation for propagation of random electromagnetic beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 24, 2728–2736 (2007).
[CrossRef]

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

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

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21, 2382–2385 (2004).
[CrossRef]

Lahiri, M.

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef]

Lajunen, H.

Lancis, J.

Lin, Q.

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

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[CrossRef]

Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219, 65–70 (2003).
[CrossRef]

Lindfors, K.

T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett. 34, 3394–3396 (2009).
[CrossRef]

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photon. 1, 228–231 (2007).
[CrossRef]

Liu, X. L.

Lü, B. D.

C. L. Ding, L. Z. Pan, and B. D. Lü, “Characterization of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” New J. Phys. 11, 083001 (2009).
[CrossRef]

L. Z. Pan, Z. G. Zhao, C. L. Ding, and B. D. Lü, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett. 95, 181112 (2009).
[CrossRef]

C. L. Ding, B. D. Lü, and L. Z. Pan, “Phase singularities and spectral changes of spectrally partially coherent higher-order Bessel-Gauss pulsed beams,” J. Opt. Soc. Am. A 26, 2654–2661 (2009).
[CrossRef]

L. Z. Pan, M. L. Sun, C. L. Ding, Z. G. Zhao, and B. D. Lü, “Effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic beams passing through an astigmatic optical system,” Opt. Express 17, 7310–7321 (2009).
[CrossRef]

Mandel, L.

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

Mínguez-Vega, G.

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. Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Pääkkönen, P.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

Pan, L. Z.

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. Pure Appl. Opt. 3, 1–9 (2001).
[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]

Ponomarenko, S. A.

Priimagi, A.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photon. 1, 228–231 (2007).
[CrossRef]

Rath, S.

B. Kanseri, S. Rath, and H. C. Kandpal, “Direct determination of the generalized Stokes parameters from the usual Stokes parameters,” Opt. Lett. 34, 719–721 (2009).
[CrossRef]

B. Kanseri, S. Rath, and H. C. Kandpal, “Determination of thebeam coherence-polarization matrix of a random electromagnetic beam,” IEEE J. Quantum Electron. 45, 1163–1167 (2009).
[CrossRef]

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.

Sahin, S.

S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78, 063815(2008).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688–690 (2006).
[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. Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Setälä, T.

T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett. 34, 3394–3396 (2009).
[CrossRef]

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. 11, 073004 (2009).
[CrossRef]

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photon. 1, 228–231 (2007).
[CrossRef]

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

Shevchenko, A.

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. 11, 073004 (2009).
[CrossRef]

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photon. 1, 228–231 (2007).
[CrossRef]

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

Shirai, T.

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef]

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

Silvestre, E.

Simon, R.

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

Sun, M. L.

Tervo, J.

Tong, Z. S.

Z. S. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
[CrossRef]

Torres-Company, V.

Tsang, L.

L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley-Interscience, 2000).

Turunen, J.

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

Vahimaa, P.

Van Dijk, T.

T. Van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant Intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef]

Visser, T. D.

T. Van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant Intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef]

Wang, F.

Wang, L. G.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[CrossRef]

Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219, 65–70 (2003).
[CrossRef]

Wang, T.

Wang, Z. Y.

Wolf, E.

T. Van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant Intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef]

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef]

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

D. M. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32, 3483–3485 (2007).
[CrossRef]

H. Roychowdhury, G. P. Agrawal, and E. Wolf, “Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber,” J. Opt. Soc. Am. A 23, 940–948 (2006).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688–690 (2006).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. Pure Appl. Opt. 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, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21, 2382–2385 (2004).
[CrossRef]

E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078–1080 (2003).
[CrossRef]

D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11, 1128–1135 (1994).
[CrossRef]

E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989).
[CrossRef]

J. T. Foley and E. Wolf, “Frequency shifts of spectral lines generated by scattering from space-time fluctuations,” Phys. Rev. A 40, 588 (1989).
[CrossRef]

M. Born and E. Wolf, Principles of Optics7th ed. (Cambridge University, 1999).

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

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light1st ed. (Cambridge University, 2007).

Wu, G. F.

Wyrowski, F.

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

Xin, Y.

Y. Xin, Y. R. Chen, Q. Zhao, and M. C. Zhou, “Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media,” Opt. Commun. 278, 247–252 (2007).
[CrossRef]

Yao, M.

Zhang, Y. T.

Zhao, D. M.

Zhao, Q.

Y. Xin, Y. R. Chen, Q. Zhao, and M. C. Zhou, “Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media,” Opt. Commun. 278, 247–252 (2007).
[CrossRef]

Zhao, Z. G.

L. Z. Pan, Z. G. Zhao, C. L. Ding, and B. D. Lü, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett. 95, 181112 (2009).
[CrossRef]

L. Z. Pan, M. L. Sun, C. L. Ding, Z. G. Zhao, and B. D. Lü, “Effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic beams passing through an astigmatic optical system,” Opt. Express 17, 7310–7321 (2009).
[CrossRef]

Zhou, M. C.

Y. Xin, Y. R. Chen, Q. Zhao, and M. C. Zhou, “Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media,” Opt. Commun. 278, 247–252 (2007).
[CrossRef]

Zhu, S. J.

Zhu, S. Y.

Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219, 65–70 (2003).
[CrossRef]

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[CrossRef]

Appl. Phys. Lett. (1)

L. Z. Pan, Z. G. Zhao, C. L. Ding, and B. D. Lü, “Effect of polarization on spectral switches in the diffraction of stochastic electromagnetic beams,” Appl. Phys. Lett. 95, 181112 (2009).
[CrossRef]

IEEE J. Quantum Electron. (1)

B. Kanseri, S. Rath, and H. C. Kandpal, “Determination of thebeam coherence-polarization matrix of a random electromagnetic beam,” IEEE J. Quantum Electron. 45, 1163–1167 (2009).
[CrossRef]

J. Opt. Pure Appl. Opt. (2)

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

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

J. Opt. Soc. Am. A (12)

D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11, 1128–1135 (1994).
[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]

E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989).
[CrossRef]

H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21, 2117–2123 (2004).
[CrossRef]

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21, 2382–2385 (2004).
[CrossRef]

H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536–1545 (2005).
[CrossRef]

H. Roychowdhury, G. P. Agrawal, and E. Wolf, “Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber,” J. Opt. Soc. Am. A 23, 940–948 (2006).
[CrossRef]

O. Korotkova and G. Gbur, “Angular spectrum representation for propagation of random electromagnetic beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 24, 2728–2736 (2007).
[CrossRef]

W. H. Huang, S. A. Ponomarenko, M. Cada, and G. P. Agrawal, “Polarization changes of partially coherent pulses propagating in optical fibers,” J. Opt. Soc. Am. A 24, 3063–3068(2007).
[CrossRef]

X. Y. Du and D. M. Zhao, “Changes in the polarization and coherence of a random electromagnetic beam propagating through a misaligned optical system,” J. Opt. Soc. Am. A 25, 773–779 (2008).
[CrossRef]

C. L. Ding, B. D. Lü, and L. Z. Pan, “Phase singularities and spectral changes of spectrally partially coherent higher-order Bessel-Gauss pulsed beams,” J. Opt. Soc. Am. A 26, 2654–2661 (2009).
[CrossRef]

H. C. Jacks and O. Korotkova, “Intensity-intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A 28, 1139–1144 (2011).
[CrossRef]

Nat. Photon. (1)

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photon. 1, 228–231 (2007).
[CrossRef]

New J. Phys. (2)

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. 11, 073004 (2009).
[CrossRef]

C. L. Ding, L. Z. Pan, and B. D. Lü, “Characterization of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” New J. Phys. 11, 083001 (2009).
[CrossRef]

Opt. Commun. (5)

Y. Xin, Y. R. Chen, Q. Zhao, and M. C. Zhou, “Beam radiated from quasi-homogeneous uniformly polarized electromagnetic source scattering on quasi-homogeneous media,” Opt. Commun. 278, 247–252 (2007).
[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]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219, 65–70 (2003).
[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]

Opt. Express (4)

Opt. Lett. (13)

C. L. Ding, Y. J. Cai, O. Korotkova, Y. T. Zhang, and L. Z. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36, 517–519 (2011).
[CrossRef]

T. Wang and D. M. Zhao, “Scattering theory of stochastic electromagnetic light waves,” Opt. Lett. 35, 2412–2414 (2010).
[CrossRef]

T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett. 34, 3394–3396 (2009).
[CrossRef]

W. R. Gao, “Spectral changes of the light produced by scattering from tissue,” Opt. Lett. 35, 862–864 (2010).
[CrossRef]

V. Torres-Company, G. Mínguez-Vega, J. Lancis, and A. T. Friberg, “Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper,” Opt. Lett. 32, 1608–1610 (2007).
[CrossRef]

M. Yao, Y. J. Cai, H. T. Eyyuboglu, 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]

B. Kanseri, S. Rath, and H. C. Kandpal, “Direct determination of the generalized Stokes parameters from the usual Stokes parameters,” Opt. Lett. 34, 719–721 (2009).
[CrossRef]

D. M. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32, 3483–3485 (2007).
[CrossRef]

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

J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett. 30, 2973–2975 (2005).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688–690 (2006).
[CrossRef]

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

E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078–1080 (2003).
[CrossRef]

Phys. Rev. A (4)

Z. S. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
[CrossRef]

J. T. Foley and E. Wolf, “Frequency shifts of spectral lines generated by scattering from space-time fluctuations,” Phys. Rev. A 40, 588 (1989).
[CrossRef]

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77, 043811 (2008).
[CrossRef]

S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78, 063815(2008).
[CrossRef]

Phys. Rev. E (3)

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[CrossRef]

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

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

Phys. Rev. Lett. (2)

M. Lahiri, E. Wolf, D. G. Fischer, and T. Shirai, “Determination of correlation functions of scattering potentials of stochastic media from scattering experiments,” Phys. Rev. Lett. 102, 123901 (2009).
[CrossRef]

T. Van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant Intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef]

Other (6)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

M. Born and E. Wolf, Principles of Optics7th ed. (Cambridge University, 1999).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light1st ed. (Cambridge University, 2007).

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

L. Tsang, J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley-Interscience, 2000).

G. P. Agrawal, Nonlinear Fiber Optics4th ed. (Academic, 2007).

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Figures (7)

Fig. 1.
Fig. 1.

Illustrating the notation relating to scattering of a stochastic electromagnetic plane-wave pulse.

Fig. 2.
Fig. 2.

DOP of the scattered stationary stochastic electromagnetic light wave versus the scattering angle θ for different values of the coefficients Ai and Bij; PE and PS denote the Ellis and Setälä definitions of 3D DOP, respectively.

Fig. 3.
Fig. 3.

Normalized intensity I of the scattered electromagnetic GSM plane-wave pulse versus the scattering angle θ for different values of the initial DOP P(0), pulse duration T0, temporal coherence length Tcxx of the incident pulse, and the size of spherical medium σ. The calculation parameters are (a) T0=5fs, Tcxx=5fs, σ=5λ0; (b) P(0)=0.9, Tcxx=5fs, σ=5λ0; (c) T0=5fs, P(0)=0.9, σ=5λ0; (d) T0=5fs, Tcxx=5fs, P(0)=0.9.

Fig. 4.
Fig. 4.

DOP of the scattered electromagnetic GSM plane-wave pulse versus the scattering angle θ for different values of the initial DOP P(0) of the incident pulse. PS and PE denote the Setälä and Ellis definitions of 3D DOP, respectively.

Fig. 5.
Fig. 5.

(a) and (d) DOP of the scattered electromagnetic GSM plane-wave pulse versus the scattering angle θ for different pulse durations T0=3, 5, and 7 fs. (b) and (e) DOP of the scattered field as a function of θ and T0. (c) and (f) Corresponding contour graphs. PS and PE denote the Setälä and Ellis definitions of 3D DOP, respectively.

Fig. 6.
Fig. 6.

(a) and (d) DOP of the scattered electromagnetic GSM plane-wave pulse versus the scattering angle θ for different temporal coherence length Tcxx=2, 5, and 15 fs. (b) and (e) DOP of the scattered field as a function of θ and Tcxx. (c) and (f) Corresponding contour graph. PS and PE denote the Setälä and Ellis definitions of 3D DOP, respectively.

Fig. 7.
Fig. 7.

Contour graph of the DOP of the scattered electromagnetic GSM plane-wave pulse versus the scattering angle θ for different values of the size of the spherical medium. (a) and (d) σ=λ0, (b) and (e) 5λ0, (c) and (f) 15λ0. PS and PE denote the Setälä and Ellis definitions of 3D DOP, respectively.

Equations (72)

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

Γ(t1,t2)=(Γxx(t1,t2)Γxy(t1,t2)Γyx(t1,t2)Γyy(t1,t2)),
Γij(t1,t2)=AiAjBijexp[t12+t222T02(t1t2)22Tcij2+iω0(t1t2)],
Wij(ω1,ω2)=1(2π)2++Γij(t1,t2)exp[i(ω1t1ω2t2)]dt1dt2,
W(ω1,ω2)=(Wxx(ω1,ω2)Wxy(ω1,ω2)Wyx(ω1,ω2)Wyy(ω1,ω2)),
Wij(ω1,ω2)=T0AiAjBij2πΩ0ijexp[(ω1ω0)2+(ω2ω0)22Ω0ij2(ω1ω2)22Ωcij2],
Ω0ij=1T02+2Tcij2(Spectral width),
Ωcij=TcijT0Ω0ij(Spectral coherence width).
W(i)(r1,r2,ω1,ω2)=(Wxx(i)(r1,r2,ω1,ω2)Wxy(i)(r1,r2,ω1,ω2)Wyx(i)(r1,r2,ω1,ω2)Wyy(i)(r1,r2,ω1,ω2)),
Wij(i)(r1,r2,ω1,ω2)=T0AiAjBij2πΩ0ijexp[i(k2s0·r2k1s0·r1)]×exp[(ω1ω0)2+(ω2ω0)22Ω0ij2(ω1ω2)22Ωcij2],
E(s)(rs,ω)=s×[s×DF(r,ω)E(i)(r,ω)G(rs,r,ω)d3r],
G(rs,r,ω)=exp[ikr]rexp[iks·r].
ai=as=s×s0|s×s0|,bi=s0×ai,bs=s×as.
Ex(s)(rs,ω)=fx(θ)DF(r,ω)Ex(i)(r,ω)G(rs,r,ω)d3r,
Ey(s)(rs,ω)=fy(θ)DF(r,ω)Ey(i)(r,ω)G(rs,r,ω)d3r,
Ez(s)(rs,ω)=fz(θ)DF(r,ω)Ey(i)(r,ω)G(rs,r,ω)d3r,
W(s)(rs1,rs2,ω1,ω2)=(Wxx(s)(rs1,rs2,ω1,ω2)Wxy(s)(rs1,rs2,ω1,ω2)Wxz(s)(rs1,rs2,ω1,ω2)Wyx(s)(rs1,rs2,ω1,ω2)Wyy(s)(rs1,rs2,ω1,ω2)Wyz(s)(rs1,rs2,ω1,ω2)Wzx(s)(rs1,rs2,ω1,ω2)Wzy(s)(rs1,rs2,ω1,ω2)Wzz(s)(rs1,rs2,ω1,ω2)),
Wij(s)(rs1,rs2,ω1,ω2)=T0AiAjBij2πΩ0ijfi(θ1)fj(θ2)exp[ir(k2k1)]r2×exp[(ω1ω0)2+(ω2ω0)22Ω0ij2(ω1ω2)22Ωcij2]×DDF*(r1,ω1)F(r2,ω2)exp{i[K1·r1+K2·r2]}d3r1d3r2(i=x,y;j=x,y),
Wxz(s)(rs1,rs2,ω1,ω2)=T0AxAyBxy2πΩ0xyfx(θ1)fz(θ2)exp[ir(k2k1)]r2×exp[(ω1ω0)2+(ω2ω0)22Ω0xy2(ω1ω2)22Ωcxy2]×DDF*(r1,ω1)F(r2,ω2)exp{i[K1·r1+K2·r2]}d3r1d3r2,
Wzx(s)(rs1,rs2,ω1,ω2)=T0AxAyByx2πΩ0yxfz(θ1)fx(θ2)exp[ir(k2k1)]r2×exp[(ω1ω0)2+(ω2ω0)22Ω0yx2(ω1ω2)22Ωcyx2]×DDF*(r1,ω1)F(r2,ω2)exp{i[K1·r1+K2·r2]}d3r1d3r2,
Wyz(s)(rs1,rs2,ω1,ω2)=T0Ay22πΩ0yyfy(θ1)fz(θ2)exp[ir(k2k1)]r2×exp[(ω1ω0)2+(ω2ω0)22Ω0yy2(ω1ω2)22Ωcyy2]×DDF*(r1,ω1)F(r2,ω2)exp{i[K1·r1+K2·r2]}d3r1d3r2,
Wzy(s)(rs1,rs2,ω1,ω2)=fz(θ1)fy(θ2)fy(θ1)fz(θ2)Wyz(s)(rs1,rs2,ω1,ω2),
Wzz(s)(rs1,rs2,ω1,ω2)=fz(θ1)fy(θ1)Wyz(s)(rs1,rs2,ω1,ω2),
F(r,ω0)=F0exp[r22σ2],
DDF*(r1,ω1)F(r2,ω2)exp{i[K1·r1+K2·r2]}d3r1d3r2=F02(2π)3σ6exp[12σ2(K12+K22)].
Γij(s)(rs1,rs2,t1,t2)=Wij(s)(rs1,rs2,ω1,ω2)exp[i(ω1t1ω2t2)]dω1dω2.
Γij(s)(rs,rs,t1,t2)=T0AiAjBij2πr21T02+2Tcij2fi(θ)fj(θ)F02(2π)4σ6(T02Tcij2Tcij2+2T02+2m)(T02+2m)exp[(t1r/c)2+(t2r/c)22(T02+2m)(t1t2)22Tcij2T02(1+2mTcij2+2T02T02Tcij2)(T02+2m)+iω0(t1t2)(1+2mTcij2+2T02T02Tcij2)2mω021+2m(Tcij2+2T02T02Tcij2)](i=x,y;j=x,y),
Γxz(s)(rs,rs,t1,t2)=T0AxAyBxy2πr21T02+2Tcxy2fx(θ)fz(θ)F02(2π)4σ6(T02Tcxy2Tcxy2+2T02+2m)(T02+2m)exp[(t1r/c)2+(t2r/c)22(T02+2m)(t1t2)22Tcxy2T02(1+2mTcxy2+2T02T02Tcxy2)(T02+2m)+iω0(t1t2)(1+2mTcxy2+2T02T02Tcxy2)2mω021+2m(Tcxy2+2T02T02Tcxy2)],
Γyz(s)(rs,rs,t1,t2)=T0Ay22πr21T02+2Tcyy2fy(θ)fz(θ)F02(2π)4σ6(T02Tcyy2Tcyy2+2T02+2m)(T02+2m)exp[(t1r/c)2+(t2r/c)22(T02+2m)(t1t2)22Tcyy2T02(1+2mTcyy2+2T02T02Tcyy2)(T02+2m)+iω0(t1t2)(1+2mTcyy2+2T02T02Tcyy2)2mω021+2m(Tcyy2+2T02T02Tcyy2)],
Γzx(s)(rs,rs,t1,t2)=ByxBxyΓxz(s)(rs,rs,t1,t2),
Γzy(s)(rs,rs,t1,t2)=Γyz(s)(rs,rs,t1,t2),
Γzz(s)(rs,rs,t1,t2)=fz(θ)fy(θ)Γyz(s)(rs,rs,t1,t2),
m=2σ2sin2θ2/c2.
I(rs,t)=TrΓ(s)(rs,rs,t,t)=Γxx(s)(rs,rs,t,t)+Γyy(s)(rs,rs,t,t)+Γzz(s)(rs,rs,t,t).
PS(rs,t)=32{tr[Γ(s)2(rs,rs,t,t)]tr2[Γ(s)(rs,rs,t,t)]13}.
PE(rs,t)=λ1(rs,t)λ2(rs,t)λ1(rs,t)+λ2(rs,t)+λ3(rs,t),
Γij(s)(rs,rs,t,t)=T0AiAjBij2πr2fi(θ)fj(θ)F02(2π)4σ6[1+2m(Tcij2+2T02T02Tcij2)](T02+2m)exp[2mω021+2m(Tcij2+2T02T02Tcij2)(tr/c)2T02+2m](i=x,y;j=x,y),
Γxz(s)(rs,rs,t,t)=T0AxAyBxy2πr2fx(θ)fz(θ)F02(2π)4σ6[1+2m(Tcxy2+2T02T02Tcxy2)](T02+2m)exp[2mω021+2m(Tcxy2+2T02T02Tcxy2)(tr/c)2T02+2m],
Γyz(s)(rs,rs,t,t)=T0Ay22πr2fy(θ)fz(θ)F02(2π)4σ6[1+2m(Tcyy2+2T02T02Tcyy2)](T02+2m)exp[2mω021+2m(Tcyy2+2T02T02Tcyy2)(tr/c)2T02+2m],
Γzx(s)(rs,rs,t,t)=ByxBxyΓxz(s)(rs,rs,t,t),
Γzy(s)(rs,rs,t,t)=Γyz(s)(rs,rs,t,t),
Γzz(s)(rs,rs,t,t)=fz(θ)fy(θ)Γyz(s)(rs,rs,t,t).
Wij(i)(r1,r2,ω1,ω2)=0,ij.
PE(rs,t)=Ax2Ay2cos2θAx2+Ay2cos2θ,
PS(rs,t)=(Ax2Ay2cos2θ)2+Ax2Ay2cos2θAx2+Ay2cos2θ.
θ=±π2,
PE(rs,t)=(Ax2Ay2cos2θ)2+4Ax2Ay2BxyByxcos2θAx2+Ay2cos2θ.
PS(rs,t)=(Ax2Ay2cos2θ)2+Ax2Ay2cos2θ+3Ax2Ay2BxyByxcos2θAx2+Ay2cos2θ.
θ=±π2,
PS2=14(1+3PE2).
Γ(S)=(Ax200000000).
Γ(S)=(Ax2000Ax20000).
Bij=|Bij|exp[iφij],
|Bij|=1,i=j,|Bij|1,ij,φij0,i=j.
Γij(t1,t2)=AiAj|Bij|exp[t12+t222T02(t1t2)22Tcij2+iω0(t1t2)]exp[iφij].
Γij(t1,t2)=AiAj|Bij|exp[t12+t222T02(t1t2)22Tcij2].
Γyx(t1,t2)=Γxy*(t2,t1).
Tcxy=Tcyx,|Bxy|=|Byx|.
fx*(t1)fx(t2)Γxx(t1,t2)+fx*(t1)fy(t2)Γxy(t1,t2)+fy*(t1)fx(t2)Γyx(t1,t2)+fy*(t1)fy(t2)Γyy(t1,t2)dt1dt20,
gj(t)=Ajfj(t)exp[t22T02](j=x,y).
fi*(t1)fj(t2)Γij(t1,t2)=|Bij|gi*(t1)gj(t2)exp[(t1t2)22Tcij2](i=x,y;j==x,y).
{gx*(t1)gx(t2)exp[(t1t2)22Tcxx2]+|Bxy|gx*(t1)gy(t2)exp[(t1t2)22Tcxy2]+|Byx|gy*(t1)gx(t2)exp[(t1t2)22Tcyx2]+gy*(t1)gy(t2)exp[(t1t2)22Tcyy2]}dt1dt20.
exp[(t1t2)22Tcij2]=2πTcijexp[2πTcij2κ2]exp[2πi(t1t2)κ]dκ.
2π{gx*(t1)gx(t2)Tcxxexp[2πTcxx2κ2]+[gx*(t1)gy(t2)+gy*(t1)gx(t2)]|Bxy|Tcxyexp[2πTcxy2κ2]+gy*(t1)gy(t2)Tcyyexp[2πTcyy2κ2]}exp[2πi(t1t2)κ]dt1dt2dκ0.
2π{|Gx(κ)|2Tcxxexp[2πTcxx2κ2]+2Re[Gx*(κ)Gy(κ)]|Bxy|Tcxyexp[2πTcxy2κ2]+|Gy(κ)|2Tcyyexp[2πTcyy2κ2]}exp[2πi(t1t2)κ]dt1dt2dκ0,
Gj(κ)=12πgj(t)exp(2πitκ)dt.
s1(κ)=Tcxxexp[2πTcxx2κ2],s2(κ)=|Bxy|Tcxyexp[2πTcxy2κ2],s3(κ)=Tcyyexp[2πTcyy2κ2],
|Gx|2s1+2Re[Gx*Gy]s2+|Gy|2s3=|Gx+Gy|2s2+|Gx|2(s1s2)+|Gy|2(s3s2),
Tcxxexp[2πTcxx2κ2]|Bxy|Tcxyexp[2πTcxy2κ2],
Tcyyexp[2πTcyy2κ2]|Bxy|Tcxyexp[2πTcxy2κ2].
Tcxx|Bxy|Tcxy,Tcyy|Bxy|Tcxy,
TcxxTcxy,TcyyTcxy.
max{Tcxx,Tcyy}Tcxymin{Tcxx|Bxy|,Tcyy|Bxy|}.

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