Abstract

The scattering of a partially-coherent wave from a statistically rough material surface is investigated via derivation of the scattered field cross-spectral density function. Two forms of the cross-spectral density are derived using the physical optics approximation. The first is applicable to smooth-to-moderately rough surfaces and is a complicated expression of source and surface parameters. Physical insight is gleaned from its analytical form and presented in this work. The second form of the cross-spectral density function is applicable to very rough surfaces and is remarkably physical. Its form is discussed at length and closed-form expressions are derived for the angular spectral degree of coherence and spectral density radii. Furthermore, it is found that, under certain circumstances, the cross-spectral density function maintains a Gaussian Schell-model form. This is consistent with published results applicable only in the paraxial regime. Lastly, the closed-form cross-spectral density functions derived here are rigorously validated with scatterometer measurements and full-wave electromagnetic and physical optics simulations. Good agreement is noted between the analytical predictions and the measured and simulated results.

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T. Wang and D. Zhao, “Stokes parameters of an electromagnetic light wave on scattering,” Opt. Commun.285, 893–895 (2012).
[CrossRef]

S. Basu, S. J. Cusumano, M. W. Hyde, M. A. Marciniak, and S. T. Fiorino, “Validity of using Gaussian Schell model for extended beacon studies,” Proc. SPIE8380 (2012).

M. W. Hyde, S. Basu, S. J. Cusumano, and M. F. Spencer, “Scalar wave solution for the scattering of a partially coherent beam from a statistically rough metallic surface,” Proc. SPIE8550 (2012).
[CrossRef]

2011

C. A. Mack, “Analytic form for the power spectral density in one, two, and three dimensions,” J. Micro/Nanolithogr. MEMS MOEMS10, 040501 (2011).

H. T. Yura and S. G. Hanson, “Digital simulation of an arbitrary stationary stochastic process by spectral representation,” J. Opt. Soc. Am. A28, 675–685 (2011).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Condition for invariant spectral degree of coherence of an electromagnetic plane wave on scattering,” Opt. Commun.284, 724–728 (2011).
[CrossRef]

2010

S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun.283, 4512–4518 (2010).
[CrossRef]

G. Gbur and T. Visser, “The structure of partially coherent fields,” Progress in Optics55, 285–341 (2010).
[CrossRef]

2008

O. Korotkova, T. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun.281, 515–520 (2008).
[CrossRef]

M.-J. Wang, Z.-S. Wu, and Y.-L. Li, “Investigation on the scattering characteristics of Gaussian beam from two dimensional dielectric rough surfaces based on the Kirchhoff approximation,” Prog. Electromagn. Res. B4, 223–235 (2008).
[CrossRef]

2007

2006

2004

T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media14, R1–R40 (2004).
[CrossRef]

T. Leskova, A. Maradudin, and J. Munoz-Lopez, “The design of one-dimensional randomly rough surfaces that act as Collett-Wolf sources,” Opt. Commun.242, 123–133 (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]

2003

H. Yura and S. Hanson, “Variance of intensity for Gaussian statistics and partially developed speckle in complex ABCD optical systems,” Opt. Commun.228, 263–270 (2003).
[CrossRef]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312, 263–267 (2003).
[CrossRef]

2002

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng.41, 988–993 (2002).
[CrossRef]

2001

G. Guo, S. Li, and Q. Tan, “Statistical properties of laser speckles generated from far rough surfaces,” Int. J. Infrared Millimeter Waves22, 1177–1191 (2001).
[CrossRef]

K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media11, R1–R30 (2001).
[CrossRef]

1999

G. Gbur and E. Wolf, “Determination of density correlation functions from scattering of polychromatic light,” Opt. Commun.168, 39–45 (1999).
[CrossRef]

1997

1996

T. Shirai and T. Asakura, “Multiple light scattering from spatially random media under the second-order Born approximation,” Opt. Commun.123, 234–249 (1996).
[CrossRef]

A. Ishimaru, C. Le, Y. Kuga, L. A. Sengers, and T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” Prog. Electromagn. Res.14, 1–36 (1996).

1995

J. Huttunen, A. T. Friberg, and J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E52, 3081–3092 (1995).
[CrossRef]

A. T. Friberg and E. Wolf, “Relationships between the complex degrees of coherence in the space-time and in the space-frequency domains,” Opt. Lett.20, 623–625 (1995).
[CrossRef] [PubMed]

1994

R. E. Collin, “Full wave theories for rough surface scattering: an updated assessment,” Radio Sci.29, 1237–1254 (1994).
[CrossRef]

R. Collin, “Scattering of an incident Gaussian beam by a perfectly conducting rough surface,” IEEE Trans. Antennas Propag.42, 70–74 (1994).
[CrossRef]

1992

1991

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” SIGGRAPH Computer Graphics25, 175–186 (1991).
[CrossRef]

1990

1989

1988

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am.83, 78–92 (1988).
[CrossRef]

1981

E. Bahar, “Full-wave solutions for the depolarization of the scattered radiation fields by rough surfaces of arbitrary slope,” IEEE Trans. Antennas Propag.29, 443–454 (1981).
[CrossRef]

1978

R. Axline and A. Fung, “Numerical computation of scattering from a perfectly conducting random surface,” IEEE Trans. Antennas Propag.26, 482–488 (1978).
[CrossRef]

1975

H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt.6, 5–14 (1975).
[CrossRef]

1974

H. Pedersen, “The roughness dependence of partially developed, monochromatic speckle patterns,” Opt. Commun.12, 156–159 (1974).
[CrossRef]

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun.12, 32–38 (1974).
[CrossRef]

1970

J. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta17, 761–772 (1970).
[CrossRef]

1967

1951

S. O. Rice, “Reflection of electromagnetic wave by slightly rough surfaces,” Commun. Pure Appl. Math.4, 351–378 (1951).
[CrossRef]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2 ed. (SPIE, 2005).
[CrossRef]

Asakura, T.

T. Shirai and T. Asakura, “Multiple light scattering from spatially random media under the second-order Born approximation,” Opt. Commun.123, 234–249 (1996).
[CrossRef]

H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt.6, 5–14 (1975).
[CrossRef]

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun.12, 32–38 (1974).
[CrossRef]

Axline, R.

R. Axline and A. Fung, “Numerical computation of scattering from a perfectly conducting random surface,” IEEE Trans. Antennas Propag.26, 482–488 (1978).
[CrossRef]

Bahar, E.

E. Bahar, “Full-wave solutions for the depolarization of the scattered radiation fields by rough surfaces of arbitrary slope,” IEEE Trans. Antennas Propag.29, 443–454 (1981).
[CrossRef]

Basu, S.

M. W. Hyde, S. Basu, S. J. Cusumano, and M. F. Spencer, “Scalar wave solution for the scattering of a partially coherent beam from a statistically rough metallic surface,” Proc. SPIE8550 (2012).
[CrossRef]

S. Basu, S. J. Cusumano, M. W. Hyde, M. A. Marciniak, and S. T. Fiorino, “Validity of using Gaussian Schell model for extended beacon studies,” Proc. SPIE8380 (2012).

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

Chan, T. K.

A. Ishimaru, C. Le, Y. Kuga, L. A. Sengers, and T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” Prog. Electromagn. Res.14, 1–36 (1996).

Chen, F.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Condition for invariant spectral degree of coherence of an electromagnetic plane wave on scattering,” Opt. Commun.284, 724–728 (2011).
[CrossRef]

Chen, K. S.

A. K. Fung and K. S. Chen, Microwave Scattering and Emission Models for Users (Artech House, 2010).

Chen, Y.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Condition for invariant spectral degree of coherence of an electromagnetic plane wave on scattering,” Opt. Commun.284, 724–728 (2011).
[CrossRef]

Chew, W. C.

K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media11, R1–R30 (2001).
[CrossRef]

Collin, R.

R. Collin, “Scattering of an incident Gaussian beam by a perfectly conducting rough surface,” IEEE Trans. Antennas Propag.42, 70–74 (1994).
[CrossRef]

Collin, R. E.

R. E. Collin, “Full wave theories for rough surface scattering: an updated assessment,” Radio Sci.29, 1237–1254 (1994).
[CrossRef]

Cusumano, S. J.

M. W. Hyde, S. Basu, S. J. Cusumano, and M. F. Spencer, “Scalar wave solution for the scattering of a partially coherent beam from a statistically rough metallic surface,” Proc. SPIE8550 (2012).
[CrossRef]

S. Basu, S. J. Cusumano, M. W. Hyde, M. A. Marciniak, and S. T. Fiorino, “Validity of using Gaussian Schell model for extended beacon studies,” Proc. SPIE8380 (2012).

Dainty, J.

J. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta17, 761–772 (1970).
[CrossRef]

Elfouhaily, T. M.

T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media14, R1–R40 (2004).
[CrossRef]

Fiorino, S. T.

S. Basu, S. J. Cusumano, M. W. Hyde, M. A. Marciniak, and S. T. Fiorino, “Validity of using Gaussian Schell model for extended beacon studies,” Proc. SPIE8380 (2012).

Foley, J. T.

Friberg, A. T.

J. Huttunen, A. T. Friberg, and J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E52, 3081–3092 (1995).
[CrossRef]

A. T. Friberg and E. Wolf, “Relationships between the complex degrees of coherence in the space-time and in the space-frequency domains,” Opt. Lett.20, 623–625 (1995).
[CrossRef] [PubMed]

Fujii, H.

H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt.6, 5–14 (1975).
[CrossRef]

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun.12, 32–38 (1974).
[CrossRef]

Fung, A.

R. Axline and A. Fung, “Numerical computation of scattering from a perfectly conducting random surface,” IEEE Trans. Antennas Propag.26, 482–488 (1978).
[CrossRef]

Fung, A. K.

A. K. Fung and K. S. Chen, Microwave Scattering and Emission Models for Users (Artech House, 2010).

F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech House, 1986, vol. 2).

Gamiz, V. L.

Gbur, G.

G. Gbur and T. Visser, “The structure of partially coherent fields,” Progress in Optics55, 285–341 (2010).
[CrossRef]

G. Gbur and E. Wolf, “Determination of density correlation functions from scattering of polychromatic light,” Opt. Commun.168, 39–45 (1999).
[CrossRef]

Goodman, J.

J. Goodman, “Statistical properties of laser speckle patterns,” in “Laser Speckle and Related Phenomena,” vol. 9 of Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, 1975, pp. 9–75).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Ben Roberts & Company, 2007).

Gori, F.

F. Gori, C. Palma, and M. Santarsiero, “A scattering experiment with partially coherent light,” Opt. Commun.74, 353–356 (1990).
[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. A6, 1142–1149 (1989).
[CrossRef]

Greenberg, D. P.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” SIGGRAPH Computer Graphics25, 175–186 (1991).
[CrossRef]

Guérin, C.-A.

T. M. Elfouhaily and C.-A. Guérin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media14, R1–R40 (2004).
[CrossRef]

Guo, G.

G. Guo, S. Li, and Q. Tan, “Statistical properties of laser speckles generated from far rough surfaces,” Int. J. Infrared Millimeter Waves22, 1177–1191 (2001).
[CrossRef]

Hansen, R. S.

Hansen, T.

T. Hansen and A. Yaghjian, Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications (IEEE, 1999).
[CrossRef]

Hanson, S.

H. Yura and S. Hanson, “Variance of intensity for Gaussian statistics and partially developed speckle in complex ABCD optical systems,” Opt. Commun.228, 263–270 (2003).
[CrossRef]

Hanson, S. G.

Harrington, R.

R. Harrington, Field Computation by Moment Methods (IEEE, 1993).
[CrossRef]

He, X. D.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” SIGGRAPH Computer Graphics25, 175–186 (1991).
[CrossRef]

Hoover, B. G.

Huttunen, J.

J. Huttunen, A. T. Friberg, and J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E52, 3081–3092 (1995).
[CrossRef]

Hyde, M. W.

M. W. Hyde, S. Basu, S. J. Cusumano, and M. F. Spencer, “Scalar wave solution for the scattering of a partially coherent beam from a statistically rough metallic surface,” Proc. SPIE8550 (2012).
[CrossRef]

S. Basu, S. J. Cusumano, M. W. Hyde, M. A. Marciniak, and S. T. Fiorino, “Validity of using Gaussian Schell model for extended beacon studies,” Proc. SPIE8380 (2012).

Ishimaru, A.

A. Ishimaru, C. Le, Y. Kuga, L. A. Sengers, and T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” Prog. Electromagn. Res.14, 1–36 (1996).

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

Kato, K.

Knotts, M. E.

Korotkova, O.

S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun.283, 4512–4518 (2010).
[CrossRef]

O. Korotkova, T. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun.281, 515–520 (2008).
[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, Partially Coherent Beam Propagation in Turbulent Atmosphere with Applications (VDM Verlag Dr. Müller, 2009).

Kuga, Y.

A. Ishimaru, C. Le, Y. Kuga, L. A. Sengers, and T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” Prog. Electromagn. Res.14, 1–36 (1996).

Le, C.

A. Ishimaru, C. Le, Y. Kuga, L. A. Sengers, and T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” Prog. Electromagn. Res.14, 1–36 (1996).

Leskova, T.

T. Leskova, A. Maradudin, and J. Munoz-Lopez, “The design of one-dimensional randomly rough surfaces that act as Collett-Wolf sources,” Opt. Commun.242, 123–133 (2004).
[CrossRef]

Li, J.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Condition for invariant spectral degree of coherence of an electromagnetic plane wave on scattering,” Opt. Commun.284, 724–728 (2011).
[CrossRef]

Li, S.

G. Guo, S. Li, and Q. Tan, “Statistical properties of laser speckles generated from far rough surfaces,” Int. J. Infrared Millimeter Waves22, 1177–1191 (2001).
[CrossRef]

Li, Y.-L.

M.-J. Wang, Z.-S. Wu, and Y.-L. Li, “Investigation on the scattering characteristics of Gaussian beam from two dimensional dielectric rough surfaces based on the Kirchhoff approximation,” Prog. Electromagn. Res. B4, 223–235 (2008).
[CrossRef]

Mack, C. A.

C. A. Mack, “Analytic form for the power spectral density in one, two, and three dimensions,” J. Micro/Nanolithogr. MEMS MOEMS10, 040501 (2011).

Mandel, L.

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

Mandrosov, V. I.

V. I. Mandrosov, Coherent Fields and Images in Remote Sensing (SPIE, 2004).
[CrossRef]

Maradudin, A.

T. Leskova, A. Maradudin, and J. Munoz-Lopez, “The design of one-dimensional randomly rough surfaces that act as Collett-Wolf sources,” Opt. Commun.242, 123–133 (2004).
[CrossRef]

Marciniak, M. A.

S. Basu, S. J. Cusumano, M. W. Hyde, M. A. Marciniak, and S. T. Fiorino, “Validity of using Gaussian Schell model for extended beacon studies,” Proc. SPIE8380 (2012).

Meier, S. R.

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng.41, 988–993 (2002).
[CrossRef]

Michel, T. R.

Mittra, R.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE, 1998).

Moore, R. K.

F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech House, 1986, vol. 2).

Munoz-Lopez, J.

T. Leskova, A. Maradudin, and J. Munoz-Lopez, “The design of one-dimensional randomly rough surfaces that act as Collett-Wolf sources,” Opt. Commun.242, 123–133 (2004).
[CrossRef]

Nakagawa, K.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2 ed. (World Scientific, 2006).
[CrossRef]

O’Donnell, K. A.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (IOP Publishing, 1991).

Palma, C.

F. Gori, C. Palma, and M. Santarsiero, “A scattering experiment with partially coherent light,” Opt. Commun.74, 353–356 (1990).
[CrossRef]

Parry, G.

G. Parry, “Speckle patterns in partially coherent light,” in “Laser Speckle and Related Phenomena,” vol. 9 of Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, 1975, pp. 77–121).
[CrossRef]

Pedersen, H.

H. Pedersen, “The roughness dependence of partially developed, monochromatic speckle patterns,” Opt. Commun.12, 156–159 (1974).
[CrossRef]

Peterson, A. F.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE, 1998).

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2 ed. (SPIE, 2005).
[CrossRef]

Priest, R. G.

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng.41, 988–993 (2002).
[CrossRef]

Ramo, S.

S. Ramo, J. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3 ed. (John Wiley & Sons, 1994).

Ray, S. L.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE, 1998).

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic wave by slightly rough surfaces,” Commun. Pure Appl. Math.4, 351–378 (1951).
[CrossRef]

Sahin, S.

S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun.283, 4512–4518 (2010).
[CrossRef]

Salem, M.

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]

Santarsiero, M.

F. Gori, C. Palma, and M. Santarsiero, “A scattering experiment with partially coherent light,” Opt. Commun.74, 353–356 (1990).
[CrossRef]

Sengers, L. A.

A. Ishimaru, C. Le, Y. Kuga, L. A. Sengers, and T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” Prog. Electromagn. Res.14, 1–36 (1996).

Shirai, T.

T. Shirai and T. Asakura, “Multiple light scattering from spatially random media under the second-order Born approximation,” Opt. Commun.123, 234–249 (1996).
[CrossRef]

Sillion, F. X.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” SIGGRAPH Computer Graphics25, 175–186 (1991).
[CrossRef]

Sparrow, E. M.

Spencer, M. F.

M. W. Hyde, S. Basu, S. J. Cusumano, and M. F. Spencer, “Scalar wave solution for the scattering of a partially coherent beam from a statistically rough metallic surface,” Proc. SPIE8550 (2012).
[CrossRef]

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

Sun, Y.

Tan, Q.

G. Guo, S. Li, and Q. Tan, “Statistical properties of laser speckles generated from far rough surfaces,” Int. J. Infrared Millimeter Waves22, 1177–1191 (2001).
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E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am.83, 78–92 (1988).
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Tong, Z.

S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun.283, 4512–4518 (2010).
[CrossRef]

Torrance, K. E.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” SIGGRAPH Computer Graphics25, 175–186 (1991).
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F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech House, 1986, vol. 2).

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S. Ramo, J. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3 ed. (John Wiley & Sons, 1994).

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G. Gbur and T. Visser, “The structure of partially coherent fields,” Progress in Optics55, 285–341 (2010).
[CrossRef]

O. Korotkova, T. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun.281, 515–520 (2008).
[CrossRef]

Voelz, D.

Wang, M.-J.

M.-J. Wang, Z.-S. Wu, and Y.-L. Li, “Investigation on the scattering characteristics of Gaussian beam from two dimensional dielectric rough surfaces based on the Kirchhoff approximation,” Prog. Electromagn. Res. B4, 223–235 (2008).
[CrossRef]

Wang, T.

T. Wang and D. Zhao, “Stokes parameters of an electromagnetic light wave on scattering,” Opt. Commun.285, 893–895 (2012).
[CrossRef]

Wang, Y.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Condition for invariant spectral degree of coherence of an electromagnetic plane wave on scattering,” Opt. Commun.284, 724–728 (2011).
[CrossRef]

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K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media11, R1–R30 (2001).
[CrossRef]

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S. Ramo, J. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3 ed. (John Wiley & Sons, 1994).

Wolf, E.

O. Korotkova, T. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun.281, 515–520 (2008).
[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. A312, 263–267 (2003).
[CrossRef]

G. Gbur and E. Wolf, “Determination of density correlation functions from scattering of polychromatic light,” Opt. Commun.168, 39–45 (1999).
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A. T. Friberg and E. Wolf, “Relationships between the complex degrees of coherence in the space-time and in the space-frequency domains,” Opt. Lett.20, 623–625 (1995).
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E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

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M.-J. Wang, Z.-S. Wu, and Y.-L. Li, “Investigation on the scattering characteristics of Gaussian beam from two dimensional dielectric rough surfaces based on the Kirchhoff approximation,” Prog. Electromagn. Res. B4, 223–235 (2008).
[CrossRef]

Xiao, X.

Xin, Y.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Condition for invariant spectral degree of coherence of an electromagnetic plane wave on scattering,” Opt. Commun.284, 724–728 (2011).
[CrossRef]

Xu, S.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Condition for invariant spectral degree of coherence of an electromagnetic plane wave on scattering,” Opt. Commun.284, 724–728 (2011).
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T. Hansen and A. Yaghjian, Plane-Wave Theory of Time-Domain Fields: Near-Field Scanning Applications (IEEE, 1999).
[CrossRef]

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Yura, H.

H. Yura and S. Hanson, “Variance of intensity for Gaussian statistics and partially developed speckle in complex ABCD optical systems,” Opt. Commun.228, 263–270 (2003).
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Yura, H. T.

Zhao, D.

T. Wang and D. Zhao, “Stokes parameters of an electromagnetic light wave on scattering,” Opt. Commun.285, 893–895 (2012).
[CrossRef]

Zhao, Q.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Condition for invariant spectral degree of coherence of an electromagnetic plane wave on scattering,” Opt. Commun.284, 724–728 (2011).
[CrossRef]

Zhou, M.

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Condition for invariant spectral degree of coherence of an electromagnetic plane wave on scattering,” Opt. Commun.284, 724–728 (2011).
[CrossRef]

Commun. Pure Appl. Math.

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Int. J. Infrared Millimeter Waves

G. Guo, S. Li, and Q. Tan, “Statistical properties of laser speckles generated from far rough surfaces,” Int. J. Infrared Millimeter Waves22, 1177–1191 (2001).
[CrossRef]

J. Acoust. Soc. Am.

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am.83, 78–92 (1988).
[CrossRef]

J. Micro/Nanolithogr. MEMS MOEMS

C. A. Mack, “Analytic form for the power spectral density in one, two, and three dimensions,” J. Micro/Nanolithogr. MEMS MOEMS10, 040501 (2011).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

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H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt.6, 5–14 (1975).
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Opt. Acta

J. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta17, 761–772 (1970).
[CrossRef]

Opt. Commun.

H. Pedersen, “The roughness dependence of partially developed, monochromatic speckle patterns,” Opt. Commun.12, 156–159 (1974).
[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, T. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun.281, 515–520 (2008).
[CrossRef]

T. Leskova, A. Maradudin, and J. Munoz-Lopez, “The design of one-dimensional randomly rough surfaces that act as Collett-Wolf sources,” Opt. Commun.242, 123–133 (2004).
[CrossRef]

J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Condition for invariant spectral degree of coherence of an electromagnetic plane wave on scattering,” Opt. Commun.284, 724–728 (2011).
[CrossRef]

T. Wang and D. Zhao, “Stokes parameters of an electromagnetic light wave on scattering,” Opt. Commun.285, 893–895 (2012).
[CrossRef]

T. Shirai and T. Asakura, “Multiple light scattering from spatially random media under the second-order Born approximation,” Opt. Commun.123, 234–249 (1996).
[CrossRef]

G. Gbur and E. Wolf, “Determination of density correlation functions from scattering of polychromatic light,” Opt. Commun.168, 39–45 (1999).
[CrossRef]

F. Gori, C. Palma, and M. Santarsiero, “A scattering experiment with partially coherent light,” Opt. Commun.74, 353–356 (1990).
[CrossRef]

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun.12, 32–38 (1974).
[CrossRef]

H. Yura and S. Hanson, “Variance of intensity for Gaussian statistics and partially developed speckle in complex ABCD optical systems,” Opt. Commun.228, 263–270 (2003).
[CrossRef]

S. Sahin, Z. Tong, and O. Korotkova, “Sensing of semi-rough targets embedded in atmospheric turbulence by means of stochastic electromagnetic beams,” Opt. Commun.283, 4512–4518 (2010).
[CrossRef]

Opt. Eng.

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng.41, 988–993 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A312, 263–267 (2003).
[CrossRef]

Phys. Rev. E

J. Huttunen, A. T. Friberg, and J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E52, 3081–3092 (1995).
[CrossRef]

Proc. SPIE

M. W. Hyde, S. Basu, S. J. Cusumano, and M. F. Spencer, “Scalar wave solution for the scattering of a partially coherent beam from a statistically rough metallic surface,” Proc. SPIE8550 (2012).
[CrossRef]

S. Basu, S. J. Cusumano, M. W. Hyde, M. A. Marciniak, and S. T. Fiorino, “Validity of using Gaussian Schell model for extended beacon studies,” Proc. SPIE8380 (2012).

Prog. Electromagn. Res.

A. Ishimaru, C. Le, Y. Kuga, L. A. Sengers, and T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” Prog. Electromagn. Res.14, 1–36 (1996).

Prog. Electromagn. Res. B

M.-J. Wang, Z.-S. Wu, and Y.-L. Li, “Investigation on the scattering characteristics of Gaussian beam from two dimensional dielectric rough surfaces based on the Kirchhoff approximation,” Prog. Electromagn. Res. B4, 223–235 (2008).
[CrossRef]

Progress in Optics

G. Gbur and T. Visser, “The structure of partially coherent fields,” Progress in Optics55, 285–341 (2010).
[CrossRef]

Radio Sci.

R. E. Collin, “Full wave theories for rough surface scattering: an updated assessment,” Radio Sci.29, 1237–1254 (1994).
[CrossRef]

SIGGRAPH Computer Graphics

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Waves Random Media

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[CrossRef]

K. F. Warnick and W. C. Chew, “Numerical simulation methods for rough surface scattering,” Waves Random Media11, R1–R30 (2001).
[CrossRef]

Other

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[CrossRef]

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (IOP Publishing, 1991).

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2 ed. (World Scientific, 2006).
[CrossRef]

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

A. K. Fung and K. S. Chen, Microwave Scattering and Emission Models for Users (Artech House, 2010).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Ben Roberts & Company, 2007).

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

J. Goodman, “Statistical properties of laser speckle patterns,” in “Laser Speckle and Related Phenomena,” vol. 9 of Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, 1975, pp. 9–75).
[CrossRef]

G. Parry, “Speckle patterns in partially coherent light,” in “Laser Speckle and Related Phenomena,” vol. 9 of Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, 1975, pp. 77–121).
[CrossRef]

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

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

F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech House, 1986, vol. 2).

Under quasi-monochromatic conditions (Δω ≪ ω̄), S (ρ, ω) ≈ Γ(ρ, ρ, 0)δ(ω − ω̄) and μ (ρ1, ρ2, ω) ≈ γ (ρ1, ρ2, τ) exp (jω̄τ). Here, Γ is the mutual coherence function and γ is the complex degree of coherence [59].

B. Balling, “A comparative study of the bidirectional reflectance distribution function of several surfaces as a mid-wave infrared diffuse reflectance standard,” Master’s thesis, Graduate School of Engineering and Management, Air Force Institute of Technology (AETC), Wright-Patterson AFB OH (2009). http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA495933 .

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[CrossRef]

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE, 1998).

RefractiveIndex.INFO, “RefractiveIndex.INFO Refractive index database” (RefractiveIndex.INFO, 2012). http://refractiveindex.info .

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[CrossRef]

S. Ramo, J. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3 ed. (John Wiley & Sons, 1994).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2 ed. (SPIE, 2005).
[CrossRef]

O. Korotkova, Partially Coherent Beam Propagation in Turbulent Atmosphere with Applications (VDM Verlag Dr. Müller, 2009).

V. I. Mandrosov, Coherent Fields and Images in Remote Sensing (SPIE, 2004).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Scattering geometry of a one-dimensional (the surface and source excitation are invariant in the z direction) rough surface of length 2L. The medium below the rough interface is electrically defined by the permittivity ε and permeability μ; the medium above the rough interface is vacuum (ε0, μ0). The rough surface height is described by the function h(x); the mean, standard deviation, and correlation length of the surface are 0, σh, and h, respectively. The point (−xs, ys) denotes the location of the source plane origin. The observation vector ρ = x̂x +ŷy points from the rough surface origin to the observation point. The vector ρs = xsŷys points from the source plane origin to the rough surface origin.

Fig. 2
Fig. 2

CASI measurement, MoM simulation, analytical PO solution, and PO simulation results of normalized Ss versus θr for Infragold—(a) θi = 20°, (a) θi = 40°, and (c) θi = 60°.

Fig. 3
Fig. 3

MoM simulation, analytical PO solution, and PO simulation results of normalized Ss versus θr [(a)] and |μs| versus Δθr [(b) and (c)] for hypothetical Infragold G-G surfaces. In (a) and (b), the hypothetical Infragold surface possessed the measured σh and h values of 11.09 μm and 116.9 μm, respectively. Since this hypothetical surface qualifies as a very rough surface, θ 1 / e r and |Δθr|1/e, given in Eqs. (30) and (26), are also plotted as vertical dashed line in (a) and (b), respectively. In (c), σh is varied while h is held constant at 8λ.

Fig. 4
Fig. 4

Ss versus θr at normal incidence, i.e., θi = 0°—(a) α = 2, (b) α = 1, (c) α = 0.5, and (d) α = 0.25. The solid traces are the PO analytical predictions, i.e., Eq. (20) or Eq. (23) (whichever is applicable); the circles are the PO simulation results. The vertical dashed lines in the figures mark the locations of θ 1 / e r, namely, Eq. (30).

Fig. 5
Fig. 5

|μs| versus Δθr—(a) σh = 0.01 rad, (b) σh = 0.05 rad, (c) σh = 0.1 rad, and (d) σh = 0.25 rad. The solid traces are the PO analytical predictions, i.e., Eq. (20) or Eq. (23) (whichever is applicable); the circles are the PO simulation results. The vertical dashed lines in the figures mark the locations of |Δθr|1/e, namely, Eq. (26).

Equations (34)

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W i ( u 1 , u 2 ) = E i ( u 1 ) E i * ( u 2 ) = E 0 2 exp ( u 1 2 + u 2 2 4 w s 2 ) exp [ ( u 1 u 2 ) 2 2 s 2 ] ,
E i = z ^ E z i ( u ) .
E i = 1 2 π T e i ( k u i ) exp ( j k i ρ s ) exp ( j k i . ρ ) d k u i v > 0 H i = 1 2 π k i × T e i ( k u i ) ω μ 0 exp ( j k i . ρ s ) exp ( j k i ρ ) d k u i v > 0 ,
T e i ( k u i ) = z ^ E z i ( u ) exp ( j k u i u ) d u .
E s j k 0 8 π exp ( j k 0 ρ ) ρ [ μ 0 ε 0 N t + L t ] N t = ( ϕ ^ ϕ ^ + z ^ z ^ ) C J ( ρ ) exp ( j k 0 ρ ^ ρ ) d c L t = ( ϕ ^ ϕ ^ + z ^ z ^ ) C M ( ρ ) exp ( j k 0 ρ ^ ρ ) d c ,
J ( 1 r ) n ^ × H i | x = x y = h ( x ) M ( 1 + r ) E i × n ^ | x = x y = h ( x ) .
n ^ = y ^ x ^ h ( x ) 1 + [ h ( x ) ] 2 = y ^ x ^ h x 1 + h x 2
E z s = exp ( j k 0 ρ + j π 4 ) 2 π 8 π k 0 ρ T e z i ( k u i ) e j k i ρ s L L ( x , k u i ) e j ν ρ d x d k u i ,
( x , k u i ) = ( 1 r ) ( k ν i u ^ k u i v ^ ) ( x ^ + y ^ h x ) + k 0 ( 1 + r ) ρ ^ ( y ^ x ^ h x ) .
E z s = exp ( j k 0 ρ + j π 4 ) 2 π 8 π k 0 ρ L L ( k u i ) T e z i ( k u i ) e j k i ρ s e j ν ρ d x d k u i .
W s ( ρ 1 ρ 2 ) = E z s ( ρ 1 ) E z s * ( ρ 2 ) = e j k 0 ( ρ 2 ρ 1 ) 32 π 3 k 0 ρ 1 ρ 2 L L L L 1 2 * e j ( k 2 i k 1 i ) ρ s T e z 1 i T e z 2 i * e j ν 1 ρ 1 e j ν 2 ρ 2 d k u 1 i d k u 2 i d x 1 d x 2 ,
W s ( ρ 1 , ρ 2 ) = E 0 2 k 0 2 e j k 0 ( ρ 2 ρ 1 ) 𝒮 16 π ρ s ρ 1 ρ 2 ( a 2 b 2 ) L L L L exp [ a ˜ k 0 2 ( x ^ u ^ ) 2 ρ s 2 ( x 1 2 + x 2 2 ) ] exp [ 2 b ˜ k 0 2 ( x ^ u ^ ) 2 ρ s 2 x 1 x 2 ] exp [ j k 0 ( x ^ u ^ ) 2 2 ρ s ( x 2 2 x 1 2 ) ] e j k 0 ( x ^ v ^ ) ( x 1 x 2 ) χ ( k 0 ( x ^ u ^ ) x 1 ρ s , k 0 ( x ^ u ^ ) x 2 ρ s ; x 1 , x 2 ) e j k 0 [ ( x ^ ρ ^ 1 ) x 1 ( x ^ ρ ^ 2 ) x 2 ] d x 1 d x 2 ,
p H 1 , H 2 ( h 1 , h 2 ) = 1 2 π σ h 2 1 Γ 2 exp [ h 1 2 2 Γ h 1 h 2 + h 2 2 2 σ h 2 ( 1 Γ 2 ) ] Γ ( x 1 x 2 ) = exp [ ( x 1 x 2 ) 2 h 2 ] .
χ ( k u 1 i , k u 2 i ; x 1 , x 2 ) = exp [ σ h 2 2 ( ν y 1 2 + ν y 2 2 ) ] exp { σ h 2 ν y 1 ν y 2 exp [ ( x 1 x 2 ) 2 h 2 ] } .
W s ( ρ 1 , ρ 2 ) = E 0 2 k 0 2 e j k 0 ( ρ 2 ρ 1 ) 𝒮 32 π ρ s ρ 1 ρ 2 ( a 2 b 2 ) exp [ k 0 2 σ h 2 2 ( ϑ y 1 2 + ϑ y 2 2 ) ] 2 L 2 L exp [ ( a ˜ b ˜ ) k 0 2 ( x ^ u ^ ) 2 2 ρ s 2 x a 2 ] exp [ j k 0 2 ( ϑ x 1 ϑ x 2 ) x a ] | x a | 2 L 2 L | x a | exp [ ( a ˜ + b ˜ ) k 0 2 ( x ^ u ^ ) 2 2 ρ s 2 x d 2 ] exp [ k 0 2 σ h 2 ϑ y 1 ϑ y 2 exp ( x d 2 h 2 ) ] exp [ j k 0 2 ( ϑ x 1 + ϑ x 2 ) x d ] exp [ j k 0 ( x ^ u ^ ) 2 x a 2 ρ s x d ] d x d d x a .
exp [ k 0 2 σ h 2 ϑ y 1 ϑ y 2 exp ( x d 2 h 2 ) ] = m = 0 ( k 0 2 σ h 2 ϑ y 1 ϑ y 2 ) m m ! exp ( m x d 2 h 2 ) .
W s ( ρ 1 , ρ 2 ) = E 0 2 k 0 2 h e j k 0 ( ρ 2 ρ 1 ) 𝒮 16 2 π ρ 1 ρ 2 ( a 2 b 2 ) exp [ k 0 2 σ h 2 2 ( ϑ y 1 2 + ϑ y 2 2 ) ] m = 0 ( k 0 2 σ h 2 ϑ y 1 ϑ y 2 ) m m ! exp [ k 0 2 8 h 2 ρ s 2 𝒟 m ( ϑ x 1 + ϑ x 2 ) 2 ] 𝒟 m 2 L 2 L exp { k 0 2 ( x ^ u ^ ) 2 x a 2 2 [ ( a ˜ b ˜ ) ρ s 2 + h 2 ( x ^ u ^ ) 2 4 𝒟 m ] } exp { k 0 x a 2 [ k 0 ρ 2 h 2 ( x ^ u ^ ) 2 ( ϑ x 1 + ϑ x 2 ) 2 𝒟 m + j ( ϑ x 1 ϑ x 2 ) ] } Re { erf ( ( 2 L | x a | ) D m 2 ρ s h + j k 0 ρ s h 2 2 𝒟 m [ ( x ^ u ^ ) 2 ρ s x a ( ϑ x 1 + ϑ x 2 ) ] ) } d x a
exp { x d 2 [ ( a ˜ + b ˜ ) k 0 2 ( x ^ u ^ ) 2 2 ρ s 2 + m h 2 ] } > δ ,
L > ρ s k 0 w s 2 | x ^ u ^ | ln δ .
W s ( ρ 1 , ρ 2 ) = E 0 2 k 0 h ρ s e j k 0 ( ρ 2 ρ 1 ) 𝒮 16 | x ^ u ^ | ρ 1 ρ 2 ( a 2 b 2 ) exp [ k 0 2 σ h 2 2 ( ϑ y 1 2 + ϑ y 2 2 ) ] m = 0 ( k 0 2 σ h 2 ϑ y 1 ϑ y 2 ) m m ! 𝒜 m exp [ k 0 2 ρ s 2 h 2 4 𝒟 m ( 1 ρ s 2 h 2 ( x ^ u ^ ) 2 𝒜 m ) ( ϑ x 1 2 + ϑ x 2 2 ) ] exp { ρ s 2 [ k 0 2 h 2 ( x ^ u ^ ) 2 b ˜ + m ρ s 2 ] 𝒜 m ( x ^ u ^ ) 2 ( ϑ x 1 ϑ x 2 ) 2 } exp [ j k 0 h 2 ρ s 3 2 𝒜 m ( ϑ x 1 2 ϑ x 2 2 ) ] { erf [ L k 0 | x ^ u ^ | ρ s 𝒜 m 2 𝒟 m + k 0 | x ^ u ^ | ρ s 2 h 2 2 2 𝒟 m 𝒜 m ( ϑ x 1 + ϑ x 2 ) + j ρ s | x ^ u ^ | 𝒟 m 2 𝒜 m ( ϑ x 1 ϑ x 2 ) ] + erf [ L k 0 | x ^ u ^ | ρ s 𝒜 m 2 𝒟 m k 0 | x ^ u ^ | ρ s 2 h 2 2 2 𝒟 m 𝒜 m ( ϑ x 1 + ϑ x 2 ) j ρ s | x ^ u ^ | 𝒟 m 2 𝒜 m ( ϑ x 1 ϑ x 2 ) ] ,
L > ρ s 1 + ( 2 / α ) 2 k 0 w s 2 | x ^ u ^ | ln δ ,
χ = exp [ k 0 2 σ h 2 2 ( ϑ y 1 2 + ϑ y 2 2 ) ] exp [ k 0 2 σ h 2 ϑ y 1 ϑ y 2 exp ( x d 2 h 2 ) ] = exp { k 0 2 σ h 2 2 ϑ y 1 ϑ y 2 [ ϑ y 1 ϑ y 2 + ϑ y 2 ϑ y 1 2 exp ( x d 2 h 2 ) ] } .
W s ( ρ 1 , ρ 2 ) = E 0 2 k 0 h ρ s e j k 0 ( ρ 2 ρ 1 ) 𝒮 16 | x ^ u ^ | ρ 1 ρ 2 ( a 2 b 2 ) exp [ k 0 2 σ h 2 2 ( ϑ y 1 ϑ y 2 ) 2 ] 1 𝒜 ˜ exp [ k 0 2 ρ s 2 h 2 4 𝒟 ˜ ( 1 ρ s 2 h 2 ( x ^ u ^ ) 2 𝒜 ˜ ) ( ϑ x 1 2 + ϑ x 2 2 ) ] exp [ j k 0 h 2 ρ s 3 2 𝒜 ˜ ( ϑ x 1 2 ϑ x 2 2 ) ] exp { k 0 2 ρ s 2 [ h 2 ( x ^ u ^ ) 2 b ˜ + σ h 2 ρ s 2 ϑ y 1 ϑ y 2 ] 𝒜 ˜ ( x ^ u ^ ) 2 ( ϑ x 1 ϑ x 2 ) 2 } { erf [ L k 0 | x ^ u ^ | ρ s 𝒜 ˜ 2 𝒟 ˜ + k 0 | x ^ u ^ | ρ s 2 h 2 2 2 𝒟 ˜ 𝒜 ˜ ( ϑ x 1 + ϑ x 2 ) + j ρ s | x ^ u ^ | 𝒟 ˜ 2 𝒜 ˜ ( ϑ x 1 ϑ x 2 ) ] + erf [ L k 0 | x ^ u ^ | ρ s 𝒜 ˜ 2 𝒟 ˜ k 0 | x ^ u ^ | ρ s 2 h 2 2 2 𝒟 ˜ 𝒜 ˜ ( ϑ x 1 + ϑ x 2 ) j ρ s | x ^ u ^ | 𝒟 ˜ 2 𝒜 ˜ ( ϑ x 1 ϑ x 2 ) ] } ,
| ϑ x 1 ϑ x 2 | 1 / e = | sin θ 1 r sin θ 2 r | 1 / e | x ^ u ^ | ρ s 8 w s 2 s 2 s 2 + 4 w s 2 + 2 ρ s 2 h 2 ( x ^ u ^ ) 2 k 0 2 1 h 2 w s 2 ( x ^ u ^ ) 2 + 2 ρ s 2 σ h 2 ϑ y 1 ϑ y 2 .
| Δ θ r | 1 / e cos θ i ρ s cos θ r 8 w s 2 s 2 s 2 + 4 w s 2 + 2 k 0 2 1 σ h 2 ( 1 + cos θ r / cos θ i ) 2 1 1 + [ Ω σ h ( 1 + cos θ r / cos θ i ) ] 2 ,
| Δ θ r | 1 / e 2 Ω 2 1 + ( 2 / α ) 2 cos θ i cos θ r + Ω 2 k 0 2 w s 2 1 + ( 2 / α ) 2 4 σ h 2 ( 1 + cos θ r / cos θ i ) 2 cos θ i cos θ r .
sin 2 θ 1 / e r = 2 Ω 2 + 2 σ h 2 ( 1 + cos θ 1 / e r ) 2 + 1 + ( 2 / α ) 2 2 k 0 2 w s 2 ,
cos 4 ( θ 1 / e r / 2 ) 1 1 + 2 σ h 2 cos 2 ( θ 1 / e r / 2 ) + 2 Ω 2 + 1 + ( 2 / α ) 2 2 k 0 2 w s 2 4 + 8 σ h 2 = 0 .
ψ = 1 2 + 4 σ h 2 [ 1 ± 1 ( 1 + 2 σ h 2 ) ( 2 Ω 2 + ( 1 + ( 2 / α ) 2 ) 2 k 0 2 w s 2 ) ] .
θ 1 / e r 2 cos 1 [ 1 1 + 2 σ h 2 1 1 4 ( 1 + 2 σ h 2 ) ( 2 Ω 2 + 1 + ( 2 / α ) 2 2 k 0 2 w s 2 ) ] 2 cos 1 [ 1 1 + 2 σ h 2 1 8 1 + 2 σ h 2 ( 2 Ω 2 + 1 + ( 2 / α ) 2 2 k 0 2 w s 2 ) ] .
E z i ( ρ ) = μ ε J z ( ρ ) + j k 0 μ 0 ε 0 A z ( ρ ) + ( F y ( ρ ) x F x ( ρ ) y ) ρ C ,
𝒮 = 1 2 * 1 , 2 = ( x ^ u ^ ϑ x 1 , 2 ϑ y 1 , 2 y ^ u ^ + y ^ ρ ^ 1 , 2 + ϑ x 1 , 2 ϑ y 1 , 2 x ^ ρ ^ 1 , 2 ) r 1 , 2 ( x ^ u ^ ϑ x 1 , 2 ϑ y 1 , 2 y ^ u ^ y ^ ρ ^ 1 , 2 ϑ x 1 , 2 ϑ y 1 , 2 x ^ ρ ^ 1 , 2 ) ,
ϑ x 1 , 2 = x ^ ρ ^ 1 , 2 x ^ v ^ ϑ y 1 , 2 = y ^ ρ ^ 1 , 2 y ^ v ^ a = 1 4 w s 2 + 1 2 s 2 b = 1 2 s 2 a ˜ = a 4 ( a 2 b 2 ) b ˜ = b 4 ( a 2 b 2 ) .
𝒮 | | = | | 1 | | 2 * | | 1 , 2 = ( y ^ ρ ^ 1 , 2 + ϑ x 1 , 2 ϑ y 1 , 2 x ^ ρ ^ 1 , 2 x ^ u ^ + ϑ x 1 , 2 ϑ y 1 , 2 y ^ u ^ ) r | | 1 , 2 ( y ^ ρ ^ 1 , 2 + ϑ x 1 , 2 ϑ y 1 , 2 x ^ ρ ^ 1 , 2 + x ^ u ^ ϑ x 1 , 2 ϑ y 1 , 2 y ^ u ^ ) ,

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