Abstract

The case of a partially-coherent wave scattered from a material circular cylinder is investigated. Expressions for the TMz and TEz scattered-field cross-spectral density functions are derived by utilizing the plane-wave spectrum representation of electromagnetic fields and cylindrical wave transformations. From the analytical scattered-field cross-spectral density functions, the mean scattering widths are derived and subsequently validated via comparison with those computed from Method of Moments Monte Carlo simulations. The analytical relations as well as the simulation results are discussed and physically interpreted. Key insights are noted and subsequently analyzed.

© 2013 Optical Society of America

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2013

J. Geng, R. W. Ziolkowski, R. Jin, and X. Liang, “Active cylindrical coated nano-particle antennas: polarization-dependent scattering properties,” J. Electromagn. Waves Appl.27, 1392–1406 (2013).
[CrossRef]

H. Zhang, Z. Huang, and Y. Shi, “Internal and near-surface electromagnetic fields for a uniaxial anisotropic cylinder illuminated with a Gaussian beam,” Opt. Express21, 15645–15653 (2013).
[CrossRef] [PubMed]

2012

2011

M. A. Ashraf and A. A. Rizvi, “Electromagnetic scattering from a random cylinder by moments method,” J. Electromagn. Waves Appl.25, 467–480 (2011).
[CrossRef]

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

2010

X. M. Sun, H. H. Wang, and H. Y. Zhang, “Scattering by an infinite cylinder arbitrarily illuminated with a couple of Gaussian beams,” J. Electromagn. Waves Appl.24, 1329–1339 (2010).
[CrossRef]

M. L. Marasinghe, M. Premaratne, and D. M. Paganin, “Coherence vortices in Mie scattering of statistically stationary partially coherent fields,” Opt. Express18, 6628–6641 (2010).
[CrossRef] [PubMed]

2008

H. Zhang and Y. Han, “Scattering of shaped beam by an infinite cylinder of arbitrary orientation,” J. Opt. Soc. Am. B25, 131–135 (2008).
[CrossRef]

S. Ahmed and Q. A. Naqvi, “Electromagnetic scattering from parallel perfect electromagnetic conductor cylinders of circular cross-sections using an iterative procedure,” J. Electromagn. Waves Appl.22, 987–1003 (2008).
[CrossRef]

2007

B. H. Henin, A. Z. Elsherbeni, and M. H. Al Sharkawy, “Oblique incidence plane wave scattering from an array of circular dielectric cylinders,” Prog. Electromagn. Res.68, 261–279 (2007).
[CrossRef]

2006

1998

Z. W. L. Guo, “Electromagnetic scattering from a multilayered cylinder arbitrarily located in a Gaussian beam, a new recursive algorithms,” Prog. Electromagn. Res.18, 317–333 (1998).
[CrossRef]

1997

1996

M. Tateiba and Z. Q. Meng, “Wave scattering from conducting bodies embedded in random media—theory and numerical results,” Prog. Electromagn. Res.14, 317–361 (1996).

Z. Q. Meng and M. Tateiba, “Radar cross sections of conducting elliptic cylinders embedded in strong continuous random media,” Waves Random Complex Media6, 335–345 (1996).
[CrossRef]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: cylindrical-wave approach,” J. Opt. Soc. Am. A13, 483–493 (1996).
[CrossRef]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a set of perfectly conducting circular cylinders in the presence of a plane surface,” J. Opt. Soc. Am. A13, 2441–2452 (1996).
[CrossRef]

1995

1993

A. Elsherbeni, M. Hamid, and G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,” J. Electromagn. Waves Appl.7, 1323–1342 (1993).
[CrossRef]

1991

M. Kluskens and E. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag.39, 91–96 (1991).
[CrossRef]

1986

1982

S. Kozaki, “A new expression for the scattering of a Gaussian beam by a conducting cylinder,” IEEE Trans. Antennas Propag.30, 881–887 (1982).
[CrossRef]

S. Marin, “Computing scattering amplitudes for arbitrary cylinders under incident plane waves,” IEEE Trans. Antennas Propag.30, 1045–1049 (1982).
[CrossRef]

1979

T. Kojima and Y. Yanagiuchi, “Scattering of an offset two-dimensional Gaussian beam wave by a cylinder,” J. Appl. Phys.50, 41–46 (1979).
[CrossRef]

1965

J. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag.13, 334–341 (1965).
[CrossRef]

1961

1955

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys.33, 189–195 (1955).
[CrossRef]

Ahmed, S.

S. Ahmed and Q. A. Naqvi, “Electromagnetic scattering from parallel perfect electromagnetic conductor cylinders of circular cross-sections using an iterative procedure,” J. Electromagn. Waves Appl.22, 987–1003 (2008).
[CrossRef]

Al Sharkawy, M. H.

B. H. Henin, A. Z. Elsherbeni, and M. H. Al Sharkawy, “Oblique incidence plane wave scattering from an array of circular dielectric cylinders,” Prog. Electromagn. Res.68, 261–279 (2007).
[CrossRef]

Alonso, M. A.

Ashraf, M. A.

M. A. Ashraf and A. A. Rizvi, “Electromagnetic scattering from a random cylinder by moments method,” J. Electromagn. Waves Appl.25, 467–480 (2011).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012).

Borghi, R.

Dändliker, R.

Elsherbeni, A.

A. Elsherbeni, M. Hamid, and G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,” J. Electromagn. Waves Appl.7, 1323–1342 (1993).
[CrossRef]

Elsherbeni, A. Z.

B. H. Henin, A. Z. Elsherbeni, and M. H. Al Sharkawy, “Oblique incidence plane wave scattering from an array of circular dielectric cylinders,” Prog. Electromagn. Res.68, 261–279 (2007).
[CrossRef]

Fischer, D. G.

Frezza, F.

Friberg, A. T.

Fukumitsu, O.

Gamiz, V. L.

Geng, J.

J. Geng, R. W. Ziolkowski, R. Jin, and X. Liang, “Active cylindrical coated nano-particle antennas: polarization-dependent scattering properties,” J. Electromagn. Waves Appl.27, 1392–1406 (2013).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Gori, F.

Gouesbet, G.

Gréhan, G.

Guo, Z. W. L.

Z. W. L. Guo, “Electromagnetic scattering from a multilayered cylinder arbitrarily located in a Gaussian beam, a new recursive algorithms,” Prog. Electromagn. Res.18, 317–333 (1998).
[CrossRef]

Hamid, M.

A. Elsherbeni, M. Hamid, and G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,” J. Electromagn. Waves Appl.7, 1323–1342 (1993).
[CrossRef]

Han, Y.

Hansen, T.

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

Harrington, R.

R. Harrington, Time-Harmonic Electromagnetic Fields (IEEE, 2001).
[CrossRef]

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

Henin, B. H.

B. H. Henin, A. Z. Elsherbeni, and M. H. Al Sharkawy, “Oblique incidence plane wave scattering from an array of circular dielectric cylinders,” Prog. Electromagn. Res.68, 261–279 (2007).
[CrossRef]

Hoover, B. G.

Huang, Z.

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice Hall, 1991).

Jin, R.

J. Geng, R. W. Ziolkowski, R. Jin, and X. Liang, “Active cylindrical coated nano-particle antennas: polarization-dependent scattering properties,” J. Electromagn. Waves Appl.27, 1392–1406 (2013).
[CrossRef]

Kaivola, M.

Kerker, M.

Kluskens, M.

M. Kluskens and E. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag.39, 91–96 (1991).
[CrossRef]

Kojima, T.

T. Kojima and Y. Yanagiuchi, “Scattering of an offset two-dimensional Gaussian beam wave by a cylinder,” J. Appl. Phys.50, 41–46 (1979).
[CrossRef]

Kozaki, S.

S. Kozaki, “A new expression for the scattering of a Gaussian beam by a conducting cylinder,” IEEE Trans. Antennas Propag.30, 881–887 (1982).
[CrossRef]

Krattiger, B.

Liang, X.

J. Geng, R. W. Ziolkowski, R. Jin, and X. Liang, “Active cylindrical coated nano-particle antennas: polarization-dependent scattering properties,” J. Electromagn. Waves Appl.27, 1392–1406 (2013).
[CrossRef]

Lindberg, J.

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).
[CrossRef]

Madrazo, A.

Mandel, L.

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

Marasinghe, M. L.

Marin, S.

S. Marin, “Computing scattering amplitudes for arbitrary cylinders under incident plane waves,” IEEE Trans. Antennas Propag.30, 1045–1049 (1982).
[CrossRef]

Matijevic, E.

Meng, Z. Q.

Z. Q. Meng and M. Tateiba, “Radar cross sections of conducting elliptic cylinders embedded in strong continuous random media,” Waves Random Complex Media6, 335–345 (1996).
[CrossRef]

M. Tateiba and Z. Q. Meng, “Wave scattering from conducting bodies embedded in random media—theory and numerical results,” Prog. Electromagn. Res.14, 317–361 (1996).

Mittra, R.

A. Peterson, S. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE, 1997).
[CrossRef]

Naqvi, Q. A.

S. Ahmed and Q. A. Naqvi, “Electromagnetic scattering from parallel perfect electromagnetic conductor cylinders of circular cross-sections using an iterative procedure,” J. Electromagn. Waves Appl.22, 987–1003 (2008).
[CrossRef]

Newman, E.

M. Kluskens and E. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag.39, 91–96 (1991).
[CrossRef]

Ngo, D.

Nieto-Vesperinas, M.

Paganin, D. M.

Peterson, A.

A. Peterson, S. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE, 1997).
[CrossRef]

Premaratne, M.

Ray, S.

A. Peterson, S. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE, 1997).
[CrossRef]

Ren, K. F.

Richmond, J.

J. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag.13, 334–341 (1965).
[CrossRef]

Rizvi, A. A.

M. A. Ashraf and A. A. Rizvi, “Electromagnetic scattering from a random cylinder by moments method,” J. Electromagn. Waves Appl.25, 467–480 (2011).
[CrossRef]

Santarsiero, M.

Schettini, G.

Setälä, T.

Shi, Y.

Souli, N.

Sun, X. M.

X. M. Sun, H. H. Wang, and H. Y. Zhang, “Scattering by an infinite cylinder arbitrarily illuminated with a couple of Gaussian beams,” J. Electromagn. Waves Appl.24, 1329–1339 (2010).
[CrossRef]

Takenaka, T.

Tateiba, M.

Z. Q. Meng and M. Tateiba, “Radar cross sections of conducting elliptic cylinders embedded in strong continuous random media,” Waves Random Complex Media6, 335–345 (1996).
[CrossRef]

M. Tateiba and Z. Q. Meng, “Wave scattering from conducting bodies embedded in random media—theory and numerical results,” Prog. Electromagn. Res.14, 317–361 (1996).

Tian, G.

A. Elsherbeni, M. Hamid, and G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,” J. Electromagn. Waves Appl.7, 1323–1342 (1993).
[CrossRef]

van Dijk, T.

Videen, G.

Visser, T. D.

Voelz, D.

Wait, J. R.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys.33, 189–195 (1955).
[CrossRef]

Wang, H. H.

X. M. Sun, H. H. Wang, and H. Y. Zhang, “Scattering by an infinite cylinder arbitrarily illuminated with a couple of Gaussian beams,” J. Electromagn. Waves Appl.24, 1329–1339 (2010).
[CrossRef]

Wolf, E.

D. G. Fischer, T. van Dijk, T. D. Visser, and E. Wolf, “Coherence effects in Mie scattering,” J. Opt. Soc. Am. A29, 78–84 (2012).
[CrossRef]

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

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

Xiao, X.

Yaghjian, A.

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

Yanagiuchi, Y.

T. Kojima and Y. Yanagiuchi, “Scattering of an offset two-dimensional Gaussian beam wave by a cylinder,” J. Appl. Phys.50, 41–46 (1979).
[CrossRef]

Yokota, M.

Zhang, H.

Zhang, H. Y.

X. M. Sun, H. H. Wang, and H. Y. Zhang, “Scattering by an infinite cylinder arbitrarily illuminated with a couple of Gaussian beams,” J. Electromagn. Waves Appl.24, 1329–1339 (2010).
[CrossRef]

Zimmermann, E.

Ziolkowski, R. W.

J. Geng, R. W. Ziolkowski, R. Jin, and X. Liang, “Active cylindrical coated nano-particle antennas: polarization-dependent scattering properties,” J. Electromagn. Waves Appl.27, 1392–1406 (2013).
[CrossRef]

Appl. Opt.

Can. J. Phys.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys.33, 189–195 (1955).
[CrossRef]

IEEE Trans. Antennas Propag.

M. Kluskens and E. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag.39, 91–96 (1991).
[CrossRef]

S. Kozaki, “A new expression for the scattering of a Gaussian beam by a conducting cylinder,” IEEE Trans. Antennas Propag.30, 881–887 (1982).
[CrossRef]

J. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag.13, 334–341 (1965).
[CrossRef]

S. Marin, “Computing scattering amplitudes for arbitrary cylinders under incident plane waves,” IEEE Trans. Antennas Propag.30, 1045–1049 (1982).
[CrossRef]

J. Appl. Phys.

T. Kojima and Y. Yanagiuchi, “Scattering of an offset two-dimensional Gaussian beam wave by a cylinder,” J. Appl. Phys.50, 41–46 (1979).
[CrossRef]

J. Electromagn. Waves Appl.

J. Geng, R. W. Ziolkowski, R. Jin, and X. Liang, “Active cylindrical coated nano-particle antennas: polarization-dependent scattering properties,” J. Electromagn. Waves Appl.27, 1392–1406 (2013).
[CrossRef]

X. M. Sun, H. H. Wang, and H. Y. Zhang, “Scattering by an infinite cylinder arbitrarily illuminated with a couple of Gaussian beams,” J. Electromagn. Waves Appl.24, 1329–1339 (2010).
[CrossRef]

A. Elsherbeni, M. Hamid, and G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,” J. Electromagn. Waves Appl.7, 1323–1342 (1993).
[CrossRef]

M. A. Ashraf and A. A. Rizvi, “Electromagnetic scattering from a random cylinder by moments method,” J. Electromagn. Waves Appl.25, 467–480 (2011).
[CrossRef]

S. Ahmed and Q. A. Naqvi, “Electromagnetic scattering from parallel perfect electromagnetic conductor cylinders of circular cross-sections using an iterative procedure,” J. Electromagn. Waves Appl.22, 987–1003 (2008).
[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).
[CrossRef]

J. Opt.

G. Gouesbet, “Scattering of higher-order Gaussian beams by an infinite cylinder,” J. Opt.28, 45–65 (1997).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

B. G. Hoover and V. L. Gamiz, “Coherence solution for bidirectional reflectance distributions of surfaces with wavelength-scale statistics,” J. Opt. Soc. Am. A23, 314–328 (2006).
[CrossRef]

J. Lindberg, T. Setälä, M. Kaivola, and A. T. Friberg, “Spatial coherence effects in light scattering from metallic nanocylinders,” J. Opt. Soc. Am. A23, 1349–1358 (2006).
[CrossRef]

D. G. Fischer, T. van Dijk, T. D. Visser, and E. Wolf, “Coherence effects in Mie scattering,” J. Opt. Soc. Am. A29, 78–84 (2012).
[CrossRef]

G. Videen and D. Ngo, “Light scattering from a cylinder near a plane interface: theory and comparison with experimental data,” J. Opt. Soc. Am. A14, 70–78 (1997).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz-Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A14, 3014–3025 (1997).
[CrossRef]

M. Yokota, T. Takenaka, and O. Fukumitsu, “Scattering of a Hermite-Gaussian beam mode by parallel dielectric circular cylinders,” J. Opt. Soc. Am. A3, 580–586 (1986).
[CrossRef]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: cylindrical-wave approach,” J. Opt. Soc. Am. A13, 483–493 (1996).
[CrossRef]

R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, “Plane-wave scattering by a set of perfectly conducting circular cylinders in the presence of a plane surface,” J. Opt. Soc. Am. A13, 2441–2452 (1996).
[CrossRef]

E. Zimmermann, R. Dändliker, N. Souli, and B. Krattiger, “Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach,” J. Opt. Soc. Am. A12, 398–403 (1995).
[CrossRef]

A. Madrazo and M. Nieto-Vesperinas, “Scattering of electromagnetic waves from a cylinder in front of a conducting plane,” J. Opt. Soc. Am. A12, 1298–1309 (1995).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Prog. Electromagn. Res.

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It should be noted that no source can be spatially completely incoherent, i.e., ℓs= 0. The physical minimum value of ℓs is on the order of λ [1].

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Figures (6)

Fig. 1
Fig. 1

Two-dimensional scattering geometry (invariant in the z direction) of a material (ε, μ) circular cylinder (of radius a) illuminated by a partially-coherent wave. The partially-coherent incident field is emitted from the η-z plane whose origin is located at (−xs, ys). The spatial-domain vectors emanating from the source-plane origin, ρξη = xsŷys and s = ρξη + ρ, point to the center of the cylinder and to any location in space ξ ≥ 0, respectively. The wavenumber vector k i = ξ ^ k ξ i + η ^ k η i denotes the direction of a plane wave emanating from the source-plane origin. The vector ki makes the angle α with respect to the ξ axis.

Fig. 2
Fig. 2

TMz σ 2 D ¯ results for an aluminum cylinder of radius a = 10λ. The plots compare the σ 2 D ¯ obtained using the analytical expression derived in Section 2.3 (dashed traces labeled “Theory”) with those obtained from the MoM Monte Carlo simulations (circles and bars labeled “Simulation”). The color of the trace denotes the value of s. For (a), s = 1λ, 5λ, and 10λ; for (b), s = 10λ, 30λ, and 50λ. Included for reference is the scattering width for a fully-coherent plane wave (solid black trace labeled “F-C Plane Wave”).

Fig. 3
Fig. 3

TEz σ 2 D ¯ results for an aluminum cylinder of radius a = 10λ. The plots compare the σ 2 D ¯ obtained using the analytical expression derived in Section 2.5 (dashed traces labeled “Theory”) with those obtained from the MoM Monte Carlo simulations (circles and bars labeled “Simulation”). The color of the trace denotes the value of s. For (a), s = 1λ, 5λ, and 10λ; for (b), s = 10λ, 30λ, and 50λ. Included for reference is the scattering width for a fully-coherent plane wave (solid black trace labeled “F-C Plane Wave”).

Fig. 4
Fig. 4

TMz σ 2 D ¯ results for a germanium cylinder of radius a = 10λ. The plots compare the σ 2 D ¯ obtained using the analytical expression derived in Section 2.3 (dashed traces labeled “Theory”) with those obtained from the MoM Monte Carlo simulations (circles and bars labeled “Simulation”). The color of the trace denotes the value of s. For (a), s = 1λ, 5λ, and 10λ; for (b), s = 10λ, 30λ, and 50λ. Included for reference is the scattering width for a fully-coherent plane wave (solid black trace labeled “F-C Plane Wave”).

Fig. 5
Fig. 5

TEz σ 2 D ¯ results for an germanium cylinder of radius a = 10λ. The plots compare the σ 2 D ¯ obtained using the analytical expression derived in Section 2.5 (dashed traces labeled “Theory”) with those obtained from the MoM Monte Carlo simulations (circles and bars labeled “Simulation”). The color of the trace denotes the value of s. For (a), s = 1λ, 5λ, and 10λ; for (b), s = 10λ, 30λ, and 50λ. Included for reference is the scattering width for a fully-coherent plane wave (solid black trace labeled “F-C Plane Wave”).

Fig. 6
Fig. 6

Physical picture of role played by coherence in the forward scatter phenomenon. Depicted is one instance of a partially-coherent incident field (dashed red traces on left) which, when scattered from the cylinder, yields a σ2D instance (dashed red trace on right). When all σ2D instances are averaged, the resulting σ 2 D ¯ (solid red trace) in the forward scatter direction is reduced with the forward scatter lobe subsequently broadened. To serve as a reference, the case of fully-coherent plane-wave illumination is also depicted (solid blue traces).

Equations (24)

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W i ( η 1 , η 2 , ω ) = E i ( η 1 , ω ) E i * ( η 2 , ω ) = 1 2 π Γ i ( η 1 , η 2 , τ ) e j ω τ d τ
E i = z ^ E z i ( η ) ,
E i = 1 2 π T e i ( k η i ) e j k i s d k η i ξ 0 = z ^ k 0 2 π Γ T e z i ( α ) cos ( α ) e j k 0 x s cos ( α ) e j k 0 y s sin ( α ) e j k 0 ρ cos ( ϕ α ) d α ,
T e z i ( k η i ) = E z i ( η ) e j k η i η d η .
E i = z ^ k 0 2 π n = j n J n ( k 0 ρ ) e j n ϕ π / 2 π / 2 T e z i ( α ) cos ( α ) e j k 0 x s cos ( α ) e j k 0 y s sin ( α ) e j n α d α ,
H ϕ i = k 0 2 π 1 j η 0 n = j n J ˙ n ( k 0 ρ ) e j n ϕ n TM ,
E z s = n = b n H n ( 2 ) ( k 0 ρ ) H ϕ s = 1 j η 0 n = b n H ˙ n ( 2 ) ( k 0 ρ ) E z t = n = c n J n ( k ρ ) H ϕ t = 1 j η n = c n J ˙ n ( k ρ ) ,
b n = k 0 2 π j n e j n ϕ R n TM n TM ,
R n TM = η J n ( k a ) J ˙ n ( k 0 a ) η 0 J n ( k 0 a ) J ˙ n ( k a ) η J n ( k a ) H ˙ n ( 2 ) ( k 0 a ) η 0 H n ( 2 ) ( k 0 a ) J ˙ n ( k a )
W s ( ρ 1 , ρ 2 , ω ) = n = m = b n ( ϕ 1 ) b m * ( ϕ 2 ) H n ( 2 ) ( k 0 ρ 1 ) H m ( 1 ) ( k 0 ρ 2 ) = k 0 2 ( 2 π ) 2 n = m = n TM m TM * j m n e j ( n ϕ 1 m ϕ 2 ) R n TM R m TM * H n ( 2 ) ( k 0 ρ 1 ) H m ( 1 ) ( k 0 ρ 2 ) .
n TM m TM * = π / 2 π / 2 π / 2 π / 2 cos ( α 1 ) cos ( α 2 ) T e z i ( α 1 ) T e z i * ( α 2 ) e j k 0 x s [ cos ( α 1 ) cos ( α 2 ) ] e j k 0 y s [ sin ( α 1 ) sin ( α 2 ) ] e j ( n α 1 m α 2 ) d α 1 d α 2 = 1 k 0 2 k 0 k 0 k 0 k 0 T e z i ( k η 1 i ) T e z i * ( k η 2 i ) e j x s ( k ξ 1 i k ξ 2 i ) e j y s ( k η 1 i k η 2 i ) e j [ n sin 1 ( k η 1 i / k 0 ) m sin 1 ( k η 2 i / k 0 ) ] d k η 1 i d k η 2 i
W PW i ( η 1 , η 2 , ω ) = E 0 2 C ( η 1 η 2 ; s ) ,
W FF s ( ρ , ρ , ω ) = S FF s ( ρ , ω ) = E 0 2 π 2 k 0 ρ n = m = R n TM R m TM * e j ( n m ) ϕ k 0 k 0 e j ( n m ) sin 1 ( k η i / k 0 ) C ( η ; s ) e j k η i η d η d k η i .
σ 2 D ¯ ( ω ) = lim ρ 2 π ρ | E z s ( ρ , ω ) | 2 | E z i ( η , ω ) | 2 = lim ρ 2 π ρ S FF s ( ρ , ω ) W PW i ( η , η , ω ) .
H i = z ^ H z i ( η ) = z ^ 1 2 π T h z i ( k η i ) e j k i s d k η i ξ 0 = z ^ k 0 2 π n = j n J n ( k 0 ρ ) e j n ϕ n TE ,
E ϕ i = k 0 2 π η 0 j n = j n J ˙ n ( k 0 ρ ) e j n ϕ n TE .
H z s = n = b n H n ( 2 ) ( k 0 ρ ) E ϕ s = η 0 j n = b n H ˙ n ( 2 ) ( k 0 ρ ) H z t = n = c n J n ( k ρ ) E ϕ t = η j n = c n J ˙ n ( k ρ ) .
b n = k 0 2 π j n e j n ϕ R n TE n TE ,
R n TE = η J n ( k 0 a ) J ˙ n ( k a ) η 0 J n ( k a ) J ˙ n ( k 0 a ) η H n ( 2 ) ( k 0 a ) J ˙ n ( k a ) η 0 J n ( k a ) H ˙ n ( 2 ) ( k 0 a ) .
W s ( ρ 1 , ρ 2 , ω ) = η 0 2 n = m = b n ( ϕ 1 ) b m * ( ϕ 2 ) H ˙ n ( 2 ) ( k 0 ρ 1 ) H ˙ m ( 1 ) ( k 0 ρ 2 ) = η 0 2 k 0 2 ( 2 π ) 2 n = m = n TE m TE * j m n e j ( n ϕ 1 m ϕ 2 ) R n TE R m TE * H ˙ n ( 2 ) ( k 0 ρ 1 ) H ˙ m ( 1 ) ( k 0 ρ 2 ) .
T h i = k i × T e i ω μ 0 = z ^ T e η i η 0 cos ( α ) ,
n TE m TE * = 1 η 0 2 k 0 k 0 k 0 k 0 T e η i ( k η 1 i ) T e η i * ( k η 2 i ) k ξ 1 i k ξ 2 i e j x s ( k ξ 1 i k ξ 2 i ) e j y s ( k η 1 i k η 2 i ) e j [ n sin 1 ( k η 1 i / k 0 ) m sin 1 ( k η 2 i / k 0 ) ] d k η 1 i d k η 2 i .
S FF s ( ρ , ω ) = E 0 2 k 0 π 2 ρ n = m = R n TE R m TE * e j ( n m ) ϕ k 0 k 0 exp [ j ( n m ) sin 1 ( k η i / k 0 ) ] k 0 2 ( k η i ) 2 C ( η ; s ) e j k η i η d η d k η i .
C ( η ; s ) = e η 2 / s 2 .

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