Abstract

A rigorous method to analyze the electromagnetic scattering of an elliptically polarized plane wave by a sphere buried in a dielectric half-space, is presented. The electric field components of the incident and the scattered monochromatic plane waves are expanded in series of vectorial spherical harmonics, with unknown expansion coefficients. The scattered–reflected and scattered–transmitted fields are computed by exploiting the plane-wave spectrum of the scattered field, considering the reflection and transmission of each elementary plane wave by the interface. The boundary-condition imposition leads to a linear system that returns the unknown coefficients of the scattered field. To achieve a numerical solution, a code has been implemented, and a truncation criterion for the involved series has been proposed. Comparisons with the literature and simulations performed with a commercial software are presented. A generalization of the method to the case of a short pulse scattered by a buried sphere is presented, taking into account the dispersive properties of the involved media.

© 2013 Optical Society of America

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References

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    [CrossRef]
  2. P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2008 (1)

2006 (3)

I. C. Khoo, D. H. Werner, X. Liang, and A. Diaz, “Nanosphere dispersed liquid crystals for tunable negative-zero-positive index of refraction in the optical and terahertz regimes,” Opt. Lett. 31, 2592–2594 (2006).
[CrossRef]

R. W. Ziolkowski and A. Erentok, “Metamaterial-based efficient electrically small antennas,” IEEE Trans. Antennas Propag. 54, 2113–2130 (2006).
[CrossRef]

P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B 110, 7238–7248 (2006).
[CrossRef]

2004 (2)

G. L. Wang and W. C. Chew, “Formal solution to the electromagnetic scattering by buried dielectric and metallic spheres,” Radio Sci. 39, RS5004 (2004).
[CrossRef]

B. Esen, I. Akkaya, and A. Yapar, “Scattering of a plane wave from a perfectly conducting sphere buried in a conducting dielectric,” Electromagnetics 24, 607–621 (2004).
[CrossRef]

2003 (2)

Y. Feng, H. Zheng, Z. Zhub, and F. Zua, “The microstructure and electrical conductivity of aluminum alloy foams,” Mater. Chem. Phys. 78, 196–201 (2003).
[CrossRef]

T. Takenaka, H. Zhou, and T. Tanaka, “Inverse scattering for a three-dimensional object in the time domain,” J. Opt. Soc. Am. A 20, 1867–1874 (2003).
[CrossRef]

1999 (1)

K. Asami, E. Gheorghiu, and T. Yonezawa, “Real-time monitoring of yeast cell division by dielectric spectroscopy,” Biophys. J. 76, 3345–3348 (1999).
[CrossRef]

1998 (1)

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

1996 (2)

Y. Eremin and N. Orlov, “Simulation of light scattering from a particle upon a wafer surface,” Appl. Opt. 35, 6599–6604 (1996).
[CrossRef]

S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried PEC bodies,” IEEE Trans. Antennas Propag. 44, 143–151 (1996).
[CrossRef]

1993 (2)

1991 (1)

1988 (1)

R. C. Wittmann, “Spherical wave operators and the translation formulas,” IEEE Trans. Antennas Propag. 36, 1078–1087 (1988).
[CrossRef]

1986 (1)

P. A. Bobbert and J. Vlieger, “Light scattering by a sphere on a substrate,” Phys. A 137, 209–242 (1986).
[CrossRef]

1980 (1)

1919 (1)

H. Weyl, “Ausbreitung elektromagnetischer Wellen Ãijber einem ebenen Leiter,” Ann. Phys. 365, 481–500 (1919).
[CrossRef]

Akkaya, I.

B. Esen, I. Akkaya, and A. Yapar, “Scattering of a plane wave from a perfectly conducting sphere buried in a conducting dielectric,” Electromagnetics 24, 607–621 (2004).
[CrossRef]

Asami, K.

K. Asami, E. Gheorghiu, and T. Yonezawa, “Real-time monitoring of yeast cell division by dielectric spectroscopy,” Biophys. J. 76, 3345–3348 (1999).
[CrossRef]

Barber, P. W.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).

Bobbert, P. A.

P. A. Bobbert and J. Vlieger, “Light scattering by a sphere on a substrate,” Phys. A 137, 209–242 (1986).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Carin, L.

S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried PEC bodies,” IEEE Trans. Antennas Propag. 44, 143–151 (1996).
[CrossRef]

Chew, W. C.

G. L. Wang and W. C. Chew, “Formal solution to the electromagnetic scattering by buried dielectric and metallic spheres,” Radio Sci. 39, RS5004 (2004).
[CrossRef]

Diaz, A.

Doicu, A.

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

A. Doicu, R. Schuh, and T. Wriedt, “Scattering by particles on or near a plane surface,” in Light Scattering Reviews 3, A. A. Kokhanovsky, ed. (Springer Praxis Books, 2008), pp. 109–130.

El-Sayed, I. H.

P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B 110, 7238–7248 (2006).
[CrossRef]

El-Sayed, M. A.

P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B 110, 7238–7248 (2006).
[CrossRef]

Eremin, Y.

Eremin, Y. A.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

Erentok, A.

R. W. Ziolkowski and A. Erentok, “Metamaterial-based efficient electrically small antennas,” IEEE Trans. Antennas Propag. 54, 2113–2130 (2006).
[CrossRef]

Esen, B.

B. Esen, I. Akkaya, and A. Yapar, “Scattering of a plane wave from a perfectly conducting sphere buried in a conducting dielectric,” Electromagnetics 24, 607–621 (2004).
[CrossRef]

Feng, Y.

Y. Feng, H. Zheng, Z. Zhub, and F. Zua, “The microstructure and electrical conductivity of aluminum alloy foams,” Mater. Chem. Phys. 78, 196–201 (2003).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Gheorghiu, E.

K. Asami, E. Gheorghiu, and T. Yonezawa, “Real-time monitoring of yeast cell division by dielectric spectroscopy,” Biophys. J. 76, 3345–3348 (1999).
[CrossRef]

Hill, S. C.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Jain, P. K.

P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B 110, 7238–7248 (2006).
[CrossRef]

Khoo, I. C.

Lee, K. S.

P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B 110, 7238–7248 (2006).
[CrossRef]

Liang, X.

Orlov, N.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Rother, T.

T. Rother, Electromagnetic Wave Scattering on Nonspherical Particles (Springer, 2009).

Schuh, R.

A. Doicu, R. Schuh, and T. Wriedt, “Scattering by particles on or near a plane surface,” in Light Scattering Reviews 3, A. A. Kokhanovsky, ed. (Springer Praxis Books, 2008), pp. 109–130.

Stefani, F. D.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Sturgess, K.

S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried PEC bodies,” IEEE Trans. Antennas Propag. 44, 143–151 (1996).
[CrossRef]

Takenaka, T.

Taminiau, T. H.

Tanaka, T.

Taubenblatt, M. A.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Tran, T. K.

van Hulst, N. F.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Videen, G.

Vitebskiy, S.

S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried PEC bodies,” IEEE Trans. Antennas Propag. 44, 143–151 (1996).
[CrossRef]

Vlieger, J.

P. A. Bobbert and J. Vlieger, “Light scattering by a sphere on a substrate,” Phys. A 137, 209–242 (1986).
[CrossRef]

Wang, G. L.

G. L. Wang and W. C. Chew, “Formal solution to the electromagnetic scattering by buried dielectric and metallic spheres,” Radio Sci. 39, RS5004 (2004).
[CrossRef]

Werner, D. H.

Weyl, H.

H. Weyl, “Ausbreitung elektromagnetischer Wellen Ãijber einem ebenen Leiter,” Ann. Phys. 365, 481–500 (1919).
[CrossRef]

Wiscombe, W. J.

Wittmann, R. C.

R. C. Wittmann, “Spherical wave operators and the translation formulas,” IEEE Trans. Antennas Propag. 36, 1078–1087 (1988).
[CrossRef]

Wriedt, T.

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

A. Doicu, R. Schuh, and T. Wriedt, “Scattering by particles on or near a plane surface,” in Light Scattering Reviews 3, A. A. Kokhanovsky, ed. (Springer Praxis Books, 2008), pp. 109–130.

Yapar, A.

B. Esen, I. Akkaya, and A. Yapar, “Scattering of a plane wave from a perfectly conducting sphere buried in a conducting dielectric,” Electromagnetics 24, 607–621 (2004).
[CrossRef]

Yonezawa, T.

K. Asami, E. Gheorghiu, and T. Yonezawa, “Real-time monitoring of yeast cell division by dielectric spectroscopy,” Biophys. J. 76, 3345–3348 (1999).
[CrossRef]

Zheng, H.

Y. Feng, H. Zheng, Z. Zhub, and F. Zua, “The microstructure and electrical conductivity of aluminum alloy foams,” Mater. Chem. Phys. 78, 196–201 (2003).
[CrossRef]

Zhou, H.

Zhub, Z.

Y. Feng, H. Zheng, Z. Zhub, and F. Zua, “The microstructure and electrical conductivity of aluminum alloy foams,” Mater. Chem. Phys. 78, 196–201 (2003).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski and A. Erentok, “Metamaterial-based efficient electrically small antennas,” IEEE Trans. Antennas Propag. 54, 2113–2130 (2006).
[CrossRef]

Zua, F.

Y. Feng, H. Zheng, Z. Zhub, and F. Zua, “The microstructure and electrical conductivity of aluminum alloy foams,” Mater. Chem. Phys. 78, 196–201 (2003).
[CrossRef]

Ann. Phys. (1)

H. Weyl, “Ausbreitung elektromagnetischer Wellen Ãijber einem ebenen Leiter,” Ann. Phys. 365, 481–500 (1919).
[CrossRef]

Appl. Opt. (2)

Biophys. J. (1)

K. Asami, E. Gheorghiu, and T. Yonezawa, “Real-time monitoring of yeast cell division by dielectric spectroscopy,” Biophys. J. 76, 3345–3348 (1999).
[CrossRef]

Electromagnetics (1)

B. Esen, I. Akkaya, and A. Yapar, “Scattering of a plane wave from a perfectly conducting sphere buried in a conducting dielectric,” Electromagnetics 24, 607–621 (2004).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

R. C. Wittmann, “Spherical wave operators and the translation formulas,” IEEE Trans. Antennas Propag. 36, 1078–1087 (1988).
[CrossRef]

S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried PEC bodies,” IEEE Trans. Antennas Propag. 44, 143–151 (1996).
[CrossRef]

R. W. Ziolkowski and A. Erentok, “Metamaterial-based efficient electrically small antennas,” IEEE Trans. Antennas Propag. 54, 2113–2130 (2006).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Phys. Chem. B (1)

P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B 110, 7238–7248 (2006).
[CrossRef]

Mater. Chem. Phys. (1)

Y. Feng, H. Zheng, Z. Zhub, and F. Zua, “The microstructure and electrical conductivity of aluminum alloy foams,” Mater. Chem. Phys. 78, 196–201 (2003).
[CrossRef]

Opt. Commun. (1)

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. A (1)

P. A. Bobbert and J. Vlieger, “Light scattering by a sphere on a substrate,” Phys. A 137, 209–242 (1986).
[CrossRef]

Radio Sci. (1)

G. L. Wang and W. C. Chew, “Formal solution to the electromagnetic scattering by buried dielectric and metallic spheres,” Radio Sci. 39, RS5004 (2004).
[CrossRef]

Other (7)

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

T. Rother, Electromagnetic Wave Scattering on Nonspherical Particles (Springer, 2009).

A. Doicu, R. Schuh, and T. Wriedt, “Scattering by particles on or near a plane surface,” in Light Scattering Reviews 3, A. A. Kokhanovsky, ed. (Springer Praxis Books, 2008), pp. 109–130.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

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Figures (7)

Fig. 1.
Fig. 1.

Statement of the problem.

Fig. 2.
Fig. 2.

Elementary plane wave d E s of the scattered field with incident angles d α i and d β i .

Fig. 3.
Fig. 3.

Amplitude of the coefficients e m n in the case of an air sphere with radius 15 cm buried in glass, ε 2 = 2.25 , at a depth of 30 cm, for an incident wave with φ i = 0 at a frequency f = 300 MHz , as a function of the incident angle ϑ i and of the index .

Fig. 4.
Fig. 4.

Amplitude of the coefficients e m n in the same scenario of Fig. 3, as a function of the index , for three different incident angles: ϑ i = 0 (dashed line), ϑ i = π / 4 (solid line), and ϑ i = π / 2 (dashed-dotted line).

Fig. 5.
Fig. 5.

Comparison of the numerical results of our code, simulations performed by the commercial software Comsol and the results obtained in [11,26]. A PEC sphere with radius 15 cm is buried at a depth of 30 cm.

Fig. 6.
Fig. 6.

Comparison of the numerical results obtained with our Matlab code (solid line) and with Comsol simulations (dashed line). A normal incidence is considered, with the electric field linearly polarized along y , on an air-glass interface ( ε 1 = 1 , ε 2 = 2.1054 ), at a wavelength λ 0 = 1 μm . The scatterer is an air sphere of radius a = 200 nm buried at depth h = 5 a . The scattered field is calculated along a line parallel to the x axis at a distance 3 a from the center of the sphere.

Fig. 7.
Fig. 7.

Comparison of numerical results obtained with our Matlab code (solid line) and Comsol simulations (dashed line). An oblique incidence ϑ i = φ i = π / 8 , with the electric field elliptically polarized E ϑ i = 0.5 V / m and E φ i = 0.75 V / m at a wavelength λ 0 = 1 μm , is considered. The scatterer is an NaCl sphere ( ε NaCl = 2.347 ). All the other parameters are the same as in Fig. 6.

Equations (49)

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

E i ( r ) = ( E ϑ i ϑ 0 i + E φ i φ 0 i ) e i k i · r ,
k i = k 1 ( sin ϑ i cos φ i x 0 + sin ϑ i sin φ i y 0 + cos ϑ i z 0 ) ,
ϑ 0 i = cos ϑ i cos φ i x 0 + cos ϑ i sin φ i y 0 sin ϑ i z 0 ,
φ 0 i = sin φ i x 0 + cos φ i y 0 .
E r ( r ) = E r 0 e i k r · r ,
E t ( r ) = E t 0 e i k t · r ,
E r 0 = ( R H 12 E ϑ i ϑ 0 r + R E 12 E φ i φ 0 r ) e i 2 h k 1 cos ϑ i ,
E t 0 = ( T H 12 E ϑ i ϑ 0 t + T E 12 E φ i φ 0 t ) e i h ( k 2 cos ϑ t k 1 cos ϑ i ) ,
M m n ( 1 ) ( k r ) = j n ( k r ) m m n ( ϑ , φ ) ,
N m n ( 1 ) ( k r ) = j n ( k r ) k r p m n ( ϑ , φ ) + 1 k r d [ r j n ( k r ) ] d r n m n ( ϑ , φ ) ,
m m n ( ϑ , φ ) = e i m φ [ i π n m ( cos ϑ ) ϑ 0 τ n m ( cos ϑ ) φ 0 ] ,
n m n ( ϑ , φ ) = e i m φ [ τ n m ( cos ϑ ) ϑ 0 + i π n m ( cos ϑ ) φ 0 ] ,
p m n ( ϑ , φ ) = e i m φ n ( n + 1 ) P n m ( cos ϑ ) r 0 ,
π n m ( cos ϑ ) = m sin ϑ P n m ( cos ϑ ) ,
τ n m ( cos ϑ ) = d P n m ( cos ϑ ) d ϑ .
E t ( r ) = n = 1 m = n n [ a m n M m n ( 1 ) ( k 2 r ) + b m n N m n ( 1 ) ( k 2 r ) ] .
a m n = γ m n E t 0 · m m n * ( ϑ t , φ t ) ,
b m n = i γ m n E t 0 · n m n * ( ϑ t , φ t ) ,
γ m n = ( 1 ) m i n 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) !
E s ( r ) = n = 1 m = n n [ e m n M m n ( 3 ) ( k 2 r ) + f m n N m n ( 3 ) ( k 2 r ) ] ,
E sr ( r ) = n = 1 m = n n i n 2 π ( e m n 0 2 π 0 π 2 i E Re m n e i k r · r d α r d β r + f m n 0 2 π 0 π 2 i E Rf m n e i k r · r d α r d β r )
E R e m n = e i m β i sin α i ( i R H 21 π n m α 0 r R E 21 τ m n β 0 r ) e i 2 k 2 h cos α r ,
E R f m n = e i m β i sin α i ( i R H 21 τ n m α 0 r R E 21 π m n β 0 r ) e i 2 k 2 h cos α r ,
E st ( r ) = n = 1 m = n n i n 2 π ( e m n 0 2 π 0 π 2 i E Te m n e i k t · r d α t d β t + f m n 0 2 π 0 π 2 i E Tf m n e i k t · r d α t d β t ) ,
E Te m n = e i m β i sin α i ( i T H 21 π n m α 0 t T E 21 τ m n β 0 t ) e i 2 k 2 h cos α t ,
E Tf m n = e i m β i sin α i ( i T H 21 τ n m α 0 t T E 21 π m n β 0 t ) e i 2 k 2 h cos α t .
E sr ( r ) = q = 1 p = q q [ M p q ( 1 ) ( r ) n = 1 m = n n ( e m n C p q m n + f m n G p q m n ) + N p q ( 1 ) ( r ) n = 1 m = n n ( e m n D p q m n + f m n H p q m n ) ] ,
C p q m n = i n γ p q δ m p 0 π 2 i ( R H 21 π n m π q p R E 21 τ n m τ q p ) sin α e i 2 h k 2 cos α d α D p q m n = i n γ p q δ m p 0 π 2 i ( R H 21 π n m τ q p + R E 21 τ n m π q p ) sin α e i 2 h k 2 cos α d α G p q m n = i n γ p q δ m p 0 π 2 i ( R H 21 τ n m π q p R E 21 π n m τ q p ) sin α e i 2 h k 2 cos α d α H p q m n = i n γ p q δ m p 0 π 2 i ( R H 21 τ n m τ q p + R E 21 π n m π q p ) sin α e i 2 h k 2 cos α d α ,
( E t + E s + E sr ) | r = a × r 0 = 0 .
m m n ( ϑ , φ ) × r 0 = n m n ( ϑ , φ ) n m n ( ϑ , φ ) × r 0 = m m n ( ϑ , φ ) ,
n = 1 m = n n ( F p q m n e m n + G p q m n f m n ) = a p q n = 1 m = n n ( D p q m n e m n + L p q m n f m n ) = b p q ,
F p q m n = δ m p δ n q h n ( 1 ) ( k 2 a ) j n ( k 2 a ) + C p q m n ,
L p q m n = δ m p δ n q h n ( 1 ) ( k 2 a ) j n ( k 2 a ) + H p q m n ,
e m n = a m n j n ( k 2 a ) h n ( 1 ) ( k 2 a ) ,
f m n = b m n j n ( k 2 a ) h n ( 1 ) ( k 2 a ) .
E sp ( r ) = n = 1 m = n n [ r m n M m n ( 1 ) ( k 3 r ) + s m n N m n ( 1 ) ( k 3 r ) ] .
( E t + E s + E sr E sp ) | r = a × r 0 = 0 ,
[ × ( E t + E s + E sr E sp ) ] | r = a × r 0 = 0 .
[ × M m n ( 1 , 3 ) ( k r ) ] × r 0 = k m m n ( ϑ , φ ) z ˙ n ( 1 , 3 ) ( k r ) ,
[ × N m n ( 1 , 3 ) ( k r ) ] × r 0 = k n m n ( ϑ , φ ) z n ( 1 , 3 ) ( k r ) ,
n = 1 m = n n ( F p q m n e m n + G p q m n f m n R p q m n r m n ) = a p q n = 1 m = n n ( D p q m n e m n + L p q m n f m n S p q m n s m n ) = b p q n = 1 m = n n ( F p q m n e m n + G p q m n f m n S p q m n r m n ) = a p q n = 1 m = n n ( D p q m n e m n + L p q m n f m n R p q m n s m n ) = b p q ,
R p q m n = j n ( k 3 a ) j n ( k 2 a ) δ m p δ n q ,
S p q m n = j n ( k 3 a ) j n ( k 2 a ) δ m p δ n q ,
I ( q ) = 0 π 2 i f ( cos α ) e i 2 q cos α sin α d α ,
I ( q ) = I R + I I ,
I R = 0 π 2 f ( cos α ) sin α e i 2 q cos α d α ,
I I = π 2 π 2 i f ( cos α ) sin α e i 2 q cos α d α .
I I = i 1 0 f ( i u ) e 2 qu d u .
N = k 2 a + 4 ( k 2 a ) 1 / 3 + 2 .

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