Abstract

The light scattered by a sphere behind a plane surface is solved with an extension of Mie theory. I solve the boundary conditions at the sphere and at the surface simultaneously and develop the scattering amplitude and Mueller scattering matrices. This approach involves the multiplication of the fields in the half-space region not including the sphere by an appropriate Fresnel reflection coefficient and the projection of these fields onto the half-space region including the sphere. The scattered fields from the sphere, reflecting off the surface and interacting with the sphere, are assumed to be incident upon the surface at near-normal incidence. The exact scatter approaches this limit when (1) the sphere is at a large distance from the surface, (2) the sphere radius is small compared with the incident wavelength, or (3) the difference between the refractive indices of the two half-space regions is either small or great.

© 1993 Optical Society of America

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References

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  1. N. A. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
    [Crossref]
  2. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  3. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  4. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  5. C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).
  6. J. H. Brunning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I—Multipole expansion and ray-optical solution,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
    [Crossref]
  7. V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
    [Crossref]
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    [Crossref]
  9. G. O. Olaofe, “Scattering by two cylinders,” Radio Sci. 5, 1351–1360 (1970).
    [Crossref]
  10. H. A. Yousif, S. Köhler, “Scattering by two penetrable cylinders at oblique incidence. I. The analytical solution,” J. Opt. Soc. Am. A 5, 1085–1096 (1988).
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    [Crossref] [PubMed]
  14. P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–241 (1986).
  15. T. C. Rao, R. Barakat, “Plane-wave scattering by a conducting cylinder partially buried in a ground plane. 1. TM case,” J. Opt. Soc. Am. A 6, 1270–1280 (1989).
    [Crossref]
  16. I. V. Lindell, A. H. Sihvola, K. O. Muinonen, P. W. Barber, “Scattering by a small object close to an interface. I. Exact-image theory formulation,” J. Opt. Soc. Am. A 8, 472–476 (1991).
    [Crossref]
  17. K. O. Muinonen, A. H. Sihvola, I. V. Lindell, K. A. Lumme, “Scattering by a small object close to an interface. II. Study of backscattering,” J. Opt. Soc. Am. A 8, 477–482 (1991).
    [Crossref]
  18. A. N. Sommerfeld, “Ueber die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–695 (1909).
  19. A. N. Sommerfeld, “Ueber die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 81, 1135–1153 (1926).
  20. H. Weyl, “Ausbreitung elektromagnetischer Wellen ueber einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
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    [Crossref]
  22. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  23. B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).
  24. S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).
  25. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions.” Q. Appl. Math. 20, 33–40 (1962).
  26. E. K. Miller, A. J. Poggio, G. J. Burke, E. S. Seldon, “Analysis of wire antennas in the presence of a conducting half-space. Part I. The vertical antenna in free space,” Can.J. Phys. 50, 879–888 (1972).
    [Crossref]
  27. E. K. Miller, A. J. Poggio, G. J. Burke, E. S. Seldon, “Analysis of wire antennas in the presence of a conducting half-space. Part II. The horizontal antenna in free space,” J. Phys. Can. 50, 2614–2627 (1972).
    [Crossref]
  28. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  29. J. R. Wait, Electromagnetic Wave Theory (Harper & Row, New York, 1985).

1991 (3)

1989 (1)

1988 (4)

1986 (1)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–241 (1986).

1972 (2)

E. K. Miller, A. J. Poggio, G. J. Burke, E. S. Seldon, “Analysis of wire antennas in the presence of a conducting half-space. Part I. The vertical antenna in free space,” Can.J. Phys. 50, 879–888 (1972).
[Crossref]

E. K. Miller, A. J. Poggio, G. J. Burke, E. S. Seldon, “Analysis of wire antennas in the presence of a conducting half-space. Part II. The horizontal antenna in free space,” J. Phys. Can. 50, 2614–2627 (1972).
[Crossref]

1971 (1)

J. H. Brunning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I—Multipole expansion and ray-optical solution,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[Crossref]

1970 (1)

G. O. Olaofe, “Scattering by two cylinders,” Radio Sci. 5, 1351–1360 (1970).
[Crossref]

1967 (1)

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

1965 (1)

N. A. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
[Crossref]

1962 (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions.” Q. Appl. Math. 20, 33–40 (1962).

1961 (1)

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

1955 (1)

R. V. Row, “Theoretical and experimental study of electromagnetic scattering by two identical conducting cylinders,” J. Appl. Phys. 26, 666–675 (1955).
[Crossref]

1954 (1)

B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).

1952 (1)

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[Crossref]

1926 (1)

A. N. Sommerfeld, “Ueber die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 81, 1135–1153 (1926).

1919 (1)

H. Weyl, “Ausbreitung elektromagnetischer Wellen ueber einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).

1909 (1)

A. N. Sommerfeld, “Ueber die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–695 (1909).

Barakat, R.

Barber, P. W.

Bobbert, P. A.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–241 (1986).

Bohren, C. F.

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

Brunning, J. H.

J. H. Brunning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I—Multipole expansion and ray-optical solution,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[Crossref]

Burke, G. J.

E. K. Miller, A. J. Poggio, G. J. Burke, E. S. Seldon, “Analysis of wire antennas in the presence of a conducting half-space. Part I. The vertical antenna in free space,” Can.J. Phys. 50, 879–888 (1972).
[Crossref]

E. K. Miller, A. J. Poggio, G. J. Burke, E. S. Seldon, “Analysis of wire antennas in the presence of a conducting half-space. Part II. The horizontal antenna in free space,” J. Phys. Can. 50, 2614–2627 (1972).
[Crossref]

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions.” Q. Appl. Math. 20, 33–40 (1962).

Friedman, B.

B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).

Fuller, K. A.

Huffman, D. R.

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

Kattawar, G. W.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Köhler, S.

Liang, C.

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Lindell, I. V.

Lo, Y. T.

J. H. Brunning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I—Multipole expansion and ray-optical solution,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[Crossref]

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Logan, N. A.

N. A. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
[Crossref]

Lumme, K. A.

Miller, E. K.

E. K. Miller, A. J. Poggio, G. J. Burke, E. S. Seldon, “Analysis of wire antennas in the presence of a conducting half-space. Part I. The vertical antenna in free space,” Can.J. Phys. 50, 879–888 (1972).
[Crossref]

E. K. Miller, A. J. Poggio, G. J. Burke, E. S. Seldon, “Analysis of wire antennas in the presence of a conducting half-space. Part II. The horizontal antenna in free space,” J. Phys. Can. 50, 2614–2627 (1972).
[Crossref]

Muinonen, K. O.

Olaofe, G. O.

G. O. Olaofe, “Scattering by two cylinders,” Radio Sci. 5, 1351–1360 (1970).
[Crossref]

Poggio, A. J.

E. K. Miller, A. J. Poggio, G. J. Burke, E. S. Seldon, “Analysis of wire antennas in the presence of a conducting half-space. Part II. The horizontal antenna in free space,” J. Phys. Can. 50, 2614–2627 (1972).
[Crossref]

E. K. Miller, A. J. Poggio, G. J. Burke, E. S. Seldon, “Analysis of wire antennas in the presence of a conducting half-space. Part I. The vertical antenna in free space,” Can.J. Phys. 50, 879–888 (1972).
[Crossref]

Rao, T. C.

Row, R. V.

R. V. Row, “Theoretical and experimental study of electromagnetic scattering by two identical conducting cylinders,” J. Appl. Phys. 26, 666–675 (1955).
[Crossref]

Russek, J.

B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).

Seldon, E. S.

E. K. Miller, A. J. Poggio, G. J. Burke, E. S. Seldon, “Analysis of wire antennas in the presence of a conducting half-space. Part I. The vertical antenna in free space,” Can.J. Phys. 50, 879–888 (1972).
[Crossref]

E. K. Miller, A. J. Poggio, G. J. Burke, E. S. Seldon, “Analysis of wire antennas in the presence of a conducting half-space. Part II. The horizontal antenna in free space,” J. Phys. Can. 50, 2614–2627 (1972).
[Crossref]

Sihvola, A. H.

Sommerfeld, A. N.

A. N. Sommerfeld, “Ueber die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 81, 1135–1153 (1926).

A. N. Sommerfeld, “Ueber die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–695 (1909).

Stein, S.

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

Stratton, J. A.

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

Twersky, V.

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Videen, G.

Vlieger, J.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–241 (1986).

Wait, J. R.

J. R. Wait, Electromagnetic Wave Theory (Harper & Row, New York, 1985).

Weyl, H.

H. Weyl, “Ausbreitung elektromagnetischer Wellen ueber einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).

Yousif, H. A.

Ann. Phys. (Leipzig) (3)

A. N. Sommerfeld, “Ueber die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–695 (1909).

A. N. Sommerfeld, “Ueber die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 81, 1135–1153 (1926).

H. Weyl, “Ausbreitung elektromagnetischer Wellen ueber einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).

Can.J. Phys. (1)

E. K. Miller, A. J. Poggio, G. J. Burke, E. S. Seldon, “Analysis of wire antennas in the presence of a conducting half-space. Part I. The vertical antenna in free space,” Can.J. Phys. 50, 879–888 (1972).
[Crossref]

IEEE Trans. Antennas Propag. (1)

J. H. Brunning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I—Multipole expansion and ray-optical solution,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[Crossref]

J. Acoust. Soc. Am. (1)

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[Crossref]

J. Appl. Phys. (1)

R. V. Row, “Theoretical and experimental study of electromagnetic scattering by two identical conducting cylinders,” J. Appl. Phys. 26, 666–675 (1955).
[Crossref]

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

J. Phys. Can. (1)

E. K. Miller, A. J. Poggio, G. J. Burke, E. S. Seldon, “Analysis of wire antennas in the presence of a conducting half-space. Part II. The horizontal antenna in free space,” J. Phys. Can. 50, 2614–2627 (1972).
[Crossref]

Opt. Lett. (2)

Physica (Utrecht) (1)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–241 (1986).

Proc. IEEE (1)

N. A. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
[Crossref]

Q. Appl. Math. (3)

B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions.” Q. Appl. Math. 20, 33–40 (1962).

Radio Sci. (2)

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

G. O. Olaofe, “Scattering by two cylinders,” Radio Sci. 5, 1351–1360 (1970).
[Crossref]

Other (6)

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

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

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

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

J. R. Wait, Electromagnetic Wave Theory (Harper & Row, New York, 1985).

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Figures (5)

Fig.1
Fig.1

Geometry of the scattering system. A sphere of radius a is located a distance d above a surface. A plane wave travels in the xz plane at angle α1 with respect to the z axis below the surface. It is refracted at the surface and travels at angle α2 above the surface.

Fig.2
Fig.2

The image coordinate surface is located a distance 2d from the sphere coordinate system along the negative z axis. Fields in the image coordinate system are inverted; e.g., the image of the incident plane wave travels in the xz plane at angle πα2 with respect to the z axis.

Fig.3
Fig.3

Light-scattering Mueller matrix elements for an a = 0.5λ spherical air bubble (nsph = 1.0) suspended a distance d = 0.5λ from the plane surface inside a glass substrate (n2 = 1.5), where the developed theory (*) and the approximations of Eqs. (6.2) (○) are used. The angle of incidence is α1 = 30°.

Fig.4
Fig.4

Light-scattering Mueller matrix elements for an a = 0.5λ spherical air bubble (nsph = 1.0) suspended a distance d = 0.5λ from the plane surface inside a glass substrate (n2 = 1.5), where the developed theory (*) and the approximations of Eqs. (6.2) (○) are used. The angle of incidence is α1 = 30°.

Fig.5
Fig.5

Light-scattering Mueller matrix elements for an a = 0.05λ spherical air bubble (nsph = 1.0) suspended a distance d = 0.5λ from the plane surface inside a glass substrate (n2 = 1.5), where the developed theory (*) and the approximations of Eqs. (6.2) (○) are used. The angle of incidence is α1= 30°.

Equations (57)

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2 E μ 2 E t 2 μ σ E t = 0 , 2 H μ 2 H t 2 μ σ H t = 0 ,
1 r 2 r ( r 2 u r ) + 1 r 2 sin θ θ [ sin θ u θ ] + 1 r 2 sin 2 θ 2 u ϕ 2 + k 2 u = 0 .
u ( r , θ , ϕ ) = R ( r ) Θ ( θ ) Φ ( ϕ ) .
Φ ( ϕ ) = exp ( i m ϕ ) , Θ ( θ ) = P n m ( cos θ ) = [ ( 2 n + 1 ) ( n m ) ! 2 ( n + m ) ! ] 1 / 2 P n m ( cos θ ) , R ( r ) = z n ( k r ) = ( π / 2 r ) 1 / 2 Z n + 1 / 2 ( k r ) ,
E = n , m p n m M n m ( i ) + q n m N n m ( i ) , H = k i ω μ n , m p n m N n m ( i ) + q n m M n m ( i ) ,
M n m ( i ) = θ ̂ [ i m sin θ z n ( k r ) P n m ( cos θ ) exp ( i m ϕ ) ] ϕ ̂ { z n ( k r ) d d θ [ P n m ( cos θ ) ] exp ( i m ϕ ) } , N n m ( i ) = r ̂ [ 1 k r z n ( k r ) n ( n + 1 ) P n m ( cos θ ) exp ( i m ϕ ) ] + θ ̂ { 1 k r d d r [ r z n ( k r ) ] d d θ [ P n m ( cos θ ) ] exp ( i m ϕ ) } + ϕ ̂ { 1 k r d d r [ r z n ( k r ) ] i m sin θ P n m ( cos θ ) exp ( i m ϕ ) } ,
E inc = n = 0 m = n n a n m M n m ( 1 ) + b n m N n m ( 1 ) , H inc = k i ω μ n = 0 m = n n b n m M n m ( 1 ) + a n m N n m ( 1 ) .
E sca = n = 0 m = n n e n m M n m ( 3 ) + f n m N n m ( 3 ) , H sca = k i ω μ n = 0 m = n n f n m M n m ( 3 ) + e n m N n m ( 3 ) .
E sph = n = 0 m = n n c n m M n m ( 1 ) + d n m N n m ( 1 ) , H sph = k sph i ω μ n = 0 m = n n d n m M n m ( 1 ) + c n m N n m ( 1 ) .
E int = n = 0 m = n n g n m M n m ( 1 ) + h n m N n m ( 1 ) , H int = k i ω μ n = 0 m = n n h n m M n m ( 1 ) + g n m N n m ( 1 ) .
e n m = ( a n m + g n m ) × k sph μ 2 ψ n ( k sph a ) ψ n ( k 2 a ) k 2 μ sph ψ n ( k 2 a ) ψ n ( k sph a ) k sph μ 2 ψ n ( k sph a ) ξ n ( k 2 a ) k 2 μ sph ξ n ( k 2 a ) ψ n ( k sph a ) = ( a n m + g n m ) Q e n , f n m = ( b n m + h n m ) × k sph μ 2 ψ n ( k 2 a ) ψ n ( k sph a ) k 2 μ sph ψ n ( k sph a ) ψ n ( k 2 a ) k sph μ 2 ξ n ( k 2 a ) ψ n ( k sph a ) k 2 μ sph ψ n ( k sph a ) ξ n ( k 2 a ) = ( b n m + h n m ) Q f n ,
ψ n ( r ) = r j n ( r ) , ξ n ( r ) = r h n ( 1 ) ( r )
P n m [ cos ( π θ ) ] = ( 1 ) n + m P n m ( cos θ ) .
R TE ( i , j ) ( θ i ) = Z j cos θ i + Z i [ 1 ( n i / n j ) 2 sin 2 θ i ] 1 / 2 Z j cos θ i + Z i [ 1 ( n i / n j ) 2 sin 2 θ i ] 1 / 2 , T TE ( i , j ) ( θ i ) = 2 Z j cos θ i Z j cos θ i + Z i [ 1 ( n i / n j ) 2 sin 2 θ i ] 1 / 2 , R TM ( i , j ) ( θ i ) = Z i cos θ i Z j [ 1 ( n i / n j ) 2 sin 2 θ i ] 1 / 2 Z i cos θ i + Z j [ 1 ( n i / n j ) 2 sin 2 θ i ] 1 / 2 , T TM ( i , j ) ( θ i ) = 2 Z j cos θ i Z i cos θ i + Z j [ 1 ( n i / n j ) 2 sin 2 θ i ] 1 / 2 ,
Z i / Z j = μ i k j / μ j k i ,
M n m ( 3 ) = n = | m | C n ( n , m ) M n m ( 1 ) + D n ( n , m ) N n m ( 1 ) ,
N n m ( 3 ) = n = | m | D n ( n , m ) M n m ( 1 ) + C n ( n , m ) N n m ( 1 ) ,
C n ( n , m ) = c n ( n , m ) + 2 k d 2 n + 3 n + m + 1 n + 1 c n + 1 ( n , m ) + 2 k d 2 n 1 n m n c n 1 ( n , m )
D n ( n , m ) = 2 i k d n ( n + 1 ) m c n ( n , m ) .
c n ( 0 , 0 ) = ( 1 ) n ( 2 n + 1 ) 1 / 2 h n ( 1 ) ( 2 k 2 d ) ,
c n ( 1 , 0 ) = ( 1 ) n ( 2 n + 1 ) 1 / 2 h n ( 1 ) ( 2 k 2 d ) ,
[ ( n m + 1 ) ( n + m ) ( 2 n + 1 ) ] 1 / 2 c n ( n , m ) = [ ( n m + 1 ) ( n + m ) ( 2 n + 1 ) ] 1 / 2 c n ( n , m 1 ) + 2 k 2 d [ ( n m + 2 ) ( n m + 1 ) 2 n + 3 ] 1 / 2 c n + 1 ( n , m 1 ) + 2 k 2 d [ ( n + m ) ( n + m 1 ) 2 n 1 ] 1 / 2 c n 1 ( n , m 1 ) ,
c n ( n , m ) = c n ( n , m ) ,
n c n ( n 1 , 0 ) ( 2 n + 1 2 n 1 ) 1 / 2 ( n + 1 ) c n ( n + 1 , 0 ) ( 2 n + 1 2 n + 3 ) 1 / 2 = ( n + 1 ) c n + 1 ( n , 0 ) ( 2 n + 1 2 n + 3 ) 1 / 2 n c n 1 ( n , 0 ) ( 2 n + 1 2 n 1 ) 1 / 2 .
E int = n = 0 m = n n e n m ( 1 ) n + m + 1 R ( 0 ° ) M n m ( 3 ) + f n m ( 1 ) n + m R ( 0 ° ) N n m ( 3 ) = n = 0 m = n n { e n m ( 1 ) n + m + 1 R ( 0 ° ) × [ n = | m | C n ( n , m ) M n m ( 1 ) + D n ( n , m ) N n m ( 1 ) ] + f n m ( 1 ) n + m R ( 0 ° ) [ n = | m | D n ( n , m ) M n m ( 1 ) + C n ( n , m ) N n m ( 1 ) ] } = n = 0 m = n n n = | m | ( 1 ) n + m R ( 0 ° ) × { [ f n m D n ( n , m ) e n m C n ( n , m ) ] M n m ( 1 ) + [ f n m C n ( n , m ) e n m D n ( n , m ) ] N n m ( 1 ) } .
g n m = n = | m | R ( 0 ° ) ( 1 ) n + m [ f n m D n ( n , m ) e n m C n ( n , m ) ] ,
h n m = n = | m | R ( 0 ° ) ( 1 ) n + m [ f n m C n ( n , m ) e n m D n ( n , m ) ] ,
e n m = { a n m + n = | m | R ( 0 ° ) ( 1 ) n + m × [ f n m D n ( n , m ) e n m C n ( n , m ) ] } Q e n ,
f n m = { b n m + n = | m | R ( 0 ° ) ( 1 ) n + m × [ f n m C n ( n , m ) e n m D n ( n , m ) ] } Q f n .
n 1 sin α 1 = n 2 sin α 2 .
E TE inc = T TE ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] × n = 0 m = n n a n m TE M n m ( 1 ) + b n m TE N n m ( 1 ) , H TE inc = k 2 i ω μ 2 T TE ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] × n = 0 m = n n b n m TE M n m ( 1 ) + a n m TE N n m ( 1 ) , E TM inc = T TM ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] × n = 0 m = n n a n m TM M n m ( 1 ) + b n m TM N n m ( 1 ) , H TM inc = k 2 i ω μ 2 T TM ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] × n = 0 m = n n b n m TM M n m ( 1 ) + a n m TM N n m ( 1 ) .
a n m TE = i n n ( n + 1 ) { [ ( n m ) ( n + m + 1 ) ] 1 / 2 P n m + 1 ( cos α 2 ) [ ( n m + 1 ) ( n + m ) ] 1 / 2 P n m 1 ( cos α 2 ) } = 2 i n + 2 n ( n + 1 ) P n m ( cos α 2 ) α 2 ,
b n m TE = i n + 2 ( 2 n + 1 ) n ( n + 1 ) { P n + 1 m 1 ( cos α 2 ) × [ ( n m + 1 ) ( n m + 2 ) ( 2 n + 1 ) ( 2 n + 3 ) ] 1 / 2 + P n + 1 m + 1 ( cos α 2 ) [ ( n + m + 1 ) ( n + m + 2 ) ( 2 n + 1 ) ( 2 n + 3 ) ] 1 / 2 } = 2 i n + 2 n ( n + 1 ) m P n m ( cos α 2 ) sin α 2 ,
a n m TM = i b n m TE ,
b n m TM = i a n m TE .
e n m TE = { T TE ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] a n m TE + R TE ( 2 , 1 ) ( 0 ° ) n = | m | ( 1 ) n + m [ f n m TE D n ( n , m ) e n m TE C n ( n , m ) ] } Q e n ,
e n m TM = { T TM ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] a n m TM + R TM ( 2 , 1 ) ( 0 ° ) n = | m | ( 1 ) n + m [ f n m TM D n ( n , m ) e n m TM C n ( n , m ) ] } Q e n ,
f n m TE = { T TE ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] b n m TE + R TE ( 2 , 1 ) ( 0 ° ) n = | m | ( 1 ) n + m [ f n m TE C n ( n , m ) e n m TE D n ( n , m ) ] } Q f n ,
f n m TM = { T TM ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] b n m TM + R TM ( 2 , 1 ) ( 0 ° ) n = | m | ( 1 ) n + m [ f n m TM C n ( n , m ) e n m TM D n ( n , m ) ] } Q f n .
h n ( 1 ) ( k r ) [ ( i ) n / i k r ] exp ( i k r ) .
n 1 sin ( π θ 1 ) = n 2 sin ( π θ 2 ) .
( E θ sca E ϕ sca ) = exp ( i k r ) i k r [ S 2 S 3 S 4 S 1 ] ( E TM inc E TE inc ) .
S 1 = n = 0 m = n n ( i ) n exp ( i m ϕ ) { [ 1 + R TE ( π θ 2 ) ( 1 ) n + m exp ( 2 i k d cos θ 2 ) ] f n m TM m sin θ 2 P n m ( cos θ 2 ) + [ 1 R TE ( π θ 2 ) ( 1 ) n + m exp ( 2 i k d cos θ 2 ) ] e n m TE θ 2 P n m ( cos θ 2 ) } , S 2 = i n = 0 m = n n ( i ) n exp ( i m ϕ ) { [ 1 R TM ( π θ 2 ) ( 1 ) n + m exp ( 2 i k d cos θ 2 ) ] e n m TM m sin θ 2 P n m ( cos θ 2 ) + [ 1 + R TM ( π θ 2 ) ( 1 ) n + m exp ( 2 i k d cos θ 2 ) ] f n m TM θ 2 P n m ( cos θ 2 ) } , S 3 = i n = 0 m = n n ( i ) n exp ( i m ϕ ) { [ 1 R TM ( π θ 2 ) ( 1 ) n + m exp ( 2 i k d cos θ 2 ) ] e n m TE m sin θ 2 P n m ( cos θ 2 ) + [ 1 + R TM ( π θ 2 ) ( 1 ) n + m exp ( 2 i k d cos θ 2 ) ] f n m TE θ 2 P n m ( cos θ 2 ) } , S 4 = n = 0 m = n n ( i ) n exp ( i m ϕ ) { [ 1 + R TE ( π θ 2 ) ( 1 ) n + m exp ( 2 i k d cos θ 2 ) ] f n m TM m sin θ 2 P n m ( cos θ 2 ) + [ 1 R TE ( π θ 2 ) ( 1 ) n + m exp ( 2 i k d cos θ 2 ) ] e n m TM θ 2 P n m ( cos θ 2 ) } ,
S 1 = n = 0 m = n n ( i ) n exp ( i m ϕ ) exp [ i ( k 2 k 1 ) d cos θ 2 ] T TE ( 2 , 1 ) ( π θ 2 ) [ f n m TM m sin θ 2 P n m ( cos θ 2 ) + e n m TE θ 2 P n m ( cos θ 2 ) ] , S 2 = i n = 0 m = n n ( i ) n exp ( i m ϕ ) exp [ i ( k 2 k 1 ) d cos θ 2 ] T TM ( 2 , 1 ) ( π θ 2 ) [ e n m TM m sin θ 2 P n m ( cos θ 2 ) + f n m TM θ 2 P n m ( cos θ 2 ) ] , S 3 = i n = 0 m = n n ( i ) n exp ( i m ϕ ) exp [ i ( k 2 k 1 ) d cos θ 2 ] T TM ( 2 , 1 ) ( π θ 2 ) [ e n m TE m sin θ 2 P n m ( cos θ 2 ) + f n m TE θ 2 P n m ( cos θ 2 ) ] , S 4 = n = 0 m = n n ( i ) n exp ( i m ϕ ) exp [ i ( k 2 k 1 ) d cos θ 2 ] T TE ( 2 , 1 ) ( π θ 2 ) [ f n m TE m sin θ 2 P n m ( cos θ 2 ) + e n m TM θ 2 P n m ( cos θ 2 ) ] .
e n m ¯ TE = ( 1 ) m e n m TE , e n m ¯ TE = ( 1 ) m + 1 e n m TM , f n m ¯ TE = ( 1 ) m + 1 f n m TE , f n m ¯ TM = ( 1 ) m f n m TM ,
P n m ( x ) = ( 1 ) m P n m ( x ) .
S 11 = 1 2 ( | S 1 | 2 + | S 2 | 2 ) , S 12 = 1 2 ( | S 2 | 2 | S 1 | 2 ) , S 33 = Re ( S 1 S 2 * ) , S 34 = Im ( S 2 S 1 * ) .
R TE ( i , j ) ( θ ) = R TM ( i , j ) ( θ ) = 0 , T TE ( i , j ) ( θ ) = T TM ( i , j ) ( θ ) = 1 ,
e n m TE = a n m TE Q e n , e n m TM = a n m TM Q e n , f n m TE = b n m TE Q f n , f n m TM = b n m TM Q f n .
S 1 = n = 0 m = n n ( i ) n × exp ( i m ϕ ) [ f n m TE m sin θ 2 P n m ( cos θ 2 ) + e n m TE θ 2 P n m ( cos θ 2 ) ] , S 2 = i n = 0 m = n n ( i ) n × exp ( i m ϕ ) [ e n m TM m sin θ 2 P n m ( cos θ 2 ) + f n m TM θ 2 P n m ( cos θ 2 ) ] , S 3 = i n = 0 m = n n ( i ) n × exp ( i m ϕ ) [ e n m TE m sin θ 2 P n m ( cos θ 2 ) + f n m TE θ 2 P n m ( cos θ 2 ) ] , S 4 = n = 0 m = n n ( i ) n × exp ( i m ϕ ) [ f n m TM m sin θ 2 P n m ( cos θ 2 ) + e n m TM θ 2 P n m ( cos θ 2 ) ] ,
e n m TE = T TE ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] a n m TE Q e n , e n m TM = T TM ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] a n m TM Q e n , f n m TE = T TE ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] b n m TE Q f n , f n m TM = T TM ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] b n m TM Q f n .
j n ( ρ ) ρ n 1 1 3 5 ( 2 n + 1 ) , h n ( 1 ) ( ρ ) ρ ( n + 1 ) × 1 3 5 ( 2 n 1 ) .
Q e n 0 , Q f n 2 3 ( μ 2 k sph 2 μ sph k 2 2 μ 2 k sph 2 + 2 μ sph k 2 2 ) ( k 2 a ) 3 .
S 1 = i T TE ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] × [ 1 + R TE ( 2 , 1 ) ( θ 2 ) exp ( i 2 k 2 d cos θ 2 ) ] × ( μ 2 k sph 2 μ sph k 2 2 μ 2 k sph 2 + 2 μ sph k 2 2 ) ( k 2 a ) 3 , S 2 = i T TM ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] × [ cos ( θ 2 α 2 ) + cos ( θ 2 + α 2 ) R TE ( 2 , 1 ) ( θ 2 ) × exp ( i 2 k 2 d cos θ 2 ) ] ( μ 2 k sph 2 μ sph k 2 2 μ 2 k sph 2 + 2 μ sph k 2 2 ) ( k 2 a ) 3 , S 3 = S 4 = 0 .
S 1 = i T TE ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] × T TE ( 2 , 1 ) ( π θ 2 ) exp [ i ( k 2 k 1 ) d cos θ 2 ) ] × ( μ 2 k sph 2 μ sph k 2 2 μ 2 k sph 2 + 2 μ sph k 2 2 ) ( k 2 a ) 3 , S 2 = i T TM ( 1 , 2 ) ( α 1 ) exp [ i ( k 2 k 1 ) d cos α 2 ] × T TM ( 2 , 1 ) ( π θ 2 ) exp [ i ( k 2 k 1 ) d cos θ 2 ) ] × ( μ 2 k sph 2 μ sph k 2 2 μ 2 k sph 2 + 2 μ sph k 2 2 ) ( k 2 a ) 3 cos ( θ 2 α 2 ) , S 3 = S 4 = 0 .
E θ = k 2 2 I l 4 π 2 ω [ 1 R TM ( 2 , 1 ) ( θ 2 ) × exp ( 2 i k 2 d cos θ 2 ) ] exp ( i k 2 r ) i k 2 r sin θ 2 ,
E ϕ = k 2 2 I l 4 π 2 ω [ 1 + R TE ( 2 , 1 ) ( θ 2 ) exp ( 2 i k 2 d cos θ 2 ) ] exp ( i k 2 r ) i k 2 r ,

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