Abstract

The Debye series decomposition of the partial-wave scattering amplitudes of a multilayer sphere is derived. The partial-wave transmission and reflection terms appearing in the Debye series are multiple-scattering amplitudes written in terms of four basic quantities and combined together layer by layer in an identical way. The resulting expressions are then used to calculate the scattered intensity of a spherical Bragg grating covering a dielectric core particle and to analyze a number of new structures appearing in the scattered intensity.

© 2005 Optical Society of America

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  1. K. A. Fuller, D. W. Mackowski, “Electromagnetic scattering by compounded spherical particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, 2000), pp. 225–272.
    [Crossref]
  2. P. Massoli, “Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets,” Appl. Opt. 37, 3227–3235 (1998).
    [Crossref]
  3. A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [Crossref]
  4. M. Kerker, J. P. Kratohvil, E. Matijevic, “Light scattering functions for concentric spheres. Total scattering coefficients, m1= 2.1050, m2= 1.4821,” J. Opt. Soc. Am. 52, 551–561 (1962).
    [Crossref]
  5. R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960–1967 (1985).
    [Crossref] [PubMed]
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  7. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990), pp. 45–53.
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    [Crossref] [PubMed]
  9. D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551–1559 (1990).
    [Crossref] [PubMed]
  10. Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
    [Crossref]
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  13. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, G. Grehan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36, 5188–5198 (1997).
    [Crossref] [PubMed]
  14. Ref. 7, pp. 187–193.
  15. K. A. Fuller, “Scattering of light by coated spheres,” Opt. Lett. 18, 257–259 (1993).
    [Crossref] [PubMed]
  16. D. D. Smith, K. A. Fuller, “Photonic bandgaps in Mie scattering by concentric stratified spheres,” J. Opt. Soc. Am. B 19, 2449–2455 (2002).
    [Crossref]
  17. P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908) [reprinted and translated into English in P. L. Marston, ed., Selected Papers on Geometrical Aspects of Scattering, Vol. 89 of SPIE Milestone Series, SPIE1994), pp. 198–204].
  18. B. van der Pol, H. Bremmer, “Diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).
  19. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
    [Crossref]
  20. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
    [Crossref]
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    [Crossref]
  22. J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677–4690, 4960 (1994).
    [Crossref] [PubMed]
  23. J. A. Lock, C. L. Adler, “Debye-series analysis of the first-order rainbow produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A 14, 1316–1328 (1997).
    [Crossref]
  24. E. A. Hovenac, J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781–795 (1992).
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  26. A. E. Siegman, “Bragg diffraction of a Gaussian beam by a crossed-Gaussian volume grating,” J. Opt. Soc. Am. 67, 545–550 (1977).
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  27. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964), Sec. 10.3, p. 445.
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  29. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), p. 47.
  30. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 111–113.
  31. Ref. 29, pp. 189–195.
  32. Ref. 30, pp. 181–183, 483–489.
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    [Crossref]
  34. Ref. 28, pp. 208–209.

2002 (1)

1998 (1)

1997 (2)

1996 (1)

1994 (2)

1993 (1)

1992 (1)

1991 (1)

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[Crossref]

1990 (1)

1988 (1)

1985 (1)

1981 (1)

1977 (3)

1969 (2)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[Crossref]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[Crossref]

1962 (1)

1951 (1)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[Crossref]

1937 (1)

B. van der Pol, H. Bremmer, “Diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

1908 (1)

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908) [reprinted and translated into English in P. L. Marston, ed., Selected Papers on Geometrical Aspects of Scattering, Vol. 89 of SPIE Milestone Series, SPIE1994), pp. 198–204].

Ackerman, T. P.

Aden, A. L.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[Crossref]

Adler, C. L.

Altenkirch, R. A.

Bhandari, R.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 111–113.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), pp. 51–70.

Bremmer, H.

B. van der Pol, H. Bremmer, “Diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990), pp. 45–53.

Debye, P.

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908) [reprinted and translated into English in P. L. Marston, ed., Selected Papers on Geometrical Aspects of Scattering, Vol. 89 of SPIE Milestone Series, SPIE1994), pp. 198–204].

Fuller, K. A.

D. D. Smith, K. A. Fuller, “Photonic bandgaps in Mie scattering by concentric stratified spheres,” J. Opt. Soc. Am. B 19, 2449–2455 (2002).
[Crossref]

K. A. Fuller, “Scattering of light by coated spheres,” Opt. Lett. 18, 257–259 (1993).
[Crossref] [PubMed]

K. A. Fuller, D. W. Mackowski, “Electromagnetic scattering by compounded spherical particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, 2000), pp. 225–272.
[Crossref]

Gouesbet, G.

Grehan, G.

Guo, L. X.

Hong, C.-S.

Hovenac, E. A.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 111–113.

Jamison, J. M.

J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677–4690, 4960 (1994).
[Crossref] [PubMed]

Johnson, B. R.

Kai, L.

Kerker, M.

M. Kerker, J. P. Kratohvil, E. Matijevic, “Light scattering functions for concentric spheres. Total scattering coefficients, m1= 2.1050, m2= 1.4821,” J. Opt. Soc. Am. 52, 551–561 (1962).
[Crossref]

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[Crossref]

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), p. 47.

Kratohvil, J. P.

Lin, C.-Y.

J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677–4690, 4960 (1994).
[Crossref] [PubMed]

Lock, J. A.

Mackowski, D. W.

D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551–1559 (1990).
[Crossref] [PubMed]

K. A. Fuller, D. W. Mackowski, “Electromagnetic scattering by compounded spherical particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, 2000), pp. 225–272.
[Crossref]

Massoli, P.

Matijevic, E.

Menguc, M. P.

Nussenzveig, H. M.

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236(4), 116–127 (1977).
[Crossref]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[Crossref]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[Crossref]

Ren, K. F.

Siegman, A. E.

Smith, D. D.

Toon, O. B.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), p. 125.

van der Pol, B.

B. van der Pol, H. Bremmer, “Diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Wang, Y. P.

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), pp. 51–70.

Wu, Z. S.

Yariv, A.

Yeh, P.

Appl. Opt. (8)

J. Appl. Phys. (1)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[Crossref]

J. Math. Phys. (2)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[Crossref]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Philos. Mag. (1)

B. van der Pol, H. Bremmer, “Diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Phys. Z. (1)

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908) [reprinted and translated into English in P. L. Marston, ed., Selected Papers on Geometrical Aspects of Scattering, Vol. 89 of SPIE Milestone Series, SPIE1994), pp. 198–204].

Radio Sci. (1)

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[Crossref]

Sci. Am. (1)

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236(4), 116–127 (1977).
[Crossref]

Other (11)

Ref. 28, pp. 208–209.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964), Sec. 10.3, p. 445.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981), p. 125.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), p. 47.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 111–113.

Ref. 29, pp. 189–195.

Ref. 30, pp. 181–183, 483–489.

K. A. Fuller, D. W. Mackowski, “Electromagnetic scattering by compounded spherical particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, 2000), pp. 225–272.
[Crossref]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, 1980), pp. 51–70.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990), pp. 45–53.

Ref. 7, pp. 187–193.

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Figures (4)

Fig. 1
Fig. 1

Scattered intensity as a function of the scattering angle θ for an unpolarized plane wave in air with λ = 0.575 μm incident on a homogeneous sphere of radius 26.3 μm and refractive index 1.50. The solid line is the scattered intensity and the open circles are the diffraction-plus-external reflection intensity.

Fig. 2
Fig. 2

Scattered intensity as a function of the scattering angle θ for an unpolarized plane wave in air with λ = 0.575 μm incident on the homogeneous sphere of Fig. 1 (solid curve) and on a 63-layer grating covering a core particle of radius 20.0 μm and refractive index 1.50 (open circles). Each grating layer is 0.1 μm thick, and the refractive index in successive layers alternates between 1.55 and 1.50.

Fig. 3
Fig. 3

Scattered intensity as a function of the scattering angle θ for an unpolarized plane wave in air with λ = 0.575 μm incident on the 63-layer grating covering the core particle of Fig. 2. Solid curve, scattered intensity; open circles, diffraction-plus-external reflection intensity; filled circles, transmitted intensity; filled diamonds, intensity for transmission following one internal multiple reflection between the layers.

Fig. 4
Fig. 4

Scattered intensity as a function of the scattering angle θ for an unpolarized plane wave in air with (a) λ = 0.50 μm, (b) λ = 0.55 μm, (c) λ = 0.60 μ, and (d) λ = 0.65 μm incident on the homogeneous sphere of Fig. 1 (solid curve) and the 63-layer grating covering the core particle of Fig. 2 (open circles). The filled circles are the multilayer sphere diffraction-plus-external reflection intensity.

Equations (73)

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N 12 , l = n 1 ψ l ( X 12 ) ψ l ( Y 12 ) - n 2 ψ l ( X 12 ) ψ l ( Y 12 ) ,
D 12 , l = n 1 χ l ( X 12 ) ψ l ( Y 12 ) - n 2 χ l ( X 12 ) ψ l ( Y 12 ) ,
P 12 , l = n 1 ψ l ( X 12 ) χ l ( Y 12 ) - n 2 ψ l ( X 12 ) χ l ( Y 12 ) ,
Q 12 , l = n 1 χ l ( X 12 ) χ l ( Y 12 ) - n 2 χ l ( X 12 ) χ l ( Y 12 ) ,
ψ l ( w ) = w j l ( w ) ,
χ l ( w ) = w n l ( w ) ,
X 12 = 2 π n 2 A 1 / λ ,
Y 12 = 2 π n 1 A 1 / λ .
ξ l ( 1 ) ( w ) = w h l ( 1 ) ( w ) ,
ξ l ( 2 ) ( w ) = w h l ( 2 ) ( w ) ,
N 12 , l Q 12 , l - D 12 , l P 12 , l = n 1 n 2 ,
Ψ 1 , l ( r , θ , ϕ ) = l = 1 n 1 { ( i l ) ( 2 l + 1 ) / [ l ( l + 1 ) ] } d 12 , l × ψ l ( n 1 k r ) P l 1 [ cos ( θ ) ] sin ( ϕ ) ,
Ψ 2 , l ( r , θ , ϕ ) = l = 1 n 2 { ( i l ) ( 2 l + 1 ) / [ l ( l + 1 ) ] } [ ψ l ( n 2 k r ) - b 12 , l ξ l ( 1 ) ( n 2 k r ) ] P l 1 [ cos ( θ ) ] sin ( ϕ ) ,
a 12 , l , b 12 , l = N 12 , l / ( N 12 , l + i D 12 , l ) ,
c 12 , l , d 12 , l = - i n 2 / ( N 12 , l + i D 12 , l ) .
N 123 = D 12 N 23 - N 12 P 23 ,
D 123 = D 12 D 23 - N 12 Q 23 ,
P 123 = Q 12 N 23 - P 12 P 23 ,
Q 123 = Q 12 D 23 - P 12 Q 23 ,
N 123 Q 123 - D 123 P 123 = n 1 n 2 2 n 3 ,
a 123 , b 123 = N 123 / ( N 123 + i D 123 ) .
N 12 J + 1 = D 12 J N J , J + 1 - N 12 J P J , J + 1 ,
D 12 J + 1 = D 12 J D J , J + 1 - N 12 J Q J , J + 1 ,
P 12 J + 1 = Q 12 J N J , J + 1 - P 12 J P J , J + 1 ,
Q 12 J + 1 = Q 12 J D J , J + 1 - P 12 J Q J , J + 1 ,
N 12 J + 1 Q 12 J + 1 - D 12 J + 1 P 12 J + 1 = n 1 ( j = 2 J n j 2 ) n J + 1 .
a 12 M + 1 , b 12 M + 1 = N 12 M + 1 / ( N 12 M + 1 + i D 12 M + 1 ) .
N 12345 = D 123 N 345 - N 123 P 345 ,
D 12345 = D 123 D 345 - N 123 Q 345 ,
P 12345 = Q 123 N 345 - P 123 P 345 ,
Q 12345 = Q 123 D 345 - P 123 Q 345 .
Ψ 1 , l ( r , θ , ϕ ) = l = 1 n 1 { ( i l ) ( 2 l + 1 ) / [ l ( l + 1 ) ] } × T 21 , l ξ l ( 2 ) ( n 1 k r ) P l 1 [ cos ( θ ) ] sin ( ϕ ) ,
Ψ 2 , l ( r , θ , ϕ ) = l = 1 n 2 { ( i l ) ( 2 l + 1 ) / [ l ( l + 1 ) ] } [ ξ l ( 2 ) ( n 2 k r ) + R 212 , l ξ l ( 1 ) ( n 2 k r ) ] P l 1 [ cos ( θ ) ] sin ( ϕ )
T 21 , l = - 2 i n 2 / [ ( N 12 , l + Q 12 , l ) + i ( D 12 , l - P 12 , l ) ]
R 212 , l = [ - ( N 12 , l - Q 12 , l ) + i ( D 12 , l + P 12 , l ) ] / [ ( N 12 , l + Q 12 , l ) + i ( D 12 , l - P 12 , l ) ]
Ψ 1 , l ( r , θ , ϕ ) = l = 1 n 1 { ( i l ) ( 2 l + 1 ) / [ l ( l + 1 ) ] } [ ξ l ( 1 ) ( n 1 k r ) + R 121 , l ξ l ( 2 ) ( n 1 k r ) ] P l 1 [ cos ( θ ) ] sin ( ϕ ) ,
Ψ 2 , l ( r , θ , ϕ ) = l = 1 n 2 { ( i l ) ( 2 l + 1 ) / [ l ( l + 1 ) ] } × T 12 , l ξ l ( 1 ) ( n 2 k r ) P l 1 [ cos ( θ ) ] sin ( ϕ ) .
T 12 , l = - 2 i n 1 / [ ( N 12 , l + Q 12 , l ) + i ( D 12 , l - P 12 , l ) ]
R 121 , l = [ - ( N 12 , l - Q 12 , l ) - i ( D 12 , l + P 12 , l ) ] / [ ( N 12 , l + Q 12 , l ) + i ( D 12 , l - P 12 , l ) ]
a 12 , l , b 12 , l = ( ½ ) [ 1 - R 212 , l - T 21 , l T 12 , l / ( 1 - R 121 , l ) ] ,
c 12 , l , d 12 , l = T 21 , l / ( 1 - R 121 , l ) .
T 31 = 2 i n 2 n 3 / [ ( N 123 + Q 123 ) + i ( D 123 - P 123 ) ] ,
T 13 = 2 i n 1 n 2 / [ ( N 123 + Q 123 ) + i ( D 123 - P 123 ) ] ,
R 3 γ 3 = [ - ( N 123 - Q 123 ) + i ( D 123 + P 123 ) ] / [ ( N 123 + Q 123 ) + i ( D 123 - P 123 ) ] ,
R 1 γ 1 = [ - ( N 123 - Q 123 ) - i ( D 123 + P 123 ) ] / [ ( N 123 + Q 123 ) + i ( D 123 - P 123 ) ]
a 123 , b 123 = ( ½ ) [ 1 - R 3 γ 3 - T 31 T 13 / ( 1 - R 1 γ 1 ) ] ,
c 123 , d 123 = T 31 / ( 1 - R 1 γ 1 ) .
T 31 = T 32 T 21 / ( 1 - R 212 R 232 ) ,
T 13 = T 12 T 23 / ( 1 - R 212 R 232 ) ,
R 3 γ 3 = R 323 + T 32 R 212 T 23 / ( 1 - R 212 R 232 ) ,
R 1 γ 1 = R 121 + T 12 R 232 T 21 / ( 1 - R 212 R 232 ) ,
T M + 1 , 1 = 2 i n 2 n 3 n M + 1 / [ ( N 12 M + 1 + Q 12 M + 1 ) + i ( D 12 M + 1 - P 12 M + 1 ) ] ,
T 1 , M + 1 = 2 i n 1 n 2 n M / [ ( N 12 M + 1 + Q 12 M + 1 ) + i ( D 12 M + 1 - P 12 M + 1 ) ]
R M + 1 , Γ , M + 1 = - ( N 12 M + 1 - Q 12 M + 1 ) + i ( D 12 M + 1 + P 12 M + 1 ) ( N 12 M + 1 + Q 12 M + 1 ) + i ( D 12 M + 1 - P 12 M + 1 ) ,
R 1 , Γ , 1 = - ( N 12 M + 1 - Q 12 M + 1 ) - i ( D 12 M + 1 + P 12 M + 1 ) ( N 12 M + 1 + Q 12 M + 1 ) + i ( D 12 M + 1 - P 12 M + 1 )
a 12 M + 1 , b 12 M + 1 = ( ½ ) [ 1 - R M + 1 , Γ , M + 1 - T M + 1 , 1 T 1 , M + 1 / ( 1 - R 1 , Γ , 1 ) ] .
T 51 = T 53 T 31 / ( 1 - R 3 α 3 R 3 β 3 ) ,
T 15 = T 13 T 35 / ( 1 - R 3 α 3 R 3 β 3 ) ,
R 5 Γ 5 = R 5 β 5 + T 53 R 3 α 3 T 35 / ( 1 - R 3 α 3 R 3 β 3 ) ,
R 1 Γ 1 = R 1 α 1 + T 13 R 3 β 3 T 31 / ( 1 - R 3 α 3 R 3 β 3 ) ,
Ψ 1 , l ( r , θ , ϕ ) = l = 1 n 1 { ( i l ) ( 2 l + 1 ) / [ l ( l + 1 ) ] } [ ξ l ( 1 ) ( n 1 k r ) + R 121 , l ψ l ( n 1 k r ) ] P l 1 [ cos ( θ ) ] sin ( ϕ ) ,
Ψ 1 , l ( r , θ , ϕ ) = l = 1 n 2 { ( i l ) ( 2 l + 1 ) / [ l ( l + 1 ) ] } × T 12 , l ξ l ( 1 ) ( n 2 k r ) P l 1 [ cos ( θ ) ] sin ( ϕ ) .
T 12 , l = - i n 1 / ( N 12 , l + i D 12 , l ) ,
R 121 , l = [ - ( N 12 , l - Q 12 , l ) - i ( D 12 , l + P 12 , l ) ] / ( N 12 , l + i D 12 , l )
a 123 = a 23 + a 12 c 23 T 23 / ( 1 + a 12 R 232 ) ,
b 123 = b 23 + b 12 d 23 T 23 / ( 1 + b 12 R 232 )
λ r = 4 n ave Δ cos ( θ t ) ,
sin ( θ i ) = n ave sin ( θ t )
L ref ( θ ) = ( π A 1 / λ ) 2 ( R 212 ray , TE + R 212 ray , TM ) / 2 ,
R 212 ray , TE = [ n 2 cos ( θ i ) - n 1 cos ( θ t ) ] 2 / [ n 2 cos ( θ i ) + n 1 cos ( θ t ) ] 2 ,
R 212 ray , TM = [ n 1 cos ( θ i ) - n 2 cos ( θ i ) ] 2 / [ n 1 cos ( θ i ) + n 1 cos ( θ t ) ] 2 ,
n 2 sin ( θ i ) = n 1 sin ( θ t ) ,
θ = π - 2 θ i .

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