Abstract

We develop a semiclassical theory to explain the rapid ripple fluctuations in the extinction efficiency of light scattering by a transparent prolate spheroid. The theory is based on uniform asymptotic expansion of spheroidal radial functions. We have calculated the extinction efficiency for normal and oblique incidence. Our results suggest that the excitation of resonant electromagnetic modes inside a spheroidal particle is an important factor in the ripple structure. To verify this assumption and based on a Breit–Wigner formula, we develop a method to fit the peaks that appear in the spheroid’s extinction cross section when some scattering parameters vary. In other words, our calculations suggest that narrow resonances are related to ripple fluctuations, whereas broad resonances contribute to extinction cross-sectional background.

© 2005 Optical Society of America

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References

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  1. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [CrossRef] [PubMed]
  2. S. Asano, “Light-scattering properties of spheroidal particles,” Appl. Opt. 18, 712–723 (1979).
    [CrossRef] [PubMed]
  3. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  4. L. G. Guimarães, H. M. Nussenzveig, “Theory of Mie resonances and ripple fluctuations,” Opt. Commun. 89, 363–369 (1992).
    [CrossRef]
  5. H. M. Nussenzveig, “Light tunneling in clouds,” Appl. Opt. 42, 1588–1593 (2003).
    [CrossRef] [PubMed]
  6. V. A. Markel, V. M. Shalaev, “Absorption of light by soot particles in micro-droplets of water,” J. Quant. Spectrosc. Radiat. Transfer 63, 321–339 (1999).
    [CrossRef]
  7. V. A. Markel, V. M. Shalaev, “Absorption of light by soot particles in micro-droplets of water: erratum,” J. Quant. Spectrosc. Radiat. Transfer 66, 591 (2000).
    [CrossRef]
  8. V. A. Markel, “The effects of averaging on the enhancement factor for absorption of light by carbon particles in microdroplets of water,” J. Quant. Spectrosc. Radiat. Transfer 72, 765–774 (2002).
    [CrossRef]
  9. P. Chýlek, G. Videen, D. Ngo, R. G. Pinnick, J. D. Klett, “Effect of black carbon on the optical properties and climate forcing of sulfate aerosols,” J. Geophys. Res. Atmos. 100, 16,325–16,332 (1995).
    [CrossRef]
  10. A. Kokhanovsky, “Optical properties of terrestrial clouds,” Earth Sci. Rev. 64, 189–241 (2004).
    [CrossRef]
  11. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
    [CrossRef] [PubMed]
  12. M. Cai, O. Painter, K. J. Vahala, P. C. Sercel, “Fiber-coupled microsphere laser,” Opt. Lett. 25, 1430–1432 (2000).
    [CrossRef]
  13. P. C. G. de Moraes, L. G. Guimarães, “Semiclassical theory to optical resonant modes of a transparent dielectric spheroidal cavity,” Appl. Opt. 41, 2955–2961 (2002).
    [CrossRef] [PubMed]
  14. H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
    [CrossRef] [PubMed]
  15. M. L. Gorodetsky, V. S. Ilchenko, “High-Q optical whispering gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
    [CrossRef]
  16. N. V. Voshchinnikov, V. G. Farafonov, “Optical-properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
    [CrossRef]
  17. P. T. Leung, S. W. Ng, K. M. Pang, K. M. Lee, “Morphology-dependent resonances in dielectric spheres with many tiny inclusions,” Opt. Lett. 27, 1749–1751 (2002).
    [CrossRef]
  18. M. L. Goldberger, K. M. Watson, Collision Theory (Wiley, New York, 1964).
  19. M. I. Mishchenko, A. A. Lacis, “Morphology-dependent resonances of nearly spherical particles in random orientation,” Appl. Opt. 42, 5551–5556 (2003).
    [CrossRef] [PubMed]
  20. J. A. Lock, “Excitation of morphology-dependent resonances and van de Hulst’s localization principle,” Opt. Lett. 24, 427–429 (1999).
    [CrossRef]
  21. A. G. Simão, L. G. Guimarães, G. Videen, “A comparative study in resonant light scattering between spherical and cylindrical dielectric hosts with a metallic inclusion,” J. Quant. Spectrosc. Radiat. Transfer 70, 777–786 (2001).
    [CrossRef]

2004 (1)

A. Kokhanovsky, “Optical properties of terrestrial clouds,” Earth Sci. Rev. 64, 189–241 (2004).
[CrossRef]

2003 (2)

2002 (4)

V. A. Markel, “The effects of averaging on the enhancement factor for absorption of light by carbon particles in microdroplets of water,” J. Quant. Spectrosc. Radiat. Transfer 72, 765–774 (2002).
[CrossRef]

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[CrossRef] [PubMed]

P. C. G. de Moraes, L. G. Guimarães, “Semiclassical theory to optical resonant modes of a transparent dielectric spheroidal cavity,” Appl. Opt. 41, 2955–2961 (2002).
[CrossRef] [PubMed]

P. T. Leung, S. W. Ng, K. M. Pang, K. M. Lee, “Morphology-dependent resonances in dielectric spheres with many tiny inclusions,” Opt. Lett. 27, 1749–1751 (2002).
[CrossRef]

2001 (1)

A. G. Simão, L. G. Guimarães, G. Videen, “A comparative study in resonant light scattering between spherical and cylindrical dielectric hosts with a metallic inclusion,” J. Quant. Spectrosc. Radiat. Transfer 70, 777–786 (2001).
[CrossRef]

2000 (2)

M. Cai, O. Painter, K. J. Vahala, P. C. Sercel, “Fiber-coupled microsphere laser,” Opt. Lett. 25, 1430–1432 (2000).
[CrossRef]

V. A. Markel, V. M. Shalaev, “Absorption of light by soot particles in micro-droplets of water: erratum,” J. Quant. Spectrosc. Radiat. Transfer 66, 591 (2000).
[CrossRef]

1999 (2)

V. A. Markel, V. M. Shalaev, “Absorption of light by soot particles in micro-droplets of water,” J. Quant. Spectrosc. Radiat. Transfer 63, 321–339 (1999).
[CrossRef]

J. A. Lock, “Excitation of morphology-dependent resonances and van de Hulst’s localization principle,” Opt. Lett. 24, 427–429 (1999).
[CrossRef]

1995 (1)

P. Chýlek, G. Videen, D. Ngo, R. G. Pinnick, J. D. Klett, “Effect of black carbon on the optical properties and climate forcing of sulfate aerosols,” J. Geophys. Res. Atmos. 100, 16,325–16,332 (1995).
[CrossRef]

1994 (1)

M. L. Gorodetsky, V. S. Ilchenko, “High-Q optical whispering gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
[CrossRef]

1993 (1)

N. V. Voshchinnikov, V. G. Farafonov, “Optical-properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

1992 (1)

L. G. Guimarães, H. M. Nussenzveig, “Theory of Mie resonances and ripple fluctuations,” Opt. Commun. 89, 363–369 (1992).
[CrossRef]

1990 (1)

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

1979 (1)

1975 (1)

Asano, S.

Barber, P. W.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

Bohren, C. F.

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

Cai, M.

Chýlek, P.

P. Chýlek, G. Videen, D. Ngo, R. G. Pinnick, J. D. Klett, “Effect of black carbon on the optical properties and climate forcing of sulfate aerosols,” J. Geophys. Res. Atmos. 100, 16,325–16,332 (1995).
[CrossRef]

de Moraes, P. C. G.

Farafonov, V. G.

N. V. Voshchinnikov, V. G. Farafonov, “Optical-properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

Goldberger, M. L.

M. L. Goldberger, K. M. Watson, Collision Theory (Wiley, New York, 1964).

Gorodetsky, M. L.

M. L. Gorodetsky, V. S. Ilchenko, “High-Q optical whispering gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
[CrossRef]

Guimarães, L. G.

P. C. G. de Moraes, L. G. Guimarães, “Semiclassical theory to optical resonant modes of a transparent dielectric spheroidal cavity,” Appl. Opt. 41, 2955–2961 (2002).
[CrossRef] [PubMed]

A. G. Simão, L. G. Guimarães, G. Videen, “A comparative study in resonant light scattering between spherical and cylindrical dielectric hosts with a metallic inclusion,” J. Quant. Spectrosc. Radiat. Transfer 70, 777–786 (2001).
[CrossRef]

L. G. Guimarães, H. M. Nussenzveig, “Theory of Mie resonances and ripple fluctuations,” Opt. Commun. 89, 363–369 (1992).
[CrossRef]

Hill, S. C.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

Huffman, D. R.

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

Ilchenko, V. S.

M. L. Gorodetsky, V. S. Ilchenko, “High-Q optical whispering gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
[CrossRef]

Kippenberg, T. J.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[CrossRef] [PubMed]

Klett, J. D.

P. Chýlek, G. Videen, D. Ngo, R. G. Pinnick, J. D. Klett, “Effect of black carbon on the optical properties and climate forcing of sulfate aerosols,” J. Geophys. Res. Atmos. 100, 16,325–16,332 (1995).
[CrossRef]

Kokhanovsky, A.

A. Kokhanovsky, “Optical properties of terrestrial clouds,” Earth Sci. Rev. 64, 189–241 (2004).
[CrossRef]

Lacis, A. A.

Lai, H. M.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

Lee, K. M.

Leung, P. T.

P. T. Leung, S. W. Ng, K. M. Pang, K. M. Lee, “Morphology-dependent resonances in dielectric spheres with many tiny inclusions,” Opt. Lett. 27, 1749–1751 (2002).
[CrossRef]

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

Lock, J. A.

Markel, V. A.

V. A. Markel, “The effects of averaging on the enhancement factor for absorption of light by carbon particles in microdroplets of water,” J. Quant. Spectrosc. Radiat. Transfer 72, 765–774 (2002).
[CrossRef]

V. A. Markel, V. M. Shalaev, “Absorption of light by soot particles in micro-droplets of water: erratum,” J. Quant. Spectrosc. Radiat. Transfer 66, 591 (2000).
[CrossRef]

V. A. Markel, V. M. Shalaev, “Absorption of light by soot particles in micro-droplets of water,” J. Quant. Spectrosc. Radiat. Transfer 63, 321–339 (1999).
[CrossRef]

Mishchenko, M. I.

Ng, S. W.

Ngo, D.

P. Chýlek, G. Videen, D. Ngo, R. G. Pinnick, J. D. Klett, “Effect of black carbon on the optical properties and climate forcing of sulfate aerosols,” J. Geophys. Res. Atmos. 100, 16,325–16,332 (1995).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, “Light tunneling in clouds,” Appl. Opt. 42, 1588–1593 (2003).
[CrossRef] [PubMed]

L. G. Guimarães, H. M. Nussenzveig, “Theory of Mie resonances and ripple fluctuations,” Opt. Commun. 89, 363–369 (1992).
[CrossRef]

Painter, O.

Pang, K. M.

Pinnick, R. G.

P. Chýlek, G. Videen, D. Ngo, R. G. Pinnick, J. D. Klett, “Effect of black carbon on the optical properties and climate forcing of sulfate aerosols,” J. Geophys. Res. Atmos. 100, 16,325–16,332 (1995).
[CrossRef]

Sercel, P. C.

Shalaev, V. M.

V. A. Markel, V. M. Shalaev, “Absorption of light by soot particles in micro-droplets of water: erratum,” J. Quant. Spectrosc. Radiat. Transfer 66, 591 (2000).
[CrossRef]

V. A. Markel, V. M. Shalaev, “Absorption of light by soot particles in micro-droplets of water,” J. Quant. Spectrosc. Radiat. Transfer 63, 321–339 (1999).
[CrossRef]

Simão, A. G.

A. G. Simão, L. G. Guimarães, G. Videen, “A comparative study in resonant light scattering between spherical and cylindrical dielectric hosts with a metallic inclusion,” J. Quant. Spectrosc. Radiat. Transfer 70, 777–786 (2001).
[CrossRef]

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[CrossRef] [PubMed]

Vahala, K. J.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[CrossRef] [PubMed]

M. Cai, O. Painter, K. J. Vahala, P. C. Sercel, “Fiber-coupled microsphere laser,” Opt. Lett. 25, 1430–1432 (2000).
[CrossRef]

Videen, G.

A. G. Simão, L. G. Guimarães, G. Videen, “A comparative study in resonant light scattering between spherical and cylindrical dielectric hosts with a metallic inclusion,” J. Quant. Spectrosc. Radiat. Transfer 70, 777–786 (2001).
[CrossRef]

P. Chýlek, G. Videen, D. Ngo, R. G. Pinnick, J. D. Klett, “Effect of black carbon on the optical properties and climate forcing of sulfate aerosols,” J. Geophys. Res. Atmos. 100, 16,325–16,332 (1995).
[CrossRef]

Voshchinnikov, N. V.

N. V. Voshchinnikov, V. G. Farafonov, “Optical-properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

Watson, K. M.

M. L. Goldberger, K. M. Watson, Collision Theory (Wiley, New York, 1964).

Yamamoto, G.

Young, K.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

Appl. Opt. (5)

Astrophys. Space Sci. (1)

N. V. Voshchinnikov, V. G. Farafonov, “Optical-properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

Earth Sci. Rev. (1)

A. Kokhanovsky, “Optical properties of terrestrial clouds,” Earth Sci. Rev. 64, 189–241 (2004).
[CrossRef]

J. Geophys. Res. Atmos. (1)

P. Chýlek, G. Videen, D. Ngo, R. G. Pinnick, J. D. Klett, “Effect of black carbon on the optical properties and climate forcing of sulfate aerosols,” J. Geophys. Res. Atmos. 100, 16,325–16,332 (1995).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (4)

V. A. Markel, V. M. Shalaev, “Absorption of light by soot particles in micro-droplets of water,” J. Quant. Spectrosc. Radiat. Transfer 63, 321–339 (1999).
[CrossRef]

V. A. Markel, V. M. Shalaev, “Absorption of light by soot particles in micro-droplets of water: erratum,” J. Quant. Spectrosc. Radiat. Transfer 66, 591 (2000).
[CrossRef]

V. A. Markel, “The effects of averaging on the enhancement factor for absorption of light by carbon particles in microdroplets of water,” J. Quant. Spectrosc. Radiat. Transfer 72, 765–774 (2002).
[CrossRef]

A. G. Simão, L. G. Guimarães, G. Videen, “A comparative study in resonant light scattering between spherical and cylindrical dielectric hosts with a metallic inclusion,” J. Quant. Spectrosc. Radiat. Transfer 70, 777–786 (2001).
[CrossRef]

Nature (1)

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[CrossRef] [PubMed]

Opt. Commun. (2)

L. G. Guimarães, H. M. Nussenzveig, “Theory of Mie resonances and ripple fluctuations,” Opt. Commun. 89, 363–369 (1992).
[CrossRef]

M. L. Gorodetsky, V. S. Ilchenko, “High-Q optical whispering gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

Other (2)

M. L. Goldberger, K. M. Watson, Collision Theory (Wiley, New York, 1964).

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

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Figures (4)

Fig. 1
Fig. 1

Effective potential Ueff, which has the form of a well surrounded by a potential barrier.

Fig. 2
Fig. 2

For three values of c, α = 0, and N = 1.5, (a) the behavior of the ripple fluctuations in Qext and (b) the resonance calculations as size parameter β varies are shown. The resonances are highly sensitive to small variations in the spheroid axis ratio. For instance, in (a), (b), and (c) the normalized difference in axis length, 100 ( a b ) / a = 100 ( 1 1 [ 2 c / β ] 2 ), assumes values in the ranges (0.138%–0.175%), (2.242%–2.818%), and (5.115%–6.430%), respectively.

Fig. 3
Fig. 3

For fixed values of lr = 23, α = 0, and N = 1.5, TE and TM polarization, and c = 0.5, 2.0, 3.0, near a given resonance the ripple fluctuation is well fitted by a BWF (open circles).

Fig. 4
Fig. 4

For values of l = 23, c = 3.0, α = 45°, and N = 1.5, (a) shows the behavior of the ripple fluctuations in Qext as a function of size parameter β and (b) shows the resonance calculations. The left vertical axis is resonant angular momentum mr (squares); the right vertical axis is related to resonance width γm (circules). (c) Overlap of the two consecutive BWFs (solid curve). Notice here that the overlap from left to right is almost complete for the resonant modes related to low partial waves (mr = 1 and mr = 2; filled and open squares) and gets weaker for higher partial waves as shown for (mr = 15 and mr = 16; filled and empty circles) and (mr = 22 and mr = 23; filled and open triangles, respectively).

Tables (1)

Tables Icon

Table 1 Comparison of Elements of the UAE and BWF Methods of Calculating Resonance for Several Values of ca

Equations (23)

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[ 2 + k 2 N 2 ( ξ ) ] [ E H ] = 0 ,
N ( ξ ) = { 1 ξ > ξ r N ξ < ξ r .
ψ l m = S l m ( c , η ) R l m ( c , ξ ) exp ( ± i m ϕ ) ,
Q ext = 4 c 2 [ ( ξ r 2 1 ) ( ξ r 2 cos 2 α ) ] 1 / 2 × R { l = 1 i 1 a l ( 1 ) S 1 l ( c , cos α ) + sin α m = 1 l = m i ( l 1 ) [ k a m l ( 1 ) S m l ( c , cos α ) + i b m l ( 1 ) S m l ( c , cos α ) ] } ,
l = 1 T n l J q l J = 2 l = 1 T n l J ( 1 U n l J ) × [ i l N 1 l 2 ( c ) S 1 l ( c , cos α ) ] .
T n l J { ln [ ( ξ r 2 1 ) 1 / 2 R 1 l ( 3 ) ( c , ξ r ) ] ɛ J ln [ ( ξ r 2 1 ) 1 / 2 R 1 n ( 1 ) ( N c , ξ r ) ] } R 1 l ( 3 ) ( c , ξ r ) × 1 1 d η S 1 l ( c , η ) S 1 n ( N c , η ) ,
U n l J R 1 l ( 4 ) ( c , ξ r ) R 1 l ( 3 ) ( c , ξ r ) { ln [ ( ξ r 2 1 ) 1 / 2 R 1 l ( 4 ) ( c , ξ r ) ] ɛ J ln [ ( ξ r 2 1 ) 1 / 2 R 1 n ( 1 ) ( N c , ξ r ) ] ln [ ( ξ r 2 1 ) 1 / 2 R 1 l ( 3 ) ( c , ξ r ) ] ɛ J ln [ ( ξ r 2 1 ) 1 / 2 R 1 n ( 1 ) ( N c , ξ r ) ] } ,
ln { [ ( β ¯ / c ) 2 ] 1 / 2 1 R m l ( 3 ) ( c , β ¯ / c ) } = ɛ J ln { [ ( β ¯ / c ) 2 ] 1 / 2 1 R m l ( 1 ) ( N c , β ¯ / c ) .
{ [ ( ξ 2 1 ) ] 1 / 2 R m l ( k ) } + U eff ( ξ ) { [ ( ξ 2 1 ) ] 1 / 2 R m l ( k ) } = × c 2 { [ ( ξ 2 1 ) ] 1 / 2 R m l ( k ) } .
U eff ( ξ ) λ l m 2 [ N ( ξ ) c ] [ N ( ξ ) c ] 2 ξ 2 1 + ( m 2 1 ) ( ξ 2 1 ) 2 [ N ( ξ ) 2 1 ] c 2 ,
β r L ( c ) Ξ [ arctan Ξ + π ( n + 1 / 4 ) / L ( N c ) + υ ( N c ) 2 ( arctan Ξ + [ 3 3 3 + 2 ] / 3 Ξ ) / 4 ] ,
γ λ l m ( c ) N ɛ J exp [ 2 L ( c ) ln { L ( c ) / [ c ξ r ] } ,
Δ U ~ ( N 2 1 ) c 2 [ 1 + ( 1 + m 2 / l 2 ) / ( 2 ξ r 2 ) ] .
U l r l r J ( β ) 1 2 i γ ( β β r ) + i γ .
Q ext ( β , α ) ~ 2 + A 0 ( α ) + 2 β 2 m = 1 l r γ m 2 ( β β m ) 2 + γ m 2 A m ( α ) ,
ξ ext ξ r ~ l ξ r 1 8 l ( 1 l + m 2 l 3 ) c 2 .
J l r 0
TE 23 0
TM 23 0
TE 23 0
TM 23 0
TE 23 0
TM 23 0

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