Abstract

We study the resonant scattering of light by a transparent dielectric spheroid. We try to understand the features of the resonant modes of a spheroidal optical cavity. In this way, we use an analogy between optics and quantum mechanics. Through this analogy it is possible to interpret resonances as quasi-bound states of light. Using semiclassical methods such as the WKB method and a uniform asymptotic expansion for spheroidal radial functions, we developed algorithms that permit us to calculate the resonance position as well as the resonance width.

© 2002 Optical Society of America

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  3. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. 1. Diffraction and specular reflection,” Appl. Opt 35, 500–514 (1996).
    [CrossRef] [PubMed]
  4. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. 2. Transmission and cross-polarization effects,” Appl. Opt 35, 515–531 (1996).
    [CrossRef] [PubMed]
  5. S. Asano, G. Yamamoto, “Light-scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [CrossRef] [PubMed]
  6. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).
  7. J. A. Stratton, P. M. Morse, L. J. Chu, D. C. Little, F. J. Corbato, Elliptic and Spheroidal Wave Functions (Wiley, New York, 1956).
  8. N. V. Voshchinnikov, V. G. Farafonov, “Optical-properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
    [CrossRef]
  9. S. Asano, “Light-scattering properties of spheroidal particles,” Appl. Opt. 18, 712–723 (1979).
    [CrossRef] [PubMed]
  10. B. P. Sinha, R. H. MacPhie, “On the computation of the prolate spheroidal radial functions of the second kind,” J. Math. Phys. 16, 2378–2381 (1975).
    [CrossRef]
  11. D. B. Hodge, “Eigenvalues and eigenfunctions of spheroidal wave equation,” J. Math. Phys. 11, 2308–2312 (1970).
    [CrossRef]
  12. H. A. Eide, J. J. Stamnes, K. Stamnes, F. M. Schulz, “New method for computing expansion coefficients for spheroidal functions,” J. Quant. Spectrosc. Radiat. Transfer 63, 191–203 (1999).
    [CrossRef]
  13. L. G. Guimarães, “Explicit asymptotic formulas for the spheroidal angular eigenvalues,” J. Phys. A 28, L233–L237 (1995).
    [CrossRef]
  14. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  15. F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
    [CrossRef]
  16. M. I. Mishchenko, “Extinction of light by randomly-oriented nonspherical grains,” Astrophys. Space Sci. 164, 1–13 (1990).
    [CrossRef]
  17. M. I. Mishchenko, “Light-scattering by randomly oriented axially symmetrical particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).
    [CrossRef]
  18. M. I. Mishchenko, “Light-scattering by randomly oriented axially symmetrical particles,” J. Opt. Soc. Am. A 9, 497–497 (1992).
    [CrossRef]
  19. J. J. Goodman, B. T. Draine, P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett. 16, 1198–1200 (1991).
    [CrossRef] [PubMed]
  20. J. I. Hage, J. M. Greenberg, R. T. Wang, “Scattering from arbitrarily shaped particles—theory and experiment,” Opt. Lett. 30, 1141–1152 (1991).
  21. 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, New York, 2000).
    [CrossRef]
  22. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, England, 1992).
    [CrossRef]
  23. P. C. G. de Moraes, “Análise semiclássica do espalhamento da luz por um esferóide,” Master’s thesis (Universidade Federal do Rio de Janeiro, Brazil, 1999).
  24. 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]
  25. 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]
  26. F. W. J. Olver, Asymptotic and Special Functions (Academic, New York, 1974).
  27. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  28. L. G. Guimarães, H. M. Nussenzveig, “Uniform approximation to Mie resonances,” J. Mod. Opt. 41, 625–647 (1994).
    [CrossRef]

1999

H. A. Eide, J. J. Stamnes, K. Stamnes, F. M. Schulz, “New method for computing expansion coefficients for spheroidal functions,” J. Quant. Spectrosc. Radiat. Transfer 63, 191–203 (1999).
[CrossRef]

1998

1996

J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. 1. Diffraction and specular reflection,” Appl. Opt 35, 500–514 (1996).
[CrossRef] [PubMed]

J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. 2. Transmission and cross-polarization effects,” Appl. Opt 35, 515–531 (1996).
[CrossRef] [PubMed]

1995

L. G. Guimarães, “Explicit asymptotic formulas for the spheroidal angular eigenvalues,” J. Phys. A 28, L233–L237 (1995).
[CrossRef]

1994

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]

L. G. Guimarães, H. M. Nussenzveig, “Uniform approximation to Mie resonances,” J. Mod. Opt. 41, 625–647 (1994).
[CrossRef]

1993

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

1992

1991

1990

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]

M. I. Mishchenko, “Extinction of light by randomly-oriented nonspherical grains,” Astrophys. Space Sci. 164, 1–13 (1990).
[CrossRef]

1979

1975

B. P. Sinha, R. H. MacPhie, “On the computation of the prolate spheroidal radial functions of the second kind,” J. Math. Phys. 16, 2378–2381 (1975).
[CrossRef]

S. Asano, G. Yamamoto, “Light-scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
[CrossRef] [PubMed]

1971

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

1970

D. B. Hodge, “Eigenvalues and eigenfunctions of spheroidal wave equation,” J. Math. Phys. 11, 2308–2312 (1970).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

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).

Chu, L. J.

J. A. Stratton, P. M. Morse, L. J. Chu, D. C. Little, F. J. Corbato, Elliptic and Spheroidal Wave Functions (Wiley, New York, 1956).

Corbato, F. J.

J. A. Stratton, P. M. Morse, L. J. Chu, D. C. Little, F. J. Corbato, Elliptic and Spheroidal Wave Functions (Wiley, New York, 1956).

de Moraes, P. C. G.

P. C. G. de Moraes, “Análise semiclássica do espalhamento da luz por um esferóide,” Master’s thesis (Universidade Federal do Rio de Janeiro, Brazil, 1999).

Draine, B. T.

Eide, H. A.

H. A. Eide, J. J. Stamnes, K. Stamnes, F. M. Schulz, “New method for computing expansion coefficients for spheroidal functions,” J. Quant. Spectrosc. Radiat. Transfer 63, 191–203 (1999).
[CrossRef]

Farafonov, V. G.

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

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

Flatau, P. J.

Fuller, K. A.

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, New York, 2000).
[CrossRef]

Goodman, J. J.

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]

Greenberg, J. M.

Guimarães, L. G.

L. G. Guimarães, “Explicit asymptotic formulas for the spheroidal angular eigenvalues,” J. Phys. A 28, L233–L237 (1995).
[CrossRef]

L. G. Guimarães, H. M. Nussenzveig, “Uniform approximation to Mie resonances,” J. Mod. Opt. 41, 625–647 (1994).
[CrossRef]

Hage, J. I.

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]

Hodge, D. B.

D. B. Hodge, “Eigenvalues and eigenfunctions of spheroidal wave equation,” J. Math. Phys. 11, 2308–2312 (1970).
[CrossRef]

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]

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]

Leung, P. T.

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]

Little, D. C.

J. A. Stratton, P. M. Morse, L. J. Chu, D. C. Little, F. J. Corbato, Elliptic and Spheroidal Wave Functions (Wiley, New York, 1956).

Lock, J. A.

J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. 1. Diffraction and specular reflection,” Appl. Opt 35, 500–514 (1996).
[CrossRef] [PubMed]

J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. 2. Transmission and cross-polarization effects,” Appl. Opt 35, 515–531 (1996).
[CrossRef] [PubMed]

Mackowski, D. W.

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, New York, 2000).
[CrossRef]

MacPhie, R. H.

B. P. Sinha, R. H. MacPhie, “On the computation of the prolate spheroidal radial functions of the second kind,” J. Math. Phys. 16, 2378–2381 (1975).
[CrossRef]

Mishchenko, M. I.

Morse, P. M.

J. A. Stratton, P. M. Morse, L. J. Chu, D. C. Little, F. J. Corbato, Elliptic and Spheroidal Wave Functions (Wiley, New York, 1956).

Nussenzveig, H. M.

L. G. Guimarães, H. M. Nussenzveig, “Uniform approximation to Mie resonances,” J. Mod. Opt. 41, 625–647 (1994).
[CrossRef]

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, England, 1992).
[CrossRef]

Olver, F. W. J.

F. W. J. Olver, Asymptotic and Special Functions (Academic, New York, 1974).

Schulz, F. M.

H. A. Eide, J. J. Stamnes, K. Stamnes, F. M. Schulz, “New method for computing expansion coefficients for spheroidal functions,” J. Quant. Spectrosc. Radiat. Transfer 63, 191–203 (1999).
[CrossRef]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[CrossRef]

Sinha, B. P.

B. P. Sinha, R. H. MacPhie, “On the computation of the prolate spheroidal radial functions of the second kind,” J. Math. Phys. 16, 2378–2381 (1975).
[CrossRef]

Stamnes, J. J.

H. A. Eide, J. J. Stamnes, K. Stamnes, F. M. Schulz, “New method for computing expansion coefficients for spheroidal functions,” J. Quant. Spectrosc. Radiat. Transfer 63, 191–203 (1999).
[CrossRef]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[CrossRef]

Stamnes, K.

H. A. Eide, J. J. Stamnes, K. Stamnes, F. M. Schulz, “New method for computing expansion coefficients for spheroidal functions,” J. Quant. Spectrosc. Radiat. Transfer 63, 191–203 (1999).
[CrossRef]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[CrossRef]

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Stratton, J. A.

J. A. Stratton, P. M. Morse, L. J. Chu, D. C. Little, F. J. Corbato, Elliptic and Spheroidal Wave Functions (Wiley, New York, 1956).

van de Hulst, H. C.

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

Voshchinnikov, N. V.

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

Wang, R. T.

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

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

J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. 1. Diffraction and specular reflection,” Appl. Opt 35, 500–514 (1996).
[CrossRef] [PubMed]

J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. 2. Transmission and cross-polarization effects,” Appl. Opt 35, 515–531 (1996).
[CrossRef] [PubMed]

Appl. Opt.

Astrophys. Space Sci.

M. I. Mishchenko, “Extinction of light by randomly-oriented nonspherical grains,” Astrophys. Space Sci. 164, 1–13 (1990).
[CrossRef]

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

J. Math. Phys.

B. P. Sinha, R. H. MacPhie, “On the computation of the prolate spheroidal radial functions of the second kind,” J. Math. Phys. 16, 2378–2381 (1975).
[CrossRef]

D. B. Hodge, “Eigenvalues and eigenfunctions of spheroidal wave equation,” J. Math. Phys. 11, 2308–2312 (1970).
[CrossRef]

J. Mod. Opt.

L. G. Guimarães, H. M. Nussenzveig, “Uniform approximation to Mie resonances,” J. Mod. Opt. 41, 625–647 (1994).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

L. G. Guimarães, “Explicit asymptotic formulas for the spheroidal angular eigenvalues,” J. Phys. A 28, L233–L237 (1995).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

H. A. Eide, J. J. Stamnes, K. Stamnes, F. M. Schulz, “New method for computing expansion coefficients for spheroidal functions,” J. Quant. Spectrosc. Radiat. Transfer 63, 191–203 (1999).
[CrossRef]

Opt. Commun

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.

Phys. Rev. A

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]

Phys. Rev. D

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Other

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

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

F. W. J. Olver, Asymptotic and Special Functions (Academic, New York, 1974).

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

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, New York, 2000).
[CrossRef]

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, England, 1992).
[CrossRef]

P. C. G. de Moraes, “Análise semiclássica do espalhamento da luz por um esferóide,” Master’s thesis (Universidade Federal do Rio de Janeiro, Brazil, 1999).

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

J. A. Stratton, P. M. Morse, L. J. Chu, D. C. Little, F. J. Corbato, Elliptic and Spheroidal Wave Functions (Wiley, New York, 1956).

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Figures (4)

Fig. 1
Fig. 1

Scattering geometry of a transparent prolate spheroid.

Fig. 2
Fig. 2

λ lm level diagram for c = 1.2. The level separations are defined by Δλ l ≡ λ l+1 m - λ l m and δλ m ≡ λ l m+1 - λ l m ; δλ m increases as m increases; and Δλ l ≫ |δλ m |. Note that, for general finite eccentricity values (c ≠ 0), the total angular momentum symmetry is broken and the (2l + 1) spectrum degeneracy is lifted so that the λ l m level diagram is very similar to a band structure of levels.

Fig. 3
Fig. 3

Effective potential U eff.

Fig. 4
Fig. 4

For N = 1.33, l = 65, and TE polarization, shown in a WKB approximation, some resonance features are calculated as c or m varies. In (a), (b) and (c), (d) the behavior of the resonance position and width are shown, respectively. Panels (e) and (f) compare the numerical calculation (dashed curves) of the derivative of the resonance position β with formulas (29) (solid curves).

Tables (1)

Tables Icon

Table 1 Comparison of WKB and UAE Calculations of Resonance Position and Resonance Widtha

Equations (31)

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

2+k2N2ξEH=0,
Nξ=1ξ>ξrNξ<ξr.
ψlm=Slmc, ηRlmc, ξe±imϕ,
δδη1-η2δSlmδη+λlm2c-c2η2-m21-η2×Slm=0,
δδξξ2-1δRlmδξ-λlm2c-c2ξ2+m2ξ2-1×Rlm=0.
lnRβ¯+iηβ¯-Nj ln RNβ¯=0.
NjRNβjRNβj  ηβjηβj
wj  Rβj/ηβj-Rβj/ηβj/NRNβj/RNβj-RNβj/RNβj-ηβj/ηβj-ηβj/ηβj.
Δββc=0=c212β2c=03m2ll+1-1.
Δββc=0=c24β2c=0m2l2-1.
-Fξ+UeffFξ=c2Fξ,
Fξ=Rξξ2-1.
Ueffξ  λlm2-Nξc2ξ2-1+m2-1ξ2-12-Nξ2-1c2.
ΔU  N2-1c21+1+m2/l2/2ξr2.
ξint=1+12l˜2c˜2+l˜4c˜4+4 M2c˜21/2,
ξext=1+12l2c2+l4c4+4 M2c21/2,
1Nl˜2+M2c2l˜21/2<cξr<l2+M2c2l21/2.
pb2πb1+1-m2l2c24l2lλ¯.
Δββ-Δpbpbm2l2-1c24l2.
Φ=n+14π+limδ0tan-11jK˜ξr+δK˜ξr-δ, n=0, 1, , nmax,
wjβξj exp-2Ψlimδ0K˜ξr+δK˜ξr+δ2+jK˜ξr-δ2.
Φ  ξintξr|K˜ξ|dξ,
Ψξrξext|K˜ξ|dξ.
K˜2ξ=c2-Ueffξ-14ξ4+6ξ2-3ξ2-12ξ2.
ΦLNcζNc2-1-arctanζNc2-1-vNc24arctanζNc2-1+ζNc2-1ζNc2,
ΨLcln1+1-ζc2ζc-1-ζc2-vc241-ζc2ζc2+ln1+1-ζc2ζc.
βnLcΞtg-1Ξ+n+14πLNc+tg-1Ξ+33-3+2ϰ3ΞvNc24.
βn-δβn-ρπN2ρ2-1 δ.
Δβ  ρtg-1N2ρ2-1N2ρ2-1×δLNc/δcΔcLNc|m+Δm-LNc|mLNc|l+Δl-LNc|l.
RNβNRt1/4 Ai-tlimδ01K˜ξr-δ1/2, ηβjNητ1/4Biτlimδ01K˜(ξr+δ1/2,
-Njln Ai-tt1/2limδ0K˜ξr-δlnBiττ1/2limδ0K˜ξr+δ,

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