Abstract

This work deals with some aspects of the resonant scattering of electromagnetic waves by a metallic sphere covered by a dielectric layer, in the weak-absorption approximation. We carry out a geometrical optics treatment of the scattering and develop semiclassical formulas to determine the positions and widths of the system resonances. In addition, we show that the mean lifetime of broad resonances is strongly dependent on the polarization of the incident light.

© 2003 Optical Society of America

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References

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  1. H. Scharfman, “Scattering from dielectric coated spheres in the region of the 1st resonance,” J. Appl. Phys. 25, 1352–1356 (1954).
    [CrossRef]
  2. V. H. Weston, R. Hemenger, “High-frequency scattering from a coated sphere,” J. Res. Natl. Bur. Stand. 66D, 613–619 (1962).
  3. J. Rheinstein, “Scattering of electromagnetic waves from dielectric coated conducting spheres,” IEEE Trans. Antennas Propag. AP-12, 334–340 (1964).
    [CrossRef]
  4. E. L. Murphy, “Reduction of electromagnetic backscatter from a plasma-clad conducting body,” J. Appl. Phys. 36, 1918–1927 (1965).
    [CrossRef]
  5. A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from 2 concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  6. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  7. H. M. Nussenzweig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, UK, 1992).
  8. L. G. Guimarães, H. M. Nussenzveig, “Theory of Mie resonances and ripple fluctuations,” Opt. Commun. 89, 363–369 (1992).
    [CrossRef]
  9. L. G. Guimarães, H. M. Nussenzveig, “Uniform approximation to Mie resonances,” J. Mod. Opt. 41, 625–647 (1994).
    [CrossRef]
  10. L. G. Guimarães, “Theory of Mie caustics,” Opt. Commun. 103, 339–344 (1993).
    [CrossRef]
  11. T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a coated sphere—investigation of the volume-averaged source function and resonance positions,” Appl. Opt. 33, 7789–7797 (1994).
    [CrossRef] [PubMed]
  12. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10, 343–352 (1993).
    [CrossRef]
  13. T. M. Bambino, L. G. Guimarães, “Resonances of a coated sphere,” Phys. Rev. E 53, 2859–2863 (1996).
    [CrossRef]
  14. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  15. S. C. Hill, R. E. Benner, Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).
  16. S. C. Hill, R. K. Chang, Nonlinear Optics in Droplets (Nova Science, New York, 1995).
  17. 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]
  18. M. Cai, O. Painter, K. J. Vahala, P. C. Sercel, “Fiber-coupled microsphere laser,” Opt. Lett. 25, 1430–1432 (2000).
    [CrossRef]
  19. J. A. Lock, J. M. Jamison, C. Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677–4690 (1994).
    [CrossRef] [PubMed]
  20. G. Roll, T. Kaiser, S. Lange, G. Schweiger, “Ray interpretation of multipole fields in spherical dielectric cavities,” J. Opt. Soc. Am. A 15, 2879–2891 (1998).
    [CrossRef]
  21. C. J. Joachain, Quantum Collision Theory (North-Holland, Amsterdam, 1987).
  22. J. R. Taylor, Scattering Theory: The Quantum Theory on Nonrelativistic Collisions (Wiley, New York, 1972).
  23. G. Videen, J. Li, P. Chýlek, “Resonances and poles of weakly absorbing spheres,” J. Opt. Soc. Am. A 12, 916–921 (1995).
    [CrossRef]
  24. E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
    [CrossRef]
  25. M. L. Goldberger, K. M. Watson, Collision Theory (Wiley, New York, 1964).
  26. H. M. Nussenzveig, “Time delay in electromagnetic scattering,” Phys. Rev. A 55, 1012–1019 (1997).
    [CrossRef]
  27. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1968).
  28. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  29. J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. 9, 24–75 (1960).
    [CrossRef]
  30. R. W. Robinett, “Periodic orbit theory analysis of the circular disk or annular billiard: nonclassical effects and the distribution of energy eigenvalues,” Am. J. Phys. 67, 67–77 (1999).
    [CrossRef]
  31. 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, 16325–16332 (1995).
    [CrossRef]

2002 (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]

2000 (1)

1999 (1)

R. W. Robinett, “Periodic orbit theory analysis of the circular disk or annular billiard: nonclassical effects and the distribution of energy eigenvalues,” Am. J. Phys. 67, 67–77 (1999).
[CrossRef]

1998 (1)

1997 (1)

H. M. Nussenzveig, “Time delay in electromagnetic scattering,” Phys. Rev. A 55, 1012–1019 (1997).
[CrossRef]

1996 (1)

T. M. Bambino, L. G. Guimarães, “Resonances of a coated sphere,” Phys. Rev. E 53, 2859–2863 (1996).
[CrossRef]

1995 (2)

G. Videen, J. Li, P. Chýlek, “Resonances and poles of weakly absorbing spheres,” J. Opt. Soc. Am. A 12, 916–921 (1995).
[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, 16325–16332 (1995).
[CrossRef]

1994 (3)

1993 (2)

1992 (1)

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

1965 (1)

E. L. Murphy, “Reduction of electromagnetic backscatter from a plasma-clad conducting body,” J. Appl. Phys. 36, 1918–1927 (1965).
[CrossRef]

1964 (1)

J. Rheinstein, “Scattering of electromagnetic waves from dielectric coated conducting spheres,” IEEE Trans. Antennas Propag. AP-12, 334–340 (1964).
[CrossRef]

1962 (1)

V. H. Weston, R. Hemenger, “High-frequency scattering from a coated sphere,” J. Res. Natl. Bur. Stand. 66D, 613–619 (1962).

1960 (1)

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. 9, 24–75 (1960).
[CrossRef]

1955 (1)

E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
[CrossRef]

1954 (1)

H. Scharfman, “Scattering from dielectric coated spheres in the region of the 1st resonance,” J. Appl. Phys. 25, 1352–1356 (1954).
[CrossRef]

1951 (1)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from 2 concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Aden, A. L.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from 2 concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Bambino, T. M.

T. M. Bambino, L. G. Guimarães, “Resonances of a coated sphere,” Phys. Rev. E 53, 2859–2863 (1996).
[CrossRef]

Benner, R. E.

S. C. Hill, R. E. Benner, Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).

Bohren, C. F.

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

Cai, M.

Chang, R. K.

S. C. Hill, R. K. Chang, Nonlinear Optics in Droplets (Nova Science, New York, 1995).

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, 16325–16332 (1995).
[CrossRef]

G. Videen, J. Li, P. Chýlek, “Resonances and poles of weakly absorbing spheres,” J. Opt. Soc. Am. A 12, 916–921 (1995).
[CrossRef]

Goldberger, M. L.

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

Guimarães, L. G.

T. M. Bambino, L. G. Guimarães, “Resonances of a coated sphere,” Phys. Rev. E 53, 2859–2863 (1996).
[CrossRef]

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

L. G. Guimarães, “Theory of Mie caustics,” Opt. Commun. 103, 339–344 (1993).
[CrossRef]

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

Hemenger, R.

V. H. Weston, R. Hemenger, “High-frequency scattering from a coated sphere,” J. Res. Natl. Bur. Stand. 66D, 613–619 (1962).

Hill, S. C.

S. C. Hill, R. E. Benner, Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).

S. C. Hill, R. K. Chang, Nonlinear Optics in Droplets (Nova Science, New York, 1995).

Huffman, D. R.

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

Jamison, J. M.

Joachain, C. J.

C. J. Joachain, Quantum Collision Theory (North-Holland, Amsterdam, 1987).

Johnson, B. R.

Kaiser, T.

Keller, J. B.

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. 9, 24–75 (1960).
[CrossRef]

Kerker, M.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from 2 concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

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

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, 16325–16332 (1995).
[CrossRef]

Lange, S.

Li, J.

Lin, C. Y.

Lock, J. A.

Murphy, E. L.

E. L. Murphy, “Reduction of electromagnetic backscatter from a plasma-clad conducting body,” J. Appl. Phys. 36, 1918–1927 (1965).
[CrossRef]

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, 16325–16332 (1995).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, “Time delay in electromagnetic scattering,” Phys. Rev. A 55, 1012–1019 (1997).
[CrossRef]

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

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

Nussenzweig, H. M.

H. M. Nussenzweig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, UK, 1992).

Painter, O.

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, 16325–16332 (1995).
[CrossRef]

Rheinstein, J.

J. Rheinstein, “Scattering of electromagnetic waves from dielectric coated conducting spheres,” IEEE Trans. Antennas Propag. AP-12, 334–340 (1964).
[CrossRef]

Robinett, R. W.

R. W. Robinett, “Periodic orbit theory analysis of the circular disk or annular billiard: nonclassical effects and the distribution of energy eigenvalues,” Am. J. Phys. 67, 67–77 (1999).
[CrossRef]

Roll, G.

Rubinow, S. I.

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. 9, 24–75 (1960).
[CrossRef]

Scharfman, H.

H. Scharfman, “Scattering from dielectric coated spheres in the region of the 1st resonance,” J. Appl. Phys. 25, 1352–1356 (1954).
[CrossRef]

Schweiger, G.

Sercel, P. C.

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]

Taylor, J. R.

J. R. Taylor, Scattering Theory: The Quantum Theory on Nonrelativistic Collisions (Wiley, New York, 1972).

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]

van de Hulst, H. C.

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

Videen, 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, 16325–16332 (1995).
[CrossRef]

G. Videen, J. Li, P. Chýlek, “Resonances and poles of weakly absorbing spheres,” J. Opt. Soc. Am. A 12, 916–921 (1995).
[CrossRef]

Watson, K. M.

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

Weston, V. H.

V. H. Weston, R. Hemenger, “High-frequency scattering from a coated sphere,” J. Res. Natl. Bur. Stand. 66D, 613–619 (1962).

Wigner, E. P.

E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
[CrossRef]

Am. J. Phys. (1)

R. W. Robinett, “Periodic orbit theory analysis of the circular disk or annular billiard: nonclassical effects and the distribution of energy eigenvalues,” Am. J. Phys. 67, 67–77 (1999).
[CrossRef]

Ann. Phys. (1)

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. 9, 24–75 (1960).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

J. Rheinstein, “Scattering of electromagnetic waves from dielectric coated conducting spheres,” IEEE Trans. Antennas Propag. AP-12, 334–340 (1964).
[CrossRef]

J. Appl. Phys. (3)

E. L. Murphy, “Reduction of electromagnetic backscatter from a plasma-clad conducting body,” J. Appl. Phys. 36, 1918–1927 (1965).
[CrossRef]

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from 2 concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

H. Scharfman, “Scattering from dielectric coated spheres in the region of the 1st resonance,” J. Appl. Phys. 25, 1352–1356 (1954).
[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, 16325–16332 (1995).
[CrossRef]

J. Mod. Opt. (1)

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

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

J. Res. Natl. Bur. Stand. (1)

V. H. Weston, R. Hemenger, “High-frequency scattering from a coated sphere,” J. Res. Natl. Bur. Stand. 66D, 613–619 (1962).

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]

L. G. Guimarães, “Theory of Mie caustics,” Opt. Commun. 103, 339–344 (1993).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev. 98, 145–147 (1955).
[CrossRef]

Phys. Rev. A (1)

H. M. Nussenzveig, “Time delay in electromagnetic scattering,” Phys. Rev. A 55, 1012–1019 (1997).
[CrossRef]

Phys. Rev. E (1)

T. M. Bambino, L. G. Guimarães, “Resonances of a coated sphere,” Phys. Rev. E 53, 2859–2863 (1996).
[CrossRef]

Other (10)

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

S. C. Hill, R. E. Benner, Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).

S. C. Hill, R. K. Chang, Nonlinear Optics in Droplets (Nova Science, New York, 1995).

C. J. Joachain, Quantum Collision Theory (North-Holland, Amsterdam, 1987).

J. R. Taylor, Scattering Theory: The Quantum Theory on Nonrelativistic Collisions (Wiley, New York, 1972).

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

H. M. Nussenzweig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, UK, 1992).

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1968).

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

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

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Figures (7)

Fig. 1
Fig. 1

Behavior of the Ueff potential as a function of radial distance r for a given impact parameter pλ/k and two distinct sizes of metallic core. Resonances located below or above the barrier top are such that b<(λ/kbelow)<Nb or a<(λ/kabove)<b, respectively. (a) Mie-like modes are excited if a<b/N; (b) for a thinner dielectric layer with a>b/N, the Fabry–Perot-like modes can propagate. In the limit of weak absorption N>1m, the barrier height behaves as ΔU=k2(N2-1-m2). In addition, all narrow resonances located below the top of the barrier are excited by tunneling.

Fig. 2
Fig. 2

Polygon trajectories of the rays in the interior of the dielectric spherical layer for a given impact parameter p. (a) In the Mie-like mode case, there are no reflections in the metallic sphere, and the trajectories are regular polygons whose size is the chord length AB¯. (b) In the Fabry–Perot-like case, the reflections in the metallic sphere are important, and the allowed trajectories are covered with star polygons where the optical path connecting points A and B is VAB.

Fig. 3
Fig. 3

Behavior of the resonance position β when the ratio λ/β varies, for N=1.33, TE and TM polarizations, n=0, 1 ,, 6, and ρ=1.1, N, 1.6. β is the real part of the complex zeros of Eq. (15). For high values of λ the related deeper Mie-like resonance tends to βλ/N.

Fig. 4
Fig. 4

Behavior of the resonance width Γ/2 when the ratio λ/β varies, for N=1.33, TE and TM polarizations, n=0, 1,, 6, and ρ=1.1, N, 1.6. Γ/2 is the imaginary part of the complex zeros of Eq. (15). For TE polarization the vertical axis is a logarithm scale, whereas for polarization TM polarization the vertical axis is linear. Note that the narrow resonances have a behavior similar to exponential decay, and the broad ones seem to tend to a fixed value. Moreover, for TM polarization the Γ/2 width presents a peak in the limit λ/βsin θB, where θB is Brewster’s incidence angle.

Fig. 5
Fig. 5

Exact numerical results obtained by solving transcendental Eq. (15) with semiclassical estimate (27) for the resonance width. The position at which the peak of Γ/2 occurs is in accordance with the position predicted by semiclassical formulas (27) and (28).

Fig. 6
Fig. 6

Relative absorption effects in the resonance position and width for narrow resonance. In the left-hand graph, the right-hand-side vertical axis shows β˙, the values of the derivative of β with respect to m. For NR modes, the resonance position β and the width Γ/2 vary quadratically and linearly, respectively, with m, in accordance with semiclassical statements (42) and (43).

Fig. 7
Fig. 7

Absorption effects in broad resonances. Note that the resonance position β seems to vary linearly with m as much as the width Γ/2 does, in accordance with semiclassical statements (42) and (43).

Equations (51)

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

Qext=l=1(Qext,l(1)+Qext,l(2)),
Qext,l(j)=2β2(2l+1)Re(al(j)),j=1, 2.
al(j)=12(1-Sl(j)).
Sl(j)=Shs,lSpot,l(j),
Shs,l=-ζl(2)(β)ζl(1)(β),
Spot,l(j)=-R22,l(j)+T21,l(j)r11,l(j)T12,l(j)ζl(1)(α)/ζl(2)(α)1-ρl(j),
R22,l(j)=T21,l(j)-1,
T21,l(j)=ln ζl(1)(β)-ln ζl(2)(β)ln ζl(1)(β)-Nj ln ζl(2)(α),
R11,l(j)=T12,l(j)-1,
T12,l(j)=Njln ζl(1)(α)-ln ζl(2)(α)ln ζl(1)(β)-Nj ln ζl(2)(α),
r11,l(j)=-ζl(2)(γ)ζl(1)(γ)δj1-ζl(2)(γ)ζl(1)(γ)δj2,
ρl(j)=r11,l(j)R11,l(j)ζl(1)(α)ζl(2)(α),
j=δj1/N˜2+δj2.
Spot,l(j)(1-mMl2(β))exp[2iδpot,l(j)]+O(m2),
Spot,l(j)(β)1-iΓBW(β-βr)+iΓBW/2+O(m),
cos(2δpot,l(j))|β=βr-1+O(m2).
ΓBW2δpot,l(j)β-11+mMl2(β)4β=βr+O(m2).
4ΓBW-i ln[Spot,l(j)]ββ=βr.
ρl(j)(β)=1.
-d2Fdr2+Ueff(r)F=k2F,
Ueff(r)=λ2r2-k2(N2-m2-1)-2ik2mN.
tan[Φj(β, λ, ρ)]Ξ(β, λ)jα2-λ2,
Φj(β, λ, ρ)ϕ(α, λ)-π4-H(γ-λ)ϕ(γ, λ)-π212-δj,2
Ξ(β, λ)H(λ-β)λ2-β2{1-i exp[-2Ψ(β, λ)]}-iH(β-λ)β2-λ2.
ϕ(kN˜r, λ)=λ/kNrk2-Ueff(r)dr=(kN˜r)2-λ2-λ arccosλkN˜r,
Ψ(β, λ)=bλ/kUeff(r)-k2dr=λ ln[(λ+λ2-β2)/β]-λ2-β2.
Φj(βp, λ, ρ)(n+1/2)π+H(λ-βp)×[arctan Πj-π/2];n=0, 1,,
Γp(β)Φjβ-1H(λ-β)sin 2Φj exp(-2Ψ)-H(β-λ)ln1+Πj1-Πj,
Δj(β, λ, ρ)2 exp(2Ψ)H(λ-β)Πj-tan ΦjΠj+tan Φj-H(β-λ)tan ΦjΠj.
ΓBW(β)2δpot,l(j)β-1Γp(β)2H(λ-β)1-Φjβ-1sin(2Φj)λ2(1+N2)2(λ2-β2)(α2-λ2)-1+H(β-λ)ΠjΦjβ-1.
[H(λ-β)(tan Φj-Πj)+H(β-λ)tan Φj]0,
Γp(β, λ, ρ)Φjβ-1[H(λ-β)T12(j)/2-H(β-λ)ln(|R11,l(j)|)].
R11,l(j)jα2-λ2-β2-λ2jα2-λ2+β2-λ2;
Γp(β, λ, ρ)sec θtNH(λ-β)T12(j)/2+H(β-λ)p=1[T12(j)]pp.
ψAB¯exp[i(knAB¯-π/2)],
ΞA͡Bexp(iλΩ).
kNAB¯-π/2=2nπ+λΩ,n=0, 1, 2,,
AB¯=2b cos θt
Ω=2π2-θt.
n+14π=α2-λ2-λ arccosλα.
Ω=2 arccosbasin2 θt+cos θt1-b2a2sin2 θt1/2=2arccosλα-arccosλγ,
ψ VABexpikNVAB.
VAB=2bα(α2-λ2-γ2-λ2).
α2-λ2-arccosλα-γ2-λ2+arccosλγ=nπ.
kN-2ππ=λΩ,
τN2 sin Ωπbc[2H(λ-β)/T12(j)-H(β-λ)/ln(|R11,l(j)|)].
[Φj(β, λ, ρ)-(n+1/2)π-H(λ-β)(arctan Πj-π/2)]m=0=H(β-λ)ejmNΠj1-Πj2m=0+O(m2),
ΓΦjβ-1H(λ-β)sin 2Φj exp(-2Ψ)-H(β-λ)ln1+Πj1-Πjm=0+H(λ-β)×Φjβ-1sin 2Φjm=0×ejmN+2βmN+O(m2),
βn(λ, ρ)λ[arctanN2-1-H(N-ρ)arctan(N2/ρ2-1)]+(n+1/4)πN2-1-H(N-ρ)N2/ρ2-1.
βλN+N2-1jN21+1.0887λ2/3.
βn(λ, ρ1)λN2+N2-1N2(ρ-1)+2N2-32N2N2-1(n+1/4)π.

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