Abstract

A generic perturbation theory for the morphology-dependent resonances (MDR’s) of dielectric spheres is developed based on the dyadic formulation of a completeness relation established previously [J. Opt. Soc. Am. B 16, 1409 (1999)]. Unlike other perturbation methods proposed previously, the formulation presented here takes full account of the vector nature of MDR’s and hence does not limit its validity to perturbations that preserve spherical symmetry. However, the second-order frequency correction obtained directly from the theory, which is expressed as a sum of contributions from individual MDR’s, converges slowly. An efficient scheme, based on the dyadic form of the completeness relation, is thus constructed to accelerate the rate of convergence. As an example illustrating our theory, we apply the perturbation method to study MDR’s of a dielectric sphere that contains another smaller spherical inclusion and compare the results with those obtained from an exact diagonalization method.

© 1999 Optical Society of America

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References

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  1. See, e.g., M. Kerker, ed., Selected Papers on Light Scattering, Proc. SPIE951, (1988), and references therein.
  2. P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).
  3. R. K. Chang and A. J. Campillo, eds., Optical Processes in Microcavities (World Scientific, Singapore, 1996).
  4. R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
    [Crossref]
  5. J. B. Snow, S.-X. Qian, and R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985).
    [Crossref] [PubMed]
  6. J.-Z. Zhang and R. K. Chang, “Generation and suppression of stimulated Brillouin scattering in single liquid droplets,” J. Opt. Soc. Am. B 6, 151–153 (1989).
    [Crossref]
  7. H. M. Tzeng, K. F. Wall, M. B. Long, and R. K. Chang, “Laser emission from individual droplets at wavelengths corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499–501 (1984).
    [Crossref] [PubMed]
  8. L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
    [Crossref]
  9. See, e.g., P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).
  10. M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964).
  11. L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
    [Crossref]
  12. D. Brady, G. Papen, and J. E. Sipe, “Spherical distributed dielectric resonators,” J. Opt. Soc. Am. B 10, 644–657 (1993).
    [Crossref]
  13. F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327–1335 (1992); R. L. Armstrong, J.-G. Xie, T. Ruekgauer, J. Gu, and R. G. Pinnick, “Effects of submicrometer-sized particles on microdroplet lasing,” Opt. Lett. 18, 119–121 (1993).
    [Crossref] [PubMed]
  14. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
    [Crossref] [PubMed]
  15. K. M. Lee, P. T. Leung, and K. M. Pang, “Dyadic formulation of morphology-dependent resonances. I. Completeness relation,” J. Opt. Soc. Am. B 16, 1409–1417 (1999).
    [Crossref]
  16. We consider MDR’s of stable optical systems in the present paper; hence the imaginary part of the complex wave number of a MDR is always less than zero.
  17. H. M. Lai, P. T. Leung, K. Young, S. C. Hill, and P. W. Barber, “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]
  18. H. M. Lai, C. C. Lam, P. T. Leung, and K. Young, “The effect of perturbations on the widths of narrow morphology-dependent resonances in Mie scattering,” J. Opt. Soc. Am. B 8, 1962–1973 (1991).
    [Crossref]
  19. P. T. Leung and K. M. Pang, “Completeness and time-independent perturbation of morphology-dependent resonances in dielectric spheres,” J. Opt. Soc. Am. B 13, 805–817 (1996).
    [Crossref]
  20. K. M. Lee, P. T. Leung, and K. M. Pang, “Iterative perturbation scheme for morphology-dependent resonances in dielectric spheres,” J. Opt. Soc. Am. A 15, 1383–1393 (1998).
    [Crossref]
  21. C.-T. Tai, Dyadic Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, New York, 1993).
  22. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, New York, 1995).
  23. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  24. M. M. Mazumder, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in inhomogeneous spheres: comparison of layered T-matrix and time-independent perturbation method,” J. Opt. Soc. Am. A 9, 1844–1853 (1992).
    [Crossref]
  25. A. L. Fetter and J. D. Walecka, Quantum Theory of Many Body Systems (McGraw-Hill, New York, 1971).
  26. In the present paper we assume that either the MDR’s of the unperturbed system are nondegenerate or the perturbation does not couple degenerate MDR’s by symmetry arguments. Otherwise a degenerate perturbation theory would have to be formulated, which is out of place here.
  27. S. C. Hill, H. I. Saleheen, and K. A. Fuller, “Volume current method for modeling light scattering by inhomogeneously perturbed spheres,” J. Opt. Soc. Am. A 12, 905–915 (1995).
    [Crossref]
  28. We notice that Qlml′m′ of Eq. (A12) is nonzero only if m≠ 0. Thus we concentrate on the m≠0 cases here.
  29. By mode order we mean, for TE modes, that the mode with the smallest positive Re(ωa) is of mode order 1, the second smallest is of mode order 2, and so on. The TM modes are similarly defined, except that the mode on the imaginary frequency axis is of mode order 0.
  30. See, e.g., D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols (World Scientific, Singapore, 1988).
  31. K. M. Pang, “Completeness and perturbation of morphology-dependent resonances in dielectric spheres,” Ph.D. thesis dissertation (Chinese University of Hong Kong, Hong Kong, 1999).

1999 (1)

1998 (1)

1996 (2)

1995 (1)

1993 (2)

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[Crossref]

D. Brady, G. Papen, and J. E. Sipe, “Spherical distributed dielectric resonators,” J. Opt. Soc. Am. B 10, 644–657 (1993).
[Crossref]

1992 (2)

1991 (1)

1990 (1)

H. M. Lai, P. T. Leung, K. Young, S. C. Hill, and P. W. Barber, “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]

1989 (1)

1985 (2)

1984 (1)

1980 (1)

R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[Crossref]

Arnold, S.

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[Crossref]

Barber, P. W.

M. M. Mazumder, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in inhomogeneous spheres: comparison of layered T-matrix and time-independent perturbation method,” J. Opt. Soc. Am. A 9, 1844–1853 (1992).
[Crossref]

H. M. Lai, P. T. Leung, K. Young, S. C. Hill, and P. W. Barber, “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]

R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[Crossref]

See, e.g., P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Benner, R. E.

R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[Crossref]

Borghese, F.

Brady, D.

Brune, M.

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[Crossref]

Chang, R. K.

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, New York, 1995).

Collot, L.

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[Crossref]

Denti, P.

Druger, D.

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[Crossref]

Fetter, A. L.

A. L. Fetter and J. D. Walecka, Quantum Theory of Many Body Systems (McGraw-Hill, New York, 1971).

Folan, L. M.

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[Crossref]

Fuller, K. A.

Goldberger, M. L.

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

Gorodetsky, M. L.

Haroche, S.

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[Crossref]

Hill, S. C.

S. C. Hill, H. I. Saleheen, and K. A. Fuller, “Volume current method for modeling light scattering by inhomogeneously perturbed spheres,” J. Opt. Soc. Am. A 12, 905–915 (1995).
[Crossref]

M. M. Mazumder, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in inhomogeneous spheres: comparison of layered T-matrix and time-independent perturbation method,” J. Opt. Soc. Am. A 9, 1844–1853 (1992).
[Crossref]

H. M. Lai, P. T. Leung, K. Young, S. C. Hill, and P. W. Barber, “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]

See, e.g., P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Ilchenko, V. S.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Khersonskii, V. K.

See, e.g., D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols (World Scientific, Singapore, 1988).

Lai, H. M.

H. M. Lai, C. C. Lam, P. T. Leung, and K. Young, “The effect of perturbations on the widths of narrow morphology-dependent resonances in Mie scattering,” J. Opt. Soc. Am. B 8, 1962–1973 (1991).
[Crossref]

H. M. Lai, P. T. Leung, K. Young, S. C. Hill, and P. W. Barber, “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]

Lam, C. C.

Lee, K. M.

Lefevre-Seguin, V.

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[Crossref]

Leung, P. T.

Long, M. B.

Mazumder, M. M.

Moskalev, A. N.

See, e.g., D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols (World Scientific, Singapore, 1988).

Owen, J. F.

R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[Crossref]

Pang, K. M.

Papen, G.

Qian, S.-X.

Raimond, J. M.

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[Crossref]

Saija, R.

Saleheen, H. I.

Savchenkov, A. A.

Sindoni, O. I.

Sipe, J. E.

Snow, J. B.

Tai, C.-T.

C.-T. Tai, Dyadic Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, New York, 1993).

Tzeng, H. M.

Varshalovich, D. A.

See, e.g., D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols (World Scientific, Singapore, 1988).

Walecka, J. D.

A. L. Fetter and J. D. Walecka, Quantum Theory of Many Body Systems (McGraw-Hill, New York, 1971).

Wall, K. F.

Watson, K. M.

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

Young, K.

H. M. Lai, C. C. Lam, P. T. Leung, and K. Young, “The effect of perturbations on the widths of narrow morphology-dependent resonances in Mie scattering,” J. Opt. Soc. Am. B 8, 1962–1973 (1991).
[Crossref]

H. M. Lai, P. T. Leung, K. Young, S. C. Hill, and P. W. Barber, “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]

Zhang, J.-Z.

Chem. Phys. Lett. (1)

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[Crossref]

Europhys. Lett. (1)

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[Crossref]

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (5)

Opt. Lett. (3)

Phys. Rev. A (1)

H. M. Lai, P. T. Leung, K. Young, S. C. Hill, and P. W. Barber, “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. Lett. (1)

R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[Crossref]

Other (15)

See, e.g., M. Kerker, ed., Selected Papers on Light Scattering, Proc. SPIE951, (1988), and references therein.

P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).

R. K. Chang and A. J. Campillo, eds., Optical Processes in Microcavities (World Scientific, Singapore, 1996).

See, e.g., P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

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

We consider MDR’s of stable optical systems in the present paper; hence the imaginary part of the complex wave number of a MDR is always less than zero.

A. L. Fetter and J. D. Walecka, Quantum Theory of Many Body Systems (McGraw-Hill, New York, 1971).

In the present paper we assume that either the MDR’s of the unperturbed system are nondegenerate or the perturbation does not couple degenerate MDR’s by symmetry arguments. Otherwise a degenerate perturbation theory would have to be formulated, which is out of place here.

We notice that Qlml′m′ of Eq. (A12) is nonzero only if m≠ 0. Thus we concentrate on the m≠0 cases here.

By mode order we mean, for TE modes, that the mode with the smallest positive Re(ωa) is of mode order 1, the second smallest is of mode order 2, and so on. The TM modes are similarly defined, except that the mode on the imaginary frequency axis is of mode order 0.

See, e.g., D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols (World Scientific, Singapore, 1988).

K. M. Pang, “Completeness and perturbation of morphology-dependent resonances in dielectric spheres,” Ph.D. thesis dissertation (Chinese University of Hong Kong, Hong Kong, 1999).

C.-T. Tai, Dyadic Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, New York, 1993).

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, New York, 1995).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

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Figures (6)

Fig. 1
Fig. 1

Dielectric sphere containing a spherical inclusion. The radii of the host sphere and of the inclusion are, respectively, a and b, and the separation between their centers O and O, respectively, is d.

Fig. 2
Fig. 2

Plot of the MDR’s of a uniform dielectric sphere with an inclusion (nI=1.5, nII=1.43, b=0.5a, and d=0.45a). These are the l=10 and m=2 modes. The filled circles show the MDR’s of the unperturbed system, the centers of the open circles are the exact positions, and the crosses are the results of the second-order perturbation theory.

Fig. 3
Fig. 3

Errors in the MDR frequencies as calculated by the perturbation theory plotted against mode order. The parameters of the system are the same as in Fig. 2. Circles, the differences between the unperturbed frequencies and the exact numerical results; squares and diamonds, errors of the first-order and the second-order perturbation results, respectively.

Fig. 4
Fig. 4

Effect of the refractive index of the inclusion nII on the frequencies of the MDR’s. We consider the TE and the TM leading modes. nII varies from 1.4 to 1.6 with step size 0.1. Other parameters are the same as in Fig. 2. Notice that, if we decrease nII, the real part of the frequency increases and the imaginary part becomes more negative.

Fig. 5
Fig. 5

Similar to Fig. 4, but we now vary the size of inclusion b from 0.05a to 0.5a with step size 0.05. The refractive index of the inclusion is restored to nII=1.43. We see that the larger the inclusion, the more negative the imaginary part of the frequency becomes.

Fig. 6
Fig. 6

We now vary the separation d from 0 to 0.45a with step size 0.05a; b=0.5a. As in the previous cases, a larger separation makes the real part of the frequency larger and the imaginary part more negative.

Equations (92)

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×1(r)×b-ω2b=0,
×1(r)×G¯tb(r, r; ω)-ω2G¯tb(r, r; ω)
=I¯t(r, r).
×1(0)(r)×D¯tb(r, r; ω)-ω2D¯tb(r, r; ω)
=I¯t(r, r),
D¯tb(r, r; ω)=α-bα(0)(r)bα(0)(r)2ωα(0)(ω-ωα(0)),
δρ(r)1(0)(r)-1(r).
G¯tb(r, r; ω)=D¯tb(r, r; ω)+δG¯tb(r, r; ω).
δG¯tb(r,r; ω)=s<ad3sD¯tb(r, s; ω)·×[δρ(s)×G¯tb(s, r; ω)].
G¯tb=D¯tb+D¯tb·ΔD¯tb+D¯tb·ΔD¯tb·ΔD¯tb+ ,
[D¯tb·ΔD¯tb](r, r; ω)=s<ad3sD¯tb(r, s; ω)·×[δρ(s)D¯tb(s, r; ω)].
G¯tb(r, r; ω)=αβbα(0)(r)Gαβbβ(0)(r).
G=D+DΔD+DΔDΔD+ ,
Dαβ=-121ωα(0)[ω-ωα(0)]δαβ,
Δαβ=s<ad3sbα(0)(s)·×[δρ(s)×bβ(0)(s)].
Wαβ{γ}(ω)Δαβ+ζγΔαζDζζΔζβ+ζγξγΔαζDζζΔζξDξξΔξβ+ .
Gαβ=Dαβ+DααWαβ{γ}Dββ+DααWαγ{γ}Dγγn=0(Wγγ{γ}Dγγ)nWγβ{γ}Dββ=Dαβ+DααWαβ{γ}Dββ-DααWαγ{γ}Wγβ{γ}Dββ2ωγ(0)[ω-ωγ(0)]+Wγγ{γ}.
Fγ(ω)-2ωγ(0)[ω-ωγ(0)]-Wγγ{γ}(ω).
ωγ=ωγ(0)+s=1μsωγ(s).
ωγ(1)=-ωγ(0)2V2γγ,
ωγ(2)=ωγ(0)4βγV2γβ ωβ(0)[ωγ(0)-ωβ(0)]V2βγ,
ωγ(3)=-ωγ(0)8αγβγ V2γαV2αβV2βγωα(0)ωβ(0)[ωγ(0)-ωα(0)][ωγ(0)-ωβ(0)]-V2γγαγ V2γαV2αγωα(0)ωγ(0)[ωγ(0)-ωα(0)]2,
V2γβ=1ωγ(0)ωβ(0)s<ad3sbγ(0)(s)·×δρ(s)×bβ(0)(s)=s<ad3sδρ(s)-iωγ(0)×bγ(0)(s)·iωβ(0)×bβ(0)(s)=s<ad3sδρ(s)(0)eγ(0)(s)·(0)eβ(0)(s),
ωβ(0)ωγ(0)-ωβ(0)[ωγ(0)]2ωβ(0)[ωγ(0)-ωβ(0)]-ωγ(0)ωβ(0)-1,
ωγ(2)=ωγ(0)42V2γγ2+βγ [ωγ(0)]2V2γβV2βγωβ(0)[ωγ(0)-ωβ(0)]-β ωγ(0)ωβ(0)V2γβV2βγ-βV2γβV2βγ.
β eβ(r)eβ(r)ωβ=0,r, r<a,
ωγ(2)=ωγ(0)42V2γγ2+βγ [ωγ(0)]2V2γβV2βγωβ(0)[ωγ(0)-ωβ(0)]-βV2γβV2βγ.
βV2γβV2βγ=s<ad3ss<ad3sδρ(s)(0)(s)2eγ(0)(s)·βeβ(0)(s)eβ(0)(s)·(0)(s)2×δρ(s)eγ(0)(s).
βV2γβV2βγ=2d3s(0)(s)Λ2γ(s)·Λ2γ(s)-2d3s(0)(s)Λ2γL(s)·Λ2γL(s).
V1γβs<adsδ(s)eγ(0)(s)·eβ(0)(s).
ωγ(1)=-ωγ(0)2V1γγ,
ωγ(2)=ωγ(0)42V1γγ2+βγ [ωγ(0)]2V1γβV1βγωβ(0)[ωγ(0)-ωβ(0)]+2d3s(0)(s)Λ1γL(s)·Λ1γL(s),
d3s(0)(s)Λ1γL(s)·Λ1γL(s)=4πd3s[ρI(s)]Ψ1γ(s),
ϱrjl(nIωr)d[rjl(nIωr)]drr=a=1rhl(1)(ωr)d[rhl(1)(ωr)]drr=a,
b1jlm=1N1jlω1j×[jl(nIω1jlr)Xlm],
b2jlm=1N2jljl(nIω2jlr)Xlm,
N1jl2=(nI2-1)(a3/2)jl2(nIω1jla),
N2jl2=1-1nI2jl(nIω2jla)jl(nIω2jla)+1nIω2jla2+l2+l(ω2jla)2 a32jl2(nIω2jla).
eIII=lmAlmIIIhl(nIIIωr)Xlm(Ω)+BlmIIInIII2×[hl(nIIIωr)Xlm(Ω)],
eII=lmAlmIIjl(nIIωr)Xlm(Ω)+BlmIInII2×[jl(nIIωr)Xlm(Ω)],
eI|r=a-=lmAlmIIIhl(nIIIωa)Xlm(Ω)+BlmIIInIII2×[hl(nIIIωa)Xlm(Ω)]+nIII2-nI2nIII2nI2BlmIII ia[l(l+1)]1/2×hl(nIIIωa)Ylmrˆ,
×eI|r=a-=lm{AlmIII×[hl(nIIIωa)Xlm(Ω)]+ω2BlmIIIhl(nIIIωa)Xlm(Ω)},
eI|r=b+=lmAlmIIjl(nIIωb)Xlm(Ω)+BlmIInII2×[jl(nIIωb)Xlm(Ω)]+nII2-nI2nII2nI2BlmII ib[l(l+1)]1/2×jl(nIIωb)Ylmrˆ,
×eI|r=b+=lm{AlmII×[jl(nIIωb)Xlm(Ω)]+ω2BlmIIjl(nIIωb)Xlm(Ω)}.
Vd3r[(××P)·Q¯-P·××Q¯]
=Sd2s·[P××Q¯+(×P)×Q¯]
××D¯e(x, y; λ)-λ2D¯e(x, y; λ)=I¯(x, y)
D¯e=iλlmjl(λx)Xlm(Ωx)hl(λy)Xlm*(Ωy)-rˆrˆλ2δ3(x-y)+1λ2×[jl(λx)Xlm(Ωx)]×[hl(λy)Xlm*(Ωy)]y>x,
D¯e=iλimhl(λx)Xlm(Ωx)jl(λy)Xlm*(Ωy)-rˆrˆλ2δ3(x-y)+1λ2×[hl(λx)Xlm(Ωx)]×[jl(λy)Xlm*(Ωy)]y<x,
eI(y)
=-r=ad2srˆ·[eI××D¯e+(×eI)×D¯e]+r=bd2srˆ·[eI××D¯e+(×eI)×D¯e],
eI(y)=lm{AlmIIIClIIIjl(nIωy)Xlm(Ωy)+BlmIIIDlIII×[jl(nIωy)Xlm(Ωy)]}+lm{AlmIIClIIhl(nIωy)Xlm(Ωy)+BlmIIDlII×[hl(nIωy)Xlm(Ωy)]},
ClIII=inIωa[nIIIωahl(nIωa)hl(nIIIωa)-nIωahl(nIIIωa)hl(nIωa)],
DlIII=inIωahl(nIωa)hl(nIIIωa)1nIII2-1nI2+ωanIIIhl(nIωa)hl(nIIIωa)-ωanIhl(nIIIωa)hl(nIωa),
ClII=inIωb[nIωbjl(nIIωb)jl(nIωb)-nIIωbjl(nIωb)jl(nIIωb)],
DlII=inIωbjl(nIωb)jl(nIIωb)1nI2-1nII2+ωbnIjl(nIIωb)jl(nIωb)-ωbnIIjl(nIωb)jl(nIIωb).
hl(nIωx)Xlm(Ω)
=lm{Plmlmhl(nIωx)Xlm(Ω)+Qlmlm×[hl(nIωx)Xlm(Ω)]},
×[hl(nIωx)Xlm(Ω)]
=lm{Plmlm×[hl(nIωx)Xlm(Ω)]+nI2ω2Qlmlmhl(nIωx)×Xlm(Ω)},
Plmlm=(-1)mil-lδmm2(2l+1)(2l+1)l(l+1)l(l+1)1/2×l(-i)l[l(l+1)+l(l+1)-l(l+1)](2l+1)lll000×lllm-m0jl(nIωd),
Qlmlm=(-1)l+l+m+1il-l+1×dmδmm(2l+1)(2l+1)l(l+1)l(l+1)1/2×lil(2l+1)lll000×lllm-m0jl(nIωd).
lll000
jl(nIωx)Xlm(Ω)=lmPlmlmjl(nIωx)Xlm(Ω)+lm(-1)l+l+1Qlmlm×[jl(nIωx)Xlm(Ω)],
×[jl(nIωx)Xlm(Ω)]=lmPlmlm×[jl(nIωx)Xlm(Ω)]+lm(-1)l+l+1nI2ω2×Qlmlmjl(nIωx)Xlm(Ω).
AlmIIIhl(nIIIωa)=AlmIIIClIIIjl(nIωa)+lm[AlmIIClIIPlmlmhl(nIωa)+BlmIIDlIInI2ω2Qlmlmhl(nIωa)],
BlmIII hl(nIIIωa)nI2=BlmIIIDlIIIjl(nIωa)+lm[AlmIIClIIQlmlmhl(nIωa)+BlmIIDlIIPlmlmhl(nIωa)],
AlmIIjl(nIIωb)=AlmIIClIIhl(nIωb)+lm[AlmIIIClIIIPlmlmjl(nIωb)+BlmIIIDlIII(-1)l+l+1nI2ω2×Qlmlmjl(nIωb)],
BlmII jl(nIIωb)nI2=BlmIIDlIIhl(nIωb)+lm[AlmIIIClIII(-1)l+l+1×Qlmlmjl(nIωb)+BlmIIIDlIIIPlmlmjl(nIωb)],
M(ω)AlmIIIBlmIIIAlmIIBlmII=0,
e1jlm(r)=1N1jljl(nIω1jlr)Xlm(Ω),
e2jlm(r)=1N2jliIω2jl×[jl(nIω2jlr)Xlm(Ω)].
V1γβ=r<adrδ(r)eγ(r)·eβ(r),
V1γβ=lII-IN1j1l1N1j2l2{Plm,l1m(k1j1l1)Plm,l2m(k1j2l2)×K1[jl(k1j1l1r), jl(k1j2l2r)]-(-1)l1+l2Qlm,l1m(k1j1l1)Qlm,l2m(k1j2l2)×K2[jl(k1j1l1r), jl(k1j2l2r)]}.
V1γβ=lII-IN2j1l1N2j2l2Plm,l1m(k2j1l1)Plm,l2m(k2j2l2)×K2[jl(k2j1l1r), jl(k2j2l2r)]I2ω2j1l1ω2j2l2-(-1)l1+l2Qlm,l1m(k2j1l1)Qlm,l2m(k2j2l2)×ω2j1l1ω2j2l2K1[jl(k2j1l1r), jl(k2j2l2r)].
V1γβ=-li2l+1(II-I)N1j1l1N1j2l2(-1)l2Plm,l1m(k1j1l1)×Qlm,l2m(k2j2l2)ω2j2l2×K1[jl(k1j1l1r), jl(k1j2l2r)]-(-1)l1Qlm,l1m(k1j1l1)Plm,l2m(k2j2l2)×K2[jl(k1j1l1r), jl(k2j2l2r)]Iω2j2l2.
K1[zl(ξ1r),zl(ξ2r)]
=r<bd3rzl(ξ1r)Xlm*(Ω)·zl(ξ2r)Xlm(Ω)=b3zl(ξ1b)zl(ξ2b)(ξ1b)2-(ξ2b)2ξ2bzl(ξ2b)zl(ξ2b)-ξ1bzl(ξ1b)zl(ξ1b),
K2[zl(ξ1r),zl(ξ2r)]
=r<bd3r×[zl(ξ1r)Xlm*(Ω)]·×[zl(ξ2r)Xlm(Ω)]=bξ1ξ2zl(ξ1b)zl(ξ2b)ξ12-ξ22×ξ1bzl(ξ2b)zl(ξ2b)-ξ2bzl(ξ1b)zl(ξ1b)+ξ12-ξ22ξ1ξ2,
d3r(0)(r)Λ1γL(r)·Λ1γL(r)
=4πd3r[ρI(r)]Ψ1γ(r),
·[(0)(r)Ψ1γ]=-·[(0)(r)Λ1γL]=(II-I)δ(r-b)rˆ·eγ.
[ρI]=I4πδ(r-b)blClYlm*(Ω),
Cl=-i2(l+l)+1 [l(l+1)]1/2N1jlII-IIjl(k1jlb)×Qlm,lm(k1jl)
Cl=-[l(l+1)]1/2N2jlII-II2ω2jlmjl(k2jlb)Plm,lm(k2jl)
2Ψ1γ=-δ(r-b)blClYlm(Ω)
Ψ1γ=lAlrblYlm(Ω).
Al=bal I-1Is=l (s+1)/(2s+1)Slm,sm(2s+1)-(s+1)(1-1/I)×t=0sbat+1CtStm,sm+Cl2l+1.
(r/a)lYlm(Ω)=l=0lSlm,lm(r/a)lYlm(Ω),
Slm,lm=(-1)mdal-l×[(2l+1)(2l+1)]1/2(2l-2l-1)!!(2l+1)!!(2l+1)!!×lll-l000lll-l-mm0.
d3r(0)(r)Λ1γL(r)·Λ1γL(r)=IblAlCl.

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