Abstract

The resonance locations and quality factors (Q’s) of morphology-dependent resonances in an inhomogeneous sphere with a small refractive-index perturbation are computed by using the T-matrix method for layered axisymmetric objects and a time-independent perturbation method. The resonance locations computed are similar. The changes in the Q computed with the two methods are typically within a factor of 2 or 3 of each other when the change from the unperturbed Q is less than 50%. For the type of perturbation that we consider here, an increase in the refractive index in a nonconcentric spherical region inside the larger sphere, the resonance frequencies always decrease, but the Q’s decrease or increase depending on the unperturbed Q and the location and shape of the perturbation. The change in frequency and the change in Q depend on the overlap of the perturbation with the energy-density distribution of the morphology-dependent resonance. For the same overlap, the change in Q is much larger for higher-Q modes than for lower-Q modes. A refractive index perturbation that causes a relatively small change in Q may cause the resonance frequency of a high-Q MDR to shift by many linewidths.

© 1992 Optical Society of America

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  31. A. Biswas, H. Latifi, R. L. Armstrong, R. G. Pinnick, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40, 7413–7416 (1989).
    [Crossref] [PubMed]
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    [Crossref]
  37. P. W. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
    [Crossref] [PubMed]
  38. 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]
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    [Crossref]

1991 (3)

1990 (2)

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]

P. Chylek, “Resonance structure of Mie scattering: distance between resonances,” J. Opt. Soc. Am. A 7, 1609–1613 (1990).
[Crossref]

1989 (3)

1988 (4)

1987 (2)

1986 (1)

1985 (5)

1984 (3)

1983 (1)

1982 (2)

J. F. Owen, R. K. Chang, P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,” Aerosol Sci. Technol. 1, 293–302 (1982).
[Crossref]

P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,”IEEE Trans Antennas Propag. 30, 168–172 (1982).
[Crossref]

1981 (4)

1980 (1)

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

1979 (1)

1978 (3)

1977 (1)

A. Ashkin, J. M. Dziedzic, “Observations of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[Crossref]

1976 (1)

1975 (1)

1973 (1)

P. Affolter, B. Eliasson, “Electromagnetic resonances and Q-factors of lossy dielectric spheres,” IEEE Trans. Microwave Theory Tech. MTT-21, 573–578 (1973).
[Crossref]

1971 (1)

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

1968 (2)

1967 (1)

M. Gastine, L. Courtois, J. L. Dormann, “Electromagnetic resonances of free dielectric spheres,” IEEE Trans. Microwave Theory Tech. MTT-25, 694–700 (1967).
[Crossref]

1958 (1)

P. J. Mevel, “Étude de la structure détaillée des courbes de diffusion des ondes electromagnétiques par les spheres diélectriques,”J. Phys. Radium 19, 630–636 (1958).
[Crossref]

Affolter, P.

P. Affolter, B. Eliasson, “Electromagnetic resonances and Q-factors of lossy dielectric spheres,” IEEE Trans. Microwave Theory Tech. MTT-21, 573–578 (1973).
[Crossref]

Armstrong, R. L.

A. Biswas, H. Latifi, R. L. Armstrong, R. G. Pinnick, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40, 7413–7416 (1989).
[Crossref] [PubMed]

Ashkin, A.

Barber, P. W.

D. Q. Chowdhury, S. C. Hill, P. W. Barber, “Morphology-dependent resonances in radially-inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702–1705 (1991).
[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]

J. F. Owen, R. K. Chang, P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,” Aerosol Sci. Technol. 1, 293–302 (1982).
[Crossref]

P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,”IEEE Trans Antennas Propag. 30, 168–172 (1982).
[Crossref]

J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, “Enhancement of fluorescence induced by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078 (1981).
[Crossref]

J. F. Owen, R. K. Chang, P. W. Barber, “Determination of optical fiber diameter from resonances in the elastic scattering spectrum,” Opt. Lett. 6, 272–274 (1981).
[Crossref] [PubMed]

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

D. S. Wang, P. W. Barber, “Scattering by inhomogeneous nonspherical objects,” Appl. Opt. 18, 1190–1197 (1979).
[Crossref] [PubMed]

P. W. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
[Crossref] [PubMed]

Benner, R. E.

Bennett, H. S.

Biswas, A.

A. Biswas, H. Latifi, R. L. Armstrong, R. G. Pinnick, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40, 7413–7416 (1989).
[Crossref] [PubMed]

Bryant, H. C.

Campillo, A. J.

A. J. Campillo, J. D. Eversole, H.-B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

R. L. Hightower, C. B. Richardson, H.-B. Lin, J. D. Eversole, A. J. Campillo, “Measurement of scattering of light from layered microspheres,” Opt. Lett. 13, 946–948 (1988).
[Crossref] [PubMed]

Chang, R. K.

J.-Z. Zhang, R. K. Chang, “Generation and suppression of stimulated Brillouin scattering in single liquid droplets,” J. Opt. Soc. Am. B 6, 151–153 (1989).
[Crossref]

J. R. Snow, S.-X. Qian, R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985).
[Crossref] [PubMed]

S.-X. Qian, J. B. Snow, R. K. Chang, “Coherent Raman mixing and coherent anti-Stokes Raman scattering from micrometer-size droplets,” Opt. Lett. 10, 499–501 (1985).
[Crossref] [PubMed]

H. M. Tzeng, K. F. Wall, M. B. Long, R. K. Chang, “Laser emission from individual droplets at wavelengths corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499–501 (1984).
[Crossref] [PubMed]

P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,”IEEE Trans Antennas Propag. 30, 168–172 (1982).
[Crossref]

J. F. Owen, R. K. Chang, P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,” Aerosol Sci. Technol. 1, 293–302 (1982).
[Crossref]

J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, “Enhancement of fluorescence induced by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078 (1981).
[Crossref]

J. F. Owen, R. K. Chang, P. W. Barber, “Determination of optical fiber diameter from resonances in the elastic scattering spectrum,” Opt. Lett. 6, 272–274 (1981).
[Crossref] [PubMed]

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

Chew, H.

H. Chew, “Radiation lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[Crossref] [PubMed]

Ching, S. C.

Chowdhury, D. Q.

Chylek, P.

Conwell, P. R.

Courtois, L.

M. Gastine, L. Courtois, J. L. Dormann, “Electromagnetic resonances of free dielectric spheres,” IEEE Trans. Microwave Theory Tech. MTT-25, 694–700 (1967).
[Crossref]

Dorain, P. B.

J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, “Enhancement of fluorescence induced by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078 (1981).
[Crossref]

Dormann, J. L.

M. Gastine, L. Courtois, J. L. Dormann, “Electromagnetic resonances of free dielectric spheres,” IEEE Trans. Microwave Theory Tech. MTT-25, 694–700 (1967).
[Crossref]

Dziedzic, J. M.

Eliasson, B.

P. Affolter, B. Eliasson, “Electromagnetic resonances and Q-factors of lossy dielectric spheres,” IEEE Trans. Microwave Theory Tech. MTT-21, 573–578 (1973).
[Crossref]

Eversole, J. D.

A. J. Campillo, J. D. Eversole, H.-B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

R. L. Hightower, C. B. Richardson, H.-B. Lin, J. D. Eversole, A. J. Campillo, “Measurement of scattering of light from layered microspheres,” Opt. Lett. 13, 946–948 (1988).
[Crossref] [PubMed]

Fahlen, T. S.

Fuchs, R.

Fuller, K. A.

Gastine, M.

M. Gastine, L. Courtois, J. L. Dormann, “Electromagnetic resonances of free dielectric spheres,” IEEE Trans. Microwave Theory Tech. MTT-25, 694–700 (1967).
[Crossref]

Hightower, R. L.

Hill, S. C.

Jackson, J. D.

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

Kattawar, G. W.

Kiefer, W.

R. Thurn, W. Kiefer, “Structural resonances observed in the Raman spectra of optically levitated liquid droplets,” Appl. Opt., 24, 1515–1519 (1985).
[Crossref] [PubMed]

R. Thurn, W. Kiefer, “Raman-microsampling technique applying optical levitation by radiation pressure,” Appl. Spec-trosc. 38, 78–83 (1984).
[Crossref]

Kiehl, J. T.

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[Crossref]

Kliewar, K. L.

Ko, M. K. W.

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[Crossref]

Lai, H. M.

Lam, C. C.

Latifi, H.

A. Biswas, H. Latifi, R. L. Armstrong, R. G. Pinnick, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40, 7413–7416 (1989).
[Crossref] [PubMed]

Lettieri, T. R.

T. R. Lettieri, R. L. Preston, “Observation of sharp resonances in the spontaneous Raman spectrum of a single optically levitated microdroplet,” Opt. Commun. 54, 349–352 (1985).
[Crossref]

Leung, P. T.

H. M. Lai, C. C. Lam, P. T. Leung, K. Young, “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, 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]

Lin, H.-B.

A. J. Campillo, J. D. Eversole, H.-B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

R. L. Hightower, C. B. Richardson, H.-B. Lin, J. D. Eversole, A. J. Campillo, “Measurement of scattering of light from layered microspheres,” Opt. Lett. 13, 946–948 (1988).
[Crossref] [PubMed]

Long, M. B.

Mevel, P. J.

P. J. Mevel, “Étude de la structure détaillée des courbes de diffusion des ondes electromagnétiques par les spheres diélectriques,”J. Phys. Radium 19, 630–636 (1958).
[Crossref]

Owen, J. F.

J. F. Owen, R. K. Chang, P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,” Aerosol Sci. Technol. 1, 293–302 (1982).
[Crossref]

P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,”IEEE Trans Antennas Propag. 30, 168–172 (1982).
[Crossref]

J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, “Enhancement of fluorescence induced by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078 (1981).
[Crossref]

J. F. Owen, R. K. Chang, P. W. Barber, “Determination of optical fiber diameter from resonances in the elastic scattering spectrum,” Opt. Lett. 6, 272–274 (1981).
[Crossref] [PubMed]

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

Pinnick, R. G.

A. Biswas, H. Latifi, R. L. Armstrong, R. G. Pinnick, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40, 7413–7416 (1989).
[Crossref] [PubMed]

P. Chylek, V. Srivastava, R. G. Pinnick, R. T. Wang, “Scattering of electromagnetic waves by composite spherical particles: experiment and effective medium approximations,” Appl. Opt. 27, 2396–2404 (1988).
[Crossref]

Preston, R. L.

T. R. Lettieri, R. L. Preston, “Observation of sharp resonances in the spontaneous Raman spectrum of a single optically levitated microdroplet,” Opt. Commun. 54, 349–352 (1985).
[Crossref]

Qian, S.-X.

Ramaswamy, V.

Richardson, C. B.

Rosasco, G. J.

Rushforth, C. K.

Snow, J. B.

Snow, J. R.

Srivastava, V.

Stolen, R. H.

Thurn, R.

R. Thurn, W. Kiefer, “Structural resonances observed in the Raman spectra of optically levitated liquid droplets,” Appl. Opt., 24, 1515–1519 (1985).
[Crossref] [PubMed]

R. Thurn, W. Kiefer, “Raman-microsampling technique applying optical levitation by radiation pressure,” Appl. Spec-trosc. 38, 78–83 (1984).
[Crossref]

Tzeng, H. M.

Wall, K. F.

Wang, D. S.

D. S. Wang, P. W. Barber, “Scattering by inhomogeneous nonspherical objects,” Appl. Opt. 18, 1190–1197 (1979).
[Crossref] [PubMed]

D. S. Wang, “Light scattering by nonspherical multilayered particles,” Ph.D. dissertation (University of Utah, Salt Lake City, 1979).

Wang, R. T.

Waterman, P. C.

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

Yeh, C.

Young, K.

Zhang, J.-Z.

Aerosol Sci. Technol. (1)

J. F. Owen, R. K. Chang, P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,” Aerosol Sci. Technol. 1, 293–302 (1982).
[Crossref]

Appl. Opt. (13)

R. Thurn, W. Kiefer, “Structural resonances observed in the Raman spectra of optically levitated liquid droplets,” Appl. Opt., 24, 1515–1519 (1985).
[Crossref] [PubMed]

A. Ashkin, J. M. Dziedzic, R. H. Stolen, “Outer diameter measurement of low birefringence optical fibers by a new resonant backscatter technique,” Appl. Opt. 20, 2299–2303 (1981).
[Crossref] [PubMed]

A. Ashkin, J. M. Dziedzic, “Observations of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803–01814 (1981).
[Crossref] [PubMed]

P. Chylek, V. Ramaswamy, A. Ashkin, J. M. Dziedzic, “Simultaneous determination of refractive index and size of spherical dielectric particles from light scattering data,” Appl. Opt. 22, 2302–2307 (1983).
[Crossref] [PubMed]

S. C. Hill, R. E. Benner, C. K. Rushforth, P. R. Conwell, “Structural resonances observed in the fluorescence emission from small spheres on substrates,” Appl. Opt. 23, 1680–1683 (1984).
[Crossref] [PubMed]

S. C. Hill, R. E. Benner, C. K. Rushforth, P. R. Conwell, “Sizing dielectric spheres and cylinders by aligning measured and computed structural resonance locations: algorithm for multiple orders,” Appl. Opt., 24, 2380–2390 (1985).
[Crossref] [PubMed]

H. S. Bennett, G. J. Rosasco, “Resonances in the efficiency factors for absorption: Mie scattering theory,” Appl. Opt. 17, 491–493 (1978).
[Crossref] [PubMed]

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Figures (14)

Fig. 1
Fig. 1

Geometries used for the computations of the shift in the resonance size parameter x and in the normalized quality factor Qp/Qo. In (a) the offset length l is 0.167a, where a is the radius of the outer sphere. The radius of the enclosed sphere is 0.667a. In (b) the offset length is 0.8a and the radius of the enclosed sphere is 0.2a.

Fig. 2
Fig. 2

Shift in the resonance location as a function of the refractive index perturbation for the first-order TE(22,1) mode, which has an unperturbed refractive index m = 2.0, xo = 13.4127, Qo = 1.32 × 107, and FWHM = 1.013 × 10−6, where FWHM is the full width at half-maximum of the unperturbed MDR. The shift is expressed in number of linewidths from the unperturbed resonance. The results are for the geometry shown in Fig. 1(a).

Fig. 3
Fig. 3

Shift in the resonance location as a function of m for the same case as studied in Fig. 2. The change in refractive index is 0.003.

Fig. 4
Fig. 4

Shift in the resonance location as a function of the mode number n for first-order TE modes having an azimuthal mode number m = 1. The unperturbed refractive index is 2.0, and the change in the refractive index is 0.002. Curve (a) shows the results for the geometry shown in Fig. 1(a), and curve (b) shows the results for the geometry shown in Fig. 1(b). The shift is expressed in absolute units. The shift is calculated by the TIPM.

Fig. 5
Fig. 5

Contour plots of the modal energy density inside the sphere. The contour plots are superimposed upon the geometry of the droplet. The energy density is calculated at the plane = 45°. (a) and (b), respectively, show the results for the TE(22, m) modes having azimuthal mode numbers m = 1 and m = 16. (c) shows the results for TE(10, 1); (d) and (e) show the same contour plots as (a) and (c), respectively, but are superimposed upon the geometry of Fig. 1(b). In all the figures, the ratio of the maximum contour level to the minimum contour level is 5.

Fig. 6
Fig. 6

Qp/Qo as a function of the change in refractive index for the first-order TE(10, 1) mode, with Qo = 1.34 × 103 and xo = 6.8263. The results are for the geometry shown in Fig. 1(a).

Fig. 7
Fig. 7

Qp/Qo as a function of the change in refractive index for the first-order TE(16, 1) mode, with Qo = 1.19 × 105 and xo = 10.1591. The results are for the geometry shown in Fig. 1(a).

Fig. 8
Fig. 8

Qp/Qo as a function of the change in refractive index for the first-order TE(22, 1) mode, with Qo = 1.32 × 107 and xs = 13.4127. The results are for the geometry shown in Fig. 1(a).

Fig. 9
Fig. 9

Qp/Qo as a function of the change in refractive index for the first-order TE(18, 1) mode, with Qo = 5.05 × 108 and xo = 9.0369. The unperturbed refractive index is 2.5, and the results are for the geometry shown in Fig. 1(a).

Fig. 10
Fig. 10

Qp/Qo as a function of the azimuthal mode number m for the first-order TE(10, m) mode. The unperturbed refractive index m = 2.0, the change in refractive index Δm = 0.1, Qo = 1.34 × 103, and xo = 6.8263. The results are for the geometry shown in Fig. 1(a).

Fig. 11
Fig. 11

Qp/Qo as a function of the azimuthal mode number m for the first-order TE(16, m) mode, with m = 2.0, Δm = 0.05, Qo = 1.19 × 105, and xo = 10.1591. The results are for the geometry shown in Fig. 1(a).

Fig. 12
Fig. 12

Qp/Qo as a function of the azimuthal mode number m for the first-order TE(22, m) mode, with m = 2.0, Δm = 0.001, Qo = 1.32 × 107, and xo = 13.4127. The results are for the geometry shown in Fig. 1(a).

Fig. 13
Fig. 13

Qp/Qo as a function of the change in refractive index for first-order TE modes, with n = 10, n = 16, n = 22, and n = 28, and azimuthal mode number m = 1. The unperturbed Q’s are 1.34 × 103, 1.19 × 105, 1.32 × 107, and 1.65 × 109 for n = 10, 16, 22, and 28, respectively. The results are calculated by the TIPM, using the geometry of Fig. 1(b).

Fig. 14
Fig. 14

Qp/Qo and the shift in resonance location as a function of the mode number n and the unperturbed Q for first-order TE modes, with n = 10–26 and azimuthal mode number m = 1. The unperturbed Q’s are 1.34 × 103 for n = 10 and 3.27 × 108 for n = 26. The results are calculated by the TIPM, using the geometry of Fig. 1(b).

Tables (1)

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Table 1 First-Order TE Modes

Equations (29)

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( r ) = { 1 r > a m 2 + μ F ( r , θ ) r a ,
F ( r , θ ) = { 1 inside the enclosed sphere 0 otherwise .
[ T ] = [ B ] [ A ] - 1 ,
[ B ] = [ Q 2 11 ] [ D ] [ Q 1 31 ] - [ Q 2 13 ] [ D ] [ Q 1 11 ] ,
[ A ] = [ Q 2 31 ] [ D ] [ Q 1 31 ] - [ Q 2 33 ] [ D ] [ Q 1 11 ] .
× × E n ( o ) m - ω o 2 c 2 0 E n ( o ) m = 0 ,
E n ( o ) m = ϕ ( r ) X n , m ( θ , ϕ ) ,
ϕ ( r ) = { h n ( 1 ) ( x o ) j n ( m x o ) j n ( m k r ) r a h n ( 1 ) ( k r ) r > a ,
X n , m = 1 n ( n + 1 ) L ^ Y n , m ( θ , ϕ ) ,
E n ( o ) m = ϕ ( r ) X n , m * ( θ , ϕ ) ,
( r ) = 0 + 1 ( r ) ,
F n λ ( r ) = m a m λ E n ( o ) m + F n ( 1 ) λ ( r ) ,
× × F n λ - ( ω λ ) 2 c 2 ( r ) F n λ = 0 ,
ω λ = ω o + ω 1 λ
m V m m a m λ = - ( 2 ω 1 λ ω o ) m G m m a m λ ,
V m m = R d V 1 ( r ) E n ( o ) m · E n ( o ) m ,
G m m = R d V o E n ( o ) m · E n ( o ) m + i 2 ω o R d S E n ( o ) m · E n ( o ) m
ω 1 m ω o = - V m m 2 G ,
V m m = R d V μ F ( r , θ ) [ ϕ ( r ) ] 2 X n , m 2 ,
X n , m 2 = 1 n ( n + 1 ) [ ½ ( n - m ) ( n + m + 1 ) Y n , m + 1 2 + ½ ( n + m ) ( n - m + 1 ) Y n , m + 1 2 + m 2 Y n , m 2 ] ,
G = ( m 2 - 1 ) a 3 2 [ h n ( 1 ) ( x o ) ] 2 .
1 Q p = 1 Q o + μ C 1 + μ 2 C 2 + ,
C 1 = - 2 ( m 2 - 1 ) x o Im [ D n n ( 1 ) ( z o ) ] ,
C 2 = 2 ( m 2 - 1 ) x o 3 n n D n n ( 1 ) 2 | j n ( m x o ) w n ( x o ) | 2 ,
w n ( x o ) = j n ( m x o ) h n ( 1 ) ( x o ) - m j n ( m x o ) h n ( 1 ) ( x o ) ,
D n n ( 1 ) = - { 1 x o 2 j n ( m x o ) j n ( m x o ) } ϕ = 0 2 π θ = 0 π x = 0 x ( θ ) x 2 F ( x , θ ) × j n ( m x ) j n ( m x ) d x Y n , m * ( θ , ϕ ) Y n , m ( θ , ϕ ) sin θ d θ d ϕ ,
z o = x o - i x o 2 Q o .
Q p Q o 1 1 + 4 m 2 ( Δ m ) 2 C 2 Q o 1 - 4 m 2 ( Δ m ) 2 C 2 Q o ,
Q p Q o 1 - 4 m 2 ( Δ m ) 2 C 2 Q o ,

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