Abstract

Based on the completeness of morphology-dependent resonances (MDRs) in a dielectric sphere and the associated MDR expansion of the transverse dyadic Green’s function, a generic perturbation theory is formulated. The method is capable of handling cases with degeneracies in the MDR frequencies, which are ubiquitous in systems with a specific symmetry. One then applies the perturbation scheme to locate the MDRs of a dielectric sphere that contains several smaller spherical inclusions. To gauge the accuracy and efficiency of the perturbation scheme, we also use a transfer-matrix method to obtain an eigenvalue equation for MDRs in these systems. The results obtained from these two methods are compared, and good agreement is found.

© 2002 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, M. Kerker, ed., SPIE Milestone Series Bellingham, Wash., (SPIE, 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. G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metaalosungen,” Ann. Phys. (Leipzig) 25, 377–442 (1908).
    [CrossRef]
  10. M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964).
  11. 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]
  12. K. M. Lee, P. T. Leung, and K. M. Pang, “Dyadic formulation of morphology-dependent resonances. II. Perturbation theory,” J. Opt. Soc. Am. B 16, 1418–1430 (1999).
    [CrossRef]
  13. E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
    [CrossRef]
  14. 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]
  15. 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]
  16. 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).
    [CrossRef]
  17. R. L. Armstrong, J.-G. Xie, T. E. Ruekgauer, J. Gu, and R. G. Pinnick, “Effects of submicrometer-sized particles on microdroplet lasing,” Opt. Lett. 18, 119–121 (1993).
    [CrossRef] [PubMed]
  18. D. Ngo and R. G. Pinnick, “Suppression of scattering resonances in inhomogeneous microdroplets,” J. Opt. Soc. Am. A 11, 1352–1359 (1994).
    [CrossRef]
  19. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
    [CrossRef] [PubMed]
  20. 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]
  21. See, e.g., A. L. Fetter and J. D. Walecka, Quantum Theory of Many Body Systems (McGraw-Hill, New York, 1971).
  22. 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]
  23. 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]
  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. See, e.g., P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).
  26. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).
  27. C.-T. Tai, Dyadic Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, New York, 1993).
  28. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, New York, 1995).
  29. Our definition for the transverse dyadic Green’s function is slightly different from the conventional one, which is related to ours through direct differentiation. In addition to yielding the magnetic field of a localized current source, the Green’s function defined by us is symmetric for the transposition of the field and source points and can be expanded in terms of the tensor products of relevant MDR fields.
  30. In general, both G and Δ are non-Hermitian and sometimes cannot be diagonalized. However, it is always possible to transform these matrices into a triangular form, and our derivation in this paper still holds.
  31. See, e.g., S. S. M. Wong Computational Methods in Physics and Engineering (Prentice-Hall, Englewood Cliffs, N. J., 1992).
  32. 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).
  33. K. M. Pang, “Completeness and perturbation of morphology-dependent resonances in dielectric spheres,” Ph.D. dissertation (Chinese University of Hong Kong, Shatin, Hong Kong, China, 2000).
  34. S. W. Ng, “Mie’s scattering: a morphology-dependent resonance approach,” M. Phil. thesis (Chinese University of Hong Kong, Shatin, Hong Kong, China, 2000).

1999 (2)

1998 (2)

E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

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]

1996 (2)

1994 (1)

1993 (2)

R. L. Armstrong, J.-G. Xie, T. E. Ruekgauer, J. Gu, and R. G. Pinnick, “Effects of submicrometer-sized particles on microdroplet lasing,” Opt. Lett. 18, 119–121 (1993).
[CrossRef] [PubMed]

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]

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]

1908 (1)

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metaalosungen,” Ann. Phys. (Leipzig) 25, 377–442 (1908).
[CrossRef]

Armstrong, R. L.

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]

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.

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.

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]

Emily, E. S. C.

E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

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]

Gorodetsky, M. L.

Gu, J.

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.

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]

Ilchenko, V. S.

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.

Maassen van den Brink, A.

E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Mazumder, M. M.

Mie, G.

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metaalosungen,” Ann. Phys. (Leipzig) 25, 377–442 (1908).
[CrossRef]

Ngo, D.

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.

Pinnick, R. 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]

Ruekgauer, T. E.

Saija, R.

Savchenkov, A. A.

Sindoni, O. I.

Snow, J. B.

Suen, W. M.

E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Tong, S. S.

E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Tzeng, H. M.

Wall, K. F.

Xie, J.-G.

Young, K.

E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

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.

Ann. Phys. (Leipzig) (1)

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metaalosungen,” Ann. Phys. (Leipzig) 25, 377–442 (1908).
[CrossRef]

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. (4)

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]

Rev. Mod. Phys. (1)

E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Other (15)

See e.g., M. Kerker, ed., Selected Papers on Light Scattering, M. Kerker, ed., SPIE Milestone Series Bellingham, Wash., (SPIE, 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).

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

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

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

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).

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).

Our definition for the transverse dyadic Green’s function is slightly different from the conventional one, which is related to ours through direct differentiation. In addition to yielding the magnetic field of a localized current source, the Green’s function defined by us is symmetric for the transposition of the field and source points and can be expanded in terms of the tensor products of relevant MDR fields.

In general, both G and Δ are non-Hermitian and sometimes cannot be diagonalized. However, it is always possible to transform these matrices into a triangular form, and our derivation in this paper still holds.

See, e.g., S. S. M. Wong Computational Methods in Physics and Engineering (Prentice-Hall, Englewood Cliffs, N. J., 1992).

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. dissertation (Chinese University of Hong Kong, Shatin, Hong Kong, China, 2000).

S. W. Ng, “Mie’s scattering: a morphology-dependent resonance approach,” M. Phil. thesis (Chinese University of Hong Kong, Shatin, Hong Kong, China, 2000).

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Figures (4)

Fig. 1
Fig. 1

Schematic diagrams showing a host dielectric sphere doped with three latex inclusions. Two different configurations are shown here.

Fig. 2
Fig. 2

(a), (b) Real and imaginary parts, respectively, of the shift in the MDR frequencies δω (in units of a-1) versus perturbation parameter δn for a TE MDR with ω(0)=(2.68186,-0.42285), l=2, and j=2. The three inclusions are centered at {0.17a, 0, 0}, {0.49a, π/2, 0}, and {0.81a, π/2,-π/2} [i.e., Fig. 1(a)]. Dotted curves and cross, results obtained from the first-order perturbation theory and the T-matrix method, respectively.

Fig. 3
Fig. 3

Similar to Fig. 2, except that the third inclusion is now centered at {0.81a,-π, 0} instead of at {0.81a, π/2,-π/2} [i.e., Fig. 1(b)].

Fig. 4
Fig. 4

Schematic diagram showing a layered-inclusion system. The large circle with center O is the host sphere, and the smaller circles are the inclusions. The ith inclusion, centered at a distance di from O and inscribed between the spherical shells B1i and B2i, has a radius bi and a spherical surface B3i.

Equations (82)

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

×1(r)×b-ω2b=0.
b1jlm=1iω1jl×f1jl(r)r Xlm,
b2jlm=f2jl(r)r Xlm,
-ddrρ(r) ddr+ρ(r) l(l+1)r2-ρ(r)(r)ωνjl2fνjl=0,
b1jlmiω1jl×f1jl(r)r Xlm*,
b2jlmf2jl(r)r Xlm*,
bνjlm|bνjlm
limXr<X d3r bνjlm·bνjlm+iωνjl+ωνjl(X)-1/2× d3rδ(r-X)bνjlm·bνjlm.
 I¯t(r, r)·V(r)d3r=Vt(r),
I¯t(r, r)=νjlm bνjlm(r)bνjlm(r)2.
×1×G¯tb(r, r;ω)-ω2G¯tb(r, r;ω)=I¯t(r, r),
G¯tb(r, r;ω)=νjlm-bνjlm(r)bνjlm(r)2ωνjl(ω-ωνjl),r, r<a.
b(r)= G¯tb(r, r;ω)·r×4πj(r)(r)d3r,
eνjlm=i(r)ωνjl×bνjlm,
eνjlm=-i(r)ωνjl×bνjlm.
×10(r)×D¯tb(r, r)-ω2D¯tb(r, r)=I¯t(r, r)
D¯tb(r, r)=-αm bαm(0)(r)bαm(0)(r)2ωα(0)[ω-ωα(0)],
Dαm,βn=-12 1ωα(0)[ω-ωα(0)]δαβδmn.
Bαm(0)=bαm(0)(r),
Bαm(0)=bαm(0)(r).
D¯tb(r, r)=B(0)DB(0).
×1(r)×G¯tb(r, r;ω)-ω2G¯tb(r, r;ω)=I¯t(r, r),
G¯tb(r, r)D¯tb(r, r)+δG¯tb(r, r)
δG¯tb(r, r)=s<a d3s D¯tb(r, r)·×[δρ(s)×G¯tb(s, r)],
G¯tb=D¯tb+D¯tb·ΔD¯tb+D¯tb·ΔD¯tb·ΔD¯tb+ ,
(D¯tb·ΔD¯tb)(r, r)=s<a d3s D¯tb(r, s)·×[δρ(s)×D¯tb(s, r)]
G¯tb=αm,βn bαm(0)(r)Gαm,βnbβn(0)(r),
G=D+DΔD+DΔDΔD+.
Δαm,βn=s<ad3sbαm(0)(s)·×[δρ(s)×bβn(0)(s)].
Kαm,βn=-12 1ωγ(0)[ω-ωγ(0)]δmn
Hαm,βn=-12 1ωα(0)[ω-ωα(0)]δmn
W=Δ+ΔHΔ+ΔHΔHΔ+.
G=D[I+WK+(WK)2+](I+WH).
G=D(I-WK)-1(I+WH).
G˜=P-1GP,
G¯tb(r, r)=BG˜B,
G=(D-1-WKD-1)-1(I+WH).
[D-1]αm,βn=-2ωα(0)[ω-ωα(0)]δαβδmn,
Γ=Iγ000,
WWγW2W3W4.
D-1-WKD-1=D-1-Wγ0W30,
Pγ-1001(D-1-WKD-1)Pγ001
=D-1-Pγ-1WγPγ0W3Pγ0.
ω=ωγ(0)-[λp/2ωγ(0)],
ωγp=ωγ(0)+s=1 μsωγp(s).
λp=μΔ˜pp+μ2(Δ˜HΔ˜)pp+O(μ3)+.
ωγp(1)=-ωγ(0)2Vγp,γp,
ωγp(2)=ωγ(0)4 βγq Vγp,βq ωβ(0)[ωγ(0)-ωβ(0)]Vβq,γp,
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) a32jl2(nIω1jla),
N2jl2=1-1nI2jl(nIω2jla)jl(nIω2jla)+1nIω2jla2+l2+l(ω2jla)2 a32jl2(nIω2jla).
Δαm,βn=ωαωβ(II-I)i=1N Iαm,βn(i),
Iαm,βn(i)vi d3r eαm(r)·eβn(r)
eγm(r)=1Nγjl(kγr)Xlm*(Ω),
eγn(r)=1Nγjl(kγr)Xln(Ω),
Iγm,γn(i)=1Nγ2 mm Dmml*(i)Dmnl(i)×vi d3r·jl(kγri)Xlm*(Ωi)·jl(kγri)Xlm(Ωi),
jl(kγr)Xlm*(Ω)=l Tlmlm(i) jl(kγr)Xlm*(Ω)-Vlmlm(i)×[jl(kγr)Xlm*(Ω)],
jl(kγr)Xlm(Ω)=l Tlmlm(i) jl(kγr)Xlm(Ω)+Vlmlm(i)×[jl(kγr)Xlm(Ω)],
Tlmlm(i)=δmm(-1)mil-l2(2l+1)(2l+1)l(l+1)l(l+1)1/2×lil[l(l+1)+l(l+1)-l(l-1)](2l+1)lll000×lllm-m0jl(kγdi),
Vlmlm(i)=δmm(-1)mil-l+1dim(2l+1)(2l+1)l(l+1)l(l+1)1/2×l il(2l+1)lll000×lllm-m0jl(kγdi),
Iγm,γn(i)=1Nγ2m Dmml*(i)Dmnl(i)×l[Tlmlm(i)]2K1(kγbi)-[Vlmlm(i)]2K2(kγbi),
K1(kγbi)=ri<bi d3rijl(kγri)Xlm*(Ωi)·jl(kγri)Xlm(Ωi)=bi32[jl2(kγbi)-jl+1(kγbi)jl-1(kγbi)],
K2(kγbi)=ri<bi d3ri×[jl(kγri)Xlm*(Ωi)]·×[jl(kγri)Xlm(Ωi)]=bi(kγbi)22+1jl2(kγbi)+kγbi jl(kγbi)jl(kγbi)-(kγbi)22jl+1(kγbi)jl-1(kγbi).
eI|1=lmalm(1)jl(nIky)jl(nIka)Xlm(Ωy)+blm(1)×jl(nIky)jl(nIka) Xlm(Ωy)+lm clm(1) hl(nIky)hl(nIka) Xlm(Ωy)+dlm(1)×hl(nIky)hl(nIka) Xlm(Ωy);
eI|2=lmalm(2)jl(nIky)jl(nIka)Xlm(Ωy)+blm(2)×jl(nIky)jl(nIka)Xlm(Ωy)+lm clm(2) hl(nIky)hl(nIka) Xlm(Ωy)+dlm(2)×hl(nIky)hl(nIka) Xlm(Ωy);
eII=lm AlmII jl(nIIky)jl(nIIkb1) Xlm(Ωy)+BlmIInII2×jl(nIIky)jl(nIIkb1) Xlm(Ωy),
eI(y)=lm alm(2) jl(nIky)jl(nIka) Xlm(Ωy)+blm(2)×jl(nIky)jl(nIka) Xlm(Ωy)+lm clm(2) hl(nIky)hl(nIka) Xlm(Ωy)+dlm(2)×hl(nIky)hl(nIka) Xlm(Ωy)+lm AlmIIGl hl(nIIky)jl(nIIka) Xlm(Ωy)+BlmIIHl×hl(nIIky)jl(nIIka) Xlm(Ωy),
Gl=inIka[nIkb1 jl(nIIkb1)jl(nIkb1)-nIIkb1 jl(nIkb1)jl(nIIkb1)],
Hl=inIkb1jl(nIkb1)jl(nIIkb1)1nI2-1nII2+kb1nIjl(nIIkb1)jl(nIkb1)-kb1nIIjl(nIkb1)jl(nIIkb1).
Qal(2)bl(2)cl(2)dl(2)=Pal(1)bl(1)cl(1)dl(1).
al(2)bl(2)cl(2)dl(2)=Q-1Pal(1)bl(1)cl(1)dl(1)=M(ω)al(1)bl(1)cl(1)dl(1).
alm(n)blm(n)clm(n)dlm(n)=MnDn,n-1M3D3,2M2D2,1M1alm(1)blm(1)clm(1)dlm(1),
eout=lm alm(o) jl(nokr)jl(noka) Xlm(Ω)+blm(o)×jl(nokr)jl(noka) Xlm(Ω)+lm clm(o) hl(nokr)hl(noka) Xlm(Ω)+dlm(o)×hl(nokr)hl(noka) Xlm(Ω).
alm(o)blm(o)clm(o)dlm(o)=S(ω)alm(n)blm(n)clm(n)dlm(n),
alm(o)blm(o)clm(o)dlm(o)=SMnDn,n-1M3D3,2M2D2,1M1alm(1)blm(1)clm(1)dlm(1)=T(ω)alm(1)blm(1)clm(1)dlm(1).
clm(1)=0,
dlm(1)=0,
alm(o)=0,
blm(o)=0,
T(ω)=T11(ω)T12(ω)T21(ω)T22(ω),

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