Abstract

We derive formulas for whispering gallery mode resonances and bending losses in infinite cylindrical dielectric shells and sets of concentric cylindrical shells. The formulas also apply to spherical shells and to sections of bent waveguides. The derivation is based on a Wentzel-Kramers-Brillouin (WKB) treatment of Helmholtz equation and can in principle be extended to any number of concentric shells. A distinctive limit analytically arises in the analysis when two shells are brought at very close distance to one another. In that limit, the two shells act as a slot waveguide. If the two shells are sufficiently apart, we identify a structural resonance between the individual shells, which can either lead to a substantial enhancement or suppression of radiation losses.

© 2016 Optical Society of America

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References

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    [Crossref]
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  14. D.-P. Cai, J.-H. Lu, C.-C. Chen, C.-C. Lee, C.-E. Lin, and T.-J. Yen, “High Q-factor microring resonator wrapped by the curved waveguide,” Sci. Rep. 5, 10078 (2015).
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    [Crossref] [PubMed]
  16. K. Malmir, H. Habibiyan, and H. Ghafoorifard, “An ultrasensitive optical label-free polymeric biosensor based on concentric triple microring resonators with a central microdisk resonator,” Opt. Commun. 365, 150–156 (2016).
    [Crossref]
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    [Crossref]
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2016 (1)

K. Malmir, H. Habibiyan, and H. Ghafoorifard, “An ultrasensitive optical label-free polymeric biosensor based on concentric triple microring resonators with a central microdisk resonator,” Opt. Commun. 365, 150–156 (2016).
[Crossref]

2015 (1)

D.-P. Cai, J.-H. Lu, C.-C. Chen, C.-C. Lee, C.-E. Lin, and T.-J. Yen, “High Q-factor microring resonator wrapped by the curved waveguide,” Sci. Rep. 5, 10078 (2015).
[Crossref] [PubMed]

2013 (1)

2011 (2)

G. Kozyreff, J. L. Dominguez Juarez, and J. Martorell, “Nonlinear optics in spheres: from second harmonic scattering to quasi-phase matched generation in whispering gallery modes,” Laser Photon. Rev. 5, 737 (2011).
[Crossref]

A. Kargar and C.-Y. Chao, “Design and optimization of waveguide sensitivity in slot microring sensors,” J. Opt. Soc. Am. A 28, 596–603 (2011).
[Crossref]

2009 (1)

2008 (1)

G. Kozyreff, J. L. Dominguez Juarez, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77, 043817 (2008).
[Crossref]

2007 (4)

M. L. Gorodetsky and A. E. Fomin, “Eigenfrequencies and q factor in the geometrical theory of whispering-gallery modes,” Quantum Electron. 37, 167 (2007).
[Crossref]

C.-Y. Chao, “Simple and effective calculation of modal properties of bent slot waveguides,” J. Opt. Soc. Am. B 24, 2373–2377 (2007).
[Crossref]

R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).
[Crossref]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express 15, 6768–6773 (2007).
[Crossref] [PubMed]

2006 (3)

I. H. Agha, J. E. Sharping, M. A. Foster, and A. L. Gaeta, “Optimal sizes of silica microspheres for linear and nonlinear optical interaction,” Appl. Phys. B 83, 303–309 (2006).
[Crossref]

Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74, 063804 (2006).
[Crossref]

M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering-gallery modes,” IEEE J. Sel. Topics Quantum Electon. 12, 33–39 (2006).
[Crossref]

2005 (3)

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[Crossref]

R. Jedidi and R. Pierre, “Efficient analytical and numerical methods for the computation of bent loss in planar waveguides,” J. Lightwave Technol. 23, 2278 (2005).
[Crossref]

S. Spillane, T. Kippenberg, K. Vahala, K. Goh, E. Wilcut, and H. Kimble, “Ultrahigh-q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

2004 (2)

L. Prkna, J. Čtyrokỳ, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
[Crossref]

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004).
[Crossref] [PubMed]

1999 (1)

1993 (1)

1992 (1)

1976 (1)

1960 (1)

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

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbooks of Mathematical Functions (Dover, 1972), 10th ed.

Agha, I. H.

I. H. Agha, J. E. Sharping, M. A. Foster, and A. L. Gaeta, “Optimal sizes of silica microspheres for linear and nonlinear optical interaction,” Appl. Phys. B 83, 303–309 (2006).
[Crossref]

Almeida, V. R.

Barnes, F. S.

D. C. Chang and F. S. Barnes, “Reduction of radiation loss in a curved dielectric slab waveguide,” Scientific Report No. 2, AFOSR-72-2417, Electromagnetic Laboratory, Dept. Electr. Eng., Univ. of Colorado Boulder (1973).

Barrios, C. A.

Bender, C. M.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer-Verlag, 1999).
[Crossref]

Cai, D.-P.

D.-P. Cai, J.-H. Lu, C.-C. Chen, C.-C. Lee, C.-E. Lin, and T.-J. Yen, “High Q-factor microring resonator wrapped by the curved waveguide,” Sci. Rep. 5, 10078 (2015).
[Crossref] [PubMed]

Chang, D. C.

D. C. Chang and F. S. Barnes, “Reduction of radiation loss in a curved dielectric slab waveguide,” Scientific Report No. 2, AFOSR-72-2417, Electromagnetic Laboratory, Dept. Electr. Eng., Univ. of Colorado Boulder (1973).

Chao, C.-Y.

Chen, C.-C.

D.-P. Cai, J.-H. Lu, C.-C. Chen, C.-C. Lee, C.-E. Lin, and T.-J. Yen, “High Q-factor microring resonator wrapped by the curved waveguide,” Sci. Rep. 5, 10078 (2015).
[Crossref] [PubMed]

Cole, J. H.

R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).
[Crossref]

Ctyrok?, J.

L. Prkna, J. Čtyrokỳ, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
[Crossref]

Ctyroký, J.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[Crossref]

Demchenko, Y. A.

Dominguez Juarez, J. L.

G. Kozyreff, J. L. Dominguez Juarez, and J. Martorell, “Nonlinear optics in spheres: from second harmonic scattering to quasi-phase matched generation in whispering gallery modes,” Laser Photon. Rev. 5, 737 (2011).
[Crossref]

G. Kozyreff, J. L. Dominguez Juarez, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77, 043817 (2008).
[Crossref]

Dumeige, Y.

Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74, 063804 (2006).
[Crossref]

Féron, P.

Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74, 063804 (2006).
[Crossref]

Fomin, A. E.

M. L. Gorodetsky and A. E. Fomin, “Eigenfrequencies and q factor in the geometrical theory of whispering-gallery modes,” Quantum Electron. 37, 167 (2007).
[Crossref]

M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering-gallery modes,” IEEE J. Sel. Topics Quantum Electon. 12, 33–39 (2006).
[Crossref]

Foster, M. A.

I. H. Agha, J. E. Sharping, M. A. Foster, and A. L. Gaeta, “Optimal sizes of silica microspheres for linear and nonlinear optical interaction,” Appl. Phys. B 83, 303–309 (2006).
[Crossref]

Gaeta, A. L.

I. H. Agha, J. E. Sharping, M. A. Foster, and A. L. Gaeta, “Optimal sizes of silica microspheres for linear and nonlinear optical interaction,” Appl. Phys. B 83, 303–309 (2006).
[Crossref]

Ghafoorifard, H.

K. Malmir, H. Habibiyan, and H. Ghafoorifard, “An ultrasensitive optical label-free polymeric biosensor based on concentric triple microring resonators with a central microdisk resonator,” Opt. Commun. 365, 150–156 (2016).
[Crossref]

Goh, K.

S. Spillane, T. Kippenberg, K. Vahala, K. Goh, E. Wilcut, and H. Kimble, “Ultrahigh-q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

Gorodetsky, M. L.

Y. A. Demchenko and M. L. Gorodetsky, “Analytical estimates of eigenfrequencies, dispersion, and field distribution in whispering gallery resonators,” J. Opt. Soc. Am. B 30, 3056–3063 (2013).
[Crossref]

M. L. Gorodetsky and A. E. Fomin, “Eigenfrequencies and q factor in the geometrical theory of whispering-gallery modes,” Quantum Electron. 37, 167 (2007).
[Crossref]

M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering-gallery modes,” IEEE J. Sel. Topics Quantum Electon. 12, 33–39 (2006).
[Crossref]

Habibiyan, H.

K. Malmir, H. Habibiyan, and H. Ghafoorifard, “An ultrasensitive optical label-free polymeric biosensor based on concentric triple microring resonators with a central microdisk resonator,” Opt. Commun. 365, 150–156 (2016).
[Crossref]

Hammer, M.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[Crossref]

Haus, H.

Hiremath, K. R.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[Crossref]

Hubálek, M.

L. Prkna, J. Čtyrokỳ, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
[Crossref]

Ilchenko, V. S.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999), 3rd ed.

Jedidi, R.

Kargar, A.

Keller, J. B.

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

Kimble, H.

S. Spillane, T. Kippenberg, K. Vahala, K. Goh, E. Wilcut, and H. Kimble, “Ultrahigh-q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

Kippenberg, T.

S. Spillane, T. Kippenberg, K. Vahala, K. Goh, E. Wilcut, and H. Kimble, “Ultrahigh-q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

Kozyreff, G.

G. Kozyreff, J. L. Dominguez Juarez, and J. Martorell, “Nonlinear optics in spheres: from second harmonic scattering to quasi-phase matched generation in whispering gallery modes,” Laser Photon. Rev. 5, 737 (2011).
[Crossref]

G. Kozyreff, J. L. Dominguez Juarez, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77, 043817 (2008).
[Crossref]

Laine, J. P.

Lam, C. C.

Lee, C.-C.

D.-P. Cai, J.-H. Lu, C.-C. Chen, C.-C. Lee, C.-E. Lin, and T.-J. Yen, “High Q-factor microring resonator wrapped by the curved waveguide,” Sci. Rep. 5, 10078 (2015).
[Crossref] [PubMed]

Leung, P. T.

Li, X.

Lin, C.-E.

D.-P. Cai, J.-H. Lu, C.-C. Chen, C.-C. Lee, C.-E. Lin, and T.-J. Yen, “High Q-factor microring resonator wrapped by the curved waveguide,” Sci. Rep. 5, 10078 (2015).
[Crossref] [PubMed]

Lipson, M.

Little, B. E.

Liu, F.

Lu, J.-H.

D.-P. Cai, J.-H. Lu, C.-C. Chen, C.-C. Lee, C.-E. Lin, and T.-J. Yen, “High Q-factor microring resonator wrapped by the curved waveguide,” Sci. Rep. 5, 10078 (2015).
[Crossref] [PubMed]

Maleki, L.

Malmir, K.

K. Malmir, H. Habibiyan, and H. Ghafoorifard, “An ultrasensitive optical label-free polymeric biosensor based on concentric triple microring resonators with a central microdisk resonator,” Opt. Commun. 365, 150–156 (2016).
[Crossref]

Marcuse, D.

Martorell, J.

G. Kozyreff, J. L. Dominguez Juarez, and J. Martorell, “Nonlinear optics in spheres: from second harmonic scattering to quasi-phase matched generation in whispering gallery modes,” Laser Photon. Rev. 5, 737 (2011).
[Crossref]

G. Kozyreff, J. L. Dominguez Juarez, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77, 043817 (2008).
[Crossref]

Matsko, A. B.

Orszag, S. A.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer-Verlag, 1999).
[Crossref]

Pierre, R.

Prkna, L.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[Crossref]

L. Prkna, J. Čtyrokỳ, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
[Crossref]

Qin, S.

Qiu, M.

Rubinow, S.

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

Savchenkov, A. A.

Schermer, R. T.

R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).
[Crossref]

Schiller, S.

Sharping, J. E.

I. H. Agha, J. E. Sharping, M. A. Foster, and A. L. Gaeta, “Optimal sizes of silica microspheres for linear and nonlinear optical interaction,” Appl. Phys. B 83, 303–309 (2006).
[Crossref]

Spillane, S.

S. Spillane, T. Kippenberg, K. Vahala, K. Goh, E. Wilcut, and H. Kimble, “Ultrahigh-q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbooks of Mathematical Functions (Dover, 1972), 10th ed.

Stoffer, R.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[Crossref]

Su, Y.

Vahala, K.

S. Spillane, T. Kippenberg, K. Vahala, K. Goh, E. Wilcut, and H. Kimble, “Ultrahigh-q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

Wang, T.

Wilcut, E.

S. Spillane, T. Kippenberg, K. Vahala, K. Goh, E. Wilcut, and H. Kimble, “Ultrahigh-q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

Xu, Q.

Yen, T.-J.

D.-P. Cai, J.-H. Lu, C.-C. Chen, C.-C. Lee, C.-E. Lin, and T.-J. Yen, “High Q-factor microring resonator wrapped by the curved waveguide,” Sci. Rep. 5, 10078 (2015).
[Crossref] [PubMed]

Young, K.

Zhang, Z.

Ann. Phys. (1)

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

Appl. Opt. (2)

Appl. Phys. B (1)

I. H. Agha, J. E. Sharping, M. A. Foster, and A. L. Gaeta, “Optimal sizes of silica microspheres for linear and nonlinear optical interaction,” Appl. Phys. B 83, 303–309 (2006).
[Crossref]

IEEE J. Quantum Electron. (1)

R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).
[Crossref]

IEEE J. Sel. Topics Quantum Electon. (1)

M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering-gallery modes,” IEEE J. Sel. Topics Quantum Electon. 12, 33–39 (2006).
[Crossref]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

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

Laser Photon. Rev. (1)

G. Kozyreff, J. L. Dominguez Juarez, and J. Martorell, “Nonlinear optics in spheres: from second harmonic scattering to quasi-phase matched generation in whispering gallery modes,” Laser Photon. Rev. 5, 737 (2011).
[Crossref]

Opt. Commun. (1)

K. Malmir, H. Habibiyan, and H. Ghafoorifard, “An ultrasensitive optical label-free polymeric biosensor based on concentric triple microring resonators with a central microdisk resonator,” Opt. Commun. 365, 150–156 (2016).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (2)

L. Prkna, J. Čtyrokỳ, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
[Crossref]

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[Crossref]

Phys. Rev. A (3)

Y. Dumeige and P. Féron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74, 063804 (2006).
[Crossref]

S. Spillane, T. Kippenberg, K. Vahala, K. Goh, E. Wilcut, and H. Kimble, “Ultrahigh-q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[Crossref]

G. Kozyreff, J. L. Dominguez Juarez, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77, 043817 (2008).
[Crossref]

Quantum Electron. (1)

M. L. Gorodetsky and A. E. Fomin, “Eigenfrequencies and q factor in the geometrical theory of whispering-gallery modes,” Quantum Electron. 37, 167 (2007).
[Crossref]

Sci. Rep. (1)

D.-P. Cai, J.-H. Lu, C.-C. Chen, C.-C. Lee, C.-E. Lin, and T.-J. Yen, “High Q-factor microring resonator wrapped by the curved waveguide,” Sci. Rep. 5, 10078 (2015).
[Crossref] [PubMed]

Other (4)

M. Abramowitz and I. A. Stegun, Handbooks of Mathematical Functions (Dover, 1972), 10th ed.

D. C. Chang and F. S. Barnes, “Reduction of radiation loss in a curved dielectric slab waveguide,” Scientific Report No. 2, AFOSR-72-2417, Electromagnetic Laboratory, Dept. Electr. Eng., Univ. of Colorado Boulder (1973).

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer-Verlag, 1999).
[Crossref]

J. D. Jackson, Classical Electrodynamics (Wiley, 1999), 3rd ed.

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Figures (3)

Fig. 1
Fig. 1 Cavities and their parameters. a) cylindrical shell, b) spherical shell, c) two concentric cylindrical shell, d) two spherical shells. The orange arrows indicate the direction of propagation. In the case of two concentric shells, light can be concentrated in the inner shell but also in the outer shell or in the low-index region between the two shells (slot waveguide). The refractive index is n in the blue region and unity elsewhere.
Fig. 2
Fig. 2 Shape factor for a single annular cylinder, with outer radius R = 6.4µm and varying thickness h. Refractive index n = 1.65, orbital number ν = 60. Solid line: analytical formula. Dots: numerical calculation.
Fig. 3
Fig. 3 Shape factor σ′ of the two-shell problem, when light is confined in the inner shell. R1 = 3.6µm, h1 = 1µm, h2 = 0.23µm, n = 1.65 Top left: ν = 25 and variable outer shell position. Top right: R3 = 5.78µm and variable ν. Bottom: field distribution corresponding to the minimum (left) and maximum (right) of σ′. The vacuum wavelength in both cases is found to be 1.27µm.

Equations (115)

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exp i ( φ k c t ) , k = k r i k i ,
exp i [ i ( β s k r c t ) α s ] β = / R , α = k i / ( k r R ) ,
ν = { for cylindrical geometries , + 1 / 2 for spherical geometries .
x = k r R ν ,
Γ 0 k i R 2 e 2 ν S ( x ) ξ n 2 1 , S ( x ) = 1 x 2 ln 1 + 1 x 2 x ,
k i R = σ × Γ 0 ,
x 1 + ϵ τ n + O ( ϵ 3 / 2 ) , ϵ = 2 1 / 3 ν 2 / 3 ,
D ( τ , h ^ ) = 0 , h ^ = ϵ 1 ( h / R ) ,
D ( τ , h ^ ) A i ( τ ) B i ( h ^ τ ) A i ( h ^ τ ) B i ( τ ) .
D ( τ , h ^ 1 + h ^ 2 ) + π Δ D ( τ , h ^ 2 ) D ( τ h ^ 1 , h ^ 2 ) = 0 , Δ = 2 1 / 3 ν 4 / 3 ( n 2 1 ) R 2 R 1 R 1 ,
d 2 ψ d r 2 + 1 r d ψ d r + ( n i 2 k 2 ν 2 r 2 ) ψ = 0 , n i = { n R h < r < R , 1 elsewhere
[ ψ ] + = [ 1 ξ d ψ d r ] + = 0
y = k r / ν ,
d 2 ψ d y 2 + 1 y d ψ d y = ν 2 q ( y ) ψ , q ( y ) = { y 2 n 2 x ( 1 h / R ) < y < x , y 2 1 elsewhere ,
ψ ± const . × e ± ν S ( y ) | 1 y 2 | 1 / 4 ( 1 + j 1 ν j f j ) ,
f 1 = 2 + 3 y 2 24 ( 1 y 2 ) 3 / 2 , f 2 = 4 + 300 y 2 + 81 y 4 1152 ( 1 y 2 ) 3 ,
S ( y ) = { i ( y 2 1 arccos ( 1 / y ) ) y > 1 , 1 y 2 ln 1 + 1 y 2 y y < 1 .
ψ A e ν S ( y ) + B e ν S ( y ) | 1 y 2 | 1 / 4 ( 1 + j 1 ν j f i ) , x < y < 1
ψ 2 C e i π / 4 e ν S ( y ) + B e ν S ( y ) ( y 2 1 ) 1 / 4 ( 1 + j 1 ν j f i ) , y < 1 .
ψ 2 5 / 6 π 1 / 2 ν 1 / 6 C [ i A i ( w ) + B i ( w ) ] , w = ( y 1 ) / ϵ .
A = i C , B = C .
ψ C i e ν S ( y ) + e ν S ( y ) | 1 y 2 | 1 / 4 ( 1 + j 1 ν j f i ) , x < y < 1 .
ψ G e ν S ( y ) | 1 y 2 | 1 / 4 ( 1 + j 1 ν j f i ) , y < x ( 1 h / R )
y = 1 + ϵ t n t = n y 1 ϵ = n k r / ν 1 ϵ ,
ψ = F ( t ) , F ( t ) + t F ( t ) ϵ ( 3 2 t 2 F ( t ) F ( t ) ) + O ( ϵ 2 ) .
A ( t ) ( 1 ϵ t 5 ) A i ( t + 3 ϵ t 2 10 ) , ( t ) ( 1 ϵ t 5 ) B i ( t + 3 ϵ t 2 10 ) .
ψ = H A ( t ) + I ( t ) ,
d ψ d y = n ϵ [ H A ( t ) + I ( t ) ] n ν [ H A i ( t ) + I B i ( t ) 2 1 / 3 ν 1 / 3 + O ( ν 1 ) ] .
x = k R ν = 1 + ϵ t 1 n .
h ^ = h ϵ R ,
H A ( t 1 ) + I ( t 1 ) = C i e ν S ( x ) + e ν S ( x ) | 1 x 2 | 1 / 4 ,
H A ( t 0 ) + I ( t 0 ) = G e ν S ( x 0 ) | 1 x 0 2 | 1 / 4 ,
n ξ [ H A i ( t 1 ) + I B i ( t 1 ) 2 1 / 3 ν 1 / 3 ] = C q ( x ) i e ν S ( x ) e ν S ( x ) | 1 x 2 | 1 / 4 ,
n ξ H A i ( t 0 ) + I B i ( t 0 ) 2 1 / 3 ν 1 / 3 = G q ( x 0 ) e ν S ( x 0 ) | 1 x 0 2 | 1 / 4 .
q ( x ) , q ( x 0 ) n 2 1 + O ( ν 2 / 3 )
H A ( t 1 ) + I ( t 1 ) ~ α ϵ [ H A i ( t 1 ) + I B i ( t 1 ) ] ( 1 + 2 i e 2 ν S ( x ) ) ,
H A ( t 0 ) + I ( t 0 ) ~ α ϵ [ H A i ( t 0 ) + I B i ( t 0 ) ] ,
α = 2 n ξ n 2 1 .
D ( τ , h ^ ) A i ( τ ) B i ( h ^ τ ) A i ( h ^ τ ) B i ( τ ) = 0 .
x ~ 1 + 2 1 / 3 ν 2 / 3 τ n ν 1 ξ n 2 1 1 + P 2 1 P 2 + 2 2 / 3 ν 4 / 3 n × [ 3 τ 2 10 P 2 1 P 2 ( 3 h ^ 2 + 4 h ^ τ 10 8 n 2 ξ 2 ( n 2 1 ) ( 1 P P ˜ ) B i ( h ^ τ ) ( 1 P 2 ) 2 B i ( h ^ τ ) ) ] + O ( ν 5 / 3 ) ,
P = B i ( τ ) B i ( h ^ τ ) , P ˜ = B i ( τ ) B i ( h ^ τ ) .
σ = 1 1 P 2 ,
t 3 t 2 ~ h 2 ϵ R 2 h ^ 2 and t 1 t 0 ~ h 1 ϵ R 1 h ^ 1 .
ψ ~ H A ( t ) + I ( t ) , R 0 < r < R 1 ,
ψ ~ J A ( t ) + K ( t ) , R 2 < r < R 3 ,
ψ ~ L cosh ν ( S ( y ) S ( x 1 ) ) + M cosh ν ( S ( y ) S ( x 2 ) ) q 1 / 4 , R 1 < r < R 2 ,
M sinh ν S 21 q 1 / 4 ~ α ϵ [ H A i ( t 1 ) + I B i ( t 1 ) ] ,
L sinh ν S 21 q 1 / 4 ~ α ϵ [ J A i ( t 2 ) + K B i ( t 2 ) ] ,
J A ( t 3 ) + K ( t 3 ) ~ α ϵ [ J A i ( t 3 ) + K i ( t 3 ) ] ( 1 + 2 i e 2 ν S ( x 3 ) ) ,
J A ( t 2 ) + K ( t 2 ) ~ α ϵ [ H A i ( t 1 ) + I B i ( t 1 ) cosh ( ν S 21 ) ( J A i ( t 2 ) + K B i ( t 2 ) ) ] sinh ν S 21 ,
H A ( t 1 ) + I ( t 1 ) ~ α ϵ [ cosh ( ν S 21 ) ( H A i ( t 1 ) + I B i ( t 1 ) ) J A i ( t 2 ) K B i ( t 2 ) ] sinh ν S 21 ,
H A ( t 0 ) + I ( t 0 ) ~ α ϵ [ H A i ( t 0 ) + I B i ( t 0 ) ] ,
{ H A i ( t 1 ) + I B i ( t 1 ) = 0 H A i ( h ^ 1 t 1 ) + I B i ( h ^ 1 t 1 ) = 0 { J A i ( t 3 ) + K B i ( t 3 ) = 0 J A i ( h ^ 2 t 3 ) + K B i ( h ^ 2 t 3 ) = 0
D ( t 1 , h ^ 1 ) = 0
D ( t 3 , h ^ 2 ) = 0 .
J A i ( t 3 ) + K B i ( t 3 ) ~ 2 i α ϵ [ J A i ( t 3 ) + K B i ( t 3 ) ] e 2 ν S ( x 3 ) ,
J A i ( t 2 ) + K B i ( t 2 ) ~ 0 .
σ = 1 1 P 2 .
H = 1 ,
I = A i ( τ ) / B i ( τ ) + ϵ I 1 + + 2 α ϵ e 2 ν S ( x 1 ) δ I ,
t 1 ~ τ + ϵ τ 1 + + 2 α ϵ e 2 ν S ( x 1 ) δ τ ,
J = ϵ J 1 + + 2 α ϵ e 2 ν S ( x 1 ) δ J ,
K = ϵ K 1 + + 2 α ϵ e 2 ν S ( x 1 ) δ K ,
t 3 = R 3 R 1 ϵ R 1 + R 3 R 1 τ .
( J 1 K 1 ) = α π sinh ( ν S 21 ) B i ( τ 0 ) D ( t 3 , h ^ 2 ) ( B i ( t 3 ) A i ( t 3 ) ) .
A i ( t 3 ) δ J + B i ( t 3 ) δ K ~ ϵ [ J 1 A i ( t 3 ) + K 1 B i ( t 3 . ) ] ( i e 2 ν S 31 + R 3 R 1 δ τ ) ,
A i ( t 2 ) δ J + B i ( t 2 ) δ K ~ α ϵ B i ( t 1 ) δ I sinh ν S 21 + ϵ [ J 1 A i ( t 2 ) + K 1 B i ( t 2 ) ] R 2 R 1 δ τ ,
B i ( t 1 ) δ I ~ α ϵ [ A i ( t 2 ) δ J B i ( t 2 ) δ K ] sinh ν S 21 + [ H A i ( t 1 ) + I B i ( t 1 ) ] δ τ ,
B i ( t 0 ) δ I ~ [ H A i ( t 0 ) + I B i ( t 0 ) ] R 0 R 1 δ τ ,
δ τ = σ × i ,
σ = ( ϵ α 2 / π 2 ) e 2 ν S 31 ( 1 P 2 ) sinh 2 ( ν S 21 ) D ( t 3 , h ^ 2 ) 2 , P = B i ( τ ) B i ( h ^ 1 τ ) .
σ = ( ϵ α 2 / π 2 ) e 2 ν S 31 sinh 2 ( ν S 21 ) D ( t 3 , h ^ 2 ) 2 .
Δ sinh ( ν S 21 ) α ϵ ~ ν S ( x 1 ) ( x 2 x 1 ) α ϵ ~ 2 1 / 3 ν 4 / 3 ( n 2 1 ) R 2 R 1 R 1 ~ n k ( R 2 R 1 ) ( n k R 1 / 2 ) 1 / 3 ( n 2 1 )
t 1 ~ t 3 h ^ 2 , t 0 ~ t 3 h ^ 1 h ^ 2 .
J A ( t 3 ) + K ( t 3 ) ~ α ϵ [ J A i ( t 3 ) + K B i ( t 3 ) ] ( 1 + 2 i e 2 ν S ( x 3 ) ) ,
J A ( t 1 ) + K ( t 1 ) ~ ( H J ) A i ( t 1 ) + ( I K ) B i ( t 1 ) Δ ,
H A ( t 1 ) + I ( t 1 ) ~ ( H J ) A i ( t 1 ) + ( I K ) B i ( t 1 ) Δ ,
H A ( t 0 ) + I ( t 0 ) α ϵ [ H A i ( t 0 ) + I B i ( t 0 ) ] .
J [ A i ( t 3 ) + 2 i α ϵ e 2 ν S π B i ( t 3 ) ] + K B i ( t 3 ) ~ 0 ,
J A i ( t 1 ) + K B i ( t 1 ) ~ ( H J ) A i ( t 1 ) + ( I K ) B i ( t 1 ) Δ ,
( H J ) A i ( t 1 ) + ( I K ) B i ( t 1 ) ~ 0 ,
H A i ( t 0 ) + I B i ( t 0 ) ~ 0 .
D ( t 3 , h ^ 1 + h ^ 2 ) + π Δ D ( t 3 , h ^ 2 ) D ( t 1 , h ^ 1 ) + 2 i α ϵ e 2 ν S ( x 3 ) π B i ( t 3 ) [ B i ( t 0 ) + π Δ B i ( t 1 ) D ( t 1 , h ^ 1 ) ]
D ( t 3 , h ^ 1 + h ^ 2 ) + π Δ D ( t 3 , h ^ 2 ) D ( t 1 , h ^ 1 ) 2 i α ϵ e 2 ν S π D ( t 1 , h ^ 1 ) D ( t 3 , h ^ 2 ) = 0
t 3 ~ τ σ × 2 i α ϵ e 2 ν S ,
( τ ) D ( τ , h ^ 1 + h ^ 2 ) + π Δ D ( τ , h ^ 2 ) D ( τ h ^ 2 , h ^ 1 ) = 0
σ = 1 ( τ ) D ( τ h ^ 2 , h ^ 1 ) π D ( τ , h ^ 2 ) .
τ ( j ) ( j π / h ^ ) 2 ,
n 2 k 2 β 2 + k 2 , β = ν / R , k = j π / h ,
k i R ~ σ × Γ 0 , Γ 0 = 2 e 2 ν S ( x ) ξ n 2 1 ,
Γ 0 ~ 2 ξ ( n 2 1 ) [ ν x n l ( ν x ) ] 2 , n l ( z ) = π 2 z Y l + 1 / 2 ( z ) .
ψ = ϕ ( y ) e ν S ( y ) , S ( y ) = ± q ( y ) .
2 S ϕ + ( S + S y ) ϕ = 1 ν y ( y ϕ )
ϕ ( y ) = f ( y ) y S ( y ) , f ( y ) = ( y ϕ ( y ) ) 2 ν y S ( y ) .
f j + 1 ( y ) = ( y ϕ j ( y ) ) 2 y S ( y ) , ϕ j = f j y S ( y ) .
ψ ± ~ const . × e ± ν S ( y ) | 1 y 2 | 1 / 4 ( 1 + j 1 ν j f j ) .
H = 1 ,
I ~ i 0 + ϵ 1 / 2 i 1 + ϵ i 2 + + 2 α ϵ e 2 ν S ( x ) δ i ,
t 1 ~ τ + ϵ 1 / 2 τ 1 + ϵ τ 2 + + 2 α ϵ e 2 ν S ( x ) δ τ ,
t 0 = ( 1 ϵ h ^ ) t 1 h ^ .
τ 1 = 1 + P 2 1 P 2 α , τ 2 = 3 τ 2 10 P 2 1 P 2 ( 3 h ^ 2 + 4 h ^ τ 10 4 α 2 ( 1 P P ˜ ) B i ( h ^ τ ) ( 1 P 2 ) 2 B i ( h ^ τ ) )
P = B i ( τ ) B i ( h ^ τ ) , P ˜ = B i ( τ ) B i ( h ^ τ ) .
δ i B i ( τ ) = [ H A i ( τ ) + I B i ( τ ) ] ( i + δ τ ) ,
δ i B i ( h ^ τ ) = [ H A i ( h ^ τ ) + I B i ( h ^ τ ) ] δ τ ,
A i ( z ) B i ( z ) A i ( z ) B i ( z ) = 1 / π ,
δ τ = 1 1 P 2 × i .
k i R 1 1 P 2 × 2 e 2 ν S ξ n 2 1 .
A i ( t 3 ) δ J + B i ( t 3 ) δ K ϵ J 1 π B i ( t 3 ) ( i e 2 ν S 31 + R 3 R 1 δ τ ) ,
A i ( t 2 ) δ J + B i ( t 2 ) δ K α ϵ B i ( t 1 ) δ I sinh ν S 21 + ϵ [ J 1 A i ( t 2 ) + K 1 B i ( t 2 ) ] R 2 R 1 δ τ ,
B i ( t 1 ) δ I α ϵ [ A i ( t 2 ) δ J B i ( t 2 ) δ K ] sinh ν S 21 δ τ π B i ( t 1 ) ,
B i ( t 0 ) δ I ( R 0 / R 1 ) δ τ π B i ( t 0 ) ,
A i ( t 3 ) δ J + B i ( t 3 ) δ K i ϵ J 1 π B i ( t 3 ) e 2 ν S 31 ,
A i ( t 2 ) δ J + B i ( t 2 ) δ K 0 ,
B i ( t 1 ) δ I α ϵ δ J π sinh ν S 21 δ τ π B i ( t 1 ) ,
B i ( t 0 ) δ I δ τ π B i ( t 0 ) ,

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