Abstract

A general model is presented for coupling of high-Q whispering-gallery modes in optical microsphere resonators with coupler devices that possess a discrete and continuous spectrum of propagating modes. By contrast to conventional high-Q optical cavities, in microspheres the independence of high intrinsic quality-factor and controllable parameters of coupling via an evanescent field offer a variety of regimes similar to those that are already available in rf devices. The theory is applied to data reported earlier on different types of couplers to microsphere resonators and is complemented by the experimental demonstration of enhanced coupling efficiency (∼80%) and variable loading regimes with Q>108 fused-silica microspheres.

© 1998 Optical Society of America

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References

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  1. V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–396 (1989).
    [CrossRef]
  2. H. Mabuchi and H. J. Kimble, “Atom galleries for whispering atoms: binding atoms in stable orbits around a dielectric cavity,” Opt. Lett. 19, 749–751 (1994).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  7. L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
    [CrossRef]
  8. P. W. Barber and R. K. Chang, Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).
  9. S. P. Vyatchanin, M. L. Gorodetsky, and V. S. Ilchenko, “Tunable narrowband optical filters with whispering gallery modes,” Zh. Prikl. Spektrosk. 56, 274–280, 1992
  10. M. L. Gorodetsky and V. S. Ilchenko, “High-Q optical whispering gallery microresonators: precession approach for spherical mode analysis and emission patterns,” Opt. Commun. 113, 133–143 (1994).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, “Phase-matched excitation of whispering gallery mode resonances using a fiber taper,” Opt. Lett. 22, 1129–1131 (1997).
    [CrossRef] [PubMed]
  14. D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” Proc. Inst. Electr. Eng. Part J. 140, 177–188 (1993).
  15. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  16. H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and 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]
  17. S. Shiller, “Asymptotic expansion of morphological resonance frequencies in Mie scattering,” Appl. Opt. 32, 2181–2185 (1993).
    [CrossRef]
  18. N. Dubreuil, Institut d’Optique, Centre Universitaire d’Orsay, B.P. 147, 91403 Orsay cedex, France (personal communication, 1997).
  19. M. L. Gorodetsky and V. S. Ilchenko, “Thermal nonlinear effects in optical whispering-gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).
  20. D. S. Starodubov, V. Grubsky, J. Feinberg, B. Kobrin, and S. Juma, “Bragg grating fabrication in germanosilicate fibers by use of near-UV light: a new pathway for refractive index changes,” Opt. Lett. 22, 1086–1088 (1997).
    [CrossRef] [PubMed]

1997 (2)

1996 (2)

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “A very low threshold whispering gallery mode microsphere laser,” Phys. Rev. B 54, R1777–R1780 (1996).
[CrossRef]

V. V. Vasiliev, V. L. Velichansky, M. L. Gorodetsky, V. S. Ilchenko, L. Hollberg, and A. V. Yarovitsky, “High-coherence diode laser with optical feedback via a microcavity with ‘whispering gallery’ modes,” Quantum Electron. 26, 657–658 (1996).
[CrossRef]

1995 (2)

1994 (3)

1993 (3)

L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
[CrossRef]

D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” Proc. Inst. Electr. Eng. Part J. 140, 177–188 (1993).

S. Shiller, “Asymptotic expansion of morphological resonance frequencies in Mie scattering,” Appl. Opt. 32, 2181–2185 (1993).
[CrossRef]

1992 (2)

M. L. Gorodetsky and V. S. Ilchenko, “Thermal nonlinear effects in optical whispering-gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

S. P. Vyatchanin, M. L. Gorodetsky, and V. S. Ilchenko, “Tunable narrowband optical filters with whispering gallery modes,” Zh. Prikl. Spektrosk. 56, 274–280, 1992

1991 (1)

1990 (1)

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and 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]

1989 (1)

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–396 (1989).
[CrossRef]

Arnold, S.

Barber, P. W.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and 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]

Birks, T. A.

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–396 (1989).
[CrossRef]

Brune, M.

L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
[CrossRef]

Byer, R. L.

Cheung, G.

Collot, L.

L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
[CrossRef]

Connoly, J.

Dubreuil, N.

Feinberg, J.

Gorodetsky, M. L.

V. V. Vasiliev, V. L. Velichansky, M. L. Gorodetsky, V. S. Ilchenko, L. Hollberg, and A. V. Yarovitsky, “High-coherence diode laser with optical feedback via a microcavity with ‘whispering gallery’ modes,” Quantum Electron. 26, 657–658 (1996).
[CrossRef]

M. L. Gorodetsky and V. S. Ilchenko, “High-Q optical whispering gallery microresonators: precession approach for spherical mode analysis and emission patterns,” Opt. Commun. 113, 133–143 (1994).
[CrossRef]

S. P. Vyatchanin, M. L. Gorodetsky, and V. S. Ilchenko, “Tunable narrowband optical filters with whispering gallery modes,” Zh. Prikl. Spektrosk. 56, 274–280, 1992

M. L. Gorodetsky and V. S. Ilchenko, “Thermal nonlinear effects in optical whispering-gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–396 (1989).
[CrossRef]

Griffel, G.

Grubsky, V.

Hare, J.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “A very low threshold whispering gallery mode microsphere laser,” Phys. Rev. B 54, R1777–R1780 (1996).
[CrossRef]

N. Dubreuil, J. C. Knight, D. Leventhal, V. Sandoghdar, J. Hare, V. Lefévre-Seguin, J. M. Raimond, and S. Haroche, “Eroded monomode optical fiber for whispering-gallery mode excitation in fused-silica microspheres,” Opt. Lett. 20, 1515–1517 (1995).
[CrossRef]

Haroche, S.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “A very low threshold whispering gallery mode microsphere laser,” Phys. Rev. B 54, R1777–R1780 (1996).
[CrossRef]

N. Dubreuil, J. C. Knight, D. Leventhal, V. Sandoghdar, J. Hare, V. Lefévre-Seguin, J. M. Raimond, and S. Haroche, “Eroded monomode optical fiber for whispering-gallery mode excitation in fused-silica microspheres,” Opt. Lett. 20, 1515–1517 (1995).
[CrossRef]

L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
[CrossRef]

Hill, S. C.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and 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]

Hollberg, L.

V. V. Vasiliev, V. L. Velichansky, M. L. Gorodetsky, V. S. Ilchenko, L. Hollberg, and A. V. Yarovitsky, “High-coherence diode laser with optical feedback via a microcavity with ‘whispering gallery’ modes,” Quantum Electron. 26, 657–658 (1996).
[CrossRef]

Ilchenko, V. S.

V. V. Vasiliev, V. L. Velichansky, M. L. Gorodetsky, V. S. Ilchenko, L. Hollberg, and A. V. Yarovitsky, “High-coherence diode laser with optical feedback via a microcavity with ‘whispering gallery’ modes,” Quantum Electron. 26, 657–658 (1996).
[CrossRef]

M. L. Gorodetsky and V. S. Ilchenko, “High-Q optical whispering gallery microresonators: precession approach for spherical mode analysis and emission patterns,” Opt. Commun. 113, 133–143 (1994).
[CrossRef]

S. P. Vyatchanin, M. L. Gorodetsky, and V. S. Ilchenko, “Tunable narrowband optical filters with whispering gallery modes,” Zh. Prikl. Spektrosk. 56, 274–280, 1992

M. L. Gorodetsky and V. S. Ilchenko, “Thermal nonlinear effects in optical whispering-gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–396 (1989).
[CrossRef]

Jacques, F.

Juma, S.

Kimble, H. J.

Knight, J. C.

Kobrin, B.

Lai, H. M.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and 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]

Lefévre-Seguin, V.

Lefèvre-Seguin, V.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “A very low threshold whispering gallery mode microsphere laser,” Phys. Rev. B 54, R1777–R1780 (1996).
[CrossRef]

L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
[CrossRef]

Leung, P. T.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and 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]

Leventhal, D.

Love, J. D.

D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” Proc. Inst. Electr. Eng. Part J. 140, 177–188 (1993).

Mabuchi, H.

Morris, N.

Raimond, J. M.

Raimond, J.-M.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “A very low threshold whispering gallery mode microsphere laser,” Phys. Rev. B 54, R1777–R1780 (1996).
[CrossRef]

L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
[CrossRef]

Rowland, D. R.

D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” Proc. Inst. Electr. Eng. Part J. 140, 177–188 (1993).

Sandoghdar, V.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “A very low threshold whispering gallery mode microsphere laser,” Phys. Rev. B 54, R1777–R1780 (1996).
[CrossRef]

N. Dubreuil, J. C. Knight, D. Leventhal, V. Sandoghdar, J. Hare, V. Lefévre-Seguin, J. M. Raimond, and S. Haroche, “Eroded monomode optical fiber for whispering-gallery mode excitation in fused-silica microspheres,” Opt. Lett. 20, 1515–1517 (1995).
[CrossRef]

Schiller, S.

Serpengüzel, A.

Shiller, S.

Starodubov, D. S.

Taskent, D.

Treussart, F.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “A very low threshold whispering gallery mode microsphere laser,” Phys. Rev. B 54, R1777–R1780 (1996).
[CrossRef]

Vasiliev, V. V.

V. V. Vasiliev, V. L. Velichansky, M. L. Gorodetsky, V. S. Ilchenko, L. Hollberg, and A. V. Yarovitsky, “High-coherence diode laser with optical feedback via a microcavity with ‘whispering gallery’ modes,” Quantum Electron. 26, 657–658 (1996).
[CrossRef]

Velichansky, V. L.

V. V. Vasiliev, V. L. Velichansky, M. L. Gorodetsky, V. S. Ilchenko, L. Hollberg, and A. V. Yarovitsky, “High-coherence diode laser with optical feedback via a microcavity with ‘whispering gallery’ modes,” Quantum Electron. 26, 657–658 (1996).
[CrossRef]

Vyatchanin, S. P.

S. P. Vyatchanin, M. L. Gorodetsky, and V. S. Ilchenko, “Tunable narrowband optical filters with whispering gallery modes,” Zh. Prikl. Spektrosk. 56, 274–280, 1992

Yarovitsky, A. V.

V. V. Vasiliev, V. L. Velichansky, M. L. Gorodetsky, V. S. Ilchenko, L. Hollberg, and A. V. Yarovitsky, “High-coherence diode laser with optical feedback via a microcavity with ‘whispering gallery’ modes,” Quantum Electron. 26, 657–658 (1996).
[CrossRef]

Young, K.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and 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]

Appl. Opt. (1)

Europhys. Lett. (1)

L. Collot, V. Lefèvre-Seguin, M. Brune, J.-M. Raimond, and S. Haroche, “Very high-Q whispering gallery modes resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–333 (1993).
[CrossRef]

Laser Phys. (1)

M. L. Gorodetsky and V. S. Ilchenko, “Thermal nonlinear effects in optical whispering-gallery microresonators,” Laser Phys. 2, 1004–1009 (1992).

Opt. Commun. (1)

M. L. Gorodetsky and V. S. Ilchenko, “High-Q optical whispering gallery microresonators: precession approach for spherical mode analysis and emission patterns,” Opt. Commun. 113, 133–143 (1994).
[CrossRef]

Opt. Lett. (7)

Phys. Lett. A (1)

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–396 (1989).
[CrossRef]

Phys. Rev. A (1)

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and 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]

Phys. Rev. B (1)

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “A very low threshold whispering gallery mode microsphere laser,” Phys. Rev. B 54, R1777–R1780 (1996).
[CrossRef]

Proc. Inst. Electr. Eng. Part J. (1)

D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” Proc. Inst. Electr. Eng. Part J. 140, 177–188 (1993).

Quantum Electron. (1)

V. V. Vasiliev, V. L. Velichansky, M. L. Gorodetsky, V. S. Ilchenko, L. Hollberg, and A. V. Yarovitsky, “High-coherence diode laser with optical feedback via a microcavity with ‘whispering gallery’ modes,” Quantum Electron. 26, 657–658 (1996).
[CrossRef]

Zh. Prikl. Spektrosk. (1)

S. P. Vyatchanin, M. L. Gorodetsky, and V. S. Ilchenko, “Tunable narrowband optical filters with whispering gallery modes,” Zh. Prikl. Spektrosk. 56, 274–280, 1992

Other (3)

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

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

N. Dubreuil, Institut d’Optique, Centre Universitaire d’Orsay, B.P. 147, 91403 Orsay cedex, France (personal communication, 1997).

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Figures (4)

Fig. 1
Fig. 1

Schematic of the excitation of WG modes in a high-Q microsphere.

Fig. 2
Fig. 2

Efficiency of three kinds of coupler in terms of the loading quality factor Qc with optimized parameters at d=0 as a function of sphere radius a; numerical results are obtained for the TMll1 mode. Critical coupling is possible when the intrinsic quality factor of the WG mode, Q0, is larger than Qc [see Eq. (9)].

Fig. 3
Fig. 3

Output intensity of the prism coupler observed under variable loading (successively increasing microsphere-prism gap). Fused-silica sphere with 270-µm diameter.

Fig. 4
Fig. 4

Resonance contrast as a function of the loaded quality factor.

Equations (61)

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

R=Rk=1-Tk2/2=1-T2/2.
A0(t)=iTkBkin(t)+RA0(t-τ0)×exp(i2πnsL/λ-αL/2),
Bkout(t)=RkBkin(t)+iTkA0(t),
dA0dt+(δc+δ0+iΔω)A0=iCBin,
δ0=αc2ns,δc=1-RRτ0=T22τ0,C=TΓτ0.
Γ=TBinTBin.
A0=i2δcBinδ0+δc+iΔωΓT=iΓBinδ0+δc+iΔω2δcτ01/2.
Bout=Bin-Bin2δcΓδ0+δc+iΔωTT,
(Bout)2=(Bin)21-4δcδ0Γ2(δ0+δc)2+(Δω)2.
K=4Q0QcΓ2(Q0+Qc)2=4QΓ2Q0+Qc,
1Q=2δ0ω+2δcω=1Q0+1Qc.
dA0dz=iΔβ0(z)A0+iCk(z)exp[i(βk-β0)z]Bk,
dBkdz=iΔβk(z)Bk+iCk(z)exp[-i(βk-β0)z]A0.
Δβ0=ω(nc2-1)8πC|e0|2ds,
Δβk(z)=ω(ns2-1)8πS|ek|2ds,
Ck2=ω264π2C(nc2-1)ek*e0dsS(ns2-1)e0*ekds.
A0out=RA0in exp(iΦ)+iTkBkin,
Bkout=Bkin+iTkA0,
Tk=-Ck exp[i(β0-βk)z]dz,Φ=-Δβ0dz.
ω0-ω0=Φτ0=ω(nc2-1)8πτ0C|e0|2dv,
δc=Tk22τ0=ω2(nc2-1)2128π2τ0Cek*e0 exp[i(β0-βk)z]dv2.
Es(r, t)=exp(-iωt)jAˆj(t)eˆj(r),
14πseˆj1eˆj2*dv=δj1,j2
|eˆj|2|ej|2=c4πS(eˆj×hˆj*)zds=1τj.
××Es+s(r)c2+c(r)-1c2+i2δ0s(r)ω0c22Est2
=-s(r)-1c22Ect2,
dAˆ0dt+Aˆ0(δ0+iΔω)
=iω(ns2-1)8πexp(iωt)SEceˆ0*dv,
ω0=ω0-ω8πC(nc2-1)|eˆ0|2dv
Ec(r, t)=exp(-iωt)Bβ(z, t)eβ(r)exp(iβz)dβ.
Bβ=kBkδ(β-βk)+B˜β.
c4π(eβ1×hβ2*)zds=δ(β1-β2)
××Ec+c(r)c2+s(r)-1c22Ect2
=-c(r)-1c22Est2,
βBβz-iω2(s-1)2c2Bβeβ exp(iβz)dβ
=i(c-1)ω22c2Es exp(iωt).
Bβ=Bβin exp(iΔβz)+iω28πcβ×-z exp[iωt-iβz+iΔβ(z-z)]×C(nc2-1)(Es×hβ*)zdsdz,
Δβ(z)=ω2(ns2-1)8πcβS(eβ×hβ*)zds.
dAˆ0dt+(δ0+δc+iΔω)Aˆ0
=iω(nc2-1)8πBβinCeˆ0eβ* exp(-iβz)dvdβ,
δc=ω364π2cS--zC(ns2-1)(nc2-1)β×exp[i(β+Δβ)(z-z)]×(eˆ0×hβ*)z(eβeˆ0*)dsdzdvdβω2(nc2-1)2128π2Ceˆ0eβ* exp(-iβz)dv2dβ,
Tβ=ω(nc2-1)8πCe0eβ* exp[i(β0-β)z]dv.
Tβ=ω(nc2-1)8π2πaγ1/2×C exp[-(βa-m)2/2γa]e0eβ*ds.
eˆsx(r, θ, ϕ)2ns2-1ns2a3/2lπ1/4×exp[-l(π/2-θ)2/2+ilϕ]×jl(knsr)/jl(knsa)raexp[-γ(r-a)]r>a,
ecx(ρ, φ)2ηγbnccJ0(ηρ)/J0(ηb)ρbexp[-γ(ρ-b)]ρ>b,
ηb2.405 exp-1+1/nc22bγ2.4051-1+1/nc22bγ,
γ[(l+1/2)2/a2-k2]1/2kns2-1.
jl(knsr)jl(knsa)+kns(r-a)jl(knsa)jl(knsa)+kns(r-a)jl(tl+1/2,q),
J0(ηρ)J0(ηb)+η(ρ-b)J0(ηb)J0(ηb)+η(ρ-b)J0(t0,1),
Qc162π5ns4nc(ns2-1)29(nc2-1)a3/2b3λ9/2×exp[2γd+(l-βa)2/γa],
knsatl+1/2,q-nsns2-1l+1/2+(l+1/2)1/3[3π(q-1/4)]2/3/2-nsns2-1,
b2.3a[(nka)2-l2]1/20.51aλ2n2(4q-1)1/3,
Qc102aλ5/2n3(n2-1)4q-1exp(2γd).
fπ[nc2k2-β2-γ2-(π/g)2]1/2.
ecx(y)2π5/4γf3/2ncgexp(-x2/2g2)×sin[α(1/γ-y)]/sin(α/γ)y0e-γyy>0,
α2=nc2k2-β2,
Qc8π2ns3nc(ns2-1+ns)(ns2-1)3/2(nc2-1)2af3λ4×exp[2γd+(l-βa)2/γa].
sin Φ0=lncka,ΔΦ2=ns2-1np2ka cos2 Φ0,
ΔΘ2=ns+ns2-1np2ka.
Qc2π5/2ns1/2(ns2-1)(nc2-ns2)1/2aλ3/2 exp(2γd).
Γ=TBinTBin,

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