Abstract

A perturbation theory is presented for the frequency shift of highly resonant photonic whispering gallery modes in a transparent sphere. Using a vector wave equation, we derive a general formula for the shifts in TE and TM polarization by adsorption of another dielectric medium. The adsorbed medium can have an arbitrary shape and refractive-index profile. The formula is applied to adsorption of a thin layer and deposition of a small spherical particle, many such particles, and thin cylindrical particles on the resonator surface. We found that the ratio of the TM mode shift to the TE mode shift is sensitive to the shape of the adsorbates and their orientation. Calculation results are discussed in terms of a dipolar field.

© 2006 Optical Society of America

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  1. A. Serpengüzel, S. Arnold, and G. Griffel, "Excitation of resonances of microspheres on an optical fiber," Opt. Lett. 20, 654-656 (1995).
    [CrossRef] [PubMed]
  2. E. Krioukov, D. J. W. Klunder, A. Driessen, J. Greve, and C. Otto, "Sensor based on an integrated optical microcavity," Opt. Lett. 27, 512-514 (2002).
    [CrossRef]
  3. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
    [CrossRef]
  4. F. Vollmer, S. Arnold, D. Braun, I. Teraoka, and A. Libchaber, "Multiplexed DNA detection by optical resonances in microspheres," Biophys. J. 85, 1974-1979 (2003).
    [CrossRef] [PubMed]
  5. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, "Shift of whispering gallery modes in microspheres by protein adsorption," Opt. Lett. 28, 272-274 (2003).
    [CrossRef] [PubMed]
  6. L. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, "Tunable delay line with interacting whispering-gallery-mode resonators," Opt. Lett. 29, 626-628 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  10. I. Teraoka and S. Arnold, "Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium," J. Opt. Soc. Am. B 20, 1937-1946 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  14. C. C. Lam, P. T. Leung, and K. Young, "Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering," J. Opt. Soc. Am. B 9, 1585-1592 (1992).
    [CrossRef]
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    [CrossRef] [PubMed]
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2005 (2)

M. Noto, F. Vollmer, D. Keng, I. Teraoka, and S. Arnold, "Nanolayer characterization through wavelength multiplexing of a microsphere resonator," Opt. Lett. 30, 510-512 (2005).
[CrossRef] [PubMed]

A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, "Ultra-high-Q microcavity operation in H2O and D2O," Appl. Phys. Lett. 87, 151118 (2005).
[CrossRef]

2004 (1)

2003 (3)

2002 (2)

E. Krioukov, D. J. W. Klunder, A. Driessen, J. Greve, and C. Otto, "Sensor based on an integrated optical microcavity," Opt. Lett. 27, 512-514 (2002).
[CrossRef]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

1996 (1)

1995 (1)

1993 (1)

1992 (2)

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]

Armani, A. M.

A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, "Ultra-high-Q microcavity operation in H2O and D2O," Appl. Phys. Lett. 87, 151118 (2005).
[CrossRef]

Armani, D. K.

A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, "Ultra-high-Q microcavity operation in H2O and D2O," Appl. Phys. Lett. 87, 151118 (2005).
[CrossRef]

Arnold, S.

M. Noto, F. Vollmer, D. Keng, I. Teraoka, and S. Arnold, "Nanolayer characterization through wavelength multiplexing of a microsphere resonator," Opt. Lett. 30, 510-512 (2005).
[CrossRef] [PubMed]

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, "Shift of whispering gallery modes in microspheres by protein adsorption," Opt. Lett. 28, 272-274 (2003).
[CrossRef] [PubMed]

F. Vollmer, S. Arnold, D. Braun, I. Teraoka, and A. Libchaber, "Multiplexed DNA detection by optical resonances in microspheres," Biophys. J. 85, 1974-1979 (2003).
[CrossRef] [PubMed]

I. Teraoka and S. Arnold, "Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium," J. Opt. Soc. Am. B 20, 1937-1946 (2003).
[CrossRef]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

A. Serpengüzel, S. Arnold, and G. Griffel, "Excitation of resonances of microspheres on an optical fiber," Opt. Lett. 20, 654-656 (1995).
[CrossRef] [PubMed]

I. Teraoka and S. Arnold, "Dielectric property of particles at interface in random sequential adsorption and its application to whispering gallery mode resonance-shift sensors" (submitted to J. Appl. Phys.).

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]

Braun, D.

F. Vollmer, S. Arnold, D. Braun, I. Teraoka, and A. Libchaber, "Multiplexed DNA detection by optical resonances in microspheres," Biophys. J. 85, 1974-1979 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Driessen, A.

Folan, L. M.

Gorodetsky, M. L.

Greve, J.

Griffel, G.

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1978).

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]

Holler, S.

Ilchenko, V. S.

Ilchenko, V. V.

Johnson, B. R.

Keng, D.

Khoshsima, M.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, "Shift of whispering gallery modes in microspheres by protein adsorption," Opt. Lett. 28, 272-274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Klunder, D. J. W.

Krioukov, E.

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]

Lam, C. C.

Landau, L. D.

E. M. Lifshitz, L. D. Landau, and L. P. Pitaevskii, Electrodynamics of Continuous Media, Course of Theoretical Physics, 2nd ed. (Butterworth-Heinemann, 1984), Vol. 8.

Leung, P. T.

C. C. Lam, P. T. Leung, and K. Young, "Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering," J. Opt. Soc. Am. B 9, 1585-1592 (1992).
[CrossRef]

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]

Libchaber, A.

F. Vollmer, S. Arnold, D. Braun, I. Teraoka, and A. Libchaber, "Multiplexed DNA detection by optical resonances in microspheres," Biophys. J. 85, 1974-1979 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Lifshitz, E. M.

E. M. Lifshitz, L. D. Landau, and L. P. Pitaevskii, Electrodynamics of Continuous Media, Course of Theoretical Physics, 2nd ed. (Butterworth-Heinemann, 1984), Vol. 8.

Maleki, L.

Matsko, A. B.

Min, B.

A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, "Ultra-high-Q microcavity operation in H2O and D2O," Appl. Phys. Lett. 87, 151118 (2005).
[CrossRef]

Noto, M.

Otto, C.

Pitaevskii, L. P.

E. M. Lifshitz, L. D. Landau, and L. P. Pitaevskii, Electrodynamics of Continuous Media, Course of Theoretical Physics, 2nd ed. (Butterworth-Heinemann, 1984), Vol. 8.

Savchenkov, A. A.

Serpengüzel, A.

Spillane, S. M.

A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, "Ultra-high-Q microcavity operation in H2O and D2O," Appl. Phys. Lett. 87, 151118 (2005).
[CrossRef]

Teraoka, I.

M. Noto, F. Vollmer, D. Keng, I. Teraoka, and S. Arnold, "Nanolayer characterization through wavelength multiplexing of a microsphere resonator," Opt. Lett. 30, 510-512 (2005).
[CrossRef] [PubMed]

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, "Shift of whispering gallery modes in microspheres by protein adsorption," Opt. Lett. 28, 272-274 (2003).
[CrossRef] [PubMed]

F. Vollmer, S. Arnold, D. Braun, I. Teraoka, and A. Libchaber, "Multiplexed DNA detection by optical resonances in microspheres," Biophys. J. 85, 1974-1979 (2003).
[CrossRef] [PubMed]

I. Teraoka and S. Arnold, "Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium," J. Opt. Soc. Am. B 20, 1937-1946 (2003).
[CrossRef]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

I. Teraoka and S. Arnold, "Dielectric property of particles at interface in random sequential adsorption and its application to whispering gallery mode resonance-shift sensors" (submitted to J. Appl. Phys.).

Vahala, K. J.

A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, "Ultra-high-Q microcavity operation in H2O and D2O," Appl. Phys. Lett. 87, 151118 (2005).
[CrossRef]

Vollmer, F.

M. Noto, F. Vollmer, D. Keng, I. Teraoka, and S. Arnold, "Nanolayer characterization through wavelength multiplexing of a microsphere resonator," Opt. Lett. 30, 510-512 (2005).
[CrossRef] [PubMed]

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, "Shift of whispering gallery modes in microspheres by protein adsorption," Opt. Lett. 28, 272-274 (2003).
[CrossRef] [PubMed]

F. Vollmer, S. Arnold, D. Braun, I. Teraoka, and A. Libchaber, "Multiplexed DNA detection by optical resonances in microspheres," Biophys. J. 85, 1974-1979 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Young, K.

C. C. Lam, P. T. Leung, and K. Young, "Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering," J. Opt. Soc. Am. B 9, 1585-1592 (1992).
[CrossRef]

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)

Appl. Phys. Lett. (2)

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, and S. M. Spillane, "Ultra-high-Q microcavity operation in H2O and D2O," Appl. Phys. Lett. 87, 151118 (2005).
[CrossRef]

Biophys. J. (1)

F. Vollmer, S. Arnold, D. Braun, I. Teraoka, and A. Libchaber, "Multiplexed DNA detection by optical resonances in microspheres," Biophys. J. 85, 1974-1979 (2003).
[CrossRef] [PubMed]

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

J. Opt. Soc. Am. B (2)

Opt. Lett. (6)

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]

Other (3)

E. M. Lifshitz, L. D. Landau, and L. P. Pitaevskii, Electrodynamics of Continuous Media, Course of Theoretical Physics, 2nd ed. (Butterworth-Heinemann, 1984), Vol. 8.

I. Teraoka and S. Arnold, "Dielectric property of particles at interface in random sequential adsorption and its application to whispering gallery mode resonance-shift sensors" (submitted to J. Appl. Phys.).

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1978).

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Figures (7)

Fig. 1
Fig. 1

Left, particle V p placed near the surface of a transparent microsphere V 1 . The rest of the microsphere is V 2 . A fiber is side coupled to the sphere. Right, cross-sectional view of the electric field (red online) and the particle, shown for the first-order mode.

Fig. 2
Fig. 2

Factor R l plotted as a function of refractive index n p 1 in the immediate neighborhood of the resonator’s surface in the adsorption layer. The parameters used are n 1 = 1.47 , n 2 = 1.33 , and l = 250 . The solid curve (red online) is for the first-order mode ( v = 1 ) , and the dotted curve, nearly overlapping the solid curve, represents the approximation formula by Eq. (29). The factors for v = 2 , 3 are shown as a dashed and a dashed–dotted curve, respectively.

Fig. 3
Fig. 3

Plot of X l m 2 W l m , which represents the sensitivity of the resonance shift of a WGM mode with l = 500 , v = 1 to deposition of a spherical particle, as a function of θ, the polar angle of the spot where the particle lands. The numbers adjacent to the curves are denoted m . The mode of m = 500 has the most sensitive zone at the equator of the resonator.

Fig. 4
Fig. 4

TM-to-TE shift ratio of the WGM for deposition of a single spherical particle, plotted as a function of the polar angle for the spot of deposition. The WGM modes are l = 500 , m = 496 500 . The dashed curves are insensitive to the deposition.

Fig. 5
Fig. 5

TM-to-TE shift ratio for monolayer coverage by spherical particles plotted as a function of refractive index n p of the particles for n 1 = 1.47 , n 2 = 1.33 and l = 500 . Solid and dashed curves are for v = 1 , 2 . For reference, the ratio for l = 150 , v = 1 is shown as a dashed–dotted curve. The horizontal lines are the ratios when a single spherical particle is adsorbed.

Fig. 6
Fig. 6

TM-to-TE shift ratios for deposition of a spherical particle (short-dashed curve) and a cylindrical particle (lying and standing cylinders) and adsorption of a packed layer of spherical particles plotted as a function of n p , the refractive index of the particles. Parameters used are n 1 = 1.47 , n 2 = 1.33 , l = 500 , v = 1 .

Fig. 7
Fig. 7

Quasi-static TE field E p in the microsphere. The fields far from the sphere are E 01 and E 02 , respectively, in media 1 and 2. The gray (orange online) hatching represents the top surface of 1 in the TM mode.

Equations (64)

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ϵ r ( r ) = { n 1 2 , r V 1 n 2 2 , otherwise } .
δ ϵ r ( r ) = { n p 2 n 2 2 , r V p 0 , otherwise } .
× × E 0 = k 0 2 ϵ r E 0 .
× × E = k 2 ( ϵ r + δ ϵ r ) E .
V E 0 * × × E d r = k 2 V E 0 * ϵ r E d r + k 2 V p E 0 * δ ϵ r E d r .
V E 0 * × × E d r = V E × × E 0 * d r = k 0 2 V E ϵ r E 0 * d r .
( k 0 2 k 2 ) V E ϵ r E 0 * d r = k 2 V 0 E 0 * δ ϵ r E d r .
δ k k 0 = V p δ ϵ r E 0 * E p d r 2 V ϵ r E 0 * E 0 d r ,
E 0 = exp ( i m φ ) k 0 r S 0 ( r ) X l m ( θ ) ,
[ 2 r 2 l ( l + 1 ) r 2 ] S 0 = ϵ r ( r ) k 0 2 S 0 .
X l m ( θ ) = i m sin θ P l m ( cos θ ) e ̂ θ θ P l m ( cos θ ) e ̂ φ ,
S 0 ( r ) = { ψ l ( n 1 k 0 r ) , r < a B l χ l ( n 2 k 0 r ) , r > a } ,
n 1 ψ l ( n 1 k 0 a ) ψ l ( n 1 k 0 a ) = n 2 χ l ( n 2 k 0 a ) χ l ( n 2 k 0 a )
W l m 0 π sin θ d θ 0 2 π d φ X l m 2 = 4 π ( l + m ) ! l ( l + 1 ) ( l m ) ! ( 2 l + 1 ) .
ϵ r E 0 * E 0 d r = W l m k 0 2 0 ϵ r ( r ) [ S 0 ( r ) ] 2 d r .
ϵ r E 0 * E 0 d r = W l m a 2 k 0 2 ( n 1 2 n 2 2 ) [ S 0 ( a ) ] 2 .
E 0 = exp ( i m φ ) k 0 2 n 2 ( r ) [ 1 r T 0 ( r ) Y l m ( θ ) + 1 r 2 T 0 ( r ) Z l m ( θ ) ] .
Y l m ( θ ) = e ̂ r × X l m ( θ ) ,
Z l m ( θ ) = l ( l + 1 ) P l m ( cos θ ) e ̂ r ,
1 n 1 ψ l ( n 1 k 0 a ) ψ l ( n 1 k 0 a ) = 1 n 2 χ l ( n 2 k 0 a ) χ l ( n 2 k 0 a ) .
ϵ r E 0 * E 0 d r = W l m k 0 4 0 1 n 4 ( r ) { [ T 0 ( r ) ] 2 + l ( l + 1 ) r 2 [ T 0 ( r ) ] 2 } d r .
ϵ r E 0 * E 0 d r = W l m a 2 k 0 2 ( n 1 2 n 2 2 1 ) [ T 0 ( a ) ] 2 { [ χ l ( n 2 k 0 a ) χ l ( n 2 k 0 a ) ] 2 + l ( l + 1 ) ( n 1 k 0 a ) 2 } .
V p δ ϵ r E 0 * E p d r = W l m k 0 2 a a + t δ ( n 2 ) [ S 0 ( r ) ] 2 d r .
( δ k k 0 ) TE = 1 a ( n 1 2 n 2 2 ) a a + t δ ( n 2 ) d r .
E p = exp ( i m φ ) k 0 2 n p 2 [ 1 r T ( r ) Y l m ( θ ) + 1 r 2 T ( r ) Z l m ( θ ) ] ,
V p δ ϵ r E 0 * E p d r = W l m k 0 4 a a + t δ ( n 2 ) [ T ( r ) T 0 ( r ) + l ( l + 1 ) r 2 T ( r ) T 0 ( r ) ] d r ,
V p δ ϵ r E 0 * E p d r = W l m k 0 2 [ T 0 ( a ) ] 2 [ n p 1 2 ( χ l χ l ) 2 + l ( l + 1 ) ( k 0 a ) 2 ] a a + t δ ( n 2 ) d r ,
( δ k k 0 ) TM = R l ( k 0 a ; n 1 , n 2 , n p 1 ) n 2 2 n p 1 2 a ( n 1 2 n 2 2 ) a a + t δ ( n 2 ) d r ,
R l ( x ; n 1 , n 2 , n p ) 1 + l ( l + 1 ) ( n p 2 n 1 2 ) x 2 ( χ l χ l ) 2 + l ( l + 1 ) ( n 1 x ) 2 .
R l ( x ; n 1 , n 2 , n p ) 1 + n p 2 n 1 2 n 2 2 ( k 0 a l ) 2 + n 1 2 .
( δ k k 0 ) TM ( δ k k 0 ) TE = n 2 2 n p 1 2 a a + t δ ( n 2 ) d r a a + t δ ( n 2 ) d r R l ( k 0 a ; n 1 , n 2 , n p 1 ) .
( δ k k 0 ) TE = n p 2 n 2 2 n 1 2 n 2 2 t a ,
( δ k k 0 ) TM ( δ k k 0 ) TE = R l ( k 0 a ; n 1 , n 2 , n p ) .
E p = 3 n 2 2 2 n 2 2 + n p 2 E 02 .
V p δ ϵ r E 0 * E p d r = ( n p 2 n 2 2 ) 3 n 2 2 2 n 2 2 + n p 2 V p E 0 2 ,
V p δ ϵ r E 0 * E p d r = ( n p 2 n 2 2 ) 3 n 2 2 2 n 2 2 + n p 2 [ S 0 ( a ) ] 2 k 0 2 a 2 V p X l m ( θ ) 2 .
( δ k k 0 ) TE = ( η W l m ) ( V p a 3 ) X l m ( θ ) 2 ,
η n p 2 n 2 2 n 1 2 n 2 2 3 n 2 2 2 n 2 2 + n p 2 .
V p δ ϵ r E 0 * E p d r = ( n p 2 n 2 2 ) 3 n 2 2 2 n 2 2 + n p 2 [ T 0 ( a ) ] 2 ( n 2 k 0 a ) 2 V p [ ( χ l χ l ) 2 Y l m ( θ ) 2 + Z l m ( θ ) 2 ( n 2 k 0 a ) 2 ] ,
( δ k k 0 ) TM = η W l m V p a 3 ( χ l χ l ) 2 Y l m ( θ ) 2 + ( n 2 k 0 a ) 2 Z l m 2 ( χ l χ l ) 2 + l ( l + 1 ) ( n 1 k 0 a ) 2 .
( δ k k 0 ) TM ( δ k k 0 ) TE = ( χ l χ l ) 2 + ( n 2 k 0 a ) 2 Z l m 2 X l m 2 ( χ l χ l ) 2 + l ( l + 1 ) ( n 1 k 0 a ) 2 ,
( δ k k 0 ) TE = ( η 4 π ) ( V p a 3 ) N .
( δ k k 0 ) TE = ( η 4 π ) ( V p a 2 ) N R l ( k 0 a ; n 1 , n 2 , n 2 ) .
δ ( n 2 ) = 2 π 3 1 2 ( n p 2 n 2 2 ) ( r a ) ( 2 b r + a ) ( 2 b ) 2
( δ k k 0 ) TE , MF = 2 π 3 3 2 n p 2 n 2 2 n 1 2 n 2 2 b a .
( δ k k 0 ) TM , MF = 2 n 2 2 n 1 2 n 2 2 b a 1 n 2 2 B C ln n 2 C B R l ( k 0 , a ; n 1 , n 2 , n 2 ) ,
( δ k k 0 ) TE , NI = ( 2 π 3 3 2 ) η ( b a ) ,
( δ k k 0 ) TM , NI = ( 2 π 3 3 2 ) η ( b a ) R l ( k 0 a ; n 1 , n 2 , n 2 ) .
E p = 2 n p 2 n 2 2 + n p 2 E 02 .
( δ k k 0 ) TM ( δ k k 0 ) TE = ( χ l χ l ) 2 + [ l ( l + 1 ) ( n 2 k 0 a ) 2 ] 4 n 2 2 ( 3 n 2 2 + n p 2 ) ( χ l χ l ) 2 + [ l ( l + 1 ) ( n 1 k a ) 2 ] .
( δ k k 0 ) TM ( δ k k 0 ) TE = ( χ l χ l ) 2 + [ l ( l + 1 ) ( n 2 k 0 a ) 2 ] ( n 2 2 + n p 2 ) 2 n 2 2 ( χ l χ l ) 2 + [ l ( l + 1 ) ( n 1 k a ) 2 ] .
I i = 1 , 2 , p V i ( E 0 * × × E E × × E 0 * ) d r .
V i ( E 0 * × × E E × × E 0 * ) d r = S i d S ( E × × E 0 * E 0 * × × E ) ( i = 1 , 2 , p ) ,
I = I 1 I 2 + I 3 I 4 ,
I 1 = S 1 d S ( E 1 × × E 01 * E 2 × × E 02 * ) ,
I 2 = S 1 d S ( E 01 * × × E 1 E 02 * × × E 2 ) ,
I 3 = S p d S ( E p E 2 ) × × E 02 * ,
I 4 = S p d S E 02 * × × ( E p E 2 ) .
I 1 = i ω 0 S 1 d S ( E 1 E 2 ) × B 0 * .
Φ P ( r , θ , φ ) = Φ P 0 + l , m A l m r l Y l m ( θ , φ ) ,
E P z = l , m A l m l r l 1 cos θ Y l m ( θ , φ ) + l , m A l m r l 1 sin θ θ Y l m ( θ , φ ) .
V p E 02 * E p d r = ( 4 π 3 ) 1 2 b 3 E 02 A 1 0 .
Φ 2 ( r , θ , φ ) = ( 4 π 3 ) 1 2 E 02 r Y 1 0 ( θ , φ ) + l , m C l m r l 1 Y l m ( θ , φ ) ,
V p E 02 * E p d r = 3 n 2 2 2 n 2 2 + n P 2 V p E 02 2 ,

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