Abstract

A theory is presented that considers the resonance shift of degenerate whispering-gallery modes (WGMs) in sensor applications. The theory is then applied to a pair of counterpropagating waves in a spheroidal resonator. Adsorption of a particle lifts the twofold degeneracy, resulting in a pair of standing waves with a symmetric field around the particle, a standing symmetric wave (SSW) and an antisymmetric wave (ASW). The shift for a SSW is twice as large as the one for a nondegenerate WGM when the particle radius is sufficiently smaller than the wavelength, whereas the shift for an ASW is nearly zero. The ratio of the split to the mean shift gives an estimate of the particle size, whereas the mean shift is sensitive to its polarizability. With an increasing particle radius, the ratio of the split to the mean shift decreases. There is a particle radius that maximizes the split.

© 2009 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]

2008 (1)

2007 (5)

M. Noto, D. Keng, I. Teraoka, and S. Arnold, “Detection of protein orientation on silica microsphere surface using TE/TM whispering gallery modes,” Biophys. J. 92, 4466-4472 (2007).
[CrossRef] [PubMed]

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, “Integrated refractive index optical ring resonator detector for capillary electrophoresis,” Anal. Chem. 79, 930-937 (2007).
[CrossRef] [PubMed]

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783-787 (2007).
[CrossRef] [PubMed]

I. Teraoka and S. Arnold, “Estimation of surface density of molecules adsorbed on a whispering gallery mode resonator: utility of isotropic polarizability,” J. Appl. Phys. 102, 076109 (2007).
[CrossRef]

M. Mohageg, A. Savchenkov, and L. Maleki, “Coherent backscattering in lithium niobate whispering-gallery-mode resonators,” Opt. Lett. 32, 2574-2576 (2007).
[CrossRef] [PubMed]

2006 (3)

2005 (1)

N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87, 201107 (2005).
[CrossRef]

2003 (3)

2002 (4)

1996 (1)

1995 (1)

1993 (1)

Armani, A. M.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783-787 (2007).
[CrossRef] [PubMed]

Arnold, S.

I. Teraoka and S. Arnold, “Estimation of surface density of molecules adsorbed on a whispering gallery mode resonator: utility of isotropic polarizability,” J. Appl. Phys. 102, 076109 (2007).
[CrossRef]

M. Noto, D. Keng, I. Teraoka, and S. Arnold, “Detection of protein orientation on silica microsphere surface using TE/TM whispering gallery modes,” Biophys. J. 92, 4466-4472 (2007).
[CrossRef] [PubMed]

O. Gaathon, J. Culic-Viskota, M. Mihnev, I. Teraoka and S. Arnold, “Enhancing sensitivity of a whispering gallery mode bio-sensor by subwavelength confinement,” Appl. Phys. Lett. 89, 223901 (2006).
[CrossRef]

I. Teraoka and S. Arnold, “Theory on resonance shifts in TE and TM whispering-gallery modes by non-radial perturbations for sensing applications,” J. Opt. Soc. Am. B 23, 1381-1389 (2006).
[CrossRef]

I. Teraoka and S. Arnold, “Enhancing sensitivity of a whispering gallery mode microsphere sensor by a high-refractive index surface layer,” J. Opt. Soc. Am. B 23, 1434-1441 (2006).
[CrossRef]

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]

I. Teraoka, S. Arnold, and F. Vollmer, “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, 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, 4057-4049 (2002).
[CrossRef]

Astratov, V. N.

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, 4057-4049 (2002).
[CrossRef]

Culic-Viskota, J.

O. Gaathon, J. Culic-Viskota, M. Mihnev, I. Teraoka and S. Arnold, “Enhancing sensitivity of a whispering gallery mode bio-sensor by subwavelength confinement,” Appl. Phys. Lett. 89, 223901 (2006).
[CrossRef]

Driessen, A.

Fan, X.

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, “Integrated refractive index optical ring resonator detector for capillary electrophoresis,” Anal. Chem. 79, 930-937 (2007).
[CrossRef] [PubMed]

N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87, 201107 (2005).
[CrossRef]

Flagan, R. C.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783-787 (2007).
[CrossRef] [PubMed]

Fraser, S. E.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783-787 (2007).
[CrossRef] [PubMed]

Gaathon, O.

O. Gaathon, J. Culic-Viskota, M. Mihnev, I. Teraoka and S. Arnold, “Enhancing sensitivity of a whispering gallery mode bio-sensor by subwavelength confinement,” Appl. Phys. Lett. 89, 223901 (2006).
[CrossRef]

Gorodetsky, M. L.

Greve, J.

Hanumegowda, N. M.

N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87, 201107 (2005).
[CrossRef]

Hare, J.

Haroche, S.

Hiremath, K. R.

Holler, S.

Ilchenko, V. S.

Johnson, B. R.

Keng, D.

M. Noto, D. Keng, I. Teraoka, and S. Arnold, “Detection of protein orientation on silica microsphere surface using TE/TM whispering gallery modes,” Biophys. J. 92, 4466-4472 (2007).
[CrossRef] [PubMed]

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, 4057-4049 (2002).
[CrossRef]

Kippenberg, T. J.

Klunder, D. J. W.

Krioukov, E.

Kulkarni, R. P.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783-787 (2007).
[CrossRef] [PubMed]

Lee, K.-M.

Lefèvre-Seguin, V.

Leung, P.-T.

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, 4057-4049 (2002).
[CrossRef]

Maleki, L.

Mihnev, M.

O. Gaathon, J. Culic-Viskota, M. Mihnev, I. Teraoka and S. Arnold, “Enhancing sensitivity of a whispering gallery mode bio-sensor by subwavelength confinement,” Appl. Phys. Lett. 89, 223901 (2006).
[CrossRef]

Mohageg, M.

Ng, S.-W.

Noto, M.

M. Noto, D. Keng, I. Teraoka, and S. Arnold, “Detection of protein orientation on silica microsphere surface using TE/TM whispering gallery modes,” Biophys. J. 92, 4466-4472 (2007).
[CrossRef] [PubMed]

Otto, C.

Patel, B. C.

N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87, 201107 (2005).
[CrossRef]

Raimond, J.-M.

Sandoghdar, V.

Savchenkov, A.

Savchenkov, A. A.

Spillane, S. M.

Stica, C. J.

N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87, 201107 (2005).
[CrossRef]

Suter, J. D.

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, “Integrated refractive index optical ring resonator detector for capillary electrophoresis,” Anal. Chem. 79, 930-937 (2007).
[CrossRef] [PubMed]

Teraoka, I.

M. Noto, D. Keng, I. Teraoka, and S. Arnold, “Detection of protein orientation on silica microsphere surface using TE/TM whispering gallery modes,” Biophys. J. 92, 4466-4472 (2007).
[CrossRef] [PubMed]

I. Teraoka and S. Arnold, “Estimation of surface density of molecules adsorbed on a whispering gallery mode resonator: utility of isotropic polarizability,” J. Appl. Phys. 102, 076109 (2007).
[CrossRef]

I. Teraoka and S. Arnold, “Enhancing sensitivity of a whispering gallery mode microsphere sensor by a high-refractive index surface layer,” J. Opt. Soc. Am. B 23, 1434-1441 (2006).
[CrossRef]

O. Gaathon, J. Culic-Viskota, M. Mihnev, I. Teraoka and S. Arnold, “Enhancing sensitivity of a whispering gallery mode bio-sensor by subwavelength confinement,” Appl. Phys. Lett. 89, 223901 (2006).
[CrossRef]

I. Teraoka and S. Arnold, “Theory on resonance shifts in TE and TM whispering-gallery modes by non-radial perturbations for sensing applications,” J. Opt. Soc. Am. B 23, 1381-1389 (2006).
[CrossRef]

I. Teraoka, S. Arnold, and F. Vollmer, “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, 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]

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, 4057-4049 (2002).
[CrossRef]

Vahala, K. J.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783-787 (2007).
[CrossRef] [PubMed]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. 27, 1669-1671 (2002).
[CrossRef]

Vollmer, F.

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, S. Arnold, and F. Vollmer, “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]

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, 4057-4049 (2002).
[CrossRef]

Weiss, D. S.

White, I.

N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87, 201107 (2005).
[CrossRef]

White, I. M.

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, “Integrated refractive index optical ring resonator detector for capillary electrophoresis,” Anal. Chem. 79, 930-937 (2007).
[CrossRef] [PubMed]

Zhu, H.

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, “Integrated refractive index optical ring resonator detector for capillary electrophoresis,” Anal. Chem. 79, 930-937 (2007).
[CrossRef] [PubMed]

Zourob, M.

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, “Integrated refractive index optical ring resonator detector for capillary electrophoresis,” Anal. Chem. 79, 930-937 (2007).
[CrossRef] [PubMed]

Anal. Chem. (1)

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, “Integrated refractive index optical ring resonator detector for capillary electrophoresis,” Anal. Chem. 79, 930-937 (2007).
[CrossRef] [PubMed]

Appl. Phys. Lett. (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, 4057-4049 (2002).
[CrossRef]

N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87, 201107 (2005).
[CrossRef]

O. Gaathon, J. Culic-Viskota, M. Mihnev, I. Teraoka and S. Arnold, “Enhancing sensitivity of a whispering gallery mode bio-sensor by subwavelength confinement,” Appl. Phys. Lett. 89, 223901 (2006).
[CrossRef]

Biophys. J. (2)

M. Noto, D. Keng, I. Teraoka, and S. Arnold, “Detection of protein orientation on silica microsphere surface using TE/TM whispering gallery modes,” Biophys. J. 92, 4466-4472 (2007).
[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]

J. Appl. Phys. (1)

I. Teraoka and S. Arnold, “Estimation of surface density of molecules adsorbed on a whispering gallery mode resonator: utility of isotropic polarizability,” J. Appl. Phys. 102, 076109 (2007).
[CrossRef]

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

J. Opt. Soc. Am. B (4)

Opt. Express (1)

Opt. Lett. (6)

Science (1)

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783-787 (2007).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Two resonance shifts caused by adsorption of two meridional strips plotted as a function of ϕ p , the angular distance between the two strips. The signs for symmetric standing wave (SSW, −) and antisymmetric standing wave (ASW,+) correspond to the plus–minus sign in Eq. (17) Also shown are the eigenfunctions (amplitude of E θ at the peak of | P l m ( cos θ ) | ) for m ϕ p = π 3 and 4 π 3 . For each ϕ p , the center positions of the strips ( ϕ = 0 and ϕ p ) are indicated by gray lines.

Fig. 2
Fig. 2

Mean and split of fractional resonance shifts of degenerate WGM in adsorption of a spherical particle of R I = 1.58 onto the equator of a spherical resonator of radius 200 μ m and R I = 1.452 , immersed in a medium of R I = 1.32 , plotted as a function of particle radius R (solid curves). The wavelength is 1.32 μ m , and the WGM has m = l and ν = 1 . Also shown are the mean shift and split by a Gaussian permittivity profile (dotted curves).

Fig. 3
Fig. 3

Mean and split of fractional resonance shifts of degenerate WGM in adsorption of a spherical particle of R I = 1.58 onto the equator of a spherical resonator of radius 200 μ m and R I = 1.452 immersed in a medium of R I = 1.32 , plotted as a function of particle radius R. The means and splits are compared for three radial modes ( ν = 1 , solid curve; 2, dotted curve; 3, dash-dotted curve). The wavelength is 1.32 μ m , and the WGM has m = l .

Fig. 4
Fig. 4

Mean and split of fractional resonance shifts of degenerate WGM, in adsorption of a spherical particle of R I = 1.58 onto the equator of a spherical resonator of radius 100 μ m and R I = 1.452 , immersed in a medium of R I = 1.32 , plotted as a function of particle radius R. The means and splits are compared for three wavelengths ( 1.32 μ m , solid curve; 1.06 μ m , dotted curve; 0.78 μ m , dash-dotted curve). The WGM has m = l and ν = 1 .

Fig. 5
Fig. 5

The ratio of the split to the mean of the shifts for two degenerate WGM, plotted as a function of k 0 R is compared for different radial modes ν, values of m, resonator radii a, and wavelengths λ. The curves were calculated for landing of a spherical adsorbate of R I = 1.58 onto the equator of a spherical resonator of R I = 1.452 immersed in a medium of R I = 1.32 , but landing on the other parts of the resonator produce overlapping curves.

Fig. 6
Fig. 6

Split between SSW and ASW resonances (in terms of k ) in adsorption of a spherical particle of R I = 1.58 onto the equator of a spherical resonator of R I = 1.452 immersed in a medium of R I = 1.32 , plotted as a function of the resonator radius a (solid lines). The wavelength is indicated adjacent to each line. Also shown are the lines for the intrinsic linewidth of WGM (dotted lines) and the curves for the total linewidth (dash-dotted curve) assuming a nonintrinsic linewidth of 1 × 10 8 relative to k 0 .

Equations (41)

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

× × E i ( 0 ) = k 0 2 ɛ r E i ( 0 ) ,
× × i = 1 N c i E i = k 2 ( ɛ r + δ ɛ r ) i = 1 N c i E i ,
i = 1 N c i E j ( 0 ) * × × E i d r = k 2 i = 1 N c i ( ɛ r + δ ɛ r ) E j ( 0 ) * E i d r .
E j ( 0 ) * × × E i d r = E i × × E j ( 0 ) * d r = k 0 2 ɛ r E i E j ( 0 ) * d r .
k 0 2 i c i ɛ r E j ( 0 ) * E i d r = k 2 i c i ( ɛ r + δ ɛ r ) E j ( 0 ) * E i d r .
δ ( k 2 ) i c i ɛ r E j ( 0 ) * E i ( 0 ) d r + k 0 2 i c i δ ɛ r E j ( 0 ) * E i d r = 0 .
δ ( k 2 ) c j ( ɛ r ) j j + k 0 2 i = 1 N c i ( δ ɛ r ) j i = 0 ,
E f ( 0 ) = exp ( i m ϕ ) k 0 r S l ( r ) X l m ( θ ) ,
E b ( 0 ) = exp ( i m ϕ ) k 0 r S l ( r ) X l m * ( θ ) ,
X l m ( θ ) i e ̂ θ m sin θ P l m ( cos θ ) e ̂ ϕ P l m ( cos θ ) θ ,
( ɛ r ) ff = ( ɛ r ) bb = 1 k 0 2 r 2 n 2 S l 2 | X l m | 2 d r = a 2 k 0 2 ( n 1 2 n 2 2 ) W l m [ S l ( a ) ] 2 ,
δ ɛ r = ( n p 2 n 2 2 ) V p a 2 δ ( r a + ) 1 2 ( β p π ) 1 2 exp ( β p ϕ 2 ) ,
( δ ɛ r ) ff = ( δ ɛ r ) bb = ( n p 2 n 2 2 ) V p k 0 2 a 2 [ S l ( a ) ] 2 W l m 4 π ,
( δ ɛ r ) fb = ( δ ɛ r ) bf * = ( n p 2 n 2 2 ) V p k 0 2 a 2 [ S l ( a ) ] 2 L l m 4 π exp ( m 2 β p ) .
| δ ( k 2 ) ( ɛ r ) ff + k 0 2 ( δ ɛ r ) ff k 0 2 ( δ ɛ r ) fb k 0 2 ( δ ɛ r ) fb δ ( k 2 ) ( ɛ r ) ff + k 0 2 ( δ ɛ r ) ff | = 0
δ ( k 2 ) k 0 2 = ( δ ɛ r ) ff ± ( δ ɛ r ) fb ( ɛ r ) ff = n p 2 n 2 2 n 1 2 n 2 2 V p 2 π a 3 [ 1 ± L l m W l m exp ( m 2 β p ) ] ,
δ ɛ r = ( n p 2 n 2 2 ) V p a 2 δ ( r a + ) 1 2 ( β p π ) 1 2 { exp ( β p ϕ 2 ) + exp [ β p ( ϕ ϕ p ) 2 ] } .
( δ ɛ r ) ff = ( δ ɛ r ) bb = ( n p 2 n 2 2 ) V p k 0 2 a 2 [ S l ( a ) ] 2 W l m 2 π ,
( δ ɛ r ) fb = ( δ ɛ r ) bf * = ( n p 2 n 2 2 ) V p k 0 2 a 2 [ S l ( a ) ] 2 L l m 4 π exp ( m 2 β p ) [ 1 + exp ( 2 i m ϕ p ) ] .
δ ( k 2 ) k 0 2 = n p 2 n 2 2 n 1 2 n 2 2 V p π a 3 [ 1 ± L l m W l m exp ( m 2 β p ) cos m ϕ p ] .
E ± ( r , θ , ϕ ) = S l ( r ) 2 1 2 k r { exp ( i m ϕ ) X l m ( θ ) ± exp [ i m ( ϕ p ϕ ) ] X l m * ( θ ) } ,
δ ( k 2 ) k 0 2 = n p 2 n 2 2 n 1 2 n 2 2 V p 2 π a 3 [ N p ± L l m W l m exp ( m 2 β p ) | i = 1 N p exp ( 2 i m ϕ i ) | ] ,
δ ɛ r = α ex ɛ 0 1 a 2 sin θ p δ ( r a + ) δ ( θ θ p ) δ ( ϕ ϕ p ) ,
( δ ɛ r ) ff = ( δ ɛ r ) bb = α ex ɛ 0 1 k 0 2 a 2 [ S l ( a ) ] 2 | X l m ( θ p ) | 2 ,
( δ ɛ r ) fb = ( δ ɛ r ) bf * = α ex ɛ 0 1 k 0 2 a 2 [ S l ( a ) ] 2 [ X l m ( θ p ) ] 2 exp ( 2 i m ϕ p ) .
δ ( k 2 ) k 0 2 = 2 α ex ɛ 0 ( n 1 2 n 2 2 ) a 3 | X l m ( θ p ) | 2 ± [ X l m ( θ p ) ] 2 W l m ,
α ex ɛ 0 V p = ( n p 2 n 2 2 ) 3 n 2 2 n p 2 + 2 n 2 2 ,
δ ( k 2 ) k 0 2 = 2 α ex ɛ 0 ( n 1 2 n 2 2 ) a 3 × | X l m ( θ p 1 ) | 2 + | X l m ( θ p 2 ) | 2 ± { [ X l m ( θ p 1 ) ] 2 + [ X l m ( θ p 2 ) ] 2 cos m ( ϕ p 1 ϕ p 2 ) } W l m .
δ ɛ r = α ex ɛ 0 r p 3 ( β π ) 3 2 exp [ β r p 2 ( r r p ) 2 β ( θ θ p ) 2 β ϕ 2 sin 2 θ p ] ,
( δ ɛ r ) ff = ( δ ɛ r ) bb α ex ɛ 0 k 0 2 r p 3 β π [ S l ( r ) ] 2 exp [ β r p 2 ( r r p ) 2 ] d r exp [ β ( θ θ p ) 2 ] | X l m ( θ ) | 2 d θ .
( δ ɛ r ) ff = α ex ɛ 0 k 0 2 a 2 [ S l ( a ) ] 2 exp [ 2 Γ R + Γ 2 ( a + R ) 2 β ] × { | X l m ( θ p ) | 2 + 1 2 β [ | X l m ( θ p ) | 2 + X l m * ( θ p ) X l m ( θ p ) ] } ,
( δ ɛ r ) fb = α ex ɛ 0 k 0 2 a 2 [ S l ( a ) ] 2 exp [ 2 Γ R + Γ 2 ( a + R ) 2 β m 2 β sin 2 θ p ] × ( [ X l m ( θ p ) ] 2 + 1 2 β { [ X l m ( θ p ) ] 2 + X l m ( θ p ) X l m ( θ p ) } ) .
δ k mean k 0 = α ex ɛ 0 ( n 1 2 n 2 2 ) a 3 [ 2 Γ R + Γ 2 ( a + R ) 2 β ] | X l m ( θ p ) | 2 W l m ,
Δ k sp k 0 = α ex ɛ 0 ( n 1 2 n 2 2 ) a 3 exp [ 2 Γ R + Γ 2 ( a + R ) 2 β ] 2 exp ( m 2 β sin 2 θ p ) [ X l m ( θ p ) ] 2 W l m .
δ k mean k 0 = 4 π n p 2 n 2 2 n 1 2 n 2 2 n 2 2 n p 2 + 2 n 2 2 ( R a ) 3 exp [ 2 Γ R + ( Γ R ) 2 C ] | X l m ( θ p ) | 2 W l m ,
Δ k sp k 0 = 4 π n p 2 n 2 2 n 1 2 n 2 2 n 2 2 n p 2 + 2 n 2 2 ( R a ) 3 exp [ 2 Γ R + ( Γ R ) 2 C 1 C ( R m a sin θ p ) 2 ] 2 [ X l m ( θ p ) ] 2 W l m ,
R peak = 3 Γ ( 1 + { 1 + ( 6 C ) [ m 2 ( Γ a sin θ p ) 2 1 ] } 1 2 ) 1 .
k 0 R peak = 3 { ( n eff 2 n 2 2 ) 1 2 + [ n eff 2 + ( 6 C 1 ) n 2 2 ] 1 2 } 1 ,
( δ ɛ r ) ff = 2 α ex ɛ 0 k 0 2 V p r p R r p + R [ S l ( r ) ] 2 d r θ θ + | X l m ( θ ) | 2 ϕ + sin θ d θ ,
( δ ɛ r ) fb = α ex ɛ 0 k 0 2 V p r p R r p + R [ S l ( r ) ] 2 d r θ θ + | X l m ( θ ) | 2 1 m sin ( 2 m ϕ + ) sin θ d θ ,
4 π R 3 3 I = 1 + 4 π 1 2 y exp ( y 2 ) + 2 erfc y + 1 3 ( 2 π ) 1 2 y 3     ,

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