Abstract

We present a variational method for whispering gallery modes (WGMs) and apply it to sensor responses of WGMs in a spherical resonator to two types of perturbation for which we know the exact answers. The perturbations are adsorption of a thin dielectric layer and a uniform change in the surroundings’ refractive index. The variational method gives the perturbed wave function and, if a suitable trial function is chosen, the resonance wavelength shift up to the second order in the perturbation. The linear part is identical to the result of the first-order perturbation theory.

© 2008 Optical Society of America

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References

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  1. 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]
  2. 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]
  3. A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Review of applications of whispering-gallery mode resonators in photonics and nonlinear optics,” in The Interplanetary Network Progress Report, Vol. 42-162 (NASA, 2005), pp. 1-51.
  4. F. Baldini and A. Giannetti, “Optical chemical and biochemical sensors: new trends,” Proc. SPIE 5826, 485-496 (2005).
    [CrossRef]
  5. 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]
  6. 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]
  7. 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]
  8. 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]
  9. C. Boozer, G. Kim, S. Cong, H. W. Guan, and T. Londergan, “Looking towards label-free biomolecular interaction analysis in a high-throughput format: a review of new surface plasmon resonance technologies,” Curr. Opin. Biotechnol. 17, 400-405 (2006).
    [CrossRef] [PubMed]
  10. A. Janshoff and C. Steinem, “Label-free detection of protein-ligand interactions by the quartz crystal microbalance,” Methods Mol. Biol. 305, 47-64 (2005).
    [PubMed]
  11. 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]
  12. S. Arnold and S. Holler, “Microparticle photophysics: fluorescence microscopy and spectroscopy of a photonic atom,” in Cavity-Enhanced Spectroscopies, R.D.van Zee and J.P.Looney, eds., Vol. 40 of Experimental Methods in the Physical Sciences (Academic, 2002).
  13. 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]
  14. I. M. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett. 31, 1319-1321 (2006).
    [CrossRef] [PubMed]
  15. I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33-38 (2006).
    [CrossRef]
  16. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10, 343-352 (1993).
    [CrossRef]

2007 (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]

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]

2006 (5)

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]

I. M. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett. 31, 1319-1321 (2006).
[CrossRef] [PubMed]

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33-38 (2006).
[CrossRef]

C. Boozer, G. Kim, S. Cong, H. W. Guan, and T. Londergan, “Looking towards label-free biomolecular interaction analysis in a high-throughput format: a review of new surface plasmon resonance technologies,” Curr. Opin. Biotechnol. 17, 400-405 (2006).
[CrossRef] [PubMed]

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]

2005 (3)

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]

A. Janshoff and C. Steinem, “Label-free detection of protein-ligand interactions by the quartz crystal microbalance,” Methods Mol. Biol. 305, 47-64 (2005).
[PubMed]

F. Baldini and A. Giannetti, “Optical chemical and biochemical sensors: new trends,” Proc. SPIE 5826, 485-496 (2005).
[CrossRef]

2003 (2)

2002 (1)

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]

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.

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, “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]

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]

S. Arnold and S. Holler, “Microparticle photophysics: fluorescence microscopy and spectroscopy of a photonic atom,” in Cavity-Enhanced Spectroscopies, R.D.van Zee and J.P.Looney, eds., Vol. 40 of Experimental Methods in the Physical Sciences (Academic, 2002).

Baldini, F.

F. Baldini and A. Giannetti, “Optical chemical and biochemical sensors: new trends,” Proc. SPIE 5826, 485-496 (2005).
[CrossRef]

Boozer, C.

C. Boozer, G. Kim, S. Cong, H. W. Guan, and T. Londergan, “Looking towards label-free biomolecular interaction analysis in a high-throughput format: a review of new surface plasmon resonance technologies,” Curr. Opin. Biotechnol. 17, 400-405 (2006).
[CrossRef] [PubMed]

Braun, D.

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]

Cong, S.

C. Boozer, G. Kim, S. Cong, H. W. Guan, and T. Londergan, “Looking towards label-free biomolecular interaction analysis in a high-throughput format: a review of new surface plasmon resonance technologies,” Curr. Opin. Biotechnol. 17, 400-405 (2006).
[CrossRef] [PubMed]

Fan, X.

I. M. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett. 31, 1319-1321 (2006).
[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]

Giannetti, A.

F. Baldini and A. Giannetti, “Optical chemical and biochemical sensors: new trends,” Proc. SPIE 5826, 485-496 (2005).
[CrossRef]

Grudinin, I. S.

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33-38 (2006).
[CrossRef]

Guan, H. W.

C. Boozer, G. Kim, S. Cong, H. W. Guan, and T. Londergan, “Looking towards label-free biomolecular interaction analysis in a high-throughput format: a review of new surface plasmon resonance technologies,” Curr. Opin. Biotechnol. 17, 400-405 (2006).
[CrossRef] [PubMed]

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]

Holler, S.

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]

S. Arnold and S. Holler, “Microparticle photophysics: fluorescence microscopy and spectroscopy of a photonic atom,” in Cavity-Enhanced Spectroscopies, R.D.van Zee and J.P.Looney, eds., Vol. 40 of Experimental Methods in the Physical Sciences (Academic, 2002).

Ilchenko, V. S.

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33-38 (2006).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Review of applications of whispering-gallery mode resonators in photonics and nonlinear optics,” in The Interplanetary Network Progress Report, Vol. 42-162 (NASA, 2005), pp. 1-51.

Janshoff, A.

A. Janshoff and C. Steinem, “Label-free detection of protein-ligand interactions by the quartz crystal microbalance,” Methods Mol. Biol. 305, 47-64 (2005).
[PubMed]

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]

Kim, G.

C. Boozer, G. Kim, S. Cong, H. W. Guan, and T. Londergan, “Looking towards label-free biomolecular interaction analysis in a high-throughput format: a review of new surface plasmon resonance technologies,” Curr. Opin. Biotechnol. 17, 400-405 (2006).
[CrossRef] [PubMed]

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]

Libchaber, A.

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]

Londergan, T.

C. Boozer, G. Kim, S. Cong, H. W. Guan, and T. Londergan, “Looking towards label-free biomolecular interaction analysis in a high-throughput format: a review of new surface plasmon resonance technologies,” Curr. Opin. Biotechnol. 17, 400-405 (2006).
[CrossRef] [PubMed]

Maleki, L.

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33-38 (2006).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Review of applications of whispering-gallery mode resonators in photonics and nonlinear optics,” in The Interplanetary Network Progress Report, Vol. 42-162 (NASA, 2005), pp. 1-51.

Matsko, A. B.

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33-38 (2006).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Review of applications of whispering-gallery mode resonators in photonics and nonlinear optics,” in The Interplanetary Network Progress Report, Vol. 42-162 (NASA, 2005), pp. 1-51.

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]

Oveys, H.

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]

Savchenkov, A. A.

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33-38 (2006).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Review of applications of whispering-gallery mode resonators in photonics and nonlinear optics,” in The Interplanetary Network Progress Report, Vol. 42-162 (NASA, 2005), pp. 1-51.

Steinem, C.

A. Janshoff and C. Steinem, “Label-free detection of protein-ligand interactions by the quartz crystal microbalance,” Methods Mol. Biol. 305, 47-64 (2005).
[PubMed]

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]

Strekalov, D.

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33-38 (2006).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Review of applications of whispering-gallery mode resonators in photonics and nonlinear optics,” in The Interplanetary Network Progress Report, Vol. 42-162 (NASA, 2005), pp. 1-51.

Teraoka, I.

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]

Vollmer, F.

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.

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, 4057-4049 (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]

Biophys. J. (1)

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]

Curr. Opin. Biotechnol. (1)

C. Boozer, G. Kim, S. Cong, H. W. Guan, and T. Londergan, “Looking towards label-free biomolecular interaction analysis in a high-throughput format: a review of new surface plasmon resonance technologies,” Curr. Opin. Biotechnol. 17, 400-405 (2006).
[CrossRef] [PubMed]

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

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

Methods Mol. Biol. (1)

A. Janshoff and C. Steinem, “Label-free detection of protein-ligand interactions by the quartz crystal microbalance,” Methods Mol. Biol. 305, 47-64 (2005).
[PubMed]

Opt. Commun. (1)

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun. 265, 33-38 (2006).
[CrossRef]

Opt. Lett. (2)

Proc. SPIE (1)

F. Baldini and A. Giannetti, “Optical chemical and biochemical sensors: new trends,” Proc. SPIE 5826, 485-496 (2005).
[CrossRef]

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]

Other (2)

S. Arnold and S. Holler, “Microparticle photophysics: fluorescence microscopy and spectroscopy of a photonic atom,” in Cavity-Enhanced Spectroscopies, R.D.van Zee and J.P.Looney, eds., Vol. 40 of Experimental Methods in the Physical Sciences (Academic, 2002).

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Review of applications of whispering-gallery mode resonators in photonics and nonlinear optics,” in The Interplanetary Network Progress Report, Vol. 42-162 (NASA, 2005), pp. 1-51.

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Figures (5)

Fig. 1
Fig. 1

Radial function S l ( r ) (solid curve, vertically shifted) of the first-order radial mode for WGM with l = 677 in a sphere of radius a = 100 μ m , n 1 = 1.452 , surrounded by a medium of n 2 = 1.32 at λ 1.32 μ m . Also shown is S l ( r ) (dashed curve) and the difference in S l ( r ) when the sphere acquires a layer of t = 1 nm , n p = 1.6 , magnified by 1000 (dashed-dotted curve).

Fig. 2
Fig. 2

Shift in the peak positions (peak 1, outer peak; peak 2, inner peak) of S l ( r ) of the second-order WGM, Δ r peak , in adsorption of a layer of n p = 1.6 , plotted as a function of layer thickness t (solid curve, black). The estimates relying on the numerical result of k at resonance and a constant number of waves (crosses) are superimposed.

Fig. 3
Fig. 3

Fractional shift in the peak position of the radial function, Δ r peak Δ r peak , 0 , of the first-order mode in adsorption of a thin layer of n p = 1.6 , plotted as a function of layer thickness t. The estimate by the variational method (dashed curve) is compared with the result of exact numerical analysis (solid curve, black) and the estimate relying on the numerical result of k at resonance and a constant optical path length (crosses). Inset: S l ( r ) on the plain sphere (black curve), S l ( r ) in the coated sphere ( n p = 1.6 , t = 30 nm ; curve), and the estimate by the variational method (crosses).

Fig. 4
Fig. 4

Fractional shift in the resonance wavelength, Δ k k 0 , in adsorption of a thin layer of n p = 1.6 , plotted as a function of layer thickness t. The result of exact numerical analysis (solid curve, black) is compared with the estimate by the first-order perturbation (dotted curve), the estimates by the variational method with thin-layer approximation (dashed curve), and with the evanescent field decay taken into account (dashed-dotted curve).

Fig. 5
Fig. 5

Fractional shift in the peak position of the radial function, Δ r peak Δ r peak , 0 , of the first-order mode in the surroundings’ RI change Δ n 2 , plotted as a function of Δ n 2 . The estimate by the variational method (dashed line) is compared with the result of exact numerical analysis (solid line, black) and the estimate relying on the numerical result of k at resonance and a constant number of waves (crosses).

Tables (2)

Tables Icon

Table 1 Definite Integrals in [ 0 , )

Tables Icon

Table 2 Definite Integrals in ( a , )

Equations (53)

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

× × E = n 2 k 2 E ,
n o ( r ) = { n 1 ( r < a ) n 2 ( r > a ) .
E l m = exp ( i m φ ) k r S l ( r ) X l m ( θ ) ,
X l m ( θ ) i m sin θ P l m ( cos θ ) e ̂ θ θ P l m ( cos θ ) e ̂ φ ,
[ d 2 d r 2 + k 2 n 2 l ( l + 1 ) r 2 ] S l ( r ) = 0 .
S l ( r ) = { A l ψ l ( n 1 k r ) ( r < a ) B l χ l ( n 2 k r ) ( r > a ) ,
A l ψ l ( n 1 k a ) = B l χ l ( n 2 k a ) ,
A l n 1 ψ l ( n 1 k a ) = B l n 2 χ l ( n 2 k a ) ,
0 [ n o ( r ) ] 2 S l ( i ) ( r ) S l ( j ) ( r ) d r = 0 ( i j ) .
VAR E * × × E d r [ n ( r ) ] 2 E * E d r ,
n ( r ) = { n 1 ( r < a ) n p ( a < r < a + t ) . n 2 ( r > a )
E = exp ( i m φ ) k r S l [ r ( 1 + b ) ] X l m ( θ ) ,
E exp ( i m φ ) k r [ S l ( r ) + b r S l ( r ) + 1 2 b 2 r 2 S l ( r ) ] X l m .
Δ k k 0 = Δ r peak i r peak i , 0 ( i = 1 , 2 ) ,
n 2 E * E d r = n o 2 E * E d r + ( n 2 n o 2 ) E * E d r ,
n o 2 E * E d r = W l m k 2 0 n o 2 [ S l 2 + 2 b r S l S l + b 2 r 2 ( S l 2 + S l S l ) ] d r ,
W l m 2 π 0 π sin θ X l m 2 d θ = 4 π ( l + m ) ! l ( l + 1 ) ( l m ) ! ( 2 l + 1 ) .
n o 2 E * E d r W l m k 2 ( n 1 2 n 2 2 ) a 2 [ S l ( a ) ] 2 { 1 + b + b 2 [ 2 a S l ( a ) S l ( a ) 1 ] } .
Γ S l ( a ) S l ( a ) = n 2 k χ l ( n 2 k a ) ,
E n 2 n o 2 E W l m t k 2 ( n p 2 n 2 2 ) [ S l ( a ) ] 2 ( 1 + 2 b a S l ( a ) S l ( a ) + b 2 a 2 { [ S l ( a ) S l ( a ) ] 2 + S l ( a + ) S l ( a ) } ) ,
Λ S l ( a + ) S l ( a ) = l ( l + 1 ) a 2 n 2 2 k 2 .
E n 2 E W l m k 2 a 2 [ S l ( a ) ] 2 { ( n 1 2 n 2 2 ) [ 1 + b b 2 ( 1 + 2 a Γ ) ] + 2 t a ( n p 2 n 2 2 ) [ 1 2 b a Γ + b 2 a 2 ( Γ 2 + Λ ) ] } .
VAR k 2 1 + b 1 + b b 2 ( 1 + 2 a Γ ) + p [ 1 2 b a Γ + b 2 a 2 ( Γ 2 + Λ ) ] ,
p 2 t a n p 2 n 2 2 n 1 2 n 2 2 .
k 2 VAR 1 2 b 2 a Γ + p ( 1 2 b a Γ + 2 b 2 a 2 Γ 2 ) .
b m p 2 .
Δ k k 0 = Δ ( k 2 ) 2 k 0 2 = p 2 + 1 4 p 2 a Γ = t a n p 2 n 2 2 n 1 2 n 2 2 + ( t a n p 2 n 2 2 n 1 2 n 2 2 ) 2 a Γ .
Δ r peak r peak , 0 = b min = p 2 = t a n p 2 n 2 2 n 1 2 n 2 2 .
p = 2 n p 2 n 2 2 n 1 2 n 2 2 1 exp ( 2 t Γ ) 2 a Γ ,
[ n ( r ) ] 2 = { n 1 2 ( r < a ) n 2 2 + Δ ( n 2 2 ) ( r > a ) .
( n 2 n o 2 ) E * E d r W l m k 2 Δ ( n 2 2 ) a d r [ S l 2 + 2 b r S l S l + b 2 r 2 ( S l 2 + S l S l ) ] .
( n 2 n o 2 ) E * E d r Δ ( n 2 2 ) W l m k 2 a S l 2 [ I a S l 2 b ( 1 + I a S l 2 ) + b 2 ( 1 + a Γ + I a S l 2 ) ] ,
I a S l 2 = Λ Γ 2 Γ a 2 n 2 2 k 2 .
Γ = Λ 1 2 l ( l + 1 ) 2 Λ a 3 ,
I a S l 2 = Γ 2 Λ a 1 2 Γ a .
( n 2 n o 2 ) E * E d r W l m k 2 Δ ( n 2 2 ) a 2 S l 2 [ 1 a Γ b ( 2 + 1 a Γ ) + b 2 ( 2 a Γ + 2 + 1 a Γ ) ] .
VAR k 2 1 + b 1 + b 2 b 2 a Γ + q [ ( a Γ ) 1 2 b + 2 b 2 a Γ ] ,
q Δ ( n 2 2 ) n 1 2 n 2 2 ,
b m q 2 a Γ .
Δ k k 0 = 1 2 Δ ( k 2 ) k 0 2 = 1 2 a Γ [ Δ ( n 2 2 ) n 1 2 n 2 2 + 1 2 ( Δ ( n 2 2 ) n 1 2 n 2 2 ) 2 ] .
q 2 a Γ = Δ ( n 2 2 ) n 1 2 n 2 2 1 2 a Γ .
ψ l 2 d z = z 2 [ ( 1 l ( l + 1 ) z 2 ) ψ l 2 1 z ψ l ψ l + ψ l 2 ] ,
E = exp ( i m φ ) k r U ( r ) [ 0 i m P l m ( cos θ ) sin θ P l m θ ] ,
U ( r ) S l ( r ) + b r S l ( r ) + 1 2 b 2 r 2 S l ( r ) .
( × × E ) r = 0 ,
( × × E ) θ = exp ( i m φ ) k r [ U l ( l + 1 ) r 2 U ] i m sin θ P l m ,
( × × E ) φ = exp ( i m φ ) k r [ U U l ( l + 1 ) r 2 ] θ P l m .
E × × E = 1 k 2 r 2 [ ( m P l m sin θ ) 2 + ( P l m θ ) 2 ] U [ U l ( l + 1 ) r 2 U ] .
2 π 0 π sin θ d θ [ ( m P l m sin θ ) 2 + ( P l m θ ) 2 ] = W l m .
U [ U l ( l + 1 ) r 2 U ] k 2 n 2 S l 2 + k 2 b [ 2 n 2 ( S l 2 + r S l S l ) + d n 2 d r r S l 2 ] + k 2 b 2 { n 2 [ S l 2 + 4 r S l S l + r 2 ( S l 2 + S l S l ) ] + 2 d n 2 d r ( r S l 2 + r 2 S l S l ) + 1 2 d 2 n 2 d r 2 r 2 S l 2 } ,
[ d 3 d r 3 + k 2 d n 2 d r + k 2 n 2 d d r + 2 l ( l + 1 ) r 3 l ( l + 1 ) r 2 d d r ] S l = 0 ,
[ d 4 d r 4 + k 2 d 2 n 2 d r 2 + 2 k 2 d n 2 d r d d r + k 2 n 2 d 2 d r 2 6 l ( l + 1 ) r 4 + 4 l ( l + 1 ) r 3 d d r l ( l + 1 ) r 2 d 2 d r 2 ] S l = 0 ,
E * × × E d r = W l m ( n 1 2 n 2 2 ) 1 2 ( 1 + b ) a [ S l ( a ) ] 2 .

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