Abstract

This study proposes a thermostable refractive index (RI) sensor consisting of a silica microsphere coated with a poly(methyl methacrylate) (PMMA) layer. The first-order and second-order whispering gallery modes (WGMs) of both TE and TM polarizations are considered theoretically. The layer thickness is carefully optimized to eliminate the thermal drift and enhance the RI sensitivity and detection limit. In various WGMs, at the thermostable thickness of the PMMA layer, the first-order TM mode corresponds to the highest sensitivity and the smallest detection limit. The theoretical predictions provide guidelines for the design and fabrication of thermostable RI sensors.

© 2011 Optical Society of America

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2010 (2)

C. H. Dong, F. W. Sun, C. L. Zou, X. F. Ren, G. C. Guo, and Z. F. Han, “High-Q silica microsphere by poly(methyl methacrylate) coating and modifying,” Appl. Phys. Lett. 96, 061106 (2010).
[CrossRef]

B. B. Li, Q. Y. Wang, Y. F. Xiao, X. F. Jiang, Y. Li, L. X. Xiao, and Q. H. Cong, “On chip, high-sensitivity thermal sensor based on high-Q polydimethylsiloxane-coated microresonator,” Appl. Phys. Lett. 96, 251109 (2010).
[CrossRef]

2008 (3)

F. Vollmer and S. Arnord, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[CrossRef] [PubMed]

I. M. White and X. Fan, “On the performance quantification of resonant refractive index sensors,” Opt. Express 16, 1020–1028 (2008).
[CrossRef] [PubMed]

L. He, Y. F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93, 201102 (2008).
[CrossRef]

2007 (2)

2006 (6)

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

I. Teraoka and S. Arnold, “Enhancing the sensitivity of a whispering-gallery mode microsphere sensor by a high-refractive-index surface layer,” J. Opt. Soc. Am. B 23, 1434–1442 (2006).
[CrossRef]

N. M. Hanumegowda, I. M. White, and X. Fan, “Aqueous mercuric ion detection with microsphere optical ring resonator sensors,” Sens. Actuators B Chem. 120, 207–212 (2006).
[CrossRef]

H. Y. Zhu, J. D. Suter, I. M. White, and X. D. Fan, “Aptamer based microsphere biosensor for thrombin detection,” Sensors 6, 785–795 (2006).
[CrossRef]

C. Y. Chao and L. J. Guo, “Design and optimization of microring resonators in biochemical sensing applications,” J. Lightwave Technol. 24, 1395–1402 (2006).
[CrossRef]

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes—Part I: Basics,” IEEE J. Quantum Electron. 12, 3–14 (2006).
[CrossRef]

2005 (1)

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

2003 (2)

2002 (3)

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]

E. S. Kang, T. H. Lee, and B. S. Bae, “Measurement of the thermo-optic coefficients in sol-gel derived inorganic-organic hybrid material films,” Appl. Phys. Lett. 81, 1438–1440(2002).
[CrossRef]

1999 (1)

1996 (1)

1992 (1)

Y. Takezawa, N. Taketani, S. Tanno, and S. Ohara, “Empirical estimation method of intrinsic loss spectra in transparent amorphous polymers for plastic optical fibers,” J. Appl. Polym. Sci. 46, 1835–1841 (1992).
[CrossRef]

1988 (1)

1986 (1)

Arnold, S.

Arnord, S.

F. Vollmer and S. Arnord, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[CrossRef] [PubMed]

Bae, B. S.

E. S. Kang, T. H. Lee, and B. S. Bae, “Measurement of the thermo-optic coefficients in sol-gel derived inorganic-organic hybrid material films,” Appl. Phys. Lett. 81, 1438–1440(2002).
[CrossRef]

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, 4057–4059 (2002).
[CrossRef]

Cariou, J. M.

Chao, C. Y.

Cong, Q. H.

B. B. Li, Q. Y. Wang, Y. F. Xiao, X. F. Jiang, Y. Li, L. X. Xiao, and Q. H. Cong, “On chip, high-sensitivity thermal sensor based on high-Q polydimethylsiloxane-coated microresonator,” Appl. Phys. Lett. 96, 251109 (2010).
[CrossRef]

Corodetsky, M. L.

Dong, C.

L. He, Y. F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93, 201102 (2008).
[CrossRef]

Dong, C. H.

C. H. Dong, F. W. Sun, C. L. Zou, X. F. Ren, G. C. Guo, and Z. F. Han, “High-Q silica microsphere by poly(methyl methacrylate) coating and modifying,” Appl. Phys. Lett. 96, 061106 (2010).
[CrossRef]

Driessen, A.

Dugas, J.

Fan, X.

I. M. White and X. Fan, “On the performance quantification of resonant refractive index sensors,” Opt. Express 16, 1020–1028 (2008).
[CrossRef] [PubMed]

N. M. Hanumegowda, I. M. White, and X. Fan, “Aqueous mercuric ion detection with microsphere optical ring resonator sensors,” Sens. Actuators B Chem. 120, 207–212 (2006).
[CrossRef]

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

Fan, X. D.

H. Y. Zhu, J. D. Suter, I. M. White, and X. D. Fan, “Aptamer based microsphere biosensor for thrombin detection,” Sensors 6, 785–795 (2006).
[CrossRef]

Gaddam, V.

L. He, Y. F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93, 201102 (2008).
[CrossRef]

Greve, J.

Guo, G. C.

C. H. Dong, F. W. Sun, C. L. Zou, X. F. Ren, G. C. Guo, and Z. F. Han, “High-Q silica microsphere by poly(methyl methacrylate) coating and modifying,” Appl. Phys. Lett. 96, 061106 (2010).
[CrossRef]

Guo, L. J.

Han, M.

Han, Z. F.

C. H. Dong, F. W. Sun, C. L. Zou, X. F. Ren, G. C. Guo, and Z. F. Han, “High-Q silica microsphere by poly(methyl methacrylate) coating and modifying,” Appl. Phys. Lett. 96, 061106 (2010).
[CrossRef]

Hanumegowda, N. M.

N. M. Hanumegowda, I. M. White, and X. Fan, “Aqueous mercuric ion detection with microsphere optical ring resonator sensors,” Sens. Actuators B Chem. 120, 207–212 (2006).
[CrossRef]

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

Haus, H. A.

He, L.

L. He, Y. F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93, 201102 (2008).
[CrossRef]

Hightower, R. L.

Holler, S.

Ilchenko, V. S.

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes—Part I: Basics,” IEEE J. Quantum Electron. 12, 3–14 (2006).
[CrossRef]

M. L. Corodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
[CrossRef]

Jiang, X. F.

B. B. Li, Q. Y. Wang, Y. F. Xiao, X. F. Jiang, Y. Li, L. X. Xiao, and Q. H. Cong, “On chip, high-sensitivity thermal sensor based on high-Q polydimethylsiloxane-coated microresonator,” Appl. Phys. Lett. 96, 251109 (2010).
[CrossRef]

Kang, E. S.

E. S. Kang, T. H. Lee, and B. S. Bae, “Measurement of the thermo-optic coefficients in sol-gel derived inorganic-organic hybrid material films,” Appl. Phys. Lett. 81, 1438–1440(2002).
[CrossRef]

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.

Laine, J. P.

Lee, T. H.

E. S. Kang, T. H. Lee, and B. S. Bae, “Measurement of the thermo-optic coefficients in sol-gel derived inorganic-organic hybrid material films,” Appl. Phys. Lett. 81, 1438–1440(2002).
[CrossRef]

Li, B. B.

B. B. Li, Q. Y. Wang, Y. F. Xiao, X. F. Jiang, Y. Li, L. X. Xiao, and Q. H. Cong, “On chip, high-sensitivity thermal sensor based on high-Q polydimethylsiloxane-coated microresonator,” Appl. Phys. Lett. 96, 251109 (2010).
[CrossRef]

Li, Y.

B. B. Li, Q. Y. Wang, Y. F. Xiao, X. F. Jiang, Y. Li, L. X. Xiao, and Q. H. Cong, “On chip, high-sensitivity thermal sensor based on high-Q polydimethylsiloxane-coated microresonator,” Appl. Phys. Lett. 96, 251109 (2010).
[CrossRef]

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, 4057–4059 (2002).
[CrossRef]

Little, B. E.

Martin, L.

Matsko, A. B.

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes—Part I: Basics,” IEEE J. Quantum Electron. 12, 3–14 (2006).
[CrossRef]

Michel, P.

Ohara, S.

Y. Takezawa, N. Taketani, S. Tanno, and S. Ohara, “Empirical estimation method of intrinsic loss spectra in transparent amorphous polymers for plastic optical fibers,” J. Appl. Polym. Sci. 46, 1835–1841 (1992).
[CrossRef]

Otto, C.

Patel, B. C.

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

Ren, X. F.

C. H. Dong, F. W. Sun, C. L. Zou, X. F. Ren, G. C. Guo, and Z. F. Han, “High-Q silica microsphere by poly(methyl methacrylate) coating and modifying,” Appl. Phys. Lett. 96, 061106 (2010).
[CrossRef]

Richardson, C. B.

Savchenkov, A. A.

Stica, C. J.

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

Sun, F. W.

C. H. Dong, F. W. Sun, C. L. Zou, X. F. Ren, G. C. Guo, and Z. F. Han, “High-Q silica microsphere by poly(methyl methacrylate) coating and modifying,” Appl. Phys. Lett. 96, 061106 (2010).
[CrossRef]

Suter, J. D.

H. Y. Zhu, J. D. Suter, I. M. White, and X. D. Fan, “Aptamer based microsphere biosensor for thrombin detection,” Sensors 6, 785–795 (2006).
[CrossRef]

Taketani, N.

Y. Takezawa, N. Taketani, S. Tanno, and S. Ohara, “Empirical estimation method of intrinsic loss spectra in transparent amorphous polymers for plastic optical fibers,” J. Appl. Polym. Sci. 46, 1835–1841 (1992).
[CrossRef]

Takezawa, Y.

Y. Takezawa, N. Taketani, S. Tanno, and S. Ohara, “Empirical estimation method of intrinsic loss spectra in transparent amorphous polymers for plastic optical fibers,” J. Appl. Polym. Sci. 46, 1835–1841 (1992).
[CrossRef]

Tanno, S.

Y. Takezawa, N. Taketani, S. Tanno, and S. Ohara, “Empirical estimation method of intrinsic loss spectra in transparent amorphous polymers for plastic optical fibers,” J. Appl. Polym. Sci. 46, 1835–1841 (1992).
[CrossRef]

Teraoka, I.

Vollmer, F.

F. Vollmer and S. Arnord, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[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, 4057–4059 (2002).
[CrossRef]

Wang, A.

Wang, Q. Y.

B. B. Li, Q. Y. Wang, Y. F. Xiao, X. F. Jiang, Y. Li, L. X. Xiao, and Q. H. Cong, “On chip, high-sensitivity thermal sensor based on high-Q polydimethylsiloxane-coated microresonator,” Appl. Phys. Lett. 96, 251109 (2010).
[CrossRef]

White, I. M.

I. M. White and X. Fan, “On the performance quantification of resonant refractive index sensors,” Opt. Express 16, 1020–1028 (2008).
[CrossRef] [PubMed]

H. Y. Zhu, J. D. Suter, I. M. White, and X. D. Fan, “Aptamer based microsphere biosensor for thrombin detection,” Sensors 6, 785–795 (2006).
[CrossRef]

N. M. Hanumegowda, I. M. White, and X. Fan, “Aqueous mercuric ion detection with microsphere optical ring resonator sensors,” Sens. Actuators B Chem. 120, 207–212 (2006).
[CrossRef]

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

Xiao, L. X.

B. B. Li, Q. Y. Wang, Y. F. Xiao, X. F. Jiang, Y. Li, L. X. Xiao, and Q. H. Cong, “On chip, high-sensitivity thermal sensor based on high-Q polydimethylsiloxane-coated microresonator,” Appl. Phys. Lett. 96, 251109 (2010).
[CrossRef]

Xiao, Y. F.

B. B. Li, Q. Y. Wang, Y. F. Xiao, X. F. Jiang, Y. Li, L. X. Xiao, and Q. H. Cong, “On chip, high-sensitivity thermal sensor based on high-Q polydimethylsiloxane-coated microresonator,” Appl. Phys. Lett. 96, 251109 (2010).
[CrossRef]

L. He, Y. F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93, 201102 (2008).
[CrossRef]

Yang, L.

L. He, Y. F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93, 201102 (2008).
[CrossRef]

Zhu, H. Y.

H. Y. Zhu, J. D. Suter, I. M. White, and X. D. Fan, “Aptamer based microsphere biosensor for thrombin detection,” Sensors 6, 785–795 (2006).
[CrossRef]

Zhu, J.

L. He, Y. F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93, 201102 (2008).
[CrossRef]

Zou, C. L.

C. H. Dong, F. W. Sun, C. L. Zou, X. F. Ren, G. C. Guo, and Z. F. Han, “High-Q silica microsphere by poly(methyl methacrylate) coating and modifying,” Appl. Phys. Lett. 96, 061106 (2010).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (6)

C. H. Dong, F. W. Sun, C. L. Zou, X. F. Ren, G. C. Guo, and Z. F. Han, “High-Q silica microsphere by poly(methyl methacrylate) coating and modifying,” Appl. Phys. Lett. 96, 061106 (2010).
[CrossRef]

L. He, Y. F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett. 93, 201102 (2008).
[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]

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

B. B. Li, Q. Y. Wang, Y. F. Xiao, X. F. Jiang, Y. Li, L. X. Xiao, and Q. H. Cong, “On chip, high-sensitivity thermal sensor based on high-Q polydimethylsiloxane-coated microresonator,” Appl. Phys. Lett. 96, 251109 (2010).
[CrossRef]

E. S. Kang, T. H. Lee, and B. S. Bae, “Measurement of the thermo-optic coefficients in sol-gel derived inorganic-organic hybrid material films,” Appl. Phys. Lett. 81, 1438–1440(2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes—Part I: Basics,” IEEE J. Quantum Electron. 12, 3–14 (2006).
[CrossRef]

J. Appl. Polym. Sci. (1)

Y. Takezawa, N. Taketani, S. Tanno, and S. Ohara, “Empirical estimation method of intrinsic loss spectra in transparent amorphous polymers for plastic optical fibers,” J. Appl. Polym. Sci. 46, 1835–1841 (1992).
[CrossRef]

J. Lightwave Technol. (2)

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

Nat. Methods (1)

F. Vollmer and S. Arnord, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (4)

Sens. Actuators B Chem. (1)

N. M. Hanumegowda, I. M. White, and X. Fan, “Aqueous mercuric ion detection with microsphere optical ring resonator sensors,” Sens. Actuators B Chem. 120, 207–212 (2006).
[CrossRef]

Sensors (1)

H. Y. Zhu, J. D. Suter, I. M. White, and X. D. Fan, “Aptamer based microsphere biosensor for thrombin detection,” Sensors 6, 785–795 (2006).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

d λ R / d T for TE modes (solid curves) and TM modes (dashed curves) of two orders ( v = 1 , 2) as a function of the PMMA layer thickness h 0 .

Fig. 2
Fig. 2

Resonant wavelength shift δ λ R as a function of the temperature change δ T for TE modes of v = 1 at various PMMA layer thicknesses. The discrete spots represent δ λ R calculated by Eq. (1). The solid curves represent δ λ R = ( d λ R / d T ) × δ T .

Fig. 3
Fig. 3

d λ R / d T for TE modes of two orders ( v = 1 , 2) in microspheres of different radii as a function of the PMMA layer thickness h 0 .

Fig. 4
Fig. 4

RI Sensitivity S for TE modes (solid curves) and TM modes (dashed curves) of two orders ( v = 1 , 2) as a function of the PMMA layer thickness h 0 .

Fig. 5
Fig. 5

Resonant wavelength shift δ λ R as a function of the refractive index change δ n 0 of the surrounding medium for WGMs of v = 1 at a 0 = 50 μm . The curves represent δ λ R = δ n 0 × S . The symbols represent δ λ R calculated by Eq. (1).

Fig. 6
Fig. 6

Energy fraction η p in PMMA and Q-factor for TE modes (solid curves) and TM modes (dashed curves) of v = 1 as a function of the PMMA layer thickness h 0 .

Fig. 7
Fig. 7

S × Q for the WGMs of v = 1 as a function of the PMMA layer thickness h 0 . The solid and dashed curves represent the values for the TE and TM modes respectively.

Equations (17)

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N 0 χ l ( n 0 k a 1 ) χ l ( n 0 k a 1 ) = B l ψ l ( n p k a 1 ) + χ l ( n p k a 1 ) B l ψ l ( n p k a 1 ) + χ l ( n p k a 1 ) , N 0 = { n 0 / n p , TE modes n p / n 0 , TM modes ,
B l = N s ψ l ( n s k a 0 ) χ l ( n p k a 0 ) ψ l ( n s k a 0 ) χ l ( n p k a 0 ) ψ l ( n p k a 0 ) ψ l ( n s k a 0 ) N s ψ l ( n p k a 0 ) ψ l ( n s k a 0 ) , N s = { n s / n p , TE modes n p / n s , TM modes .
d λ R d T = λ R ( 1 n eff d n eff d T + 1 D d D d T ) ,
d λ R d T λ R ( η s ( d n s / d T ) + η p ( d n p / d T ) η s n s + η p n p + η 0 n 0 + α s a 0 + α p h 0 a 0 + h 0 ) .
η s = n s 2 I s n s 2 I s + n p 2 I p + n 0 2 I 0 , η p = n p 2 I p n s 2 I s + n p 2 I p + n 0 2 I 0 , η 0 = n 0 2 I 0 n s 2 I s + n p 2 I p + n 0 2 I 0 ( TE modes ) ,
η s = I s I s + I p + I 0 , η p = I p I s + I p + I 0 , η 0 = I 0 I s + I p + I 0 ( TM modes ) ,
I s = 0 a 0 [ A l ψ l ( n s k 0 r ) ] 2 d r I 0 = a 1 [ C l χ l ( n 0 k 0 r ) ] 2 d r , I p = a 0 a 1 [ B l ψ l ( n p k 0 r ) + χ l ( n p k 0 r ) ] 2 d r .
A l = B l ψ l ( n p k a 0 ) + χ l ( n p k a 0 ) ψ l ( n s k a 0 ) , C l = B l ψ l ( n p k a 1 ) + χ l ( n p k a 1 ) χ l ( n 0 k a 1 ) .
( d λ R d T ) TE λ R ( n s 2 ( d n s / d T ) I s + n p 2 ( d n p / d T ) I p n s 3 I s + n p 3 I p + n 0 3 I 0 + α s a 0 + α p h 0 a 0 + h 0 ) ,
( d λ R d T ) TM λ R ( ( d n s / d T ) I s + ( d n p / d T ) I p n s I s + n p I p + n 0 I 0 + α s a 0 + α p h 0 a 0 + h 0 ) .
S = δ λ R δ n 0 = λ R δ n 0 · δ k k 0 ,
( δ k k 0 ) TE = δ ( n 0 2 ) I 0 2 ( n s 2 I s + n p 2 I p + n 0 2 I 0 ) , TE modes ,
( δ k k 0 ) TM = δ ( n 0 2 ) [ T 0 ( a 1 ) T 0 ( a 1 + ) + n 0 2 k 0 2 I 0 ] 2 n 0 4 k 0 2 ( I s + I p + I 0 ) , TM modes ,
T ( r ) = { A l ψ l ( n s k r ) r < a 0 B l ψ l ( n p k r ) + χ l ( n p k r ) a 0 < r < a 1 C l χ l ( n 0 k r ) r > a 1 .
S TE = n 0 λ R I 0 n s 2 I s + n p 2 I p + n 0 2 I 0 = λ R n 0 η 0 ,
S TM = λ R 2 [ n 0 k 0 I 0 C l 2 χ l ( n 0 k 0 a 1 ) χ l ( n 0 k 0 a 1 ) ] 2 π n 0 2 ( I s + I p + I 0 ) = λ R n 0 η 0 C l 2 λ R 2 χ l ( n 0 k 0 a 1 ) χ l ( n 0 k 0 a 1 ) 2 π n 0 2 ( I s + I p + I 0 ) .
1 Q η s σ s λ R 2 π n s + η p σ p λ R 2 π n p = η s 1 Q 0 + η p σ p λ R 2 π n p ,

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