Abstract

This study presents a chemical vapor sensor based on polymer-coated microsphere resonators. A theoretical simulation of the sensor response is performed, and optimization of the polymer layer thickness is investigated. Results show that the sensor exhibits a good linearity and a low detection limit of the refractive index change. Especially at the thermostable thickness of the polymer layer, the refractive index detection limit of the wavelength around 780nm can be as low as 2×108 refractive index unit for a spectral resolution of 10fm, without any temperature control. Because of the good sensing performance and simple manipulation, the proposed sensor is a very promising platform for chemical vapor detections.

© 2011 Optical Society of America

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2010 (3)

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]

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]

N. Lin, L. Jiang, S. M. Wang, L. Yuan, H. Xiao, Y. F. Lu, and H. L. Tsai, “Ultrasensitive chemical sensors based on whispering gallery modes in a microsphere coated with zeolite,” Appl. Opt. 49, 6463–6471 (2010).
[CrossRef] [PubMed]

2009 (2)

2008 (4)

2007 (5)

2006 (3)

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]

2004 (2)

W. Fang, D. B. Buchholz, R. C. Bailey, J. T. Hupp, R. P. H. Chang, and H. Cao, “Detection of chemical species using ultraviolet microdisk lasers,” Appl. Phys. Lett. 85, 3666–3668(2004).
[CrossRef]

A. Ksendzov, M. L. Homer, and A. M. Manfreda, “Integrated optics ring-resonator chemical sensor with polymer transduction layer,” Electron. Lett. 40, 63–65 (2004).
[CrossRef]

2003 (1)

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, 4057–4059 (2002).
[CrossRef]

2001 (1)

1999 (1)

1996 (1)

1988 (1)

Adamovsky, G.

Arnold, S.

Bailey, R. C.

W. Fang, D. B. Buchholz, R. C. Bailey, J. T. Hupp, R. P. H. Chang, and H. Cao, “Detection of chemical species using ultraviolet microdisk lasers,” Appl. Phys. Lett. 85, 3666–3668(2004).
[CrossRef]

Beckmann, T.

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]

Buchholz, D. B.

W. Fang, D. B. Buchholz, R. C. Bailey, J. T. Hupp, R. P. H. Chang, and H. Cao, “Detection of chemical species using ultraviolet microdisk lasers,” Appl. Phys. Lett. 85, 3666–3668(2004).
[CrossRef]

Buse, K.

Cao, H.

W. Fang, D. B. Buchholz, R. C. Bailey, J. T. Hupp, R. P. H. Chang, and H. Cao, “Detection of chemical species using ultraviolet microdisk lasers,” Appl. Phys. Lett. 85, 3666–3668(2004).
[CrossRef]

Chang, R. P. H.

W. Fang, D. B. Buchholz, R. C. Bailey, J. T. Hupp, R. P. H. Chang, and H. Cao, “Detection of chemical species using ultraviolet microdisk lasers,” Appl. Phys. Lett. 85, 3666–3668(2004).
[CrossRef]

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.

Curley, M. J.

De Leonardis, F.

V. M. N. Passaro, F. Dell’Olio, and F. De Leonardis, “Ammonia optical sensing by microring resonators,” Sensors 7, 2741–2749 (2007).
[CrossRef]

Dell’Olio, F.

V. M. N. Passaro, F. Dell’Olio, and F. De Leonardis, “Ammonia optical sensing by microring resonators,” Sensors 7, 2741–2749 (2007).
[CrossRef]

Diggs, D. E.

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]

Fan, X.

Y. Sun and X. Fan, “Analysis of ring resonators for chemical vapor sensor development,” Opt. Express 16, 10254–10267(2008).
[CrossRef] [PubMed]

Y. Sun, S. I. Shopova, G. F. Mason, and X. Fan, “Rapid chemical-vapor sensing using optofluidic ring resonators,” Opt. Lett. 33, 788–790 (2008).
[CrossRef] [PubMed]

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, “Overview of novel integrated optical ring resonator bio/chemical sensors,” Proc. SPIE 6452, 64520M (2007).
[CrossRef]

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. M. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87, 201107 (2005).
[CrossRef]

Fan, X. D.

Fang, W.

W. Fang, D. B. Buchholz, R. C. Bailey, J. T. Hupp, R. P. H. Chang, and H. Cao, “Detection of chemical species using ultraviolet microdisk lasers,” Appl. Phys. Lett. 85, 3666–3668(2004).
[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]

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.

Haertle, D.

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, 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.

Homer, M. L.

A. Ksendzov, M. L. Homer, and A. M. Manfreda, “Integrated optics ring-resonator chemical sensor with polymer transduction layer,” Electron. Lett. 40, 63–65 (2004).
[CrossRef]

Hupp, J. T.

W. Fang, D. B. Buchholz, R. C. Bailey, J. T. Hupp, R. P. H. Chang, and H. Cao, “Detection of chemical species using ultraviolet microdisk lasers,” Appl. Phys. Lett. 85, 3666–3668(2004).
[CrossRef]

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, L.

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]

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]

Ksendzov, A.

A. Ksendzov, M. L. Homer, and A. M. Manfreda, “Integrated optics ring-resonator chemical sensor with polymer transduction layer,” Electron. Lett. 40, 63–65 (2004).
[CrossRef]

Laine, J. P.

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]

Lin, N.

Ling, T.

Little, B. E.

Lu, Y. F.

Manfreda, A. M.

A. Ksendzov, M. L. Homer, and A. M. Manfreda, “Integrated optics ring-resonator chemical sensor with polymer transduction layer,” Electron. Lett. 40, 63–65 (2004).
[CrossRef]

Mason, G. F.

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]

Oveys, H.

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, “Overview of novel integrated optical ring resonator bio/chemical sensors,” Proc. SPIE 6452, 64520M (2007).
[CrossRef]

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

Passaro, V. M. N.

V. M. N. Passaro, F. Dell’Olio, and F. De Leonardis, “Ammonia optical sensing by microring resonators,” Sensors 7, 2741–2749 (2007).
[CrossRef]

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.

Sarkisov, S. S.

Savchenkov, A. A.

Schwesyg, J. R.

Shopova, S. I.

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]

Sun, Y.

Suter, J. D.

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, “Overview of novel integrated optical ring resonator bio/chemical sensors,” Proc. SPIE 6452, 64520M (2007).
[CrossRef]

Teraoka, I.

Tsai, H. L.

Vollmer, F.

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]

Wang, S. M.

White, I. M.

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

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, “Overview of novel integrated optical ring resonator bio/chemical sensors,” Proc. SPIE 6452, 64520M (2007).
[CrossRef]

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. M. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87, 201107 (2005).
[CrossRef]

Xiao, H.

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]

Yuan, L.

Zhu, H.

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, “Overview of novel integrated optical ring resonator bio/chemical sensors,” Proc. SPIE 6452, 64520M (2007).
[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]

Zimmermann, A. S.

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. (3)

Appl. Phys. Lett. (6)

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]

W. Fang, D. B. Buchholz, R. C. Bailey, J. T. Hupp, R. P. H. Chang, and H. Cao, “Detection of chemical species using ultraviolet microdisk lasers,” Appl. Phys. Lett. 85, 3666–3668(2004).
[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]

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]

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]

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]

Electron. Lett. (1)

A. Ksendzov, M. L. Homer, and A. M. Manfreda, “Integrated optics ring-resonator chemical sensor with polymer transduction layer,” Electron. Lett. 40, 63–65 (2004).
[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. Lightwave Technol. (1)

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

Opt. Express (4)

Opt. Lett. (5)

Proc. SPIE (1)

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, “Overview of novel integrated optical ring resonator bio/chemical sensors,” Proc. SPIE 6452, 64520M (2007).
[CrossRef]

Sensors (1)

V. M. N. Passaro, F. Dell’Olio, and F. De Leonardis, “Ammonia optical sensing by microring resonators,” Sensors 7, 2741–2749 (2007).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Polymer-coated microresonator chemical vapor sensor configuration.

Fig. 2
Fig. 2

RI sensitivity S at the wavelength around (a)  780 nm and (b)  1550 nm , for TE modes (solid curves) and TM modes (dashed curves) of the first three orders as a function of the polymer layer thickness h 0 .

Fig. 3
Fig. 3

Resonance shift δ λ R at the wavelength around (a)  780 nm and (b)  1550 nm as a function of the polymer layer RI change δ n 2 . The curves represent δ λ R = S × δ n 2 . The symbols represent δ λ R calculated by Eq. (1).

Fig. 4
Fig. 4

Q factor at the wavelength around (a)  780 nm and (b)  1550 nm as a function of the polymer layer thickness h 0 .

Fig. 5
Fig. 5

Amplitude noise standard deviation σ ampl at the wavelength around (a)  780 nm and (b)  1550 nm as a function of the polymer layer thickness h 0 , for a SNR of 60 dB .

Fig. 6
Fig. 6

Thermal noise standard deviation σ temp at the wavelength around (a)  780 nm and (b)  1550 nm as a function of the polymer layer thickness h 0 , for a temperature fluctuation of σ T = 10 3 K .

Fig. 7
Fig. 7

Sensor resolution R at the wavelength around (a)  780 nm and (b)  1550 nm as a function of the polymer layer thickness h 0 , for SNR = 60 dB , σ spect = 2.9 fm , and σ T = 10 3 K .

Fig. 8
Fig. 8

RI DL at the wavelength around (a)  780 nm and (b)  1550 nm as a function of the polymer layer thickness h 0 , for SNR = 60 dB , σ spect = 2.9 fm and σ T = 10 3 K .

Fig. 9
Fig. 9

RI DL of the TE v = 1 mode around 780 nm as a function of the polymer layer thickness h 0 , at σ T = 10 3 , 5 × 10 3 , and 10 2 K respectively.

Equations (15)

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N 0 χ l ( k 0 n 3 a 1 ) χ l ( k 0 n 3 a 1 ) = B l ψ l ( k 0 n 2 a 1 ) + χ l ( k 0 n 2 a 1 ) B l ψ l ( k 0 n 2 a 1 ) + χ l ( k 0 n 2 a 1 ) , N 0 = { n 3 / n 2 , TE modes n 2 / n 3 , TM modes ,
B l = N 1 ψ l ( k 0 n 1 a 0 ) χ l ( k 0 n 2 a 0 ) ψ l ( k 0 n 1 a 0 ) χ l ( k 0 n 2 a 0 ) ψ l ( k 0 n 1 a 0 ) ψ l ( k 0 n 2 a 0 ) N 1 ψ l ( k 0 n 1 a 0 ) ψ l ( k 0 n 2 a 0 ) , N 1 = { n 1 / n 2 , TE modes n 2 / n 1 , TM modes .
S TE = λ R n 2 I 2 i = 1 3 n i 2 I i = λ R n 2 η 2 ,
S TM = λ R 2 [ n 1 n 3 k 0 I 2 n 3 A l 2 ψ l ( k 0 n 1 a 0 ) ψ l ( k 0 n 1 a 0 ) + n 1 C l 2 χ l ( k 0 n 3 a 1 ) χ l ( k 0 n 3 a 1 ) ] 2 π n 1 n 2 n 3 i = 1 3 I i = λ R n 2 η 2 λ R 2 [ n 3 A l 2 ψ l ( k 0 n 1 a 0 ) ψ l ( k 0 n 1 a 0 ) n 1 C l 2 χ l ( k 0 n 3 a 1 ) χ l ( k 0 n 3 a 1 ) ] 2 π n 1 n 2 n 3 i = 1 3 I i ,
I 1 = 0 a 0 [ A l ψ l ( k 0 n 1 r ) ] 2 d r , I 3 = a 1 [ C l χ l ( k 0 n 3 r ) ] 2 d r , I 2 = a 0 a 1 [ B l ψ l ( k 0 n 2 r ) + χ l ( k 0 n 2 r ) ] 2 d r ,
A l = B l ψ l ( k 0 n 2 a 0 ) + χ l ( k 0 n 2 a 0 ) ψ l ( k 0 n 1 a 0 ) , C l = B l ψ l ( k 0 n 2 a 1 ) + χ l ( k 0 n 2 a 1 ) χ l ( k 0 n 3 a 1 ) ,
η i = n i 2 I i n 1 2 I 1 + n 2 2 I 2 + n 3 2 I 3 ( TE modes ) , η i = I i I 1 + I 2 + I 3 ( TM modes ) , i = 1 , 2   and   3 .
DL = R S ,
R = 3 σ = 3 σ spect 2 + σ ampl 2 + σ temp 2 ,
σ ampl λ R 4.5 ( SNR 0.25 ) × 1 Q ,
1 Q in 1 ( Q abs ) silica + 1 ( Q abs ) polymer ,
σ temp λ R ( 1 n eff d n eff d T + 1 a 1 d a 1 d T ) σ T ,
σ temp λ R ( i = 1 3 η i ( d n i / d T ) i = 1 3 ( η i n i ) + κ 1 a 0 + κ 2 h 0 a 0 + h 0 ) σ T ,
( σ temp ) TE λ R ( i = 1 3 I i n i 2 ( d n i / d T ) i = 1 3 ( I i n i 3 ) + κ 1 a 0 + κ 2 h 0 a 0 + h 0 ) σ T ,
( σ temp ) TM λ R ( i = 1 3 I i ( d n i / d T ) i = 1 3 ( I i n i ) + κ 1 a 0 + κ 2 h 0 a 0 + h 0 ) σ T .

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