Abstract

This study proposes a thermal sensor based on whispering gallery modes (WGMs) in a polymer core optical ring resonator (PCORR). The thermal sensitivity and detection limit (i.e., the temperature resolution) for WGMs of various orders and polarizations are theoretically studied as a function of the ring wall thickness. The results show that the temperature detection limits can be as low as 4×105 and 6×106K for laser linewidths of 2 and 0.3MHz, respectively. The ultrahigh temperature resolution makes the PCORR a very promising platform for temperature measurement. The analysis also shows that the WGM of a lower order has better thermal sensing performance and a thinner optimal thickness of the ring resonator.

© 2011 Optical Society of America

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    [CrossRef]

2011

2010

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]

Q. L. Ma, T. Rossmann, and Z. X. Guo, “Whispering-gallery mode silica microresonators for cryogenic to room temperature measurement,” Meas. Sci. Technol. 21, 025310 (2010).
[CrossRef]

Y. F. Xiao, C. L. Zou, P. Xue, L. X. Xiao, Y. Li, C. H. Dong, Z. F. Han, and Q. H. Gong, “Quantum electrodynamics in a whispering-gallery microcavity coated with a polymer nanolayer,” Phys. Rev. A 81, 053807 (2010).
[CrossRef]

B. B. Li, Q. Y. Wang, Y. F. Xiao, X. F. Jiang, Y. Li, L. X. Xiao, and Q. H. Gong, “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]

B. Ozel, R. Nett, T. Weigel, G. Schweiger, and A. Ostendorf, “Temperature sensing by using whispering gallery modes with hollow core fibers,” Meas. Sci. Technol. 21, 094015(2010).
[CrossRef]

2009

2008

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

Q. L. Ma, T. Rossmann, and Z. X. Guo, “Temperature sensitivity of silica micro-resonators,” J. Phys. D 41, 245111 (2008).
[CrossRef]

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]

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

2007

2005

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]

1996

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering gallery mode microsphere laser,” Phys. Rev. A 54, R1777–R1780 (1996).
[CrossRef] [PubMed]

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

1991

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]

Beckmann, T.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

Buse, K.

Byer, R. L.

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.

Y. F. Xiao, C. L. Zou, P. Xue, L. X. Xiao, Y. Li, C. H. Dong, Z. F. Han, and Q. H. Gong, “Quantum electrodynamics in a whispering-gallery microcavity coated with a polymer nanolayer,” Phys. Rev. A 81, 053807 (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]

C. H. Dong, L. He, Y. F. Xiao, V. R. Gaddam, S. K. Ozdemir, Z. F. Han, G. C. Guo, and L. Yang, “Fabrication of high-Q polydimethylsiloxane optical microspheres for thermal sensing,” Appl. Phys. Lett. 94, 231119 (2009).
[CrossRef]

Fan, X.

Fan, X. D.

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

Gaddam, V. R.

C. H. Dong, L. He, Y. F. Xiao, V. R. Gaddam, S. K. Ozdemir, Z. F. Han, G. C. Guo, and L. Yang, “Fabrication of high-Q polydimethylsiloxane optical microspheres for thermal sensing,” Appl. Phys. Lett. 94, 231119 (2009).
[CrossRef]

Gong, Q. H.

Y. F. Xiao, C. L. Zou, P. Xue, L. X. Xiao, Y. Li, C. H. Dong, Z. F. Han, and Q. H. Gong, “Quantum electrodynamics in a whispering-gallery microcavity coated with a polymer nanolayer,” Phys. Rev. A 81, 053807 (2010).
[CrossRef]

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

C. H. Dong, L. He, Y. F. Xiao, V. R. Gaddam, S. K. Ozdemir, Z. F. Han, G. C. Guo, and L. Yang, “Fabrication of high-Q polydimethylsiloxane optical microspheres for thermal sensing,” Appl. Phys. Lett. 94, 231119 (2009).
[CrossRef]

Guo, L. J.

Guo, Z. X.

Q. L. Ma, T. Rossmann, and Z. X. Guo, “Whispering-gallery mode silica microresonators for cryogenic to room temperature measurement,” Meas. Sci. Technol. 21, 025310 (2010).
[CrossRef]

Q. L. Ma, T. Rossmann, and Z. X. Guo, “Temperature sensitivity of silica micro-resonators,” J. Phys. D 41, 245111 (2008).
[CrossRef]

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]

Y. F. Xiao, C. L. Zou, P. Xue, L. X. Xiao, Y. Li, C. H. Dong, Z. F. Han, and Q. H. Gong, “Quantum electrodynamics in a whispering-gallery microcavity coated with a polymer nanolayer,” Phys. Rev. A 81, 053807 (2010).
[CrossRef]

C. H. Dong, L. He, Y. F. Xiao, V. R. Gaddam, S. K. Ozdemir, Z. F. Han, G. C. Guo, and L. Yang, “Fabrication of high-Q polydimethylsiloxane optical microspheres for thermal sensing,” Appl. Phys. Lett. 94, 231119 (2009).
[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]

Hare, J.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering gallery mode microsphere laser,” Phys. Rev. A 54, R1777–R1780 (1996).
[CrossRef] [PubMed]

Haroche, S.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering gallery mode microsphere laser,” Phys. Rev. A 54, R1777–R1780 (1996).
[CrossRef] [PubMed]

He, L.

C. H. Dong, L. He, Y. F. Xiao, V. R. Gaddam, S. K. Ozdemir, Z. F. Han, G. C. Guo, and L. Yang, “Fabrication of high-Q polydimethylsiloxane optical microspheres for thermal sensing,” Appl. Phys. Lett. 94, 231119 (2009).
[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]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

Ilchenko, V. S.

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. Gong, “On chip, high-sensitivity thermal sensor based on high-Q polydimethylsiloxane-coated microresonator,” Appl. Phys. Lett. 96, 251109 (2010).
[CrossRef]

Lefevre-Seguin, V.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering gallery mode microsphere laser,” Phys. Rev. A 54, R1777–R1780 (1996).
[CrossRef] [PubMed]

Li, B. B.

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

Y. F. Xiao, C. L. Zou, P. Xue, L. X. Xiao, Y. Li, C. H. Dong, Z. F. Han, and Q. H. Gong, “Quantum electrodynamics in a whispering-gallery microcavity coated with a polymer nanolayer,” Phys. Rev. A 81, 053807 (2010).
[CrossRef]

Lin, N.

Ling, T.

Lu, Y. F.

Ma, Q. L.

Q. L. Ma, T. Rossmann, and Z. X. Guo, “Whispering-gallery mode silica microresonators for cryogenic to room temperature measurement,” Meas. Sci. Technol. 21, 025310 (2010).
[CrossRef]

Q. L. Ma, T. Rossmann, and Z. X. Guo, “Temperature sensitivity of silica micro-resonators,” J. Phys. D 41, 245111 (2008).
[CrossRef]

Mason, G. F.

Nett, R.

B. Ozel, R. Nett, T. Weigel, G. Schweiger, and A. Ostendorf, “Temperature sensing by using whispering gallery modes with hollow core fibers,” Meas. Sci. Technol. 21, 094015(2010).
[CrossRef]

Ostendorf, A.

B. Ozel, R. Nett, T. Weigel, G. Schweiger, and A. Ostendorf, “Temperature sensing by using whispering gallery modes with hollow core fibers,” Meas. Sci. Technol. 21, 094015(2010).
[CrossRef]

Oveys, H.

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

Ozdemir, S. K.

C. H. Dong, L. He, Y. F. Xiao, V. R. Gaddam, S. K. Ozdemir, Z. F. Han, G. C. Guo, and L. Yang, “Fabrication of high-Q polydimethylsiloxane optical microspheres for thermal sensing,” Appl. Phys. Lett. 94, 231119 (2009).
[CrossRef]

Ozel, B.

B. Ozel, R. Nett, T. Weigel, G. Schweiger, and A. Ostendorf, “Temperature sensing by using whispering gallery modes with hollow core fibers,” Meas. Sci. Technol. 21, 094015(2010).
[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]

Raimond, J. M.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering gallery mode microsphere laser,” Phys. Rev. A 54, R1777–R1780 (1996).
[CrossRef] [PubMed]

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]

Rossmann, T.

Q. L. Ma, T. Rossmann, and Z. X. Guo, “Whispering-gallery mode silica microresonators for cryogenic to room temperature measurement,” Meas. Sci. Technol. 21, 025310 (2010).
[CrossRef]

Q. L. Ma, T. Rossmann, and Z. X. Guo, “Temperature sensitivity of silica micro-resonators,” J. Phys. D 41, 245111 (2008).
[CrossRef]

Sandoghdar, V.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering gallery mode microsphere laser,” Phys. Rev. A 54, R1777–R1780 (1996).
[CrossRef] [PubMed]

Savchenkov, A. A.

Schiller, S.

Schweiger, G.

B. Ozel, R. Nett, T. Weigel, G. Schweiger, and A. Ostendorf, “Temperature sensing by using whispering gallery modes with hollow core fibers,” Meas. Sci. Technol. 21, 094015(2010).
[CrossRef]

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.

J. D. Suter, I. M. White, H. Zhu, and X. Fan, “Thermal characterization of liquid core optical ring resonator sensors,” Appl. Opt. 46, 389–396 (2007).
[CrossRef] [PubMed]

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

Treussart, F.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering gallery mode microsphere laser,” Phys. Rev. A 54, R1777–R1780 (1996).
[CrossRef] [PubMed]

Tsai, H. L.

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]

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. Gong, “On chip, high-sensitivity thermal sensor based on high-Q polydimethylsiloxane-coated microresonator,” Appl. Phys. Lett. 96, 251109 (2010).
[CrossRef]

Wang, S. M.

Weigel, T.

B. Ozel, R. Nett, T. Weigel, G. Schweiger, and A. Ostendorf, “Temperature sensing by using whispering gallery modes with hollow core fibers,” Meas. Sci. Technol. 21, 094015(2010).
[CrossRef]

White, I. M.

J. D. Suter, I. M. White, H. Zhu, and X. Fan, “Thermal characterization of liquid core optical ring resonator sensors,” Appl. Opt. 46, 389–396 (2007).
[CrossRef] [PubMed]

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

Xiao, L. X.

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

Y. F. Xiao, C. L. Zou, P. Xue, L. X. Xiao, Y. Li, C. H. Dong, Z. F. Han, and Q. H. Gong, “Quantum electrodynamics in a whispering-gallery microcavity coated with a polymer nanolayer,” Phys. Rev. A 81, 053807 (2010).
[CrossRef]

Xiao, Y. F.

Y. F. Xiao, C. L. Zou, P. Xue, L. X. Xiao, Y. Li, C. H. Dong, Z. F. Han, and Q. H. Gong, “Quantum electrodynamics in a whispering-gallery microcavity coated with a polymer nanolayer,” Phys. Rev. A 81, 053807 (2010).
[CrossRef]

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

C. H. Dong, L. He, Y. F. Xiao, V. R. Gaddam, S. K. Ozdemir, Z. F. Han, G. C. Guo, and L. Yang, “Fabrication of high-Q polydimethylsiloxane optical microspheres for thermal sensing,” Appl. Phys. Lett. 94, 231119 (2009).
[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]

Xue, P.

Y. F. Xiao, C. L. Zou, P. Xue, L. X. Xiao, Y. Li, C. H. Dong, Z. F. Han, and Q. H. Gong, “Quantum electrodynamics in a whispering-gallery microcavity coated with a polymer nanolayer,” Phys. Rev. A 81, 053807 (2010).
[CrossRef]

Yang, L.

C. H. Dong, L. He, Y. F. Xiao, V. R. Gaddam, S. K. Ozdemir, Z. F. Han, G. C. Guo, and L. Yang, “Fabrication of high-Q polydimethylsiloxane optical microspheres for thermal sensing,” Appl. Phys. Lett. 94, 231119 (2009).
[CrossRef]

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[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Polymer core ring resonator (PCORR)-based thermal sensor configuration. The PCORR is shown in cylindrical coordinates (r, θ, and z). n 1 , n 2 , and n 3 are the refractive indices of the polymer core, ring wall and surrounding medium, respectively. FSR is defined as the difference between two adjacent resonant wavelengths. δ λ R denotes the resonance shift induced by the temperature change.

Fig. 2
Fig. 2

Thermal sensitivity S for TM modes (solid curves) and TE modes (dashed curves) of the first three orders as a function of the ring wall thickness h. The symbols denote the highest thermal sensitivity S MAX for the TM l = 1 (solid circle), TE l = 1 (open circle), TM l = 2 (solid star), TE l = 2 (open star), TM l = 3 (solid diamond), and TE l = 3 (open diamond) modes, respectively.

Fig. 3
Fig. 3

Light energy intensity distribution along the radial direction for TM modes of the first three orders, at (a)  h = 1 and (b)  2 μm .

Fig. 4
Fig. 4

Resonant wavelength shift δ λ R for TM modes of different orders as a function of the temperature change δ T at h = 1 and 2 μm .

Fig. 5
Fig. 5

Q factors for TM modes (solid curves) and TE modes (dashed curves) of the first three orders as a function of the ring wall thickness h.

Fig. 6
Fig. 6

Temperature detection limit ( δ λ R ) min for TM modes (solid curves) and TE modes (dashed curves) of the first three orders as a function of the ring wall thickness h. The value of ( δ λ R ) min / Δ λ R limited by the detector noise is assumed to be 1 / 100 .

Fig. 7
Fig. 7

Temperature detection limit of the first-order TM mode restricted by laser linewidths of 15.5 and 2.4 fm , respectively, as a function of the ring wall thickness h.

Equations (13)

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N 0 H m ( 1 ) ( k 0 n 3 R 2 ) H m ( 1 ) ( k 0 n 3 R 2 ) = B m J m ( k 0 n 2 R 2 ) + H m ( 1 ) ( k 0 n 2 R 2 ) B m J m ( k 0 n 2 R 2 ) + H m ( 1 ) ( k 0 n 2 R 2 ) , N 0 = { n 2 / n 3 , TE   modes n 3 / n 2 , TM   modes ,
B m = N 1 J m ( k 0 n 1 R 1 ) H m ( 1 ) ( k 0 n 2 R 1 ) J m ( k 0 n 1 R 1 ) H m ( 1 ) ( k 0 n 2 R 1 ) J m ( k 0 n 1 R 1 ) J m ( k 0 n 2 R 1 ) N 1 J m ( k 0 n 1 R 1 ) J m ( k 0 n 2 R 1 ) , N 0 = { n 1 / n 2 , TE   modes n 2 / n 1 , TM   modes .
R 1 = R 1 ( 1 + α 1 δ T ) , h = h ( 1 + α 2 δ T ) , n i = n i + ( d n i / d T ) δ T ,
S = δ λ R δ T = λ R ( 1 n eff d n eff d T + α ) ,
α = α 1 R 1 + α 2 h R 2 .
S λ R ( i = 1 3 η i ( d n i / d T ) i = 1 3 ( η i n i ) + α 1 R 1 + α 2 h R 2 ) .
η i = { I i I 1 + I 2 + I 3 , TE   Modes n i 2 I i n 1 2 I 1 + n 2 2 I 2 + n 3 2 I 3 , TM   Modes , i = 1 , 2 , 3 ,
I 1 = 0 R 1 | A m J m ( k 0 n 1 r ) | 2 d r , I 3 = R 2 | C m H m ( 1 ) ( k 0 n 3 r ) | 2 d r , I 2 = R 1 R 2 | B m J m ( k 0 n 2 r ) + H m ( 1 ) ( k 0 n 2 r ) | 2 d r .
A m = B m J m ( k 0 n 2 R 1 ) + H m ( 1 ) ( k 0 n 2 R 1 ) J m ( k 0 n 1 R 1 ) , C m = B m J m ( k 0 n 2 R 2 ) + H m ( 1 ) ( k 0 n 2 R 2 ) H m ( 1 ) ( k 0 n 3 R 2 ) .
S TE λ R ( i = 1 3 I i ( d n i / d T ) i = 1 3 ( I i n i ) + α 1 R 1 + α 2 h R 2 ) , S TM λ R ( i = 1 3 I i n i 2 ( d n i / d T ) i = 1 3 ( I i n i 3 ) + α 1 R 1 + α 2 h R 2 ) .
( δ T ) min = ( δ λ R ) min Δ λ R × λ R Q × S ,
1 Q 1 Q sca + 1 ( Q abs ) wall + 1 ( Q abs ) core ,
S ( 1 n 2 d n 2 d T + α 1 ) λ R λ R R 2 ( α 1 α 2 ) h ,

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