Abstract

Recently there is a growing interest in developing few-mode fiber (FMF) based distributed sensors, which can attain higher spatial resolution and sensitivity compared with the conventional single-mode approaches. However, current techniques require two lightwaves injected into both ends of FMF, resulting in their complicated setup and high cost, which causes a big issue for geotechnical and petroleum applications. In this paper, we present a single-end FMF-based distributed sensing system that allows simultaneous temperature and strain measurement by Brillouin optical time-domain reflectometry (BOTDR) and heterodyne detection. Theoretical analysis and experimental assessment of multi-parameter discriminative measurement techniques applied to distributed FMF sensors are presented. Experimental results confirm that FM-BOTDR has similar performance with two-end methods such as FM-BOTDA, but with simpler setup and lower cost. The temperature-induced expansion strain (TIES) in response to different modes is discussed as well. Furthermore, we optimized the FMF design by exploiting modal profile and doping concentration, which indicates up to fivefold enhancement in measurement accuracy. This novel distributed FM-sensing system endows with good sensitivity characteristics and can prevent catastrophic failure in many applications.

© 2015 Optical Society of America

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References

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  1. X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors (Basel) 11(12), 4152–4187 (2011).
    [Crossref] [PubMed]
  2. T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
    [Crossref]
  3. W. R. Habel and K. Krebber, “Fiber-optic sensor applications in civil and geotechnical engineering,” Photonic Sensors 1(3), 268–280 (2011).
    [Crossref]
  4. M. Alahbabi, Y. T. Cho, and T. P. Newson, “Comparison of the methods for discriminating temperature and strain in spontaneous Brillouin-based distributed sensors,” Opt. Lett. 29(1), 26–28 (2004).
    [Crossref] [PubMed]
  5. S. M. Maughan, H. H. Kee, and T. P. Newson, “Simultaneous distributed fiber temperature and strain sensor using microwave coherent detection of spontaneous Brillouin backscatter,” Meas. Sci. Technol. 12(7), 834–842 (2001).
    [Crossref]
  6. T. R. Parker, M. Farhadiroushan, V. A. Handerek, and A. J. Rogers, “Temperature and strain dependence of the power level and frequency of spontaneous Brillouin scattering in optical fibers,” Opt. Lett. 22(11), 787–789 (1997).
    [Crossref] [PubMed]
  7. M. N. Alahbabi, Y. T. Cho, and T. P. Newson, “Simultaneous temperature and strain measurement with combined spontaneous Raman and Brillouin scattering,” Opt. Lett. 30(11), 1276–1278 (2005).
    [Crossref] [PubMed]
  8. X. Liu and X. Bao, “Brillouin spectrum in LEAF and simultaneous temperature and strain measurement,” J. Lightwave Technol. 30(8), 1053–1059 (2012).
    [Crossref]
  9. X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).
  10. G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photon. 6(4), 413–487 (2014).
    [Crossref]
  11. A. Li, Y. Wang, Q. Hu, and W. Shieh, “Few-mode fiber based optical sensors,” Opt. Express 23(2), 1139–1150 (2015).
    [Crossref]
  12. K. Y. Song and Y. H. Kim, “Characterization of stimulated Brillouin scattering in a few-mode fiber,” Opt. Lett. 38(22), 4841–4844 (2013).
    [Crossref] [PubMed]
  13. S. Li, M.-J. Li, and R. S. Vodhanel, “All-optical Brillouin dynamic grating generation in few-mode optical fiber,” Opt. Lett. 37(22), 4660–4662 (2012).
    [Crossref] [PubMed]
  14. A. Li, Q. Hu, and W. Shieh, “Characterization of stimulated Brillouin scattering in a circular-core two-mode fiber using optical time-domain analysis,” Opt. Express 21(26), 31894–31906 (2013).
    [Crossref] [PubMed]
  15. D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
    [Crossref]
  16. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).
  17. H. H. Kee, G. P. Lees, and T. P. Newson, “All-fiber system for simultaneous interrogation of distributed strain and temperature sensing by spontaneous Brillouin scattering,” Opt. Lett. 25(10), 695–697 (2000).
    [Crossref] [PubMed]
  18. A. B. Ruffin, M.-J. Li, X. Chen, A. Kobyakov, and F. Annunziata, “Brillouin gain analysis for fibers with different refractive indices,” Opt. Lett. 30(23), 3123–3125 (2005).
    [Crossref] [PubMed]
  19. M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007).
    [Crossref] [PubMed]
  20. K. Oh and U.-C. Paek, Silica Optical Fiber Technology for Devices and Components: Design, Fabrication, and International Standards (Wiley, 2012).
  21. E. Ip, N. Bai, Y.-K. Huang, E. Mateo, F. Yaman, M.-J. Li, S. Bickham, S. Ten, J. Linares, C. Montero, V. Moreno, X. Prieto, Y. Luo, G.-D. Peng, G. Li, and T. Wang, “6×6 MIMO transmission over 50+25+10 km heterogeneous spans of few-mode fiber with inline erbium-doped fiber amplifier,” in Optical Fiber Communication Conference, 2012 OSA Technical Digest Series (Optical Society of America, 2012), paper OTu2C.4.
  22. X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
    [Crossref]

2015 (1)

2014 (2)

G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photon. 6(4), 413–487 (2014).
[Crossref]

X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).

2013 (3)

2012 (3)

X. Liu and X. Bao, “Brillouin spectrum in LEAF and simultaneous temperature and strain measurement,” J. Lightwave Technol. 30(8), 1053–1059 (2012).
[Crossref]

S. Li, M.-J. Li, and R. S. Vodhanel, “All-optical Brillouin dynamic grating generation in few-mode optical fiber,” Opt. Lett. 37(22), 4660–4662 (2012).
[Crossref] [PubMed]

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

2011 (2)

X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors (Basel) 11(12), 4152–4187 (2011).
[Crossref] [PubMed]

W. R. Habel and K. Krebber, “Fiber-optic sensor applications in civil and geotechnical engineering,” Photonic Sensors 1(3), 268–280 (2011).
[Crossref]

2007 (1)

2005 (2)

2004 (1)

2001 (1)

S. M. Maughan, H. H. Kee, and T. P. Newson, “Simultaneous distributed fiber temperature and strain sensor using microwave coherent detection of spontaneous Brillouin backscatter,” Meas. Sci. Technol. 12(7), 834–842 (2001).
[Crossref]

2000 (1)

1997 (1)

1995 (1)

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Alahbabi, M.

Alahbabi, M. N.

Annunziata, F.

Bai, N.

Bao, X.

X. Liu and X. Bao, “Brillouin spectrum in LEAF and simultaneous temperature and strain measurement,” J. Lightwave Technol. 30(8), 1053–1059 (2012).
[Crossref]

X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors (Basel) 11(12), 4152–4187 (2011).
[Crossref] [PubMed]

Burgess, D. T.

X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).

Chen, L.

X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors (Basel) 11(12), 4152–4187 (2011).
[Crossref] [PubMed]

Chen, X.

Cho, Y. T.

Crowley, A. M.

Demeritt, J. A.

Farhadiroushan, M.

Fini, J. M.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Gray, S.

Habel, W. R.

W. R. Habel and K. Krebber, “Fiber-optic sensor applications in civil and geotechnical engineering,” Photonic Sensors 1(3), 268–280 (2011).
[Crossref]

Handerek, V. A.

Hines, M.

X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).

Horiguchi, T.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Hu, Q.

Kee, H. H.

S. M. Maughan, H. H. Kee, and T. P. Newson, “Simultaneous distributed fiber temperature and strain sensor using microwave coherent detection of spontaneous Brillouin backscatter,” Meas. Sci. Technol. 12(7), 834–842 (2001).
[Crossref]

H. H. Kee, G. P. Lees, and T. P. Newson, “All-fiber system for simultaneous interrogation of distributed strain and temperature sensing by spontaneous Brillouin scattering,” Opt. Lett. 25(10), 695–697 (2000).
[Crossref] [PubMed]

Kim, Y. H.

Kobyakov, A.

Koyamada, Y.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Krebber, K.

W. R. Habel and K. Krebber, “Fiber-optic sensor applications in civil and geotechnical engineering,” Photonic Sensors 1(3), 268–280 (2011).
[Crossref]

Kurashima, T.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Lees, G. P.

Li, A.

Li, G.

Li, J.

X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).

Li, M.-J.

Li, S.

Li, X.

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

Liang, J.

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

Lin, S.

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

Liu, A.

Liu, X.

Maughan, S. M.

S. M. Maughan, H. H. Kee, and T. P. Newson, “Simultaneous distributed fiber temperature and strain sensor using microwave coherent detection of spontaneous Brillouin backscatter,” Meas. Sci. Technol. 12(7), 834–842 (2001).
[Crossref]

Nelson, L. E.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Newson, T. P.

Oigawa, H.

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

Parker, T. R.

Richardson, D. J.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Rogers, A. J.

Ruffin, A. B.

Shieh, W.

Shimizu, K.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Song, K. Y.

Sun, X.

X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).

Tateda, M.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Ueda, T.

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

Vodhanel, R. S.

Walton, D. T.

Wang, J.

Wang, Y.

Xia, C.

Zenteno, L. A.

Zhang, Y.

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

Zhao, N.

Zhu, B.

X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).

Adv. Opt. Photon. (1)

IEEE Photonics J. (1)

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

J. Lightwave Technol. (2)

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

X. Liu and X. Bao, “Brillouin spectrum in LEAF and simultaneous temperature and strain measurement,” J. Lightwave Technol. 30(8), 1053–1059 (2012).
[Crossref]

Meas. Sci. Technol. (1)

S. M. Maughan, H. H. Kee, and T. P. Newson, “Simultaneous distributed fiber temperature and strain sensor using microwave coherent detection of spontaneous Brillouin backscatter,” Meas. Sci. Technol. 12(7), 834–842 (2001).
[Crossref]

Nat. Photonics (1)

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Opt. Express (3)

Opt. Lett. (7)

Photonic Sensors (1)

W. R. Habel and K. Krebber, “Fiber-optic sensor applications in civil and geotechnical engineering,” Photonic Sensors 1(3), 268–280 (2011).
[Crossref]

Proc. SPIE (1)

X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).

Sensors (Basel) (1)

X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors (Basel) 11(12), 4152–4187 (2011).
[Crossref] [PubMed]

Other (3)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).

K. Oh and U.-C. Paek, Silica Optical Fiber Technology for Devices and Components: Design, Fabrication, and International Standards (Wiley, 2012).

E. Ip, N. Bai, Y.-K. Huang, E. Mateo, F. Yaman, M.-J. Li, S. Bickham, S. Ten, J. Linares, C. Montero, V. Moreno, X. Prieto, Y. Luo, G.-D. Peng, G. Li, and T. Wang, “6×6 MIMO transmission over 50+25+10 km heterogeneous spans of few-mode fiber with inline erbium-doped fiber amplifier,” in Optical Fiber Communication Conference, 2012 OSA Technical Digest Series (Optical Society of America, 2012), paper OTu2C.4.

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Figures (14)

Fig. 1
Fig. 1 (a) Operational Principle of BOTDR Sensing System; (b) Schematic of Brillouin frequency shift.

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Fig. 2
Fig. 2 The intensity profiles of optical/acoustic modes in FMF. (a) Normalized Intensity Profile for LP01; (b) Normalized Intensity Profile for LP11.

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Fig. 3
Fig. 3 Brillouin Gain Spectrum of BOTDR in FMF for LP01/11 modes.

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Fig. 4
Fig. 4 Experimental Setup of Few-mode Brillouin Sensing System. DFB-LD, Distributed Feedback Laser Diode; OS, Optical Switch; EPG, Electrical Pulse Generator; PM-EDFA, Polarization Maintaining Erbium-doped Fiber Amplifier; FPC, Fiber Polarization Controller; MC, Mode Converter; BS, Beam Splitter; MMUX, Mode Multiplexer; FUT, Fiber Under Test; MDEMUX, Mode De-Multiplexer; RE: Reflective End; LO, Local Oscillator; OH, 90° Optical Hybrid; BR, Balanced Receiver; TDS, Time Domain Oscilloscope.

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Fig. 5
Fig. 5 (a) The operation principle of few-mode Brillouin sensing systems; (b) Sample of experimental Brillouin spectrum.

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Fig. 6
Fig. 6 (a) Output Spectra of SPBS in FMF for LP01 (left) and LP11 (right) modes, with the corresponding mode patterns shown in the inset; (b) SNR comparison between LP01 and LP11 modes along the sensing fiber for FM-BOTDR system after 20 times averaging.

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Fig. 7
Fig. 7 Calibration of Temperature Coefficient for different modes in FMF.

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Fig. 8
Fig. 8 Calibration of Strain Coefficient for different modes in FMF.

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Fig. 9
Fig. 9 Differential BFS Dependence on Temperature in FMF.

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Fig. 10
Fig. 10 Differential BFS Dependence on Strain in FMF.

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Fig. 11
Fig. 11 Three-Dimensional BGS Diagrams (upper), Received signal strength vs. sensing distance (bottom); (a) for LP01 mode; (b) for LP11 mode.

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Fig. 12
Fig. 12 3D plots of Brillouin frequency shift υB vs. simultaneous temperature T and total strain ∑ε changes, where the surface stands for numerical prediction, and the red dots represent the experimental data when we change temperature and strain simultaneously. (a) υB vs.T vs.∑ε for LP01; (b) υB vs.T vs.∑ε for LP11. The nonlinear relation between υB and ∑ε is due to the temperature-induced expansion strain.

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Fig. 13
Fig. 13 Refractive Index Profiles of Various Designs; n vs. Radius r (μm).

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Fig. 14
Fig. 14 Specifications of HNL-FMF core. (a) Cross section and profile for HNL-FMF core; (b) Refractive index and modal profiles of HNL-FMF.

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Tables (3)

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Table 1 Parameters for Custom-designed Step-index FMF [21]

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Table 2 Comparison of f-T and f-ε coefficients in FMF

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Table 3 Comparison of various mode profiles and doping design in FMF

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Equations (20)

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d 2 f o d r 2 + 1 r ⋅ d f o dr + k o 2 ⋅( n o 2 ( r )− n o​​ ​eff 2 )⋅ f o =0.
n a ( r )= V Clad V L ( r ) .
λ a = λ 2 n o eff .
k a = 2π λ a = 4π n o eff λ .
n a eff = V Clad V eff .
d 2 f a d r 2 + 1 r ⋅ d f a dr + k a 2 ⋅( n a 2 ( r )− n a​​ ​eff 2 )⋅ f a =0.
Δ o = n o 2 ( r )− n o Clad 2 2 n o 2 ( r ) ×100%.
Δ a = n a 2 ( r )− n a Clad 2 2 n a 2 ( r ) ×100%.
ν B = 2 n o eff λ ⋅ V eff = V Clad λ ⋅ 2 n o eff n a eff .
I u = ( ∫ E o E o * ρ u * r dr dθ ) 2 ∫ ( E o E o * ) 2 r dr dθ⋅ ∫ ρ  ρ * r dr dθ .
n o ( ΔT,Δε, ω Ge , ω F )= n o Clad ∗[1+( 1× 10 −3 +3× 10 −6 ΔT+1.5× 10 −7 Δε )∗ ω Ge +( −3.3× 10 −3 +3.6× 10 −6 ΔT+7.5× 10 −7 Δε )* ω F ].
n a ( ΔT,Δε, ω Ge , ω F )= n a Clad ∗[1+( 7.2× 10 −3 −4.7× 10 −5 ΔT−2.1× 10 −6 Δε )∗ ω Ge +( 2.7× 10 −3 −1.8× 10 −5 ΔT−3.8× 10 −6 Δε )* ω F ].
( Δ ν B Mode 1 Δ ν B Mode 2 )=( C ν T Mode  1 C ν ε Mode  1 C ν T Mode  2 C ν ε Mode  2 )⋅( ΔT Δε ).
( Δ ν B Mode 1 ⋮ Δ ν B Mode N )=( C ν T Mode  1 … C ν ε Mode  1 ⋮ ⋱ ⋮ C ν T Mode  N ⋯ C ν ε Mode  N )⋅( ΔT Δε ).
ΔT= C ν ε Mode 2 ⋅Δ ν B Mode 1 − C ν ε Mode 1 ⋅Δ ν B Mode 2 C ν ε Mode 2 ⋅ C ν T Mode 1 − C ν ε Mode 1 ⋅ C ν T Mode 2 .
Δε= C ν T Mode 2 ⋅Δ ν B Mode 1 − C ν T Mode 1 ⋅Δ ν B Mode 2 C ν T Mode 2 ⋅ C ν ε Mode 1 − C ν T Mode 1 ⋅ C ν ε Mode 2 .
δT= | C ν ε Mode 2 |⋅δ ν B Mode 1 +| C ν ε Mode 1 |⋅δ ν B Mode 2 | C ν ε Mode 2 ⋅ C ν T Mode 1 − C ν ε Mode 1 ⋅ C ν T Mode 2 | .
δε= | C ν T Mode 2 |⋅δ ν B Mode 1 +| C ν T Mode 1 |⋅δ ν B Mode 2 | C ν T Mode 2 ⋅ C ν ε Mode 1 − C ν T Mode 1 ⋅ C ν ε Mode 2 | .
RMS( ΔT )= { [ C ν ε Mode 2 ⋅RMS( Δ ν B Mode 1 ) C ν ε Mode 2 C ν T Mode 1 − C ν ε Mode 1 C ν T Mode 2 ]  2 + [ C ν ε Mode 1 ⋅RMS( Δ ν B Mode 2 ) C ν ε Mode 2 C ν T Mode 1 − C ν ε Mode 1 C ν T Mode 2 ]  2 } 1/2
RMS( Δε )= { [ C ν T Mode 2 ⋅RMS( Δ ν B Mode 1 ) C ν T Mode 2 C ν ε Mode 1 − C ν T Mode 1 C ν ε Mode 2 ]  2 + [ C ν T Mode 1 ⋅RMS( Δ ν B Mode 2 ) C ν T Mode 2 C ν ε Mode 1 − C ν T Mode 1 C ν ε Mode 2 ]  2 } 1/2

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