Abstract

An in-fiber integrated quasi-distributed high temperature sensor array is proposed and demonstrated. The sensor array consists of some weakly reflective joint surfaces which are welded by single mode fiber (SMF) and double-clad fiber (DCF). The characteristics of the reflected signal of the sensor array are analyzed, and the relationship between the signal intensity and the number of sensors is simulated for evaluating sensor multiplex capacity. Due to its all-silica structure, the sensor array could test temperature up to 1000°C for a long time. This sensor array is flexible and easy to be fabricated only by splicing without any connector, which will be beneficial to space constrained quasi-distributed high temperature sensing applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (2)

H. Chen, M. Buric, P. R. Ohodnicki, J. Nakano, B. Liu, and B. T. Chorpening, “Review and perspective: Sapphire optical fiber cladding development for harsh environment sensing,” Appl. Phys. Rev. 5(1), 011102 (2018).
[Crossref]

A. Yan, S. Li, Z. Peng, R. Zou, P. Ohodnicki, P. Lu, K. Byerly, M.-J. Li, and K. P. Chen, “Multi-point fiber optic sensors for real-time monitoring of the temperature distribution on transformer cores,” Proc. SPIE 10639, 1063912 (2018).

2017 (4)

X. Qiao, Z. Shao, W. Bao, and Q. Rong, “Fiber Bragg Grating Sensors for the Oil Industry,” Sensors (Basel) 17(3), 429 (2017).
[Crossref] [PubMed]

C. Hill, D. Homa, Z. Yu, Y. Cheng, B. Liu, A. Wang, and G. Pickrell, “Single Mode Air-Clad Single Crystal Sapphire Optical Fiber,” Appl. Sci. (Basel) 7(5), 473 (2017).
[Crossref]

G. Liu, Q. Sheng, D. Dam, J. Hua, W. Hou, and M. Han, “Self-gauged fiber-optic micro-heater with an operation temperature above 1000°C,” Opt. Lett. 42(7), 1412–1415 (2017).
[Crossref] [PubMed]

S. Yang, D. Hu, and A. Wang, “Point-by-point fabrication and characterization of sapphire fiber Bragg gratings,” Opt. Lett. 42(20), 4219–4222 (2017).
[Crossref] [PubMed]

2016 (3)

2015 (1)

H. Bartelt, T. Elsmann, T. Habisreuther, K. Schuster, and M. Rothhardt, “Optical Bragg grating sensor fibers for ultra-high temperature applications,” Proc. SPIE 9655, 96552S (2015).
[Crossref]

2014 (2)

2013 (3)

X. Tan, Y. Geng, X. Li, R. Gao, and Z. Yin, “High temperature microstructured fiber sensor based on a partial-reflection-enabled intrinsic Fabry-Perot interferometer,” Appl. Opt. 52(34), 8195–8198 (2013).
[Crossref] [PubMed]

C. L. Lee, J. M. Hsu, J. S. Horng, W. Y. Sung, and C. M. Li, “Microcavity Fiber Fabry Prot Interferometer With an Embedded Golden Thin Film,” IEEE Photonic. Tech. Lett. 25(9), 833–836 (2013).
[Crossref]

M. Ratuszek, M. J. Ratuszek, and J. Hejna, “The study of thermal connecting of telecommunication optical fibers (SiO2: GeO2) and EDF (SiO2: Al2O3, Er) fibers,” B. Pol. Acad. Sci. Tech. (Paris) 61(1), 279–286 (2013).
[Crossref]

2012 (2)

S. Farsinezhad, F. E. Seraji, S. Farsinezhad, and F. E. Seraji, “Analysis of Fresnel Loss at Splice Joint Between Single-Mode Fiber and Photonic Crystal Fiber,” International Journal of Optics and Applications 2(1), 17–21 (2012).
[Crossref]

P. Rugeland and W. Margulis, “Revisiting twin-core fiber sensors for high-temperature measurements,” Appl. Opt. 51(25), 6227–6232 (2012).
[Crossref] [PubMed]

2010 (3)

2008 (1)

2006 (1)

K. T. Kim, K. H. Lee, E. S. Shin, H. S. Song, K. B. Hong, S. Hwangbo, and K. R. Sohn, “Characteristics of side-polished thermally expanded core fiber and its application as a band-edge filter with a high cut-off property,” Opt. Commun. 261(1), 51–55 (2006).
[Crossref]

2005 (1)

T. L. Lowder, K. H. Smith, B. L. Ipson, A. R. Hawkins, R. H. Selfridge, and S. M. Schultz, “High-temperature sensing using surface relief fiber Bragg gratings,” IEEE Photonic. Tech. Lett. 17(9), 1926–1928 (2005).
[Crossref]

1998 (1)

L. Yuan, “Optical path automatic compensation low-coherence interferometric fibre-optic temperature sensor,” Opt. Laser Technol. 30(1), 33–38 (1998).
[Crossref]

1990 (1)

K. Shiraishi, Y. Aizawa, and S. Kawakami, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
[Crossref]

Aizawa, Y.

K. Shiraishi, Y. Aizawa, and S. Kawakami, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
[Crossref]

Amezcua-Correa, R.

Antonio-Lopez, J. E.

Bao, W.

X. Qiao, Z. Shao, W. Bao, and Q. Rong, “Fiber Bragg Grating Sensors for the Oil Industry,” Sensors (Basel) 17(3), 429 (2017).
[Crossref] [PubMed]

Bartelt, H.

Bierlich, J.

Buric, M.

H. Chen, M. Buric, P. R. Ohodnicki, J. Nakano, B. Liu, and B. T. Chorpening, “Review and perspective: Sapphire optical fiber cladding development for harsh environment sensing,” Appl. Phys. Rev. 5(1), 011102 (2018).
[Crossref]

Byerly, K.

A. Yan, S. Li, Z. Peng, R. Zou, P. Ohodnicki, P. Lu, K. Byerly, M.-J. Li, and K. P. Chen, “Multi-point fiber optic sensors for real-time monitoring of the temperature distribution on transformer cores,” Proc. SPIE 10639, 1063912 (2018).

Chen, H.

H. Chen, M. Buric, P. R. Ohodnicki, J. Nakano, B. Liu, and B. T. Chorpening, “Review and perspective: Sapphire optical fiber cladding development for harsh environment sensing,” Appl. Phys. Rev. 5(1), 011102 (2018).
[Crossref]

Chen, K. P.

A. Yan, S. Li, Z. Peng, R. Zou, P. Ohodnicki, P. Lu, K. Byerly, M.-J. Li, and K. P. Chen, “Multi-point fiber optic sensors for real-time monitoring of the temperature distribution on transformer cores,” Proc. SPIE 10639, 1063912 (2018).

Cheng, Y.

C. Hill, D. Homa, Z. Yu, Y. Cheng, B. Liu, A. Wang, and G. Pickrell, “Single Mode Air-Clad Single Crystal Sapphire Optical Fiber,” Appl. Sci. (Basel) 7(5), 473 (2017).
[Crossref]

Chiang, K. S.

T. Zhu, T. Ke, Y. Rao, and K. S. Chiang, “Fabry–Perot optical fiber tip sensor for high temperature measurement,” Opt. Commun. 283(19), 3683–3685 (2010).
[Crossref]

Chorpening, B. T.

H. Chen, M. Buric, P. R. Ohodnicki, J. Nakano, B. Liu, and B. T. Chorpening, “Review and perspective: Sapphire optical fiber cladding development for harsh environment sensing,” Appl. Phys. Rev. 5(1), 011102 (2018).
[Crossref]

Dai, N.

Dam, D.

Dellith, J.

Dong, B.

Ebendorff-Heidepriem, H.

Elsmann, T.

Eznaveh, Z. S.

Farsinezhad, S.

S. Farsinezhad, F. E. Seraji, S. Farsinezhad, and F. E. Seraji, “Analysis of Fresnel Loss at Splice Joint Between Single-Mode Fiber and Photonic Crystal Fiber,” International Journal of Optics and Applications 2(1), 17–21 (2012).
[Crossref]

S. Farsinezhad, F. E. Seraji, S. Farsinezhad, and F. E. Seraji, “Analysis of Fresnel Loss at Splice Joint Between Single-Mode Fiber and Photonic Crystal Fiber,” International Journal of Optics and Applications 2(1), 17–21 (2012).
[Crossref]

Gao, R.

Geng, Y.

Gong, J.

Habisreuther, T.

Han, M.

Hawkins, A. R.

T. L. Lowder, K. H. Smith, B. L. Ipson, A. R. Hawkins, R. H. Selfridge, and S. M. Schultz, “High-temperature sensing using surface relief fiber Bragg gratings,” IEEE Photonic. Tech. Lett. 17(9), 1926–1928 (2005).
[Crossref]

Hejna, J.

M. Ratuszek, M. J. Ratuszek, and J. Hejna, “The study of thermal connecting of telecommunication optical fibers (SiO2: GeO2) and EDF (SiO2: Al2O3, Er) fibers,” B. Pol. Acad. Sci. Tech. (Paris) 61(1), 279–286 (2013).
[Crossref]

Hill, C.

C. Hill, D. Homa, Z. Yu, Y. Cheng, B. Liu, A. Wang, and G. Pickrell, “Single Mode Air-Clad Single Crystal Sapphire Optical Fiber,” Appl. Sci. (Basel) 7(5), 473 (2017).
[Crossref]

Homa, D.

C. Hill, D. Homa, Z. Yu, Y. Cheng, B. Liu, A. Wang, and G. Pickrell, “Single Mode Air-Clad Single Crystal Sapphire Optical Fiber,” Appl. Sci. (Basel) 7(5), 473 (2017).
[Crossref]

Hong, K. B.

K. T. Kim, K. H. Lee, E. S. Shin, H. S. Song, K. B. Hong, S. Hwangbo, and K. R. Sohn, “Characteristics of side-polished thermally expanded core fiber and its application as a band-edge filter with a high cut-off property,” Opt. Commun. 261(1), 51–55 (2006).
[Crossref]

Horng, J. S.

C. L. Lee, J. M. Hsu, J. S. Horng, W. Y. Sung, and C. M. Li, “Microcavity Fiber Fabry Prot Interferometer With an Embedded Golden Thin Film,” IEEE Photonic. Tech. Lett. 25(9), 833–836 (2013).
[Crossref]

Hou, W.

Hsu, J. M.

C. L. Lee, J. M. Hsu, J. S. Horng, W. Y. Sung, and C. M. Li, “Microcavity Fiber Fabry Prot Interferometer With an Embedded Golden Thin Film,” IEEE Photonic. Tech. Lett. 25(9), 833–836 (2013).
[Crossref]

Hu, D.

Hu, X.

Hua, J.

Hwangbo, S.

K. T. Kim, K. H. Lee, E. S. Shin, H. S. Song, K. B. Hong, S. Hwangbo, and K. R. Sohn, “Characteristics of side-polished thermally expanded core fiber and its application as a band-edge filter with a high cut-off property,” Opt. Commun. 261(1), 51–55 (2006).
[Crossref]

Ipson, B. L.

T. L. Lowder, K. H. Smith, B. L. Ipson, A. R. Hawkins, R. H. Selfridge, and S. M. Schultz, “High-temperature sensing using surface relief fiber Bragg gratings,” IEEE Photonic. Tech. Lett. 17(9), 1926–1928 (2005).
[Crossref]

Kawakami, S.

K. Shiraishi, Y. Aizawa, and S. Kawakami, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
[Crossref]

Ke, T.

T. Zhu, T. Ke, Y. Rao, and K. S. Chiang, “Fabry–Perot optical fiber tip sensor for high temperature measurement,” Opt. Commun. 283(19), 3683–3685 (2010).
[Crossref]

Kido, L.

Kim, K. T.

K. T. Kim, K. H. Lee, E. S. Shin, H. S. Song, K. B. Hong, S. Hwangbo, and K. R. Sohn, “Characteristics of side-polished thermally expanded core fiber and its application as a band-edge filter with a high cut-off property,” Opt. Commun. 261(1), 51–55 (2006).
[Crossref]

Lally, E.

Lang, C.

Lee, C. L.

C. L. Lee, J. M. Hsu, J. S. Horng, W. Y. Sung, and C. M. Li, “Microcavity Fiber Fabry Prot Interferometer With an Embedded Golden Thin Film,” IEEE Photonic. Tech. Lett. 25(9), 833–836 (2013).
[Crossref]

Lee, K. H.

K. T. Kim, K. H. Lee, E. S. Shin, H. S. Song, K. B. Hong, S. Hwangbo, and K. R. Sohn, “Characteristics of side-polished thermally expanded core fiber and its application as a band-edge filter with a high cut-off property,” Opt. Commun. 261(1), 51–55 (2006).
[Crossref]

Li, C. M.

C. L. Lee, J. M. Hsu, J. S. Horng, W. Y. Sung, and C. M. Li, “Microcavity Fiber Fabry Prot Interferometer With an Embedded Golden Thin Film,” IEEE Photonic. Tech. Lett. 25(9), 833–836 (2013).
[Crossref]

Li, H.

Li, J.

Li, M.-J.

A. Yan, S. Li, Z. Peng, R. Zou, P. Ohodnicki, P. Lu, K. Byerly, M.-J. Li, and K. P. Chen, “Multi-point fiber optic sensors for real-time monitoring of the temperature distribution on transformer cores,” Proc. SPIE 10639, 1063912 (2018).

Li, S.

A. Yan, S. Li, Z. Peng, R. Zou, P. Ohodnicki, P. Lu, K. Byerly, M.-J. Li, and K. P. Chen, “Multi-point fiber optic sensors for real-time monitoring of the temperature distribution on transformer cores,” Proc. SPIE 10639, 1063912 (2018).

Li, X.

LiKamWa, P.

Liu, B.

H. Chen, M. Buric, P. R. Ohodnicki, J. Nakano, B. Liu, and B. T. Chorpening, “Review and perspective: Sapphire optical fiber cladding development for harsh environment sensing,” Appl. Phys. Rev. 5(1), 011102 (2018).
[Crossref]

C. Hill, D. Homa, Z. Yu, Y. Cheng, B. Liu, A. Wang, and G. Pickrell, “Single Mode Air-Clad Single Crystal Sapphire Optical Fiber,” Appl. Sci. (Basel) 7(5), 473 (2017).
[Crossref]

Liu, G.

Lorenz, A.

Lowder, T. L.

T. L. Lowder, K. H. Smith, B. L. Ipson, A. R. Hawkins, R. H. Selfridge, and S. M. Schultz, “High-temperature sensing using surface relief fiber Bragg gratings,” IEEE Photonic. Tech. Lett. 17(9), 1926–1928 (2005).
[Crossref]

Lu, P.

A. Yan, S. Li, Z. Peng, R. Zou, P. Ohodnicki, P. Lu, K. Byerly, M.-J. Li, and K. P. Chen, “Multi-point fiber optic sensors for real-time monitoring of the temperature distribution on transformer cores,” Proc. SPIE 10639, 1063912 (2018).

Margulis, W.

Monro, T. M.

Nakano, J.

H. Chen, M. Buric, P. R. Ohodnicki, J. Nakano, B. Liu, and B. T. Chorpening, “Review and perspective: Sapphire optical fiber cladding development for harsh environment sensing,” Appl. Phys. Rev. 5(1), 011102 (2018).
[Crossref]

Nguyen, L. V.

Ohodnicki, P.

A. Yan, S. Li, Z. Peng, R. Zou, P. Ohodnicki, P. Lu, K. Byerly, M.-J. Li, and K. P. Chen, “Multi-point fiber optic sensors for real-time monitoring of the temperature distribution on transformer cores,” Proc. SPIE 10639, 1063912 (2018).

Ohodnicki, P. R.

H. Chen, M. Buric, P. R. Ohodnicki, J. Nakano, B. Liu, and B. T. Chorpening, “Review and perspective: Sapphire optical fiber cladding development for harsh environment sensing,” Appl. Phys. Rev. 5(1), 011102 (2018).
[Crossref]

Peng, J.

Peng, Z.

A. Yan, S. Li, Z. Peng, R. Zou, P. Ohodnicki, P. Lu, K. Byerly, M.-J. Li, and K. P. Chen, “Multi-point fiber optic sensors for real-time monitoring of the temperature distribution on transformer cores,” Proc. SPIE 10639, 1063912 (2018).

Pickrell, G.

C. Hill, D. Homa, Z. Yu, Y. Cheng, B. Liu, A. Wang, and G. Pickrell, “Single Mode Air-Clad Single Crystal Sapphire Optical Fiber,” Appl. Sci. (Basel) 7(5), 473 (2017).
[Crossref]

Qiao, X.

X. Qiao, Z. Shao, W. Bao, and Q. Rong, “Fiber Bragg Grating Sensors for the Oil Industry,” Sensors (Basel) 17(3), 429 (2017).
[Crossref] [PubMed]

Rao, Y.

T. Zhu, T. Ke, Y. Rao, and K. S. Chiang, “Fabry–Perot optical fiber tip sensor for high temperature measurement,” Opt. Commun. 283(19), 3683–3685 (2010).
[Crossref]

Ratuszek, M.

M. Ratuszek, M. J. Ratuszek, and J. Hejna, “The study of thermal connecting of telecommunication optical fibers (SiO2: GeO2) and EDF (SiO2: Al2O3, Er) fibers,” B. Pol. Acad. Sci. Tech. (Paris) 61(1), 279–286 (2013).
[Crossref]

Ratuszek, M. J.

M. Ratuszek, M. J. Ratuszek, and J. Hejna, “The study of thermal connecting of telecommunication optical fibers (SiO2: GeO2) and EDF (SiO2: Al2O3, Er) fibers,” B. Pol. Acad. Sci. Tech. (Paris) 61(1), 279–286 (2013).
[Crossref]

Rong, Q.

X. Qiao, Z. Shao, W. Bao, and Q. Rong, “Fiber Bragg Grating Sensors for the Oil Industry,” Sensors (Basel) 17(3), 429 (2017).
[Crossref] [PubMed]

Rothhardt, M.

Rugeland, P.

Schultz, S. M.

T. L. Lowder, K. H. Smith, B. L. Ipson, A. R. Hawkins, R. H. Selfridge, and S. M. Schultz, “High-temperature sensing using surface relief fiber Bragg gratings,” IEEE Photonic. Tech. Lett. 17(9), 1926–1928 (2005).
[Crossref]

Schülzgen, A.

Schuster, K.

Schwuchow, A.

Selfridge, R. H.

T. L. Lowder, K. H. Smith, B. L. Ipson, A. R. Hawkins, R. H. Selfridge, and S. M. Schultz, “High-temperature sensing using surface relief fiber Bragg gratings,” IEEE Photonic. Tech. Lett. 17(9), 1926–1928 (2005).
[Crossref]

Seraji, F. E.

S. Farsinezhad, F. E. Seraji, S. Farsinezhad, and F. E. Seraji, “Analysis of Fresnel Loss at Splice Joint Between Single-Mode Fiber and Photonic Crystal Fiber,” International Journal of Optics and Applications 2(1), 17–21 (2012).
[Crossref]

S. Farsinezhad, F. E. Seraji, S. Farsinezhad, and F. E. Seraji, “Analysis of Fresnel Loss at Splice Joint Between Single-Mode Fiber and Photonic Crystal Fiber,” International Journal of Optics and Applications 2(1), 17–21 (2012).
[Crossref]

Shao, Z.

X. Qiao, Z. Shao, W. Bao, and Q. Rong, “Fiber Bragg Grating Sensors for the Oil Industry,” Sensors (Basel) 17(3), 429 (2017).
[Crossref] [PubMed]

Shen, X.

Sheng, Q.

Shin, E. S.

K. T. Kim, K. H. Lee, E. S. Shin, H. S. Song, K. B. Hong, S. Hwangbo, and K. R. Sohn, “Characteristics of side-polished thermally expanded core fiber and its application as a band-edge filter with a high cut-off property,” Opt. Commun. 261(1), 51–55 (2006).
[Crossref]

Shiraishi, K.

K. Shiraishi, Y. Aizawa, and S. Kawakami, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
[Crossref]

Smith, K. H.

T. L. Lowder, K. H. Smith, B. L. Ipson, A. R. Hawkins, R. H. Selfridge, and S. M. Schultz, “High-temperature sensing using surface relief fiber Bragg gratings,” IEEE Photonic. Tech. Lett. 17(9), 1926–1928 (2005).
[Crossref]

Sohn, K. R.

K. T. Kim, K. H. Lee, E. S. Shin, H. S. Song, K. B. Hong, S. Hwangbo, and K. R. Sohn, “Characteristics of side-polished thermally expanded core fiber and its application as a band-edge filter with a high cut-off property,” Opt. Commun. 261(1), 51–55 (2006).
[Crossref]

Song, H. S.

K. T. Kim, K. H. Lee, E. S. Shin, H. S. Song, K. B. Hong, S. Hwangbo, and K. R. Sohn, “Characteristics of side-polished thermally expanded core fiber and its application as a band-edge filter with a high cut-off property,” Opt. Commun. 261(1), 51–55 (2006).
[Crossref]

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C. L. Lee, J. M. Hsu, J. S. Horng, W. Y. Sung, and C. M. Li, “Microcavity Fiber Fabry Prot Interferometer With an Embedded Golden Thin Film,” IEEE Photonic. Tech. Lett. 25(9), 833–836 (2013).
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[Crossref]

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A. Yan, S. Li, Z. Peng, R. Zou, P. Ohodnicki, P. Lu, K. Byerly, M.-J. Li, and K. P. Chen, “Multi-point fiber optic sensors for real-time monitoring of the temperature distribution on transformer cores,” Proc. SPIE 10639, 1063912 (2018).

Appl. Opt. (2)

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[Crossref]

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C. Hill, D. Homa, Z. Yu, Y. Cheng, B. Liu, A. Wang, and G. Pickrell, “Single Mode Air-Clad Single Crystal Sapphire Optical Fiber,” Appl. Sci. (Basel) 7(5), 473 (2017).
[Crossref]

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C. L. Lee, J. M. Hsu, J. S. Horng, W. Y. Sung, and C. M. Li, “Microcavity Fiber Fabry Prot Interferometer With an Embedded Golden Thin Film,” IEEE Photonic. Tech. Lett. 25(9), 833–836 (2013).
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[Crossref]

T. Zhu, T. Ke, Y. Rao, and K. S. Chiang, “Fabry–Perot optical fiber tip sensor for high temperature measurement,” Opt. Commun. 283(19), 3683–3685 (2010).
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[Crossref] [PubMed]

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Figures (9)

Fig. 1
Fig. 1 The schematic configuration of the in-fiber sensor array.
Fig. 2
Fig. 2 The images captured by microscope and refractive index analyzer. (a,b) The cross-image of the DCF and the SMF, (c,d) the refractive index profile of the DCF and the SMF.
Fig. 3
Fig. 3 (a,b) The relationship of the fusion splicing time versus the joint reflectivity and insertion loss, (c) the relationship between the normalized output power of sensor array and the number of sensors in different splicing time, (d) the maximum number of sensors in series versus splicing time.
Fig. 4
Fig. 4 The configuration of the experimental setup. The OPD is the Optical path difference, and the OPC is the optical path compensation. The optical path correlation condition is satisfied when the OPD is equal to the OPC.
Fig. 5
Fig. 5 The calibration results of the DCF and SMF sensors.
Fig. 6
Fig. 6 (a) The experimental setup of quasi-distributed high temperature sensing, (b) the SiC heating rod after heating and its temperature distribution.
Fig. 7
Fig. 7 (a) The scanning output of the sensor array scanned by the WLIDI before and after heating, (b) the single sensor S3 trend with temperature.
Fig. 8
Fig. 8 The results of the quasi-distributed temperature sensing by the sensor array and thermocouple. The solid lines are measured by the thermocouple at the different condition, and the scatter plots are measured by the fiber sensor array. The position relationship between the SiC and the fiber sensor array is shown in Fig. 6.
Fig. 9
Fig. 9 The result of the temperature stability experiment of the sensor array.

Equations (9)

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

R= ( n 2 n 1 ) 2 ( n 2 + n 1 ) 2
I R (k)={ I 0 R 0 k=0 I 0 R k k=0 k1 (1 R k ) η k k=1,2,3,4
I T (k)={ I 0 k=0 I 0 k=0 k1 (1 R k ) η k k=1,2,3,4
X k = n k l k k=1,2,3
O P k = n k l k .
dO P k =[ n k l k T + l k n k T ]dT= n k l k [ 1 l k l k T + 1 n k n k T ]dT= n k l k [ α T + C T ]dT
α T = 1 l k l k T
C T = 1 n k n k T .
T ik = ΔO P k n k l k ( α T + C T ) + T 0k

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