光学仪器  2023, Vol. 45 Issue (1): 73-79 PDF

Simulation of optical fiber coating for OFDR temperature sensing
WANG Lujun, XIN Wei, LIU Yu, ZHANG Xuedian, LIU Xuejing
School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: In order to design coated fiber for optical frequency domain reflectometer (OFDR) temperature sensing, improve the temperature sensitivity of OFDR and increase its applicable scenarios, this paper theoretically analyzes and simulates the effects of one and two layers of coating on the temperature sensitivity of Rayleigh frequency shift in single-mode fiber. Firstly, the effects are analyzed by using the Lame solution theory in term of the geometrical, thermal and mechanical properties of the coating. Secondly, a simplified solution with only one layer of coating is proposed based on the relationship of axial strain in optical fiber caused by temperature and the force balance between optical fiber and one layer of coating. Finally, the temperature sensitivity of Rayleigh frequency shift of the one-layer and two-layer coated optical fiber is simulated on the basis of the established theoretical model. The results show that the sensitivity increases with the increase of Young's modulus, radius and thermal expansion coefficient of the outer coating. However, it is almost independent of Poisson's ratio of the coating. The results of this paper could broaden the application of OFDR in high temperature sensitive and cryogenic temperature scenarios.
Key words: fiber    coating    optical frequency domain reflectometer    temperature sensitivity

1 热应变理论

 $\dfrac{{\text{δ}}_{\lambda} }{{\lambda }_{c}}=-\dfrac{{\text{δ}} \upsilon }{{\upsilon }_{c}}={K}_{T}{\text{δ}}_ T+{K}_{\varepsilon }{\text{δ}} {\varepsilon }_{a}$ (1)

${C}_{T}={\lambda }_{c}{K}_{T}$ ${{C}_{\varepsilon }=\lambda }_{c}{K}_{\varepsilon }$ ，由式（1），可以定义瑞利频移温度灵敏度为

 $\begin{array}{c}\dfrac{{\text{δ}}_\lambda }{{\text{δ}}_ T}={C}_{T}+{C}_{\varepsilon }\dfrac{{\text{δ}} {\varepsilon }_{a}}{{\text{δ}}_ T}\end{array}$ (2)

 $\begin{split} \left[\begin{array}{c}\text{δ} {\sigma }_{r}^{i}\\ \text{δ} {\sigma }_{\theta }^{i}\\ \text{δ} {\sigma }_{a}^{i}\end{array}\right]=&\left[\begin{array}{c}{\lambda }_{i}+2{\mu }_{i}\\ {\lambda }_{i}\\ {\lambda }_{i}\end{array}\begin{array}{c}{\lambda }_{i}\\ {\lambda }_{i}+2{\mu }_{i}\\ {\lambda }_{i}\end{array}\begin{array}{c}{\lambda }_{i}\\ {\lambda }_{i}\\ {\lambda }_{i}+2{\mu }_{i}\end{array}\right] \times\\ & \left[\begin{array}{c}\text{δ} {\varepsilon }_{r}^{i}-{\alpha }_{i} \text{δ}_T\\ \text{δ} {\varepsilon }_{\theta }^{i}-{\alpha }_{i} \text{δ}_T\\ \text{δ} {\varepsilon }_{a}^{i}-{\alpha }_{i} \text{δ}_T\end{array}\right] \end{split}$ (3)

 $\begin{array}{c}{\lambda }_{i}=\dfrac{{\eta }_{i}{E}_{i}}{\left(1+{\eta }_{i}\right)\left(1-2{\eta }_{i}\right)}\text{，}{\mu }_{i}=\dfrac{{E}_{i}}{2\left(1+{\eta }_{i}\right)}\end{array}$ (4)
 $\begin{array}{c}\text{δ} {\varepsilon }_{r}^{i}={U}_{i}+\dfrac{{V}_{i}}{{r}^{2}}\text{，}\text{δ}{\varepsilon }_{\theta }^{i}={U}_{i}-\dfrac{{V}_{i}}{{r}^{2}}\text{，}\text{δ} {\varepsilon }_{a}^{i}={W}_{i}\end{array}$ (5)

 $\begin{array}{c}\text{δ} {\sigma }_{r}^{i}\left({r}_{i}\right)=\text{δ} {\sigma }_{r}^{i+1}\left({r}_{i}\right)\quad \left(i=0, 1,\cdots ,m-1\right)\end{array}$ (6)
 $\begin{array}{c}\text{δ} {u}_{r}^{i}\left({r}_{i}\right)=\text{δ} {u}_{r}^{i+1}\left({r}_{i}\right)\quad \left(i=0, 1,\cdots ,m-1\right)\end{array}$ (7)
 $\begin{array}{c}\text{δ} {\sigma }_{r}^{m}\left({r}_{m}\right)=0\end{array}$ (8)
 $\begin{array}{c}\displaystyle\sum _{i=0}^{m}\text{δ} {\sigma }_{a}^{i}\cdot {A}_{i}=0\end{array}$ (9)
 $\begin{array}{c}\text{δ} {\varepsilon }_{a}^{0}=\text{δ} {\varepsilon }_{a}^{1}=\cdot \cdot \cdot =\text{δ} {\varepsilon }_{a}^{m}=\text{δ} {\varepsilon }_{a}\end{array}$ (10)

 图 1 涂层光纤示意图 Figure 1 Schematic diagram of coated fiber

 $\begin{array}{c}{F}_{0}={E}_{0}{A}_{0}\text{δ} {\varepsilon }_{0}\end{array}$ (11)
 $\begin{array}{c}{F}_{1}={E}_{1}{A}_{1}\text{δ} {\varepsilon }_{1}\end{array}$ (12)
 $\begin{array}{c}{F}_{0}={F}_{1}\end{array}$ (13)

${F}_{0}$ ${F}_{1}$ 带来的应变关系由图1（a）可得

 $\begin{array}{c}\text{δ} {\varepsilon }_{a}={\alpha }_{0}\text{δ}_ T+\text{δ} {\varepsilon }_{0}={\alpha }_{1}\text{δ}_ T-\text{δ} {\varepsilon }_{1}\end{array}$ (14)

 $\begin{array}{c}\dfrac{\text{δ} {\varepsilon }_{a}}{\text{δ}_ T}=\left(\dfrac{{E}_{1}{A}_{1}}{{E}_{1}{A}_{1}+{E}_{0}{A}_{0}}\left({\alpha }_{1}-{\alpha }_{0}\right)+{\alpha }_{0}\right)\end{array}$ (15)

 $\begin{array}{c}\dfrac{\text{δ} \lambda }{\text{δ}_ T}={C}_{T}+{C}_{\varepsilon }\left(\dfrac{{E}_{1}{A}_{1}}{{E}_{1}{A}_{1}+{E}_{0}{A}_{0}}\left({\alpha }_{1}-{\alpha }_{0}\right)+{\alpha }_{0}\right)\end{array}$ (16)

2 仿真结果及讨论

 图 2 瑞利频移温度灵敏度（ $\text{δ}_ \lambda /\text{δ}_ T$ ）与 1 层外涂层参数（ ${\eta }_{1}$ ， ${\alpha }_{1}$ ， ${E}_{1}$ ， ${r}_{1}$ ）的关系 Figure 2 Relationship between temperature sensitivity of Rayleigh frequency shift（ $\text{δ}_ \lambda /\text{δ}_ T$ ）and the external first coating parameters （ ${\eta }_{1}，{\alpha }_{1}，{E}_{1}，{r}_{1}$ ）

 图 3 光纤瑞利频移随温度变化趋势 Figure 3 Rayleigh frequency shift of optical fiber varies with temperature

 图 4 瑞利频移温度灵敏度（ $\text{δ}_ \lambda / \text{δ}_T$ ）与 2 层外涂层参数（ ${\eta }_{2}$ ， ${\alpha }_{2}$ ， ${E}_{2}$ ， ${r}_{2}$ ）的关系 Figure 4 Relationship between temperature sensitivity of Rayleigh frequency shift（ $\text{δ}_ \lambda / \text{δ}_T$ ）and the external second coating parameters （ ${\eta }_{2}，{\alpha }_{2}，{E}_{2}，{r}_{2}$ ）
3 结　论

 [1] BAO X Y, CHEN L. Recent progress in distributed fiber optic sensors[J]. Sensors, 2012, 12(7): 8601–8639. DOI:10.3390/s120708601 [2] LIANG C S, BAI Q, YAN M, et al. A comprehensive study of optical frequency domain reflectometry[J]. IEEE Access, 2021, 9: 41647–41668. DOI:10.1109/ACCESS.2021.3061250 [3] 崔楠楠. 高空间分辨率的光频域反射关键技术研究[D]. 成都: 电子科技大学, 2019: 4. [4] YANG Y J, XIANG Y, BU A M, et al. Liquid-phase deposition of Al2O3 coating on quartz fiber for enhanced strength at elevated temperature [J]. Composite Interfaces, 2022, 29(5): 487–501. DOI:10.1080/09276440.2021.1979751 [5] LI P F, FU C L, ZHONG H J, et al. A nondestructive measurement method of optical fiber young’s modulus based on OFDR[J]. Sensors, 2022, 22(4): 1450. DOI:10.3390/s22041450 [6] ZHANG M, LIU Z H, MA Y W, et al. Gelatin-coated long period fiber grating humidity sensor with temperature compensation[J]. Optical Engineering, 2022, 61(2): 027104. [7] WANG Z, CHEN D L, YANG X C, et al. Temperature sensor of single-mode-no-core-single-mode fiber structure coated with PDMS[J]. Optical Fiber Technology, 2022, 68: 102793. DOI:10.1016/j.yofte.2021.102793 [8] SERAJI F E, TOUTIAN G. Effect of temperature rise and hydrostatic pressure on microbending loss and refractive index change in double-coated optical fiber[J]. Progress in Quantum Electronics, 2006, 30(6): 317–331. DOI:10.1016/j.pquantelec.2007.01.001 [9] WANG X Y, SUN X Y, HU Y W, et al. Highly-sensitive fiber Bragg grating temperature sensors with metallic coatings[J]. Optik, 2022, 262: 169337. DOI:10.1016/j.ijleo.2022.169337 [10] HABISREUTHER T, HAILEMICHAEL E, ECKE W, et al. ORMOCER coated fiber-optic Bragg grating sensors at cryogenic temperatures[J]. IEEE Sensors Journal, 2012, 12(1): 13–16. DOI:10.1109/JSEN.2011.2108280 [11] LIU Y Q, GUO Z Y, ZHANG Y, et al. Simultaneous pressure and temperature measurement with polymer-coated fibre Bragg grating[J]. Electronics Letters, 2000, 36(6): 564–566. DOI:10.1049/el:20000452 [12] GU H D, DONG H J, ZHANG G Y, et al. Effects of polymer coatings on temperature sensitivity of brillouin frequency shift within double-coated fibers[J]. IEEE Sensors Journal, 2013, 13(2): 864–869. DOI:10.1109/JSEN.2012.2230438 [13] LU X, SOTO M A, THÉVENAZ L. Impact of the fiber coating on the temperature response of distributed optical fiber sensors at cryogenic ranges[J]. Journal of Lightwave Technology, 2018, 36(4): 961–967. DOI:10.1109/JLT.2017.2757843 [14] KWON Y S, NAEEM K, JEON M Y, et al. Enhanced sensitivity of distributed-temperature sensor with Al-coated fiber based on OFDR[J]. Optical Fiber Technology, 2019, 48: 229–234. DOI:10.1016/j.yofte.2019.01.021 [15] 赵梦梦. 基于光频域反射技术的分布式光纤传感器研究[D]. 合肥: 安徽大学, 2020: 14. [16] FROGGATT M, MOORE J. High-spatial-resolution distributed strain measurement in optical fiber with Rayleigh scatter[J]. Applied Optics, 1998, 37(10): 1735–1740. DOI:10.1364/AO.37.001735 [17] TUR M, SOVRAN I, BERGMAN A, et al. Structural health monitoring of composite-based UAVs using simultaneous fiber-optic interrogation by static Rayleigh-based distributed sensing and dynamic fiber Bragg grating point sensors[C]//Proceedings of SPIE 9634, 24th International Conference on Optical Fibre Sensors. Curitiba, Brazil: SPIE, 2015: 96340P. [18] 张洪艺. 基于OFDR的分布式光纤应变传感系统设计与数据解调关键技术研究[D]. 北京: 北京邮电大学, 2019: 9. [19] KREGER S T, GIFFORD D K, FROGGATT M E, et al. High resolution distributed strain or temperature measurements in single-and multi-mode fiber using swept-wavelength interferometry[C]//Proceedings of Optical Fiber Sensors 2006. Cancun Mexico: Optica Publishing Group, 2006: ThE42. [20] BOUTEN P C P, BROER D J, JOCHEM C M G, et al. Optical fiber coatings: high modulus coatings for fibers with a low microbending sensitivity[J]. Polymer Engineering and Science, 1989, 29(17): 1172–1176. DOI:10.1002/pen.760291706 [21] MA G M, ZHOU H Y, LI Y B, et al. High-resolution temperature distribution measurement of GIL spacer based on OFDR and ultraweak FBGs[J]. IEEE Transactions on Instrumentation and Measurement, 2020, 69(6): 3866–3873. DOI:10.1109/TIM.2019.2937408