## 2 Experiment and Theory  
The Rayleigh scattering-based distributed optical fiber temperature/strain sensing system is usually called coherent OTDR (COTDR) [18; 21]. Figure 1 shows a typical COTDR structure. A narrow linewidth laser (NLL) outputs continuous laser light, and the laser light is split by an optical coupler 1 (OC1) with a ratio of 90/10. The output light with lower power comes into the acousto-optic modulator (AOM) and then the AOM generates an optical probe pulse, and through the Erbium-doped fiber amplifier (EDFA), the peak power of the probe pulse is improved to about 30 dBm. The probe pulse is then launched into the 1 st port of the optical fiber circulator, and then it outputs from the 2 nd port that links the fiber under testing, and the Rayleigh scattering light of the probe pulse propagating in the fiber under test comes back, and through the 3 rd port, it is input into the optical coupler 2 (OC2). The OC2 is an X-type 3 dB coupler, in which the laser light with higher power from the OC1 mixes with the Rayleigh scattering light and then coherent intermediate frequency signals are generated by heterodyne. The coherent intermediate frequency signal is then converted into a radio frequency (RF) signal by the balanced photodetector (BPD).  
In the experiment, a polarization scrambler (PS) is adopted to randomize the optical polarization state of the input light to reduce the polarization noise in the OTDR trace. As the AOM shifts the frequency of its input light by 200 MHz, the center frequency of the RF signal is at 200 MHz. Then, a band pass filter (BPF) with a center frequency of 200 MHz and bandwidth of 10 MHz purifies the RF signal, and then the power of the RF signal is then converted into a voltage signal by the radio frequency power detector (RFPD). A data acquisition card (DAQ) collects the voltage signal and sends the data to the personal computer (PC). Finally, the OTDR trace is presented on the screen of the PC, as shown in Figure 2. By linearly changing the driving current or laser diode temperature, the laser frequency output from the NLL will be linearly changed. Therefore, the three-dimensional (3D) Rayleigh scattering spectra about the fiber length, Rayleigh scattering power, and frequencies are obtained.  
To demodulate the temperature/strain, data about the 3D Rayleigh spectrum from two measurements are needed. Then, for each position along the sensing fiber, two 2D arrays containing the frequency and power data of Rayleigh scattering light are extracted,  
Figure 1: The schematic diagram of COTDR.  
Figure 2: The OTDR trace at a certain laser frequency.  
respectively, from the data about the two 3D Rayleigh spectra. By the cross-correlation algorithm in Formula (1), the Rayleigh frequency shift can be determined [18].  
\[R_{u,v}(\Delta f,L)=\frac{\sum\limits_{i=1}^{m}{(p_{u}(f_{i},L)-\overline{p}_{u}( L))(p_{v}(f_{i}+\Delta f,L)-\overline{p}_{v}(L))}}{\sqrt{\sum\limits_{i=1}^{m}{(p_{u}( f_{i},L)-\overline{p}_{u}(L))^{2}}\cdot\sum\limits_{i=1}^{m}{(p_{v}(f_{i}+ \Delta f,L)-\overline{p}_{v}(L))^{2}}}} \tag{1}\]  
In Formula (1), \(u\) and \(v\) represent two independent measurements; \(R_{u,v}(\Delta f,L)\) is the cross-correlation coefficient at position \(L\) with a Rayleigh frequency shift of \(\Delta f\); \(p_{u}(f_{i},L)\) and \(p_{v}(f_{i}+\Delta f,L)\) are the 2D arrays by the \(u\)th measurement and the \(v\)th measurement, respectively; and \(\overline{p}_{u}(L)\) and \(\overline{p}_{v}(L)\) are the average values of the 2D arrays \(p_{u}(f_{i},L)\) and \(p_{v}(f_{i}+\Delta f,L)\) with array length \(m\) that is also the number of frequency sweeping points. At last, the temperature or strain can be obtained by the conversion relationships in Formulas (2) and (3) between the Rayleigh frequency shift and temperature change \(\Delta T\) or strain change \(\Delta\varepsilon\).  
\[\Delta T =(\Delta f/1.34\ \mathrm{GHz})\ ^{\circ}\mathrm{C} \tag{2}\] \[\Delta\varepsilon =(\Delta f/151.0\ \mathrm{MHz})\mu\varepsilon \tag{3}\]