# A Novel Dual-Wavelength Method for Evaluating Temperature Effect in Fiber-Optic SPR Sensors
## 3 The Principle of Dual-Wavelength Method and Theoretical Calculation of Fiber-Optic SPR Sensor
The dependence of output signal on temperature and RI is analyzed based on the model analysis in Section 2. Hence, the theoretical analysis and numerical calculation are carried  
Figure 4: The fiber-optic SPR resonance spectra at different RI (core: 200 μm, 0.36 NA, sensor length: 10 mm, 50 nm gold film at 300 K). Figure 3: Dual-wavelength signal by shifting of the resonance curve. out to quantitative analysis of the variations of temperature and RI simultaneously. The dependence of reflectance signals on RI and temperature are simulated at the wavelengths \(\lambda_{1}=495\) nm and \(\lambda_{2}=575\) nm, as shown in Figure 5. It is obvious that the reflectance signals \(S(\lambda_{1})\) and \(S(\lambda_{2})\) are linear with both RI and temperature. For example, the reflectance signal \(S(\lambda_{1})\) increases linearly with RI at a temperature of 300 K, while the linear decrease of the signal \(S(\lambda_{2})\) is with RI at the temperature of 300 K. Besides, the absolute slope of \(S(\lambda_{2})\) is larger than that of \(S(\lambda_{1})\), which reveals that the change of reflectance signal at \(\lambda_{2}\) is faster than \(\lambda_{1}\) to RI variation. In other words, the constructed fiber-optic SPR sensor is more sensitive to RI at the \(\lambda_{2}\). Meanwhile, another feature is that the linear variation of the reflectance signals \(S(\lambda)\) with temperature is less significant than RI. It demonstrates that the reflectance signal is less sensitive to temperature. The variation of the output signal mainly depends on the change of RI. In summary, the relationship between the output signal change versus RI and temperature variation is approximately linear with different slopes depending upon the light wavelength used in the measurement. For the detailed scenario, Figure 6 shows the output signal changes versus the RI at different temperatures. It is obvious that the dependence of the \(S(\lambda_{1})\) (or \(S(\lambda_{2})\)) on the RI is nearly the same at \(T=290\) K (\(T\) is the value of temperature) and \(T\)= 320 K. The difference in the slope of the signal versus RI is negligible compared to either of the two slopes. It clearly suggests that, for the two defended sensor outputs, the dependence of the output signal on RI is almost independent with temperature. Another important observation is that the spacings between the curve pairs of \(T=290\) K and \(T=320\) K are different. The spacing of \(\lambda_{1}\) is larger than the spacing of \(\lambda_{2}\), which reveals that the signal is less sensitive to temperature variation at the signal of \(S(\lambda_{2})\). As the curves show in Figure 6, the linear sensitivities between output signals versus the RI are \(-21.6454/\) RIU and \(9.2836/\)RIU for \(S(\lambda_{2})\) and \(S(\lambda_{1})\), respectively. Figure 7 shows the change of the output signal versus the temperature with RI equalling \(n=1.333\) RIU (\(n\) is the value of RI) and \(n=1.338\) RIU. Similarly, two important key features can be obviously found from these curves. Firstly, the spacings between the curve pairs of \(n=1.333\) RIU and \(n=1.338\) RIU for \(S(\lambda_{1})\) and \(S(\lambda_{2})\) are larger than its counterpart in Figure 6 because of the significant RI dependence and low temperature sensitivity. Secondly, it is clearly observed that, for both of the output signals, the slopes of the output signal versus temperature for \(n=1.333\) RIU and \(n=1.338\) RIU  
Figure 5: The SPR reflected intensity signal shifts versus temperature and RI for the dual wavelengths (495 nm and 575 nm). are nearly equal. This means that, for the two signals, the temperature dependence of the output is approximately independent with RI. The small detectable sensitivities between output signals versus the temperature are \(-2.8672/10^{4}\) K and \(-5.3535/10^{4}\) K, respectively. Based on the above analysis, the dependence of the two output signals, defined by the dual-wavelength technique, on RI and temperature can be viewed as linear and independent with each other.