# 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  
Based on the model of fiber-optic SPR sensor illustrated in Section 2.1, the functional relationship between the output light intensity and the wavelength of incident light is fitted as shown in Figure 3. The solid curve represents a typical resonance spectrum of the fiber-optic SPR sensor. The resonance curve exhibits a valley and the signal of light reflectance gets its minimum value in a characteristic wavelength. However, the surface-plasmon wave of the fiber-optic SPR sensor will be changed due to the variation of RI or temperature of an analyte, which in turn leads to the variation of resonance incident wave vector. The original resonance curve could shift to the dotted line as shown in Figure 3 with a resonance wavelength drift. The conventional method to obtain the variation is to detect the resonance dip in the transmission spectrum. However, the temperature effect cannot be ruled out, which results in the bias of detection result. Inspired by the dual-incident angle technique for the prism-based SPR sensor [37; 38; 39; 40], a dual-wavelength method is presented for the fiber-optic SPR sensor in this paper. Based on the theoretical analysis and the resonance characteristics of the fiber-optic SPR sensor, we present a linear model between the change of reflectance transmission signal and the change of RI/temperature of the analyte with two specific incident wavelengths. The two specific incident wavelengths (\(\lambda_{1}\) and \(\lambda_{2}\)) are chosen on the opposite sides of the resonance curve. The light reflectance signal at wavelength \(\lambda_{1}\) is denoted as \(S(\lambda_{1})\), and the light reflectance signal at wavelength \(\lambda_{2}\) is denoted as \(S(\lambda_{2})\). Compared with the traditional fiber-optic SPR detection method which just exploits the resonance wavelength obtained at the dip of transmission spectrum, the presented dual-wavelength measurement takes full advantage of the data obtained at both \(\lambda_{1}\) and \(\lambda_{2}\) and offers two output signals to indicate the variations in temperature and RI. Based on the assumption of the original dual-wavelength method, the selection principle of the wavelengths is that the dependence of the defined output signals on RI and temperature are linear and independent of each other in the detection range of both RI and temperature. In order to find two fixed wavelengths that meet these conditions for practical applications, the RI and temperature effects are analyzed in detail. Figure 4 shows the resonance spectra of the analyte with RI range from 1.333 to 1.340, and the temperature is set at 310 K. As shown in the Figure 4, the resonancespectrum shifts to the right side as RI increasing. In wavelength range 1, the lines of the resonance spectra are almost parallel to each other. The same phenomenon appears in the wavelength range 2. Therefore, a linear relationship between the RI and reflectance signal can be established in the two wavelength ranges. The right fixed wavelength can be chosen in range 1, and the left fixed wavelength can be chosen in range 2. In order to investigate the feasibility of the dual-wavelength method for the simultaneous measurement of the changes of RI and temperature, a theoretical calculation and numerical analysis are conducted to quantify the output signal, i.e., light intensity, of a fiber-optic SPR sensor with different temperature and RI. In our simulation, the core of a multimode fiber is coated with gold film. The thickness of the gold sensor film is assumed to be 50 nm. The numerical aperture (NA) of the optical fiber is set to 0.36. Temperature variation is in the range of 290-320 K and the interval of variation of the RI is set to 1.333-1.338 RIU. The two characteristic wavelengths \(\lambda_{1}=\) 495 nm and \(\lambda_{2}=\) 575 nm are sought for the evaluation of temperature effect and RI on the fiber-optic SPR sensor.