## 3 Results and Discussion  
We consider an optical irradiance of 100 W/cm\({}^{2}\) and an initial temperature of 293.15 K. In this case Figure 3a shows the maximum temperature in terms of the wavelength of the illumination source. As expected from our design, the maximum response appears at \(\lambda_{\text{res}}=1.351\)\(\upmu\)m. For this case, the temperature reaches a maximum of 294.9 K. Figure 3b represents the temperature maps at 1.351 \(\upmu\)m and 1.4 \(\upmu\)m wavelengths, resonant and non-resonant respectively. To isolate the environmental changes in temperature from those caused by a change in the refractive index, we can use a Wheatstone bridge where our sensor is combined with a dummy element that responds to the environmental temperature but it is not exposed to the analyte. This approach has been previously used with bolometric antenna-coupled detectors [29]. To read out the device's response, the metal layer is connected to an external electric circuit (e.g., a voltage divider). The output from the bolometric element--the gold layer--also depends on the value of the resistance of the thin metal layer, \(R_{0}\). This value depends on the geometry as \(R_{0}=\rho_{0}L/(t_{m}\times w)\), where \(\rho_{0}\) is the resistivity of the metal at room temperature, \(L\) is the length, and the denominator contains the transversal section as the product of the thickness, \(t_{m}\), and the transversal width, \(w\). For generality, we may approximate this value as \(R_{0}=10\)\(\Omega\). The proposed device uses a 35 nm-thick layer of gold with resitivity \(\rho_{\text{Au}}=2.2\times 10^{-8}\)\(\Omega\).m [30]. As an example, an element with width \(w=10\)\(\upmu\)m, and length \(L=160\)\(\upmu\)m, generates a resistance of \(R_{0}=10.05\)\(\Omega\). Therefore, we can see that the proposed element is small enough to allow a very compact design of the sensor, which is desiderable to measure tiny volumes of analyte, like human tear samples. In these conditions, the spectral variation of resistance also shows the peak at the resonant frequency (see Figure 4a). Therefore, by considering the physical mechanisms involved in the sensor, we obtain an electric signal related with the variation in the index of refraction of the analyte. As an example, we have calculated the spectral variation in resistance, \(\Delta R(\lambda)\), for two values of the index of refraction of the analyte, \(n_{a,1}=1.33\), and \(n_{a,2}=1.34\). We show how the resonant peaks move from \(\lambda_{1}=1.351\)  
Figure 3: (a) Maximum temperature reached at the structure in terms of the wavelength. (b) Temperature distribution within the structure for the resonant (\(\lambda=1.351\)\(\upmu\)m) and non-resonant (\(\lambda=1.400\)\(\upmu\)m) wavelengths. to \(\lambda_{2}=1.361\)\(\upmu\)m respectively. Figure 4b shows the change in resistance for a fixed value of the optical excitation at \(\lambda=1.351\)\(\upmu\)m; while the index of refraction of the analyte varies from 1.33 to 1.34. The response is symmetric when moving towards a lower index of refraction. For a practical operation of the device, as a bolometric system, its requires a connection to an external biasing source. Depending on the biasing strategy (voltage bias or current bias), the voltage change, \(\Delta V_{\text{out}}\), due to a change in resistance, \(\Delta R\), is given as:  
\[\Delta V_{\text{out},V_{\text{bias}}}=V_{\text{bias}}\frac{R_{\text{ext}}}{ \left(R_{0}+R_{\text{ext}}\right)^{2}}\Delta R, \tag{4}\]  
for a voltage bias, \(V_{\text{bias}}\), that sources a voltage divider with an external resistance \(R_{\text{ext}}\), and  
\[\Delta V_{\text{out},b_{\text{bias}}}=I_{\text{bias}}\Delta R, \tag{5}\]  
for a current bias \(I_{\text{bias}}\). In both cases, the signal \(\Delta V_{\text{out}}\) is proportional to \(\Delta R\). The plots in Figure 4 represent the dependence of \(\Delta V_{\text{out}}\) with respect to the wavelength (see Figure 4a), and with respect to the index of refraction (Figure 4b).