# Refractive Index Sensing through Surface Plasmon Resonance in Light-Diffusing Fibers
## 3 Results and Discussion  
As a first test, we have recorded the changes in the transmission spectrum of the fiber, with the uncoated fiber region immersed in water-glycerol mixtures with RI ranging from 1.332 to 1.443. From the measurements, it resulted that the external medium's RI only influenced the total power transmitted through the LDF, while its spectral distribution kept constant. Therefore, we report in Figure 3a only the changes in the transmitted power, as a function of the SRI. We see that, for RI ranging from 1.332 to -1.38 the transmitted power remained constant, while for RI larger than 1.38 the transmitted power dropped sensibly (\(\Delta\)P--4 dB when varying the RI from 1.332 to 1.443). This behavior is consistent with a ray-optic description of light transmission through a multimode fiber [16]: as long as the external medium's RI is lower than the cladding index (about 1.38 in our case), the power guided in the fiber is only affected by evanescent wave absorption (EWA), which is negligible in our case due to the short sensing region. Vice versa, when the RI of the external medium is situated between the cladding RI and the core RI, the variation in the transmitted power is mostly due to the reduction of the number of propagation modes satisfying the total internal reflection (TIR) condition at the boundary between the fiber core and the external medium (see Figure 1b). These arguments are confirmed by a numerical analysis carried out using the model described in Reference [17]: in brief, the normalized power transmitted over a sensing length \(L\) is calculated by summing the power contributions over all the acceptance angles \((\theta_{c},\pi/2)\), where \(\theta_{c}=sin^{-1}(n_{cl}/n_{c0}))\)  
Figure 2: Outline of the LDF sensor systems: (**a**) Intensity-based LDF sensor; (**b**) SPR-LDF sensor. is the critical angle outside the sensing region. Assuming an unpolarized collimated light source, we can express the normalized transmitted power as [17]:  
\[P_{L}=\frac{1}{2}\left(\frac{\sum_{\theta_{i}=\theta_{c}}^{\pi/2}P_{0}(\theta_{i} )\cdot R_{p}(\theta_{i})^{N(\theta_{i})}}{\sum_{\theta_{i}=\theta_{c}}^{\pi/2}P_ {0}(\theta_{i})}+\frac{\sum_{\theta_{i}=\theta_{c}}^{\pi/2}P_{0}(\theta_{i}) \cdot R_{s}(\theta_{i})^{N(\theta_{i})}}{\sum_{\theta_{i}=\theta_{c}}^{\pi/2}P_ {0}(\theta_{i})}\right) \tag{1}\]  
where \(P_{0}(\theta_{i})\) is the initial power propagated by TIR, \(N(\theta_{i})=cot(\theta_{i})L/d\) is the number of reflections, \(d\) is the core diameter, \(R_{p}\) and \(R_{s}\) are the Fresnel reflectance coefficients at the boundary between the core and the sensing medium for p-polarization and s-polarization, respectively. As regards to the distribution of power among the various modes, the typical assumption about multimode fibers is a Lambertian source, which is actually the most accurate when the fiber is illuminated through a lens with a numerical aperture (NA) larger than that of the fiber. Under this assumption, the power distribution can be expressed as [17; 18; 19]:  
\[P_{0}(\theta_{i})\propto\frac{n_{co}^{2}sin\theta_{i}cos\theta_{i}}{\left(1-n _{co}^{2}cos^{2}\theta_{i}\right)^{2}} \tag{2}\]  
From Figure 3, however, we see that the loss computed under the Lambertian source assumption does not match the experimentally observed behavior. Instead, an excellent agreement is found if assuming a perfectly scrambled angular distribution (\(P_{0}(\theta_{i})=const\) in Equation (1)). This finding is reasonable considering the scrambling of optical energy operated by the helical voids running along the core. To best illustrate this point, we compare in Figure 3b the angular distribution of power as expressed by Equation (2) (Lambertian source) with the one speculated for our LDF. It can be seen that, in the case of the Lambertian source, the power density of rays is higher near the critical angle (\(\theta_{i}\approx 72^{\circ}\) in our case), therefore the loss of TIR for these rays, consequent to the increase of the SRI, has a greater impact on the transmitted power.