## 2 Results
By precisely evaluating this ratio, photonic sensor designs are crafted to enhance sensitivity and responsiveness, thereby contributing to the refinement and efficacy of sensor technologies in diverse applications, such as environmental monitoring, biomedical diagnostics, and industrial process control. The EFR is defined as the calculated ratio of the integration of intensity within the specified region, typically the upper cladding,  
Figure 1: (**a**) Effective refractive index (n\({}_{\text{eff}}\)) of the Si\({}_{3}\)N\({}_{4}\) ridge waveguide versus different dimensions of the core, (**b**) EFR versus different dimensions of the waveguide core at an operational wavelength of 1545 nm, (**c**) sensitivity of the Si\({}_{3}\)N\({}_{4}\) waveguide versus different dimensions of the core in the presence of indexes 1.30 and 1.31, (**d**) normalized E-field distribution in the ridge waveguide of W = 1000 nm and H = 340 nm at an operational wavelength of 1545 nm. to the comprehensive integration of intensity across the entire waveguide structure as follows [33]:  
\[\text{EFR}=\frac{\iint_{\text{upper cladding}}|\text{E(x,y,z)}|^{2}\text{dxdydz}}{ \iint_{\text{total}}|\text{E(x,y,z)}|^{2}\text{dxdydz}}; \tag{1}\]  
In Figure 1b, the trend becomes evident as the core dimension increases--the n\({}_{\text{eff}}\) of the mode rises, leading to a concurrent decrease in the EFR. Notably, for TE-polarized light, the width (W) of the core exerts a more pronounced impact on the EFR compared to the height (H) of the core. Therefore, designing a sensing device with a higher EFR value is suggested for enhanced sensitivity. Furthermore, the EFR has a direct relationship with the sensitivity of the waveguide. To demonstrate this, the sensitivity of Si\({}_{3}\)N\({}_{4}\) straight waveguides with different core dimensions is calculated by utilizing the following formula. \[\text{S}_{\text{waveguide}}=\frac{\Delta\text{n}_{\text{eff}}}{\Delta\text{n }}; \tag{2}\]  
Here, \(\Delta\text{n}_{\text{eff}}\) represents the variation in the effective index of the mode resulting from a shift in the ambient refractive index (\(\Delta\text{n}=\text{n}_{1}-\text{n}_{2}\)). In this context, we assign values of 1.30 and 1.31 to n\({}_{1}\) and n\({}_{2}\), respectively. The waveguide sensitivity exhibits a decrease from 0.8 n\({}_{\text{eff}}\)/RIU to 0.5 n\({}_{\text{eff}}\)/RIU, corresponding to an increase in the core width (W) and height (H). Specifically, this trend is observed as the core dimensions vary from W = 1000 nm (and H = 340 nm) to W = 1600 nm (and H = 370 nm), as illustrated in Figure 1c. To visually elucidate the characteristics of the fundamental mode within the waveguide core with dimensions of W = 1000 nm and H = 340 nm at 1545 nm, refer to the normalized E-field distribution portrayed in Figure 1d. This representation provides valuable insights into the spatial distribution and intensity profile of the electric field associated with the fundamental mode under consideration. A schematic of the RR structure is presented in Figure 2a, and the transmission spectra of RRs with radii of 50 \(\upmu\)m, 75 \(\upmu\)m, and 100 \(\upmu\)m are meticulously computed across the wavelength spectrum of 1540 nm to 1550 nm with a step size of 0.1 nm, all within the ambient medium of air (n = 1.0). The gap (g) between the bus waveguide and the ring is fixed at 200 nm. Given that the numerical simulations are conducted using the 2D-FEM, the n\({}_{\text{eff}}\) values for the particular geometry are extracted from Figure 1a. In this context, n\({}_{\text{eff}}\) values are chosen to pertain to a waveguide geometry with dimensions of W = 1000 nm and H = 340 nm. This selection is pivotal, as it aligns with the subsequent fabrication of the RR structure, ensuring consistency in the geometric dimensions. By employing these n\({}_{\text{eff}}\) values, the simulations establish a foundation for accurate predictions and assessments in the context of the fabricated RR structure.