## 2 Results  
The effective refractive index (n\({}_{\text{eff}}\)) of the Si\({}_{3}\)N\({}_{4}\) ridge waveguide is meticulously computed across the wavelength spectrum of 1540 nm to 1550 nm, employing a fine step size of 0.1 nm. To ascertain the influence of the waveguide core dimensions on the mode propagating within, variations in the width (W) and height (H) are systematically investigated. Analysis of Figure 1a reveals a discernible trend: the n\({}_{\text{eff}}\) of the mode exhibits a consistent increase with the enlargement of the waveguide core dimensions. The n\({}_{\text{eff}}\) of a waveguide is intricately linked to the dimensions of its core. As the dimensions of the core change, the n\({}_{\text{eff}}\) of the guided modes within the waveguide also undergoes variation. Fundamentally, the n\({}_{\text{eff}}\) is a measure of the propagation of the speed of light within the waveguide structure relative to the speed of light in a vacuum. When the core dimensions increase, the n\({}_{\text{eff}}\) tends to rise due to the increased confinement of light within the core material, leading to a longer interaction length between the light and the core. This enhanced interaction promotes a higher n\({}_{\text{eff}}\), as more light energy is confined within the core region, resulting in a slower phase velocity compared to the surrounding cladding material. Conversely, decreasing core dimensions can lead to a reduction in the n\({}_{\text{eff}}\) as the confinement of light weakens, allowing for more energy to spread into the lower refractive index cladding material in the form of an evanescent field. In consideration of potential fabrication inaccuracies arising from over-etching or under-etching, specific values for the W are judiciously chosen as 1000 nm and 1200 nm, while the H is systematically altered within the range of 340 nm to 370 nm. Importantly, it is emphasized that this range of dimensions is tailored to exclusively support the TE\({}_{0}\) fundamental mode within the specified wavelength range of 1540 nm to 1550 nm. As an illustration, consider the n\({}_{\text{eff}}\) of the mode within a waveguide characterized by dimensions of W = 1000 nm and H = 340 nm, yielding a value of 1.5267. In contrast, a waveguide with larger dimensions, specifically W = 1200 nm and H = 340 nm, can sustain a fundamental mode with a higher n\({}_{\text{eff}}\) = 1.5637. This results in superior mode confinement within the core, precluding the potential to design highly sensitive devices because of the decrease in the evanescent field around the waveguide, which could otherwise interact with the ambient medium. Consequently, formulating photonic sensors with the capacity to provide a more substantial evanescent field is suggested. The evanescent field plays a crucial role in various photonic sensing applications due to its unique property of decaying exponentially outside the core of an optical waveguide. This phenomenon allows the evanescent field to interact sensitively with the surrounding medium, making it a valuable tool for sensing minute changes in the refractive index, concentration, or surface properties of the adjacent materials. The evanescent field can penetrate the analyte, enabling label-free and non-invasive detection methods. The Evanescent Field Ratio (EFR) assumes pivotal significance in the advancement of photonic sensors, particularly those reliant on leveraging the evanescent field for interaction with the surrounding environment. In this context, the evanescent field serves as a crucial element in facilitating the detection and monitoring of variations within the ambient medium [32]. The EFR emerges as a critical parameter because it sums up the intricate connection between the intensity of the evanescent field and its function in detecting dynamic changes.