## 3 Results and Discussion  
Due to the relatively large grating layer thickness, far from being treated within perturbation approximation, a numerical modeling approach is needed to obtain precise results. Here, the performance of the designed fiber-tip RWG is first studied with a 2D finite-difference time domain (FDTD) method using commercial software (FullWave, RSoft 2021), and the design parameters are listed in Table 1. The Au mirrors are located at the ninth grating bar on both sides, with the center of the fiber core in the middle of the central grating notch. In the simulations, the TE-polarized core fiber mode (electric field  
Figure 2: Schematic of the designed fiber-tip RWG sensor: (**a**) cross-section perpendicular to the grating lines, (**b**) top view, and (**c**) multiplexed sensor array on the facet of a typical 19-core fiber. (**a**) The black dashed line represents the central (core) axis of the fiber, and only half of the sensor’s geometry is presented for simplicity. along grating grooves), located 0.6 \(\upmu\)m below the fiber/waveguide interface, was used as the incident beam. Two power monitors in the xy-plane were located 1.15 \(\upmu\)m below the fiber/waveguide interface to collect the reflected power, one for the total reflected power and another one for the mode reflection, with the latter defined by the overlapping integral between the fiber core mode and the field at that specific xy-plane. The mesh sizes were set to be 20 nm for both x- and z-direction. Within the waveguide, for the grating and Au layers, as well as for areas around all structural edges, non-uniform meshes were applied with finer grid sizes assuring results with good convergence. Perfectly matched layer (PML) boundaries were employed for \(\pm\)z and +x, while symmetric boundaries were employed for \(\
eg\)x, to save computational resources. The calculated reflection spectrum is shown in Figure 3a. The main resonant peak at \(\sim\)1529.5 nm is blue-shifted compared to the RCWA result (green curve). This blue shift may be attributed to both the finite beam size of the core mode and the grating length, as discussed in Reference [26]. With the use of Au layers to create a FP cavity, the achieved resonant mode reflectance was \(\sim\)74.8%, which is quite close to the total reflectance of \(\sim\)79.4% (see Figure 3a). The mode reflectance is specified as the part of the total incident energy coupled back to the core mode of the fiber, while the total reflectance is the part of the total incident energy diffracted back by the grating. Apparently, most of the reflected power can be coupled back to the core mode. It is worth noticing that, besides the main resonant peak, a secondary peak is observed at \(\sim\)1634.0 nm for both the mode and the total reflectance. This side peak cannot be found in the RCWA curve for the uniform grating, and this could be attributed to the FP resonance behavior. The mode reflectance for a uniform grating on the fiber facet with the same design parameters is presented in Figure 3a by a red curve. In the corresponding simulation, the +x domain, expanded as grating bars far away from the core area, may also contribute to mode reflectance because the grating itself also serves as a second-order Bragg grating, and coupling occurs into waveguide modes propagating along both positive and negative x-directions. The mode reflectance for the uniform grating was found to be \(\sim\)25.4%. In comparison, the use of a FP cavity improved the peak intensity approximately three times, and, at the same time, the peak FWHM (full width at half maximum) was reduced from \(\sim\)23.0 nm to 13.7 nm due to the improved Q factor of the FP cavity. Such an increase in peak intensity in conjunction with a decreased FWHM is crucial for signal identification in sensing applications.