## 3 Results and Discussion
### Sensing Performance Simulation  
The tilt angle, grating period, grating length, and refractive index modulation depth are the most important structural parameters for a TMFBG. When the structural parameters change, the optical properties of the TMFBG will also change. The sensing model of the TMFBG is established using OptiFDTD 8.0 to study the influence of the cladding radius from the perspective of the energy distribution. In addition, the OptiGrating simulation module of the OptiWave 8.0 series software is used to study the transmission spectral response of the TMFBG under different refractive indices. Figure 6 shows the simulation results regarding the refractive index sensing characteristics of the TMFBG when the refractive index of the external environment is changed from 1.34, 1.36, 1.38, and 1.40 to 1.42. The grating period is set to 0.53 \(\upmu\)m. The tilt angle of the grating is 4\({}^{\circ}\), and the modulation depth is 0.0001. When the ambient refractive index increases from 1.34 to 1.42, the cladding modes decrease gradually, and the high-order cladding modes disappear. This is because, as the ambient refractive index increases and gradually approaches the effective refractive index of the cladding, the higher-order cladding modes will leak into the external environment. When the ambient refractive index increases, the core mode of the TMFBG remains constant due to its wavelength location. This is consistent with mode coupling theory and the sensing characteristics of the TFBG structure. The core mode is insensitive to the change in the external refractive index, while the cladding mode is sensitive. As shown in Figure 6b, when the ambient refractive index increases from 1.34 to 1.42, the cutoff wavelength is red-shifted. In practical applications, the refractive index can be detected by calibrating the drift value of the cutoff wavelength. When the refractive index ranges from 1.34 to 1.42, the cutoff wavelength shift of the TMFBG with the cladding diameter of 20 \(\upmu\)m is -40 nm. The refractive index sensitivity can be calculated as \(\Delta\lambda/\Delta\text{m}\approx 5\text{ nm}/\text{RIU}\). The influence of various structural parameters of the TMFBG on the transmission spectra was studied and calculated using OptiGrating. It was seen that as the tilt angle of the TMFBG gradually increases, the core mode couples into radiation mode; the transmission depth of the core mode decreases; and the number of cladding modes increases. When the grating period of the TMFBG gradually increases, the resonance peaks will shift towards to the longer wavelengths, without any impact on the transmission intensity. As the length of the TMFBG gradually increases, the transmission depth of both core mode and cladding mode continuously strengthens. When the depth of refractive index modulation increases, it only affects the resonant depth of the TMFBG, without affecting the wavelength position. Simulation study of the sensing characteristics of the TMFBG on the refractive index indicates that as the refractive index increases, the core mode is unchanged, while the higher-order cladding mode continuously disappears. By calibrating the location of  
Figure 6: Simulation of refractive index sensing characteristics. (**a**) Transmission spectra of TMFBG and (**b**) the cutoff wavelength under different ambient refractive indices. the cutoff wavelength, refractive index detection can be achieved. The spectra quality and sensing performance are determined by the structural parameters. To obtain a bigger signal-to-noise ratio or deeper resonance dips, it is needed for using a smaller grating inclination, the bigger refractive index modulation depth or the longer grating length. These three parameters do not need to be met simultaneously and can be appropriately balanced based on processing conditions. While the central wavelength is determined by the grating period. The refractive index sensing can be realized by tracing the locations of cutoff wavelength.