## 3 Results and Discussion  
### Structural Parameters and Transmission Spectrum  
The structural parameters will determine the spectrum parameters of the TMFBG, namely, its intensity, bandwidth, and central wavelength. Using the OptiGrating 8.0 simulation software of the OptiWave Company (Ottawa, ON, Canada), the transmission spectrum was studied to optimize the tilt angle, grating length, grating period, and refractive index modulation depth of the TMFBG. When the inclination angles of the TMFBG were \(4^{\circ}\), \(5^{\circ}\), \(6^{\circ}\), \(7^{\circ}\), and \(8^{\circ}\), the transmission spectra were as shown in Figure 2. In the simulation model, the external environment was a cylindrical boundary coaxial with the optical fiber, with a radius of 100 \(\upmu\)m and a refractive index of 1.33. The grating period was 0.53 \(\upmu\)m; the grating length was 10,000 \(\upmu\)m; and the modulation depth was 0.0001. The grating inclination changed from \(4^{\circ}\), \(5^{\circ}\), \(6^{\circ}\), and \(7^{\circ}\) to \(8^{\circ}\). In the transmission spectra of the TMFBG, the core mode at the right side has the longest wavelength value and transmits inside the fiber core. When the light signals escape out from the fiber core, they will transmit in the cladding layer of optical fiber in the higher-ordered modes and be named cladding modes. The number of cladding modes increases with the increasing grating inclination. Because the coupling between the core modes becomes weaker as the angle increases, the coupling between the forward transmitted core mode and the backward transmitted cladding mode is enhanced, which produces more cladding modes [26]. Comparing the results for the \(4^{\circ}\) and \(8^{\circ}\) tilt angles, in addition to the increasing number of cladding modes, there is also the disappearance of the low-order cladding modes. As the angle decreases, more energy from the forward core modes is coupled to the radiation mode. In addition to the cladding mode, the transmission depth of the core mode also decreases with the increase in the tilt angle. A greater tilt angle mainly affects the grating visibility, and it ultimately affects the transmission power. Therefore, with an increase in the grating inclination, the visibility of the grating becomes lower, and the transmission depth becomes smaller. The simulation results clearly describe the relationship between the grating inclination and the transmission spectrum of the TMFBG. In Figure 3, the grating periods of the TMFBG are, respectively, 0.526 \(\upmu\)m, 0.528 \(\upmu\)m, 0.530 \(\upmu\)m, 0.532 \(\upmu\)m, and 0.534 \(\upmu\)m. The corresponding transmission spectra are compared. Here, the grating inclination is fixed at 4\({}^{\circ}\). As the grating period gradually increases from 0.526 \(\upmu\)m to 0.534 \(\upmu\)m, the wavelengths of the core mode and the cladding mode in the transmission spectrum are red-shifted as a whole. The transmission depths of the core mode and the cladding mode scarcely change. Therefore, when keeping the other morphological parameters unchanged, the grating period has a linear dependence on the wavelength location. The change in the grating period only affects the position of the resonant peak for each order of the grating, with no effect on the transmission intensity. When the grating length is, respectively, set to 10 mm, 20 mm, 30 mm, 40 mm, and 50 mm, the transmission spectrum of the TMFBG is shown in Figure 4. The grating inclination, the grating period, and the modulation depth are 4\({}^{\circ}\), 0.53 \(\upmu\)m, and 0.0001, respectively. With the increase in the grating length from 10 mm to 50 mm, the transmission depth of  
Figure 3: Transmission spectrum of TMFBG at grating periods of 0.526 \(\upmu\)m, 0.528 \(\upmu\)m, 0.530 \(\upmu\)m, 0.532 \(\upmu\)m, and 0.534 \(\upmu\)m.  
Figure 2: Transmission spectrum of TMFBG with grating inclinations of 4\({}^{\circ}\), 5\({}^{\circ}\), 6\({}^{\circ}\), 7\({}^{\circ}\), and 8\({}^{\circ}\), respectively.