## 2 Principle and Device Fabrication  
Figure 1a depicts a schematic diagram of the proposed hybrid grating structure, containing an Ex-TFG and an FBG. It is well known that in a standard FBG, the only strong coupling permitted occurs between the forward- and backward-propagating core modes, for which the light is confined to near the fiber axis and is isolated from the surroundings by the relative thick cladding [36]. However, for the Ex-TFG with largely tilted fringes, the forward-propagating core mode was coupled into forward-propagating cladding modes according to the phase-match condition [37; 38]. Hence, a Bragg resonance was embedded in a series of cladding mode resonances at discrete wavelengths, so that each cladding mode resonance had two orthogonal TE and TM polarization states. The Bragg wavelength \(\lambda_{\text{B}}\) of the FBG and the resonant wavelength \(\lambda_{i,\text{clad}}^{m}\) of the \(m\)th cladding for the Ex-TFG, are determined by the phase-match condition and can be expressed as follows  
\[\lambda_{\text{B}}=2n_{\text{eff,core}}\Lambda_{\text{B}}, \tag{1}\]  
\[\lambda_{i,\text{clad}}^{m}=\left[n_{\text{eff,core}}-n_{i,\text{eff,clad}}^{m }\right]\Lambda_{\text{G}}/\cos\theta. \tag{2}\]  
where, \(i\) represents TE or TM modes, \(n_{\text{eff,core}}\) and \(n_{i,\text{eff,clad}}^{m}\) are the effective indices of the core mode and the \(m\)th cladding mode, respectively, \(\Lambda_{\text{B}}\) and \(\Lambda_{\text{G}}\) are the grating period of the FBG and nominal period of the Ex-TFG, respectively, and \(\theta\) is the tilt angle. The resonant wavelengths of the TE and TM cladding modes of Ex-TFG with the same order could have separation depending on the difference of their effective indices. From Equations (1) and (2), the resonant wavelength \(\lambda_{i,\text{clad}}^{m}\) of the Ex-TFG is affected by the surrounding medium's RI. However, any changes of temperature and strain to the hybrid grating will produce shift in both \(\lambda_{\text{B}}\) and \(\lambda_{i,\text{clad}}^{m}\). In the device fabrication, the B/Ge co-doped silica RDF with 6.8/80 \(\upmu\)m core and cladding diameters was initially hydrogen loaded at 80 \({}^{\circ}\)C and 150 bar pressure for 48 h to enhance its photosensitivity. An FBG was fabricated in a hydrogenated RDF by scanning a phase-mask with a period of 1064 nm using a frequency doubled continuous wave argon-ion laser. By switching to an amplitude-mask and rotating it by an angle with respect to the fiber axis, the Ex-TFG was then inscribed on top of the FBG. The hybrid grating with the FBG and Ex-TFG was annealed at 80 \({}^{\circ}\)C for 48 h to release the residual hydrogen and stabilize the grating structure [33]. The lengths of FBG and Ex-TFG are \(L_{\text{FBG}}\) = 1 mm and \(L_{\text{Ex-TFG}}\) = 5 mm, respectively. Inspected with a high-resolution microscope, the optical microscopic images shown in Figure 1b demonstrate that the periods of the FBG and Ex-TFG are 536 nm and 22.96 \(\upmu\)m, respectively, and the tilted angle of Ex-TFG is around 80\({}^{\circ}\). The spectrum and sensing performance of the hybrid grating were examined by the experimental setup shown in Figure 2, which includes a supercontinuum (SuperK) light source, an in-fiber linear polarizer, polarization controller (PC), and an optical spectrum analyzer (OSA). Launched from the SuperK source, light was polarized and adjusted by the polarizer and the controller, respectively. The transmission spectrum was monitored by the OSA with a wavelength resolution of 0.02 nm. Figure 1c shows the full spectrum of the hybrid grating in the range of 1300-1680 nm, including a sharp Bragg resonance dip at the FBG and a set of cladding mode resonance dips at the Ex-TFG in two orthogonal polarization states. For a specific order of cladding mode such as the peak at around 1560 nm, the separation of dual-peak is approximately 9.50 nm, which is larger than that of the Ex-TFG in standard single-mode fiber (SMF) [35; 38].