## 2 Theory  
An FBG, by definition, is a regular perturbation of the RI within the fiber core [9] that reflects the wavelengths satisfying the Bragg criterion as in Equation (1), where \(\lambda_{B}\) is the Bragg wavelength, \(\eta_{eff}\) is effective RI of the grating, and \(\Lambda\) is the pitch of the grating. \[\lambda_{B}=2.\eta_{eff}.\Lambda \tag{1}\]  
As shown by [PERSON] et. al. [42], when two separate and identical Bragg gratings are written in the fiber core, with a small gap between, they form an FP structure. This structure manifests unique properties of having two spectral peaks within its main reflection band [22] and has better measurement resolution than a single direct grating [43]. The structure of a grating-based FP in a standard fiber (125 \(\upmu\)m cladding diameter), is shown in Figure 1, which consists of two identical uniform Bragg gratings of length \(g\), separated by a distance \(e\). Following are the coupled mode equations derived using the coupled mode theory [44], to study the behavior of the grating and related FP structure. Consider \(r_{g}(\lambda)\) and \(t_{g}(\lambda)\) to be the complex amplitude reflection and transmission coefficients, respectively, for a uniform grating, given by Equations (2) and (3):  
\[r_{g}(\lambda)=\frac{-ksinh(\sqrt{(k^{2}g^{2}-\sigma^{2}g^{2})})}{\sigma sinh (\sqrt{(k^{2}g^{2}-\sigma^{2}g^{2})})+j\sqrt{(k^{2}-\sigma^{2})}cosh(\sqrt{(k^ {2}g^{2}-\sigma^{2}g^{2})})} \tag{2}\]  
\[t_{g}(\lambda)=\frac{1}{cosh(\sqrt{(k^{2}g^{2}-\sigma^{2}g^{2})})-j(\frac{ \sigma}{\sqrt{(k^{2}-\sigma^{2})}})sinh(\sqrt{(k^{2}g^{2}-\sigma^{2}g^{2})})} \tag{3}\]  
where \(g\) is the grating length. \(k\) and \(\sigma\) are the \(ac\) and \(dc\) coupling coefficients of the grating [45], respectively. The reflection spectrum \(r_{f/p1}(\lambda)\) of an FP structure with two identical Bragg gratings, separated by a distance \(e\) (the cavity length or gap), is given by Equation (4) [22]:  
\[r_{f/p1}(\lambda)\propto r_{g}(\lambda).f(\phi) \tag{4}\]  
where \(f(\phi)\) is an interference function of the cavity, which depends on phase difference \(\phi(\lambda)\) between the light reflected from the gratings, given by Equation (6), assuming \(\eta_{c}\) is the RI of the cavity region (gap). Figure 2 shows the normalized reflection spectrum of the FP structure in a standard optical fiber  
\[f(\phi)=1+cos(\phi(\lambda)) \tag{5}\]  
\[\phi(\lambda)=\frac{4\pi\eta_{c}e}{\lambda} \tag{6}\]  
[PERSON] et al. [24] also provided a set of modeling equations for designing an FP structure based on complex reflection and transmission coefficients of the uniform grating, i.e., Equations (2) and (3). The amplitude reflection coefficient of the FP, \(r_{f/p2}(\lambda)\) is given by Equation (7), and its power reflection  
Figure 1: Schematic of grating-based Fabry–Perot. Figure 2: Normalized reflection spectrum of the FP structure. Footnote 2: The reflection coefficient is plotted against the wavelength. \[r_{fp2}(\lambda)=r_{g}(\lambda).\left[\frac{(1+[t_{g}^{2}-r_{g}^{2}].\exp{(-j \phi)})}{(1-r_{g}^{2}.\exp{(-j\phi)})}\right] \tag{7}\]  
As can be seen in Figures 2 and 3, the FP structure results in extremely narrow reflection peaks within the grating spectrum. If the minimum of the interference function \(\lambda_{min}\) lies in the grating reflection spectrum, the peak splits into two. There are two cases mentioned by [PERSON] et.al. [22]: (i) when \(\lambda_{min}\) coincides with \(\lambda_{B}\), the two peaks are of equal intensity; and (ii) when \(\lambda_{B}>\lambda_{min}\), Peak 1 (shorter wavelength) has lower intensity than Peak 2 (longer wavelength), as shown in Figure 3. With the applied strain and temperature, the spectrum of the grating and interference function both vary as a function of the phase difference (\(\phi(\lambda)\)). Furthermore, the change of wavelengths for a grating can be derived from Equation (1) as in Equation (8), where the first term constitutes the strain effect and the latter is the temperature effect.