## 2 Operation Principle  
The spectral characteristics of an interferometer (including the free spectra range and the chirp rate) are strongly dependent on the wavelength-to-phase relationship at the interference port. The key point of the fabrication of the chirped F-P interferometer is to realize the highly nonlinearly changing wavelength-to-phase relationship. Thus, the optical path difference should depend on the wavelength. The optical microfiber can perfectly meet this condition. When the microfiber diameter reduces to only several micrometers, the effective refractive index \(n_{\textit{eff}}\) strongly depends on the microfiber diameter [16]. The microfiber diameter-to-\(n_{\textit{eff}}\) relationship calculated by the COMSOL software is shown in Figure 1a. As the diameter decreases, the effective refractive index of the microfiber \(n_{\textit{eff}}\) also declines and drops dramatically with the decreasing of the fiber diameter when the diameter is less than 7 nm [16]. Figure 2 shows the fabrication and sensing process of the proposed chirped F-P interferometer. A standard multimode fiber (62.5 \(\upmu\)m/125 \(\upmu\)m) is drawn and tapered by butane flame brushing. The horizontal cross section of the microfiber conforms to a certain outline shape. In traditional tapered fiber analysis models, the microfiber is divided into two tapers with linearly changing diameters and a uniform region to simplify the simulation process, as shown in Figure 1b. However, the existence of a  
Figure 1: **(a)**The microfiber diameter-to-\(n_{\textit{eff}}\) relationship curve calculated by the COMSOL software; **(b)** the traditional tapered fiber analysis model (up) with 3 regions and the proposed tapered fiber analysis model (down) with 2 slowly changing regions (\(d\) is the diameter of the tapered microfiber and x is the position along the fiber). uniform region does not satisfy the real situation, especially for the analyzing of interferometer-type optical sensors. Thus, we consider the microfiber to be two slowly changing regions with a specific outlined shape. In the grating fabrication step, by adding periodical refractive index modulation of the microfiber with a phase mask and excimer laser, a broadband fiber Bragg grating can be achieved. The detailed analyses are shown as follows. Here, we consider the two taper regions to be symmetrical and the origin of the coordinates is at the position of the minimum radius. As the red curve shown in Figure 2, the diameter \(d\) of the tapered microfiber is the function of the position along the fiber \(x\) that can be expressed as follows:  
\[d=f(x). \tag{1}\]  
According to the coupled-mode theory, the reflected optical wavelength satisfies the phase matching condition when the Bragg grating is formed. Then, the wavelength at position x is also related to x, which is expressed as follows:  
\[\lambda(x)=n_{\mathit{eff}}(f(x))\cdot\Lambda \tag{2}\]  
where \(\wedge\) is the grating period of the phase mask. Due to the monotonicity of \(f(x)\) and \(n_{\mathit{eff}}\), (2) can also be expressed as follows:  
\[x=f^{-1}(n_{\mathit{eff}}^{-1}(\frac{\lambda}{\Lambda})) \tag{3}\]  
where \(f^{-1}(x)\) and \(n^{-1}_{\mathit{eff}}\left(d\right)\) are the inverse functions of \(f(x)\) and \(n_{\mathit{eff}}\left(d\right)\) (the curve in Figure 1a), respectively. Then, the phase difference of \(\lambda\) when the light propagates back and forth in the cavity can expressed as follows:  
\[\Delta\varphi(\lambda)=4x/\lambda\cdot 2\pi+\pi=f^{-1}(n_{\mathit{eff}}^{-1}( \frac{\lambda}{\Lambda}))/\lambda\cdot 8\pi+\pi. \tag{4}\]  
The extra \(\pi\) in Equation (4) is caused by the half-wave loss. The phase change between the different wavelengths will keep changing slowly and monotonously.