# High-Resolution, Broad-Range Detection Setup for Polarimetric Optical Fiber Sensors
## 2 Materials and Methods  
Let us consider the polarization interferometer presented in Figure 1a. A light beam from a single-mode polarization-maintaining (SMPM) fiber is collimated by lens L and impinges on Wollaston prism WP. As the axes of the Wollaston prism are aligned along the axes of the fiber, polarization modes \({}_{\mathrm{o}}\)HE\({}_{11}\) and \({}_{\mathrm{e}}\)HE\({}_{11}\) do not mix; they deviate from each other after propagating through the prism. They then pass through analyzer A, whose axis forms an angle of \(45^{\circ}\) with the axes of Wollaston prism WP. Finally, the polarization modes interfere on area image sensor AIS, forming a set of equidistant and straight fringes, as shown in Figure 1b. When one side of the area image sensor AIS is parallel to the direction of fringes, as shown in Figure 1b, the interferometer is equivalent to that described in Ref. [36]. Consider the modification to the interferometer presented in Figure 1 in which the area image sensor AIS is aligned at an angle \(\alpha\) to the fringe pattern, as shown in Figure 2. An \(xy\) coordinate system associated with the image sensor is introduced for the purpose of subsequent analysis. Figure 1: Common-path interferometer for retardation measurement: (**a**) schematic diagram; (**b**) fringe pattern on the detector. L—lens; WP—Wollaston prism; A—analyzer; AIS—area image sensor. In order to obtain insight into operation of such a setup without dealing with excessively complicated formulas, let us introduce four simplifying assumptions: (i) the AIS consists of equally spaced pixels of a width \(x_{p}\) and a height \(y_{p}\); (ii) the AIS is aligned in such a way that the phase difference \(\varphi\) between the two beams interfering at the origin of the coordinate system \(xy\) equals the phase difference of interfering beams; (iii) power density \(p_{0}\) is constant on the area of the detector; and (iv) visibility of the fringe pattern is 1.0. Then, power density of the interference pattern \(p(\varphi,x,y)\) can be expressed as:  
\[p(\varphi,x,y)=p_{0}\bigg{[}1+\cos\bigg{(}\varphi+\frac{2\pi}{\Lambda}(x\ \sin\alpha+y\ \cos\alpha)\bigg{)}\bigg{]}, \tag{1}\]  
where \(\Lambda\) represents the fringe spacing, \(\alpha\) represents the orientation angle, \(p_{0}\) represents the maximum power density, and \(\varphi\) represents the phase difference of the polarization modes. The power incident on a pixel from the bottom row of pixels located at a distance \(x_{N}\) from the origin of the \(xy\) coordinate system, as shown in Figure 2, can be expressed using (1) as:  
\[P(\varphi,x_{N})=\int\limits_{x_{N}}^{x_{N}+x_{p}}\int\limits_{0}^{y_{p}}p_{0 }\bigg{[}1+\cos\bigg{(}\varphi+\frac{2\pi}{\Lambda}(x\ \sin\alpha+y\ \cos\alpha)\bigg{)}\bigg{]}\text{dydx}. \tag{2}\]  
By setting:  
\[ \begin{array}{l}k_{0}=\frac{2\pi}{\Lambda}\ \sin\alpha,\\ k_{1}=\frac{2\pi}{\Lambda}\ \cos\alpha\end{array}  \tag{3}\]  
using these in (2), we arrive at:  
\[P(\varphi,x_{N})=p_{0}\int\limits_{x_{N}}^{x_{N}+x_{p}}\bigg{(}y_{p}+\frac{1}{ k_{1}}\sin\big{(}\varphi+k_{0}x+k_{1}y_{p}\big{)}-\frac{1}{k_{1}}\sin(\varphi+k_{0}x )\bigg{)}\text{d}x. \tag{4}\]  
Rewriting (4) as the sum of three integrals and integrating and employing some trigonometric identities, we obtain  
\[P(\varphi,x_{N})=P_{0}+P_{1}\cos\big{(}\varphi+k_{0}x_{N}+\gamma\big{)}, \tag{5}\]  
where  
\[P_{0}=p_{0}x_{p}y_{p}, \tag{6}\]  
\[P_{1}=\frac{4p_{0}}{k_{0}k_{1}}\sin\bigg{(}\frac{k_{0}x_{p}}{2}\bigg{)}\sin \bigg{(}\frac{k_{1}y_{p}}{2}\bigg{)}, \tag{7}\]  
Figure 2: Fringe pattern on the rotated image sensor. AIS—area image sensor.