# High-Resolution, Broad-Range Detection Setup for Polarimetric Optical Fiber Sensors
## 2 Materials and Methods
The output is vector \(\mathbf{v_{c}}\), which contains values of power incident on each pixel, corresponding to a given phase difference \(\varphi\) and rounded down to the nearest level of the \(N\)-bit A/D converter. The complex fast Fourier transform of vector \(\mathbf{v_{c}}\) produces complex spectrum vector \(\mathbf{s_{c}}\). The phase of the second component of the calculated spectrum \(\mathbf{v_{c}}\)[1] is phase difference \(\varphi_{F}\), corresponding to the rounded data (Please note that the first component of \(\mathbf{v_{c}}\), i.e., \(\mathbf{v_{c}}\)[0] is the constant component). The phase error, arising from quantization, is calculated as \(\varphi_{F}-\varphi\). The results of the calculations are presented in Section 3.1. Testing of the proposed interferometer should focus on its resolution, as the operation of similar interferometers in the fringe counting mode is already well-researched. This requires a means of introducing small amounts of phase difference or optical path length difference into the light incident on the Wollaston prism. A method used in ellipsometry that employs a quarter-wave (\(\lambda/4\)) plate rotated by angle \(\theta\) was selected, and a test setup was designed. A schematic diagram of the test setup is presented in \[P(\varphi,x_{N})=P_{0}+P_{1}\cos\big{(}\varphi+k_{0}x_{N}+\gamma\big{)}, \tag{5}\]6 Light from source LS (a laser diode or a superluminescent diode) is collimated by lens L, polarized by a linear polarizer P, and passes through a quarter-wave plate \(\lambda/4\). The light then passes through Wollaston prism WP and analyzer A. Finally, the light interferes on the area detector array in camera C, forming the fringe pattern. The Jones vector \(\mathbf{v_{o}}\) of light incident on the Wollaston prism can be expressed as:  
\[P(\varphi,x_{N})=P_{0}+P_{1}\cos\big{(}\varphi+k_{0}x_{N}+\gamma\big{)}, \tag{5}\]7  
where \(\mathbf{R}0\) represents the rotation matrix, \(\theta\) is the rotation angle of the quarter-wave plate, \(\mathbf{M}_{QWP}\) is the matrix of the quarter-wave plate, and \(\mathbf{v}_{i}\) is the Jones vector of the light after the polarizer P. Substituting for \(\mathbf{R}\), \(\mathbf{M}_{QWP}\), and \(\mathbf{v}_{i}\) in Equation (18) and performing matrix multiplications and some rearrangements, we arrive at:  
\[\mathbf{v_{o}}(\theta)=\frac{1}{\sqrt{2}} \begin{bmatrix}1-(1-i)\;(\sin( \theta))^{2}-\frac{1}{2}(1-i)\;\sin(2\theta)\\ 1-(1-i)\;(\sin(\theta))^{2}+\frac{1}{2}(1-i)\;\sin(2\theta)\end{bmatrix} =\frac {1}{\sqrt{2}} \begin{bmatrix}m_{x}\\ m_{y}\end{bmatrix} . \tag{19}\]  
Figure 3: Schematic diagram of the measurement setup. Explanations in the text. The phase difference \(\Delta\varphi\) between the \(m_{x}\) and \(m_{y}\) components of \(\mathbf{v_{o}}\) can be expressed as the argument of \((m_{x}/m_{y})\), viz. :  
\[\Delta\varphi(\theta)=\arg\biggl{(}\frac{m_{x}}{m_{y}}\biggr{)}=\arg\Bigl{(}( \cos(2\theta))^{2}+i\cdot\sin(2\theta)\Bigr{)}. \tag{20}\]  
For small angles (i.e., \(\,|\,\theta\,|\,<0.105\) rad (\(6^{\circ}\))), \(\Delta\varphi\) can be conveniently approximated by  
\[\Delta\varphi(\theta)=2\theta \tag{21}\]  
with an approximation error below 2%. If values greater than 0.1 rad of the phase difference \(\Delta\varphi\) are needed, a Soleil-Babinet compensator can be used instead of a rotating quarter-wave plate. A measurement setup was built in the configuration schematically presented in Figure 3. A laser diode or a superluminescent diode was used as the light source LS. The polarizer P and analyzer A were made from a high-contrast polarizing film on a glass substrate. A zero-order quarter-wave plate \(\lambda/4\), mounted in a rotation mount (PRM1/M, Thorlabs, Newton, NJ, USA), was used to change the retardation. A Wollaston prism WP made from quartz was used. A monochrome 1280 \(\times\) 1024 camera (EO-1312M-GL, Edmund Optics, Mainz, Germany) acquired the fringe pattern, which was sent by a USB link to a PC for image processing.