# High-Resolution, Broad-Range Detection Setup for Polarimetric Optical Fiber Sensors
## 2 Materials and Methods
and offset angle \(\gamma\) is given by  
\[\gamma=\frac{1}{2}\big{(}k_{0}x_{p}+k_{1}y_{p}\big{)}, \tag{8}\]  
the power \(P_{i}\) incident on the \(m\)-th pixel can be expressed using (5) as:  
\[P_{i}(m,\phi)=P\big{(}\phi,(m-1)\ x_{p}\big{)}=P_{0}+P_{1}\cos\big{(}\phi+(m-1 )\ k_{0}x_{p}+\gamma\big{)}, \tag{9}\]  
From (9), it can be seen that for a given phase difference \(\varphi\) and the offset angle \(\gamma\), the power incident on subsequent pixels of the bottom row of the detector array varies as a sampled harmonic function. If angle \(\alpha\) is adjusted to fulfill the condition  
\[k_{0}x_{D}\geq 2\pi, \tag{10}\]  
where \(x_{D}\) is the width of the detector array, at least one period of this harmonic function can be registered. Let us calculate the phase difference change \(\Delta\varphi\) needed to shift the detected pattern \(P_{m}\) by one pixel. This means that  
\[P_{i}(m,\phi)=P_{i}(m-1,\phi+\Delta\phi), \tag{11}\]  
from (9), we have  
\[\cos(\varphi+\gamma+k(i-1)x_{P})=\cos\big{(}\varphi+\Delta\varphi+\gamma+k(i- 2)x_{P}\big{)}, \tag{12}\]  
which reduces to  
\[P(\varphi,x_{N})=P_{0}+P_{1}\cos\big{(}\varphi+k_{0}x_{N}+\gamma\big{)}, \tag{5}\]0  
If \(k_{0}x_{D}=2\pi\), and a detector with \(N\) pixels in the row is used, Equation (13) becomes  
\[P(\varphi,x_{N})=P_{0}+P_{1}\cos\big{(}\varphi+k_{0}x_{N}+\gamma\big{)}, \tag{5}\]1  
This means that when a detector with 1024 pixels in the row is used, the resolution of phase measurement can be \(2\pi/1024\) rad. It may be expected that the resolution can be even better when data proceeding techniques that enable sub-pixel resolution are used. The resolution of the presented method may degrade when the amplitude \(P_{1}\) of the harmonic component in Equation (9) is much smaller than \(P_{0}\), which is similar to working with low-visibility interference fringes. In order to investigate this issue, let us introduce a parameter \(V\), defined as:  
\[P(\varphi,x_{N})=P_{0}+P_{1}\cos\big{(}\varphi+k_{0}x_{N}+\gamma\big{)}, \tag{5}\]2  
It should be noted that \(V\) is not the visibility of fringes described by Equation (1); it can be regarded as the visibility in a single row of pixels. From Equations (3), (6), (7) and (9), we obtain:  
\[P(\varphi,x_{N})=P_{0}+P_{1}\cos\big{(}\varphi+k_{0}x_{N}+\gamma\big{)}, \tag{5}\]3  
which is a product of two \(\sin(x)/x\) functions. Since the dimensions \(x_{p}\) and \(y_{p}\) of a pixel are much smaller than the fringe spacing \(\Lambda\), \(\sin\alpha\) is much smaller than one and \(\cos\alpha\approx 1\), \(V\) becomes:  
\[P(\varphi,x_{N})=P_{0}+P_{1}\cos\big{(}\varphi+k_{0}x_{N}+\gamma\big{)}, \tag{5}\]4  
For five pixels per fringe, i.e., \(\Lambda=5y_{p}\), \(V\) equals 0.93. This means that the method is capable of operation even when the fringes are closely spaced. The quantization of the signal in an image sensor is another factor that introduces an error, often referred to as quantization error. In the case of the detection setup presented, this error affects the accuracy of the phase difference measurement. Most commonly, image sensors use one or more analog-to-digital (A/D) converters that have a resolution ranging from eight to sixteen bits (cf., e.g., [43; 44]). In the most prevalent CMOS image sensors, these A/D converters are integrated on the detector chip. The quantization error is difficult to investigate analytically; therefore, numerical modeling was conducted. First, an analytical description of an ideal image sensor, followed by an ideal \(N\)-bit A/D converter, was created. This image sensor is illuminated by the fringe pattern shown in \[P(\varphi,x_{N})=P_{0}+P_{1}\cos\big{(}\varphi+k_{0}x_{N}+\gamma\big{)}, \tag{5}\]5 The power incident on each pixel in a row is described by (9). The orientation angle \(\alpha\) of the fringe pattern is selected in such a way that the period of the cosine function in (9) is equal to the number of pixels in the row (e.g., 512), i.e., condition (10) becomes \(k_{0}x_{D}=2\pi\).