# Characteristic Analysis and Experiment of Adaptive Fiber Optic Current Sensor Technology
## 2 Open Loop Mechanism of Polarization-Type Fiber Optic Current Sensor  
Figure 1 shows the polarization-type fiber optic current sensor. The fiber optic current sensor can be divided into three parts: a light source, a sensing system, and a photoelectric conversion system. During light propagation in the absence of birefringence, refractive index \(n_{11}=n_{22}\), \(\chi_{11}=\chi_{22}\), incident linearly polarized light waves can be decomposed into left and right circularly polarized light corresponding to the different propagation coefficients \({k_{0}}^{+}\) and \({k_{0}}^{-}\):  
\[\left\{ \begin{array}{c}{k_{0}}^{+}=\frac{\omega}{c}\sqrt{1+\chi_{11}+\chi_{1 2}}\\ {k_{0}}^{-}=\frac{\omega}{c}\sqrt{1+\chi_{11}-\chi_{12}}\end{array} \right. \tag{1}\]  
where \(\chi\) is the polarizability tensor of the medium \(\chi=\left( \begin{array}{ccc}\chi_{11}&-i\chi_{12}&0\\ i\chi_{21}&\chi_{22}&0\\ 0&0&\chi_{33}\end{array} \right)\), \(c\) is the speed of light in the vacuum, and \(\omega\) is the optical frequency. Figure 1: Structure of the polarization-type fiber optic current sensor. After light passes through an ideal fiber of length dl, the rotation angle of the electric vector is  
\[\theta=\frac{\varphi}{2}=\frac{k^{+}-k^{-}}{2}dl\approx\frac{1}{2}\frac{\omega}{c }\frac{\chi_{12}}{\sqrt{1+\chi_{11}}}dl=\frac{\omega K}{2 cn}Hdl=VHdl \tag{2}\]  
In the presence of birefringence, \(\chi_{11}\
eq\chi_{22}\), refractive index \(n_{11}^{2}=n_{0}^{2}-\eta,n_{22}^{2}=n_{0}^{2}+\eta\), the propagation coefficients are:  
\[\left\{ \begin{array}{l}k^{+}=\frac{\omega}{c}\sqrt{1+\chi_{11}+\sqrt{\eta^{ 2}+\chi_{12}^{2}}}\\ k^{-}=\frac{\omega}{c}\sqrt{1+\chi_{11}-\sqrt{\eta^{2}+\chi_{12}^{2}}}\end{array} \right. \tag{3}\]  
After light passes through an ideal fiber of length dl, the rotation angle of the electric vector is:  
\[\frac{\varphi}{2}=\frac{k^{+}-k^{-}}{2}dl\approx\frac{1}{2}\frac{\omega}{c} \frac{\sqrt{\eta^{2}+\chi_{12}^{2}}}{\sqrt{1+\chi_{11}}}dl \tag{4}\]  
The photoelectric conversion system converts the light signal into an electric signal. Then the electrical signal is input to a computer system through an interface for post-processing, so as to obtain the magnitude of the measured current. A PIN photodetector is often used. The signal output from the photodetector can be expressed as:  
\[I_{P}=h_{f}\cdot J_{P}+h_{0}\cdot V_{P} \tag{5}\]  
where \(I_{P}\) is the output current, \(J_{P}\) is the luminous flux, \(V_{P}\) is the voltage across the photoelectric converter, \(h_{f}\) is the sensitivity of photodetector, and \(h_{0}\) is the conductance parameters of the photodetector. The output signal from the signal processing circuit is:  
\[U_{out}=\frac{J_{s}-J_{save}}{J_{save}}-\frac{J_{p}-J_{pure}}{J_{pure}} \tag{6}\]  
where \(U_{out}\) is the output signal from the signal processing circuit, \(J_{s}\) and \(J_{p}\) are the output signals of the two linearly polarized beams passing through the photodetector, respectively, and \(J_{save}\) and \(J_{pure}\) are the DC components corresponding to the two optical paths, respectively. The influence of external factors like temperature drift on the fiber optic current sensor can be equivalently converted into an additional magnetic field. The temperature-dependent magnetic field can be written as follows:  
\[H_{v}=\frac{C\upsilon}{T-T_{P}}H \tag{7}\]  
where \(H_{\upsilon}\) is the temperature-dependent equivalent magnetic field, \(H\) is the magnetic field induced by the measured current, \(\upsilon\) is a coefficient related to the molecular field, \(C\) is the Curie constant, and \(T_{P}\) is the paramagnetic Curie temperature. The open loop system diagram of a polarization-type fiber optic current sensor can be derived from this model, as shown in Figure 2.  
One can see from Figure 2 that the input variables \(I_{1}\) and \(T\) have only positive feedback, thus the fiber optic current sensor is an open loop system [15]. In fact, the difference between the refractive indices of the core and cladding is much less than 1% in a general sensing fiber.