# Electro-Optic Sensor for Measuring Electrostatic Fields in the Frequency Domain
## 2 Method and Experimental Results
### Analytical Model  
The mathematical model developed in MATLAB mimics the behavior of the optical chain shown in Figure 2 and was used for parametric simulations to find the optimal configuration for the test bench. Jones notation [PERSON] (1984) was used to describe mathematically the components of the system and to evaluate the output signal. The optical beam propagates along \(y\)-axis and has components only in the \(x\)-\(z\) plane, therefore can be represented by the following polarisation vector:  
\[E^{Opt}(t)=\left( \begin{array}{c}E^{Opt}_{x}e^{i(ky-wt)}\\ 0\\ E^{Opt}_{z}e^{i(ky-wt)}\end{array} \right) \tag{1}\]  
where \(ky-wt\) is the phase of the light wave. In the specific case shown in Figure 2, the initial optical beam is characterized only by a component along \(z\)-axis. Moreover, the model exploits the normalised polarisation vector, which yields the following description of the optical beam:  
\[\frac{E^{Opt}(t)}{|E^{Opt}(t)|}=E^{Opt}_{Norm}(t)=\left( \begin{array}{c}0\\ 1\end{array} \right) \tag{2}\]  
All the other optical components are known elements of the Jones calculus and the electromagnetic wave that reaches the photodetector \(E^{Opt}_{Out\_Norm}\) can be computed as:  
\[E^{Opt}_{Out\_Norm}(t)=\frac{1}{2\sqrt{2}}\left( \begin{matrix}0\\ 1\end{matrix} \right)\cdot\underbrace{\left[ \begin{matrix}-1&1\\ 1&-1\\ 0&e^{i\frac{T_{b}}{2}}\end{matrix} \right]}_{Polarizer}\cdot\underbrace{\left[ \begin{matrix}e^{-i\frac{T_{b}}{2}}&0\\ 0&e^{i\frac{T_{b}}{2}}\end{matrix} \right]}_{Bias\ Crystal}\cdot\underbrace{\left[ \begin{matrix}e^{-i\frac{T_{b}}{2}}&0\\ 0&e^{i\frac{T_{b}}{2}}\end{matrix} \right]}_{Sensing\ crystal}\cdot\underbrace{ \left[ \begin{matrix}1&-1\\ -1&1\end{matrix} \right]}_{Polarizer} \tag{3}\]  
Figure 2: Schematic overview of the EO sensor for measurements of a DC electric field. where the amplitude factor \(\frac{1}{2\sqrt{2}}\) comes from the analyser and the polariser. Particularly important are the factors of \(\Gamma_{b}\) and \(\Gamma_{s}\), which represent the phase retardation \(\Delta\phi=\phi_{x}-\phi_{z}\) between the two components \(E_{x}\) and \(E_{z}\) of the laser beam propagating through each crystal. The phase retardation between the two components of the optical beam is due to the birefringence of the EO crystals and is expressed as:  
\[\Gamma=\Delta\phi=\phi_{x}-\phi_{z}=\frac{2\pi}{\lambda}(n_{e}-n_{o})L_{y} \tag{4}\]  
where \(\lambda\) is the wavelength of the optical beam and \(L_{y}\) is the longitudinal dimension of the crystal. The Pockels effect results in a linear modulation of the refractive indices of the crystal. In Figure 2, both crystals experience an electric field only along \(z\)-axis. The first crystal (_Electro-optic Bias crystal_ in Figure 2) is intentionally modulated with a sinusoidal modulation field offset by a DC component. The second crystal, or the _sensing crystal_, is instead used to couple the external electric field. Given the dependence of the refractive indices of \(LiNbO_{3}\) from the electric field, Equation (4) can be expanded, taking into account the Pockels effect, and applied to the two crystals as:  
\[ \begin{cases}\Gamma_{s}=\ \frac{\overbrace{2\pi}(n_{e}-n_{o})L_{yy}}{ \overbrace{2\pi}(n_{e}-n_{o})L_{yy}}-\frac{\pi}{\pi}(n_{e}^{3}r_{33}-n_{o}^{3} r_{13})L_{yy}[E_{ext}]\\ \Gamma_{b}=\ \underbrace{\frac{2\pi}{\lambda}(n_{e}-n_{o})L_{yy}}_{ Natural\ birefringence}-\underbrace{\pi}_{\lambda}(n_{e}^{3}r_{33}-n_{o}^{3}r_{13})L_{yy}[E_{ mod}+E_{dc\_bias}]}_{Pockels\ effect}\end{cases}  \tag{5}\]  
where \(r_{33}\) and \(r_{13}\) are the electro-optic coefficients of lithium niobate, \(E_{dc\_bias}\) is the DC component of the bias voltage, \(E_{mod}=A\sin(2\pi ft)\) is the sinusoidal modulation and \(E_{ext}\) is the field under measurement. To ease the notation, the dependence of the refractive indices and the electro-optic coefficients from the wavelength of the laser beam and the crystal's temperature [20] are not written down explicitly. However, these dependencies have been taken into account in the mathematical model.