# Electro-Optic Sensor for Measuring Electrostatic Fields in the Frequency Domain
## 2 Method and Experimental Results
### DC Measurement in the Frequency Domain  
Figure 3 shows the system's transfer function, obtained by sweeping the DC bias electric field acting on the bias crystal while measuring the relative output light intensity. An important parameter of the crystals, annotated in Figure 3, is \(E_{\pi}\) which is the electric field acting on the crystal for which the phase retardation between the two light components is \(\delta=\pi\). The value of \(E_{\pi}\) is proportional to the optical beam wavelength \(\lambda\) and given by:  
\[E_{\pi}=\frac{\lambda}{L_{y}n_{e}^{3}r_{33}-n_{o}^{3}r_{13}} \tag{11}\]  
where \(L_{y}\) is the crystal's length, \(n_{e}\) and \(n_{o}\) are the refractive indices, and \(r_{13}\) and \(r_{33}\) are the corresponding electro-optic coefficients. Since it is a property of the crystal, we will distinguish between and \(E_{\pi B}\), for the bias crystal, and \(E_{\pi S}\), for the sensing crystal, as marked in Figure 2. Figure 3 shows the transfer function of the bias crystal with \(E_{\pi B}\) being the electric field difference between the function's consecutive minimum and maximum. Table 1 summarizes the properties of the crystals used for the simulations and in the developed test bench. \begin{table}
\begin{tabular}{c|c c} \hline \hline  & **Bias Crystal** & **Sensing Crystal** \\ \hline
**Material** & \(LiNbO_{3}\) (5\% \(MgO\) doped) & \\ \(\mathbf{L_{x}}\) [mm] & 3 & 5 \\ \(\mathbf{L_{y}}\) [mm] & 40 & 50 \\ \(\mathbf{L_{z}}\) [mm] & 3 & 2 \\ \(\mathbf{E_{\pi}}\) [\(\frac{\mathrm{kV}}{\mathrm{m}}\)] & 159 & 203.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the crystals used in the EO sensor. Figure 3: Transfer function of the system. Conceptually, it is feasible to build the sensor with only a single EO crystal. However, using two separate crystals brings some practical advantages. The first crystal can be placed further away from the incident electric field and be unaffected by it. Since the crystal is affixed between two Printed Circuit Boards (PCBs), it becomes possible to apply an arbitrary value of a static electric field, referred to henceforth as the DC bias. Such a solution allows us to fully control the system's working point regardless of the incident electric field strength. To perform DC measurements in the frequency domain, we supplement the DC bias with a sinusoidal modulation \(\mathrm{E_{MOD}}\). The DC bias is chosen on purpose to exploit the symmetric shape of the transfer function at Working Point (WP) 1 or 2, as indicated in Figure 3. Sinusoidal modulation around either of these two points would result in an output signal containing only even (for WP1) or odd (for WP2) harmonics of the modulation frequency. Measurements of an incident field become possible by observing the change to the lowest harmonic, which is absent for a given WP, i.e., the first harmonic (H1) for WP1 and the second harmonic (H2) for WP2. The choice of working point, as well as the modulation's frequency and amplitude, is discretionary and not physically limited by the sensor construction. Modulation around WP1 (as well as the other crests and troughs of the transfer function) results in a higher sensitivity of the developed sensor. To explain the sensor's principle of operation we assume a DC bias set to reach WP1 and a sinusoidal modulation with a peak-to-peak modulation amplitude \(E_{pp}=E_{\pi B}\). Figure 4 illustrates the system behaviour with no incident electric field on the sensing crystal. The horizontal axis is normalized by \(E_{\pi S}\) to highlight the possibility of scaling the system's sensitivity by varying the geometrical and physical properties of the crystals. As the electric field across the bias crystal modulates, so does the output light intensity. However, due to the transfer function's symmetry the latter happens at twice the modulation frequency, as shown in Figure 5a,b. The two PCBs surrounding the bias crystal would unavoidably interact with the external electric field to be measured.