# Three-Dimensional Broadband Electric Field Sensor Based on Integrated Lithium Niobate on Insulator
## 2 Design and Simulation
Figure 4a,b show the simulation mode of the two sensor chips with a construction of 60 degrees. The excitation source is a plane wave with an amplitude of 1 V/m at 1 GHz. The wave vector \(k\) of the plane-wave radiation is along the x-axis, and \(E_{0}\) is the  
Figure 3: **(a)** Simulation model in the HFSS software; **(b)** axial ratio of each pair of antennas on a single sensor chip. Figure 2: **(a)** Lateral view and **(b)** perspective view of the packaged 3D sensor. polarization direction along the z-axis. Ideally, the induced voltage between the modulation electrodes of chip 1 should only be affected by the original excitation source. However, the antennas of chip 2 will partly reflect the incident electric field to chip 1. Therefore, the induced voltage between the modulation electrodes of each chip is not only from the original source but also the reflected electric field from the antennas of other chips. Between the modulation electrodes, the simulated induced electric field on chip 1 is shown in Figure 4c. For comparison, the simulation results without chip 2 are depicted in Figure 4d. The amplitudes of the induced electric field on chip 1 under the two-chip mode and the one-chip mode are 0.4-0.9 V/m and 0.01-0.5 V/m, respectively. The reflection obviously affects the induced electric field of the two-chip mode compared with the one-chip mode. In practical terms, the unknown incident direction, amplitude, and polarization of the electric field are far more complicated than the simulation. The quantitative simulation is hard to produce. A 3 \(\times\) 3 matrix is used to calibrate the measured electric field amplitude from each sensor chip. Ideally, each sensor chip is mutually orthometric without interaction. Thus, all the elements of the matrix are zero except for the diagonal elements. Mathematically, the response of the 3D sensor under an electric field is expressed as  
\[\left( \begin{array}{ccc}A&B&C\\ a_{11}&0&0\\ 0&a_{22}&0\\ 0&0&a_{33}\end{array} \right)\left( \begin{array}{c}b_{1}\\ b_{2}\\ b_{3}\end{array} \right)=\left( \begin{array}{c}c_{1}\\ c_{2}\\ c_{3}\end{array} \right) \tag{1}\]  
Figure 4: (**a**) Perspective view and (**b**) lateral view of the simulation mode of the two-chip mode; (**c**) simulation results of the induced electric field of the two-chip mode and (**d**) the one-chip mode. where the vector \(B\) is the original electric field. The vector \(C\) is the output voltage of the 3D sensor. The matrix \(A\) is the response matrix of the 3D sensor. In practice, the nondiagonal elements are nonzero due to the interaction among sensor chips. Therefore, the 3D sensor can be calibrated by solving all the unknown elements of the matrix \(A\), where \(A\) is indicated as  
\[A= \begin{pmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{pmatrix}  \tag{2}\]  
As expressed in Equation (1), the measurement result of one sensor chip is the product of the corresponding row of \(A\) and the vector \(B\). Hence, to solve the unknown elements of \(A\) in Equation (2), three linearly independent vectors \(B\) should be realized in the experiment to build a linearly independent equation set.