# Polarization Effect on the Performance of On-Chip Wireless Optical Point-to-Point Links
## 2 Design of the Dielectric Vivaldi Antenna  
The proposed integrated antenna is schematized in Figure 1, where the relevant geometrical parameters are highlighted. As anticipated, the antenna is constructed with a SiN slot waveguide tapered according to a Vivaldi profile, and it is directly coupled to an input SiN strip waveguide. This geometry has been chosen to guarantee wideband performance of the antenna, thus maximizing the bandwidth of the on-chip network.  
The Vivaldi profile can be calculated according to the following equations:  
\[\mathrm{y}=\mathrm{C}_{1}\mathrm{e}^{\mathrm{R}\;\mathrm{x}}+\mathrm{C}_{2} \tag{1}\]  
with  
\[\mathrm{C}_{1}=\frac{\mathrm{y}_{1}-\mathrm{y}_{2}}{\mathrm{e}^{\mathrm{R}\; \mathrm{x}_{1}}-\mathrm{e}^{\mathrm{R}\;\mathrm{x}_{2}}},\;\mathrm{C}_{2}= \frac{\mathrm{y}_{2}\mathrm{e}^{\mathrm{R}\;\mathrm{x}_{1}}-\mathrm{y}_{1} \mathrm{e}^{\mathrm{R}\;\mathrm{x}_{2}}}{\mathrm{e}^{\mathrm{R}\;\mathrm{x}_{ 1}}-\mathrm{e}^{\mathrm{R}\;\mathrm{x}_{2}}} \tag{2}\]  
In Equation (2), R is the opening rate coefficient, whereas x\({}_{1}\), y\({}_{1}\) and x\({}_{2}\), y\({}_{2}\) are the edge coordinates of the Vivaldi profile in the xy plane.  
The Vivaldi antenna is fed by a strip waveguide thanks to the direct coupling mechanism between the strip and the slot waveguides. The input optical signal of the strip waveguide is coupled directly to the slot mode, and then, it is radiated by the Vivaldi antenna. The radiation occurs predominantly toward the x direction in Figure 1, i.e., in front of the antenna.  
The numerical model of the antenna was implemented through the three-dimensional Finite Difference Time Domain (3D-FDTD) method [45]. For the design of the antenna and the analysis of its radiation characteristics, the structure was assumed to be immersed in a homogenous medium with refractive index \(n_{\text{SiO2}}=1.445\) at the wavelength \(\lambda=1.55\)\(\upmu\)m. This assumption eases the calculation of the far-field radiation diagrams because a homogeneous medium is required when the near-to-far-field transformations are performed.  
In the proposed structure, the thickness of the waveguides was considered equal to t = 300 nm, as is the case in commercially available SiN samples. The dispersion of the refractive index of SiN [46; 47] was included in the model, and the simulation domain was delimited by Perfectly-Matched-Layer (PML) boundary conditions.  
The gain of the Vivaldi antenna depends on different geometrical parameters such as the width of the strip waveguide w, the slot gap g, and the length of the Vivaldi directors L\({}_{\text{a}}\). The initial total width of the slot waveguide, which opens according to the Vivaldi profile, is geometrically constrained by the width w of the input strip waveguide. Specifically, these geometrical parameters are linked according to the equation w = g + 2s, where w is the strip width, g is the initial gap of the slot Vivaldi antenna, and s is the width of the slot rails. The antenna gain can be optimized by changing the cross-section geometrical parameters (i.e., w, g, and s). For a given value of the strip width w, the gap g can be suitably chosen to maximize the gain. Moreover, once the antenna cross section has been optimized, the antenna gain can be further improved by changing the length L\({}_{\text{a}}\) of the Vivaldi radiator.  
In the following, we will consider three arbitrarily chosen values of the strip width w = 600 nm, w = 900 nm, and w = 1100 nm, which correspond to three different configurations identified by the letters A, B, and C, respectively. All the three considered values of the strip width w give the excitation of either a single TE or a single TM mode in the strip waveguide. By considering three different configurations, we aim to analyze the effect on the antenna gain induced by the cross-section geometrical parameters (i.e., w, s, and g) and to demonstrate that, given a certain strip width w, the gap g can be optimized to maximize the gain.