# Polarization Effect on the Performance of On-Chip Wireless Optical Point-to-Point Links
## 2 Design of the Dielectric Vivaldi Antenna
### Vivaldi Antenna Radiation Analysis  
In order to investigate the radiation characteristics of the proposed Vivaldi antenna, the far-field of the Vivaldi antenna coupled to the strip waveguide was evaluated by simulating the overall structure via the 3D-FDTD method. For this purpose, near-to-far field projections of the DFT-transformed electromagnetic fields were calculated on a closed surface surrounding the antenna and the strip waveguide [45]. In the evaluation of the radiation characteristics, the antenna was considered to be embedded in a homogeneous medium (with refractive index \(\text{n}_{\text{SiO2}}\) = 1.445) as required by near-to-far field projection calculation. The antenna gain G(\(\theta\),\(\phi\)) is defined as:  
\[\text{G}(\theta,\phi)=4\pi\frac{\text{I}(\theta,\phi)}{\text{P}_{\text{in}}} \tag{4}\]  
Figure 2: Overlap integral calculated according to Equation (3) as a function of the initial slot gap value g. The blue, red, and yellow curves correspond to configurations A (w = 600 nm), B (w = 900 nm), and C (w = 1100 nm), respectively. For each value of the slot gap g, the slot rail width s can be calculated from the formula s = (w \(-\) g)/2. The wavelength is \(\lambda\) = 1.55 μm. where I(0, \(\phi\)) is the radiation intensity in spherical coordinates, and P\({}_{\text{in}}\) is the input power launched into the strip waveguide [49]. As an example of the radiation characteristics of the Vivaldi antenna, Figure 3a,b show the gain as a function of (a) the angle \(\theta\) (with \(\Phi\) = 0) and (b) the angle \(\Phi\) (with \(\theta\) = \(\pi/2\)), for the TE (solid curves) and TM (dashed curves) modes, in the case of the Vivaldi antenna with strip width w = 900 nm, slot gap g = 300 nm, slot rail width s = 300 nm, and antenna length L\({}_{\text{a}}\) = 25 \(\upmu\)m. Moreover, Figure 3c,d report the electric field pattern of the Vivaldi antenna, which is calculated via 3D-FDTD in the xy plane for z = 0 \(\upmu\)m (i.e., in the middle of the antennas) for the TE and TM polarizations. The geometry of the [PERSON] antenna is also evidenced by the black lines in Figure 3c,d. The color bar represents the normalized amplitude of the electric field in logarithmic scale. It is worth pointing out that the field patterns of Figure 3c,d are calculated in the antenna region at the wavelength \(\lambda\) = 1.55 \(\upmu\)m via DFT of the time domain fields. Therefore, they are not far-fields, but they can help to qualitatively explain the far-field radiation diagrams. For the sake of comparison, we also consider the radiation characteristics of a simple truncated waveguide with width w = 900 nm (Figure 4a,b). The truncated waveguide can also be considered as an antenna because it radiates the propagating mode in the surrounding medium due to the abrupt termination. The gain curves in the vertical (a) and in the horizontal (b) planes are reported in Figure 4a,b for the truncated strip waveguide of width w = 900 nm for the TE (solid curve) and TM (dashed curve) modes. Moreover, the field patterns calculated for the truncated waveguide in the xy plane are also reported in Figure 4c,d. As can be seen in Figure 3a,b, in the case of the analyzed Vivaldi antenna, the gain curves for the TE and the TM polarizations are very similar, especially in the main radiation lobe. This does not happen for the truncated waveguide (Figure 4a,b). Moreover, as the field patterns in Figure 3c,d show, the Vivaldi-shaped directors contribute to narrowing the radiated beam and to increasing the maximum gain with respect to the simple truncated waveguide structure (Figure 4c,d). Spurious radiation also occurs along the directors, causing sidelobes in the Vivaldi antenna radiation diagram.