# Boosting Second Harmonic Radiation from AlGaAs Nanoantennas with Epsilon-Near-Zero Materials
## 2 Results and Discussion
In the calculations, we used \(\chi^{(2)}=100\) pm/V as reported for Al\({}_{0.18}\)Ga\({}_{0.82}\) as in [48]. We thus solved the problem of the SH wavelength using the calculated fields at the FW to define the second harmonic source as external current density. The \(i\)-th component of the external current density \(J_{i}\) is expressed as:  
\[J_{i}=j\omega_{SH}\varepsilon_{0}\chi^{(2)}E_{FF,j}E_{FF,k}, \tag{2}\]  
where \(\varepsilon_{0}\) is the vacuum permittivity and \(E_{FF,i}\) is the \(i\)-th component of the electric field at the FW. Note that here we are assuming that no SH signal is coming from the substrate. Figure 2d-f shows the electric field distribution at the Second Harmonic wavelength SH. Figure 2a,d suggest that the use of the ENZ enlarged the mode-matching between the mode at FW and SH wavelengths. The overlap integral estimation [5] clarified that the maximum overlap (about 46%) was reached for the ENZ substrate whereas was only about 42% for the gold substrate and 39% for the dielectric substrate. Figure 1: (**a**) Sketch of the proposed structure; (**b**) simulated model with the discretization mesh; (**c**) scattering power at varying cylinder radius (\(h=400\) nm) for the different considered substrates, \(I_{0}=1.6\) GW/cm\({}^{2}\). The geometrical conversion efficiency is defined as:  
\[\eta_{SHG}=\frac{\int_{A}\overset{\rightarrow}{S}_{SH}\cdot\hbar da}{I_{0}\times \pi r^{2}} \tag{3}\]  
where \(\overset{\rightarrow}{S}_{SH}\) is the Poynting vector of the SH field, \(\hbar\) is the unit vector normal to a surface A enclosing the antenna, and \(I_{0}\) is the incident field intensity. We numerically found a SH conversion efficiency of up to 3 \(\times\) 10\({}^{-3}\) in the case of nanodisks placed over the ENZ substrate. We also evaluated how much SH power was redirected in the air region by defining \(\Gamma\) as the ratio between the SH power radiated in the air and the SH power radiated in the substrate region through surface A enclosing the nanodisk (see Figure 1b). For the ENZ substrate, \(\Gamma\) reached the maximum value of 1191. This means that in the proposed structure we were able to obtain a very high SH efficiency and at the same time, the second harmonic signal was redirected into the upper region where the air was present. On the other hand, in the case of the dielectric substrate, \(\Gamma\) is equal to 0.28 meaning that most of the SH signal was redirected into the substrate, as shown in Figure 3b, and the SHG efficiency was about 4 \(\times\) 10\({}^{-7}\). A good SH signal redirection was instead obtained in the case of a gold substrate, where \(\Gamma\) was equal to 836, see Figure 3c. However, the achieved efficiency did not exceed 8 \(\times\) 10\({}^{-7}\). In this case, the SH efficiency limiting factor was represented by the poor electric field enhancement inside the cylinder at the fundamental wavelength. The second harmonic fields in Figure 2d-f indicate that higher order modes were excited inside the structure at the SH wavelength. Such higher order multipoles were responsible for the obtained far field. We stress that the analysis performed did not take surface SHG effects into account, but we only considered the quadratic bulk nonlinearity contribution since the latter term has been shown to be dominant in these kinds of dielectric structures [5; 6; 7; 8; 9]. A complete analysis and comparison between surface and volume nonlinear contributions was postponed to a future effort. We note that, although all the calculations have been performed for a cylinder optimized on the ENZ substrate, even if we consider the other substrates with the respective optimized cylinder radius, we obtained a maximum SH conversion efficiency of less than 10\({}^{-6}\) for the AlO\({}_{x}\) substrate (\(r\) = 205 nm) and 5.5 \(\times\) 10\({}^{-5}\) for the gold substrate (\(r\) = 260 nm).