# Boosting Second Harmonic Radiation from AlGaAs Nanoantennas with Epsilon-Near-Zero Materials
## 2 Results and Discussion
Moreover, the proposed ENZ substrate could be realized by using transparent conductive oxide, such as ITO, GzO, AZO or doped-CdO, where the plasma frequency can be tuned by varying the doping of these materials. We performed our simulation by means of a commercial software based on the finite element method (COMSOL Multiphysics@v. 5.2.--www.comsol.com.--COMSOL AB, Stockholm, Sweden). The model geometry is shown in Figure 1b. The mesh was customized to be more detailed in the cylindrical nanoantenna and in the air-substrate interface close to the nanodisk. Doing so enabled us to focus on the key phenomena of the proposed structure. The minimum mesh element size was less than 1 nm. The domain was truncated by a perfectly matched layer (PML), and far-field calculations were done on the inner boundary of the PML domain. Surface A was used to calculate total scattered power. We evaluated three different scenarios. First, we considered the case of an ENZ substrate. We investigated the optical scattering of the AlGaAs nanodisk by changing its radius between 200 nm and 290 nm. The maximum scattered power was reached for a nanocylinder with a radius of \(r\) = 232 nm, as shown in Figure 1c. Figure 2a shows the electric field distribution in the case of \(r\) = 232 nm, when the substrate was the ENZ material. At 1550 nm, the ENZ material had a real part of the permittivity equal to \(-\)16.48 and an imaginary part equal to 0.4217 (see Table 1). It can be seen how the electric field inside the cylinder reached an enhancement factor \(\left|\mathrm{E}\right|/\left|\mathrm{E}_{0}\right|\) equal to 35. The high enhancement of the field at the interface between the substrate and the cylinder can be attributed to the excitation of surface plasmon polaritons of the underlying ENZ material by the dielectric nanoparticle on top [45]. Different is the case of the dielectric substrate. In the latter circumstance, the field circulated inside the disk (see Figure 2b) because a low refractive index (\(n_{\mathrm{s}}\approx 1.6\)), enabling sub-wavelength optical confinement in the nanocavity by total internal reflection. The multipole analysis deduced from the distributions of the induced electric current, density had been carried out [46]. For \(r\) = 232 nm the scattering behaviour was dominated by the electric and magnetic dipolar contributions as can be also concluded by looking at the field distribution. Finally, the case of gold substrate was considered. See Table 1 for the gold permittivity at the FW. The mode excited inside the nanoantenna resembled the one of the ENZ substrate but the field enhancement was more than 8 times lower. As the real part of the permittivity of the substrate became more negative, the electric field generated by the antenna was reduced because the substrate behaves as an electric conductor, cancelling the dipolar moment at the direct surface by an imaginary mirror dipole in the substrate. This is similar to the case of increasing the imaginary part of the permittivity [47]. We stress that, even though the radius of the cylinder was optimized for the ENZ substrate, no significant field enhancement was obtained for the other substrates using a different radius. We then simulated the second harmonic generation from the proposed structure. Since AlGaAs is a non-centrosymmetric material, we had a strong volume contribution which could enhance the SH signal. We first note that, among the different resonant modes that were observed in the nanodisks in Figure 1c, the resonance at \(r=232\) nm in the case of ENZ substrate, showed the strongest scattering power and electric field enhancement inside the structure. This can potentially enhance the nonlinear optical response achievable from these nanoantennas. We recall that the second harmonic (SH) wavelength falls at the ENZ condition wavelength of the optimized substrate. The second-order nonlinear susceptibility tensor of (100)-grown AlGaAs is anisotropic and has only off-diagonal elements \(\chi^{(2)}_{ijk}\) with \(i\
eq j\
eq k\).