# Boosting Second Harmonic Radiation from AlGaAs Nanoantennas with Epsilon-Near-Zero Materials
## 2 Results and Discussion  
We considered a nanocylinder with a height of 400 nm made of Al\({}_{0.18}\)Ga\({}_{0.82}\)As placed over a semi-infinite substrate layer. The excitation is assumed to be a \(x\)-polarized plane wave (see Figure 1) at a fundamental wavelength (FW) of 1550 nm with incident amplitude E\({}_{0}\) corresponding to a pump intensity of 1.6 GW/cm\({}^{2}\). A sketch of the proposed structure is reported in Figure 1a. The considered nanoantenna height is compatible with other AlGaAs structures [6; 9; 13] already fabricated and optimized for the efficient generation of the second harmonic signal. Al\({}_{x}\)Ga\({}_{1-x}\)As is a III-V semiconductor which is gaining more and more popularity for nonlinear photonics due to its non-centrosymmetric structure. Moreover, it has a large band gap enabling two photons absorption-free operation at 1.55 \(\upmu\)m, a high non-resonant quadratic susceptibility (\(d_{14}\approx 100\) pm/V for GaAs in the near infrared) [40], and a broad spectral window of transparency in the mid-infrared (up to 17 \(\upmu\)m), which allows for the generation of intense second-order nonlinear optical effects [41; 42]. For the dispersion of the refractive index of AlGaAs, we use the analytical model proposed in Reference [43]. At the FW it has a refractive index n\({}_{\text{Al0.18 Ga0.82 As}}\) equal to 3.275 that is Re(\(\epsilon\)) = 10.729, Im(\(\epsilon\)) = 0, see Table 1.  
In the following we considered three different substrates where the AlGaAs nanoantenna was placed on an ENZ, a dielectric and a gold substrate. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & **Re(\(\epsilon\)) + j Im(\(\epsilon\))** & **Re(\(\epsilon\)) + j Im(\(\epsilon\))** \\  & **FW** & **SH** \\ \hline ENZ & \(-16.48\) + j 0.4217 & 0.0006 + j 0.05 \\ Gold & \(-115.13\) + j 11.259 & \(-22.068\) + j 1.3708 \\ Dielectric & \(2.56\) + j 0 & \(2.56\) + j 0 \\ AlGaAs & \(10.729\) + j 0 & \(12.583\) + j 0.086 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the permittivities of the materials used in this work. For the dispersion of the ENZ substrate we considered a material modelled with a single Lorentz oscillator:  
\[\varepsilon_{ENZ}(\omega)=\varepsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega^{2}+i \omega\gamma}, \tag{1}\]  
with parameters \(\varepsilon_{\infty}=5.5\), \(\omega_{p}=5.7535\)\(\times\)\(10^{15}\) rad/s and \(\gamma=2.3327\)\(\times\)\(10^{13}\) rad/s. The optimized substrate has the epsilon near zero condition around \(\lambda\) = 775 nm, where the dielectric permittivity assumes a value \(\varepsilon=0.0006+\)i0.05. The dielectric substrate had instead a refractive index \(n_{\mathrm{s}}\) fixed to 1.6 that is Re(\(\varepsilon\)) = 2.56, Im(\(\varepsilon\)) = 0 as listed in Table 1. These values correspond to those of AlO\({}_{\mathrm{x}}\)[6]. They were chosen because of the realization that AlGaAs nanostructures on an AlO\({}_{\mathrm{x}}\) layer are easily obtained in modern manufacturing processes [6]. The optical properties of gold are obtained by the Johnson and [PERSON] interpolation [44]. For completeness, Table 1 summarizes the permittivities of the materials used in this study both at the Fundamental Wavelength (FW) and at Second Harmonic (SH) wavelength. The experimental realization of such kind of structure will probably face various problems. However, a novel procedure for fabricating nanodisks from III\(-\)V nanowires has been recently proposed in [42]. The main advantage of this new approach of slicing nanowires with Focused Ion Beam (FIB) milling is the possibility to create disk-based configurations on any substrate. In fact, the method of FIB nanowire slicing permits one to easily create nanoantenna chains of different designs without relying on the substrate material and to make them compatible with Complementary Metal-Oxide Semiconductor (CMOS) technologies.