# Boosting Second Harmonic Radiation from AlGaAs Nanoantennas with Epsilon-Near-Zero Materials
## 1 Introduction  
Dielectric nanoantennas have recently gained increasing interest for many applications such as directional scattering and emission, nonlinear spectroscopy and microscopy [1; 2; 3; 4]. Recently, a nanoscale system based on AlGaAs nanodisks placed over an AlOx substrate pumped in the telecom range was theoretically proposed as an efficient system to enhance second-order nonlinear effects in nanoscale optics [5]. The conversion efficiencies reported were higher than \(10^{-5}\)[6; 7; 8], which is more than four orders of magnitude greater than optimized centrosymmetric plasmonic nanoantennas pumped at similar intensities. Indeed, this dramatic enhancement is due the non-centrosymmetric nature of AlGaAs. Moreover, in contrast to plasmonics, resonances of high-index dielectric nanoparticles provide a mode volume that is not limited to interfaces and thus lead to greater nonlinear conversion efficiencies. Several strategies have been employed to enhance harmonic nonlinearities including dielectric nanopillars in a dimer configuration [9] and a metal-dielectric hybrid nanoantenna configuration [10; 11] exploiting the anapole condition. Promising studies on enhanced second harmonic generation (SHG) at the nanoscale have also been performed on perovskite nanoparticles [12]. However, the generated nonlinear signal tends to remain more confined in the substrate since it typically has a greater refractive index than the surrounding air. Recently several authors have discussed some possible strategies to engineer the radiation pattern of nonlinear nano-antennas, using the properties of the fundamental frequency field (i.e., polarization state and propagation direction) as tuning parameters [13; 14; 15]. However, an unexplored possibility to optimize the nonlinear radiation pattern is the use of epsilon near zero (ENZ) substrates. ENZ materials have attracted lots of attention because of their peculiar linear properties, such as their ability to realize perfect electromagnetic tunneling [16; 17] and to re-direct antennas electromagnetic radiation [18; 19]. In particular, it has already been demonstrated that when an antenna is placed on top of an extremely low-permittivity substrate, the air is the \"denser\" medium withhigher relative permittivity and therefore the antenna pattern will be predominantly backscattered into the air [20; 21]. Their ability to be good reflectors and, at the same time, provide high electric field values, makes them also good candidates for energy harvesting purposes [22] and photocatalysis [23]. ENZ conditions, which occur when the real part of the dielectric permittivity crosses the zero, may be found in natural (e.g., low loss noble metals, some semiconductors and some polar dielectrics) and artificial materials (metamaterials) alike, where the latter are more flexible both in terms of the zero-crossing wavelength and effective losses [24; 25; 26; 27]. Another possible path to tune the ENZ wavelength is the use of oxide materials, such as Indium Tin Oxide (ITO), Aluminum Zinc Oxide (AZO) or Dysprosium-doped Cadmium Oxide (Dy:CdO) [28; 29; 30], whose doping determines the spectral position of the ENZ crossing point. ENZ materials have been also shown to be good platforms for nonlinear processes like harmonic generation, optical bistability and soliton excitation [31; 32; 33; 34; 35; 36; 37]. In this framework, ENZ substrates have been also shown to enhance harmonic generation processes when the ENZ mode is coupled with localized surface plasmon resonances [38]. Moreover, a new approach to engineer an optical medium with an unprecedentedly large intensity-dependent refractive index using nanoantennas coupled to a thin ENZ material has been reported [39]. Here we demonstrate how one can exploit the peculiar ability of ENZ substrates to re-direct the electromagnetic radiation to increase the efficiency of nonlinear processes generated in AlGaAs nanoantennas.