# Second-Harmonic Generation in Mie-Resonant GaAs Nanowires
## 1 Introduction  
Optical and electronic properties of semiconductors such as Si and GaAs are suitable for the development of highly-efficient nonlinear and tunable nanophotonic devices. Semiconductors possess a high refractive index, which is a fundamental requirement to slow-down light and to increase light confinement and field-enhancement at the nano- or sub-wavelength-scale. In addition, semiconductors have low losses at telecom wavelengths, they are chemically and electrically tunable, and they show a strong nonlinear optical response from visible to far-infrared wavelengths. The bulk quadratic nonlinear response is very strong in anisotropic crystals such as GaAs [1], in which a \(\chi^{(2)}\) effect is also present at the surface due to crystal-symmetry breaking and electric-field discontinuity [2]. On the other hand, in isotropic crystals such as Si, \(\chi^{(2)}\) is only strong on the surface. Cubic nonlinearities are very large both in the bulk of Si and GaAs. Enhanced nonlinear optical effects have been predicted and experimentally observed in a variety of nanoscale structures based on semiconductors: At the band-edge of photonic crystals [3; 4; 5; 6; 7], in leaky-mode-resonant gratings or photonic-crystal slabs [8; 9; 10], and more recently in Mie-resonant nanoantennas [11; 12] and metasurfaces [13; 14; 15; 16]. Besides the large nonlinearity, the fabrication processes of nanoscale semiconductor devices are mature and reliable, and in some cases compatible with existing technologies for photonic integrated circuits, e.g., silicon photonics. The recent popularity gained by semiconductors in nonlinear nanophotonics is also due to the modest amount of absorption losses at telecom wavelengths, at photon energies below the electronic bandgap. Indeed, nanostructures based on semiconductors offer an all-dielectric solutionfor the design of nanophotonic devices with enhanced light-matter interactions [17], and they are considered valid alternatives to plasmonic-based systems [18], especially for nonlinear optics which require high light-intensities and, in particular, for applications in the visible and near-infrared, where metals absorb light more efficiently. It has been shown that the combination of high material nonlinearity, low absorption losses and the presence of Mie resonances in semiconductor nanoparticles creates ideal conditions to induce highly-efficient nonlinear interactions. Conversion efficiencies as high as \(10^{-5}\) for second-harmonic (SH) generation in AlGaAs nanocylinders [13; 14] and \(10^{-6}\) for third-harmonic (TH) generation in Si nanospheres [11] have been experimentally demonstrated by tuning the pump signal on the magnetic-dipole resonance. SH generation with record-high efficiency has been predicted in AlGaAs resonators that host bound-in-continuum states [19]. Moreover, the possibility to tailor polarization and radiation patterns has been demonstrated for SH light scattered by AlGaAs nanocylinders [20; 21]. The key factor to achieve strong nonlinear interactions in semiconductor nanoparticles is to couple light to Mie scattering modes [22]. Nonlinear effects, such as SH and TH generation, have been so far investigated in isolated and arrayed Si- and GaAs-based resonant nanoparticles with three-dimensional (3D) light confinement, such as spheres and finite cylinders (nanodisks or nanopillars). Here we analyze SH generation in Mie-resonant GaAs nanowires, or infinite cylinders. Mie theory is used to describe the response in the linear regime, which is dominated by the normal modes of the nanowire. Next, an average nonlinear susceptibility, derived by the linear Mie-scattering coefficients and normal modes, is introduced to predict the far-field SH scattering efficiency and finite-element simulations are performed to calculate SH light absorption, to examine near fields, and to test the validity of the average nonlinear susceptibility.