# Second-Harmonic Generation in Mie-Resonant GaAs Nanowires
## 4 Absorption of SH Light  
The usual metrics for SH conversion efficiency involves the SH light that one may collect in a certain direction in the far-field region, or the total light scattered \(Q_{SH}\) in all directions in the far-field, as defined in Equation (16). However, a portion of SH energy is inexorably dissipated in the near field via one-photon absorption. The rate of this SH absorption can be calculated as follows:  
\[A_{SH}=\frac{\omega e_{0}/2}{2r_{NW}l_{inc}}\mathrm{Im}(\epsilon_{GaAs})\int_{ NW}\big{|}\mathbf{E}_{1}^{T}\big{|}^{2}dS, \tag{20}\]  
\(A_{SH}\) can be significant in enhanced nonlinear interactions at the nanoscale. For example, in plasmonic resonators, in addition to the pump energy lost as heat in the metal, reabsorption of SH light is orders of magnitude larger than far-field scattered SH light [32; 33; 34]. In other words, \(A_{SH}>>Q_{SH}\) in many nanoplasmonic systems designed to resonate at the pump and/or at the harmonic frequency.  
Figure 6: Pump and SH field distribution for a GaAs nanowire with radius 268 nm illuminated with TM-polarized pump source tuned at 1550 nm.  
Dielectric resonators are considered immune to pump absorption if illuminated in the transparency region below the bandgap (i.e., if the pump wavelength \(\lambda_{P}\) is larger than \(\lambda_{BG}\)), although Mie theory predicts that light absorption on the Mie resonances above the bandgap can reach values as high as 5% of the total light extinction for GaAs nanowires, as one can infer by comparing absorption maps and scattering maps in Figure 3. In addition, a study of SH light absorption in dielectric nanoantennas (isolated or in metasurfaces) has not been reported so far, taking for granted that most of the SH light is scattered in the far-field by dielectric resonators, and not absorbed in the near field. Here we demonstrate that this is not the case, at least for nanowires with 2D light-confinement. In Figure 7, we report a side-by-side comparison of SH scattering efficiency spectra, \(Q_{SH}\), and SH absorption spectra, \(A_{SH}\), as functions of the nanowire radii.  
SH scattering peaks are associated with SH absorption peaks, as in other resonant photonic structure, including plasmonic nanoantennas and gratings. \(A_{SH}\) is non-negligible with respect to \(Q_{SH}\). Indeed, \(A_{SH}\) is on the same order or even larger than \(Q_{SH}\) for most resonant peaks, including those peaks that entirely fall in the transparency region of GaAs (\(\lambda_{SH}\geq\lambda_{BG}\)). These facts are summarized in Figure 8, where the SH scattering and absorption spectra are plotted for a nanowire radius \(r_{NW}=220\) nm illuminated with a TM polarized pump plane wave with intensity \(I_{p}=2\) GW/cm\({}^{2}\). In the transparent region (\(\lambda_{SH}\geq\lambda_{BG}\)), in which GaAs is a poor absorber at both pump and SH wavelengths, scattering and absorption of SH light show similar efficiencies on the order of \(10^{-5}\) on the doubly-resonant peak at \(\lambda_{p}\simeq 1.933\) um. In the intermediate wavelength range in which GaAs is mostly transparent for the pump but absorbing for the SH (\(\lambda_{SH}<\lambda_{BG}<\lambda_{p}\)), SH absorption gets somewhat larger than SH scattering but the scattering efficiency reaches almost the same level that one has in the fully transparent region (see, e.g., the peak at \(\lambda_{p}\simeq 1.29\) um). In the opaque range (\(\lambda_{p}\leq\lambda_{BG}\)), SH absorption dominates over scattering by at least one order of magnitude.  
Figure 7: **(a)** Same as Figure 4b. **(b)** SH absorption \(A_{SH}\), as a function of pump/SH wavelength and nanowire size. The dark-shaded portion of the SH wavelength axis indicates the opaque region for both pump and SH light (\(\lambda_{p}\leq\lambda_{BG}\)); the medium-shaded portion indicates a region that is transparent for the pump but absorptive for the SH (\(\lambda_{SH}<\lambda_{BG}<\lambda_{p}\)); the light-shaded portion indicates the transparent range (\(\lambda_{SH}\geq\lambda_{BG}\)).