# Second-Harmonic Generation in Mie-Resonant GaAs Nanowires
## 3 Second-Harmonic Generation: Scattering and Absorption Efficiencies  
We consider an input pump with TM polarization and intensity \(I_{p}=2\,\mathrm{GW}/\,\mathrm{cm}^{2}\). The second-order bulk response due to \(\chi^{(2)}_{xyz}\) will induce a TE-polarized SH signal. The normal modes available in the spectrum play a key role in determining the far-field emission efficiency and radiation pattern for SH light. We have investigated far-field SH radiation by varying the pump wavelength across the whole spectrum shown in Figure 3. The total SH scattered efficiency is calculated in two ways: (i) By introducing an effective nonlinear susceptibility retrieved from the linear response at both the pump and SH wavelengths; (ii) with a finite-element solver (COMSOL). The SH scattering efficiency is defined as  
\[Q_{\mathrm{SH}}=\frac{\oint\mathbf{S}_{SH}\cdot\hat{n}d\ell}{2r_{NW}I_{p}}, \tag{16}\]  
where \(\mathbf{S}_{SH}\) is the Poynting vector at the SH wavelength in the far-field zone, and the integral is performed along a closed line surrounding the nanowire in the far-field with \(\hat{n}\) being the unit vector normal to the line and pointing in the outward direction. Neglecting pump depletion and third-order nonlinear effects, it is straightforward to prove that the SH scattering efficiency shows the usual linear dependence on the pump intensity \(I_{p}\) and a simple dependence on an _average nonlinear susceptibility_, \(\chi^{(2)}_{xy}\), as follows:  
Figure 3: Scattering (**a,b**) and absorption (**c,d**) efficiency spectra for TE-polarized incident light (**a,c**) and TM-polarized incident light (**b,d**) for GaAs nanowires with variable radius, \(r_{NW}\). The white dashed lines in (**a,b**) follow the maxima of the scattering coefficients. The white dotted lines indicate the electronic-bandgap wavelength of GaAs (\(\sim\)870 nm). \[Q_{SH}=\frac{\eta_{b}}{4\epsilon_{b}^{2}}\pi^{2}(k_{SH}r_{NW})^{3}\Big{|}\chi_{av} ^{(2)}\Big{|}^{2}I_{p} \tag{17}\]  
where \(\eta_{b}\) is the background impedance, \(k_{SH}=2k=\frac{2\omega_{p}}{c}\sqrt{\epsilon_{b}}\) is the SH wavenumber in the background medium and the average susceptibility is calculated as the average of \(\left|\chi_{eff}^{(2)}(\phi)\right|^{2}\), across all the possible SH scattering angles between 0 and \(2\pi\) in the \(xy\) plane, i.e.,  
\[\left|\chi_{av}^{(2)}\right|^{2}=\left<\left|\chi_{eff}^{(2)}(\phi)\right|^{2} \right>=\frac{\int_{0}^{2\pi}\left|\chi_{eff}^{(2)}(\phi^{\prime})\right|^{2} d\phi^{\prime}}{2\pi}. \tag{18}\]  
The derivation of Equation (17) is based on the Lorentz reciprocity theorem, and it is reported in the Appendix A section. The angle-dependent, effective nonlinear coefficient, \(\chi_{eff}^{(2)}(\phi^{\prime})\), is related to the field enhancement at both pump and SH and to the overlap integral in the volume of the nanowire (\(NW\)):  
\[\chi_{eff}^{(2)}(\phi^{\prime})=\frac{\int_{NW}\chi_{xyz}^{(2)}\xi_{x}\xi_{y} \xi_{z}(\phi^{\prime})dS}{\pi r_{NW}^{2}}. \tag{19}\]  
In Equation (19), \(\xi_{x}\) is the \(E_{x}\) field distribution in the nanowire under TM plane wave excitation at the pump frequency with amplitude \(E_{inc}=1\) V/m and incident at \(\varphi=0\), \(\xi_{y}\) is the \(E_{y}\) field distribution at the pump frequency under TM plane wave excitation with amplitude \(E_{inc}=1\) V/m and incident at \(\varphi=0\), and \(\xi_{z}(\phi^{\prime})\) is the \(E_{z}\) field under TE plane wave excitation at the SH frequency with amplitude \(E_{inc}=1\) V/m and incident at \(\varphi=\phi^{\prime}\). The fields in Equation (19) are calculated in the linear regime and therefore they are evaluated by using the Mie-theory expressions presented in Equations (12)-(13). We have verified that truncating the cylindrical expansion of the fields in order to include only four normal modes in Equation (19) is enough to predict with good accuracy the SH scattering efficiency using the expression in Equation (17).