# Second-Harmonic Generation in Mie-Resonant GaAs Nanowires
## 2 Mie Scattering in the Linear Regime and Normal Modes of GaAs Nanowires  
The nanowire geometry and the electromagnetic fields have translational invariance in the direction of the nanowire axis (\(z\) axis in our coordinate system) and therefore we analyze linear and nonlinear optical scattering only in the \(xy\) plane. Light is incident as a plane wave with either TM polarization (electric field parallel to the scattering \(xy\)-plane) or TE polarization (electric field perpendicular to the scattering plane). We assume that the principal axes of the crystal are oriented as follows: \(\hat{x}\parallel[100],\ \hat{y}\parallel[010]\), and \(\hat{z}\parallel[001]\). The second-order nonlinear tensor contains only elements of type \(\chi_{ijk}^{(2)}=\chi^{(2)}\) with \(i\
eq j\
eq k\), so that the nonlinear current induced at the SH frequency can be written as:  
\[\left( \begin{array}{c}J_{x}^{NL}\\ J_{y}^{NL}\\ J_{z}^{NL}\end{array} \right)=-i2\omega_{p}e_{0}\chi^{(2)}\left( \begin{array}[ ]{c}E_{y}^{p}E_{z}^{p}\\ E_{x}^{p}E_{y}^{p}\\ E_{x}^{p}E_{y}^{p}\end{array} \right), \tag{1}\]  
where \(\omega_{p}\) is the pump angular frequency, \(E_{x,y,z}^{p}\) indicate the pump electric-field components, and \(J_{x,y,z}^{NL}\) indicate the SH induced current components. Both the relative permittivity of GaAs, \(e_{GaAs}\), and the nonlinear susceptibility \(\chi^{(2)}\) are weakly dispersive at infrared wavelengths, in particular for photon energies smaller than the GaAs energy bandgap, \(E_{g}=h\frac{c}{\lambda_{BG}}=1.42\) eV, where \(\lambda_{BG}\simeq 870\) nm. The frequency-dependence becomes significant at shorter wavelengths (\(\lambda\leq\lambda_{BG}\)), where interband transitions play a non-trivial role in the spectra of SH light scattering and absorption. In Figure 1, we plot the real and imaginary parts of \(e_{GaAs}\) and \(\chi^{(2)}\) that we use for calculations of linear and nonlinear scattering and absorption efficiencies. Linear and nonlinear optical data are taken from [23; 24; 25; 26],respectively. Both models account for the electronic band structure of GaAs, and therefore they include the \(E_{0},E_{0}+\Delta_{0},\ E_{1},E_{1}+\Delta_{1},\ E_{0}^{\prime}\), and \(E_{2}\) transitions of GaAs. Maps of scattering and absorption efficiency spectra for plane wave excitation are reported in Figure 2 as a function of the nanowire radius. The efficiencies are calculated in the linear regime, i.e., assuming the input light intensity induces negligible nonlinear polarization sources in GaAs. Under these circumstances the problem can be approached by using Mie theory and the scattering and absorption efficiencies can be evaluated as normalized cross-sections. The normalization factor is the power (per unit length) flowing through the geometric cross-section, \(2r_{NN}l_{inc}\), where \(r_{NN}\) is the nanowire radius and \(I_{inc}\) is the input light intensity [27]. In other words, scattering efficiencies, \(Q_{sat}^{TE,TM}\), and absorption cross sections, \(Q_{abs}^{TE,TM}\), are evaluated, for TE- and TM-polarized incident plane waves, as follows:  
\[Q_{sat}^{TE}=\frac{2}{x}\bigg{(}|b_{0}|^{2}+2\sum_{1}^{\infty}|b_{n}|^{2} \bigg{)} \tag{2}\]  
\[Q_{sat}^{TM}=\frac{2}{x}\bigg{(}|a_{0}|^{2}+2\sum_{1}^{\infty}|a_{n}|^{2} \bigg{)} \tag{3}\]  
\[Q_{abs}^{TE}=\frac{2}{x}\text{Re}\bigg{(}b_{0}+2\sum_{1}^{\infty}b_{n}\bigg{)} -\frac{2}{x}\bigg{(}|b_{0}|^{2}+2\sum_{1}^{\infty}|b_{n}|^{2}\bigg{)} \tag{4}\]  
\[Q_{abs}^{TM}=\frac{2}{x}\text{Re}\bigg{(}a_{0}+2\sum_{1}^{\infty}a_{n}\bigg{)} -\frac{2}{x}\bigg{(}|a_{0}|^{2}+2\sum_{1}^{\infty}|a_{n}|^{2}\bigg{)} \tag{5}\]  
where \(x=k_{NN}\) and \(k\) is the wavenumber in the background medium.