# Second-Harmonic Generation in Mie-Resonant GaAs Nanowires
## 2 Mie Scattering in the Linear Regime and Normal Modes of GaAs Nanowires
TE light scattering only depends on the \(b_{n}\) (\(c_{n}\) inside the nanowire) coefficients, while TM scattering is only ruled by the \(a_{n}\) (\(d_{n}\) inside the nanowire) coefficients. Although the expansions in Equations (2)-(5) are valid for plane waves, it is possible to expand the fields with the same Mie coefficients in the presence of any kind of source, including SH sources induced in the volume and on the surface of the nanowire. In other words, TE(TM) scattered fields are always equal to the superposition of normal modes of an increasing order number, each weighted by the corresponding coefficient \(b_{n}\) (\(a_{n}\)). The normal modes of an open cavity must radiate into the far-field [28], and therefore it is common to associate them with cartesian multipoles [29; 30]. For example, the Mie coefficient \(a_{0}\) is related to a magnetic-dipole moment (or electric quadrupole [29; 31]) and the associated electric field is of type \(a_{0}H_{0}^{\prime}(kr)\hat{\phi}\); the coefficient \(a_{1}\) is associated with a superposition of an electric dipole moment and a combination of electric-octupole moments. Higher-order coefficients \(a_{n}\), with \(n>1\), are related to higher-order multipoles. TE-polarized normal modes are instead weighted by the \(b_{n}\) coefficient and they are z-polarized. The coefficient \(b_{0}\) is related to a TE mode with electric field distribution associated with an electric dipole, therefore of type \(a_{0}H_{0}(r)\hat{\varepsilon}\). Similarly, in the TM case, coefficients \(b_{n}\) with \(n\geq 1\) are associated with higher-order modes. In Figure 2, we summarize the electric and magnetic field distributions associated with the first four TE- and TM-polarized modes with the corresponding Mie coefficients. In Figure 3, we report scattering and absorption linear efficiencies, calculated with Equations (2)-(5), in response to either TE or TM polarized plane waves impinging on the nanowire. The scattering peaks are associated with the maxima of the Mie scattering coefficients, therefore, they correspond to maxima of either \(a_{n}\) for the TM case or \(b_{n}\) for the TE case--the maxima occur when the denominators of the scattering coefficients, \(d_{n}\) and \(a_{n}\) for TM polarization and \(c_{n}\) and \(b_{n}\) for TE polarization approach zero. The Mie coefficients peaks overlapped with the scattering and absorption maxima are reported in \[Q_{sat}^{TM}=\frac{2}{x}\bigg{(}|a_{0}|^{2}+2\sum_{1}^{\infty}|a_{n}|^{2} \bigg{)} \tag{3}\]6 On each maximum of a Mie coefficient, light is efficiently coupled to the corresponding normal mode and intensity maxima of internal near-fields and scattered far-fields are observable. For this reason, maxima of scattering and absorption are virtually overlapped. The abrupt increase of absorption cross-section above the bandgap (\(\lambda<\lambda_{\text{BG}}\)) is due to the electronic transitions of GaAs. Although quenched by absorption, resonances associated with higher-order modes appear at shorter wavelengths within the visible range. It is worth noticing that the response of the nanowire is dominated by four normal modes, namely the order \(0,\ \pm 1,\ \pm 2\), and \(\pm 3\), across the whole range of nanowire radii considered, from 50 to 300 nm, and across a broad wavelength range that covers visible and near-infrared.