# Second-Harmonic Generation in Mie-Resonant GaAs Nanowires
## 2 Mie Scattering in the Linear Regime and Normal Modes of GaAs Nanowires
In our simple geometry, with a cylindrical nanowire, the scattering coefficients for TE-polarized light scattering, i.e., \(b_{n}\), and TM-polarized light scattering, i.e., \(a_{n}\), retrieved by imposing continuity of the tangential fields at the nanowire surface, are written as follows [27]:  
\[a_{n}=\frac{mJ_{n}^{\prime}(x)J_{n}(mx)-J_{n}^{\prime}(mx)J_{n}(x)}{mH_{n}^{ \prime}(x)J_{n}(mx)-J_{n}^{\prime}(mx)H_{n}(x)}, \tag{6}\]  
\[b_{n}=\frac{J_{n}^{\prime}(x)J_{n}(mx)-mJ_{n}^{\prime}(mx)J_{n}(x)}{H_{n}^{ \prime}(x)J_{n}(mx)-mJ_{n}^{\prime}(mx)H_{n}(x)}, \tag{7}\]  
where \(m=\frac{\sqrt{G_{\text{GaAs}}}}{\sqrt{E_{0}}}\) is the nanowire index contrast, \(J_{n}\) are the Bessel functions of first kind and order \(n\) and \(J_{n}^{\prime}\) their derivatives, whereas \(H_{n}\) are the Hankel functions of first kind and \(H_{n}^{\prime}\) their derivatives. Figure 1: (**a**) Real and imaginary parts of the GaAs relative permittivity, \(\epsilon_{GaAs}\), in the wavelength range of interest. (**b**) Real and imaginary parts, and amplitude, of the GaAs second-order nonlinear susceptibility, \(\chi^{(2)}\) for second-harmonic (SH) generation. The \(x\)-axis in (**b**) refers to pump wavelength. It is worth mentioning that TE and TM problems are fully decoupled in our \(z\)-invariant structure with scattering only occurring in the \(xy\)-plane. Incident, scattered and internal fields are superpositions of vector cylindrical harmonics, namely superpositions of Hankel or Bessel functions. These harmonics are the normal modes of the particle. In particular, for an incident TM-polarized plane wave with electric-field amplitude \(E_{inc}\), the scattered field outside the nanowire (\(r>r_{NW}\)) can be written as follows:  
\[\mathbf{E_{S}^{TM}}(r,\phi)=E_{inc}\sum_{n=-\infty}^{\infty}(-i)^{n}i\mathbf{a}_{n} \Big{[}\frac{in}{kr}H_{n}(kr)\hat{\rho}-H_{n}^{\prime}(kr)\hat{\phi}\Big{]}e^{in \phi}, \tag{8}\]  
\[\mathbf{H_{S}^{TM}}(r,\phi)=\frac{-iE_{inc}}{\eta}\sum_{n=-\infty}^{\infty}(-i)^{n }i\mathbf{a}_{n}H_{n}(kr)\mathbf{\varepsilon}e^{in\phi}, \tag{9}\]  
where \(\eta=\eta_{0}/\sqrt{\epsilon_{b}}\) is the impedance of the background medium and \(\eta_{0}\) the impedance in vacuo, and \(\hat{r}\bot\hat{\phi}\bot\hat{z}\) are the unit vectors of a cylindrical coordinate system in which the axial coordinate, \(z\), is aligned with the nanowire axis. By duality, for a TE-polarized plane wave the scattered field outside the nanowire is as follows:  
\[\mathbf{E_{S}^{TE}}(r,\phi)=-E_{inc}\sum_{n=-\infty}^{\infty}(-i)^{n}b_{n}H_{n}( kr)\mathbf{\varepsilon}e^{in\phi} \tag{10}\]  
\[Q_{sat}^{TM}=\frac{2}{x}\bigg{(}|a_{0}|^{2}+2\sum_{1}^{\infty}|a_{n}|^{2} \bigg{)} \tag{3}\]0  
\[Q_{sat}^{TM}=\frac{2}{x}\bigg{(}|a_{0}|^{2}+2\sum_{1}^{\infty}|a_{n}|^{2} \bigg{)} \tag{3}\]1 Field distribution in the scattering plane (\(xy\)) associated with the first four normal modes with TE and TM polarization. The vector cylindrical harmonics inside the nanowire can be evaluated by substitution of \(H_{n}(kr)\) and \(H_{n}^{\prime}(kr)\) with \(J_{n}(mkr)\) and \(J_{n}^{\prime}(mkr)\), respectively. For example, the TM and TE-polarized electric fields inside the nanowire can be calculated as follows:  
\[Q_{sat}^{TM}=\frac{2}{x}\bigg{(}|a_{0}|^{2}+2\sum_{1}^{\infty}|a_{n}|^{2} \bigg{)} \tag{3}\]2  
\[Q_{sat}^{TM}=\frac{2}{x}\bigg{(}|a_{0}|^{2}+2\sum_{1}^{\infty}|a_{n}|^{2} \bigg{)} \tag{3}\]3  
with internal scattering coefficients:  
\[Q_{sat}^{TM}=\frac{2}{x}\bigg{(}|a_{0}|^{2}+2\sum_{1}^{\infty}|a_{n}|^{2} \bigg{)} \tag{3}\]4  
\[Q_{sat}^{TM}=\frac{2}{x}\bigg{(}|a_{0}|^{2}+2\sum_{1}^{\infty}|a_{n}|^{2} \bigg{)} \tag{3}\]5  
It is clear that electric (magnetic) fields in TM polarization have the same shape and symmetry of magnetic (electric) fields in TE polarization, with the only difference being the scattering coefficients. Figure 2 summarizes this concept.