# Second-Harmonic Generation in Mie-Resonant GaAs Nanowires
## 3 Second-Harmonic Generation: Scattering and Absorption Efficiencies
The differences in the calculated \(Q_{SH}\) values with respect to the finite-element numerical simulations (COMSOL) are on the order of 1%, and they are mainly due to the order of truncation of cylindrical harmonics in the fields' expansions. The maxima of scattering coefficients are plotted in Figure 4 for a large range of nanowire radii next to the SH generation efficiency \(Q_{SH}\) map. It is clear that the peaks in the SH scattering spectrum are associated with the excitation of nanowire modes at the pump or at the SH wavelengths, or at both wavelengths if the system is doubly-resonant. In Figure 4, we have indicated with circles several SH peaks. In one case--see the black circle--the radius is equal to 220 nm and the pump wavelength is equal to \(\sim\)1933 nm, therefore the SH is generated at 966.5 nm, in the transparency region of GaAs (\(\lambda>\lambda_{BG}\)). In this scenario, the nanowire is doubly resonant for both the pump and the SH field, and this condition only occurs in a very narrow band. In particular, the pump plane wave is exciting the TM\({}_{0}\) mode, which is related to a broadband peak of the Mie coefficient \(a_{0}\), whereas the SH is resonant with the mode TE\({}_{3}\), which is associated with a narrowband peak of the Mie coefficient \(b_{3}\). The result of this doubly-resonant, TM\({}_{0}+\) TM\({}_{0}\rightarrow\) TE\({}_{3}\) interaction is that SH light is scattered with high efficiency, larger than \(10^{-5}\). In Figure 5, we plot the field distribution of the pump and SH on this peak, when the nanowire radius is \(r_{NW}=220\) nm. Figure 4: (**a**) Maximum peaks of the Mie scattering coefficients monitored as a function of nanowire radius. For TM-scattering coefficients (\(a_{n}\)), the wavelength axis is on the bottom (pump wavelength); for TE-scattering coefficients (\(b_{n}\)), the wavelength axis in on the top (SH wavelength). (**b**) Color map representation of SH scattering efficiency as a function of wavelength and nanowire size. The circles in (**a,b**) highlight scenarios in which either the pump or the SH, or both pump and SH, are coupled to normal modes of the nanowire. Figure 5: Pump and SH field distribution for a GaAs nanowire with radius 220 nm illuminated with TM-polarized pump source tuned at 1933 nm. If the pump is slightly detuned from this double resonance condition, then the structure turns out to be singly-resonant with the only TM\({}_{0}\) mode resonantly excited by the pump and a 4-fold reduction of conversion efficiency--the dotted black circle belongs to this singly-resonant region. This explains the spectral behavior of the SH efficiency near this region, with a sharp peak (due to SH coupling to the TE\({}_{3}\) mode) emerging from a shallow peak (due to pump coupling to the TM\({}_{0}\) mode). Owing to a combination of factors, including the chromatic dispersion of GaAs as well as the dispersion of the scattering coefficients, the double resonance condition persists over a wide range of nanowire sizes, form radii smaller than 200 nm to radii larger than 300 nm. Other peaks, indicated in Figure 4 with circles of different colors, are due to mode coupling of either the pump or SH to normal modes of the nanowire. It is remarkable that the conversion efficiency remains at similar levels, between \(10^{-6}\) and \(10^{-5}\), even in cases in which the SH wavelength, \(\lambda_{SH}\), is smaller than \(\lambda_{BG}\) and therefore absorption losses are expected to inhibit up-conversion. In particular, on the peak indicated with a red circle--corresponding to a nanowire with radius 268 nm and pumped at 1550 nm--the efficiency is as high as in the doubly resonant case, even though only the pump signal is here resonantly coupled to the TM\({}_{1}\) mode and the SH is tuned above the bandgap (\(\lambda_{SH}<\lambda_{BG}\)).