# Second-Harmonic Generation in Mie-Resonant GaAs Nanowires

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## Abstract

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## 1. Introduction

^{−5}for second-harmonic (SH) generation in AlGaAs nanocylinders [13,14] and 10

^{−6}for third-harmonic (TH) generation in Si nanospheres [11] have been experimentally demonstrated by tuning the pump signal on the magnetic-dipole resonance. SH generation with record-high efficiency has been predicted in AlGaAs resonators that host bound-in-continuum states [19]. Moreover, the possibility to tailor polarization and radiation patterns has been demonstrated for SH light scattered by AlGaAs nanocylinders [20,21]. The key factor to achieve strong nonlinear interactions in semiconductor nanoparticles is to couple light to Mie scattering modes [22]. Nonlinear effects, such as SH and TH generation, have been so far investigated in isolated and arrayed Si- and GaAs-based resonant nanoparticles with three-dimensional (3D) light confinement, such as spheres and finite cylinders (nanodisks or nanopillars). Here we analyze SH generation in Mie-resonant GaAs nanowires, or infinite cylinders. Mie theory is used to describe the response in the linear regime, which is dominated by the normal modes of the nanowire. Next, an average nonlinear susceptibility, derived by the linear Mie-scattering coefficients and normal modes, is introduced to predict the far-field SH scattering efficiency and finite-element simulations are performed to calculate SH light absorption, to examine near fields, and to test the validity of the average nonlinear susceptibility. Although light is confined in only two dimensions, defined by the scattering plane orthogonal to the nanowires’ axis, we show that, if the nanowires are judiciously designed, conversion efficiencies for SH generation are similar to those achievable in nanoparticles with 3D light confinement. Finally, we discuss the role of absorption losses in SH generation, and demonstrate that: (i) Nanowires produce intense SH light even when the SH photon energy is tuned above the electronic bandgap of GaAs, where the material is opaque and it is commonly assumed that harmonic generation is not efficient, and (ii) absorption of SH light is not negligible even when the SH photon energy is tuned below the electronic bandgap of GaAs.

## 2. Mie Scattering in the Linear Regime and Normal Modes of GaAs Nanowires

## 3. Second-Harmonic Generation: Scattering and Absorption Efficiencies

_{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, ${\mathrm{TM}}_{0}+{\mathrm{TM}}_{0}\to {\mathrm{TE}}_{3}$ 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 ${r}_{NW}=220\text{}\mathrm{nm}.$

_{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 (${\lambda}_{SH}<{\lambda}_{BG}$). The field distributions for pump and SH in this scenario are reported in Figure 6. While the pump takes on the typical shape of a TM

_{1}mode, the SH is coupled to a superposition of modes, with the stronger components being the TE

_{1}and the TE

_{2}, i.e., the modes associated with the ${b}_{1}$ an ${b}_{2}$ Mie coefficients, respectively.

## 4. Absorption of SH Light

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**(

**a**) SH scattering problem, with a volume current distribution ${J}_{NL}$ inside the nanowire radiating into the background medium and inducing a field ${E}_{SH}$ on a detector located in a point in the far-field. (

**b**) Reciprocal scattering problem, with a radiating line current source located at the detector point that induces a field ${E}_{1}^{TE}$ inside the nanowire. (

**c**) The equivalent of (

**b**) with a plane wave source instead of a line current (the approximation is valid if ${k}_{SH}R\gg 1$).

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**Figure 1.**(

**a**) Real and imaginary parts of the GaAs relative permittivity, ${\u03f5}_{GaAs}$, in the wavelength range of interest. (

**b**) Real and imaginary parts, and amplitude, of the GaAs second-order nonlinear susceptibility, ${\chi}^{\left(2\right)}$ for second-harmonic (SH) generation. The $x$-axis in (

**b**) refers to pump wavelength.

**Figure 2.**Field distribution in the scattering plane ($xy$) associated with the first four normal modes with TE and TM polarization.

**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 (~870 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.

**Figure 6.**Pump and SH field distribution for a GaAs nanowire with radius 268 nm illuminated with TM-polarized pump source tuned at 1550 nm.

**Figure 7.**(

**a**) Same as Figure 4b. (

**b**) SH absorption ${A}_{SH}$, as a function of pump/SH wavelength and nanowire size. The dark-shaded portion of the SH wavelength axis indicates the opaque region for both pump and SH light (${\lambda}_{p}\le {\lambda}_{BG}$); the medium-shaded portion indicates a region that is transparent for the pump but absorptive for the SH (${\lambda}_{SH}<{\lambda}_{BG}<{\lambda}_{p}$); the light-shaded portion indicates the transparent range (${\lambda}_{SH}\ge {\lambda}_{BG}$).

**Figure 8.**SH scattering and absorption efficiency for a GaAs nanowire with ${r}_{NW}=220\text{}\mathrm{nm}$ with an input pump intensity ${I}_{p}=2\text{}\mathrm{GW}/{\mathrm{cm}}^{2}$. The three background colors indicate the three regions described in the text with the same color code of Figure 7.

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**MDPI and ACS Style**

de Ceglia, D.; Carletti, L.; Vincenti, M.A.; De Angelis, C.; Scalora, M.
Second-Harmonic Generation in Mie-Resonant GaAs Nanowires. *Appl. Sci.* **2019**, *9*, 3381.
https://doi.org/10.3390/app9163381

**AMA Style**

de Ceglia D, Carletti L, Vincenti MA, De Angelis C, Scalora M.
Second-Harmonic Generation in Mie-Resonant GaAs Nanowires. *Applied Sciences*. 2019; 9(16):3381.
https://doi.org/10.3390/app9163381

**Chicago/Turabian Style**

de Ceglia, Domenico, Luca Carletti, Maria Antonietta Vincenti, Costantino De Angelis, and Michael Scalora.
2019. "Second-Harmonic Generation in Mie-Resonant GaAs Nanowires" *Applied Sciences* 9, no. 16: 3381.
https://doi.org/10.3390/app9163381