Nonlinear nanophotonics is a rapidly developing research field with various applications including nonlinear light sources [

1], ultrafast chip-based optoelectronic devices [

2], nonlinear microscopy and spectroscopy techniques [

3] and bioimaging and sensing [

4]. Exploiting nonlinear optical effects in nanostructures plays an important role in the implementation of miniature nonlinear photonic components for further integration of multiple optical functionalities into a single compact optical chip. To this purpose, nonlinear nanoplasmonics has been widely studied both theoretically and experimentally during the last decade [

5,

6,

7,

8,

9,

10]. However, its performance is restricted by the high ohmic losses, small mode volumes and low laser damage threshold. High-index all-dielectric nanostructures provide a powerful platform for controlling light at the nanoscale [

11,

12]. They offer unique opportunities to boost the nonlinear effects due to the strong near-field enhancement associated with the excitation of Mie-type resonances [

13,

14]. In recent years, various fabrication techniques, including lithography, chemical methods, dewetting, etc., have been developed for the realization of all-dielectric nanostructures, greatly paving the way for low-cost manufacturing of all-dielectric nanostructures [

15,

16,

17]. Via engineering and control over the optically-induced electric and magnetic resonances in all-dielectric nanostructures, both high nonlinear conversion efficiency and directivity of the harmonic radiation pattern can be achieved. Nanostructures made of high-index semiconductors with a strong nonlinear response, such as Si, Ge, which possess a large third-order susceptibility, have been investigated widely for third-harmonic generation, showing huge nonlinear enhancement when exciting the nanostructure in the vicinity of Mie resonances, particularly magnetic dipole (MD) resonance [

18], Fano resonance or collective modes [

19,

20,

21] and anapole states [

22,

23,

24,

25]. In contrast to the third-order nonlinearity, Si and Ge do not possess bulk second-order nonlinearity due to their centrosymmetric crystalline structure [

26]. However, III-V semiconductors, such as GaAs or AlGaAs, show a strong second-order nonlinear response due to their large bulk second-order optical susceptibility

${\chi}^{\left(2\right)}$, representing a great deal of interest because of their strong nonlinear effects and optoelectronic properties [

27]. By engineering the AlGaAs alloy composition, two-photon absorption can be avoided at the telecommunication wavelengths, enabling high transparency in a broad spectral window from visible to far infrared. Both theoretical and experimental studies have been performed to enhance the second-harmonic generation (SHG) process in GaAs or AlGaAs Mie resonators [

28,

29,

30,

31,

32,

33,

34]. The SHG from AlGaAs nanoantennas has been predicted to reach conversion efficiency of

${10}^{-3}$, and later, efficiency of

${10}^{-4}$ was measured experimentally by exploiting the magnetic dipole (MD) resonance [

28,

31]. However, when employing AlGaAs structures, despite the high conversion efficiency, there were no emitted second-harmonic (SH) signal observed in both the forward and backward normal directions. The absence of the emission at normal directions is due to the specific nonlinear susceptibility tensor of [100]-grown zinc-blend AlGaAs crystalline structures, which only contain off-diagonal elements

${\chi}_{ijk}^{\left(2\right)}$ with

$i\ne j\ne k$. It limits the collected nonlinear signal and restricts many photonic applications, e.g., highly-efficient nonlinear light sources and nonlinear spectroscopy.

In this work, we present a theoretical study on the resonant multipolar effects in AlGaAs nanoantennas in both linear and nonlinear responses. Then, we propose and design an asymmetric AlGaAs nanoantenna composed of a nanodisc and an adjacent nanobar. Under normally incident pump, our nanoantenna supports resonant responses at both the fundamental and harmonic wavelengths, enabling a high second-harmonic conversion efficiency of the order of ${10}^{-3}$ at the pump intensity of ${\mathrm{I}}_{0}=1$ $\mathrm{GW}/{\mathrm{cm}}^{2}$ and also normal SH emission due to the the specifically distributed induced nonlinear currents and resonant modes supported by both nanodisc and nanobar. Such highly-efficient longitudinal SH emission has not been realized in [100]-grown AlGaAs nanoantennas to date. Our results may offer new opportunities for the design of new types of novel nonlinear photonic metadevices.