^{(2)}Nonlinear Optics

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Edited by: Stefan Wabnitz, Sapienza University of Rome, Italy

Reviewed by: Venu Gopal Achanta, Tata Institute of Fundamental Research, India; Daria Smirnova, Australian National University, Australia

This article was submitted to Optics and Photonics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Metal-less nanophotonics offers new opportunities for non-linear optics with respect to optical waveguides and microresonators, taking advantage of the progress within nanofabrication to boost the development of subwavelength Mie-type structures. Here, we review recent results on second harmonic generation with semiconductor nanoresonators, focusing on their scattering features in terms of efficiency and control over radiation patterns. First, two theoretical models are comparatively discussed with a view to possible improvements in analysis and design. Then, some relevant experiments are reported, and the origin of the χ^{(2)} generation is discussed, outlining the main open topics to investigate in the near future and the advantages offered by these nanostructures to the development of novel photonic devices.

Light manipulation at the nanoscale has been a central topic in nanophotonics for the last two decades. The control of light–matter interaction through subwavelength-resonating elements has led to the possibility of engineering the optical properties of nanopatterned materials, and, in turn, this has led to peculiar phenomena and opportunities for striking applications, including optical magnetism, negative refraction [

Many of these applications originated from the observation of a magnetic response produced by subwavelength structures. In metallic nanostructures, the response to external electromagnetic excitation arises from surface plasmon polaritons or localized surface plasmons. The resonant behavior of these nanostructures allows for the imprint of an abrupt phase discontinuity to propagating light. Control over optical properties allows for the design of 2D arrays of resonators with a spatially varying phase response and, consequently, the fabrication of metasurfaces for beam shaping [

Nanophotonics has also attracted the interest of the scientific community for intensity-dependent phenomena. Tight field confinement boosts nonlinear generation efficiency in the medium, allowing for the presence of subwavelength nonlinear light sources with amplitude and phase control [

An alternative way to solve this problem is offered by all-dielectric nanostructures. According to Mie theory, high-refractive-index dielectric nanoparticles exhibit strong electric and magnetic resonances, demonstrated for the first time in the visible spectral range with silicon nanoparticles [

The linear resonant behavior of high-refractive-index dielectric nanoparticles is typically obtained via multipolar decomposition of the scattered electromagnetic fields, which consists of the approximation of complex charge distributions with point multipoles [

For nonlinear generation processes in the visible range, high-refractive-index nanoparticles are even more advantageous than their plasmonic counterparts due to (a) the absence of ohmic losses and (b) strong field confinement inside the resonator volume, thus enabling the exploitation of the large bulk nonlinearity of these materials. These features are promising for the integration of flat metasurfaces in optoelectronic devices for efficient parametric generation of light with controlled amplitude, phase, and polarization. In this context, the first relevant results have been obtained with third harmonic generation (THG) in silicon nanodisks. In 2014, Shcherbakov et al. demonstrated that local enhancement of an electromagnetic field in silicon nanodisks, due to the magnetic resonance response in the near IR, gives rise to two-order-of-magnitude enhancement of THG with respect to bulk silicon [

Since second-order bulk nonlinear processes are inhibited in centrosymmetric materials like silicon, the quest for high-χ^{(2)} nonlinear Mie resonators drew the attention on non-centrosymmetric III-V alloys. Among them, promising results have been obtained with aluminum gallium arsenide (AlGaAs) [

This review focuses on three axes of AlGaAs Mie nanoresonators. Firstly, we illustrate two state-of-the-art methods for deriving the linear scattering cross section: multipolar decomposition, and QNM. Secondly, we report on two tools to predict the second harmonic generation (SHG): the first (numerical) consists of finite-elements (FEM) calculations followed by multipolar decomposition, and the second (analytical) relies on Green's functions. Finally, we focus on SHG shaping by highlighting several experimental works.

AlGaAs is a zinc-blend alloy exhibiting a high refractive index in the visible/near-IR range (_{g} ~ 1.42 ^{(2)} ~ 200 _{0.18}Ga_{0.82}As nanocylinders. This alloy composition prevents two-photon absorption in the telecom range while preserving a high refractive index [

AlGaAs nanocylinder scattering efficiency computed with COMSOL for radius r = 225 nm and height h = 400 nm. External excitation is approximated with a plane wave with the k vector parallel to the cylinder axis (see inset). Blue solid line represents total scattering, while dashed line is the expansion on spherical harmonics multipoles: electric and magnetic dipoles (ED and MD, respectively) and quadrupoles (EQ and MQ, respectively). Reprinted with permission from Carletti et al. [

The inset of the same figure reports the electric field distribution inside the resonator at λ = 1640 nm, which testifies to the presence of rotating displacement currents typical of a magnetic dipole. A quantitative understanding of this resonant behavior can be obtained by replacing the complex distribution of charges and currents with an equivalent set of point multipoles generating the same fields [

The lowest-order multipole contributions to this expansion are reported as dashed lines in

In

This is a non-Hermitian eigenvalue problem. QNMs are therefore modes of an open system with a complex eigenfrequency

where _{m} coefficients are known analytically. The scattering spectrum in

The proposed case immediately shows the potential of highlighting constructive or destructive interference among QNMs, thereby enabling a good understanding of the physics of the resonating behavior of nanoantennas. It is not surprising to see negative contributions to the total scattering as this is the footprint of interference between quasinormal modes. It is important to appreciate the change in paradigm from multipole expansion to QNM: in the case of the former, once computed, the scattered field was projected on multipoles distributions; in the latter case, instead, eigenmodes of the structures, i.e., independent of the external source, are analyzed, and then the response to an incident field is retrieved. Furthermore, the electric field distribution of every QNM in the near and far field can be studied separately, which offers a powerful opportunity to apply coupled-mode theory and study mode overlap in the frequency generation processes.

The exhaustive comprehension of linear properties enabled the development of models for nonlinear optical phenomena in nanoantennas. In the following section, we have focused on second harmonic generation in AlGaAs resonators. They can be studied numerically with FEM, reducing the analysis to a two-step scattering problem: (1) an external excitation at the fundamental frequency (FF) ω is imposed, and electric fields in the nanostructure are retrieved; and (2) a second-harmonic (SH) polarization vector is computed as ^{SH}(^{(2)}:[^{SH}(^{SH} (

The above requirements imply that having highly resonant modes at FF and SH is necessary, but this is not enough if their distributions do not spatially overlap with respect to symmetry conditions imposed by the χ^{(2)} tensor. Indeed, it is immediately possible to highlight that, when moving to shorter wavelengths, the number of excited modes increases. An example is summarized in

A different strategy to study the SHG efficiency from high-index dielectric nanoparticles (resumed in

The numerical model treated in the previous section enables the investigation of resonance overlap at FF and SH to optimize the nanoantennas design for enhanced SHG. Although the theoretical analyses reported above refer to air-suspended nanocylinders, two slightly different types of (100) high-contrast AlGaAs nanoantennas on a low refractive index have been studied the most: (a) a monolithic device on an aluminum oxide substrate with _{FF} = 1.55 μ^{−5} could be expected for a nanocylinder pumped with a field intensity of 1 GW/cm^{2} at normal incidence. This numerical prediction was confirmed experimentally. Please notice that no SHG occurs under normal incidence on (100) bulk zinc-blend crystals, other than a contribution from surface χ^{(2)} that is negligible for FF below half bandgap. Furthermore, experimental results for nanocylinders with radii ranging from 175 to 225 nm pumped at telecom wavelengths displayed good agreement with a theoretical model considering just bulk χ^{(2)} contribution. A summary of those result is reported in

The same outcome also holds for polarization properties, as summarized in

Similar results regarding the polarization properties of SHG in AlGaAs nanocylinders were detailed by Camacho-Morales et al. [^{(2)}. The comprehension of single element SHG properties, in terms of polarization and directionality, successfully enabled the development of antenna arrays for the controlled generation of SH polarization states [

A quite different landscape was explored by Liu et al. [

In contrast to the previous cases, the SH polarization pattern could not be exhaustively explained when considering just bulk χ^{(2)} contribution. This is the case particularly when it involves resonances at FF that exhibit strong field intensity close to the resonator surface or when the SH signal is above the energy bandgap, as is displayed in ^{(2)} contribution remains quantitatively challenging due to the unexplored value of this quantity, thereby urging for further and deeper investigations.

Even if these reviewed works confirm three-order-of-magnitude higher SHG efficiency in AlGaAs nanocylinders compared to plasmonic structures [^{(2)} symmetry in [100] grown AlGaAs.

In reference [

A different approach that maintains normal excitation consists of the integration of a dielectric grating around a nanoparticle so as to control the phase of the nonlinear generation [

An alternative solution to the absence of normal SHG is to rotate the AlGaAs crystalline structure, and thus the χ^{(2)} tensor, by fabricating nanoantennas from (111)-GaAs [

Advances in integrated photonics require nonlinear compact devices, like metasurfaces, with high conversion efficiencies and tailorable emission features. After having drawn the attention of the scientific community on the subject of linear applications, nanopatterned dielectric materials have become promising also for nonlinear optics. Here, we reviewed recent progress in high-index dielectric χ^{(2)} nanoresonators, which proved advantageous over their plasmonic counterparts because of lower optical losses and a higher conversion efficiency stemming from to the overlap of cavity modes with a strong bulk χ^{(2)}. Since their multi-resonant behavior enabled the background-free nonlinear displacement current distribution to be tailored locally, a complete understanding of their features is imperative for the SH wavefront design. In this respect, we benchmarked the numerical calculations of the linear scattering cross section against the QNM theory. The latter provides complementary physical insight into Mie-like resonances in complex structures that cannot be studied analytically. We then focused on fully vectorial numerical calculations, which allowed to model nonlinear effects in good agreement with experimental results. In conclusion, we would like to stress that a better understanding of the relative contribution of bulk vs. surface nonlinear susceptibility would significantly help future progress regarding subwavelength frequency conversion devices. Finally, when compared to their guided-wave counterparts, nanoresonators offer a larger number of spatial degrees of freedom for the engineering of nonlinear generation processes. The development of more powerful design tools, combined with the technological advances of the last years, will probably lead to the creation of unprecedented nonlinear photonic devices.

CG and GM wrote the paper draft. GL was the scientific supervisor. All the others have contributed to its evolution to the final form.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The handling editor declared a past co-authorship with one of the authors GL.

The authors thank C. De Angelis, L. Carletti, D. Rocco, and T. Wu for fruitful discussions.

The Supplementary Material for this article can be found online at: