## Abstract

Magnetic dipole resonances supported by high refractive index dielectric nanoparticles have appeared recently as a new alternative to plasmonic nanostructures to manipulate light–matter interactions. The volume field enhancement within the particles significantly boosts some nonlinear processes, mitigating the need for conventional phase-matching conditions. In this paper, we show that by using an AlGaAs nanocylinder array whose lattice resonance spectrally matches the magnetic resonance of individual particles, the two resonances will couple with each other, leading to their strong hybridization and helping achieve a stronger circulation of the electric fields within the particles. As a result of this mode hybridization, at a low pump laser power intensity of ${0.{053\;\;{\rm GW/cm}}^{2}}$, the efficiency of second-harmonic generation can be further enhanced by more than 2 orders of magnitude compared to that from an individual nanocylinder.

© 2020 Optical Society of America

## 1. INTRODUCTION

Dielectric nanoparticles made from high refractive index materials support the electric and magnetic types of multipolar Mie resonances, at which both the far-field scattering properties and near-field characteristics are substantially modified. The uses of Mie resonances have represented a new means of steering light–matter interactions [1], and the exploration of them have received widespread attention recently in many new applications, including, e.g., Huygens’ type of dielectric metasurface for wavefront manipulation with high transmission efficiency [2,3] and enhanced nonlinear optical processes by the magnetic response [4]. In particular, the use of Mie resonance to enhance nonlinear effects and mitigate the phase-matching conditions in conventional nonlinear optics has been of intensive interest in recent years. Due to the low nonlinear susceptibility of most materials, optically thick materials are necessary to achieve considerable nonlinear generation. As a result, the nonlinear signals generated at different positions along the propagation of the pump laser within the materials should be in phase so that the overall output will be constructively added. This requirement, which is also called the phase-matching condition, plays a stringent role and is the bottleneck for nonlinear optics at the nanoscale. Compared to the plasmonic counterpart of metallic nanostructures [5], these all-dielectric nanoantennas have many advantages in terms of applications in nonlinear optics. Although the maximum local field enhancement in dielectric nanoantennas is typically not as high as those in the hotspots in plasmonic antennas, the enhancement happens within the volume of the dielectric particles, and the nonlinear process happens at a longer interaction scale compared to the plasmonic case, where the field enhancement is only at the metal interface [6]. Next, it is usually some semiconductors that are used as high refractive index materials to support the Mie resonances. Compared to the metallic nanoantennas, these semiconductors usually have low dissipative loss, leading to a higher pump laser damage threshold. Most semiconductors possess higher nonlinear efficient than noble metals, and a substantial higher nonlinear conversion efficiency can be achieved for the same pump power. Considering these effects, all-dielectric nanoantennas have been widely investigated in the past several years as the top choice for resonant optical structures to free the nonlinear optics from the limitation of phase-matching conditions [7]. To date, many nonlinear optical phenomena with enhanced efficiency based on the Mie resonances supported by dielectric nanoantennas have been investigated, including the second-harmonic generation (SHG) [8], third-harmonic generation (THG) [9], and even higher order of frequency mixing [10]. Nonlinear conversion efficiently enhanced by several orders of magnitude compared to those from the bare semiconductor film with the same thickness has been achieved.

Among the mainly used semiconductors, GaAs/AlGaAs has been well investigated for the application of SHG due to its superior material properties, including the high second-order nonlinear susceptibility among regular semiconductor materials, and the ease of integrating a high-quality crystalline film on a low-index substrate to achieve a high refractive index contrast [11]. Carletti *et al*. first numerically investigated the SHG enhanced by the magnetic resonance of individual AlGaAs cylinder nanoantenna suspended in air [12] and predicted an SHG efficiency exceeding ${{10}^{- 3}}$ at the laser pump density of ${{1\;{\rm GW/cm}}^2}$ at telecom wavelengths. Later, the same group experimentally fabricated the AlGaAs cylinder array on the AlGaAs-on-${{\rm AlO}_x}$ platform using the selective oxidation technique [13]. The array pitch was chosen as high as 3 µm to eliminate the coupling between adjacent cylinders to mimic the case of individual cylinders, and the SHG conversion efficiency up to ${6} \times {{10}^{- 5}}$ at the pump laser power intensity of ${1.6}\;{{{\rm GW/cm}}^2}$ was experimentally demonstrated. At the same time, Liu *et al*. used the same fabrication technique to produce a similar AlGaAs cylinder array with the pitch as small as 600 nm, which has a lattice resonance around 960 nm while the pump laser wavelength is at 1020 nm [14]. As a result, the whole array worked in the subwavelength period regime. An absolute SHG conversion efficiency of ${\sim}{2} \times {{10}^{- 5}}$ was reported at the pump laser intensity ${3.4}\;{{{\rm GW/cm}}^2}$ when the pump laser wavelength was tuned to match the MD resonance of the AlGaAs cylinder array. However, a deep investigation into the dependence of the overall SHG efficiency on the dielectric array periodicity is still missing. It is known that the localized surface plasmon (LSP) mode supported by individual metallic nanostructures with relatively broad resonance will experience significant narrowing of the bandwidth when the nanostructures are arranged into an ordered array with proper pitch. The new sharp resonance is due to the coupling between adjacent nanostructures, which results in a hybridization of the original plasmonic mode [15]. As a result, the incident wave at resonance will have more interactions with the array, leading to many potential applications in optical sensing, photocatalysis, and nonlinear optics, etc. As we will show in this work, the periodicity is also quite important in an all-dielectric nanoantenna array supporting Mie resonances, and it will strongly affect the interactions between the adjacent nanoparticles and the subsequent nonlinear responses.

In this paper, we numerically investigate the influence of the AlGaAs antenna array pitch on the SHG efficiency. Our emphasis is focused on the special case when the first-order diffraction resonance matches the MD resonance of individual antennas. It is well known that in plasmonic systems, when the lattice resonance of the plasmonic nanostructure array meets the scattering resonance of individual plasmonic antennas, the interaction of the two resonances will lead to a new hybridization state with a significantly narrowed bandwidth and a stronger light–array interaction [16]. We note that the hybridization between individual antenna resonance and lattice resonance is more pronounced in the all-dielectric case because of the special mode characteristics of the MD resonance. Our results demonstrate that the SHG conversion efficiency at the new hybridized resonance will be further enhanced by more than 2 orders of magnitude compared to that from an individual AlGaAs antenna at the same pump laser power density.

## 2. STRUCTURE AND RESULTS

For comparison purposes, the linear scattering property and the SHG conversion efficiency from an individual AlGaAs antenna are first presented. In practice, the electromagnetic response from such a single antenna depends on geometrical parameters, such as shapes and spatial dimension, and one can adjust these parameters to tune the spectral resonances of the individual dielectric antennas. We choose a set of geometrical parameters for an individual AlGaAs antenna in the shape of a nanocylinder and present its scattering efficiency and SHG conversion efficiency as a function of wavelengths. Then the same cylinder geometry is used to form an antenna array, whose forward-scattering characteristics, i.e., the transmission spectra and the SHG conversion efficiency, are investigated for different array pitches.

#### A. SHG from a Single Nanoantenna

The geometry of the nanocylinder is schematically illustrated in Fig. 1. The cylinder made of AlGaAs material has a radius and height fixed as $R = {200}\;{\rm nm}$ and $h = {450}\;{\rm nm}$, sitting on top of the low-index substrate, which is assumed to be ${{\rm SiO}_2}$. The adhesion of the AlGaAs layer to the ${{\rm SiO}_2}$ substrate can be achieved using the wafer-bonding technique. To simplify both the analysis and numerical calculations, all the material dispersions are ignored in the near infrared band and we assume the refractive indices of these media as 3.2 (AlGaAs) and 1.45 (${{\rm SiO}_2}$). The nanocylinder is top-cladded by air, ensuring a high refractive index contrast around it so that the Mie types of resonances will be supported.

The electromagnetic responses of an individual nanocylinder were numerically investigated using the finite-element method working in the frequency domain implemented in the commercial software, COMSOL Multiphysics [17]. A plane wave polarized along the *x* axis was incident normally to the nanocylinder surface from the top and the scattered field was calculated with the reflected field from a bare air/${{\rm SiO}_2}$ interface used as the background. By scanning the excitation wavelength, the total scattering efficiency of a single cylinder can be calculated as a function of wavelengths; the results are presented as the blue line in Fig. 2. The scattering efficiency is defined in Eq. (1),

One can see from the results that two resonances are present at the wavelengths of 1265 nm and 1541 nm, respectively. Detailed simulation results of the resonance mode distributions at the two wavelengths reveal that the former is an electric dipole (ED) resonance [Fig. 2(b)], where the electric field lines point from one side to the other, while the latter is a MD [Fig. 2(c)]. An evident feature of the MD resonance is that the electric field lines circulate both inside and surrounding the cross section of the cylinder, forming a loop and a right-hand relation with the magnetic field. We will show later that vertical parts of the circulating electric field lines have a large overlap with the electric field of the lattice resonance, which is the origin of the hybridization between MD and lattice resonances.

The SHG from an individual AlGaAs antenna is calculated based on the coupled-wave equations [18] between the fields at the fundamental frequency (FF) and the SH frequency. Instead of using the undepleted pump approximation [18], the coupling is achieved taking into account the dependence of the polarizations on the electric fields at both frequencies. Thanks to the zinc-blend crystal structure of AlGaAs material [14], the second-order nonlinear susceptibility tensor component $\chi _{\textit{ijk}}^{(2)}$ is nonzero only when $i \ne j \ne k$. Then the polarizations at the two frequencies can be simplified as

where the numbers 1 and 2 in the subscripts denote the fields at the fundamental and SH frequencies, respectively; $i,j,k$ are related with field components along different directions; and ${\chi ^{(2)}}$ is assumed as 200 pm/V for the pump laser wavelength in the near-infrared [13]. The polarizations ${P_2}$ in Eq. (3) work as the sources for the SH frequency and the generated fields at the SH frequency can then been calculated. The red line in Fig. 2 presents the calculated absolute SHG conversion efficiency, which only considers the generated SH power along the same direction as the pump laser and is calculated as follows:#### B. SHG from MD and Lattice Resonance Coupling

As described above, the enhancement in the SHG from the MD resonance originates from a circulation of the electric field inside the nanocylinder. We further show that a stronger circulation of the electric fields can be achieved by using a nanocylinder array. The basic idea is sketched in Fig. 3(a). It is known that a grating will experience a transmission drop at certain wavelengths depending on the grating pitch arising from grating diffraction. This phenomenon happens at wavelengths, usually referred to as lattice resonances, where the diffraction angles reach zero, leading to the diffracted beam propagating along the grating surface while interacting with the grating elements. For the AlGaAs nanocylinder array, the lattice resonance can be tuned by optimizing the array pitch to match the MD resonance of individual nanocylinders. The two resonances will couple and interact with each other, leading to a strong hybridization. An evident phenomenon with the hybridization is associated with the narrowed bandwidth of the new resonance [15]. The hybrid lattice–MD resonance will see a further strengthening of the electric field circulation inside the AlGaAs cylinders, and then an increased SHG efficiency can be expected.

The condition for the lattice resonance along the $x$ direction is governed by Eq. (5) [19],

*et al*. investigated the SHG from metallic metasurfaces by changing the incident angle to tune the lattice resonance targeting at the localized plasmonic resonance by V-shaped plasmonic nanoantennas [20]. Unfortunately, when $\theta$ is not zero, the surface wave at the lattice resonance will propagate along only one direction along the array. To have more interactions between the lattice resonance and the AlGaAs nanocylinder, it is better to keep normal incidence and change the value of ${P}$ instead. Then lattice resonance with the power flow propagating along two directions in the array surface is present, as Fig. 3(a) illustrates.

Interactions between adjacent nanocylinders in the array will also induce a spectral shift in the MD resonance. To find the appropriate array pitch, we map the zeroth-order transmittance at normal incidence as a function of the fundamental wavelengths and the pitch values. The results are shown in Fig. 3(b), where the dark region on the left is due to the excitation of ED resonances while the one on the right is from MD resonance. The white dashed lines denote the positions where the $\pm {1}$st-order lattice resonance happens, and it can be simply calculated, assuming the diffraction angle to be zero.

Several behaviors are present in the 2D mapping results. Both the ED and MD resonances experience a redshift when the array pitch increases, approaching the corresponding resonances for an individual nanocylinder (1265 nm for ED and 1541 nm for MD). There is an evident crossing between the lattice resonance (the dashed line) and the MD resonance, at which the MD resonance bandwidth sees a minimum. However, for the ED resonance, there is no clear crossing with the lattice resonance. We believe these two distinct behaviors are related to the different mode distributions with ED and MD resonances. The main electric fields of the ED resonance are along the ${\rm AlGaAs}/{{\rm SiO}_2}$ interface, while the electric fields from the lattice resonance are perpendicular to the surface (the energy flow is along it). As a result, the lattice resonance may even disturb the ED electric fields. The MD resonance, however, is different and has strong components perpendicular to the interface. As a result, it can be coupled to the lattice resonance significantly and has a strong hybridization. To clearly demonstrate the spectral behavior of the MD–lattice hybridization, we plot in Fig. 4(a) the zero-order transmission spectra through the nanocylinder array at three different pitch values: 1055 nm where the lattice resonance matches the MD, and two more values, each 100 nm, deviating from 1055 nm. As can be seen, all the transmission spectra at these three pitches have the MD resonance close to 1500 nm, but the one at the pitch value of 1055 nm has, indeed, the narrowest resonances. The transmission spectra with the other two pitch values have slightly broader MD resonance, and the lattice resonance are present somewhere else with a sudden change of the transmittance (1385 nm for the pitch of 955 nm, and 1675 nm for the pitch of 1155 nm).

Using the same approach described above for the SHG calculation for individual nanocylinders, we calculated the SHG conversion efficiency for the array when the MD–lattice resonance hybridization happens. Only the Poynting vector at the same direction as the pump laser is considered (in the transmission), to be assistant with the individual antenna case. The blue line in Fig. 4(b) presents the calculated results, where one can see that the SHG is achieved at its maximum when the pump wavelength is around 1544 nm. Note that the fundamental (pump) wavelength is used in the $x$ axis and the SHG is actually at their half wavelengths. One can see that a stronger SHG efficiency that achieve the order of ${{10}^{- 4}}$ from MD–lattice resonance is achieved, which is more than 2 orders of magnitude higher than that from pure MD resonance in an individual nanocylinder. Furthermore, the SHG results from the other two pitch values of 955 nm and 1155 nm are also presented, and we can see that they are much weaker with broader bandwidth. Since the nanocylinder geometry is all the same for the three cases and the only difference is the pitch values, we can conclude that the hybridization of the MD-lattice plays an important role in the further enhancement of the SHG.

To reveal the underlying physics for the SHG enhancement associated with the MD–lattice resonance, we plot in Fig. 4(c) the magnetic field as well as the vector distributions for the electric fields at the MD–lattice resonance for the fundamental (pump) wavelength. All the fields are taken from the cross section of the nanocylinder (in the $x{-}z$ plane through the center). In comparison with that in an individual nanocylinder [cf. Figs. 4(c) and 2(c)], one can see that although the shapes of the magnetic fields resemble each other in the two cases, the field strength is roughly 2 times stronger in the MD–lattice resonance. Furthermore, there is a distinct difference between the electric fields. For the MD–lattice resonance, there is a stronger vortex or circulation of the E fields around the nanocylinder, arising from the coupling between the lattice resonance and the MD resonance. As a result of the stronger local field and interaction between the pump laser and nanocylinder, a more pronounced nonlinear response is therefore obtained. These field distributions further confirm the schematic figure sketched in Fig. 3(a). To show the power dependence of SHG on the pump power intensity, we keep the pump laser wavelength as 1544 nm and then calculate the SH power as a function of the pump power intensity. It is apparent that there exists a quadratic relationship, and the value of SHG power can reach up to ${10^{- 3}}$ when the ${I_{0}}$ is around ${0.35}\;{{{\rm GW/cm}}^2}$.

## 3. CONCLUSION

We have numerically investigated MD–lattice resonance hybridization resulting from the coupling of collective Mie resonance and the first-order diffraction in a periodic array of AlGaAs nanoantennas. Due to this hybridization, the SHG from the all-dielectric antenna array has an enhancement by 2 orders of magnitude over individual antennas, which is more significant than that from a similar array composed of plasmonic antennas [20], where 10-fold enhancement was experimentally observed. We attribute the higher enhancement to the special mode distributions of the electric fields at the MD resonances, which experience stronger circulation boosted by the lattice resonance. The MD–lattice resonance significantly boosts the local electromagnetic fields of the pump laser and the corresponding SHG responses. At a pump laser power density of ${0.053}\;{{{\rm GW/cm}}^2}$, an SHG conversion efficiency is enhanced by 2 orders of magnitude compared to that of an individual nanocylinder having the same geometrical parameters. We note that both materials exhibit nonnegligible dispersion in the spectral range of our interest, while in our calculations, the refractive index of materials is kept constant and only corresponds to material dispersion at the MD wavelength. As a result, ignoring the dispersion may lead to a slight mismatch of resonances for other wavelengths. However, since the SHG enhancement is obtained by normalizing the result with that from individual AlGaAs structures, we expect the conclusion will not be affected by ignoring the dispersion. Our results represent a further improvement of the nonlinear responses from the Mie resonances supported by high refractive index nanostructures and will help push forward the applications of the nonlinear optics based on all-dielectric nanoantennas in various fields.

## Funding

Natural Science Foundation of Zhejiang Province (LY15F050008); Taishan Scholar Foundation of Shandong Province (tsqn201909079); National Natural Science Foundation of China (11974221, 91750201); Innovation Group of Jinan (2018GXRC010).

## Disclosures

The authors declare no conflicts of interest.

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