## Abstract

We put forward the multipolar model which captures the physics behind linear and nonlinear response driven by high-quality (high-*Q*) supercavity modes in subwavelength particles. We show that the formation of such trapped states associated with bound states in the continuum (quasi-BIC) can be understood through multipolar transformations of coupled leaky modes. The quasi-BIC state appears with increasing the order of the dominating multipole, where dipolar losses are completely suppressed. The efficient optical coupling to this state in the AlGaAs nanodisk is implemented via azimuthally polarized beam illumination matching its multipolar origin. We establish a one-to-one correspondence between the standard phenomenological non-Hermitian coupled-mode theory and multipolar models. The derived multipolar composition of the generated second-harmonic radiation from the AlGaAs nanodisk is then validated with full-wave numerical simulations. Back-action of the second-harmonic radiation onto the fundamental frequency is taken into account in the coupled nonlinear model with pump depletion. A hybrid metal-dielectric nanoantenna is proposed to augment the conversion efficiency up to tens of per cent due to increasing quality factors of the involved resonant states. Our findings delineate novel promising strategies in the design of functional elements for nonlinear nanophotonics applications.

## 1 Introduction

Controlling light at the nanoscale has been a vibrant field of research for many years motivated by its various applications for optical nanoantennas, integrated photonic circuitry, optical computing, and high-speed ultrathin photonic devices [1], [2], [3], [4], [5]. High-index dielectric nanoparticles have emerged as a promising platform to enhance light–matter interactions at the nanoscale based on optically induced electric and magnetic Mie-type resonances [4], [5]. Due to the strong near-field confinement and tailorable field distributions in the subwavelength regime, optically resonant dielectric nanostructures offer powerful tools to facilitate various nonlinear processes including nonlinear frequency conversion, wave mixing, and ultrafast all-optical switching [4], [6], [7], [8], [9], [10], [11]. Resonant mechanisms of light localization in dielectric nanostructures, such as magnetic dipole resonance [12], [13], [14], nonradiating anapole state [15], [16], [17], magnetic Fano resonance [18], [19], and topologically protected edge states [20], [21], have been widely utilized for applications of nonlinear nanophotonics, such as nanoscale light sources [22], imaging [23], sensing, and advanced optoelectronic devices [3].

Methods typically employed to describe nonlinear harmonic generation at the nanoscale are based on the multipolar decomposition of the fields in spherical multipoles [4], [24], [25], [26], [27]. This approach provides a transparent interpretation for the measurable far-field characteristics, such as conversion efficiency and radiation patterns, since they are essentially determined by the interference of dominating multipolar modes [4]. The nonlinear response depends on both incident polarization and symmetry of the specific material or composite structures, such as nanoparticle oligomers [27], [28], [29]. For instance, the nonlinear response of disk-shaped nanoantennas made of noncentrosymmetric III–V semiconductors (GaAs or AlGaAs) grown along (100), (110), and (111) crystallographic directions exhibits four-fold, two-fold, and continuous rotational symmetries, respectively [26], [30]. Specifics of the radiation characteristics in each case can be explained by different parities of nonlinearly generated multipoles, so-called nonlinear multipolar interference. This represents an example where multipolar analysis has been proven to be a useful instrument for design of directional nonlinear light emission [31].

A recently suggested approach to trap light in individual subwavelength dielectric nanoresonators is based on high-quality supercavity modes associated with the physics of quasi-bound states in the continuum (quasi-BIC) [32], [33], [34]. These states are in some sense similar to BICs in infinite periodic dielectric structures: their high finesse is due to the destructive interference of several far-field radiation channels. Two interpretations of quasi-BIC formation in an open resonator, based on (i) leaky modes supported by the particle [33] and (ii) multipoles of the electromagnetic field [35], were discussed; however, no direct correspondence between these two models has been established up to now. Lately, it has been proposed to utilize these high-quality states to enhance the classical nonlinear process of second-harmonic generation in dielectric nanodisks [36].

In this paper, we develop a comprehensive multipolar theory of the second-harmonic generation (SHG) from high-quality quasi-BIC states in an AlGaAs cylindrical nanoresonator (nanodisk). The process of SHG is schematically illustrated in Figure 1. We show that the formation of quasi-BIC states can be naturally understood through multipolar transformations of coupled leaky axially symmetric modes supported by the nanodisk. The strong SHG can be expected in the case of efficient excitation of the mode with the high quality factor. To achieve efficient coupling to the mode at the pump frequency, the multipolar composition of the pump source should match the multipolar composition of this mode. We show that the azimuthally polarized beam can be used to couple to the quasi-BIC state most efficiently because it contains magnetic multipoles with zero azimuthal numbers only that matches the quasi-BIC multipolar origin and maximizes the modal overlap. Analyzing the spatial distribution of the induced nonlinear source with the approach put forward in our earlier works [25], [26], we reveal that the complex multipolar composition of the second-harmonic radiation coincides with the multipolar composition of a particular high-quality eigenmode of the disk at the second-harmonic (SH) frequency. We obtain that the efficiency of SHG is strongly enhanced up to several per cent, provided the generated frequency matches the supported resonance, for parameters of the nanoresonator corresponding to the nearly resonant excitation of the quasi-BIC state at the fundamental frequency (FF). We perform full-wave numerical simulations of SHG taking into account nonlinear effects of back-action and propose the BIC-inspired design of a hybrid metal-dielectric nanoantenna where the effect of pump depletion is further increased suggesting a promising application for the frequency downconversion.

## 2 Results and discussion

### 2.1 Multipolar model of quasi-BIC formation

We consider a high-index cylindrical dielectric resonator that supports leaky modes (modes of an open resonator) that may hybridize (couple) when tuning geometric parameters. The particle is characterized by a frequency-dependent dielectric permittivity

Here, we focus on rotationally symmetric TE-polarized modes in the cylindrical coordinate system *φ*. With the use of the finite element method (FEM) in COMSOL Multiphysics, we perform the eigenmode analysis numerically and plot dispersion as a function of the normalized geometric parameters defined as the disk aspect ratio *r*/*h* and the size parameter *r*/*λ*_{0}, where *λ*_{0} is the mode wavelength. The size parameter defined as a ratio of the disk radius to the mode wavelength can also be referred to as a normalized eigenfrequency. The results of calculations for the AlGaAs nanodisk with the fixed height *h* = 645 nm are summarized in Figure 2. Two dispersion curves depicted as colored dots exhibit characteristic avoided crossing in the plane of parameters, which is a signature of the strong-coupling regime. It is accompanied by modification of the modes quality factors and formation of the quasi-BIC state that can be naturally understood through multipolar transformations of coupled modes as we describe below.

For a nonspherical shape of the resonator, the interacting modes can be in general viewed as a superposition of the spherical multipoles distinguished by orbital *l* and azimuthal *m* indices. Multipolar analysis of the considered TE-polarized axially symmetric eigenmodes suggests that the basis of parent multipoles mainly consists of a longitudinal magnetic dipole (MD) *l* = 1, *m* = 0 and a magnetic octupole (MO) *l* = 3, *m* = 0 so that each mode has two multipolar radiation channels, dipolar and octupolar. In Figure 2, sizes of circles illustrate their relative contributions in multipolar expansions of the modes (line plots are provided in Supplementary Material). The occurrence of high-*Q* supercavity mode is accompanied by an increase in the order of a dominating multipole from *l* = 1 (MD) to *l* = 3 (MO) and corresponds to the pure magnetic octupole. In the quasi-BIC condition (*r*/*h* = 0.71 and *r*/*λ*_{0} = 0.29), two magnetic dipoles interfere destructively in the coupling to the octupole, thus, restoring its high-quality factor. Insets show how the electric field distributions inside the disk and far-field diagrams of these modes change as we move along the dispersion curves. While far from the BIC point, the radiation patterns are dipolar, near the BIC point the pattern of the high-*Q* mode turns to the rotationally symmetric magnetic octupole with three lobes. In addition, the black line in Figure 2 shows the dispersion of the mode near the second-harmonic frequency. The eigenmode analysis is performed for the AlGaAs disk taking into account the material dispersion. The refractive index in the fundamental wavelength range 1500–1700 nm

We then corroborate our rigorous numerical results by an analytical model setting the correspondence of the non-Hermitian coupled-mode theory and the multipolar analysis. Coupling of two leaky eigenmodes shown in Figure 2 can be described by the Hamiltonian [33], [34]

being a square matrix, where *W* and *V* are complex coupling parameters including the interaction between the modes via free space, *Q* factor for one of the modes (BIC state). The avoided crossing of two modes in the two-level system qualitatively captures the numerically calculated dispersion depicted in Figure 2.

For our system, we derive the dynamic equations for two modes with slowly varying amplitudes

where the matrix

The *d* coefficients govern both the dipolar and octupolar radiative decays of the modes and their coupling to each other and to the external field,

with the amplitude column-vector

The non-Hermitian Hamiltonian of the structure in Eq. (4) assumes the form

where

The excitation of the quasi-BIC mode depends on how efficiently the incident source couples to the BIC state, that is on their spatial overlap. The external source can be divided into the octupolar component *m* = 0 that matches the multipolar composition of the modes, while the plane wave consists of electric and magnetic multipoles with different azimuthal indices *m*. Furthermore, the relative contribution of MO is significantly higher in the AP vector beam than in the plane wave, so that the azimuthally polarized cylindrical vector beam is favorable for excitation of the high-quality resonance in the dielectric nanodisk. Thus, hereafter we focus on the study of SHG in a AlGaAs nanodisk excited by an AP cylindrical vector beam in the vicinity of the BIC point.

### 2.2 Nonlinear response driven by quasi-BIC state

For noncentrosymmetric materials, the induced nonlinear polarization is defined by the second-order polarizability tensor

where

We assume that the main axes of the crystalline lattice are oriented along the axis of the Cartesian coordinate system:

Here,

where *l* of generated multipoles are even for electric and odd for magnetic multipoles, while the azimuthal number takes the values

In the case when a nanoantenna supports both a quasi-BIC resonance near the pump frequency and an eigenmode near the SH frequency, we expect a resonant increase in the up-conversion. In addition to the eigenmodes at the fundamental frequency, Figure 4 features the dispersion of the high-quality mode at the second-harmonic frequency lying in the same range of parameters (SH mode). The *Q*-factor of this mode is large (*m* = ±2. This mode is composed of the multipoles that the induced nonlinear current at the quasi-BIC conditions generates. The multipolar decomposition of the SH mode is visualized by circles of different colors. The right inset additionally shows relative contributions of multipoles in the line plot. The SH mode has a more complex multipolar content than the FF modes, and a more sophisticated near-field distribution (top inset), which modifies slightly in the considered parameter range. The radiation pattern of the SH mode is generally multi-lobed and noticeably changes depending on the multipolar composition. The most efficient SHG can be reached at the resonant conditions, close to the dispersion crossing of the quasi-BIC mode and the SH mode.

As a pump, we choose an AP beam in order to maximize the overlaps of the modes with the incident field. We numerically calculate dependencies of multipolar compositions of the linear scattering and SH radiation on the disk radius in the case of AP beam excitation (for comparison with the case of PW excitation see Supplementary Material). The multipolar contents of both FF and SH fields are found to be in excellent agreement with the results of our theory that justifies its validity. Figure 5 shows two-dimensional maps of linear scattering of the AP beam and the SH power. We recover maps of linear scattering, MD and MO contributions in linear scattering on the nanodisk radius by calculating slices at fixed pump wavelengths. We observe strong excitation of a magnetic octupole (MO) in the proximity of the BIC point so that the high-*Q* mode branch can be clearly traced in the map of linear scattering. Next, we model SHG and obtain a sharp enhancement peak in the SH intensity. The resonant area in the parameter space is visualized with high contrast in Figure 5D. These our results fully explain the recent experimental observations reported in Refs. [39], [40].

Figure 6 shows two slices of 2D color maps in Figure 5 for two close wavelengths: the slice at *λ*_{0} = 1580 nm contains the BIC point and the slice at *λ*_{0} = 1600 nm contains the intersection of the SH and FF dispersion curves. If we consider the fixed pump wavelength *λ*_{0} = 1580 nm, we see that MO dominates in the linear scattering and the maxima of the scattering cross-section and the SH power for this slice correspond to the BIC point (*r* = 457 nm, *h* = 645 nm). We obtain that in this case magnetic quadrupole (MQ) *λ*_{0} = 1600 nm corresponding to the intersection of the FF high-*Q* mode and SH mode dispersion branches, the SHG enhancement is even higher. In this case major multipoles are

Because the distribution of the AP beam intensity is inhomogeneous, here we define the scattering efficiency as the ratio of the scattering power at the fundamental frequency *P*_{FF} to the energy flux through the geometrical cross section of the particle in the waist plane

Here, *S* (

We obtain that the absolute maximum of SHG corresponds to the intersection point and ideally, it may reach about 1% conversion. Also we calculate the conversion efficiency that doesn’t depend on the incident power

The coupled-mode considerations suggest scaling the SHG efficiency (15) depending on detunings from the resonances at the fundamental and SH frequencies:

Here,

We compare near-field profiles, far-field diagrams and multipolar compositions of the SH mode and the SH radiation near the peak SHG for two types of incident radiation. We obtain that at the parameters of the maximum SHG, the multipolar composition of the SH generated field, the SH electric field distribution and the SH radiation pattern are almost the same as those of the SH eigenmode, see Figures 4, 5 and 7. This explicitly shows that even though the generated SH radiation is composed of several different multipoles, the only one SH mode characterized by

### 2.3 Coupled model of SHG with pump depletion

In our numerical modeling, we take into account the nonlinear effects of back-action of the second-harmonic radiation on the fundamental frequency radiation. In the case of highly efficient double-resonant excitation, the electric field amplitudes inside the particle at the fundamental and second-harmonic frequencies may become comparable, and the nonlinear correction of the polarization

The expression for the nonlinear polarization at the fundamental frequency

The nonlinear source at the second-harmonic frequency is given by Eq. (8).

In the conditions of the resonant excitation of the magnetic-octupole quasi-BIC mode and the high-quality mode at the second-harmonic frequency, the electric field can be represented as a superposition

where

where the coefficient

is proportional to the overlap integral,

We calculate the scattering and SHG conversion efficiencies taking into account coupling of the excited SH and FF modes and compare them with the numbers obtained in the undepleted pump approximation. The results for two types of incident radiation for parameters corresponding to the maximum SHG efficiency (*r* = 470 nm, *h* = 645 nm) are summarized in Table 1.

ρ | |||
---|---|---|---|

AP undepleted pump | 34.14 | 0.02 | 1.33·10^{−4} |

AP back-action | 32.93 | 0.0182 | 1.25·10^{−4} |

PW undepleted pump | 3.97 | 0.0031 | 5.49·10^{−5} |

PW back-action | 3.95 | 0.0031 | 5.44·10^{−5} |

Values of the pump intensity and the energy flux through the geometric cross section of the disk used in our calculations are given in Table 2. *S* denotes the geometrical cross section of the particle. In the case of AP beam excitation, *P* is the energy flux through the geometrical cross section of the particle in the focal plane. The characteristics of AP beam are illustrated in Fig. S3 of Supplementary Material. The plane-wave intensity ^{−1} for 2 ps lasers pulses is approximately twice higher than the estimate in Ref. [40] based on the experimental data.

AP | 0.89 | 2.16 | 6.94 | 4.404 |

PW | 2.37 | 2.37 | 6.06 | 14.36 |

We obtain that in the case of AP excitation, nonlinear effects of back-action become noticeable due to efficient excitation of the high-quality modes. Thus, coupling of electromagnetic fields at the fundamental and second-harmonic frequencies should be taken into consideration when the conversion efficiency reaches 1%. In the case of PW excitation, the maximum SHG is also reached for the parameters corresponding to the intersection of the eigenmodes dispersion curves, however, the total generated power turns to be lower than in the case of the AP beam pump. As noted above, it can be explained using multipolar analysis of the incident radiation. The multipolar composition of the incident plane wave is more complicated than multipolar composition of the AP beam, the spectrum of linear scattering contains several multipoles with different values of the azimuthal index *m*. As a result, the electric field distribution at the BIC point is distorted and appears dependent on the azimuthal angle *φ*. We still observe a resonance at the BIC point in the linear scattering spectrum but the corresponding scattering efficiency is an order of magnitude lower than in the case of AP excitation. Since the magnetic octupolar relative contribution with

In the next section, we consider the BIC-inspired nanoantenna design where the second-harmonic radiation significantly influences the linear scattering. Such nanoresonators can be particularly promising for realization of the efficient frequency downconversion at the nanoscale.

### 2.4 BIC-inspired nonlinear enhancement in a hybrid metal-dielectric nanoantenna

One type of BICs which has been widely studied is the symmetry-protected BIC occurring at *Q*-factor [42]. In particular, by properly designing a nanodisk metasurface, it can support the longitudinal magnetic dipole resonance at

A schematic illustration of the proposed principle is shown in Figure 8A. As was described in Ref. [16], when an electric current is placed near a perfect-electric-conductor (PEC) surface, the excited free electron oscillations in the adjacent PEC affect the near-field and far-field properties of such composite system. The effect can be reproduced with an oppositely oriented image of the electric current by using the image dipole model. Using this analogy, a periodic system of MD resonators can be transformed to a single MD resonator surrounded by a PEC-like box. As an example, we employ a dielectric disk surrounded by gold (Au) nanobars, as shown in Figure 8B. Remarkably, by shrinking a periodic system into a single nanoresonator, the quasi-BIC modes can be directly and efficiently excited under structured beam illumination that facilitates nonlinear frequency conversion.

We then take the AlGaAs nanodisk supporting a magnetic dipole resonance near 1550 nm and study the nonlinear performance of our proposed design. The geometric parameters of our structure are determined to be the following: the radius and height of AlGaAs nanodisk are *r*_{0} = 237.5 nm, *Q* mode. The quality factor of this cavity mode is estimated to be

We further examine the SHG process from the designed AlGaAs-Gold hybrid nanoresonator. In order to show the importance of our advanced simulation method introduced in Section 2.3, we compare the calculated results based on the undepleted pump and the coupled back-action models in Figure 8C.

In the undepleted pump model, we follow two steps to simulate the nonlinear response. First, we calculate the linear response at the fundamental wavelength and obtain the nonlinear polarization induced inside the structure based on Eq. (8). We then employ the nonlinear polarization as a source to simulate the electromagnetic response at the harmonic wavelength. In this two-step procedure, we disregard the influence of the harmonic waves on the pump field. As seen in Figure 8C, the SHG emission is significantly enhanced when exciting the designed high-*Q* resonance at the pump frequency. Near the resonance, we observe a clear difference in the nonlinear emission power when using the coupled back-action model as compared to the undepleted pump model: at the pump wavelength of 1549 nm, the total SHG power is around 0.593 W and 0.546 W for these two cases, respectively. This indicates that neglecting coupling from SH field to the pump field in the undepleted pump model results in more than 8% error in the nonlinear simulation. Under AP pump illumination, the SHG signal can be boosted in our proposed configuration leading to nearly 2000-fold enhancement of the SHG emission power as compared to the case of SHG driven by the conventional MD resonance in a single free-standing nanodisk.

## 3 Conclusions

We have explicated the multipolar nature of quasi-BIC states in subwavelength dielectric resonators supporting high-quality supercavity modes called quasi-bound states in the continuum and discussed their applications for nonlinear nanophotonics. Using our multipolar model, we have analyzed optimal conditions for the efficient excitation of quasi-BIC states in high-index dielectric nanodisks under structured light illumination. In particular, we have explained the multifold increase of the second-harmonic conversion efficiency for the case of azimuthally polarized cylindrical vector beam illumination compared to the linearly polarized plane wave excitation. Implementing numerically the coupled nonlinear model, we have computationally refined values of SHG efficiencies and discussed pump depletion in high-quality nanoresonators.

**Funding source: **Russian Foundation for Basic Research

**Award Identifier / Grant number: **18-02-00381

**Award Identifier / Grant number: **19-02-00261

**Funding source: **Australian Research Council

**Award Identifier / Grant number: **DE190100430

**Funding source: **University of New South Wales

## Acknowledgment

This work is supported by the Russian Foundation for Basic Research (Grant Nos. 18-02-00381 and 19-02-00261) and the Australian Research Council (grant DE190100430). I.V. acknowledges partial support from the Foundation for the Advancement of Theoretical Physics and Mathematics BASIS. D.S. thanks Y. Kivshar and A. Poddubny for valuable discussions. The work of A.E.M. is supported by the UNSW Scientia Fellowship.

**Author contribution:**All authors contributed extensively to the work presented in this paper.**Research funding:**None declared.**Employment or leadership:**None declared.**Honorarium:**None declared.**Competing interest statement:**The authors declare no conflicts of interest regarding this article.

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## Supplementary material

Supplementary material to this article can be found online at https://doi.org/10.1515/nanoph-2020-0156.

**Received:**2020-02-28

**Accepted:**2020-05-01

**Published Online:**2020-08-17

© 2020 Irina Volkovskaya et al., published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.