Strong Purcell effect of magnetic quasi-BICs in the dielectric metasurface

The ultra-strong light field trapped in micro- and nanostructures, and derived from optical bound states in the continuums (BICs), provides a new platform for the facilitation of light–matter interactions. In this work, the spontaneous emission enhancement of quantum emitters using magnetic quasi-BICs metasurfaces is numerically studied. It is found that ultrahigh Purcell factor (PF) ( >105 ) can be easily achieved when the asymmetry parameter decreases. In theory, infinite PF may emerge due to the infinite Q-factor at the BICs. In addition, the PF exhibits a strong dependence on the polarization direction of dipole emitter and the local field distribution in the metasurface. Finally, the photoluminescence enhancement for different perturbation parameters is calculated to confirm strong Purcell effect in the quasi-BICs metasurface. These results show that the magnetic dipole BICs mode holds an enormous potential in manipulating spontaneous emission.


Introduction
Electromagnetic spontaneous emission is an important effect in the quantum optics and nanophotonics [1,2]. In 1946, Purcell et al found that the spontaneous emission rate can be tailored by changing the electromagnetic environment of the quantum source, named as the Purcell effect [3]. Nowadays, the resonant micro-or nanostructures accompanied by a large local field are frequently used to modulate the electromagnetic environment [4]. In order to facilitate the enhancement of spontaneous emission, these common micro-or nanostructures, including whispering gallery microcavities [5,6], photonic crystals defect cavities [7,8], surface plasmon polaritons nanostructures [9][10][11], dielectric nanoparticles [12][13][14], metamaterials (metasurfaces) [15][16][17], have been widely used. However, the Purcell factor (PF), a quantitative description of the enhancement of spontaneous emission, is usually within the range of 10-10 5 due to scattering loss and absorption loss in these structures [4,18]. To date, some attempts are also made to further improve the PF. For example, Qian et al achieved a large PF of 2 × 10 4 by employing the mechanism of topological state-led mode coupling recently [18].
According to Purcell effect principle (PF ∝ Q/V), a larger PF enhancement requires a higher Q-factor and a smaller effective mode volume in micro-or nanostructures [1,4]. Optical bound states in the continuums (BICs), may serve as an excellent solution for giant PF enhancement due to its infinitely high Q-factor and zero radiation loss [19][20][21]. The BICs will transform into a quasi-BICs mode with ultra-high Q-factor when the micro-and nanostructure are perturbed [22][23][24]. And the quasi-BIC metasurfaces with high Q-factor have two important properties, namely, sharp resonance spectrum [25] and significant local field enhancement [26,27]. Therefore, they are widely used in various nanophotonic devices with high-performance, such as ultralow-threshold lasers [28][29][30], chemical and biological sensors [31,32], nonlinear optical devices [33][34][35], efficient optical absorbers [36,37], and so on. Recent year, Koshelev et al analyzed and summarized the high Q-factor properties of various metal or dielectric metasurfaces with symmetry breaking. They found that the inverse quadratic dependence of Q-factor for quasi-BICs on The distribution of electric and magnetic field vectors inside the metasurface, where red arrows represent the electric vectors and the yellow arrows represent the magnetic vector. (c) Transmission spectra of quasi-BICs, the dashed circles are the simulation data, and the solid line is the fitting results of the Fano formula. The inset shows the schematic diagram of unit cell. The period of the structure P = 940 nm, the height of the silicon nanodisk h = 220 nm, the radius of nanodisk R = 240 nm, and the radius of the air-hole on the edge of the nanodisk r = 130 nm. (d) Normalized scattering energy of multipole. The inset shows magnetic dipole components. The ED, MD, TD, EQ and MQ are the responses of electric dipole, magnetic dipole, toroidal dipole, electric quadrupole, and magnetic quadrupole, respectively. Here, the material of the metasurface device is Si and the substrate is SO2. The optical constants of Si and SO2 are taken from the Palik Handbook. In the wavelength range we studied, the refractive index (n, k) of Si is about (3.48, 0) and that of SO2 is (1.445, 0), see appendix A. The resonance is excited by a y-polarized incident light. perturbation parameter (α) allow for tailoring the Q-factor by adjusting α [23]. The smaller the α, the higher the Q-factor, and thus the stronger field confinement.
In this work, the Purcell effect of quasi-BICs in the dielectric metasurface is symmetrically studied. By introducing an air-hole at the edge of the symmetric silicon nanodisk, the magnetic dipole BICs mode is transformed into a quasi-BICs with high Q-factor. It is found that the calculated PF depends on the location of quantum emitter and reaches its maximum at resonance wavelength when the quantum emitter is located in the position of field maximum. Besides, the PF shows polarization dependence on the direction of magnetic dipole emitter, and can be tailored by the asymmetric parameter α through changing the radius of the air-hole. Finally, the photoluminescence (PL) enhancement is calculated as a function of the air-hole radius. Our results demonstrate that the PF can be effectively regulated by the asymmetric parameters of quasi-BICs, and the giant PF can be achieved as α value is a very small.

Result and discussion
A circular air-hole introduced at the edge of the nanodisk to break the symmetry of the structure opens a coupling channel between the nonradiative BICs mode and free space. This operation accompanied the emergence of quasi-BICs with high Q-factor which usually possesses a strong magnetic response. The characteristics of and zero linewidth and infinite Q-factor for the radius of air-hole r = 0 confirm the existence of BICs. As r increase, the decreased Q-factor and enlarged linewidth gradually confirms the transformation of a perfect true BICs mode to the quasi-BICs, as illustrated by in appendix B.
As shown in figure 1(a), the notched nanodisk array can strongly trap the magnetic field in the interior of the nanodisk under the illumination of y-polarized light. The reason is that the symmetry perturbation of structure excites a circular electric field distribution in the xy-plane of the nanodisk, such that a magnetic dipole oscillation in the z-direction is observed at the center of the nanodisk, as shown in figure 1(b). The excited quasi-BICs usually exhibits an asymmetric Fano profile in the transmission spectrum, which is formed by the coupling of discrete bound states supported by nanodisks and the free space continuum [23,38,39]. Next, the formation of Fano resonance is analyzed in detail. After cutting a notch at the edge of the nanodisk, the electric dipole along the y direction (P y ) produced by the incident polarization will exhibit slightly different dipole strengths in the notch part and the opposite side. These two asymmetric dipoles will generate a z-directed magnetic field near the center of the nanodisk, which gives rise to a z-polarized magnetic dipole (M z ). It serves as a dark mode because near-field coupling of nanopillars dramatically suppresses the radiative loss. Finally, the broken symmetry induces interference between the bright in-plane electric dipole P y and the dark magnetic dipole M z , so that the Fano resonance is observed. The resonance curve can be fitted by the Fano formula [23]: Here Ω = 2(ω − ω 0 )/γ, ω 0 is the resonance frequency, γ is the inverse lifetime. q is the Fano asymmetry parameter. T bg and T 0 describe the background contribution of nonresonant modes to the resonant peak amplitude and the offset, respectively. Figure 1(c) shows that the calculated transmission spectrum (dashed circles) of the silicon notched disk metasurface is almost consistent with the Fano formula fitting (solid line). The multipole decomposition in Cartesian coordinates [40,41] confirms that the mode is dominated by a magnetic dipole in the z-direction, while other multipole radiations are significantly suppressed, as shown in figure 1(d), which is in agreement with the discussion on the field vectors. In this work, the finite-element method (FEM) (COMSOL Multiphysics) and finite-difference time-domain method (FDTD Solutions) are employed to analyze the optical responses of metasurfaces. The basic principles of FDTD and COMSOL are different, while they can calculate the optical response characteristics of nanodevices. The complex eigenfrequency (ω = ω 0 − iγ) can be calculated by commercial software Comsol Multiphysics based on the FEM. The Q-factor is calculated by Q = ω 0 /2γ. Meanwhile, the field distributions of quasi-BICs are also calculated by COMSOL. The transmission spectra, scattered power of different multipoles, PF and PL enhancement are calculated by FDTD. In the calculations of both software, the x-and y-directions are set as periodic boundary conditions, and the z-direction is set as a perfectly matched layer boundary condition. In transmission spectrum calculations and multipole decompositions, a y-polarized incident light along the z-axis is used. The dipole light source is used for the PF calculation. In addition, the multiple randomly placed dipole sources are used to simulate actual PL emission. The optical constants of Si and SiO 2 are taken from Palik Handbook [42], as shown in figure 6.
The light-matter interaction of a system which means an emitter is coupled to a resonant cavity is described by cavity quantum electrodynamics (CQED) [4]. The spontaneous emission is modulated by the resonant cavity mode, the PF is defined as the ratio of the decay rate Γ in the resonant cavity environment to the decay rate Γ 0 in free space [13,43,44]: where P and P 0 are the radiated power of the dipole emitter in the resonant cavity environment and in free space without resonators, respectively. The PL emission is proportional to the PF, thus PL enhancement can be expressed as follows: where, QE is the quantum efficiency in resonant metasurface, and QE 0 is the intrinsic quantum efficiency. It has been clearly demonstrated that such a notched disk metasurface excites a strong magnetic dipole resonance in the previous part of the paper. Herein, the spontaneous emission from the magnetic dipole is the focus of our discussion. According to the CQED theory, the strong Purcell effect can be observed when the dipole emitter overlaps well with the enhanced local magnetic field. The magnetic field distributions of the quasi-BICs mode in the xy-, yz-and xz-planes are shown in figures 2(a)-(c), respectively. The magnetic field is mainly localized in the central region of the nanodisk. Five different locations in the nanodisk are marked as A, B, C, D, and E to represent the different field enhancement regions, as shown in figure 2(d).
Here, the red point A holds the strongest field enhancement, and the field intensity at points B, C, E, and D gradually decreases. The PF values of five positions are calculated when the magnetic dipole emitter is placed at the corresponding point. According to the discussions of multipole response and field vector distribution (see figures 1(b) and (c)), the magnetic dipole oscillates along the z-axis. The Purcell effect is effectively improved when the magnetic dipole emitter overlaps with magnetic hot-spot in the space. So the same setting of the polarization of magnetic dipole emitter is also performed along z-direction. The calculated PF spectra are shown in figure 2(e). The PF value reaches a maximum of 1900 at point A which is contributed to processing the strongest field enhancement among the five positions. While points B, C, D, and E have decreasing PF as the field intensity decreases gradually. It is worth noting that the maximum value of PF in the given wavelength range overlaps with the resonant wavelength where the local field is strongest. These results demonstrate that the PF depends on the interaction of the dipole emitter with the resonance mode. The PF is calculated when an electric dipole emitter is located at the selected five positions above, and the PF is close to 900 (see appendix C). In addition, the PF is calculated with a magnetic dipole emitter placed at the surface of the metasurface (above point A), the maximum values is 320 (see appendix D).
As previously discussed, this quasi-BICs mode is dominated by a magnetic dipole oscillating in the z-direction. Next, the effect of the polarization directions of dipole emitter on the PF is discussed. As shown in the inset of figure 3(a), the direction angle θ is the angle between the magnetic dipole emitter and the z-axis. The radius of the air-hole r is set to be 130 nm, and the dipole emitter is placed at the point A with the maximum field intensity. Figure 3(a) shows the PF mappings under different direction angles and wavelengths. It is clear that the PF decreases gradually as the magnetic dipole emitter goes from parallel to the z-axis (θ = 0 • or θ = 180 • ) to perpendicular to the z-axis (θ = 90 • ). The PF reaches a maximum when θ = 0 • or θ = 180 • , while it is close to 0 when θ = 90 • . This reason is that the dipole emitter placed at the direction angle θ of 0 • or 180 • overlaps almost completely with the magnetic dipole mode excited by the device. While, when θ = 90 • , the component of the magnetic dipole in the z-direction is 0, so the PF is almost 0. The polar plot of the maximum PF of the magnetic dipole emitter at different direction angles is shown in figure 3(b), where the direction angle is the pole angle and the PF corresponds to the pole diameter. As the direction angle changes from parallel to perpendicular to the z-axis, the PF gradually decreases from the maximum value 1900. These results also fully confirm that the quasi-BICs mode is a magnetic dipole resonance in the z-direction, and the PF is very sensitive to the polarization direction of the dipole emitter.  For the symmetry-protected BICs, the important feature is that the Q-factor of the excited quasi-BICs can be adjusted by asymmetric parameters, thus its local field intensity can also be effectively tuned. In our design, the smaller perturbation is obtained when the radius r of the air-hole decreases, which leads to a larger Q-factor and a stronger magnetic field intensity. The dependence of the PF value on the different air-hole radius r is calculated with the magnetic magnetic dipole emitter located at point A. As shown in figure 4(a), it can be observed that as the air-hole radius r gradually decreases (from 150 nm to 60 nm), the PF gradually increases. When r = 60 nm, it almost reaches 10 5 which is higher than many current research systems [18]. In figure 4(b), the PF and Q-factor are given with respect to the asymmetry parameter α. Here, α is defined as the ratio of the notch area to the nanodisk area. It is well known that Q-factor is proportional to α −2 . It is clear that the PF and Q-factor are similar, with deviations arising from slight differences in the effective mode volume. The PF value gradually increases as the Q-factor increases. Theoretically, the Q-factor tends to infinity as r decreases to 0, which induces an infinite PF. Therefore, the excitation of BICs is an effective method to enhance the PF.
Finally, the PL enhancement brought by quasi-BICs can be directly confirmed. In figure 5, the PL intensity is gradually enhanced as the air-hole radius r becomes smaller. The enhancement trend is consistent with that of the PF. It is noted that when r = 60 nm, the PL enhancement reaches up to 9.4 × 10 4 , which is attributed to the high optical state density of quasi-BICs. According to the symmetry-protected BICs theory, the Q-factor of quasi-BICs continues to increases as the r decreases, so the PL intensity continues to increase theoretically. Therefore, BICs can effectively promote light emission.

Conclusion
In summary, in this work, the Purcell effect of quasi-BICs in dielectric metasurface is investigated. A magnetic quasi-BICs with high Q-factor is excited by introducing air-holes at the edge of a highly symmetric nanodisks. With the excitation of a magnetic dipole in the z-direction, a huge magnetic field is trapped inside the nanodisk. The Purcell effect of quasi-BICs is studied by embedding a dipole source in the resonant nanodisk. The PF reaches the maximum when the dipole emitter overlaps well with the localized field. The PF is modulated when the polarization direction of the dipole emitter is offset from the z-axis. Moreover, as the asymmetric parameter of quasi-BICs decreases, the PF gradually increases, and its growth trend is similar to the that of the Q-factor. Therefore, the PF theoretically tends to infinity at the BICs state. Finally, the PL characteristics of the quasi-BICs with different perturbed air-hole radius are studied, and the PL enhancement factor increases with the decrease of air-hole radius. Thus, the PF and the PL intensity can be effectively tuned by changing the asymmetric parameters of the quasi-BICs, which provides an exciting platform for the realization of ultra-high PF in nanostructures.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Appendix B. Transmission spectra and Q-factors versus air-hole radius r
In figure 7, the transmission spectra and Q-factors at different air-hole radius r are calculated. It can be clearly seen that the resonant linewidth vanishes when r = 0, as r gradually increase, and a narrow resonant peak in the transmission spectrum appears and broadens, which indicates an effective excitation of a leakage mode, as shown in figure 7(a). Figure 7(b) shows the Q-factors of this mode versus r. As r decreases, the Q-factor gradually increases. When r = 0, the calculated Q-factor will tend to be infinite. These results confirm the conversion of a non-radiative BICs mode into a radiative quasi-BIC mode due to the perturbation of the air-hole. The essence is that the embedding of the air-hole transforms the high-order group C 4v into the low-order group, which opens the radiation channel, so that these non-radiative BICs transform into leaky mode quasi-BICs.