1 Introduction

Enhancing magnetic responses in all-dielectric nanostructures and metamaterials with high-refractive-index offer a powerful platform for efficient manipulation and localization of light at the nanoscale [1,2,3]. Magnetic dipole responses supported by all-dielectric metamaterials are particularly intriguing, as it can efficiently concentrate magnetic energy into subwavelength scales due to very low radiation loss compared with the electric dipole responses [4,5,6]. Various types of promising functions in dielectric nanostructures have been realized via involving MD responses, including enhanced magnetic nonlinearity [7], controllable light absorption [8], boosting MD transitions [9], dynamic nonlinear image [10] and directional scattering of light [11]. Therefore, how to fully exploit MD responses is extremely crucial for realizing and functionalizing the dielectric metamaterials.

Many mechanisms in dielectric metamaterials have been applied to enhance magnetic responses by minimizing the radiative loss including using trapped mode [12], Fano interference [13] and high-order modes [14], especially optical bound states in the continuum (BICs) [15,16,17,18], which make it possible to further improve the performance of magnetic responses in nanostructures. Ideal BIC in an optical system cannot directly be excited by incident wave and observed as a dark mode [19] due to infinitely long lifetime and infinitely high Q-factor, while quasi-BICs with finite lifetime and Q-factors allow the leakage of low radiation into free space and have been nicely observed in dielectric metamaterials [20,21,22,23]. Especially, the MD induced by quasi-BIC is known for its ability to boost the nonlinear process due to the associated string near-field enhancement and larger mode volume compared to other multipoles, which provide an approach to realize extreme light localization, nonlinear optical effects and radiation engineering [8, 10, 24, 25].

Recently, quasi-BICs allow small radiative leakage which results in finite but high Q-factors and can be realized by tuning structural symmetry and excitation parameters to distort the original ideal BICs [26,27,28]. Magnetic responses are highly dependent on the geometry of dielectric nanostructures, which broadens the horizon to manipulate MD responses induced by quasi-BICs by introducing geometric asymmetry [10, 24, 25, 29]. More importantly, magnetic responses are also highly dependent on choice of the structure’s material, such as the permittivity or refractive index, offering an equally promising but a much less studied approach to control the MD quasi-BICs [30, 31].

In this work, we study multiple MD excitation supported by an all-dielectric metamaterial composed of permittivity-asymmetric nanodisk dimer. Through breaking permittivity symmetry in the nanodisk dimer, symmetry-protected dual quasi-BICs with Fano line-shape can be observed in the transmission, whose Q-factors show the inverse quadratic dependence on asymmetry parameters. The calculated far-field contributions from multipole components and magnetic near-field distributions show the dominant multiple MD excitation, which preserve asymmetric magnetic confinement with large localized field enhancement. Further, multiple MDs with high Q-factors are designed as a multiwavelength sensor. Our work provides a useful insight into the multiple MD excitation induced by quasi-BICs in a permittivity-asymmetric all-dielectric metamaterial.

2 Structures description

Figure 1 shows the all-dielectric metamaterials of periodic nanodisk dimer embedded in a thin glass layer of thickness H. The lattice constant of the dimer unit cell is P. Geometric parameters of the dimer consist of the disk radius R and the gap width G. In our simulation, all transmission spectra are calculated by FDTD Solutions based on the finite-difference time-domain (FDTD) method, multipolar decomposition and near-field enhancement are calculated by COMSOL Multiphysics based on finite element methods, where periodical boundary conditions are set in the x and y directions, perfectly matched layers are set in the z direction and the x-polarized plane wave propagated along the z axis is normally incident on the structure. And we use the lossless silicon and glass with the permittivity ε0 = 13.7 and εg = 1.94 in the calculation, respectively. The introduced permittivity-asymmetric ε can be realized by control the proportion of the silicon and doping impurities in the nanodisk. The asymmetric parameter δ = \(\sqrt{{\varepsilon }_{0}}- \sqrt{\varepsilon }\) is introduced to specify the structural asymmetry of unit cell.

Fig. 1
figure 1

a Schematic of all-dielectric metamaterials of nanodisk dimer embedded in a thin glass layer of thickness H. b Top-view cross section of the unit cell and definition of the geometrical parameters of the dimer. The structural symmetry is broken by introducing different permittivities in the dimer

Full size image

3 Simulation results and discussions

Figure 2a shows the calculated transmission spectra at different δ. When the δ = 0.43, we can observe three modes marked as I, II, III, respectively. As the δ decreases, corresponding to decreasing the doping impurities, it can be seen that resonant wavelength of mode I slightly increases due to the increased effective refractive index of the dimer, while modes II and III undergo an obvious red shift and become sharper and finally vanish when the permittivity is symmetric (δ = 0), which confirms the existence of two symmetry-protected BIC states. Along with the permittivity asymmetry in the dimer, such two BIC states can be transformed into two quasi-BICs with Fano line-shape due to the interference between the free space continuum and discrete BIC states supported by dimer. This Fano line-shape can be fitted by a typical Fano formula given by [32]

$${T}_{\mathrm{F}{\text{ano}}}(\omega )={\left|{a}_{1}+i{a}_{2}+\frac{b}{\omega -{\omega }_{0}+i\gamma }\right|}^{2}$$
(1)

where a1, a2 and b are constant real numbers, γ is the overall damping rate and ω0 is the central resonant frequency of the mode. And the Q-factor Qrad of the mode can be extracted by Qrad = ω0/2γ. The fitted results for the δ = 0.43 are shown in Fig. 2(b). The Q-factors of mode I at 1734 nm, mode II at 1651 nm, mode III at 1552.7 nm reach 2357, 462 and 11,930, respectively. Figure 2c and d shows the extracted Q-factors of modes II and III at different δ. The Q-factors for the mode II and III exhibits the inverse quadratic trend and diverges as the δ approaches 0, which demonstrates that such two modes are indeed BICs.

Fig. 2
figure 2

a Evolution of simulated transmission spectra versus the δ changing from 0 to 0.43. Three modes are marked by I, II and III, respectively. Structural parameters: R = 230 nm, T = 200 nm, G = 40 nm, Px = Py = 1200 nm. b Fano fitted spectrum near the three modes for the δ = 0.43. (c, d) Dependence of Q-factors on the δ for the mode II and mode III, respectively

Full size image

Figure 3 shows the far-field scattered powers of the induced electric dipole, magnetic dipole, toroidal dipole, electric quadrupole and magnetic quadrupole moments based on the displacement current density under the Cartesian coordinate,

$$ {\overrightarrow{P}}= \frac{1}{i\omega }\smallint {\overrightarrow{j}}d^{3} r, $$
(2)
$$ {\overrightarrow{M}} = \frac{1}{2c}\smallint ({\overrightarrow{r}} \times {\overrightarrow{j}})d^{3} r, $$
(3)
$$ {\overrightarrow{T}} = \frac{1}{10c}\smallint [({\overrightarrow{r}} \cdot {\overrightarrow{j}}){\overrightarrow{r}} - 2r^{2} {\overrightarrow{j}}]d^{3} r, $$
(4)
$$ Q_{\alpha \beta }^{(e)} = \frac{1}{2i\omega }\smallint [r_{\alpha } j_{\beta } + r_{\beta } j_{\alpha } - \frac{2}{3}({\overrightarrow{r}} \cdot {\overrightarrow{j}})\delta_{\alpha \beta } ]d^{3} r, $$
(5)
$$ Q_{\alpha \beta }^{(m)} = \frac{1}{3c}\smallint [({\overrightarrow{r}} \times {\overrightarrow{j}})_{\alpha } r_{\beta } + ({\overrightarrow{r}} \times {\overrightarrow{j}})_{\beta } r_{\alpha } ]d^{3} r, $$
(6)

where c is the speed of light in a vacuum and the subscript α, β = x, y, z [33]. The \(\overrightarrow{\text{P}}\), \(\overrightarrow{\text{M}}\),

Fig. 3
figure 3

Scattered powers of the Cartesian ED, MD, TD, EQ, and MQ for mode I at a δ = 0 and b δ = 0.43, respectively. (c, d) mode II and III at δ = 0.43

Full size image

\(\overrightarrow{\text{T}}\), \({\overrightarrow{Q}}^{(e)}\)and\({\overrightarrow{Q}}^{(m)}\)are the electric dipole (ED), magnetic dipole (MD), toroidal dipole (TD), electric quadrupole (EQ) and magnetic quadrupole (MQ) moments, respectively, and the corresponding scattered power is calculated by \({I}_{P}= \frac{2{\omega }^{4}}{3{c}^{3}}{\left|\overrightarrow{\text{P}}\right|}^{2}\), \({I}_{M}=\frac{2{\omega }^{4}}{3{c}^{3}}{\left|\overrightarrow{\text{M}}\right|}^{2}\), \({I}_{T}=\frac{2{\omega }^{6}}{3{c}^{5}}{\left|\overrightarrow{\text{T}}\right|}^{2}\), \({I}_{{Q}^{(e)}}=\frac{{\omega }^{6}}{5{c}^{5}}\sum {\left|{\overrightarrow{Q}}_{\alpha \beta }^{\text{(e)}}\right|}^{2}\) and \({I}_{{Q}^{(m)}}=\frac{{\omega }^{6}}{40{c}^{5}}\sum {\left|{\overrightarrow{Q}}_{\alpha \beta }^{\text{(e)}}\right|}^{2}\), respectively. For mode I in Fig. 3a, the TD moment plays a dominant role at 1758 nm in comparison with other multipoles when the δ = 0, while the MD moment is the strongest contributor for the mode I in Fig. 3b due to the introduced permittivity asymmetry when the δ = 0.43. For mode II and III, MD contribution dominates across the wavelength region when the δ = 0.43. This further confirms multiple MD excitation in permittivity-asymmetric all-dielectric metamaterials.

The dominant contributions of three MD modes can also be verified in magnetic near-field enhancement at resonant wavelengths in Fig. 4. The dominant TD moment in mode I manifests itself in the symmetric near-field distribution in Fig. 4a, where the magnetic field can be firmly constrained within the dimer and no radiation energy transmits outward at 1758 nm, and the electric field in the x–y plane forms two peculiar reversed loops, indicating a typical TD feature. However, asymmetric magnetic field enhancement in mode I is excited in Fig. 4b due to the introduced permittivity asymmetry when the δ = 0.43. The asymmetric MD here is mainly aligned with the longitudinal direction (z-axis) and confined in the nanodisk with the high permittivity (ε0 = 13.7). On the other hand, the dominant asymmetric magnetic near-field enhancement in mode III is mainly confined in the nanodisk with the low permittivity (ε = 10.7) in Fig. 4d. For mode II, the dominant MD enhancement in Fig. 4c is mainly localized in the nanodisk with the low permittivity (ε = 10.7), while weak TD moment is excited and confined in the nanodisk with the high permittivity (ε0 = 13.7). It should be noted that MD moments in three modes cannot radiate along the z direction and radiation in the x–y plane can be ignored due to the array effect, which results in high Q-factors of multiple MD modes.

Fig. 4
figure 4

Normalized magnetic field amplitudes |H/H0| in the middle x–y plane of the nanodisk dimer for different modes. a Mode I for the δ = 0. (b–d) Modes I, II, III for the δ = 0.43. White arrows show the electric field directions

Full size image

Figure 5a shows the calculated transmission spectra when the nanodisk dimer metamaterials (δ = 0.43) are immersed in different refractive index of gaseous medium from 1.00 to 1.06. Thanks to high Q-factors of the three modes, transmission spectra show a remarkable red shift for modes I, II and III, even though a small refractive index fluctuation (Δn = 0.02). To have a detailed view of this trend, the shift of extracted resonant wavelengths Δλ corresponding to the different refractive indices n is shown in Fig. 5b, where the fitted lines show a good linearity. The sensitivity (S) and figure of merit (FoM) are calculated as

$${S}_{\lambda }(n)= \frac{\Delta \lambda }{\Delta n}$$
(7)
$$\mathrm{FoM}(n) = \frac{{S}_{\lambda }(n)}{\mathrm{FWHM}(n)}$$
(8)

where FWHM is the full width at half maximum of mode [34]. As shown in Fig. 5b, the theoretical bulk refractive index sensitivity \({S}_{\lambda }(n)\) for modes I, II and III reaches 206, 363 and 298 nm/RIU, respectively. It is noteworthy that the sensitivity of modes II and III is higher than that of mode I due to the properties of quasi-BICs. Given the line width of the modes, the corresponding FoM of modes I, II and III reaches 229, 108 and 2292, respectively. Compared with the sensitivity with other published papers, the proposed metamaterial with high sensitivity and FoM shows its great superiority in multiwavelength biosensing application over the previous works [35,36,37,38] and can modify lines at several spectral positions simultaneously.

Fig. 5
figure 5

a Transmission spectra of the proposed metamaterials (δ = 0.43) for gaseous medium with different refractive index n. b Wavelength shifts as a function of the refractive index n

Full size image

4 Conclusions

In conclusion, we have shown that multiple magnetic dipole excitation can be realized in an all-dielectric metamaterial composed of permittivity-asymmetric nanodisk dimer with symmetric geometric parameters. In addition to the original MD mode, symmetry-protected dual quasi-BICs induced by introducing the permittivity asymmetry in the nanodisk dimer are observed and their Q-factors show inverse square dependence on the asymmetric parameter. The calculating far-field radiations and magnetic near-field distributions explicate the dominant multiple MD excitation in the metamaterials. Finally, the sensing characteristics of multiple MD modes are analyzed. This in-depth analysis of multiple MD excitation in permittivity-asymmetric all-dielectric metamaterials induced by quasi-BICs may facilitate the realization of efficient light–matter interaction with better performance.