Toroidal Dipole Excitation in Metamaterial Perfect Absorber Consisting of Dielectric Nanodisks Quadrumer Clusters and Spacer on Metal Substrate

Abstract

We proposed an infrared narrowband metamaterial perfect absorber (MPA) which is induced by toroidal dipole resonance in a dielectric-metal hybrid system. The MPA is composed of amorphous-silicon (a-Si) nanodisk quadrumer clusters, dielectric spacer, and Au substrate, where the dielectric spacer is inserted between Si disk quadrumer and Au substrate. Near field distribution and multipole decomposition of far-field, scattering powers show that toroidal dipole mode is formed by opposite phase magnetic dipoles in neighboring Si nanodisks. The effects of geometric and material parameters on absorption characteristics were explored. The sensing performance of the MPA was also evaluated. The proposed MPA has potential applications in air sensing applications. Since the nanodisks quadrumer of the MPA retains C_4v symmetry, perfect absorption band is polarization independent. Furthermore, the absorption quality factor of the hybrid dielectric-metal hybrid absorber is higher than that of all-metal perfect absorbers, thanks to the low loss feature of the dielectric resonator.

1. Introduction

Metamaterial perfect absorbers (MPA), which are artificial material structures usually periodically arranged, i.e., metamaterial, that can absorb incident electromagnetic waves totally or near totally at the designed wavelength [1]. MPAs can find promising applications in many regions including thermophotovoltaics [2,3], infrared stealth [4,5], radiative cooling [6,7], infrared photodetectors [8,9], sensors [10,11,12,13], and modulators [14,15] up to date since the first MPA was demonstrated in microwave frequency [16]. Physical mechanism to design MPA’s perfect absorption can usually be attributed to electromagnetic resonance including electric and magnetic multipoles [17,18,19], coupling effects between multipoles [20], and critical coupling in the bound states in the continuum [21]. Recently, a new kind of resonance dominated by toroidal dipole has attracted considerable and growing attention owing to it violates both the space-inversion and time-reversal symmetries [22], which may facilitate to design new kind of MPA.

Toroidal dipole results from poloidal currents flowing on the torus surface along its meridians with the characterization of a closed head-to-tail magnetic loop [23]. Toroidal dipole is usually overlooked in the natural medium due to a much weaker electromagnetic response compared with electric and magnetic multipoles [24]. Fortunately, toroidal dipole response can be greatly enhanced in metamaterials [25]. As a demonstration, a structure comprising of a 3D array of toroidal solenoids was firstly proved numerically in 2007 [26]. Experimental observations were taken from microwave region [24] to optical frequency [27] utilizing a subwavelength four asymmetric split ring (SRR) resonators array. Since subwavelength SRRs are not easy to fabricate especially in the optical region, a simple structure which supports magnetic resonances including plasmonic cavities [28], Metal/Dielectric/Metal structure [29] and high refractive index dielectric particle [30] were used to construct toroidal dipole resonators. Considering that all dielectric metasurfaces consisting of high refractive index dielectric particles ire easy to fabricate and has low intrinsic losses, all dielectric metasurfaces that support toroidal dipole have intrigued great attention in research about the response enhancement of toroidal dipole [31,32].

MPAs which support toroidal dipole were proposed utilizing different kinds of resonators including L-shaped metal strip [33], depth asymmetry metallic circular groove [34], and C-shaped groove array [35]. However, most of them suffer from low quality factors and polarization dependence, which is not desired for sensing application. Dielectric-metal hybrid systems are promising in designing narrow band MPAs since their bandwidth is often narrow [36,37]. Inspired by the aforementioned all dielectric metasurface which supports toroidal dipole resonance and narrow band perfect absorption in a dielectric-plasmonic system, we proposed a dielectric-metal hybrid metamaterial narrowband toroidal dipole induced perfect absorber consisting of Si disks quadrumer clusters and dielectric spacer on a metal substrate in a near-infrared wavelength range. It is found that toroidal dipole is caused by the coupling effect of magnetic dipole with opposite phase in neighboring Si nanodisks. Toroidal dipole resonance can be excited under arbitrary polarization angle at normal incidence conditions. Geometric parameters on toroidal dipole resonance perfect absorption were also studied in detail. Considering its narrow bandwidth and sensitivity to environmental medium, the performance of the proposed MPA in refractive index sensing application was also discussed.

2. Model and Method

The designed MPA is composed of amorphous-Silicon (a-Si) nanodisk quadrumer clusters, plasma-deposited Silicon Nitride (SiNx) spacer and Au substrate from top to bottom as shown in Figure 1. To keep forth-fold symmetry, the MPA is arranged periodically in both x and y directions with the same geometric parameters. Quadrumer is composed of four squarely distributed disks with diameter d and height h. The gaps between neighboring disks along x or y direction in a unit cell are labeled as g. The thickness of dielectric spacer and metal substrate are set as t₁ and t₂, respectively. The permittivity of Au is modeled by Drude equation with plasma frequency ω_p = 1.37 × 10¹⁶ rad/s and collision frequency γ = 7.31 × 10¹³ rad/s [38]. The refractive index of SiNx spacer and a-Si were set as 2 [39] and 3.42 [40]. Unless otherwise specified, the geometric parameters are set as p = 1110 nm, d = 464 nm, h = 232 nm, g = 55 nm, t₁ = 38 nm and t₂ = 400 nm.

Absorption spectra and electromagnetic field distribution are calculated by COMSOL Multiphysics. Single unit cell is simulated with periodic condition applied in x and y directions. Perfect matched layer condition is applied in z direction above the MPA distance about one-half of maximum wavelength. Since thickness of Au substrate is thick enough to prevent light from transmitting, the absorption A of whole structure can be obtained with A = 1 − R with R representing the reflection. Multipolar decomposition of scattered powers is performed in Cartesian coordinate system using Equations (1)–(11) to clarify the resonance contribution [29].

$$\mathit{J}=i\omega {\epsilon }_0\left({\epsilon }_a-{\epsilon }_b\right){\mathit{E}}_{(r)}$$

(1)

$$p=\frac{i}{\omega }{\displaystyle \int J{\mathrm{d}}^{3}r}$$

(2)

$$m=\frac{1}{2}{\displaystyle \int \left(\mathit{r}\times \mathit{J}\right){\mathrm{d}}^{3}r}$$

(3)

$$\mathit{t}=\frac{1}{10}{\displaystyle \int \left[\left(\mathit{r}\cdot \mathit{J}\right)\mathit{r}-2{\mathit{r}}^2\mathit{J}\right]{\mathrm{d}}^{3}r}$$

(4)

$$e{q}_{\alpha \beta }=\frac{i}{\omega }{\displaystyle \int \left[{\mathit{r}}_{\alpha }{\mathit{J}}_{\beta }+{\mathit{r}}_{\beta }{\mathit{J}}_{\alpha }-\frac{2}{3}{\delta }_{\alpha ,\beta }\left(\mathit{r}\cdot \mathit{J}\right)\right]{\mathrm{d}}^{3}r}$$

(5)

$$m{q}_{\alpha \beta }=\frac{1}{3}{\displaystyle \int \left[{\left(\mathit{r}\times \mathit{J}\right)}_{\alpha }{\mathit{r}}_{\beta }+{\left(\mathit{r}\times \mathit{J}\right)}_{\beta }{\mathit{r}}_{\alpha }\right]{\mathrm{d}}^{3}r}$$

(6)

$$I_p=\frac{2{\omega }^4}{3c^3}{\left|p\right|}^2$$

(7)

$$I_m=\frac{2{\omega }^4}{3c^3}{\left|m\right|}^2$$

(8)

$$I_t=\frac{2{\omega }^4}{3c^5}{\left|t\right|}^2$$

(9)

$$I_{eq}=\frac{{\omega }^6}{5c^5}\Sigma {\left|e{q}_{\alpha \beta }\right|}^2$$

(10)

$$I_{mq}=\frac{{\omega }^6}{40c^5}\Sigma {\left|m{q}_{\alpha \beta }\right|}^2$$

(11)

where J is the induced polarization current density, r is the position vector, ω is the circular frequency of incident wave, ε_a is the permittivity of corresponding material in MPA, ε_b is the permittivity of surrounding background medium and c is the speed of light in background medium. Dipole moments and quadrupole moments including electric dipole p, magnetic dipole m, toroidal dipole t, electric quadrupole eq, magnetic quadrupole mq are derived by integral the vector operation of r and J over a unit cell as described by Equations (2)–(6), and the subscripts α, β = x, y or z. The scattered power I_p, I_m, I_t, I_eq and I_mq are calculated from p, m, t, eq and mq by Equations (7)–(11).

3. Results and Discussion

Figure 2 depicts the absorption spectra (black solid line) of our designed MPA when the structure is impinged by x polarized light at normal incidence conditions. In fact, the absorption spectra at y polarized light normal incidence condition overlap with the black solid line which can attribute to the structure’s 90° rotational symmetry. Resonance absorption can be found at an optical communication wavelength 1550 nm with A larger than 0.99, and full width at half maximum (FWHM) about 5 nm. To understand the perfect absorption mechanism at 1550 nm, coupled mode theory (CMT) was used to describe the absorption spectrum theoretically [41,42]. According to CMT, the reflection coefficient r of single-port system as an analog of the MPA can be expressed as

$$r=\frac{s_{-}}{s_{+}}=\frac{{\gamma }_e-{\gamma }_i+j(\omega -{\omega }_0)}{{\gamma }_e+{\gamma }_i+j(\omega -{\omega }_0)}$$

(12)

where s₋ and s₊ represent the output and input waves of amplitudes respectively. γ_e and γ_i are external leakage rate and intrinsic loss rate of the MPA respectively. ω₀ is resonance circular frequency. Absorption A = 1 − |r|² thus A can be expressed as

$$A=\frac{4{\gamma }_e{\gamma }_i}{{(\omega -{\omega }_0)}^2+{({\gamma }_e+{\gamma }_i)}^2}$$

(13)

When the MPA is driven on resonance (ω = ω₀), and the external leakage and intrinsic loss rates are the same (γ_e = γ_i), A equals to 1 which means critical coupling states is fulfilled, and all incident energy can be absorbed totally. The Equation (13) is used to fit the COMSOL calculated absorption spectrum, and they fit well with each other, as shown in Figure 2. Fitting parameters are: ω₀ = 1.26 × 10¹⁵ rad/s and γ_e = γ_i = 4.05 × 10¹² rad/s respectively.

To gain deep insight into the physical mechanism of resonance absorption, we plot the scattering powers of different multipoles in a Cartesian coordinate system in Figure 3. It can be observed clearly that the contribution of toroidal dipole moment is dominated at resonance wavelength 1550 nm which indicates the resonance absorption is mainly induced by toroidal dipole in Figure 3a. Simultaneously, magnetic quadruple mq also provides a noticeable contribution which is consistent with previous study [39,40]. By decomposing the x, y and z components of scattered power in Figure 3b–d, we find that the x component of scatter power from toroidal dipole I_t,x is predominant in toroidal dipole scattered power and nearly equals to it in value, whereas y and z components of scatter power from toroidal dipole, I_t,_y and I_t,z approach zero compared to I_t,x. This indicates the toroidal dipole moment is along the x-direction when the MPA is illuminated by x polarized light.

To further demonstrate the absorption peak is induced by toroidal dipole resonance, we plot the electromagnetic field of the MPA at resonance absorption peak wavelength in Figure 4. The black solid lines sketch the geometry of the MPA’s unit cell. Figure 4a depicts the magnetic field distribution of the nanodisks quadrumer at x-y plane with black cones represent the displacement current. It can be observed clearly that, circulation orientations of displacement currents in y direction a-Si neighboring disks are opposite, and the same in x-direction a-Si neighboring disks, which indicates opposite phase magnetic dipoles along the z-direction were induced in y-direction disks pair. Figure 4b depicts the magnetic field distribution at the y-z plane with black arrows representing the magnetic field vector. Opposite magnetic dipoles caused a closed magnetic vortex in the y-z plane. Magnetic field vector distributed in a head-to-tail manner, which is the characterization of toroidal dipole resonance, and the direction of toroidal dipole moment is along x-direction according to the right-hand screw rule. Electromagnetic power is mainly dissipated between two y-direction coupled neighboring a-Si disk pairs, while no coupling effect caused power dissipation in x-direction as shown in Figure 4c,d. This also indicates that resonant absorption is induced by opposite phase magnetic dipole in y-direction a-Si disk pairs, which we name as toroidal dipole resonance as aforementioned. Also, the toroidal dipole resonance electromagnetic field enhancement is mainly located at lossless Si nanodisks rather than lossy metal substrate, and this is the mechanism of narrow absorption bandwidth.

Most toroidal dipole resonance induced perfect absorption in MPAs are sensitive to polarization state. Our proposed MPA has the attractive property of being polarization-independent at normal incidence condition which can attribute to its 90° rotational symmetry. As shown in Figure 5a, the shape of absorption spectra under polarization angle from 15° to 75° are almost the same as absorption spectra under x polarized incidence condition with nearly unchanged resonance wavelength. While the direction of toroidal dipole moment is polarization-dependent. When the polarization angle increases from 15° to 75°, I_t,x (solid line) increases and I_t,y (dotted line) decreases gradually as depicted in Figure 5b, which indicates the direction of t shifts from x-direction to y-direction gradually. Since I_t,z (dashed line) nearly approaches to 0 compared with I_t,x and I_t,y, direction angle θ_t of t be obtained by ${\theta }_{\mathrm{t}}=\mathrm{atan}(t_y/t_x)=\mathrm{atan}(\sqrt{I_{t,x}/I_{t,y}})$. As shown in Figure 5c, the slope of linear fit line nearly equals 1 which indicates the direction of toroidal dipole moment is parallel to polarization direction at resonance wavelength.

The effects of geometric parameters on the resonance absorption of the proposed MPA is also investigated respectively. As shown in Figure 6a,b increasing p or g causes a blueshift of resonant wavelength. Both p and g have influence on the coupling strength on neighboring Mie resonators thus when p or g were changed from critical coupled state, resonance absorption would be weakened [43]. The difference is that increasing p causes resonant state changing from under coupling state (γ_i > γ_e) to critical coupling state (γ_i = γ_e) and then to over coupling state (γ_i < γ_e) successively, whereas increasing g causes resonant state changing from over coupling state (γ_i < γ_e) to critical coupling state (γ_i = γ_e) and then to under coupling state (γ_i > γ_e) successively [44]. The resonant wavelength position is also sensitive to the height h and diameter d of Si nanodisks, because the electromagnetic scattering property of Si nanodisks is governed by the Mie resonance. In sphere Mie resonators, magnetic dipole resonance wavelength is proportional to its radius. Increasing h or d of nanodisks is equal to increasing the radius of its equivalent sphere Mie resonator thus causing a redshift of resonance wavelength as shown in Figure 6c,d.

The resonance absorption wavelength can also be tuned by the spacer’s refractive index and thickness since the dielectric spacer affects the coupling between nanodisk resonators and metal substrate [45]. As shown in Figure 7a, λ_r redshifts occur from 1542 nm to 1559 nm when increasing the refractive index of dielectric spacer ns from 1.5 to 2.5. This can be understood by the perturbation theory that (Δω/ω) = −(∫∫∫Δε|E|²dV/2∫∫∫ε|E|²dV) [43], where Δω is change of resonance circular frequency caused by material perturbation, Δε is the change of material’s dielectric permittivity. Increasing of refractive index and permittivity of spacer causes Δε > 0. Thereby, the resonance shifts towards a long wavelength (Δω < 0) according to perturbation theory. Effects of spacers thickness on resonance are shown in Figure 7b, showing that resonance wavelength redshifts from 1537 nm to 1562 nm when the t₁ changes from 30 nm to 46 nm, which may provide a new route to design the resonance wavelength of MPA.

The resonant wavelength of the MPA is dependent on the surrounding background medium, which is promising for air sensing applications. Sensing performance can be evaluated by the following formula with sensitivity S and figure of merit (FOM)

$$S=\frac{\Delta \lambda }{\Delta n},\mathrm{FOM}=\frac{S}{\mathrm{FWHM}}$$

(14)

where Δλ is the resonance peak wavelength shift and Δn is the refractive index change of the surrounding background medium. As depicted in Figure 8a, increasing the refractive index of the background medium from 1 to 1.05 can cause a red shift of resonance wavelength from 1550 nm to 1561 nm with nearly unchanged FWHM about 5 nm and step interval about 2.2 nm. The slope of the linear fitting line in Figure 8b indicates a sensitivity S = 226 nm/RIU and obtained FOM = 44.1.

4. Conclusions

In summary, MPA composed of a-Si disk symmetric quadrumer clusters, dielectric spacer, and Au substrate was proposed and the mechanism of the toroidal dipole excitation was analyzed by near field distribution and multipole decomposition of far field scattering powers. Effects of the geometrical parameters on the toroidal dipole resonance absorption were studied. The direction of the toroidal dipole moment can be tuned linearly by changing the polarization angle without weakening the absorption ability. Narrow bandwidth, polarization-independent, and high sensitivity to the refractive index fluctuation of the surrounding background medium may facilitate this metamaterial absorber to be promising in the gas sensing application area.