Resonances in finite-size all-dielectric metasurfaces for light trapping and propagation control

We investigate the development and tuning of resonant optical effects in finite-size periodic arrays (metasurfaces) of silicon nanoparticles. By applying Green's tensor formalism and the coupled dipole approximation while incorporating electric and magnetic dipole moments, we outline a theoretical framework to model the optical response of such nanoparticle arrays. We consider the resonant optical response of finite-size arrays as a function of the nanoparticle (unit cell) number in two distinct scenarios of collective resonances: the lattice resonant Kerker effect, which is a complete suppression of the backward scattering, and the quasi-bound state in the continuum. Our developed models and findings provide a pathway for extracting crucial details about the lattice period and the required array size for the experimental observation of collective resonances. These resonances are typically predicted under the assumption of an infinite periodic lattice. By bridging the theoretical predictions with practical considerations, our results contribute to better understanding of specific conditions needed to experimentally observe these collective resonances in finite-size arrays.

Figure 1

Extinction cross section ${\sigma }_{\mathrm{ext}}$ (ECS), scattering cross section ${\sigma }_{\mathrm{sca}}$ (SCS), and contributions to the SCS from electric dipole (ED), magnetic dipole (MD), electric quadrupole (EQ), and magnetic quadrupole (MQ) for a silicon nanoparticle illuminated by a plane wave propagating in the medium with $n_S=\sqrt{{\varepsilon }_S}=1.4$ (see the inset). The blue circle indicates the wavelength ${\lambda }_K=834$ nm at which the first Kerker condition is satisfied. The values of the wavelength $\lambda$ are indicated for vacuum.

Figure 2

Illustration of the considered collective resonant effects in a periodic finite-size metasurface of subwavelength-separated Si particles ($d<\lambda /n_S$) with a square unit cell. (a) Lattice Kerker resonance: at the Kerker wavelength, a normally incident plane wave can resonantly induce electric and magnetic dipoles with the same amplitude in each particle due to the overlapping of collective ED and MD resonances. The scattered field from these dipole moments yields zero reflection from the metasurface. (b) Nonradiant trapped eigenmode: dipole bound state in the continuum. For the magnetic dipole BIC, all particles have the same MD moment with the out-of-plane orientation ${\mathbf{m}}_0\parallel {\mathbf{e}}_z$.

Figure 3

Numerically evaluated lattice sum for in-plane ($\parallel$) dipoles (blue) and the real part of the inverse single-particle dipole polarizabilities (red) as a function of the lattice period $d<{\lambda }_K/n_S$ at the Kerker wavelength ${\lambda }_K=834$ nm (see Fig. 1). The quantities are normalized by $6\pi /k_S^3$. The blue circle indicates the lattice period such that the ED and MD lattice resonances overlap, and the lattice resonant Kerker effect emerges according to Eq. (10).

Figure 4

(a) Effective ED and MD polarizabilities of a nanoparticle in the infinite square lattice for in-plane dipoles (9) as a function of wavelength of normally incident plane wave. (b) Scattering cross section per particle ${\sigma }_0$ for (blue) the infinite and (red dashed) $35\times 35$ particle array. The values of the wavelength $\lambda$ are indicated for vacuum.

Figure 5

(a) Scattering cross section per particle ${\sigma }_{0,N}\equiv {\sigma }_{\mathrm{sca}}\left(N\right)/N^2$, where ${\sigma }_{\mathrm{sca}}\left(N\right)$ is the total scattering cross section (6) for the $N\times N$ array with lattice period $d=577$ nm. The array is illuminated by an external plane wave as shown in Fig. 3. The green points, labeled c and d, indicate $N$ and $\lambda$ for which the radi- ation patterns are plotted in (c) and (d), respectively. (b) The quality factor of the collective Kerker resonance as a function of $N$, which is normalized by the quality factor for the infinite lattice $Q_{\infty }=70$ calculated via Fig. 4. (c) Far-field radiation pattern, given by Eq. (4), for the $9\times 9$ array with lattice period $d=577$ nm at the Kerker wavelength ${\lambda }_K=834$ nm. (d) The same but at the wavelength $\lambda =900$ nm, which is far from resonances. The values of the wavelength $\lambda$ are indicated for vacuum.

Figure 6

Numerically evaluated real part of lattice sum for out-of-plane ($\perp$) dipoles (red), and the real part of the inverse single-particle MD polarizability (green) as a function of lattice period $d<\lambda /n_S$ at the wavelength $\lambda =808$ nm. The quantities are normalized by $6\pi /k_S^3$. The blue circle indicates the lattice period such that the lattice supports a magnetic dipole BIC according to Eq. (C1).

Figure 7

Magnetic dipole quasi-BIC resonance in the finite-size array of $13\times 13$ particles with period $d=495$ nm. (a) Mean values $|\langle m_x\rangle |$ (dashed) and $|\langle m_z\rangle |$ (solid) of MD components [Eq. (12)] that can be induced by an oblique incident plane wave with the polarization shown by the inset. The components are normalized by $|{\alpha }_m{\mathbf{H}}_{\mathrm{inc}}|$. The angle of incidence is ${\theta }_{\mathrm{inc}}=2^{\circ }$. (b) Total scattering cross section (6) per particle ${\sigma }_{0,N}\equiv {\sigma }_{\mathrm{sca}}\left(N\right)/N^2$ (gray solid), and independent contributions of EDs and MDs to the scattering per particle (red dashed-dotted and green dotted, respectively). The values of the wavelength $\lambda$ are indicated for vacuum.

Figure 8

Mean value $|\langle m_z\rangle |$ of the out-of-plane MD component, given by Eq. (12), that is induced by an oblique incident plane wave with the polarization as shown by the inset in Fig. 7. The angle of incidence is ${\theta }_{\mathrm{inc}}=2^{\circ }$. The $N\times N$ array has a lattice period of $d=495$ nm. The resonant phenomenon is associated with the excitation of magnetic dipole quasi-BIC in the finite array of nanoparticles. The values of the wavelength $\lambda$ are indicated for vacuum.

Figure 9

Normalized magnetic energy density determined by the total magnetic field in the $xy$ plane at position $z=110$ nm above the array. The array has (a) $35\times 35$ particles and (b) $21\times 21$ particles. An oblique incident plane wave, shown by the inset in Fig. 7, has a wavelength of the resonant magnetic dipole quasi-BIC excitation $\lambda =808$ nm.

Figure 10

(a) Mean value $|\langle p_z\rangle |$ of the out-of-plane ED component induced from the near field by an electric dipole ${\mathbf{p}}_{\mathrm{inc}}$ as shown in the inset. The electric dipole is placed at $g=150$ nm above the $13\times 13$ array. The pink markers indicate the pairs of parameters $(d,\lambda )$, which obey Eq. (C1) corresponding to the electric dipole BIC observation in the infinite array. (b) Total scattered power (5) for the $N\times N$ array with lattice period $d=423$ nm as a function of $N$ and wavelength of incident dipole radiation. The power $P_{\mathrm{sca}}$ is normalized by the power of the incident dipole radiation $P_{\mathrm{inc}}=\frac{4{\pi }^3{\left|{\mathbf{p}}_{\mathrm{inc}}\right|}^2}{3{\varepsilon }_0^2{c}_S{\mu }_0{\lambda }^4}$ where ${\mu }_0$ is the vacuum magnetic permeability [9]. The values of the wavelength $\lambda$ are indicated for vacuum.

Figure 11

Inverse normalized (multiplied by factor $k_S^3$) dipole polarizabilities (A3) for the considered spherical particle. Their imaginary parts are $\mathrm{Im}\left[{\left(k_S^3\alpha \right)}^{-1}\right]=1/\left(6\pi \right)$, where $\alpha ={\alpha }_p/\left({\varepsilon }_0{\varepsilon }_S\right)$ or $\alpha ={\alpha }_m$, in the considered spectral range. The blue circle indicates wavelength ${\lambda }_K=834$ nm when the first Kerker condition ${\alpha }_p/\left({\varepsilon }_0{\varepsilon }_S\right)={\alpha }_m$ is satisfied; see Sec. 3. The values of the wavelength $\lambda$ are indicated for vacuum.

Figure 12

(a) An array with in-plane disorder of nanoparticle positions. The largest shift of nanoparticle position $\mathrm{\Delta }{r}_j$ is randomly selected from range $\left[0,\mathrm{\Delta }{r}_{\max }\right]$, where the angle ${\phi }_j$ is randomly selected from 0 to $2\pi$. Here, integer $j$ runs from 1 to $N_{\mathrm{tot}}$. (b) Averaged scattering cross-section per particle $\langle {\sigma }_{0,N}\rangle \equiv \langle {\sigma }_{\mathrm{sca}}\left(N\right)\rangle /N^2$ where $\langle {\sigma }_{\mathrm{sca}}\left(N\right)\rangle$ is the averaged total scattering cross section (6) for the $N\times N$ array with period $d=577$ nm and $\mathrm{\Delta }{r}_{\max }=70$ nm. The array is illuminated by an external plane wave, as shown in Fig. 3. The averaging procedure is performed for $N_{\mathrm{iter}}=10$ configurations of the array. The values of the wavelength $\lambda$ are indicated for vacuum.

Figure 13

Development of the total quality factor of the magnetic dipole quasi-BIC in finite-size arrays as a function of $N$ for (a) different values of absorption losses in particles, and (b) different angles of incidence. The inset shows the spectrum of the normalized (by a maximal value for certain $N$) mean $z$ component of MD moments for $N=35$ at ${\theta }_{\mathrm{inc}}=2^{\circ }$ and with material losses.