Dipole-quadrupole model and multipole analysis of resonant membrane metasurfaces

TL;DR

We introduce a semi-analytical Dipole-Quadrupole Model (DQM) to analyze resonance spectra of membrane metasurfaces under arbitrary incidence by incorporating electric and magnetic quadrupoles alongside dipoles. The framework derives TE and TM reflectance/transmittance expressions in terms of dipole and quadrupole moments, with careful consideration of unit-cell localization to ensure convergence. The approach explains a range of phenomena, including lattice anapole and generalized Kerker effects, Fano resonances, quasi-BICs, and anti-Fano behavior, while remaining applicable to non-centrosymmetric unit cells such as conical or partially perforated holes. Validation against full-wave simulations shows excellent agreement, providing a practical design and interpretation tool for both conventional and membrane metasurfaces with potential applications in sensing and nonlinear optics.

Abstract

Membrane-metasurfaces, formed by periodic arrangements of holes in a dielectric layer, are gaining attention for their easier manufacturing via subtractive techniques, unnecessity of substrates, and access to resonant near-fields. Despite their practical relevance, their theoretical description remains elusive. Here, we present a semi-analytical dipole-quadrupole model for the multipole analysis of numerically-obtained reflection and transmission spectra in metasurfaces excited at arbitrary angles. Dipole models are generally sufficient to study traditional metasurfaces made of solid nanostructures. However, the inclusion of electric and magnetic quadrupoles is necessary to study membrane-metasurfaces, which offer an ideal platform to showcase our method. We demonstrate the importance of choosing the optimal position of a symmetric membrane-metasurface's unit cell to ensure the sufficiency of the dipole-quadrupole approximation. We show that our formalism can explain complex phenomena arising from inter-multipole interference, including lattice anapole and generalized Kerker effects, Fano resonances, and quasi-bound states in the continuum. We also present the applicability of the method to membrane-metasurfaces with non-centrosymmetric unit cells (e.g., conical holes and surface voids). By enabling a deeper insight into the coupling mechanisms leading to the formation of local and collective resonances, our method expands the electromagnetics toolbox to study, understand, and design conventional and membrane-metasurfaces.

Figures (10)

• Figure 1: Schematics of a membrane metasurface (alias "holey-metasurface") made of perforated circular (diameter $D$) periodic (period $P$) holes on dielectric layer (thickness $H$) and respective irradiation condition. The refractive index of the dielectric membrane and surrounding environment is $n_{\rm d}$ and $n_{\rm s}$, respectively. The metasurface's possible symmetric unit cell (UC) configurations are indicated by (UC1) red, (UC2) blue, (UC3) black and (UC4) purple dashed squares. The unit cell can have an asymmetric shape, e.g, the yellow dashed square illustrates one of the possible asymmetric unit cell (AUC). The arrows ($x$,$y$,$z$) indicate the direction of the Cartesian coordinate axes and the $xy$-plane is at $z=0$, i.e., the middle of the membrane layer. Note that the origin of the coordinate system has to coincide with the center of mass of the UC.
• Figure 2: (a) Reflectance and (b) its absolute multipole contributions of the membrane metasurface shown in Fig. \ref{['fig:str']} with $H=200$ nm, $D=200$ nm, $P=300$ nm and the normal incidence excitation ($\theta=\varphi=0$). The inset in (b1-b4) shows the considered unit cell configuration, UC1-UC4, respectively. The multipole contributions are plotted in a unitless form as the absolute values of corresponding terms in Eq. \ref{['rrr']}. The refractive index of the dielectric layer and surrounding environment is $n_{\rm d}=2.45$ and $n_{\rm s}=1$, respectively.
• Figure 3: (a) Reflectance and its (b) absolute dipole and (c) absolute quadrupole contributions of the membrane metasurface under the conditions considered in Fig. \ref{['fig:rmuc']}. The inset in (b) shows the considered unit cell configuration: (1) symmetric and (2) asymmetric unit cell. The black dot indicates the center of multipole decomposition. In (d1) and (d2), it is shown the distribution of the real part of the $z$-component of the electric field within symmetric and asymmetric unit cells ($xz$-plane, $y=0$), respectively, at the wavelength $\lambda=545$ nm. The black arrows indicate the direction of the electric field. (e) Schematic presentation of the scattering of an array of out-of-plane dipole moments in the $xz$-plane.
• Figure 4: Comparison of (a) full numerical, (b) our dipole-quadrupole model (DQM), and (c) coupled-dipole model (CDM) based specular reflectance of the structure considered in Fig. \ref{['fig:rmuc']} as a function of the incidence angle $\theta$ at $\varphi=0$ for both (top panels, 1) TE and (bottom panels, 2) TM polarizations. In DQM and CDM calculations, the unit cell configuration UC1 is considered. sBIC: symmetry-protected BIC, aBIC: accidental BIC. The vertical dashed lines show the position of the sections shown in Fig. \ref{['fig:Rt10']}.
• Figure 5: Multipole decomposition of (a) metasurface resonances (red dashed: numerical, blue solid: DQM) into (b) dipole and (c) quadrupole contributions for (1) TE and (2) TM polarized illumination at $\theta=10^{\circ}$ and $\varphi=0$. The absolute dipole and quadrupole contributions [moments with tilde in (b1--c2)] are plotted in a unitless form, i.e., they include all coefficients (including angular dependencies) of the moments in Eqs. \ref{['rTE']} and \ref{['rTM']}. In (c1), $\tilde{M}_{xx-zz} = \tilde{M}_{xx} - \tilde{M}_{zz} = 2(\tilde{M}_{xx} + \tilde{M}_{yy}/2)$. The metasurface parameters are as in Fig. \ref{['fig:rtuc1']}.