Temporal coupled mode theory for high-$Q$ resonances in dielectric metasurfaces

TL;DR

The paper addresses predicting and controlling high-$Q$ resonances in dielectric metasurfaces by developing a symmetry-aware temporal coupled mode theory (TCMT). It derives invariant constraints on coupling and decoupling coefficients from energy conservation and time-reversal symmetry, and applies them to grating geometries to obtain a generic Fano line-shape formula in 2D. The authors classify how unit-cell symmetries govern Fano profiles, robust reflection/transmission zeros, bound states in the continuum (BICs), and a duality between uniguided resonances (UGR) and unicoupled/coupled resonances (UCR), verified by numerical simulations. The framework provides analytic insight into designing metasurfaces with tailored spectral features and suggests extensions to lossy systems and critical coupling regimes.

Abstract

In this work, we propose a coupled mode theory for resonant response from quasi-guided modes in periodic dielectric metasurfaces. First, we derived a generic set of constraints imposed onto the parameters of the temporal coupled mode theory by energy conservation and time-reversal symmetry in an invariant form that allows for asymmetry between the coupling and decoupling coefficients. The proposed approach is applied to the problem of Fano resonances induced by isolated quasi-guided modes in the regime of specular reflection. Our central result is a generic formula for the line-shape of the Fano resonance in transmittance for the lossless metasurfaces in the framework of 2D electrodynamics. We consider all possible symmetries of the metasurface elementary cell and uncover the effects that the symmetry incurs on the profile of the Fano resonance induced by an isolated high-$Q$ mode. It is shown that the proposed approach correctly describes the presence of robust reflection and transmission zeros in the spectra as well as the spectral signatures of bound states in the continuum. The approach is applied to uniderictionally guided resonant modes in metasurfaces with an asymmetric elementary cell. It is found that the existence of such modes and the transmittance in their spectral vicinity are consistent with the theoretical predictions. Furthermore, the theory predicts that a uniderictionally guided resonant mode is dual to a counter-propagating mode of a peculiar type which is coupled with the outgoing wave on both sides of the metasurface but, nonetheless, exhibits only a single-sided coupling with incident waves.

Figures (10)

• Figure 1: Dielectric metasurface in the form of a ruled grating. (a) The light blue bars show the ruled grating with refractive index $n_1 = 3.5$. The plane of incidence is shaded in gray. The magenta arrows shows the wave vector $k$ of the incident field. The $y$-component of the electric field and the $x$-component of $k$ are shown as vectors. The refractive index of ambient medium $n_0 = 1.5$. (b) Diagram of diffraction orders. The colorbar on the right indicates the number of channels open for diffraction on either side of the metasurface.
• Figure 2: Definition of the scattering channels for the structure shown in Fig. \ref{['Fig1']} (a). $\theta$ is the angle of incidence.
• Figure 3: Grating with a unit cell of various symmetries. A -- rectangular unit cell (D$_{2h}$), B -- isosceles trapezoid unit cell (C$_{2v}^x$), C -- isosceles trapezoid unit cell (C$_{2v}^z$), D -- parallelogram unit cell (C$_{2h}^y$), E -- right trapezoid unit cell (C$_{s}^y$).
• Figure 4: Transmission through the grating with rectangular unit cell (D$_{2h}$). (a) The spectra of the high-$Q$ leaky band calculated by FDFD with the mode profiles shown on top. (b) Transmittance and the energy stored by the FDFD and the TCMT methods. (c) Transmittance against the incident frequency; TCMT -- dotted line, FDFD -- solid line. The scattering solutions at $\omega=\omega_0$ are shown on the right of each plot. The geometry is described in the inset in subplot (a); $w = 0.675 \Lambda$, $h = 0.3 \Lambda$.
• Figure 5: Transmission through the grating with isosceles trapezoid unit cell (C$_{2v}^x$). (a) The spectra of the high-$Q$ leaky band calculated by FDFD with the mode profiles shown on top. (b) Transmittance and the energy stored by the FDFD and the TCMT methods. (c) Transmittance against the incident frequency; TCMT – dotted line, FDFD – solid line. The geometry is described in the inset in subplot (a). The scattering solutions at $\omega=\omega_0$ are shown on the right of each plot. The in-plane mirror symmetry is broken by setting $\phi = 80^{\circ}$ according to the scheme shown in Fig. \ref{['Fig3']}.