A Non-diffracting Resonant Angular Filter

Abstract

We conceptualise and numerically simulate a resonant metamaterial interface incorporating non-local, or beyond nearest neighbour, coupling that acts as a discrete angular filter. It can be designed to yield perfect transmission at customizable angles of incidence, without diffraction, allowing for tailored transmission in arbitrarily narrow wavenumber windows. The theory is developed in the setting of discrete, infinitely periodic quantum graphs and we realise it numerically as an acoustic meta-grating. The theory is then applied to continuous acoustic waveguides, first for the medium surrounding the interface and then for the interface itself, showing the efficacy of quantum graph theory in interface design.

Figures (3)

• Figure 1: Graph representation. (a) The $k$-space filter. Inset illustrates a top down view of a vertex with in- and out-going amplitudes $b_{e}^{\text{out/in}}$ on each edge $e$. (b) Transmission $T = |t_\mu|^2$ and reflection $R = |t_\mu - 1|^2$ amplitudes as function of $\kappa_y$ (defining incidence angle) near resonance. Field plots (i) and (ii) show the real component of the wave field $\Re\{\psi_{m,e}(z_{m,e})\}$ within the filter bonds with $\mu = 4$ and $\ell_\mu = 3\pi$ at a frequency $k = 1/\ell$. (i)/(ii) show the field off/on resonance respectively at $\kappa_y$ marked in (b).
• Figure 2: Discrete and continuous spaces, coupled by the filter. (a) A square periodic lattice with period $\ell$ with the filter attached at the midpoint of the lattice as helices that facilitate BNN connections - one helix is highlighted in black resembling the bond in Fig. \ref{['fig:Graph Setup']}(a). (b) Two rectangular acoustic waveguides coupled by a grating of thin tubes with period $\ell$, coupled at the midpoint to the filter. (c) A rectangular acoustic waveguide with periodic holes spaced by length $\ell$, the openings of which are coupled to the filter.
• Figure 3: Graph and FEM simulations. (a,b) FEM simulations of a discrete lattice with point source excitation at the frequencies marked by horizontal dashed lines in (c). The interface is marked by the vertical dashed black line. (c) Transmission coefficient as a function of frequency and angle as predicted from the graph model. (d,e) as (a,b), but for the continuous waveguides joined by the filter (Fig. \ref{['fig:3D Models']}(b)). (f) as (c) but from FEM predictions. (g,h,i) as (d,e,f) but for the continuous waveguide (Fig. \ref{['fig:3D Models']}(c)).