A Flux-Tunable Discrete Angular Filter

TL;DR

Problem: discrete angular filtering in wave systems is topology-constrained; Approach: model as a periodic quantum graph with magnetic flux $\Phi$ along helix bonds and apply $\delta$-type boundary conditions with parameter $\lambda$, employing a gauge transformation to move $A$ into a phase and derive the one-dimensional magnetic Schrödinger equation on each edge; derive analytic lead-to-lead transmission $t_\mu(k,\kappa_y)$ and resonance conditions $\kappa_y^{(q)} = (\Phi + q\pi)/(\mu \ell)$. Findings: at resonance the transmission is nonzero only when $k \ell_\mu$ equals $p\pi$ and $\kappa_y=\kappa_y^{(q)}$, with amplitude $(2ik)/(2ik-\lambda)$; tuning $\Phi$ shifts pass angles and setting $\lambda=0$ recovers the topology-fixed case. Significance: this yields a programmable, flux-tunable angular filter for non-reciprocal transport and beam shaping with potential applications in analogue wave processing, imaging, and sensing.

Abstract

Recent work by Lawrie et al. [PRR 7, 023209 (2025)] introduced a non-diffracting resonant angular filter on a network of thin channels (modelled via quantum graph theory) that exhibits unit transmission of acoustic waves at a discrete, symmetry-paired set of incidence angles determined solely by the graph topology, while transmission at all other angles is strictly forbidden. In the present work, we study the same filtering geometry for waves governed by the magnetic Schrödinger equation rather than the classical wave equation. Using a phase shift induced by non-reciprocal wave propagation due to the presence of the magnetic potential and tuning $δ$-type vertex boundary conditions, we make the previously topology-fixed discrete pass directions continuously tunable: both the transmission angle and the transmission coefficient become control parameters. The resulting flux-tunable angular filtering device replaces topology-constrained passbands with a programmable steering device, broadening the scope for wave-filter and beam-shaping applications.

Figures (2)

• Figure 1: (a) shows a schematic of the filter placed between two semi-infinite half spaces (shown in grey), where waves are free to travel. Illustrated via an arrow is a wave of amplitude $1$ with incidence angle $\theta$ scattering at the filter into a reflected wave with amplitudes $r_{\mu}$ and a transmitted wave with amplitude $t_{\mu}$. (b) shows a schematic of the filter itself which is formed of an array of vertices with period $\ell$ and discrete index $m$ coupled to the scattering environment by leads to the left ($l$) and right ($r$) of the vertex and to one another in the down ($d$) and up ($u$) directions via a bond of length $\ell_{\mu}$, here $\mu = 1$. The connected bonds form a helix, through which a solenoid is placed which will induce a magnetic potential through the bonds. (c) shows a side view of the filter where the solenoid is a constant distance $r$ away from the graph bonds; emphasised are the local edge wave amplitudes heading in and out of vertex $m$.
• Figure 2: (a) The squared magnitude of the transmission coefficient from Eq. (\ref{['eq: Transmission Definition']}) for a filter with unit period $\ell = 1$, nearest-neighbor vertex connections ($\mu = 1$), and bond length $\ell_{\mu} = 2\pi/k + \epsilon$ where $\epsilon = 0.01$ (off-resonance). The red curve corresponds to $\Phi = \lambda = 0$, showing unit transmission at zero angle. The blue curve illustrates the effect of changing the magnetic potential to $\Phi = 1$, while the green curve shows the effect of changing the vertex parameter to $\lambda = 2$. (b--d) Effects of different boundary configurations on two incident beams at angles $\theta = 0$ and $\pi/4$ radians. (b) Unit transmission is achieved for the zero-angle beam, while transmission is excluded for the beam at incident angle $\theta = \pi/4$. (c) Unit transmission is achieved for the beam at incident angle $\theta = \pi/4$. (d) The transmission coefficient is chosen such that an ideal incident angle is split evenly at the boundary.