Boundary-Driven Exceptional Points in Photonic Waveguide Lattices

Abstract

We predict and analyze boundary-driven exceptional points in semi-infinite Hermitian photonic waveguide lattices with a side-coupled defect. The exceptional points arise from coherent reflections at the lattice termination, which induce strong memory effects in the defect dynamics. Using an exact analytic approach, we derive the defect's non-Markovian memory kernel, revealing the trajectories and coalescence conditions of the resonances, which can be precisely tuned by the defect position and the coupling strength. Our results provide a simple and experimentally accessible platform for exploring memory-enabled non-Hermitian physics in Hermitian photonic lattices.

Figures (4)

• Figure 1: (a) Schematic of a semi-infinite waveguide lattice with a side-coupled defect waveguide, and (b) corresponding single-level Fano Anderson model. (c) Contour paths in the complex $s$-plane for Eq. (\ref{['bsolution']}). The solid segment $\mathbf{I}$ along ${\rm Re}(s)=0$ is the branch cut of the self-energy $\Sigma(s)$. The Bromwich path $\mathbf{B}$ can be deformed into the Hankel paths $h_1$ and $h_2$, and the pole contributions $s_k$ on the second Riemann sheet (resonant states). The shaded region indicates the domain of analytic continuation for $\hat{b}(s)$ on the second Riemann sheet
• Figure 2: Behavior of the resonance poles $s_1$ and $s_2$ as a function of the normalized coupling constant $g/J$ for increasing odd values of $n_0$: (a) $n_0=3$, (b) $n_0=5$, and (c) $n_0=7$. The solid blue curves (or blue circles) correspond to the exact numerically computed poles, whereas the dashed red curves (or red asterisks) indicate the approximate poles obtained via the Lambert functions. The upper panels show the loci of the two poles in the complex $s$ plane as $g/J$ is varied from $0.1$ to $0.5$ (arrows indicate the direction of increasing $g/J$). The middle and lower panels display the real and imaginary parts of the two poles, respectively, as functions of $g/J$. The boundary-driven non-Markovian exceptional point occurs at the eigenvalue crossing.
• Figure 3: (a) Numerically-computed light intensity decay curves in the defect waveguide for $g/J=0.35$ and for a few increasing values of odd index $n_0$. The EP is crossed as $n_0$ is varied from below to above $n_0=5$. The curves have been obtained by numerically solving the coupled-mode equations (1) and (2) on a lattice comprising 150 waveguides to avoid right edge effects. The inset shows an enlargement of the decay dynamics in the last stage. As one can clearly seen, crossing the EP as $n_0$ is increased above $n_0=5$ corresponds to the emergence of an oscillatory dynamics, while the fastest decay is obtained at the EP. (b) Same as (a), where the decaying curves are computed using the approximate expression given by Eq.(14) in the text.
• Figure 4: Same as Fig.3(a), but with the additional coupling $g_1= \alpha g$ for (a) $\alpha=0$, (b) $\alpha=0.1$, (c) $\alpha=0.2$, (d) $\alpha=0.3$, and (e) $\alpha=0.4$. For the sake of visibility, only the last-stage relaxation dynamics is plotted.