Edge mode resonance in the dark state by finite beam

We treat edge-mode resonance that may exist at boundaries of transversely finite beams illuminating a photonic lattice. The lattice is in the dark state signifying a perfect bound state in the continuum (BIC). The dark state is non-radiative in symmetric systems because lateral waves cannot couple to the lattice due to destructive interference between counter-propagating waves. However, if the incident beam is spatially finite, the balance is broken at beam boundaries, and edge modes are induced supporting local electromagnetic fields. Numerical studies illustrate key properties including coupled-wave near fields, spectral signatures at resonance, and local field profiles for representative beam geometries. The work contributes fundamentally to the understanding of resonant metasurfaces that are of high scientific interest worldwide.