Wave Scattering and Guided Modes in Periodic Pillars

We investigate the scattering of scalar harmonic source fields by a periodic pillar, that is, a spatial structure that is periodic in one dimension and of finite extent in the other two. Uniqueness of scattering solutions can be abstracted by guided modes. Extending results for periodic slabs to pillars, we give conditions under which "inverse" pillars cannot admit guided modes. In addition, we present a new construction of guided modes at frequency $\omega$ and Bloch wavenumber $\kappa$ with $\omega$ embedded in the continuous spectrum for each $\kappa$. These guided modes have period larger than the period of the structure, and possess a real dispersion relation $\omega=\omega(\kappa)$, which is atypical of modes at embedded frequencies.