Frenet-Serret analysis of helical Bloch modes in N-fold rotationally symmetric rings of coupled spiralling optical waveguides

The behavior of electromagnetic waves in chirally twisted structures is a topic of enduring interest, dating back at least to the invention in the 1940s of the microwave travelling wave tube amplifier and culminating in contemporary studies of chiral metamaterials, metasurfaces, and photonic crystal fibers (PCFs). Optical fibers with chiral microstructures, drawn from a spinning preform, have many useful properties, exhibiting for example circular birefringence and circular dichroism. It has recently been shown that chiral fibers with N fold rotationally symmetric (symmetry group CN) transverse microstructures support families of helical Bloch modes (HBMs), each of which consists of a superposition of azimuthal Bloch harmonics (or optical vortices). An example is a fiber with N coupled cores arranged in a ring around its central axis (N core single ring fiber). Although this type of fiber can be readily modelled using scalar coupled mode theory, a full description of its optical properties requires a vectorial analysis that takes account of the polarization state of the light particularly important in studies of circular and vortical birefringence. In this paper we develop, using an orthogonal two dimensional helicoidal coordinate system embedded in a cylindrical surface at constant radius, a rigorous vector coupled mode description of the fields using local Frenet Serret frames that rotate and twist with each of the N cores. The analysis places on a firm theoretical footing a previous HBM theory in which a heuristic approach was taken, based on physical intuition of the properties of Bloch waves. We believe this study provides clarity in what can sometimes be a rather difficult field, and will facilitate further exploration of real-world applications of these fascinating waveguiding systems.