Vector Helical Bloch Modes in N-Fold Rotationally Symmetric Waveguiding Structures: Spin and Azimuthal Order

Many different techniques have been used to model the electromagnetic response of photonic crystal fibres (PCFs) and waveguide arrays. Here we introduce a new vectorial approach based on helical Bloch waves (hBWs) that appear in structures (such as many PCFs) that are N-fold rotationally symmetric (symmetry group C
_n
), i.e., they repeat perfectly in N sectors over 2π, when viewed in a cylindrical coordinate frame [1]. Helical Bloch waves can also carry optical angular momentum: a complex Bloch wave field propagates azimuthally around the axis of the structure, forming helical Bloch modes (hBMs) when a certain azimuthal resonance condition is satisfied.