Geometric Frequency Mixing in Helical Waveguides via a One-Dimensional Covariant Helmholtz Model: Gauge Reduction and Spectral Splitting

Abstract

This study develops a one-dimensional covariant Helmholtz model for a vector-valued wave field transported along a circular helical centerline and represented in the Frenet-Serret frame. For a helix with constant curvature kappa>0 and torsion tau not equal 0 , the geometric coupling is described by a constant skew-symmetric connection matrix Omega is an element of so(3) . The covariant Helmholtz operator is shown to admit an exact gauge reduction to the flat componentwise Helmholtz operator through u(s)=e(-Omega s)y(s) . Thus, within the one-dimensional centerline formulation, the helix preserves the operator spectrum while redistributing the observed Frenet components through parallel transport. The closed-form solutions show that a monochromatic input with wavenumber k is decomposed into a carrier and two geometric sidebands governed by the Darboux rotation rate lambda= root kappa(2)+tau(2) . In the sub-geometric regime k