A new geometric approximation scheme, designed to solve the covariant Maxwell equations inside a rotating hollow slender conducting cavity (modelling a ring-laser) on a background spacetime, is constructed. It is shown that for well-defined conditions the electromagnetic spectrum splits into TE- and TM-type modes. A twisted mode spectrum is found to depend on the integrated Frenet torsion of the cavity and this in turn may affect the Sagnac beat frequency induced by a non-zero rotation of the cavity.