Biaxial Gaussian Beams, Hermite-Gaussian Beams, and Laguerre-Gaussian Vortex Beams in Isotropy-Broken Materials

We develop the paraxial approximation for electromagnetic fields in arbitrary isotropy-broken media in terms of the ray-wave tilt and the curvature of materials Fresnel wave surfaces. We obtain solutions of the paraxial equation in the form of biaxial Gaussian beams, which is a novel class of electromagnetic field distributions in generic isotropy-broken materials. Such beams have been previously observed experimentally and numerically in hyperbolic metamaterials but evaded theoretical analysis in the literature up to now. The biaxial Gaussian beams have two axes: one in the direction of Abraham momentum, corresponding to the ray propagation, and another in the direction of Minkowski momentum, corresponding to the wave propagation, in agreement with the recent theory of refraction, ray-wave tilt, and hidden momentum [Durach, 2024, Ref. 1]. We show that the curvature of the wavefronts in the biaxial Gaussian beams correspond to the curvature of the Fresnel wave surface at the central wave vector of the beam. We obtain the higher-order modes of the biaxial beams, including the biaxial Hermite-Gaussian and Laguerre-Gaussian vortex beams, which opens avenues toward studies of optical angular momentum (OAM) in isotropy-broken media, including generic anisotropic and bianisotropic materials.