General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states

Rigorous solutions of the Maxwell equations describing propagation invariant optical fields are presented in general; the elements for their specific applications to the Bessel, Mathieu and Weber families are also provided. Electric and magnetic transverse modes, and several polarization state solutions, are constructed; the connections between them are explicitly established. Their respective energy densities and Poynting vectors are also evaluated, in order to exhibit their propagation invariant nature. The experience with Bessel beams allows us to recognize that vector modes exhibit new and important features compared with the corresponding scalar fields; the results of this work constitute a first step towards the analysis of the dynamical properties of vector Mathieu and Weber beams.