Modes of Open Resonators and Waveguides: Connection of their Description as Gaussian Beams with More Accurate Representation

Using a separation of variables in ad hoc system of curvilinear coordinates, an analytic description of modes of open resonators and waveguides is obtained without using a semi-classic approximation in solving the wave equation. It is shown on what condition and in what approximation this description transforms into that of Gaussian-Hermitian beams.