Poynting theorem for the natural modes of open dielectric resonators with active regions

We study the general properties of the natural modes of dielectric resonators containing active regions filled with a gain material. Such resonators serve as models of THz microcavity lasers, where the pumping is achieved by the injection of carriers. Using the Maxwell equations with rigorous radiation condition and Poynting theorem, we establish the correspondence between the self-excitation thresholds of the natural modes and their Q factors and overlap coefficients of the mode field with the active region. This gives the firm ground to the empiric “gain=loss” expression that is in the core of the semiclassical, i.e. based on the ray description, theory lasers. The derived expressions can be used for the validation of the numerical analyses of the lasing modes. Numerical example is presented for a two-dimensional (2D) model of a disk laser with radially non-uniform material gain.