Lasing-threshold condition for oblique TE and TM modes, spectral singularities, and coherent perfect absorption

We study spectral singularities and their application in determining the threshold gain coefficient g((E/M)) for oblique transverse electric or magnetic (TE/TM) modes of an infinite planar slab of homogenous optically active material. We show that g((E)) is a monotonically decreasing function of the incidence angle theta (measured with respect to the normal direction to the slab), while g((M)) has a single maximum, theta(c), where it takes an extremely large value. We identify theta(c) with the Brewster's angle and show that g((E)) and g((M)) coincide for theta = 0 (normal incidence), tend to zero as theta -> 90 degrees, and satisfy g((E)) < g((M)) for 0 < theta < 90 degrees. We therefore conclude that lasing and coherent perfect absorption are always more difficult to achieve for the oblique TM waves and that they are virtually impossible for the TM waves with theta approximate to theta(c). We also give a detailed description of the behavior of the energy density and the Poynting vector for spectrally singular oblique TE and TM waves. This provides an explicit demonstration of the parity-invariance of these waves and shows that the energy density of a spectrally singular TM wave with theta > theta(c) is smaller inside the gain region than outside it. The converse is true for the TM waves with theta < theta(c) and all TE waves.