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/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 $\theta\to 90^\circ$, and satisfy $g^{(E)}<g^{(M)}$ for $0<\theta<90^\circ$. 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 $\theta\approx\theta_c$. 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.