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

Abstract

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 $\ensuremath{\theta}$ (measured with respect to the normal direction to the slab), while ${g}^{(M)}$ has a single maximum, ${\ensuremath{\theta}}_{c}$, where it takes an extremely large value. We identify ${\ensuremath{\theta}}_{c}$ with the Brewster's angle and show that ${g}^{(E)}$ and ${g}^{(M)}$ coincide for $\ensuremath{\theta}=0$ (normal incidence), tend to zero as $\ensuremath{\theta}\ensuremath{\rightarrow}{90}^{\ensuremath{\circ}}$, and satisfy ${g}^{(E)}<{g}^{(M)}$ for $0<\ensuremath{\theta}<{90}^{\ensuremath{\circ}}$. 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 $\ensuremath{\theta}\ensuremath{\approx}{\ensuremath{\theta}}_{c}$. 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 $\ensuremath{\theta}>{\ensuremath{\theta}}_{c}$ is smaller inside the gain region than outside it. The converse is true for the TM waves with $\ensuremath{\theta}<{\ensuremath{\theta}}_{c}$ and all TE waves.