Nonlinear spectral singularities of a complex barrier potential and the lasing threshold condition

A spectral singularity is a mathematical notion with an intriguing physical realization in terms of certain zero-width resonances. In optics it manifests as lasing at the threshold gain. We explore the recently developed nonlinear generalization of spectral singularities for a complex barrier potential and study the associated resonance effect for an infinite planar slab of gain medium. In particular, for a Kerr nonlinearity, we show that the first-order perturbative equation that determines the nonlinear spectral singularities provides an explicit expression for the intensity of the emitted waves from the slab. This is a mathematical derivation of the known linear relationship between the output intensity of the slab and its gain coefficient. We also discuss the implications of our results for the time-reversed system which acts as a coherent perfect absorber.