We describe one-dimensional stationary scattering of a two-component wave field by a non-Hermitian matrix potential which features odd-$\mathcal{P}\mathcal{T}$ symmetry, i.e., symmetry with ${\left(\mathcal{P}\mathcal{T}\right)}^2=-1$. The scattering is characterized by a $4\times 4$ transfer matrix. The main attention is focused on spectral singularities which are classified into two types. Weak spectral singularities are characterized by the zero determinant of a diagonal $2\times 2$ block of the transfer matrix. This situation corresponds to the lasing or coherent perfect absorption of pairs of oppositely polarized modes. Strong spectral singularities are characterized by a zero diagonal block of the transfer matrix. We show that in odd-$\mathcal{P}\mathcal{T}$-symmetric systems any spectral singularity is self-dual, i.e., lasing and coherent perfect absorption occur simultaneously. A detailed analysis is performed for a case example of a $\mathcal{P}\mathcal{T}$-symmetric coupler consisting of two waveguides, one with localized gain and another with localized absorption, which are coupled by a localized antisymmetric medium. For this coupler, we discuss weak self-dual spectral singularities and their splitting into complex-conjugate eigenvalues which represent bound states characterized by propagation constants with real parts lying in the continuum. A rather counterintuitive restoration of the unbroken odd-$\mathcal{P}\mathcal{T}$-symmetric phase subject to the increase of the gain-and-loss strength is also revealed. A comparison between odd- and even-$\mathcal{P}\mathcal{T}$-symmetric couplers, the latter characterized by ${\left(\mathcal{P}\mathcal{T}\right)}^2=1$, is also presented.

• Received 13 November 2018

DOI:https://doi.org/10.1103/PhysRevA.99.013823