Antisymmetric PT-photonic structures with balanced positive and negative index materials

We propose a new class of synthetic optical materials in which the refractive index satisfies $n(-\text{bx})=-n^{\ast }(\text{bx})$. We term such systems antisymmetric parity-time (APT) structures. Unlike PT-symmetric systems which require balanced gain and loss, i.e. $n(-\text{bx})=n^{\ast }(\text{bx})$, APT systems consist of balanced positive and negative index materials. Despite the seemingly PT-symmetric optical potential $V(\text{bx})\equiv n(\text{bx}{)}^2{\omega }^2\mathrm{/}{c}^2$, APT systems are not invariant under combined PT operations due to the discontinuity of the spatial derivative of the wavefunction. We show that APT systems can display intriguing properties such as spontaneous phase transition of the scattering matrix, bidirectional invisibility, and a continuous lasing spectrum.