$\mathcal{P}\mathcal{T}$ symmetry and anti-symmetry by anti-Hermitian wave coupling and nonlinear optical interactions

Light propagation in systems with anti-Hermitian coupling, described by a spinor-like wave equation, provides a general route for the observation of anti parity-time ($\mathcal{P}\mathcal{T}$ ) symmetry in optics. Remarkably, under a different definition of parity operator, a $\mathcal{P}\mathcal{T}$ symmetry can be found as well in such systems. Such symmetries are ubiquitous in nonlinear optical interactions and are exemplified by considering modulation instability in optical fibers and optical parametric amplification.