Oblique-incidence-induced phase transition in parity-time symmetric optical bilayers

We investigate the scattering of optical waves from one-dimensional parity-time ($PT$) symmetric structures of arbitrary complexity. We prove that it is feasible to observe the transition from an unbroken $PT$-symmetric phase to a broken one when the angle of incidence increases. Analytical formulae are developed that emphasize the key roles played by the interference between the exponential increase and decayed fields in the gain and loss media, and the phase shifts when reflected/transmitted from the gain-loss boundary. A specific layer thickness is proposed. Distributions of field and energy flux under both phases are studied. This investigation shows that we can observe optical $PT$-symmetric phase transition in a single structure that obeys the causality principle. (paper)