Absence of Exceptional Points in Square Waveguide Arrays with Apparently Balanced Gain and Loss

The concept of parity-time (PT) symmetry in the field of optics has been intensively explored. This study shows the absence of exceptional points in a three-dimensional system composed of a square waveguide array with diagonally-balanced gain/loss distribution. More specifically, we show that an array of four coupled waveguides supports eight fundamental propagation supermodes, four of which are singlet and the other two pairs are double degenerated. It is found that the singlet states follow the routine PT phase transition; however, the double-degenerated modes never coalesce as the gain/loss-to-coupling strength level varies, showing no actual PT symmetry-derived behavior. This is evident in the phase rigidity which does not approach zero. The absence of exceptional points is ascribed to the coupling of non-symmetric supermodes formed in the diagonal waveguide pairs. Our results suggest comprehensive interplay between the mode pattern symmetry, the lattice symmetry and the PT-symmetry, which should be carefully considered in PT-phenomena design in waveguide arrays.