Absence and recovery of exceptional points in coupled waveguides with complex conjugate distribution of refractive index

Non-Hermitian systems with parity-time (PT) symmetry complex potential can exhibit phase transition when the degree of non-Hermiticity is varied. The existence of exceptional point (EP), at which two eigenstates coalesce, gives rise to a great variety of novel phenomena in various fields of physics. In real optical system, e.g., waveguides and photonic crystals, the complex conjugate distribution of refractive index is used to implement the configuration of PT-symmetry. But this is neither a sufficient condition, nor a necessary one. To demonstrate that, we propose a structure with four coupled waveguides in a square array bearing diagonally distributed gain/loss distribution and show that there are four singlet supermodes that exhibit general PT-symmetry phase transition from unbroken phase (with completely real eigenvalues) to broken phase (with conjugate eigenvalues), however, the other two pairs of degenerated supermodes lose their EPs, always bearing complex conjugate imaginary part. Moreover, the phase rigidity and their projection to the original states are used to depict the behavior of phase transition and absence of EP. It is noted that the annihilation of EP origins from the unequal propagation constants of the hybrid modes, by viewing the whole structure as two diagonally coupled double-waveguides. All results are theoretically analyzed by coupled mode theory and numerically verified by finite element method. Finally, we show that an appropriate choice of the ratio between gain and loss can be used to get balanced energy distribution in each channel, in order to recover the stable propagation mode without attenuation, i.e., to get EP healing from the previous annihilation.