Paraxial linear propagation of light in an optical waveguide with material gain and loss is governed by a Schrödinger equation with a complex potential. In this Letter, new classes of non-parity-time ()-symmetric complex potentials featuring conjugate-pair eigenvalue symmetry are constructed by operator symmetry methods. Due to this eigenvalue symmetry, it is shown that the spectrum of these complex potentials is often all-real. Under parameter tuning in these potentials, a phase transition can also occur, where pairs of complex eigenvalues appear in the spectrum. A distinctive feature of the phase transition here is that the complex eigenvalues may bifurcate out from an interior continuous eigenvalue inside the continuous spectrum; hence, a phase transition takes place without going through an exceptional point. In one spatial dimension, this class of non--symmetric complex potentials is of the form , where is an arbitrary -symmetric complex function. These potentials in two spatial dimensions are also derived. Diffraction patterns in these complex potentials are further examined, and unidirectional propagation behaviors are demonstrated.
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