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A method for constructing optical potentials with an arbitrary distributionof gain and loss and completely real spectrum is presented. For each arbitrarydistribution of gain and loss, several classes of refractive-index profileswith freely tunable parameters are obtained such that the resulting complexpotentials, although being non-parity-time-symmetric in general, still featureall-real spectra for a wide range of tuning parameters. When these refractiveindices are tuned below certain thresholds, phase transition can occur, wherecomplex-conjugate pairs of eigenvalues appear in the spectrum. Thesenon-parity-time-symmetric complex potentials generalize the concept ofparity-time-symmetric potentials to allow for more flexible gain and lossdistributions while still maintaining all-real spectra and the phenomenon ofphase transition.

All-real spectra in optical systems with arbitrary gain and loss distributions