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