  The spectral and localization properties of $\mathcal{PT}$-symmetric optical
superlattices, either infinitely extended or truncated at one side, are
theoretically investigated, and the criteria that ensure a real energy spectrum
are derived. The analysis is applied to the case of superlattices describing a
complex ($\mathcal{PT}$-symmetric) extension of the Harper Hamiltonian in the
rational case.
