We introduce a family of non-Hermitian optical potentials that are given in terms of double-exponential periodic functions. The center of $\mathcal{PT}$ symmetry is not around zero and the potential satisfies a shifted $\mathcal{PT}\text{-symmetry}$ relation at two distinct locations. Motivated by wave transmission through thin phase screens and gratings, we examine these refractive index modulations from the perspective of optical lattices that are homogeneous along the propagation direction. The diffraction dynamics, abrupt phase transitions in the eigenvalue spectrum, and exceptional points in the band structure are examined in detail. In addition, the nonlinear properties of wave propagation in Kerr nonlinearity media are studied. In particular, coherent structures such as lattice solitons are numerically identified by applying the spectral renormalization method. The spatial symmetries of such lattice solitons follow the shifted $\mathcal{PT}\text{-symmetric}$ relations. Furthermore, such lattice solitons have a power threshold and their linear and nonlinear stabilities are critically dependent on their spatial symmetry point.

• Received 2 September 2015

DOI:https://doi.org/10.1103/PhysRevA.93.013803