A unidirectionally invisible PT-symmetric complex crystal with arbitrary thickness

We introduce a new class of parity-time ( )-symmetric complex crystals which are almost transparent and one-way reflectionless over a broad frequency range around the Bragg frequency, i.e., unidirectionally invisible, regardless of the thickness L of the crystal. The  -symmetric complex crystal is synthesized by a supersymmetric (SUSY) transformation of a Hermitian square well potential, and exact analytical expressions of transmission and reflection coefficients are given. As L is increased, the transmittance and reflectance from one side remain close to one and zero, respectively, whereas the reflectance from the other side secularly grows like ∼L2 owing to unidirectional Bragg scattering. This is a distinctive feature as compared to the previously studied case of the complex sinusoidal  -symmetric potential V (x)=V0 exp (−2ikox) at the symmetry breaking point, where transparency breaks down as L → ∞.