A unidirectional invisible PT-symmetric complex crystal with arbitrary thickness

We introduce a new class of $\mathcal{P}\mathcal{T}$-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 $\mathcal{P}\mathcal{T}$-symmetric complex crystal is synthesized by a supersymmetric transformation of an 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 $\sim L^2$ owing to unidirectional Bragg scattering. This is a distinctive feature as compared to the previously studied case of the complex sinusoidal $\mathcal{P}\mathcal{T}$-symmetric potential $V(x)=V_0\exp (-2i{k}_{o}x)$ at the symmetry breaking point, where transparency breaks down as $L\to \mathrm{\infty }$.