Unidirectional invisibility and PT symmetry with graphene

We investigate the reflectionlessness and invisibility properties in the transverse electric (TE) mode solution of a linear homogeneous optical system which comprises the $\mathcal{PT}$-symmetric structures covered by graphene sheets. We derive analytic expressions, indicate roles of each parameter governing optical system with graphene and justify that optimal conditions of these parameters give rise to broadband and wide angle invisibility. Presence of graphene turns out to shift the invisible wavelength range and to reduce the required gain amount considerably, based on its chemical potential and temperature. We substantiate that our results yield broadband reflectionless and invisible configurations for realistic materials of small refractive indices, usually around $\eta = 1$, and of small thickness sizes with graphene sheets of rather small temperatures and chemical potentials. Finally, we demonstrate that pure $\mathcal{PT}$-symmetric graphene yields invisibility at small temperatures and chemical potentials.