Tunneling of electromagnetic waves through bilayers: Special cases and point approximations

We develop a transfer matrix approach for the investigation of tunneling properties of obliquely incident plane waves with the transverse-magnetic polarization on non-magnetic bilayer structures. These structures are composed of two adjacent layers with constant complex-valued permittivities having a jump discontinuity at the interface. We focus on the two cases: (i) the bilayer with real permittivities and (ii) the bilayer with balanced gain and loss regions having parity-time symmetry. The key point is that the reflection-transmission amplitudes can be expressed in terms of the matrix that connects the boundary conditions of the magnetic field at the external sides of the layers. These expressions are relatively simple and allow to analyze analytically tunneling properties in some special cases, as well as to apply point approximations for extremely thin bilayers.