Reflection from a semi-infinite stack of layers using homogenization

• A canonical reflection and transmission problem for a periodic medium is treated.
• An effective homogenized medium that captures the main features of reflection is found for a wide range of frequencies.
• The long-wave behaviour at low frequencies can, under some circumstances, be replicated at high frequencies.

A canonical scattering problem is that of a plane wave incident upon a periodic layered medium. Our aim here is to replace the periodic medium by a homogenized counterpart and then to investigate whether this captures the reflection and transmission behaviour accurately at potentially high frequencies.We develop a model based upon high frequency homogenization and compare the reflection coefficients and full fields with the exact solution. For some material properties it is shown that the asymptotic behaviour of the dispersion curves are locally linear near critical frequencies and that low frequency behaviour is replicated at these critical, high, frequencies. The homogenization approach accurately replaces the periodic medium and the precise manner in which this is achieved then opens the way to future numerical implementation of this technique to scattering problems.