On the relation between the propagation constant of Bloch surface waves and the thickness of the upper layer of a photonic crystal

We consider the derivation of a dispersion relation of Bloch surface waves supported by interfaces between a semi-infinite one-dimensional photonic crystal and a homogeneous medium. From the derived dispersion relation, we obtain an explicit analytical expression that defines the relationship between the propagation constant and the thickness of the upper layer of the photonic crystal.