Wave spectrum of multilayers with finite thicknesses of interfaces

To describe a multilayer structure with arbitrary thicknesses of the interfaces between layers, we introduce a model in which the dependence of a material parameter along the axis of such a superlattice is described by a Jacobian elliptic sine function. Depending on the value of the modulus {κ} of the elliptic function, the model describes the limiting cases of multilayers with sharp interfaces (κ=1,d/l=0, where d is the thickness of the interface, l is the period of the superlattice) and of sinusoidal superlattices (κ=0,d/l=1/4), as well as all intermediate situations. We investigate the wave spectrum in such a superlattice. The dependences of the widths of the gaps in the spectrum at the boundaries of all odd Brillouin zones on the ratio $d/l$ are found. It is shown that the thicknesses of the interfaces can be determined if the experimental value of the relation between the widths of the first gap Δν₁ and any other gap Δνₙ is known.