Wave spectrum of multilayers with finite thicknesses of interfaces

Abstract

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 \ensuremath{\kappa} of the elliptic function, the model describes the limiting cases of multilayers with sharp interfaces $(\ensuremath{\kappa}=1,d/l=0,$ where d is the thickness of the interface, l is the period of the superlattice) and of sinusoidal superlattices $(\ensuremath{\kappa}=0,d/l=1/4),$ 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 $\ensuremath{\Delta}{\ensuremath{\nu}}_{1}$ and any other gap $\ensuremath{\Delta}{\ensuremath{\nu}}_{n}$ is known.