Waves in a superlattice with anisotropic inhomogeneities

Dependences of the dispersion laws and damping of waves in an initially sinusoidal superlattice on inhomogeneities with anisotropic correlation properties are studied for the first time. The period of the superlattice is modulated by the random function described by the anisotropic correlation function $K_\phi({\mathbf r})$ that has different correlation radii, $k_\parallel^{-1}$ and $k_\perp^{-1}$, along the axis of the superlattice $z$ and in the plane $xy$, respectively. The anisotropy of the correlation is characterized by the parameter $\lambda=1-k_\perp/k_\parallel$ that can change from $\lambda=0$ to $\lambda=1$ when the correlation wave number $k_\perp$ changes from $k_\perp=k_\parallel$ (isotropic 3D inhomogeneities) to $k_\perp=0$ (1D inhomogeneities). The correlation function of the superlattice $K(r)$ is developed. Its decreasing part goes to the asymptote $L$ that divides the correlation volume into two parts characterized by finite and infinite correlation radii. The dependences of the width of the gap in the spectrum at the boundary of the Brillouin zone $\Delta\nu$ and the damping of waves $\xi$ on the value of $\lambda$ are studied. It is shown that decreasing $L$ leads to the decrease of $\Delta\nu$ and increase of $\xi$ with the increase of $\lambda$.