The wave-vector power spectrum of the local tunnelling density of states: ripples in a d-wave sea

A weak scattering potential imposed on a $CuO_2$ layer of a cuprate superconductor modulates the local density of states $N(x,\omega)$. In recently reported experimental studies scanning-tunneling maps of $N(x,\omega)$ have been Fourier transformed to obtain a wave-vector power spectrum. Here, for the case of a weak scattering potential, we discuss the structure of this power spectrum and its relationship to the quasi-particle spectrum and the structure factor of the scattering potential. Examples of quasi-particle interferences in normal metals and $s$- and d-wave superconductors are discussed.