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The supplementary material provides the description how to calculate the interference pattern, caused by the interaction with a line defect, like a step edge, for electronic states of a nearly free electron system embedded in a weak 1D potential.

The electronic state at the interface between the Ag(111) surface and NaCl island has been studied. Due to the lattice mismatch, islands of NaCl grow on Ag(111) with various moiré patterns which lead to a weak periodic one-dimensional (1D) potential for the electronic state. In contrast to the well-known situation for a purely 1D system, the ideal two-dimensional system should cause a modulation of the local electronic density for energies only below and not above the mini band gap corresponding to the periodic potential. However, the experimental observation by scanning tunneling microscopy and spectroscopy of states reveals such a modulation also in the forbidden range of energy. Since the particular system allows us to disentangle the effects of local scattering and those caused by the periodic potential, it could be revealed that the loss of coherence and the mechanism of the detection by the STM are responsible for the experimentally observed modulation, which should be suppressed in the case of an ideal system.