fits to Eq. (\[Gauss\]). The insets show samples of the measured phase shifts. The data used to obtain the histograms amounts 5 to 10 times the date shown in the insets.*](FluctPaper.eps){width="47.50000%"}

The histogram in Fig. \[SubstrateHologram\](b) shows the normalized phase shift distribution of the a-C substrate film [ *without*]{} Au adsorbate. The Gaussian fit from Eq. \[Gauss\] as shown by the full line has a width of $\sigma_{\rm a-C} = 0.13$ rad. Since $\sigma_{\rm a-C} > 2 \sigma_{\rm vac}$ the fluctuations must reflect properties of the substrate. Using Eq. \[thick\] and the value of $V_{\rm subs} = 27.4$ V the change in thickness corresponds to $\Delta d_{\rm subs} = 2 \sigma_{\rm subs}/(C_{\rm
  E}V_{\rm subs}) = 1.3$ nm, for $V_{\rm HL} = 10.7$ V even $\Delta
d_{\rm subs} = 3.3$ nm. The graph in the inset Fig.\[SubstrateHologram\](b) reveals that phase fluctuations of comparable depth occur on length scales of 0.3 nm. Such narrow intrusions with a depth of up to 1/5 of the substrate thickness would make the substrate very unstable. Moreover, the edges of the intrusions would inevitably lead to strong pinning centers similar to those of nanopits in HOPG,[@HBB+97; @Hoev01] which are inconsistent with the observed island formation.

Since 