include a section of the vacuum for calibration.[@Reim89; @LL02] A closer analysis of the data is given in Sec. \[SubstrateHolography\] concerning the substrate and in Sec. \[sectionFilmChar\] concerning the Au adsorbate.

![\[SampleHologram\] *Holographic phase shift $\Delta\phi$ of a segment of the a-C substrate with part of an island as shown in Fig. \[expfig1\](b) in the far right corner. The image has the size of 28.2$\times$28.2 nm$^2$. The $z$-axis value is proportional to the observed phase shift. In the foreground the almost linear decrease of the a-C film thickness towards its edge is visible. (For alternative representations of the data see Figs. \[holoscans\] and \[holo20diag\].)* ](SampleHologram.eps){width="48.00000%"}

The relation between the change in phase shift $\Delta \phi$ observed in the holographic images and the corresponding thickness $d$ of the observed object can be derived from scattering theory[@Reim89] and is expressed via the standard formula $$\label{thick}
  d = \frac{\Delta\phi}{V_0\ C_{\rm E}}\,,$$ where the interaction constant depends on the acceleration voltage resulting in $C_{\rm E} = 7.29 \times 10^{6}$ rad/(V m) in our case and where $V_0$ denotes the effective internal potential of the sample.

For bulk gold the theoretical values[@Buhl59; @Kell61] for $V_0$ are $V_{\rm Au,theo} = 28\pm2$ V, while experimental values[@GC99] are 