  The branched pattern typically observed through the scanning gate microscopy
(SGM) of two dimensional electron gases in the presence of weak, smooth
disorder has recently been found to be robust against a very large shift in the
Fermi energy of the electron gas. We propose a toy model, where the potential
landscape reduces to a single localized feature, that makes it possible to
recast the understanding of branch formation through the effect of caustics in
an appropriate set of classical trajectories, and it is simple enough to allow
for a quantitative analysis of the energy and spatial dependence of the
branches. We find the energy stability to be extremely generic, as it rests
only upon the assumptions of weak disorder, weak scattering, and the
proportionality of the SGM response to the density of classical electron
trajectories. Therefore, the robustness against changes of the electron's Fermi
energy remains when adopting progressively realistic models of smooth disorder.
