Friedel oscillations in microwave billiards

Friedel oscillations of electron densities near step edges have an analog in microwave billiards. A random plane-wave model, normally only appropriate for the eigenfunctions of a purely chaotic system, can be applied and is tested for non-purely-chaotic dynamical systems with measurements on pseudointegrable and mixed dynamics geometries. It is found that the oscillations in the pseudointegrable microwave cavity match the random plane-wave modeling. Separating the chaotic from the regular states for the mixed system requires incorporating an appropriate phase-space projection into the modeling in multiple ways for good agreement with experiment.

Figure 1

(Color online) Average intensity distribution of even eigenmodes. In the frequency range 2–6 GHz: (a) symmetric eigenmodes of the barrier billiard and (b) chaotic eigenmodes of the desymmetrized mushroom billiard. Red color (dark-colored) corresponds to high intensity and blue (light-colored) to low intensity. The dotted and dashed areas were used to evaluate: (a) the average intensity profile at Dirichlet and Neumann boundary conditions and (b) the straight and circular boundary segments.

Figure 2

Experimental intensity profile of even eigenmodes of the barrier billiard (points) perpendicular to the Dirichlet boundary segments (upper panel) and to the Neumann boundary segment beyond the barrier (lower panel). The random plane-wave model predictions are the solid lines.

Figure 3

Similar to the upper panel of Fig. 2 except for only 19 eigenmodes in a narrow frequency window 5.51–5.77 GHz.

Figure 4

Average intensity profile of chaotic eigenmodes (points) of the mushroom billiard perpendicular to its straight boundary segment (upper panel) and to its circular boundary segment (lower panel) and model predictions of the unrestricted and the restricted random plane-wave model (dashed and solid lines, respectively).