Snapshots of Anderson localization beyond the ensemble average

We study $(1+1)D$ transverse localization of electromagnetic radiation at microwave frequencies directly by two-dimensional spatial scans. Since the longitudinal direction can be mapped onto time, our experiments provide unique snapshots of the buildup of localized waves. The evolution of the wave functions is compared with semianalytical calculations. Studies beyond ensemble averages reveal counterintuitive surprises. Oscillations of the wave functions are observed in space and explained in terms of a beating between the eigenstates.

Figure 1

Experimental setup. Nylon bars (red) are placed on top of an oxygen-free copper plate, the distance between the bars is random in the transverse $x$ direction. The $z$ direction is disorder free. One of the two microwave antennas (black disks) is scanned over the sample and a vector network analyzer is used to measure the transmitted spectrum between the two antennas. Each scan along the $x$ axis is equivalent to a snapshot in time.

Figure 2

Experimentally determined false color images of the amplitude distribution for (a) an ordered and (b) a disordered sample at 9.2 GHz. Every row is normalized independently. The scale bar denotes 100 mm.

Figure 3

Participation ratio versus propagation distance at 9.2 GHz for an ordered sample (blue) and a disordered ensemble (red). Red line: calculation for an ensemble of 100 disordered samples. Blue line: linear fit.

Figure 4

(a) Experimental and (b) calculated oscillations in the intensity profile for one sample excited at 10.2 GHz. Every column is normalized independently. (c) Expansion coefficients for the different eigenstates. Only two eigenstates, indicated by the stars, contribute significantly. (d) Center of mass of the intensity versus propagation direction for both the calculated and experimental data shown in (a) and (b).

Figure 5

(a) Experimental and (b) numerically calculated false color plots of the wave function intensity in transverse direction after 365 mm of propagation along the $z$ direction for different positions of the excitation antenna at 9.2 GHz. The white lines indicate the position of the excitation antenna. The dashed box marks an antidiagonal wave profile. Scale bar denotes 100 mm. (c)–(e) Calculations using mode decomposition for the area marked with the dashed box in (a) for 180, 365, and 1950 mm of propagation, respectively. Beating of eigenmodes can result in (c) circular, (d) antidiagonal, or (e) diagonal patterns.