Double-slit experiments with microwave billiards

Single and double-slit experiments are performed with two microwave billiards with the shapes of a rectangle and a quarter stadium, respectively. The classical dynamics of the former is regular, whereas that of the latter is chaotic. Microwaves can leave the billiards via slits in the boundary, forming interference patterns on a screen. The aim is to determine the effect of the billiard dynamics on their structure. For this the development of a method for the construction of a directed wave packet by means of an array of multiple antennas was crucial. The interference patterns show a sensitive dependence not only on the billiard dynamics but also on the initial position and direction of the wave packet.

Figure 1

(a) Perspective depiction of the modular assembly of a rectangular microwave billiard (not to scale). The microwave resonator is composed of a frame, 5 mm high, defining the boundary of the billiard squeezed between a top and bottom plate, each 5 mm thick. These are screwed together via the holes along the boundaries. Small holes in the top plate (indicated by the crosses) are used to introduce the antennas. (b) Sketch of the experimental setup. The interference pattern of the waves emanating from a billiard with two slits are detected on a “screen.” A rectangular billiard with regular dynamics and a tilted stadium billiard with chaotic dynamics are used in the experiments. The positions of the antennas used to excite microwaves inside the resonators are indicated by the crosses. The size of the slits is enlarged by a factor of $4$ for their better visibility.

Figure 2

Schematic view (not to scale) of the experimental setup for the measurement of interference patterns. The antenna $a$ is fixed inside the billiard and the antenna $b$ is moved in small steps ($\Delta x$ $=$ $\Delta y$ $=$ 5 mm) in the vicinity of the slits on a grid (crosses) to measure the electric field strengths. The distance between the midpoints of the slits $d$ and the slit size $s$ are indicated. The vertical distance between the billiard edge and the lines of measuring points parallel to the lower billiard edge is denoted by $l$. The coordinate system ($x$, $y$) used in the following is indicated. Its point of origin is the midpoint between the slits. The microwave power is generated and measured, respectively, by the VNA.

Figure 3

Photograph of the measurement setup in the anechoic chamber. The microwave billiard is placed on a table and the receiving antenna is carefully aligned perpendicularly to it. The cable connected to the receiving antenna is coated by a cylindrical double layer of absorber and guided by a positioning unit. The microwave power is coupled from the vectorial network analyzer through a coaxial cable into the cavity.

Figure 4

Frequency spectrum of the open two-slit rectangular (top) and stadium (bottom) billiard measured with the emitting antenna $a$ inside the resonator at the point $x=0$ mm, $y=-400$ mm and the receiving antenna $b$ placed outside at the point $x=0$, $y=l=320$ mm. The slit size is $s=9.5$ mm and the distance between the slits equals $d=78$ mm. The cut-off frequency $f_{\mathrm{co}}=c/(2s)=15.79$ GHz is determined by the slit size and marked by an arrow in both panels.

Figure 5

Intensity on the screen (solid line) for the resonance frequency indicated at the top of the panel for the rectangular billiard (top panel) and the stadium billiard (bottom panel). The position of the emitting antenna is chosen on the line $x=0$ at $y=-400$ mm. The screen is located at $l=155$ mm. The slit sizes and their distance are $s=9.5$ mm and $d=78$ mm, respectively. The dashed line in the top panel results from the Fraunhofer formula [Eq. (4)] and the one in the bottom panel from the formula in Eq. (7).

Figure 6

Two-dimensional intensity patterns ($I\propto |S_{\mathit{ba}}(x,y)|{}^2$) on a grid of measuring points for the rectangular (left panels) and stadium billiard (right panels) at a resonance frequency indicated at the top of each panel. Blue (dark) color corresponds to low intensity and yellow (bright) color to high intensity. The slit sizes and their distance are $s=20$ mm and $d=240$ mm, respectively.

Figure 7

Joint probability distributions of the correlators ${\Delta }_{\rho }$ and ${\Delta }_{\varphi }$ of the electric fields defined by Eqs. (8) and (9) outside the rectangular billiard (left panel) and the stadium billiard (right panel). Yellow (bright) color corresponds to high and blue (dark) color to low probability. The pairs (${\Delta }_{\rho }$, ${\Delta }_{\varphi }$) are evaluated for $30$ excitation frequencies on a grid of $10\hspace{0.5em}000$ measuring points. The slit sizes are $s=20$ mm and their distance is $d=240$ mm as in Fig. 6.

Figure 8

Distributions of the correlators ${\Delta }_{\rho }$ (top panel) and ${\Delta }_{\varphi }$ (bottom panel) for the rectangular (solid lines) and the stadium billiard (dashed lines) in semilogarithmic scale. The data set is the same as that in Fig. 7.

Figure 9

Time evolution (upper panels) of the intensity measured on a screen and its average (lower panels) for the rectangular (left panels) and the stadium billiard (right panels). The upper panels and the solid lines in the lower panels result from the double-slit experiment, and the dashed lines in the lower panels show the time-averaged superposition of the intensities (average taken up to $75$ ns) of the two corresponding single-slit experiments. The time evolution is obtained from the measured scattering matrix $S_{\mathit{ba}}(f)$ via a Fourier transform [cf. Eq. (10)]. For the color scale, see Fig. 6. The slit sizes are $s=9.5$ mm and their distance is set to $d=78$ mm. The distance of the screen from the billiard is $l=320$ mm.

Figure 10

Array of $5\times 5$ positions (crosses) of the emitting antenna for the generation of a directional wave packet in free space. For each of its $25$ positions the field distribution around it is measured with a moving receiving antenna (see text for details).

Figure 11

Three snapshots of the propagation in free space of a wave packet constructed according to Eq. (14), where the time spectra were obtained from measurements for the $25$ positions of the emitting antenna shown in Fig. 10 with the center position at $(x,y)=(0,0)$. The direction of the wave packet is indicated by the angle $\alpha$. For the color scale see Fig. 6.

Figure 12

Sketch of the experimental setup for the construction of a directed wave packet inside the resonator (not to scale). An array of $5\times 5$ holes for the emitting antenna is drilled into the top billiard plate.

Figure 13

Time spectra (left panels) with antenna $b$ at $x=0$ and time-averaged intensity (right panels) for a wave packet constructed according to Eq. (14) which starts with an initial angle of ${270}^{°}$ (top), ${51}^{°}$ (middle), and ${22}^{°}$ (bottom) in the rectangular billiard. The periodic orbits corresponding to these angles are shown as insets. The dashed lines in the top left panel indicate the periodicity of the escape times. The parameters $d$, $s$, and $l$ are the same as in Fig. 9.

Figure 14

Time spectra (left panels) with antenna $b$ at $x=0$ and time-averaged intensity (right panels) for a wave packet constructed according to Eq. (14) which starts with an initial angle of ${270}^{°}$ (top), ${51}^{°}$ (middle), and ${22}^{°}$ (bottom) in the stadium billiard. The dashed lines in the top left panel indicate the periodicity of the escape times. The parameters $d$, $s$, and $l$ are the same as in Fig. 9.