Double-Slit Experiments with Microwave Billiards

Abstract

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.

Full text

Double-Slit Experiments with Microwave Billiards
S. Bittner,1B. Dietz,1, ∗M. Miski-Oglu,1P. Oria Iriarte,1A. Richter,1, 2, †and F. Sch¨afer3
1Institut f¨ur Kernphysik, Technische Universit¨at Darmstadt, D-64289 Darmstadt, Germany
2ECT*, Villa Tambosi, I-38123 Villazzano (Trento), Italy
3LENS, University of Florence, I-50019 Sesto-Fiorentino (Firenze), Italy
(Dated: July 28, 2011)
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.
PACS numbers: 05.45.Mt, 03.65.Yz, 03.65.Ta
I. INTRODUCTION
At the beginning of the 19th century, Thomas Young
performed for the first time an experiment that still at-
tracts strong interest in physics: the interference of light
beams passing through a double slit [1]. This experiment
and its outcome have played a profound role in the deve-
lopment of optics and quantum mechanics and have been
used since as a paradigm to unveil the wave nature of a
number of physical entities, in particular single electrons
[2, 3], neutrons [4], atoms [5, 6] and molecules [7].
Recently, Casati and Prosen carried out a numeri-
cal simulation of a double-slit experiment with a well
directed Gaussian wave packet initially confined inside
quantum billiards [8], whose classical dynamics are regu-
lar or chaotic. The time evolution of the wave packet is
governed by the first-order time-dependent Schr¨odinger
equation. The wave packets may leak out of the billiard
via two slits in its sidewall. The interference patterns
resulting from diffraction of the wave packet at the slit
openings have been investigated. For the billiard with re-
gular dynamics, the interference pattern is similar to the
well-known result obtained from a double-slit experiment
with plane waves in the Fraunhofer regime (far field re-
gion); i.e., the intensity encountered on a screen is equal
to the sum of the intensities from the two corresponding
single-slit experiments plus an interference term [9]. On
the other hand, for the billiard with chaotic dynamics,
the intensity on the screen is equal to the superposition
of the diffraction maxima radiated from each of the sin-
gle slits. As a consequence, the intensity pattern becomes
unimodal.
The experiments described in the present work were
initially motivated by these numerical studies. They
were performed with flat cylindrical microwave cavities
∗dietz@ikp.tu-darmstadt.de
†richter@ikp.tu-darmstadt.de
(also called microwave billiards) that have one or two
openings in their side walls. Below a certain excitation
frequency, only the lowest transverse magnetic modes
with electric field perpendicular to the top and bot-
tom plate of the cavity exist. Accordingly, there the
related Helmholtz equation is identical with the time-
independent Schr¨odinger equation of the quantum bil-
liard of corresponding shape [10, 11]. An appropriate
choice of the shape of the cavity allows the investigation
of wave phenomena in quantum billiards with regular,
chaotic, or mixed dynamics. In contrast to the quan-
tum case, the electromagnetic wave equation governing
the time evolution of the waves in microwave billiards
is of second order. Still, as will be outlined below, the
experiments provide insight into the dependence of the
interference patterns resulting from such double-slit ex-
periments on the classical dynamics.
The paper is organized as follows. Section II is devoted
to the description of the experimental setup. Stationary
wave patterns originating from the diffraction of electro-
magnetic waves emitted from a single antenna, which is
situated inside the microwave billiard, at the openings
are presented in Sec. III. In Sec. IV we investigate the
temporal evolution of the diffracted waves in the exterior
close to the slits. In Sec. V we present a new method
developed for the construction of a directed initial wave
packet with an array of emitting antennas. It is used to
realize the situation considered in the numerical simula-
tion of Ref. [8].
II. EXPERIMENTS
A. Billiards
The principle of the construction of the microwave
billiards is illustrated in Fig. 1(a), which exemplifies a
billiard of rectangular shape. They are composed of
three plates, 5 mm thick, made of copper as depicted
in Fig. 1(a). The middle plate has a hole in the form
arXiv:1105.1137v2 [nlin.CD] 27 Jul 2011

2
(a)
(b)
FIG. 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 (in-
dicated by the crosses) are used to introduce the antennas.
(b) Sketch of the experimental setup. The interference pat-
tern 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.
of the shape of the billiard. This yields a microwave
cavity with a height of 5 mm. One of its sides [bottom
side in Fig. 1(b)] is composed of three modular bars of
copper enabling the variation of the slits size sand dis-
tance d(see also Fig. 2) between the slits, respectively.
Microwaves are excited inside the resonator via an an-
tenna located at different positions (crosses). The in-
terference pattern resulting from microwaves leaving the
billiard via the slits is recorded on a “screen,” which con-
sists of a second antenna moving in the exterior (see be-
low). In order to investigate the relation between the
dynamics of the corresponding classical billiard and the
interference patterns, a regular billiard with rectangu-
lar form and a chaotic one with the shape of a desym-
metrized tilted stadium, in the following simply called
stadium, were used in the experiments. The three plates
FIG. 2. Schematic view (not to scale) of the experimental
setup for the measurement of interference patterns. The an-
tenna ais fixed inside the billiard and the antenna bis moved
in small steps (∆x= ∆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 dand the slit size s
are indicated. The vertical distance between the billiard edge
and the lines of measuring points parallel to the lower bil-
liard edge is denoted by l. The coordinate system (x,y) used
in the following is indicated. Its point of origin is the mid-
point between the slits. The microwave power is generated
and measured, respectively, by the VNA.
are screwed together and wires of solder are inserted into
grooves along the contour of the billiard [not shown in
Fig. 1(a)] in order to improve the electrical contact be-
tween the plates. The rectangular microwave billiard has
dimensions 768 ×475 mm. The ratio of the side lengths
is taken close to the golden ratio (√5 + 1)/2 in order to
avoid degeneracies of the eigenmodes. The stadium bil-
liard is composed of a quarter circle and a trapezoid [12].
The radius of the quarter circle is 431 mm, the length of
the left side of the stadium is 347 mm, and the length of
the bottom side 872 mm.
B. Measurements
The microwave power is coupled into and out of the
setup with wire antennas. They consist of a straight
thin wire of about 0.5 mm diameter that is soldered in a
holder. The electromagnetic signal is generated by a vec-
torial network analyzer (VNA) of type PNA-L N5230A
by Agilent Technologies. It is led to one antenna (port a)
via a coaxial cable and fed into the cavity. This antenna
penetrates partially into the cavity via a hole in the top
plate. The output signal is received by another antenna
(port b) in front of the slits outside the billiards. The
ratio of the output power (Pout,b) and the input power
(Pin,a) as well as the relative phase of both signals are
measured. They yield the complex-valued scattering (S)-
matrix elements. The squared modulus of the scattering
matrix is given as
|Sba(f)|2=Pout,b (f)
Pin,a(f).(1)

3
A graphical representation of |Sba(f)|2versus the exci-
tation frequency is referred to as a frequency spectrum.
Resonance frequencies of the cavity are obtained from
the locations of its peaks i.e. resonances. Close to the re-
sonance frequency fnof the nth isolated resonance, the
S-matrix element Sba can be expressed in good approxi-
mation [13, 14] as
Sba ∝δba −iγn,bγn,a
f−fn+iΓn
2
,(2)
where Γnis the full width at half maximum of the re-
sonance. The numerator γn,bγn,a is proportional to the
product of the electric field strengths at the positions ~ra,b
of the antennas, i.e.
Sba(fn)∝Ez(~rb)Ez(~ra).(3)
Accordingly, the electric field distribution Ez(~rb) may be
determined [11] by fixing the position of antenna a, vary-
ing the position ~rbof antenna band measuring Sba for
each position of antenna b, see below in Sec. III. The
electric field intensity Iat antenna bis proportional to
|Ez(~rb)|2and thus to |Sba|2.
Figure 2 illustrates schematically the experimental
setup. Microwave power is coupled into the billiard
through antenna a, which penetrates 4 mm into the cav-
ity. In order to measure the leakage of microwave power
through the slits to the exterior, the receiving antenna
bof 15 mm length is moved outside of the billiard close
to its edge with the slits (see grid of positions in Fig. 2)
in steps of 5 mm which is one third of the minimal con-
sidered wavelength. It is connected to the VNA via a
flexible cable. The distance lbetween the billiard edge
and the lines of measuring points parallel to the edge (in
the following referred to as the screen) varies between 20
and 500 mm.
The measurements were carried out in a so-called an-
echoic chamber (see Fig. 3). Pyramidal polyurethane
foam structures VHP-12 NRL (from Emerson & Cum-
ing) cover the whole surface of the chamber and ensure
attenuation of the reflected microwave power by −50 dB
over the whole frequency range of the measurements [15].
Antenna bis guided by a carriage attached to a position-
ing unit. This unit is driven by bipolar step motors and
connected to a computerized numerical control (CNC)
module. A PC program communicates with the VNA
and with the CNC, enabling the control of the position-
ing unit and the simultaneous gathering of data. An-
tenna bis aligned perpendicularly to the billiard plane
to ensure that only the zcomponent of the electric field
is detected. To prevent a change of the polarization of
the electromagnetic field when it escapes into the three-
dimensional free space, and to avoid parasitic reflections
between the metallic parts of the measurement device
and the billiard, all components are covered with mi-
crowave absorption material that consists of urethane
foam sheets impregnated with carbon (EPP-51 material
from ARC Technologies [16]). Moreover, the copper cable
FIG. 3. (Color online) Photograph of the measurement setup
in the anechoic chamber. The microwave billiard is placed on
a table and the receiving antenna is carefully aligned perpen-
dicularly 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.
connected to the receiving antenna is coated with an ab-
sorptive double layer consisting of urethane foam EPP-51
and urethane flat plate EPF-11. The VNA is calibrated
in order to remove the effects of the cables and the con-
nectors on the measured spectra. These effects include
reflections at connectors, attenuation in the cables and
the time delay of the signal accumulated during its pas-
sage through the cables. For each position of antenna b,
the complex S-matrix element Sba is measured between
antennas aand bfrom 0.5 to 20 GHz with a frequency
step of 5 or 10 MHz. As a consequence the measurement
of the frequency spectrum takes a few seconds for each
of the 20 000 positions of the receiving antenna.
Figure 4 shows a frequency spectrum for the rectangu-
lar (top) and the stadium (bottom) billiard. The emit-
ting antenna awas positioned inside the resonator sym-
metrically with respect to the slits at a distance 400 mm
from the lower billiard edge and the receiving antenna b
outside at the point x= 0, y=l= 320 mm (see coordi-
nate system introduced in Fig. 2). In both billiards the
slit size sis set to 9.5 mm and the distance dbetween the
slits is 78 mm. The spectrum shows many sharp peaks
at resonance frequencies corresponding to quasibound
modes. Below the cut-off frequency fco =c/(2s) = 15.79
GHz defined by the width sof the slits, which have the
effect of a short waveguide (see Fig. 1) on microwaves lea-
ving the billiard, the transmission amplitude is notably
suppressed. Here, cdenotes the velocity of light. Note
that, due to the short length of the waveguides, the cut-
off frequency provides only an approximate value for the
onset of the leakage of microwave power through the slits

4
FIG. 4. Frequency spectrum of the open two-slit rectangular
(top) and stadium (bottom) billiard measured with the emit-
ting antenna ainside the resonator at the point x= 0 mm,
y=−400 mm and the receiving antenna bplaced 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 fco =c/(2s) = 15.79 GHz is determined by
the slit size and marked by an arrow in both panels.
to the exterior.
III. STATIONARY PATTERNS OF THE
INTENSITY ON THE SCREEN
In Fig. 5 we present the squared modulus of the mea-
sured transmission scattering matrix element |Sba|2at a
resonance frequency indicated at the top of the respective
panels as function of the position xof antenna bon the
screen (solid lines) for the rectangular (top panel) and the
stadium billiard (bottom panel). The screen is located
at a distance l= 155 mm from the bottom edge of the
billiard. The slit size and the distance between the slits
are s= 9.5 mm and d= 78 mm, respectively. The posi-
tion of the emitting antenna is symmetrical with respect
to the slits, i.e. x= 0. For the rectangular billiard, we
observe an interference pattern with a central maximum
located at x= 0 and two symmetrical lateral maxima.
If the emitting antenna is placed nonsymmetrically with
respect to the slits, we obtain a very similar interference
pattern, i.e., the measured patterns are nearly indepen-
dent of the position of the emitting antenna as expected
for a stationary field distribution inside the resonator.
It should be noted that such interference patterns are
observed only at frequencies of sharp resonances. The
interference pattern for the rectangular billiard is well
described by the Fraunhofer formula,
I(x) = I1(x) + I2(x)+2pI1(x)I2(x) cos (kdx/l),(4)
which is valid in the far field region if s2/(lλ)1 is
fulfilled [9]. For the parameters of the setup presented
in Fig. 5, s2/(lλ)≈0.01. Here, I1(x) and I2(x) are
the intensities radiated by each individual slit and the
oscillatory term describes the interferences between the
waves emitted through the slits [9]. Experimentally,
I1(x) and I2(x) are obtained from measurements with,
respectively, one of the slits closed. They are well de-
scribed by I(˜x) = sin2(˜x)/˜x2, where ˜x=ksx/l. The
curve obtained by inserting these intensities into Eq. (4)
is plotted as dashed line in the upper panel of Fig. 5,
showing very good agreement with the measured interfe-
rence pattern. Similar results of a double-slit experiment
with water surface waves instead of microwaves have been
published in Ref. [17] where the motion of water waves
in a tank with a shallow bottom was investigated. As in
the case of microwaves the spatial propagation of the wa-
ter waves is governed by the two-dimensional Helmholtz
equation and is equivalent to the Schr¨odinger equation
describing the dynamics in a quantum billiard of corres-
ponding shape.
An interference pattern measured with the stadium bil-
liard is shown as solid line in the bottom panel of Fig. 5.
At x= 0 we observe a maximum in the interference pat-
tern obtained for the rectangular billiard, whereas that
for the stadium billiard exhibits a minimum. According
to Berry [18], the wave functions of a generic chaotic bil-
liard can be modeled by a superposition of plane waves
with the same wave number |~
kn|=kbut random direc-
tions and random amplitudes an, i.e.,
ψk(~r) = X
n
anei~
kn·~r ,(5)
yielding a Bessel function of the first kind of order 0 for
the spatial correlator [11],
hψk(~r)ψk(~r 0)i=J0(k|~r −~r 0|).(6)
This implies that in the vicinity of the slits the waves are
correlated as J0(kd) such that the interference term in
Eq. (4) is modified [17]. Accordingly,
I(x) = I1(x) + I2(x)+
+2pI1(x)I2(x)J0(kd) cos (kdx/l).(7)
For wave numbers kwith J0(kd) = 0 the interference
term vanishes and the intensity should be equal to the
sum of the intensities of the single-slit experiments. The
intensity corresponding to Eq. (7) is plotted as dashed
line in the lower panel of Fig. 5. No agreement between

5
FIG. 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).
Eq. (7) and the experimental result is found. For the ex-
ample shown in Fig. 5, the visibility (Imax −Imin)/Imax
of the measured pattern is comparable to that for the
rectangular billiard. In general the patterns exhibit only
for the rectangular billiard such a clear spatial symme-
try. However, for all considered cases the interference
structure does not disappear, even not for wave numbers
for which the interference term should vanish according
to Eq. (7). It should be noted also that the results for
chaotic billiards with water surface waves [17] do not pro-
vide convincing evidence that the total intensity equals
the sum of the intensities for the single-slit experiments
as predicted in Ref. [8].
Examples of intensity distributions in the plane close
to the slits are shown in Fig. 6 for the rectangular (left
panels) and stadium (right panels) billiard. The slit size
is s= 20 mm and the distance between the slits is set to
d= 240 mm. Antenna ais positioned at x= 0. The grid
of the positions of antenna bcovers an area of 0.5 m2.
In the upper two left panels the intensity patterns at fre-
quencies f= 5.758 GHz and 7.731 GHz corresponding to
isolated resonances are shown. They are symmetric with
respect to the line x= 0, as expected due to the symme-
try of the setup. In the lower left panels the intensities
for frequencies chosen in the regime of overlapping res-
onances are shown. We observe a slightly asymmetric
interference pattern. This is explicable because several
modes, which are symmetric or antisymmetric with re-
spect to the symmetry line, are excited simultaneously
with different strengths such that the resulting field dis-
tribution has no spatial symmetry. Furthermore, exper-
imental imperfections might also contribute to the ob-
served asymmetry. For the stadium billiard (right panels
of Fig. 6) the observation of non-symmetric interference
patterns both for frequencies in the regimes of isolated
and of overlapping resonances is attributed to the ran-
dom field distribution of the corresponding modes. There
are even cases where microwave power leaks out through
just one slit, as, for instance, for f= 4.557 GHz in Fig. 6.
These results provide the first insight into the effect of
the shape of the billiards on the interference patterns.
In order to analyze in more detail the interference pat-
terns of the waves leaving the billiards we define the
quantities
∆ρ= ln |Ez(x0+δx)|2
|Ez(x0−δx)|2,(8)
∆φ= arg[Ez(x0+δx)] −arg[Ez(x0−δx)] ,(9)
which correlate the intensities and the phases, respec-
tively, of the electric field Ezat points x0+δx and
x0−δx, where x0denotes the central position on the
screen. These correlators have been introduced and dis-
cussed in Ref. [19] and provide information about the
symmetry and visibility of the interference structure. In
a double-slit experiment with plane waves the phase dif-
ference of the exiting waves at the slits determines the
symmetry of the obtained pattern whereas the ampli-
tude difference should account for the visibility. If the
amplitudes and the phases of the fields Ezat the posi-
tions x0+δx and x0−δx coincide, i.e. if ∆ρ= ∆φ= 0,
the pattern is symmetric with visibility of 100%.
These correlators were evaluated for the measurements
shown in Fig. 6. Therefore, x0= 0 and the screen is lo-
cated at the bottom side of the billiards at a distance
from 5 mm up to 500 mm with a spatial resolution of 5
mm in yand δx. For each value of ythe pair (∆ρ,∆φ)
was determined for 100 values of δx (see Fig. 2), yielding
a total number of 10 000 measuring points. The parame-
ters sand dare 20 and 240 mm, respectively. Thirty re-
sonance frequencies of the corresponding closed billiards
are chosen as excitation frequencies. Thus we obtain
300 000 pairs of (∆ρ, ∆φ). The resulting joint probabil-
ity distribution of (∆ρ, ∆φ) is shown in Fig. 7. For the
rectangular billiard (left panel), the probability distribu-
tion is strongly concentrated around (∆ρ,∆φ) = (0,0).
This yields an evidence for the high (though not perfect)

6
FIG. 6. (Color online) Two-dimensional intensity patterns (I∝ |Sba(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.
spatial symmetry of the measured interference patterns
(cf. the left panels of Fig. 6). For an ideal double slit
experiment with plane waves, one would expect a nonva-
nishing distribution only at (0,0). The distribution for
the stadium billiard is shown in the right panel. It is
also centered around (0,0), but in comparison to the left
panel much less concentrated in the direction of the ∆φ
coordinate. This is expected due to the slight asymme-
try in the interference patterns of the stadium billiard
observed in the right panels of Fig. 6. In the top and
bottom panels of Fig. 8 the distributions of, respectively,
the intensity and phase correlator are presented. The
distributions of ∆ρare almost identical for the rectan-
gular (solid line) and the stadium (dashed line) billiard

7
FIG. 7. (Color online) Joint probability distributions of the
correlators ∆ρand ∆φof the electric fields defined by Eqs. (8)
and (9) outside the rectangular billiard (left panel) and the
stadium billiard (right panel). Yellow (bright) color corre-
sponds to high and blue (dark) color to low probability. The
pairs (∆ρ, ∆φ) are evaluated for 30 excitation frequencies on a
grid of 10 000 measuring points. The slit sizes are s= 20 mm
and their distance is d= 240 mm as in Fig. 6.
and decay exponentially with |∆ρ|, i.e., the visibility of
the measured interference patterns is similar for both bil-
liards (cf. Fig. 5). In contrast, the distributions of ∆φ
(bottom panel) differ markedly. While the distribution
is peaked at ∆φ= 0 for the rectangular billiard, i.e.,
there most patterns are symmetric, it is bell shaped and
much broader around ∆φ= 0 for the stadium billiard.
As above, this is attributed to the strongly nonsymmet-
rical patterns caused by the random field distributions
as shown in the right panels of Fig. 6. In conclusion, the
distributions of the correlators show a clear difference for
the interference patterns of the regular rectangular and
the chaotic stadium billiard.
IV. TEMPORAL EVOLUTION OF THE FIELDS
OUTSIDE THE BILLIARDS
In this section we analyze and discuss the time evolu-
tion of the field in the vicinity of the double slit. The
emitting antenna ais placed inside the billiard symmet-
rically with respect to the slits at a distance of 400 mm
to the edge with the slits whereas the distance lof the
screen to this edge equals 320 mm. Close to an isolated
resonance, Sba is essentially given by a modified Green
function up to a factor that varies slowly with the fre-
quency [20]. Consequently, the propagator K(~rb, ~ra, t, 0)
is related to the Fourier transform of the experimentally
obtained scattering matrix [21], i.e.,
K(~rb, ~ra, t, 0) ∼˜
Sba(~rb, ~ra, t, 0)
=R∞
−∞ Sba(f)e−2πitf df .
(10)
The experimental spectra are measured at Ndiscrete
frequencies in the range from fmin to fmax and, corres-
pondingly, the discrete Fourier transform is evaluated in
FIG. 8. Distributions of the correlators ∆ρ(top panel) and
∆φ(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.
Eq. (10). The maximal time accessible is determined by
1/∆f, where ∆fis the frequency step used in the ex-
periment. In the present measurements ∆f= 10 MHz,
which corresponds to a maximal time of 100 ns.
The upper panels of Fig. 9 show the time evolution of
the intensity on the screen for the rectangular (left) and
the stadium (right) billiard, i.e., |˜
Sba(x, t)|2versus the
position xof antenna b. The time a wave emitted from
antenna aneeds to reach antenna bon the screen is called
escape time. At each escape time, an interference pattern
is observed as an intensity variation along x. It results
from the interference between the waves emitted from
the two slits. As a result of the spatial symmetry of the
rectangular billiard and the choice of the position of the
emitting antenna aat x= 0, every part of the cylindrical
wave pulse emitted by this antenna with initial momen-
tum direction ~
k= (kx, ky) travels the same path as its
counterpart with ~
k= (−kx, ky). This is a consequence of
the omnidirectionality of the initial wave pulse. Accord-
ingly, these waves hit the slits with no phase difference.
The positions of the maxima and minima observed in
the intensity pattern in the upper left panel of Fig. 9 do
not change with time. For the stadium billiard, however,
there is no spatial symmetry and, consequently, the posi-

8
FIG. 9. (Color online) 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 pa-
nels result from the double-slit experiment, and the dashed
lines in the lower panels show the time-averaged superpo-
sition of the intensities (average taken up to 75 ns) of the
two corresponding single-slit experiments. The time evolu-
tion is obtained from the measured scattering matrix Sba(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.
tions of the maxima and minima in the upper right part
of Fig. 9 change with time.
In order to quantify how the interference patterns are
influenced by the time evolution of the waves emitted
from the two slits, for each position xof antenna b
the intensities |˜
Sba(x, t)|2determined at discrete times
tiare averaged over time up to tN= 75 ns, yield-
ing h|˜
Sba(x, t)|2it=1
NPN
i=1 |˜
Sba(x, ti)|2. It should be
noted that this quantity corresponds to the averaged cur-
rent evaluated in the numerical simulation by Casati and
Prosen [8]. The time average is performed up to 75 ns
since |˜
Sba(x, t)|2is negligibly small for longer times. The
resulting time-averaged intensities are shown in the lower
panels of Fig. 9 as solid lines. For the rectangular billiard
the interference structure is still clearly visible, whereas it
disappears for the stadium billiard. In addition, the sum
of the time-averaged intensities h|˜
Sba(x, t)|2itobtained
from two single-slit experiments in which the left, respec-
tively, right slit was closed (dashed lines) is shown for
comparison. It has no interference structure and shows a
qualitative agreement with the time-averaged intensity
resulting from the double-slit experiment only for the
fully chaotic stadium billiard.
If the emitting antenna is not placed on the symmetry
line, the interference pattern of the time-averaged inten-
sity distribution disappears also for the rectangular bil-
liard and resembles that for the stadium cavity. Indeed,
waves emitted with initial wave vector ~
k= (kx, ky) and
~
k= (−kx, ky) do not cover the same path length until
they reach the slits and thus they have a phase relation
varying in time. This finding confirms that the mech-
anism leading to the emergence of the interference pat-
terns observed in the left panels of Fig. 9 is not linked
to the classical dynamics of the billiard but to spatial
symmetry properties of the system consisting of the bil-
liard with the slits and the emitting antenna. We may
expect from these results that no interference patterns
are observed when the slits are not positioned symmetri-
cally. This was not tested experimentally, though. Only
recently our attention was drawn to a work by Fonte and
Zerbo, where similar results as observed in our measure-
ments were obtained with numerical simulations [22].
We also performed ray-tracing simulations, where clas-
sical pointlike particles were injected from the position
of antenna awith the angles of initial direction sweeping
the whole interval from 0 to 2π. The time of flight tf
the particles need to reach the position of antenna bwas
computed. We observe a good agreement between the
measured time spectra |˜
Sba(x, t)|2and the classical ones
up to a time t∗≈15 ns which correponds to a length
of ct∗≈4.5 m. The latter are obtained by counting the
number of particles that need a certain time of flight and
plotting it versus tf.
The results of the experiments with a single emitting
antenna presented in this section thus lead to the follow-
ing conclusions:
•Clear interference patterns that are symmetric with
respect to the two slits are observed only for the rec-
tangular billiard with the emitting antenna placed
on the symmetry line x= 0. These are well de-
scribed by the Fraunhofer formula, Eq. (4). Fur-
thermore, the time averaged intensities show clear
interference patterns for the rectangular billiard if
the emitting antenna is positioned symmetrically.
•Interference patterns are not observed in the time-
averaged intensities resulting from the stadium bil-
liard and also not for the rectangular billiard with
the emitting antenna placed asymmetrically with
respect to the slits.
•For the stadium billiard, the sum of the time-
averaged intensities of the two single-slit experi-
ments is roughly equal to that of the double-slit
experiment.
The experiments presented in this section were per-
formed with an omnidirectional wave pulse emitted from
a single antenna. However, as mentioned in Sec. I, the
initial states considered in the numerical simulations pre-
sented in Ref. [8] are localized wave packets with a well-
defined direction. For the realization of a similar exper-
imental situation we developed a method for the con-
struction of directional wave packets using an array of

9
antennas, which is presented in the following section. It
was used recently in Ref. [23] to generate plane waves
from a superposition of cylindrical waves.
V. EXPERIMENTS WITH A DIRECTIONAL
WAVE PACKET
In this section we present a new method for the con-
struction of directional wave packets. Their time evolu-
tion is used for the analysis of the experimental spectra
in order to study the dependence of the interference pat-
terns on the billiard geometry as well as on the initial
angle. An arbitrary initial state Ez(~r0, t0) evolves ac-
cording to
Ez(~r, t)∝RGd2~r0K(~r , ~r0, t, t0)∂Ez(~r0,t0)
∂t −
RGd2~r0∂ K(~r,~r0,t,t0)
∂t Ez(~r0, t0),
(11)
where K(~r, ~r0, t, t0) is the electromagnetic propagator be-
tween two antennas at positions ~r0and ~r [24]. The second
term in Eq. (11) can be set to zero by a proper choice of
initial conditions, i.e., Ez(~r0, t0) = 0. For the analysis
of the experimental data we replace K(~r, ~r0, t, t0) by the
Fourier transform ˜
Sba(~r, ~r0, t, t0) of the measured trans-
mission scattering matrix element [see Eq. (10)]. Here ~r
denotes the position of the receiving antenna band ~r0
that of the emitting antenna a. Experimentally, we can
obtain ˜
Sba(~r, ~r0, t, t0) only for discrete positions ~r0iof
the emitting antenna. Accordingly, the integral entering
Eq. (11) has to be replaced by a sum,
Ez(~r, t)∝X
i
˜
Sba(~r, ~r0i, t, t0)∂Ez(~r0i, t0= 0)
∂t .(12)
In analogy to Ref. [8], we set
∂Ez
∂t (~r0,0) = −iωE0exp −(~r0−~
R)2
2σ2!exp[i~
k·(~r0−~
R)] ,(13)
where ~
Rdenotes the central position of the initial wave packet, σits width and ~
kthe wave vector. Inserting Eq. (13)
into Eq. (12) we obtain
E(WP)
z(~r, t)∝ −iωE0X
i
exp −(~r0i−~
R)2
2σ2!exp[i~
k·(~r0i−~
R)] ˜
Sba(~r, ~r0i, t, t0).(14)
Equation (14) is the final expression for the time evo-
lution of the constructed wave packet, E(WP)
z(~r, t). It is
proportional to a sum over the Fourier transforms of the
spectra weighted by complex coefficients. The initial di-
rection and the width of the wave packet can be varied
FIG. 10. Array of 5 ×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).
arbitrarily since ~
k=k(cos α, sin α) and σare free pa-
rameters. However, the width σshould be chosen larger
than the minimal wavelength used in the experiments,
i.e. σ > 15 mm. On the other hand, if it is chosen to be
much larger than the size of the domain of emitting an-
tennas, the coefficients exp{−(~r0i−~
R)2/(2σ2)}would all
be approximately equal to 1 such that all antennas would
provide the same contribution to the sum in Eq. (14).
Thus σis chosen of the order of the size of this domain.
We tested the propagation of the wave packet con-
structed according to Eq. (14) experimentally in free
space. A single emitting antenna ahung down from the
ceiling of an empty room at a position ~ra=~r0iand
the position ~rb=~r of the receiving antenna was var-
ied in the vicinity of ~r0iin a plane perpendicular to the
emitting wire antenna. For each position ~r the transmis-
sion spectrum Sba(~r, ~r0i, f ) was measured for frequencies
from 0.5 to 20 GHz. These transmission measurements
were repeated for altogether 25 positions ~r0iof the emit-
ting antenna aon a square grid (see schematic sketch in
Fig. 10). Using the principle of linear superposition, for
each position ~r of antenna bthe time spectra for the 25
different positions ~r0iof the emitting antenna are added
up according to Eq. (14). Three snapshots for the resul-

10
FIG. 11. (Color online) 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 α. For the color
scale see Fig. 6.
ting wave packet propagation are shown in Fig. 11, with
the width of the of the wave packet and its initial angle
chosen as σ= 20 mm, respectively, α= 31◦. As ob-
served in the upper panels the wave packet moves into
the expected direction and spreads out fast. However,
the spatial structure reveals propagation in additional
directions. We attribute this effect to the discretization
of the initial positions ~r0in Eq. (11).
For the double slit experiments 5 ×5 antenna holes
arranged in a square array at distances of 7.5 mm were
drilled into the top plates of the billiards as shown in the
FIG. 12. Sketch of the experimental setup for the construction
of a directed wave packet inside the resonator (not to scale).
An array of 5×5 holes for the emitting antenna is drilled into
the top billiard plate.
sketch of Fig. 12. The position of the central antenna of
the squared array is located on the line x= 0 at a dis-
tance of 400 mm from the edge with the slits. In analogy
to the measurement in free space presented above, one
emitting antenna ais put consecutively into the 25 holes
at positions ~r0iand the field distributions outside the bil-
liards are obtained by measuring frequency spectra with
the moving antenna b. The computed time spectra are
then added according to Eq. (14). The initial angle of
propagation αof the wave packet constructed this way
is indicated in Fig. 12. It is chosen such that the wave
packet is emitted along periodic orbits of the rectangular
billiard.
Three cases corresponding to different values of the
initial angle αare analyzed in the following. Figure 13
shows for the rectangular billiard in the left panels the
time spectra and in the right panels the time-averaged
intensity patterns for an initial wave packet with width
σ= 30 mm and α= 270◦(top panels), α= 51◦(mid-
dle panels), and α= 22◦(bottom panels) and antenna b
positioned at x= 0. The corresponding periodic orbits
are shown in the insets. The parameters d,s, and lare
the same as in Fig. 9. The time average was computed
up to 75 ns which corresponds to ct = 22.5 m. In the
time spectra the positions of the peaks correspond to the
time the initial wave pulse needs to reach the slits. For
an initial angle α= 270◦the wave packet moves toward
the slits along the vertical bouncing ball orbit, it trav-
els back and forth reflecting at the upper and the lower
edges of the billiard. The time spectrum exhibits a clear
periodicity. The peaks appear in pairs since one part of
the wave packet is sent to the back with an initial an-
gle of α= 90◦and the other to the slits. The period
of the escape times in the upper left panel agrees well

11
FIG. 13. Time spectra (left panels) with antenna bat 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 lare the same as in Fig. 9.
with the length of the bouncing ball orbit, 0.95 m. In
the top right panel of Fig. 13 the time-averaged spatial
intensity displays an interference pattern with visibility
(Imax −Imin)/Imax close to 100%.
The initial angle α= 51◦corresponds to the perio-
dic orbit shown in the inset of the left middle panel of
Fig. 13. The peak structure observed in the middle left
panel does not show a clear periodicity. This is attributed
to the spreading of the wave packet in additional direc-
tions (cf. Fig. 11). The visibility of the corresponding
interference pattern (see middle right panel of Fig. 13)
is approximately 40%. The initial angle α= 22◦cor-
responds to the orbit depicted in the inset of the bot-
tom left panel of Fig. 13. For this case the peaks appear
stochastically. Still, the time-averaged spatial intensity
depicted in the bottom right panel shows interference
patterns with visibility close to 80%. The presence of
a few prominent peaks, especially that at about 1.5 m,
in the time spectrum suggests that these dominate the
interference pattern. However, the shape of the inter-
ference pattern is barely modified if the largest peak is
removed.
The same initial angles are chosen in the stadium bil-
liard. The results are shown in Fig. 14. The peaks in
the top left panel (α= 270◦) again appear periodically,
where the period is equal to the length of the vertical
diffractive orbit shown in the inset, which is 0.86 m. The
visibility of the interferences is less than that for the rec-
tangular billiard and is approximately 60%. In addition,
the peaks are slightly nonsymmetrical with respect to
x= 0. Thus, in spite of remains of interferences, there
are visible differences to the rectangular billiard in the
interference patterns (see the top right panel of Fig. 13).
The dominant peaks in the middle left panel (α= 51◦)
display a periodic structure that again corresponds to the
period of the vertical diffractive orbit, indicating that a
part of the wave packet sticks in the vicinity of this perio-
dic orbit. In addition, further narrow peaks in between
are observed. The visibility of the interference pattern
shown in the middle right panel of Fig. 14 is moderately
lower than the one for the rectangular billiard (see the
middle right panel of Fig. 13). In the bottom left panel
(α= 22◦) the escape times exhibit no periodicity. The
interferences disappear almost completely and the visibil-
ity is approximately 15%. Generally, the differences be-
tween the intensity patterns originating from the rectan-
gular and the stadium billiard with regular and chaotic
dynamics, respectively, are especially large for initial an-
gles close to 20◦. A question that immediately arises is
why in the regular case interferences are well pronounced
only for certain initial angles.

12
FIG. 14. Time spectra (left panels) with antenna bat 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
lare the same as in Fig. 9.
VI. CONCLUSIONS
We performed double-slit experiments with waves lea-
ving a microwave billiard whose dynamics is either regu-
lar or chaotic. Microwaves were coupled into the resona-
tors with differently arranged antennas, and the depen-
dence of the interference patterns on a screen outside the
billiard on its dynamics, the mode of excitation and the
initial direction of the signal were analyzed.
The emission of a wave pulse from a single antenna
generates an omnidirectional pulse. Unless the system
consisting of the billiard with the two slits and the emit-
ting antenna is fully symmetric, the waves travel along
different paths in the billiard before exiting through the
slits. For an asymmetric setup the interference pattern
measured on a screen at a certain distance from the lower
edge of the billiard changes with time. Consequently,
when averaging over time the interference patterns va-
nish. Thus, their emergence does not only depend on
whether the dynamics inside the billiard is regular or
chaotic, whereas the choice of a definite direction of the
initial momentum of the wave packet seems to be crucial
[22].
Furthermore, experiments with a square array confi-
guration of 5 ×5 emitting antenna positions were car-
ried out. The superposition of the Fourier transforms
of the spectra measured successively for each of these
positions provides a method to create a directed wave
packet. A test of it in free space indeed yielded propaga-
tion of the wave packet in a specific direction, although
it spreads out quickly and has additional propagation
directions. For the rectangular billiard with regular dy-
namics a clear interference pattern with 100% visibility
is only observed if the initial wave packet is sent along its
symmetry line. In the stadium billiard with chaotic dy-
namics remnants of interferences are observed practically
for all initial angles of the directed wave packet although
the visibility is in general smaller than for the rectangu-
lar cavity. For angles close to 20◦the interferences are
almost completely suppressed, whereas they appear with
high visibility (about 80%) for the rectangular one. The
underlying mechanism for the formation of interferences
is not fully understood since the high sensitivity of the
patterns to the choice of the initial parameters of the
wave packet does not allow to draw final conclusions.
In conclusion, the numerical results presented by
Casati and Prosen [8] are only approximately reproduced
for certain initial directions for both the rectangular and
the stadium billiard. These discrepancies cannot be at-
tributed to the different orders in time of the electromag-
netic wave equation and the Schr¨odinger equation since
the interference patterns always result from a time aver-

13
age. It should be noted that in the simulations performed
by Casati and Prosen the sharp distinction between pat-
terns from billiards with regular and chaotic dynamics
was only obtained for certain initial conditions, too [25].
This demonstrates that further investigations are neces-
sary for an understanding of the interference patterns.
Repeating the present measurements with visible light
is desirable because a higher ratio between the cavity
size and the wavelength can be realized, i.e., the system
will be closer to the semiclassical limit. Preparations of
appropriately shaped optical cavities with regular and
chaotic dynamics, respectively, are underway.
ACKNOWLEDGMENTS
We acknowledge T. Prosen and H.-J. St¨ockmann for
numerous suggestions and useful discussions. Special
thanks go to R. Jakoby and his group from the Institute
of Microwave Engineering at the TU Darmstadt for pro-
viding the anechoic chamber and helping with the mea-
surements in it. One of us (P. O. I.) gratefully thanks
the Deutscher Akademischer Austausch Dienst (DAAD)
and the Fundaci´on La Caixa for financial support. This
work was supported by the DFG through SFB634.
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