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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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