Microwave transmission spectra in regular and irregular one-dimensional scattering arrangements
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Physica E 9 (2001) 384–388
www.elsevier.nl/locate/physe
Microwave transmission spectra in regular and irregular
one-dimensional scattering arrangements
Ulrich Kuhl, Hans-Jurgen Stockmann ∗
Fachbereich Physik, Philipps-Universitat, Renthof 5, D-35032 Marburg, Germany
Abstract
There is a close correspondence between one-dimensional tight-binding systems, and the propagation of microwaves
through a single-mode waveguide with inserted scatterers. Varying the lengths of the scatterers arbitrary sequences of site
potentials can be realized. Exemplary results on the transmission through regular and random arrangements of scatterers as
well as through sequences with correlated disorder are presented. ? 2001 Elsevier Science B.V. All rights reserved.
PACS: 42.25.Dd; 42.70.Qs; 71.20.−b; 71.23.−k
Keywords: Anderson localization; Photonic crystal; Harper equation; Hofstadter butter y; Correlated disorder
1. Introduction regular allowed and forbidden transmission bands are
observed, in complete analogy to electronic Bloch
Since the pioneering paper of Anderson [1] a lot of bands in crystalline solids. Because of this correspon-
work has been done in the theoretical studies of the dence it is a common practice to speak of photonic
one-dimensional tight-binding Schrodinger equation crystals and photonic band gaps in this context [2].
For a random sequence of site potentials we have
n+1 + Vn n + n−1 =E n; (1) the one-dimensional Anderson model with site dis-
where Vn are the potentials at site n, and n is the order [1]. In the context of dynamical localization
amplitude of the wave function. All transfer matrix the interest focussed on so-called pseudo-random
elements have been assumed to be equal and have sequences where the site potentials are given by
been normalized to one. Only nearest-neighbour in- Vn = V0 cos(2 n ) [3]. For the special case = 1
teractions have been considered. the corresponding Schrodinger equation is known
Depending on the site potentials a number of as the Harper equation. It has been studied already
di erent situations can be found. For constant Vn 1976 by Hofstadter in the context of an electron
in a two-dimensional crystalline lattice with a per-
∗
pendicularly applied magnetic eld [4]. Depending
Corresponding author.
E-mail address: stoeckmann@physik.uni-marburg.de (H.-J.
on whether , corresponding to the number of ux
Stockmann). quanta per unit cell, is rational or irrational, the trans-
1386-9477/01/$ - see front matter ? 2001 Elsevier Science B.V. All rights reserved.
PII: S 1 3 8 6 - 9 4 7 7 ( 0 0 ) 0 0 2 3 3 - 2U. Kuhl, H.-J. Stockmann / Physica E 9 (2001) 384–388 385
mission shows Bloch bands or can be described by The experiments were performed in the frequency
a Cantor set. The observed two-dimensional trans- range where only the rst mode can propagate, rang-
mission pattern in the ( ; E) plane is known as ing from the cuto frequency of min = c=2a= 7.5 GHz
the Hofstadter butter y. According to Anderson’s up to max = c=2b = c=a= 15 GHz, where the prop-
work the existence of transmission bands should be agation of the second mode becomes possible.
q The
impossible in one-dimensional disordered systems, dispersion relation is given by k = (2=c) 2 − 2min .
but recently it was shown by Izrailev and Krokhin
All transmission data presented below are plotted
[5] that for a peculiar type of correlated disorder
as a function of the wave number k in units of
even here allowed bands and mobility edges can be
=d, where d = 20:5 mm is the distance between the
observed.
scatterers.
In this letter we give a review on microwave
In the single-mode regime the propagation of
analogue experiments on the one-dimensional
the waves can be described by a 2×2 transfer ma-
tight-binding model. After introducing the idea of the
trix. Let an , bn be the amplitudes of the waves pro-
experimental approach a number of recent results are
pagating to the right and to the left, respectively,
presented.
between scatterers n − 1 and n (see Fig. 1). Then
the amplitudes in the subsequent section are ob-
tained as
2. Idea of the experimental approach
! !
an+1 an
The tight-binding Hamiltonian (1) can be rewritten = Tn ; (4)
in form of a transfer matrix equation bn+1 bn
n+1 n where Tn is the transfer matrix describing the proper-
= Tn ; (2)
n n−1 ties of scatterer n. From time-reversal symmetry fol-
lows that the transfer can be written as
where the transfer matrix is given by
E − Vn −1
1 (+ n )
|tn | e |r|tnn|| e−
Tn =
1 0
: (3) Tn = ; (5)
− |r|tnn|| e |t1n | e−(+ n )
This reformulation has the advantage that now the
amplitudes of the wave function along the chain are where |tn |, |rn | are the moduli of transmission
obtained by ordinary matrix multiplication, provided and re ection amplitudes, respectively, obeying
2 2
that the initial values 0 , 1 are known. |tn | + |rn | = 1 (in reality about 0.3% of the energy
We shall see in a moment that a very similar trans- is absorbed by each scatterer). n is the phase of the
fer matrix equation governs the propagation of elec- transmission amplitude, and = kd=2 is the phase
tromagnetic waves through a one-dimensional array of shift from the free propagation between the scatterers
scatterers. This is the starting point of the experimen- which has been included into the transfer matrix for
tal approach to the study of tight-binding Schrodinger convenience.
equations [6]. A comparison of Eqs. (2) and (4) shows the
Fig. 1 shows the experimental set-up. One hundred close analogy of the one-dimensional tight-binding
cylindrical scatterers can be introduced into a wave- Schrodinger equation with the wave propagation
guide with dimensions a = 20 mm, b = 10 mm and a through a single-mode waveguide with inserted scat-
total length of 2.1 m. The lengths of all scatterers can terers. The analytical form of the respective transfer
be varied individually with the help of micrometer matrices is di erent, however, and it is not immedi-
screws. The upper part of the waveguide can be ro- ately clear how to relate the site potentials to the screw
tated against the lower one thus varying the position lengths. We proceeded quite pragmatically by map-
of antenna 2. This feature enabled us to study not only ping the minimum potential value to a screw length
the total transmission through the system, but also to of 0 mm, and the maximum value to a screw length
measure the eld intensities within the waveguide. of 3 mm, and interpolating linearly in between.386 U. Kuhl, H.-J. Stockmann / Physica E 9 (2001) 384–388
Fig. 1. (Top) Schematic view of the waveguide. The microwaves are coupled in through antenna 1 on the left and coupled out through
antenna 2 on the right. (Bottom) Photograph of the apparatus.
The optimum maximum screw length of 3 mm had
been determined before in a preliminary step. Though
lacking a sound justi cation the procedure proved to
be successful.
3. Experimental results
We now turn to the presentation of some typical
results. For lack of space this can be done only curso-
rily. To give an impression of what can be done, one
example is presented for each of the situations listed
in the introduction.
(i) Vn = const. Fig. 2 shows two transmission pat- Fig. 2. Transmission through an array with every third (a) and
tern for a situation where only every third (a) and ev- every fourth (b) scatterer introduced. The plotted wave number
range corresponds to a frequency range from 7.5 to 15 GHz.
ery fourth (b) scatterer was introduced 3 mm [6]. The
forbidden and allowed Bloch bands are clearly dis- zones are =3d and =4d, in accordance with the ex-
cernible. Since the lattice constants for the two cases periment. The experimental set-up allows the determi-
are 3d and 4d, respectively, the widths of the Brioullin nation of the wave function within the waveguide asU. Kuhl, H.-J. Stockmann / Physica E 9 (2001) 384–388 387
Fig. 4. Transmission spectra for a periodic arrangement of scatter-
ers with ranging from 0 to 1 in steps of 0.005. The transmission
intensities were converted to a gray scale. The rst two Bloch
bands are seen, showing two copies of the Hofstadter butter y.
Fig. 3. (a) Bloch function for the case that every fth scatterer is
introduced at 3 mm. (b) Localized wave function in a scattering
arrangement where half of the scatterers, chosen at random, were
introduced at 3 mm.
well, as was explained above. Fig. 3(a) shows a Bloch
function thus obtained.
(ii) Vn = random. This is the situation of the
one-dimensional Anderson model with site disorder.
Hence localization is expected. Fig. 3(b) shows an
example of a localized wave function. Similar results
have been obtained in superconducting cavities using
the perturbing bead method [7].
(iii) Vn = V0 cos(2 n). Depending on whether is Fig. 5. (Top) Sequence of screw lengths with hidden correlated
disorder. (Bottom) Transmission spectrum obtained with this se-
rational or irrational Bloch bands or Cantor-set spectra
quence.
are expected. Fig. 4 shows the result [6]. In the exper-
iment the cosine function was replaced by a Heaviside
step function, which was much easier to realize. In (iv) Vn = correlated disordered. In a recent work
the accessible frequency range the rst two Brillouin Izrailev and Krokhin developed a technique to calcu-
zones are visible, both of them unfortunately blurred late from an arbitrary prescribed transmission struc-
by absorption at the low and the high-frequency ends, ture a sequence of site potentials reproducing this
respectively. Nevertheless, the similarity with the Hof- transmission structure. Fig. 5 shows a preliminary ex-
stadter butter y [4] is clearly recognizable. This was perimental example. In the upper part the used site
the rst experimental realization of this exotic object. potential is shown [8]. It looks completely random, but388 U. Kuhl, H.-J. Stockmann / Physica E 9 (2001) 384–388
actually there is an intricate hidden correlation be- We thank the organizers of these workshops for the
tween the sites. In the lower part the observed trans- invitations and the institutions for their hospitality
mission spectrum is plotted, showing transmission for which made this work possible. The experiments were
k=(=d) below 0.3, and in the range 0.5 – 0.8, with a supported by the DFG via the SFB 185 “Nichtlineare
gap in between. This is a rst experimental demon- Dynamik”.
stration of the fact that contrary to common wisdom,
transmission bands may exist in one-dimensional dis-
ordered systems. A more complete account of these References
results has been published elsewhere [9].
[1] P. Anderson, Phys. Rev. 109 (1958) 1492.
[2] C. Soukoulis (Ed.), Photonic Band Gaps and Localization,
Proceedings of the NATO Advanced Study Institute, 1991,
Acknowledgements Plenum Press, New York, 1993.
[3] M. Griniasty, S. Fishman, Phys. Rev. Lett. 60 (1988) 1334.
The experiment on correlated disordered systems [4] D. Hofstadter, Phys. Rev. B 14 (1976) 2239.
have been performed in cooperation with F. Izrailev [5] F. Izrailev, A. Krokhin, Phys. Rev. Lett. 82 (1999) 4062.
[6] U. Kuhl, H.-J. Stockmann, Phys. Rev. Lett. 80 (1998)
and A. Krokhin, Puebla. The idea to the cooperation 3232.
was developed during a workshop at the International [7] C. Dembowski et al., Phys. Rev. E 60 (1999) 3942.
Center for Sciences in Cuernavaca in November 1998. [8] F. Izrailev, private communication.
Final discussions took place at the workshop at the [9] U. Kuhl, F. Izrailev, A. Krokhin, H.-J. Stockmann, Appl. Phys.
MPI for Complex Systems in Dresden in May 1999. Lett. 77 (2000) 633.You can also read
