Microwave transmission spectra in regular and irregular one-dimensional scattering arrangements

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1 Physica E 9 (2001) Microwave transmission spectra in regular and irregular one-dimensional scattering arrangements Ulrich Kuhl, Hans-Jurgen Stockmann Fachbereich Physik, Philipps-Universitat, Renthof 5, D 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: Dd; Qs; b; k Keywords: Anderson localization; Photonic crystal; Harper equation; Hofstadter buttery; Correlated disorder 1. Introduction Since the pioneering paper of Anderson [1] a lot of work has been done in the theoretical studies of the one-dimensional tight-binding Schrodinger equation n+1 + V n n + n 1 = E n ; (1) where V n are the potentials at site n, and n is the amplitude of the wave function. All transfer matrix elements have been assumed to be equal and have been normalized to one. Only nearest-neighbour interactions have been considered. Depending on the site potentials a number of dierent situations can be found. For constant V n Corresponding author. address: stoeckmann@physik.uni-marburg.de (H.-J. Stockmann). regular allowed and forbidden transmission bands are observed, in complete analogy to electronic Bloch bands in crystalline solids. Because of this correspondence it is a common practice to speak of photonic crystals and photonic band gaps in this context [2]. For a random sequence of site potentials we have the one-dimensional Anderson model with site disorder [1]. In the context of dynamical localization the interest focussed on so-called pseudo-random sequences where the site potentials are given by V n = V 0 cos(2n ) [3]. For the special case =1 the corresponding Schrodinger equation is known as the Harper equation. It has been studied already 1976 by Hofstadter in the context of an electron in a two-dimensional crystalline lattice with a perpendicularly applied magnetic eld [4]. Depending on whether, corresponding to the number of ux quanta per unit cell, is rational or irrational, the trans /01/$ - see front matter? 2001 Elsevier Science B.V. All rights reserved. PII: S (00)

2 U. Kuhl, H.-J. Stockmann / Physica E 9 (2001) mission shows Bloch bands or can be described by a Cantor set. The observed two-dimensional transmission pattern in the (; E) plane is known as the Hofstadter buttery. According to Anderson s work the existence of transmission bands should be impossible in one-dimensional disordered systems, but recently it was shown by Izrailev and Krokhin [5] that for a peculiar type of correlated disorder even here allowed bands and mobility edges can be observed. In this letter we give a review on microwave analogue experiments on the one-dimensional tight-binding model. After introducing the idea of the experimental approach a number of recent results are presented. 2. Idea of the experimental approach The tight-binding Hamiltonian (1) can be rewritten in form of a transfer matrix equation ( ) ( ) n+1 n = T n ; (2) n n 1 where the transfer matrix is given by ( ) E Vn 1 T n = : (3) 1 0 This reformulation has the advantage that now the amplitudes of the wave function along the chain are obtained by ordinary matrix multiplication, provided that the initial values 0, 1 are known. We shall see in a moment that a very similar transfer matrix equation governs the propagation of electromagnetic waves through a one-dimensional array of scatterers. This is the starting point of the experimental approach to the study of tight-binding Schrodinger equations [6]. Fig. 1 shows the experimental set-up. One hundred cylindrical scatterers can be introduced into a waveguide with dimensions a = 20 mm, b = 10 mm and a total length of 2.1 m. The lengths of all scatterers can be varied individually with the help of micrometer screws. The upper part of the waveguide can be rotated against the lower one thus varying the position of antenna 2. This feature enabled us to study not only the total transmission through the system, but also to measure the eld intensities within the waveguide. The experiments were performed in the frequency range where only the rst mode can propagate, ranging from the cuto frequency of min = c=2a= 7.5 GHz up to max = c=2b = c=a= 15 GHz, where the propagation of the second mode becomes possible. The dispersion relation is given by k =(2=c) 2 2 min. All transmission data presented below are plotted as a function of the wave number k in units of =d, where d =20:5 mm is the distance between the scatterers. In the single-mode regime the propagation of the waves can be described by a 2 2 transfer matrix. Let a n, b n be the amplitudes of the waves propagating to the right and to the left, respectively, between scatterers n 1 and n (see Fig. 1). Then the amplitudes in the subsequent section are obtained as ( ) ( ) an+1 an = T n ; (4) b n+1 b n where T n is the transfer matrix describing the properties of scatterer n. From time-reversal symmetry follows that the transfer can be written as T n = 1 t n e(+n) rn t n e rn t n e 1 t n e (+n) ; (5) where t n, r n are the moduli of transmission and reection amplitudes, respectively, obeying t n 2 + r n 2 = 1 (in reality about 0.3% of the energy is absorbed by each scatterer). n is the phase of the transmission amplitude, and = kd=2 is the phase shift from the free propagation between the scatterers which has been included into the transfer matrix for convenience. A comparison of Eqs. (2) and (4) shows the close analogy of the one-dimensional tight-binding Schrodinger equation with the wave propagation through a single-mode waveguide with inserted scatterers. The analytical form of the respective transfer matrices is dierent, however, and it is not immediately clear how to relate the site potentials to the screw lengths. We proceeded quite pragmatically by mapping the minimum potential value to a screw length of 0 mm, and the maximum value to a screw length of 3 mm, and interpolating linearly in between.

3 386 U. Kuhl, H.-J. Stockmann / Physica E 9 (2001) 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 justication 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 cursorily. To give an impression of what can be done, one example is presented for each of the situations listed in the introduction. (i) V n = const. Fig. 2 shows two transmission pattern for a situation where only every third (a) and every fourth (b) scatterer was introduced 3 mm [6]. The forbidden and allowed Bloch bands are clearly discernible. Since the lattice constants for the two cases are 3d and 4d, respectively, the widths of the Brioullin Fig. 2. Transmission through an array with every third (a) and every fourth (b) scatterer introduced. The plotted wave number range corresponds to a frequency range from 7.5 to 15 GHz. zones are =3d and =4d, in accordance with the experiment. The experimental set-up allows the determination of the wave function within the waveguide as

4 U. Kuhl, H.-J. Stockmann / Physica E 9 (2001) Fig. 4. Transmission spectra for a periodic arrangement of scatterers with ranging from 0 to 1 in steps of The transmission intensities were converted to a gray scale. The rst two Bloch bands are seen, showing two copies of the Hofstadter buttery. 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) V n = 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) V n = V 0 cos(2n). Depending on whether is rational or irrational Bloch bands or Cantor-set spectra are expected. Fig. 4 shows the result [6]. In the experiment the cosine function was replaced by a Heaviside step function, which was much easier to realize. In the accessible frequency range the rst two Brillouin zones are visible, both of them unfortunately blurred by absorption at the low and the high-frequency ends, respectively. Nevertheless, the similarity with the Hofstadter buttery [4] is clearly recognizable. This was the rst experimental realization of this exotic object. Fig. 5. (Top) Sequence of screw lengths with hidden correlated disorder. (Bottom) Transmission spectrum obtained with this sequence. (iv) V n = correlated disordered. In a recent work Izrailev and Krokhin developed a technique to calculate from an arbitrary prescribed transmission structure a sequence of site potentials reproducing this transmission structure. Fig. 5 shows a preliminary experimental example. In the upper part the used site potential is shown [8]. It looks completely random, but

5 388 U. Kuhl, H.-J. Stockmann / Physica E 9 (2001) actually there is an intricate hidden correlation between the sites. In the lower part the observed transmission spectrum is plotted, showing transmission for k=(=d) below 0.3, and in the range , with a gap in between. This is a rst experimental demonstration of the fact that contrary to common wisdom, transmission bands may exist in one-dimensional disordered systems. A more complete account of these results has been published elsewhere [9]. Acknowledgements The experiment on correlated disordered systems have been performed in cooperation with F. Izrailev and A. Krokhin, Puebla. The idea to the cooperation was developed during a workshop at the International Center for Sciences in Cuernavaca in November Final discussions took place at the workshop at the MPI for Complex Systems in Dresden in May We thank the organizers of these workshops for the invitations and the institutions for their hospitality which made this work possible. The experiments were supported by the DFG via the SFB 185 Nichtlineare Dynamik. References [1] P. Anderson, Phys. Rev. 109 (1958) [2] C. Soukoulis (Ed.), Photonic Band Gaps and Localization, Proceedings of the NATO Advanced Study Institute, 1991, Plenum Press, New York, [3] M. Griniasty, S. Fishman, Phys. Rev. Lett. 60 (1988) [4] D. Hofstadter, Phys. Rev. B 14 (1976) [5] F. Izrailev, A. Krokhin, Phys. Rev. Lett. 82 (1999) [6] U. Kuhl, H.-J. Stockmann, Phys. Rev. Lett. 80 (1998) [7] C. Dembowski et al., Phys. Rev. E 60 (1999) [8] F. Izrailev, private communication. [9] U. Kuhl, F. Izrailev, A. Krokhin, H.-J. Stockmann, Appl. Phys. Lett. 77 (2000) 633.