Quasiperiodic Bloch-Like States in a Surface-Wave Experiment

Abstract

Bloch-like surface waves associated with a quasiperiodic structure are observed in a classic wave propagation experiment which consists of pulse propagation with a shallow fluid covering a quasiperiodically drilled bottom. We show that a transversal pulse propagates as a plane wave with quasiperiodic modulation, displaying the characteristic undulatory propagation in this quasiperiodic system and reinforcing the idea that analogous concepts to Bloch functions can be applied to quasicrystals under certain circumstances.PACS numbers: 47.35.+i, 47.54.+r, 71.23.Ft The main difficulty towards the development of a systematic analytic approach to the transport properties of quasiperiodic systems has been the absence of an analogous Bloch theorem approach as used in the periodic case. In the first efforts to apply a modified version of the Bloch theorem, it was noticed that the dense spectrum of quasiperiodic systems is dominated by only a few special reciprocal lattice points that may be taken to construct a quasi-Brillouin zone [1]. Thus, by considering only the dominant Fourier components, the atomic distribution can be expanded in terms of a discrete aperiodic lattice. Wave functions of the form Ψ k = u k (r)e ik·r will therefore solve the Schrodinger equation. In this case u k is quasiperiodic and should formally be defined on a countable dense set of reciprocal lattice vectors. But, by the above considerations, this expansion is useful since the Fourier development of the modulation function u k can be restricted to the few special reciprocal vectors that dominate the spectra. Thus, Bloch-like states could describe the plane wave propagation in so schematized quasicrystals and free-electron-like bands are expected. Recently this idea was experimentally tested showing that analogous concepts to Bloch functions can be applied to quasicrystals [2].The classic wave propagation in quasicrystalline systems was addressed in a first seminal acoustic experiment of He and Maynard [3] by the feature that acoustical waves are ideal tools to investigate formally similar * Electronic address: manolo@iec.csic.es † Electronic address: aragon@fata.unam.mx quantum propagation effects [4]. On the other hand, appearance of the quasicrystalline symmetry in fluids dynamics was firstly predicted theoretically by Zaslavsky and co-workers [5] and a simulation similar to the conditions of the present experiments was reported in Ref. [6]. Finally, compressible quasisymmetric flows were considered in Ref. [7], whereas a general outlook on order and disorder in fluid motion can be found in the experiments of Gollub [8].In this letter we shall see that a discrete restricted spectral scenario can be displayed by means of impulsive waves in hydrodynamic quasicrystals, where we observe Bloch-like surface waves. The waves are generated at the frequencies corresponding to the Fourier components of the quasiperiodic structure at the dominant diffraction spots. The observed Bloch-like waves are plane waves with quasiperiodic modulation generated ...

Anomalous properties of the band-edge states in large two-dimensional photonic quasicrystals

“…Obviously, this new band-edge state has a twofold symmetry like state H. However, its intensity map is very different from that of H. To the best of our knowledge, bandedge states with reduced symmetry have not been reported before. In the previous studies on lasing 17 and surface wave functions 16 in quasiperiodic structures, only symmetric bandedge states have been observed. This case shows that bandedge states with reduced symmetry is possible in quasiperiodic systems when the sample size is sufficiently large.…”

“…24 By observing the liquid surface wave on a quasiperiodically drilled bottom, the wave function at the band edge is shown to be a plane wave with quasiperiodic modulation. 16 Strongly suppressed group velocity has also been observed at the band-edge states of Fibonacci photonic quasicrystals by ultrashort pulse interferometry. 4 Secondly, great achievements have been made in the applications of the band-edge states in photonic crystals.…”

“…16,17 Thus, how the properties of the band-edge states evolve with sample size were not studied, and the effects due to the quasiperiodic long-range order were not fully manifested in these studies. In order to fully explore these effects, large-sized samples are required.…”

“…[11][12][13][14][15] Recently, a wide interest has been attracted to the bandedge states in quasiperiodic composites. 4,[16][17][18][19][20][21][22][23] Firstly, the property of wave functions at band edges in quasiperiodic structures remains an interesting and unresolved topic. 24 By observing the liquid surface wave on a quasiperiodically drilled bottom, the wave function at the band edge is shown to be a plane wave with quasiperiodic modulation.…”

Spectral and diffusive properties of silver-mean quasicrystals in one, two, and three dimensions

“…The discovery of quasicrystalline solid-state structures, characterized by the presence of long-range order but no translational symmetry 1 , which can be viewed as intermediates between crystalline and amorphous structures 2 with icosahedral 1 , dodecagonal 2 , decagonal 3 , and octagonal 4 orientational symmetry, as well as the classical wave propagation in quasiperiodic structures 5 , renewed the interest in studying quasiperiodic systems, in particular their transport properties [6][7][8][9][10] . Quantities commonly studied to characterize the electron dynamics in this context are the return probability C(t) and the mean-square displacement w(t), defined as…”