Experimental Evidence of Rainbow Trapping and Bloch Oscillations of Torsional Waves in Chirped Metallic Beams

Abstract

The Bloch oscillations (BO) and the rainbow trapping (RT) are two apparently unrelated phenomena, the former arising in solid state physics and the latter in metamaterials. A Bloch oscillation, on the one hand, is a counter-intuitive effect in which electrons start to oscillate in a crystalline structure when a static electric field is applied. This effect has been observed not only in solid state physics but also in optical and acoustical structured systems since a static electric field can be mimicked by a chirped structure. The RT, on the other hand, is a phenomenon in which the speed of a wave packet is slowed down in a dielectric structure; different colors then arrive to different depths within the structure thus separating the colors also in time. Here we show experimentally the emergence of both phenomena studying the propagation of torsional waves in chirped metallic beams. Experiments are performed in three aluminum beams in which different structures were machined: one periodic and two chirped. For the smaller value of the chirping parameter the wave packets, with different central frequencies, are back-scattered at different positions inside the corrugated beam; the packets with higher central frequencies being the ones with larger penetration depths. This behavior represents the mechanical analogue of the rainbow trapping effect. This phenomenon is the precursor of the mechanical Bloch oscillations, which are here demonstrated for a larger value of the chirping parameter. It is observed that the oscillatory behavior observed at small values of the chirp parameter is rectified according to the penetration length of the wave packet.

Figure 1

Scheme of the structured metallic beam and the experimental setup: (1) NI-PXI for generating, recording and analyzing signals, (2) high-fidelity audio amplifier, (3) electromagnetic-acoustic transducer (EMAT), (4) machined beam and (5) Doppler interferometer. The beam with rectangular section can be separated in three regions. On the right, a vibration isolation system (A) consisting of a wedge covered by an absorbing mastic seal. The central region is uniform while the region on the left contains the chirped structure. The lengths ${\ell }_n$ of the different cells n defining the structuration of the beam is determined by an analytical expression (see Eq. (1) in the text). The (red) spots indicate the position of the laser measurements.

Figure 2

(a) Calculated band structure of the torsional modes propagating in a structured aluminum beam with rectangular cross section. The levels are given as function of the chirp intensity γ, a dimensionless parameter representing the elastic analogue of a DC electric field. The vertical dashed lines define the two chirped structures manufactured to demonstrate the Bloch oscillations and the rainbow trapping. They correspond to γ = 0.03 and 0.06, respectively. (b) Frequency interval, δf, between adjacent levels within the second pass band of the frequency spectrum represented in (a). The intervals are represented for several values of the chirp parameter γ; from γ = 0 (periodic) to 0.07. Notice that for γ > 0.03 the intervals between levels are approximately constant at the band center, defining the regime where Bloch oscillations can be observed.

Figure 3

Measured time evolution of a wave packet of torsional waves propagating in the elastic beam containing a periodically structured region, corresponding to the sample γ = 0. (a) Results for a wave packet whose central frequency lies in the first gap (f_C = 8 kHz). It is observed that the wave packet is totally reflected at the interface. (b) Results when the central frequency lies in the second band (f_C = 11.5 kHz). Now, partial transmission and partial reflection of the packet is observed each time that the packet crosses the interface. The horizontal red lines are guides for the eye defining the interface between the uniform and the periodic parts of the beam. The beam is schematically drawn on the right-hand side. The color scale defines the measured amplitude of displacement (in arbitrary units).

Figure 4

Experimental characterization of the rainbow trapping of torsional waves in a chirped aluminum beam. The propagation of a wave packet with central frequency, f_C, is shown as a function of time and position in the sample with chirp intensity γ = 0.03. (a) Results for f_C = 9 kHz; (b) 10 kHz; (c) 11 kHz; and (d) 12 kHz. The beam is schematically drawn on the right-hand side. The color scale defines the measured amplitude of displacement (in arbitrary units).

Figure 5

(a) Levels of the different minibands locally defined at the cells defining the structured beam with a chirp parameter γ = 0.03. White spaces between two consecutive set of levels define the local bandgaps. The colored lines describe qualitatively the behavior wave-packets centered at the frequencies analyzed in Fig. 4: 9 kHz (red), 10 kHz (yellow), 11 kHz (green) and 12 kHz (blue). The vertical arrows define the cells where the upper edge of the forbidden gap has the corresponding frequency, establishing the penetration lengths at which the wave-packets is reflected. Notice that wave-packets with higher frequencies are reflected at deeper distances inside the chirped beam, a feature defining the rainbow trapping effect.

Figure 6

Experimental characterization of the Bloch oscillations of torsional waves propagating in a chirped structure with γ = 0.06. (a) Oscillatory behavior observed for a wave packet centered at f_C = 14.5 kHz. (b) Oscillations corresponding to a packet with f_C = 15 kHz. The horizontal red lines define the interface between the uniform and structured parts of the beam, which is schematically drawn on the right-hand side. The color scale defines the amplitude of displacement (in arbitrary units).

Figure 7

Levels of the different minibands locally defined at the cells of the structured beam with a parameter γ = 0.06. White spaces between two consecutive set of levels define the local bandgaps. The horizontal red lines defines the central frequencies of the wave-packets analyzed in Fig. 5: 14.5 kHz and 15 kHz. The arrows describes qualitatively the Bloch oscillations of the wave-packets. The packet centered at 14.5 kHz (continuous line) bounces between the cells defined by the vertical black arrows while that centered at 15 kHz (dashed line) does between cells defined by the dashed arrows. The red dotted lines indicates the leaking of energy through the interface with the uniform part of the beam as it is observed in Fig. 5. The amplitude of the oscillations for a given frequency is defined by the distance between vertical arrows.

Figure 8

Experimental characterization of the rectified rainbow-Bloch oscillations in a chirped aluminum beam. The propagation of a wave packet with central frequency, f_C, is shown as a function of time and position in the sample with chirp intensity γ = 0.06. (a) Results for f_C = 11 kHz; (b) 12 kHz; (c) 13 kHz; and (d) 14 kHz. The beam is schematically drawn on the right-hand side. The color scale defines the amplitude of displacement (in arbitrary units).

Figure 9

(Upper panel) Scheme of the chirped beam corresponding to γ = 0. It consists of a periodic distributions of large cuboids with length ${\ell }_W$ separated by small notches with a much smaller length ${\ell }_C$. The lattice period is $\ell ={\ell }_W+{\ell }_C$. (Bottom panel) Plot of the function $\cos (k_Z\ell )$ obtained from the analytical dispersion relation of torsional waves propagating in the periodic beam. The regions where $\left|\cos (k_Z\ell )\right|>1$ define the bandgaps of the periodic beam.