Experimental observation of roton-like dispersion relations in metamaterials

Abstract

Previously, rotons were observed in correlated quantum systems at low temperatures, including superfluid helium and Bose-Einstein condensates. Here, following a recent theoretical proposal, we report the direct experimental observation of roton-like dispersion relations in two different three-dimensional metamaterials under ambient conditions. One experiment uses transverse elastic waves in microscale metamaterials at ultrasound frequencies. The other experiment uses longitudinal air-pressure waves in macroscopic channel–based metamaterials at audible frequencies. In both experiments, we identify the roton-like minimum in the dispersion relation that is associated to a triplet of waves at a given frequency. Our work shows that designed interactions in metamaterials beyond the nearest neighbors open unprecedented experimental opportunities to tailor the lowest dispersion branch—while most previous metamaterial studies have concentrated on shaping higher dispersion branches.

Fig. 1.. Two blueprints for roton metamaterials.

(A) 3D microstructure unit cell of a metamaterial beam (26) supporting transverse-like rotons for elastic wave propagation along the z-direction. The blue and red cylinders are responsible for the nearest-neighbor interactions and third-nearest-neighbor interactions, respectively, between the yellow masses. Different colors are for illustration only; the entire structure is made from a single polymer material. The geometrical parameters are indicated. (B) Unit cell of the channel-based metamaterial beam supporting rotons for airborne longitudinal pressure waves along the z-direction in the channel system. This unit cell is roughly complementary to the unit cell in (A) and is composed of a bottom piece and an upper piece, whose front half is intentionally removed to show the inner compartment (yellow). The cylindrical channels for air pressure propagation are rendered semitransparent in red and blue, respectively, in analogy to (A). Here, the masses in (A) correspond to cylindrical compartments. A microphone is installed in the through hole with diameter d on the front wall of the lower piece. The other holes are for alignment and assembly. Only geometrical parameters different from those in (A) are given in (B).

Fig. 2.. Roton metamaterial microstructures for elastic waves.

The shown polymer samples, which have been manufactured in one piece by multiphoton 3D laser microprinting, follow the blueprint shown in Fig. 1A. (A) Overview of a sample imaged with a wide-field microscope. Photo credit: M. F. Groß (Karlsruhe Institute of Technology) and T. Frenzel (Karlsruhe Institute of Technology). (B) 3D iso-intensity surface acquired with a confocal fluorescence optical microscope (LSM 800, Zeiss) using the autofluorescence of the polymer. Scale bars and labels added in postprocessing using blender. Parts of the unit cell frames are made to appear transparent to reveal the interior. The third-nearest-neighbor coupling is colored in red, the nearest-neighbor coupling is colored in blue. (C to E) Scanning electron micrographs of (C) the unit cell frames, (D) the uppermost layers, and (E) a view along the center axis of one column of unit cells.

Fig. 3.. Roton metamaterial sample for airborne sound.

(A) The 3D-printed polymer sample follows the blueprint shown in Fig. 1B. It has been assembled from 100 individual pieces, 2 for each of the N_z = 50 unit cells. The metamaterial sample has a length of 2 m along the z-direction. Therefore, only the bottom part and the top part are shown here. (B) One of the two pieces for one unit cell. (C) Zoom-in view of the highlighted rectangle region in (B) showing an installed microphone on the side wall. Photo credit: J. A. Iglesias Martínez (Université de Bourgogne Franche-Comté).

Fig. 4.. Measured and calculated roton dispersions.

(A) Measured raw data (left) for the sample in Fig. 2 versus position and frequency and derived roton band structure (right). (B) Corresponding numerically calculated behavior for the same finite sample length and including damping. (C) As in (A) but for the sample in Fig. 3. (D) Numerically calculated behavior corresponding to the measurements in (C). The white solid curves are the calculated roton band structures for a lossless metamaterial beam that is infinitely extended along the z-direction. For the elastic metamaterial, we use the geometrical parameters: a_xy = 200 μm, a_z = 100 μm, 2r₁ = 16.8 μm, 2r₂ = 25.2 μm, and t₂ = 60 μm. For the airborne metamaterial, we use the geometrical parameters: a_xy = 100 mm, a_z = 50 mm, 2r₁ = 10 mm, 2r₂ = 16 mm, 2r₃ = 30 mm, t₂ = 30 mm, d = 9.8 mm, r₄ = 7.5 mm, and L = 120 mm. The gray curves in (B) and (D) correspond to the approximate analytical dispersion relations of the higher-order gradient effective medium model with parameters c₂, c₄, and c₆ fitted to the interval k_z ∈ [0,0.6 × π/a_z].