Predicting double negativity using transmitted phase in space coiling metamaterials

Abstract

Metamaterials are engineered materials that offer the flexibility to manipulate the incident waves leading to exotic applications such as cloaking, extraordinary transmission, sub-wavelength imaging and negative refraction. These concepts have largely been explored in the context of electromagnetic waves. Acoustic metamaterials, similar to their optical counterparts, demonstrate anomalous effective elastic properties. Recent developments have shown that coiling up the propagation path of acoustic wave results in effective elastic response of the metamaterial beyond the natural response of its constituent materials. The effective response of metamaterials is generally evaluated using the 'S' parameter retrieval method based on amplitude of the waves. The phase of acoustic waves contains information of wave pressure and particle velocity. Here, we show using finite-element methods that phase reversal of transmitted waves may be used to predict extreme acoustic properties in space coiling metamaterials. This change is the difference in the phase of the transmitted wave with respect to the incident wave. This method is simpler when compared with the more rigorous 'S' parameter retrieval method. The inferences drawn using this method have been verified experimentally for labyrinthine metamaterials by showing negative refraction for the predicted band of frequencies.

Keywords: acoustics; metamaterials; negative refractive index.

Figure 1.

Schematic of the experimental set-up used for the demonstration of negative refraction for 2D and 3D labyrinthine structures. The blue dots are representatives of microphones arranged in an array interfaced to a DAQ card via amplifiers.

Figure 2.

(a) Schematic of simulation set-up for determining the transmission properties. Panel (1) shows the surrounding shell, panel (2) shows the complete outer shell, while panel (3) shows the meta unit inside a waveguide. The source is placed at left (red) end, while the detector is placed at the right (green) end of the waveguide. Transmitted wave characteristics for (b) outer surrounding shell (excluding the front and rear surface), (c) complete outer shell and (d) single meta unit in the waveguide. Schematic of unit cell for (e) 2D (w = 0.8 mm, d = 1.5 mm, L = 10 mm) and (f) 3D labyrinthine space coiling structures.

Figure 3.

Transmission wave characteristics for (a) 2D maze and (b) 3D labyrinthine space coiling metamaterials for different frequencies. The positions of the metamaterial unit in the simulation box are represented by yellow region.

Figure 4.

Effective refractive index (n), impedance (Z) and change in phase obtained for (a) 2D maze and (b) 3D labyrinthine space coiling metamaterials. Refractive index and impedance have been obtained using the ‘S’ parameter retrieval method. The inset shows equivalent cells for 2D and 3D structures. AOB (in (a) inset) represents the effective path travelled by the acoustic waves in 2D maze, while A'OB' (in (b) inset) represents analogous path in 3D equivalent cell.

Figure 5.

(a) Schematic showing the positive and negative refractive index region of transmitted wave in prismatic arrangement of 3D meta units. (b) Polar plot of normalized transmitted wave amplitude when the wave travels through the prismatic arrangement of meta units. The radial direction represents normalized transmitted amplitude.