# Recent Advances in Non-Traditional Elastic Wave Manipulation by Macroscopic Artificial Structures
## 2 Elastic Metamaterials
### Elastic Gradient-Index Lens Using Phononic Crystals  
Phononic crystals (PCs) are artificial structures that are composed of periodic structures, and that exhibit complete or partial frequency bandgaps of elastic or acoustic waves. The main difference between a PC and an MTM arises from the geometric size of the structure [59; 60]. In the case of MTM,  
Figure 1: (**a**) Unit cell of the locally resonant elastic metamaterial (EMTM); (**b**) Effective medium parameters of proposed EMTM. Black line: mass density, \(\rho\). Blue line: shear modulus, \(\mu\). Red line: bulk modulus, \(\kappa\). Green line: \(\kappa+\mu\); (**c**) Total flux distribution for a transverse wave incidence on the wedge. Colors: the magnitude. Black Arrows: directions; (**d**) Unit cell design of a locally resonant EMTM; (**e**) The band structure of z-axis alligned EMTM. Blue shaded region: a flexural bandgap. Red shaded region: a longitudinal bandgap; (**f**) The band structure of x-axis alliged EMTM. Orange shaded region: a torsional bandgap. (**a–c**) adapted from [56]. (**d–f**) adapted from [58]. the size and periodic spacing of the micro-components are very small compared to the wavelength of waves passing through them. However, for PCs, the periodic spacing of the structures is similar to the wavelength. As mentioned in the introduction, these gaps can be used to create band gaps through Bragg scattering. Due to the importance of bandgap properties, investigating the frequency bandgaps has been a large part of work on PCs. Since the pioneering work by [PERSON] et al. [61], PCs have been studied due to their unique physical properties and their ability to manipulate the propagation of elastic waves. The properties of PCs have potential applications such as sensors [62; 63; 64; 65], filters [66; 67; 68] and waveguides [69; 70; 71]. In particular, gradient-index (GRIN) PCs designed as another application have a refractive index (\(n\)) that gradually decreases outwards from the center (Figure 2a); they were developed for elastic focusing. In a GRIN PC, an incident wave gradually tends to bend toward the center axis at which \(n\) is highest, and eventually converges to the focal point with no aberration following the Lagrangian viewpoint in terms of particle tracking. This focusing phenomenon has many practical applications such as flat lenses for detection and imaging. In comparison, some examples of GRIN lenses that do not exploit PCs will be introduced. The first GRIN PCs lens was composed of solid cylinders embedded in an epoxy medium [72]. The refractive index profiles along the direction transverse to the wave propagation has a formula with hyperbolic secant function; this gradient enables redirection of incident waves and focusing. The GRIN PC can be realized by controlling \(n\) by adjusting the radii of cylinders in each layer or by changing elastic properties of cylinders (Figure 2b). The GRIN PC allows acoustic waves to converge over a broad range of operating frequencies (Figure 2c). The following studies of GRIN PCs for elastic  
Figure 2: **(a)** Schematic diagram of the principle of a gradient-index (GRIN) lens. Red arrow: incident beams; **(b)** GRIN Phononic crystal (PC) realized by adjusting radius of cylinders (left). GRIN PC realized by changing elastic properties of cylinders (right); **(c)** Finite-Difference Time-Domain (FDTD) simulation of displacement fields when wave propagates in the GRIN PC at a reduced frequency of \(\Omega\) = 0.05 (top), \(\Omega\) = 0.10 (center) and \(\Omega\) = 0.20 (bottom); **(d)** Schematic of the GRIN PC plate. Green arrow: wave propagation direction; **(e)** Displacement amplitude along the \(x\) direction measured at 3 MHz. Green contour: area where amplitude \(>\) 0.95 times maximum value. **(a–c)** adapted from [72]. **(d,e)** adapted from [73]. waves mainly considered the propagation of Lamb waves, which propagate in solid plates. The particle motion in Lamb waves is perpendicular to the direction of propagation.