# Recent Advances in Non-Traditional Elastic Wave Manipulation by Macroscopic Artificial Structures
## 2 Elastic Metamaterials
### Elastic Gradient-Index Lens Using Phononic Crystals
Many modes of particle motion exist, but the two most common modes are the symmetric S0 mode and the antisymmetric A0 mode. These two modes are important because they carry more energy than the higher-order modes in most cases. A GRIN PC in a silicon plate (Figure 2d) can focus the lowest-order antisymmetric A0 Lamb wave (Figure 2e) [73]. This work was extended to optimize GRIN PC plate design to get a refractive index gradient that has an approximately parabolic profile and to focus the zero-order antisymmetric A0 Lamb wave within a broadband frequency region; the finite-element method and a laser-ultrasonic technique were used to numerically and experimentally demonstrate sub-wavelength focusing of Lamb waves that had frequencies of 7-10 MHz [74]. GRIN PC plates for focusing and waveguiding of the Lamb wave A0 without a waveguide (Figure 3a) and with a waveguide (Figure 3b) have been demonstrated numerically and experimentally [75]; a piezoelectric AT-cut quartz was used as the material of the plane. The inaccuracy of the location of maximum focusing between the simulation and experiment may occur from the hole etching. This study may enable development of an active GRIN PC plate device based on a piezoelectric substrate. GRIN PCs in plates can control both S0 and A0 Lamb waves simultaneously (Figure 3c) in a wide frequency range [76]. Another device that uses a GRIN PC in which air holes were scattered in a silicon substrate (Figure 3d) has been proposed [77]; the focusing of a Rayleigh wave was experimentally demonstrated (Figure 3e) using a laser ultrasonic technique. Figure 3: (**a**) GRIN PC plate on an AT-cut quartz plate (top). Arrow: wave propagation direction (\(x\) axis). The simulation of focusing of the lowest A0 mode in the designed GRIN PC lens (center). The measured amplitude distributions of the GRIN PC plate lens at 10 MHz (bottom); **(b)** Same as (**a**) but a linear waveguide is added; (**c**) Upper side: Normalized field distribution for a A0-polarized (left) and a S0-polarized (right) plane waves where the thickness of the plate is designed to focus the two Lamb modes simultaneously. Lower side: same as the upper side but the thickness of the plate is constant; (**d**) GRIN PC on a silicon plate with a thickness 500 \(\upmu\)m (left) with simulated (blue squares) and experimental (green circles) values of the effective refractive index (right); (**e**) Experimental maximum of out-of-plane displacement in GRIN PC lens. Dashed black lines: ray trajectories. (**a,b**) adapted from [75]. (**c**) adapted from [76]. (**d,e**) adapted from [77]. One GRIN lens without using PCs was made by adjusting the thickness of the plate [78]. The desired \(n\) was obtained by designing \(n\) to increase as the thickness of the lens decreases. Three kinds of GRIN lens, Luneburg lens, Maxwell fish-eye lens, \(90^{\circ}\) lens were introduced and numerically demonstrated [79]. Unlike PCs or using thickness variation of the plate, this study uses resonant EMTM to design GRIN lenses. A Luneburg lens is a GRIN lens in which \(n\) decreases radially from the center to the perimeter, expressed as  
\[n(r)=\sqrt{(2-(r/R)^{2}}, \tag{1}\]  
where \(R\) and \(r\) denotes the radius of the sphere and the radial distance measured from the center of the sphere. A Maxwell fish-eye lens is a GRIN lens that focuses the point on the spherical edge to another point on the opposite edge when a point source is located at the edge of the sphere. \(n\) varies according to the relation expressed as  
\[n(r)=\frac{2}{1+(r/R)^{2}}, \tag{2}\]  
where \(R\) and \(r\) have the same meaning as in Equation (1). \(90^{\circ}\) rotating lens is a GRIN lens which rotates a ray about \(90^{\circ}\).