# Recent Advances in Non-Traditional Elastic Wave Manipulation by Macroscopic Artificial Structures
## 2 Elastic Metamaterials
### Elastic Hyperlens  
Light that is emitted or scattered from an object, is composed of two components: a propagating component that is composed of low wavevectors, and an evanescent component that is composed of high wavevectors. The propagating wave encodes large feature information, and can propagate to the far-field, whereas the evanescent wave encodes fine features, but its power decays exponentially in a conventional environment and cannot reach the far-field. Because of this property of the evanescent component, realization of sub-wavelength imaging is a big challenge in a general sense. This problem can be solved using hyperlenses constructed by MTMs with strong anisotropy. A hyperlens has very large or unbounded cutoff wavevectors, so it can support the propagation of an evanescent wave to the far-field [85; 86; 87]. This hyperlens shows a remarkable potential for applications, such as biomedical imaging, nanolithography and optical sensing. The first hyperlens concept was derived from EM-MTMs [88]. The device is non-resonant, so is unconstrained in terms of material loss. One acoustic hyperlens does not use local resonance accompanied by negative effective parameters [89]; this design has been used to demonstrate hyperlenses that have elliptical equi-frequency contours (EFCs) in the elastic regime, and therefore have very strong anisotropy [90]. However, elliptical EFCs have limited resolution because their wavevectors are still bounded despite having higher cutoff  
Figure 4: **(a)** Non-dimensionalized mean-squared integral of velocity generated by line source excitation for 20 kHz (top), 40 kHz (center), and 60 kHz (bottom). The focal points match the ray-theory simulation in the center figure; (**b**) Corresponding experiment results using a 100-k\(\Omega\) resistor at (top) 20 kHz, (center) 40 kHz, and (bottom) 60 kHz. Dashed white line: outer edges of the lens; (**c**) Average power generated by the GRIN lens of load resistance at 40 kHz; (**d**) (top) Experimental apparatus, (lower left) close-up view of the PC lens sample. (lower right) Locations of the Luneburg harvesters and the baseline harvester; (**e**) Experimental root mean square (RMS) wave fields for incidence of (top) \(0^{\circ}\); (**f**) Average power generated by the PC Luneburg harvesters and baseline harvester. (**a-c**) adapted from [83]. (**d–f**) adapted from [84]. wavevectors than spherical EFCs. One solution to this problem is to use hyperbolic EFCs that have unbounded wavevectors. EFCs shows the wave vectors along the propagation direction of waves at the same frequency. The propagation direction of each wave is perpendicular to EFCs with distinct figuration, so diffraction characteristics can be predicted. The characteristics of spherical EFCs, elliptic EFCs and hyperbolic EFCs are distinct (Figure 5). In spherical EFCs in an isotropic medium, a cutoff wavevector (\(k_{c}\)) limits the spatial frequency along the tangential direction (\(\theta\)), so a diffraction limit occurs. In a hyperlens that uses an elliptic EFC, the magnitude of \(k_{c}\) increases due to the strong anisotropy, so sub-wavelength imaging is possible. However, when the magnitude of the circumferential wavevector (\(k_{\theta}\)) becomes larger than the magnitude of \(k_{c}\), the information contained in wavevector cannot be transmitted along the radial direction \(r\) because radial wavevector (\(k_{r}\)) becomes imaginary. In a hyperlens that uses hyperbolic EFCs, the real-valued \(k_{r}\) enables \(k_{\theta}\) to become very large, because the cutoff wavevector is unbounded, so the wave components of the large circumferential wavevectors \(k_{\theta}\) can propagate along the \(r\) direction. Therefore, evanescent waves that have high spatial frequency and cannot be resolved by a conventional lens, are converted to propagating waves. Thus, the hyperbolic EFC hyperlens enables super-resolution sub-wavelength imaging to strengthen the evanescent waves and imaging features.