# Recent Advances in Non-Traditional Elastic Wave Manipulation by Macroscopic Artificial Structures
## 3 Elastic Metasurfaces  
Metasurfaces (MSs) are artificially-designed one-dimensional or two-dimensional equivalents of bulk MTMs. MSs are much smaller, more compact, and simpler than conventional MTMs, but can still manipulate waves. The physical mechanism of the MSs exploits adjustment of wave phase shifts, unlike bulk MTMs, in which waves are manipulated by changing the effective parameters of the material. MSs can be used in applications such as anomalous reflection and refraction [132, 133], polarization conversion [134, 135], focal lenses [136, 137, 138] and holograms [139, 140, 141, 142]. MSs are based on the generalized Snell's law with phase discontinuity [143]. The generalized [PERSON]'s law of refraction is written as  
\[n_{t}\sin(\theta_{t})-n_{i}\sin(\theta_{i})=\frac{1}{k_{0}}\frac{d\Phi}{dx}, \tag{11}\]  
and for reflection,  
\[\sin(\theta_{r})-\sin(\theta_{i})=\frac{1}{n_{i}k_{0}}\frac{d\Phi}{dx}, \tag{12}\]where \(k_{0}\) denotes the wavenumber of the wave and \(d\Phi/dx\) denotes the phase gradient parallel to the plane, \(\theta_{i}\), \(\theta_{t}\) and \(\theta_{r}\) are the incident, transmission and reflection angle, respectively, and \(n_{i}\) and \(n_{t}\) are the refractive indices of two media (Figure 10). This law indicates that the transmitted and reflected beams can have an arbitrary direction depending on \(d\Phi/dx\) as well as \(n_{i}\) and \(n_{t}\). An elastic MS that can control both same-mode and mode-converted Lamb waves in plates has been demonstrated [144]. It is composed of locally-resonant geometric tapers; this design has the ability to control the phase shift accurately. However, the transmitted field contains unwanted wave types because of the mode conversion. This mode conversion at the material interface complicates the modulation. An elastic MS to split SV-waves and P-waves in elastic solids into different propagation directions without involving mode conversion has been proposed [145]. Transmission of SV-waves are different from those of P-waves, because the phase speed of the SV-wave only depends on the material properties while that of P-wave is susceptible to the thickness of the MS. Therefore, SV-waves can be steered while the MS remains transparent to P waves (Figure 11a,b). An elastic MS composed of wave guides can steer the incident elastic SH waves by using an electrorheological elastomer [146]. The deflected elastic SH waves were tuned by adjusting the phase velocity that depends on the applied electric fields in the waveguides. An elastic phase-controlling elastic MS using parallel and periodically arranged-micro-sections has been demonstrated analytically and numerically [147]. Wave propagation properties depend on the periodic length of the meta-layer element at the sub-wavelength scale. The phase-controlling properties was accomplished by the frequency dependent elastic properties. The above three studies on MSs were conducted by changing the wave velocity. However, other studies have used methods such as changing the propagation distance and adjusting local resonances. An elastic MS has been produced by changing the heights of three zigzag unit cells [148] (Figure 11c); this MS can be used in source illusion devices for shifting, transforming, and splitting a point-source (Figure 11d). Figure 10: Schematics of the generalized [PERSON]’s law of refraction (**a**) and reflection (**b**). \(dx\) is the distance between two beams at the plane of incidence. \(\Phi\) is the phase shift of the blue beam and \(\Phi+d\Phi\) is the phase shift of the red beam. An elastic MS has been produced by adjusting the propagation distance [149] (Figure 12a). This concept extends the concept of an acoustic MS that compensates for the lack of MTM and PCs. The designed refraction angles obtained by the numerical simulations were in excellent agreement with the analytical predictions (Figure 12b,c). An elastic MS that exploits local resonance of 3D column structures can support high transmission and full phase control to deflect flexural waves [150].