## 3 Metasurfaces
### Phase Jumps and Generalized Law of Refraction  
Conventional devices (e.g., lenses, holograms etc.) shape the wavefronts over a distance larger than the operating wavelength [38; 39]. Lenses make use of their shape and the material properties (refractive index) to focus energy by gradually varying the phase of an electromagnetic wave, holograms generate images in the far-field through constructive interference [39]. The same is also true in the case of DNG materials, which also gradually temper the phase over a distance. Metasurfaces provide a promising alternative to this approach because an electromagnetic wave passing through a metasurface undergoes a phase jump. This is due to the fact that the phase is discontinuous at a metasurfaces [40]. This anomalous behaviour decreases the reliance on wave propagation and allows the shaping of wavefronts and focussing of energy over sub-wavelength distance. Therefore, metasurfaces can potentially replace the bulky and heavy devices used traditionally for such purposes.  
This phase discontinuity (or jump) owes its existence to the resonant behaviour of the miniaturized (sub-wavelength) metasurface building blocks. When an electromagnetic wave strikes an individual element of a metasurface, it induces a surface electromagnetic wave. This causes the charge present in this element to oscillate. This whole phenomenon is known as surface plasmon [16]. The interaction of the impinging electromagnetic wave and the surface plasmon leads to the phase discontinuity across the metasurface. Due to this discontinuity, the [PERSON]'s law has been replaced by a \"generalized law\" for metasurfaces [41].  
The generalized law has been derived on the basis of [PERSON]'s principle of the least time as stationary phase [41]. [PERSON]'s principle states that out of all the possible paths between two points, light travels on the path which takes the least amount of time [42]. Consider a plane wave at an incidence angle of \(\theta\) impinging on a metasurface. Two possible paths that the wave can take are shown in Figure 1. Since the metasurface causes an abrupt change in the phase (phase discontinuity), this change at point A is shown by \(\phi\), while on A' is it is represented by \(\phi+d\phi\). The distance between A and A' is assumed to be \(dx\). It is also assumed that \(\phi\) is a continuous function of \(x\). It is assumed that the two paths are infinitesimally close such that the phase difference between them to reach point A is zero. This leads to the generalized refraction law given by the following Equation (1) [41].  
\[n\sin(\theta)-n^{\prime}\sin(\theta^{\prime})=\frac{\lambda_{0}}{2\pi}\frac{ dx}{d\phi} \tag{1}\]  
where \(\lambda_{0}\) is the free space wavelength, \(n\) and \(n^{\prime}\) are the refractive indices of medium 1 and medium 2 as shown in Figure 1.  
Similarly the generalised law of reflection for metasurfaces is given by Equation (2).  
\[\sin(\theta)-\sin\Bigl{(}\theta_{refl}\Bigr{)}=\frac{\lambda_{0}}{2\pi n}\left| \frac{dx}{d\phi}\right| \tag{2}\]  
where \(\theta_{refl}\) is the angle of reflection. The non-linear relation between the incidence and the reflection is worth noting since the angle of incidence and reflection are no longer equal for a metasurface. Equations (1) and (2) can be easily reduced to the original [PERSON]'s law of refraction and reflection if there is no phase discontinuity across the surface (\(d\phi/dx=0\)). The generalised laws of reflection and refraction have been demonstrated by reflectarrays and v-shaped antennas, respectively [41; 43; 44].