IntroductionMetasurfaces and surface waves are two booming research branches in modern optics and wave physics. As 2D counterparts of 3D metamaterials, metasurfaces have inherited many unusual properties of artificially structured metamaterials, e.g., negative refraction, super-resolution imaging, and invisibility. [1][2][3][4][5][6] Nevertheless, owing to the reduced dimension, metasurfaces are much easier to fabricate, thus providing possibilities to transform theoretical innovations to practical applications. [7,8] In principle, the mechanisms of light-matter interaction in metasurfaces are related to the exotic physical properties of surface waves on metasurfaces. There are many kinds of surface waves propagating along interfaces of two kinds of media, such as surface plasmons, Dyakonov, Bloch, and Tamm surface waves. [9,10] Surface plasmons require that one of the media has negative permittivity; Dyakonov surface waves require linear anisotropy of at least one of the media. Bloch and Tamm surface waves require periodic permittivity of one of the media truncated by the interface or along the interface. In this paper, we would like to highlight the Dispersion is one common yet critical property of all kinds of waves. It is related to the spatial and spectral evolution of waves in structured materials. This review gives a summary of the methodologies used for the on-demand dispersion engineering of waves in metasurfaces, which possess many unusual properties such as extremely short wavelength, diffraction-less directional propagation, and tunable phase shift, resulting in many exciting applications with performances beyond the limitations set by traditional geometric and physical optics. Since most metasurfaces could be continuously transformed from simple thin films, the methodologies proposed here are universal and may be applied to analyze and design various functional metasurfaces.
Dispersion EngineeringWithout loss of generality, the geometry of almost all metasurfaces could be continuously transformed from simple thin films. The transformation process is illustrated in Figure 1. First, by stacking and bending multilayers, one obtain the superlens, planar hyperlens, and curved hyperlenses. [26][27][28] The hyperbolic dispersion in anisotropic metal-dielectric structures may lead to the formation of Dyakonov plasmons, which are similar to Dyakonov waves on anisotropic dielectric materials. [29][30][31][32][33][34] Second, a 90° rotation of the multilayer would result a 1D grating, which can be either periodic or gradient. Third,