Extraordinary light focusing and Fourier transform properties of gradient-index metalenses

We propose and demonstrate a new type of metalenses that are phase compensated by gradient index (GRIN) or inhomogeneous permittivity metamaterials. Both elliptically and hyperbolically dispersive GRIN metalenses for both internal and external focusing are studied. The requirements for the GRIN metalenses and the light focusing characteristics are analyzed and numerically verified. The GRIN metalenses can achieve super resolution and have ordinary or extraordinary Fourier transform functions, thus enabling exotic applications.