A review of gradient index metamaterials lenses

As a kind of vibration propagation phenomenon existing in nature, waves have been captured and studied by people for a long time. With the concept of metamaterials proposed in recent decades, the manipulation and regulation of waves have reached a new height. Metamaterials are composites with artificially designed structures and supernormal physical properties that natural materials do not have. Generally, most of them have special characteristics such as negative equivalent mass densities, negative equivalent elastic modulus, negative refractive index, and so on. As a kind of metamaterials, phononic crystals are periodic structures whose Bragg band gaps or local resonant band gaps forbid the propagation of sound waves or elastic waves. The band characteristics can be adjusted by elaborate design, so that the acoustic waves or elastic waves under the passband frequency can achieve extreme control functions of wave propagation such as imaging, focusing and directional transmission. Gradient-index metamaterials are composites, whose refractive index varies with space, and have been employed in various applications. Gradient index lenses are composed of locally inhomogeneous materials, in which the refractive index is a function of spatial coordinates and the waves propagate along the curved trajectory. After proper design, they can have the functions of bending, deflection or focusing of the waves. The core of the design is to distribute the effective refractive index of the lens reasonably, which can be changed locally by changing the properties of each cell, such as the lattice size of phononic crystals, the filling rate of scatters, the material of scatters, and so on. In addition, Gradient index metamaterials can be designed to accurately focus waves over a wide frequency range for several engineering applications such as traveling wave energy harvesting for self-powered electronic devices. In this review, we firstly introduce the concept of gradient lens in optics and its history during about two hundred years. The relationship between refractive index distribution and the trajectory of wave propagation is analyzed theoretically. The typical hyperbolic refraction trajectories are listed in detail. The refractive index distribution formulas and focusing schematic diagram of several commonly known lenses (Luneburg, Eaton, etc.) are given. Then, three design methods of gradient index lens are introduced. First of them is the equivalent medium method, which helps to deduce the equivalent material parameters for the calculation of the effective speed of waves. Secondly, some research is based on the band characteristics of phononic crystals. The final method is using the size distribution of elastic plates to design structures with gradient index. Each method is introduced not only in theory but also in cases. In addition, we also introduce several research of applying gradient lenses to wave control. Finally, the conclusions and drawbacks of the above gradient index metamaterials are summarized. It is pointed out that although the present lens designs have achieved some encouraging results, they are still confronted with several key problems such as the realizations of low-frequency and broadband, nanoscale, minor-error, lightweight and the manufacture and testing of samples, and the environmental adaptations requirements. Therefore, there is still a long way for the current gradient index metamaterials toward the real application. The trend and outlook toward future are also presented. It is hoped that this review could provide guidelines for the design and realization of gradient index lenses.