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Using the equifrequency surfaces (EFS) to describe negative refractions in left-handed materials (LHMs) and photonic crystals (PCs), negative phase and negative group refractive indexes in LHMs were compared with positive phase and negative group refractive indexes in PCs. The refractive indexes in PCs were dependent on frequencies and incident angles of electromagnetic wave, while indexes in LHMs were constant in the left-handed region. Furthermore, the phase compensating effect resulting from the negative phase refractive index was addressed to distinguish the perfect lens made of LHMs from the superlens realized in the all angle negative refraction (AANR) region of PCs.

An additional concave incident surface comprised of two-dimensional (2D) sonic crystals (SCs) is employed to tune the acoustic image in the far-field region. The tunability is realized through changing the curvature of the concave surface. To explain the tuning mechanism, a simple ray-trace analysis is demonstrated based on the wave-beam negative refractive law. Then, a numerical confirmation is carried out. Results show that both the position and the intensity of the image can be tuned by the introduced concave surface.

In this article, we present an experimental and numerical study of novel optical properties of two-dimensional dielectric photonic crystals (PCs) which exhibit negative refraction. We investigate two mechanisms which utilize the band structure of the PC to generate a negative effective index of refraction (n_eff<0) and demonstrate the negative refraction experimentally. To the isotropic extend of n_eff, different PC slab structures are employed to focus the radiation of a point source. It is shown experimentally that the PC can generate an image of the source with subwavelength resolution in the vicinity of the PC interface. Using a different PC, one can also obtain a far field focusing. In the latter case, we explicitly show the flat lens behavior of the structure. These examples indicate that PC-based lenses can surpass limitations of conventional lenses and lead to novel optics applications.

Recently there has been a great deal of interest in an unusual category of material, that is, a material that exhibits negative refractive index or more generally negative group velocity. Perhaps the most immediate application of this type of material is in an area known as total and negative refraction, which may potentially lead to many novel optical devices. The reason that the phenomenon of total and negative refraction has become so interesting to the physics community is also due largely to the notion that this phenomenon would never occur in conventional materials with positive refractive index. It turns out that total and negative refraction can be realized even in natural crystalline materials or in artificial materials (e.g. photonic crystals) without negative (effective) refractive index. In this brief review, after providing a brief historic account for the research related to finding materials with negative group velocity and achieving negative refraction, we discuss the three primary approaches that have yielded experimental demonstrations of negative refraction, in an effort to clarify the underlying physics involved with each approach. A brief discussion on the subwavelength resolution application of the negative (effective) refractive index material is also given.

In the transfer matrix framework of electromagnetic transport, we calculate the group velocities in a one-dimensional photonic crystal composed of a biaxially anisotropic chiral material and dielectric material. According to the isofrequency curves, it is unravelled that two polarized waves possess negative group velocities. As a result, negative refraction can be achieved in the present configuration. In addition, the chirality has a significant influence on the group velocities in the biaxially anisotropic chiral photonic crystal, which even causes the group velocities to change from positive to negative. Our investigations may have potential applications in the chiral-based negative refraction devices.

In recent years, significant attention has been attracted to hyperbolic (or indefinite) materials resulting from their extraordinary optical properties, such as all-angle negative refraction. Here, tunable biaxial hyperbolic dispersion and negative refraction were found in graphite crystal. First principles were used to calculate the permittivity of graphite in $x$-, $y$- and $z$-directions, the results indicate that graphite crystals exhibit indefinite dielectric constant tensor in certain frequency range. By adjusting strain, the optical properties of materials changed, and even switched from hyperbolic to elliptical materials in some frequency. This work demonstrates the feasibility of achieving high-performance tunable biaxial natural hyperbolic materials which can be used in various ways.

Complex structures, large size and limited manipulation of acoustic waves are the problems that restrict the development of acoustic metasurfaces. Here, we report a transmission-type acoustic metasurface based on local resonance mechanism, which is composed of meta-atomic units called dumbbell-shaped double-split hollow spheres (DSDSHS). This metasurface with subwavelength scale has the advantage of simple structure and easy preparation, and can realize the full manipulation of sound waves. Negative refraction with different transmission angles and high intensity plate focusing lens are realized in the air environment of audible frequency. The proposed metasurface has potential applications in the miniaturization and integration of sound transmission and sound energy collection, opening a new opportunity for manipulation of acoustic wavefront.

A recipe is given for creating material with a desired refraction coefficient by embedding many small particles in a given material. To implement practically this recipe, some technological problems are to be solved. These problems are formulated. Applications of the materials with a desired refraction coefficient are mentioned. One of the possible applications is preparing materials with desired wave-focusing properties. These are materials which have a desired radiation pattern at a fixed frequency and at a fixed incident direction. The other application is preparing materials with negative refraction.

Many-body quantum-mechanical scattering problem is solved asymptotically when the size of the scatterers (inhomogeneities) tends to zero and their number tends to infinity.

A method is given for calculation of the number of small inhomogeneities per unit volume and their intensities such that embedding of these inhomogeneities in a bounded region results in creating a new system, described by a desired potential. The governing equation for this system is a non-relativistic Schrödinger's equation described by a desired potential.

Similar ideas were developed by the author for acoustic and electromagnetic (EM) wave scattering problems.