A double Dirac cone is realized at the center of the Brillouin zone of a two-dimensional phononic crystal (PC) consisting of a triangular array of core-shell-structure cylinders in water. The double Dirac cone is induced by the accidental degeneracy of two double-degenerate Bloch states. Using a perturbation method, we demonstrate that the double Dirac cone is composed of two identical and overlapping Dirac cones whose linear slopes can also be accurately predicted from the method. Because the double Dirac cone occurs at a relatively low frequency, a slab of the PC can be mapped onto a slab of zero refractive index material by using a standard retrieval method. Total transmission without phase change and energy tunneling at the double Dirac point frequency are unambiguously demonstrated by two examples. Potential applications can be expected in diverse fields such as acoustic wave manipulations and energy flow control.

/pdf/double-dirac-cones-in-phononic-crystals-27ouhu5vog.pdf

Double Dirac cones in phononic crystals

We present an effective medium theory that can predict the effective permittivity and permeability of a geometrically anisotropic two-dimensional metamaterial composed with a rectangular array of elliptical cylinders. It is possible to obtain a closed-form analytical solution for the anisotropic effective medium parameters if the aspect ratio of the lattice and the eccentricity of elliptical cylinder satisfy certain conditions. The derived effective medium theory can recover the well-known Maxwell-Garnett results in the quasi-static regime. More importantly, it is valid beyond the long-wavelength limit, where the wavelength in the host medium is comparable to the size of the lattice so that previous anisotropic effective medium theories fail. The validity of the derived effective medium theory is verified by band structure calculations. A real sample of a recent theoretically proposed anisotropic medium with near-zero index to control flux is achieved from the derived effective medium theory and control of electromagnetic waves in the sample is clearly demonstrated. The accuracy of the theory is further systematically studied as material and geometric parameter varies, and we find the derived effective medium theory applies over a large range in the parameter space. Such applicability greatly broadens the applicable realm of the effective medium theory and opens many vistas in the design of structures with desired anisotropic material characteristics.

/pdf/effective-medium-theory-for-anisotropic-metamaterials-2ye2wggauc.pdf

Effective medium theory for anisotropic metamaterials

A simple core-shell two-dimensional photonic crystal is studied where the triangle lattice symmetry and $C_{6v}$ rotation symmetry leads to rich physics in the study of accidental degeneracy's in photonic bands. We systematically evaluate different types of accidental nodal points, depending on the dispersions around them and their topological properties, when the geometry and permittivity are continuously changed. These accidental nodal points can be the critical states lying between a topological phase and a normal phase and are thus important for the study of topological photonic states. In time-reversal systems, this leads to the photonic quantum spin Hall insulator where the spin is defined upon the orbital angular momentum for transverse-magnetic polarization. We study the topological phase transition as well as the properties of the edge and bulk states and their application potentials in optics.

Accidental degeneracy and topological phase transitions in two-dimensional core-shell dielectric photonic crystals

Materials with anisotropic material parameters can be utilized to fabricate many fascinating devices, such as hyperlenses, metasolids, and one-way waveguides. In this study, we analyze the effects of geometric anisotropy on a two-dimensional metamaterial composed of a rectangular array of elliptic cylinders and derive an effective medium theory for such a metamaterial. We find that it is possible to obtain a closed-form analytical solution for the anisotropic effective medium parameters, provided the aspect ratio of the lattice and the eccentricity of the elliptic cylinder satisfy certain conditions. The derived effective medium theory not only recovers the well-known Maxwell-Garnett results in the quasi-static regime, but is also valid beyond the long-wavelength limit, where the wavelength in the host medium is comparable to the size of the lattice so that previous anisotropic effective medium theories fail. Such an advance greatly broadens the applicable realm of the effective medium theory and introduces many possibilities in the design of structures with desired anisotropic material characteristics. A real sample of a recently theoretically proposed anisotropic medium, with a near-zero index to control the flux, is achieved using the derived effective medium theory, and control of the electromagnetic waves in the sample is clearly demonstrated.

/pdf/effective-medium-theory-for-anisotropic-metamaterials-42dc1fq9m7.pdf

By exploiting the accidental degeneracy of the doubly-degenerate dipolar and quadrupolar modes, we show that a two-dimensional dielectric photonic crystal (PC) can exhibit the double Dirac cone dispersion relation at the Γ point. Using a perturbation method and group theory, we demonstrate that the double cone is composed of two identical and overlapping Dirac cones with predictable linear slopes, and the linearity of the dispersion is guaranteed by the spatial symmetry of the Bloch eigenstates. Numerical simulations including wave-front shaping, unidirectional transmission and perfect tunneling show that the corresponding PC structure can be characterized by a zero effective refractive index.

Double Dirac cones in two-dimensional dielectric photonic crystals.

We developed a selection rule for Dirac-like points in two-dimensional dielectric photonic crystals. The rule is derived from a perturbation theory and states that a non-zero, mode-coupling integral between the degenerate Bloch states guarantees a Dirac-like point, regardless of the type of the degeneracy. In fact, the selection rule can also be determined from the symmetry of the Bloch states even without computing the integral. Thus, the existence of Dirac-like points can be quickly and conclusively predicted for various photonic crystals independent of wave polarization, lattice structure, and composition.

Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals.

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Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals.