We show how one may obtain conical (Dirac) dispersions in photonic crystals, and in some cases, such conical dispersions can be used to create a metamaterial with an effective zero refractive index. We show specifically that in two-dimensional photonic crystals with

The Dirac equation is the wave equation formulated to describe relativistic spin 1/2 particles [

It turns out that the Dirac dispersion has interesting relationship with metamaterials, which are artificial composite materials that have novel wave manipulation capabilities. Since the theoretical proposal of materials with negative refractive indices proposed by Veselago in 1968 [

While Dirac cone dispersion and zero-index materials may seem unrelated, there is a subtle relationship between them. If we have a homogeneous material with isotropic dispersive permittivity

We will show that if a 2D photonic crystal has

Let us first consider an example of a 2D photonic crystal that exhibits a Dirac-like cone in the zone center and in this specific structure, effective medium theory can be used to relate the system to an effectively zero-refractive-index system [

(a) The band structure of a two-dimensional photonic crystal composing of alumina cylinders arranged in a square lattice for the TM polarization. The radii of the cylinders are

In order to demonstrate that the photonic crystal with a band dispersion shown in Figure

Numerical simulation that demonstrates the

The realization of

As the necessary (but not sufficient) condition to get an

For a periodic structure with a permittivity distribution

These basis functions have different symmetry properties. For instance, modes of the

We now have a recipe to create Dirac dispersion at a finite frequency at

From symmetry considerations, we know that as long as there is an accidental degeneracy of a doubly degenerate state with a nondegenerate state, the Dirac-like point can be formed. And we have already shown in Figure

(a) The band structure of a core-shell photonic crystal arranged in square lattice for the TE polarization. The radii of the shell and core to be

For completeness, we also show the properties of photonic crystals consisting of low dielectric cylinders and we examine whether such systems can be used to mimic a

(a) The band structure of PMMA photonic crystal arranged in a square lattice for the TM polarization. The radii of the cylinders are

Some remarks are in order here. We note that the band dispersion is not just linear in one direction, but it is isotropic and linear in all directions of

The square lattice has the special property that the group of the M point (

(a) The band structure of a alumina vein structure for the TE polarization, with the thickness of the vein equal to

The above discussions show that Dirac-like cone dispersion at the

In the above discussion, we considered the relationship between Dirac-like point and zero-index material. The zero-index property is related to a triply degenerate state, with a flat band of a quasi-longitudinal mode crossing the Dirac-like point formed by cones generated by two linear bands. At a first glance, the longitudinal flat band with a zero group velocity in homogeneous material has no role in the Dirac-like point physics and in the literature, there are indeed calculations that ignored the longitudinal mode at all [

The MST equation can be written as [

The secular equation of (

Neglecting the

Solving (

Therefore, the Berry phase of the eigenstate

From the above analysis, we can see that the Berry phase is equal to zero, which is caused by the existence of the longitudinal flat band. This result is consistent with the discussion shown in [

In summary, we show that a Dirac-like point formed by a triply degenerate state can exist at the

This work is supported by Hong Kong RGC GRF Grant 600311.