Gapless surface states originating from accidentally degenerate quadratic band touching in a three-dimensional tetragonal photonic crystal

A tetragonal photonic crystal composed of high-index pillars can exhibit a frequency-isolated accidental degeneracy at a high-symmetry point in the first Brillouin zone. A photonic band gap can be formed there by introducing a geometrical anisotropy in the pillars. In this gap, gapless surface and domain-wall states emerge under a certain condition. We analyze their physical properties in terms of an effective Hamiltonian, and a good agreement between the effective theory and numerical calculation is obtained.

Figure 1

(a) Tetragonal photonic crystal composed of circular pillars. The pillar axis is taken to be parallel to the $z$ direction. (b) Tetragonal photonic crystal composed of two-step circular pillars. The parity symmetry in the $z$ direction is broken. (c) The first Brillouin zone for the bulk photonic crystals of panels (a) and (b), and the surface Brillouin zone relevant to the system without the translational invariance along the $z$ direction. Points of high symmetry are also indicated.

Figure 2

[(a), (b)] Photonic band structure of a tetragonal photonic crystal composed of circular pillars of a high-index material before (a) and after (b) introducing the $z$-parity symmetry breaking. In panel (a), the dielectric constant $\epsilon$, radius $r$, and height $h$ of the pillars are taken to be 100, $0.295a$, and $0.259a$, respectively, where $a$ is the lattice constant in plane. The lattice constant $a_z$ in the $z$ direction is $0.5a$. The outer medium is air ($\epsilon =1$). In panel (b), the pillar is an coaxial two-step one with inner-core radius of $0.75r$ and height $h$. The outer shell has radius $r$ and height $0.75h$. The dielectric constant of the pillar is 100. (c) Electric and magnetic field intensities of the four Bloch modes (ordered in frequency) near the accidental degeneracy in panel (a) (marked by arrow). The Bloch wave vector is $\mathit{k}=(0.45,0.45,0)2\pi /a$, and the intensities are plotted on a mirror $xy$ plane in between two consecutive layers of the pillars. The maximal field intensity on the plane is normalized to be 1. The white circle represents the in-plane position of the pillar edge.

Figure 3

Schematic illustration of domain walls used in the numerical calculation. Two types of the domain walls are considered.

Figure 4

Dispersion relation of the domain-wall states in the composite structure with the two domains of the symmetry-broken tetragonal photonic crystal. The pillar axis of one domain is inverted from that in the other domain. The parameters are the same as in the Fig. 2. The domain-wall states of type I (II) are indicated by red filled (blue open) circle. The shaded region is the projection of the bulk band structure.

Figure 5

[(a), (b)] Dispersion curves of the surface states in the 16-layer-thick tetragonal photonic crystal without the $z$-parity symmetry. Projection of the bulk band structure is also plotted. The parameters are the same as in the Fig. 2. In panel (a), the PhC is capped by the perfect-electric-conductor (PEC) wall placed with distance $(a_z-h)/2$ from the top of the PhC. The bottom surface is open. In panel (b), the PhC is capped by the perfect-magnetic-conductor (PMC) wall with distance $(a_z-h)/2$ from the bottom of the PhC. The top surface is open. The surface states at the top (bottom) are indicated by red filled (blue open) circle. (c) Electric and magnetic field intensities of the surface modes near the $\overline{\mathrm{M}}$ point in the surface Brillouin zone [marked by arrows in panels (a) and (b)]. The Bloch wave vector is $\mathit{k}=(0.45,0.45)2\pi /a$, and the intensities are plotted on a PEC-PMC wall. The maximal field intensity on the wall is normalized to be 1. The white circle represents the in-plane position of the two-step pillar edge.