Extremal transmission through a microwave photonic crystal and the observation of edge states in a rectangular Dirac billiard

This paper presents experimental results on properties of waves propagating in an unbounded and a bounded photonic crystal consisting of metallic cylinders that are arranged in a triangular lattice. First, we present transmission measurements of plane waves traversing a photonic crystal. The experiments are performed in the vicinity of a Dirac point, i.e., an isolated conical singularity of the photonic band structure. There, the transmission shows a pseudodiffusive $1/L$ dependence, with $L$ being the thickness of the crystal, a phenomenon also observed in graphene. Second, eigenmode intensity distributions measured in a microwave analog of a relativistic Dirac billiard, a rectangular microwave billiard that contains a photonic crystal, are discussed. Close to the Dirac point, states have been detected that are localized at the straight edge of the photonic crystal corresponding to a zigzag edge in graphene.

Figure 1

Schematic view of the triangular lattice of metallic cylinders (black). The voids between the cylinders are marked by colored circles. For graphene, the red (bright) and the blue (dark) voids correspond to the two atoms that generate the two independent triangular sublattices. The arrows indicate the two directions $\Gamma \mathrm{M}$ and $\Gamma \mathrm{K}$ considered in the transmission experiments. The radius of the cylinders equals $R=0.25a$, where $a$ is the lattice constant. The dashed lines indicate the positions of the metallic walls used in the experiments described in Sec. 2b to realize a microwave Dirac billiard.

Figure 2

Left: band structure $f(k_x,k_y)$ of the triangular lattice shown in Fig. 1 as a function of the quasimomentum components $(k_x,k_y)$. It was computed by numerically solving the scalar Helmholtz equation for the electric field strength imposing the Dirichlet boundary condition at the metallic cylinders. Shown are the first and the second bands. Blue (darkest) color corresponds to the lowest, red (brightest) color to the largest values of $f$ within a band. The hexagonal Brillouin zone with the two nonequivalent $K$ points $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ is indicated. Right: band structure computed numerically by varying the quasimomentum vector along the path $\Gamma$MK$\Gamma$ defining the boundary of the irreducible Brillouin zone depicted together with the Brillouin zone in the inset. Adopted from Ref. 17.

Figure 3

Experimental setup (not to scale). The photonic crystal consisted of 960 copper cylinders arranged in a triangular lattice. A microwave horn antenna with an opening angle of $2\Delta \theta ={28}^{\circ }$ was used as emitting antenna and the signal transmitted through the crystal was received by a waveguide antenna. The crystal was placed symmetrically with respect to the antennas and oriented such that normal incidence of the emitted waves coincided with the $\Gamma \mathrm{K}$ direction, in distinction to the experiments described in Ref. 17 where it corresponded to the $\Gamma \mathrm{M}$ direction.

Figure 4

Examples of transmission spectra measured with the setup shown in Fig. 3 in $\Gamma \mathrm{K}$ direction through a lattice with, respectively, 8, 18, and 36 rows of cylinders. The dashed line marks the Dirac frequency.

Figure 5

The product of the transmission $|S_{ba}{\left(f\right)|}^2$ through the photonic crystal and its thickness $L$ in units of the lattice constant $a$ as function of $L$ for different frequencies, from top to bottom for 13.18, 13.797, and 18.7 GHz. The filled circles are experimental data. They are joined with dashed lines to guide the eye. The solid lines result from the fits of a linear, a constant, and an exponential function, respectively, to the data points.

Figure 6

Double-logarithmic plot of the transmission $|S_{ba}{\left(f\right)|}^2$ through the photonic crystal as function of $L$ at the Dirac frequency $f_D=13.797$ GHz. The filled circles are experimental data. The solid line shows the prediction for the average behavior obtained by evaluating Eqs. (4)– (6) of Ref. 20. The fit of Eq. (2) describing the oscillations[18] yields the empty squares. They are joined with a dashed line to guide the eye.

Figure 7

Photograph of the Dirac billiard with the top plate removed. The photonic crystal with a triangular lattice geometry is enclosed by a rectangular brass frame with side lengths of, respectively, 218 and 420 mm. It consisted of 273 copper cylinders of radius $R=5$ mm and height $h=8$ mm had a lattice constant $a=4R$ and was squeezed between two copper plates. The two white crosses mark the positions of the antennas.

Figure 8

Top panel: transmission spectrum measured between two wire antennas. Bottom panel: magnification of the spectrum in a region around the Dirac frequency. The arrows mark the resonance frequencies for which the measured electric field intensity distributions are shown on Fig. 9. The labels of the resonances refer to the panels in Fig. 9.

Figure 9

The electric field intensity distributions measured at the resonance frequencies in Fig. 8. These are marked with arrows and the labels ranging from a$\to$j. The blue (darkest) color corresponds to the lowest, the red (brightest) to the highest intensity.