Abnormal $K$-point dependent electromagnetic beam propagation in a graphenelike triangular metallic photonic structure

We study the electromagnetic wave propagation in a graphenelike triangular metallic photonic structure and show that, associated to trigonally warped Dirac cones, the transport properties resulting from an incident beam at the armchair edge display drastically different behaviors as compared to the results obtained for other analogous systems. Namely, at $K$ point, only the center beam associated to the lower Dirac cone and the side beams to the upper cone and, at $K^{\prime }$ point, only the center beam associated to the lower cone display significant intensities. The other expected beams are all strongly inhibited. This can be related to the metallic nature of the structure, where the low frequency electromagnetic bands are formed by local resonance modes confined inside the structural units. A propagating wave should follow the structure local patterns that govern its field distribution, whose symmetry properties determine its excitation by an incident wave.

Figure 1

Metallic triangular photonic structure (a) and the corresponding EM frequency band diagram (b). Dirac cones are formed by the first two bands (c). A 3D view of the Dirac cone at $K$ ($4\pi /3a$, 0) is shown in (d). Isofrequency contours of the upper and lower cone at the same $K$ point are depicted in (e) and (f), where the frequency level (${\omega }_1$ and ${\omega }_2$) as well as the contour interval ($\mathrm{\Delta }\omega$) values are normalized to $\omega a/2\pi c$.

Figure 2

Electric field distributions corresponding to the upper (a) and lower (b) Dirac band branch at $k=(\pi /a,0)$, and in an individual triangle (c). The corresponding field magnitudes along a slice passing by the A and B triangle centers are shown in (d) and (e). The $\pm$ signs indicate the field polarities.

Figure 3

Electric field distribution of a monochromatic wave of frequency ${\omega }_1$, propagating in an $18\times 15$ period sample. The incident beam is injected through a wave guide at the left side. The inset in the lower part indicates the incident (thick arrows) and expected propagation (thin arrows) directions. The inset in the upper part shows a magnified zone around the unit cell in the incident region. The $\pm$ signs indicate the field polarities and dashed lines the regions where transmissions are considered.

Figure 4

Same configuration as in Fig. 3, but for incident wave frequency ${\omega }_2$.

Figure 5

Transmission spectra in the center and side beam directions in the vicinity of the Dirac frequency ${\omega }_D$. The incidence and transmission regions are depicted in Figs. 3 and 4. The frequency levels ${\omega }_1$ and ${\omega }_2$ are also indicated.