Configurable Dirac-like conical dispersions in complex photonic crystals

We investigate Dirac-like conical dispersions in photonic crystals with complex unit cells. Comparing with photonic crystals with simple unit cells, the complex-unit-cell design can provide extra degrees of freedom to engineer the frequency of the Dirac-like point in a broad frequency regime. Interestingly, we find that many functionalities of double zero media associated with the Dirac-like point are well preserved in such complex photonic crystals, such as wave tunneling, cloaking, wave front control, etc. Different transmission behaviors, e.g., total reflection and negative refraction, can be achieved by shifting the frequency of the Dirac-like point.

Figure 1

Band structure and eigenfields of a two-dimensional complex PC. (a) Band structure for a engineered Dirac-like conical dispersion in a complex unit cell. The lattice constant is $a$. The unit cell is constructed by dielectric cylinders with $\varepsilon =12.6$ (silicon). The radius of cylinders at the corners and the center of the unit cell are set as $r_1=0.25a$ and $r_2=0.076a$, respectively. The eigenfields of dipolar modes and the monopolar mode at the Dirac-like point are shown in (b)–(d).

Figure 2

Frequency-configurable Dirac-like conical dispersions. Dashed curve: One-to-one relationship between the radius $r_1$ and $r_2$, corresponding to different frequency-configurable Dirac-like points. Solid curve: The dependence of the normalized eigenfrequency of the Dirac-like points on the radius $r_1$. Curves of different colors correspond to different materials ($\varepsilon =4.3$, $\varepsilon =6.7$, $\varepsilon =9.8$, and $\varepsilon =12.6$). The points in bold represent the Dirac-like points when cylinder 2 is absent.

Figure 3

The functionalities of double zero medium associated with Dirac-like points are well preserved in such complex PCs. (a) Electromagnetic wave incident from the lower-left channel experiences a ${90}^{\circ }$ bending waveguide composed of PC with DLCD and emerges through the upper-right channel. The boundaries of waveguide are set as PMCs. (b) An electromagnetic wave passes through the bending waveguide when an object with PMC boundaries is inserted into the waveguide. At ${\omega }_D=0.449\left(2\pi c/a\right)$, the simulations of a concave “lens” made of (c) the PC with DLCD and (d) double zero medium are plotted.

Figure 4

Different transmission behaviors can be achieved with different ${\omega }_D$. The inset graphs are the enlarged views of the band structures around the DLCDs. A Gaussian beam with frequency $\omega =0.429\left(2\pi c/a\right)$ is sent from the left bottom towards the sample with incident angle $\theta =3^{\circ }$. (a) When the sample is set as PC 2 with ${\omega }_{D2}=0.429\left(2\pi c/a\right)$ sandwiched by PC 1 with ${\omega }_{D1}=0.412\left(2\pi c/a\right)$, the Gaussian beam experiences a total reflection at the interface between PC 1 and PC 2. (b) The Gaussian beam experiences positive and negative group velocity in PC 1 with ${\omega }_{D1}=0.412\left(2\pi c/a\right)$ and PC 2 with ${\omega }_{D2}=0.450\left(2\pi c/a\right)$, respectively. The negative refraction is exhibited in the tunneling process.