Klein tunneling and supercollimation of pseudospin-1 electromagnetic waves

Pseudospin plays a central role in many novel physical properties of graphene and other artificial systems which have pseudospins of $1/2$. Here we show that in certain photonic crystals (PCs) exhibiting conical dispersions at $\mathbf{k}=0$, the eigenmodes near the “Dirac-like point” can be described by an effective spin-orbit Hamiltonian with a higher dimension value $S=1$, treating the wave propagation in positive index (upper cone), negative index (lower cone), and zero index (flat band) media within a unified framework. The three-component spinor gives rise to boundary conditions distinct from those of pseudospin $1/2$, leading to wave transport behaviors as manifested in super Klein tunneling and supercollimation. For example, collimation can be realized more easily with pseudospin 1 than pseudospin $1/2$. The effective medium description of the PCs allows us to further understand the physics of pseudospin-1 electromagnetic (EM) waves from the perspective of complementary materials. The special wave scattering properties of pseudospin-1 EM waves, in conjunction with the discovery that the effective photonic potential can be varied by a simple change of length scale, offer ways to control photon transport. As a useful platform to study pseudospin-1 physics, dielectric PCs are much easier to fabricate and characterize than ultracold atom systems proposed previously. The system also provides a platform to realize the concept of “complementary medium” using dielectric materials and has the unique advantage of low loss.

Figure 1

The band dispersion of a 2D photonic crystal (PC) with dielectric cylinders having radius $r=0.2a$ and $\varepsilon =12.5$ arranged in a square lattice in air (see inset). (a) The band structure with a region of linear dispersions intercepted by a flat band near the $\mathrm{\Gamma }$ point $\left(\mathbf{k}=0\right)$ marked by the dashed rectangle. (b) The 3D band surfaces for the linear region marked in (a) exhibiting a conical dispersion near $\mathbf{k}=0$.

Figure 2

Klein tunneling of pseudospin-1 EM waves. (a) Schematic diagram of the scattering of pseudospin-1 EM waves by a square photonic potential barrier and definitions of the wave vectors ${\mathbf{q}}_0$ and ${\mathbf{q}}_1$ with their respective angles $\theta$ and $\varphi$. (b) The realization of a square photonic potential barrier by a PC sandwich structure. The two PCs, PC1 and PC2, have the same structure as shown in Fig. 1 but differ by a length scale $(a_1=15a_2/14$ and $r_1=15r_2/14)$. The thicknesses of PC1 and PC2 are $21a_1$ and $45a_2$, respectively. The change in length scale shifts the “Dirac-like point” while preserving the conical dispersion. Length-scale change in the pseudospin-1 photonic system is hence an analog of potential change (gating) in pseudospin-$1/2$ graphene.

Figure 3

Klein tunneling for pseudospin-1 EM waves and pseudospin-$1/2$ electrons. Transmission amplitudes are plotted for EM waves in a PC sandwich structure (blue open circles), in an effective spin-orbit Hamiltonian of Eq. (9) (red solid line), and electrons in graphene (green open triangles). The geometric parameters of the sandwich structure are shown in Fig. 2. The reduced frequency, photonic potential, and group velocity in the effective Hamiltonian are chosen as $\delta \omega =V_0/2,V_0={\omega }_{D2}/15$, and $v_g=0.2962c$. For graphene, the electron energy $E$ and potential $U_0$ are taken as $E/\hslash v_F=\delta \omega /v_g$ and $U_0/\hslash v_F=V_0/v_g$, respectively, where $v_F$ is the Fermi velocity of electrons in graphene. Both pseudospin-1 EM waves and pseudospin-$1/2$ electrons have a barrier width $45a_2$.

Figure 4

Schematic of a superlattice of PCs. (a) A superlattice realized by stacking alternate layers of two PCs with different length scales. The layers have equal thickness $d$ and the spatial period $L=2d$. (b) The pseudospin-1 photonic states feel a Kronig-Penney type of photonic potential formed by the superlattice in (a). The barrier height $V_0$ is the difference of the Dirac-like point frequencies of two PCs.

Figure 5

Photonic dispersion of pseudospin-1 EM waves in a superlattice (Kronig-Penney photonic potential). (a) Band dispersion near $\delta \omega =V_0/2$ for the superlattice realized by layers of PC1 and PC2 with equal thickness $d=15a_2$ $(a_2$ is the lattice constant of PC2) and the spatial period $L=30a_2$. The “photonic potential” is shown in Fig. 4, with $V_0={\omega }_{D2}/15$. (b) Band dispersion relation, $\delta \omega$ vs $k_x$, for different values of $k_y$. Red solid lines, blue open circles, and green solid triangles correspond to $k_y=0,0.48\pi /L$, and $0.96\pi /L$, respectively.

Figure 6

Supercollimation of pseudospin-1 EM wave packets traveling in the superlattice. (a) Electric field amplitude distribution of a Gaussian wave packet at $t=0$ with the reduced center frequency of $\overline{\delta {\omega }_c}=0.06\pi v_g/L$ and a half width of $r_0=30d$. (b) and (c) Electric field amplitude distributions at $t=1200d/v_g$ in a single PC (left panel) and in the PC superlattice (right panel) with the initial propagation direction of the wave packet in an angle $\theta =0^{\circ }$ (b) and ${45}^{\circ }$ (c), respectively, with respect to the $x$ axis. The right-hand panels show collimation behavior in the presence of the superlattice.