Type-II Dirac photonic lattices

If a photonic lattice possesses Dirac cones Leykam and Desyatnikov (2016) in its band structure, it can be called a Dirac photonic lattice. One of the most famous Dirac photonic lattices is the photonic graphene Zandbergen and de Dood (2010); Rechtsman et al. (2013a, b); Crespi et al. (2013); Zeuner et al. (2014); Plotnik et al. (2014); Song et al. (2015); Nalitov et al. (2015), which is also known as the honeycomb lattice. In addition, the Lieb lattice Taie et al. (2015); Vicencio et al. (2015); Mukherjee et al. (2015); Diebel et al. (2016); Ozawa et al. (2017); Xia et al. (2018), the kagome lattice El Hassan et al. (2019); Li et al. (2019a), the superhoneycomb lattices Zhong et al. (2017); Kang et al. (2019), and some others Zhong et al. (2019) can be also classified into Dirac photonic lattices. It is worth mentioning that the Dirac cones of these photonic lattices aforementioned are mostly type-I, even though some of them are tilted. Actually, there are also type-II and type-III Dirac cones Milićević et al. (2019). The main difference between them is that the Fermi surface of the type-I Dirac cone is a point, that of the type-II Dirac cone is a pair of crossing lines, and that of the type-III Dirac cone is a line. In other words, the isofrequency contours of the type-I Dirac cones are closed, while those of the type-II counterparts are open and have hyperbolic profiles (e.g., on both $+k_{y}$ and $-k_{y}$ directions). The isofrequency contours of the type-III ones are also open, but only have hyperbolic profiles on one side (e.g., on either $+k_{y}$ or $-k_{y}$ direction). Note that there is also a different definition on the type-III degeneracy Li et al. (2019b). As a plethora of efforts being implemented to explore the type-II Weyl-like features Soluyanov et al. (2015); Xu et al. (2015); Chen et al. (2016); Autès et al. (2016); Noh et al. (2017a); Chen et al. (2018); Xie et al. (2019) both in condensed matter physics and photonics, type-II Dirac objects attract ongoing attentions too Yan et al. (2017); Noh et al. (2017b); Chang et al. (2017); Fei et al. (2018); Politano et al. (2018); Kim et al. (2018); Wang et al. (2018), especially in photonics Lin et al. (2017); Pyrialakos et al. (2017); Wang et al. (2017); Hu et al. (2018); Mann et al. (2018); Milićević et al. (2019). Different from ubiquitous type-I Dirac cones, appearance of type-II Dirac cones demands either high anisotropic lattice arrangement or accurate manipulation of the lattice environment. Despite the difference, quasiparticles corresponding to both the two types of Dirac cones are massless, and can be described by the massless Dirac Hamiltonian Li et al. (2017), thereby Dirac materials become unique paradigms to explore relativistic Dirac-related phenomena, such as Klein tunneling Katsnelson et al. (2006).

Being a relativistic phenomenon, Klein tunneling means that a massless particle can overpass a barrier higher than its energy freely. Surprisingly, this nonintuitive prediction was also explained successfully in the frame of classical electromagnetic theory and simulated and observed in classical systems, such as deformed honeycomb lattice Bahat-Treidel et al. (2010); Bahat-Treidel and Segev (2011), waveguide superlattices Longhi (2010); Dreisow et al. (2012), matamaterials Sun et al. (2017), and pseudospin-1 photonic crystals Fang et al. (2019). As far as we know, Klein tunneling in type-II Dirac photonic lattices is still open to be explored.

In this work, we unveil type-II Dirac cones in novel but simple lattice waveguide arrays constructed by identical waveguide channels that are transversely arranged in morphable lattice profiles with three sites in one unit cell, via a controllable angle parameter $\theta$ in the range $[\pi/6,\pi]$ (see Figure 1). When the angle reaches its supreme value, a dislocated Lieb lattice Li et al. (2018) that has both type-I and type-III Dirac cones Milićević et al. (2019) is created. With the angle value decreases, the lattice deforms with the type-III Dirac cones reducing into tilted type-I Dirac cones. Further deformation of the lattice by minishing the angle value, the type-I Dirac cones tilt even further and evolve into the type-II Dirac cones gradually. Light modulation and manipulation based on these various Dirac cones are elucidated by the corresponding conical diffractions, and for the first time, the Klein tunneling is demonstrated in the type-II Dirac photonic lattice. Our results provide a new design for type-II Dirac photonic lattices, which has advantage of simplicity since the appearance of the type-II Dirac cones is only dependent on the spatial symmetry of the lattice.

Propagation of a light beam in a photonic lattice can be faithfully described by the Schrödinger-like paraxial wave equation:

where the transverse $(x,y)$ and longitudinal $z$ coordinates are normalized to the characteristic transverse scale $r_{0}$ and the diffraction length $L_{\rm dif}=kr_{0}^{2}$, respectively; $k=2\pi n_{0}/\lambda$ is the wavenumber with $n_{0}$ being the background refractive index and $\lambda$ the wavelength. The lattice potential $\mathcal{R}(x,y)=p\sum_{n,m}\mathcal{Q}(x-x_{n},y-y_{m})$ is composed of Gaussian waveguides $\mathcal{Q}=\exp(-x^{2}/a_{x}^{2}-y^{2}/a_{y}^{2})$ with $p$ being the depths of two sublattices, $a_{x,y}$ waveguide widths, and $(x_{n},y_{m})$ the transverse location of each waveguide channel. The lattice constant is labeled as $d$ which is the distance between two nearest-neighbor sites. The photonic lattice can be prepared by using, for instance, the femtosecond laser writing technique in fused silica material Rechtsman et al. (2013c); Stützer et al. (2018); Lustig et al. (2019) and the optically induced technique in photorefractive crystals Song et al. (2015); Xia et al. (2018); Wang et al. (2020). Taking the former technique as an example, one can use the following parameters: $\lambda=633\,\rm nm$, $d=15\,\mu\rm m$, $a_{x}=a_{y}=10\,\mu\rm m$. If we choose the transverse scale $r_{0}=10\,\mu\rm m$, the diffraction length $L_{\rm dif}=1.4\,\rm mm$. The lattice depth $p=10$ corresponds to refractive index change of $7\times 10^{-4}$ in a real physical system.

The solution of Equation (1) can be written as $\psi(x,y,z)=u(x,y)\exp(i\beta z)$ with $\beta$ being the propagation constant (or “energy” of the quasiparticle) and $u(x,y)$ the Bloch mode. Plugging this solution into Equation (1), one obtains

which can be solved numerically by using the plane-wave expansion method. As a periodic function of Bloch momenta $k_{x,y}$, $\beta(k_{x},k_{y})$ is the band structure of the photonic lattice ${\mathcal{R}}(x,y)$. Clearly, photonic lattices with different geometries possess different band structures Tarruell et al. (2012). One may see the lattice transverse profile shown in Figure 1(a), which possesses three sites (labeled as A, B and C) in one unit cell. Now, we introduce another controlling parameter, the angle $\theta$ between sites B and C along vertical $y$ direction, as shown by the zigzag line. Here in Figure 1(a), the angle $\theta=2\pi/3$. If the angle $\theta=\pi$, one obtains the dislocated Lieb lattice Li et al. (2018), as shown in Figure 1(b), which is different from the traditional Lieb lattice Vicencio et al. (2015); Mukherjee et al. (2015). One can change the angle continuously to deform the lattice, and in Figure 1(c), the lattice with $\theta=\pi/3$ is shown. It is convenient to check the far-field diffraction patterns Bartal et al. (2005) of the lattices, which can show the corresponding Brillouin zones directly. According to numerical simulations, one may find that the first Brillouin zones are deformed hexagons, and the smaller the angle, the larger the deformation of the hexagons. In Figure 1(d), we only show the far-field diffraction pattern of the lattice shown in Figure 1(c). Since the first Brillouin zone is always a hexagon, there must be always three sites in one unit cell no matter what the value of the angle. Even though there are still three sites in one unit cell in the lattice with $\theta=\pi/3$, the property may change greatly in comparison with those with $\theta=2\pi/3$ and $\theta=\pi$. An explicit fact is that under the tight-binding approximation, there are 2, 3 and 3 nearest-neighbor sites for sites A, B and C, respectively, both in Figure 1(a) and 1(b). However in Figure 1(c), the numbers are 4, 5 and 5. According to the geometry of the lattice in Figure 1(c), the locations of the six corners of the first Brillouin zone can be obtained theoretically

which are in accordance with the numerical results in Figure 1(d).

By solving Equation (2) numerically, we obtain the band structures for the lattices in Figure 1, which are displayed in Figure 2. In Figure 2(a) that corresponds to the lattice with $\theta=2\pi/3$ in Figure 1(a), all the Dirac cones are tilted type-I. While in Figure 2(b) that corresponds to Figure 1(b), the Dirac cones between the top and bottom bands are tilted type-I, but those between the middle and bottom bands are type-III. When the angle decreases successively to $\theta=\pi/3$, as shown in Figure 2(c), all the Dirac cones become type-II, because the group velocity along $y$ (i.e., $-d\beta/dk_{y}$) of the mode around the Dirac points does not change sign.

To understand the emergence of the type-II Dirac cones, we theoretically analyze the couplings among sites in Figure 1(c) by adopting the tight-binding approximation method and only considering the nearest-neighbor interaction. The corresponding Hamiltonian can be written as

in which ${\bf k}=[k_{x},k_{y}]$, ${\bf e}_{1}=[a,0]$, ${\bf e}_{2}=[0,a]$, ${\bf e}_{3}=[\sqrt{3}a/2,a/2]$, ${\bf e}_{4}=[\sqrt{3}a/2,-a/2]$, and $t$ the coupling strength. The eigenvalues of Equation (3) are the band structure, but the analytical solution is hard to obtain. Even so, one still can obtain the locations of the Dirac points, which are $[0,\pm 4\pi/3a]$ and $[\pm 2\pi/(4+\sqrt{3})a,\pm 2\pi/3a]$ between the top and middle bands, and $[0,\pm 2\pi/3a]$ and $[\pm 2\pi/(4+\sqrt{3})a,\pm 4\pi/3a]$ between the middle and bottom bands. Numerical band structure is displayed in Figure 3(a), which looks visually the same with that in Figure 2(c) although quantitatively there are difference in the value of $\beta$. Since all the Dirac cones are type-II for this case, we take the Dirac point at $[0,4\pi/3a]$ as an example without loss of generality, and this Dirac cone is shown in the inset of Figure 3(a). Before going into a subtle theoretical analysis on this type-II Dirac cone, it is necessary to have a look at the corresponding cross sections in the $(k_{x},k_{y})$ plane with $\beta=0$ [Figure 3(b)], the $(k_{y},\beta)$ plane with $k_{x}=0$ [Figure 3(c)], and the $(k_{x},\beta)$ plane with $k_{y}=4\pi/3a$ [Figure 3(d)]. In the $\beta=0$ plane, the intersection of the top and middle bands, as shown in Figure 3(b), exhibits two crossing lines Soluyanov et al. (2015); Milićević et al. (2019). In Figure 3(c), the red and blue lines show the profile of the Dirac cone in the cross section $k_{x}=0$ which clearly elucidates that the sign of the slope $d\beta/dk_{y}$ does not change along $k_{y}$ direction. While in Figure 3(d), the profile of the Dirac cone in the cross section $k_{y}=0$ is symmetric about $k_{x}=0$ (i.e., sign of $d\beta/dk_{x}$ changes), which is similar to that for a (tilted) type-I Dirac points. To this regard, if we expand the Hamiltonian in the infinitesimal region $(p_{x}=k_{x}-k_{x}^{\rm D},p_{y}=k_{y}-k_{y}^{\rm D})$ around the Dirac point $(k_{x}^{\rm D},k_{y}^{\rm D})$, we can only consider the component along $k_{y}$ direction and let $p_{x}=0$ safely. As a result, the corresponding Hamiltonian can be written as

The eigenvalues of this Hamiltonian are

Clearly, one may find the relation $\beta_{1}|_{p_{y}=0}=\beta_{2}|_{p_{y}=0}=0$, which indicates that the two bands are degenerated at the point $(p_{x}=0,p_{y}=0)$ — the location of the Dirac point. One also finds that $d\beta_{1}/dp_{y}$ and $d\beta_{2}/dp_{y}$ always have the same sign, therefore the velocity does not change its sign around the Dirac point, and the Dirac cone is type-II definitely.

Summarizing, we have constructed a type-II Dirac photonic lattice from the Lieb-like lattice by simply adjusting a angle parameter which only changes the spatial symmetry of the lattice. Conical diffraction and Klein tunneling in this novel type-II Dirac photonic lattice are discussed in detail. We believe that our results may not only provide a feasible avenue on light manipulation, but also help realize other topological photonic and analogize nonrelativistic phenomena in photonic lattices. In addition to photonics, we believe that the developed type-II Dirac lattice in this work may also provide a completely new flatform for cold atoms and acoustics.

Note added: A paper Wu et al. (2020) on constructing type-II Dirac points by proposing a band-folding scheme, and a paper Pyrialakos et al. (2020) on symmetry-controlled edge states in the type-II phase of Dirac photonic lattices, were published after submission of this paper. We would like to emphasize that, in our work for the first time, we find the simplest way to construct type-II Dirac points which only depends on the spatial geometry of the photonic lattice.

This work was supported by Guangdong Basic and Applied Basic Research Foundation (2018A0303130057), National Natural Science Foundation of China (U1537210, 11534008), and Fundamental Research Funds for the Central Universities (xzy012019038, xzy022019076). The authors acknowledge the computational resources provided by the HPC platform of Xi an Jiaotong University.