A Dirac point is a linear band crossing point originally used to describe unusual transport properties of materials like graphene. In recent years, there has been a surge of exploration of type-II Dirac/Weyl points using various engineered platforms including photonic crystals, waveguide arrays, metasurfaces, magnetized plasma and polariton micropillars, aiming toward relativistic quantum emulation and understanding of exotic topological phenomena. Such endeavors, however, have focused mainly on linear topological states in real or synthetic Dirac/Weyl materials. We propose and demonstrate nonlinear valley Hall edge (VHE) states in laser-written anisotropic photonic lattices hosting innately the type-II Dirac points. These self-trapped VHE states, manifested as topological gap quasi-solitons that can move along a domain wall unidirectionally without changing their profiles, are independent of external magnetic fields or complex longitudinal modulations, and thus are superior in comparison with previously reported topological edge solitons. Our finding may provide a route for understanding nonlinear phenomena in systems with type-II Dirac points that violate the Lorentz invariance and may bring about possibilities for subsequent technological development in light field manipulation and photonic devices.

- Advanced Photonics
- Vol. 3, Issue 5, 056001 (2021)
Abstract
Video Introduction to the Article
1 Introduction
Loosely speaking, topological insulators in condensed matter physics refer to materials that only allow electrons to conduct along the surface but not in bulk.1,2 The concept was introduced into the realm of photonics about a dozen years ago, and has since led to the burgeoning development in photonic topological insulators (PTIs)3
Recently, nonlinear topological photonics has attracted increasing attention,16 as nonlinearity exists inherently in many photonic topological systems such as topological insulator lasers17
It is natural to ask if topological edge solitons can exist in a TRS-preserved system, or systems without complex modulation or external magnetic fields, which is essential for practical applications. For instance, coupling and locking of semiconductor laser arrays to produce coherent high-power laser sources is one of the key motivations behind the development of topological insulator lasers.17
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A general consensus is that many topological and valley Hall effects are mediated by materials with nontrivial degeneracy in momentum space, characterized by Dirac/Weyl cones. In the two-dimensional case, a well-celebrated example is graphene, in which electrons around the Dirac cones behave as massless Dirac fermions. Different from type-I Dirac cones in graphene where the corresponding Fermi surface is a point, there are also so-called tilted type-II Dirac cones with the Fermi surface being a pair of crossing lines.53
In this work, as one typical example, we propose and demonstrate experimentally a scheme to establish type-II Dirac photonic lattices in a nonlinear medium and, more importantly, to reveal the existence of topologically protected nonlinear VHE states. Such nonlinear VHE states in Dirac systems have inherited topological features from their linear counterparts,65 thus benefiting from easy implementation without any need of “time” modulation or external magnetic fields.
2 Principles and Methods
The propagation of a light beam in a photonic lattice is described by the Schrödinger-like paraxial wave equation:
An exemplary type-II Dirac photonic lattice is displayed in Fig. 1(a) with , and its Brillouin zone (BZ) spectrum is shown in Fig. 1(b) in which the first BZ is indicated by the shaded hexagon. The design process of type-II Dirac photonic lattices can be found in the Supplementary Material. There are three sites in each unit cell of the type-II lattice, and they are labeled as , , and in Fig. 1(a). Such a lattice can be considered as the direct outcome from properly stretching a dislocated Lieb lattice in the vertical direction.67 The experimental lattice structure and measured BZ spectrum shown in Figs. 1(c) and 1(d) match perfectly with the numerical results in Figs. 1(a) and 1(b). Moreover, we calculate the band structure of the type-II lattice as shown in Fig. 1(e) based on the plane-wave expansion method. Evidently, under proper lattice design, the bands are tilted and connected at some points—the type-II Dirac points, which are induced solely by the spatial geometry of the lattice.67 To see more clearly, we zoom in on one Dirac cone indicated by a red circle in Fig. 1(e), and let one horizontal plane (corresponding to the Fermi surface) go across the Dirac point. The intersection at the Dirac cone forms a pair of crossing lines in the momentum space [see the inset in Fig. 1(f)]—the characteristic feature of the type-II Dirac cone.61
Figure 1.Photonic lattices with type-II Dirac cones. (a) Numerically designed lattice structure with
3 Results
3.1 Linear Valley Hall Edge States
In Fig. 1(a), the lattice is uniform, but if we intentionally introduce a detuning to one site in the unit cell, the inversion symmetry of the lattice will be broken. In Fig. 2(a), we intentionally increase the depth of site from to , and in Fig. 2(b), we increase the depth of site from to , while keeping that of other sites still at . The band crossings in the band structure indeed disappear as the gap opens, as shown in Fig. 2(c). We also display the calculated Berry curvature of the first band in the plane shown in Fig. 2(d), with the red and blue colors representing the positive and negative values. The valley Chern number can be numerically obtained for each valley, which is . One finds that the Berry curvatures of the first band corresponding to Figs. 2(a) and 2(b) are opposite. Therefore, a domain wall (DW) can be established between two lattices with different index detunings, as shown in Fig. 2(e), where the DW is outlined by a rectangle. Crossing the DW in the direction, the difference of the valley Chern numbers is , which indicates that a topologically nontrivial edge state emerges along the DW. Note that the lattice in Fig. 2(e) is periodic along the direction. The corresponding band structure is displayed in Fig. 2(f). The black curves in the band structure represent the bulk states, whereas the red curve in the bandgap is the VHE state that distributes along the DW. We would like to note that the edge state may also appear in the bandgap between the second and third bulk bands if different inversion symmetries are applied, as discussed in detail in the Supplementary Material.
Figure 2.Linear topological VHE states at the DW between two type-II Dirac cone photonic lattices. (a) and (b) Inversion-symmetry-broken photonic lattices with the depth of site
3.2 Nonlinear Valley Hall Edge States and Quasi-Solitons
The appearance of topological edge solitons inherited from their linear counterparts strongly depends on the dispersion condition of the linear topological edge states.27 One may obtain bright solitons or dark solitons depending on the different dispersions for given self-focusing nonlinearity. Indeed, the inversion-symmetry-breaking perturbation lifts the degeneracy at Dirac points. However, the bands do not lose their symmetry properties around the Dirac points and affect the dispersions of the VHE states. According to our numerical analysis, the VHE states arising from type-II Dirac cones fulfill the dispersion condition (, i.e., the normal dispersive regime), which is a necessary condition for the existence of bright VHE solitons, but such a condition is not satisfied for the type-I Dirac photonic lattices. However, we do not claim that VHE solitons are not supported in lattices with type-I Dirac cones. In fact, the realization of VHE solitons from type-I Dirac cones is still an open question and merits further exploration. We thus solve for the first-order derivative (group velocity of the edge state) and the second-order derivative of the linear VHE state in Fig. 2(f), and the results are displayed in Fig. 3(a). Here the VHE state with moves along the positive direction. Notedly, as shown in Fig. 3(a), the normal dispersion () region is , where defines the width of the first BZ. By adopting the Newton method, we found a family of such nonlinear solutions: the peak amplitude and power of the nonlinear VHE states at are plotted in Fig. 3(b) as a function of the nonlinearity-dependent propagation constant . One finds that both and decrease as increases, reducing to nearly zero at where the nonlinear localized states resemble or reduce to the linear ones.
Figure 3.Numerically obtained nonlinear topological VHE states and robust transport of quasi-solitons from MI. (a) Dispersion spectrum of the linear edge state in
Let us now examine the modulational instability (MI) of nonlinear VHE states under the action of self-focusing nonlinearity, which is possible since at . To do so, we add a random noise to the nonlinear VHE state obtained at , with a noise amplitude about 5% of the nonlinear edge state shown in Fig. 3(c). Representative propagation of the peak amplitude of the perturbed state is shown in Fig. 3(d). The amplitude profiles of the nonlinear edge state at two selected distances [marked by two red dots in Fig. 3(d)] are displayed in Fig. 3(c), showing growth of MI during propagation which leads to formation of quasi-soliton filaments along the DW. The larger the noise amplitude is (e.g., 10% of the nonlinear edge state), the faster the growth of the MI is, and the quicker the formation of quasi-soliton filaments. These MI-induced soliton filaments are considered as precursors for the formation of optical solitons,26 but here they also benefit from the topological protection inherited from the corresponding linear VHE states. Without loss of generality, we take out one of such bright filaments at [indicated by a red circle in Fig. 3(c)] as the input and investigate its long-distance propagation dynamics. To this end, we introduce another physical quantity—the barycenter of the filament defined as —to record its movement during propagation. We first shift the selected filament in Fig. 3(c) to the center of the window () and then track its propagation. The peak amplitude and barycenter of the filament are displayed in Fig. 3(e), showing the stability of a moving VHE quasi-soliton. Even over an extremely long propagation distance (), the peak amplitude remains nearly invariant and the barycenter exhibits a saw-tooth-like oscillating behavior. The appearance of the saw-tooth behavior in the center of mass of the wavepacket is mainly due to the simulation method (the split-step Fourier method) we adopted: for a chosen window along , part of the transporting VHE quasi-soliton appears at the left end of the numerical window when it reaches the right end, resulting in an apparent periodic jump of the barycenter between two ends. The same reason holds for the panel with in Fig. 3(g). In Fig. 3(f), snapshots of the quasi-soliton taken at different propagation distances are displayed. We observe clearly that the quasi-soliton moves along the positive direction with a constant speed (it is same as ), and it remains localized with negligible radiation loss either along the DW or into the bulk—a result of interplay between nonlinearity and topological protection. For direct comparison, we propagate the same input filament in the linear lattice, i.e., removing the nonlinear term in Eq. (4). As expected, without the balance from the nonlinearity, the filament spreads quickly along the DW because of diffraction [see Fig. 3(g)], yet remains localized in the direction perpendicular to the DW. Here we only discuss the case of VHE solitons with , but the approach and analysis also apply to other parameter cases. Since there is no power threshold for the nonlinear VHE state [see the continuous peak amplitude and power curves in Fig. 3(b)], the corresponding quasi-soliton possesses a higher intensity if it bifurcates from the nonlinear VHE state with a larger amplitude. Since the nonlinearity can induce a defect in the topological structure, the topological property will be broken if the strength of the defect is larger than the bandgap width. From this point of view, one cannot seek for the VHE solitons with very high intensity. In the region with , the obtained solitons have a smaller peak amplitude and intensity. It is impossible to design corners for the DW of type-II Dirac photonic lattices due to the spatial symmetry, yet artificial corners can play a role of disorder but would result in intervalley scattering due to weak valley protection. Such issues are further addressed in the Supplementary Material, where we use a large-scale random disorder to check the robustness of the VHE solitons.68
3.3 Experimental Observation of Nonlinear Valley Hall Edge States
In the experiment, an inversion-symmetry-broken type-II Dirac photonic lattice with a DW [corresponding to Fig. 2(e)], as shown in Fig. 4(a1), is established by employing the CW-laser-writing technique (similar for writing the uniform lattice in Fig. 1(c) but with a judiciously controlled writing process).71
Figure 4.Experimental observation of nonlinear topological VHE states. (a1) Experimentally established type-II Dirac lattices with a center DW (marked by the white line), where the inset (bottom-right) shows discrete diffraction from single-site excitation. (b)–(d) Linear (first and third rows) and nonlinear (second and fourth rows) outputs in real (first and second rows) and momentum (third and fourth rows) space obtained from excitation by two superimposed out-of-phase elliptical beams (circled by the dashed ellipse). The initial Bloch momenta of the probe beam are (b)
To appreciate the formation of nonlinear VHE states presented in Fig. 3(e), two out-of-phase elliptical Gaussian beams (due to the property of the VHE state that has a staggered phase structure along the axis) are superimposed as a probe beam with an input power of only , whose position is marked by the white dashed oval in Fig. 4(a1). The linear and nonlinear output intensity patterns in real space and their corresponding spectra in momentum space for three different excitation conditions (e.g., with Bloch momentum ) are shown in Figs. 4(b)–4(d). When the bias field is off, the probe beam itself does not exhibit nonlinear self-action, but it is localized in the vertical direction without spreading into the bulk due to the excitation of the linear VHE states, although somewhat extended along the DW after 1 cm of propagation. However, the probe beam undergoes self-focusing when the bias field is turned on () and, as a result, the corresponding nonlinear output also becomes more localized along the DW direction [Figs. 4(b2)–4(d2)]. This can be seen more clearly from the overall beam width [full width at half maximum (FWHM)], which decreased approximately from (linear case) to (nonlinear case) [Figs. 4(e2)–4(e4)]. Interestingly, we found that the output pattern moves slightly to the left (right) even in the linear condition for the Bloch momentum , thanks to the initial transverse velocity of the VHEs, while that for remains invariant [Fig. 4(e1)]. Due to the nonlinear action, an appreciable portion (about 20%) is “diminished” from the initial position of the probe beam (marked by a solid ellipse) as compared with the linear case. This is a direct signature of nonlinearity-induced transport of the VHEs. Indeed, by using the barycenter of the beam defined earlier, and plotting the superimposed linear and nonlinear output profiles along the direction [Figs. 4(e2)–4(e4)], we can see clearly that the center is further shifted away from the initial input position under nonlinear propagation. Furthermore, nonlinearity-induced spectral reshaping is evident due to formation of nonlinear VHEs by comparing nonlinear [Figs. 4(b4)–4(d4)] with linear [Figs. 4(b3)–4(d3)] output spectra. Our results indicate that self-trapped nonlinear VHE states indeed exist in type-II Dirac photonic lattices. Due to the limited crystal length, it is not feasible to experimentally show the long-distance transport of the quasi-solitons as demonstrated in our theoretical analysis. However, these experimental results in Fig. 4 are corroborated by numerical simulations, as detailed in the Supplementary Material. Here we would like to note that the waveguide is assumed uniform without appreciable losses along the propagation direction. Thus the transmission efficiency is nearly invariant for the probe beam.
4 Discussion and Conclusion
We have proposed and experimentally demonstrated photonic lattices exhibiting type-II Dirac points, thereby unveiling the existence of nonlinear VHE states and the formation of topological quasi-solitons. We have shown theoretically that it is crucial to have the type-II Dirac dispersion and a DW between two lattices of opposite Berry curvatures in order to achieve self-trapping of the VHE solitons in the TRS-preserving topological systems. We would like to note that the nonlinear experiments with topological VHE states may also be implemented in femtosecond laser writing photonic lattices,74 now that optical nonlinearity has been demonstrated in such a platform.22,32,75 We believe that our results may prove relevant to other type-II systems such as nonlinear effects and high-frequency rectification in type-II topological semimetals76 and may also enlighten new ideas in nonlinear non-Hermitian topological systems.77 Moreover, there is still a plethora of interesting topics yet to be explored in nonlinear systems that could involve type-II Dirac points, including higher-order topological phases,73,75,78 new physics arising from engineered longitudinal modulation,79,80 synthetic dimensions,81 and even the innovation of topological semiconductor laser technologies.17
Hua Zhong received her BS degree from Qingdao University of Technology in 2015 and her master’s degree from Xi’an Jiaotong University in 2018. Currently, she is pursuing her PhD as a student under the supervision of Dr. Yiqi Zhang at the School of Electronic Science and Engineering of Xi’an Jiaotong University. Her focus is topological and waveguide array physics.
Shiqi Xia received his PhD from Nankai University, Tianjin, China, in 2021. He is currently a postdoc fellow at the School of Physics of Nankai University. His research interests include nonlinear optics, topological photonics, non-Hermitian system, as well as synthetic dimensions.
Yiqi Zhang received his PhD from Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an, China, in 2011. He is currently an associate professor at the School of Electronic Science and Engineering of Xi’an Jiaotong University. His current research interests include topological photonics and quantum-optical analogies.
Yongdong Li received his PhD from Xi’an Jiaotong University, Xi’an, China, in 2005. He is currently a professor at the School of Electronic Science and Engineering of Xi’an Jiaotong University. His research interests include modeling and applications of plasmas.
Daohong Song received his PhD from Nankai University, Tianjin, China, in 2009. He is currently a professor at the TEDA Applied Physics Institute and School of Physics of Nankai University. His current research interests include topological photonics and nonlinear optics.
Chunliang Liu received his BS and MS degrees from Xi’an Jiaotong University, Xi’an, China, in 1982 and 1987, respectively, and his PhD from the China Institute of Atomic Energy, Beijing, China, in 1992. He is currently a professor at the School of Electronic Science and Engineering of Xi’an Jiaotong University. He actively conducts research in flat panel displays, plasma discharge physics, and high-power microwave technology.
Zhigang Chen earned his PhD from Bryn Mawr College in 1995. After two years of postdoctoral research, he was promoted to the rank of Senior Research Staff Member at Princeton University before joining the faculty at San Francisco State University in 1998. He is currently a specially appointed chair professor at Nankai University. His research interests include nonlinear optics, topological photonics, and optical manipulation. He is a fellow of OSA and APS.
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