image: FIG.1. (a) Schematic representation of the third generation of the Sierpinski gasket waveguide arrays with r=0.3a, 0.5a, 0.6a. Blue and orange sectors of these arrays represent the first and second-generation structures respectively. Microphotographs of the corresponding fs-laser written waveguide arrays are presented in the insets. (b) Eigenvalues of the stationary linear states vs distortion parameter. (c) Intensity distributions for four representative eigenstates.
Credit: by Hua Z. et al.
Fractals are self-similar and aperiodic structures which are widely represented in universe. The main characteristics of fractals are non-integer effective Hausdorff dimension and the lack of bulk. Fractal HOTIs have been proposed in electronic and acoustics systems, but have not been experimentally observed in topological photonic systems. In photonic systems, nonlinearity can not only effectively tune propagation constants of nonlinear modes within the spectrum of the topological photonic system, which leads to qualitative and complex changes of its internal structure, but also may result in the mobility control around the Fermi level and geometrical frustration. Therefore, the adjustability in the construction of topological systems can be increased by tuning the nonlinear power. Introducing this unusual fractal structure into nonlinear photonic systems can open up new research prospects for the control and localization of light.
In a new paper published in Light: Science & Applications, a team led by Professor Yiqi Zhang from Xi’an Jiaotong University, and Professor Yaroslav V. Kartashov from Russian Academy of Sciences, worked together to experimentally write two different types of Sierpinski gasket fractal waveguide arrays in fused silica samples using femtosecond laser direct writing technology for the first time. They realized the aperiodic photonic fractal HOTI experimentally and numerically, and studied the interplay between topological modes and nonlinear effects to obtain topological corner solitons. The reported method and technique not only produce much richer soliton families and dynamics, but also open new avenues for future light control on aperiodic system.
The research team build aperiodic fractal HOTIs on two different types of Sierpinski gasket waveguide arrays. The structures are shown in Fig. 1. Due to the method of construction, fractal arrays have multiple holes, inner corners and edges, and the effective Hausdorff dimension is a fraction. Further, the research team introduced a shift parameter r to adjust the coupling strength between the adjacent waveguides while fixing the distance between the fractal arrays, which can substantially change the energy spectrum and obtain the topological corner states. Figure 1b displays the linear spectrum versus distortion parameter, different from the periodic HOTIs, the topological states in this case are encountered in both r>0.5a and r<0.5a domains (see the colored curves). The red branch corresponds to hybrid corner states, the others correspond to co-existing outer corner states with different internal structure. In addition, because fractal waveguide arrays are aperiodic structures and lacking bulk, the research team calculated the real-space polarization index to characterize topological properties of this system, the corresponding results indicate that fractal structure can support topological nontrivial modes in whole space. Intensity distributions of four typical localized states corresponding to circles in Fig. 1b are shown in Fig. 1c. This rule applies to the next and higher order structures.
In the presence of nonlinearity and localized corner states in fractal arrays, the research team investigated the focusing nonlinear topological corner spatial soliton families by Newton method. They proved the localization of thresholdless corner solitons is closely related to the nonlinear propagation constants. It is found that with rapid increase of power, the solitons typically gradually broaden and the localization becomes weaker. When the propagation constants reach the border of the extended states, the solitons may spread to the entire array.
Besides, the research team selected four representative locations to study excitation dynamics of the nonlinear corner states under variable energy pulses. They investigated that the structure of r=0.3a can support well localized thresholdless hybrid corner states in any inner or outer corners. While in the array with r=0.5a, the gap width is rather narrow, even moderate variations of pulse energy can cause considerable variations of the output intensity distribution, which may need higher pulse energies to achieve comparable degree of localization. Array with r=0.6a supports 3 types of linear outer corner modes which can only form thresholdless outer corner solitons confined in three closely spaced outer corner waveguides. By comparison, the theoretical results agree well with the experimental results.
In summary, the research team reported the first example of nonlinear aperiodic photonic fractal HOTI that supports a rich variety of topological corner states. The remarkable new feature of fractal structures considered here is that they possess corner states for a very broad range of distortion parameters, substantially exceeding the range, where higher-order topological phase emerges in periodic HOTIs. These results may be used in new designs of topological lasers or on-chip lasers, not only extend the class of HOTIs, but also highlight new prospects for exploration and practical utilization of nonlinear phenomena in photonic fractals.
Journal
Light Science & Applications
Article Title
Observation of nonlinear fractal higher-order topological insulator