Fractal photonic topological insulators

Ultra-broadband all-optical nonlinear activation function enabled by MoTe2/optical waveguide integrated devices

All-optical nonlinear activation functions (NAFs) are crucial for enabling rapid optical neural networks (ONNs). As linear matrix computation advances in integrated ONNs, on-chip all-optical NAFs face challenges such as limited integration, high latency, substantial power consumption, and a high activation threshold. In this work, we develop an integrated nonlinear optical activator based on the butt-coupling integration of two-dimensional (2D) MoTe2 and optical waveguides (OWGs). The activator exhibits an ultra-broadband response from visible to near-infrared wavelength, a low activation threshold of 0.94 μW, a small device size (~50 µm2), an ultra-fast response rate (2.08 THz), and high-density integration. The excellent nonlinear effects and broadband response of 2D materials have been utilized to create all-optical NAFs. These activators were applied to simulate MNIST handwritten digit recognition, achieving an accuracy of 97.6%. The results underscore the potential application of this approach in ONNs. Moreover, the classification of more intricate CIFAR-10 images demonstrated a generalizable accuracy of 94.6%. The present nonlinear activator promises a general platform for three-dimensional (3D) ultra-broadband ONNs with dense integration and low activation thresholds by integrating a variety of strong nonlinear optical (NLO) materials (e.g., 2D materials) and OWGs in glass.

Observation of nonlinear fractal higher order topological insulator

Higher-order topological insulators (HOTIs) are unique materials hosting topologically protected states, whose dimensionality is at least by 2 lower than that of the bulk. Topological states in such insulators may be strongly confined in their corners which leads to considerable enhancement of nonlinear processes involving such states. However, all nonlinear HOTIs demonstrated so far were built on periodic bulk lattice materials. Here, we demonstrate the first nonlinear photonic HOTI with the fractal origin. Despite their fractional effective dimensionality, the HOTIs constructed here on two different types of the Sierpiński gasket waveguide arrays, may support topological corner states for unexpectedly wide range of coupling strengths, even in parameter regions where conventional HOTIs become trivial. We demonstrate thresholdless spatial solitons bifurcating from corner states in nonlinear fractal HOTIs and show that their localization can be efficiently controlled by the input beam power. We observe sharp differences in nonlinear light localization on outer and multiple inner corners and edges representative for these fractal materials. Our findings not only represent a new paradigm for nonlinear topological insulators, but also open new avenues for potential applications of fractal materials to control the light flow.

From individuals to families: design and application of self-similar chiral nanomaterials

Establishing an intimate relationship between similar individuals is the beginning of self-extension. Various self-similar chiral nanomaterials can be designed using an individual-to-family approach, accomplishing self-extension. This self-similarity facilitates chiral communication, transmission, and amplification of synthons. We focus on describing the marriage of discrete cages to develop self-similar extended frameworks. The advantages of utilizing cage-based frameworks for chiral recognition, enantioseparation, chiral catalysis and sensing are highlighted. To further promote self-extension, fractal chiral nanomaterials with self-similar and iterated architectures have attracted tremendous attention. The beauty of a fractal family tree lies in its ability to capture the complexity and interconnectedness of a family's lineage. As a type of fractal material, nanoflowers possess an overarching importance in chiral amplification due to their large surface-to-volume ratio. This review summarizes the design and application of state-of-the-art self-similar chiral nanomaterials including cage-based extended frameworks, fractal nanomaterials, and nanoflowers. We hope this formation process from individuals to families will inherit and broaden this great chirality.

Hyperbolic photonic topological insulators

Topological photonics provides a new degree of freedom to robustly control electromagnetic fields. To date, most of established topological states in photonics have been employed in Euclidean space. Motivated by unique properties of hyperbolic lattices, which are regular tessellations in non-Euclidean space with a constant negative curvature, the boundary-dominated hyperbolic topological states have been proposed. However, limited by highly crowded boundary resonators and complicated site couplings, the hyperbolic topological insulator has only been experimentally constructed in electric circuits. How to achieve hyperbolic photonic topological insulators is still an open question. Here, we report the experimental realization of hyperbolic photonic topological insulators using coupled ring resonators on silicon chips. Boundary-dominated one-way edge states with pseudospin-dependent propagation directions have been observed. Furthermore, the robustness of edge states in hyperbolic photonic topological insulators is also verified. Our findings have potential applications in the field of designing high-efficient topological photonic devices with enhanced boundary responses.

Dynamical Detection of Topological Spectral Density

Local density of states (LDOS) is emerging as powerful means of exploring classical-wave topological phases. However, the current LDOS detection method remains rare and merely works for static situations. Here, we introduce a generic dynamical method to detect both the static and Floquet LDOS, based on an elegant connection between dynamics of chiral density and local spectral densities. Moreover, we find that the Floquet LDOS allows to measure out Floquet quasienergy spectra and identify topological π modes. As an example, we demonstrate that both the static and Floquet higher-order topological phase can be universally identified via LDOS detection, regardless of whether the topological corner modes are in energy gaps, bands, or continuous energy spectra without band gaps. Our study opens a new avenue utilizing dynamics to detect topological spectral densities and provides a universal approach of identifying static and Floquet topological phases.

Corner Reflectors: Fractal Analysis and Integrated Single-Photon Sources

In this work, the properties of the radiation emitted by a corner reflector with an electric dipole feeder are analyzed in the optical domain, where the distance between the dipole and the corner apex can be large in terms of the wavelength. A comprehensive study of the fractal properties of the radiated intensity patterns is presented. The use of this setup for the realization of single-photon sources in photonic integrated circuits is also put forward, and a detailed study of the emission properties of the device and its optimal configurations is presented.

Fractal photonic anomalous Floquet topological insulators to generate multiple quantum chiral edge states

Anomalous Floquet topological insulators with vanishing Chern numbers but supporting chiral edge modes are attracting more and more attention. Since the existing anomalous Floquet topological insulators usually support only one kind of chiral edge mode even at a large lattice size, they are unscalable and unapplicable for multistate topological quantum systems. Recently, fractal topological insulators with self-similarity have been explored to support more nontrivial modes. Here, we demonstrate the first experimental realization of fractal photonic anomalous Floquet topological insulators based on dual Sierpinski carpet consisting of directional couplers using the femtosecond laser direct writing. The fabricated lattices support much more kinds of chiral edge states with fewer waveguides and enable perfect hopping of quantum states with near unit transfer efficiency. Instead of zero-dimensional bound modes for quantum state transport in previous laser direct-written topological insulators, we generate multiple propagating single-photon chiral edge states in the fractal lattice and observe high-visibility quantum interferences. These suggest the successful realization of highly indistinguishable single-photon chiral edge states, which can be applied in various quantum operations. This work provides the potential for enhancing the multi-fold manipulation of quantum states, enlarging the encodable quantum information capacity in a single lattice via high-dimensional encoding and many other fractal applications.

Observation of nonlinear disclination states

Introduction of controllable deformations into periodic materials that lead to disclinations in their structure opens novel routes for construction of higher-order topological insulators hosting topological states at disclinations. Appearance of these topological states is consistent with the bulk-disclination correspondence principle, and is due to the filling anomaly that results in fractional charges to the boundary unit cells. So far, topological disclination states were observed only in the linear regime, while the interplay between nonlinearity and topology in the systems with disclinations has been never studied experimentally. We report here on the experimental observation of the nonlinear photonic disclination states in waveguide arrays with pentagonal or heptagonal disclination cores inscribed in transparent optical medium using the fs-laser writing technique. The transition between nontopological and topological phases in such structures is controlled by the Kekulé distortion coefficient r with topological phase hosting simultaneously disclination states at the inner disclination core and spatially separated from them corner-I, corner-II, and extended edge states at the outer edge of the structure. We show that the robust nonlinear disclination states bifurcate from their linear counterparts and that location of their propagation constants in the gap and, hence, their spatial localization can be controlled by their power. Nonlinear disclination states can be efficiently excited by Gaussian input beams, but only if they are focused into the waveguides belonging to the disclination core, where such topological states reside. Our results open new prospects for investigation of nonlinear effects in topological systems with disclinations and are relevant for different areas of science, including Bose-Einstein and polariton condensates, where potentials with the disclinations can be created.

Observation of bulk quadrupole in topological heat transport

The quantized bulk quadrupole moment has so far revealed a non-trivial boundary state with lower-dimensional topological edge states and in-gap zero-dimensional corner modes. In contrast to photonic implementations, state-of-the-art strategies for topological thermal metamaterials struggle to achieve such higher-order hierarchical features. This is due to the absence of quantized bulk quadrupole moments in thermal diffusion fundamentally prohibiting possible band topology expansions. Here, we report a recipe for generating quantized bulk quadrupole moments in fluid heat transport and observe the quadrupole topological phases in non-Hermitian thermal systems. Our experiments show that both the real- and imaginary-valued bands exhibit the hierarchical features of bulk, gapped edge and in-gap corner states-in stark contrast to the higher-order states observed only on real-valued bands in classical wave fields. Our findings open up unique possibilities for diffusive metamaterial engineering and establish a playground for multipolar topological physics.

A second wave of topological phenomena in photonics and acoustics

Light and sound are the most ubiquitous forms of waves, associated with a variety of phenomena and physical effects such as rainbows and echoes. Light and sound, both categorized as classical waves, have lately been brought into unexpected connections with exotic topological phases of matter. We are currently witnessing the onset of a second wave of active research into this topic. The past decade has been marked by fundamental advances comprising two-dimensional quantum Hall insulators and quantum spin and valley Hall insulators, whose topological properties are characterized using linear band topology. Here, going beyond these conventional topological systems, we focus on the latest frontiers, including non-Hermitian, nonlinear and non-Abelian topology as well as topological defects, for which the characterization of the topological features goes beyond the standard band-topology language. In addition to an overview of the current state of the art, we also survey future research directions for valuable applications.

Photonic elementary cellular automata for simulation of complex phenomena

Cellular automata are a class of computational models based on simple rules and algorithms that can simulate a wide range of complex phenomena. However, when using conventional computers, these 'simple' rules are only encapsulated at the level of software. This can be taken one step further by simplifying the underlying physical hardware. Here, we propose and implement a simple photonic hardware platform for simulating complex phenomena based on cellular automata. Using this special-purpose computer, we experimentally demonstrate complex phenomena, including fractals, chaos, and solitons, which are typically associated with much more complex physical systems. The flexibility and programmability of our photonic computer present new opportunities to simulate and harness complexity for efficient, robust, and decentralized information processing using light.

Direct extraction of topological Zak phase with the synthetic dimension

Measuring topological invariants is an essential task in characterizing topological phases of matter. They are usually obtained from the number of edge states due to the bulk-edge correspondence or from interference since they are integrals of the geometric phases in the energy band. It is commonly believed that the bulk band structures could not be directly used to obtain the topological invariants. Here, we implement the experimental extraction of Zak phase from the bulk band structures of a Su-Schrieffer-Heeger (SSH) model in the synthetic frequency dimension. Such synthetic SSH lattices are constructed in the frequency axis of light, by controlling the coupling strengths between the symmetric and antisymmetric supermodes of two bichromatically driven rings. We measure the transmission spectra and obtain the projection of the time-resolved band structure on lattice sites, where a strong contrast between the non-trivial and trivial topological phases is observed. The topological Zak phase is naturally encoded in the bulk band structures of the synthetic SSH lattices, which can hence be experimentally extracted from the transmission spectra in a fiber-based modulated ring platform using a laser with telecom wavelength. Our method of extracting topological phases from the bulk band structure can be further extended to characterize topological invariants in higher dimensions, while the exhibited trivial and non-trivial transmission spectra from the topological transition may find future applications in optical communications.

Top-down patterning of topological surface and edge states using a focused ion beam

The conducting boundary states of topological insulators appear at an interface where the characteristic invariant ℤ2 switches from 1 to 0. These states offer prospects for quantum electronics; however, a method is needed to spatially-control ℤ2 to pattern conducting channels. It is shown that modifying Sb2Te3 single-crystal surfaces with an ion beam switches the topological insulator into an amorphous state exhibiting negligible bulk and surface conductivity. This is attributed to a transition from ℤ2 = 1 → ℤ2 = 0 at a threshold disorder strength. This observation is supported by density functional theory and model Hamiltonian calculations. Here we show that this ion-beam treatment allows for inverse lithography to pattern arrays of topological surfaces, edges and corners which are the building blocks of topological electronics.

Effectively writing low propagation and bend loss waveguides in the silica glass by using a femtosecond laser

We report writing low-loss waveguides (WGs) by using a femtosecond laser in silica glass. A record low propagation loss of 0.07 dB/cm is achieved, and the lowest bend loss reaches 0.001 dB/mm with the bend radius of 30 mm. The optimal effective writing speed reaches 125 µm/s, which is two orders higher than the previous reported value. Fan-out devices with well controllable low loss for three-dimensional photonic integration are also fabricated. This work provides an effective strategy to create WG devices for 3D high-density photonic integration.

Fractal photonic topological insulators

Topological insulators constitute a novel state of matter with scatter-free edge states surrounding an insulating bulk. Conventional wisdom regards the insulating bulk as essential, since the invariants describing the topological properties of the system are defined therein. Here, we study fractal topological insulators based on exact fractals comprised exclusively of edge sites. We present experimental proof that, despite the lack of bulk bands, photonic lattices of helical waveguides support topologically protected chiral edge states. We show that light transport in our topological fractal system features increased velocities compared to the corresponding honeycomb lattice. By going beyond the confines of the bulk-boundary correspondence, our findings pave the way toward an expanded perception of topological insulators and open a new chapter of topological fractals.