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The statistical mechanical behavior of weakly nonlinear multimoded optical settings is attracting increased interest during the last few years. The main purpose of this work is to numerically investigate the main factors that affect the thermalization process in photonic lattices. In particular, we find that lattices with identically selected properties (such as temperature, coupling coefficient, lattice size, and excitation conditions) can exhibit very different thermalization dynamics and thus thermalization distances. Our investigation is focused on two different two-dimensional lattices: the honeycomb lattice and the triangular lattice. Our numerical results show that, independently of the excitation conditions, the honeycomb lattice always thermalizes faster than the triangular lattice. We mainly explain this behavior to the quasilinear spectrum that promotes wave-mixing in the honeycomb lattice in comparison to the power-like spectrum of the triangular lattice. In addition, we investigate the combined effects of temperature as well as the sign and magnitude of the nonlinearity. Switching either the sign of the Kerr nonlinear coefficient or the sign of the temperature can lead to significant differences in the thermalization dynamics, a phenomenon that can be physically explained in terms of wave instabilities. Larger absolute values of the temperature |T| result in more uniform distributions for the power occupation numbers and faster thermalization speeds. Finally, as expected, increasing the magnitude of the nonlinearity results in accelerated thermalization. Our findings provide valuable insights into optical thermalization in discrete systems where experimental realization may bring about new possibilities for light manipulation and applications., Comment: 11 pages, 4 figures

Kagome lattices represent an archetype of intriguing physics, attracting a great deal of interest in different branches of natural sciences, recently in the context of topological crystalline insulators. Here, we demonstrate two distinct classes of corner states in breathing Kagome lattices (BKLs) with "bearded" edge truncation, unveiling their topological origin. The in-phase corner states are found to exist only in the topologically nontrivial regime, characterized by a nonzero bulk polarization. In contrast, the out-of-phase corner states appear in both topologically trivial and nontrivial regimes, either as bound states in the continuum or as in-gap states depending on the lattice dimerization conditions. Furthermore, the out-of-phase corner states are highly localized, akin to flat-band compact localized states, and they manifest both real- and momentum-space topology. Experimentally, we observe both types of corner states in laser-written photonic bearded-edge BKLs, corroborated by numerical simulations. Our results not only deepen the current understanding of topological corner modes in BKLs, but also provide new insight into their physical origins, which may be applied to other topological BKL platforms beyond optics., Comment: 20 pages,4 figures

Vortex phenomena are ubiquitous in nature, from vortices of quantum particles and living cells [1-7], to whirlpools, tornados, and spiral galaxies. Yet, effective control of vortex transport from one place to another at any scale has thus far remained a challenging goal. Here, by use of topological disclination [8,9], we demonstrate a scheme to confine and guide vortices of arbitrary high-order charges10,11. Such guidance demands a double topological protection: a nontrivial winding in momentum space due to chiral symmetry [12,13] and a nontrivial winding in real space arising from collective complex coupling between vortex modes. We unveil a vorticity-coordinated rotational symmetry, which sets up a universal relation between the topological charge of a guided vortex and the order of rotational symmetry of the disclination structure. As an example, we construct a C3-symmetry photonic lattice with a single-core disclination, thereby achieving robust transport of an optical vortex with preserved orbital angular momentum (OAM) that corresponds solely to one excited vortex mode pinned at zero energy. Our work reveals a fundamental interplay of vorticity, disclination and higher-order topological phases14-16, applicable broadly to different fields, promising in particular for OAM-based photonic applications that require vortex guides, fibers [17,18] and lasers [19]., Comment: 11 pages, 4 figures

The orbital degrees of freedom play a pivotal role in understanding fundamental phenomena in solid-state materials as well as exotic quantum states of matter including orbital superfluidity and topological semimetals. Despite tremendous efforts in engineering synthetic cold-atom, electronic and photonic lattices to explore orbital physics, thus far high orbitals in an important class of materials, namely, the higher-order topological insulators (HOTIs), have not been realized. Here, we demonstrate p-orbital corner states in a photonic HOTI, unveiling their underlying topological invariant, symmetry protection, and nonlinearity-induced dynamical rotation. In a Kagome-type HOTI, we find that topological protection of the p-orbital corner states demands an orbital-hopping symmetry, in addition to the generalized chiral symmetry. Due to orbital hybridization, the nontrivial topology of the p-orbital HOTI is hidden if bulk polarization is used as the topological invariant, but well manifested by the generalized winding number. Our work opens a pathway for the exploration of intriguing orbital phenomena mediated by higher band topology applicable to a broad spectrum of systems., Comment: 11 pages, 4 figures

A hallmark of symmetry-protected topological phases (SPTs) are topologically protected boundary states, which are immune to perturbations that respect the protecting symmetry. It is commonly believed that any perturbation that destroys an SPT phase simultaneously destroys the boundary states. However, by introducing and exploring a weaker sub-symmetry (SubSy) requirement on perturbations, we find that the nature of boundary state protection is in fact more complex. We demonstrate that the boundary states are protected by only the SubSy using prototypical Su-Schrieffer-Heeger (SSH) and breathing Kagome lattice (BKL) models, even though the overall topological invariant and the SPT phase are destroyed by SubSy preserving perturbations. By employing judiciously controlled symmetry breaking in photonic lattices, we experimentally demonstrate such SubSy protection of topological states. Furthermore, we introduce a long-range hopping symmetry in BKLs, which resolves a debate on the topological nature of their corner states. Our results apply to other systems beyond photonics, heralding the possibility of exploring the intriguing properties of SPT phases in the absence of full symmetry in different physical contexts., Comment: Main text & supplementary material

Higher-order topological insulators (HOTIs) are recently discovered topological phases, possessing symmetry-protected corner states with fractional charges. An unexpected connection between these states and the seemingly unrelated phenomenon of bound states in the continuum (BICs) was recently unveiled. When nonlinearity is added to a HOTI system, a number of fundamentally important questions arise. For example, how does nonlinearity couple higher-order topological BICs with the rest of the system, including continuum states? In fact, thus far BICs in nonlinear HOTIs have remained unexplored. Here, we demonstrate the interplay of nonlinearity, higher-order topology, and BICs in a photonic platform. We observe topological corner states which, serendipitously, are also BICs in a laser-written second-order topological lattice. We further demonstrate nonlinear coupling with edge states at a low nonlinearity, transitioning to solitons at a high nonlinearity. Theoretically, we calculate the analog of the Zak phase in the nonlinear regime, illustrating that a topological BIC can be actively tuned by both focusing and defocusing nonlinearities. Our studies are applicable to other nonlinear HOTI systems, with promising applications in emerging topology-driven devices., Comment: Comments are welcome

We demonstrate, both analytically and experimentally, free-space pin-like optical vortex beams (POVBs).Such angular-momentum-carrying beams feature tunable peak intensity and undergo robust anti-diffracting propagation, realized by judiciously modulating both the amplitude and the phase profile of a standard laser beam. Specifically, they are generated by superimposing a radially-symmetric power-law phase on a helical phase structure, which allows the inclusion of an orbital angular momentum term to the POVBs. During propagation in free-space, these POVBs initially exhibit autofocusing dynamics, and subsequently, their amplitude patterns morph into a high-order Bessel-like profile characterized by a hollow-core and an annular main-lobe with a constant or tunable width during propagation. In contrast with numerous previous endeavors on Bessel beams, our work represents the first demonstration of long-distance free-space generation of optical vortex 'pins', with their peak intensity evolution controlled by the impressed amplitude structure. Both the Poynting vectors and the optical radiation forces associated with these beams are also numerically analyzed, revealing novel properties that may be useful for a wide range of applications., Comment: 12 pages, 4 figures

We demonstrate dynamical topological phase transitions in evolving Su-Schrieffer-Heeger (SSH) lattices made of interacting soliton arrays, which are entirely driven by nonlinearity and thereby exemplify emergent nonlinear topological phenomena. The phase transitions occur from topologically trivial-to-nontrivial phase in periodic succession with crossovers from topologically nontrivial-to-trivial regime. The signature of phase transition is gap-closing and re-opening point, where two extended states are pulled from the bands into the gap to become localized topological edge states. Crossovers occur via decoupling of the edge states from the bulk of the lattice., Comment: are welcome!

We predict that Bessel-like beams of arbitrary integer order can exhibit a tunable self-similar behavior (that take an invariant form under suitable stretching transformations). Specifically, by engineering the amplitude and the phase on the input plane in real space, we show that it is possible to generate higher-order vortex Bessel-like beams with fully controllable radius of the hollow core and maximum intensity during propagation. In addition, using a similar approach, we show that it is also possible to generate zeroth order Bessel-like beams with controllable beam width and maximum intensity. Our numerical results are in excellent agreement with our theoretical predictions.

We introduce the concept of spatial spectral phase gradient, and demonstrate, both theoretically and experimentally, how this concept could be employed for generating single- and multi-path self-accelerating beams. In particular, we show that the trajectories of the accelerating beams are determined a priori by different key spatial frequencies through direct spectrum-to-distance mapping. In the non-paraxial regime, our results clearly illustrate the breakup of Airy beams from a different perspective, and demonstrate how circular, elliptic or hyperbolic accelerating beams can be created by judiciously engineering the spectral phase. Furthermore, we found that the accelerating beams still follow the predicted trajectory also for vectorial wavefronts. Our approach not only generalizes the idea of Fourier-space beam engineering along arbitrary convex trajectories, but also offers new possibilities for beam/pulse manipulation not achievable through standard direct real-space approaches or by way of time-domain phase modulation., Comment: 10 pages, 4 figures