The recent discovery of higher-order topological insulators (TIs) [1-5] has opened new possibilities in the search for novel topological materials and metamaterials. Secondorder TIs have been implemented in two-dimensional (2D) systems [6-19] exhibiting topological 'corner states', as well as three-dimensional (3D) systems having one-dimensional (1D) topological 'hinge states' [20]. Third-order TIs, which have topological states three dimensions lower than the bulk (which must thus be 3D or higher), have not yet been reported. Here, we describe the realization of a third-order TI in an anisotropic diamondlattice acoustic metamaterial. The bulk acoustic bandstructure has nontrivial topology characterized by quantized Wannier centers. By direct acoustic measurement, we observe corner states at two corners of a rhombohedron-like structure, as predicted by the quantized Wannier centers. This work extends topological corner states from 2D to 3D, and may find applications in novel acoustic devices.Higher-order TIs are a new class of topological materials supporting a generalization of the bulk-boundary correspondence principle, in which topological states are guaranteed to exist along boundaries two or more dimensions smaller than that of the bulk [1][2][3][4][5]. In standard TIs, topological edge states occur at one lower dimension than the bulk [21,22]; for instance, a quantum Hall insulator has a 2D bulk and topological states on 1D edges. By contrast, a 2D second-order TI supports zero-dimensional (0D) topological 'corner states'. Such a lattice was first devised based on quantized quadrupole moments [1,2] and quickly realized in mechanical [6], electromagnetic [7], and electrical [9] metamaterials. Later, another type of 2D second-order TIs based on quantized Wannier centers, was proposed [23][24][25] and demonstrated in acoustic metamaterials [10,11]. In 3D materials, second-order TI behavior has also been observed in the form of 1D topological 'hinge states' in bismuth [20].According to theoretical predictions, TIs of arbitrarily high order are possible. However, in real materials the bulk is at most 3D. Thus, barring the use of 'synthetic' dimensions [26,27], the only remaining class of high-order TI is a third-order TI with 3D bulk and 0D corner states.As of this writing, no such material has been reported in the literature, although there exists a theoretical proposal based on quantized octupole moments [1,2].Here, we realize a third-order TI in a 3D acoustic metamaterial, observing topological states at the corners of a rhombohedron-like sample. This third-order TI is based on the extension of Wannier-type second-order TIs to 3D [23,25], and can be regarded as a 3D generalization of the classic 1D Su-Schrieffer-Heeger (SSH) model [28]. Just as in the SSH case, the eigenmode polarizations are quantized by lattice symmetries, and the Wannier centers are pinned to highsymmetry points; the mismatch between the Wannier centers and lattice truncations gives rise to charge fractionalization and hence lower-dimensio...

Abstract:In this review paper, we report recent progress on Pancharatnam-Berry (PB) phase optical elements, such as lens, grating, and de ector. PB lenses exhibit a fast switching time between two or more focal lengths with large diopter change and aperture size, which is particularly attractive for addressing the accommodation mismatch in head-mounted display devices. On the other hand, PB gratings and de ectors o er a large-angle beam de ection with wide acceptance cone and high e ciency, as compared to conventional volume gratings. Such merits provide great advantages for waveguide-coupling augmented reality headsets. Moreover, the thickness of PB optical elements is only a few micrometers, thus they can be conveniently integrated into modern wearable display systems.

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The current understanding of topological insulators and their classical wave analogs, such as photonic topological insulators, is mainly based on topological band theory. However, standard band theory does not apply to amorphous phases of matter, which are formed by non-crystalline lattices with no long-range positional order but only shortrange order, exhibiting unique phenomena such as the glass-to-liquid transition. Here, we experimentally investigate amorphous variants of a Chern number-based photonic topological insulator. By tuning the disorder strength in the lattice, we demonstrate that photonic topological edge states can persist into the amorphous regime prior to the glass-to-liquid transition. After the transition to a liquid-like lattice configuration, the signatures of topological edge states disappear. This interplay between topology and short-range order in amorphous lattices paves the way for new classes of non-crystalline topological photonic bandgap materials.

At photonic Dirac points, electromagnetic waves are governed by the same equations as twocomponent massless relativistic fermions. However, photonic Dirac points are known to occur in pairs in "photonic graphene" and other similar photonic crystals, which necessitates special precautions to excite only states near one of the Dirac points. Systems hosting unpaired photonic Dirac points are significantly harder to realize, as they require broken time-reversal symmetry.Here, we report on the first observation of an unpaired Dirac point in a planar two-dimensional photonic crystal. The structure incorporates gyromagnetic materials, which break time-reversal symmetry; the unpaired Dirac point occurs when a parity-breaking parameter is fine-tuned to a topological transition between a photonic Chern insulator and a conventional photonic insulator phase. Evidence for the unpaired Dirac point is provided by transmission and field-mapping experiments, including a demonstration of strongly non-reciprocal reflection. This photonic crystal is suitable for investigating the unique features of two-dimensional Dirac states, such as one-way Klein tunneling.