Polarization and Weak Topology in Chern Insulators.

Chern insulators, and more broadly, topological insulators, present an obstruction to the construction of exponentially localized electronic Wannier functions. This implies a fundamental difficulty in determining whether such insulators exhibit electric polarization. Here, we show that these insulators can indeed exhibit bound charges and adiabatic currents consistent with changes in bulk polarization over space and time, respectively. We also show that the change in polarization across crystalline domains within these strong topological insulators is quantized in the presence of crystalline symmetries.

Realization of a Z-Classified Chiral-Symmetric Higher-Order Topological Insulator in a Coupling-Inverted Acoustic Crystal.

Higher-order topological band theory has transformed the landscape of topological phases in quantum and classical systems. Here, we experimentally demonstrate a two-dimensional higher-order topological phase, referred to as the multiple chiral topological phase, which is protected by a multipole chiral number (MCN). Our realization differs from previous higher-order topological phases in that it possesses a larger-than-unity MCN, which arises when the nearest-neighbor couplings are weaker than long-range couplings. Our phase has an MCN of 4, protecting the existence of 4 midgap topological corner modes at each corner. The multiple topological corner modes demonstrated here could lead to enhanced quantum-inspired devices for sensing and computing. Our study also highlights the rich and untapped potential of long-range coupling manipulation for future research in topological phases.

Photonic quadrupole topological insulator using orbital-induced synthetic flux.

The rich physical properties of multiatomic crystals are determined, to a significant extent, by the underlying geometry and connectivity of atomic orbitals. The mixing of orbitals with distinct parity representations, such as s and p orbitals, has been shown to be useful for generating systems that require alternating phase patterns, as with the sign of couplings within a lattice. Here we show that by breaking the symmetries of such mixed-orbital lattices, it is possible to generate synthetic magnetic flux threading the lattice. We use this insight to experimentally demonstrate quadrupole topological insulators in two-dimensional photonic lattices, leveraging both s and p orbital-type modes. We confirm the nontrivial quadrupole topology by observing the presence of protected zero-dimensional states, which are spatially confined to the corners, and by confirming that these states sit at mid-gap. Our approach is also applicable to a broader range of time-reversal-invariant synthetic materials that do not allow for tailored connectivity, and in which synthetic fluxes are essential.

Observation of Degenerate Zero-Energy Topological States at Disclinations in an Acoustic Lattice.

Building upon the bulk-boundary correspondence in topological phases of matter, disclinations have recently been harnessed to trap fractionally quantized density of states (DOS) in classical wave systems. While these fractional DOS have associated states localized to the disclination's core, such states are not protected from deconfinement due to the breaking of chiral symmetry, generally leading to resonances which, even in principle, have finite lifetimes and suboptimal confinement. Here, we devise and experimentally validate in acoustic lattices a paradigm by which topological states bind to disclinations without a fractional DOS but which preserve chiral symmetry. The preservation of chiral symmetry pins the states at the midgap, resulting in their protected maximal confinement. The integer DOS at the defect results in twofold degenerate states that, due to symmetry constraints, do not gap out. Our study provides a fresh perspective about the interplay between symmetry protection in topological phases and topological defects, with possible applications in classical and quantum systems alike.

Chiral-Symmetric Higher-Order Topological Phases of Matter.

We introduce novel higher-order topological phases of matter in chiral-symmetric systems (class AIII of the tenfold classification), most of which would be misidentified as trivial by current theories. These phases are protected by "multipole chiral numbers," bulk integer topological invariants that in 2D and 3D are built from sublattice multipole moment operators, as defined herein. The integer value of a multipole chiral number indicates how many degenerate zero-energy states localize at each corner of a system. These higher-order topological phases of matter are generally boundary-obstructed and robust in the presence of chiral-symmetry-preserving disorder.

Observation of bound states in the continuum embedded in symmetry bandgaps.

In the past decade, symmetry-protected bound states in the continuum (BICs) have proven to be an important design principle for creating and enhancing devices reliant upon states with high-quality (Q) factors, such as sensors, lasers, and those for harmonic generation. However, as we show, current implementations of symmetry-protected BICs in photonic crystal slabs can only be found at the center of the Brillouin zone and below the Bragg diffraction limit, which fundamentally restricts their use to single-frequency applications. By microprinting a three-dimensional (3D) photonic crystal structure using two-photon polymerization, we demonstrate that this limitation can be overcome by altering the radiative environment surrounding the slab to be a 3D photonic crystal. This allows for the protection of a line of BICs by embedding it in a symmetry bandgap of the crystal. This concept substantially expands the design freedom available for developing next-generation devices with high-Q states.

Point-Defect-Localized Bound States in the Continuum in Photonic Crystals and Structured Fibers.

We show that point defects in two-dimensional photonic crystals can support bound states in the continuum (BICs). The mechanism of confinement is a symmetry mismatch between the defect mode and the Bloch modes of the photonic crystal. These BICs occur in the absence of band gaps and therefore provide an alternative mechanism to confine light. Furthermore, we show that such BICs can propagate in a fiber geometry and exhibit arbitrarily small group velocity which could serve as a platform for enhancing nonlinear effects and light-matter interactions in structured fibers.

Observation of a Higher-Order Topological Bound State in the Continuum.

Higher-order topological insulators are a recently discovered class of materials that can possess zero-dimensional localized states regardless of the dimension of the system. Here, we experimentally demonstrate that the topological corner-localized modes of higher-order topological systems can be symmetry-protected bound states in the continuum; these states do not hybridize with the surrounding bulk states of the lattice even in the absence of a bulk band gap. This observation expands the scope of bulk-boundary correspondence by showing that protected boundary-localized states can be found within topological bands, in addition to being found in between them.

A fractional corner anomaly reveals higher-order topology.

Spectral measurements of boundary-localized topological modes are commonly used to identify topological insulators. For high-order insulators, these modes appear at boundaries of higher codimension, such as the corners of a two-dimensional material. Unfortunately, this spectroscopic approach is only viable if the energies of the topological modes lie within the bulk bandgap, which is not required for many topological crystalline insulators. The key topological feature in these insulators is instead fractional charge density arising from filled bulk bands, but measurements of such charge distributions have not been accessible to date. We experimentally measure boundary-localized fractional charge density in rotationally symmetric two-dimensional metamaterials and find one-fourth and one-third fractionalization. We then introduce a topological indicator that allows for the unambiguous identification of higher-order topology, even without in-gap states, and we demonstrate the associated higher-order bulk-boundary correspondence.

Robust temporal pumping in a magneto-mechanical topological insulator.

The transport of energy through 1-dimensional (1D) waveguiding channels can be affected by sub-wavelength disorder, resulting in undesirable localization and backscattering phenomena. However, quantized disorder-resilient transport is observable in the edge currents of 2-dimensional (2D) topological band insulators with broken time-reversal symmetry. Topological pumps are able to reduce this higher-dimensional topological insulator phenomena to lower dimensionality by utilizing a pumping parameter (either space or time) as an artificial dimension. Here we demonstrate a temporal topological pump that produces on-demand, robust transport of mechanical energy using a 1D magneto-mechanical metamaterial. We experimentally demonstrate that the system is uniquely resilient to defects occurring in both space and time. Our findings open a path towards exploration of higher-dimensional topological physics with time as a synthetic dimension.

Strong Nonreciprocity in Modulated Resonator Chains through Synthetic Electric and Magnetic Fields.

We study nonreciprocity in spatiotemporally modulated 1D resonator chains from the perspective of equivalent 2D resonator arrays with a synthetic dimension and transverse synthetic electric and magnetic fields. The synthetic fields are respectively related to temporal and spatial modulation of the resonator chain, and we show that their combination can induce strong transmission nonreciprocity, i.e., complete isolation with only a weak perturbative modulation. This nonreciprocal effect is analogous to the Hall effect for charged particles. We experimentally implement chains of two and three spatiotemporally modulated resonators and measure over 58 dB of isolation contrast.

A quantized microwave quadrupole insulator with topologically protected corner states.

The theory of electric polarization in crystals defines the dipole moment of an insulator in terms of a Berry phase (geometric phase) associated with its electronic ground state. This concept not only solves the long-standing puzzle of how to calculate dipole moments in crystals, but also explains topological band structures in insulators and superconductors, including the quantum anomalous Hall insulator and the quantum spin Hall insulator, as well as quantized adiabatic pumping processes. A recent theoretical study has extended the Berry phase framework to also account for higher electric multipole moments, revealing the existence of higher-order topological phases that have not previously been observed. Here we demonstrate experimentally a member of this predicted class of materials-a quantized quadrupole topological insulator-produced using a gigahertz-frequency reconfigurable microwave circuit. We confirm the non-trivial topological phase using spectroscopic measurements and by identifying corner states that result from the bulk topology. In addition, we test the critical prediction that these corner states are protected by the topology of the bulk, and are not due to surface artefacts, by deforming the edges of the crystal lattice from the topological to the trivial regime. Our results provide conclusive evidence of a unique form of robustness against disorder and deformation, which is characteristic of higher-order topological insulators.

Quantized electric multipole insulators.

The Berry phase provides a modern formulation of electric polarization in crystals. We extend this concept to higher electric multipole moments and determine the necessary conditions and minimal models for which the quadrupole and octupole moments are topologically quantized electromagnetic observables. Such systems exhibit gapped boundaries that are themselves lower-dimensional topological phases. Furthermore, they host topologically protected corner states carrying fractional charge, exhibiting fractionalization at the boundary of the boundary. To characterize these insulating phases of matter, we introduce a paradigm in which "nested" Wilson loops give rise to topological invariants that have been overlooked. We propose three realistic experimental implementations of this topological behavior that can be immediately tested. Our work opens a venue for the expansion of the classification of topological phases of matter.

Aberration characterization for the optimal design of high-resolution endoscopic optical coherence tomography catheters.

We study the major factors causing degradation in the lateral resolution of gradient-index-lens-based catheters used for high-resolution optical coherence tomography. Chromatic aberration and astigmatism were taken into account in the propagation of broadband single-mode Gaussian beams through the catheter geometry. It was found that, while chromatic aberration did not preclude achieving high resolution, astigmatism posed a major technical difficulty, because its correction requires a very sensitive adjustment of parameters, especially for catheters with long working distances.

Multimodal Nonlinear Microscopy by Shaping a Fiber Supercontinuum From 900 to 1160 nm.

Nonlinear microscopy has become widely used in biophotonic imaging. Pulse shaping provides control over nonlinear optical processes of ultrafast pulses for selective imaging and contrast enhancement. In this study, nonlinear microscopy, including two-photon fluorescence, second harmonic generation, and third harmonic generation, was performed using pulses shaped from a fiber supercontinuum (SC) spanning from 900 to 1160 nm. The SC generated by coupling pulses from a Yb:KYW pulsed laser into a photonic crystal fiber was spectrally filtered and compressed using a spatial light modulator. The shaped pulses were used for nonlinear optical imaging of cellular and tissue samples. Amplitude and phase shaping the fiber SC offers selective and efficient nonlinear optical imaging over a broad bandwidth with a single-beam and an easily tunable setup.

Nonlinear interferometric vibrational imaging for fast label-free visualization of molecular domains in skin.

The most prevalent molecular constituents of skin are spatially mapped by the use of nonlinear interferometric vibrational imaging, a coherent anti-Stokes Raman scattering (CARS)-based technique. Raman-like profiles over the range from 2,800 to 3,000 cm(-1) are acquired by means of completely suppressing the non-resonant background, allowing the generation of images based on the molecule-specific spectral profiles over the probed region with high spatial resolution. A simple algorithm that maps spectral content to color allows the visualization of histology in a manner analogous to that obtained with more conventional staining procedures (e.g., hematoxylin-eosin), but faster and with the benefit of having access to localized spectra, which could further enhance the potential for diagnosis of diseases, especially during the early stages of development.

Molecular histopathology by spectrally reconstructed nonlinear interferometric vibrational imaging.

Sensitive assays for rapid quantitative analysis of histologic sections, resected tissue specimens, or in situ tissue are highly desired for early disease diagnosis. Stained histopathology is the gold standard but remains a subjective practice on processed tissue taking from hours to days. We describe a microscopy technique that obtains a sensitive and accurate color-coded image from intrinsic molecular markers. Spectrally reconstructed nonlinear interferometric vibrational imaging can differentiate cancer versus normal tissue sections with greater than 99% confidence interval in a preclinical rat breast cancer model and define cancer boundaries to ± 100 µm with greater than 99% confidence interval, using fresh unstained tissue sections imaged in less than 5 minutes. By optimizing optical sources and beam delivery, this technique can potentially enable real-time point-of-care optical molecular imaging and diagnosis.

High speed nonlinear interferometric vibrational analysis of lipids by spectral decomposition.

Unlike other CARS-based (coherent anti-Stokes Raman scattering) spectroscopy techniques, nonlinear interferometric vibrational spectroscopy (NIVS) is linear in analyte concentration and has a Raman line shape free of nonresonant background distortions. We use spontaneous Raman scattering as a high accuracy benchmark for NIVS. As a challenging comparison, we examine spectra in the CH stretching region of six lipid samples. Singular value decomposition and reference to an independent chemical assay are used to directly compare NIVS and spontaneous Raman scattering. We demonstrate that NIVS can determine the relative degree of unsaturation in six different lipid samples as accurately as spontaneous Raman spectroscopy, but 200 times faster. A skin tissue sample is mapped out to demonstrate quantitative lipid-protein differentiation with spatial resolution.

High-Speed Nonlinear Interferometric Vibrational Imaging of Biological Tissue With Comparison to Raman Microscopy.

Vibrational contrast imaging of the distribution of complex biological molecules requires the use of techniques that provide broadband spectra with sufficient resolution. Coherent anti-Stokes Raman scattering (CARS) microscopy is currently limited in meeting these requirements due to the presence of a nonresonant background and its inability to target multiple resonances simultaneously. We present nonlinear interferometric vibrational imaging (NIVI), a technique based on CARS that uses femtosecond pump and Stokes pulses to retrieve broadband vibrational spectra over 200 cm(-1) (full-width at half maximum). By chirping the pump and performing spectral interferometric detection, the anti-Stokes pulses are resolved in time. This phase-sensitive detection allows suppression of not only the nonresonant background, but also of the real part of the nonlinear susceptibility χ((3)), improving the spectral resolution and features to make them comparable to those acquired with spontaneous Raman microscopy, as shown for a material sample and mammary tissue.