Geometric Phase Research Articles

Observation of Kekulé vortices around hydrogen adatoms in graphene

Fractional charges are one of the wonders of the fractional quantum Hall effect. Such objects are also anticipated in two-dimensional hexagonal lattices under time reversal symmetry—emerging as bound states of a rotating bond texture called a Kekulé vortex. However, the physical mechanisms inducing such topological defects remain elusive, preventing experimental realization. Here, we report the observation of Kekulé vortices in the local density of states of graphene under time reversal symmetry. The vortices result from intervalley scattering on chemisorbed hydrogen adatoms. We uncover that their 2π winding is reminiscent of the Berry phase π of the massless Dirac electrons. We can also induce a Kekulé pattern without vortices by creating point scatterers such as divacancies, which break different point symmetries. Our local-probe study thus confirms point defects as versatile building blocks for Kekulé engineering of graphene’s electronic structure.

Atomic‐scale strain analysis for advanced Si/SiGe heterostructure by using transmission electron microscopy

AbstractThree‐dimensional stacked transistors based on Si/SiGe heterojunction are a potential candidate for future low‐power and high‐performance computing in integrated circuits. Observing and accurately measuring strain in Si/SiGe heterojunctions is critical to increasing carrier mobility and improving device performance. Transmission electron microscopy (TEM) with high spatial resolution and analytical capabilities provides technical support for atomic‐scale strain measurement and promotes significant progress in strain mapping technology. This paper reviews atomic‐scale strain analysis for advanced Si/SiGe heterostructure based on TEM techniques. Convergent‐beam electron diffraction, nano‐beam electron diffraction, dark‐field electron holography, and high‐resolution TEM with geometrical phase analysis, are comprehensively discussed in terms of spatial resolution, strain precision, field of view, reference position, and data processing. Also, the advantages and critical issues of these strain analysis methods based on the TEM technique are summarized, and the future direction of TEM techniques in the related areas is prospected.

On-chip optical wavefront shaping by transverse-spin-induced Pancharatanam-Berry phase.

Pancharatnam-Berry (PB) metasurfaces can be applied to manipulate the phase and polarization of light within subwavelength thickness. The underlying mechanism is attributed to the geometric phase originating from the longitudinal spin of light. Here, we demonstrate, to the best of our knowledge, a new type of PB geometric phase derived from the intrinsic transverse spin of guided light. Using full-wave numerical simulations, we show that the rotation of a metallic nano-bar sitting on a metal substrate can induce a geometric phase covering 2π full range for the surface plasmons carrying an intrinsic transverse spin. Especially, the geometric phase is different for the surface plasmons propagating in opposite directions due to spin-momentum locking. We apply the geometric phase to design metasurfaces to manipulate the wavefront of surface plasmons to achieve steering and focusing. Our work provides a new mechanism for on-chip light manipulations with potential applications in designing ultra-compact optical devices for imaging and sensing.

High transmission of circularly polarized light and circular dichroism with all-dielectric metamaterial

The optical chirality of metamaterials gives birth to distinct responses of left circularly polarized (LCP) and right circularly polarized (RCP) light, leading to asymmetric transmission and circular dichroism (CD) of circularly polarized light. In this work, meta-atom structure composed of square and semicircular combinations were designed, which exhibits extreme high ratio of transmission, reflection and CD values in blue light range. In compare to other reported structures, the structure composed of semicircular and square combinations can achieve better asymmetric transmission. At the wavelength of 460 nm, the incident LCP light can be mostly converted to RCP light, and the incident RCP light can be almost completely reflected. Furthermore, the geometric phase can be controlled by designing the orientation angle of the meta-atom, and the metamaterial can function under any linear or circular polarization and achieving the regulation of the light field. The designed metamaterial provides promising insights into the high CD and possible applications in optical integrated devices, such as anomalously refracted, focusing, and holograms, etc.

High efficiency independent modulation at dual-wavelength based on Pancharatnam–Berry and propagation phases

Metasurfaces capable of controlling multiple wavelengths independently have attracted broad interests these years due to their significance in multi-channel information processing applications. Previous solving strategies include spatial multiplexing or extensive searching for appropriate structures, both of which have their own disadvantageous, such as low efficiency, large computer resource requirement, or time consumption. In this paper, by combining the Pancharatnam–Berry (PB) phase and propagation phase, we propose a strategy to simplify the design complexity in a dual-wavelength metasurface system, in which two simple rectangular-shaped dielectric pillars (T1 and T2) with different aspect ratios are chosen as basic structures and crossed at the geometric center to achieve manipulation. The larger pillar T2 controls the longer wavelength through the PB phase while the smaller T1 acts as a perturbation to T2. The crossed T1&T2 is studied as a whole to tune the short wavelength. The investigations by the multipole expansion method reveal that the polarization conversion ratio of the meta-atoms is dependent on the interference of the formed multipoles. To validate the proposed strategy, a dual-wavelength achromatic metalens and a wavelength-multiplexed holographic metasurface operating at the infrared thermal imaging band are designed. Our design strategy can find widespread applications in metasurfaces where multiple objectives are required to be realized.

Multifunctional beam modulation based on gratings combined with additive metasurfaces

A design scheme combining grating and element additive metasurface is proposed in this paper. Multi-dimensional joint control of the phase, amplitude, angle and polarization and multi-functional integrated metasurface applications were realized by using the inherent characteristics of the light wave properties (polarization and wavelength). Firstly, suitable structural parameters are obtained based on the reflection response of the unit structure, and a 2-bit encoded metasurface array is constructed by utilizing the combinatorial operation between the unit structure. Combined with the polarization selectivity of the lower grating, the dual-function integration of asymmetric transmission (the contrast ratio is 98%) and abnormal reflections (operating bandwidth is 300 nm) was realized. Next, two new half-wave plates were obtained by addition and rotating the elemental structure, and the polarization conversion efficiency of 95%–97% was achieved thanks to the Fabry–Perot cavity formed by the upper metasurface and lower grating. Finally, encoded metasurface based on the geometric phase principle to achieve 300 nm bandwidth cross-polarized wave deflection characteristics with polarization conversion efficiency close to 100%. Our work combines grating structures with element additive metasurfaces, which solves the problem that element structure needs to be redesigned with the change of function, and also paves the way for the application of multifunctional integrated metasurfaces in optical sensing, optical imaging and other fields.

Geometric optimisation of a multiple coiled-up resonator for broadband and octave band acoustic absorption

This study presents a comprehensive approach to designing sound absorption metamaterials, aiming to achieve broadband absorption capabilities while maintaining a compact thickness, starting from an exceptionally low frequency of 100 Hz. By employing a parallel arrangement of resonators with varying lengths, the proposed approach exploits the envelopes of absorption frequencies to realize broad-spectrum sound absorption.An analytical model is utilized in the geometric optimization phase, accounting for inlet losses and thermo-viscous losses near the walls. The absorption for multiple ducts coupled in parallel is derived using the series and parallel impedance method. A geometric parameterization procedure is implemented to maximize normal incidence acoustic absorption, leading to design optimization for both broadband and octave-band absorbers. Design maps are established to assess acoustic performance and physical space requirements, facilitating informed decision-making in metamaterial design.Prototype fabrication and experimental validation involve 3D printing of optimized geometries using polylactic acid, followed by testing within impedance tubes according to ISO 10534–2 standards. The results demonstrate the effectiveness of the proposed method in achieving broadband sound absorption across specified frequency ranges.

Atomic-Scale Multimodal Characterization of Self-Assembled InAs/InGaAlAs Quantum Dots.

Self-assembled quantum dots (QDs) are potential candidates for photoelectric and photovoltaic devices, because of their discrete energy levels. The characterization of QDs at the atomic level using a multimodal approach is crucial to improving device performance because QDs are nanostructures with highly correlated structural parameters. In this study, scanning transmission electron microscopy, geometric phase analysis, and atom probe tomography were employed to characterize structural parameters such as the shape, strain, and composition of self-assembled InAs-QDs with InGaAlAs spacer layers. The measurements revealed characteristic AlAs-rich regions above the QDs and InAs-rich regions surrounding the QD columns, which can be explained by the relationship between the effect of strain and surface curvature around the QD. The methodology described in this study accelerates the development of future QD devices because its multiple perspectives reveal phenomena such as atomic-scale segregations and allow for more detailed discussions of the mechanisms of these phenomena.

Exceptional classifications of non-Hermitian systems

Eigenstate coalescence in non-Hermitian systems is widely observed in diverse scientific domains encompassing optics and open quantum systems. Recent investigations have revealed that adiabatic encircling of exceptional points (EPs) leads to a nontrivial Berry phase in addition to an exchange of eigenstates. Based on these phenomena, we propose in this work an exhaustive classification framework for EPs in non-Hermitian physical systems. In contrast to previous classifications that only incorporate the eigenstate exchange effect, our proposed classification gives rise to finer Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb{Z}}}_{2}$$\\end{document} classifications depending on the presence of a π Berry phase after the encircling of the EPs. Moreover, by mapping arbitrary one-dimensional systems to the adiabatic encircling of EPs, we can classify one-dimensional non-Hermitian systems characterized by topological phase transitions involving EPs. Applying our exceptional classification to various multiband systems, we expect to enhance the understanding of topological phases in non-Hermitian systems.

Nonadiabatic Conical Intersection Dynamics in the Local Diabatic Representation with Strang Splitting and Fourier Basis.

We develop and implement an exact conical intersection nonadiabatic wave packet dynamics method that combines the local diabatic representation, Strang splitting for the total molecular propagator, and discrete variable representation with uniform grids. By employing the local diabatic representation, this method captures all nonadiabatic effects, including nonadiabatic transitions, electronic coherences, and geometric phase. Moreover, it is free of singularities in the first and second derivative couplings and does not require the electronic wave function to be continuous with respect to the nuclear coordinates. We further show that in contrast to the adiabatic representation, the split-operator method can be directly applied to the full molecular propagator with the locally diabatic ansatz. The Fourier series, employed as the primitive nuclear basis functions, is universal and can be applied to all types of reactive coordinates. The combination of local diabatic representation, Strang splitting, and Fourier basis allows numerically exact modeling of conical intersection quantum dynamics directly with adiabatic electronic states that can be obtained from standard electronic structure computations.

Beyond braid statistics: Constructing a lattice model for anyons with exchange statistics intrinsic to one dimension

Anyons obeying fractional exchange statistics arise naturally in two dimensions: Hard-core two-body constraints make the configuration space of particles not simply-connected. The braid group describes how topologically-inequivalent exchange paths can be associated to non-trivial geometric phases for abelian anyons. Braid-anyon exchange statistics can also be found in one dimension (1D), but this requires broken Galilean invariance to distinguish different ways for two anyons to exchange. However, recently it was shown that an alternative form of exchange statistics can occur in 1D because hard-core three-body constraints also make the configuration space not simply-connected. Instead of the braid group, the topology of exchange paths and their associated non-trivial geometric phases are described by the traid group. In this article we propose a first concrete model realizing this alternative form of anyonic exchange statistics. Starting from a bosonic lattice model that implements the desired geometric phases with number-dependent Peierls phases, we then define anyonic operators so that the kinetic energy term in the Hamiltonian becomes local and quadratic with respect to them. The ground-state of this traid-anyon-Hubbard model exhibits several indications of exchange statistics intermediate between bosons and fermions, as well as signs of emergent approximate Haldane exclusion statistics. The continuum limit results in a Galilean invariant Hamiltonian with eigenstates that correspond to previously constructed continuum wave functions for traid anyons. This provides not only an a-posteriori justification of our lattice model, but also shows that our construction serves as an intuitive approach to traid anyons, i.e. anyons intrinsic to 1D.

3D nano-printed geometric phase metasurfaces for generating accelerating beams with complex amplitude manipulation

3D nano-printed geometric phase metasurfaces for generating accelerating beams with complex amplitude manipulation

Geometric phase for two-mode entangled squeezed-coherent states

In this paper, we investigate the geometric phase (GP) of two-mode entangled squeezed-coherent states (ESCSs), undergoing unitary cyclic evolution. Results show that increasing the squeezing parameter of either mode of the balanced ESCS compresses the GP elliptically with respect to the coherence parameter of the corresponding mode. While in the case of unbalanced ESCS, the GP is compressed hyperbolically by increasing the squeezing parameters of the either mode. By generalizing the approach to higher constituting-state dimensions, it is found that the GPs of both balanced and unbalanced ESCSs, increase for a specific value of the coherence parameter. By analyzing the states through the Schmidt decomposition method, we find that, locally, the balanced and unbalanced ESCSs are unitarily equivalent. Finally, based on the interferometry technique, we suggest a theoretical scheme for the physical generation of ESCSs.

Longitudinal encoding of focused vector vortex beams empowered by silicon metasurfaces

Vector vortex beams (VVBs) have attracted considerable applications due to their unique properties, such as vortex filtering, particle trapping, and quantum information. Recently, information coding methods based on metasurface strategies show excellent performance in remote sensing and communication, but the implementation of the longitudinal dimension of the optical field in information coding remains to be developed. Here, we provide a strategy for constructing all-dielectric terahertz (THz) metasurface designs for longitudinal codecs, with the individual focal planes carrying specific encrypted information. Benefiting from the joint modulation of the geometrical and propagation phases, the physical mechanism of spin-polarization multiplexing realizes the coherent superposition of multidimensional focused orbital angular momentum (OAM) modes. As a bridge between Poincaré sphere (PS) and the higher-order Poincaré sphere (HOPS), the proposed design produces a longitudinally controllable variation of OAM mode superposition states. A series of metasurface examples for generating specific coding sequences are characterized by manipulating topological charges and phase factors embedded within orthogonal circularly polarized channels. The longitudinal dimension, as a new coding degree of freedom, can be further expanded for meta-optical applications with the assistance of the VVB pattern.

Tunable broadband terahertz graphene metasurface for complex-amplitude vortex beam generator and hologram

The independent controls of phase and amplitude, known as complex-amplitude modulation, play a crucial role for modulating the electromagnetic wavefronts. In this paper, a tunable broadband terahertz (THz) graphene metasurface with complex-amplitude modulation is proposed. A novel hybrid modulation mechanism that combines the absorption, polarization conversion, and Pancharatnam–Berry (PB) phase is employed to achieve a wide complex-amplitude modulation range over a broad frequency band. In addition, the metasurface can be dynamically tuned from the electrically tunable nature of the graphene in the THz range. To reveal the tunability characteristics of the metasurface as the graphene chemical potential varies, we study the Kubo formula of the graphene conductivity model and combine it with the equivalent circuit model. Two THz graphene meta-devices are designed for complex-amplitude vortex beam generator and hologram. By varying the graphene chemical potentials of 0 and 1 eV as the ON and OFF states, a THz quad-vortex-beam generator with switchable modes is designed and verified. We also demonstrate a THz hologram with complex-amplitude modulation by adjusting the chemical potential, exhibiting low imaging noise and high image quality. The proposed method is promised to be applied to the design of more complex-amplitude THz graphene meta-devices.

Nanoscale surface metrology with a liquid crystal-based phase-shifting angular shearing interferometer.

In this Letter, a phase-shifting angular shearing interferometer has been proposed for the application in optical surface metrology (SM) by using a combination of a wedge-shaped liquid crystal (LC) cell and a polarization phase shifter. The demonstration of this angular shearing interferometer for step height measurement is accomplished with the help of a phase-shifting technique. Four phase-shifted interferograms produced by a geometrical phase shifter are subjected to a simplified Wiener deconvolution method, which resembles a simple analysis technique for shearing interferograms in comparison to alternative approaches. A simulation study has been conducted to validate the proposed technique. The experimental results show an accuracy of 5.56% for determining the step height, which also agrees with the results obtained by atomic force microscopy. Owing to the tunability of birefringence, the proposed LC-based angular shearing interferometry technique will be useful to control spatial resolution in optical metrology.

Computational adaptive telescope imaging via self-interference digital holography

Conventional optical telescope only collect the amplitude of optical wavefront with incoherent light, while self-interference incoherent digital holography offers a unique method of coding both the optical wavefront amplitude and phase information by interference patterns. We developed an adaptive non-scanning three-dimensional (3D) telescopic imaging technique based on self-interference incoherent holography with a geometric phase lens (GP-SIDH). The phase recorded in the point spread hologram and the incoherent self-interference extended holograms are employed to detect spatial positions, correct optical aberration and extract edges by computational reconstruction. The 3D imaging and position detection performances of our proposed method are investigated. Both simulations and experimental results show that the proposed method exhibits better performances for different depth objects imaging and space position detection. Furthermore, this system can be easily modularization and implanted into any existing optical telescopic system owing to its simplicity and convenience. The proposed approach can be a powerful tool for the multi-objects long-working-distance imaging, which is more valuable for telescope application.

Spin Unlocked Vortex Beam Generation on Nonlinear Chiroptical Metasurfaces.

Optical vortices with spin and orbital angular momentum (SAM and OAM) states offer multiple degrees of freedom for manipulating optical fields and thus enable great potentials in optical information processing. Recently, the optical metasurface has become an important platform for vortex beam generation and steering. However, the strong spin-orbit interaction on such metasurfaces usually leads to spin locked OAM generation, which limits the complete control of the angular momentum state of light. Here, we propose to solve this constraint using geometric phase controlled nonlinear chiroptical metasurfaces. The metasurface consists of two types of plasmonic meta-atoms which have opposite handedness and exhibit a strong spin-dependent circular dichroism effect. By encoding specific phase singularities and phase gradients to different channels, we experimentally demonstrate the spin unlocked second harmonic beam steering. The proposed nonlinear chiroptical metasurfaces may have important applications in developing multifunctional nonlinear optical devices.

Q&A of liquid crystal alignment: theory and practice

We selected and summarized the most important aspects of liquid crystal alignment. Alignment by the surface profile and flat surface alignment are discussed as the main factors leading to the high-quality orientation of liquid crystals. The advanced level of understanding and formulation allows linking the alignment layer properties to the molecular structure of the alignment material. The paper reviews and combines the latest knowledge about liquid crystal alignment, dwells upon the relationship between the polar and azimuthal anchoring energy, explains the major particularities regarding anchoring measurements in the case of birefringent alignment materials with truly high azimuthal anchoring energy, and observes the current trends of water-friendly alignment materials and their application in geometrical phase device fabrication.

Ultrafast quantum control of atomic excited states via interferometric two-photon Rabi oscillations

Quantum-state manipulation through coherent interaction with a radiation field is a fundamental process with broad implications in quantum optics and quantum information processing. However, current quantum control methods are limited by their operation at Rabi frequencies below the gigahertz range, which restricts their applicability to systems with long coherence times. To overcome this limitation, alternative approaches utilizing ultrafast driving lasers have garnered great interest. In this work, we demonstrate two-photon Rabi oscillations in the excited states of argon operating at terahertz frequencies driven by ultrafast laser pulses. Leveraging quantum-path interferometry, we are able to measure and manipulate both the amplitudes and phases of the transition dipoles by exploiting the intensity and polarization state of the driving laser. This precise control enables femtosecond population transfer and coherent accumulation of geometric phase. Our findings provide valuable insights into the all-optical manipulation of extreme-ultraviolet radiations and demonstrate the possibility of ultrafast quantum control through interferometric multiphoton transitions.