Topological Charge Number Research Articles

Ideal unconventional charge-2 Dirac fermions in an ultralightweight chiral crystal

When topology meets chirality, it is of great interest to search for possible topological properties in chiral crystals, thereby achieving the effect of making $1+1$ greater than 2. Here, we accomplish a complete list of topological charge-2 (C-2) Dirac fermions based on 65 S\"ohncke space groups. Such emergent fermions occurring in chiral crystals enjoy a higher topological charge number and exhibit more unique characteristics compared to C-0 cases, including multiple Fermi arcs extending to most of the Brillouin zone. For material realization, we propose that an ultralightweight chiral crystal ${\mathrm{N}}_{4}\mathrm{Cl}$ is a high-quality C-2 Dirac semimetal, where the seductive nodal points reside at the Fermi level and exhibit a large linear energy range to overcome the interruption of irrelevant bands. More interestingly, the doubly degenerate Landau levels are described by a Kramers-like degeneracy in such a C-2 Dirac fermion, featuring a doubly degenerate zero mode. Furthermore, the topological phase transitions from Dirac to Weyl states of ${\mathrm{N}}_{4}\mathrm{Cl}$ can be manipulated by applying shear strains. These fascinating findings will certainly evoke continued exploration of unconventional Dirac semimetals, in turn leading to extensive application prospects and uncommon insights.

Topological Faraday Effect for Optical Vortices in Magnetic Films.

Here we experimentally demonstrate the topological Faraday effect-the polarization rotation caused by the orbital angular momentum of light. It is found that the Faraday effect of the optical vortex beam passing through a transparent magnetic dielectric film differs from the Faraday effect for a plane wave. The additional contribution to the Faraday rotation depends linearly on the topological charge and radial number of the beam. The effect is explained in terms of the optical spin-orbit interaction. These findings underline the importance of using the optical vortex beams for studies of magnetically ordered materials.

40.1: Liquid Crystal Polymer Phase Plates Fabricated by Direct Laser Writing

Optical vortex is a type of structured beam that carry orbital‐angular momentum (OAM) with different topological charge numbers. In this paper, we fabricate liquid crystal polymer phase plates to generate vortex beams by femtosecond 3D direct laser writing. Instead of using traditional photoresists, we compound liquid crystal monomers to create different structures. As a two‐photon direct laser writing technique, Nanoscribe is of high resolution of 100 nm, which implements high accuracy of our structures. We design spiral phase plate (SPP) with diameter of 25µm and a total height difference h of 1.25um to generate vortex beams with different topological charge numbers. We also set up the interference system to characterize the vortex beam. With the interference beam coupled, the presence of the vortex is experimentally indicated by the fringe “dislocation defects” at the center of the vortex in visible region. It presents a new way to fabricate liquid crystal polymer structures with unique birefringent properties, indicating new possibilities in photonics application.

Differential Frequency Exploration of Vortex Light in Lithium Niobate Crystals

In recent years, Orbital Angular Momentum (OAM) beams have been applied in optical communications to improve channel capacity and spectral efficiency. However, in practical applications, OAM information is often imprinted on short-wavelength light beams. How to completely transfer this information to the O-band to achieve long-distance transmission has not been conveniently achieved through most traditional methods. We studied the differential frequency experiment of OAM-carrying beams from both theoretical and experimental facets. In the periodic polarization 0 class matched lithium niobate crystal, the difference in frequency between the incident 1950 nm strong pump light and the 780 nm weak input light is achieved, resulting in output light in the O band. The polarization period of the crystal is 20 μm, and the best phase matching is achieved when the temperature is maintained at 41.2 °C. At this time, 780 nm vortex light produces 1300 nm vortex light, and the nonlinear conversion efficiency reaches 0.1387% (topological charge number l = 5). During the experiment, momentum, energy, and topological charge are all conserved. Our experiment successfully converted vortex light at 780 nm into vortex light at 1300 nm, paving the way for the subsequent conversion of 780 nm single photons generated by quantum dots carrying OAM into OAM photons in the communication band.

Optical memory for arbitrary perfect Poincaré states in an atomic ensemble.

Inherent spin angular momentum (SAM) and orbital angular momentum (OAM), which manifest as polarization and spatial degrees of freedom (DOFs) of photons, hold a promise of large capability for applications in classical and quantum information processing. To enable these photonic spin and orbital dynamic properties strongly coupled with each other, Poincaré states have been proposed and offer advantages in data multiplexing, information encryption, precision metrology, and quantum memory. However, since the transverse size of Laguerre-Gaussian beams strongly depends on their topological charge numbers | l |, it is difficult to store asymmetric Poincaré states due to the significantly different light-matter interaction for distinct spatial modes. Here, we experimentally realize the storage of perfect Poincaré states with arbitrary OAM quanta using the perfect optical vortex, in which 121 arbitrarily selected perfect Poincaré states have been stored with high fidelity. The reported work has great prospects in optical communication and quantum networks for dramatically increased encoding flexibility of information.

Asymmetric spin splitting of Laguerre-Gaussian beams in chiral PT-symmetric metamaterials.

We systematically study the spin Hall effect of light (SHEL) in chiral PT-symmetric metamaterials when Laguerre Gaussian beams (LG beams) are incident and discover that cross-polarization (rs p, rp s) and intrinsic orbital angular momentum (IOAM) result in an asymmetric splitting of left-spin circularly polarized (LCP) light and right-spin circularly polarized (RCP) light. Additionally, there are spin Hall shift peaks near |rpp | ≪ |rss | (rs s and rp p are Fresnel reflection coefficients). Altering the topological charge number ℓ, the chiral parameter κ, the dimensionless frequency M, and the incident angle θ may also influence the asymmetric spin splitting and displacement peak. We believe that this research will provide new ways to manipulate and enhance the asymmetric spin splitting of light and provide new applications for spin photonic devices.

From concept to reality: computing visual vortex beam interferometer for displacement measurement.

In addition to the concept of picometer resolution, we discuss macro displacement measurement with a vortex beam interferometer. Three factors limiting large displacement measurement are resolved. Small topological charge numbers promise both high sensitivity and large displacement measurements. With a computing visual method, a virtual moiré pointer image immune to beam misalignment is proposed to calculate displacements. Interestingly, the absolute benchmark is found for cycle counting in the moiré pointer image of fractional topological charge. The vortex beam interferometer would not stop at the tiny displacement measurement in simulations. We report experimental measurements of nanoscale to hundred millimeter displacement in a vortex beam displacement measurement interferometer (DMI) for the first time, to the best of our knowledge.

Controllable Self‐Focusing Circular Vortex Pearcey Gaussian Beam with Low Spatial Coherence

AbstractA new type of controllable self‐focusing circular vortex Pearcey Gaussian beam, which can create an optical potential well with adjustable depth and width at the center of the beam by adjusting the coherent width and the number of topological charges, is introduced. As the coherent width increases, the intensity becomes lower in the dark notch caused by the topological charges, which are consistent with the experimental results. In addition, the self‐focusing location and the strength of the circular vortex Pearcey Gaussian beam can be controlled by the spatial coherence in the simulation.

Circularly Polarized Antenna Array with Decoupled Quad Vortex Beams.

Achieving multiple vortex beams with different modes in a planar microstrip array is pivotal, yet still extremely challenging. Here, a hybrid method combining both Pancharatnam−Berry (PB) phase that is induced by the rotation phase and excitation phase of a feeding line has been proposed for decoupling two orthogonal circularly polarized vortex beams. Theoretical analysis is derived for array design to generate quad vortex beams with different directions and an arbitrary number of topological charges. On this basis, two 8 × 8 planar arrays were theoretically designed in an X band, which are with topological charges of l1 = −1, l2 = 1, l3 = −1, and l4 = 1 in Case I and topological charges of l1 = −1, l2 = 1, l3 = −1, and l4 = 1 in Case II. To further verify the above theory, the planar array in Case I is fabricated and analyzed experimentally. Dual-LP beams are realized by using rectangular patch elements with two orthogonally distributed feeding networks on different layers based on two types of feeding: proximity coupling and aperture coupling. Both the numerical simulation and experimental measurement results are in good agreement and showcase the corresponding quad-vortex-beam characteristics within 8~12 GHz. The array achieves a measured S11 < −10 dB and S22 < −10 dB bandwidth of more than 33.4% and 29.2%, respectively. In addition, the isolation between two ports is better than −28 dB. Our strategy provides a promising way to achieve large capacity and high integration, which is of great benefit to wireless and radar communication systems.

Experimental Realization of Weyl Exceptional Rings in a Synthetic Three-Dimensional Non-Hermitian Phononic Crystal.

Weyl points-topological monopoles of quantized Berry flux-are predicted to spread to Weyl exceptional rings in the presence of non-Hermiticity. Here, we use a one-dimensional Aubry-Andre-Harper model to construct a Weyl semimetal in a three-dimensional parameter space comprising one reciprocal dimension and two synthetic dimensions. The inclusion of non-Hermiticity in the form of gain and loss produces a synthetic Weyl exceptional ring (SWER). The topology of the SWER is characterized by both its topological charge and non-Hermitian winding numbers. We experimentally observe the SWER and synthetic Fermi arc in a one-dimensional phononic crystal with the non-Hermiticity introduced by active acoustic components. Our findings pave the way for studying the high-dimensional non-Hermitian topological physics in acoustics.

Single-Frame Characterization of Ultrafast Pulses with Spatiotemporal Orbital Angular Momentum.

Light that carries spatiotemporal orbital angular momentum (ST-OAM) makes possible new types of optical vortices arising from transverse OAM. ST-OAM pulses exhibit novel properties during propagation, transmission, refraction, diffraction, and nonlinear conversion, attracting growing experimental and theoretical interest and studies. However, one major challenge is the lack of a simple and straightforward method for characterizing ultrafast ST-OAM pulses. Using spatially resolved spectral interferometry, we demonstrate a simple, stationary, single-frame method to quantitatively characterize ultrashort light pulses carrying ST-OAM. Using our method, the presence of an ST-OAM pulse, including its main characteristics such as topological charge numbers and OAM helicity, can be identified easily from the unique and unambiguous features directly seen on the raw data—without any need for a full analysis of the data. After processing and reconstructions, other exquisite features, including pulse dispersion and beam divergence, can also be fully characterized. Our fast characterization method allows high-throughput and quick feedback during the generation and optical alignment processes of ST-OAM pulses. It is straightforward to extend our method to single-shot measurement by using a high-speed camera that matches the pulse repetition rate. This new method can help advance the field of spatially and temporally structured light and its applications in advanced metrologies.

Generation of elliptical airy vortex beams based on all-dielectric metasurface

Elliptical airy vortex beams (EAVBs) can spontaneously form easily identifiable topological charge focal spots. They are used for topological charge detection of vortex beams because they have the abruptly autofocusing properties of circular airy vortex beams and exhibit unique propagation characteristics. We study the use of the dynamic phase and Pancharatnam–Berry phase principles for generation and modulation of EAVBs by designing complex-amplitude metasurface and phase-only metasurface, at an operating wavelength of 1500 nm. It is found that the focusing pattern of EAVBs in the autofocusing plane splits into |m|+1 tilted bright spots from the original ring, and the tilted direction is related to the sign of the topological charge number m. Due to the advantages of ultra-thin, ultra-light, and small size of the metasurface, our designed metasurface device has potential applications in improving the channel capacity based on orbital angular momentum communication, information coding, and particle capture compared to spatial light modulation systems that generate EAVBs.

Propagation Characteristics of Circular Airy Vortex Beams in a Uniaxial Crystal along the Optical Axis

Circular airy vortex beams (CAVBs) have attracted much attention due to their “abruptly autofocusing” effect, phase singularity, and their potential applications in optical micromanipulation, communication, etc. In this paper, we numerically investigated the propagation properties of circular airy beams (CABs) imposed with different optical vortices (OVs) along the optical axis of a uniaxial crystal for the first time. Like other common beams, a left-hand circular polarized (LHCP) CAVB, propagating along the optical axis in a uniaxial crystal, can excite a right-hand circular polarized (RHCP) component superimposed with an on-axis vortex of topological charge (TC) number of 2. When the incident beam is an LHCP CAB imposed with an on-axis vortex of TC number of l = 1, both of the two components have an axisymmetric intensity distribution during propagation and form hollow beams near the focal plane because of the phase singularity. The phase pattern shows that the LHCP component carries an on-axis vortex of TC number of l = 1, while the RHCP component carries an on-axis vortex of TC number of l = 3. With a larger TC number (l = 3), the RHCP component has a larger hollow region in the focal plane compared to the LHCP component. We also studied cases of CABs imposed with one and two off-axis OVs. The off-axis OV makes the CAVB’s profile remain asymmetric throughout the propagation. As the propagation distance increases, the off-axis OVs move near the center of the beam and overlap, resulting in a special intensity and phase distribution near the focal plane.

Propagation properties of partially coherent radially and azimuthally polarized vortex beams in turbulent atmosphere

Analytical formulas for the average intensity of the partially coherent radially and azimuthally polarized vortex (PCRPV and PCAPV) beams propagating through the turbulent atmosphere are derived based on the extended Huygens-Fresnel principle. The effect of the different topological charge number m of PCRPV beam and PCAPV beam in turbulence on average intensity and degree of hollowness (DOH) has been studied in detail. It can be found that PCRPV beam and PCAPV beam in turbulence with the larger topological charge m can greatly keep dark hollow constructure on propagation. And we have studied the orientation angles of the polarization ellipse of the PCRPV and PCAPV beams with different m in atmospheric turbulence. The results show that the capacity to keep radial polarization decline with the increase of topological charge m. In addition, we combined the complex screen method and multi-phase screens method to simulate the propagation of PCRPV beam and PCAPV beam in atmospheric turbulence for verifying our theoretical results. It is shown that the simulated results are consistent with theoretical results. Our study could be important for free-space optical communications.

Topological charges of fullerenes

We defined the topological charge and bump number for fullerenes. All even fullerene isomers from C_{20} up to C_{70} were constructed. After optimizing the geometry of the molecules the relation between the topological charge, the bump number and the atomization energy were analysed.

Collapse Dynamics of Vortex Beams in a Kerr Medium with Refractive Index Modulation and PT-Symmetric Lattices

Using the two-dimensional nonlinear Schrödinger equation, the collapse dynamics of vortex beams in a Kerr medium with refractive index modulation and parity–time (PT) symmetric lattices are explored. The critical power for the collapse of vortex beams in a Kerr medium with real optical lattices (i.e., refractive index modulation lattices) was obtained and discussed. Numerical calculations showed that the number of self-focusing points, the locations of the collapse, and the propagation distances for collapse are sensitively dependent on the modulation factors, topological charge numbers, and initial powers. When the vortex optical field propagates in a Kerr medium with real optical lattices, the optical field will collapse into a symmetrical shape. However, the shape of the vortex beam will be chaotically distorted and collapse in asymmetric patterns during propagation in a Kerr medium with PT-symmetric lattices because of the presence of the complex refraction index. Introducing PT-symmetric lattices into nonlinear Kerr materials may offer a new approach to controlling the collapse of vortex beams.

Propagation dynamics of vector vortex beams in a strongly nonlocal nonlinear medium with parity-time-symmetric potentials

We demonstrate the dynamical properties of a vector vortex optical field (VVOF) in a strongly nonlocal nonlinear medium (SNNM) with sine and cosine parity-time-symmetric potentials (SCPT) by using the coupled vector Snyder-Mitchell model. Our study shows that the shape of the optical field is chaotically distorted in different propagation distances due to the modulation of complex refractive index. Despite the distorted optical field, the VVOF reciprocally evolves in a periodic stretch and shrink behavior during propagation in the SNNM-SCPT. The reciprocal conversions between the linear and circular polarizations periodically occur during propagation. The evolution of VVOF and the linear and circular polarization conversions are strongly dependent on the modulation of the complex refractive index, the initial powers and the vortex topological charge numbers. These results can provide a new way to complexly manipulate the VVOF in a SNNM-SCPT.

Security-enhanced multiple-image encryption based on modified iterative phase retrieval algorithm with structured phase mask in Fresnel domain

A security-enhanced multiple-image encryption method is proposed based on modified iterative phase retrieval algorithm with structured phase mask in Fresnel domain in this paper, where each plaintext is encoded to only one independent phase-only mask with the help of the used structured phase mask, and a group of phase-only masks retrieved by multiple plaintexts are integrated into the ciphertext with the noise-like distribution by using the phase mask multiplexing technique. In the proposed method, the Fresnel diffraction distances and the wavelength of light as well as the optical parameters of the structured phase mask such as focal length, topological charge number are all used as digital keys to expand the key space. Most importantly, the sensitivities of the digital keys are substantially improved by mapping the digital keys with the initial values of chaotic system thanks to the nonlinear, pseudo-random, high sensitivity of initial value of special phase generated by chaotic system, contributing to an extremely high security of multiple-image encryption method. Moreover, computational simulations are performed to demonstrate the feasibility of the proposed security-enhanced multiple-image encryption method, and the simulation results confirm that the proposed method exhibits high effectiveness and strong robustness.

Switchable frequency terahertz vortex beam generator

Most of the reported vortex beam generators generate vortex beams at a fixed frequency, which limits the practical applications. Therefore, it is inevitable to explore a vortex beam generator, which can actively control the operating frequency. We propose a switchable frequency terahertz vortex beam metasurface, it is freely switchable under single-frequency mode and dual-frequency mode by changing the external temperature, the phase state of vanadium dioxide (VO&lt;sub&gt;2&lt;/sub&gt;) is also changeable. External temperature changes will cause VO&lt;sub&gt;2&lt;/sub&gt; to transform from insulating state to metallic state. Generally, VO&lt;sub&gt;2&lt;/sub&gt; conductivity can increase by several orders of magnitude as operating temperature changes. By using the phase change property of VO&lt;sub&gt;2&lt;/sub&gt;, we can obtain a metasurface with switchable operating frequencies. For operating at room temperature, the proposed metasurface behaves as a single-frequency terahertz vortex generator. When (left-handed circularly polarized, LCP) terahertz wave is vertically incident on the metasurface, it generates vortex beams with different topological charge numbers at a frequency of 1.1 THz, and the mode purity is above 85%. The simulation results show that the mode purity of the vortex beam with the topological charge &lt;i&gt;l&lt;/i&gt; = 1 is 90%, and the mode purity is about 91.1% for the vortex beam with &lt;i&gt;l&lt;/i&gt; = 2, and 85.4% for the vortex beam with &lt;i&gt;l&lt;/i&gt; = 3. When the external temperature is of 68 ℃, the designed metasurface becomes a dual-frequency vortex beam generator. At this time, the operating frequencies of vortex beams with different topological charges (&lt;i&gt;l&lt;/i&gt; = 1, 2, 3) are 0.7 and 1.23 THz, whose mode purities are both above 60%. That is to say, the corresponding mode purities at topological charge with&lt;i&gt; l&lt;/i&gt; = 1 for two operating frequencies are 89.1% and 71.6%, respectively. The mode purities are 83.2% and 94.4% with topological charge &lt;i&gt;l&lt;/i&gt; = 2, respectively. The mode purities are 62.4% and 68.2% with topological charge&lt;i&gt; l&lt;/i&gt; = 3, respectively. Therefore, the proposed switchable frequency terahertz vortex generator provides a new design idea for working frequency modulation in wireless terahertz communication.

Nonreciprocal transmission of vortex beam in double Laguerre-Gaussian rotational cavity system

By constructing an optorotational system composed of two linearly coupled Laguerre-Gaussian rotational cavities, we realize the nonreciprocal transmission of the vortex beam with the orbital angular momentum. Two vortex beam cavity modes driven by strong fields are coupled with a rotational mirror via the torsion, and two cavity modes interact with each other via the optical fiber. A weak probe field is incident from one side of the system for examining the optical response along one propagating direction. With the Hamiltonian of the system and the Heisenberg-Langevin equation, we can obtain the transmission of the output light field from the input-output theory. The result shows that the optical nonreciprocity of the vortex beam arises from the quantum interference between the optorotational interaction and the linear coupling interaction between two vortex beam modes, and the phase difference can be used to adjust the optical nonreciprocity. The phase difference can determine not only the occurrence of the nonreciprocity but also the direction of nonreciprocity. Moreover, the ratio of the topological charges carried by the two vortex beam fields has an influence on the transmission. Under an appropriate topological charge ratio, the unidirectional transmission of the vortex beam can be realized in such a system. It is found that whether the topological charge ratio is positive or negative, i.e. whether the vortex beam is left-hand beam or right-hand beam, does not affect the transmission; the influence of the topological charge on the transmission amplitude actually comes from the topological charge number carried by the vortex beam, due to the fact that the coupling strength between the rotating mirror mode and the cavity mode depends on the topological charge number. In addition, we also obtain the condition that the system damping rates should meet for realizing the perfect nonreciprocal propagation of the vortex beam. Finally, we can achieve the nonreciprocal group velocity of the slow light. The direction of the nonreciprocal slow light can be controlled via phase modulation. Our work provides a possible application in manipulating the vortex beam propagation. Furthermore, we extend the nonreciprocity of ordinary beams in the optomechanical system to the nonreciprocity of the vortex beam in the optorotational system. The results are expected to be applied to fabricating the ideal optical isolators for the vortex beam carrying the orbital angular momentum in optical communication.