Fractional Vortex Beams Research Articles

Generation of terahertz radiation with fractional or integer OAMs from a fractional-order vortex two-color field

This paper investigates the generation of the ultra-broadwidth (0.1–30 THz) terahertz (THz) radiation carrying fractional/integer orbital angular momentums (OAMs) via the interaction of the two-color (ω 0 and 2ω 0) laser field carrying initial fractional topological charges (TCs) with air in a moderate pump intensity regime (20 TW cm−2 < I pump < 50 TW cm−2). The two four-wave mixing (FWM) processes (i.e., ω 0 + ω′0 − 2ω 0 → ω THz and 2ω 0 − ω 0 − ω′0 → ω THz) are responsible for THz generation. The two processes can produce two THz pulses. They interfere with each other and THz interference vortex beams are obtained. More importantly, the generation probability from the first FWM process grows while that of the second process declines in the positive frequency region over distance. This is largely due to the combined action of phase mismatch and the blue shift of the THz center frequency. For a longer distance, THz fractional vortex beams (FVBs)/integer vortex beams (IVBs) are produced by the dominant FWM process (ω 0 + ω′0 − 2ω 0 → ω THz). Therefore, via employing different combinations of the initial TCs of the ω 0 and 2ω 0 pulses, one can manipulate the generation of the THz vortex beams with arbitrary fractional-order or integer-order TCs at some specific propagation distances. What is even more interesting is that, when employing half-integer TCs, THz FVBs with varying TC over distance can be produced, companied with birth and annihilation of the alternative vortex pair. This is principally due to diffraction-related effects and the unstable nature of the fractional vortex structures. This simple manipulation for THz waves carrying arbitrary fractional or integer TCs in this scheme encourages the applications for optically rotation, manipulation of molecular or cell assays and image edge enhancement in the field of biomedicine.

Deep-learning-assisted communication capacity enhancement by non-orthogonal state recognition of structured light.

In light of pending capacity crunch in information era, orbital-angular-momenta-carrying vortex beams are gaining traction thanks to enlarged transmission capability. However, high-order beams are confronted with fundamental limits of nontrivial divergence or distortion, which consequently intensifies research on new optical states like low-order fractional vortex beams. Here, we experimentally demonstrate an alternative mean to increase the capacity by simultaneously utilizing multiple non-orthogonal states of structured light, challenging a prevailing view of using orthogonal states as information carriers. Specifically, six categories of beams are jointly recognized with accuracy of >99% by harnessing an adapted deep neural network, thus providing the targeted wide bandwidth. We then manifest the efficiency by sending/receiving a grayscale image in 256-ary mode encoding and shift keying schemes, respectively. Moreover, the well-trained model is able to realize high fidelity recognition (accuracy >0.8) onto structured beams under unknown turbulence and restricted receiver aperture size. To gain insights of the framework, we further interpret the network by revealing the contributions of intensity signals from different positions. This work holds potential in intelligence-assisted large-capacity and secure communications, meeting ever growing demand of daily information bandwidth.

Analysis of practical fractional vortex beams at far field

The subject of calculating the topological charge (TC) or vortex strength of optical vortices has generated divided opinions among scientists. This is due to the fact that proper analytical results are hard to support from the experimental point of view, leading to different results and conclusions. In this work we will present numerical data that shows the limits of TC measurements for practical fractional vortices and the possible challenges that high order measurements may pose. By analyzing the far field phase and the behavior of the transitions we have shown that they follow specific curves that depend not only on the TC but also on the beam waist. This leads us to present a new “strength staircase” for practical vortices. Our aim is to give some insight in practical scenarios that have not been taken into account in previous results.

Dynamics of Fractional Vortex Beams at Fraunhofer Diffraction Zone

Fractional vortex beams (FVBs) possess unique topological properties that are manifested in the vortex distribution. However, there are still discrepancies in the value of the vortex strength of FVBs at the far field. In this work we present a complete picture of the behavior of the phase singularities of non-integer (commonly known as fractional) beams in the Fraunhofer diffraction region and demonstrate a very good correspondence between experiments and simulations. As shown in the text, the original beam waist ω0 was found to be a key factor relating to the beam profile topology. This variable was measured in the process of calibrating the experiment. Finally, an experimental method to obtain the non-integer topological charge is proposed. This method only requires an analysis of the intensity, knowledge of the transition behaviors, and the beam waist.

Measuring obscured OAM spectrum using Stokes fluctuations in a non-interferometric approach

A novel technique to measure the orbital angular momentum (OAM) spectrum of the beam obscured by a complex random media is proposed and experimentally demonstrated. This is realized by measuring the complex correlation polarization function (CPCF) with the help of a two-point Stokes fluctuations correlation of the random light. The OAM spectrum analysis is implemented by projecting a complex field of the CPCF into spiral harmonics. A detailed theoretical framework is developed to measure complex amplitude and to decompose different integer OAM states of the obscured beam. The developed theoretical framework is verified by numerical simulation and tested by experimental demonstration of OAM mode compositions of the fractional optical vortex (FOV) beam.

Precise measurement of trapping and manipulation properties of focused fractional vortex beams.

Fractional vortex beams (FVBs) were believed to be hard to rotate microparticles at a half-integer topological charge due to the unique radial opening (low-intensity gap) in their intensity ring. However, recent research discovered more symmetric intensity structures with less intensity inhomogeneity of practical FVBs at the focal plane. Here, we experimentally demonstrated the manipulation of trapped microparticles and precisely measured their rotation periods at the focal plane of practical FVBs by using a high-speed camera. We verified that the measured orbital angular momentum (OAM) derived from the collective microparticle rotation is roughly proportional to the fractional OAM of practical FVBs. Furthermore, we also experimentally obtained the trapped microparticles' power spectra under the illumination of FVBs, from which we achieved the average trap stiffness to evaluate the two-dimensional trapping strength of the practical focused FVB intensity ring. Our results provide a new insight and an efficient tool on finely trapping and rotating microparticles and bio-cells by using fractional vortex beams.

Review on fractional vortex beam

Abstract As an indispensable complement to an integer vortex beam, the fractional vortex beam has unique physical properties such as radially notched intensity distribution, complex phase structure consisting of alternating charge vortex chains, and more sophisticated orbital angular momentum modulation dimension. In recent years, we have noticed that the fractional vortex beam was widely used for complex micro-particle manipulation in optical tweezers, improving communication capacity, controllable edge enhancement of image and quantum entanglement. Moreover, this has stimulated extensive research interest, including the deep digging of the phenomenon and physics based on different advanced beam sources and has led to a new research boom in micro/nano-optical devices. Here, we review the recent advances leading to theoretical models, propagation, generation, measurement, and applications of fractional vortex beams and consider the possible directions and challenges in the future.

Intensity interference characteristics of fractional Laguerre–Gaussian optical vortex

The paper reports the intensity characteristics of fractional vortex beam, and controls the coupling between photon orbital angular momentum and spin angular momentum by coherently manipulating Laguerre–Gaussian vortex beam with linearly polarized plane light and its conjugate beam. By comparing theory with experiment, fractional vortex beams with different topological charges are generated by using pure phase liquid crystal spatial light modulator, and their intensity characteristics and interference characteristics are observed. The results show that the radial notch size of the spot is related to the fractional part of the topological charge value, and the closer the fractional part of the topological charge value is to the half integer, the more obvious the radial notch is. With the increase of integer part of topological charge, the spot radius also increases. In addition, interference with linearly polarized plane light reveals that the number of branches is equal to the number of topological charges. Interfere with its conjugate beam, and judge the topological charge number of vortex beam by the relationship between petal number and topological charge number. The petal number of integer vortex beam is twice the topological charge number, and fractional order vortex beam destroys the ring structure of Laguerre–Gaussian beam.

Recognizing fractional orbital angular momentum using feed forward neural network

Fractional vortex beam (FVB) possessing helical phase can be applied in the shift-keying communication due to its fractional orbital angular momentum (FOAM) mode, which theoretically allows an infinite increase of the transmitted capacity. However, the discontinuity of spiral phase makes FVB more likely to be disturbed in turbulence environment, and the precise measurement of distorted FOAM modes is crucial for practical FOAM-based communication application. Here, we proposed a FOAM mode recognition method with feedforward neural network (FNN). Employing the diffraction preprocessing of a two-dimensional fork grating, the original optical features of FVBs can be extended along the far-field diffraction order, endowing FNN more feature information and saving calculation time, and enlarging the detection range to conjugate FOAM modes. The simulation results show that the 9-layer FNN can identify FOAM mode with interval of 0.1 with an accuracy of 99.1% under the turbulences of Cn2=1×10-14m-2/3 and Δz=10m. Furthermore, we experimentally constructed a 102-ary FOAM shift-keying communication link to transmit gray image, and the signals are successfully demodulated by the FNN model with the pixel-error-rate of 0.07160. It is anticipated that the proposed FNN-based FOAM recognition method will break the limitation of precision measurement under turbulence environment in practical FOAM applications.

Quasi-stable fractional vortex solitons in nonlocal nonlinear media

This paper studied the propagations of fractional vortex beams in nonlocal nonlinear media, and found that quasi-stable solitons can form when the nonlocality is strong enough and the initial power is equal to the critical power. The propagation properties, including intensity patterns, phase structures, transverse energy flows, and rotation period, were all investigated. The critical power of solitons with different topological charges is the same, however, the orbital angular momentum (OAM) of that increases with the increase of fractional topological charge. In addition, we revealed that fractional vortex solitons have lateral shifts in the y-direction, and the shift values can be controlled by the fractional topological charge. Obviously, the quasi-stable fractional vortex solitons have important academic value and potential application to optical switches, optical wrenches and optical communications, etc.

Simulated generation of fractional polarization singularities based on fractional vortex beam

Polarization singularity and fractional vortex beam have been researched separately for decades. In this article, we connected these two topics by generating fractional polarization singularities with fractional vortex beam. The topological charges of fractional polarization singularities are non-integer and not just ±1/2, and we proposed “quasi-Lemon” and “quasi-Star” structure. Our work broadened the topological structure of polarization singularities.

Anomalous multi-ramp fractional vortex beams with arbitrary topological charge jumps

Traditional fractional vortex beams are well-known “jump” beams: that is, their net topological charge jumps by unity as the effective topological charge of the source passes a half-integer value. Here, we propose an anomalous multi-ramp fractional vortex (AMRFV) beam. Unlike the traditional fractional vortex beams, an AMRFV beam can be designed to have arbitrary jumps in topological charge at any critical threshold of the source charge. We walk through some examples of AMRFV beams using simulations and present a clear interpretation of the multi-jump characteristic based on the evolution of phase singularities.

Study of fractional optical vortex beam in the near-field

The near-field amplitude and phase characteristics of fractional optical vortex beam from a spiral phase plate fabricated on the tip of an optical fiber is reported here. A near-field scanning optical microscope is used to measure the intensity and phase structure of the fractional optical vortex beam as a function of propagation distance, Z and fractionality, l. The measured intensity-phase structure of the fractional optical vortex beam as a function of Z, from near- to far-field and l are complex and dominated by diffraction-related smoothening of the structure. Theoretical simulations carried out using the angular spectrum method matches well with the experimentally measured behavior. The trajectory of singular point in the vortex beam, as its charge is changed from l = – 1 to +1 in fractional steps, is simulated and measured using an interferometer setup. Such studies enable us to understand the complex nature of the fractional optical vortex beam in the near-field, its modifications during propagation to far-field, and as a function of fractionality.

Observation of multiramp fractional vortex beams and their total vortex strength in free space

Fractional vortex beams (FVBs) have increasingly attracted attention in recent years due to their complex but interesting topological structures and orbital angular momentum. However, the issue about the total vortex strength of practical FVBs propagating in free space has so far proved inconclusive. Here, we have further experimentally investigated the evolution properties of multiramp fractional vortex beams (MFVBs) including FVBs in free space, and we have systematically revealed the dependence of the total vortex strength on both the non-integer topological charge α and the multiramp number m, which are the parameters of the initial multiramp phase structures of MFVBs. In near-field regions, the vortices contained in MFVBs are unstable, and they change and evolve as the propagation distance increases, therefore it is hard to confirm effectively their vortex strength. However, in far-field regions, the evolution of vortices in fields becomes stable, thus it is considerably easy to determine the value of the vortex strength experimentally via measuring vortex structures by the interference method. Our result shows that the vortex strength for practical FVBs and MFVBs has the different behavior compared with the ideal plane-wave FVBs and MFVBs. These findings give us an understanding of such light fields of both FVBs and MFVBs and may be useful to potential applications related to light signal process and propagation.

Partially coherent radially polarized fractional vortex beam.

A new kind of partially coherent vector beam, named a partially coherent radially polarized fractional vortex (PCRPFV) beam, is introduced as a natural extension of the recently introduced scalar partially coherent fractional vortex beams [Zeng et al., Opt. Express26, 26830 (2018)10.1364/OE.26.026830]. Realizability conditions and propagation formulas for a PCRPFV beam are derived. Statistical properties of a focused PCRPFV beam, such as average intensity, degree of polarization, state of polarization and cross-spectral density matrix, are illustrated in detail and compared with that of a partially coherent radially polarized integer vortex beam and a scalar partially coherent fractional vortex beam. It is found that the statistical properties of a PCRPFV beam are qualitatively different from these simpler beam classes and are strongly determined by the vortex phase (i.e., fractional topological charge) and initial coherence width. We demonstrate experimental generation of PCRPFV beams and confirm their behavior. Our results will be useful for the rotating and trapping of particles, the detection of phase objects, and polarization lidar systems.

Localized geometric phase analogue associated with Poincaré beams due to unfolding of fractional optical vortex beams through a birefringent crystal

Fractional optical vortex beam is generated by the diffraction of a Gaussian beam using computer generated hologram embedded with mixed screw-edge dislocation. Unfolding of the generated fractional vortex beam into eigen-polarization components with orthogonal polarization results in the conversion of scalar phase singularity to vector polarization singularities in the beam cross-section. The evolution of the singularities of the ellipse field namely C-points (points of undefined major axis) and L-lines (lines of undefined handedness) in the state of polarization distribution on a transverse plane quantifies the transformation. The effect of the phase morphology dictated by the fractional order of the dislocation, transverse spatial separation and longitudinal relative phase of the two eigen-polarization components on determining the complex transverse polarization structure is investigated. The nature of the generated Poincaré beam is also indicated by projecting the states of polarization on to the Poincaré sphere. With increasing order of dislocation from 0.0 to 1.0 in fractional steps and with increasing relative phase, the partial Poincaré beam is transformed to a full Poincaré beam. The transformation of the local structure around the C-points is measured through a localized phase parameter defined similar to Pancharatnam–Berry geometric phase by the projected Poincaré sphere contour for different dynamic phase difference of the unfolded beams generated from different fractional order dislocation.

Investigation on magnetization induced by tightly focused azimuthally polarized fractional vortex beam

All-optical magnetic recording based on the inverse Faraday effect has become a research hotspot in recent years due to its ultra-high storage density and ultra-fast magnetization reversal rate. Existing studies have shown that by optimizing the phase modulation or performing 4π tight focusing of azimuthally polarized vortex beams, high-resolution longitudinal magnetization fields with different axisymmetric intensity patterns can be generated. In order to meet the requirements of more complex all-optical magnetic recording and asymmetric magnetic particle capture and manipulation, it is particularly important to generate an asymmetric light-induced magnetization field with adjustable center position. Studies have shown that the fractional vortex phase could lead to the asymmetric focal field distribution generation under tight focusing conditions, which means that the tightly-focused azimuthally polarized light carrying the fractional vortex phase can produce a novel asymmetric light-induced magnetization field. As a new degree of freedom for the regulation of the magnetization field, the fractional topological charge will bring more new phenomena, new effects and new applications in the field of interaction between light and matter. In this work, for the first time to our knowledge, the magnetization induced by tightly focused azimuthally polarized fractional vortex beam is studied based on the Richard-Wolf vector diffraction theory and the inverse Faraday effect. The equivalent approximation of the magnetization induced by azimuthally polarized fractional vortex beam regarded as a weighted superposition of magnetization induced and crossly induced by a finite number of azimuthally polarized adjacent integer-order vortex beams, where the number of the equivalent terms is chosen by using the histogram intersection method of the intensity distribution image of the magnetization field. The magnetization field distribution under different values of &lt;i&gt;α&lt;/i&gt; are also numerically simulated. Studies have shown that magnetization induced by the azimuthally polarized fractional vortex beam is asymmetrically distributed. When the fractional vortex topological charge α belongs to [0.5,1.5], as the vortex topological load increases, the splitting phenomenon of the transverse distribution of magnetization field appears with the magnetization spot position shift in the direction perpendicular to the optical axis. When &lt;i&gt;α&lt;/i&gt; equals 0.5 or 1.5, the maximum offset of the center of the magnetization spot is 0.24&lt;i&gt;λ&lt;/i&gt;. When the fractional vortex topological charge α belongs to [2,3], the transverse distribution of magnetization field splits two hot intensity spots with gradually growing outer ring diameter. When the fractional vortex topological charge &lt;i&gt;α&lt;/i&gt; tends to be an integer 3, the transverse distribution of magnetization field also gets round and symmetrical. In particular, when the fractional vortex topological charge α is a half-integer, especially larger than 3. The number of hot spots (&lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$\alpha - 0.5 $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20200269_M8.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20200269_M8.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;) of the intensity of the magnetization field and the number of dark spots (&lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ \alpha - 1.5 $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20200269_M9.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20200269_M9.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;) surrounded by them all have a positive correlation with the number of vortex order. The research in this paper is expected to have new applications in the fields of all-optical magnetic recording and the capture and manipulation of magnetic particles.

Focus shaping of partially coherent radially polarized vortex beam with tunable topological charge

In this paper, we have introduced a new class of partially coherent vector vortex beams, named radially polarized multi-Gaussian Schell-model (MGSM) vortex beam, carrying the vortex phase with tunable topological charges (i.e., both integral and fractional values) as a natural extension of the radially polarized MGSM beam. The tight focusing properties of the radially polarized MGSM vortex beam passing through a high numerical aperture (NA) objective lens are investigated numerically based on the vectorial diffraction theory. Numerical results show that the focal intensity distributions of the radially polarized MGSM vortex beam can be shaped by regulating the structure of the correlation functions and the topological charge of vortex phase. In contrast with the integral vortex beam, the most intriguing property of the fractional vortex beam is that the focal intensity distribution at the focal plane can be nonuniformity and asymmetry, while such unique characteristics will vanish when the spatial coherence length is sufficiently small. Furthermore, some focal fields with novel structure, such as a focal spot with nonuniform asymmetric or an anomalous asymmetric hollow focal field, can be formed by choosing suitable fractional values of topological charge and spatial coherence length. Our results will be useful for optical trapping, especially for trapping of irregular particles or manipulation of absorbing particles.

Gouy phase-assisted topological transformation of vortex beams from fractional fork holograms.

The topological transformation of optical vortex beams from fractional fork holograms induced by the modulation of controlled Gouy phase (GP) is demonstrated. The GP change is tuned by varying the wavefront curvature of the input beam on the plane of the fractional phase generating optic. The locus of the point of singularity traces a semi-circle about the beam axis for maximum possible change of the associated GP. The morphology parameters describing the anisotropic vortex phases of generated optical vortices are tuned in the experiment by varying the input wavefront curvature. Through the transformation of the transverse Poynting vector of the fractional vortex beams, control of the extrinsic orbital angular momentum is demonstrated for the first time, to the best of our knowledge. This could enable better manipulation of an optically trapped micro-particle and be used in optically driven micro-machines.

Vortex strength and beam propagation factor of fractional vortex beams.

Fractional vortex beams (FVBs) with non-integer topological charges attract much attention due to unique features of propagations, but different viewpoints still exist on the change of their total vortex strength. Here we have experimentally demonstrated the distribution and number of vortices contained in FVBs at the Fraunhofer diffraction region. We have verified that the jumps of total vortex strength for FVBs happen only when non-integer topological charge is before and after (but very close to) any even integer number that originates from two different mechanisms for generation and movement of vortices on focal plane. Meanwhile, we have also measured the beam propagation factor (BPF) of such FVBs and have found that their BPF values almost increase linearly in the x component (along the initial edge dislocation) and oscillate increasingly in the y component (vertical to the initial edge dislocation). Our experimental results are in good agreement with numerical results.