Review on partially coherent vortex beams

Acquiring topological charge in real-time for vortex beams encounters numerous challenges due to the turbulent atmosphere and coherence degradation. We propose an experimental scheme employing the strong detail extraction capability of deep neural networks to recognize the topological charge of partially coherent vortex beams propagating through the turbulent atmosphere and encountering unknown obstacles. Notably, coherence modulation has demonstrated advantages in deep neural network-based recognition. By comparing with high-coherence vortex beams, the deep neural network accurately recognizes topological charges for low-coherence vortex beams using only half of the available dataset. Furthermore, when the turbulent atmosphere and obstacles were considered, the accuracy of low-coherence vortex beams surpassed that of high-coherence vortex beams with equal amounts of training data. Additionally, the encrypted optical communication using partially coherent vortex beams was demonstrated. The coherence parameter significantly enhanced the channel capacity. This study holds potential for applications in free-space optical communication.

Traditional optical measurement techniques primarily rely on interference fringes formed by planar wave interference. Due to their helical phase front and phase singularity, vortex beams exhibit high sensitivity to minor wavefront distortions. When vortex beams carrying non-zero orbital angular momentum (OAM) are employed as either the object or reference beam in optical measurements, they can significantly enhance the speed, accuracy, and adaptability of these measurements. The petal-like fringes generated by conjugated vortex beams serve as a source for extracting displacement information. Based on the principle of conjugated vortex beams, an experimental scheme for measuring in-plane displacement was designed. This scheme was implemented using conjugated Laguerre-Gaussian (LG) beams with different topological charges. Simulation results demonstrated that for in-plane displacements of 13.32nm and 10.656nm, the relative errors in the displacement reconstructed using the Hough transform algorithm were 0.12% and 0.19%, respectively, at a topological charge of 12. The findings indicate that the proposed experimental scheme enables nanometer-scale in-plane displacement measurements with high precision.

A photon can carry orbital angular momentum equal to an integer number of the reduced Planck’s constant. This principle expresses itself in geometrical quantization of optical vortex beams, which thus can propagate only in the form of fields having a helically wavefront characterized by an integer valued topological charge. However, one can create an optical vortex beam of an effective fractional charge by combining multiple integer vortices. Here, we investigate this apparent violation of the geometrical quantization of orbital angular momentum of light. Our approach relies on observation of the light-induced motion of a microscopic particle, which thus acts as an optomechanical probe for the optical vortex beam. A fractional topological charge corresponds to an abrupt jump in the helical phase front of the beam. This singularity expresses itself as an off-axis disturbance in the intensity profile, and thus complicates the optomechanical probing. We overcome this problem by distributing the disturbance along the vortex ring, so that a microparticle can continuously orbit due to the orbital angular momentum transfer. We demonstrate theoretically that whatever effort is put into smoothing the fractional vortex ring (as long as the net topological charge is fixed), the particle’s orbital motion cannot be as uniform as in the case of an integer vortex beam. We support this prediction by experimental proof. The experimental technique benefits from the recently introduced “perfect” vortex beams which allow an optically trapped particle to orbit along a constant trajectory irrespective of any topological charge.

Terahertz (THz) vortex beams carrying orbital angular momentum (OAM) with high purity and tunable topological charge (TC) will undoubtedly bring extraordinary capacities for advanced imaging or communication systems. We propose a convenient and efficient method to generate a tunable and broadband vortex beam that is excited by superradiant Smith Purcell radiation (SSPR) on a helical grating. This scheme fully explores the advantages of natural broadband evanescent wave carried by the electron beam and the Bloch’s theorem regulated helical periodic systems. An explicit relation is established between the index of the spatial harmonic wave on the grating and the topological charge of the vortex beam. The electron energy in the SSPR can be customized to ignite the specific spatial harmonics and manipulate the OAM beam, accordingly. The separated radiation region also promises the high purity of the OAM spectrum. The harmonic excitation reduces the communality between the wavelength and the device size and alleviates the difficulty of device fabrication in the THz band. The proposed vortex scheme can not only be handily scaled to microwave and mid‑infrared regions, but also bring possibilities to applications based on compact tunable vortex beam sources.

The crosstalk of orbital angular momentum (OAM) modes is one of the most important issues for free-space optical communication. We theoretically derive a crosstalk model for Laguerre–Gaussian (LG) vortex beams propagating in turbulent atmosphere. The calculated crosstalk of different OAM modes in different turbulence by analytical expression is consistent with experimental measurements. Both theoretical and experimental results reveal that the crosstalk of high order LG vortex beams is more obvious in atmospheric turbulence. Different from the numerical calculation of the integral expression of the previous studies, this analytical expression can directly and quantitatively characterize the crosstalk and the law of different topological charges in vortex beams. Based on this analytical expression, the crosstalk widths of vortex beams with different topological charges can be calculated accurately, so that the optimal sets of OAM modes can be found more easily under each atmospheric turbulence condition.

The generating of vortex beams in the terahertz (THz) band attracts significant attention due to their applications in high-speed communication and high-resolution imaging. In this article, a novel reflective metasurface working in the THz band is designed to generate four vortex beams with different topological charges in different directions. The unit cell is designed based on the geometric phase, and it consists of two metallic (gold) layers and one dielectric layer in between. The top layer of the unit cell includes an elliptic patch and a circular ring, and the bottom layer of the unit cell is a metallic ground. The reflection efficiency of the unit cell is very high due to the presence of metallic ground. To break through the limitations of traditional methods, the metasurface is a good choice to generate beams that carry orbital angular momentum (OAM). Using the concept of geometric phase, the reflection phase of reflective circular polarization (CP) electromagnetic waves can be controlled in an ingenious way. Owing to the property of the geometric phase, inverse phase shift can be achieved for left-handed circular polarization and right-handed circular polarization waves. By utilizing this trait of geometric phase, one can decompose a linear polarization wave into two orthogonal circular polarization waves and control their properties respectively. By rotating the top layer of the unit cell, 360-degree phase shift and the phase distribution satisfying the requirement for generating the multi-beam multi-mode vortex beam can be achieved. In order to control the direction and the topological charge of each beam, based on the geometric phase, the theory of reflectarray and the phase composition principle, the phase distribution of the reflective metasurface is calculated to provide the phase compensation to make the vortex beams point to certain directions. It is worthwhile to point out that the method presented in this paper provides a way to generate complex multi-mode vortex beams in the THz band. The simulations and measurements show that the metasurface can generate four vortex beams with topological charges <i>l</i> = ± 1 and ± 2 in different directions in the THz band. These results also indicate that our design has great potential applications in wireless communication and high-resolution imaging.

We show both theoretically and numerically that if an optical vortex beam has a symmetric or almost symmetric angular harmonics spectrum [orbital angular momentum (OAM) spectrum], then the order of the central harmonic in the OAM spectrum equals the normalized-to-power OAM of the beam. This means that an optical vortex beam with a symmetric OAM spectrum has the same topological charge and the normalized-to-power OAM has an optical vortex with only one central angular harmonic. For light fields with a symmetric OAM spectrum, we give a general expression in the form of a series. We also study two examples of form-invariant (structurally stable) vortex beams with their topological charges being infinite, while the normalized-to-power OAM is approximately equal to the topological charge of the central angular harmonic, contributing the most to the OAM of the entire beam.

Optical vortex beams, carrying orbital angular momentum (OAM), are of great interest for applications in optical communications, particle manipulation, and quantum information processing. However, achieving high‐efficiency generation and manipulation of vortex beams with a single compact element, particularly in the visible range, remains challenging. Here, we demonstrate laser‐written geometric phase fork gratings in silica glass that enable simultaneous generation and propagation control of vortex beams with an efficiency up to 98.4% at 515 nm. By tailoring the phase‐gradient period and topological charge ( l = 1–100), we achieve tunable control over the vortex beam's topological charge, diameter, and diffraction position. The fabricated gratings maintain high efficiency across a broad spectral range from 450 to 690 nm. Furthermore, we demonstrate spin–orbit coupling between incident vortices and fork gratings, verifying that the output OAM values follow the expected addition rule l out = l in ± l fork . In addition, the fork gratings exhibit excellent thermal stability (up to 800°C) and optical damage threshold (close to pristine silica glass). These results establish nanopore‐based geometric‐phase fork gratings as a scalable, broadband, high‐performance platform with outstanding thermal and optical tolerance for next‐generation OAM photonics.

The Orbital Angular Momentum (OAM) carried by light beams with helical phasefront (vortex beams) has been widely employed in many applications such as optical tweezers, optical drives of micro-machines, atom trapping, and optical communication. OAM provides an additional dimension (diversity) to multiplexing techniques, which can be utilized in addition to conventional multiplexing methods to achieve higher data rates in wireless communication. OAM beams have been thoroughly studied and used in the optical regime but in the mm-wave and THz-wave region, they are still under investigation. In these frequency bands, there are difficulties associated with beam-splitting and beam-combining processes as well as with the use of spiral phase plates and other methods for OAM generation, since the wavelength is much larger compared to those at optical frequencies, leading to higher diffraction losses. The present paper describes the natural generation of high-power OAM modes by gyro-type vacuum electron devices with cylindrical interaction circuit and axial output of the generated rotating higher-order transverse electric mode TEm,n, where m > 1 and n are the azimuthal and radial mode index, respectively. The ratio between the total angular momentum (TAM) JN and total energy WN of N photons is given by m/ω, where ω is the angular frequency of the operating mode, which in a gyrotron oscillator is close to the TEm,n-mode cutoff frequency in the cavity. Therefore, m/ω = Rc/c, where Rc is the caustic radius and c the velocity of light in vacuum. This means that the OAM is proportional to the caustic radius and at a given frequency the same for all modes with the same azimuthal index m. Right-hand rotation (co-rotation with the electrons) corresponds to a positive value of m and left-hand rotation to negative m. The corresponding OAM mode number (topological charge) is l = m – 1. Circularly polarized TE1n modes only possess a Spin Angular Momentum (SAM: s = ±1). TE0n modes have neither SAM nor OAM. This is the result of the photonic (quasi-optical) approach to derive the TAM of modes generated in gyrotrons. The same result follows from the electromagnetic (EM) wave approach for the TAM within a given waveguide volume per total energy of the EM wave in the same volume. Such high-power output beams with very pure higher-order OAM, generated by gyrotron oscillators or amplifiers (broadband) could be used for multiplexing in long-range wireless communications. The corresponding mode and helical wavefront sensitive detectors for selective OAM-mode sorting are available and described in the present paper.

The partially coherent Lorentz-Gauss vortex beam generated by a Schell-model source has been introduced. Based on the extended Huygens-Fresnel principle, the cross-spectral density function of a partially coherent Lorentz-Gauss vortex beam propagating in oceanic turbulence is derived. The influences of coherence length, topological charge M, and oceanic turbulence on the spreading properties and position of the coherence vortex for a partially coherent Lorentz-Gauss vortex beam are analyzed in detail. The results show that a partially coherent Lorentz-Gauss vortex beam propagating in stronger oceanic turbulence will evolve into a Gaussian-like beam more rapidly as the propagation distance increases, and the number of coherent vortices will change.

Orbital angular momentum (OAM) vortex beam has a phase singularity with a twisted wave-front, whose complex amplitude comprises the helical term exp (ilθ). OAM vortex beam has been widely used in optical manipulation, imaging, optical communication, sensing and so on. Ultra-short pulse OAM vortex beams have the advantages of both vortex beams and ultra-short pulses, and can be applied to chiral material processing, long distance transmission, strong field physics and nonlinear frequency conversion. Direct generation of ultra-short OAM vortex beams has the advantages of compact and simple system and good beam quality. The research progress on direct generation of ultra-short OAM vortex beams was summarized. At present, the pulse width of the ultra-short OAM vortex beam generated by the active method is still limited to a few hundred femtoseconds. How to obtain the pulsed vortex beam output within 100 femtoseconds or even with a few cycles through the direct output method will be an important future development direction.

We investigate the orbital angular momentum of partially coherent beams which are constructed by a superposition of mutually incoherent vortex modes, each mode having a different beam width and topological charge. It is shown that these simple beams nevertheless provide great flexibility in controlling orbital angular momentum through adjustment of the beam parameters and have significant potential for particle rotation and trapping.

X‐ray magnetic circular dichroism is a well‐established method to study element specific magnetic properties of a material, while electron energy‐loss magnetic chiral dichroism (EMCD), which is the electron wave analogue to XMCD, is scarcely used today. Recently discovered electron vortex beams, which carry quantized orbital angular momenta (OAM) L, promise to also reveal magnetic signals [1]. Since electron beams can be easily focused down to sub‐nanometer diameters, this novel technique provides the possibility to quantitatively determine local magnetic properties with unrivalled lateral resolution. In order to generate the spiralling wave front of an electron vortex beam with an azimuthally growing phase shift of up to 2p and a phase singularity in its axial centre, specially designed apertures are needed [2,3]. Dichroic signals on the L 2 and L 3 edge are expected to be of the order of 5% [4,5]. The generation of EVBs in the double aberration‐corrected FEI Titan 3 80‐300 transmission electron microscope (TEM) is achieved by the implementation of a dislocation‐type apertures into the condenser lens system. The setup allows for scanning TEM investigations (STEM) with vortex beams, whose OAM is selected by means of an additional discriminator aperture. New FIB cutting strategies facilitate the production for 50 µm wide and 1 µm thick high quality vortex apertures (see fig. 1a). However, in the case of a fork‐type aperture, the EVB are dispersed in the x‐y plane resulting in mixed probe that interacts with the magnetic sample. We have recently devised an escape route to this problem by blocking any partial beams that carry other but the desired OAM prior to the interaction of the beam with the ferromagnetic sample. This is achieved by using a special condensor aperture in combination with a fork‐type aperture to select a single partial beam with the chosen OAM (s. fig 1b). This approach allows to generate atom‐sized EVB with angstrom‐sized probes and a well‐defined OAM by which atomic resolution HR‐STEM is achieved (see. fig 2). Although this discretization results in an increased signal‐to‐noise ratio, this novel technique is capable of atomic resolution EELS measurements which is the prerequisite for atomic EMCD measurements. First experiments using this new optical setup show very promising EMCD results on ferromagnetic FePt nanoparticles.

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<sub>2</sub>) is also changeable. External temperature changes will cause VO<sub>2</sub> to transform from insulating state to metallic state. Generally, VO<sub>2</sub> conductivity can increase by several orders of magnitude as operating temperature changes. By using the phase change property of VO<sub>2</sub>, 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 <i>l</i> = 1 is 90%, and the mode purity is about 91.1% for the vortex beam with <i>l</i> = 2, and 85.4% for the vortex beam with <i>l</i> = 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 (<i>l</i> = 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<i> l</i> = 1 for two operating frequencies are 89.1% and 71.6%, respectively. The mode purities are 83.2% and 94.4% with topological charge <i>l</i> = 2, respectively. The mode purities are 62.4% and 68.2% with topological charge<i> l</i> = 3, respectively. Therefore, the proposed switchable frequency terahertz vortex generator provides a new design idea for working frequency modulation in wireless terahertz communication.

The propagation performance of high-power partially coherent fibre laser beams in a real environment is investigated and the theoretical model of a high-power fibre laser propagating in a real environment is established. The influence of a collimating system and thermal blooming is considered together with atmospheric turbulence and mechanical jitter. The laser energy concentration of partially coherent beams in the far field is calculated and analysed based on the theoretical model. It is shown that the propagation performance of partially coherent beams depends on the collimating system, atmospheric turbulence, mechanical jitter and thermal blooming. The propagation performance of partially coherent beams and fully coherent beams is studied and the results show that partially coherent beams are less sensitive to the influence of thermal blooming, which results in that the energy degeneration for partially coherent beams is only 50% of that for fully coherent beams. Both partially coherent beams and fully coherent beams become less sensitive to thermal blooming when the average structural constant of the refraction index fluctuations increases to 1.7 × 10−14m−2/3. The investigation presents a reference for applications of a high-power fibre laser system.