Topological charge of a linear combination of optical vortices: topological competition

Where is the orbital angular momentum in vortex superposition states?

In this paper, we explore the distribution of the orbital angular momentum (OAM) in the coaxial vortex superposition states based on the independent propagation principle of light in this interference process. We find that in this case, some specific singular points exist in the spatial intensity distribution. The first type of singular point is located at the center point of the spatial intensity distribution. The second type of specific singular point is at the critical location of the overlapping area in angular direction. By analogy with the angular momentum superposition of two axially rotating homogeneous disks with different radius in rigid body, We present a suggestion: the center point is located at the overlapping area of all the superposed components. Therefore, the topological charge value in the center point should be doubled by the actual number of superposition field components. The singular point at the critical location of the overlapping area in angular direction should also be co-owned by the superposition components outside the position of the ring (including the corresponding component of the ring). The total OAM is exactly equal to the sum of those two types contained in the superposition states, which is equal to the input OAM of the superposition state components. The conservation of the OAM in the coaxial interference process is demonstrated.

Generating a hollow twisted correlated beam using correlated perturbations

In this study, a twisted correlated optical beam with a dark hollow center in its average intensity is synthesized by correlated correlation perturbation and incoherent mode superposition. This new hollow beam has a topological charge (TC) mode with a zero value compared with a coherence vortex that has a TC mode with a nonzero value. We transform the twisted correlated beam from solid centered to dark hollow centered by constructing a correlation between the twist factor and the spot structure parameter. Theoretical and experimental results show that twist correlation makes the random optical beam an asymmetric orbital angular momentum spectral distribution and a tunable intensity center. Controlling the correlation parameters can make the focal spot of the twisted beam a dark core when the dominant mode of the TC is still zero. The new nontrivial beams and their proposed generation method provide important technical preparations for the optical particle manipulation with low coherence environment.

Tailoring the Topological Charge of a Superposition of Identical Parallel Laguerre-Gaussian Beams

In optical computing machines, data can be transmitted by optical vortices, and the information can be encoded by their topological charges. Thus, some optical mechanisms are needed for performing simple arithmetic operations with the topological charges. Here, a superposition of several parallel identical Laguerre-Gaussian beams with single rings is studied. It is analytically and numerically shown that if the weighting coefficients of the superposition are real, then the total topological charge of the superposition is equal to the topological charge of each component in the initial plane and in the far field. We prove that the total topological charge of the superposition can be changed by the phase delay between the beams. In the numerical simulation, we demonstrate the incrementing and decrementing the topological charge. Potential application areas are in optical computing machines and optical data transmission.

Dividing the Topological Charge of a Laguerre-Gaussian Beam by 2 Using an Off-Axis Gaussian Beam

In optical computing machines, many parameters of light beams can be used as data carriers. If the data are carried by optical vortices, the information can be encoded by the vortex topological charge (TC). Thus, some optical mechanisms are needed for performing typical arithmetic operations with topological charges. Here, we investigate the superposition of a single-ringed (zero-radial-index) Laguerre-Gaussian (LG) beam with an off-axis Gaussian beam in the waist plane. Analytically, we derive at which polar angles intensity nulls can be located and define orders of the optical vortices born around these nulls. We also reveal which of the vortices contribute to the total TC of the superposition and which are compensated for by the opposite-sign vortices. If the LG beam has a TC of m, TC of the superposition is analytically shown to equal [m/2] or [m/2] + 1, where [] means an integer part of the fractional number. Thus, we show that the integer division of the TC by two can be done by superposing the LG beam with an off-axis Gaussian beam. Potential application areas are in optical computing machines and optical data transmission.

Feature recognition of a 2D array vortex interferogram using a convolutional neural network

A vortex array has important applications in scenarios where multiple vortex elements with the same or different topological charges are required simultaneously. Therefore, the detection of the vortex array is vital. Here, the interferogram between the off-axis Walsh-phase plate and the vortex array is first obtained and then decoded through a convolution neural network (CNN), which can simultaneously determine the topological charge, chirality, and the initial angle. Both the theory and experiment prove that a CNN has a remarkable effect on the classification and detection of vortex arrays.

Control of orbital angular momentum of optical vortex beams with complex wandering perturbations

This work investigates how independent perturbations and cross-correlation perturbations affect optical vortex beams. Theoretical and experimental results show that both perturbations cause the intensity, average orbital angular momentum (OAM), and the OAM spectrum of the vortex beam to vary periodically with the perturbation direction, but with different periods. When the beam is subjected to independent perturbations, the average OAM changes periodically with θ in every π/2; when the beam is subjected to cross-correlation perturbations, the average OAM varies with θ in every π. The results of this work provide a method to control the OAM and regulate low-coherence vortex beams in turbulent environments.

Fast oscillations of orbital angular momentum and Shannon entropy caused by radial numbers of structured vortex beams

We address theoretical and experimental considerations of two-parameter excitation of each Hermite-Gaussian (HG) mode in composition of a structured Laguerre-Gaussian (sLG) beam. The complex amplitude of the sLG beam is shaped in such a way that the radial and azimuthal numbers of eigenmodes are entangled with each other. As a result, variations in the amplitude and phase parameters of mode excitation, although dramatically changing the intensity and phase patterns, do not change the structural stability of the beam. We reveal that the radial number of the sLG beam can cause fast oscillations of the orbital angular momentum and Shannon entropy, dramatically increasing the uncertainty of detecting the beam in some particular state. We found that despite the fast oscillations, the sLG beam has an invariant in the form of a module of the total topological charge (TC), with the exception of narrow intervals of the phase parameter, where the measurement error does not allow us to accurately measure the sign of the TC. The difference between the interpretation of informational entropy as a measure of uncertainty and a measure of information capacity is considered on the example of the measurement of Shannon entropy in the bases of LG and HG modes.

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.

Propagation of a multi-vortex beam: far-field diffraction of a Gaussian beam from a multi-fork phase grating

In this work, the far-field propagation of multi-vortex beams is investigated. We consider diffraction of a Gaussian wave from a spatial light modulator (SLM) in which a multi-fork grating is implemented on it at the waist plane of the Gaussian wave. In the first-order diffraction pattern a multi-vortex beam is produced, and we consider its evolution under propagation when different multi-fork gratings are implemented on the SLM. We consider two different schemes for the phase singularities of the implemented grating. A topological charge (TC) equal to l₁ is considered at the center of the grating, and four similar phase singularities all having a TC equal to l₂=l₁4 (or l₂=-l₁4) are located on the corners of a square where the l₁ singularity is located on the square center. Some cases with different values of l₁, and consequently l₂, are investigated. Experimental and simulation results show that if signs of the TCs at the corners and center of the square are the same, the radius of the central singularity on the first-order diffracted beam increases, and it convolves the other singularities. If their signs are opposite, the total TC value equals zero, and at the far-field, the light beam distribution becomes a Gaussian beam. For determining the TCs of the resulting far-field beams, we interfere experimentally and by simulation the resulting far-field beams with a plane wave and count the forked interference fringes. All the results are consistent.

Geometric Progression of Optical Vortices

We study coaxial superpositions of Gaussian optical vortices described by a geometric progression. The topological charge (TC) is obtained for all variants of such superpositions. The TC can be either integer or half-integer in the initial plane. However, it always remains integer when the light field propagates in free space. In the general case, the geometric progression of optical vortices (GPOV) has three integer parameters and one real parameter, values which define its TC. The GPOV does not conserve its intensity structure during propagation in free space. However, the beam can have the intensity lobes whose number is equal to one of the family parameters. If the real GPOV parameter is equal to one, then all angular harmonics in the superposition are of the same energy. In this case, the TC of the superposition is equal to the order of the average angular harmonic in the progression. Thus, if the first angular harmonic in the progression has the TC of k and the last harmonic has the TC of n, then the TC of the entire superposition in the initial plane is equal to (n + k)/2, but the TC is equal to n during propagation. The experimental results on generating of the GPOVs by a spatial light modulator are in a good agreement with the simulation results.

Control of the orbital angular momentum via radial numbers of structured Laguerre-Gaussian beams

We found that the internal perturbations of the structured Laguerre-Gaussian beam in the form of two-parametric harmonic excitations of the Hermite-Gaussian (HG) modes in its composition mix up the radial and azimuthal numbers. The harmonic excitation is characterized by two parameters, one of them controls the amplitude of the HG modes, and the second parameter controls the phases of each HG mode. It was revealed that this mixing of the beam quantum numbers leads to the possibility of controlling the orbital angular momentum (OAM) by means of radial numbers. Non-zero radial numbers lead to rapid OAM oscillations as the phase parameter changes, while oscillations disappear if the radial number is zero. We have also shown that the variation of the phase parameter in a wide range of values does not change the modulus of the total topological charge of the structured beam, despite the fast OAM oscillations.

Influence of optical "dipoles" on the topological charge of a field with a fractional initial charge

In this work, using the Rayleigh-Sommerfeld integral and Berry formula, a topological charge (TC) of a Gaussian optical vortex with an initial fractional TC in the far field was calculated. It was found that, for diverse fractional parts of the TC, the beam contained different numbers of screw dislocations, which determined the TC of the entire beam. If a fractional part of the TC was small, the beam consisted of the main optical vortex centered on the optical axis, with the TC equal to the nearest integer (say n>0) and two edge dislocations located on the vertical axis (one above and the other below the center). When the fractional part of the initial TC increased, a "dipole" was formed from the upper edge dislocation, consisting of two vortices with TCs equal to +1 and -1. With a further increase in the fractional part, the additional vortex with TC=+1 moved to the center of the beam, and the vortex with TC=-1 moved to the periphery. When the fractional part of the TC increased further, another "dipole" was formed from the lower edge dislocation, in which, on the contrary, the vortex with TC=-1 was displaced to the optical axis (to the center of the beam) and the vortex with TC=+1 moved to the beam periphery. When the fractional part of the TC became equal to 1/2, the lower vortex with a TC=-1, which was earlier displaced to the center of the beam, began to shift to the periphery, and the upper vortex with a TC=+1 moved closer and closer to the center of the beam, eventually merging with the main vortex when the fractional part approached 1. Such dynamics of additional vortices with TCs above +1 and below -1 determined which whole TC the beam would have (n or n+1) for different values of the fractional part from the segment [n,n+1]. Our analysis has shown that, for any value of the fractional part of the initial topological charge, the TC of the beam in the far field will not be determined. Since, with an increase in the radius of the circle in the beam section on which the TC is calculated, more optical "dipoles" will appear, and the TC will be either n or n+1.

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.

Propagation and Transformation of Vortexes in Linear and Nonlinear Radio-Photon Systems

The article is devoted to issues related to the propagation and transformation of vortexes in the optical range of frequency. Within the framework of the traditional and modified model of slowly varying envelope approximation (SVEA), the process of converting vortex beams of the optical domain into vortex beams of the terahertz radio range based on nonlinear generation of a difference frequency in a medium with a second-order susceptibility is considered. The modified SVEA splits a slowly varying amplitude into two factors, which makes it possible to more accurately describe the three-wave mixing process. The theoretical substantiation of the rule of vortex beams topological charges conversion is given—the topological charge of the output radio-vortex beam is equal to the difference between the topological charges of the input optical vortex beams. A numerical simulation model of the processes under consideration has been implemented and analyzed.

Structural stability of spiral vortex beams to sector perturbations

Conditions of breaking down the structural stability of a spiral vortex beam subject to sector perturbations were considered. Employing methods of computer simulation and processing experimental results, we have shown that the spiral vortex beam has a caustic surface, the intersection of which sharply changes a shape of the Poynting vector streamlines and critical points of the spiral beam. Nevertheless, the beam propagation (scaling and rotation) does not change the perturbed streamline's shape and phase pattern. We also revealed that strong beam perturbations can cause the conversion of the circulation direction of streamlines in the perturbation region, which entails the appearance of a network of optical vortices with negative topological charges. However, the beam's orbital angular momentum remains unchanging, despite increasing the information entropy (growing a number of vortex modes), so that the perturbed beam keeps new stable states.

Physical meaning of the deviation scale under arbitrary turbulence strengths of optical orbital angular momentum

The recently so-called deviation scale [Phys. Rev. A99, 013828 (2019)PLRAAN1050-294710.1103/PhysRevA.99.013828] bridges the connection between the result of the infinitesimal propagation equation (IPE) prediction and that of the single phase screen (SPS) approximation. Thanks to the multiple phase screen (MPS) approach, in this paper we elaborate the physical meaning of the deviation scale: the spatial accumulation of slight intensity modulation of incident orbital angular momentum (OAM)-carrying beam splits the original vortex into multiple individual vortices with a topological charge (TC) of +1 and regenerates the vortex-antivortex pairs with a TC of +1 and with a TC of -1, leading to a significant deviation between these two different results only when the disruption of this compound effect on the phase distribution of the incident OAM-carrying beam becomes more significant. Other than that, we also show that the appearance of the deviation scale cannot be predicted only by the Rytov variance, which can be predicted through the vortex-splitting ratio of the received optical field alone or with the help of the normalized propagation distance.

Spatial-Light-Modulator-Based Multichannel Data Transmission by Vortex Beams of Various Orders

We report an atmospheric multichannel data transmission system with channel separation by vortex beams of various orders, including half-integer values. For the demultiplexing of the communication channels, a multichannel diffractive optical element (DOE) is proposed, being matched with the used vortex beams. The considered approach may be realized without digital processing of the output images, but only based on the numbers of informative diffraction orders, similar to sorting. The system is implemented based on two spatial light modulators (SLMs), one of which forms a multiplexed signal on the transmitting side, and the other implements a multichannel DOE for separating the vortex beams on the receiving side. The stability of the communication channel to atmospheric interference and the crosstalk between the channels are investigated.

Polarization singularity index determination by using a tilted lens

The superposition of spin and orbital angular momentum states of light generates polarization singularities. By perturbing and disintegrating their component orbital angular momentum (OAM) states, the polarization singularity indices can be determined. The spatially varying polarization distribution of these beams possesses information about the helical wavefront structures of the component OAM states, although they have plane wavefronts. The polarization singular beam (PSB) is focused using a tilted lens, and the intensity distribution at a predicted position in the direction of propagation is used to determine the component OAM content in the beam. Astigmatism introduced by the tilt of the lens modulates the vortex beam to introduce intensity nulls in the propagated beam. We demonstrate by simulations and experiments the index determination of the V points and C points using a tilted lens. This method is effective in the index determination of V points and C points formed by the superposition of component scalar vortices having opposite-sign topological charges. The degeneracy of C points with the same Stokes indices can be lifted through this technique.

Generation and evolution of different terahertz singular beams from long gas-plasma filaments

We theoretically and numerically investigate the generation and evolution of different pulsed terahertz (THz) singular beams with an ultrabroad bandwidth (0.1-40 THz) in long gas-plasma filaments induced by a shaped two-color laser field, i.e., a vortex fundamental pulse (ω₀) and a Gaussian second harmonic pulse (2ω₀). Based on the unidirectional propagation model under group-velocity moving reference frame, the simulating results demonstrate that three different THz singular beams, including the THz necklace beams with a π-stepwise phase profile, the THz angular accelerating vortex beams (AAVBs) with nonlinear phase profile, and the THz vortex beams with linear phase profile, are generated. The THz necklace beams are generated first at millimeter-scale length. Then, with the increase of the filament length, THz AAVBs and THz vortex beams appear in turn almost periodically. Our calculations confirm that all these different THz singular beams result from the coherent superposition of the two collinear THz vortex beams with variable relative amplitudes and conjugated topological charges (TCs), i.e., +2 and -2. These two THz vortex beams could come from the two four-wave mixing (FWM) processes, respectively, i.e., ω₀+ω₀-2ω₀→ω_THz and -(ω₀+ω₀) + 2ω₀→ω_THz. The evolution of the different THz singular beams depends on the combined effect of the pump ω₀-2ω₀ time delay and the separate, periodical, and helical plasma channels. And the TC sign of the generated THz singular beams can be easily controlled by changing the sign of the ω₀-2ω₀ time delay. We believe that these results will deepen the understanding of the THz singular beam generation mechanism and orbital angular momentum (OAM) conversion in laser induced gas-filamentation.

Two-dimensional Talbot effect of the optical vortices and their spatial evolution

We report on the experimental and theoretical study of the near-field diffraction of optical vortices (OVs) at a two-dimensional diffraction grating. The Talbot effect for the optical vortices in the visible range is experimentally observed and the respective Talbot carpets for the optical vortices are experimentally obtained for the first time. It is shown that the spatial configuration of the light field behind the grating represents a complex three-dimensional lattice of beamlet-like optical vortices. A unit cell of the OV lattice is reconstructed using the experimental data and the spatial evolution of the beamlet intensity and phase singularities of the optical vortices is demonstrated. In addition, the self-healing effect for the optical vortices, which consists in flattening of the central dip in the annular intensity distribution, i.e., restoring the image of the object plane predicted earlier is observed. The calculated results agree well with the experimental ones. The results obtained can be used to create and optimize the 3D OV lattices for a wide range of application areas.

Fine structure of perturbed Laguerre-Gaussian beams: Hermite-Gaussian mode spectra and topological charge

We found that small perturbations of the optical vortex core in Laguerre-Gaussian (LG) beams generate a fine structure of the Hermite-Gaussian (HG) mode spectrum in the form of weak variations of amplitudes and phases of the HG modes. We developed and implemented the intensity moments technique for measuring the HG mode spectra. We also theoretically justified and experimentally implemented a technique for measuring the topological charge of the LG beams with an arbitrary number of ring dislocations. Theoretical discussion and experimental study are accompanied by examples of estimating the orbital angular momentum and the topological charge of perturbed LG beams as well as the algorithm for plotting the HG mode spectra.

Estimation of dislocated phases in wavefronts through intensity measurements using a Gerchberg-Saxton type algorithm

Estimation of the phase of a singular paraxial light field from experimentally measured intensities using a Gerchberg-Saxton type algorithm is demonstrated. A combination of cylindrical lenses which does not conserve the orbital angular momentum of the light field is used in obtaining the measured intensities. Consistent extraction of the phases in regard of the orbital angular momentum is demonstrated both at the input and output transverse planes, using the measured intensities.

Tailoring diffraction of light carrying orbital angular momenta

A unified approach to controlling the diffraction of light carrying orbital angular momenta (OAM) is developed and experimentally verified in this Letter. This approach allows one to specify not only the number of diffraction maxima, their spatial frequencies, and the intensity distribution between them, but also the OAM in each maximum. It is verified that the approach can be used for structuring both single and multiple beams carrying OAMs. Simulations reveal phase singularities in structured beams. In addition, the approach makes it possible to shape the light in regular and irregular two-dimensional arrays with addressing the OAMs at each site. This approach offers new opportunities for singular optics.

Topological charge of asymmetric optical vortices

We obtain theoretical relationships to define topological charge (TC) of vortex laser beams devoid of radial symmetry, namely asymmetric Laguerre-Gaussian (LG), asymmetric Bessel-Gaussian (BG), and asymmetric Kummer beams, as well as Hermite-Gaussian (HG) vortex beams. Although they are obtained as superposition of respective conventional LG, BG, and HG beams, these beams have the same TC equal to that of a single mode, n. At the same time, the normalized orbital angular momentum (OAM) that the beams carry is different, differently responding to the variation of the beam's asymmetry degree. However, whatever the asymmetry degree, TC of the beams remains unchanged and equals n. Although separate HG beam does not have OAM and TC, superposition of only two HG modes with adjacent numbers (n, n + 1) and a π/2-phase shift produces a modal beam whose TC is -(2n + 1). Theoretical findings are validated via numerical simulation.

Spectral control of the orbital angular momentum of a laser beam based on 3D properties of spiral phase plates fabricated for an infrared wavelength

This paper examines the spectral properties of a spiral phase plate (SPP) generating orbital angular momentum (OAM) beams. A simple method is proposed for calculating the resulting OAM by measuring only two maximum expansion coefficients. A comparative numerical simulation of the proposed and traditional methods is performed. An SPP is fabricated for generation of an OAM with integer values at infrared and visible wavelengths. Qualitative experimental studies of the changes in a generated OAM with a change in the operating wavelength are performed using the spatial filtering method. The experimental results are found to agree with the results of numerical simulation. Beams with integer and fractional OAM values are obtained experimentally by changing the wavelength.

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.