Variable Spatial Dynamics of Optical Vortices Produced by a Double Fork‐Shaped Grating

In this work, it is theoretically and numerically demonstrated that an astigmatic transformation of a νth-order edge dislocation (shaped as a zero-intensity straight line) of a coherent light field—where ν =n + α is a real positive number, n is integer, and 0 <α <1 is fractional—produces n optical elliptic vortices (screw dislocations) with topological charge (TC) −1, which are arranged on a straight line perpendicular to the edge dislocation and found at Tricomi function zeros. We also reveal that at a distance from the said optical vortices (OV), an extra OV with charge −1 is born on the same straight line, which departs to the periphery with α tending to zero, or gets closer to the n OVs with α tending to 1. Additionally, we find that a countable number of OVs (intensity nulls) with charge −1 are produced at the field periphery and arranged on diverging hyperbolic curves equidistant from the straight line of the n main intensity nulls. These additional OVs, which we term as ‘escort’, either approach the beam center, accompanying the extra ‘companion’ OV if 0 <α <0.5, or depart to the periphery, whereas the ‘companion’ keeps close to the main OVs if 0.5 <α <1. At α =0 or α = 1, the ‘escort’ OVs are shown to be at infinity. At fractional ν, the TC of the whole optical beam is theoretically shown to be infinite. Numerical simulation results are in agreement with the theoretical findings.

Through theoretical calculation, the analytical expression for the cross-spectral density function of vortex beam with and without edge dislocation during transmission in turbulent atmosphere and free space is obtained. The calculation result is used for researching the influences of edge dislocation on optical vortex transmission. The research shows that due to the edge dislocation, when the optical vortex's topological charge is greater than +1, the optical vortex will no longer carry out steady transmission in the free space transmission. Instead, it will divide into two optical vortices, and the distance between them will gradually increase as the transmission distance increases. Optical vortex will split in turbulent atmosphere propagation. Due to the edge dislocation, when the topological charge of optical vortex is greater than +2, it is found that the distance between one optical vortex and other optical vortices is much larger than that between other optical vortices. Besides, when there's an edge dislocation, the greater the light wavelength and the structure constant are, the smaller the distance between the optical vortex and the edge dislocation on the source plane is, and the evolution of the optical vortex will be accelerated.

This work presents an investigation of tight focusing of optical vortices, which produce regions where the Poynting vector is directed opposite to the beam's propagation direction-known as areas of reverse energy flow. We study both phase optical vortices and polarization optical vortices (cylindrical vector beams). For phase optical vortices, it is shown that the emergence of reverse energy flow as well as splitting of the initial optical vortex depends on the handedness of the circular polarization and the phase of the initial optical vortex. An optical vortex with an initial topological charge of $m\; \le \; - {3}$ splits into three optical vortices with topological charges $m + {2}$, ${-}{1}$, and ${-}{1}$ when the polarization is right circular. In contrast, under left-hand circular polarization, a vortex with the topological charge $m$ is preserved at the focus. The reverse-flow regions are located between the centers of phase singularities (screw dislocation points). As the topological charge of the phase vortex increases, the region of reverse flow along the radial coordinate widens, and the distance from the vortex center to the optical axis also increases. In the focusing of cylindrical vector beams (polarization optical vortices), the reverse energy flow region is also observed, but screw dislocations are absent.

Fork-shaped gratings are periodic structures containing a spatial dislocation known to be used for the production of optical vortices in a far field. Spatial overlapping of diffraction orders in a near field results in complex spatial evolution of optical vortices. In this paper, we report the results of near-field diffraction on fork-shaped gratings with different topological charges and analyze the evolution of specific optical vortices during propagation. Optical vortices have been shown to form two-dimensional well-ordered spatial configurations in specific transverse planes. The locus of points of optical singularities has been shown to form two helical lines twisted around the ±1 diffraction order directions. Our results demonstrate that the spatial behaviour of optical vortices is in close connection with the spatial ordering arising from the Talbot effect. The quantity of optical vortices demonstrates complex spatial dynamics, which includes spatial oscillations and decreasing along the propagation direction. These results provide a foundation towards a deeper understanding of near-field singular optics phenomena.

Formation of optical vortices with topological charge [formula omitted] and [formula omitted] by use of the S-waveplate

Nonlinear propagation effects on high-power optical-vortex pulses in anisotropic crystals is reported. Nonlinear effects on optical-vortex pulse generation through the spin-orbit interaction and its propagation in an anisotropic crystal is investigated.The beam of the left-circularly-polarized optical vortex pulse is focused on one of the surfaces of a LiNbO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> or rutile crystal (NC) by a convex lens L1. After propagating along the optic axis of the crystal, the beam is collimated by another convex lens L2.

Optical vortices are a new kind of laser beam and receiving more and more attention currently.The complex amplitudes of optical vortices comprise a helical term exp (il),with l being the topological charge and the azimuthal angle.Each photon in optical vortices carries the orbital angular momentum (OAM) with a value of lħ,where ħ is the Planck's constant divided by 2.The topological charge l is the eigenvalue of optical vortices,and determines the helical wavefront distribution,thus also known as OAM state.Moreover,such an OAM state can be an infinite integer state. And vortices with various OAM states are orthogonal to each other,making it possible to be employed in high capacity data-transmission system.In addition,the above unique features contribute to their widely applications in lots of areas such as optical tweezers and spanners,rotation detection,quantum entanglement,etc.In these applications,detecting OAM states is basic,and greatly significant.Recently researchers have developed lots of approaches to detecting the OAM states,including the methods of interference,diffraction gratings,metasurface,etc.Of such approaches,the scheme of diffraction gratings is the simplest and most widely used,where one or more diffraction gratings are employed. When optical vortices propagate through such gratings,the OAM states are acquired immediately through capturing and analyzing the distinct OAM-related diffraction patterns.In this review,we focus on the techniques of detecting OAM states through diffraction gratings,which have been demonstrated by our group and other researchers.Some of the main detection gratings,including double-slit,triangle aperture and slit,angular slit,cylindrical lens,graduallychanging-period grating,annular grating,are introduced.In addition,schemes like composite fork grating,Dammann vortex grating and integrated Dammann grating,are presented to detect the OAM state for coaxial multiplexed vortices. Besides diagnosing OAM state,measuring the intensity proportion of each OAM channel,known as OAM spectrum, in multiplexed vortices is also necessary in some cases.Therefore we also introduce the ways to measure the OAM spectrum,e.g.the OAM mode sorter,the gray-scale algorithm.

Optical vortices have special wave front structures and determined orbital angular momentum, which have huge prospect to apply. In this paper, a vortex beam with different topological charges is generated by using reflected type phase only liquid crystal spatial light modulator (LCSLM), and the interference characters of a vortex beam and a plane wave and a spherical wave are analyzed. Then, a radially polarized beam and azimuthally polarized beam are experimentally obtained. Experimental results show that, using the LCSLM, vortex beams with different topological charges can be generated dynamically in real time. Energy conversion efficiency is up to 60%.

Topological reactions of optical correlation vortices

The number of zero lines of the real and imaginary parts of the optical vortex (OV), both are the same as the topological charge (TC), and all of these lines intersect at one point where the phase singularity is. Furthermore, zero crossings distribute regularly on the transverse plane of the OV lattice. Zero lines of the real and imaginary parts of the non-diffracting fields without OV that generated by multi-waves interference are periodic but coincident. We stack two groups of these kind of zero lines which can be regarded as a set of zero lines of the real part and a set of zero lines of the imaginary part respectively, to satisfy the cross state of zero lines of an OV lattice. Then two groups of multi-waves corresponding to the two fields can be obtained. The expected OV lattice that generated by the two groups of engineered waves interference together is validated through both numerical simulations and experiments.

Vertical-cavity surface-emitting lasers (VCSELs) represent an attractive light source to integrate with OAM structures to realize chip-scale vortex lasers. Although pioneering endeavors of VCSEL-based vortex lasers have been reported, they cannot achieve large topological charges (less than l = 5) due to the insufficient space-bandwidth product (SBP) caused by the inherent limited device size. Here, by integrating a nanoprinted OAM phase structure on the VCSELs, we demonstrate a vortex microlaser with a low threshold and simple structure. A monolithic microlaser array with addressable control of vortex beams with different topological charges (l = 1 to l = 5) was achieved. Nanoprinting offers high degrees of freedom for the manipulation of spatial structures. To address the challenge of insufficient SBP, two-layer cascaded spiral phase plates were designed. Thereby, a vortex beam with l = 15 and mode purity of 83.7% was obtained. Our work paves the way for future chip-scale OAM-based information multiplexing with more channels.

In a number of theoretical (J. Opt. 6, 259 (2004)) and experimental (Opt Express 19, 5760 (2011)) works, an original fractional-TC optical vortex (with TC standing for topological charge) was shown to evolve into an integer optical vortex whose TC is equal to the nearest (i) smaller integer, if the fractional part is smaller than 1/2, and (ii) larger we show that the initial fractional optical vortex evolves into an integer optical vortex with TC equal to the nearest (i) smaller integer, if the fractional part is smaller than 0.12, and (ii) larger integer, if the fractional part is larger than 0.12. This can be explained by the fact that the additional center integer, if the fractional part is larger than 1/2. In this work, using numerical simulation, of singularity is generated on the beam periphery characterized by near zero-intensity (a millionth of the maximum), thus prohibiting the experimental detection, but allowing a numerical assessment.

We present a model based on the Fresnel diffraction scheme for the spatial coherence function of random fields created by scattering optical vortex and perfect vortex beams. By using the spatial coherence function we showed analytically, numerically, and experimentally the dependence and independence of the speckle size of an optical vortex and perfect optical vortex (POV) with a topological charge, respectively. We also showed in both cases the linear dependence of speckle size on the distance of propagation. Furthermore, we describe a regime in which the spatial coherence function is nonevolving for the optical vortex beam and the POV beam with the propagation distance.

In this work, by providing comprehensive theoretical foundations, we revisit and improve a simple and efficient method that has been used for generation of 2D orthogonal arrays of optical vortices with components having different topological charges (TCs). This method has been implemented by the diffraction of a plane wave from 2D gratings where the gratings' profiles are determined by iterative computational process. Here, based on the theoretical predictions, specifications of the diffraction gratings can be easily adjusted in a way to generate experimentally a heterogeneous vortex array with the desired power shares among different elements of the array. We use the diffraction of a Gaussian beam from a class of pure phase 2D orthogonal periodic structures having sinusoidal or binary profiles possessing a phase singularity, calling pure phase 2D fork-shaped gratings (FSGs). The transmittance of each of the introduced gratings is obtained by multiplying the transmittance of two pure phase 1D FSGs along x and y directions, having topological defect numbers lx and ly and phase variation amplitudes γx and γy, respectively. By solving the Fresnel integral, we show that the diffraction of a Gaussian beam from a pure phase 2D FSG leads to generation of a 2D array of vortex beams having different TCs and power shares. The power distribution among the generated optical vortices over the different diffraction orders can be adjusted by γx and γy, and it strongly depends on the profile of the grating. Meanwhile the TCs of the generated vortices depend on lx and ly and the corresponding diffraction orders, namely lm,n = -(mlx + nly) presents the TC of (m, n)th diffraction order. We recorded the intensity patterns of the experimentally generated vortex arrays which are fully consistent with the theoretically predicted results. Furthermore, the TCs of the experimentally generated vortices are measured individually by the diffraction of each of them through a pure amplitude quadratic curved-line (parabolic-line) grating. The absolute values and signs of the measured TCs are consistent with the theoretical prediction. The generated configuration of vortices with adjustable TC and power sharing features might find many applications such as non-homogeneous mixing of a solution consisting trapped particles.

Vortex beams, known as a typical form of structured light, possess numerous applications in various fields. Their widespread application prospects have then sparked an in-depth analysis of the generation and manipulation of vortex modes in an active cavity, as well as the development of high-performance vortex lasers. In this paper, we report on a new class of highly efficient and high-power Nd:YAG vortex lasers in a robust and compact configuration, which allows direct generation of vortex beams with an easily controllable topological charge both in the continuous-wave and pulsed operation regimes. The on-demand generation of intracavity vortex modes is realized based on a Q-plate by controlling the geometric phase inside the laser resonator. The maximum output power in the continuous-wave regime is 4.11 W with a slope efficiency of 37.9%. Besides, the vortex pulses are also achieved by including a Cr:YAG crystal in the cavity as a saturable absorber. The shortest pulse width is 142.8 ns at a pulse repetition rate of 232.6 kHz, with a maximum average output power of 1.05 W. Vortex modes with other topological charges can be obtained by simply changing the corresponding Q-plate without sacrificing the lasing efficiency. The experimental results can shed some light on the design and building of highly efficient and high-power vortex lasers together with a well-defined controllable topological charge, aiming at some specific applications.