Birth and evolution of an optical vortex.

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.

A full comparative analysis of the chain of phase singularities generated when a quasi-plane wave and a Gaussian beam pass a double-phase-ramp (DPR) converter is presented based on the theoretical and experimental data. The overall output beam structure includes a system of interrelated optical vortices (OVs) whose linear trajectories form a threedimensional singular skeleton which can be applied for the trapping and guiding of microparticles. An internal structure of each individual phase singularity is characterized by the OV topological charge and by the morphology parameters of the equal intensity ellipses in the OV-core area. The rectilinear equidistant OV trajectories form a chain in the transverse cross section, and their identical morphology parameters can be useful for the applications to metrology and micromanipulation. As a separate result, we consider the DPR-induced transformations of the incident Laguerre-Gaussian beams of the lowest orders and show that the incident multicharged OVs are transformed into small sub-chains of the OVs located in the near-axial region.

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.

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.

This paper outlines a new research project (CREST) for structure health monitoring. The purpose is to develop a networked sensor for static, full-axis deformation of large and long structures such as bridges, roads, and railways. We propose a novel technique based on the combined use of an optical vortex (OV) beam and correlation image sensors (CIS). Since the beam profile of OV involves definitive features known as zeros or phase singularities and they are invariant through propagation, we can encode and transmit information to determine geometrical relations between the sender and receiver nodes. By demodulating an amplitude and phase profile of OV using the CIS, all (three lateral and three rotational) displacement components can be determined. The basic optical setup is presented and the results of the fundamental experiments are shown.

In this paper the concept of optical vortex scanning microscope (OVSM) is presented. In the OVSM a sample is scanned by the focused laser beam with optical vortex. The beam possessing an optical vortex contains a line along which the phase is undetermined. At image plane this line is seen as the single point. The position of such special singular point (vortex points) changes when the beam is diffracted by a sample. The vortex point sensitivity is higher than the sensitivity of the whole focused beam. In the OVSM the position of vortex point is traced while scanning the sample. In this paper the behavior of focused vortex beam, when scanning the phase step is studied. This simple example illustrates both the OVSM idea and the method for data analysis.

Conventional correlation-based tracking methods are limited by the correlation window size and a lack in ability to track individual speckles or particles. Wang et al.[1] in their definition of Optical Vortex Metrology (OVM) propose that phase singularities are optimal encoders for position marking and can be localized with accuracy. OVM is widely used to process real-life and pseudo optical vortices [1, 2, 3], but so far the accuracy of the method has not been determined. Optical Vortex Metrology (OVM) uses phase singularities, produced in the complex signal generated by a Laguerre-Gauss filter operation applied to an image, for displacement measurements. Each phase singularity is essentially a pseudo-phase vortex, which can be uniquely identified from its structure, enabling it to be tracked between images. OVM is of interest here as not only the technique has potential to provide accurate position tracking but also pointwise rotation measurement.

Optical vortices, having doughnut-shaped spatial profile with zero intensity around the phase singularity, are of great interest due to its wide range of applications in variety of fields in science and technology. Two spatially separated entwined vortices with opposite topological charge forms a vortex dipoles. Due to their entwining nature, the vortex dipoles are created and annihilated in pairs of opposite charge, and carry a very stable topological structures. In recent times, vortex dipoles have created a lot of interest in studying in propagation dynamics [1] and optical vortex metrology [2]. Typically, optical vortex dipoles are created using holographic techniques. Recently, a new technique has been demonstrated to generate vortex dipole through coupling between the amplitude and phase differences of the superposing beams [3]. Here we report, for the first time to the best of our knowledge, on the generation of tunable vortex dipole directly from an optical parametric oscillator (OPO). Pumping a continuous wave (cw), doubly resonant OPO with Gaussian beam and carefully adjustment of the cavity mirrors, we have generated vortex dipole of power as much as 84 mW tunable across 968 nm-1181 nm. The generic scheme can used to generate vortex dipole at any wavelength and at all time scales.

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.

Chapter 4 Singular optics

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.

It is shown in this work that, with strong focusing of a beam with optical vortex and circular polarization, three energy flows take place in the focal plane: direct longitudinal, reverse longitudinal and azimuthal transverse flows. Calculations are made analytically using the Richards–Wolf formalism and by numerical simulation. Moreover, the energy rotation at different lengths from the optical axis occurs in different directions. Therefore, the focal plane intersects along the optical axis only part of the initial beam energy per unit time. The same energy part (other things being equal) intersects the focal plane along the positive direction of the optical axis when an optical vortex with cylindrical polarization is focused. The difference is that, if an optical vortex is present, then the transverse energy flux at the focus rotates around the optical axis. If an optical vortex is not present (a beam with only cylindrical polarization), then the average transverse flow in the focal plane is zero, though, in some regions in the focal plane, the flow is directed towards the optical axis and, in other regions, away from it. This behavior of the transverse energy flow at the focus (flow direction towards the optical axis and away from the optical axis) of a cylindrical vector beam can be deemed another kind of Hall effect.

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.

Topological reactions of optical correlation vortices

A Cooke triplet antenna used for near-field focusing is designed by a quasi-optical method. To design the Cooke triplet, we propose the quasi-optical method to reduce the aberration spot radius in the design step. The ray tracing method, calculating only the coordinates in the image plane rather than the field intensity, is used without considering the diffraction effect. The diffraction effect is taken into consideration only in the verification step. We get a conclusion that the essence of the aberration optimization is to optimize the phase distribution in the diffraction integral formula. In this case, only optimizing the aberrations without considering the diffraction effect is acceptable in the design step. While in the verification step, the diffraction is used to calculate the resolution of the Cooke triplet antenna. The field of view of the Cooke triplet is −18°∼18° in simulation and −14°∼14° in experiment. The feed sources can be positioned vertically to the optical axis in the image plane without the focal surface. The resolution of the off-axis beam is only 17% lower than the resolution of the central beam whose resolution is less than 0.5 cm at the distance 0.37 m. The experiment agrees well with the simulation.