Chapter 4 Singular optics

A qualitative analysis of the mechanical response of rate-dependent media caused by a one-dimensional plane smooth wave front and by a continuous wave front attenuating in the media is performed by an underdetermined system of nonlinear partial differential equations. The analysis reveals that smooth strain, particle velocity, and stress profiles, which the smooth wave front has, are not similar and that the wave front is composed of some partial waves having different properties. The property is represented by a set of strain rate, acceleration, and stress rate. The wave front derived here from the analysis is composed of four different partial waves. The front of the wave front is necessarily a contraction wave in which strain, particle velocity, and stress increase with time, while the rear is a rarefaction wave where they all decrease with time. Between these two wave fronts there are two remaining wave fronts. We call these wave fronts mesocontraction waves I and II. Wave front I is a wave in which stress decreases notwithstanding the increase in strain and particle velocity with time, which is followed by the other, i.e., wave front II, where with time, particle velocity, and stress decrease in spite of the increase in strain. The continuous wave front having continuous and nonsmooth profiles of strain, particle velocity, and stress can also be composed of four waves. These waves possess the same property as the corresponding waves in the smooth wave front mentioned above. The velocities at three boundaries that the waves have are discontinuous. Therefore, these four wave fronts are independent waves, just as a shock wave and a rarefraction wave. Specifically, the front wave, i.e., a contraction wave front is being outrun by a second wave front, the second one is being outrun by a third wave front, and the third is being outrun by a fourth wave front, i.e., a rarefaction wave. We call the second wave front degenerate contraction wave I. We also call the third one degenerate contraction wave II. The stress-strain path and the stress-particle velocity path at a position in a rate-dependent medium which is passed by the continuous wave front are schematically shown.

The spatial dynamics of vortex dipole nested in a Gaussian opti cal beam is analyzed with scal ar diffraction theory. It is shown that the generation and annihilation of vortices are accompanied by extrem e wave front distortions. Process of nucleation and annihilation of vortex dipole is considered to confirm the vortex aftereffect in the process of transformation of an aberration wave front into a singular one. After the vortex dipole annihilation the absolute values of the average and Gaussian curvatures increase greatly in the local areas of wave front. The phase reconstruction error increases in these areas in the process of wave front sensing. To investigate the features of the vortex aftereffect for simple field the light rays which take part in the combined translational and rotary motion of energy around the field zero-lines are constructed. The phase is computed as a potential of the optical field using the light rays. Integration of the phase gradient along the ray or energy-stream line allows a uniq ue value of the phase to be connected with a point of the ray. Wave front of the beam is constructed from the computed phase. As in the vicinity of the vortex core the energy-stream line takes a spiral shape the phase incursion along th e line increases. After the vortex annihilation and vanishing of the spiral shape the incursion remains and creates the vortex aftereffect in the form of the extreme wave front distortions. The revealed effect should be taken into accoun t when constructing the systems of adaptive optics aimed at functioning in strong turbulence conditions. Keywords: Singular phase, optical vortex, strong turbulence, phase reconstruction

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.

For the first time we report the study of self-induced nonlinear lens in a liquid crystal cell with azo dye and the phase dislocations on a wave front of the beam passing through the lens. Depending on the experimental conditions, circular edge dislocation or optical vortices pairs may be created on the wave front.

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 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%.

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.

Radially polarized beams represent an important member of the family of vector beams, in particular due to the possibility of using them to create strong and tightly focused longitudinal fields, a fundamental property that has been exploited by applications ranging from microscopy to particle acceleration. Since the properties of such a focused beam are intimately related to the Gouy phase shift, proper knowledge of its behavior is crucial. Terahertz microscopic imaging is used to extract the Gouy phase shift of the transverse and longitudinal field components of a tightly focused, radially polarized beam. Since the applied terahertz time-domain approach is capable of mapping the amplitude and phase of an electromagnetic wave in space, we are able to directly trace the evolution of the geometric phase as the wave propagates through the focus. We observe a Gouy phase shift of 2π for the transverse and of π for the longitudinal component. Our experimental procedure is universal and may be applied to determine the geometric phase of other vector beams, such as optical vortices, or even arbitrarily shaped and polarized propagating waves.

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.

Features of the formation of an x-ray diffraction image by x-ray section topography are considered for a strongly distorted region near the dislocation axis in silicon single crystals. The results of experimental investigations and numerical calculations of the diffraction contrast and section topographs of rectilinear dislocations are presented for their different orientations and positions in the scattering triangle in silicon single crystals. A comparison and an analysis of the experimental topographs and the simulated images lead to the conclusion that the structure of the image of a dislocation strongly depends on its position and orientation in the scattering triangle. It has been found that each point of the strongly distorted region of the elastic field of a dislocation becomes a source of a new wave field propagating under the dislocation in a new scattering triangle. This new field interferes with the primary wave field forming the observed diffraction image of a dislocation. The addition of these waves with regard to their amplitudes and phases results in a large variety of images of defects. A comparison of different dislocation orientations in the Borrmann triangle allowed us to evaluate the role of different effects determined by the interference of the initial and newly formed wave fields, to determine on this basis the main dislocation parameters, and to optimize the diffraction conditions of the topographic measurement for the investigation of elastic-field characteristics.

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.

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.

When a phase singularity is suddenly imprinted on the axis of an ordinary Gaussian beam, an optical vortex appears and starts to grow radially, by effect of diffraction. This radial growth and the subsequent evolution of the optical vortex under focusing or imaging can be well described in general within the recently introduced theory of circular beams, which generalize the hypergeometric-Gaussian beams and which obey novel kinds of ABCD rules. Here, we investigate experimentally these vortex propagation phenomena and test the validity of circular-beam theory. Moreover, we analyze the difference in radial structure between the newly generated optical vortex and the vortex obtained in the image plane, where perfect imaging would lead to complete closure of the vortex core.

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.