The paper deals with the edge diffraction of the single charged Laguerre-Gaussian beam outside the waist. Based on the Kirchhoff-Fresnel integral, the behavior of the optical vortex (OV) migration during sequential beam blocking by the straight edge of the screen is performed analytically. Universal combination of the diffracting-beam parameters determining the shape for the OV spiral trajectories is revealed. Parametric equations describing the OV dislocation dependent on the beam screening degree are derived. In addition, the spiral trajectory, which describes the OV positions in the case of the screen-edge translatory motion across the incident beam under conditions of the weak diffraction perturbation, is obtained. As a result, the equation for a wider region of diffraction perturbation is selected as a best matching function for a certain variation interval of the beam parameters. The work has prepared the basis for solving the inverse diffraction problem: determining parameters of the beam diffractive interaction with obstacles from the migration trajectory of the optical vortex.
Based on the Kirchhoff-Fresnel approximation, we consider behavior of the optical vortices (OV) upon propagation of the diffracted Laguerre-Gaussian (LG) beams with topological charge |m| = 1, 2. Under conditions of weak diffraction perturbation (i.e. the diffraction obstacle covers only the far transverse periphery of the incident LG beam), the OVs describe almost perfect 3D spirals within the diffracted beam body, which is an impressive demonstration of the helical nature of an OV beam. The far-field OV positions within the diffracted beam cross section depend on the wavefront curvature of t…
We study, both theoretically and by experiment, migration of the amplitude zeros within a fixed cross section of the edge-diffracted optical-vortex beam, when the screen edge performs permanent translation in the transverse plane from the beam periphery towards the axis. Generally, the amplitude zeros (optical-vortex cores) describe spiral-like trajectories. When the screen edge advances uniformly, the motion of the amplitude zeros is not smooth and sometimes shows anomalously high rates, which make an impression of instantaneous "jumps" from one position to another. We analyze the nature, con…
Edge diffraction of a circular optical vortex (OV) beam transforms its singular structure: a multicharged axial OV splits into the set of single-charged ones that form the 'singular skeleton' of the diffracted beam. The OV positions in the beam cross section depend on the propagation distance as well as on the edge position with respect to the incident beam axis, and the OV cores describe regular trajectories when one or both of these parameters change. However, the trajectories are not always continuous; they may be accompanied with topological reactions, including emergence of new singularit…
When a circularly-symmetric light beam with optical vortex (OV) diffracts at an opaque screen with the sharp edge, the OV core is displaced from the beam axis and, in case of the m-charged incident OV, decomposed into |m| single-charged ones. By means of numerical simulations and based on examples of incident beams with topological charges |m| =1, 2, 3 we show that, while the screen edge monotonously advances towards the beam axis, the OVs in the diffracted beam cross section move away from the incident beam axis along spiral-like trajectories. The trajectories contain fine structure details t…
Reformulation of conventional beam definitions into their bidirectional versions and use of Hertz potentials make beam fields exact vector solutions to Maxwell's equations. This procedure is applied to higher-order elegant Laguerre-Gaussian beams of TM and TE polarization. Their vortex and anti-vortex co-axial compositions of equal and opposite topological charges are given in a closed analytic form. Polarization components of the composed beams are specified by their radial and azimuthal indices. The longitudinal components are common for beam compositions of both types, meanwhile their trans…
Based on the Kirchhoff-Fresnel approximation, we numerically analyze spatial characteristics of the light field formed after a circular Laguerre-Gaussian beam with a single-charged optical vortex (OV) passes the transparent screen with a rectilinear phase step. The main attention is paid to the localization and interactions of the OVs, which form the singular skeleton of the transformed field. The phase-step influence depends on its value and position with respect to the beam axis. Upon "weak perturbation" (low phase step) the main effect is that the OV is shifted from the initial axial positi…
We propose a generalization of the Laguerre Gauss Transform to include elliptical kernels in vortices metrology. To that end, we broaden the original pseudo field obtained using Laguerre Gauss Transform (LGT) that usually includes circularly a symmetric Gaussian core to include two parameters, the widths of the Gaussian filter in two perpendicular directions, thus generating a family of slightly elliptical kernels. This permits the generation of a family of very close vortices that improves the identification of homologous ones and the precision and locality of the measurements. We show exampl…
A generalized family of scalar structured Gaussian modes including helical-Ince--Gaussian (HIG) and Hermite--Laguerre--Gaussian (HLG) beams is presented with physical insight upon a hybrid topological evolution nature of multi-singularity vortex beams carrying orbital angular momentum (OAM). Considering the physical origins of intrinsic coordinates aberration and the Gouy phase shift, a closed-form expression is derived to characterize the general modes in astigmatic optical systems. Moreover, a graphical representation, Singularities Hybrid Evolution Nature (SHEN) sphere, is proposed to visua…
The recognition in the 1990s that vortex beams (VBs), paraxial light beams with optical vortices, carry orbital angular momentum (OAM), has benefited applications ranging from optical manipulation to high-dimensional classical and quantum information communications. The transverse profiles of common VBs, e.g., Laguerre-Gaussian beam and high-order Bessel beam, are hollow donuts whose radii grow up with OAM inevitably. The inherently unperfect character of the VBs that the radius is always positively correlated with OAM, restricts the application of the VBs in many scenarios like fiber optic da…
In optics, we can generate vortex beams using specific methods such as spiral phase plates or computer generated holograms. While, in nature, it is worth noting that water can produce vortices by a circularly symmetrical hole. So, if a light beam can generate vortex when it is diffracted by an aperture? Here, we show that the light field in the Fresnel region of the diffracted circularly polarized beam carries orbital angular momentum, which can transfer to the trapped particles and make orbital rotation.
Wuhan Institute of Technology（武汉工程大学）；Louisiana State University（路易斯安那州立大学）；Southern University of Science and Technology（南方科技大学）
Orbital angular momentum of light is regarded as a valuable resource in quantum technology, especially in quantum communication and quantum sensing and ranging. However, the OAM state of light is susceptible to undesirable experimental conditions such as propagation distance and phase distortions, which hinders the potential for the realistic implementation of relevant technologies. In this article, we exploit an enhanced deep learning neural network to identify different OAM modes of light at multiple propagation distances with phase distortions. Specifically, our trained deep learning neural…
In both light optics and electron optics, the amplitude of a wave scattered by an object is an observable that is usually recorded in the form of an intensity distribution in a real space image or a diffraction image. In contrast, retrieval of the phase of a scattered wave is a well-known challenge, which is usually approached by interferometric or numerical methods. In electron microscopy, as a result of constraints in the lens setup, it is particularly difficult to retrieve the phase of a diffraction image. Here, we use a defocused beam generated by a nanofabricated hologram to form a refere…
In this paper we consider the photon vortex beam in the rotating medium, where the rotating velocity acts as an effective vector potential. Using the Riemann-Silberstein vector, we construct the photon wave function. Using the Maxwell equations and the first-order Minkowski constitutive relations we get the dynamic equations of photon in moving medium. For the stationary states, the dynamic equations can be written as a Dirac-like equation. We obtain the approximate photon vortex beam solutions by the given the medium's different velocity distribution and find the diffracting and nondiffractin…
It is shown that the vector-vortex Laguerre-Gauss modes with polarization-orbital angular momentum (OAM) entanglement are the vector solutions of the Maxwell equations in a graded-index medium. Focusing of linearly and circularly polarized vortex light beams with nonzero azimuthal and radial indices in a cylindrical graded-index medium is investigated. The wave shape variation with distance taking into account the spin-orbit and nonparaxial effects is analyzed. Effect of long-term periodical revival of wave packets due to mode interference in a graded-index cylindrical optical waveguide is dem…
In this work, we report calculation for Compton scattering of a gamma-ray vortex with Laguerre Gaussian wave function on an electron in the framework of the relativistic quantum mechanics. We have found the following unexpected feature. The momentum of scattered photon distributes outside of the reaction plane determined by the incident photon and the scattered electron, and hence the energy of the scattered photon also distributes. This novel result indicates that one can identify a gamma-ray vortex by measuring coincidentally the scattered angles of the electron and photon.
We use the T-matrix formalism in combination with the method of far-field matching to evaluate the optical force exerted by Laguerre-Gaussian (LG) light beams on a spherical (Mie) particle. For both non-vortex and optical vortex LG beams, the theoretical results are used to analyze the optical-force-induced dynamics of the scatterer near the trapping points represented by the equilibrium (zero-force) positions. The regimes of linearized dynamics are described in terms of the stiffness matrix spectrum and the damping constant of the ambient medium. For the purely azimuthal LG beams, the dynamic…
Laguerre-Gauss (LG) modes represent an orthonormal basis set of solutions of the paraxial wave equation. LG are characterized by two integer parameters $n$ and $\ell$ that are related to the radial and azimuthal profile of the beam. The physical dimension of the mode is instead determined by the beam waist parameter $w_0$: only LG modes with the same $w_0$ satisfy the orthogonality relation. Here, we derive the scalar product between two LG modes with different beam waists and show how this result can be exploited to derive different expansions of a generic beam in terms of LG modes. In partic…
To exploit the full potential of the transverse spatial structure of light using the Laguerre-Gaussian basis, it is necessary to control the azimuthal and radial components of the photons. Vortex phase elements are commonly used to generate these modes of light, offering precise control over the azimuthal index but neglect the radially dependent amplitude term which defines their associated corresponding transverse profile. Here we experimentally demonstrate the generation of high purity Laguerre-Gaussian beams with a single step on-axis transformation implemented with a dielectric phase-ampli…
Improving axial resolution is of paramount importance for three-dimensional optical imaging systems. Here, we investigate the ultimate precision in axial localization using vortex beams. For Laguerre-Gauss beams, this limit can be achieved with just an intensity scan. The same is not true for superpositions of Laguerre-Gauss beams, in particular for those with intensity profiles that rotate on defocusing. Microscopy methods based on rotating vortex beams may thus benefit from replacing traditional intensity sensors with advanced mode-sorting techniques.
We theoretically and experimentally studied a novel class of vortex beams named open vortex beams (OVBs). Such beams are generated using Gaussian beams diffracted by partially blocked fork-shaped gratings (PB-FSGs).The analytical model of OVBs in the near field and far field is given by superpositions of Hypergeometric (HyG) modes. Unlike conventional integer and fractional vortex beams, the OVBs can have both an open ring structure and an integer topological charge (TC). The TC is decided by the circumference covered by the open ring. It is also quantitatively shown that a $π/2$ rotation of t…
We propose and experimentally verify a novel scheme for diagnosing the order of a perturbed optical vortex using its 2-D spatial autocorrelation function. The order of a vortex was found to be equal to the number of dark rings or zero points present in the correlation function. We have provided a compact analytical expression for the correlation function in the form of Laguerre polynomials. Further, we have utilized the divergence of the first zero point of Laguerre polynomials upon propagation to obtain information about higher-order spatial modes, and compared it with our experimental result…
In the past few years, orbital angular momentum beams have contributed significantly to many applications in the optical-related fields due to their unique phase characteristics. Here, a novel approach about phase-helices structure for Laguerre-Gaussian beams carrying orbital angular momentum is presented. This proposal is based on numerical simulations to track the rotation of each helix in the phase and shows an impact on the rotational speed of the Laguerre-Gaussian phase distribution by using the characteristic of distinguishability of each helix. If the helices in the phase of a vortex ca…
Thus, the results of our studies lie in developing and implementing the basic principles of digital sorting the Laguerre-Gauss modes by radial numbers both for a non-degenerate and a degenerate state of a vortex beam subject to perturbations in the form of a hard-edged aperture of variable radius. The digital sorting of LG beams by the orthogonal basis involves the use of higher-order intensity moments, and subsequent scanning of the modulated beam images at the focal plane of a spherical lens. As a result, we obtain a system of linear equations for the squared mode amplitudes and the cross am…
We use phase-shifting digital holography to measure the amplitude and phase of twisted light. In our experiment, a spatial light modulator generates the studied vortex beams in addition to a co-propagating reference beam with a controllable relative phase. We show complex field measurements for single and superposition Laguerre Gaussian (LG) modes, demonstrate full modal decompositions into LG and orbital angular momentum (OAM) power spectral bases, provide error analysis and demonstrate high insensitivity to detector misalignment to show the robustness of the technique. This enables rapid det…
Neglecting the electric field component along the light's propagation direction is a common practice, known as paraxial approximation. However, experimental evidence and theory on the head-on excitation of atoms by Laguerre-Gaussian beams reveal that the full vector character of the light field has to be taken into account. Optical vortices are a large family of fields, being Laguerre-Gaussian only one particular case. Now we extend the study of the applicability of the paraxial approximation to a broader set of optical vortices by considering the interaction of atoms with generalized Laguerre…
In this work, an analytic expression is derived for the optical field generated by an off-axis Gaussian beam diffracted by a Discrete Vortex-Producing Lens. With this system, a multifocal arrangement of asymmetric optical vortices is obtained whose topological charge values change with the position along the optical axis. This scheme allows obtaining a principal asymmetric vortex corresponding with the topological charge value of the phase mask and secondary ones with different charges. With the analytical expression, the effects induced by the discretization and misalignment on the irradiance…
The edge diffraction of a homogeneously polarized light beam is studied theoretically based on the paraxial optics and Fresnel-Kirchhoff approximation, and the dependence of the diffracted beam pattern of the incident beam polarization is predicted. If the incident beam is circularly polarized, the trajectory of the diffracted beam centre of gravity experiences a small angular deviation from the geometrically expected direction. The deviation is parallel to the screen edge and reverses the sign with the polarization handedness; it is explicitly calculated for the case of a Gaussian incident be…
Perfect vortex beams are the orbital angular momentum (OAM)-carrying beams with fixed annular intensities, which provide a better source of OAM than traditional Laguerre- Gaussian beams. However, ordinary schemes to obtain the perfect vortex beams are usually bulky and unstable. We demonstrate here a novel generation scheme by designing planar Pancharatnam-Berry (PB) phase elements to replace all the elements required. Different from the conventional approaches based on reflective or refractive elements, PB phase elements can dramatically reduce the occupying volume of system. Moreover, the PB…
A simple and high-efficiency method for direct generation of all-solid-state single- and dual-vortex lasers operating at eye-safe wavelengths is reported in a compact diode-end-pumped Nd:YAG laser cavity. By off-axis pumping of the Nd:YAG in two orthogonal directions, the original cavity mode, fundamental Gaussian mode (TEM00), is directly transformed into first-order Laguerre-Gaussian (LG01) mode with single- or dual-vortex structures depending on the degree of the off-axis pumping. Moreover, the single- and dual-vortex lasers can be produced in simultaneous eye-safe dual-wavelength operation…
Far-field slit-diffraction of circular optical-vortex (OV) beams is efficient for measurement of the topological charge (TC) magnitude but does not reveal its sign. We show that this is because in the common diffraction schemes the diffraction plane coincides with the incident OV waist plane. With explicit involvement of the incident beam spherical wavefront and based on the examples of Laguerre-Gaussian modes we show that the far-field profile possesses an asymmetry depending on the wavefront curvature and the TC sign. These features enable simple and efficient ways for the simultaneous diagn…
Spatiotemporal optical vortex (STOV) is a unique optical vortex with phase singularity in the space-time domain and the photons in a STOV can carry transverse orbital angular momentum (OAM). The STOV shows many fantastic properties which are worth exploring. Here, we theoretically and experimentally study the diffraction property of STOV, which is a fundamental wave phenomenon. The diffraction behaviors of STOVs are obviously affected by the transverse OAM. The diffraction patterns of STOV pulses diffracted by a grating show multi-lobe structure with each gap corresponding to 1 topological cha…
Laguerre-Gaussian (LG) beams carrying orbital angular momentum (OAM) have shown promise in deep tissue imaging, medical diagnostics, and optical communication due to their robust propagation properties through scattering media. However, an exact model that provides a mechanism for the enhanced scattering properties of LG beams over Gaussian beams has not been established till date. Here, we examine this issue by studying the propagation of polarized vortex beams transmitted through tissue-like turbid scattering media. We demonstrate that the intensity profile has a much more profound effect on…
A novel type of discrete basis for paraxial beams is proposed, consisting of monomial vortices times polynomials of Gaussians in the radial variable. These bases have the distinctive property that the effective size of their elements is roughly independent of element order, meaning that the optimal scaling for expanding a localized field does not depend significantly on truncation order. This behavior contrasts with that of bases composed of polynomials times Gaussians, such as Hermite-Gauss and Laguerre-Gauss modes, where the scaling changes roughly as the inverse square root of the truncatio…
Analytical forms of the optical helicity and optical chirality of monochromatic Laguerre-Gaussian optical vortex beams are derived up to second order in the paraxial parameter $kw_0$. We show that input linearly polarised optical vortices which possess no optical chirality, helicity or spin densities can acquire them at the focal plane for values of a beam waist $w_0 \approx λ$ via an OAM-SAM conversion which is manifest through longitudinal (with respect to the direction of propagation) fields. We place the results into context with respect to the intrinsic and extrinsic nature of SAM and OAM…
In optics, mode conversion is an elegant way to switch between Hermite Gaussian and Laguerre Gaussian beam profiles and thereby impart orbital angular momentum onto the beam and to create vortices. In optics such vortex beams can be produced in a setup consisting of two identical cylinder lenses. In electron optics, quadrupole lenses can be used for the same purpose. Here we investigate generalized asymmetric designs of a quadrupole mode converter that may be realized within the constraints of existing electron microscopes and can steer the development of dedicated vortex generators for high b…
