Orbital angular momentum and informational entropy in perturbed vortex beams

Fragmental optical vortex for the detection of rotating object based on the rotational Doppler effect

Rotational Doppler effect (RDE), as a counterpart of the conventional linear Doppler effect in the rotating frame, has attracted increasing attention in recent years on rotational object detection. Many previous works have investigated the RDE based on the whole optical vortex field. In this work, we report on the RDE of the partially obstructed optical vortex and the corresponding rotational speed extraction method. Based on the orbital angular momentum (OAM) mode analysis theory, we establish the relationship between the OAM spectrum and the RDE frequency shift of fragmental optical vortex (FOV). The mechanism of the rotational speed extraction is analysed and validated by the numerical simulation and experiments. Further, a dual Fourier transformation method is proposed to accurately obtain the rotational speed which successfully overcomes the problem of the discrete distribution of the RDE signals. Our work may be useful for practical remote sensing based on the optical RDE metrology.

Self-healing property of the self-rotating beam

In this study, we demonstrate the self-healing of self-rotating beams with asymmetric intensity profiles. The proposed self-rotating beam exhibits an asymmetric intensity profile and self-healing properties in free-space propagation. In addition, the rotation direction and beam intensity profile of the self-rotating beam can be adjusted using the parameters a and b in the phase function. The effects of the position and size of the obstruction on the self-healing property of a self-rotating beam were studied both experimentally and numerically. The simulation and experimental results demonstrate that a self-rotating beam can overcome a block of obstacles and regenerate itself after a characteristic distance. Transverse energy flows were used to explain the self-healing properties. Moreover, the beam rotates during propagation, which can be used to capture and manipulate microscopic particles in a three-dimensional space. It is expected that these rotating beams with self-healing properties will be useful in penetrating obstacles for optical trapping, transportation, and optical therapy.

Orbital Angular Momentum of Superpositions of Optical Vortices Perturbed by a Sector Aperture

In optical communications, it is desirable to know some quantities describing a light field, which are conserved on propagation or resistant to some distortions. Typically, optical vortex beams are characterized by their orbital angular momentum (OAM) and/or topological charge (TC). Here, we show analytically that the OAM of a single rotationally symmetric optical vortex is not affected by an arbitrary-shape aperture or by other amplitude perturbations. For a superposition of two or several optical vortices (with different TCs), we studied what happens to its OAM when it is distorted by a hard-edge sector aperture. We discovered several cases when such perturbation does not violate the OAM of the whole superposition. The first case is when the incident beam consists of two vortices of the same power. The second case is when the aperture half-angle equals π multiplied by an integer number and divided by the difference between the topological charges. For more than two incident beams, this angle equals π multiplied by an integer number and divided by the greatest common divisor of all possible differences between the topological charges. We also show that such a sector aperture also conserves the orthogonality between the complex amplitudes of the constituent vortex beams. For two incident vortex beams with real-valued radial envelopes of the complex amplitudes, the OAM is also conserved, when there is a ±π/2 phase delay between the beams. When two beams with the same power pass through a binary radial grating, their total OAM is also conserved. We hope that these findings could be useful for optical communications since they allow for the identification of incoming optical signals by their OAM by registering only part of the light field within a sector aperture, thus reducing the cost of the receiving devices.

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.

Phase memory of optical vortex beams

Optical vortex beams are under considerable scrutiny due to their demonstrated potential for applications ranging from quantum optics to optical communications and from material processing to particle trapping. However, upon interaction with inhomogeneous material systems, their deterministic properties are altered. The way these structured beams are affected by different levels of disturbances is critical for their uses. Here, for the first time, we quantify the degradation of perfect optical vortex beams after their interaction with localized random media. We developed an analytical model that (1) describes how the spatial correlation and the phase variance of disturbance affect the phase distribution across the vortex beams and (2) establishes the regimes of randomness for which the beams maintain the memory of their initial vorticity. Systematic numerical simulations and controlled experiments demonstrate the extent of this memory effect for beams with different vorticity indices.

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.

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.

Optical vortex beams with a symmetric and almost symmetric OAM spectrum

We show both theoretically and numerically that if an optical vortex beam has a symmetric or almost symmetric angular harmonics spectrum [orbital angular momentum (OAM) spectrum], then the order of the central harmonic in the OAM spectrum equals the normalized-to-power OAM of the beam. This means that an optical vortex beam with a symmetric OAM spectrum has the same topological charge and the normalized-to-power OAM has an optical vortex with only one central angular harmonic. For light fields with a symmetric OAM spectrum, we give a general expression in the form of a series. We also study two examples of form-invariant (structurally stable) vortex beams with their topological charges being infinite, while the normalized-to-power OAM is approximately equal to the topological charge of the central angular harmonic, contributing the most to the OAM of the entire beam.

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.

Digital sorting perturbed Laguerre-Gaussian beams by radial numbers

We developed a new alterative technique of the digital sorting of Laguerre-Gaussian beams (LG) by radial numbers resorting to algebra of the high-order intensity moments. The term "digital mode sorting" involves sorting the main mode characteristics (in the form of a mode spectrum) by the computer cells. If necessary, the spatial mode spectrum can be reproduced, for example, by means of a spatial light modulator. In the experiment, we investigated both a single LG mode and a composition of LG modes with the same topological charge but different radial numbers subjected to perturbations via a hard-edged circular aperture. The LG beams sorting was accomplished by monitoring the amplitude spectrum of the triggered secondary LG modes then recovering the sorted modes and the perturbed beam as a whole. We have revealed degenerate states of the perturbed LG beam composition when the one kth mode in the amplitude spectrum can be related to a set of LG modes with the same radial numbers. In order to decrypt and to sort beams in such a degenerate state, it is necessary to know several keys, the number of which is equal to the number of LG modes in the initial wave composition. We were also able to analyze and to sort such degenerate mode states. For monitoring the measure of uncertainty arising in the perturbed beam, we measured informational entropy (Shannon entropy).

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

We theoretically show that optical vortices conserve the integer topological charge (TC) when passing through an arbitrary aperture or shifted from the optical axis of an arbitrary axisymmetric carrier beam. If the beam contains a finite number of off-axis optical vortices with same-sign different TC, the resulting TC of the beam is shown to equal the sum of all constituent TCs. If the beam is composed of an on-axis superposition of Laguerre-Gauss modes (n, 0), the resulting TC equals that of the mode with the highest TC. If the highest positive and negative TCs of the constituent modes are equal in magnitude, the "winning" TC is the one with the larger absolute value of the weight coefficient. If the constituent modes have the same weight coefficients, the resulting TC equals zero. If the beam is composed of two on-axis different-amplitude Gaussian vortices with different TC, the resulting TC equals that of the constituent vortex with the larger absolute value of the weight coefficient amplitude, irrespective of the correlation between the individual TCs. In the case of equal weight coefficients of both optical vortices, TC of the entire beam equals the greatest TC by absolute value. We have given this effect the name "topological competition of optical vortices".