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.

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.

Super bursts of the orbital angular momentum in astigmatic-invariant structured LG beams.

A structured Laguerre-Gaussian (sLG) beam in an optical system with an astigmatic element acquires additional degrees of freedom in the form of changing the fine structure of the beam, its orbital angular momentum (OAM), and topological charge. We have theoretically and experimentally revealed that at a certain ratio between the beam waist radius and the focal length of the cylindrical lens, the beam turns into an astigmatic-invariant one, and such a transition does not depend on the beam radial and azimuthal numbers. Moreover, in the vicinity of the OAM zero, its sharp bursts occur, the magnitude of which significantly exceeds the initial beam OAM and grows rapidly as the radial number increases.

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).

Orbital angular momentum and informational entropy in perturbed vortex beams.

We theoretically and experimentally investigated transformations of vortex beams subjected to sector perturbations in the form of a hard-edged aperture. The transformations of the vortex spectra, the orbital angular momentum (OAM), and the informational entropy of the perturbed beam were measured. We found that relatively small angular sector perturbations have almost no effect on the OAM, although the informational entropy is rapidly increasing due to the birth of new optical vortices caused by diffraction at the diaphragm edges. At large perturbation angles, the uncertainty principle between the angle and OAM involves vortices, with both positive and negative topological charges, so that the OAM decreases to almost zero, and the entropy increases sharply.

Measurement of the vortex and orbital angular momentum spectra with a single cylindrical lens.

A new technique for measuring the degenerate spectra of optical vortices and orbital angular momentum (OAM) of singular beams is theoretically studied and experimentally verified. The technique is based on measuring the intensity moments of higher orders of a beam containing vortices with both positive and negative topological charges. The appropriate choice of the vortex mode amplitudes of the combined beam forms anomalous regions in the form of resonant dips and bursts in the OAM spectrum. Since the intensity moments for vortices with positive and negative topological charges are the same (degenerate) for an axially symmetric beam, it was necessary to break the symmetry of the beam, so measurements were taken at the plane of the double focus of a cylindrical lens. The calibration measurements showed that the experimental error is not higher than 3.5%. The technique was implemented for measuring and analyzing combined beams with OAM anomalies. It was found that the dips and bursts in the OAM spectrum are caused by the vortex avalanche induced by weak perturbations of the holographic grating responsible for shaping the beam. The OAM dips or bursts are controlled by the ratio between the energy fluxes of the vortex avalanche with positive or negative topological charges.

Vortex avalanche in the perturbed singular beams.

We have theoretically and experimentally considered transforming a single optical vortex beam into the vortex avalanche caused by weak local perturbations of the holographic grating responsible for the beam shaping. The vortex avalanche is accompanied by a sharp change of the orbital angular momentum (OAM) so that its dependence on the holographic grating perturbations forms the OAM spectrum. We revealed that the restored vortex beam has anomalous regions of the OAM spectrum in a form of sharp dips and bursts (resonances) that occur when the integer perturbation parameter coincides with a topological charge of one of the vortex modes. We found also that the intensity of the perturbed beam is nonuniformly distributed among the vortex modes with positive and negative topological charges. They form two groups of satellites with clearly marked intensity maxima. As the grating perturbation increases, the initial beam intensity is almost completely pumped over into the vortex avalanche with nearly the same energy redistribution among the satellites.

Measurement of the vortex spectrum in a vortex-beam array without cuts and gluing of the wavefront.

We present a new technique for measuring the squares of the amplitudes and phases of partial vortex-beams in a complex beam array in real time. The technique is based on measuring the high-order intensity moments and analyzing the solutions of a system of linear equations. Calibration measurements have shown that the measurement error at least for a beam array of 10-15 beams does not exceed 4%.

Super pulses of orbital angular momentum in fractional-order spiroid vortex beams.

We consider optical properties of hypergeometric-Gaussian beam compositions with spiral-like intensity and phase distributions called spiroid beams. Their orbital angular momentum as a function of a fractional-order topological charge has a chain of super-pulses (bursts and dips). The form of the super-pulses can be controlled by the spiral parameters. Such a phenomenon can be used in optical switches and triggers for optical devices and communication systems.

Transmission of fractional topological charges via circular arrays of anisotropic fibers.

We demonstrate that in circular arrays of anisotropic fibers at certain distribution of anisotropy directors robust transmission of optical fields with half-integer topological charges is possible. We show that this is possible because the supermodes of such arrays may contain in their circularly polarized components half-integer topological charges of opposite values. We also study the structure of singularities in these supermodes.

Helical-core fiber analog of a quarter-wave plate for orbital angular momentum.

We have studied the effect of a twist defect on the conversion of the fundamental mode (FM) into an optical vortex (OV) in a helical-core fiber (HCF). We have shown that if such a twist defect is situated in the middle of the HCF, which converts the FM into an OV, such a fiber system can continuously change the orbital angular momentum (OAM) of the output field from 0 to 1 (in a.u.). This control of the OAM is achieved by variation of the twist angle. In this action upon the OAM, this system has analogy with the quarter-wave plate, which is able to change the spin angular momentum. We also introduced the generalized Stokes parameters (SPs) and Poincaré sphere to visualize evolution of the superposition of states with zero and nonzero OAM. Connection of SPs with geometric characteristics of the location of singularity is made.

Vector erf-Gaussian beams: fractional optical vortices and asymmetric TE and TM modes.

We have considered the paraxial vector erf-Gaussian beams with field distribution in the form of the error function that are shaped by the cone of plane waves with a fractional step of the azimuthal phase distribution modulated by the Gaussian envelope. We have revealed that the initial distributions of the transverse electric and transverse magnetic fields have a far from standard form but at the far diffraction field the field distributions recover nearly the symmetric form.

Subwave spikes of the orbital angular momentum of the vortex beams in a uniaxial crystal.

We have theoretically predicted gigantic spikes of orbital angular momentum caused by conversion processes of the centered optical vortex in the circularly polarized components of an elliptic vortex beam propagating perpendicularly to the crystal optical axis. We have experimentally observed the conversion process inside subwave deviations of the crystal length. We have found that the total orbital angular momentum of the wave beam is conserved.

Natural shaping of the cylindrically polarized beams.

We have experimentally and theoretically shown that the circularly polarized beam bearing a singly charged optical vortex propagating through a uniaxial crystal can be split after focusing into the radially and azimuthally polarized beams in the vicinity of the focal area provided that the polarization handedness and the vortex topological charge have opposite signs.

Indistinguishability limit for off-axis vortex beams in uniaxial crystals.

The circular polarization components of a vortex beam in a uniaxial crystal exhibit complex propagation characteristics. We demonstrate how the amount of splitting may be distinguished by use of a vortex beam. We predict and experimentally verify a threshold angle subtending the crystal and beam axes, below which the splitting is indistinguishable.

Filter of optical vortices: highly twisted high-birefringence optical fibers.

Modes and bandgap structure of highly twisted high-birefringence weakly guiding fibers are studied in the scalar approximation. It is shown that within the gap the system can serve as a filter of circular optical vortices.

Focusing of wedge-generated higher-order optical vortices.

We experimentally show that when higher-order optical vortices generated by a system of dielectric wedges are subjected to focusing they do not lose their structural stability.

Generation of higher-order optical vortices by a dielectric wedge.

Diffraction of a Gaussian beam with a system of successively located optical wedges is considered. It is shown that the system is able to form higher-order optical vortices.