Orbital angular momentum and topological charge of a Gaussian beam with multiple optical vortices

Here we study theoretically and numerically a Gaussian beam with multiple optical vortices with unitary topological charge (TC) of the same sign, located uniformly on a circle. Simple expressions are obtained for the Gaussian beam power, its orbital angular momentum (OAM), and TC. We show that the OAM normalized to the beam power cannot exceed the number of vortices in the beam. This OAM decreases with increasing distance from the optical axis to the centers of the vortices. The topological charge, on the contrary, is independent of this distance and equals the number of vortices. The numerical simulation corroborates that after passing through a random phase screen (diffuser) and propagating in free space, the beams of interest can be identified by the number of local intensity minima (shadow spots) and by the OAM.

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Measurement of the orbital angular momentum of an astigmatic Hermite–Gaussian beam

Computer Optics ◽

10.18287/2412-6179-2019-43-3-356-367 ◽

2019 ◽

Vol 43 (3) ◽

pp. 356-367

Author(s):

V.V. Kotlyar ◽

A.A. Kovalev ◽

A.P. Porfirev

Keyword(s):

Angular Momentum ◽

Gaussian Beam ◽

Orbital Angular Momentum ◽

Topological Charge ◽

Optical Axis ◽

Optical Vortex ◽

Cylindrical Lens ◽

Optical Vortices ◽

Special Choice ◽

Different Types

Here we study three different types of astigmatic Gaussian beams, whose complex amplitude in the Fresnel diffraction zone is described by the complex argument Hermite polynomial of the order (n, 0). The first type is a circularly symmetric Gaussian optical vortex with and a topological charge n after passing through a cylindrical lens. On propagation, the optical vortex "splits" into n first-order optical vortices. Its orbital angular momentum per photon is equal to n. The second type is an elliptical Gaussian optical vortex with a topological charge n after passing through a cylindrical lens. With a special choice of the ellipticity degree (1: 3), such a beam retains its structure upon propagation and the degenerate intensity null on the optical axis does not “split” into n optical vortices. Such a beam has fractional orbital angular momentum not equal to n. The third type is the astigmatic Hermite-Gaussian beam (HG) of order (n, 0), which is generated when a HG beam passes through a cylindrical lens. The cylindrical lens brings the orbital angular momentum into the original HG beam. The orbital angular momentum of such a beam is the sum of the vortex and astigmatic components, and can reach large values (tens and hundreds of thousands per photon). Under certain conditions, the zero intensity lines of the HG beam "merge" into an n-fold degenerate intensity null on the optical axis, and the orbital angular momentum of such a beam is equal to n. Using intensity distributions of the astigmatic HG beam in foci of two cylindrical lenses, we calculate the normalized orbital angular momentum which differs only by 7 % from its theoretical orbital angular momentum value (experimental orbital angular momentum is –13,62, theoretical OAM is –14.76).

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Asymmetric hypergeometric laser beams

Computer Optics ◽

10.18287/2412-6179-2019-43-5-735-740 ◽

2019 ◽

Vol 43 (5) ◽

pp. 735-740

Author(s):

V.V. Kotlyar ◽

A.A. Kovalev ◽

E.G. Abramochkin

Keyword(s):

Exact Solution ◽

Angular Momentum ◽

Hypergeometric Function ◽

Orbital Angular Momentum ◽

Topological Charge ◽

Optical Axis ◽

Type Equation ◽

Laser Beams ◽

Propagation Equation ◽

Degenerate Hypergeometric Function

Here we study asymmetric Kummer beams (aK-beams) with their scalar complex amplitude being proportional to the Kummer function (a degenerate hypergeometric function). These beams are an exact solution of the paraxial propagation equation (Schrödinger-type equation) and obtained from the conventional symmetric hypergeometric beams by a complex shift of the transverse coordinates. On propagation, the aK-beams change their intensity weakly and rotate around the optical axis. These beams are an example of vortex laser beams with a fractional orbital angular momentum (OAM), which depends on four parameters: the vortex topological charge, the shift magnitude, the logarithmic axicon parameter and the degree of the radial factor. Changing these parameters, it is possible to control the beam OAM, either continuously increasing or decreasing it.

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Orbital angular momentum and topological charge of a multi-vortex Gaussian beam

Journal of the Optical Society of America A ◽

10.1364/josaa.401561 ◽

2020 ◽

Vol 37 (11) ◽

pp. 1740

Author(s):

Alexey A. Kovalev ◽

Victor V. Kotlyar ◽

Alexey P. Porfirev

Keyword(s):

Angular Momentum ◽

Gaussian Beam ◽

Orbital Angular Momentum ◽

Topological Charge

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Transformation of a high-order edge dislocation to optical vortices (spiral dislocations)

Computer Optics ◽

10.18287/2412-6179-co-855 ◽

2021 ◽

Vol 45 (3) ◽

pp. 319-323

Author(s):

V.V. Kotlyar ◽

A.A. Kovalev ◽

A.G. Nalimov

Keyword(s):

Angular Momentum ◽

Orbital Angular Momentum ◽

Edge Dislocation ◽

Topological Charge ◽

Hermite Polynomial ◽

High Order ◽

Cylindrical Lens ◽

Optical Vortices ◽

Focal Distance ◽

Straight Line

We theoretically show that an astigmatic transformation of an nth-order edge dislocation (a zero-intensity straight line) produces n optical elliptical vortices (spiral dislocations) with unit topological charge at the double focal distance from the cylindrical lens, located on a straight line perpendicular to the edge dislocation, at points whose coordinates are the roots of an nth-order Hermite polynomial. The orbital angular momentum of the edge dislocation is proportional to the order n.

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Topological charge of optical vortices devoid of radial symmetry

Computer Optics ◽

10.18287/2412-6179-co-719 ◽

2020 ◽

Vol 44 (4) ◽

pp. 510-518

Author(s):

A.A. Kovalev

Keyword(s):

Numerical Simulation ◽

Topological Charge ◽

Radial Symmetry ◽

Laser Beams ◽

Beam Power ◽

Optical Vortices ◽

Phase Retardation ◽

Beam Asymmetry ◽

Angular Momenta ◽

Theoretical Predictions

Here we theoretically obtain values of the topological charge (TC) for vortex laser beams devoid of radial symmetry: asymmetric Laguerre-Gaussian (LG) beams, Bessel-Gaussian (BG) beams, Kummer beams, and vortex Hermite-Gaussian (HG) beams. All these beams consist of conventional modes, namely, LG, BG, or HG modes, respectively. However, all these modes have the same TC equal to that of a single constituent mode n. Orbital angular momenta (OAM) of all these beams, normalized to the beam power, are different and changing differently with varying beam asymmetry. However, for arbitrary beam asymmetry, TC remains unchanged and equals n. Superposition of just two HG modes with the adjacent numbers (n, n+1) and with the phase retardation of (pi)/2 yields a modal beam with the TC equal to – (2n+1). Numerical simulation confirms the theoretical predictions.

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Birth of optical vortices in propagating fields with an original fractional topological charge

Computer Optics ◽

10.18287/2412-6179-co-715 ◽

2020 ◽

Vol 44 (4) ◽

pp. 493-500

Author(s):

A.A. Kovalev ◽

A.P. Porfirev

Keyword(s):

Angular Momentum ◽

Free Space ◽

Topological Charge ◽

Random Phase ◽

Phase Variation ◽

Optical Vortex ◽

Optical Vortices ◽

Integer Number ◽

Weak Phase ◽

Phase Distortions

In contrast to the orbital angular momentum (OAM), which is conserved on free space propagation, the topological charge (TC) of a paraxial optical vortex (OV) is not conserved in the general case. Here, we investigate a Gaussian beam with a fractional TC in the original plane and demonstrate both theoretically and numerically how the TC changes in the course of propagation. Depending on the proximity of the topological charge to an even or odd integer number, an optical vortex with the original fractional TC is shown to behave in a number of different ways. For simple OVs (Laguerre-Gaussian or Bessel-Gaussian modes), TC is conserved both in propagation and after weak phase distortions. An experiment shows that when scattered by a random phase screen, the integer TC of an OV is conserved right up to a random phase variation of π. Therefore, in the case of weak turbulences, it is expedient to measure a discretely varying TC instead of a continuously varying OAM.

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