Influence of optical “dipoles” on the topological charge of a field with a fractional initial charge

A. G. Nalimov; A. G. Nalimov; V. V. Kotlyar; V. V. Kotlyar

doi:10.1364/JOSAA.455744

Journal of the Optical Society of America A
Vol. 39,
Issue 5,
pp. 812-819
(2022)
•https://doi.org/10.1364/JOSAA.455744

Influence of optical “dipoles” on the topological charge of a field with a fractional initial charge

A. G. Nalimov and V. V. Kotlyar

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A. G. Nalimov and V. V. Kotlyar, "Influence of optical “dipoles” on the topological charge of a field with a fractional initial charge," J. Opt. Soc. Am. A 39, 812-819 (2022)

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Abstract

In this work, using the Rayleigh–Sommerfeld integral and Berry formula, a topological charge (TC) of a Gaussian optical vortex with an initial fractional TC in the far field was calculated. It was found that, for diverse fractional parts of the TC, the beam contained different numbers of screw dislocations, which determined the TC of the entire beam. If a fractional part of the TC was small, the beam consisted of the main optical vortex centered on the optical axis, with the TC equal to the nearest integer (say $n \gt {{0}}$) and two edge dislocations located on the vertical axis (one above and the other below the center). When the fractional part of the initial TC increased, a “dipole” was formed from the upper edge dislocation, consisting of two vortices with TCs equal to ${+}{{1}}$ and ${-}{{1}}$. With a further increase in the fractional part, the additional vortex with ${\rm{TC}} = + {{1}}$ moved to the center of the beam, and the vortex with ${\rm{TC}} = - {{1}}$ moved to the periphery. When the fractional part of the TC increased further, another “dipole” was formed from the lower edge dislocation, in which, on the contrary, the vortex with ${\rm{TC}} = - {{1}}$ was displaced to the optical axis (to the center of the beam) and the vortex with ${\rm{TC}} = + {{1}}$ moved to the beam periphery. When the fractional part of the TC became equal to 1/2, the lower vortex with a ${\rm{TC}} = - {{1}}$, which was earlier displaced to the center of the beam, began to shift to the periphery, and the upper vortex with a ${\rm{TC}} = + {{1}}$ moved closer and closer to the center of the beam, eventually merging with the main vortex when the fractional part approached 1. Such dynamics of additional vortices with TCs above ${+}{{1}}$ and below ${-}{{1}}$ determined which whole TC the beam would have ($n$ or $n + {{1}}$) for different values of the fractional part from the segment [$n,\;n + {{1}}$]. Our analysis has shown that, for any value of the fractional part of the initial topological charge, the TC of the beam in the far field will not be determined. Since, with an increase in the radius of the circle in the beam section on which the TC is calculated, more optical “dipoles” will appear, and the TC will be either $n$ or $n + {{1}}$.

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$R = 8 µ m$		$R = 138 µ m$
μ	TC	μ	TC
3.0–3.12	3	3.0–3.11	3
3.14–3.25	4	3.12–3.2	4
3.26–3.67	3	3.21–3.72	3
3.68–4	4	3.73–4	4

$μ$	TC	Comment (the $x$ and $y$ axes are shown in Fig. 1)
3.0–3.11	3	An optical vortex in the center on the optical axis with $T C = 3$ was shown on the phase picture. There were two edge dislocations of different signs on the vertical axis at $y > 0$ and at $y < 0$ . Therefore, the TC of the entire optical vortex was 3 [Fig. 3(d)].
3.11–3.12	$3 \to 4$ (transition)	The upper edge dislocation ( $y > 0$ ) generated an optical “dipole 1” [Fig. 3(b)]: two adjacent optical vortices with charges $+ 1$ and $- 1$ above the center (centered at the point $y = 20 µ m$ , $x = 0$ ). The TC of the beam did not change and remained equal to 3. When μ increased, the upper vortex in the dipole with a TC of $- 1$ moved beyond the boundary of the considered field ( $y > R = 138 µ m$ ), and the lower vortex in the dipole with $T C + 1$ descended to the center. Further, for $μ = 3.12$ , only the lower vortex with $T C + 1$ situated in $y = 9.02 µ m$ remained. The TC of the beam became equal to 4 [Fig. 3(e)].
3.12–3.18	4	The optical vortex with $T C + 1$ remaining in a circle of radius $R = 138 µ m$ still moved to the center of the beam, and the TC of the entire beam remained equal to 4.
3.18–3.21	$4 \to 3$ (transition)	Now, the lower edge dislocation ( $y < 0$ ) generated an optical “dipole 2” [Fig. 3(b)]: two close vortices with TCs of $- 1$ and $+ 1$ (centered at the point $y = - 26.1 µ m$ , $x = 0$ ). When μ increased, these two vortices diverged in different directions: the lower one with $T C + 1$ moved to the boundary of the region ( $y = - 138 µ m$ ), and the upper one with $T C - 1$ moved toward the center of the beam. At $μ = 3.21$ , a vortex with $T C + 1$ left the field boundary, and only a vortex with $T C - 1$ remained at $y = - 11 µ m$ . The TC of the entire beam again became equal to 3 [Fig. 3(a)].
3.21–3.72	3	When μ increased, the vortex above the center ( $y > 0$ ) with $T C = + 1$ continued to gradually approach the center. The vortex from below ( $y < 0$ ) with $T C = - 1$ first approached the center and then, at $μ > 3.5$ , began to move away from the center [Fig. 3(a)].
3.73–3.74	$3 \to 4$ (transition)	Two vortices with TCs of $- 1$ and $+ 1$ ( $y < 0$ ), which diverged at the μ value of 3.5, began to approach each other at $μ > 3.73$ . A vortex with $T C = - 1$ went along the $y < 0$ coordinate from the center, and a vortex with $T C = + 1$ from the beam periphery shifted to the center. These vortices connected and mutually “annihilated” approximately at a distance of $y = - 25 µ m$ . There remained only an additional vortex with $T C = + 1$ from above ( $y > 0$ ), which had practically already approached the center of the beam. Therefore, the TC of the entire beam became equal to 4 [Fig. 3(a)].
3.74–4.00	4	An additional vortex with $T C = + 1$ on top of the beam ( $y > 0$ ) merged with the original vortices in the center of the beam with $T C = 3$ . Further, the TC of the entire beam remained equal to 4.

$R = 8 µ m$		$R = 138 µ m$
μ	TC	μ	TC
3.0–3.12	3	3.0–3.11	3
3.14–3.25	4	3.12–3.2	4
3.26–3.67	3	3.21–3.72	3
3.68–4	4	3.73–4	4

$μ$	TC	Comment (the $x$ and $y$ axes are shown in Fig. 1)
3.0–3.11	3	An optical vortex in the center on the optical axis with $T C = 3$ was shown on the phase picture. There were two edge dislocations of different signs on the vertical axis at $y > 0$ and at $y < 0$ . Therefore, the TC of the entire optical vortex was 3 [Fig. 3(d)].
3.11–3.12	$3 \to 4$ (transition)	The upper edge dislocation ( $y > 0$ ) generated an optical “dipole 1” [Fig. 3(b)]: two adjacent optical vortices with charges $+ 1$ and $- 1$ above the center (centered at the point $y = 20 µ m$ , $x = 0$ ). The TC of the beam did not change and remained equal to 3. When μ increased, the upper vortex in the dipole with a TC of $- 1$ moved beyond the boundary of the considered field ( $y > R = 138 µ m$ ), and the lower vortex in the dipole with $T C + 1$ descended to the center. Further, for $μ = 3.12$ , only the lower vortex with $T C + 1$ situated in $y = 9.02 µ m$ remained. The TC of the beam became equal to 4 [Fig. 3(e)].
3.12–3.18	4	The optical vortex with $T C + 1$ remaining in a circle of radius $R = 138 µ m$ still moved to the center of the beam, and the TC of the entire beam remained equal to 4.
3.18–3.21	$4 \to 3$ (transition)	Now, the lower edge dislocation ( $y < 0$ ) generated an optical “dipole 2” [Fig. 3(b)]: two close vortices with TCs of $- 1$ and $+ 1$ (centered at the point $y = - 26.1 µ m$ , $x = 0$ ). When μ increased, these two vortices diverged in different directions: the lower one with $T C + 1$ moved to the boundary of the region ( $y = - 138 µ m$ ), and the upper one with $T C - 1$ moved toward the center of the beam. At $μ = 3.21$ , a vortex with $T C + 1$ left the field boundary, and only a vortex with $T C - 1$ remained at $y = - 11 µ m$ . The TC of the entire beam again became equal to 3 [Fig. 3(a)].
3.21–3.72	3	When μ increased, the vortex above the center ( $y > 0$ ) with $T C = + 1$ continued to gradually approach the center. The vortex from below ( $y < 0$ ) with $T C = - 1$ first approached the center and then, at $μ > 3.5$ , began to move away from the center [Fig. 3(a)].
3.73–3.74	$3 \to 4$ (transition)	Two vortices with TCs of $- 1$ and $+ 1$ ( $y < 0$ ), which diverged at the μ value of 3.5, began to approach each other at $μ > 3.73$ . A vortex with $T C = - 1$ went along the $y < 0$ coordinate from the center, and a vortex with $T C = + 1$ from the beam periphery shifted to the center. These vortices connected and mutually “annihilated” approximately at a distance of $y = - 25 µ m$ . There remained only an additional vortex with $T C = + 1$ from above ( $y > 0$ ), which had practically already approached the center of the beam. Therefore, the TC of the entire beam became equal to 4 [Fig. 3(a)].
3.74–4.00	4	An additional vortex with $T C = + 1$ on top of the beam ( $y > 0$ ) merged with the original vortices in the center of the beam with $T C = 3$ . Further, the TC of the entire beam remained equal to 4.

Abstract

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Journal of the Optical Society of America A