# Dividing the Topological Charge of a Laguerre–Gaussian Beam by 2 Using an Off-Axis Gaussian Beam

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Topological Charge of Superposition of a Laguerre-Gaussian Beam with an Off-Axis Gaussian Beam: General Theory

_{0}of both beams were supposed to be equal. Without loss of generality, we also supposed that a > 0. The complex amplitude of such superposition reads as

_{0}, η = y/w

_{0}) and denote α = a/w

_{0}:

_{0}, η

_{0}) of nulls of the complex amplitude Equation (2) can be obtained from the following equation:

_{p}the intensity nulls are at a different distance from the center (Figure 1b).

_{0}is expressed explicitly. In the first equation, to simplify the argument of the exponent, we replace ξ

_{0}by ms/(2α), where s is a scaled horizontal Cartesian coordinate of the sought-for intensity null (if this null resides in a certain point (x

_{0}, y

_{0}), then s = 2αx

_{0}/(mw

_{0})). Thus, the first equation in (5) takes the following form:

^{−1}cos φ

_{p}and

_{p}, given by Equation (6) (there can be zero, one, or two vortices depending on what value is greater: V or cos φ

_{p}).

^{s}= es has the only root s = 1. So, there are no roots if A > 1/e (curve I in Figure 2c), there is one root if A = 1/e (curve II in Figure 2c), and there are two roots if 0 < A < 1/e (curve III in Figure 2c), one of which is in the interval (0, 1), and the other are in (1, ∞).

_{p}.

_{p}< 0, there is one intensity null (ξ

_{0}< 0) in the direction φ

_{p}.

_{p}= 0, the intensity null is on the vertical axis (ξ

_{0}= 0).

_{p}< V, there are two intensity nulls, so that 0 < ξ

_{0}< m/(2α) for one null and ξ

_{0}> m/(2α) for the other.

_{p}= V, there is one null with ξ

_{0}= m/(2α).

_{p}> V, there are no intensity nulls in this direction.

_{0}, η

_{0}), and which order these vortices have. To do this, we consider a point in the vicinity of the intensity null, i.e., a point (ξ, η) = (ξ

_{0}+ r cos φ, η

_{0}+ r sin φ) with r being a small distance:

^{2}− η

^{2}) does not affect the phase distribution. In the square brackets, doing the binomial expansion of the first term and series expansion of the exponent exp(2αrcos φ) in the second term, after neglecting the terms r

^{2}, r

^{3}, etc., we obtain:

_{0}, η

_{0}), then

_{0}, η

_{0}) is defined by the expression in the square brackets, and (see Appendix A) it equals

_{p}≤ 3π/2, i.e., in one transverse half-plane (Figure 3). For other directions φ

_{p}, either there are no intensity nulls, or there is one intensity null without a vortex around it, or there are two nulls with optical vortices of opposite orders so that they do not affect the total TC.

_{p}are independent of the Gaussian beam amplitude |C|. Thus, TC depends only on the phase delay arg C between the beams. It seems strange enough, as in the absence of the Gaussian beam (|C| → 0), there is no way to change the TC of the LG beam. However, the apparent collision is explainable. According to Table 1, when |C| → 0, there are two intensity nulls for each angle φ

_{p}with cos φ

_{p}> 0. However, the second null goes to infinity at |C| → 0 and disappears completely at |C| = 0, making TC equal to m instead of [m/2] or [m/2] + 1.

## 3. Topological Charge of Superposition of a LG Beam and an Off-Axis Gaussian Beam: Particular Cases

_{0}= Ψ. If the absolute phase delay between the beams |arg C| is less than π/2, then the TC of the LG beam remains equal to 1, independently of the Gaussian beam power. Otherwise, TC becomes equal to zero, although the beam can have up to two intensity nulls, depending on the sign of the quantity cos φ

_{p}− exp(α

^{2})/(2eα|C|

^{1/m}). For example, for α = 1 (shift of the Gaussian beam equals its waist radius), arg C = π (antiphase superposition), a number of intensity nulls depend on the sign of |C| − 1/2. If C = −1, there are no intensity nulls (bottom row in Table 1). If C = −1/2 (next-to-last row in Table 1), there is an intensity null, but without the vortex (phase on a contour around the null increases and decreases without 2π jumps). If C = −1/10, the Gaussian beam affects the intensity distribution of the LG beam weakly, but two intensity nulls simple with optical vortices of the opposite sign.

_{0}= 0.5 mm; azimuthal index of LG beam m = 1; a transverse shift of the Gaussian beam a = w

_{0}; Gaussian beam weight coefficients of C = 10 (Figure 4a–c), C = 0.1 (Figure 4d–f), C = −1 (Figure 4g–i), C = −0.5 (Figure 4j–l), and C = −0.1 (Figure 4m–o); calculation domain –R ≤ x, y ≤ R (R = 2 mm); and number of points N = 1024 (along both coordinate axes).

_{0}/2 = 0.25 mm) without an optical vortex (Figure 4j–l), or there are two opposite-sign vortices (Figure 4m–o).

_{0}, φ

_{1}is in the range (−π/2, π/2) and thus TC becomes equal to 1. The number of intensity nulls can also be different, e.g., at α = 1 and arg C = π, it depends on the sign of |C| − 1/e.

_{0}= 0.5 mm; azimuthal index of LG beam m = 2; a transverse shift of the Gaussian beam a = w

_{0}; Gaussian beam weight coefficient C = 0.1 (Figure 5a–c), C = −0.1 (Figure 5d–f), C = −1/e (Figure 5g–i), and C = −0.5 (Figure 5j–l); calculation domain −R ≤ x, y ≤ R (R = 2 mm); number of points N = 1024.

_{0}= 0.5 mm) (Figure 5g–i), or there is only one vortex (Figure 5j–l).

_{p}= Ψ + πp/2 (p = 0, 1, 2, 3).

_{p}, such that cos φ

_{p}≤ 0 and thus the TC of the whole superposition equals 2. TC can be made equal to 3 if the beams have a phase delay of π (arg C = π and Ψ = π/2), as in this case, there are three angles φ

_{p}, such that cos φ

_{p}≤ 0: φ

_{0}= π/2, φ

_{1}= π, φ

_{2}= 3π/2.

_{0}= 0.5 mm, azimuthal index of LG beam m = 4, a transverse shift of the Gaussian beam a = w

_{0}, Gaussian beam weight coefficient C = 0.1 (Figure 6a–c) and C = −0.1 (Figure 6d-f), calculation domain −R ≤ x, y ≤ R with R = 2 mm in Figure 6a,b,d,e and R = 5 mm in Figure 6c,f, and number of points N = 1024.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

^{iφ}):

_{1}and z

_{2}(z

_{1}+ z

_{2}= 0 and z

_{1}⋅z

_{2}= (a + ib)/(a − ib)). So,

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**Figure 1.**Locations of the intensity nulls on the polar angles in the superposition of a LG beam and an on-axis [21] (

**a**) and off-axis (

**b**) Gaussian beam.

**Figure 3.**Contribution of the intensity nulls of the superposition (1) into the topological charge. Red circles are the intensity nulls, an orange star is the off-axis Gaussian beam, and the grey half-plane is the one where the intensity nulls contribute to TC (other nulls do not contribute).

**Figure 4.**Distributions of intensity (

**a**,

**d**,

**g**,

**j**,

**m**) and phase (

**b**,

**e**,

**h**,

**k**,

**n**), as well as normalized-to-maximum horizontal intensity cross-sections (

**c**,

**f**,

**i**,

**l**,

**o**) of the superposition of the LG beam with an off-axis Gaussian beam for the following parameters: wavelength λ = 532 nm; waist radius w

_{0}= 0.5 mm; azimuthal index of LG beam m = 1; transverse shift of the Gaussian beam a = w

_{0}; Gaussian beam weight coefficient C = 10 (

**a**–

**c**), C = 0.1 (

**d**–

**f**), C = −1 (

**g**–

**i**), C = −0.5 (

**j**–

**l**), and C = −0.1 (

**m**–

**o**); calculation domain −R ≤ x, y ≤ R (R = 2 mm); and number of points N = 1024. TC was computed along the dashed circle (

**b**,

**e**,

**h**,

**k**,

**n**). The white scale mark in the left bottom area of each Figure denotes 1 mm.

**Figure 5.**Distributions of intensity (

**a**,

**d**,

**g**,

**j**) and phase (

**b**,

**e**,

**h**,

**k**), as well as normalized-to-maximum horizontal intensity cross-sections (

**c**,

**f**,

**i**,

**l**) of the superposition of the LG beam with an off-axis Gaussian beam for the following parameters: wavelength λ = 532 nm; waist radius w

_{0}= 0.5 mm; azimuthal index of LG beam m = 2; the transverse shift of the Gaussian beam a = w

_{0}; Gaussian beam weight coefficient C = 0.1 (

**a**–

**c**), C = −0.1 (

**d**–

**f**), C = −1/e (

**g**–

**i**), and C = −0.5 (

**j**–

**l**); calculation domain −R ≤ x, y ≤ R (R = 2 mm); and number of points N = 1024. TC was computed along the dashed circle (

**b**,

**e**,

**h**,

**k**). The white scale mark in the left bottom area of each Figure denotes 1 mm. The inset (

**b**) shows the central area with 3× zoom.

**Figure 6.**Distributions of intensity (

**a**,

**d**) and phase (

**b**,

**c**,

**e**,

**f**) of the superposition of the LG beam with an off-axis Gaussian beam for the following parameters: wavelength λ = 532 nm, waist radius w

_{0}= 0.5 mm, azimuthal index of LG beam m = 4, a transverse shift of the Gaussian beam a = w

_{0}, Gaussian beam weight coefficient C = 0.1 (

**a**–

**c**) and C = −0.1 (

**d**–

**f**), calculation domain −R ≤ x, y ≤ R with R = 2 mm (

**a**,

**b**,

**d**,

**e**) and R = 5 mm (

**c**,

**f**), and number of points N = 1024. TC was computed along the dashed circle (

**b**,

**c**,

**e**,

**f**). The white scale mark in the left bottom area of each Figure denotes 1 mm. Black digits in phase distributions (

**b**,

**c**,

**e**,

**f**) denote vortices.

φ_{p} | ξ_{0} | TC |
---|---|---|

cos φ_{p} < 0 | ξ_{0} < 0 | +1 |

cos φ_{p} = 0 | ξ_{0} = 0 | +1 |

0 < cos φ_{p} < V | 0 < ξ_{0} < m/(2α) | +1 |

ξ_{0} > m/(2α) | –1 | |

cos φ_{p} = V | ξ_{0} = m/(2α) | 0 |

cos φ_{p} > V | - |

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**MDPI and ACS Style**

Kovalev, A.A.; Kotlyar, V.V.; Kozlova, E.S.; Butt, M.A.
Dividing the Topological Charge of a Laguerre–Gaussian Beam by 2 Using an Off-Axis Gaussian Beam. *Micromachines* **2022**, *13*, 1709.
https://doi.org/10.3390/mi13101709

**AMA Style**

Kovalev AA, Kotlyar VV, Kozlova ES, Butt MA.
Dividing the Topological Charge of a Laguerre–Gaussian Beam by 2 Using an Off-Axis Gaussian Beam. *Micromachines*. 2022; 13(10):1709.
https://doi.org/10.3390/mi13101709

**Chicago/Turabian Style**

Kovalev, Alexey A., Victor V. Kotlyar, Elena S. Kozlova, and Muhammad Ali Butt.
2022. "Dividing the Topological Charge of a Laguerre–Gaussian Beam by 2 Using an Off-Axis Gaussian Beam" *Micromachines* 13, no. 10: 1709.
https://doi.org/10.3390/mi13101709