# Controlling the Spin Hall Effect in the Sharp Focus of an Axial Superposition of Two Optical Vortices with Left- and Right-Handed Circular Polarization

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Projections of the Electric and Magnetic Field Strength Vectors at the Focus

_{ν,µ}in (3) are defined by the following:

_{0}is the maximum tilt angle of the rays to the optical axis, which determines the numerical aperture of the aplanatic lens NA = sin θ

_{0}, J

_{μ}(x) is the Bessel function of the first kind and μ-th order, z is a longitudinal coordinate, and the focal plane is situated at z = 0. Function (4) depends on the radial and longitudinal coordinates I

_{ν,µ}(r, z). Numbers of Function (4) can be: ν = 0,1,2; μ = n − 2, n − 1, n, n + 1, n + 2.

## 3. Density of the Longitudinal Component of the Spin Angular Momentum Vector at the Focus

## 4. Full Longitudinal SAM at the Focus

_{0}− W

_{2}. The decrease in the total SAM during focusing is due to the spin-orbit conversion, when part of the spin is converted into an “orbit”. Below we show this in more detail.

## 5. The Density of the Longitudinal Orbital Angular Momentum at the Focus

## 6. Total Longitudinal OAM at the Focus

## 7. Simulation

_{z}(second and fourth rows) for different values of the parameter α: 0 (first column), 0.5 (second column), 0.75 (third column), 0.9 (fourth column), and 1 (fifth column). The topological charges are n = 3 (first and second rows), n = 5 (third and fourth rows).

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Dependences of ${\widehat{S}}_{z}/W$ (curve 1) and ${\widehat{L}}_{z}/W$ (curve 2) on the distance z for a non-vortex field (1) with n = 0 (

**a**) and for an optical vortex with a topological charge n = 1 (

**b**) and at α = 0.

**Figure 2.**The dependence of ${\widehat{S}}_{z}/W$ (curve 1) and ${\widehat{L}}_{z}/W$ (curve 2) on α after the spherical lens (or zone plate) for the initial field (1) at n = 0 (

**a**) and n = 1 (

**b**).

**Figure 3.**Dependences of ${\widehat{S}}_{z}/W$ (curve 1), ${\widehat{L}}_{z}/W$ (curve 2) and their sum (curve 3) on the focal length f of a spherical lens or on the numerical aperture NA (n = 1, α = 0.5).

**Figure 4.**Intensity distributions I (

**a**–

**e**,

**k**–

**o**) and longitudinal SAM S

_{z}(

**f**–

**j**,

**p**–

**t**) calculated at the beam focus (1) for different α: 0 (

**a**,

**f**,

**k**,

**p**), 0.5 (

**b**,

**g**,

**l**,

**q**), 0.75 (

**c**,

**h**,

**m**,

**r**), 0.9 (

**d**,

**i**,

**n**,

**s**), and 1 (

**e**,

**j**,

**o**,

**t**). The topological charges are n = 3 (

**a**–

**j**), n = 5 (

**k**–

**t**).

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**MDPI and ACS Style**

Kotlyar, V.V.; Nalimov, A.G.; Kovalev, A.A.
Controlling the Spin Hall Effect in the Sharp Focus of an Axial Superposition of Two Optical Vortices with Left- and Right-Handed Circular Polarization. *Appl. Sci.* **2023**, *13*, 8466.
https://doi.org/10.3390/app13148466

**AMA Style**

Kotlyar VV, Nalimov AG, Kovalev AA.
Controlling the Spin Hall Effect in the Sharp Focus of an Axial Superposition of Two Optical Vortices with Left- and Right-Handed Circular Polarization. *Applied Sciences*. 2023; 13(14):8466.
https://doi.org/10.3390/app13148466

**Chicago/Turabian Style**

Kotlyar, Victor V., Anton G. Nalimov, and Alexey A. Kovalev.
2023. "Controlling the Spin Hall Effect in the Sharp Focus of an Axial Superposition of Two Optical Vortices with Left- and Right-Handed Circular Polarization" *Applied Sciences* 13, no. 14: 8466.
https://doi.org/10.3390/app13148466