# Enhancing the Spin Hall Effect of Cylindrically Polarized Beams

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Spin Angular Momentum of a Superposition of Rotationally Symmetric Beams with Cylindrical and Linear Polarization

_{C}and C

_{L}being the superposition coefficients, defining the contributions of the cylindrically and linearly polarized beams.

_{V}(ρ, θ, 0) = G(ρ)exp(imθ), the integral over the angle θ is expressed via the Bessel function, and the Fresnel transform reads as

_{C,z}and S

_{L,z}are separate SAM densities of the cylindrically and of the elliptically polarized beams. They are zero, and thus only the third term remains in Equation (9). Substituting here the field components from Equations (6) and (7), we get

_{x}or E

_{y}component becomes zero, or the expression in the second line of Equation (10) becomes real.

## 3. Energies of Cylindrically and Linearly Polarized Beams for Maximizing the Spin Angular Momentum Density

_{C}and C

_{L}. But it follows that the energy of beam (1) grows as well. Thus, we should fix the energy at some value, e.g., W

_{0}. The energy of an arbitrary paraxial vector field can be obtained as

_{C}C

_{L}, it can be maximized at an arbitrary point (r, φ, z) for all superpositions (1) with fixed energy W

_{0}by solving the following optimization problem:

_{0}is the Lagrange multiplier.

_{C}and C

_{L}yields two equations:

_{C}and C

_{L}, respectively, yields

_{0}/2. We note that we obtained a similar result (18) earlier for a particular case [12] of a superposition of a cylindrically polarized single-ringed LG beam with a linearly polarized Gaussian beam. In the current work, we generalize the results of the work [12] for a similar superposition, but with arbitrary amplitudes A(r) and B(r) in Equations (2) and (3).

## 4. Intensity and Spin Density of a Gaussian Vector Field with One-Dimensional Periodical Modulation

_{z}= 0), but as light field (22) propagates in free space, two areas of opposite-sign spin are generated, where the normalized spin density is given by (25). This is the simplest way to demonstrate the spin Hall effect and to obtain two opposite-handed circularly polarized focal regions from linearly polarized light.

_{0}.

## 5. Cylindrical Vector Beam with Spatial Carrier Frequency

_{v}(x) in Equation (29) are modified Bessel functions. According to Equation (29), the paraxial vector field is composed of two off-axis vortex beams with right- and left-handed circular polarizations, with centers of phase singularities (vortex centers) found at points ${x}_{\pm}=\pm \left(\alpha z/k\right)$. The phase singularity points and the intensity nulls of field (29) coincide. Optical vortices near these points have the opposite-sign topological charges, n and −n. Near the intensity nulls, each component of the light field is, respectively, defined by the amplitude ${\left(x+\alpha -iy\right)}^{n}$ and ${\left(x-\alpha +iy\right)}^{n}$. We note that if n = 0, then field (29) reduces exactly to field (22). In the initial plane, field (27) has neither orbital angular momentum (OAM) nor spin density. However, upon free-space propagation, the single field splits into two fields, both having the opposite-sign longitudinal SAM and the OAM. In the areas of negative spin (left-handed circular polarization), both the topological charge and the OAM are positive, and vice versa, where the spin is positive (right-handed circular polarization), both the OAM and the topological charge are negative. The total spin and OAM of the whole field remain equal to zero, as is the case for the initial field (27).

_{x}component (Figure 1b) of beam (27) with w = 1 mm, n = 3, and α = 0.001k at distance z

_{0}from the waist plane. All distributions are obtained by using a Fresnel transform for the wavelength 532 nm. The polarization distribution pattern in Figure 1a is shown by pink (S

_{z}> 0) and cyan (S

_{z}< 0) ellipses. As seen in Figure 1a, the intensity distribution contains two bright rings, with near-circular polarization inside these rings. Besides, the left and right patterns show opposite-sign polarization (right-handed circular polarization near the left ring and left-handed circular polarization near the right ring). On the right of Figure 1b, three screw dislocations are seen, confirming that the topological charge of the right ring in the field E

_{x}is equal to n = 3. On the left of Figure 1b, the phase distribution contains three screw dislocations of opposite sign, confirming that the left ring of the field E

_{x}has a topological charge of n = −3.

## 6. Numerical Simulation of Superpositions of Rotationally Symmetric Beams with Cylindrical and Linear Polarization

_{0}is the waist radius, m and p are, respectively, the azimuthal and radial indices defining the topological charge and the number of rings, ${L}_{p}^{m}(x)$ is the Laguerre polynomial, and W

_{LG}is the normalizing factor equal to the energy of the Laguerre-Gaussian beam and introduced to ensure that the energy of beam (30) is unit:

_{1}is the waist radius that should be greater than w

_{0}so that the Gaussian beam overlaps the Laguerre-Gaussian beam, and W

_{G}is the normalizing factor for reducing the energy of beam (32) to a unit value:

_{C}= C

_{L}. It also confirms that the SAM density changes with the weight coefficients only by magnitude rather than shape.

_{0}is the waist radius, m is the order of cylindrical polarization, α

_{0}is the scaling factor of the Bessel-Gaussian beam defining the radius of the light ring, J

_{m}(x) is the mth-order Bessel function of the first kind, and W

_{BG}is the normalizing factor for reducing the energy of beam (34) to a unit value:

_{m}(ξ) being the modified mth-order Bessel function.

_{01}and w

_{02}:

_{2}/q

_{1}with q

_{i}= 1 + iλz/(πw

^{2}

_{0i}), i = 1, 2, and W

_{BG}is the normalizing factor for reducing the energy of beam (36) to a unit value:

_{C}= C

_{L}. As shown in Figure 2, the SAM density changes with the weight coefficients only by magnitude rather than shape.

## 7. Designing a Metalens for Generating Two Beams with the Opposite-Sign Spins

_{x}and E

_{y}at a distance λ from the metasurface. This field was then the input field of the Rayleigh-Somerfield transform. Using this transform, the resulting light field at a distance of 50 μm was computed. Simulation by only the FDTD method at such a distance (nearly 50 μm) is impossible in 3D due to excessive computational complexity. Shown in Figure 6 is the intensity of the field, obtained at the distance λ beyond the metasurface.

^{−1}and n = 1. In addition to the spatial frequency, which is present in the metasurface from Figure 5, the topological charge is added here. Therefore, instead of the grating (Figure 5), fork grating was obtained, although only 2 periods were fitted in Figure 8a of such a fork grating (with an edge dislocation). In total, the metasurface with a size of 8 × 8 μm was split into 14 × 14 blocks, each 26 × 26 pixels (0.571 μm).

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Intensity distribution of beam (27) with w = 1 mm, n = 3, and α = 0.001k at a distance of z

_{0}from the waist plane, shown by white-yellow rings (

**a**), and polarization distribution over the beam transverse section, shown by ellipses (pink ellipses denote right-handed polarization S

_{z}> 0 and cyan ellipses denote left-handed polarization S

_{z}< 0); phase distribution of one transverse component of the light field E

_{x}(

**b**). The size of both figures is 30 × 30 mm.

**Figure 2.**Intensity (

**a**–

**e**) and SAM density (

**f**–

**j**) distributions of several superpositions of the cylindrically polarized Laguerre-Gaussian beams (30) and linearly polarized Gaussian beams (32) with different weight coefficients for the following parameters: wavelength λ = 532 nm, Gaussian beam waist radii w

_{0}= 1 mm and w

_{1}= 5 mm, radial and azimuthal orders of the cylindrically polarized Laguerre-Gaussian beam p = 2 and m = 3, propagation distance from the initial plane z = z

_{0}, superposition coefficients C

_{C}

^{2}= 0.95, C

_{L}

^{2}= 0.05 (

**a**,

**f**), C

_{C}

^{2}= 0.70, C

_{L}

^{2}= 0.30 (

**b**,

**g**), C

_{C}

^{2}= C

_{L}

^{2}= 0.50 (

**c**,

**h**), C

_{C}

^{2}= 0.30, C

_{L}

^{2}= 0.70 (

**d**,

**i**), and C

_{C}

^{2}= 0.01, C

_{L}

^{2}= 0.99 (

**e**,

**j**). The numbers near the color scales denote the minimal and maximal values.

**Figure 3.**Intensity (

**a**–

**e**) and SAM density (

**f**–

**j**) distributions of several superpositions of the cylindrically polarized Bessel-Gaussian beams (34) and linearly polarized difference of two Gaussian beams (36) with different weight coefficients for the following parameters: wavelength λ = 532 nm, waist radius of the Gaussian envelope of the Bessel-Gaussian beam w

_{0}= 1 mm, scaling factor α

_{0}= k/1000, order of cylindrical polarization m = 5, waist radii of the subtracted linearly polarized Gaussian beams w

_{01}= 5 mm and w

_{02}= 7 mm (at these radii the light ring of the difference beam has the same radius as that of the Bessel-Gaussian beam), propagation distance from the initial plane z = z

_{0}, superposition coefficients C

_{C}

^{2}= 0.95, C

_{L}

^{2}= 0.05 (

**a**,

**f**), C

_{C}

^{2}= 0.70, C

_{L}

^{2}= 0.30 (

**b**,

**g**), C

_{C}

^{2}= C

_{L}

^{2}= 0.50 (

**c**,

**h**), C

_{C}

^{2}= 0.30, C

_{L}

^{2}= 0.70 (

**d**,

**i**), and C

_{C}

^{2}= 0.01, C

_{L}

^{2}= 0.99 (

**e**,

**j**). The numbers near the color scales denote the minimal and maximal values.

**Figure 6.**Intensity (

**a**) and polarization distribution (

**b**) of the electric field at the distance λ from the metasurface.

**Figure 7.**Intensity of light at a distance of 50.633 μm from the metasurface as well as the polarization distribution. Arrows with circles indicate polarization direction in the center of each circle, and the arrow shows the rotation direction of the vector electric field with time.

**Figure 8.**Metasurface, generating the cylindrical vector beam (27) with spatial carrier frequency (

**a**), and polarization of a plane linearly polarized wave passed through this metasurface at a distance λ from it (

**b**).

**Figure 9.**Intensity of the cylindrical vector beam with the carrier frequency, generated by the metalens, and polarization of this beam, depicted as ellipses with arrows (

**a**), as well as the phase of the E

_{y}field component (

**b**). Each ellipse (

**a**) describes rotation of the electric field vector with time.

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**MDPI and ACS Style**

Kovalev, A.A.; Nalimov, A.G.; Kotlyar, V.V.
Enhancing the Spin Hall Effect of Cylindrically Polarized Beams. *Micromachines* **2024**, *15*, 350.
https://doi.org/10.3390/mi15030350

**AMA Style**

Kovalev AA, Nalimov AG, Kotlyar VV.
Enhancing the Spin Hall Effect of Cylindrically Polarized Beams. *Micromachines*. 2024; 15(3):350.
https://doi.org/10.3390/mi15030350

**Chicago/Turabian Style**

Kovalev, Alexey A., Anton G. Nalimov, and Victor V. Kotlyar.
2024. "Enhancing the Spin Hall Effect of Cylindrically Polarized Beams" *Micromachines* 15, no. 3: 350.
https://doi.org/10.3390/mi15030350