Enhancing the Spin Hall Effect of Cylindrically Polarized Beams

Abstract

Two linked gear wheels in a micromachine can be simultaneously rotated in opposite directions by using a laser beam that has in its section areas the spin angular momentum (SAM) of the opposite sign. However, for instance, a cylindrical vector beam has zero SAM in the focus. We alter a cylindrical vector beam so as to generate areas in its focus where the SAM is of opposite signs. The first alteration is adding to the cylindrical vector beam a linearly polarized beam. Thus, we study superposition of two rotationally symmetric beams: those with cylindrical and linear polarization. We obtain an expression for the SAM and prove two of its properties. The first property is that changing superposition coefficients does not change the shape of the SAM density distribution, whereas the intensity changes. The second property is that maximal SAM density is achieved when both beams in the superposition have the same energy. The second perturbation is adding a spatial carrier frequency. We study the SAM density of a cylindrical vector beam with a spatial carrier frequency. Due to periodic modulation, upon propagation in space, such a beam is split into two beams, having left and right elliptic polarization. Thus, in the beam transverse section, areas with the spin of different signs are separated in space, which is a manifestation of the spin Hall effect. We demonstrate that such light beams can be generated by metasurfaces, with the transmittance depending periodically on one coordinate.

Keywords: beam energy; carrier frequency; cylindrical vector beam; spin Hall effect; spin angular momentum.

Figure 1

Intensity distribution of beam (27) with w = 1 mm, n = 3, and α = 0.001k at a distance of z₀ 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 w₀ = 1 mm and w₁ = 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 = z₀, superposition coefficients C_C² = 0.95, C_L² = 0.05 (a,f), C_C² = 0.70, C_L² = 0.30 (b,g), C_C² = C_L² = 0.50 (c,h), C_C² = 0.30, C_L² = 0.70 (d,i), and C_C² = 0.01, C_L² = 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 w₀ = 1 mm, scaling factor α₀ = k/1000, order of cylindrical polarization m = 5, waist radii of the subtracted linearly polarized Gaussian beams w₀₁ = 5 mm and w₀₂ = 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 = z₀, superposition coefficients C_C² = 0.95, C_L² = 0.05 (a,f), C_C² = 0.70, C_L² = 0.30 (b,g), C_C² = C_L² = 0.50 (c,h), C_C² = 0.30, C_L² = 0.70 (d,i), and C_C² = 0.01, C_L² = 0.99 (e,j). The numbers near the color scales denote the minimal and maximal values.

Figure 4

Direction of linear polarization in the light field of (19).

Figure 5

Binary metasurface relief.

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.