# Spin Hall Effect of Double-Index Cylindrical Vector Beams in a Tight Focus

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**J**= [cos mφ, sin mφ] with φ being the angular polar coordinate and m being the polarization order (for m = 1, radial polarization). A further generalization is a two-index polarization singularity with the Jones vector

**J**= [cos mφ, sin nφ], where m ≠ n [12], i.e., such a generalized vector field has different orders on the different axes. Recently, we studied such fields with V-points and for many values m and n we obtained the Poincare-Hopf index analytically [13].

## 2. A Light Field with a Double-Index Polarization Singularity near the Tight Focus

_{x}and E

_{y}field components were different. The amplitude of the electric vector of such a field is given by

**E**is the strength vector of the electric field, φ is the azimuthal angle in the source plane, (m, n) is the two-index polarization order, θ is the polar angle, describing the tilt of the light rays to the optical axis, A(θ) is the amplitude of the source field as a function of the axis tilt angle. Directions of the electric vectors are illustrated in Figure 1.

_{ν,μ}are defined as follows:

_{ν}is the νth-order Bessel function of the first kind.

_{ν,μ}are also real valued in the focal plane (z = 0). Then, substituting the transverse field components from Equation (7), we get the following expression:

_{x}and E

_{y}are proportional to i

^{m}

^{+1}(or i

^{n}

^{+1}), multiplied by some real-valued function. Near the center (r << λ), if n > m ≥ 2, the transverse components E

_{x}and E

_{y}are approximately proportional to the vector

**J**= [cos (m−2)φ, –sin (m−2)φ]. If m > n ≥ 2, they are proportional to the vector

**J**= [cos (n−2)φ, –sin (n−2)φ]. Thus, a saddle-type polarization singularity is generated in the center [17].

## 3. Balance of Light Field Energy near the Tight Focus

_{ν,μ}from Equation (4). Therefore, such a harmonic has the following energy W

_{ν,μ}(see Appendix A):

_{x}consists of three angular harmonics with their amplitude being proportional to the functions I

_{0,m}, I

_{2,m+2}/2, I

_{2,m–2}/2. The component E

_{y}is a superposition of harmonics described by the functions I

_{2,m+2}/2 and I

_{2,m–2}/2. Finally, the component E

_{z}is a superposition of harmonics described by the functions I

_{1,m+1}and I

_{1,m–1}. Therefore, the total energy of the focal field is

_{x}, E

_{y}, and E

_{z}field components. Since W

_{2,m+2}= W

_{2,m–2}and W

_{1,m+1}= W

_{1,m–1}, we get

_{0,m}+ W

_{2,m+2}+ 2W

_{1,m+1}= 2πf

^{2}. Thus, the total energy W of the input field is distributed in the focal field in the proportions shown in Figure 2. One third W/3 goes into the longitudinal component E

_{z}, and 2W/3 goes into the transverse components E

_{x}(5W/8) and E

_{y}(W/24). The energy of the component E

_{x}is distributed into the mth-order angular harmonic (7W/12) and into the harmonics of the orders m − 2 and m + 2, each of the energy W/48. The energy of the component E

_{y}is distributed equally in the angular harmonics of the orders m − 2 and m + 2, each of the energy W/48. The energy of the component E

_{z}is distributed equally in the angular harmonics of the orders m − 1 and m + 1, each of the energy W/6.

_{y}.

_{2,m}/W

_{0,m}= 1/7 ≈ 0.143, but even if sin α = 0.95, then W

_{2,m}/W

_{0,m}≈ 0.057, i.e., almost all energy goes into the central mth-order harmonic.

_{x}, E

_{y}, and E

_{z}), illustrated in Figure 3.

## 4. Spin Angular Momentum of Double-Index Polarization Vortices in a Tight Focus

## 5. Simulation

_{x}|

^{2}+ |E

_{y}|

^{2}+ |E

_{z}|

^{2}and the longitudinal SAM density of a tightly focused light field with double-index polarization singularity of three different orders (m, n): (1, 2) (Figure 4a,d,g), (3, 6) (Figure 4b,e,h), (7, 14) (Figure 4c,f,i) at the following parameters: wavelength λ = 532 nm, focal length of the lens f = 10 μm, numerical aperture sin α = 0.95, amplitude apodization function is homogeneous, i.e., A(θ) ≡ 1. The SAM density distribution was computed directly by Equation (8) and then it was compared with the one computed by Equation (10). The distributions were visually the same, relative error was computed as max|S

_{z}

_{(10)}–S

_{z}

_{(8)}|/max|S

_{z}

_{(8)}| (where S

_{z}

_{(8)}and S

_{z}

_{(10)}are, respectively, SAM densities computed by Equation (8) and by Equation (10)). Maximal relative error was at (m, n) = (7, 14) and equal = 1.9 × 10

^{−15}. The third row of Figure 4 illustrates the approximate SAM distributions obtained by Equation (20). Formally, the relative error from the distributions obtained directly by Equation (8) is large (14–32%), but it is seen in Figure 4 that it almost does not affect the shape of the distribution. This is because the error was caused by neglecting the side angular harmonics of the order m ± 2 and n ± 2, and since, according to Figure 3, there are 16 such harmonics, each of them separately is weak and cannot reshape the SAM distribution significantly.

_{y}= 0 on the horizontal axis (φ = 0 and φ = π) and E

_{x}= 0 on the vertical axis (φ = ±π/2), which is consistent with Equation (7) for the complex amplitudes of the light field. It is also seen that in both cases a saddle-type polarization [17] singularity is generated in the center.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of the Energy of a Single Angular Harmonic Near the Tight Focus

_{ν,μ}from Equation (4). Below we derive the energy W

_{ν,μ}of such a separate angular harmonic:

_{ν,μ}from Equation (4), we get

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**Figure 1.**Conventional radial polarization (m = n = 1) (

**a**), third-order radial polarization (m = n = 3) (

**b**), double-index polarization singularity (m = 1, n = 3) (

**c**). For radial polarization (

**a**), there are two angles (0 and π) with a horizontal electric vector and two angles (π/2 and 3π/2) with a vertical electric vector. For 3rd-order radial polarization (

**b**), there are six angles with horizontal electric vectors and six angles with vertical electric vectors. For the field with double-index polarization singularity of the order (1, 3), there are six angles with horizontal electric vectors and two angles with vertical electric vectors.

**Figure 2.**Energy distribution of a tightly focused linearly polarized optical vortex by the field components and by the angular harmonics.

**Figure 3.**Energy distribution of a tightly focused light field with double-index polarization singularity by the field components and by the angular harmonics. Numbers above each pillar indicate the fraction of the whole energy of the focused light field that goes to the given angular harmonic of the given field component (the sum of all numbers is 100%).

**Figure 4.**Distributions of intensity |E

_{x}|

^{2}+ |E

_{y}|

^{2}+ |E

_{z}|

^{2}(

**a**–

**c**) and of longitudinal SAM density 2Im{E

_{x}*E

_{y}} (

**d**–

**i**) of tightly focused light field with double-index polarization singularity of the orders (m, n) = (1, 2) (

**a**,

**d**,

**g**), (m, n) = (3, 6) (

**b**,

**e**,

**h**), (m, n) = (7, 14) (

**c**,

**f**,

**i**) at the following parameters: wavelength λ = 532 nm, focal length of the lens f = 10 μm, numerical aperture sin α = 0.95, amplitude apodization function is homogeneous, i.e., A(θ) ≡ 1. SAM distributions were computed directly by Equation (8) (

**d**–

**f**) and by an approximate Equation (20) (

**g**–

**i**). All the figures have the size 4 × 4 μm

^{2}(scale mark shows 1 μm). Light and dark colors in the intensity distribution mean maximum and zero. Red and blue colors (

**d**–

**i**) mean, respectively, positive and negative SAM.

**Figure 5.**Distributions of intensity |E

_{x}|

^{2}+ |E

_{y}|

^{2}+ |E

_{z}|

^{2}(

**a**,

**b**) and of longitudinal SAM density 2Im{E

_{x}

^{*}E

_{y}} (

**c**–

**f**) of tightly focused light field with double-index polarization singularity of the orders (m, n) = (6, 7) (

**a**,

**c**,

**e**) and (m, n) = (16, 17) (

**b**,

**d**,

**f**) at the following parameters: wavelength λ = 532 nm, focal length of the lens f = 10 μm, numerical aperture sin α = 0.95, amplitude apodization function is homogeneous, i.e., A(θ) ≡ 1. SAM distributions were computed directly by Equation (8) (

**c**,

**d**) and by an approximate Equation (20) (

**e**,

**f**). Figures have the size 4 × 4 μm

^{2}(

**a**,

**c**,

**e**) and 6 × 6 μm

^{2}(

**b**,

**d**,

**f**) (scale mark shows 1 μm). Light and dark colors in the intensity distribution mean maximum and zero. Red and blue colors (

**c**–

**f**) mean, respectively, positive and negative SAM.

**Figure 6.**Intensity distribution and polarization directions of a tightly focused light field with double-index polarization singularity of the orders (m, n) = (3, 7) (

**a**) and (m, n) = (5, 3) (

**b**) at the following parameters: wavelength λ = 532 nm, focal length of the lens f = 10 μm, numerical aperture sin α = 0.95, radial apodization function is A(θ) ≡ 1. Scale mark shows 1 μm.

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**MDPI and ACS Style**

Kovalev, A.A.; Kotlyar, V.V.
Spin Hall Effect of Double-Index Cylindrical Vector Beams in a Tight Focus. *Micromachines* **2023**, *14*, 494.
https://doi.org/10.3390/mi14020494

**AMA Style**

Kovalev AA, Kotlyar VV.
Spin Hall Effect of Double-Index Cylindrical Vector Beams in a Tight Focus. *Micromachines*. 2023; 14(2):494.
https://doi.org/10.3390/mi14020494

**Chicago/Turabian Style**

Kovalev, Alexey A., and Victor V. Kotlyar.
2023. "Spin Hall Effect of Double-Index Cylindrical Vector Beams in a Tight Focus" *Micromachines* 14, no. 2: 494.
https://doi.org/10.3390/mi14020494