# Spin-Orbital Conversion with the Tight Focus of an Axial Superposition of a High-Order Cylindrical Vector Beam and a Beam with Linear Polarization

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Investigation

#### 2.1. Projections of Electric and Magnetic Fields Vectors in the Focal Plane

_{ν,µ}depend only on the radial variable r and are equal to the expression:

_{μ}(x) is the Bessel function of the first kind and of the μth order, NA = sin θ

_{0}is the numerical aperture of an aplanatic optical system and A(θ) is any real function that describes the input field amplitude, which has an axial symmetry (plane wave, Gaussian beam, Bessel-Gaussian beam). For the integrals I

_{ν,µ}(3), the first index ν = 0, 1, 2 describes the type of the integral, and the second index μ = 0, 1, 2, …, m is equal to the order of the Bessel function.

#### 2.2. The Intensity Distribution in the Focal Plane

_{0}(0) = 1. The arguments of the cosines are even in the expression (9) for the intensity. This means that the intensity pattern, although it does not have a radial symmetry, has an axial symmetry, i.e., $I(r,\mathsf{\phi})=I(r,\mathsf{\phi}+\mathsf{\pi})$. Additionally, it can be seen from (5) that the intensity I

_{x}will have a maximum on the optical axis due to the term ${a}^{2}{I}_{0,0}^{2}$, and it follows from (6) that I

_{y}will have a zero on the optical axis. It should also be noted that the intensity pattern I

_{y}will have 2m local maxima since the expression for I

_{y}contains the squared cos(mφ). The total intensity (9) will have 2(m − 1) local maxima (except the intensity maximum on the optical axis) since formula (9) has cos(2(m − 1)φ). These conclusions will be confirmed by modeling.

#### 2.3. The Energy Flux Density in the Focal Plane

_{x}, E

_{y}) = (cos(mφ), sin(mφ)) and in the focus (2). The longitudinal projections of the SAM and the OAM vectors in the focus are zero at each point. Below, we show that the superposition of a CVB and a light field with linear polarization (2) has a local spin and a vortex energy flux. The Poynting vector is provided by the following formula [12]:

**E**and

**H**are vectors of electric and magnetic fields, * is a complex conjugation, $\times $ is a vector multiplication and c is the light speed in a vacuum. Further, the constant c/(2π) will be ignored. We substituted the expressions for the projections of the electromagnetic field in the focus (2) into expression (10) and obtained:

#### 2.4. The Density of the Stokes Vector in the Focal Plane

**S**are calculated by the formulas [17]:

_{1}, s

_{2}, s

_{3}) denote the unnormalized components of the Stokes vector. The normalized Stokes vector, as it can be seen from (16), has a unit length ${S}_{1}^{2}+{S}_{2}^{2}+{S}_{3}^{2}=1$. Due to the cumbersomeness of the expressions, and in order to find out whether the circular polarization will be in focus, we obtained expressions only for the third Stokes projection without normalization, i.e., we calculated a function in the form ${s}_{3}=2\mathrm{Im}\left({E}_{x}^{*}{E}_{y}\right)$. It should be preliminarily noted that the third component of the Stokes vector is proportional to the longitudinal projection of the SAM [16]:

_{2}only for the even number m = 2p, p = 0, 1, 2, …:

_{2}in the focus will be axisymmetric since all arguments of the cosines and sines are even. The maximum argument in (20) has sin(2mφ) which is equal to 2m. Therefore, the number of sign changes for the function s

_{2}will be equal to 4m.

## 3. Numerical Simulations Results and Discussion

**U**(ρ, ψ, z) is an electric or magnetic field, B(θ, φ) is an electric or a magnetic field at the input of a wide-aperture optical system dependent on the exit pupil coordinates (θ is a polar angle, φ is an azimuth angle), T(θ) is a lens apodization function, f is a focal length, k = 2π/λ is a wave number, λ is a wavelength, α is the maximum polar angle defined by the lens numerical aperture (NA = sinα) and

**P**(θ, φ) is a polarization matrix. Integral (22) allows us to calculate the distribution of the electromagnetic field components in the exit pupil coordinates (Figure 1).

**P**(θ, φ) for the electric and magnetic fields has the form [18,19]:

#### 3.1. The Distribution of Linear Polarization Vectors in the Initial Plane

#### 3.2. The Intensity Distribution in the Focal Plane

_{z}at a ≠ 1 has two local intensity maxima on the horizontal axis at φ = 0 and φ = π. There are four local maxima for any a in Figure 3c, Figure 4c and Figure 5c. This is consistent with the formula (6) since I

_{y}should have 2m such maxima. Figure 6, Figure 7 and Figure 8 show intensity distributions similar to those shown in Figure 3, Figure 4 and Figure 5 but for the odd m = 3.

_{z}has four local intensity maxima for any a. Two maxima located on the horizontal x-axis are larger in magnitude than two maxima on the vertical y-axis. There are six local maxima for any a in Figure 6c, Figure 7c and Figure 8c. This is consistent with the formula (6) since I

_{y}should have 2m such maxima.

#### 3.3. The Distribution of the Stokes Vector Projections in the Focal Plane

_{1}and s

_{2}(s

_{3}= 0) for even numbers m = 2 (a, b) and m = 4 (c, d).

_{1}almost coincides with I

_{x}(Figure 4b). This is because s

_{1}= I

_{x}− I

_{y}and I

_{x}> I

_{y}.

_{1}almost coincides with I

_{x}(Figure 7c). This is because s

_{1}= I

_{x}− I

_{y}and I

_{x}> I

_{y}. It can be seen from Figure 10c that the third projection of the Stokes vector s

_{3}changes sign 2m = 6 times on circles with certain radii and with center on the optical axis. This is consistent with the formula (19), which includes sin(mφ). This function changes the sign 2m times per turn. The second Stokes projection changes sign when going around a closed trajectory around the optical axis 4m times: 8 (Figure 9b), 16 (Figure 9d) and 12 (Figure 10b). This is consistent with the formula (20), in which the term with the maximum argument has the form sin(2mφ).

_{2}(a, c) and the third s

_{3}(b, d) components of the Stokes vector of the focused vector field (1) with other odd numbers m: 1 (a, b) and 5 (c, d).

_{2}changes sign 4m times when going around the optical axis: 4 (Figure 11a) and 20 (Figure 11c). This is consistent with the formula (20). Additionally, the distribution s

_{3}changes sign 2m times: 2 (Figure 11b) and 10 (Figure 11d). This is consistent with the formula (19).

_{3}is close to +1 (red color) or −1 (blue color), decreases with the decreasing parameter a. The comparison of Figure 10c and Figure 12 show that the structures of the normalized S

_{3}and the non-normalized s

_{3}qualitatively agree.

#### 3.4. The Distribution of the Poynting Vector Projections in the Focal Plane

_{x}and P

_{y}are equal to zero (Figure 13b,c), and the longitudinal component P

_{z}does not have a radial symmetry (Figure 13a). It can be seen from Figure 13a and the formula (13) that the Poynting vector longitudinal component has a local maximum on the optical axis, and two local maxima (side lobes) are located on the vertical axis at φ = π/2 and φ = 3π/2 since the function P

_{z}(13) for m = 2 depends on the angle as cos(2φ). The calculation parameters in Figure 13 and Figure 14 are the same as in all previous Figures. Figure 13a also shows that the longitudinal component of the Poynting vector outwardly coincides with the intensity distribution in Figure 3a (m = 2). This is explained by the fact that the expression for the intensity (9) at m = 2, as well as (13), depends on the angle as cos(2φ).

_{z}is radially symmetric and has a maximum value on the optical axis. This is consistent with the Equation (13). It can be seen from Figure 14b,c that the transverse energy flow rotates in eight local subwavelength regions: counterclockwise in four regions and clockwise in the other four regions. Both transverse projections of the energy flux P

_{x}and P

_{y}change the sign eight times when going around the optical axis along a circle of some radius. This is consistent with formulas (11) and (12) since the dependence on the angle in these formulas is determined by the function cos((m + 1)φ) at m = 3. However, the number of local areas with a transverse vortex flow will be 2m. This follows from Equation (15), in which there are factors cos(mφ) and sin(mφ). The number of local regions with a vortex energy flow must be equal to the number of regions with a circular polarization, i.e., 2m. This follows from the effect of the spin-orbital interaction. There are six such regions in Figure 14d, and they lie along a circle of some radius. Their size is about 200 nm. Integrating in (15) the angular P

_{φ}and the radial P

_{r}projections of the Poynting vector over the angle φ, we obtain that the total transverse energy flux is equal to zero in the focus.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The distributions of linear polarization vectors in the cross section of the field (1) for m = 2 (

**a**,

**c**,

**e**) and m = 3 (

**b**,

**d**,

**f**), and a = 1/2 (

**a**,

**b**), a = 3/2 (

**c**,

**d**), a = 1 (

**e**,

**f**).

**Figure 3.**The intensity I (

**a**) and its components I

_{x}(

**b**), I

_{y}(

**c**) I

_{z}(

**d**) of the focused vector field (1) of the second (m = 2) order with a = 1.

**Figure 4.**The intensity I (

**a**) and its components I

_{x}(

**b**), I

_{y}(

**c**) I

_{z}(

**d**) of the focused vector field (1) of the second (m = 2) order with a = 1/2.

**Figure 5.**The intensity I (

**a**) and its components I

_{x}(

**b**), I

_{y}(

**c**) I

_{z}(

**d**) of the focused vector field (1) of the second (m = 2) order with a = 3/2.

**Figure 6.**The intensity I (

**a**) and its components I

_{x}(

**b**), I

_{y}(

**c**) I

_{z}(

**d**) of the focused vector field (1) of the second (m = 3) order with a = 1.

**Figure 7.**The intensity I (

**a**) and its components I

_{x}(

**b**), I

_{y}(

**c**) I

_{z}(

**d**) of the focused vector field (1) of the second (m = 3) order with a = 1/2.

**Figure 8.**The intensity I (

**a**) and its components I

_{x}(

**b**), I

_{y}(

**c**) I

_{z}(

**d**) of the focused vector field (1) of the second (m = 3) order with a = 3/2.

**Figure 9.**The Stokes vector components s

_{1}(

**a**,

**c**) and s

_{2}(

**b**,

**d**) of the focused vector field (1) with m = 2 (

**a**,

**b**) and m = 4 (

**c**,

**d**) for a = 1.

**Figure 10.**The Stokes vector components s

_{1}(

**a**), s

_{2}(

**b**), and s

_{3}(

**c**) of the focused vector field (1) with m = 3 and a = 1.

**Figure 11.**The Stokes vector components s

_{2}(

**a**,

**c**) and s

_{3}(

**b**,

**d**) of the focused vector field (1) with m = 1 (

**a**,

**b**) and m = 5 (

**c**,

**d**).

**Figure 12.**The third component of the normalized Stokes vector (16) S

_{3}of the focused vector field (1) with m = 3 and different parameters a: 1/2 (

**a**), 1 (

**b**) and 3/2 (

**c**). Arrows indicate the circular, elliptical or linear polarization.

**Figure 13.**The Poynting vector components in the focus for m = 2 and a = 1: P

_{z}(

**a**), P

_{x}(

**b**), P

_{y}(

**c**).

**Figure 14.**The Poynting vector components in the focus for m = 3 and a = 1: P

_{z}(

**a**), P

_{x}(

**b**), P

_{y}(

**c**). The arrows show the direction of the transverse Poynting vector in the focus (

**d**).

**Figure 15.**Optical setup for the generation of a beam with a non-uniform polarization (1). P is a polarizer; BS is a beam splitter; ND is a neutral density filter; M1 and M2 are mirrors; CVB m is a vector wave plate.

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**MDPI and ACS Style**

Kotlyar, V.; Stafeev, S.; Zaitsev, V.; Kozlova, E.
Spin-Orbital Conversion with the Tight Focus of an Axial Superposition of a High-Order Cylindrical Vector Beam and a Beam with Linear Polarization. *Micromachines* **2022**, *13*, 1112.
https://doi.org/10.3390/mi13071112

**AMA Style**

Kotlyar V, Stafeev S, Zaitsev V, Kozlova E.
Spin-Orbital Conversion with the Tight Focus of an Axial Superposition of a High-Order Cylindrical Vector Beam and a Beam with Linear Polarization. *Micromachines*. 2022; 13(7):1112.
https://doi.org/10.3390/mi13071112

**Chicago/Turabian Style**

Kotlyar, Victor, Sergey Stafeev, Vladislav Zaitsev, and Elena Kozlova.
2022. "Spin-Orbital Conversion with the Tight Focus of an Axial Superposition of a High-Order Cylindrical Vector Beam and a Beam with Linear Polarization" *Micromachines* 13, no. 7: 1112.
https://doi.org/10.3390/mi13071112