Interference Generation of a Reverse Energy Flow with Varying Orbital and Spin Angular Momentum Density

Abstract

This paper presents a novel method for generating and shaping reverse energy flow through the interference of light fields from a minimal number of point light sources. Until now, reverse energy flow has only been observed using complex light fields, such as optical vortices or cylindrical vector beams, limiting the formation of reverse energy flow near the optical axis. We demonstrate both analytically and numerically that unbounded regions of reverse energy flow can be achieved with just two point light sources, positioned asymmetrically at specific angles (e.g., 90 or 45 degrees) and with particular polarization states. The results indicate that the relative reverse energy flow can be enhanced by increasing the number of sources to three or four, adjusting their polarization, or introducing a vortex phase singularity. The presence of an initially embedded asymmetry in the fields under consideration leads to the formation of a non-uniform distribution of spin and orbital angular momentum density. Variations in the polarization state, as well as the introduction of a vortex phase singularity, allow for changing the distribution of angular momentum density while maintaining the presence of a reverse energy flow. We also explore the feasibility of implementing the obtained results using sectional phase diffractive optical elements, which will enhance the energy efficiency of the generated fields compared to point sources.

1. Introduction

Great interest in the study of a wide variety of structured laser beams [1,2,3,4,5,6,7] is associated with various features of the characteristics of these optical fields, such as amplitude-phase distribution, polarization state, coherence, etc. Various properties of laser beams are in demand in various applications. For example, laser beams with a vortex phase singularity [8,9,10,11,12] are successfully used in optical tweezers and spanners [13,14,15], for multiplexing of optical data transmission channels [16,17,18], optical imaging and microscopy [19,20,21], as well as for laser processing and patterning of various materials [22,23,24,25]. Vector beams with inhomogeneous polarization are also used to solve the problem of overcoming the diffraction limit in optical microscopy, laser material processing, optical trapping and manipulation of nano- and microparticles, holography, and optical communications [26,27,28,29,30,31,32,33,34,35].

It should be noted that for various applications, a combination of both the phase and polarization distribution of the generated light field may be important. This makes it possible to control characteristics of the electromagnetic field, such as the energy flow and Poynting vector density [36,37,38,39], orbital and spin angular momentum (OAM and SAM) density [40,41,42,43,44,45,46,47,48,49], optical forces distribution [50,51,52,53,54], as well as their interrelation [55,56,57,58,59,60,61,62,63]. Thus, controlling the amplitude-phase and polarization state of the generated light field allows one to influence various characteristics of the field required in specific situations.

Currently, one of the primary methods for shaping structured vector beams involves the use of metasurfaces [64,65,66,67]. However, these devices are still complex and expensive to manufacture for widespread use. A more common approach involves combining two orthogonally polarized light fields using diffractive optical elements (DOEs), spatial light modulators (SLMs), or digital micromirror devices (DMDs) [68,69,70,71,72]. Another method is the multi-beam interference of light fields with different polarization states from coherent sources [73,74,75,76,77]. In this scenario, various vector structured beams can be formed by adjusting the position of the sources and the polarization/focusing characteristics of the shaped light beams [78]. This method does not require the use of special optical elements but allows for dynamic control of the field’s polarization structure.

We propose a simple method for shaping a reverse energy flow caused by the interference of light fields from a small number (two or three) of point light sources. Reverse energy flow refers to the direction of energy flow being opposite to the propagation direction of the beam [79]. This method can be utilized to move dielectric nanoparticles with absorption in the opposite direction to the laser beam’s propagation, exerting a greater force compared to similar particles without absorption [80,81].

Previously, the potential for generating a reverse energy flow using circularly polarized optical vortex beams [82] and cylindrical vector beams [83,84,85] has been demonstrated. Typically, in these cases, the regions of the formation of the inverse energy flow are located near to the optical axis and are bounded (size is approximately of the order of the wavelength). To elongate the region of energy backflow, annular apertures [86,87] and specific phase or polarization distributions [88,89] have been suggested.

In this paper, we analytically and numerically demonstrate that unbounded regions of the reverse flow can be formed using only two point light sources located at equal distances from the origin of the coordinate system (see scheme in Figure 1). This is an important difference from the works [90,91], where interference patterns generated by four plane waves with linear polarization were considered. The magnitude of the reverse flow depends on both the angles between the radius vectors directed to the sources and on the polarization states of the sources. An important factor for the formation of the reverse flow is the asymmetric position of the two point sources, i.e., at an angle of 90 or 45 degrees between the radius vectors, depending on the polarization state. For two co-planar beams (located on the diameter of the circle), which were considered in the work [78], it is impossible to obtain a reverse energy flow.

Thus, the light fields under investigation initially have a broken symmetry, which leads to the formation of a non-uniform angular momentum density [92]. Introducing a phase difference does not change the fact of the presence of a reverse energy flow, but can be used to vary the distribution of angular momentum. We also study the possibility of implementing the obtained results using sectional phase DOEs. This approach will enhance the energy efficiency of the generated fields in comparison to point sources.

2. Theoretical Background

There are various approaches to calculating the propagation and focusing of vector beams, for example, the method of expansion in plane waves [93,94] and vector Rayleigh–Sommerfeld integrals [95,96,97] convenient for calculating the field in the near diffraction zone [98]. In the far diffraction zone or focal region, Richards–Wolf formulas [99,100] are often used. Obviously, in all cases, the task of calculation and analysis is simplified if there is a possibility of obtaining an analytical solution [101,102]. In particular, in the works [103,104], an analysis of the formation of a reverse energy flow for vortex beams with different polarizations was performed based on the point complex source method [105,106]. However, in this paper, we consider solutions that are simple for experimental implementation (for example, a set of holes in a screen [78] or segments in an annular slit [86], supplemented by a lens), which are difficult to describe analytically. In this case, it is logical to use the Richards–Wolf formulas.

The components of the electric and magnetic field vectors in the case of tight focusing are calculated using the Richards–Wolf formulas [99]. In a spherical coordinate system, they are given by the expressions:

$$\left[\begin{array}{c}E(r,\mathrm{\phi },z)\\ H(r,\mathrm{\phi },z)\end{array}\right]=-{\displaystyle \frac{if}{\mathrm{\lambda }}}{\displaystyle \underset{0}{\overset{\mathrm{\Theta }}{\int }}{\displaystyle \underset{0}{\overset{2\mathrm{\pi }}{\int }}T(\mathrm{\theta })F(\mathrm{\theta },\mathrm{\varphi })\left[\begin{array}{c}{P}_E(\mathrm{\theta },\mathrm{\varphi })\\ P_H(\mathrm{\theta },\mathrm{\varphi })\end{array}\right]\exp \left[ik(r\sin \mathrm{\theta }\cos (\mathrm{\varphi }-\mathrm{\phi })+z\cos \mathrm{\theta })\right]}}\sin \mathrm{\theta }d\mathrm{\theta }d\mathrm{\varphi }$$

(1)

where

$$\begin{array}{l}{P}_E(\mathrm{\theta },\mathrm{\varphi })=\left[\begin{array}{cc}A(\mathrm{\theta },\mathrm{\varphi })& C(\mathrm{\theta },\mathrm{\varphi })\\ C(\mathrm{\theta },\mathrm{\varphi })& B(\mathrm{\theta },\mathrm{\varphi })\\ -D(\mathrm{\theta },\mathrm{\varphi })& -E(\mathrm{\theta },\mathrm{\varphi })\end{array}\right]\left(\begin{array}{c}{c}_x(\mathrm{\theta },\mathrm{\varphi })\\ c_y(\mathrm{\theta },\mathrm{\varphi })\end{array}\right),\\ P_H(\mathrm{\theta },\mathrm{\varphi })=\left[\begin{array}{cc}C(\mathrm{\theta },\mathrm{\varphi })& -A(\mathrm{\theta },\mathrm{\varphi })\\ B(\mathrm{\theta },\mathrm{\varphi })& -C(\mathrm{\theta },\mathrm{\varphi })\\ -E(\mathrm{\theta },\mathrm{\varphi })& D(\mathrm{\theta },\mathrm{\varphi })\end{array}\right]\left(\begin{array}{c}{c}_x(\mathrm{\theta },\mathrm{\varphi })\\ c_y(\mathrm{\theta },\mathrm{\varphi })\end{array}\right),\end{array}$$

(2)

$$\begin{array}{l}A(\mathrm{\theta },\mathrm{\varphi })=1+{\cos }^2\mathrm{\varphi }(\cos \mathrm{\theta }-1)\hspace{0.17em},\\ B(\mathrm{\theta },\mathrm{\varphi })=1+{\sin }^2\mathrm{\varphi }(\cos \mathrm{\theta }-1)\hspace{0.17em},\\ C(\mathrm{\theta },\mathrm{\varphi })=\sin \mathrm{\varphi }\cos \mathrm{\varphi }(\cos \mathrm{\theta }-1)\hspace{0.17em},\\ D(\mathrm{\theta },\mathrm{\varphi })=\cos \mathrm{\varphi }\sin \mathrm{\theta },\hspace{1em}\\ E(\mathrm{\theta },\mathrm{\varphi })=\sin \mathrm{\varphi }\sin \mathrm{\theta }\hspace{0.17em}.\end{array}$$

(3)

In the above expressions (r, ϕ, z) are cylindrical coordinates in the focal region, (θ, $\mathrm{\varphi }$) are spherical angular coordinates at the exit of the pupil of the focusing system, sin (Θ) = NA is the numerical aperture of the system, F(θ, $\mathrm{\varphi }$) is the transmission (input) function, T(θ) is the apodization function, k = 2π/λ is the wave number, λ is the radiation wavelength, f is the focal length, and $\left(\begin{array}{c}{c}_x(\mathrm{\theta },\mathrm{\varphi })\\ c_y(\mathrm{\theta },\mathrm{\varphi })\end{array}\right)$ is the polarization vector. We do not assume that it is standardized. For the focal plane, z is 0.

The longitudinal component of the Poynting vector (we do not calculate the transverse ones in this work, since they do not form a reverse flow) is defined by the formula:

$$P_z\hspace{1em}\mathrm{Re}\left[E_x{H}_y^{\ast }-E_y{H}_x^{\ast }\right]=\mathrm{Re}\left[E_x^{\ast }{H}_y-E_y^{\ast }{H}_x\right].$$

(4)

Taking the real part in this expression corresponds to the classical definition. However, recent studies have revealed that the imaginary part of this component plays a role in the transverse force acting on optically trapped particles [107]. In most cases, the impact of the imaginary part is minimal compared to the real part [108,109]. Therefore, in this paper, we focus on the classical version represented by Equation (4).

The symmetry breaking in the beam structure leads to the formation of a non-uniform distribution of the angular momentum density. In particular, the SAM density distribution may be calculated by the following formula [110]:

$$S=\left(\begin{array}{c}{S}_x\\ S_y\\ S_z\end{array}\right)\simeq \mathrm{Im}\left(\begin{array}{c}{E}_y^{\ast }{E}_z-E_z^{\ast }{E}_y\\ E_z^{\ast }{E}_x-E_x^{\ast }{E}_z\\ E_x^{\ast }{E}_y-E_y^{\ast }{E}_x\end{array}\right).$$

(5)

Note that the longitudinal component of the SAM typically dominates and is associated with the presence of circular polarization in the generated field [111,112,113]. This allows trapped particles to rotate around their axis in the transverse plane. However, at sharp focusing, transverse components of the SAM may also exist [43,44,114,115], causing the particles to rotate around their axis in the direction of the light propagation. The coexistence of both transverse and longitudinal components [116] can result in the intricate 3D rotation of trapped particles.

Taking into account the spin–orbit interaction, we also consider the longitudinal component of the external OAM density distribution, associated with the breaking of the axisymmetric structure of the field [117,118]:

$$L_z\simeq \mathrm{Re}\left[x{E}_z^{\ast }{E}_x-y{E}_y^{\ast }{E}_z\right].$$

(6)

Note that a non-zero value for expression (6) indicates a shift in the longitudinal component of the focused field, which, in turn, results in the emergence of the spin Hall effect of light [55,62,63,116,117].

3. Analytics and Numerical Illustrations for Two Point Sources

3.1. A Single Point Source

If we consider a single point source as F(θ, $\mathrm{\varphi }$) with the center at the point $({\mathrm{\theta }}_0,{\mathrm{\varphi }}_0)$, radial dimension $\Delta \mathrm{\theta }$ and angular dimension $\Delta \mathrm{\varphi }$, then based on Equations (1) and (2) at z = 0 (in the focal plane), we can write approximate expressions (the factor in front of the integral is omitted):

$$\begin{array}{l}{E}_x^{\ast }\hspace{1em}R\Delta \mathrm{\varphi }\exp \left(-ikr\sin {\mathrm{\theta }}_0\cos ({\mathrm{\varphi }}_0-\mathrm{\phi })\right)\times \left[A({\mathrm{\varphi }}_0)c_x^{\ast }({\mathrm{\varphi }}_0)+C({\mathrm{\varphi }}_0)c_y^{\ast }({\mathrm{\varphi }}_0)\right],\\ E_y^{\ast }\hspace{1em}R\Delta \mathrm{\varphi }\exp \left(-ikr\sin {\mathrm{\theta }}_0\cos ({\mathrm{\varphi }}_0-\mathrm{\phi })\right)\times \left[C({\mathrm{\varphi }}_0)c_x^{\ast }({\mathrm{\varphi }}_0)+B({\mathrm{\varphi }}_0)c_y^{\ast }({\mathrm{\varphi }}_0)\right],\\ H_x\hspace{1em}R\Delta \mathrm{\varphi }\exp \left(+ikr\sin {\mathrm{\theta }}_0\cos ({\mathrm{\varphi }}_0-\mathrm{\phi })\right)\times \left[C({\mathrm{\varphi }}_0)c_x({\mathrm{\varphi }}_0)-A({\mathrm{\varphi }}_0)c_y({\mathrm{\varphi }}_0)\right],\\ H_y\hspace{1em}R\Delta \mathrm{\varphi }\exp \left(+ikr\sin {\mathrm{\theta }}_0\cos ({\mathrm{\varphi }}_0-\mathrm{\phi })\right)\times \left[B({\mathrm{\varphi }}_0)c_x({\mathrm{\varphi }}_0)-C({\mathrm{\varphi }}_0)c_y({\mathrm{\varphi }}_0)\right],\end{array}$$

(7)

where $R=\Delta \mathrm{\theta }T({\mathrm{\theta }}_0)\sin {\mathrm{\theta }}_0$.

If we substitute the components of the fields defined by Equation (7) into Equation (4) and take into account Equation (3), we obtain

$$P_z={(R\Delta \mathrm{\varphi })}^2\cos {\mathrm{\theta }}_0\left({\left|c_x({\mathrm{\varphi }}_0)\right|}^2+{\left|c_y({\mathrm{\varphi }}_0)\right|}^2\right).$$

(8)

This is a positive value, which means that there can be no reverse energy flow with a single point source. In addition, the value in Equation (8) does not depend on r and $\mathrm{\phi }$, i.e., it is the same across the entire focal plane.

3.2. Two Point Sources

With two or more sources, Equation (7) remains true for each of them separately (taking into account their parameters), and the total electric and magnetic fields will be equal to their sum. To simplify the expressions, we will assume that $\Delta \mathrm{\theta }$ and $\Delta \mathrm{\varphi }$ are the same for all sources, and that they themselves are located on one ring with a radius ${\mathrm{\theta }}_0$. In this case, the multiplier $R\Delta \mathrm{\varphi }$ will be the same for all fields and will not be written down in the following expressions. Exponential factors for two sources will no longer be reduced, and a dependence on r and $\mathrm{\phi }$ will appear. The right side of Equation (4) after the transformations will be equal to

$$\begin{array}{l}{E}_x^{\ast }{H}_y-E_y^{\ast }{H}_x=\left({\left|c_{x1}\right|}^2+{\left|c_{y1}\right|}^2+{\left|c_{x2}\right|}^2+{\left|c_{y2}\right|}^2\right)\cos {\mathrm{\theta }}_0+\\ +e^{ip\alpha }\left[\left(B_1{A}_2-C_1{C}_2\right)c_{x1}{c}_{x2}^{\ast }\right.+\left(B_1{C}_2-C_1{B}_2\right)c_{x1}{c}_{y2}^{\ast }-\\ -\left(C_1{A}_2-A_1{C}_2\right)c_{y1}{c}_{x2}^{\ast }\left.-\left(C_1{C}_2-A_1{B}_2\right)c_{y1}{c}_{y2}^{\ast }\right]+\\ +e^{-ip\alpha }\left[\left(A_1{B}_2-C_1{C}_2\right)c_{x1}^{\ast }{c}_{x2}\right.-\left(A_1{C}_2-C_1{A}_2\right)c_{x1}^{\ast }{c}_{y2}+\\ +\left(C_1{B}_2-B_1{C}_2\right)c_{y1}^{\ast }{c}_{x2}\left.-\left(C_1{C}_2-B_1{A}_2\right)c_{y1}^{\ast }{c}_{y2}\right]\end{array}$$

(9)

where $p=kr\sin {\mathrm{\theta }}_0;\hspace{0.33em}\hspace{0.33em}\alpha =\cos ({\mathrm{\varphi }}_1-\mathrm{\phi })-\cos ({\mathrm{\varphi }}_2-\mathrm{\phi })$. If the quantities $c_x(\mathrm{\varphi })$ and $c_y(\mathrm{\varphi })$ are real, then the procedure for taking the real part is simplified, and as a result we obtain the following expression for P_z:

$$\begin{array}{l}{P}_z=\left(c_{x1}^2+c_{y1}^2+c_{x2}^2+c_{y2}^2\right)\cos {\mathrm{\theta }}_0+\\ +\cos p\alpha \left[\left(A_1{B}_2+B_1{A}_2-2C_1{C}_2\right)\left(c_{x1}{c}_{x2}+c_{y1}{c}_{y2}\right)+\right.\\ +\left.\left(C_2(A_1-B_1)+C_1(B_2-A_2)\right)\left(c_{y1}{c}_{x2}-c_{x1}{c}_{y2}\right)\right]\end{array}$$

(10)

In its explicit form it looks like

$$\begin{array}{l}{P}_z=\left(c_{x1}^2+c_{y1}^2+c_{x2}^2+c_{y2}^2\right)\cos {\mathrm{\theta }}_0+\\ +\cos \left[2kr\sin {\mathrm{\theta }}_0\cdot \sin {\displaystyle \frac{{\mathrm{\varphi }}_1-{\mathrm{\varphi }}_2}{2}}\sin \left({\displaystyle \frac{{\mathrm{\varphi }}_1+{\mathrm{\varphi }}_2}{2}}-\mathrm{\phi }\right)\right]\times \\ \times \left[\left(c_{x1}{c}_{x2}+c_{y1}{c}_{y2}\right)\left(2\cos {\mathrm{\theta }}_0+{(1-\cos {\mathrm{\theta }}_0)}^2{\sin }^2({\mathrm{\varphi }}_1-{\mathrm{\varphi }}_2)\right)+\right.\\ +0.5\left.\left(c_{y1}{c}_{x2}-c_{x1}{c}_{y2}\right){(1-\cos {\mathrm{\theta }}_0)}^2\sin 2({\mathrm{\varphi }}_1-{\mathrm{\varphi }}_2)\right].\end{array}$$

(11)

Even after assuming that the polarization coefficients are real numbers, Equation (11) still remains quite complex. Therefore, below we will consider some specific cases where simplifications are possible.

3.3. Proportional Relationships of Polarization Coefficients

One of the simplifications of Equation (10) is provided under the condition (we assume $c_{x1}\ne 0$):

$$c_{y1}{c}_{x2}-c_{x1}{c}_{y2}=0\hspace{1em}\iff \hspace{1em}{c}_{y2}={\displaystyle \frac{c_{y1}{c}_{x2}}{c_{x1}}}.$$

(12)

Since the determinant of the components of the polarization vectors is zero, this condition corresponds to proportional relationships of the polarization coefficients. In this case, instead of Equation (11), we obtain

$$\begin{array}{l}{P}_z=\left(c_{x1}^2+c_{y1}^2+c_{x2}^2\left(1+c_{y1}^2/c_{x1}^2\right)\right)\cos {\mathrm{\theta }}_0+\\ +\cos \left[2kr\sin {\mathrm{\theta }}_0\sin {\displaystyle \frac{{\mathrm{\varphi }}_1-{\mathrm{\varphi }}_2}{2}}\sin \left({\displaystyle \frac{{\mathrm{\varphi }}_1+{\mathrm{\varphi }}_2}{2}}-\mathrm{\phi }\right)\right]\times \\ \times \left(c_{x1}{c}_{x2}+c_{y1}^2{c}_{x2}/c_{x1}\right)\left[2\cos {\mathrm{\theta }}_0+{(1-\cos {\mathrm{\theta }}_0)}^2{\sin }^2({\mathrm{\varphi }}_1-{\mathrm{\varphi }}_2)\right]\end{array}$$

(13)

For convenience, we introduce the following notations:

$$\begin{array}{l}U({\mathrm{\varphi }}_1,{\mathrm{\varphi }}_2,\mathrm{\phi })=\cos \left[2kr\sin {\mathrm{\theta }}_0\sin {\displaystyle \frac{{\mathrm{\varphi }}_1-{\mathrm{\varphi }}_2}{2}}\sin \left({\displaystyle \frac{{\mathrm{\varphi }}_1+{\mathrm{\varphi }}_2}{2}}-\mathrm{\phi }\right)\right],\\ Q_1({\mathrm{\varphi }}_1,{\mathrm{\varphi }}_2)=2\cos {\mathrm{\theta }}_0+{(1-\cos {\mathrm{\theta }}_0)}^2{\sin }^2({\mathrm{\varphi }}_1-{\mathrm{\varphi }}_2).\end{array}$$

(14)

Note that the value $Q_1({\mathrm{\varphi }}_1,{\mathrm{\varphi }}_2)$ is always positive.

Let us rewrite Equation (13) as follows:

$$P_z=\left(c_{x1}^2+c_{y1}^2+c_{x2}^2\left(1+{\displaystyle \frac{c_{y1}^2}{c_{x1}^2}}\right)\right)\cos {\mathrm{\theta }}_0+U({\mathrm{\varphi }}_1,{\mathrm{\varphi }}_2,\mathrm{\phi })Q_1({\mathrm{\varphi }}_1,{\mathrm{\varphi }}_2)\left(c_{x1}{c}_{x2}+c_{y1}^2{\displaystyle \frac{c_{x2}}{c_{x1}}}\right).$$

(15)

If $c_{x1}$, $c_{x2}$, $c_{y1}$ are positive, then $c_{y2}$ is also positive. The extrema of Equation (15) will be at ${\mathrm{\varphi }}_2-{\mathrm{\varphi }}_1=90°$ (the sources are located in perpendicular directions) and $U({\mathrm{\varphi }}_1,{\mathrm{\varphi }}_2,\mathrm{\phi })=\pm 1$. In explicit form:

$${\left(P_z\right)}_{{\displaystyle \frac{\max }{\min }}}=\left(c_{x1}^2+c_{y1}^2+c_{x2}^2\left(1+{\displaystyle \frac{c_{y1}^2}{c_{x1}^2}}\right)\right)\cos {\mathrm{\theta }}_0\pm \left(c_{x1}{c}_{x2}+c_{y1}^2{\displaystyle \frac{c_{x2}}{c_{x1}}}\right)(1+{\cos }^2{\mathrm{\theta }}_0).$$

(16)

Note that with sharp focusing, i.e., when $\cos {\mathrm{\theta }}_0\underset{{\mathrm{\theta }}_0\to 90°}{\to }0$, the minimum of this expression is negative, which corresponds to the appearance of a reverse energy flow.

Usually, when describing the phenomenon of reverse flow, an important characteristic is not only its magnitude but also its ratio to the largest possible value of the forward flow [84,86].

If we introduce the notation $q_1={\displaystyle \frac{c_{x1}{c}_{x2}+c_{y1}^2{c}_{x2}/c_{x1}}{c_{x1}^2+c_{y1}^2+c_{x2}^2\left(1+c_{y1}^2/c_{x1}^2\right)}}$, then the ratio of the minimum value of the energy flow to the maximum value will be equal to:

$$\mathrm{\eta }={\displaystyle \frac{{(P_z)}_{\min }}{{(P_z)}_{\max }}}={\displaystyle \frac{\cos {\mathrm{\theta }}_0-q_1{(1-\cos {\mathrm{\theta }}_0)}^2}{\cos {\mathrm{\theta }}_0+q_1{(1-\cos {\mathrm{\theta }}_0)}^2}}.$$

(17)

As can be seen from Equation (17), at ${\mathrm{\theta }}_0=90°$, the minimum negative relative value η = −1 (the greatest presence of reverse energy flow) is achieved regardless of q₁. If some value ${\mathrm{\theta }}_0$ is defined, then the minimum of the ratio η will be at the maximum achievable value of q₁ (the derivative with respect to q is negative). It can be shown that the maximum value q₁ = 0.5 is achieved at $c_{x1}/c_{x2}=1$ (regardless of $c_{y1}$). Then, the ratio will be equal to:

$$\mathrm{\eta }=-{\displaystyle \frac{{(1-\cos {\mathrm{\theta }}_0)}^2}{{(1+\cos {\mathrm{\theta }}_0)}^2}}.$$

(18)

Thus, the best results are achieved with equal $c_{x2}=c_{x1}$ and arbitrary $c_{y1}$. However, taking into account Equation (12), we obtain that $c_{y2}=c_{y1}{c}_{x2}/c_{x1}=c_{y1}$.

This means that for two point sources with the same polarization, located in perpendicular directions (${\mathrm{\varphi }}_2-{\mathrm{\varphi }}_1=90°$), there is always a reverse energy flow. Its relative magnitude increases with increasing angle ${\mathrm{\theta }}_0$ (i.e., focusing sharpness) and reaches an extremum of η = −1 at ${\mathrm{\theta }}_0=90°$.

Although there are studies in which the ratio η reached higher values (|η| ≥ 1) [83,84], the regions of the reverse flow in these cases had either a finite area or an infinite area, but of measure zero with respect to the entire focal plane [86,88,89]. In the case under consideration, the interference generation method provides theoretically infinite regions of the reverse flow, the fraction of which is close to half of the total focal plane. A similar situation was described in works [90,91], where the interference of four plane waves was considered and limited rectangular regions of the reverse flow were formed. The fraction of these regions was approximately half of the focal region. In this work, the difference lies not only in the exhaustive analysis of the conditions for achieving the reverse flow during the interference of fields formed by several point sources (taking into account their positions and polarization states), but also the possibility of forming unlimited strip regions of the reverse flow.

Figure 1 and Figure 2 display the results of numerical modeling for two point sources located in perpendicular directions with the same polarization. For linear polarization (Figure 1), we also showed (bottom row) a visual correspondence between the E field intensity and the distribution of $P_z$ in the focal plane (left) and along the optical axis (right). It is seen that the regions of negative values of $P_z$ coincide with the regions of minimal values of the E field intensity.

Strictly speaking, circular polarization (Figure 2) is not covered in the analytical discussion provided in this section, as one of the polarization coefficients is imaginary. However, as depicted in Figure 2, in this scenario, there is also a reverse energy flow, nearly equivalent to that achieved with linear polarization.

The results were obtained using Equations (1)–(6) with the following parameters: wavelength is λ = 1 µm, input field size is 200 µm × 200 µm, the output field size is 10 µm × 10 µm, numerical aperture NA = 0.99. Point sources in the calculations had a radius equal to the wavelength λ.

As can be seen, the distribution of all quantities in both cases is one-dimensional, i.e., it represents stripes located at an angle of 45° (=90°/2). This is quite expected based on the geometry of the location of two point sources.

Comparison of Figure 1 and Figure 2 shows that the energy flux distribution pattern $P_z$ and the η ratio are the same, although the polarization type in the sources changes. This change has a minor effect on the OAM distribution pattern and a much more significant effect on the SAM distribution. For linear polarization (Figure 1), the averaged values in each component are zero, while for circular polarization (Figure 2), they have a positive value (this is also expected, since the right-handed, i.e., “positive”, circular polarization is considered).

Note that for ${\mathrm{\varphi }}_2-{\mathrm{\varphi }}_1=180°$ (co-planarly located sources) the value $P_z$ will always be non-negative. For such an arrangement, it is equal to $P_z=\cos {\mathrm{\theta }}_0\left(c_{x1}^2+c_{x2}^2+2\cos \left(2kr\sin {\mathrm{\theta }}_0\cos \left(\mathrm{\phi }-{\mathrm{\varphi }}_2\right)\right)c_{x1}{c}_{x2}\right)$; and this expression is non-negative, since the coefficient for $c_{x1}{c}_{x2}$ is no greater than two in modulus. Various field characteristics obtained during the interference of co-planarly located sources with different polarization states were examined in detail in [78].

3.4. Orthogonal States of Polarization

Another option for simplifying Equation (10) arises when both $c_x$ and $c_y$ are nonzero and the equality holds:

$$c_{x1}{c}_{x2}+c_{y1}{c}_{y2}=0\hspace{1em}\iff \hspace{1em}{c}_{y2}=-{\displaystyle \frac{c_{x1}{c}_{x2}}{c_{y1}}}.$$

(19)

Equation (19) corresponds to orthogonal polarization states. In this case, instead of Equation (13), we obtain:

$$\begin{array}{l}{P}_z=\left(c_{x1}^2+c_{y1}^2+c_{x2}^2\left(1+{\displaystyle \frac{c_{x1}^2}{c_{y1}^2}}\right)\right)\cos {\mathrm{\theta }}_0+\\ +\cos \left[2kr\sin {\mathrm{\theta }}_0\sin {\displaystyle \frac{{\mathrm{\varphi }}_1-{\mathrm{\varphi }}_2}{2}}\sin \left({\displaystyle \frac{{\mathrm{\varphi }}_1+{\mathrm{\varphi }}_2}{2}}-\mathrm{\phi }\right)\right]\times \\ \times {\displaystyle \frac{1}{2}}\left(c_{y1}{c}_{x2}+c_{x1}^2{\displaystyle \frac{c_{x2}}{c_{y1}}}\right){(1-\cos {\mathrm{\theta }}_0)}^2\sin 2({\mathrm{\varphi }}_2-{\mathrm{\varphi }}_1)\end{array}$$

(20)

Taking into account Equation (14) and introducing the notation

$$Q_2({\mathrm{\varphi }}_1,{\mathrm{\varphi }}_2)=0.5{(1-\cos {\mathrm{\theta }}_0)}^2\sin 2({\mathrm{\varphi }}_2-{\mathrm{\varphi }}_1),$$

(21)

let us rewrite Equation (20) as follows:

$$P_z=\left[c_{x1}^2+c_{y1}^2+c_{x2}^2\left(1+{\displaystyle \frac{c_{x1}^2}{c_{y1}^2}}\right)\right]\cos {\mathrm{\theta }}_0+U({\mathrm{\varphi }}_1,{\mathrm{\varphi }}_2,\mathrm{\phi })Q_2({\mathrm{\varphi }}_1,{\mathrm{\varphi }}_2)\left(c_{y1}{c}_{x2}+c_{x1}^2{\displaystyle \frac{c_{x2}}{c_{y1}}}\right)$$

(22)

The bracket in the second term of Equation (22) is positive. The maximum of $P_z$ will be when $U({\mathrm{\varphi }}_1,{\mathrm{\varphi }}_2,\mathrm{\phi })=1$, and $Q_2({\mathrm{\varphi }}_1,{\mathrm{\varphi }}_2)$ reaches its maximum value, that is, when ${\mathrm{\varphi }}_2-{\mathrm{\varphi }}_1=45°$ (the maximum is equal to $0.5{\left(1-\cos {\mathrm{\theta }}_0\right)}^2$). Substituting into Equation (22) gives

$${\left(P_z\right)}_{\max }=\left[c_{x1}^2+c_{y1}^2+c_{x2}^2\left(1+{\displaystyle \frac{c_{x1}^2}{c_{y1}^2}}\right)\right]\cos {\mathrm{\theta }}_0+{\displaystyle \frac{1}{2}}{\left(1-\cos {\mathrm{\theta }}_0\right)}^2\left(c_{y1}{c}_{x2}+c_{x1}^2{\displaystyle \frac{c_{x2}}{c_{y1}}}\right).$$

(23)

Similarly, the minimum of $P_z$ will be when $Q_2({\mathrm{\varphi }}_1,{\mathrm{\varphi }}_2)$ is still maximum, and $U({\mathrm{\varphi }}_1,{\mathrm{\varphi }}_2,\mathrm{\phi })=-1$. This minimum is equal to

$${\left(P_z\right)}_{\min }=\left[c_{x1}^2+c_{y1}^2+c_{x2}^2\left(1+{\displaystyle \frac{c_{x1}^2}{c_{y1}^2}}\right)\right]\cos {\mathrm{\theta }}_0-{\displaystyle \frac{1}{2}}{\left(1-\cos {\mathrm{\theta }}_0\right)}^2\left(c_{y1}{c}_{x2}+c_{x1}^2{\displaystyle \frac{c_{x2}}{c_{y1}}}\right).$$

(24)

From Equation (24), it follows that the minimum will be negative only in the non-paraxial case, when $\cos {\mathrm{\theta }}_0\underset{{\mathrm{\theta }}_0\to 90°}{\to }0$.

In the general case, the expression for determining the exact boundary ${\mathrm{\theta }}_0$ required to form a reverse flow is cumbersome, so we will give it only for the case $c_{x1}=0$:

$$\cos {\mathrm{\theta }}_0<{\displaystyle \frac{2\left(c_{y1}^2+c_{x2}^2+c_{y1}{c}_{x2}\right)-2\sqrt{c_{y1}^2+c_{x2}^2}(c_{y1}+c_{x2})}{2c_{y1}{c}_{x2}}}.$$

(25)

The ratio of the minimum value to the maximum value is as follows:

$$\mathrm{\eta }={\displaystyle \frac{{(P_z)}_{\min }}{{(P_z)}_{\max }}}={\displaystyle \frac{\cos {\mathrm{\theta }}_0-q_2{(1-\cos {\mathrm{\theta }}_0)}^2}{\cos {\mathrm{\theta }}_0+q_2{(1-\cos {\mathrm{\theta }}_0)}^2}},$$

(26)

where $q_2={\displaystyle \frac{1}{2}}{\displaystyle \frac{c_{y1}{c}_{x2}+c_{x1}^2{c}_{x2}/c_{y1}}{c_{x1}^2+c_{y1}^2+c_{x2}^2\left(1+c_{x1}^2/c_{y1}^2\right)}}$.

Similar to the case considered in the previous section, at ${\mathrm{\theta }}_0=90°$, the minimum negative relative value η = −1 (the greatest presence of reverse energy flow) is achieved, regardless of q₂. If some value ${\mathrm{\theta }}_0$ is defined, then the minimum of the ratio η will be at the maximum achievable value q₂ = 0.5 (at $c_{y1}/c_{x2}=1$, regardless of $c_{x1}$). Then, instead of Equation (26), we obtain the following expression:

$$\mathrm{\eta }=-{\displaystyle \frac{{(1-\cos {\mathrm{\theta }}_0)}^2-4\cos {\mathrm{\theta }}_0}{{(1+\cos {\mathrm{\theta }}_0)}^2}}.$$

(27)

The relation defined by Equation (27) is not so good as in Equation (18). Moreover, if $\cos {\mathrm{\theta }}_0>3-2\sqrt{2}\hspace{0.33em}\hspace{0.33em}({\mathrm{\theta }}_0<80°7^{\prime })$, then the minimum will be positive, so that the reverse flow is present only when focusing at large angles $({\mathrm{\theta }}_0>80°7^{\prime })$.

Thus, for two point sources with orthogonal polarization, a reverse energy flow is formed if they are located in directions at 45 degrees (${\mathrm{\varphi }}_2-{\mathrm{\varphi }}_1=45°$) and there is a sharp focusing $({\mathrm{\theta }}_0>80°7^{\prime })$. The relative magnitude of the reverse flow reaches an extremum η = −1 at ${\mathrm{\theta }}_0=90°$. Note that this analysis is valid only for real polarization coefficients, i.e., it is not applicable to elliptical polarization.

Figure 3 and Figure 4 show the results of numerical simulation for two point sources with orthogonal polarization states. The angle between the sources is 45 degrees. The theoretical considerations provided in this section align with the results depicted in Figure 3.

In this case, the distribution of all values also takes the form of stripes located at an angle of 22.5° (=45°/2). Since both cases actually exhibit linear polarization (although in Figure 4 it is non-uniform), the distributions of the SAM density projections have a mean equal to zero. The value of η for Figure 3 is worse than for Figure 1, which corresponds to the given Equations (18) and (27).

Although the relative values of the reverse flux η in this situation are worse than those obtained in the previous paragraph (as theoretically predicted in Equations (18) and (27)), the combination of the locations of several point sources, considering both configurations, can be utilized for various variations of the generated interference fields.

4. Interference Formation of Reverse Flow in Different Configurations of Point Light Sources

From the results presented above, it follows that by arranging the point light sources and choosing their polarization states, it is possible not only to form regions with a reverse energy flow but also to vary the density of the angular momentum distribution. An additional degree of freedom is the combination of the two configurations considered above.

4.1. Three Point Light Sources

Figure 5, Figure 6 and Figure 7 show the results of numerical modeling for the case of three point sources with orthogonal polarization states. The angles between the sources are 45 degrees. The results reveal a significant increase in the complexity of the distribution structures being analyzed. The areas of reverse flow become segregated, with the exception of the circular polarization case (Figure 6).

The case shown in Figure 5 is interesting because the relative magnitude of the reverse flow η increased significantly. We can assume that there is one pair of point sources with an angle of 90 degrees between them and two sources with an angle of 45 degrees. Due to this, the value of η is better than for the previously considered cases, although it is smaller in absolute value than the algebraic sum of η from each pair. Even more important is that the stripes in the pictures are not strictly straight, which also affected the distribution of OAM and SAM density. At the same time, S_x retained a striped structure at an angle of 22.5°, and S_z at an angle of 67.5°. Note that the areas with negative values $P_z$ are closed.

Using three point sources with orthogonal circular polarizations impaired the value of the relative return flux η compared to two identical polarizations. This is most likely explained by the fact that a pair of points with opposite circular polarization with an angle of 45 degrees between the points does not give a reverse flux and, therefore, adding a third point impaired the result. The stripes in the distributions of $P_z$ and $L_z$ remained straight, while all the components of the SAM became more complex in structure. Note that the transverse components S_x and S_y have large areas with positive values, while in the case of S_z the average value is zero.

The results for the relative reverse flux η in Figure 7 are better than those in Figure 4 for the two orthogonal polarizations, but not by much. The distribution for $P_z$ and SAM has a mosaic structure, while the OAM density pattern retains straight lines. The average values of all SAM components are zero.

Note that it is possible to extend the considered configurations to other quadrants (see Figure 8). This extension also alters the interference structure, but the presence of the reverse flow η remains in roughly the same proportion as for the three sources.

Point light sources provide a low level of energy. Therefore, in the next section, we will discuss alternating orthogonal polarization states for non-point sources.

4.2. Segmented Annular-Shaped Polarization-Phase Elements

In this section, we consider ring segments instead of point sources, which can reduce the relative magnitude of the reverse energy flow but increase the total amount of energy entering the focal region. In this case, we use orthogonal polarization states similar to the previous section with continuation to other quadrants. As can be seen from the results shown in Figure 9, Figure 10 and Figure 11, the presence of the reverse energy flow is preserved everywhere, although its relative magnitude η decreased in absolute value, which is associated with the increase in point regions to ring segments.

For orthogonal linear polarizations (Figure 9), the distribution pattern of $P_z$ has a rotational symmetry of the 4th order. The OAM and SAM density distributions are noticeably less symmetrical and have zero values on average. For orthogonal circular polarizations (Figure 10), the distribution patterns $P_z$ and $L_z$ have a rotational symmetry of the 8th order. Moreover, there is a ring of negative values of $P_z$ in the central part.

When using orthogonal cylindrical polarizations (Figure 11), in contrast to previous cases, a rather high value of η is observed. This observation likely correlates with previously obtained results on the presence of a reverse flow for beams with radial and azimuthal polarizations with orders greater than one [84,85].

As can be seen from the examples provided, when a ring structure is used, the distribution of $P_z$ can be noticeably altered by changing the polarization type. However the SAM and OAM density distributions do not change qualitatively. Nevertheless, the distribution of OAM density can be significantly changed by adding a spiral phase plate (SPP) [119]. The complex transmission function of the SPP is described by the phase function exp(imφ), where m is the order of the vortex phase singularity. Consequently, the energy flow structure should also undergo changes [88]. The corresponding calculation results are presented in Figure 12, Figure 13 and Figure 14.

As can be seen, the most noticeable consequence of introducing a positive charge vortex phase was a change in the OAM density distribution; at m = 0 (left columns), an equal number of positive and negative values is observed, which leads to a zero average OAM, but in the presence of a vortex phase m = 1 (middle column), the clear majority of values became positive, and at a higher charge m = 2 (right column), the number and relative magnitude of positive values also increases.

However, a more interesting effect of introducing a vortex phase singularity is the enhancement of the relative reverse energy flux η for linear and circular polarization types (Figure 12 and Figure 13), especially in the case of m = 2. Maximization of the relative reverse flux for a second-order vortex phase was previously observed for uniform circular [83] and linear [86] polarization. Such maximization is achieved by concentrating the reverse energy flux at the central point of the focal plane (on the optical axis). A demonstration of a similar effect for alternating orthogonal polarization states is shown in this work for the first time. For orthogonal cylindrical polarizations (Figure 14), introducing a vortex phase singularity leads to a shift of regions with negative values of $P_z$ from the center to the periphery, which affects a decrease in the absolute value of η.

5. Discussion

In Section 3, it is analytically shown that the greatest relative value of the reverse flow is obtained when two points are located with a longitude difference of 90 degrees and the same linear polarization at these points. The question arises: Is it possible to improve the result by adding points? Since it has already been proven that the linear polarization can be arbitrary, and it is the difference in the directions of the points that is important, then, taking into account the form of Equations (1)–(4) and (7), for the sake of simplifying the calculations, we will place 4 points in positions with directions of 0, 90, 180, and 270 degrees, respectively. The polarization vector at all points will be taken equal to the linear x-polarization. Calculations according to Equation (7) yield the following values of the field components:

$$\begin{array}{l}{E}_y^{\ast }=H_x=0 \hspace{1em}for\hspace{0.17em}all\hspace{0.17em}dots;\\ D1:E_x^{\ast }=e^{-ipx}\cos {\mathrm{\theta }}_0,\hspace{0.33em}\hspace{0.33em}{H}_y=e^{ipx};\\ D2:E_x^{\ast }=e^{-ipy},\hspace{0.33em}\hspace{0.33em}{H}_y=e^{ipy}\cos {\mathrm{\theta }}_0;\\ D3:E_x^{\ast }=e^{ipx}\cos {\mathrm{\theta }}_0,\hspace{0.33em}\hspace{0.33em}{H}_y=e^{-ipx};\\ D4:E_x^{\ast }=e^{ipy},\hspace{0.33em}\hspace{0.33em}{H}_y=e^{-ipy}\cos {\mathrm{\theta }}_0;\\ where\hspace{0.17em}\hspace{0.17em}p=k\sin {\mathrm{\theta }}_0.\end{array}.$$

(28)

5.1. Two Sources

The main results are in Section 3.1, but for ease of analysis, we will write down all the expressions here. For a pair of points (D1, D3) and for a pair of points (D2, D4), the difference in directions is 180 degrees, and the energy flow values will only be non-negative, as mentioned above. For pairs (D1, D2) and (D3, D4), we obtain the same expressions for the flow

$$P_z=2\cos {\mathrm{\theta }}_0+\cos p(y-x)(1+{\cos }^2{\mathrm{\theta }}_0).$$

(29)

The ratio of the maximum and minimum values is

$$\mathrm{\eta }={\displaystyle \frac{\min P_z}{\max P_z}}=-{\displaystyle \frac{{(1-\cos {\mathrm{\theta }}_0)}^2}{{(1+\cos {\mathrm{\theta }}_0)}^2}}.$$

(30)

5.2. Three Sources

Let us see if adding a third point will improve things. Consider the triple (D1, D2, D3). We obtain the following expression for the energy flow

$$P_z=\cos {\mathrm{\theta }}_0+4{\cos }^{2}px\cos {\mathrm{\theta }}_0+2\cos px\cos py(1+{\cos }^2{\mathrm{\theta }}_0).$$

(31)

Finding the maximum value is obvious:

$$\cos px\cos py=1\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow \hspace{0.17em}\hspace{0.17em}\max P_z=5\cos {\mathrm{\theta }}_0+2(1+{\cos }^2{\mathrm{\theta }}_0).$$

(32)

Finding the minimum is not at all trivial. Omitting the details, we write the final expression:

$$\min P_z=\left\{\begin{array}{c}-{\displaystyle \frac{{\sin }^4{\mathrm{\theta }}_0}{4\cos {\mathrm{\theta }}_0}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\cos {\mathrm{\theta }}_0\ge 2-\sqrt{3},\hspace{0.17em}\\ 5\cos {\mathrm{\theta }}_0-2(1+{\cos }^2{\mathrm{\theta }}_0),\hspace{0.17em}\hspace{0.17em}else.\end{array}\right..$$

(33)

At the transition point, continuity is maintained.

A comparison of the ratio η (minimum of P_z to its maximum) for two and three points is provided in

Table 1. Comparison of the ratio η for two and three points located in directions at 90 degrees.

${\mathbf{\theta }}_0$	${\mathbf{\theta }}_0\to 0°$	$30°$	$45°$	$60°$	$74°27^{\prime }$	$90°$
2 points	$-{{\mathrm{\theta }}_0}^4/16$	$\begin{array}{l}56\sqrt{3}-97\approx \\ \approx -0.005\end{array}$	$\begin{array}{l}12\sqrt{2}-17\approx \\ \approx -0.029\end{array}$	$-{\displaystyle \frac{1}{9}}\approx -0.111$	$-{\displaystyle \frac{1}{3}}\approx -0.333$	–1
3 points	$-{{\mathrm{\theta }}_0}^4/36$	$\begin{array}{l}{\displaystyle \frac{7\sqrt{3}-15}{48\cdot 26}}\approx \\ \approx -0.0023\end{array}$	$\begin{array}{l}{\displaystyle \frac{3\sqrt{2}-5}{56}}\approx \\ \approx -0.013\end{array}$	$-{\displaystyle \frac{9}{160}}\approx -0.056$	$-{\displaystyle \frac{3}{13}}\approx -0.231$	–1

.

As can be seen from the results presented in Table 1, adding a third point impairs the value of the ratio η. It is important to note that the results of Section 4.1 do not contradict this statement, as they involve a different arrangement of points and polarizations. Other triplets consistently yield the same outcome. For the triplet (D1, D3, D4), the expression obtained is identical to Equation (33), and for the triplets (D1, D2, D4) and (D2, D3, D4), the same result is also achieved.

5.3. Four Sources

If we take all four points, we obtain the following expression for the flow

$$P_z=4\cos {\mathrm{\theta }}_0\left({\cos }^{2}px+{\cos }^{2}py\right)+4\cos px\cos py(1+{\cos }^2{\mathrm{\theta }}_0).$$

(34)

The maximum is

$$\cos px\cos py=1\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\max P_z=8\cos {\mathrm{\theta }}_0+4(1+{\cos }^2{\mathrm{\theta }}_0).$$

(35)

This is 4 times more than for two points. Finding the minimum is more difficult, but easier than for three points. The minimum will be under the following conditions:

$$\cos px\cos py=-1\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow \hspace{0.17em}\hspace{0.17em}\min P_z=8\cos {\mathrm{\theta }}_0-4(1+{\cos }^2{\mathrm{\theta }}_0)$$

(36)

This is also 4 times greater than for two points. Therefore, the ratio of the minimum to the maximum η is determined by the same formula (see Equation (30)) as for two points. However, here the minimum is achieved in isolated areas (Figure 15), whereas for two points, it occurs on straight lines (Figure 1).

Thus, it has been analytically shown that even with two light points located at a 90-degree angle, with the same polarization, it is possible to form a relative reverse energy flow up to η = −1 in linear unbounded regions. It has been numerically shown that it is possible to enhance the relative reverse energy flow (η < −1) by using point sources (at least three) located at a 45-degree angle, with orthogonal polarization states. Additional enhancement is possible by introducing a second-order vortex phase singularity.

6. Conclusions

In this paper, we analytically and numerically investigated the possibility of forming a reverse energy flow based on the interference of radiation from a minimum number of point sources in order to obtain unbounded areas of the reverse flow. It is analytically shown that it is possible to form a reverse energy flow in linear unbounded areas using only two point sources located at an angle of 90 degrees, with the same polarization (achievable relative reverse energy flow up to η = −1).

It is shown numerically that the relative reverse energy flow can be enhanced to obtain values η < −1 by using at least three point sources located at an angle of 45 degrees with orthogonal polarization states. It is also shown numerically that additional enhancement is possible by introducing a second-order vortex phase singularity.

An important factor for the formation of the reverse flow is the asymmetric position of several point sources, i.e., at an angle of 90 or 45 degrees. The presence of an initially embedded asymmetry in the fields under consideration leads to the formation of a non-uniform distribution of SAM and OAM density. Variations in the polarization state, as well as the introduction of a vortex phase singularity, allow for changing the distribution of angular momentum density while maintaining the presence of a reverse energy flow.

In conclusion, we would like to provide some insights into potential approaches for experimentally investigating reverse energy flow. Ref. [120] demonstrates that reverse energy flow can affect electrically neutral particles with non-zero conductivity, such as ZnO particles. In this scenario, the size of these particles must be significantly smaller than the wavelength. Furthermore, in Refs. [81,90,121], we have numerically shown that a Rayleigh particle placed in the region of reverse energy flow will move in the direction opposite to that of beam propagation. Additionally, for indirect experimental evidence of reverse energy flow, the method proposed in Ref. [90] can be employed. This method relies on measuring the energy flow distribution through the intensity distribution of the light field. The observation of a very weak local maximum, known as the Arago spot, which is caused by the diffraction of the forward flow by a circle 300 nm in diameter (the diameter of the tube containing the reverse flow), has been suggested as an explanation for the existence of reverse light energy on the optical axis.

The results obtained can be used in the field of laser manipulation of nano- and microparticles. The generation of light fields with a non-uniform distribution of the angular momentum density and the Poynting vector density can affect the nature of the motion of particles trapped in different parts of such fields. The formation of such complex optical traps is also interesting due to the fact that most real nano- and micro-objects, for which one would like to use the methods of non-invasive laser manipulation, have a non-spherical shape. These objects include various biological entities (such as spores, fungi, and cells) [122] and components of micromechanics, which are utilized in the creation of micromotors and micropumps, as well as in the optical assembly of diverse microplatforms. For instance, these microplatforms are crucial for the development of reconfigurable microenvironments for biomedical research [123,124]. Precise control of the distribution of the angular momentum of the formed optical trap will enable more accurate manipulation of such objects.