## Abstract

Based on the Richards-Wolf formalism, we obtain for the first time a set of explicit analytical expressions that completely describe a light field with a double higher-order singularity (phase and polarization), as well as distributions of its intensity and energy flux near the focus. A light field with the double singularity is an optical vortex with a topological charge *m* and with *n*th-order cylindrical polarization (azimuthal or radial). From the theory developed, rather general predictions follow. 1) For any singularity orders *m* and *n*, the intensity distribution near the focus has a symmetry of order 2(*n* – 1), while the longitudinal component of the Poynting vector has always an axially symmetric distribution. 2) If *n* = *m* + 2, there is a reverse energy flux on the optical axis near the focus, which is comparable in magnitude with the forward flux. 3) If *m* ≠_{0, forward and reverse energy fluxes rotate along a spiral around the optical axis, whereas at m = 0 the energy flux is irrotational. 4) For any values of m and n, there is a toroidal energy flux in the focal area near the dark rings in the distribution of the longitudinal component of the Poynting vector.}

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Starting from the classical Richards-Wolf paper [1], where analytical expressions have been obtained for the strength vector of a linearly polarized electric field near the focus of an aplanatic system, a lot of works appeared on the theoretical and numerical study of light fields with homogeneous (including linear, circular, elliptical) and inhomogeneous (radial, azimuthal) polarization near the sharp focus. For example, focusing of radially and azimuthally polarized light fields is studied in [2,3]. Review [4] calls the radially and azimuthally polarized beams by a united single term – cylindrical vector beams. For an elliptically polarized optical vortex of an arbitrary integer order, formulae were obtained in [5] for the components of the electric vector near the focus. In [6], higher-order cylindrical beams were investigated. However, light fields with both arbitrary phase and polarization singularities were not considered. Below, we call the higher-order cylindrical beams [6] as beams with the polarization singularity since the polarization direction on the optical axis of these beams is undetermined (a singular point). In [7–9], cylindrical vector beams were studied on a higher order Poincaré sphere. In [10], sharp focusing of a cylindrical vectorial optical vortex is numerically simulated. The cylindrical vectorial beams were generated by using micromirrors [11], diffractive optical elements [12], and a two-mode fiber with a long-period grating [13]. Propagation of a cylindrical vectorial beam shifted from the optical axis is studied in [14]. In [15], sharp focused cylindrical vectorial beams are used for drilling micro-holes. Using the Richards-Wolf formalism [1], expressions were obtained in [16–19] for the components of the electric and magnetic vectors near the focus for a linearly and circularly polarized arbitrary order optical vortex [16–18] and for a higher-order vectorial cylindrical beam [19].

In this paper, we generalize the papers [16–19] and obtain expressions for the components of the electric and magnetic fields near the focus of an optical vortex with an integer topological charge *m* and with cylindrical polarization (radial or azimuthal) of an integer order *n*. We also give an expression for the longitudinal component of the Poynting vector, from which it follows that if *n* = *m* + 2 then there is a reverse energy flux near the optical axis in the focal plane which is comparable in magnitude with the forward flux. For any values of *m* and *n*, we also derived equations for the radii of transverse rings near the focus, around which the toroidal flow of energy appears. In addition, we obtain an expression for the intensity distribution in the focal plane. It shows that the intensity distribution always has a 2(*n* - 1)-th order symmetry around the optical axis. It is also shown that a Rayleigh particle placed on the optical axis near the focus will move along the optical axis in its opposite direction.

## 2. Theoretical background

Let the transverse components of the monochromatic initial E-field (in the exit pupil of an aplanatic optical system [1]) be described by the following Jones vector:

*θ*,

*φ*) are the angles defining a unit vector or a point on a unit radius sphere,

*m*and

*n*are positive integer numbers,

*A*(

*θ*) is the amplitude of the spatial (angular) spectrum of plane waves [1] generating an optical vortex with a complex amplitude

A vectorial cylindrical field with polarization of Eq. (1) we will call a vortex field with *n*th-order azimuthal polarization, since for *n* = 1 the field (1) describes an azimuthally polarized optical vortex. Note that for *n* = 0 (*m*≠0) the field of Eq. (1) describes an optical vortex with linear polarization. The field of Eq. (1) can be represented as a superposition of two circularly polarized optical vortices:

*i*)

^{T}and (1,

*i*)

^{T}describe the left and right circular polarizations.

#### 2.1. Components of the electric and magnetic strength vectors of a light field with the double singularity in the sharp focus area

Using the Richards-Wolf formalism [1], we obtain the expressions for the components of the E-vector near the focus for the initial field (3):

In Eq. (4), the following integrals are denoted:

*f*is the focal length of the aplanatic system,

*λ*is the wavelength,

*NA*= sin

*α*is the numerical aperture,

*J*(

_{ν}*x*) is the νth-order Bessel function of the first kind,

*x*=

*kr*sin

*θ*, (

*x*,

*y*,

*z*) and (

*r*,

*φ*,

*z*) are the Cartesian and cylindrical coordinates. The integrals (5) depend only on the radial variable

*r*. As the function

*A*(

*θ*), Bessel-Gaussian functions can be used [2]:

*β*is the pupil radius of the aplanatic system divided by the waist radius of the Gaussian beam. Obviously, instead of Eq. (6) the function

*A*(

*θ*) can describe the Bessel-Gaussian beams $A(\theta )={J}_{m}(x)\mathrm{exp}(-{x}^{2})$ or the Laguerre-Gaussian beams $A(\theta )={x}^{\left|m\right|}{L}_{p}^{\left|m\right|}(x)\mathrm{exp}(-{x}^{2}/2)$ [20].

If we put *n* = 0 in Eq. (4), then we obtain the expression for the E-field of a linearly polarized optical vortex with the topological charge *m* derived in [16]:

For *m* = 0, Eq. (4) leads to the expressions we have obtained earlier [19] for the electric field with *n*th-order azimuthal polarization near the focus:

If we put *n* = 1 in Eq. (8), we get the well-known expressions [2] for the electric field with azimuthal polarization near the focus:

Similarly to Eq. (4), we give the expressions for the components of the magnetic field strength vector (H-field) near the focus of the aplanatic system for the initial light field of Eq. (3):

#### 2.2. Power flux and Intensity distribution

Based on Eqs. (4) and (10), we obtain an expression for the longitudinal component of the Poynting vector [1] **S** = [*c*/(8π)] Re[**E** × **H***], where *c* is the speed of light in vacuum, Re is the real part of a number, **E** × **H** is a cross product, * is a complex conjugation (below we omit the constant *c*/(8π)):

In the focal plane (*z* = 0) and in the particular case when *m* = 0, expression (11) coincides with a similar expression obtained in [19]. From Eq. (11) follows that if *n* = *m* + 2 then on the optical axis in the focal plane there is always negative (reverse) energy flux:

Similarly to Eq. (11), we can use Eq. (4) to derive an expression for the intensity distribution *I* = |**E**|^{2}:

If the amplitude function *A*(*θ*) of the angular spectrum is real then in the focal plane (*z* = 0) the intensity distribution reads as

We note that unlike the longitudinal component of the Poynting vector in Eq. (11), the intensity distribution is not rotationally symmetric. It has a symmetry of the order 2(*n* – 1). Equation (14) shows that for *m* = *n* > 1, the intensity contains the term ${I}_{0,0}^{2}/2$, which is nonzero on the axis (*r* = 0). It is similar to focusing a plane wave with circular polarization. That is, when *m* = *n*, phase and polarization singularities compensate each other on the optical axis.

Expressions for the transverse energy flux (transverse components of the Poynting vector) are more compact if they are written by using the polar components, i.e. radial component *S _{r}* and azimuthal component

*S*:

_{φ}It is seen in these equations that without the vortex (*m* = 0) the azimuthal component is zero. Cartesian components can be easily obtained from the polar components (*S _{x}* =

*S*cos

_{r}*φ*–

*S*sin

_{φ}*φ*,

*S*=

_{y}*S*sin

_{r}*φ*+

*S*cos

_{φ}*φ*) and if the amplitude function

*A*(

*θ*) of the angular spectrum is real then in the focal plane (

*z*= 0) we get:

Equations (17) and (18) show that if *m* ≠_{0 then the forward and reverse fluxes rotate along a spiral. If m = 0, energy flux is laminar (irrotational) for any value of n, i.e. Sx = Sy = 0. In this case, a toroidal energy flux occurs in the focus.}

#### 2.3. Toroidal flux in the focus

In the partial case, when *m* = 0 in Eq. (11) (vortex-free field), the axial energy flux in the focus (*z* = 0) is described by a simple expression:

According to Eq. (19), for any positive *n*, including linear polarization (*n* = 0), the longitudinal flux is zero on the radii that meet the condition ${I}_{0,n}^{}={I}_{2,n-2}^{}$. This means that near these radii the flux is reversed (i.e. negative, *S _{z}* < 0). Since there is no reverse flux in the initial field of Eq. (1), the reverse flux that appears at certain radii in the focal plane should change its direction from reverse to forward (in some plane before the focus). Therefore, on the radii where the Poynting vector (19) is zero, a toroidal flux of light energy is generated. Similarly, if

*n*= 0, Eq. (11) leads to an equation for the radii of transverse rings in the focus, around which the toroidal flow of energy occurs:

*I*

_{0,}

*=*

_{m}*I*

_{2,}

_{m}_{– 2}. Below we demonstrate it in numerical examples. Here, to obtain specific radii where the toroidal energy flux occurs, we suppose that the focus is generated by a narrow annular angular spectrum of plane waves:

*θ*

_{0}is the angle of a conical wave that generates the Bessel beam. Then, instead of Eq. (19) in focus we get a simple expression:

We remind that if *m* = 0 then the transverse components of the energy flux in the focus (*z* = 0) are equal to zero (*S _{x}* =

*S*= 0), whereas the longitudinal component of the Poynting vector equals zero on the rings, whose radii are determined from the equation:

_{y}Equation (22) can be solved only numerically, but it is seen that the energy flux is reverse if the first term in Eq. (21) is zero, i.e. the radius of the ring equals

*γ*are the roots of the

_{n,p}*n*th-order Bessel function. So, for any values of

*n*there is the reverse energy flux in the focus on the rings of radii (23), which forms a toroidal vortex, since all these rings make a torus surface. Presence of the toroidal flux has been studied theoretically [21–23] and experimentally [24].

#### 2.4. Circularly polarized optical vortex in the focus

An important partial case of the expressions obtained in this work is focusing of an optical vortex of an integer order *m* with homogeneous polarization (linear, circular, or elliptical). What follows from the obtained expressions directly is only the case of focusing an optical vortex with linear polarization. Indeed, if we put *n* = 0 in Eq. (1), we get a linearly polarized initial field $E=A(\theta ){e}^{im\phi}{\left(0,1\right)}^{T}$. The intensity distribution of a linearly polarized (*n* = 0) optical vortex in the focal plane follows from Eq. (14):

In addition, a longitudinal component of the Poynting vector in the focal plane of a linearly polarized optical vortex can be easily obtained from Eq. (11):

However, for a circularly polarized optical vortex similar expressions cannot be obtained directly from Eqs. (4) and (10). Therefore, we give them below. If instead of Eq. (1) the initial field is chosen in the following form:

*σ*= 1 for right circular polarization,

*σ*= –1 for left circular polarization,

*σ*= 0 for linear polarization, and

*σ*≠ 0, ± 1 for elliptical polarization (below we suppose that

*σ*is a real number), then the expressions for the components of the strength vectors of the electric and magnetic field near the focus are as follows

*σ*= 0 these values are equal (${\gamma}_{+}={\gamma}_{-}=1/\sqrt{2}$).

Based on Eq. (27), it is possible to obtain expressions for the intensity distribution in the focal plane for an optical vortex with homogeneous polarization:

According to Eq. (28), for linear and elliptical polarization the focal intensity distribution does not have the axial symmetry for it depends on the polar angle *φ*, whereas for circular polarization the intensity is circularly symmetric since the term with *φ* vanishes (*σ*^{2} = 1).

Similarly, an expression can be derived for the longitudinal component of the Poynting vector in the focal plane for a focused optical vortex with a topological charge *m* and with homogeneous polarization:

Equation (29) shows that the longitudinal energy flux in the focal plane of a homogeneously polarized optical vortex is always axially symmetric (i.e. independent of *φ*). For linear polarization (*σ* = 0), the expression for the energy flux (29) coincides up to a constant with Eq. (25) obtained from Eq. (11). For an optical vortex with left (*γ*_{+} = 0, *γ*_{–} = 1) or right (*γ*_{+} = 1, *γ*_{–} = 0) circular polarizations, expression (29) is simplified:

According to Eq. (30), if *m* = 2 (or *m* = –2) then for left (right) circular polarization there is a reverse energy flux on the optical axis in the focus. Near the rings with zero energy flux in the focal plane, whose radii satisfy $\sqrt{2}{I}_{0,m}^{}={I}_{2,m-2}^{}$ for left circular polarization and $\sqrt{2}{I}_{0,m}^{}={I}_{2,m+2}^{}$ for right circular polarization, toroidal energy fluxes are generated.

## 3. Numerical modeling

#### 3.1. Distributions of intensity and energy flux in the focus

Next, using the Richards-Wolf formulae [1] for modeling the sharp focusing of laser light with different orders of phase and polarization singularities, we show the presence of the reverse flux of Eqs. (11) and (12). The wavelength was *λ* = 532 nm, and the numerical aperture of the aplanatic lens was equal to *NA* = 0.95. Figure 1 shows distributions of polarization and phase of the considered beams with *m* = 1, *n* = 3 [Fig. 1 (a-c)] and *m* = 2, *n* = 4 [Fig. 1 (d-f)].

According to Fig. 1, adding the vortex phase converts the polarization distribution into a form with mirror symmetry with respect to the vertical axis. A cylindrical second-order vectorial beam (*m* = 0, *n* = 2) has the same mirror symmetry. Polarization distribution of such beam is shown in Fig. 2 [25].

Figure 3 shows the results of focusing an optical vortex with a topological charge *m* = 1 and with azimuthal polarization of order *n* = 3. Figure 3(a) shows the intensity distribution in the focal plane (*z* = 0), and Figs. 3(b-d) show the distributions of three components (longitudinal and two transverse) of the Poynting vector in the same plane. It is seen that the longitudinal component of the Poynting vector [Fig. 3(b)] has circular symmetry in accordance with Eq. (11) and its values near the optical axis are negative (reverse flux). The magnitude of the reverse flux is approximately 2 times lower than the maximum value of the forward flux. Note that from the intensity distribution [Fig. 3(a)] it is difficult to suppose that the longitudinal distribution of the energy flux [Fig. 3(b)] has the axial symmetry. We note that despite the intensity distribution [Fig. 3(a)] does not have the rotational symmetry, it has 4th-order symmetry with respect to Cartesian axes. According to Eq. (14) and as is shown below numerically, the intensity distribution has the order of symmetry 2(*n* – 1) = 4.

Figure 4 shows similar pictures, but with other parameters of the laser beam: *m* = 2, *n* = 4. As seen in Fig. 4, the intensity distribution has 6th order of symmetry, since 2(*n* – 1) = 6.

In Fig. 4(b), we find that the energy flux near the optical axis has the opposite propagation direction, and the distribution of the longitudinal component of the Poynting vector has the axial symmetry. It is difficult to predict from the intensity distribution in Fig. 4(a). Transverse components of the Poynting vector [Fig. 4(c,d)] look similar to the transverse components in Fig. 3(c,d).

#### 3.2. Toroidal opical flux near the focus

Using the Richards-Wolf formulae, we simulated the focusing of (i) a linearly polarized (*n* = 0) optical vortex with the topological charge *m* = 2 and of (ii) a second-order (*m* = 0, *n* = 2) cylindrical vector beam. It was supposed in the simulation that the focusing was done by a diffractive lens with a numerical aperture NA = 0.95. The wavelength of the focused light was chosen to be equal to λ = 532 nm. The results are shown in Figs. 5, 6.

(i) According to Fig. 5, there is a toroidal vortex generated in the focal plane *z* = 0 (shown in the left upper inset in Fig. 5). The torus radius is approximately 0.2 μm. In planes distant from the focus (at distances *z* = ± 0.9 μm ... ± 1.4 μm), complex structured toroidal vortices are generated shown in the right upper inset in Fig. 5.

(ii) Fig. 6 shows that, similarly to the case of focusing of a second-order (*n* = 2) cylindrical vector beam, there is a toroidal vortex generated in the focal plane (*z* = 0) (shown in the left upper inset in Fig. 6). In planes distant from the focus (at a distance *z* = ± 0.9 μm ... ± 1.3 μm), complex structured toroidal vortices are generated shown in the right upper inset in Fig. 6.

Figures 5 and 6 are obtained for a wide angular spectrum of plane waves. If we use a Bessel mode with the narrow angular spectrum *A*(*θ*) = *δ*(*θ* – *θ*_{0}) with *δ*(.) being the Dirac delta function, then according to Eqs. (11), (15) and (16) all the components of the Poynting vector *S _{z}*,

*S*and

_{r}*S*are independent of

_{φ}*z*in the area near the focus. Thus, the toroidal vortex is impossible in this case. However, if we use the angular spectrum with two narrow rings, i.e.

*A*(

*θ*) =

*C*

_{1}

*δ*(

*θ*–

*θ*

_{1}) +

*C*

_{2}

*δ*(

*θ*–

*θ*

_{2}), then

*S*becomes dependent on

_{z}*z*:

*p*= 1, 2.

According to Eq. (32), the Poynting vector is distributed with a period of λ/|cos *θ*_{1} – cos *θ*_{2}| along the longitudinal coordinate *z*. Figure 7 shows the intensity distribution and directions of the Poynting vector of a superposition of two Bessel modes near the focus. This figure was calculated for the wavelength λ = 532 nm, focal length *f* = 100λ, beam order (*m*, *n*) = (0, 2) (i.e. a vortex-free beam with second-order cylindrical polarization), cone angles of Bessel modes *θ*_{1} = 25 deg, *θ*_{2} = 85 deg, superposition coefficients *C*_{1} = *C*_{2} = 1, calculation area –2.5λ ≤ *x* ≤ 2.5λ, –λ ≤ *z* ≤ λ. The period of the distribution is λ/|cos *θ*_{1} – cos *θ*_{2}| ≈1.22λ. As seen in Fig. 7(b), there is the reverse flux on the optical axis and the Poynting vector makes a toroidal vortex around the red dots (yellow and red dots are intersections of transverse rings with zero Poynting vector with the plane *xz*). It is also seen in Fig. 7 that the toroidal vortices C and D have the opposite direction compared to the vortices A and B. This is because the vortices A and D (as well as B and C) are the sections of the same torus.

## 4. Rayleigh nanoparticle in the reverse flux in the focus

When a particle is placed in a light field, a force acts onto it. For a Rayleigh particle, this force consists of scattering and gradient forces. For Rayleigh particles with a radius *a* < λ/20 placed in a light field, the scattering force **F**_{s} and the gradient force **F**_{g} are calculated using the known formulas [26]:

*c*is the speed of light in vacuum,

*a*is the radius of a spherical Rayleigh particle,

*n*

_{1}and

*n*

_{2}are the refractive indices of the particle and of the medium. We choose the longitudinal component of the Poynting vector

*S*(11) in the focal plane (

_{z}*z*= 0) at

*m*= 0 and

*n*= 2, and obtain the intensity from Eq. (14) at

*n*= 2:

Next, using Eq. (35), we calculate the intensity gradient ∇|**E**|^{2}, and then the force vector near the focus of the polarization vortex (*m* = 0, *n* = 2) using Eqs. (33) and (34). We calculated the scattering and gradient forces for a particle with a radius *a* = 10 nm (wavelength *λ* = 532 nm) and with the refractive index *n*_{1} = 1.5 (*n*_{2} = 1). Figure 8 shows the dependence of the axial force component (in piconewtons) on the longitudinal coordinate (the particle moves along the optical axis *z*). The source beam has the power of 100 mW. Numerical aperture of the focusing lens is NA = 0.95. As seen in Fig. 8, the scattering force is greater than the gradient force by modulus and is directed against the axis *z* (i.e. the longitudinal component of the force vector is negative). Due to this force, the particle will move in the negative direction along the *z* axis.

In the center of the focal spot in Fig. 8, the total force *F _{z}* reaches −6,4*10

^{−9}pN increasing to −1,1*10

^{−10}pN at a distance of 1 μm from the focal plane.

## 5. Conclusion

In conclusion, using the Richards-Wolf formalism, we have obtained general expressions for the components of the electric and magnetic field strength vectors near the sharp focus of an optical vortex with the topological charge *m* and with *n*th-order azimuthal polarization. An expression for the longitudinal component of the Poynting vector has also been obtained. For different values of the numbers *m* and *n*, simple consequences from these formulas have been derived. If *m* = *n* > 1, there is nonzero intensity on the optical axis, like at the focusing a vortex-free circularly polarized light field, whereas if *n* = *m* + 2 then there is a reverse flux of light energy on the optical axis in the focal plane. Numerical simulation confirmed the reverse flux near the optical axis and it turned out that if *n* = *m* + 2 then the intensity distribution in the focal plane has symmetry of 2(*n* – 1)-th order around the optical axis, while the distribution of the longitudinal component of the Poynting vector has axial symmetry for any *m* and *n* (*n* = *m* + 2). In addition, we have shown that if *m* = 0 then on the rings of certain radii (proportional to the roots of the *n*th-order Bessel functions), a reverse energy flux is generated in the focus, which in 3D space near the focus propagates along the torus around these rings. This is the so-called toroidal energy flow [21–24]. If *m* ≠_{0, the toroidal flux also appears near the rings with zero flux in the focal plane (30). We have also shown numerically that near the focus of a light field with second-order azimuthal polarization the axial component of the force, acting from the field onto a dielectric spherical Rayleigh nanoparticle, is negative, i.e. directed in the backward direction.}

## Funding

Russian Science Foundation (18-19-00595); Russian Foundation for Basic Research (18-29-20003); RF Ministry of Science and Higher Education (agreement Nº 007-ГЗ/Ч3363/26).

## Acknowledgments

Section 2 “Theoretical background” was supported by the Russian Science Foundation (18-19-00595), Section 3 “Numerical modeling” was supported by the Russian Foundation for Basic Research (18-29-20003), Section 4 “Rayleigh nanoparticle in the reverse flux in the focus” was supported by RF Ministry of Science and Higher Education (State assignment).

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