## Abstract

The controlled and continuous negative energy flow (from negative to positive) on the optical axis in the focal region is obtained by adjusting the polarization distribution of the input second-order radially polarized beam (the polarization topological charge is equal to 2). Moreover, the similar evolution of negative energy flow also can be achieved for the tightly focused vector beams with polarization topological charge −2. It is because both the beams with polarization topological charges 2 and −2 can possess the same polarization and spin flow density distributions with the help of the polarization modulation. The results provide a potential method for modulating the effects induced by the spin-orbit coupling in tight focusing of optical beam.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In recent years, considerable research interests have been attracted in the study of negative energy flow of light beams because of its potential applications in the field of optical micro-manipulation. As the name implies, the negative energy flow means the direction of energy flow is opposite to the propagation direction of the beam, it can exist in some specific cases, such as the tightly focused vortex beams [1,2] and vector beams [3–6], the propagation of vector Bessel beam [7,8], non-paraxial Airy beam [9], and quantum optical system [10] etc..

In previous works [1–3], researchers showed that, the negative energy flow can be observed on the optical axis in the focal plane of the tightly focused cylindrical vector beams with polarization charge 2 or vortex beams with topological charge 2; otherwise, the energy flow on the optical axis in the focal plane is zero. Namely, the on-axis negative energy flow can be modulated by changing the polarization charge or vortex charge. One knows the controlled and continuous change of negative energy flow can provide a more flexible manipulation in real optical application. Yet, it is hard to obtain a continuous change (from negative to positive gradually) of on-axis negative energy flow in the focal region by changing the polarization charge or vortex charge. It is necessary to present a method to control the on-axis energy flow in the focal region.

The polarization is an important parameter of the optical field [11,12]. The different polarization states correspond to different kinds of optical angular momentum, such as the circular polarization is related to spin angular momentum [13–15], the hybrid polarization is related to a kind of intrinsic orbital angular momentum [16]. Moreover, the polarization determines the spin angular momentum and spin flow density distributions of vector beam [17–19]. And it has known the polarization also plays a crucial role in the field of optical micro-particle manipulation [20–24], super-resolution imaging [25–27], and optical topological structure [28,29]. This means that, in the tight focusing of vector beam, some effects induced by the spin-orbit coupling may be controlled by adjusting the polarization of input vector beam because the change of polarization can induce the change of intrinsic structure of optical angular momentum.

Here, we introduce a coefficient matrix to modulate the polarization distribution and intrinsic optical angular momentum structure of input vector beam, and obtain the controlled and continuous evolution of negative energy flow on the optical axis in the focal region of the tightly focused radially polarized beams. We find that, by adjusting the polarization distribution of input vector beam, the controllable negative energy flow on the optical axis in the focal region of the tightly focused vector beam can be obtained when the polarization topological charge is equal to 2, which can change from the negative to positive continuously. Moreover, if the input vector beams possess the same intrinsic structure (polarization and spin flow density) as the beam with polarization topological charge 2, such as the vector beam with polarization topological charge −2, there also is a continuous on-axis negative energy flow in the focal region. It means our results can provide a potential way to modulate the effects induced by the spin-orbit coupling in the tight focusing of the optical beam.

## 2. Theoretical analysis

Let us consider the tight focusing of $m$-order radially polarized beams, $m$ is the polarization topological charge of input vector beam. The Richards-Wolf diffraction integral has the form as following [30],

Generally, the polarization distribution of the radially polarized beam is described by the vectorial coefficient matrix ${\left[ {\begin{array}{cc} {{{\tilde{c}}_x}(\phi )}&{{{\tilde{c}}_y}(\phi )} \end{array}} \right]^T}$ in Eq. (1), and ${\tilde{c}_x}(\phi ) = \cos (m\phi )$, ${\tilde{c}_y}(\phi ) = \sin (m\phi )$. According to the Richards-Wolf diffraction integral, the components of tightly focused vector beam can be derived directly. According to the definition of time averaged energy flow density $\textbf{S} = (c/8\pi ){\mathop{\rm Re}\nolimits} ({\textbf{E}^\ast } \times \textbf{H})$, the longitudinal component of the energy flow density in the focal plane can be expressed as ${S_z} = {\mathop{\rm Re}\nolimits} ({E_x^\ast {H_y} - E_y^\ast {H_x}} )$. Figure 1(a) shows the energy flow density properties in the focal plane ($z = 0$) for different polarization topological charges. The calculation parameters are taken as, ${\theta _0} = {80^ \circ }$, $f = 120\lambda $, $\lambda $ is the wave length of incident light. We know that the energy flow density on the optical axis in the focal plane is negative when the polarization topological charge is equal to 2, and it is zero when the polarization topological charge is unequal to 2, which is consistent with the results obtained in previous works [1–3]. Figure 1(b) shows the evolution of energy flow density in the center (on-axis) of the focal plane as the change of polarization topological charge. Obviously, the consecutive evolution of energy flow on the optical axis can’t be obtained by changing the polarization topological charge.

It has known the polarization distribution determines the intrinsic spin angular momentum and spin flow density of vector beams [18], and it affects some effects induced by the spin-orbit coupling in the tightly focused optical beams [13–14,28]. Experimentally, the spatial polarization distribution of vector beams can be well modulated by means of the spatial light modulator and computer-generated hologram [16,31–32]. Here, in order to obtain the controlled and continuous change of the negative energy flow on the optical axis in the focal region, a coefficient matrix $\textrm{R(}{\phi _0}\textrm{)}$ is introduced to adjust the polarization distribution and intrinsic structure of optical angular momentum of input vector beam, and the effective polarization matrix ${\left[ {\begin{array}{cc} {{{\tilde{c}}_x}(\phi )}&{{{\tilde{c}}_y}(\phi )} \end{array}} \right]^T}$ in Eq. (1) can be expressed as,

*w*is the width of narrow annular region and $w = 0.5\lambda $. For the small

*w*, $\exp [{ - {{(\rho - {\rho_0})}^2}/{w^2}} ]$ can be reduced to $\delta (\rho - {\rho _0})$ [33], and it satisfies the assumption we proposed about the input vector beams approximatively. It shows that, though the polarization topological charge is unchanged, the polarization distribution and the spin flow density are well modulated by the parameters ${\phi _{01}}$ and ${\phi _{02}}$. It implies the consecutive change of on-axis negative energy flow may be obtained by means of the polarization modulation of input vector beam.

## 3. Controlled negative energy flow in the focal field

Based on the Richards-Wolf diffraction integral, the field components in the focal region can be derived directly. In previous works, researchers showed the negative energy flow can be obtained in the tight focusing of vector beam with polarization topological charge 2 [1–3]. Figure 3 shows the evolution of normalized energy flow density in the focal plane ($z = 0$) when the input vector beam with polarization topological charge 2. Obviously, the energy flow in the focal plane is well modulated by changing of the parameter ${\phi _{01}}$. Interestingly, the energy flow density on the optical axis can change from negative value to positive value with the variation of parameter ${\phi _{01}}$.

Figures 4(a) and (c) show the evolution of the energy flow density in the focal plane with the change of parameters ${\phi _{01}}$ and ${\phi _{02}}$ when the input vector beam with polarization topological charge 2, the calculation parameters are same as Fig. 1. We find that, when the parameter ${\phi _{01}}$ or ${\phi _{02}}$ is modulated independently, both the on-axis energy flow density evolution are symmetrical about ${\phi _{01}} = \pi $ (or ${\phi _{02}} = \pi $), such as the properties of energy flow density are same when ${\phi _{01}}\textrm{ = }\pi /4$ and ${\phi _{01}}\textrm{ = }7\pi /4$ (or ${\phi _{02}} = \pi /4$ and ${\phi _{02}}\textrm{ = }7\pi /4$), which is shown in Figs. 4(b) and (d). It is worth noting that, the on-axis energy flow can change from the negative to positive when the parameter ${\phi _{01}}$ is modulated independently, and it is negative value when the parameter ${\phi _{02}}$ is modulated independently. It means the negative energy flow on the optical axis can be well controlled by means of the polarization modulation of input vector beam.

In previous works, researchers showed the negative energy flow can be obtained when the input vector beam with polarization topological charge 2 [1–3]. We have shown that, by adjusting the polarization distribution, the continuous energy flow change (from negative to positive) can be obtained when the input vector beam with polarization topological charge 2. One knows the spin flow density of vector beam is determined by the polarization topological charge [18], the change of polarization distribution will induce the change of spin flow density of the vector beam. Then, there is a question: if the input vector beam possesses the same polarization and spin angular momentum distributions as the vector beam with polarization topological charge 2, but its polarization topological charge is unequal to 2, does the on-axis negative energy flow occurs in the focal region of the vector beam?

By modulating the polarization distribution, the vector beam with polarization topological charge −2 can possesses the same polarization and spin angular momentum distributions as the beam with polarization topological charge 2. It means the negative energy flow should be obtained in the tight focusing of the vector beam with polarization topological charge −2. Figure 5 shows the evolution of the energy flow density in the focal plane ($z = 0$) and on the optical axis with the change of ${\phi _{01}}$ and ${\phi _{02}}$ when the input vector beam with polarization topological charge −2. We find that, by means of the polarization modulation, the continuous on-axis negative energy flow in the focal plane also can be obtained in the case that the polarization topological charge of input vector beam is −2, which is a novel phenomenon and different from the results in Refs. [1–3].

The study of on-axis energy flow is important for the optical manipulation. Figure 6 shows the evolution of on-axis energy flow in the focal plane when the polarization topological charges of input vector beams are equal to 2 and −2. We find, by modulating the parameters ${\phi _{01}}$ and ${\phi _{02}}$, the on-axis energy flow possesses similar evolution properties in both two cases. We also know that, for the input vector beam with polarization topological charge 2, the maximal on-axis negative energy flow can be obtained when ${\phi _{01}}\textrm{ = }0$, ${\phi _{02}}\textrm{ = }0$ or ${\phi _{01}}\textrm{ = } \pm \pi $, ${\phi _{02}}\textrm{ = } \pm \pi $; for the input vector beam with polarization topological charge −2, the maximal on-axis negative energy flow can be obtained when ${\phi _{01}}\textrm{ = 0}$, ${\phi _{02}}\textrm{ = } \pm \pi $ or ${\phi _{01}}\textrm{ = } \pm \pi $, ${\phi _{02}}\textrm{ = 0}$.

In order to verify the influence of polarization distribution in the modulation of negative energy flow, Fig. 7 shows the polarization distribution and the normalized spin flow density of input vector beams when the maximal on-axis negative energy flow in the focal plane is obtained. We find that, when the maximal on-axis negative energy flow is obtained, the polarization and spin flow density distribution are same for the input vector beams with polarization topological charges 2 and −2. Generally, the negative energy flow is obtained by the change of the focal field components, which can be realized by modulating the polarization distribution of incident beam. Intrinsically, because the polarization determines the spin angular momentum and spin flow density distributions of vector beam, the change of polarization distribution induce the change of the intrinsic structure of optical angular momentum. The occurrence of negative energy flow is a manifestation of spin-orbit coupling in the tight focusing of vector beams. Namely, the effects induced by the spin-orbit coupling in the tight focusing of vector beam can be modulated with the help of the adjustment of the polarization distribution.

## 4. Conclusions

In previous works, though the on-axis negative energy flow can be obtained in the tight focusing of vector beam with polarization topological charge 2, it can’t realize continuous change from the negative to positive. In our study, by introducing a coefficient matrix to adjust the polarization distribution of input vector beam, we obtained the controlled on-axis negative energy flow in the focal plane for the beams with polarization topological charges 2 and −2 theoretically, which can change from the negative to positive continuously. We found that, by adjusting the polarization distribution, the vector beams with polarization topological charges 2 and −2 can have the same intrinsic structure including the polarization and spin angular momentum distributions. It further verifies the significance of polarization in the tight focusing of vector beams. The occurrence of the negative energy flow is a manifestation of the spin-orbit coupling in the tight focusing of vector beams. It means that the effects induced by the spin-orbit coupling can be modulated with the help of the polarization modulation. Our results provide a potential way to modulate the effects induced by the spin-orbit coupling in the tight focusing of the optical beam.

## Funding

National Natural Science Foundation of China (11974101, 11974102, 11704098).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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