## Abstract

Optical angular momenta (AM) have attracted tremendous research interest in recent years. In this paper we theoretically investigate the electromagnetic field and angular momentum properties of tightly focused arbitrary cylindrical vortex vector (CVV) input beams. An absorptive particle is placed in focused CVV fields to analyze the optical torques. The spin-orbit motions of the particle can be predicted and controlled when the influences of different parameters, such as the topological charge, the polarization and the initial phases, are taken into account. These findings will be helpful in optical beam shaping, optical spin-orbit interaction and practical optical manipulation.

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

## 1. Introduction

Optical beams have been known to carry angular momentum (AM) [1–3], which expresses the amount of dynamical rotation in the electromagnetic field. There are two distinct forms of AM: spin angular momentum (SAM) and orbital angular momentum (OAM) [4–7]. The former is related to the optical polarization and in particular to circular polarization. The latter depends on the fundamental characteristics of electromagnetic fields, and in particular on the wavefront shape. The SAM can make the particle spin around its own axis [8,9], while the OAM can make the particle orbit around the optical axis [10,11]. Generally, the two different AM forms are independent, but they can interact and be mutually coupled in specific optical processes such as in inhomogeneous or anisotropic media [12–14] and tightly focusing of circularly polarized beams [15].

In recent years, transverse optical SAM, which is perpendicular to the direction of propagation, has attracted intensive attention for its practical applications in microscopy, particle trapping [16,17] and manipulation, especially in high numerical aperture (NA) focusing system [18–21]. When tightly focusing light, one needs to take into account the polarization state of the light. Cylindrical vector vortex (CVV) optical fields have raised a great deal of interest due to their unique polarization and focusing properties [22–24]. Two extreme cases are radially polarized vortex (RPV) and azimuthally polarized vortex (APV) beams. Previous research work that was mainly on OAM can induce localized SAM in the strong focusing of spin-free linearly, radially, azimuthally, and cylindrically polarized beams [19–21,25,26]. However, when a CVV beam is focused, other optical properties including initial phase, topological charge and optical torque should be considered and exploited.

In this paper, we study the spinning dynamics of particles with highly focused CVV input beams. With the Richards and Wolf formulas for the calculation of focused field of an arbitrary CVV beam, we can theoretically obtain all components of the electromagnetic field near focus in a high NA optical system, as well as SAM densities of the focused field. The interaction and transfer between spin- and orbital AM are also shown and discussed. Furthermore, the absorptive trapped particle is introduced to discuss the influences of optical torque on the motions of the particle. We consider optical field with different parameters like topological charge, polarization state and initial phase angles, which may have potential in optical manipulation processes, practical optical trapping, laser structuring of materials [27–29] and optical communications [30,31].

## 2. Local SAM of focal field for CVV beams

A generalized CVV beam can be written as a linear superposition of a cylindrically symmetric RPV beam and a cylindrically symmetric APV beam, which are given by [32]

For tightly focusing optical systems, vectorial theory is needed to accurately describe the field near the focal plane. Originating from Ignatovsky’s diffraction theory [33], later studied in detail in [34,35], the solution is referred to the vectorial Richard-Wolf integrals. When focused in a homogeneous medium with a real-valued refractive index $n$, the electric and magnetic fields in the focal region can be expressed as [36]

For an arbitrary time-harmonic beam, the time-averaged SAM density can be written as [6,37]

Next, we discuss the influence of the topological charge $m$ on the local spin state. For comparison, we start again from the focal field profiles of CVV beams with $m=1$ and $\varphi _0=0$, $\pi /4$, $\pi /3$, $\pi /2$ and $-\pi /4$ in Fig. 4. As opposed to focusing CVV beams without OAM in Fig. 2, the CVV beams with $m=1$ generate solid focal spots. The donut-shaped longitudinal intensities disappear gradually with the increase of the initial phase $\varphi _0$, while the focal spots get increasingly sharper as seen in Figs. 4$(\textrm{d}_1)$–4$(\textrm{d}_4)$. The special cases of field distributions generated by RPV and APV beams are shown in Figs. 4$(\textrm{a}_1)$–4$(\textrm{d}_1)$ and Figs. 4$(\textrm{a}_4)$–4$(\textrm{d}_4)$, respectively. It is well known and widely used that the RP beam provides tight focus, but the focal spot of APV beam with $m=1$ shown in Fig. 4$(\textrm{d}_4)$ gives the smallest size compared with focal spots given by RPV beams as shown in Fig. 2$(\textrm{d}_1)$. This fact was noted in earlier work [39] and can be used for improving the resolution of optical imaging systems. Figure 5 gives the results of SAM density distributions of the corresponding focused CVV input beams with $m=1$ as in Fig. 4. In Fig. 5 one observes similar as in Fig. 3, but in addition one can notice that the purely transverse spin states in Fig. 3 change to three components here. Strong longitudinal SAM densities appear in the focal plane and dominate in three components, even in the special cases of RPV and APV input beams. The SAM densities of the other two components are getting weaker, which means the properties of the local spin state have been converted. Obviously, the varying ratios between transverse and longitudinal SAM densities is because of different initial phases and the induced OAM. To avoid the misunderstanding, we should point out that the global axial SAM $S_z$ is zero, which can be validated by numerically integrating the $S_z$ density over the whole focal plane.

To explore further the influences of the initial phases and OAM on the distributions of SAM density, Fig. 6 depicts the normalized cross sections of SAM densities in the focal plane of strongly focused input CVV beams with $\varphi _0=0$, $\pi /6$, $\pi /4$, $\pi /3$, $\pi /2$ and $-\pi /4$ and topological charge $m=0$ (Figs. 6$(\textrm{a}_1)$–6$(\textrm{c}_1)$), $m=1$ (Figs. 6$(\textrm{a}_2)$–6$(\textrm{c}_2)$) and $m=-1$ (Figs. 6$(\textrm{a}_3)$–6$(\textrm{c}_3)$). All distributions have been normalized to their maximum values. In Figs. 6$(\textrm{c}_2)$ and 6$(\textrm{c}_3)$, it is demonstrated that the longitudinal SAM densities $S_z$ only exist when there is OAM, i.e., $m \neq 0$. We can see that the sign of $S_z$ changes from positive to negative when the topological charge changes from $m=1$ to $m=-1$, while the absolute values of the amplitudes remain unchanged. With the increase of initial phase $\varphi _0$, the absolute values of $S_z$ increase as well. Conversely, the absolute values of $S_\phi$ decrease to zero when $\varphi _0$ reaches to $\pi /2$ no matter what the topological charge is, seen in Figs. 6$(\textrm{b}_1)$–6$(\textrm{b}_3)$. Figures 6$(\textrm{a}_1)$–6$(\textrm{a}_3)$ display that the radial SAM density values are zero when $\varphi _0=0$ or $\pi /2$ for any value of $m$. Moreover, the $S_r$ curves overlap when $\varphi _0=\pi /3$ and $\pi /6$. From the first column (Figs. 6$(\textrm{a}_1)$–6$(\textrm{a}_3)$), one can see that the sign of the initial phase $\varphi _0$ only affects the radial spin density. When changing the $\varphi _0=\pi /4$ to $-\pi /4$, the value of $S_r$ changes from negative to positive. The other two components do not change. As aforementioned discussion in Fig. 3$(\textrm{a}_1)$ and Fig. 5$(\textrm{a}_1)$, with RPV illumination ($\varphi _0=0$), the radial SAM density is always zero regardless of the value of $m$, which agrees well with the results shown in Figs. 6$(\textrm{a}_1)$–6$(\textrm{a}_3)$. With APV illumination ($\varphi _0=\pi /2$), one can only produce longitudinal spin density.

Next, we turn to the impacts of the topological charge on the SAM density. Based on the analysis in Fig. 6, the case of $\varphi _0=\pi /4$ is chosen as a general example. As presented in Fig. 7, $S_r$ and $S_\phi$ are donut-shaped for all values of topological charge that have been tested. However, for the longitudinal SAM densities $S_z$, they are donut shapes, only for $m>2$. In all three components, the density distributions expand from the center to two outer sides and the peak intensity value decreases as the value of $m$ increases. For high m, the peak intensity values are of the same order of magnitude. In fact, the redistributions of SAM densities are caused by the spin flux, which originates from the changes of OAM. As the OAM increases, the dark channel expands from the center of the SAM densities [20]. All of these observations validate the interaction between SAM and OAM in the tightly focusing process. The above results will be helpful in the application of optical trapping and tweezers.

## 3. Optical torque of particles in focused CVV fields

As discussed above, small particles can be trapped due to the localized optical SAM. In other words, the local SAM can also be detected by a small particle. Therefore, we now consider the motion of a Rayleigh sphere with permittivity $\epsilon _2$ in the CVV focal field with a medium of permittivity $\epsilon _1$. The small sphere with a radius $a$, which should be smaller than the trapping wavelength can be regarded as an electric dipole. The dipole moment is $\textbf {p}=\alpha \textbf {E}$ [40–43], where $\alpha$ is the electric polarizability given by

Lastly, the influences of the topological charge $m$ (OAM) on the local maximum values of optical torque at the hot-spot position versus the angle $\varphi _0$ are presented in Fig. 9. In Figs. 9(a)–9(c), it can be seen that for different topological charges, the three torque components vary as a function of $\phi _0$ in similar ways. The maximum longitudinal spin torque at the hot-spot dominates in the three components for any values of angle $\varphi _0$. $\Gamma _z$ starts from minimum value when $\varphi _0=0$ and increases gently to the maximum value somewhere when $\varphi _0=\pi /2$. However, the local maximum azimuthal torque decreases from maximum value to zero till $\varphi _0$ increases to $\pi /2$. Meanwhile, the local maximum value of radial torque starts from minimum value zero when $\varphi _0=0$, and increases until $\varphi _0=\pi /4$, then it decreases together with the radial torque till $\varphi _0=\pi /2$. The amplitudes of the local maximum optical torque decrease with the topological charge increases. In order to make it clear to observe, Figs. 9(d)–9(f) plot variations of each maximal optical torque component at the hot-spot separately. The colors of lines agree with the three components in the above subplots (Figs. 9(a)–9(c)), and different line styles represent different topological charges. The trends in each case are similar to the examples in Figs. 9(a)–9(c). With the increase of $m$, the values of local maximal optical torques decrease. Another common point is that the fluctuation goes more slightly when $m$ increases, which means the influences of the topological charge on the local maximal values of optical torques at the hot-spot position become tiny, and so as the influences come from the initial phase $\varphi _0$. Such kinds of examples provide a guide to trapped particle motions and forces with focused CVV beams.

## 4. Conclusions

To summarize, we have proposed theoretical investigations on electric field and AM properties of arbitrary highly focused CVV beams. With Richard-Wolf vectorial focusing theory, all components of the electric field near the focus can be obtained, as well as the SAM density produced by the focal field. It is revealed that the properties of the input beam such as the polarization state, the topological charge and initial phase can affect the SAM distribution. The interaction between the SAM and OAM is demonstrated by changing the topological charge $m$. The redistributions of SAM density are due to the spin flux, which originates from the changes of OAM. The sign of $m$ only changes the direction of longitudinal SAM density and the sign of the initial phase $\varphi _0$ only changes the direction of transverse SAM density. An absorptive sphere is placed in the focused CVV field, where it is then trapped. It can be seen that in such tight focal fields, which carry effective SAM, the particle is confined to a hot-spot position of maximal optical torque. As a result, the particle experiences spinning motion. It is found that the orientation of spinning motion depends on the sign of topological charge, the initial phase as well as the polarization of the CVV beam. Thus we can optimize the optical torques by combining different parameters based on the above theoretical simulations. These findings give more freedom for achieving good performance in optical manipulation, trapping and tweezers, beam shaping and directional coupling.

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