## Abstract

We theoretically study the propagation properties of the vector circular Airy vortex beam in detail. The results show that the orbital angular momentum can induce a localized spin angular momentum after autofocusing in the paraxial regime, which leads to an abrupt polarization transition just before the focal plane. However, there is no angular momentum conversion from orbital angular momentum to spin angular momentum during the whole propagation process. We provide an intuitive explanation for the appearance of such spin angular momentum localization. This investigation is expected to advance our understanding of the vector properties of circular Airy beam and optical spin-orbit coupling.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The circular Airy beam (CAB) was first introduced by using of the radially symmetric Airy beam in 2010 [1]. The unique property of the CAB is that this beam can maintain quite low intensity profiles before the focal point and the maximum intensity of the beam can abruptly increase by orders of magnitude just at the focal point [1,2]. Such abruptly autofocusing property holds great potential for many applications including laser fabrication, biomedicine, optical micromanipulation and high harmonics generation [3–6]. Although numerous studies on fundamental features have been devoted for CAB, most of them are based on the scalar diffraction theory [7–11] and the vector properties of CAB are rarely studied [12,13]. Therefore, the vector nature of CAB still remains largely unexplored.

Light wave carries both spin angular momentum (SAM) and orbital angular momentum (OAM). They are determined by the polarization and spatial wave front of light. In the last ten years, there has been enormous interest in the spin–orbit interactions (SOI) of light [14–16]. However, most of these SOI phenomena only involve the spin-to-orbital conversion (SOC) of angular momentum under tight focusing conditions. The inverse phenomenon is rarely reported [17]. In Ref. [12], an abrupt polarization transition was experimentally observed just before the focal plane of a radially polarized circular Airy vortex beam (CAVB). It was attributed to the orbital-to-spin conversion (OSC) of angular momentum [12,18]. However, it is worthy to note that the abrupt polarization transition was observed in the paraxial regime, where it is believed that the SOI can be ignored [16,19,20]. Furthermore, recent studies show that OSC cannot occur even under the condition of tight focusing [17,18].

In this paper, we investigate the controversial issue of SOI for the vector CAVB in the autofocusing process. The propagation properties of radially polarized CAVB are discussed numerically in detail. The simulation results show explicitly that the existence of OAM will induce a local distribution of SAM for CAVB even in the paraxial regime. However, there is no angular momentum conversion from OAM to SAM. An intuitive explanation of the reason of such SAM localization is provided.

## 2. Theoretical method

Because the longitudinal z-components of the fields are crucial for SOI [16,19,20], all of our simulations were performed by using the nonparaxial diffraction theory, even in the paraxial regime. In cylindrical coordinates, the nonparaxial propagation of light beams in free space along the optical axis can be described by the well-known Rayleigh-Sommerfeld (R-S) integrals [21,22]

Considering a radially polarized CAVB, the electric field at initial plane can be expressed as [1,2]

*w*is a scaling factor, $\alpha$ is an exponential decay factor (0<$\alpha$<1), and

*m*is the topological charge of the optical vortex. Substituting Eq. (6a) into Eqs. (1)–(3) yields

In the simulations, we assume that $\lambda = 632.8\textrm{nm}$, ${r_0} = 1\textrm{mm}$, $w = 0.08\textrm{mm}$, and $\alpha = 0.1$ throughout this paper. For the convenience of comparison, we also assume $2\pi {\int_0^\infty {|{u(r )} |} ^2}rdr = 1$.

## 3. Discussion

The propagation of a radially polarized CAB ($m = 0$) is studied firstly. Figure 1(a) shows the side view of the beam propagation from numerical simulation. Similar to the scalar CAB [1,2], the intensity maxima of the radially polarized beam follow a parabolic trajectory as the wave propagates toward the focus. The focal length is about ${z_f} = 464\textrm{mm}$, which is almost identical to the scalar one (about 465 mm). It indicates that the polarization has little influence on the focal length. Figure 1(b) shows the intensity profiles of the radial component and the longitudinal component at the focal plane. From Eqs. (8) and (11), the azimuthal component vanishes when $m = 0$. The axial intensity of the radial component is zero due to the singularity of polarization. As expected, the intensity of the longitudinal component is much less than that of the radial component because of ${z_f} \gg {r_0}$. Therefore, the initial radial polarization can be preserved after autofocusing.

We now turn to study the propagation properties of a radially polarized CAVB with the topological charge $m = \textrm{1}$. Figure 2(a) shows the side view of the beam propagation. It is found that before the focal point, the trajectory of the intensity maxima of the beam is almost as the same as that of $m = 0$. The focal length, which is about ${z_f} = 46\textrm{6mm}$, is almost unchanged. It indicates that besides the polarization, the vortex phase also has little influence on the focal length.

However, at the focal plane, the field distribution is very different from that of $m = 0$, as shown in Fig. 2(b). The axial intensities of the transverse components (including the radial and azimuthal components) are maximum at the focal plane. On the contrary, the axial intensity of the longitudinal component is zero because of the axial phase singularity of ${E_z}$. In order to investigate the abrupt polarization transition, the polarization distributions of the beam before and at the focal plane were calculated respectively. The calculation results are shown in Figs. 2(c) and 2(d). Before the focal plane, the field is almost circular polarized in the central area and radially polarized away from the central area, as shown in Fig. 2(c). The polarization distribution is more complicated at the focal plane, as shown in Fig. 2(d). The field is circular polarized in central spot, azimuthally polarized in the first dark ring, elliptically polarized in the first bright ring and radially polarized far away from the central area. Since the energy of the Airy rings is concentrated to the central area close to the focal plane, the radial polarization is converted to circular and elliptical polarizations at the focal plane. Figure 2(e) shows the calculation results of Stokes parameters ${s_3}$ at the focal plane. It can be seen that the polarization is left-handed in the central spot and right-handed in the first bright ring, which agrees well with the experimental result [12].

Then, the time-averaged OAM and SAM densities were calculated to discuss the angular momentum distribution. The expressions can be written as [14,23]

*c*is speed of light, $\omega $ is the angular frequency, ${\varepsilon _0}$ and ${\mu _0}$ are the vacuum permittivity and permeability, and the superscript * denotes the complex conjugate. Usually, for light fields in nonmagnetic media, only the electric parts of the fields are considered. From Fig. 2(b), it can be seen that the longitudinal

*z*component of the electric field is too small comparing with the transverse components. Hence, the transverse component of SAM can be ignored. Figure 3(a) and (b) show the longitudinal SAM densities ${S_z}$ and OAM densities ${L_z}$ of the beam before ($z = 420\textrm{mm}$) and at the focal plane ($z = 466\textrm{mm}$). All values are normalized to the maximum ${S_z}$ and ${L_z}$ at the focal plane. It can be easily proved that both ${S_z}$ and ${L_z}$ are $\theta $ independent by substituting Eqs. (7) and (8) into Eqs. (13) and (14). It shows that at the focal plane, the magnitudes of ${S_z}$ dominate the central area, while ${L_z}$ exhibit annular distributions. The curve of ${S_z}$ before the focal plane is qualitatively like that at the focal plane. However, the peak value of ${S_z}$ before the focal plane is much less than that at the focal plane. It clearly indicates a strong localized effect on SAM at the focal plane.

In order to study the SOI of the beam, the globe SAM and OAM were calculated. Integrating the flux density over the transverse cross-section plane yields the total flux through a plane of constant z

Then, we have [16,23] where*W*is the integral over the transverse plane of the time-averaged energy density. Here, we still only consider the electric parts of the fields. From Eqs. (17) and (18), it is seen that the field has SAM of $\sigma \hbar$ and OAM of $l\hbar $ per photon, respectively. Here, the influences of transverse SAM and OAM can be ignored reasonably, because their values are less than one ten-thousandth of the longitudinal ones.

Figure 4 shows calculated $\sigma $ and *l* of the radially polarized CAVB with $m = \textrm{1}$ at different transverse planes. It can be clearly seen that the SAM is always zero and the OAM is always $\hbar $ per photon regardless of which transverse plane is considered. Note that the total angular momentum per photon for any transverse plane should be $\hbar$ ($m = 1$) because of the conservation of angular momentum. It indicates that the OAM of the incident CAVB is not converted to a global SAM during the whole propagation process. The existence of OAM only induces a local distribution of SAM.

The aforementioned discussion shows that there is no angular momentum conversion from OAM to SAM for the radially polarized CAVB. The abrupt polarization transition arises directly from the SAM localization near the focal plane. Then, what is the cause of the SAM localization? In order to explore this question, the transverse fields of the beam need to be projected onto the circular polarization basis

When $m = 1$, the topological charge *l* of ${E_L}$ and ${E_R}$ becomes 0 and 2, respectively. Since the polarization distribution of the initial beam is radial, the difference between the OAM of RHC and LHC photons has to be equal to $2\hbar $. And since the average total angular momentum of per photon is $\hbar $, it gives 0 and $2$ for the topological charge of LHC and RHC. Figure 5(a) shows the ratios of ${W_L}/{W_R}$ at different transverse planes, where *W* was defined in Eqs. (17) and (18), subscripts *L* and *R* denote LHC and RHC, respectively. It can be clearly seen that the energy of LHC and RHC components are always equal. In other words, the photon numbers of LHC and RHC are equal across any transverse plane. So, it proves again that there is no angular momentum conversion from OAM to SAM.

Figure 5(b) shows the trajectories of the intensity maxima of LHC and RHC components. Before the focal plane, the central hollow region of the beam is large enough for the phase singularity of RHC component, so that the intensity maxima of RHC component can be follow the same parabolic trajectory with that of LHC component. Hence, the beam can maintain radial polarization before autofocusing, as shown in Fig. 2(c). However, when the beam propagates close to the focal plane, the energy of LHC component will abruptly concentrate on the axis, while the RHC component cannot because of the phase singularity. The photons with $\sigma = 1$ and $\sigma ={-} 1$ separate abruptly just before the focal plane, which leads to the appearance of the SAM localization.

When $m > 1$, it will be like the case of $m = 1$. As an example, the case of $m = 3$ will be discussed next. Figure 6(a) shows the trajectories of the intensity maxima of LHC and RHC components with $m = 3$. Before the focal plane, the RHC component can coincide well in space with the LHC one. Thus, the beam can maintain radial polarization before autofocusing. Because both topological charges of ${E_L}$ and ${E_R}$ are not zero for the case of $m > 1$, the energy of the beam cannot concentrate on the axis even at the focal plane. According to Eqs. (27) and (28), the topological charge of ${E_L}$ is always 2 less than that of ${E_R}$. Hence, when the beam propagates close to the focal plane, the energy of LHC component will concentrate on the inter ring and the energy of RHC component will concentrate on the outer ring. The photons with $\sigma = 1$ and $\sigma ={-} 1$ separate abruptly, just like the case of $m = 1$. So, the OAM can still induce a strong localized SAM after autofocusing for the case of $m > 1$, as shown in Fig. 6(b). Figure 6(c) shows calculated $\sigma $ and *l* of the beam with $m = 3$. It can be seen that the values of $\sigma $ and *l* are both unchanged at any transverse plane. It indicates that there is still no angular momentum conversion from OAM to SAM for the case of $m > 1$.

Lastly, we would like to discuss the cases of linearly and circularly polarized CAVB briefly. According to the Ref. [24], it can be easily proved that the linearly or circularly polarized CAVB can maintain its original polarization during the whole propagation process. It indicates that the orbit-induced SAM localization cannot occur for both linearly and circularly polarized CAVB.

## 4. Conclusion

In summary, the propagation properties of radially polarized CAB with and without vortex are theoretically investigated in detail. The results show that there is no angular momentum conversion from OAM to SAM whether the vortex is involved or not. The beam without vortex can maintain radial polarization during the whole propagation process, because the LHC and RHC components of the beam have the exact same intensity distribution, and the net SAM is zero everywhere in this case. When the beam carries vortex, the existence of OAM will induce a localized SAM after autofocusing, leading to an abrupt polarization transition at the focal plane. It is because that when the vortex is involved, the RHC component of the beam cannot coincide in space with the LHC one after autofocusing, and the photons of LHC and RHC will separate abruptly just before the focal plane.

## Funding

National Natural Science Foundation of China (61975125); National Key Research and Development Program of China (2017YFB0503100).

## Disclosures

The authors declare no conﬂicts of interest.

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