## Abstract

In this paper, we have introduced a new class of partially coherent vector vortex beams, named radially polarized multi-Gaussian Schell-model (MGSM) vortex beam, carrying the vortex phase with tunable topological charges (i.e., both integral and fractional values) as a natural extension of the radially polarized MGSM beam. The tight focusing properties of the radially polarized MGSM vortex beam passing through a high numerical aperture (NA) objective lens are investigated numerically based on the vectorial diffraction theory. Numerical results show that the focal intensity distributions of the radially polarized MGSM vortex beam can be shaped by regulating the structure of the correlation functions and the topological charge of vortex phase. In contrast with the integral vortex beam, the most intriguing property of the fractional vortex beam is that the focal intensity distribution at the focal plane can be nonuniformity and asymmetry, while such unique characteristics will vanish when the spatial coherence length is sufficiently small. Furthermore, some focal fields with novel structure, such as a focal spot with nonuniform asymmetric or an anomalous asymmetric hollow focal field, can be formed by choosing suitable fractional values of topological charge and spatial coherence length. Our results will be useful for optical trapping, especially for trapping of irregular particles or manipulation of absorbing particles.

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

## 1. Introduction

The shaping and construction of the desired focal field to meet the needs and improve the performance of optical manipulation is currently an important research topic [1–16]. For a long time, numerous efforts have been paid to find the effective methods for shaping the focal field. In recent years, the tight focusing of radially polarized beam has attracted considerable attention due to its peculiar properties of a strong longitudinal electric field in the focal region [2–8]. Compared with the transverse component, the longitudinal electrical field is dominant and makes no contribution to the energy ﬂow along the optical axis, which is of great importance for the trapping of metallic nanoparticles [5–7]. Furthermore, a three-dimensional orientated focal field can be obtained by configuring the relative weighting factor between the longitudinal and transverse field components [8]. In particular, the integration of conventional tight focusing with other beam engineering techniques including binary optics, diffractive optical element (DOE), 4π focusing system and planar plasmonic metalens offers additional flexibility for focus shaping [9–16]. Correspondingly, a series of novel focal field with different characteristic spatial distributions, such as flat-top focus, optical needle, optical cage, optical chain and multiple focal spots has been theoretically proposed and experimentally demonstrated [9–16].

On the other hand, as one of the most important features of light beams, the spatial coherence of partially coherent beams plays an important role in determining the focusing properties, the transverse field distribution are greatly affected by its spatial coherence width [17–19]. More recently, it also has been revealed that not only the transverse component but also the longitudinal component of the focal field distributions can be shaped by regulating the structures of the correlation functions, which provides a novel way for shaping the focal field [20,21]. In addition to the coherence, the phase also has significant effects on the tight focusing properties, and the focal field patterns can be modulated by engineering the vortex phase of the incident beam [22–24]. For instance, an equilateral-polygon-like flat-top focus can be generated by engineering the arrangement and positions of off-axis vortices of a radially polarized beam superposed with off-axis vortex arrays [24]. Nevertheless, over the past decades, previous researches were restricted to optical vortex beams with integer-order topological charges. In fact, the value of the topological charge can be non-integral [25]. A Fermat’s spiral slit can be used to modulate the topological charge of vortex beams to be an arbitrary value, both integral and fractional, within a continuous range [26]. In recent years, vortex beams with fractional topological charges called as fractional vortex beams or non-integer vortex beams have been aroused great interest due to their unique properties, such as a radial opening in the annular intensity ring, alternating charge vortices, birth and annihilation of vortices [26–30].

To our knowledge, most of previous published works are restricted to only produce circular-symmetric focus patterns, which is capable of confining spherical particles. In reality, in the field of optical trapping, particles are usually irregular in shape or diversity of property. Therefore, it is essential to generate asymmetric fields or even arbitrary shaped fields to meet diverse demands from different aspects in real environment. In this paper, we have introduced a new class of partially coherent vector vortex beam named as radially polarized multi-Gaussian Schell-model (MGSM) vortex beam with tunable topological charges (i.e., both integral and fractional values) as a nature extension of MGSM beam [31–34]. The tight focusing properties of such vortex beams passing through a high numerical aperture (NA) objective lens are investigated numerically. It is particularly interesting to find that optical vortex beams with fractional topological charge can break the circular symmetry of focal intensity distribution of vector beams allowing versatile means for focus shaping. Moreover, some focal fields with novel structure, such as a focal spot with nonuniform asymmetric or an anomalous asymmetric hollow focal field, are obtained by tailoring the fractional values of topological charge and spatial coherence length. Our results may find potential applications in optical trapping, especially for the trapping of irregular particles or manipulation of absorbing particles.

## 2. Radially polarized multi-Gaussian Schell-model vortex beam

In the Cartesian coordinate system, the vectorial electric field of a coherent radially polarized TEM beam at the source plane can be characterized by [35],

*x*and

*y*directions, respectively.

On the basis of the unified theory of coherence and polarization [35,36], the second-order spatial coherence properties of a vector partially coherent beam in the spatial frequency domain can be characterized by the 2×2 the cross-spectral density (CSD) matrix $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{{\textbf r}_1},{{\textbf r}_2},\omega } )$ with the matrix elements

To be a physically genuine CSD function, the elements of the CSD matrix should satisfy the condition of nonnegative definiteness in the form of [37]

*r*and $\phi$ are the radial and azimuthal (angle) coordinates,

*l*is the topological charge.

On the other hand, the choice of ${p_{\alpha \beta }}({\textbf v} )$ defines a family of sources with different correlation functions. Let us suppose ${p_{\alpha \beta }}({\textbf v} )$ to be of the form [20,31–34]

*M*denotes the beam index, ${B_{\alpha \beta }} = |{{B_{\alpha \beta }}} |\textrm{exp}({i{\varphi_{\alpha \beta }}} )$ is the single-point correlation coefficient and ${\delta _{\alpha \beta }}$ is the characteristic source correlations. Meanwhile, the realizability conditions of a radially polarized source are obtained as [20]

Substituting Eqs. (4)–(6) into Eq. (3), the elements of the CSD matrix take the following forms

*l*being the topological charge, it can be an arbitrary value, both integral and fractional. One can call it as the radially polarized MGSM integral vortex (IV) beam when its topological charge is integral and radially polarized MGSM fractional vortex (FV) beam when its topological charge is fractional (or non-integer). The beam source reduces to a radially polarized MGSM source under the condition of $l = 0$ [20]. When $M = 1$, the beam source reduces to the radially polarized Gaussian Schell-model (GSM) vortex source.

## 3. Tight focusing properties of a radially polarized MGSM vortex beam focused by a high NA objective lens

In this section, we consider the focusing of a radially polarized MGSM vortex beam by a high NA objective lens. The geometry scheme of problem is shown as in Fig. 1.

After being focused by a high NA objective lens, according to the Richards-Wolf vectorial diffraction, the vectorial electric field of a tightly focused cylindrical vector beam at the focal plane can be expressed as follows [19–21],

*f*is the focal length of the lens, ${k_1} = k{n_1} = 2\pi {n_1}/\lambda$ is the wave number in the surrounding medium with ${n_1}$ being the refractive index of the surrounding medium, $\theta$ is the NA angle, and ${\theta _{max }}$ is the maximal NA angle given by the formula ${\theta _{max }} = \textrm{arcsin}({{N_{NA}}/{n_1}} )$ with ${N_{NA}}$ being the NA number, ${l_x}({\theta ,\phi } )$ and ${l_y}({\theta ,\phi } )$ are the pupil apodization functions at aperture surface, which is derived by setting $x = f\;{\sin}\theta\;{\cos}\phi$ and $y = f\;{\sin}\theta\;{\sin}\phi$ in ${E_x}({x,\;y} )$ and ${E_y}({x,\;y} )$, respectively.

Considering a partially coherent radially polarized beam focused by a high NA objective lens, the statistical properties of the beam in the vicinity of the focal region can be characterized by the 3×3 CSD matrix with the elements [19–21]

By setting ${r_1} = {r_2} = r,\;{\varphi _1} = {\varphi _2} = \varphi$, one can obtain the transverse, longitudinal and total component of the focal intensity in the focal region,

## 4. Numerical results

In this section, we will explore the focal intensity distribution of a radially polarized MGSM vortex beam passing through a high NA objective lens. In the following numerical examples, the parameters are set as ${n_1} = 1,\;{N_{NA}} = 0.95,\;f = 3.0\textrm{mm},\;{w_0} = 10\textrm{mm}$, and $\lambda = 632.8\textrm{nm}$, other beam parameters are given in each captions. Note that the range of each picture is normalized in units of eight wavelengths.

Figure 2 shows the normalized focal intensity distributions of the total component ${I_{total}}$, transverse component ${I_{tra}}$, and longitudinal component ${I_{\boldsymbol z}}$ of the radially polarized MGSM beam at the focal plane for different values of beam index *M*, respectively. All the intensity distributions are normalized by the maximum value of the total focal intensity component, and the green solid line denotes the corresponding cross line of the intensity distribution at ${\rho _x}({{\rho_y}} ) = 0$. It is clearly shown that both transverse and longitudinal field distributions of a tightly focused radially polarized MGSM beam can be shaped into a flat-topped profile, thus leading to a flat-topped focal intensity distribution for small spatial coherence width ${\delta _0}$ (i.e., ${\delta _0} = 0.5\textrm{mm}$), while the focal intensity distributions of the radially polarized GSM beam (i.e., $M = 1$) all are of Gaussian profiles. Furthermore, the area of the flattop region increases as the beam index *M* increases. Our results are consistent with those reported in [20].

In this paper, we mainly consider the influences of the spatial coherence width ${\delta _0}$, the beam index *M* and the topological charge *l* (both integral and fractional value) of the radially polarized MGSM vortex beam on the focal intensity distribution. We show in Figs. 3 and 4 the normalized focal intensity distribution of a tightly focused radially polarized MGSM IV beam with integral topological charge $l = 3$ and FV beam with fractional topological charge $l = 3.5$ at the focal plane for different values of beam index *M* and spatial coherence length ${\delta _0}$, respectively. One can deduce from Fig. 3 that the normalized focal intensity distribution of the radially polarized MGSM IV beam displays an anomalous hollow shaped pattern (a full-doughnut-shaped pattern for larger topological charge $l$) when ${\delta _0}$ is large (i.e., ${\delta _0} = 5\textrm{mm}$), while it will gradually degenerates to a flat-topped profile and finally evolves into a Gaussian profile with the decrease of ${\delta _0}$. Moreover, the focal intensity distribution of the radially polarized MGSM IV beam always display circular symmetry for arbitrary spatial coherence width ${\delta _0}$. However, as shown in Fig. 4, it is particularly interesting to find that the focal intensity distribution of the radially polarized MGSM FV beam is a nonuniform asymmetric focal spot when ${\delta _0}$ is large (i.e., ${\delta _0} = 5\textrm{mm}$), which is quite different from that of the radially polarized MGSM IV beam. In particular, with the decreases of ${\delta _0}$, the focal spot gradually loses its nonuniform asymmetric characteristics, and finally becomes a Gaussian profile with circular symmetry. Thus, it is hard to distinguish the radially polarized MGSM IV beam and FV beam from their focal intensity distributions at the focal plane when ${\delta _0}$ is sufficiently small (e.g., ${\delta _0} = 0.5\textrm{mm}$). Moreover, the beam index *M* has relatively small influence on the focal intensity distributions for both of the radially polarized MGSM IV beam and FV beam. This is because that the topological charge *l* plays a decisive role in determining the focal intensity distribution, while this dominated effect will be weakened with the decreases of spatial coherence width ${\delta _0}$.

To learn more about the influences of topological charge on the focal intensity distribution, we calculate in Figs. 5 and 6 the normalized focal intensity distribution of the radially polarized MGSM IV beam and FV beam with $M = 5$ at the focal plane for different values of topological charge *l* and spatial coherence length ${\delta _0}$, respectively. In Fig. 5, the case of the radially polarized MGSM beam (i.e., $l = 0$) is also shown as a comparison. One can see from Fig. 5 that the focal intensity distribution of the radially polarized MGSM IV beam can form a focal spot or a doughnut-shaped focal field for different values of integral topological charge *l* when ${\delta _0}$ is large (i.e., ${\delta _0} = 5\textrm{mm}$). However, it will gradually degenerates to a flat-topped shape focal field and finally becomes a quasi-Gaussian profile as spatial coherence length ${\delta _0}$ decreases, which is different from that of the radially polarized MGSM beam. As shown in Fig. 6, the focal intensity distribution can exhibit a nonuniform asymmetric focal spot or an anomalous asymmetric hollow focal field at the focal plane for the radially polarized MGSM FV beam with different values of fractional topological charge *l*. Of particular note is that such a nonuniform asymmetric focal spot or anomalous asymmetric hollow focal field will be useful for trapping irregular particles or manipulating absorbing particles. A comparison of Figs. 5 and 6 shows that, the degree of asymmetry of the focal intensity distribution is associated with the fractional part of the topological charge, while the extent of hollow is closely related to its integral part of the topological charge. In particular, the pattern of the focal intensity distribution will be reversed for the radially polarized MGSM FV beam with opposite topological charge (see $l = 5.5$ and $l = - \textrm{5}.5$). However, with the decrease of ${\delta _0}$, the focal intensity distributions of the radially polarized MGSM FV beam gradually lose their nonuniform asymmetric characteristics, and finally all focal intensity profiles become quasi-Gaussian profiles with circular symmetry when ${\delta _0}$ is too small (i.e., ${\delta _0} = 0.5\textrm{mm}$). In a word, we can shape the focal intensity distribution of the radially polarized MGSM vortex beam by optimizing its beam parameters, such as the topological charge *l*, spatial coherence length ${\delta _0}$ and beam index *M*.

## 5. Summary

In conclusion, a new kind of partially coherent vector vortex beam named radially polarized MGSM vortex beam carrying vortex phase with tunable topological charge (both integral and fractional values) was introduced as a natural extension of radially polarized MGSM beam. The focal intensity distributions of the radially polarized MGSM vortex beam passing through a high NA objective lens have been explored numerically. The combined effect of both spatial coherence width and the topological charge on the tight focusing properties has been investigated in detail. Numerical results show that the focal intensity distributions of the radially polarized MGSM vortex beam can be shaped by regulating the structures of the correlation functions and the vortex phase. Some peculiar focal field with novel structure, e.g., a flat-topped focal field, a circular focal spot or a nonuniform asymmetric focal spot, a doughnut profiles or an anomalous asymmetric hollow focal field, are obtained by choosing suitable values of beam index *M*, spatial coherence length ${\delta _0}$ and topological charge *l* of the radially polarized MGSM vortex beam. In contrast with the integral vortex beam, the nonuniformity and asymmetry of a formed focal spot or anomalous hollow focal field is the most attractive property for the radially polarized MGSM FV beam. However, such nonuniform and asymmetric characteristics will be gradually vanished, and finally all focal intensity profiles become quasi-Gaussian profiles with circular symmetry as the spatial coherence length decreases. Our results will be rewarding in various applications, such as optical data storage and optical trapping, especially for trapping irregular particles or manipulating absorbing particles, where there some unique or desired focal fields are required.

## Funding

National Natural Science Foundation of China (11747065); Natural Science Foundation of Anhui Province (1808085QA10).

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