Abnormally autofocusing vortex Swallowtail Gaussian vector beam with low spatial coherence

The precondition for the application of light beams is the ability to devise light distribution with high precision. Controlling more dimensions for structured light fields is an effective method to improve the ability to devise light distribution. The Swallowtail beam, due to its rich regulatory parameters, provides the possibility to design a light field with a specific intensity distribution. Utilizing the Swallowtail beam as a foundation, we design its initial phase, polarization, and coherent structure, and propose a partially coherent azimuthally polarized circular vortex Swallowtail Gaussian beam (PCAPCVSGB) in our paper. This beam exhibits an abnormal self-focusing ability and forms an easily adjustable optical potential well at the focal plane, providing another effective tool for achieving optical manipulation. In addition, the PCAPCVSGB also shows an interesting vector property. It possesses a stable polarization singularity even with changes in coherence and topological charges, which exhibits a potential application value in optical communication.


Introduction
Coherence and polarization, the basic characteristics of any structured optical field, were considered to be independent until Wolf developed a unified theory of coherence and polarization [1]. This theory has aroused scholars' enthusiasm for studying the interaction between coherence and polarization [2], and studying the statistical properties-namely, the average intensity, degree of polarization (DOP), and degree of coherence (DOC)-of the structured optical field during propagation [3]. Besides, the vortex beams carrying orbital angular momentum and phase singularity, have attracted great attention in the past decades due to their wide applications in optical communication [4][5][6]. Afterward, the vortex beams with low spatial coherence have been confirmed to have the advantage over the coherent vortex beam or the Gaussian Schell-model beam for reducing turbulence-induced degradation and scintillation [7]. However, compared with the partially coherent vortex beam, the partially coherent vector vortex beams have a stronger anti-interference ability in free-space optical communication [8].
In addition, the swallowtail beams generated for the first time in 2017, possess higher-order catastrophic structures [9]. Compared with paraxial Airy (fold) and Pearcey (cusp) beams, the swallowtail beams with the higher-dimensional control parameter space and the stable higher-order caustic structures [9,10], have the potential research value [11,12]. Recently, the swallowtail beams have aroused scholars' study interests due to their flexibly adjustable ways-choosing corresponding cross-sections in the control parameter space to realize fundamentally different geometrical caustic structures in the initial transverse plane without propagating the light fields [10]. In 2020, Teng's work studied abruptly autofocusing circular swallowtail beams [13]. In 2021, Zhang's research generated the polycyclic tornado circular swallowtail beam with self-healing and auto-focusing [14].
The ability to engineer light distribution with high precision is the key to many applications in optics. Controlling more dimensions for structured light fields is an effective method to improve the ability to devise light distribution. The Swallowtail beam, due to its rich regulatory parameters, provides the possibility to design a light field with a specific intensity distribution. In this paper, we utilize the Swallowtail beam as a foundation and design its initial phase, polarization, and coherent structure to constitute a partially coherent azimuthally polarized circular vortex Swallowtail Gaussian beam (PCAPCVSGB). This beam exhibits an abnormal self-focusing ability and creates an easily adjustable optical potential well at the focal plane, providing another effective tool for achieving optical manipulation. In addition, this beam also shows an interesting vector property. Even with changes in coherence and topological charges, it still has a stable polarization singularity, which exhibits a potential application value in optical communication.
The paper is organized as follows: In section 2, a theoretical framework is developed to investigate the average intensity profile of the PCAPCVSGBs in the cross section at an arbitrary distance z and the experimental device for generating the PCAPCVSGBs is exhibited in figure 1. The discussions of autofocusing properties for PCAPCVSGBs under the interplay of coherence, optical vortex, and polarization singularity are given in section 3. The key findings of the study are concluded in section 4.

Theoretical model and Experimental details
The electric field of the coherent azimuthally polarized circular vortex Swallowtail Gaussian beam (APCVSGB) can be expressed in polar coordinates as where Sw(X, Y, Z) =´∞ −∞ exp[i(s 5 + Zs 3 + Ys 2 + Xs)]ds [13] , r 0 determines the initial radius of the APCVSGB, w is a factor related to the waist of the beam, m is the topological charge, and θ is the azimuthal angle. Based on the unified theory [1], the coherence and polarization properties of a statistically stationary, quasi-monochromatic PCAPCVSGB can be determined from the 2×2 cross-spectral density (CSD) matrix as where W ij (r 1 , r 2 )(i = x, y; j = x, y), defines the field correlation between two transverse points r 1 (r 1 , θ 1 ) and r 2 (r 2 , θ 2 ). The elements of the CSD matrix are expressed as follows: here µ(r 1 , θ 1 , r 2 , θ 2 ) denotes the DOC. And in polar coordinates, µ can be expressed as [15] µ( where δ is the coherent width. By the extended Huygens-Fresnel formula under the paraxial approximation, the CSD at arbitrary distance z along the propagation can be calculated, where (ρ 1 , ρ 2 ) and (r 1 , r 2 ) represent two points in the observation plane and the source plane, respectively. k = 2π λ is the wave number in a vacuum, λ is the optical wavelength. According to reference [16], the identities of the Bessel function of the first kind J l (s)and modified Bessel function of the first kind I n (s) can be expressed as e is cos φ = +∞ l=−∞ i l J l (s)e ilφ and I n (s) = 1 2π´2 π 0 e s cos θ e inθ dθ, respectively. By substituting equation (4) into equation (6) and utilizing the identities of the Bessel function, we can integrate the integrals of θ 1 . Afterward, by applying I v (s) = i −v J v (is), the integrals of θ 2 can be integrated, and the quadruple integral finally is simplified into a double integral, where and we set Then, H ij can be expressed as Notice that the result of equation (10) is nearly zero due to large L and can be left out. To simplify the calculation, we only calculate the term of |L| ⩽ 13. With these basic theories in place, the characteristics of the PCAPCVSGB can be simulated and analyzed.
The experimental setup for generating the PCAPCVSGB is sketched in figure 1(a). A Gaussian laser beam (532 nm) with linear polarization, passing through a half-wave plate (HP1) and a Gran prism which are used to control the polarization direction and the power of the beam, is focused onto the rotating ground-glass disk (RGGD) by lens L1 to generate a partially coherent beam with the Gaussian Schell-model correlation [4]. After the stray light is filtered by a CA, the beam can be collimated by lens L2 and is reflected by a mirror. Before illuminating the spatial light modulator (SLM), we adjust the polarization direction to match the requirements of the SLM (Santec SLM-200 with 1900 × 1200 pixel resolution) by the HP1. A computer-generated hologram (CGH) containing the amplitude and the phase information about the circular vortex Swallowtail Gaussian beam (CVSGB), is loaded on the SLM. The modulated beam with the Gaussian correlation is reflected toward a 4f filter system consisting of a couple of identical lenses (L3 and L4) and a CA utilized for selecting the +1 order interference fringes on the spectrum plane. A LP and a HP2 are put in the 4f filter system to filter out some unmodulated stray light and deflect the polarization direction to the correct direction of the azimuthal polarization converter (APC). The linearly polarized partially coherent circular vortex Swallowtail Gaussian beam (PCCVSGB) becomes the PCAPCVSGB after passing the APC and the transverse intensity of the PCAPCVSGB is finally detected by the charge-coupled device (CCD). The CGH can be obtained by interfering the initial field of the CVSGB with a plane wave, which are given by where f x represents the grating frequency. In order to quantitatively describe the achieved PCAPCVSGB, a typical measurement of coherent width is made by using the two-pinhole interference experiment before the beam illuminates the SLM. According to [17], the coherent width δ can be calculated by − d 2 2 ln V , where d denotes the spacing between the two pinholes (In our experiment, d = 2.85 mm) and the V is equal to the modulus of the spectral DOC µ. The measured results are shown in figures 1(b) and (c).  figure 2(a), where I c is the central intensity in the cross-section at an arbitrary distance z, and I c0 is the maximum of the central intensity for the PCAPCVSGB with the topological charge m = 0. According to the simulation results shown in figure 2(a), it can be seen intuitively that, as the value of the topological charge m increases, the autofocusing ability of the PCAPCVSGB first increases abnormally and then decreases, but the autofocusing focal length is inherently constant, which equals z f . The physical interpretation for these phenomena is that the amplitude vanishes along the helix axis (r = 0) owing to destructive interference in the vicinity of the vortex core [18] but when the topological charge m = 1, the interaction between the vortex and the polarization state weakens the destructive interference to the central intensity causing the improvement of the autofocusing ability. With increasing the topological charge, the size of the vortex core increases, and the destructive interference of the vortex core to the central intensity occupies the dominant position in the interaction between vortex and polarization state resulting in the practical disappearance of the autofocusing ability. In addition, figures 2(b1)-(d1) show the simulation results of the standardized intensity patterns for the PCAPCVSGBs in the autofocusing plane for three cases (m = 0, m = 1, m = 2), respectively. The numerical value of the color bar represents the ratio of the intensity for the PCAPCVSGB to the maximum intensity of the PCAPCVSGB with the topological charge m = 0 at the cross-section. From figures 2(b1)-(d1) we can see that the size of the intensity distribution of the PCAPCVSGB increases with increasing the number of topological charges and when the topological charge m = 1, the light energy is automatically focused on a smaller focus, resulting in the dark notch which is caused by the polarization singularity, vanishing into the center of the intensity of the PCAPCVSGB. Notice that, when the topological charge m = 2, the intensity of the PCAPCVSGB becomes lower at the center point by destructive interference in the vicinity of the vortex core, causing the reappearance of the dark hollow. Corresponding experimental results shown in figures 2(b2)-(d2) are consistent with our simulation results. These properties of the PCAPCVSGB suggest that we can selectively capture the particles, with a various refractive index which is less or greater than that of the surrounding medium, in the focal plane by adjusting the number of the topological charges, and avoid the situation that the gradient force reduces significantly in the focal plane when the topological charge m > 1 [19].

Results and discussions
Subsequently, to give a further understanding of the intensity distributions for the PCAPCVSGBs in the autofocusing plane under the influence of optical vortex, we decompose the total energy of the PCAPCVSGB into completely polarized and unpolarized parts according to the unified theory of polarization and coherence [1]. The intensity patterns of each component are exhibited in figures 3(a1)-(c1) and (a2)-(c2), respectively. The white line graphs denote the normalized intensity distributions along the cross line y = 0. The numerical value of the color bar represents the ratio of the completely polarized part (or completely unpolarized) to total intensity. In figure 3(d), we show the simulation results of the distributions (cross line, y = 0) of the DOP for the PCAPCVSGBs in the autofocusing plane. From figures 3(a1)-(c1), one can see that, with the topological charge increasing, the evolution of the intensity distribution of the completely polarized part is similar to that of the total intensity distribution shown in figures 2(b)-(d). To further analyze this phenomenon, we can find in figure 3(d) that the vortex possessing the transverse DOP and its strength governed by the number of the topological charges, which is in agreement with the previous results [20], causes the completely polarized part accounts for a large proportion of the total energy.
Next, to study the relationship between the coherence and the DOP, the line graphs in figure 4 display the simulation results for the two cases (δ = 1.5 mm and 3 mm) about the distributions (cross line, y = 0) of the DOP for the PCAPCVSGBs with different topological charges at the focal plane. From figures 4(a) and (b), we can find that the transverse DOP of the PCAPCVSGBs increases with increasing the coherent width under the condition of the same topological charge, and this phenomenon in agreement with the previous results, is called the coherence-induced polarization effects [2]. Comparing the black curves, we find that, the waist width of the DOP distribution decreases, and the value of the DOP on the optical axis r = 0 increases with the increase of the coherent width. In addition, to give a further understanding of the interplay of optical vortex, coherence, and polarization singularity, we exhibit the Stokes intensity |S 12 | 2 and the Stokes phase ϕ 12 for the PCAPCVSGBs at the focal plane, in figures 4(c1)-(h1) and (c2)-(h2), respectively. Comparing the Stokes phase for the PCAPCVSGBs possessing the same coherent width with different topological charges. As the value of the topological charge m increases, the singularity of the Stokes phase still exists and the magnitude of the phase gradient near the singularity core increases gradually. Corresponding to the Stokes intensity patterns of PCAPCVSGBs shown in figures 4(c1)-(h1), we can find that the PCAPCVSGB possesses the stable Stokes vortex structure. And the S 12 Stokes vortices are identical to the polarization singularities [21][22][23]. Based on the above theoretical analysis, we can infer that the polarization singularity owns the stronger stability than the phase singularity which can convert to the coherent singularity as the coherence decreases [24] and the polarization singularity still exists even if the value of the DOP on the optical axis r = 0 is not zero for the PCAPCVSGBs with different coherent width shown in figures 4(a) and (b).

Conclusion
In conclusion, we generate the partially coherent vortex vector beam named the PCAPCVSGB experimentally and analyze its properties from the theoretical aspects. The PCAPCVSGB exhibits abnormally autofocusing property and can create an optical potential well to selectively capture the particles with a various refractive index which is less or greater than that of the surrounding medium, in the focal plane by adjusting the value of the topological charge m. Furthermore, we study this abnormal autofocusing property. We find that the interaction between the vortex possessing transverse polarization intensity and the polarization state causes abnormal autofocusing. In addition, we exhibit the Stokes intensity and the Stokes phase for the PCAPCVSGBs with different coherent widths and further discuss the interplay of coherence, optical vortex, and polarization singularity. We believe our research results can be utilized in the fields of optical communication and optical capture.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.