Generation and Propagation of Partially Coherent Power-Exponent-Phase Vortex Beam

We report on a partially coherent power-exponent-phase vortex beam (PC-PEPV), whose spatial coherence is controllable and the initial phase exhibits a periodic power exponential change. The PC-PEPV beam was generated experimentally with various spatial coherence widths, and its propagation properties were studied both numerically and experimentally. By modulating the topological charge (TC) and power order of the PC-PEPV beam, the structure of the vortex beam can be adjusted from circular to elliptic, triangular, quadrangle, and pentagon. When the power order is odd, the PC-PEPV beam with a negative TC can be generated, and the profiles of the PC-PEPV beam can be precisely controlled via adjusting the value of the power order. For the case of high spatial coherence width, the number of the dark cores in the polygonal intensity array of the PC-PEPV beam equals the magnitude of the TC. However, when decreasing the spatial coherence width, the dark cores vanish and the intensity gradually transforms into a polygonal light spot. Fortunately, from the modulus and phase distributions of the cross-spectral density (CSD), both the magnitude and sign of the TC can be determined. In the experiment, the modulus and phase distribution of the CSD are verified by the phase perturbation method. This study has potential applications in beam shaping, micro-particle trapping, and optical tweezers.


INTRODUCTION
Since it was verified that the optical vortex can carry orbital angular momentum (OAM) [1], it became a research hot spot in the field of light manipulation. It is well known that the optical vortex possesses a helical phase term of exp(imθ), where m is the topological charge (TC) and θ is the azimuthal angle. At the center of the optical vortex, the phase is undetermined and the intensity is zero. Optical vortices are widely investigated in many fields, such as micro-particle manipulation [2][3][4][5], free-space optical communications [6][7][8], optical measurements [9,10], and super-resolution imaging [11].
The conventional optical vortex has a circular intensity profile, which limits its applications. To address this problem, various asymmetric and non-canonical models for optical vortices are proposed, providing higher freedom of control, for example, the optical vortex with a 3D freestyle structure [12], the asymmetric Bessel beam [13], remainder-phase optical vortex [14], anomalous optical vortex [15], and power-exponent-phase vortex (PEPV) [16]. It is worth noting that, compared with the classical vortex, the PEPV beam has a non-uniform phase and a spiralshaped intensity distribution. Specifically, the phase on a closed path around the center point will increase exponentially. Soon afterward, the propagation properties of the PEPV beam was investigated [17], and this kind of vortex shows more advantages in particle manipulation [18][19][20]. Recently, a new kind of PEPV (NPEPV) beam was proposed, where each part of the spiral phase increases from 0 to 2π, showing exponential growth [21]. In addition, the phase and intensity patterns both show rotational symmetry. However, these excellent studies are only focused on the field of fully coherent light.
Coherence is one of the most important characteristics of the laser beam. Compared with the fully coherent beam, the partially coherent beam [22,23] is more ubiquitous and has some unique advantages in practical applications. For instance, the partially coherent beam can increase the transfer efficiency in the process of non-linear optics [24], enhance the signal to noise ratio [25], decrease the bit error ratio [26], and realize particle trapping [27,28] and ghost imaging [29,30]. The partially coherent vortex beam was first proposed by Gori et.al., which is expressed by the incoherent superposition of a series of coherent Laguerre-Gaussian modes [31]. Subsequently, the study of the vortex beam has expanded from fully coherent to partially coherent and attracted enormous attention during the past decades [32][33][34][35][36][37][38]. The partially coherent vortex beam has several advantages over the fully coherent vortex beam. For instance, it is less disturbed when propagating through a turbulent atmosphere [39] as high-freedom beam shaping is possible [40], and it has remarkable self-reconstruction properties [41]. Different from fully coherent vortex beam with dark hollow intensity distribution, the central dark core in the partially coherent vortex beam will gradually vanish as the spatial coherence width decreases [42]. To describe the singularity of the partially coherent vortex beam, the concept of a coherence vortex, named as coherence singularity, is proposed. It reveals that the singularity of the partially coherent vortex will be hidden in the distribution of the cross-spectral density (CSD), instead of intensity [43,44]. Meanwhile, the TC of the partially coherent beam can be determined by measuring the modulus and phase of the CSD function [45,46]. However, the aforementioned conventional partially coherent vortex beams only carry the common helical phase term exp(imθ), where the structure of the beams can only be reshaped from a doughnut-like intensity pattern to a circular intensity spot, limiting its practical applications.
It inspires us to explore a special partially coherent vortex beam with a controllable structure.
To address this problem, in this study, we propose the theoretical model of a partially coherent power-exponentphase vortex (PC-PEPV) beam, and its propagation properties are studied by numerical simulations. Different from the conventional partially coherent vortex beam, the PC-PEPV beam can realize a polygonal-structured intensity distribution and be shaped freely and precisely via varying the initial spatial coherence width, power order, and topological charge. In addition, we experimentally generated the PC-PEPV beam and measured the modulus and phase distributions of its CSD function via the phase perturbation method. The coherence singularities are observed in the phase distributions, which helps the determination of magnitude and sign of the TC. Compared with the partially coherent conventional vortex beam, our study enriches the mode distributions and realizes a controllable beam shaping, which have potential applications in particle manipulation.

Principle of Generating the PC-PEPV Beam
Compared with a conventional vortex beam with a uniformly varying phase, that is, exp(imθ), the electric field of a NPEPV beam with Gaussian-distributed amplitude in the source plane (z 0) can be written as [21]: where (r, θ) denotes the polar coordinates in the source plane, r is the radial coordinate, θ is the azimuthal coordinate, w 0 is the beam width, rem(.) means the remainder function [14], m is the TC number, and n represents the power order of the spiral phase. Note that the power order n can be an integer or a fraction. It should be noted that the NPEPV beam will reduce to the conventional vortex beam if the parameters are n 1. To distinguish the difference between the conventional vortex beam and NPEPV beam, according to Eq. 1, the initial phases of them are depicted in Figure 1. Here, the value of TC is fixed as m 3. As shown in Figures 1A,B, the initial phases with power order n 1 and 2 can be regard as the initial phase of conventional vortex and NPEPV beam, respectively. Furthermore, the relationship between the initial phases and azimuth of the two phases is shown in Figure 1C. In contrast to the initial phase of the conventional vortex beam, which has a linear variation from 0 to 2π, the initial phase of the NPEPV beam shows a periodic power exponential change. However, the Eq.1 is only suitable for perfectly coherent cases which neglected the significance of coherence of the light. Hence, in the following study, we extend this fully coherent NPEPV beam to a partially coherent field.
For a partially coherent beam, the CSD function in the space-frequency domain [23,47] can be used to express its statistical characteristics as follows: where (r 1 , θ 1 ) and (r 2 , θ 2 ) represent the coordinates of two random points in the source plane, denotes the ensemble average, and * represents the complex conjugate. In the Frontiers in Physics | www.frontiersin.org November 2021 | Volume 9 | Article 781688 following derivation, for brevity, we only consider a monochromatic beam and omit the frequency ω [42,48]. By substituting Eq. 1 into Eq. 2, we find that for a Schell model PC-PEPV beam carrying the distinct power-exponent-phase shown in Figure 1B, the CSD function can be written as follows (see Supplementary Material): where σ denotes the initial spatial coherence width of the beam. Generally, the propagation properties through a paraxial ABCD optical system can be studied via the generalized Collins formula. The CSD function in the observation plane can be written as follows: Here, ρ and φ are the radial and azimuthal coordinates of the observation plane, respectively. A, B, C, and D are the transfer matrix elements of the optical system. For a focused optical system, we can set A 0, B f, C −1/f, and D 1, where f is the focal length of the lens.

OAM Spectrum of the PC-PEPV Beam
Due to the introduction of the power order, the characteristics of the PC-PEPV beam differ from the conventional partially coherent vortex, which is closely related to the OAM spectrum. In Figure 2, we calculated the OAM spectrum of the PC-PEPV beam with different initial spatial coherence and power order in the source plane. Here, TC m 3 and w 0 1 mm. Any vortex beam can be expressed as an orthogonal superposition of a series of integer OAM beams [49,50]. Hence, the Eq. 1 can be rewritten as follows: where l is an integer and the mode coefficients are given by: and the power weight is defined as |a l (r)| 2 . Likewise, the power weight of the PC-PEPV beam can be written as follows: (7) Here, the normalized power weight for each OAM mode can then be defined as follows: As depicted in Figures 2A-C, the power order n 1, which corresponds to a conventional vortex beam. Clearly, the OAM spectrum of the conventional vortex beam with high spatial coherence is shown in Figure 2A and features a unique peak at l 3. When the spatial coherence decreases, the OAM spectrum tends to a Gaussian distribution, and the main peak is located at l 3. As the power order increases, that is, n 2 and 3, the conventional vortex beam transforms into the PC-PEPV beam, and the OAM spectra with high spatial coherence are depicted in Figures 2D,G, respectively. By contrast, the OAM spectrum increases at l 0 and 6 but decreases at l 3 due to the power order in the spiral phase. Similarly, the OAM spectrum tends to a Gaussian distribution as the spatial coherence decreases.

Experimental Setup
The schematic of the experimental setup is depicted in Figure 3, which is used for generating a PC-PEPV beam and measuring the modulus and phase distribution of its CSD function. The coherent laser beam with a wavelength of λ 532 nm is emitted from a solid-state laser and then expanded by the beam expander. The expanded beam is focused on the rotating ground-glass disk (with 400 Grit and 20 Hz rotational frequency) by the lens L 1 (f 1 100 mm), and then collimated by the lens L 2 (f 2 100 mm) to get a partially coherent Gaussian Schell model beam with controllable spatial coherence. The initial spatial coherence width can be controlled via adjusting the focused beam spot size on the rotating ground-glass disk [47]. The aperture A 1 is used to get the central area of the partially coherent Gaussian Schell model beam. Then it is reflected to the first spatial light modulator (SLM 1 ) via a mirror M. As shown in Figure 3A, the fork grating loaded on SLM 1 is designed via the method of computer-generated holograms [10]. Then the lenses L 3 and L 4 and aperture A 2 are used to form a 4-f system and filter the +1 order of the output beam, named as the PC-PEPV beam. The PC-PEPV beam is focused by the lens L 5 . One path is reflected via the beam splitter, and the intensity is recorded by the charge-coupled device (CCD 1 ), which is located at the focal plane of L 5 . To verify the CSD distribution and measure the coherence singularities of the PC-PEPV beam, the other path is transmitted through the beam splitter and focused onto SLM 2 . The phase written into SLM 2 is shown in Figure 3B, containing a central square window with displacement grating, spherical wave phase for focusing, and an additional central phase perturbation [51,52]. Note that the focused input PC-PEPV beam and the center of SLM 2 need to be aligned. Finally, the output beam from SLM 2 is reflected by the beam splitter and recorded by the CCD 2 located at the Fourier plane.

Method to Measure the CSD Function of the PC-PEPV Beam
The Fourier transformation intensity recorded by the CCD 2 with no additional phase perturbation can be written as follows: where W(ρ 1 , ρ 2 ) is the CSD function of the PC-PEPV beam on the plane of SLM 2 and k represents the coordinate vector in the plane of CCD 2 . Then we introduce an on-axis phase perturbation at ρ 2 ρ 0 (0, 0) of SLM 2 . The complex-valued phase perturbation can be expressed as C [exp(iφ) -1]δ(ρ -ρ 0 ), where δ(ρ) represents the Dirac function. Thus, the perturbed Fourier transformation intensity will change as follows: Performing an inverse Fourier transform of this intensity gives the following equation: To measure the CSD function of the PC-PEPV beam, the phase assignments φ need to be assigned three times to get three equations, and the CSD function can be calculated via the following equation: Here, C 1 , C 2 , and C 3 represent three different phase perturbations with φ equal to φ 1 , φ 2 , and φ 3 , which should be chosen as far away as possible within 2π (i.e., phase working range of SLM 2 with wavelength equal to 532 nm). It can help obtain three intensities with maximum difference and enhance the signal to noise ratio of the retrieval. Here, we set φ 1 0, φ 2 −2π/3, and φ 3 2π/3.

Numerical Simulation Results With Different Coherence Width and TC
In this section, we investigate the properties of the PC-PEPV beam focused by a thin Fourier lens via numerical calculation based on the complex random screen decomposition [53][54][55]. Figure 4 illustrates the initial phase patterns of PC-PEPV beams on the source plane and the simulated distributions of PC-PEPV beams on the focal plane with high initial spatial coherence width (σ 20 mm), including the intensity, and the modulus and phase of the CSD function with on-axis reference (r 2 0, θ 2 0). The initial spatial coherence width means the transverse width between the maximum value to the point with e −1/2 the maximum value of the modulus of the degree of coherence [23]. Here, λ 532 nm, w 0 1 mm, f 300 mm, power order n 2, and TCs m 2, 3, 4, and 5. The simulated initial phase patterns and intensity patterns are shown in the first and second rows of Figure 4, respectively. It is observed that the initial phase exhibits a periodic power exponential change and the polygonal optical vortex arrays are generated, respectively. The number of the dark cores are N 2, 3, 4, and 5, which are equal to the absolute value of the TC. Note that each dark core is a unit vortex with TC +1 so that they are separated imperfect doughnut-like dark core. Moreover, the geometry of the generated PC-PEPV beam are elliptic, triangular, quadrangle, and pentagon, respectively, which is specific compared with the conventional vortex beam that has a "doughnut-like" profile. In this case where we have a high initial spatial coherence width, the number of vortices and the geometry of the PC-PEPV beam can be freely modulated via adjusting the values of the TC. However, as the TC increases, the intensity becomes higher for the central spot and lower for the outer region. The reason is that radius of a vortex beam will increase with the TC increase, which causes the total energy to disperse in the outer region. As a consequence, the dark cores in the intensity gradually become indistinguishable and the value of the TC becomes difficult to determine. To solve this problem, we calculated the modulus and the phase of the CSD function, as shown in the third and bottom rows of Figure 4, respectively. The structure of the modulus is similar to the intensity pattern when the initial spatial coherence width σ is much larger than the beam waist w 0 . Furthermore, the number of the dark cores in the modulus of the CSD function also equals the magnitude of the TC. To verify the sign of the vortex, the corresponding phase patterns are shown in Figures 4A4-D4. The phase rotation of coherence singularities is marked by the white circular arrow. Anticlockwise corresponds to a positive sign of TC and clockwise indicates that the TC is negative [56]. The locations of these phase singularities correspond to those of the dark cores in the modulus of the CSD function. Furthermore, it can be seen from the phases that a light spot appeared in the center of intensity owing to the split of the singularities.
Then we studied the characteristic of the PC-PEPV beam with a medium spatial coherence width, which is equal to the beam waist (σ 1 mm). The intensity patterns of the PC-PEPV beam are depicted in Figures 5A1-D1, which show that the focusing properties are very different between the PC-PEPV beam with different spatial coherence widths. The dark cores in the PC-PEPV beam at the focal plane disappear gradually as the coherence width σ decreases. When the TC is equal to 2 and 3, the zero-intensity dark cores in the high coherence case become low-intensity cores, as shown in Figures 5A1,B1. As the TC became larger, as shown in Figures 5C1,D1, the dark cores nearly disappeared, and the geometric structure of the vortex array disappeared at the same time. In this case, we could not determine the magnitude of the TC from its intensity. However, the modulus of the CSD function distribution in Figures 5A2-D2 remains the same as that in Figure 4 for high coherence case. Even for TC equals 5, one can also clearly tell that the number of the dark cores is equal to the number of the TC. Meanwhile, the magnitude and sign of the TC also can be determined via the phase distribution, as depicted in the bottom row of Figure 5.
In Figure 6, we consider a low spatial coherence (σ 0.5 mm) to investigate the properties of the PC-PEPV beam. The dark cores in the intensity have completely vanished due to the modes overlapping for the partially coherent beam with lower coherence width [37]. The intensity is Gaussian distributed, and the geometry of the PC-PEPV beam becomes elliptic, triangular, quadrangle, and pentagon light spots which is specific compared with the conventional partially coherent vortex beam (show a circular intensity spot with Gaussian distribution) [32]. If the coherence width decreases further, the PC-PEPV beam will degenerate to a circularly symmetric Gaussian distribution [57]. From the low coherence intensity patterns, we could not find any information about the TC. However, we can still find the magnitude and sign of the TC from the modulus and phase of the CSD function, as shown in the middle and bottom rows of Figure 6.

Regulation Characteristics of the Power Order
As well known, the TC can be negative. In the aforementioned discussion of Figures 4-6, the sign of the TC m must be positive due to the power order n 2. However, with the power order n 3, the TC of the PC-PEPV beam can be both   Figures  7C1-C3 and Figures 7D1-D3, respectively. Compared with n 2, the intensity of the central light spot increases and the intensities of the dark cores decrease. This is because the larger power order value makes the zeroth component of the spiral phase higher [21]. In  Frontiers in Physics | www.frontiersin.org November 2021 | Volume 9 | Article 781688 8 addition, the magnitude and sign of the TC can be determined by the modulus and phase of the CSD function patterns. As depicted in Figures 7A3-D3, the anticlockwise phase winding (positive TC) and clockwise phase winding (negative TC) of coherence singularities are marked by the white and black circle arrows, respectively. The positive and negative of TC is relative, which is based on the theoretical model of the vortex phase, that is, Equation 1.
To further study the influence of the power order n, the numerical simulation intensities and CSD distributions of the PC-PEPV beam with power order n increase from 4 to 10 with step 2 are shown in Figure 8. Here, the TC m equals 3 and low spatial coherence σ is equal to 0.5 mm. In contrast to the aforementioned PC-PEPV beam with small power order n, the intensity transforms from triangular to approximately a Gaussian form with the increase of power order n, as shown in Figures  8A1-D1 The reason is that the initial phase of the PC-PEPV beam is approaching to a constant with an increase power order n [21]. In addition, as shown in Figures 8A2-D2 the modulus of the CSD function remains the same structure but decreases intensity in the outer region. It is interesting to note that the three positive singularities split to three positive and negative singularity pairs, which cause the total TC approach to 0 owing to annihilate between the singularity pairs, as depicted in Figures  8A3-D3.
Corresponding Experimental Results of the PC-PEPV Beam Figure 9 shows the experimental measurements of the PC-PEPV beam with high coherence width (σ 20 mm), and the other parameters are the same as that in Figure 4. The first row of Figure 9 depicts the experimental intensity patterns of the PC-PEPV beam captured with CCD 1 , which agree well with the numerical simulation results in Figure 4. The middle row demonstrates the experimental modulus of the CSD function of the PC-PEPV beam. The green spots represent the perturbation point located in the center. For TC equals 2, 3, 4, and 5, we can observe 2, 3, 4, 5 dark cores in the modulus of the CSD function, respectively, whose location correspond to the intensity dark cores. The number of the dark cores in the intensity and modulus of the CSD function both indicate the magnitude of the TC. Furthermore, based on Eq. 12, we can also get the phase distributions of CSD function of the PC-PEPV beam, as shown in Figures 9A3-D3. When TC is positive, the phase winding is anticlockwise and consistent for all the coherence singularities. The number of coherence singularities indicates the magnitude of TC and the orientation reveals the sign of the TC. Figures 10, 11 show the experimental results of the PC-PEPV beam with medium (σ 1 mm in Figure 10) and low spatial coherences (σ 0.5 mm in Figure 11). The other parameters are the same as those in Figures 5, 6, respectively. As shown in the first row of Figures 10, 11, the dark cores in the intensity patterns fade away when the coherence width decreases, and the magnitude of the TC can no longer be determined from the intensity distributions. However, we can also obtain the magnitude of the TC from its retrieved modulus of the CSD function, as shown in the middle row of Figures 10, 11. Similarly, both the sign and magnitude of the TC can be determined from the phase patterns, as shown in the bottom row of Figures 10, 11. In the case with low coherence, the PC-PEPV beam with elliptic, triangular, quadrangle, and pentagon structures can be generated experimentally. The results may be useful in the field of beam shaping and optical trapping.
In addition, with odd power order (n 3), the PC-PEPV beam with positive and negative TC are generated experimentally, as depicted in Figure 12. All parameters of the PC-PEPV beam are same as those in Figure 7. The intensity and the modulus and phase of the CSD function are in agreement with the simulation results in Figure 7. Based on the aforementioned discussion, we know that the TC can adjust the polygonal structure of the PC-PEPV beam. Furthermore, the power order can also help realize a precise manipulation of the structure from circle to polygonal. The details of the numerical simulations and experimental intensities of the PC-PEPV beam (m 3) with high and low spatial coherence are shown in Videos 1 and 2 (mmc 1 mmc 2 of the Supplementary Material), respectively, where the power order n increase from 1 to 3 with step 0.1.

CONCLUSION
In conclusion, we have introduced the theoretical model of a PC-PEPV beam and studied its propagation properties numerically and experimentally. Different from the conventional partially coherent vortex beam where only a circular intensity distribution remains, the polygonalstructured intensity distribution of the PC-PEPV beam is manipulated by the value of the TC, and the polygonal vortex array is transformed into a polygonal light spot by decreasing the spatial coherence width. Furthermore, the value of the power order is used to control the details of the structure. For a PC-PEPV beam, the magnitude and sign of the TC can be determined by the coherence singularities in modulus and phase of CSD function, even when the optical array structure in the intensity fades away as the spatial coherence decreases. Moreover, when the power order is even, the sign of the TC is always positive. Our experimental results agree well with the numerical simulation results. We believe that the results will be useful for beam shaping and optical trapping.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.