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]: E ( r , θ ) = exp ( − r 2 w 0 2 ) exp ( i 2 π [ rem ( m θ , 2 π ) 2 π ] n ) , (1) 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: W ( r 1 , θ 1 , r 2 , θ 2 , ω ) = 〈 E ∗ ( r 1 , θ 1 , ω ) E ( r 2 , θ 2 , ω ) 〉 , (2) 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 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): W ( r 1 , θ 1 , r 2 , θ 2 ) = exp ( − r 1 2 + r 2 2 w 0 2 ) × exp [ − r 1 2 − 2 r 1 r 2 ⁡ cos ( θ 1 − θ 2 ) + r 2 2 2 σ 2 ] × exp ( i 2 π { [ rem ( m θ 1 , 2 π ) 2 π ] n − [ rem ( m θ 2 , 2 π ) 2 π ] n } ) , (3) 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: W ( ρ 1 , φ 1 , ρ 2 , φ 2 , z ) = ( 1 λ | B | ) 2 ∫ 0 ∞ ∫ 0 ∞ ∫ 0 2 π ∫ 0 2 π r 1 r 2 d r 1 d r 2 d θ 1 d θ 2 W ( r 1 , θ 1 , r 2 , θ 2 ) × exp { − i k 2 B ∗ [ A ∗ r 1 2 − 2 r 1 ρ 1 ⁡ cos ( θ 1 − φ 1 ) + D ∗ ρ 1 2 ] } × exp { i k 2 B [ A r 2 2 − 2 r 2 ρ 2 ⁡ cos ( θ 2 − φ 2 ) + D ρ 2 2 ] } . (4)

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.

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: E ( r , θ ) = 1 2 π ∑ l = − ∞ ∞ a l ( r ) exp ( i l θ ) , (5) where l is an integer and the mode coefficients are given by: a l ( r ) = 1 2 π ∫ 0 2 π E ( r , θ ) exp ( − i l θ ) d θ , (6) 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: 〈 | a l ( r ) | 2 〉 = 1 2 π exp [ − 2 r 2 w 0 2 ] ∫ 0 2 π ∫ 0 2 π exp [ i 2 π ( rem ( m θ 1 , 2 π ) 2 π ) n ] × exp [ − i 2 π ( rem ( m θ 2 , 2 π ) 2 π ) n ] × exp [ − i l ( θ 1 − θ 2 ) ] × exp [ − r 2 σ 2 ( 1 − cos ( θ 1 − θ 2 ) ) ] d θ 1 d θ 2 . (7) Here, the normalized power weight for each OAM mode can then be defined as follows: P l = ∫ 0 ∞ 〈 | a l ( r ) | 2 〉 r d r ∑ l ∫ 0 ∞ 〈 | a l ( r ) | 2 〉 r d r . (8) 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.

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 L₁ (f ₁ = 100 mm), and then collimated by the lens L₂ (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 A₁ 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 (SLM₁) via a mirror M. As shown in Figure 3A, the fork grating loaded on SLM₁ is designed via the method of computer-generated holograms [10]. Then the lenses L₃ and L₄ and aperture A₂ 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 L₅. One path is reflected via the beam splitter, and the intensity is recorded by the charge-coupled device (CCD₁), which is located at the focal plane of L₅. 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 SLM₂. The phase written into SLM₂ 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 SLM₂ need to be aligned. Finally, the output beam from SLM₂ is reflected by the beam splitter and recorded by the CCD₂ located at the Fourier plane.

The Fourier transformation intensity recorded by the CCD₂ with no additional phase perturbation can be written as follows: I 0 ( k ) = ∫ ∫ W ( ρ 1 , ρ 2 ) exp [ − i 2 π k ( ρ 1 − ρ 2 ) ] d ρ 1 d ρ 2 , (9) where W(ρ ₁, ρ ₂) is the CSD function of the PC-PEPV beam on the plane of SLM₂ and k represents the coordinate vector in the plane of CCD₂. Then we introduce an on-axis phase perturbation at ρ ₂ = ρ ₀ = (0, 0) of SLM₂. 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: I ( k ) = I 0 ( k ) + C C ∗ W ( ρ 0 , ρ 0 ) + C ∫ W ( ρ 1 , ρ 0 ) exp [ − i 2 π k ( ρ 1 − ρ 0 ) ] d ρ 1 + C ∗ ∫ W ( ρ 0 , ρ 2 ) exp [ − i 2 π k ( ρ 0 − ρ 2 ) ] d ρ 2 . (10) Performing an inverse Fourier transform of this intensity gives the following equation: F T − 1 [ I ( k ) ] ( ρ ) = F T − 1 [ I 0 ( k ) ] ( ρ ) + C C ∗ W ( ρ 0 , ρ 0 ) δ ( ρ ) + C W ( ρ 0 + ρ , ρ 0 ) + C ∗ W ( ρ 0 , ρ 0 − ρ ) . (11) 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: W ( ρ , 0 ) = F T − 1 { ( C 3 − C 1 ) [ I 2 − I 1 ] − ( C 2 − C 1 ) [ I 3 − I 1 ] ( C 3 − C 1 ) ( C 2 ∗ − C 1 ∗ ) − ( C 3 ∗ − C 1 ∗ ) ( C 2 − C 1 ) } . (12)

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 SLM₂ 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.

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 [53–55]. 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.

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 positive and negative, as shown in Figure 7. The intensity, modulus, and phase of the CSD function of the PC-PEPV beam with high coherence width (σ = 20 mm) and TC (m = 3 and −3) are depicted in Figures 7A1–A3; Figures 7B1–B3, respectively. Meanwhile, the PC-PEPV beam with low coherence width (σ = 0.5 mm) and TC (m = 3 and −3) are demonstrated in 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 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.

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 CCD₁, 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.