Generation of Perfect Vortex Beams with Complete Control over the Ring Radius and Ring Width

Abstract

We have experimentally created perfect vortex beams (PVBs) by Fourier transformation of Bessel–Gaussian vortex beams, which are generated by modulating the fundamental Gaussian beam with the spiral phase plates and the axicons, respectively. Although the method has been used many times by other authors, as far as we know, few people pay attention to the quantitative relationship between the control parameters of the PVB and ring width. The effects of the waist radius of the fundamental Gaussian beam w_g, base angle of the axicon γ, and focal length of the lens f on the spot parameters (ring radius ρ, and ring half-width Δ) of PVB are systematically studied. The beam pattern of the generated Bessel–Gaussian beam for different propagation distances behind the axicon and the fundamental Gaussian beam w_g is presented. We showed experimentally that the ring radius ρ increases linearly with the increase of the base angle γ and focal length f, while the ring half-width Δ decreases with the increase of the fundamental beam waist radius w_g, and increases with enlarging the focal length f. We confirmed the topological charge (TC) of the PVB by the interferogram between the PVB and the reference fundamental Gaussian beam. We also studied experimentally that the size of the generated PVB in the Fourier plane is independent of the TCs. Our approach to generate the PVB has the advantages of high-power tolerance and high efficiency.

1. Introduction

A vortex beam is a special light beam carrying orbital angular momentum (OAM) and taking the shape of a doughnut ring. It attracts more and more people to study from fundamental and applied perspectives, such as optical microscopy [1,2,3], hyper-entanglement [4,5], strong coupling between light and matter [6], optical trapping and optical spanners [7,8,9,10], and classical and quantum communications [11,12], etc. Its expression usually contains an exp(ilθ) term. Here, θ is the azimuthal angle, l is named as the topological charge (TC), and the OAM of the vortex beam is lħ per photon (ħ is the reduced Planck constant). The typical vortex beam includes the Laguerre–Gaussian (LG) vortex beam [13], the Bessel–Gaussian (BG) vortex beam [14,15], and the perfect vortex beam (PVB) [16], etc. For transverse modes of LG vortex beam and BG vortex beam, the mode profile and radius of peak intensity depend on the order, or TC l. This property makes coupling multiple OAM beams simultaneously into a single optical fiber with fixed annular index profile to realize multiplexed communication more difficult [17], since the ring radius of vortex beam and radial intensity profile changes with the variation in the TC it carries. Furthermore, the vortex beam with fixed radial intensity profile and radius is generally expected for trapping and manipulating the nano-particles, and it allows us to independently study the relationship between the particle rotation rate and the OAM [18]. PVB is a typical representative of a vortex beam whose radial intensity profile and radius are both independent of TC. The creation of such a beam would be of significant interest. The most commonly used method for creating the PVB is Fourier transformation of the Bessel vortex beam [19,20,21], which is usually replaced by a BG vortex beam in reality. Here, the BG vortex beam is usually created by a computer-controlled liquid-crystal spatial light modulator (SLM) [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34] or digital micromirror device (DMD) [35], a special designed hybrid phase plate [36], a transparent variable diffractive spiral axicon based on a single liquid-crystal cell [37], polymer-based phase plate [38], metasurface [39], Pancharatnam–Berry phase element [40], axicon [41,42,43], or diffraction of the BG vortex beam by using curved fork grating [44], and so on. In addition, the PVB can be directly generated by schemes including a computer-generated hologram (CGH) displayed on the reflective phase SLM [45,46,47,48] or the DMD [49,50], a radial phase shift spiral zone plate [51,52], a planar Pancharatnam–Berry (PB) phase element [53], or metasurfaces [54,55,56,57,58,59,60,61,62], etc. Recently, the coherent beam combining technology is also employed to generate the PVB [63,64]. In these studies, effects of control parameters (e.g., input beam radius, axicon parameter, etc.) on ring radius has always been the focus of scholars. Few people pay attention to the quantitative relationship between the control parameters and the ring width. Furthermore, the independent control of the ring radius and ring width in PVBs has not yet been demonstrated in its entirety.

In this work, the BG vortex beam with different order generated by combining spiral phase plates (SPPs) and axicon is Fourier transformation to the PVB at the focal plane of the Fourier lens. We demonstrate the generation of arbitrary PVBs whose transverse intensity profile (i.e., ring width and ring radius) and TCs can be independently and easily controlled via adjusting the waist radius of the input fundamental Gaussian beam w_g, choosing axicons with different base angle γ, changing the focal length of the Fourier lens f, and cascading different SPPs, respectively. Section 2 is devoted to a concise description of the formulas of each beam in every step of generating the PVB. In Section 3, we describe experimental procedures to generate the PVB. The experimental results and discussions, including effects of control parameters on the spot parameters of the PVBs, and the characteristics of PVBs with different TCs, are presented in Section 4. Finally, the conclusion is presented in Section 5. The ability of complete control over the spot parameters has implications in the field of optical tweezers or optical manipulation, and for the efficient launch of optical modes in the solid-state laser or in an annular core fiber laser, etc.

2. Theoretical Analysis

Figure 1 shows the physical procedure for transforming an input Gaussian beam into a PVB through three different stages. For the sake of simplicity, we assume the waist location of the fundamental Gaussian beam is at z = 0. The transverse field distribution of the amplitude for the input fundamental (TEM₀₀) Gaussian beam propagating along the z-axis in the cylindrical coordinate system can be mathematically written as

$$E_{\mathrm{G}}\left(r,\theta ,z\right)=A_{\mathrm{G}}\exp \left[-r^2/w_g^2(z)\right]\exp \left[i{\varphi }_{\mathrm{G}}+ik{r}^2/2R\left(z\right)+ikz\right]$$

(1)

where (r, θ, z) represents the coordinate parameters of the cylindrical coordinate system. $A_{\mathrm{G}}=\sqrt{P_{\mathrm{G}}/\left[\pi n_{0}c{\epsilon }_0{w}_g^2(z)\right]}$ (details can be found in Equations (A2) and (A3)) denotes the amplitude of the beam on the axis (r = 0). Here, P_G represents the power of the fundamental Gaussian beam, n₀ is the refractive index of the transmission medium (n₀ ≈ 1 in air), ε₀ is the permittivity of the vacuum, and c is the speed of light in the vacuum. $w_g(z)=w_g\sqrt{1+{\left(z/z_R\right)}^2}$ and w_g indicate the beam radius and waist radius of the input fundamental Gaussian beam, respectively. $z_R=k{w}_g^2/2=\pi w_g^2/\lambda$ is the Raleigh range. λ is the wavelength of the incident beam. ${\varphi }_G=-\arctan \left(z/z_R\right)$ is the Gouy phase, k is the wave vector. $R(z)=z+z_R^2/z$, is the radius of curvature of the wavefront.

After passing through the spiral phase plates (SPPs), the input fundamental Gaussian beam obtains the spiral phase function $\exp (il\theta )$, and transforms to the higher-order LG vortex beam [65,66,67,68], whose amplitude can be expressed as

$$E_{\mathrm{LG}}\left(r,\theta ,z\right)=A_{\mathrm{LG}}{\left[\frac{\sqrt{2}r}{w_g\left(z\right)}\right]}^{\left|l\right|}\exp \left[-\frac{r^2}{w_g^2\left(z\right)}\right]\exp \left[i{\varphi }_{\mathrm{LG}}+\frac{ik{r}^2}{2R\left(z\right)}+ikz+il\theta \right]$$

(2)

where $A_{\mathrm{LG}}=\sqrt{P_{\mathrm{LG}}/\left[\pi n_{0}c{\epsilon }_0\left|l\right|!w_g^2(z)\right]}$ is the amplitude (details can be found in Equations (A4)–(A6)). P_LG represents the power of the LG vortex beam, and l is the TC number of the LG vortex beam. ${\varphi }_{\mathrm{LG}}=-\left(\left|l\right|+1\right)\arctan \left(z/z_R\right)$ is the Gouy phase for the LG_0,l vortex beam. It can be seen from Equation (2), that the amplitude and phase of the LG_0,l vortex beam depend on the TC l.

Now further assume that the axicon is positioned at z = 0. An axicon possessing conical surface brings the axicon function to the LG vortex beam and transforms the LG vortex beam into the BG vortex beam [69,70], which can be simply described as

$$E_{\mathrm{BG}}\left(r,\theta ,z\right)=A_{\mathrm{BG}}{J}_l\left(k_{r}r\right)\exp \left[-r^2/w_g^{\prime 2}\left(z\right)\right]\exp (i{k}_{z}z+il\theta )$$

(3)

where $A_{\mathrm{BG}}=\sqrt{P_{\mathrm{BG}}/\left[\pi n_{0}c{\epsilon }_0{w}_g^{\prime 2}(z)\right]}$ is the amplitude (details can be found in Equations (A7)–(A9)), $w_g^{\prime }(z)=w_g^{\prime }\sqrt{1+{\left(z/z_R\right)}^2}$ and $w_g^{\prime }$ indicate the beam radius and waist radius of Gaussian beam which is used to confine the Bessel beam, respectively. P_BG represents the power of the BG vortex beam, and J_l is the lth order Bessel function of the first kind. k_r and k_z are the radial and longitudinal wavevectors, respectively. $k=\sqrt{k_r^2+k_z^2}=2\pi /\lambda$ and $k_r/k=\sin [(n-1)\gamma ]$. Here, n is the refractive index of the axicon, and γ is the base angle of the axicon. Unlike the ideal Bessel beam, the BG beam diverges to a certain distance, which is called the Rayleigh range (or diffraction-free range) and can be given by $Z_{\max }=w_g^{\prime }/\left(n-1\right)\gamma$ [71]. From Equation (3), we know that the amplitude distribution and phase of the BG vortex beam have a strong dependence on the TC l.

The intensity distribution of the BG vortex beams can be obtained using

$$I=2n_{0}c{\epsilon }_0{\left|E_{\mathrm{BG}}\right|}^2=\frac{2P_{\mathrm{BG}}}{\pi w_g^{\prime 2}\left(z\right)}{J}_l^2\left(k_{r}r\right)\exp \left[-2r^2/w_g^{\prime 2}\left(z\right)\right]$$

(4)

Finally, a plano-convex lens with focus length f is used to implement the Fourier transformation of the BG vortex beam. Set s₀ as the distance between the axicon and the lens. Only when f < s₀ < Z_max, the PVB with fixed annular intensity (ring radius ρ and ring width 2Δ) can be produced at the focal plane of the lens. For ρ >> Δ, the transverse distribution of complex amplitude for the PVB can be simply expressed by the ideal model [21]

$$E_{\mathrm{PV}}\left(r,\theta \right)=A_{\mathrm{PV}}\exp \left[-{\left(r-\rho \right)}^2/{\Delta }^2\right]\exp (il\theta )$$

(5)

where $A_{\mathrm{PV}}=\sqrt{P_{\mathrm{PV}}/\left(4\pi nc{\epsilon }_0\rho \Delta \sqrt{\pi /2}\right)}$ is the amplitude (details can be found in Equations (A10) and (A11) in Appendix A), and P_PV represents the power of the PVB. Equation (5) indicates that the TC will not affect the amplitude distribution of the PVB. The amplitude distribution only depends on two parameters, ring radius ρ and ring width 2Δ. The value of ring radius ρ can be written as

$$\rho =k_{r}f/k=f\sin [(n-1)\gamma ]\approx f(n-1)\gamma$$

(6)

Its value depends on the focal length of lens f and the radial wave vector k_r, which can be modulated by the axicon parameters (refractive index n and base angle γ). It is found that the value of ρ increases linearly with the increase of focal length f and base angle γ.

The ring half-width Δ is governed by the relation

$$\Delta =2f/k{w}_g^{\prime }$$

(7)

The value of ring width 2Δ is mainly determined by two parameters, the fundamental beam waist $w_g^{\prime }$ and the focal length f. The effect of f and $w_g^{\prime }$ are found to have reverse trends.

As evident from Equations (6) and (7) above, both the ring radius ρ and ring width 2Δ of the PVB increase with enlarging the focal length f of the Fourier transforming lens. The ring radius ρ can be independently controlled by the base angle of the axicon, while the ring width 2Δ can be independently tuned by varying the Gaussian beam waist $w_g^{\prime }$. Equations (6) and (7) are useful for guiding the experimental realization of PVBs and to analyze the results.

Finally, the intensity distribution of the PVBs can be obtained using

$$I=2n_{0}c{\epsilon }_0{\left|E_{\mathrm{PV}}\right|}^2=\frac{P_{\mathrm{PV}}}{2\pi \rho \Delta \sqrt{\pi /2}}\exp \left[-2{\left(r-\rho \right)}^2/{\Delta }^2\right]$$

(8)

3. Experimental Setup

The experimental setup for the generation of PVB is schematically shown in Figure 2. A continuous-wave, high-power (as high as 15 W), single-frequency (linewidth of <100 kHz), linearly polarized Yb-doped fiber laser and amplifier (NKT Photonics, Birkerød, Denmark, Koheras Y10) at 1064 nm is used as the laser source. To prevent the reflected beam from feeding back to the laser system, an optical isolator is employed. To adjust the pump power in the subsequent optical path, the power-control system, including a half-wave plate (HWP) and polarizing beam splitters (PBS), is inserted on the optical path. To improve the quality of the beam spot and to make the beam spot more circular, and symmetrical, a circle pinhole with a diameter of 100 µm is used to diffract the laser beam, and the Airy spot (the center bright spot) of the diffraction beam is selected by a circular diaphragm. The generated-circularly symmetric Airy spot, will be regarded as the standard spot of the fundamental Gaussian beam in the subsequent optical path. After that, the Airy spot is collimated by a lens f₁ with focal length of 200 mm. The radius of the collimated beam w_g in the experiment is measured to be 2.434 mm. An improved Mach–Zehnder interferometer (MZI), composed of M₁, M₂, NPBS₁, and NPBS₂, is employed to generate and detect the PVB. Arm 1 of the MZI (NPBS₁-M₄-NPBS₂) is used to generate the PVB under different experimental conditions. Arm 2 of the MZI (NPBS₁-M₄-NPBS₂) serves as reference light to interfere with the generated PVB and measure its TC. In arm 1, the beam expander, composed of two lenses f₂ and f₃ (focusing with lens f₂, and collimating with lens f₃), is used to adjust the waist radius w_g of the fundamental Gaussian beam radius (the combination of lenses f₂ and f₃ with different focal lengths are used here to generate different waist radius w_g (measured in the experiment), as shown in the inset). The cascaded spiral phase plates (UPO Labs, SPP₁-SPP₃) are illustrated by the collimated fundamental Gaussian beam and transform the fundamental Gaussian beam to the LG vortex beam with TC l = ±1~±6 [72]. The axicon (LBTEK, base angle γ = 0.5°, 1°, and 2°) transforms the LG vortex beam to the BG vortex beam with the same order. Then, the generated BG vortex beam is Fourier transformed by using lens f₅ so as to form a single PVB at the focus point. In arm 2, the collimated reference beam (sphere-wave reference light) is reflected by M₃ (mounted on a translation stage), and then combined with the beam of arm 1 by a NPBS₂. Finally, a CMOS camera (CinCam, CMOS-1202) is used to capture the beam patterns (intensity profiles) of the generated PVB in arm 1 and the interferograms generated by the MZI. The spiral interferograms are generated when two beams with different wavefront curvature radius interfere coaxially [73]. Here, the lens f₄ (focal length of 40 mm) is used to focus the reference Gaussian beam and generate sphere-wave reference light. The fork wire interferograms are formed when there exists an oblique angle between two interference beams [74]. At this time, the lens f₄ in the dashed frame in Figure 2 is removed, and the reference beam is a plane wave.

4. Results and Discussion

We will begin this study in three major areas. Firstly, the generated BG beam is studied for different propagation distance z behind the axicon and input fundamental Gaussian beam waist radius w_g. Secondly, the relationship between the spot parameters (ring radius ρ and ring half-width Δ) of the PVBs and the control parameters (such as the waist radius of the fundamental Gaussian beam w_g, the base angle of the axicon γ, and the focal length of the Fourier lens f) are studied in experiment. For the sake of simplicity, the PVB with a TC of zero (TC = 0) is created by Gaussian beam illumination of an axicon in combination with a lens. The spot parameters (ring radius ρ and ring half-width Δ) of the PVB can be obtained by fitting its intensity distribution with that of the standard PVB expressed by Equation (7). Thirdly, the characteristics of the PVB carrying different TCs are generated and compared. For a better comparison, we have considered the same scale for all the images.

4.1. Characteristics of the Generated BG Beam for Different Propagation Distance z behind the Axicon and the Input Fundamental Beam with Waist Radius w_g

The beam pattern of the BG beam generated by modulating the input fundamental Gaussian beam for different propagation distance z behind the axicon is shown in Figure 3. Here, the base angle of the axicon is chosen to be γ = 0.5° and the waist radius of the incident fundamental Gaussian beam is w_g = 2.434 mm. Considering the refractive index n of the axicon (material: UV fused silica) is about 1.4 at 1064 nm, and assuming $w_g^{\prime }$ = w_g, the diffraction-free distance is Z_max ≈ 700 mm. It can be seen that the spot of the generated BG beam is not ideal for propagation distance z < 150 mm. The standard BG beams are generated for propagation distance z ≥ 200 mm, and the beam spot image hardly changes with the propagation distance in experiment (200 mm ≤ z ≤ 300 mm). This is because the propagation distance z < Z_max.

The intensity distributions of the Bessel beam and BG beams are shown in Figure 4. Here, the intensity distributions of the BG beams are drawn by using Equation (4), and the intensity distribution of the Bessel beam is drawn by ignoring the Gaussian term in Equation (4). It can be seen that the higher order maxima of different beam spots ($w_g^{\prime }$) will have significant differences. Thus, the beam waist radius $w_g^{\prime }$ of the experimental generated BG beams can be obtained by fitting the experimental intensity distributions using Equation (4). Figure 5 shows the measured intensity distribution of the BG beams in experiment for z = 200 mm. Here, w_g represents the waist radius of the incident fundamental Gaussian beam, while $w_g^{\prime }$ represents the simulated one using Equation (4). We find that $w_g^{\prime }$ = w_g/2.

4.2. Effects of Waist Radius w_g, Base Angle γ and Focal Length f on the Spot Parameters of the PVB for TC = 0

Firstly, the effect of the axicon base angle on the spot parameters (ring radius ρ, and ring half-width Δ) of the generated PVB is quantitatively studied. To increase the non-diffraction range, the fundamental beam waist w_g is chosen to be 3.487 mm here ($w_g^{\prime }$ = 1.743 mm). The axicons with base angle γ = 0.5°, 1°, and 2° are employed to generate the BG beams, respectively. The diffraction-free distance Z_max for γ = 0.5°, 1°, and 2° are calculated to be about 499 mm, 250 mm, and 125 mm, respectively. The Fourier lens f₅ with a focal length of 100 mm is used to create the PVB at the focus. To generate the PVB at the focal plane of f₅, the distance between the axicon and the lens s₀ is chosen to be 120 mm in this experiment.

The measured beam patterns and their intensity distributions of the PVBs for different base angles γ are shown in Figure 6. It is obvious that the ring radius ρ of the PVB increases when enlarging the base angle γ of the axicon. The ideal PVBs are generated in rows (1) and (2) of Figure 6, while a deteriorating PVB is generated in row (3) of Figure 6. For the deteriorating PVB, there exists a weak dark ring inside the PVB, resulting in a significant asymmetry in the ring half-width for each peak. This is because the s₀ value selected in experiment (120 mm) is too close to the value of Z_max (125 mm for γ = 2°), resulting in a suboptimal generation of the PVB under this condition. In order to quantitatively describe the generated PVB, the experimental measured intensity distribution (black dots in row 2 of Figure 6) is fitted with the expression of the PVB Equation (8), and the values of ring radius ρ, and ring half-width Δ are obtained for each PVB. The ring has radius ρ = 0.316 mm, 0.671 mm, and 1.379 mm for the γ values of 0.5°, 1°, and 2°, respectively. The values of ring radius ρ and ring half-width Δ versus base angles γ are also plotted in Figure 7. The black solid line shown in Figure 7 is plotted according to Equation (6), where f is chosen to be 100 mm and the refractive index n of the axicon (material: UV fused silica) is chosen to be 1.4 at 1064 nm. Experimental results are verified with the theoretical plot obtained from Equation (6). The average value of the experimental measured ring half-width Δ is 19.3 μm, which is quite close to the theoretical value Δ = fλ/π$w_g^{\prime }$ = 2fλ/πw_g = 19.4 μm. The ring radius ρ increases linearly with the base angle γ, while the ring half-width Δ remains basically unchanged with γ, which verifies that Δ is independent of γ.

Secondly, the effect of the waist radius of the fundamental Gaussian beam w_g on the spot parameters of the PVB is also studied. Here, the axicon with the base angle of γ = 0.5° is employed, and the focal length of the Fourier lens f is fixed to 200 mm. By selecting the focal length of lenses f₂ and f₃ of the beam expander, the waist radius of the fundamental Gaussian beam w_g can be changed to f₃/f₂ of the original one.

Beam patterns and their intensity distributions of PVBs for different waist radius w_g are shown in Figure 8. It can be seen from row (a) of Figure 8 that the ring radius ρ remains almost the same for the increase of w_g, while the ring half-width Δ narrows with increasing w_g. This can be verified by the fitting results shown in row (b) of Figure 8. In order to quantitatively describe the relationship between the spot parameters of the PVB and the waist radius w_g, the spot parameters of the PVB presented in in row (b) of Figure 8 are shown again in Figure 9. It is clear that the ring half-width Δ is inversely proportional to the waist radius w_g, and the relationship between Δ and w_g can be approximately fitted by $\Delta =2f\lambda /\pi w_g$. The results of ring radius ρ shown in Figure 8 indicate the fluctuation between 0.750 mm and 0.860 mm for the value of ring radius ρ while the w_g is varied. The average ρ is 0.792 mm, which is relatively close to the theoretical value of 0.785 mm.

Next, we will consider the effect of the focal length f of the Fourier lens on the beam parameter of the PVB. Here γ = 0.5° and w_g = 2.434 mm. The effect of the focal length f of the Fourier lens on the spot parameters is analyzed.

The images of the beam pattern and intensity distribution of the PVB for different focal length f is shown in Figure 10(a1–a6,b1–b6), respectively. Both the ring radius ρ and ring half-width ∆ increase with the enlarfment of focal length f. The detailed relationship between the spot parameters of the PVB and the focal length is depicted in Figure 11. The experimental results are verified with the theoretical plots obtained by using Equation (6) and with the real experimental parameters. From the fitted lines, a linear relationship can be clearly seen for both the ring radius ρ and ring half-width ∆.

From the above analysis, it can be seen that the value of the ring radius obtained in the experiment can be well fitted by $\rho =f(n-1)\gamma$, while the ring half-width obtained in the experiment can be approximately fitted by $\Delta =2f\lambda /\pi w_g$ rather than $\Delta =f\lambda /\pi w_g$. This can be explained by the experimental fact that the beam waist radius $w_g^{\prime }$ of the experimental generated BG beam is about half of the waist radius w_g of the incident fundamental Gaussian beam ($w_g^{\prime }$ = w_g/2). The relationship between the ring half-width ∆ and $w_g^{\prime }$ is $\Delta =f\lambda /\pi w_g^{\prime }$.

4.3. Characteristics of PVBs with Different TCs

Here, we choose the base angle of the axion γ = 0.5°, the waist radius w_g = 2.434 mm, and the focal length of the Fourier lens f₅ = 75 mm. The experimental generated PVBs carrying different TCs are presented in this section. The combination of 2 pieces of SPPs with vortex order 1 and 2 pieces of SPPs with vortex order 2 allows the generated PVB carrying TCs from −6 to 6 [72]. The TCs of PVBs are verified experimentally by the interferograms generated by the MZI. The fork number is equal to the TC value of the PVB, and the fork direction is determined by the sign of the TC [74]. The number of spiral lobes depend on the TC value of the PVB, and the twist direction depends on the ring of the TC [73].

A group of beam patterns, intensity profiles and interferograms of PVBs with TC l = −2~6 is obtained as shown in Figure 12 and Figure A1, respectively. As evident from the beam patterns of the PVBs with different TCs shown in Figure 12(a1–a5) and Figure A1(a1–a4), the images of all the PVBs are similar. To determine the spot parameters (ring radius ρ and ring half-width ∆) of the generated PVB, the measured intensity distribution of each PVB is simulated using Equation (8), as shown in Figure 12(b1–b5) and Figure A1(b1–b4). The spot parameters of the PVBs for different TCs are presented in Figure 13. Both the ring radius ρ and ring half-width ∆ of the generated PVBs are almost the same for all of the TCs. The results show that the generated PVB is independent of its TC. The spiral interferograms shown in Figure 12(c1–c5) and Figure A1(c1–c4) are seen to have a spiral structure. The number of spiral lobes in the central zone is the TC value, and its rotation direction is clockwise for the negative sign of TC and vice versa. Unlike the spiral interferogram as observed in the interference of normal vortex beam with a Gaussian with a spherical wavefront [72,73], the spiral interferogram in Figure 12(c1–c5) and Figure A1(c1–c4) is faint at the center of the interferogram due to the large dark core of the PVB. The fork wire interferograms shown in Figure 12(d1–d5) and Figure A1(d1–d4) have a fork wire fringe. For the fork wire interferogram, the fork number in the central zone is the TC value, and the fork direction is downward for the negative sign of TC and upward for the positive sign of TC. Such observations clearly confirm the ability of determining the TCs of PVBs through the interferograms.

5. Conclusions

In conclusion, we demonstrated the combination of SPPs, axicons, and lens for generating PVBs with arbitrary spot parameters and TCs. The ring radius and ring width of the PVB can be completely controlled by choosing the axicon with a different base angle, adjusting the Gaussian beam waist, and changing the lens to a different focal length. We demonstrated that the ring radius increases linearly with the base angle of the axicon or focal length of the Fourier lens. The ring width decreases when increasing the Gaussian beam radius, and increases when enlarging the focal length. The approximate relation among the input fundamental Gaussian beam radius w_g, the focal length f, and the ring half-width (∆), in experiment is $\Delta =2f\lambda /\pi w_g$. The ring radius and ring width of the PVB is independent of the TC. The experiment results verify that PVBs have the advantages of a uniform ring profile and fixed radius compared with the traditional vortex beams. Our method to generate the PVB is notable for the simplicity and flexibility of its practical realization, high-power tolerance and high efficiency over the SLM-based method, and high quality of the results. The high-power PVBs have potential for excitation of OAM modes in an air-core fiber [75], or solid-state laser crystal [76].