Recently, the vortex beams which contain the phase singularity have attracted much attention,^[1–14] and some of their interesting properties are revealed. For instance, Gahagan and Swartzlander have demonstrated three-dimensional radiation pressure trapping of a low-index (hollow) spherical particle in water using a Gaussian beam containing an optical vortex.^[1] The work of Ng et al. suggested that particles trapped by optical vortices must be stabilized by ambient damping.^[3] What’s more, Soskin et al. have studied the transformations of topological charge during propagation in free space and established the general rule for angular-momentum density distribution of a vortex beam within a Gaussian envelope.^[4] The accesses to generating vortex beams have been researched as well including computer-generated holograms,^[5] two cascaded metasurfaces,^[6] three-plane-wave interference, and so on.^[7–10] Up to date, the vortex beams have been applied in many areas such as carrying information, trapping and guiding particles.^[11–14]

Laser beams with radial polarization have attracted widespread attention.^[15–20] They are featured by a totally symmetrical electric field in which the electric vector is radially distributed along the beam axis.^[15] Radially polarized beams are sharply focused light beams and can be applied in some optical instruments and devices such as lithography and confocal microscopy.^[16] In addition, radially polarized beams have a higher efficiency than the plane p-polarized and circularly polarized beams in the area of the laser cutting.^[17] There are also various studies on generation of radially polarized beams. For example, Machavariani et al. have introduced a transformation of a linearly polarized Gaussian beam to a radially polarized doughnut (0,1)^* Laguerre–Gaussian beam of high purity.^[18] Moreover, Yonezawa et al. have achieved success by use of a laser resonator including a c-cut Nd:YVO₄ crystal as the laser medium.^[20]

On the other hand, beams with special functions, such as Airy beams and Bessel beams, have been widely investigated recently.^[21–23] The Pearcey beams, based on the Pearcey function, were firstly illustrated by Pearcey in 1946. However, they have not attracted much attention until these years. In 2012, Ring et al. described the theory of Pearcey beams’ propagation and presented experimental verification of their auto-focusing and self-healing behavior.^[24] Afterwards, Deng et al. demonstrated the virtual source of a Pearcey beam.^[25] Ren et al. introduced dual Pearcey (DP) beams based on the Pearcey function of catastrophe theory.^[26] However, generating and studying Pearcey beams experimentally are not easy because of the infinite size of Pearcey beams' cross sections. Scientists tend to add a Gaussian envelope to a Pearcey beam and obtain the Pearcey–Gauss beam. The characteristics of Pearcey–Gauss beams such as the propagation properties and vectorial structures have been revealed well by researchers.^[24,27] Although theories about the properties of beams mentioned above are numerous, there are few reports on propagation properties of a radially polarized Pearcey–Gauss vortex beam (RPPGVB). Thus, we introduce this beam and investigate its propagation properties. The intensity pattern, phase pattern, spin currents and orbital currents of an RPPGVB propagating in free space are involved in our discussion.

The structure of this work is as follows. In Section 2, we derive general propagation expressions of an RPPGVB by solving the paraxial propagation equations and find out the factors influencing the propagation of the RPPGVB. Then we elicit some conclusions based on those expressions. In Section 3, we present and analyze multifarious patterns including intensity patterns, phase patterns as well as the patterns of spin currents and orbital currents, which are helpful for building deeper understanding of an RPPGVB. Finally, we draw the conclusion in Section 4.

We assume that the RPPGVB propagates along the z-axis in the Cartesian coordinate system. At z = 0, the electric field of the RPPGVB can be expressed as

From the above equations, we know that the RPPGVB focuses at z = 2ky′² when it propagates in free space and we take this distance as z_e. What’s more, when X = Y = 0, we can obtain Δx₁ = 0 and Δy₁ = y′z / (2kx′²), which represent the displacements of the beam from the initial position in the x- and y-direction, respectively, for E_x. Displacements for E_y can be expressed as Δx₂ = 0 and Δy₂ = y′z / (2kx′²).

The four displacements imply that the x-component of the RPPGVB never displaces while the y-component has the same displacements for the x- and y-components.

Figure 1 is the intensity profile of the RPPGVB at different planes as z increases in free space. When the propagation distance is not very long, there is one vortex in the central part of the profile and two faculae above the bands of main lobes. When z continues to increase, another vortex and two faculae form below the light lobes, as shown in Fig. 1(c). The vortexes expand while the faculae shrink until z = z_e, where the RPPGVB focuses at a point where x = y = 0. When z > z_e, the intensity pattern reverses. From Fig. 1, we also find that the energy of the RPPGVB primarily distributes on the main lobes while a little energy distributes on the other parts including the side lobes and faculae. This effect is even more apparent when the propagation distance approaches z_e. What is more, the peak intensity increases firstly before z = z_e and then decreases as z increases.

Fig. 1.

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	Fig. 1. Synthesizing intensity distribution as the RPPGVB travels along the z-axis in free space with parameters: λ = 500 nm, xd = yd = 0, x′ = y′ = 0.1 mm, w0 = 2 mm: (a) z = 0, (b) z = 0.3ze, (c) z = 0.8ze, (d) z = ze, (e) z = 1.2ze, (f) z = 1.8ze. The unit of the intensity in all the figures is cd.

Figure 2 depicts the intensity of the RPPGVB at z = 0.5z_e in free space with changing x′ and y′. It is quite clear that the RPPGVB’s profile shifts along the negative y-direction as scaling factors x′ and y′ increase. Meanwhile, the maximum value of the intensity decreases rapidly and the lobes of the RPPGVB become more distinct with the intensity range becoming smaller. When scaling factors reach a certain size, x′ = y′=0.4 mm, for example, the shape of overall RPPGVB’s cross section will become blurred, as shown in Fig. 2(f).

Fig. 2.

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	Fig. 2. Synthesizing intensity at z = 0.5ze when changing the values of x′ and y′ with parameters λ = 500 nm, no = ne = 1, xd = yd = 0, w0 = 2 mm: (a) x′ = y′ = 0.04 mm, (b) x′ = y′ = 0.05 mm, (c) x′ = y′ = 0.07 mm, (d) x′ = y′ = 0.1 mm, (e) x′ = y′ = 0.2 mm, (f) x′ = y′ = 0.4 mm.

All parameters of Fig. 3 are the same as those in Fig. 2 except that the propagation distance equals z_e. It is apparent that the RPPGVB focuses at z = z_e when the scaling factors are relatively small. From Figs. 3(a)–3(f), it is interesting that the focusing effect is the strongest when x′ = y′ = 0.05 mm. If we keep increasing the scaling factors, the focal point will disappear while the energy still gathers around the center of the beam’s profile. What’s more, it is shown in Figs. 3(a)–3(f) that the peak of the intensity gradually decreases to a very small value as the scaling factors increase. Comparing Fig. 3(a) with Fig. 3(f), it is also easy for us to learn that the peak intensity with x′ = y′ = 0.04 mm is almost thirty thousand times as large as that with x′ = y′ = 0.05 mm.

Fig. 3.

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	Fig. 3. Synthesizing intensity at z = ze when changing the values of x′ and y′ with parameters λ = 500 nm, xd = yd = 0, w0 = 2 mm: (a) x′ = y′ = 0.04 mm, (b) x′ = y′ = 0.05 mm, (c) x′ = y′ = 0.07 mm, (d) x′ = y′ = 0.1 mm, (e) x′ = y′ = 0.2 mm, (f) x′ = y′ = 0.4 mm.

Figure 4 exhibits the intensity of the RPPGVB with different waist sizes in free space. When w₀ = 0.4 mm, there are only three bright spots where the energy concentrates in the RPPGVB’s cross section. However, the whole cross section as well as the facula enlargers and the pattern of the intensity distribution is more and more obvious as w₀ gradually increases, which is presented by Figs. 4(b)–4(f). Moreover, in Fig. 4, the peak of the intensity exhibits increasing with an enlarging waist. Interestingly, from the peak intensities shown in Fig. 4, we also find that the increased rate gradually decreases as w₀ grows linearly. For example, the peak intensity for w₀ = 0.6 mm is five times as large as that for w₀ = 0.4 mm. However, the multiple reduces to 3 when w₀ changes from 0.6 mm to 0.8 mm, which even decreases to 1.8 when w₀ changes from 1.2 mm to 1.4 mm.

Fig. 4.

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	Fig. 4. Synthesizing intensity at z = 0.5ze when changing the value of w0 with parameters λ = 500 nm, xd = yd = 0, x′ = y′ = 0.1 mm: (a) w0 = 0.4 mm, (b) w0 = 0.6 mm, (c) w0 = 0.8 mm, (d) w0 = 1 mm, (e) w0 = 1.2 mm, (f) w0 = 1.4 mm.

Figure 5 demonstrates the intensity at z = 0 when the values of x_d and y_d are changed. From Figs. 5(a) and 5(b), we find that a part of the RPPGVB’s lobes disappear when x < 0, which means that there is a vortex at that position. From Figs. 5(a)–5(c), we find that the vortex moves at an angle to the positive x-direction. The energy of the RPPGVB mainly distributes on the lobes at x > 0. The main part of the facula appears at x > 0 and x_d = y_d < 0 in Figs. 5(a) and 5(b), and the peak intensity tends to decrease with the increase of the displacement. When x_d and y_d equal 0, as shown in Fig. 5(c), the number of the faculae is two and the vortex just locates at the center of the RPPGVB’s profile. What’s more, the whole cross section is symmetric about x = 0. The missing part of the lobe pattern reappears as well. When x_d and y_d are farther than zero, the energy mostly transfers to the lobes at x < 0 and the facula at x > 0 disappears. What’s more, the peak intensity increases when x_d and y_d get larger positive values, as shown in Figs. 5(d)–5(f).

Fig. 5.

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	Fig. 5. Intensity at z = 0 when changing the values of xd and yd with other parameters λ = 500 nm, x′ = y′ = 0.1 mm, w0 = 2 mm: (a) xd = yd = –2 mm, (b) xd = yd = –1 mm, (c) xd = yd = 0 mm, (d) xd = yd = 0.5 mm, (e) xd = yd = 1 mm, (f) xd = yd = 2 mm.

Through the previous investigations, we have learned the RPPGVB will focus on the point z = z_e and the scaling factor y′ is much smaller than the waist radius w₀. From Figs. 6(a)–6(f), we find that there is always a focal point at the center of the profile. What’s more, the faculae assume different shapes when x_d and y_d get changed. When x_d and y_d are less than zero, there are two vortexes at the cross section. When the values of x_d and y_d increase to zero, as shown in Fig. 6(c), four faculae arise and the whole beam’s profile is symmetric about the x-axis and the y-axis. When x_d and y_d keep increasing, the faculae shrink at x > 0.

Fig. 6.

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	Fig. 6. All are the same as those in Fig. 5 except z = ze.

For the intensity, the peak intensity occurs at the focal point and its value decreases at first and then grows when x_d and y_d increase to zero. When x_d > 0 and y_d > 0 with the increase of x_d and y_d, the peak intensity will become larger and larger, which is revealed by Figs. 6(d)–6(f).

Figures 7 and 8 show the phases of the x and the y components of the RPPGVB at different propagation distances. The equiphase zones are narrow bands and the boundaries between them are more orderly at z = 0 than those at other distances for both E_x and E_y. What’s more, there is a clear boundary at x = 0 and z = 0 in Fig. 7(a) and the phase is abrupt when it crosses this boundary, which makes the overall phase distribution asymmetrical. Similarly, there is a boundary at y = 0 for E_y at the initial plane from Fig. 8(a). Another attractive point is that the isophase band bends at the boundary of the lobe zone at z = 0, which cannot be found in other phase patterns. When we turn our attention to other positions, we find that the phase distribution diagrams at other planes for E_x and E_y are funnel-shaped. Furthermore, the phase boundary at x = 0 is less obvious than that in the same plane for E_x. In addition, we can learn from Figs. 8(b)–8(f) that funnel graphs are basically symmetric along x = 0 in each figure of E_y except Fig. 8(d). When the RPPGVB propagates at z = z_e, two equiphase zones appear obviously near y = 0 for E_x and E_y. The phases of the two regions equal –0.5 rad for E_x while the phases of the left and right parts are equal to 2.5 rad and –0.5 rad, respectively, for E_y.

Fig. 7.

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	Fig. 7. Phase distribution of Ex when the RPPGVB propagates along the z-direction. Parameters are the same as those in Fig. 1 and the unit of phase is rad.

Fig. 8.

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	Fig. 8. The same as Fig. 7 but Ey.

Figure 9 shows the spin currents at different planes when the RPPGVB propagates along the z-direction in free space. At z = 0, the sizes of spin currents in the central area are zero but the ones in other area are not and point at all directions regularly. Figures 9(b) and 9(c) show the spin currents at z = 0.8z_e and z_e. We find that they are similar to each other. Spin current vectors at the central area point around, while the surrounding ones have a tendency to point toward the positive direction of the y-axis. In addition, spin current vectors which are away from x = 0 are larger than those near x = 0 in size. When z equals 1.8z_e, there are two small areas whose shapes are like vortices on both sides of x = 0. The spin currents around x = 0 roughly point to the positive direction of the y-axis.

Fig. 9.

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	Fig. 9. Spin currents at different planes as the RPPGVB propagates along the z-direction. Parameters are the same as those in Fig. 1 except (a) z = 0, (b) z = 0.8ze, (c) z = ze, (d) z = 1.8ze. The lengths and directions of the arrows represent the normalized sizes and directions of the spin currents.

Figure 10 presents the orbital currents of the RPPGVB as it propagates along the z-direction. At z = 0, sizes of orbital currents around the central part equal zero. Contrary to spin currents, the orbital currents everywhere point at the center of the figure at z = 0. When z equals 0.8z_e and z_e, nearly all orbital currents point to the positive direction of the y-axis. Large orbital currents gather in the region where –2 mm ≤ x ≤ 2 mm and –2 mm ≤ y ≤ 0. When z continues to increase, the orbital currents in the upper part of the image and the gathered area of orbital currents become larger. When z equals 1.8z_e, the area extends to –2.5 mm ≤ x ≤ 2.5 mm and –1.7 mm ≤ y ≤ 2.8 mm. All of the orbital currents still point at the positive direction of the y-axis.

Fig. 10.

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	Fig. 10. All parameters are the same as those in Fig. 9 except the orbital currents of the RPPGVB. The lengths and directions of the arrows represent the normalized sizes and directions of the orbital currents.

We have introduced a new type of beam named as the RPPGVB and studied its propagation properties. Although all parameters have effects on the RPPGVB’s intensity pattern and the energy distribution, the main roles they play are different. For example, the scaling factors mainly affect the value of the intensity but have few effects on the intensity distribution. When enlarging the radius of the RPPGVB’s waist, the area of the RPPGVB’s cross-section expands accordingly, whereas the relative intensity distribution changes a little. The displacements of the vortex mainly influence the energy distribution and the shape of patterns of the RPPGVB. The spin currents and orbital currents of the RPPGVB are researched as well. We find that with the increase of the propagation distance, the spin currents tend to form two vortices at the center of the RPPGVB’s profile while the orbital currents gradually point at a specific direction orderly. Our investigation will provide a better understanding about the state of the RPPGVB propagating in vacuum and be useful for applications in information transmission.