Observation of vortex-pair dance and oscillation

Abstract

Vortex dynamics, which encompass the motion, evolution, and propagation of vortices, elicit both fascination and challenges across various domains such as fluid dynamics, atmospheric science, and physics. This study focuses on the fundamental dynamics of vortex-pair fields, specifically known as vortex-pair beams (VPBs) in optics. VPBs have gained increasing attention due to their unique properties, including vortex attraction and repulsion. Here, we explore the dynamics of pure-phase VPBs (PPVPBs) and observe intriguing helical and intertwined behaviors of vortices, resembling a vortex-pair dance. We uncover the oscillation property of the intervortex distance for PPVPBs in free space. The observed dancing and oscillation phenomena are intricately tied to the initial intervortex distance and can be explained well in the hydrodynamic picture. Notably, the vortex dancing and oscillation alter the process of vortex-pair annihilation, extending the survival range for opposite vortices. This discovery enhances our understanding of vortex interactions and sheds light on the intricate dynamics of both vortex-vortex and vortex-antivortex interactions.

Optical vortex-pair dance and oscillation uncover the intriguing and fascinating interactions of vortices.

INTRODUCTION

Vortices are prevalent phenomena observed across a spectrum of scales in nature, ranging from water eddies and atmospheric typhoon or hurricanes to majestic spiral galaxies. Vortices are also fundamental solutions within cylindrical-symmetry resonators for electromagnetic fields (1) and are pervasive in the realm of light, where they are known as optical vortices (2, 3). Optical vortices exhibit a distinctive feature—a dark core at the center, characterized by an indeterminate phase with vanishing amplitude (4, 5). These unique attributes of optical vortices give rise to a diverse array of applications, including optical micromanipulation (6–8), optical communications (9–11), quantum information (12–15), super-resolution imaging (16–18), and optical measurements (19–21).

In the presence of multiple vortices within light fields, the topological dynamics and interaction among vortices can create unique and interesting phenomena, like vortex knots (22), vortex collisions (23, 24), and the consequential process of vortex creation, annihilation, or nucleation (23–30). Among vortex interactions, a fundamental scenario involves the interaction of two vortices. In the atmosphere, the Fujiwhara effect occurs as two typhoons approach each other (31). In the domain of optics, the dynamic interplay of attraction and repulsion between two vortices has been previously observed in the dynamics of vortex-pair fields (25, 26). A vortex-pair beam (VPB) is a type of structured light fields containing a pair of vortices. It is sometimes categorized into two scenarios: an isopolar vortex pair, i.e., two vortices with identical topological charges (TCs), and a vortex dipole, characterized by two vortices with opposite TCs (32). In 1993, Indebetouw (25) discovered that the relative distance of an isopolar vortex pair remains constant during free space propagation, while a vortex dipole tends to exhibit mutual attraction. This effect was subsequently confirmed through the experiment (26). The interaction between two vortices manifests specific features, including the rotational effect observed in the isopolar vortex pair during propagation (26, 33), the reappearance of an annihilated vortex dipole in the far field (34), and an optical intrinsic orbit–orbit interaction, serving as a manifestation of the attractive and repulsive interactions within a vortex dipole (35). Furthermore, the exploration of VPB dynamics in diverse optical systems (36–49), such as those involving a graded-index medium (40, 41), a high numerical-aperture lens (42–44), an astigmatic system (45–47), a half-plane screen (48), and a knife edge (49), has been a subject of extensive discussion.

However, the aforementioned investigations (25, 26, 32–49) primarily rely on the model of complex-amplitude VPBs (CAVPBs), in which each vortex undergoes complex-amplitude modulation, constraining our comprehensive understanding of the dynamics inherent in multiple vortices. In the exploration of fractional vortex fields (50–55) and vortex arrays (56–60), researchers have found the intricate dynamics in the evolutions of vortex interactions among multiple vortices beyond the vortex attraction and repulsion process, like the birth or annihilation of vortex pairs. To further advance our understanding of the interaction between two vortices, here, we would like to address the dynamics of the pure-phase VPBs (PPVPBs), consisting of two pure-phase vortices. It is noteworthy that, to the best of our knowledge, the dynamics of PPVPBs have not been reported previously, despite their proposal and application in optical trapping and manipulation (61). Here, we elucidate the intriguing dynamics arising from vortex-vortex and vortex-antivortex interactions, resulting in a phenomenon reminiscent of vortex dance—a helical and intertwining behavior among vortices. The observed vortex dynamics can be well explained by using an optical hydrodynamic picture (62). Our experimental verification, using the interference method to trace vortex trajectories in light fields, solidifies the existence of this interesting feature. Notably, the oscillation of intervortex distance in PPVPBs represents a fundamentally unique characteristic not observed in traditional CAVPBs, where no oscillation effect had been previously identified. A comprehensive investigation reveals that the observed vortex dancing and oscillation can be precisely controlled by the initial intervortex distance, reflecting the interaction strength between the vortices. Meanwhile, the vortex dance and oscillation substantially influence the process of vortex-pair annihilation. This effect enlarges the survival range of opposite vortices, with a certain similarity to the Fujiwhara effect of two typhoons in atmosphere, presenting a distinct aspect not witnessed in CAVPBs. The observed vortex dynamics in PPVPBs here will not only promote an insightful understanding on vortex interactions in the fields such as optical physics and quantum fluids but also enrich the control way over vortex dynamics.

RESULTS

Fields of PPVPBs

We start by briefly reviewing the previous model of CAVPBs (25). The initial field of such CAVPBs embedded in a host Gaussian beam is usually expressed as $E_{\text{CAVPB}}(u,v)=M(u,v)\text{exp}(-\frac{u^2+v^2}{w_0^2})$ with $M(u,v)={[(u-u_0)+i\text{sgn}(m_1)v]}^{\mid m_1\mid }{[(u+u_0)+i\text{sgn}(m_2)v]}^{\mid m_2\mid }$, where 2u₀ is the initial distance of two vortices with TCs m₁ and m₂, w₀ is the beam width of the host Gaussian beam, and u, v refer to the transverse rectangular coordinates at the initial plane. As the magnitude of the function M(u, v) changes and deviates from unity, the two vortices in CAVPBs also undergo amplitude modulation. According to modal analysis (32, 36), these CAVPBs can be expressed as linear combinations of finite vortex modes, which, in turn, govern the dynamics of vortices. In most works, researchers only considered the cases of m₁ = m₂ = 1 for an isopolar vortex pair or m₁ = −m₂ = 1 for a vortex dipole. The vortex trajectories, illustrating the attraction or repulsion of vortices, were demonstrated in a series of prior investigations (25, 26, 32–49). Recently, one has developed the laser hydrodynamic model to explain the vortex motions in multiple complex-amplitude vortices (63, 64). Nevertheless, unraveling the underlying physics of vortex dynamics remains a challenge in many complex optical fields, surpassing the complexity observed in CAVPBs.

In contrast to the model of CAVPBs, one can have an alternative choice on two pure-phase vortices, which can be written as (61)

$$E_{\text{PPVPB}}(u,v)=F(u,v)\text{exp}(-\frac{u^2+v^2}{w_0^2})$$ (1)

with $F(u,v)={\left[\frac{u-u_0+\mathit{iv}}{\sqrt{{(u-u_0)}^2+v^2}}\right]}^{m_1}{\left[\frac{u+u_0+\mathit{iv}}{\sqrt{{(u+u_0)}^2+v^2}}\right]}^{m_2}$. Here, the magnitude of F(u, v) is always equal to unity, different from the above function M(u, v), and Eq. 1 contains two pure-phase vortices like $e^{{\mathit{im}}_1{\mathrm{\varphi }}_1}$ and $e^{{\mathit{im}}_2{\mathrm{\varphi }}_2}$ with ϕ₁ and ϕ₂ being the two local azimuthal angles at $(\pm u_0,0)$. Such fields are called PPVPBs. In Fig. 1A, it shows the phase profiles of such PPVPBs for several situations with different m₁ and m₂. One can define the initial dimensionless relative off-axis distance ũ₀ = u₀/w₀ as an indicator of external control on the interaction of vortex pair. It is noteworthy to reiterate that PPVPBs comprise two off-axis vortices with pure phase and without amplitude modulation. While this may appear similar to the aforementioned CAVPBs, it fundamentally differs from them. By examining the mode purities of PPVPBs and comparing them with CAVPBs, we ascertain that the orbital angular momentum (OAM) spectra of PPVPBs and CAVPBs are inherently distinct (refer to section A of the Supplementary Materials). When m₁ = m₂, the OAM spectra of PPVPBs consist of infinite even OAM modes, whereas for CAVPBs, their OAM spectra comprise finite even OAM modes. Conversely, when m₁ = −m₂, both PPVPBs and CAVPBs exhibit symmetrical OAM modes, with PPVPBs having an infinite set and CAVPBs having a finite set. These distinctions contribute to varied interactions between vortex-vortex and vortex-antivortex in PPVPBs, resulting in distinct behaviors of vortex dynamics.

Fig. 1. Schematic diagrams of phase distributions of PPVPBs and experimental setup.

(A) The initial phase distributions of PPVPBs with the TC values m₁ and m₂ marked on the top of subfigures. The phase singularities circled by green and yellow arrows represent positive and negative vortices, respectively. Other parameters here are ũ₀ = 0.4 and w₀ = 1.63 mm. (B) Experimental setup for generating PPVPBs and measuring the positions of phase singularities by the interference method. The position of z = 0 is the generating plane of PPVPBs, which is also called the input (or initial) plane for the subsequent optical system. Inset in (B) represents a specific focusing lens system with the lens’ position located at z = f from the input plane. Using this focusing system, at the back focal position of the lens (i.e., z = 2f here), the system becomes a 2-f lens system and optical properties at z = 2f here is similar to the situation of the far-field or Fraunhofer region in free space. HWP, half–wave plate; PBS, polarized beam splitter; BE, beam expander; BS, beam splitter; SLM, spatial light modulator; L₁ and L₂, the focusing lens with f₁ = f₂ = 300 mm; AP, aperture; MR, mirror reflector; BLK, block; RAPM, right-angle prism mirror. Here, the RAPM is movable for realizing the change of the propagation distance z by using the electrically controlled motorized system.

The evolutions of optical fields in free space or an optical system can be well predicted by using theory of matrix optics (65, 66) and a detailed description of theoretical equations can be found in Materials and Methods. Once the field evolution is achieved, the locations of vortex centers can be determined either from the phase distributions or by identifying the dark cores through taking the logarithm of the light intensities, offering an intuitive display. The comparative movies between PPVPBs and CAVPBs are available in section B of the Supplementary Materials. In the case of PPVPBs, their intensity distributions result in the formation of ripples in the light fields, akin to ripples on water caused by two falling stones, highlighting the interaction among vortices. In contrast, CAVPBs exhibit more stable and tranquil evolutions of intensity distributions during propagation, devoid of such ripples. We attribute these pronounced differences between PPVPBs and CAVPBs to the distinct dynamics of vortices, which could be seen from the vortex trajectories. We also show that the role of the host beam in PPVPBs is less important than that in CAVPBs (see the detailed discussion in section I of the Supplementary Materials).

Experimental setup

To demonstrate the vortex dynamics, we experimentally generated PPVPBs by using a phase-only SLM (Holoeye PLUTO-2-NIR-015). Figure 1B depicts the schematic of our experimental setup designed to produce PPVPBs and detect vortex locations in free space using the interference method. We use a linearly polarized He-Ne laser with a wavelength of 632.8 nm as the light source. The half–wave plate and the polarized beam splitter are used to control the horizontal polarization of the transmission light and adjust its light intensity. The beam was then expanded via a beam expander, increasing the beam waist (w₀) to approximately 1.63 mm. Subsequently, the beam underwent splitting by a beam splitter, with the reflected light serving as a reference beam for interference experiments, and the transmitted light is incident on the SLM to generate various orders of PPVPBs. Phase diagrams, as illustrated in Fig. 1A, were loaded onto the SLM. The modulated first-order diffraction beam, representing the generated PPVPB, was isolated using a suitable aperture. A 4-f lens system, composed of lenses L₁ and L₂ with focal lengths f₁ = f₂ = 300 mm, imaged the SLM plane onto the back focal plane of lens L₂. Consequently, PPVPBs were created on the rear focal plane of lens L₂, establishing the initial plane at z = 0 for studying the evolution of light fields in the subsequent optical system. A right-angle prism mirror, positioned on a motorized system, precisely adjusted the propagation distance (z) of the PPVPBs. Last, the interference patterns between PPVPBs and the reference beam were captured by a camera with 12-bit depth. Experimentally obtained fringe patterns facilitated the reconstruction of PPVPB phase distributions using the Fourier transform method (67). On the basis of this information, the vortex locations of the PPVPBs were determined through the application of the phase singularity search algorithm (68), leveraging the high-frequency characteristics of phase singularities. The methods to obtain the information of phase distributions and singularities are also introduced in Materials and Methods.

Dynamics of vortices in free space

Now, let us discuss the dynamics of vortex pair in the fields of PPVPBs. Figure 2 shows the trajectories of vortices for PPVPBs with m₁ = m₂ = 1 and m₁ = −m₂ = 1 in free space. When m₁ = m₂ = 1, the two positive vortices rotate individually, gradually repelling each other. Their trajectories exhibit central symmetry about the origin of the transverse plane; see their projection on the xy plane. This centrosymmetric characteristic is independent of the propagation distance z as shown in Fig. 2A and holds for all PPVPBs with equal TCs (i.e., m₁ = m₂). In the case of m₁ = −m₂ = 1, both positive and negative vortices rotate themselves during propagation, but their trajectories exhibit symmetry about the y axis. This reflectionally symmetric property remains unchanged across different propagation distance (z) as depicted in Fig. 2B. It is valid for all PPVPBs with opposite TCs (i.e., m₁ = −m₂). These symmetries align with the symmetry properties of the initial phase distributions of such PPVPBs, which are symmetric about the origin or the y axis, as displayed in Fig. 1A with m₁ = m₂ = 1 or Fig. 1A with m₁ = −m₂ = 1.

Fig. 2. Experimental measurements of vortex trajectories and intervortex distance for PPVPBs in free space.

(A and B) Experimental vortex trajectories for (A) m₁ = m₂ = 1 and (B) m₁ = −m₂ = 1 with ũ₀ = 0.4. The blue and red dots denote, respectively, the evolution of positive and negative vortices. The corresponding solid lines are theoretical predictions and their projections are shown by the green lines in the xy planes. (C and D) Evolution of the intervortex distance along the propagation distance for (C) m₁ = m₂ = 1 and (D) m₁ = −m₂ = 1, respectively, under different ũ₀. The corresponding theoretical predications are also displayed with the same-color curves. The experimental parameter w₀ = 1.63 mm is taken for theoretical calculations.

The evolution of each vortex within PPVPBs manifests more intriguing effects than those observed in CAVPBs. As depicted in Fig. 2 (A and B), the trajectory of each vortex in PPVPBs follows a helicoidal motion in free space. Simultaneously, the interplay among vortices induces oscillating changes in the intervortex distance, as evident in both experimental and theoretical results presented in Fig. 2 (C and D). This unique evolutionary pattern, involving simultaneous rotation and oscillation, resembles a dance and represents an interesting characteristic that has never previously observed in CAVPBs with linear polarization. The relative off-axis distance (ũ₀) plays a crucial role in the observed vortex oscillation phenomena. In Fig. 2 (C and D), it is noted that vortex oscillation persists for a longer propagation distance as ũ₀ increases. It is noteworthy that when ũ₀ is smaller than a certain value, vortices with opposite TCs can mutually annihilate each other after propagating a specific distance (see Fig. 2D). The critical value of ũ₀ for the annihilation feature in PPVPBs differs from that in CAVPBs. More information on the theoretical prediction on the intervortex distance can be found in section D of the Supplementary Materials, and further instances of vortex annihilation phenomena in PPVPBs with opposite TC vortices can be found in movies S1 and S2. A quantitative comparison between PPVPBs and CAVPBs will be addressed in subsequent discussions.

Figure 3 further presents both experimental and theoretical trajectories of vortices in the fields of PPVPBs with equal TCs (m₁ = m₂ = 2) and opposite TCs (m₁ = −m₂ = 2) in free space. As shown in Fig. 1A, the PPVPB with m₁ = m₂ = 2 showcases a central symmetry of two vortices bearing +2 TCs each, which progressively split into four distinct vortices with individual TCs of +1 during propagation. From Fig. 3A, an interesting interplay among vortices emerges, entwining them in a helicoidal dance as the propagation distance z extends. Although no oppositely signed vortices are present in the initial light field with m₁ = m₂ = 2, the evolving z engenders the generation of multiple pairs of positive and negative vortices, engaging in a mesmerizing alternation of intertwining, nucleation, and annihilation.

Fig. 3. Experimental and theoretical trajectories of vortices in PPVPBs propagating in free space.

(A) The vortex trajectories for the PPVPB with equal TCs of m₁ = m₂ = 2 and (B) the vortex trajectories for the PPVPB with opposite TCs of m₁ = −m₂ = 2. Here, the relative off-axis distance parameter is taken as ũ₀ = 0.4. The blue and red dots denote, respectively, data for the trajectories of positive and negative vortices, and their projections in the xy plane are presented by the green dots. The experimental parameter w₀ = 1.63 mm is also taken for theoretical calculations.

For PPVPBs with opposite TCs (m₁ = −m₂ = 2), as depicted in Fig. 3B, even when a pair of vortices with ±2 TCs is initially present on the plane (refer to Fig. 1A), these vortices gracefully split into two pairs, one carrying +1 TCs and the other −1 TCs. Evidently, vortices boasting high-order TCs in PPVPBs exhibit instability, consistently fragmenting into vortices with +1 or −1 TC. Analogous to the phenomena observed in PPVPBs with equal TCs (m₁ = m₂ = 2), the entanglement, helical dynamics, and the intriguing nucleation and annihilation of vortices are also observed in PPVPBs with opposite TCs (m₁ = −m₂ = 2).

On the basis of the theory of optical hydrodynamics in the recent studies (62–64), the motion of vortices, the splitting of higher-charge optical vortices, and the dynamics of vortices have been explained in complex-amplitude modulation vortex pairs, in which no helicoidal, intertwined, and oscillating vortex dynamics have been observed. Here, we use the same argument to explain the vortex dynamics, especially in PPVPBs. The initial total light field consisting of vortex pairs embedded in a host Gaussian beam can be written as a product of two fields: one for the initial tested field of one vortex under consideration and another for the initial background for the rest field that comprises the fields of other vortices and the host beam. Figure 4 shows the snapshots of the right-side tested vortices moving continuously along the direction of the background velocity field. The complete evolutions of the transverse velocity fields of the background field at any propagation distance z are demonstrated in movies S3 and S4. From Fig. 4 and the supplementary movies, in the PPVPBs with m₁ = m₂ = 1 or m₁ = −m₂ = 1, the tested vortex “surfs” in the diffraction waves from the other vortices in the background fields that induce the helical and oscillating motions (that explains the trajectories in Fig. 2). In the PPVPBs with higher-order TCs, the presence of the circulation flow from the vortex near the tested one further alters the local background velocity field. Under the diffraction ripples of the background field during propagation, the tested vortices experience not only the helical and oscillating motions but also the vortex nucleation and annihilation phenomena. In contrast, there are no complex velocity flows for CAVPBs as shown in fig. S6 (lacking the diffraction waves from other vortices); thus, there are no helical and oscillating motions of vortices. More information on the theoretical consideration of the optical hydrodynamic model and a detailed discussion on movies S3 and S4 can be found in section C of the Supplementary Materials.

Fig. 4. Snapshots of right-side tested vortices moving in the background velocity fields of PPVPBs under different propagation distances z in free space.

(A) m₁ = m₂ = 1, (B) m₁ = −m₂ = 1, (C) m₁ = m₂ = 2, and (D) m₁ = −m₂ = 2. The cyan and yellow dots denote, respectively, the locations of positive and negative vortices. The hollow and solid dots represent the previous and current vortex locations, respectively. The red arrows denote the velocity fields of the background fields. In (C) and (D), the velocity field in the empty area is not shown for better displaying the velocity field in other area because it is divergent near the right-side vortex contained in the background field. Brightness is the light intensity of background fields. All the intensities are normalized. The parameters are ũ₀ = 0.4 and w₀ = 1.63 mm. Note that we only take some snapshots from movies S3 (A and B) and S4 (A and B) for explaining the helical motions of vortices in PPVPBs, and all snapshots here are partial enlargements.

From the above, the demonstrated helical and intertwined behaviors among vortex pairs not only induce the oscillation of the intervortex distance but also change the dynamics of vortex interactions like prolonging the survival range of opposite vortices as discussed later. The characteristics of the dynamic behaviors within these PPVPBs intensify with the augmentation of the relative off-axis distance (ũ₀). We can imagine that when ũ₀ is large, the tested vortex is immersed for more “time” in the diffracted waves of other vortices and oscillates in a spiral form. Sections E and F of the Supplementary Materials provide further results, confirming the influential role of ũ₀ in regulating vortex behaviors. Consequently, the relative off-axis distance (ũ₀) emerges as a pivotal control parameter for modulating the dynamic behaviors exhibited by these PPVPBs.

Dynamics of vortices in a focusing system

It is widely recognized that the light field in far-field regions closely resembles that found at the back focal plane of a 2-f lens system (50). Therefore, to thoroughly explore the annihilation process of vortices for the PPVPBs in the far field, it is more convenient to examine the evolution within the 2-f lens system illustrated in the inset of Fig. 1B. Through changing the value of z in this focusing system, one can achieve the key dynamics of vortex pairs from the near-field region (i.e., the lens plane) to the far-field region (i.e., the focal plane). The only difference is that the beam profile of light in free space is spread out or divergent while it becomes condensed or convergent in the focusing system.

In Fig. 5A, the influence of the relative off-axis distance ũ₀ on the trajectories of vortices, when m₁ = −m₂ = 1, is depicted as they evolve from the lens (z = 500 mm) to the back focal plane (z = 2f = 1000 mm). When the value of ũ₀ is sufficiently large, such as when ũ₀ = 0.55, the positive and negative vortices undergo oscillations and persist at the back focal plane. This observation suggests that they do not annihilate each other in free space. Conversely, when ũ₀ = 0.358, the positive and negative vortex pair coincidentally merge at the focal plane, signifying their annihilation in the infinity of free space. This particular value is termed the critical value for the occurrence of vortex annihilation in PPVPBs when m₁ = −m₂ = 1. As ũ₀ decreases further, the annihilation phenomenon happens before the focal plane, indicating that this effect can be observed at a suitable distance in free space. In section H of the Supplementary Materials, we also provide the corresponding change of the intervortex distance of the vortex pair in free space. Notably, the critical value of ũ₀ in PPVPBs is much smaller than that in cases of CAVPBs with m₁ = −m₂ = 1. This distinction implies that the dynamic properties of opposite vortices in PPVPBs during evolution surpass those of CAVPBs under identical conditions. From a propagation perspective, owing to the oscillatory or dancing behaviors between opposite vortices in PPVPBs, the annihilation process becomes much slower, allowing vortices to survive over longer distances. Consequently, a slight increase in ũ₀ results in the disappearance of the annihilation process compared to that in CAVPBs. This effect in PPVPBs bears similarity to the Fujiwhara effect between two typhoons, which often prolongs the life span of typhoons (69).

Fig. 5. Evolution of vortex trajectories for different-order PPVPBs with opposite TCs in the 2-f focusing system.

(A) The influence of the relative off-axis distance ũ₀ on the vortex-trajectory evolution, where the opposite vortices happen to merge and annihilate each other at the focusing plane when ũ₀ = 0.358. (B to D) The vortex-trajectory evolutions at various critical values of ũ₀ for different-order PPVPBs, in which each plot corresponds to the situation that one pair of opposite vortices annihilate each other at the focusing plane. Here, the focal length of the 2-f focusing system is f = 500 mm and the beam parameter is also taken to be w₀ = 1.63 mm.

For m₁ = −m₂ = 2, the initial light field processes a pair of opposite vortices with ±2 TCs, but due to the unstable properties of high-order vortex pair during propagation, they rapidly split into two pairs of opposite vortices with ±1 TCs; thus, there are complex dynamics as shown in Fig. 3B. In Fig. 5B, two critical situations for vortex annihilation at the focal plane are shown. When ũ₀ = 0.569, one pair undergoes annihilation at the focal plane, while the other pair survives. When ũ₀ further reduces to ũ₀ = 0.218, the second pair also undergoes annihilation at the focal plane. At this situation, the first pair of vortices actually annihilate each other at a shorter distance or earlier. Note that the phenomenon of vortex dancing here appears before the lens plane. Thus, for cases when m₁ = −m₂ = 2, there are two critical values of ũ₀, each corresponding to the critical points of annihilation processes for the respective pairs of vortices.

Similarly, as demonstrated in Fig. 5 (C and D), for the instances of m₁ = −m₂ = 3 and m₁ = −m₂ = 4, three and four pairs of opposing vortices with ±1 TCs are observed, respectively. Each scenario is associated with a distinct critical value corresponding to the annihilation of a vortex pair at the focal plane. In addition, within Fig. 5, the focal fields of PPVPBs reveal the oscillation and dancing of vortex trajectories, featuring vortex intertwining and helical behaviors, phenomena hitherto unobserved in CAVPBs.

Table 1 enumerates critical values of ũ₀ for observing the annihilation effect of each pair of opposite vortices with ±1 TCs within PPVPBs at the focal plane in the cases of m₁ = −m₂. For comparison, corresponding critical values for observing annihilation effects in CAVPBs at the focal plane are also included. Each critical value of ũ₀ for PPVPBs is smaller than the corresponding critical value for CAVPBs. In other words, under identical conditions (for example, with the same value of ũ₀ and beam parameters), the annihilation effect occurs at a greater distance for PPVPBs than for CAVPBs. This delay is attributed to the vortex oscillation and dancing effects, which prolong the annihilation process of vortex pairs. Additional details regarding the annihilation processes of vortices in CAVPBs with opposite TCs in a 2-f lens system are provided in section G of the Supplementary Materials.

Table 1. Comparison of critical values of the relative off-axis distance ũ₀ between PPVPBs and CAVPBs under different-order opposite TCs for occurring vortex-pair annihilation at the focal plane of a 2-f lens system.

	±1 TCs	±2 TCs		±3 TCs			±4 TCs
PPVPBs	0.358	0.569	0.218	0.681	0.404	0.155	0.734	0.529	0.313	0.119
CAVPBs	0.500	0.925	0.383	1.254	0.758	0.323	1.533	1.066	0.661	0.285

DISCUSSION

Our study unveils the dynamic behaviors of PPVPBs in both free space and the focusing system. For PPVPBs featuring unit vortices, vortex trajectories form helical structures, accompanied by oscillating intervortex distances between vortices. Our experimental results are well demonstrated and confirm the theoretical predictions. In the case of PPVPBs with high-order vortices, the initial high-order vortices undergo a dynamic process, splitting into multiple unit vortices during propagation. This evolution is marked by intricate vortex intertwining and helical behaviors including vortex nucleation and annihilation. Such fast helical and intertwining behaviors resemble a dance of vortex pairs and are very common in the fields of PPVPBs. The light fields of PPVPBs with high-order TCs exhibit the nucleation, evolution, and annihilation of positive and negative vortex pairs during propagation. The vortex intertwined, and the helical dance of vortices evoke a visual effect, resembling multiple pairs of vortex dancing. The observed vortex dynamics are explained physically from the hydrodynamics of light fluids. Notably, the intervortex distance between the vortex pair at the initial plane serves as a control parameter for orchestrating vortex dancing. This vortex dancing, in turn, emerges as the primary interaction driving the prolongation of the annihilation process for vortices with opposite TCs. These results underscore the distinctiveness of vortex dynamics in PPVPBs compared to CAVPBs. Our findings offer deeper insights into vortex interactions and hold potential applications in optical micromanipulation and the transportation of optical vortex information.

MATERIALS AND METHODS

Evolutions of optical fields in paraxial systems

Here, we use theory of light diffraction in the paraxial approximation. The evolution of PPVPBs through a linear ABCD optical system, such as free space or a lens system, can be theoretically predicted from the Collins formula (65, 66)

$$\begin{array}{c}{E}_{\mathrm{o}}(x,y,z)=\frac{\text{exp}(\mathit{ikz})}{\mathrm{i\lambda B}}\iint E_{\mathrm{i}}(u,v)\text{exp}\hfill \\ \left\{\frac{\mathit{ik}}{2B}[A(u^2+v^2)-2(\mathit{xu}+\mathit{yv})+D(x^2+y^2)]\right\}\mathrm{d}u\mathrm{d}v\hfill \end{array}$$ (2)

where A, B, and D denote the elements of the ray transfer matrix $\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$ for a linear optical system, $k=2\mathrm{\pi }/\mathrm{\lambda }$ is the wave number, λ is the wavelength, and z is the propagation distance along the propagation axis. The light field of PPVPBs at the output plane (i.e., the observation plane) is obtained by substituting Eq. 1 into Eq. 2. In free space propagation, the ray transfer matrix is represented as (52) $\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)=\left(\begin{array}{cc}1& z\\ 0& 1\end{array}\right)$, where z denotes the propagation distance in free space. For the beam propagation in a 2-f lens system, the ray transfer matrix is expressed as (66) $\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)=\left[\begin{array}{cc}1-(z-f)/f& f\\ -1/f& 0\end{array}\right]$, with f denoting the focal length of the 2-f lens system, z denoting the propagation distance from the input to the output plane, and the lens located at z = f. In the 2-f lens system, it is well known that Eq. 2 becomes the two-dimensional (2D) Fourier transformation, because both A and D are equal to zero and B = f at the back focal position of the 2-f lens system. This 2D Fourier transformation is similar to the Fraunhofer diffraction equation of light at the far-field region of free space (50). The purpose of using the 2-f lens system here is to conveniently investigate the far-field behavior. To visually represent the intensity evolution of the light fields, Eq. 2 provides a theoretical basis.

Methods of achieving the phase distributions and phase singularities

Accurate positioning of vortices in PPVPBs requires the phase information of optical fields. According to the Collins formula (i.e., Eq. 2) and the ray transfer matrix, one can theoretically obtain the phase distributions of PPVPBs in free space or the 2-f lens system. On that basis, one can attain the theoretical locations of vortices for the PPVPBs during propagation by using the vortex location algorithm in (68), which is based on the high-frequency characteristics of vortices. Thus, theoretical vortex positions can be achieved by using the matrix-optics theory and the vortex search approach. In experiments, accurate measurement of vortices is more complex, because we can only directly record the intensity distribution of the light field, rather than the phase distribution. Although it is possible to determine the vortex location through the dark region of light intensity, the accuracy of this method is relatively low compared to the vortex search algorithm based on phase information (68). To accurately locate the position of vortices in the experiment, we should first attain the experimental phase information of the light field. To reconstruct the experimental phase distribution of PPVPBs, we can use phase recovery methods (67). The principle of the phase recovery method in (67) is mainly based on the Fourier transform of interference patterns. On the basis of the reconstructed phase distributions and the vortex search algorithm in (68), we can obtain the experimental vortex locations of PPVPBs.

Supplementary Materials

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Supplementary Text

Figs. S1 to S19

Legends for movies S1 to S4

References

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Movies S1 to S4