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Abstract
Tornado waves (ToWs), which refer to a light that accelerates and twists over both the radial and the angular directions, have gained a great deal of interest since the concept was introduced by Brimis et al [Opt. Lett. 45, 280 (2020) [CrossRef] ]. In this paper, we superimpose two pairs of ring swallowtail vortex beams (RSVBs) to generate ToWs and we call them tornado swallowtail waves (ToSWs). Each pair consists of RSVBs while carrying orbital angular momentum of opposite helicity and slightly different with the radius of the main ring of RSVBs. The waves spiral forward and reveal intensity maxima, exhibiting a tornado-like intensity profile during propagation. Meanwhile, the angular acceleration of the ToSWs is illustrated via tracing the angular position of the high-intensity main lobes. It is found that ToSWs present very high values of angular acceleration. Compared with typical tornado waves, ToSWs are more diverse and tunable, giving a new degree of freedom to tailor the propagation dynamics due to the flexibility of the swallowtail diffraction catastrophe. In addition, we confirm such waves experimentally and the results match well with the numerical ones. Also, we demonstrate the ability of optical manipulation of ToSWs for the first time in that they allow for particles not only to be trapped but also to be rotated. Finally, we analyze the poynting vectors and power exchange of ToSWs to demonstrate convincingly the physical mechanism.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Caustics in light are concrete exhibition of diffraction catastrophes [1]. Generally, the diffraction catastrophe causes caustics to exist as points, lines, surfaces and hypersurfaces [2]. According to the dimensionality of their control parameter spaces, seven fundamental catastrophes have been theoretically established, i.e., fold, cusp, swallowtail, butterfly, elliptical umbilic, hyperbolic umbilic, and parabolic umbilic [3,4]. The most notable representative of optical catastrophe is the fold catastrophe, referring to Airy beams in optics, which were first investigated in 2007 [5,6]. Succeeding to Airy beams, Pearcey beams, corresponding to the cusp catastrophe, were observed their unique properties of autofocusing and self-healing in 2012 [7]. Based on the two elemental beams, many derivative beams have been proposed, including symmetric Airy (Pearcey) beams [8–10], ring Airy (Pearcey) beams [11–14] and these beams carrying orbital angular momentum (OAM) [15–19]. Such structured light beams based on optical catastrophe are topical due to their many applications, for instance in particle clearing [20] and in optical imaging [21]. However, up to now, many studies are restricted to the low-order catastrophe as Airy and Pearcey beams but less pay attention to high-oder catastrophe, that is, swallowtail beams. Swallowtail beams evolve into low-order Pearcey beams in specified conditions during propagation [22–24]. Due to the flexibility of the high-order diffraction catastrophe, these swallowtail beams are more diverse and tunable.
In recent decades, shaping structured light that twists has gained a great deal of interest. Their unique properties that rotate and twist during propagation have seen them applied in a lot of fields ranging from optical trapping [25], nonlinear optics [26] and laser micromachining [27]. Lately, Schulze et al. first realized a new type of twisting light that angularly accelerates via a superposition of two Bessel vortex beams with opposite helicities [28]. These beams’ angular acceleration or deceleration can be easily controlled by adjusting a single parameter. In a like manner, the radially dependent angular accelerating light is demonstrated using Laguerre–Gaussian beams [29], which is localized around the waist plane. On the other hand, the concepts of Tornado Waves (ToWs) were first proposed by Brimis et al., which refers to a light accelerates over both the radial and the angular directions [30]. They choose ring-Airy beams (RABs) as their bases and superimpose them similar to Refs. [28,29]. Since RABs follow parabolic trajectories and tend to abruptly auto-focus [11,12], the superimposed waves evolve intensity maxima and outline a spiral profile as they propagate, which like a tornado. These tornado waves were experimentally realized recently [31].
In addition, another type of cylindrically symmetric accelerating beams, namely, ring swallowtail beams (RSBs) were verified both theoretically and experimentally in 2020 [32]. The peak intensity of RSBs sharply increases , exceeding 100 times at foci than those at the initial plane when choosing appropriate parameters. While this work provides a typical RSBs only, and RSBs carrying OAM have never been investigated. What’s more, with the similarities of RABs and RSBs, will RSBs be able to generate the ToWs successfully? If they will, can they rotate faster than the proposed ToWs? Hence, in this paper, we embed optical vortices into RSBs termed as ring swallowtail vortex beams (RSVBs) and superimpose two pairs of them to generate ToWs. Each pair consists of RSVBs while carrying OAM of opposite helicity and slightly different with the radius of the main ring of RSVBs. For the sake of convenience, we call these tornado waves which are generated via RSVBs as tornado swallowtail waves (ToSWs). We illustrate the propagation dynamics of the superimposed complex waves numerically and confirm them experimentally. In addition, we also discover the angular acceleration via tracing the angular position of the high-intensity main lobes. The influences of the parameters, including the radius of the main ring, the shift element of the swallowtail integral and the topological charge are investigated in detail. Moreover, we validate ToSWs can be applied in optical manipulating that they allow for particles not only to be trapped but also to be rotated. Finally, the analyses of poynting vectors and power exchange are employed to demonstrate convincingly the physical mechanism.
2. Theoretical background and analysis
The key to generating ToWs is to create anisotropic vortex fields [28]. Approaches for accomplishing this task are divided into two categories. The first one relies on the non-linear azimuthal vortex phase, which results in a non-constant angular velocity and also an angular acceleration [28]. In this paper, we adopt another kind of strategy, i.e., combining two radially symmetric accelerating beams with linear azimuthal vortex phase and making two beams slightly different [30]. In this manner, the wave vectors of superimposed beams vary along the propagation since these beams follow the parabolic trajectories, thus creating anisotropic vortex fields. Based on the above, we generate ToSWs by superimposing two RSVBs that carry OAM of opposite helicity and slightly different. Such beams at the initial plane can be rewritten in terms of the electric field as:
withIn the paraxial optical system, the propagation of such beams is governed by the paraxial wave equation [33]:
To make our presentation more complete, we first discuss the general dynamics of RSVBs since the propagation properties of the RSVBs are still unexplored. We set the radius of the main ring is 0.8mm and the topological charge equals 1. From the side-view and the relative intensity curve shown in Fig. 1(a), we observe that RSVBs follow a curved trajectory and inherently autofocus while forming a dark hollow channel due to the optical vortex. The transverse intensity patterns of RSVBs are shown in Figs. 1(b1)-(b4), which indicate that the beams autofocus toward the beam center and the intensity increases abruptly right before the focus. Additionally, the beams seem like to muti-focus after the foci due to the oscillation of the swallowtail integral, thus exhibiting a bottle-like structure along $z$-direction.
Fig. 1. Propagation of the typical RSVBs in the free space with $r_1 = 0.8$mm, $l = 1$, $\chi = 0$. (a) Numerical side-view, the yellow solid curve is about the relative maximum intensity along z-axis corresponding to the intensity distribution; (b1)-b4) numerical intensity distributions at different planes, which are marked by the dashed lines in (a).
Figure 2 illustrate the intensity evolution of the representative ToSWs with $r_1 = 0.8$mm, $r_2 = 0.7$mm, $l_1 = -l_2 = 1$, $\chi = 0$. Since ${\psi _1}$ and ${\psi _2}$ follow a different curved trajectory, opposite helical wavefront and abruptly autofocus individually, the superimposed waves will rotate and shrink both the radial and the angular dimensions, as shown in Fig. 2(a). In addition, the phase of each beam is also illustrated in Fig. 2(a), we can clearly identify the anisotropic vortex fields of superimposed beams. The tridimensional intensity plots of the ToSWs are presented in Fig. 2(b). We can clearly discover that two waves spiral forward and reveal intensity maxima at the "waist". Then, the waves begin to spread while the two main lobes outline tunnel shape from 3D-view. The intensity evolution patterns exhibit a spiral profile from the top view of the intensity plots, just like a tornado. For a more detailed discussion, we also plot the normalized transverse intensity patterns at different propagation planes in Figs. 2(c1)-(c8). Clearly, the beams have a circularly symmetric input pattern [Fig. 2(c1)] which consists of two relative high-intensity main rings and the maximum intensity locate at the outer ring. During the propagation, the rings spin slowly with a clockwise rotation and with two spiral side lobes and converge slightly toward the center initially [Figs. 2(c2)-(c3)]. When the beams propagate up to the focal plane [Fig. 2(c4)], the peak intensity increases more than 30 times and the beams begin to rotate abruptly. Moreover, as shown in Figs. 2(c4)-(c8), the two high intensity main lobes not only rotate but also decrease in intensity while the side lobes become conspicuous and oscillates radially, forming a series of spiral patterns with slightly shrinking width.
Fig. 2. Intensity evolution of the ToSWs propagating in free space with $r_1 = 0.8$mm, $r_2 = 0.7$mm, $l_1 = -l_2 = 1$, $\chi = 0$. (a) Visualization of the process for generating ToSWs, the red and blue twisting curves represent the trajectory of two high-intensity lobes for ToSWs; (b) 3D intensity plots of the ToSWs; (c) Normalized transverse intensity profile at different propagation distances.
As mentioned above, ToSWs angularly accelerate during propagation. To evaluate the rotation behaviour of the waves, we have marked the angular position $\tau (z)$ of the high-intensity main lobes during propagation via analyzing the transverse intensity profile [30]. We only focus on the region where the beams rotate fastest, namely, from 195mm to 239mm. We set clockwise as the positive direction of angular velocity. As plotted in Fig. 3(a), the rotation angle is monotonically increasing originally while beginning to shrink at the plane $z = 228.6$mm, which indicates that beams rotate clockwise at first and change to anticlockwise during propagation. Approximately 100 degrees that the beams rotate in the region. According to these data, the angular velocity can be estimated by $v = d \tau /dz$. As illustrated in Fig. 3(b), the angular velocity also demonstrates that the direction of rotation for the beams will change. Since the velocity is not a constant, the rotation of the beams is nonlinear, exhibiting peak value of 7.5 degrees/mm. By the same token, we can also predict the angular acceleration by $\gamma = {d ^2}\tau /d {z^2}$, as seen in Fig. 3(c). We can see the values of $\gamma$ exceed 2 degrees/mm$^2$ in the region, which indicates that ToSWs present very high values of angular acceleration. Noticed that this comparison is relative and may not entirely accurate since the values of angular acceleration are based on the data of references.
Fig. 3. Rotation behaviour of the ToSWs. (a) Rotation angle of the beams for different propagation distance; (b) angular velocity for different propagation distance; (c) angular acceleration for different propagation distance. All the parameters are the same as those in Fig. 2
Next, we concentrate on the effects of the topological charge for ToSWs. The cross-sectional intensity profile at representative planes along with the 3D intensity plots of ToSWs for diverse combinations of topological charges are shown in Fig. 4. The input profiles [Fig. 4(a2), Fig. 4(b2) and Fig. 4(c2)], are split into $M$(equals to $\left | {{l_1}} \right |$ + $\left | {{l_2}} \right |$) high-intensity main lobes and optical vortices, which are placed symmetrically. Then, the main lobes rotate and twist with $M$ spiraling sidelobes, propagating in a tornado-like fashion. However, as we increase the value of $M$, the intensity distribution of cross section will become dispersed. Figure 4(d) presents the normalized peak intensity curves for different topological charges as a function of propagation distance $z$. It is apparent that the maximum intensity will get smaller and the focal length will shift slightly with the increasing of $M$. Additionally, the total angular rotation under exact propagation distance is given by 211 degrees, 162 degrees, 108 degrees corresponding to $(l_1 = 1, l_2 = -1)$, $(l_1 = 2, l_2 = -1)$ and $(l_1 = 2, l_2 = -2)$, respectively. Thus, the total angular rotation also decreases for higher topological charges. Moreover, the transverse structure will be complex if the values of $M$ increase. By taking $l_1 = 3, l_2 = -3$ [Fig. 4(c5)] as an example, we can tell the complex pattern of secondary even tertiary lobes except for the main lobes. To sum up, we can flexibly regulate the behaviors of ToSWs including beam shape, angular rotation and focusing properties by tuning the combinations of topological charges.
Fig. 4. Intensity evolution of the ToSWs propagating in free space for different combinations of topological charges. (a1)-(a5) $l_1 = 2, l_2 = -1$; (b1)-(b5) $l_1 = 2, l_2 = -2$; (c1)-(c5) $l_1 = 3, l_2 = -3$; (d)normalized peak intensity curves for different topological charges as a function of propagation distance $z$.
Except for the topological charges, the parameters $r_i$, $\chi$ also have a great impact on the propagation of ToSWs. When we set $\chi = 0$ [the blue line in Fig. 5(a1)], there are two peak intensity along the propagation. However, if we take $\chi$ larger [the purple and the green lines in Fig. 5(a1)], the foci will move backward and the intensity of the first peak will increase while that of the second peak will decrease. On the contrary, if we take the situation of $\chi < 0$ [the black and the red lines in Fig. 5(a1)], the focal length becomes smaller while the intensity of the peak reduces and even appears to muti-focus when $\chi = -4$. The cross-sectional intensity pattern at the source plane will also be affected by the factor $\chi$, as shown in Figs. 5(a2)-(a5). We notice that when $\chi \ne 0$, the high intensity main lobes are no longer located at $y$-axis as Fig. 2(c1), but having an initial angle between the coordinate axis. Besides, if we magnify the absolute value of $\chi$, the transverse pattern presents some spiral side lobes though no propagation. Nevertheless, $\chi$ will not change the rotation behavior of ToSWs since it is the shift element for the Swallowtail integral [9] according to Eq. (3). In a similar way, we also uncover the influences of the parameters $r_i$ to ToSWs. Without loss of generality, we make $r_1=0.80$mm and turn the value of $r_2$. As described in Fig. 5(b), the factor $r_i$ will also vary the focusing properties both focal length and focusing intensity of the ToSws. Meanwhile, if the gap between the two main rings is two small, the beam will not rotate as they propagate. For the limiting case of $r_1 = r_2$, the beam will degrade into petal-like beams as Refs. [35,36] which only accelerate in radial direction. It deserves to be mentioned that the scaling factor $a$ will also determine the properties of the beams, which follows the scaling law [30]. All in all, compared with typical tornado waves [30], ToSWs are more diverse and tunable, giving a new degree of freedom $\chi$ to tailor the propagation dynamics of ToSWs due to the flexibility of the swallowtail diffraction catastrophe.
Fig. 5. Influences of the parameters $\chi$ and $r_i$ to the propagation of ToSWs. (a1) Normalized peak intensity curves for different $\chi$ as a function of propagation distance $z$; (a2)-(a5) cross-sectional intensity pattern at source plane for different $z$; (b) normalized peak intensity curves for different topological charges as a function of propagation distance $z$.
3. Experimental results
The schematic diagram of the experimental setup for generating ToSWs and optical trapping system is depicted in Fig. 6. A linearly polarized Gaussian beam is formed by a solid-stated laser (Surisetech, model MGL-V-532) with a wavelength $\lambda = 532$nm. A half-wave plate (HWP) and a Gran prism (GP) are utilized to control the power of the beam and to match the polarization requirements of the spatial light modulator (SLM). The beam will be expanded and collimated after the beam expander consisting of two thin lenses ($f_1 = 50$mm, $f_2 = 250$mm). Then, the expanded beam will be reflected by the phase-only SLM (Santec SLM-200, 1920$\times$1080 pixels, 8 $\mathrm{\mu}$m) uploaded with predesigned holograms that contain the formation of the proposed beams. These desired holograms loading SLM can be generated by the interference between the ToSWs and a plane wave [9,18]:
Fig. 6. Scheme of the experimental setup. HWP, half-wave plate; GP, Gran prism; CA, circular aperture; BS, beam splitter; SLM, spatial light modulator; DM, dichroic mirror; L1-L4, lenses; M1-M5, mirrors; BQA, beam quality analyzer.
The corresponding experimental results demonstrate the generation of the typical ToSWs in free space as shown in Fig. 7. All the parameters are the same as those in Fig. 2. We find that ToSWs will indeed rotate and exhibit a tornado-like intensity pattern in free space. Comparing the measured cross-sectional intensity profile, it is apparent to find that the experimental results are in good agreement with the numerical simulation shown in Fig. 2. To further inspect the properties of the ToSWs, we illustrate the measurements for different combinations of topological charges. The number of the total main lobes $M$ shown in Figs. 8(b1)-(b5) is coupled to topological charges for each wave, namely, $M = \left | {{l_1}} \right | + \left | {{l_2}} \right |$, which demonstrate convincingly the theoretical predictions. The corresponding phase masks of these beams are depicted in Figs. 8(a1)-(a5), respectively. It is worth noting that beams generated by the aforementioned method may distort during the propagation and suffer from a lower diffraction efficiency. Expect for the above-mentioned method, we can also utilize two SLMs to modulate the intensity and wavefront distributions of the beams to raise diffraction efficiency [42].
Fig. 7. Experimental generation of typical ToSWs. (a1) The phase mask of the ToSWs; (a2)-(a8) experimental snapshots of transverse intensity patterns corresponding to the Fig. 2.
Fig. 8. Phase masks and experimental snapshots of transverse intensity patterns with helicities (a1-a2) $l_1 = 2, l_2 = -1$; (b1-b2) $l_1 = 2, l_2 = -2$; (c1-c2) $l_1 = 3, l_2 = -2$; (d1-d2) $l_1 = 3, l_2 = -3$; (e1-e2) $l_1 = 4, l_2 = -3$.
As indicated in Refs. [28–30], such an twisting light with angular acceleration is an effective tool in optical trapping. Here, we demonstrate that the ToSWs can be applied into an optical tweezer system to manipulate particles. The sample consists of $4\mathrm{\mu}$m silica particles in water solution, which is placed on a three-dimensional adjustable platform. The image plane of the objective can be easily changed by simply moving the delay system. The trapping process of the typical ToSWs with 2 main lobes is clearly displayed in Figs. 9(a1)-(a8). The particles are moving disorderly initially. Under the action of gradient forces exerted by the transverse intensity pattern, the two particles are gradually subjected to the position of the two high-intensity main lobes. To probe whether the two particles are trapped stably, we slowly move the sample in $x-y$ plane. We choose a particle which is marked by a dashed square in Figs. 9(a5)-(a8) as a reference to the two trapped particles. It is evident that other particles can move around independently, which indicates that the two particles are trapped steadily. As mentioned above, the main lobes of ToSWs will spin as they propagate. This unique property allows for particles not only to be trapped but also to be rotated. Experimentally, we reduce the beam width to gain a higher speed of rotation and slightly move the sample along the longitudinal direction to verify the conjecture. As depicted in Figs. 9(b1)-(b3), the two trapped particles rotate clockwise slowly as we move the sample away from the image plane ($z_0$) of the objective. Since the distance of the sample and the objective increase, the image will be more and more blurry. Meanwhile, we also describe the trapping ability of ToSWs which have three lobes or four lobes in Fig. 10. We utilize the 100$\times$ oiled objective in this case to make the trapping process more clear. We observe that three particles or four particles can be trapped when the beams have three lobes or four lobes, respectively. In theory, the number of trapped particles can be adjusted by changing the number of main lobes. However, if we increase the number of main lobes, the average intensity of each lobe will decrease and the efficiency of optical trapping will also decline. In a word, these experimental results validate the optical manipulating ability including stably trapping and rotating of the ToSWs.
Fig. 9. Experimental demonstrations of optical manipulation with ToSWs. (a1)-(a4) trapping process of ToSWs with two main lobes; (a5)-(a8) demonstration of stable trapping. Dashed circles denote the particles which are trapped stably and the dashed square is the reference for the trapped particles. (b1-b5) Realization of the rotation of particles. $z_0$ represents the image plane of the objective and $\Delta z$ = 1mm.
4. Physical mechanism of ToSWs
One may legitimately ask: What physical mechanism enables such structured light to exhibit angular acceleration along propagation? Instinctively, these behaviors can be explained by investigating the local energy flow of the beams, which often refers to the poynting vectors. For a linear polarized optical field, the time-averaged poynting vector can be defined as:
Fig. 11. The Poynting vector, OAM and power exchange of ToSWs. (a) Poynting vectors at different planes. Red arrows represent the direction of the energy flow, and the length of the arrows represents the magnitude of the energy flow density; (b) OAM of the beams at different planes. Red arrows represent the direction of the OAM, and the length of the arrows represents the magnitude of the OAM flow density; (c) normalized power as a function of the propagation distance of the beams. All the other parameters are the same as those in Fig. 2.
Besides, we also observe an interesting phenomenon that the power will exchange between the main lobes and side lobes. Actually, the whole optical field of ToSWs can be divided into two individual structures which perform independent rotations [28–30]. For convenience, we regard the twisting high-intensity main lobes as a ToSWs petal region while the rest of the beam which has a lower average intensity, refers to as a ToSWs tail region. To further discover the feature of power exchange, we depict the normalized power as a function of the propagation distance as shown in Fig. 11(b), which can be calculated by:
5. Conclusion
In summary, we introduce a new class of tornado swallowtail waves which are generated by superimposing two pairs of ring swallowtail vortex beams. Each pair carries OAM of opposite helicity and slightly different with the radius of the main ring of RSVBs. The waves spiral forward and reveal intensity maxima at the "waist", exhibiting tornado-like intensity profile. Meanwhile, we also discover the angular acceleration of such waves via tracing the angular position of the high-intensity main lobes. It is found that ToSWs present very high values of angular acceleration. Moreover. the intensity pattern, focusing properties and rotating behaviours can be flexibly adjusted by varying the value of parameters $l_i$, $\chi$ and $r_i$. Compared with typical tornado waves, ToSWs are more diverse and tunable, giving a new degree of freedom $\chi$ to tailor the propagation dynamics of ToSWs due to the flexibility of the swallowtail diffraction catastrophe. In addition, we confirm such waves experimentally and the experimental results are in complete accord with numerical ones. Besides, we demonstrate the ability of optical trapping and manipulation of ToSWs for the first time. Finally, we analyze the poynting vectors and power exchange of ToSWs to demonstrate convincingly the physical mechanism. Such an angular acceleration is related to the energy exchange between the two regions of ToSWs.
Funding
Special Funds for the Cultivation of Guangdong College Students Scientific and Technological Innovation (“Climbing Program” Special Funds, pdjh2022a0129); Program of Innovation and Entrepreneurship for Undergraduates; Science and Technology Program of Guangzhou (2019050001); National Natural Science Foundation of China (11374108, 11775083, 12174122).
Disclosures
We declare that we have no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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