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Abstract
In this study, a modified interfering vortex phase mask (MIVPM) is proposed to generate a new type of self-rotating beam. The MIVPM is based on a conventional and stretched vortex phase for generating a self-rotating beam that rotates continuously with increasing propagation distances. A combined phase mask can produce multi-rotating array beams with controllable sub-region number. The combination method of this phase was analyzed in detail. This study proves that this self-rotating array beam has an effectively enhanced central lobe and reduced side lobe owing to adding a vortex phase mask compared with a conventional self-rotating beam. Furthermore, the propagation dynamics of this beam can be modulated by varying the topological charge and constant a. With an increase in the topological charge, the area crossed by the peak beam intensity along the propagation axis increases. Meanwhile, the novel self-rotating beam is used for optical manipulation under phase gradient force. The proposed self-rotating array beam has potential applications in optical manipulation and spatial localization.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Spatial light modulation and shaping can produce several types of novel beams, such as spatiotemporal optical vortices [1], structured ray-wave structured light [2], and accelerating beams [3,4]. Accelerating beams have recently attracted increasing research interest owing to their unique properties and potential applications in optical manipulation [3,4], optical bullets [5], vacuum electron acceleration [6,7], and laser processing [8]. The non-spreading Airy wave packet, predicted by Berry et al. in 1979, is a typical accelerated beam [9]. The Airy beam was first observed experimentally in 2007 [10], and its propagation path is parabolic, described by the Airy function [11,12]. Meanwhile, researchers have interpreted the self-healing property based on the internal transverse power flow [13]. This self-accelerating property is highly important because the beam can continue to propagate circumventing obstacles, which is convenient for optical communication, optical therapy, and particle manipulation. Researchers have further studied the self-accelerating properties of Airy beams and proposed control methods regarding its ballistic motion by enabling a phase plate or Gaussian beam to deviate from the optical axis [14]; however, its propagation trajectory is always a fixed parabola. Other accelerating beam types are non-paraxial Mathieu and Weber beams obtained by solving the Helmholtz equation [15], whose propagation trajectories are elliptic and parabolic, respectively. Greenfield et al. proposed non-broadening optical beams propagating along any arbitrarily chosen convex trajectory in space [16]. Froehly et al. generated arbitrary convex accelerating beams via directly applying an appropriate spatial phase profile to an incident Gaussian beam [17]. Pan et al. designed a new class of self-accelerating structured light whose wave packets exhibit a three-dimensional (3D) inhomogeneous angular velocity evolution [18]. However, the propagation trajectories of the aforementioned accelerating beams are convex in one direction, such as parabolas and arcs. Researchers have proposed several types of rotating beams by rotating a plate glass, employing moiré techniques, and employing chiral materials [19–21]. These beams cannot rotate automatically. Zhao et al. have introduced a new class of twisted partially coherent sources [22–24]. The intensities of these beams are symmetrical and the rotation angle is 90° at most. Deng et al. also proposed a rotating beam called the Airy complex variable function beam, with a maximum rotation angle of 90° [25]. The cosine-Guass beam exhabits a periodic oscillation trajectory with propagation distance [26]. In practice, beams propagating along complex paths are more generic and have many potential applications. Therefore, accelerating beams with different propagation trajectories require more detailed investigation. Recently, Tao et al. proposed a class of self-rotating beams whose intensity profile tended to self-rotate and self-bend in free-space propagation [27]. Unfortunately, the propagation trajectory of the rotating beam was constant. Vortex beams with different orbital angular momenta exhibit circular varying intensity distributions [28,29]. The radius of the circular intensity increased with the topological charge.
In this study, we design a modified interfering vortex phase mask (MIVPM) to produce a class of accelerating beams with different rotational bending trajectories during propagation. The MIVPM is constructed using a conventional and stretched vortex phase. Compared with the self-rotating beam produced by the stretched vortex phase [27], the accelerating beams generated by the MIVPM exhibit a smaller beam tail and main lobe with higher intensity. The larger the topological charge is, the larger the area crossed by the peak beam intensity becomes. The sign of a determines the rotational direction of the accelerated beam along the propagation axis. When we vary the topological charges, the sign of a, and MIVPM at rotation angles, the peak beam intensity can be delivered to any desired location, circumventing an obstacle placed in the path of the self-rotating beam. Meanwhile, a multi-rotating array beam can also be produced by a combined phase mask and the method of generating the self-rotating array beam was explained in detail. The proposed self-rotating beam with variable acceleration trajectory is applicable in the development of optical manipulation, spatial localization, and optical communication.
2. Results and discussion
The traditional vortex phase distribution is $l \cdot \theta $. Here, we introduce the power term of the radial coordinate into the vortex phase. Multiplying the power term of the radial coordinate and vortex phase yields a stretched vortex shape, which can transform the Gaussian beam into a self-rotating beam [19]. In the polar coordinate system, the phase function $\psi $ of the MIVPM with the stretched vortex phase and vortex phase can be expressed as:
Fig. 1. Phase distributions used to generate the (a) self-rotating beam, (b) vortex beam, and (c) MIVPM with ($a = 6 \times {10^6},b = 2,l = 20$), respectively. The phase profiles φ4, φ5, φ6, and φ7 can be obtained using the same method.
The phase distributions of the MIVPM and modulation intensity distributions with different topological charges were investigated in detail, the results of which are illustrated in Fig. 2. Figures 2 (a1)–(f1) illustrate the phase distributions with topological charges l = 0, 20, 40, 60, 80, and 100, respectively. When the phase profiles in Figs. 2 (a1)–(f1) are placed on the beam propagation path as a lens, the intensity results in the far field were obtained via the Fourier transform, which are shown in Figs. 2 (a2)–(f2), respectively. As the topological charge increases, the intensity tail marked with the yellow arrow in Fig. (a2) decreases, and the position of the main lobe gradually moves away from the central area, as indicated by the yellow dashed lines in Figs. 2 (a2)–(f2).
Fig. 2. Phase profiles of MIVPM with l = (a1) 0, (b1) 20, (c1) 40, (d1) 60, (e1) 80, and (f1) 100. The far-field intensity profiles (a2)–(f2) calculated by the Fourier transform of the phase distributions presented in (a1)–(f1).
Based on the angular spectrum theory of plane waves, the spatial distribution of the optical field of a self-rotating beam can be easily simulated and analyzed [30] based on the following equation:
where E, H, FT, and iFT represent the complex amplitudes of the diffracted patterns, transfer function, and forward and inverse Fourier transforms, respectively. The transfer function can be expressed as:The respective maximum intensity was the normalized object in the simulation. After passing through the MIVPM ${\psi _3}$ lens, a Gaussian beam is modulated to the intensity profiles that rotate clockwise along the z-axis, as illustrated in Figs. 3 (a1)–(d1). The intensity profiles depicted in Figs. 3 (a1)–(d1) are located at propagation distances of 1.2, 2.2, 3.2, and 4 m, respectively. Observe that the asymmetric intensity profile rotated clockwise as the propagation distance increased. Figure 3 (e1) illustrates the continuous intensity profiles of phase ${\psi _3}$ modulation that are calculated at every 10 cm between the axial positions 1.2 and 4 m on one diagram. Because the beam intensity at different axial positions is not the same, a continuous intensity arc can be observed, which represents the accelerating trajectory in 3D space. After passing through the MIVPM ${\psi _4}$ lens, a Gaussian beam is modulated to the intensity profiles that rotate clockwise along the z-axis, as illustrated in Figs. 3 (a2)–(d2). The intensity profiles depicted in Figs. 3 (a1)–(d1) are located at propagation distances of 1.2, 2.2, 3.2, and 4 m, respectively. Figure 3(e2) illustrates the continuous intensity profiles of phase ${\psi _4}$ modulation that are calculated at every 10 cm between the axial positions 1.2 and 4 m on one diagram. Similarly, Figs. 3 (a3)–(e3), (a4)–(e4), and (a5)–(e5) illustrate the intensity profiles modulated by the phases ${\psi _5}$, ${\psi _6}$, and ${\psi _7}$ presented in Fig. 1, respectively. With an increase in topological charge, the side lobes of the self-rotating beam gradually decrease. From the two-dimensional (2D) display of the 3D trajectory depicted in Figs. 3 (e1)–(e5), observe that the accelerating trajectories of the peak intensity gradually change from inside to outside with an increase in topological charge. Meanwhile, the negative constant a results in a counter-clockwise rotation direction in accelerating trajectory of the self-rotating beam. The brightest points represent peak intensities in Fig. 3. When we change the topological charges and sign of a, and rotate the MIVPM at different angles or move the MIVPM center position, the peak beam intensity can be delivered to any desired location, circumventing an obstacle placed in the path of the self-rotating beam.
Fig. 3. The calculated intensity profiles modulated by phase φ3 depicted in Fig. 1(c) at the propagation distances of (a1) 1.2, (b1) 2.2, (c1) 3.2, and (d1) 4 m. (e1) The continuous intensity profiles modulated by phase φ3 measured at every 10 cm between the axial positions 1.2 and 4 m on one diagram. The calculated intensity distributions modulated by the phase φ4 depicted in Fig. 1 at the propagation distances of (a2) 1.2, (b2) 2.2, (c2) 3.2, and (d2) 4 m. (e2) The continuous intensity profiles modulated by phase φ3 measured at every 10 cm between the axial positions 1.2 and 4 m on one diagram. Similarly, the calculated intensity distributions modulated by the phases φ5, φ6, and φ7 depicted in Fig. 1 occur at propagation distances of (a3)–(a5) 1.2, (b3)–(b5) 2.2, (b3)–(b5) 3.2, and (d3)–(d5) 4 m, respectively. (e3)–(e5) The continuous intensity profiles modulated by the phase φ3 measured at every 10 cm between the axial positions 1.2 and 4 m on one diagram.
To observe the five acceleration trajectories shown in Fig. 3, the trajectories with specific position coordinates are plotted in Fig. 4. Lines AB, CD, EF, GH, and IJ represent the acceleration trajectories at l = 20, 40, 60, 80, and 100, respectively. The three red insets on curve IJ represent the optical intensity profiles of the x-y section at three different propagation positions. The starting and ending points of the beam are indicated by blue numbers in Cartesian coordinates. The five beams propagated from the positive Y-axis to the negative Y-axis along a curved trajectory following five different arc-acceleration curves. The topological charge number determines the off-axis radius of the curved trajectory. The physical reasons for such behavior can also be understood from Fig. 2. Figure 4(b) shows the transverse energy flow when the beam propagation distance was 3.5 m. The magnitude and direction of the green arrows indicate the counterpart energy flow in the transverse plane.
Fig. 4. (a) The five acceleration trajectories versus axial propagation distance z. The lines AB, CD, EF, GH, and IJ represent the acceleration trajectories with l = 20, 40, 60, 80, and 100, respectively. The blue numbers in the parentheses represent the peak beam intensity positions in Cartesian coordinate. (b) The transverse energy flows of the two array self-rotating beams at the distances of z = 3.5m
To demonstrate the controllability of the method to a self-rotating beam, we simultaneously controlled the two beams to rotate counterclockwise with different transportation trajectories, as illustrated in Fig. 5. Figure 5 (a) shows the continuous intensity profiles modulated by phase φ with a = $- 6 \times {10^6}$ and l = -20 at axial positions between 1.2 and 4 m on one diagram. The modulated continuous intensity profiles when the topological charges are set to -40, -60, -80, and -100, the modulated continuous intensity profiles are depicted in Figs. 5 (b)–(e), respectively. Yellow arrows represent the acceleration trajectory directions. With an increase in topological charges, the acceleration trajectory of the peak beam intensity gradually increases. Therefore, we can control the bending trajectory and direction according to the magnitude of the topological charges and sign of constant a. As depicted in Figs. 3 (e1)–(e5), a single self-rotating beam propagates continuously from the positive Y-axis to the negative Y-axis along the arc-acceleration trajectory. Two self-rotating beams can also be modulated simultaneously using a combinatorial solution, as shown in Fig. 5. When the constant a described in Eq. (1) is set to a positive value and topological charge of the vortex phase is set to 20, 40, 60, 80, and 100, the resulting intensity distributions modulated by the MIVPM with the propagation distance are illustrated in Figs. 5 (f)–(j), respectively. Note that in Fig. 5, the cross-section light intensity profile is calculated at every 10 cm over a propagation distance range of 1.2–4 m; the simulated beam intensities are depicted in the figure as well. With an increase in the propagation distance, the beam gradually shifted from the center to the edge position along the curved trajectory. The topological charge number determines the range of curved trajectories. Similarly, multiple self-rotating beams can be modulated simultaneously using the same combination method.
Fig. 5. The continuous intensity profiles modulated by the MIVPM measured at every 10 cm over a propagation distance range of 1.2–4 m with constant a =$- 6 \times {10^6}$ and topological charge number l = (a) -20, (b) -40, (c) -60, (d) -80, and (e) -100 on one diagram, respectively. Similarly, the continuous intensity profiles with constant a =$6 \times {10^6}$ and the topological charge number l = (f) 20, (g) 40, (h) 60, (i) 80, and (j) 100 on one diagram, respectively. The yellow arrows represent the acceleration trajectory directions.
In this work, we proposed a new kind of modulation method to generate an array self-rotating beam along the propagation axis. The phase combination method for generating self-accelerating array structured light is illustrated in Fig. 6. In Eq. (1), ${\psi _1}$, ${\psi _2}$ and $\psi $ are used to draw the phase diagram, and only the upper half is shown in Figs. 6(a1-c1), respectively. Figure 6(c1) is only the upper part of Fig. 1(c), but it can also generate a single self-accelerating beam. Figure 6(a2) is the same as Fig. 6(c1), and Fig. 6(b2) is an inversion of Fig. 6(a2). Figure 6(c2) is the phase profile for generating the two self-accelerating beams. Meanwhile, Fig. 6(a3) is obtained by cropping Fig. 6(a2) and rotated by ±120° to obtain Fig. 6(b3) and Fig. 6(c3), respectively. Figure 6(d3) is the phase profile for generating the three self-accelerating beams. Figure 6(c2) is the same as Fig. 6(a4), and rotated by ±120° to obtain Fig. 6(b4). Figure 6(c4) is the phase profile for generating the four self-accelerating beams.
Fig. 6. Phase distributions used to generate the (a) self-rotating beam, (b) vortex beam, and (c) MIVPM with ($a = 6 \times {10^6},b = 2,l = 20$), respectively. (a2) is the same as (c1), and (b2) is an inversion of (a2). (c2) The phase profile for generating the two self-accelerating beams. (a3) The phase profile obtained by cropping (a2) and rotated by ±120° to obtain phases (b3) and (c3), respectively. (d3) The phase profile for generating the three self-accelerating beams. (c2) is the same as (a4), and rotated by ±120° to obtain (b4). (c4) The phase profile for generating the four self-accelerating beams.
The combined phase profiles in Fig. 6 are used to modulate the optical field. Figures 7(a1) and (a2), consistent with Figs. 6 (c1) and (c2), is used to generate a single and two self-rotating beams. After passing through the combined phase illustrated in Figs. 7(a1) and (a2), a Gaussian beam is modulated to the intensity profiles that rotate clockwise along the z-axis, as illustrated in Figs. 7 (b1)–(f1) and (b2)–(f2). The intensity profiles depicted in Figs. 7(b1)–(f1) and (b2)–(f2) are located at propagation distances of 1.2, 1.5, 2.2, 3.2, and 4 m, respectively. Figure 7(a3), consistent with Fig. 6 (d3), is used to generate the three array self-rotating beams. After passing through the combined phase illustrated in Fig. 7(a3), a Gaussian beam is modulated to the intensity profiles that rotate clockwise along the z-axis, as illustrated in Figs. 7(b3)–(f3). The intensity profiles depicted in Figs. 7(b3)–(f3) are located at propagation distances of 1.9, 2.5, 3.0, 3.5, and 4 m, respectively. Meanwhile, After passing through the combined phase illustrated in Fig. 7(a4), a Gaussian beam is modulated to the intensity profiles that rotate clockwise along the z-axis, as illustrated in Figs. 7(b4)–(f4). The intensity profiles depicted in Figs. 7(b4)–(f4) are located at propagation distances of 0.8, 1.0, 1.2, 1.5, and 2 m, respectively.
Fig. 7. Phase distributions for generating a single (a1), two (a2), three (a3), and four (a4) array self-rotating beams. The calculated self-rotating intensity profiles modulated by phase φ depicted in (a1) and (a2) at the propagation distances of (b1), (b2) 1.2 (c1), (c2) 1.5, (d1), (d2) 2.2, (e1), (e2) 3.2, and (f1), (f2) 4 m. The calculated three self-rotating intensity profiles modulated by phase φ depicted in (a3) at the propagation distances of (b3) 1.9 (c3) 2.5, (d3) 3.0, (e3) 3.5, and (f3) 4 m. The calculated four self-rotating intensity profiles modulated by phase φ depicted in (a4) at the propagation distances of (b4) 0.8 (c4) 1.0, (d4) 1.2, (e4) 1.5, and (f4) 2 m.
Since the combined phase has a vortex phase factor, the spiral intensity track with phase gradient can be seen in the modulated light field illustrated in Figs. 8(a2)–(d2). Figures 8(a1)–(d1) show the same combined phases as Figs. 7(a1)–(a4). After passing through the combined phase illustrated in Figs. 8(a1)–(a4), a Gaussian beam is modulated to the intensity profiles illustrated in Figs. 8(a2)–(d2). The intensity profiles depicted in Figs. 8(a2)–(d2) are located at propagation distances of 0.9, 0.9, 1.3, and 0.7 m from the phase, respectively. When parameter a and topological charge l are set to negative, the phase profiles are shown in Figs. (a3)–(d3). The corresponding modulated intensity profiles are illustrated in Figs. (a4)–(d4), respectively.
Fig. 8. Phase distributions with a = 6 × 106, l = 60 for generating a single (a1), two (b1), three (c1), and four (d1) array self-rotating beams. The calculated intensity profiles modulated by phase (a1)–(d1) at the propagation distances of (a2) 0.9, (b2) 0.9, (c2) 1.3, and (d2) 0.7 m, respectively. Phase distributions with a = -6 × 106, l = -60 for generating a single (a3), two (b3), three (c3), and four (d3) array self-rotating beams. The calculated intensity profiles modulated by phase (a3)–(d3) at the propagation distances of (a4) 0.9, (b4) 0.9, (c4) 1.3, and (d4) 0.7 m, respectively.
An optical micro-manipulation platform was built to optical manipulation via rotating beams. The key components are presented in Fig. 9. The MM-grade Gaussian spot emitted from the laser (Changchun New Industries, Semiconductor lasers, 520 nm) was covered on the liquid crystal panel of a spatial light modulator (SLM, XI'AN CAS MICROSTAR SCIENCE AND TECHNOLOGY, reflective type) after passing through a beam-expanding system composed of f1 (f1 = 30 mm) and f2 (f2 = 300 mm). The structured light modulated by the SLM must pass through a beam-shrinking system (f3 = 300 mm, f4 = 100 mm) to meet the objective pupil diameter. The half-wave plate (HWP) is the polarization modulator. The inset (the image of the particles with 5 µm) represents the image plane.
Fig. 9. Schematic of the optical platform used in optical micro-manipulation. The scale bar represents 10 µm.
In the experiment, the phases illustrated in Figs. 8(a1)–(d1) and (a3)–(d3) are loaded into SLM, and the optical beams reflected by SLM are modulated into the arc intensity profiles illustrated in Figs. 10(a1)–(d1) and (a2)–(d2), respectively. The arc direction and the number of arc-curves can be set at will. Due to the vortex phase, the arc-curve in the intensity profiles possess a phase gradient. The direction of phase gradients are along the curve direction.
Fig. 10. The intensity profiles with a clockwise curved phase gradients (a1)–(d1) modulated by the phase profiles Fig. 8(a1)–(d1), respectively. The intensity profiles with a counterclockwise arc phase gradients (a2)–(d2) modulated by the phase profiles Fig. 8(a3)–(d3), respectively.
Similar to the vortex beam driving particles, the modulated arc-curve with phase gradient can also be used to transport particles. Figure 11 depicts the continuous frames of the axial capture and transport of particles along the arc-curve trajectories. When phase with l = 20, 60, and 100 depicted in Fig. 8(a1) were loaded onto the SLM, micro-particles were continuously captured and manipulated along arc-curve trajectories in the solution pool, as shown in (a1)–(e1), (a2)–(e2), and (a3)–(e3), respectively. In the simulation, the larger the topological load is, the farther the spiral trajectory is from the center. The simulation can be verified by the optical tweezers experiment, as shown in Fig. 10. The red arrows indicate the locations of the trapped particles. The yellow dotted lines and blue rectangles represent the central areas of the CCD field of view. From the three sets of video screenshots, namely, (a1)–(e1), (a2)–(e2), and (a3)–(e3), observe that the spiral trajectory transportation process of particles gradually changing from small radius to large radius. The experimental results of the particle capture and propagation tasks were consistent with the simulation results of the beam acceleration. The white lights depicted in Fig. 11 are the external stray lights captured by the CCD after focusing.
Fig. 11. Continuous frames of axial capture and particle transport along a self-rotating beam acceleration trajectory with (a1)–(e1) a = 6 106, b = 2, l = 20; (a2)–(e2) a = 6 106, b = 2, l = 60; (a3)–(e3) a = 6 106, b = 2, l = 100.
3. Conclusions
In summary, a novel self-rotating beam type with an adjustable accelerating trajectory was proposed. A multi-rotating array beams can also be produced by the combination of phase masks. The method of generating the self-rotating beam was explained in detail, and propagation characteristics of the self-rotating beam were analyzed through simulations and experiments. The propagation dynamics of the peak beam intensity can be modulated by varying the topological charge l and constant a. The beam exhibits rotational bending trajectories during propagation and has an effectively enhanced central lobe and reduced side lobe. Furthermore, the transmission of multiple self-rotating beams with variable trajectories and directions was simultaneously modulated and realized using a combinatorial solution. The proposed modulation method for the propagation trajectory is simple, repeatable, and extensible to constructing more complex beam modes, which have potential applications in optical manipulation and spatial localization.
Funding
Innovation-Driven Project of Central South University (2020CX005); National Natural Science Foundation of China (U20A6004); State Key Laboratory of High Performance Complex Manufacturing (ZZYJKT2019-10).
Disclosures
There are no conflicts of interest related to this article.
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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