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Abstract
In this study, we demonstrate the self-healing of self-rotating beams with asymmetric intensity profiles. The proposed self-rotating beam exhibits an asymmetric intensity profile and self-healing properties in free-space propagation. In addition, the rotation direction and beam intensity profile of the self-rotating beam can be adjusted using the parameters a and b in the phase function. The effects of the position and size of the obstruction on the self-healing property of a self-rotating beam were studied both experimentally and numerically. The simulation and experimental results demonstrate that a self-rotating beam can overcome a block of obstacles and regenerate itself after a characteristic distance. Transverse energy flows were used to explain the self-healing properties. Moreover, the beam rotates during propagation, which can be used to capture and manipulate microscopic particles in a three-dimensional space. It is expected that these rotating beams with self-healing properties will be useful in penetrating obstacles for optical trapping, transportation, and optical therapy.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In several applications of laser beam technology, such as laser processing [1], optical information processing [2], and medical clinical applications [3], beams with special properties are required. Structured light has the ability to tailor a beam to a particular intensity or phase distribution. with advances in structured light, researchers can crop and modulate beams with different degrees of freedom, such as intensity, phase, polarization, and orbital angular momentum [4–7]. Modulated light fields exhibit some special properties, such as helical wavefront [8–12], self-focusing [13], and self-healing [14], which can be applied in particle trapping [15,16] and micro-fabrication [1]. It is well known that Bessel beams [17] and optical fields generated by fractal zone plates [18,19] exhibit self-healing properties when blocked by an opaque object. A Bessel beam possesses invariable spatial intensity direction features and an intensity-symmetric distribution during propagation. Another typical self-healing beam is the Airy beam, characterized by the radially symmetric or circular Airy function, which exhibits three distinguishing properties: non-diffracting [20], free acceleration [21,22], and self-healing [23]. Babinet’s criterion was used to interpret the self-healing properties of non-diffracting beams [24]. Researchers have further studied the self-healing properties of Airy beams and proposed various airy-like beams with self-healing properties [25–27]. Wang et al. investigated the self-healing properties of focused circular Airy beams [25]. Lu et al. experimentally investigated the self-healing of a heralded single-photon Airy beam [26]. Zhou et al. theoretically analyzed the self-healing properties of cosh-Airy beams using energy flow [27]. Although the Airy beam exhibited self-healing properties, its intensity direction did not change during propagation. Meanwhile, other optical beams also exhibit self-healing properties, including caustic [28], accelerating polygon [29], Pearcey [30], helico-conical optical [31,32] and spiral beams [33]. In recent years, several types of self-rotating beams have been proposed [34–38]. Researchers have generated optical beams with rotating intensity blades by employing the moiré technique [34] or rotating a plate glass [35]. However, these self-rotating beams require external conditions such as moving the gratings. Zhao et al. proposed a new class of twisted partially coherent sources for producing rotating Gaussian array profiles, whose spectral density and degree of coherence tend to rotate during propagation [36,37]. The rotation angle was approximately 90°. Undoubtedly, these results provide new insights into the twist phase and may be applied in optical trapping. A waves consisting of superpositions of different higher-order Bessel waves has an intensity pattern that rotates at a constant rate around the optical axis as it propagates [38]. These self-rotating beams did not exhibit self-healing properties. In addition to the aforementioned beams, there is a large family of spiral vortex beams with self-rotating properties [39,40]. A triangular spiral beam can quickly restore its original shape with rotational propagation if the caustic surface is not destroyed [41]. Subsequently, Volyar et al. studied the symmetric and/or asymmetric perturbation of spiral vortex beams and observed that the beam either restored its former shape or transitioned into a new structurally stable state in a far-diffraction zone [42]. However, the intensity profiles of the aforementioned self-healing beams are symmetric and/or the intensity direction does not rotate during propagation.
In this study, we demonstrate another type of beams with self-healing properties: self-rotating beams. Self-rotating beams exhibit both self-healing and self-rotating properties during propagation when blocked by an opaque object. We experimentally and numerically demonstrate the effect of the position and size of the obstruction on the self-healing properties of self-rotating beams. We observed that the self-healing ability of the side lobes was stronger than that of the main lobe, and the smaller the occlusion area, the stronger the self-healing ability. In addition, self-rotating beams with self-healing properties have been used to manipulate microscopic particles in three-dimensional (3D) layers. Some particles are trapped and transported along a 3D arc trajectory in an aqueous solution. This type of self-healing beam exhibit certain robustness. Self-rotating beams with self-healing properties can be customized and applied to optical tweezers, particle trapping, and optical communication through scattering media.
2. Analysis and simulation
It is well known that the phase distribution of a vortex beam is $\phi = a \cdot \theta $, where a and θ represent the topological charge and azimuth angle, respectively. When multiplying the vortex phase by the power term of the radial coordinate ${r^b}$, the phase distribution is expressed as $\phi = a \cdot {r^b}\theta $. As the value of b increases from 0, the vortex phase is constantly stretched and compressed, and the intensity profile becomes asymmetric, as demonstrated in a previous study [43]. The helical phase θ causes the beam to rotate during propagation. Through the simulation of varying parameter b, it was observed that the rotation was most clear when b was 2, and reached 150° [43]. The asymmetric intensity profiles are very similar to the intensities of the Airy beam, which enlightens us whether the proposed self-rotating beam also exhibits the same self-healing properties as Airy beams. In this study, the energy flow density was determined to explain the self-healing process. In the polar coordinate system, the phase profiles $\psi ({r,\theta } )$ of the self-rotating beam assume the form of the remainder of the phase $\phi $[43] as expressed by the following equation:
where (r, θ), a, b, and rem represent the polar coordinate, containment of the self-rotating beam tail and/or phase distribution of the diffractive optical element (DOE), and the remainder function, respectively. Meanwhile, a negative constant a causes the beam to rotate in the opposite direction during propagation. Herein, the constants a and b were set at ${\pm} 5 \times {10^6}$ and 2, respectively. The corresponding phase profiles are shown in Figs. 1(a) and (b), and are symmetric in the vertical direction. The sampling points were set at Nx = Ny = 1080, and pixel pitch = 8 µm. After a Gaussian beam passes through the DOEs illustrated in Figs. 1(a) and (b), the reconstructed intensity distributions at the far field obtained through Fourier transform are shown in Figs. 1(c) and (d), respectively. Only the central parts with an area of 200 × 200 pixels are shown in Figs. 1(c) and (d).
Fig. 1. Phase distributions of DOEs with (a) positive and (b) negative parameter a used to generate the self-rotating beams. The reconstructed intensity profiles (c) and (d) at the far field by the DOEs illustrated in (a) and (b), respectively. Note that the two DOEs are symmetric to each other in the vertical direction.
The beam emitted from the DOE shown in Fig. 1 is a phase-modulated Gaussian beam. When the Gaussian beam passes through the DOE, the optical field distribution in the z = 0 plane can be expressed as follows:
The optical field at the propagation distance z of the light exiting the DOE can be directly obtained by the diffraction of the angular spectrum of plane waves as follows:
where FT and FT-1 represent the Fourier transform and inverse Fourier transform, respectively. H(x,y,z) represents the transfer function obtained using Eq. (4) [44] as follows:After passing through the DOEs shown in Figs. 1(a) and (b), the beam is modulated into the intensity distribution of the self-rotating beam. As the beam continues to propagate, we add an opaque obstacle at the propagation distance ${d_1}$ from the DOE. When the beam propagates to position ${d_1}$, the optical field $U({x,y,{d_1}} )$ obtained can be expressed by a matrix (Nx = Ny = 1080). The process of adding obstacles can be regarded as adding a filter T to the optical field at the propagation distance ${d_1}$. The filter T can be expressed by Eq. (5) as follows:
Fig. 2. When the main lobe is blocked, the simulated intensity distributions are reconstructed by the DOE illustrated in Fig. 1(a) at the axial positions of (a1) 0.5, (b1) 0.7, (c1) 1.1, and (d1) 1.6 m, respectively. When the main lobe is not blocked, the simulated intensity distributions are reconstructed by the DOE illustrated in Fig. 1(a) at the axial positions of (a2) 0.5, (b2) 0.7, (c2) 1.1, and (d2) 1.6 m, respectively.
We investigated whether self-healing would occur when other parts or more of the beam are blocked. We also simulated the self-reconstruction process of the side lobes of the self-rotating beam. When an obstacle was placed at a propagation distance 0.5 m to block the side lobes of the beam, we observed the reconstruction process of the occluded part of the beam, as illustrated in Figs. 3(b1–d1). With the increase of the propagation distance, the occluded parts of the beam gradually reconstructed and self-healed, as shown in Fig. 3(a1) z = 0.5, (b1) z = 0.55, (c1) z = 0.6, and (d1) z = 0.7 m, respectively. Similarly, when the beam is more blocked or severely perturbed, it exhibits perfect self-healing properties. In the second set of simulations, the side lobes of the beam were extensively blocked by opaque obstacles, as shown in Fig. 3(a2) z = 0.5 m. As the beam continues to propagate, we simulated the intensity profiles at the propagation distances z = 0.55, 0.6 and 0.75 m, as shown in Figs. 3 (b2), (c2), and (d2), respectively. The beam self-heals its fine-intensity structure during the rotating propagation process. It can be observed from the intensity profiles illustrated in Figs. (d1) and (d2) that when the size of the obstacle is small, a small part of the side lobes is blocked, and the self-healing process is fast. In contrast, a large obstacle causes a slow reconstruction. Meanwhile, a large obstacle also causes poor reconstruction, which can be observed from the lack of upper-side lobe intensity of the beam illustrated in Fig. 3(d2).
Fig. 3. When the side lobes are blocked, the simulated intensity distributions are reconstructed by the DOE illustrated in Fig. 1(a) at axial positions of (a1) 0.5, (b1) 0.55, (c1) 0.6, and (d1) 0.7 m, respectively. When the side lobes are extensively blocked, the simulated intensity distributions reconstructed by the DOE illustrated in Fig. 1(a) at the axial positions of (a2) 0.5, (b2) 0.55, (c2) 0.6, and (d2) 0.75 m, respectively. When the side lobes are not blocked, the simulated intensity distributions are reconstructed by the DOE illustrated in Fig. 1(a) at the axial positions of (a3) 0.5, (b3) 0.55, (c3) 0.6, and (d3) 0.7 m, respectively.
3. Experimental results
Our theoretical investigation can now be realized through experiments. The optical setup for creating self-rotating beams using a reflective spatial light modulator (SLM) is shown in Fig. 4. A 532 nm laser with a Gaussian distribution was emitted by a collimated diode-pumped laser (Coherent, Genesis MX532-1000 STM, and λ=532 nm). The beam expanding system composed of lenses f1 and f2 (f1 = 30 mm and f2 = 300 mm) enlarges the laser beam by 10 times. A half wave plate (HWP) was used to modulate the polarization direction of the laser to enable the SLM to work efficiently. A SLM loaded with the phase shown in Fig. 1 can modulate the incident expanded laser beam into a target beam. The beam-splitting cube (BS) is used to deviate the beam from its original direction and into a charge-coupled device (CCD1) for observation. The distance between the BS and CCD1 is adjustable, and opaque obstacles of appropriate size are added to investigate the self-healing effect. The beam shrinking system composed of lenses F3 and F4 (f3 = 300 mm and f4 = 100 mm) reduces the modulated light spot to the same size as the entrance pupil diameter of the objective lens. The laser focused by the oil immersion objective lens (100×, N.A. 1.3) can capture particles. The focused laser trapping is an optical path and the observation optical path is used to observe the captured objects in the solution pool in real time. The solution pool was a sample chamber composed of cover glass slides.
Fig. 4. Schematic of the experimental setup for beam shaping and optical tweezers. The scale bar is 10 µm.
In the first experiment, we blocked the main corner lobe of the intensity profile of the beam at a propagation distance of 0.5 m to observe the self-healing process. Figures 5(a1)–(d1) captured by a CCD1, illustrate the intensity profiles at axial positions of 0.5, 0.7, 1.1, and 1.6 m, respectively. As the propagation distance from the SLM to CCD1 increase, the blocked lobe of the beam was gradually reconstructed while the optical intensity rotated. The intensity profiles illustrated in Figs. 5(a1)–(d1) in the experiment are consistent with the effect illustrated in Figs. 2(a1)–(d1) obtained through theoretical simulation. For comparison, the intensity distributions without an obstacle are shown in Figs. 5(a2)–(d2).
Fig. 5. (a1–d1) CCD1-captured intensity profiles, reconstructed by the DOE illustrated in Fig. 1(a), at different propagation distances when the main corner lobe of intensity profile is blocked. (a2)–(d2) The CCD1-captured intensity profiles without an obstacle, reconstructed by the DOE illustrated in Fig. 1(a), at different propagation distances.
Thereafter, we blocked the side lobes of the intensity profile of the beam at a propagation distance of 0.5 m to observe the self-healing process at different positions, which can be clearly observed in Fig. 6(a1). When the position of the fixed obstacle remained unchanged, the modulated beam continued to propagate in free space to achieve the self-healing process of the side lobes. From the intensity profiles illustrated in Fig. 6(a1) z = 0.5 m to (d1) z = 0.7 m, the occluded parts of the beam gradually reconstructed and self-healed. We can observe that the intensity profiles at the axial positions of 0.55 and 0.6 m shown in Figs. 6(b1) and (c1) in the self-healing process of the beam in the experiment are perfectly consistent with that at the same positions in the simulation shown in the Figs. 3(b1) and (c1), respectively. The intensity profiles of the beam without an obstacle at the same position are shown in Figs. 6(a2)–(d2) for comparison.
Fig. 6. (a1–d1) CCD1-captured intensity profiles, reconstructed by the DOE illustrated in Fig. 1(a), at different propagation distances when the side lobes of intensity profile are blocked. (a2–d2) The CCD1-captured intensity profiles without an obstacle, reconstructed by the DOE illustrated in Fig. 1(a), at different propagation distances.
The Poynting vector [32,45] refers to the energy flow density vector in an electromagnetic field, which can be used to illustrate the self-healing properties of optical beam. The Poynting vector can be expressed as $\vec{S} = ({{c / 4}\pi } )\vec{E} \times \vec{B},$ where c, E, B denote the speed of light, the electric field intensity, and magnetic field intensity, respectively. The direction is determined by E and B according to the right-hand screw rule. According to the Lorenz gauge, the time-average Poynting vector with x-polarized field can be expressed as follows:
Figures 7(a)–(c) show the numerically computed ${\vec{s}_ \bot }$components of the Poynting vector when the beam propagation distance was 0.8 m from the SLM. The magnitude and direction of the yellow arrows indicate the counterpart energy flow in the transverse planes. Figures 7(b) and 7(c) illustrate enlarged views of Fig. 7(a). The energy of the beam flowed from the intensity tail to the main lobe. The direction of the energy flow traces a ladder path (indicated by the yellow arrows in the figure). When the main or side lobes are blocked, energy flows from the lower right corner to the occluded area as the propagation distance z increases, as shown in Fig. 7. The energy flow from the tail to the head of a beam proves that the beam exhibits self-healing properties, as indicated by the energy flow of some confirmed Airy, Bessel and helico-conical beams.
Fig. 7. (a) Transverse energy flows of the self-rotating beam at a distance of z = 0.8 m; (b) and (c) enlarged views of the energy flows illustrated in (a).
Based on the above simulation and experimental study of the self-rotating beam, it was possible to capture and transport particles along the axial bending trajectory. Figure 8 shows that the CCD2-captured continuous frames represent the trapped and transported microparticles along the transmission trajectory of the rotating beam. The blue rectangles and red dotted lines are used to describe the observation position and central area of the objective lens, respectively. It can be observed that the particles are transported continuously from the lower part of the central position to the upper left corner, which is similar to the rotation trajectory of the beam in the simulation [34]. Initially, from the position illustrated in Figs. 8(a) to (b), the distance between the particle and the objective lens was less than the working distance of the objective lens, resulting in unclear images. When the particle is at the position shown in Fig. 8(c), it is at the working distance of the objective lens, and the image is the clearest. When the particles continue to be transported to the upper left corner along the space circular curve track, as shown in Fig. 8(d), the particles are far away from the working distance of the objective lens, and the image becomes blurred. Along the z-direction acceleration curve trajectory of the beam, the captured 5 µm diameter silica micro-spheres underwent a “defocus-focus-defocus” process. The DOE illustrated in Fig. 1(a) can realize the clockwise spatial transport of particles, and the DOE illustrated in Fig. 1(b) can also enable counterclockwisely transport of particles.
Fig. 8. CCD2-captured continuous frames showing trapped and transported micro-particles along the transmission trajectory of the rotating beam. The blue rectangles and red dotted line are used to describe the observation position and the central area of the objective lens, respectively.
4. Conclusions
In conclusion, the self-healing properties of self-rotating beams were verified both theoretically and experimentally. The reconstruction effect of the self-healing beam was affected by the size and position of the opaque obstacles. A large opaque occlusion resulted in poor self-healing of the self-rotating beam. The intensity profiles at different axial positions during the reconstruction process were simulated to demonstrate how the beams healed as they propagated. The transverse energy flow also provided an explanation for the self-healing mechanism of the self-rotating beams. The spatial arc acceleration trajectory of the self-rotating beam was used for micro-manipulation, and the particles were captured and transmitted along the axial arc trajectory in an aqueous solution. The self-rotating beams with self-healing properties can play a significant role in biomedical laser therapy penetrating biological tissue and other research fields in the future.
Funding
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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