1. Introduction

Wave fields containing invariable spatial intensity direction features during propagation have always been the interest of the scientific community. Typical examples of such fields are the Laguerre-Gaussian beams [1], Bessel beams [2], and Airy beams [3]. However, the optical beams with rotating intensity distribution along the propagation have recently attracted much attention due to both their fascinating optical properties and potential applications, such as maximizing axial resolution [4], trapping Bose-Einstein condensates [5], and high-precision spatial localization over large z ranges [6].

Some rotational properties possess relationship with the angular momentum of the beam [7,8]. Kong et al. proposed a kind of optical beams with controllable rotating intensity blades by rotating a plate glass [9]. Researchers have also proposed and demonstrated an approach to the generation of rotating intensity blades by employing moiré techniques [10]. The rotation of interference pattern is also a major scheme of a rotating beam [11]. But these beams can exhibit rotation characteristics only under special conditions, such as moving the grating, superposing two beams with frequency difference and propagating in chiral materials. Some analyses were employed to investigate general solutions with rotating intensity distributions [12]. There are many models about the rotating intensity distributions around or along the propagation axis. A class of self-steering partially coherent beams with a moving guiding center was introduced [13]. Zhao et al. have done a number of researches about rotating beams [14–20]. They introduced a new class of twisted partially coherent sources for producing rotating Gaussian array profiles [14–16]. Analogous twisted beams such as controllable rotating beams [17,18] and the extension of the basic models with twisted structured correlations [19,20] were also reported. The computer-generated holograms and binary-phase diffractive elements were also used to generate rotating beams [21,22] and some novel beams, such as annular beams [23] and helico-conical optical beams [24]. Meanwhile, some beam shaping algorithms were used to form some bending and specific shaped beams in the two-dimensional (2D) or three-dimensional (3D) space, such as the non-iterative beam shaping techniques [25,26] and the complex-amplitude beam shaping algorithms [27,28]. Hu et al. realized the optimal control of the acceleration trajectory of the Airy beam in the 3D space [29]. Some special beams with tunable intensity profiles and peaks, pin-like optical vortex beams, and Bessel-like beams, can be found applications in different areas such as optical communication and quantum information technologies [30,31]. The phase function of the proposed self-rotating beam has helical phase θ, which causes the beam to rotate in the propagation. The vortex beams also exhibit the rotation characteristics although the intensity distribution is symmetric. The spatial rotating beams demonstrate bending behaviors which tend to freely accelerate under the paraxial condition [3,32]. These bent beams can be used for the glass scribing and particle transportation [33,34].

The aim of this paper is to introduce a class of self-rotating beams whose intensity profiles tend to rotate in the free space propagation. The rotating direction and the size of the self-rotating beam are controllable, and a combined self-rotating beam can be generated by combining several phase-only holograms. The self-rotating beam can be applied for optical imaging, optical communication, and spatial localization.

2. Analysis and simulation

The self-rotating beam is evolved from a vortex beam. We start by considering the beam generation. The transmission function $T\left( {r,\theta } \right)$ of a diffractive optical element (DOE) in the polar coordinate system can be expressed in Eq. (1),

 

Fig. 1. Evolution of the phase profiles (a1-l1) and the reconstructed intensity profiles (a2-l2) of the self-rotating beams for a varying constant b. (a1) b=0, (b1) b=0.2, (c1) b=0.4, (d1) b=0.6, (e1) b=0.8, (f1) b=1, (g1) b=1.5, (h1) b=2, (i1) b=2.5, (j1) b=3, (k1) b=3.5, and (l1) b=4.

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Here we describe in detail the controllable self-rotating beam. The constant b is set as 2. When we set the constant a as $\pm 5 \times 10^6$, the DOEs with phase functions $\phi _1$ and $\phi _2$ are plotted in Figs. 2(a) and 2(b), respectively. Other parameters are set as follows, the sampling points Nx = Ny=1080, pixel pitch=8 µm, and the length (equal to width) of the hologram L=1080×pitch. The gray values between the white and the dark correspond to the phase values in the range [0, 2π]. Since they satisfy the symmetry condition of $\phi _1 = -\phi _2$, the two DOEs are reversed images of each other.

 

Fig. 2. The phase distributions of DOEs with (a) positive and (b) negative parameter a used to generate the self-rotating beams. Note that the two DOEs are symmetric to each other in the vertical direction.

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After passing through the DOE shown in Fig. 2(a), a Gaussian beam is shaped into a beam with intensity cross-sections along the z axis shown in Figs. 3(a-h), which show the intensity profiles at the axial positions of 0.38 m, 0.42 m, 0.46 m, 0.5 m, 0.6 m, 0.75 m, 0.95 m, and 1.55 m, respectively. To compare the intensity profiles of these self-rotating beams, we have normalized the intensities with respect to the peak intensity. It can be seen that the beam automatically rotates clockwisely around the optical axis. Similarly, after passing through the DOE shown in Fig. 2(b), the Gaussian beam is shaped into a beam with intensity distributions along the z axis shown in Figs. 3(i-p), respectively. The rotation direction of the beam is counterclockwise. It can be seen from Figs. 3(a-c) and 3(i-k) that, when z < 0.5 m, the beam has not been fully formed, and when z≥0.5 m, the intensity profile is relatively complete. The DOEs in Figs. 2(a) and 2(b) can modulate a Gaussian beam into the clockwise and counterclockwise self-rotating beams in the propagation, respectively. When a>0, the rotation angle versus propagation distance z of the self-rotating beams with b=0.18, 2, and 2.5 is plotted in Fig. 3(q). When a<0, the rotation angle versus propagation distance z of the self-rotating beams with b=0.18, 2, and 2.5 is plotted in Fig. 3(r). With the increase of b in the additional term r^b, where b is greater than 1, the beam is continuously stretched, which leads to asymmetrical intensity distribution. The spiral phase θ causes the beam to rotate. In the simulation, it was found that the rotation was more obvious when the parameter b was equal to 2, and the rotation angle reached 150 degree. The additional term r^b and spiral phase θ are responsible for the beam self-accelerating. Figure 3(s) vividly shows the characteristics of the 3D spatial bending. Figure 3(t) shows the calculated acceleration trajectory versus the beam propagation axis z in the simulation. The four inserted red images and their directions represent the intensity cross-sections and revolving directions of the beam during the propagation. The positions of the beam in the propagation were marked in Fig. 3(t). It is obvious that the beam rotates from the first quadrant to the fourth quadrant in the propagation. The numericals in the parentheses are the positions represented by the coordinates of x, y, and z, respectively.

 

Fig. 3. The simulated intensity distributions reconstructed by the DOE in Fig. 2 (a) at the axial positions of (a) 0.38 m, (b) 0.42 m, (c) 0.46 m, (d) 0.5 m, (e) 0.6 m, (f) 0.75 m, (g) 0.95 m, and (h) 1.55 m, respectively. The simulated intensity distributions reconstructed by the DOE in Fig. 2 (b) at the axial positions of (i) 0.38 m, (j) 0.42 m, (k) 0.46 m, (l) 0.5 m, (m) 0.6 m, (n) 0.75 m, (o) 0.95 m, and (p) 1.55 m, respectively. (q) The rotation angles versus propagation distance z of the self-rotating beams with a>0, b=0.18, 2, and 2.5, respectively. (r) The rotation angle versus propagation distance z of the self-rotating beams with a<0, b=0.18, 2, and 2.5, respectively. (s) The 3D intensity profiles of the self-rotating beams in the propagation. (t) The calculated acceleration trajectory versus the beam propagation axis z. The numericals in the parentheses are the positions represented by the coordinates of x, y, and z, respectively.

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In Fig. 3, we calculated the diffraction intensity profiles at different diffraction distances by the angular spectrum theory of plane waves. However, the influence of different parameters on intensity distribution under the same condition needs to be studied. So, we did a Fourier transform to observe the intensity profile of the far field in Fig. 4. The positive and negative signs of a determine the rotating direction of the beam, and the size of a determines the range of the self-rotating beam tail in Figs. 3 and 4. Thus, we can adjust the size of the beam by changing the constant a. When a is set as 5×10⁶, 8×10⁶, and 15×10⁶, the intensity profiles observed on the far field are shown in Figs. 4(a-c), respectively.

 

Fig. 4. Simulated intensity distributions at the focal plane reconstructed by the DOEs with b=2, (a) a=5×10⁶, (b) a=8×10⁶, and (c) a=15×10⁶, respectively.

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The phase distribution of the Airy beam can be expressed as ${{\phi ^{\prime} = ({{({x / {{s_x}}})}^3} + {{({y / {{s_y}}})}^3})} / 3},$ where ${s_x}$ and ${s_y}$ are the scaling factors of the phase in the x and y directions, respectively. The phase function of the self-rotating beam in the polar coordinates is expressed as $\phi = a \cdot {r^b} \cdot \theta ,$ where r is the radial coordinate, θ is the azimuth angle, a is a parameter to contain the self-rotating beam tail, and b is used to adjust the phase profile of the DOE. By comparing their phase functions, we can find that the both have power terms. So, with some special parameters the intensity profile of the self-rotating beam can be similar to that of the Airy beam.

3. Experimental results

The self-rotating beam can be generated experimentally. The experimental setup involving beam shaping is shown in Fig. 5, which consists of a collimated diode-pumped laser (Coherent, Genesis MX532-1000 STM, and λ=532 nm), a spatial light modulator (SLM, CAS, phase type, 1920×1080 pixels, 8 μm pitch, reflective type), a beam expander with lenses L1 and L2 (f1 = 30 mm and f2 = 300 mm), a beam splitter (BS), and a charge coupled device (CCD). The half wave plate (HWP) adjusts the polarization orientation of the beam incident to the SLM for higher diffraction efficiency.

 

Fig. 5. Schematic of the experimental setup.

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In the experiment, the DOEs shown in Fig. 2 were loaded sequentially on the SLM. The expanded laser beam passed through the SLM screen to form the required beams. The CCD behind the BS was used for the recording of the intensity distributions of the beams, and the distance between the SLM and the CCD can be adjusted. After the collimated and expanded laser beam passed through the DOE of Fig. 2(a), the reconstructed intensity distributions at different axial distances are shown in Figs. 6(a-e). With the increase of the diffraction distance, the intensity profiles rotated clockwisely. Figures 6(f-j) show the intensity distributions, reconstructed by the DOE of Fig. 2(b), in counterclockwise rotation with the increase of the diffraction distance. The graphs show the CCD-captured intensity cross-sections in the transverse direction versus the beam propagation axis z. The dotted line cross-points represent the center positions of the CCD at different propagation distances. As the distance increases, the intensity profiles of the beam rotate and deviate from the propagation direction constantly, which indicates the presence of the acceleration feature of the beam in the experiment. It can be seen that the experimental results are consistent with the simulation results.

 

Fig. 6. (a-e) The CCD-captured intensity profiles, reconstructed by the DOE in Fig. 2(a), at different propagation distances. (f-j) The CCD-captured intensity profiles, reconstructed by the DOE in Fig. 2(b), at different propagation distances. The dotted line cross-points represent the center positions of the CCD at different distances.

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A DOE in the above can reconstruct a single self-rotating beam only. However, a combined DOE can reconstruct multiple self-rotating beams in one imaging plane. The combined DOE shown in Fig. 7(a) is formed by the superposition of phase profiles of different tilting angles. The simulated intensity distribution reconstructed by the combined DOE is shown in Fig. 7(c). In the experiment, when the phase profile with a blazed grating shown in Fig. 7(b) was loaded on the SLM, the reconstructed intensity distribution is shown in Fig. 7(d). Normally, the diffraction pattern and the direct component of an SLM are mixed in the propagation. Thus, in the experiment we added a blazed grating to the hologram to separate the self-rotating beam from the zero-order diffraction component of the SLM. It can be observed that the beam rotates with the diffraction distance from Fig. 7(e). Similarly, two, three or more self-rotating beams coexisting in one diffraction plane can be formed by a combined DOE. When the constant a in Eq. (1) is set to negative, the reconstructed beam rotates counterclockwisely with the increase of diffraction distance. In the optical trapping experiment, the combined beam can be used to manipulate multiple particles at the same time. Meanwhile, due to the rotation characteristics, multiple cells or particles can be manipulated in different planes simultaneously.

 

Fig. 7. The phase profile used to generate the combined self-rotating beam in (a) simulation and (b) experiment, respectively. The combined self-rotating beam at the focal plane of the objective in (c) simulation and (d) experiment, respectively. (e) Schematic of diffraction process simulated by the angular spectrum theory of plane waves.

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4. Conclusions

In summary, the self-rotating beam was proposed in this paper. The corresponding generation method was illustrated and the reconstructed intensity distributions at different propagation distances were studied in the simulations and experiments. The beam continuously rotates in the 3D space along the optical axis. Furthermore, the self-rotating beam presents bending and self-accelerating features in the 3D space. The acceleration dynamics of the self-rotating beams is controllable and multiple self-rotating beams coexisting in one diffraction plane can be generated by a combined DOE. The rotation direction is related to the constant parameter of the DOE.

It is believed that the self-rotating beam with spatial bending may be particularly useful in micro-machining, optical sorting, optical imaging, and directional transportation in lab on a chip where the spatial bending characteristics of beams plays a significant role [4,33,34].