Annular beam with segmented phase gradients

Annular beam with segmented phase gradients Shubo Cheng, Liang Wu, and Shaohua Tao Citation: AIP Advances 6, 085322 (2016); doi: 10.1063/1.4962301 View online: http://dx.doi.org/10.1063/1.4962301 View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/6/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Constructive spin-orbital angular momentum coupling can twist materials to create spiral structures in optical vortex illumination Appl. Phys. Lett. 108, 051108 (2016); 10.1063/1.4941023 Single beam optical vortex tweezers with tunable orbital angular momentum Appl. Phys. Lett. 104, 231110 (2014); 10.1063/1.4882418 Optical fiber antenna generating spiral beam shapes Appl. Phys. Lett. 104, 031105 (2014); 10.1063/1.4862884 Photopolymerized microscopic vortex beam generators: Precise delivery of optical orbital angular momentum Appl. Phys. Lett. 97, 211108 (2010); 10.1063/1.3517519 Influence of geometric shape of optically trapped particles on the optical rotation induced by vortex beams J. Appl. Phys. 100, 043105 (2006); 10.1063/1.2260823

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AIP ADVANCES 6, 085322 (2016)

Annular beam with segmented phase gradients Shubo Cheng,1 Liang Wu,1 and Shaohua Tao1,2,a 1

School of Physics and Electronics, Central South University, Changsha, Hunan 410083, China 2 Hunan Key Laboratory for Super-microstructure and Ultrafast Process, School of Physics and Electronics, Central South University, Changsha, Hunan 410083, China

(Received 30 May 2016; accepted 24 August 2016; published online 31 August 2016) An annular beam with a single uniform-intensity ring and multiple segments of phase gradients is proposed in this paper. Different from the conventional superposed vortices, such as the modulated optical vortices and the collinear superposition of multiple orbital angular momentum modes, the designed annular beam has a doughnut intensity distribution whose radius is independent of the phase distribution of the beam in the imaging plane. The phase distribution along the circumference of the doughnut beam can be segmented with different phase gradients. Similar to a vortex beam, the annular beam can also exert torques and rotate a trapped particle owing to the orbital angular momentum of the beam. As the beam possesses different phase gradients, the rotation velocity of the trapped particle can be varied along the circumference. The simulation and experimental results show that an annular beam with three segments of different phase gradients can rotate particles with controlled velocities. The beam has potential applications in optical trapping and optical information processing. C 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4962301]

I. INTRODUCTION

Optical tweezers1 are powerful tools and extensively used in physics, biology, medical science, and other fields.2–4 A plane wave can be converted into a helical mode with an azimuthal phase of exp(imθ), where m is the topological charge and θ is the azimuthal angle. Optical beams with helical phase are known as optical vortices5 which possess orbital angular momentum (OAM). Numerous works have been done to implement the optical vortex beams for trapping and rotation of microscopic particles. For example, Laguerre-Gaussian (LG) beams6,7 and high-order Bessel beams8,9 were applied to induce rotation of particles due to the transfer of OAM from the light to the particles.10–14 An LG mode with p = 0 appears as a ring of light whose radius depends on topological charge,15 i.e., an LG beam with a smaller charge would have smaller radius, and vice versa. Circular Airy beams16–19 were generated to realize trapping and guiding of microparticles. A generated double-ring-shaped azimuthally polarized beam through an annular high numerical aperture (NA) objective lens20 can be applied for optical trapping. Although the above-mentioned ring beams can be used to trap microparticles, the microparticles cannot be transported along the circumference automatically. Roichman et al.21 showed that a holographic ring, which was designed with the shape-phase method, can have a radius of the ring independent of the topological charge of the helical phase. However, a vortex beam with multiple charges along a single ring of intensity has not been reported or generated. In recent years, multi-ring structured optical vortices have been proposed. In Ref. 22 a beam was generated with two interferenced vortex beams of unequal charges. However, the beam possesses multiple rings with different OAMs, and each ring of the beam possesses a topological charge only. A phase mask with annulus structure was designed to generate a

a Author to whom correspondence should be addressed. Electronic mail: [email protected]

2158-3226/2016/6(8)/085322/7

6, 085322-1

© Author(s) 2016.


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multi-ring vortex beam.23 Nonetheless, the width and radius of each of the optical vortex rings are still dependent on the topological charge of the carrying phase. Although the helical beams contain multiple OAM modes,24–26 each of the OAM modes still corresponds to a single ring. In this paper we will design an annular beam with segmented phase gradients. The beam has multiple OAM along the single ring of the intensity distribution and the radius of the ring is independent of the topological charges of the segments. Hence, the topological charges of the phase gradients along the annulus can be set arbitrarily for a defined radius of the intensity ring. As the proposed vortex beam allows the existence of different OAMs, it can be used to rotate trapped particles at varying velocities in a rotation period and facilitate the study for colloidal particle transport.27 Furthermore, since the beam carries OAMs and has any designed ring size, it will have greater information capacity and higher security level than the conventional vortex beam for information processing applications. We will employ the complex amplitude beam shaping method28 to generate such a beam. An experiment on the optical rotation driven by the generated beam will be carried out to verify the segmented phase gradients of the beam.

II. METHOD

We use the iterative beam shaping algorithm28 to shape simultaneously both the amplitude and phase of a reconstructed beam on the imaging plane. The computing result of the algorithm is a phase-only distribution, which will be loaded onto a phase-only spatial light modulator (SLM) to modulate the input beam and reconstructs the desired complex distribution. We can generate a target beam whose intensity distribution is a doughnut profile with uniform intensity. The phase distribution of the beam is designed to have segmented phase gradients along the intensity ring. The radius of the intensity ring can be set arbitrarily and independent of the phase gradients along the annulus. The intensity ring can be divided into n segments along the annulus. The phase distribution along the intensity ring can be written as, l 1φ        l 2φ ψ=       l n φ

0 ≤ φ ≤ φ1 φ1 < φ ≤ φ2 ··· φ n−1 < φ < 2π

(1)

where n is an integer representing the nth segment, and φ is the azimuthal angle. Theoretically, the azimuthal angles the segments cover can be different from each other. For example, we design an annular beam which possesses three equal segments and a phase distribution shown in Eq. (2) along the annulus. The radius of the intensity ring can be set arbitrarily.  3φ 0 ≤ φ ≤ 2/3π     6φ 2/3π < φ ≤ 4/3π ψ= (2)     12φ 4/3π < φ < 2π  The intensity and phase distributions of the designed target beam are shown in Figs. 1(a)-1(b), respectively. In Fig. 1(b), the Arabic numerals 1, 2, and 3 represent the first, second, third segment, respectively. Since the intensity ring of the target beam is divided into three equal segments, each segment covers the same azimuthal angle range. For example, the first segment possesses a phase distribution of ψ = 3φ and covers an azimuthal angle range of 2π/3. Thus, the phase gradients possessed by the first segment ranges from 0 to 2π. Similarly, the phase gradients possessed by the other segments range from 0 to 4π and 0 to 8π, respectively. When the target beam has been designed, the mentioned beam shaping algorithm can be used to generate a phase-only hologram. The hologram is loaded onto the SLM to reconstruct the desired beam in the imaging plane of the Fourier transform. The generation of the reconstructed annular beam is schematically shown in Fig. 2. The insets in Fig. 2 are the phase-only hologram used to generate the desired annular beam, the intensity and phase distributions of the desired beam, respectively. A blazing grating is added into the hologram to obtain an off-axis output.


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FIG. 1. (a) Intensity and (b) phase distributions of an example annular beam.

FIG. 2. Experimental setup for generation of the desired annular beam.

FIG. 3. (a) Intensity and (b) phase profiles of the target output beam, (c) the phase profile of the DOE computed by the referred algorithm, (d) intensity and (e) phase profiles of the reconstructed output beam with the DOE, and (f) three-dimensional view of the central part of the phase in the non-zero intensity region of the reconstructed beam.

III. SIMULATION AND TRAPPING EXPERIMENTS

In the simulations the parameters are set as follows, the sampling points Nx = N y = 512, wavelength λ = 532 nm, pixel pitch = 15 µm, size of the diffractive optical element (DOE) L = 512 ×pitch, and the radius of the target annular beam R = 60×pitch. The desired intensity and phase distributions of the target output beam are shown in Figs. 3(a)-3(b), respectively. The beam comprises a ring of uniform intensity and three phase gradients ranging from 0 to 4π, 0 to 8π, and


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0 to 14π, respectively. Each of the phase gradient segments covers an azimuthal angle of 2π/3. Hence, the corresponding segment of the annular beam carries a different OAM flux. We use a Fourier transform to simulate the beam propagation in the algorithm and obtain a phase distribution which will be loaded as a phase-only DOE on the SLM for beam shaping. The phase distribution of the DOE, intensity and phase distributions of the reconstructed beam are shown in Figs. 3(c)-3(e), respectively. The phases in Fig. 3 are wrapped to the range from 0 to 2π. It can be seen that both the intensity and phase distributions of the reconstructed beam are compatible with those of the target beam. For a better view we display a three-dimensional view of the phase in the non-zero intensity region of the reconstructed beam in Fig. 3(f). To illustrate the intensity and phase evolutions of the annular beam we simulate the propagation of the annular beam in free space. For a better view, we set the radius of the annular beam as R = 60 ×pitch. Figs. 4(a)-4(f) present the intensity profiles of the beam at propagating distances of 0 cm, 1 cm, 2 cm, 3 cm, 4 cm, and 5 cm, respectively. Figs. 4(g)-4(l) present the corresponding phase distributions, respectively. It is worth mentioning that Fig. 4(g) shows the phase distribution in the non-zero intensity region of the reconstructed annular beam in the imaging plane. In Figs. 4(a)-4(l),

FIG. 4. Evolutions of the intensity and phase of the reconstructed beam in free-space propagation. The intensity and phase at propagating distances of (a) and (g) 0 cm, (b) and (h) 1 cm, (c) and (i) 2 cm, (d) and (j) 3 cm, (e) and (k) 4 cm, and (f) and (l) 5 cm, respectively.


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although the annular beam diffracts with increasing propagation distances, the segmented phase gradients maintain. Meanwhile, it can be observed from Figs. 4(g)-4(l) that the locations of the phase gradients in the segments are kept unchanged in the propagation, but the phase in each segment rotates along the annulus. Furthermore, from Figs. 4(b)-4(f) we can observe that there develops an inner annular beam with weaker intensity compared with that of the original annulus. When the propagating distance is farther, i.e., at 50 cm, the original annulus of the intensity distribution in Figs. 4(b)-4(f) is no longer maintained and the intensity of the inner annular beam is enhanced. The segmented phase gradients in the inner annulus are remained as those of the designed annular beam. An optical trapping experiment was implemented to verify the gradient phase of the reconstructed annular beam. Fig. 5 shows the schematic of the optical tweezers system, which consists of an optically pumped semiconductor laser (Coherent Genesis, 1 W, with a wavelength of 532 nm and adjustable laser power ), an SLM (BNS, XY Series, 512 × 512 pixels), and an inverted microscope with a 100× Olympus oil-immersion objective lens (N.A. 1.3). The focal lengths of the lens L 1,L 2,L 3,L 4 used in the experiment are 100 mm, 300 mm, 400 mm, and 200 mm, respectively. The optical beam used in the tweezers system is the annular beam shown in Fig. 3. The topological charges of the three equal segments of the phase gradients are 6, 12, and 21, respectively. In the experiments we added a blazed grating to the phase-only DOE for an off-axis output. Polystyrene beads with a diameter of about 3 µm were immersed in deionized water of refractive index of 1.33 and used as the manipulating objects. The insets show the phase-only hologram with a blazed grating, the CCD-captured intensity distribution of the reconstructed annular beam in the imaging plane of the Fourier transform, and the exposure region of the reconstructed beam on the sample stage of the inverted microscope, respectively. It has been proved that the phase gradients in a beam of light can give rise to forces transverse to the optical axis and the generated phase-gradient force exerted on the particles points to the direction of phase gradient.19 Thus, microparticles will be firstly attracted to the ring trap by the intensity-gradient force, and then driven by the phase-gradient force along the circumference. In the trapping experiments, when the power of the laser was 650 mW, we observed that particles were trapped and rotated and the rotating velocity was obviously different in the different segments of the annular beam. Figs. 6(a)–6(g) show the video frames of trapped particles rotating automatically and counter-clockwisely along the annular beam. The particle in the top left corner of each frame is used as a reference one. The dashed circles highlight the focused region of the reconstructed annular beam. The segment with the maximum phase gradient is set as the origin, and rotation of the trapped particle is shown in Fig. 6. When the trapped particle approached the joint of two adjacent segments the velocity of the trapped particle would fluctuate owing to the difference of the phase gradients. In the experiment, we observed that in the joints the particle obviously decelerated due to the sudden decreasing of the phase gradient. The variation of the rotating velocity of the particle was compatible with the phase distribution of the beam. Furthermore, the rotating velocity of the trapped particle along the annulus in the optical trapping experiments is investigated. For simplicity

FIG. 5. Schematic of the optical tweezers system in the experiment.


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FIG. 6. (a)-(f) CCD-captured frames showing particles rotating counter-clockwisely in the high intensity region of the reconstructed annular beam with a combination of topological charges of 6, 12, and 21.

the segment with the maximum phase gradient is set as the origin and a plot of the angular positions of the particle as a function of time is shown in Fig. 7(a), where the curve is fitted one segment by one segment. It is observed that the three segments have different phase slopes resulting in different rotating velocities of the trapped particle. The particle has a maximum velocity in the segment marked with solid circle in Fig. 7(a), owing to the highest phase gradient possessed by the segment. In the rotation cycle the average angular velocity of the trapped particle along the annulus in each of the three segments can be estimated. Furthermore, the relation between the average angular velocity of the trapped particle and the corresponding phase gradient is investigated and shown in Fig. 7(b), where the average angular velocity is proportional to the corresponding phase gradient approximately. The experimental results coincide with the results in Ref. 28. When we changed the directions of the phase gradients in the computed DOE, the rotation directions of the trapped particles changed accordingly and the rotating velocities in different segments of the annular beam were also compatible with the topological charges the beam carried.

FIG. 7. (a) Plot of the angular position versus the rotating time for the trapped particle rotating counter-clockwisely in the high intensity region of the annular beam with a combination of topological charges of 6, 12, and 21. (b) Plot of the average angular velocity of the trapped particle in each segment versus the corresponding phase gradient.


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Thus, the reconstructed annular beams have been proved to possess not only the single-ring of intensity but also the varying OAMs along the annulus.

IV. CONCLUSION

Based on the iterative beam-shaping algorithm which computes a phase-only hologram to simultaneously shape both the amplitude and phase of an optical beam, we designed an annular beam with different phase gradients in different sections of the annulus. The radius of the annular beam is independent of the segmented phase gradients. A combination of different phase gradients on the annular beam can be realized with a computer-generated phase-only DOE. Thus, the reconstructed beam imposes different OAMs on the trapped particles and drives them rotate with varying velocities in a rotating cycle. Optical trapping of microspheres with the reconstructed annular beam proves that the phase structure of the reconstructed annular beam is consistent with that of the designed annular beam. The beam can find applications in optical trapping and optical information processing.

ACKNOWLEDGMENTS

The research was financially supported by the National Natural Science Foundation of China (Grant No. 61178017). 1

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