Shaping of optical vector beams in three dimensions - OSA Publishing

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Letter

Vol. 42, No. 19 / October 1 2017 / Optics Letters

Shaping of optical vector beams in three dimensions CHENLIANG CHANG,1,2 YUAN GAO,1 JIANPEI XIA,1 SHOUPING NIE,2

AND

JIANPING DING1,*

1

National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China Jiangsu Key Laboratory for Opto-Electronic Technology, School of Physics and Technology, Nanjing Normal University, Nanjing 210023, China *Corresponding author: [email protected]

2

Received 8 August 2017; revised 1 September 2017; accepted 2 September 2017; posted 6 September 2017 (Doc. ID 304382); published 25 September 2017

We present a method of shaping three-dimensional (3D) vector beams with prescribed intensity distribution and controllable polarization state variation along arbitrary curves in three dimensions. By employing a non-iterative 3D beam-shaping method developed for the scalar field, we use two curved laser beams with mutually orthogonal polarization serving as base vector components with a high-intensity gradient and controllable phase variation, so that they are collinearly superposed to produce a 3D vector beam. We experimentally demonstrate the generation of 3D vector beams that have a polarization gradient (spatially continuous variant polarization state) along 3D curves, which may find applications in polarizationmediated processes, such as to drive the motion of micro-particles. © 2017 Optical Society of America OCIS codes: (140.3300) Laser beam shaping; (260.5430) Polarization; (090.1760) Computer holography; (230.6120) Spatial light modulators. https://doi.org/10.1364/OL.42.003884

Shaping the light field by controlling its polarization state has enabled significant advances in optics. The light beams with inhomogeneous polarization distribution, referred to as vector beams, are of great interest because of their unique properties and applications in optical trapping [1,2], focusing [3], microscopy [4], surface plasma excitations [5], and optical communications [6,7]. In addition, it has been proved that light beams with a polarization gradient can exert forces and toques on particles [8,9], and exhibits the ability to drive the motion of trapped particles [10,11]. Currently, a number of approaches for generating vector beams based on a single spatial light modulator (SLM) have been proposed [12–17]. A paradigm for synthesizing vector beams via a phase-only SLM is to realize a computer-generated hologram (CGH) on the SLM so that it can independently modulate two orthogonally polarized components, each of which is imposed by prescribed intensity and phase distributions. One method to calculate the CGH is to directly encode the complex amplitude 0146-9592/17/193884-04 Journal © 2017 Optical Society of America

field into phase holographic gratings [14]. However, the generated vector beam is limited to a two-dimensional (2D) plane. An alternative method of CGH generation is calculating the inverse propagation from the target vector field using scalar or vectorial diffraction-based algorithms [15,18]. Despite its possibility of generating a vector field with three-dimensional (3D) structure [18], the polarization state is regulated at several discrete planes, and there is no continuous variation of polarization state along the beam propagation direction. Recently, a method of generating non-diffraction vector Bessel beams has been proposed [19,20], in which the polarization state varies upon the propagation of the Bessel beams. However, the non-diffraction Bessel beams have a fixed and invariant intensity distribution through the propagation distance, unable to provide a tunable polarization gradient along an arbitrary pre-designed trajectory. On the other hand, 3D beam shaping in the scalar field has been extensively studied, including iterative and non-iterative approaches. One of the attractive non-iterative methods was reported in Ref. [21] to create scalar beams with both high intensity and a phase gradient prescribed along arbitrary 3D curves. In this Letter, we extend this scheme to the generation of 3D vector beams. We propose to use two curved laser beams with mutually orthogonal polarization serving as base vector components with controllable phase variation, and then collinearly superpose them to produce a 3D vector beam. By exploiting our previously proposed vector optical field generator [15], the desired 3D vector beams with prescribed intensity and a continuous variant polarization state are experimentally generated. Figure 1(a) shows the scheme of a holographic 3D beamshaping technique in Ref. [21] that allows designing scalar complex beams whose intensity and phase distribution follow a prescribed curve in 3D geometry. Specifically, in order to generate a desired focal beam, the complex amplitude of the incident plane is given by the expression: Z 2π qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H x; y φx; y; tΦx; y; t x 00 t2 y 00 t2 dt; 0

(1) where the terms ϕx; y; t and Φx; y; t in Eq. (1) are determined by


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Fig. 1. (a) Scheme of holographic three-dimensional beam shaping technique. (b)–(e) Reconstructed intensity and phase distribution of the 2D ring curve at the focal plane with different preset initial phase. (f ) Intensity distribution of the designed 3D ring curve. (g)–(i) Reconstructed intensity of the beam before, at, and after the focal plane, respectively.

x − x 0 t2 y − y0 t2 φx; y; t exp iπ z t ; 0 λf 2 and

(2)

i φ yx 0 t − xy0 t w20 0 Z iσ t 0 0 x τy 0 τ − y 0 τx 0 τdτ ; (3) 2 w0 0 0

Φx; y; t exp

where x 0 t; y 0 t; z 0 t represents the prescribed 3D curve in the Cartesian coordinate with t ∈ 0; 2π. f and λ refer to the focal length of the Fourier lens and the wavelength, respectively. w0 is a constant. In Eq. (3), we use σ as a free parameter for controlling the phase gradient along the curve. The parameter φ0 is the initial phase factor, playing the role of dominating the starting point of the phase gradient along the curve. Equation (1) allows calculating the incident complex field that can shape a structurally stable focal beam with special intensity distribution and phase gradient. We first consider a 2D ring curve x 0 t R cost, y 0 t R sint, z 0 t 0 with different initial phase values and demonstrate the performance of this technique by simulation. The intensity distributions of the resulting beams are displayed in Figs. 1(b) and 1(d). The phase distribution of the ring is well defined along curves under the same topological charge of m 5 [see Figs. 1(c) and 1(e)]. Note that the starting points of the helical phase in Figs. 1(c) and 1(e) are spatially different from each other due to the different pre-specified initial phase φ0 , which is responsible for a specific polarization state. Furthermore, we consider the generation of a tilted ring, as shown in Fig. 1(f ). The 3D structure is revealed along the beam propagation in the focal region in simulation. The beam intensity distributions calculated before, in, and after the focal plane (z −0.6 mm, 0 mm, and 6 mm,

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respectively) are shown in Figs. 1(g)–1(i). The topological charge is determined by m σR 2 ∕w20 ; therefore, the radius of the ring (R) and the topological charge is controlled independently by introducing the free parameter σ, resulting in the shaping of kinds of “perfect helical beams.” For creating the 3D vector beam, we use the technique of [21] to shape the spatially overlapped, orthogonally polarized component beams (scalar curves) and combine this method with our proposed vector optical field generation technique [15] to create the desired 3D vector beams. First, two complex amplitude fields at the incident plane, denoted as H 1 x; y and H 2 x; y, are calculated from the corresponding component beam using Eq. (1). The experimental arrangement for creating the 3D vector beam is shown in Fig. 2. The pixelated holographic grating displayed on the SLM modulates both base complex amplitude fields [H 1 x; y and H 2 x; y] carrying their respective spatially variant offset phase along two orthogonal directions. In order to overcome the unavailability of complex modulation using the SLM, we utilize the double-phase method [22,23] as the encoding procedure to generate the phase-only holographic grating. Basically, two pure phase values θ1 x; y and θ2 x; y are extracted from the decomposition of a complex value Ax; y expiφx; y, expressed by θ1 x; y φx; y cos−1 Ax; y∕Amax θ2 x; y φx; y − cos−1 Ax; y∕Amax ;

(4)

where Amax is the maximum value of Ax; y. Here Ax; y and φx; y represent the amplitude and phase extracted from the summation of two base complex amplitude components after multiplying each offset phase as follows: i2πx sin θx Ax; y H 1 x; y exp λ i2πy sin θy H 2 x; y exp λ i2πx sin θx φx; y arg H 1 x; y exp λ i2πy sin θy ; (5) H 2 x; y exp λ

Fig. 2. Schematic of the optical setup for shaping the threedimensional vector beam, based on the overlapping of two orthogonally polarized base vector component beams.

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where θx and θy represent the diffraction angles of the holographic grating in horizontal and vertical directions, respectively. Finally, the phase-only holographic grating is expressed by expipx; y expiθ1 x; yM 1 x; y iθ2 x; yM 2 x; y: (6) In Eq. (6), M 1 and M 2 are the complementary 2D binary gratings (checkerboard patterns). The phase term px; y will be the encoding CGH loaded to the phase-only SLM. After being illuminated by a collimated plane wave, the linear polarized light passes through a 4f system consisting of a pair of lenses behind the SLM. Due to the function of the imposed offset phase terms expi2πx sin θx ∕λ and expi2πy sin θy ∕λ, the diffraction light leaving from the SLM are endowed with the required complex amplitudes H 1 x; y and H 2 x; y towards different directions (horizontal and vertical directions) under the angles of θx and θy , respectively, which are allowed to pass through a spatial filter located at the focal plane of the first lens. Two wave plates [half-wave plates (HWPs) or quarter-wave plates (QWPs)] placed at the focal plane convert the separated beams into mutually orthogonal polarization components, which serve as a pair of base vector beams for the subsequent vectorial superposition. The Ronchi grating placed at the rear focal plane of the second lens re-corrects the diffraction direction of each beam, enabling the collinear recombination of the two base vector beams. The created beams are the incident complex fields of the following focusing lens, generating two spatially overlapped 3D curves of mutually orthogonal polarization states with prescribed intensity distribution at the focal region. The variation of the polarization state at each point along the curved beam is introduced due to the relative amplitude and phase (i.e., phase difference) of the two generated base vector beams. In this way, the shaping of a vector beam in three dimensions, where the polarization state varies upon propagation, is realized. The optical experiments are carried out to verify the proposed method. The SLM used in the experiment is Holoeye Leto, which has a pixel pitch of 6.4 μm and a pixel number of 1920 × 1080. The SLM is illuminated by a collimated green laser beam with 532 nm wavelength. The values of θx and θy are both 0.38° in order to match the Ronchi gratings whose period is 83.3 μm (12 mm−1 ). We place two QWPs behind the filter to convert the two base vector beams into mutually orthogonal left and right circular polarization states. The first experiment is to generate a vector-tilted ring curve in the focal region. We adjust the topological charge m1 ; m2 and the initial phase φ01 ; φ02 of the two rings to control the polarization state along the curve. The parametric expressions of the tilted ring curve used for generation of H 1 x; y and H 2 x; y are set as x 0 t R cost, y 0 t R sint, and z 0 t sR sint, where t ∈ 0; 2π and s 1. R 0.5 mm is the radius of the ring. Figure 3(a1) shows the 3D structure of the curve. We place a CCD camera in the focal region of the focusing lens (f 250 mm) to capture the results. The CCD is moved back and forth along the optical axis to record the intensity behavior of the 3D curve at different cross sections upon propagation. Figures 3(a2)–3(a6) display the measured intensity of vector beam propagation before, at, and after the focal plane, respectively, when no polarizer is used, showing the correct 3D structure of the generated tilted ring as well as the accurate overlapping of the two base beams. The rows of

Fig. 3. Experimental results of the generated vector-tilted ring where the linear polarization state varies continuously along the curve (Visualization 1, Visualization 2, Visualization 3).

Fig. 3(b) show the generated vector-tilted ring where the topological charges of the two base vector rings are set as m1 9 and m2 1, while the initial phases are set as φ01 0 and φ02 0. The polarization direction along the ring is plotted in Fig. 3(b1), and the polarization gradient is calculated by m2 − m1 ∕2 4. When using a polarizer (whose direction is marked at the top left corner of each image), the intensity distribution exhibits the continuous linear change of the polarization state. In the case of Fig. 3(c), the initial phases of the two base vector beams are set as φ01 0 and φ02 π, while the topological charges remain the same as those in Fig. 3(b). It is obvious that the polarization direction has changed by 90 deg along the same curve compared with the results of the second row, point by point. Moreover, the intensity evolving during propagation captured by moving the CCD along the z direction in the focal region is shown in Visualization 1, Visualization 2, and Visualization 3 for each case. The second experiment is the generation of a vector focal field of an Archimedean spiral, where the polarization gradient is the same as the above tilted ring along its path. The parametric expressions of the curve are x 0 t −Rt cos10t, y 0 t −Rt sin10t, and z 0 t sR0.5 − 1 − t 2 1∕2 , where t ∈ 0; 2π, s 1, and R 0.5 mm. The 3D structure of the curve is plotted in Fig. 4(a1) and experimentally measured in Figs. 4(a2)–4(a4). The topological charge and initial phase in the results of the second and third rows in Fig. 4 are the same as those in Fig. 3. Last, we experimentally demonstrate the simultaneous generation of hybrid vector beams that are comprised of a 2D vector ring (s 0, R 1 mm) and a 3D Archimedean spiral (s 1, R 0.5 mm). The model of the hybrid beams is plotted in the first row of Fig. 5. The 2D ring curve is located at the focal plane while the Archimedean spiral has 3D structure along the optical axis. By the side is the transversal view of the beams with the designed polarization state along the curves. The corresponding intensity distributions before, at, and after the focal plane are tested by rotating a linear polarizer, whose results are shown in the third and fourth rows, respectively, in Fig. 5. The focusing and defocusing phenomenon in both the 2D ring and 3D Archimedean spiral is clearly seen as well as the variation of the polarization state along the curve with beam diffraction. The intensity distribution in the overall focusing region is also measured and stored in Visualization 4, Visualization 5, and Visualization 6 for each case. The generated 3D curves in our


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vector beams would be converted into the mutually orthogonal linear polarization; by doing so, we will generate the 3D vector curve with a polarization gradient along a meridian circle on the Poincare sphere. The overall efficiency of our experimental system is about 2.61% due to the energy loss of the SLM, filter, and Ronchi grating. In summary, we present a method of shaping vector beams with prescribed intensity distribution and continuous varied polarization states along the curves in three dimensions. The experimental results demonstrate that the generated 3D beams exhibit accurate gradient of intensity and polarization state in the focal field, which will be appreciated to yield optical gradient forces for freestyle 3D trapping and transportation of micro-sized particles similar to a scalar situation [24]. Fig. 4. Experimental results of the generated vector Archimedean spiral where the linear polarization state varies continuously along the curve.

Funding. National Key R&D Program of China (2017YFA0303700); National Natural Science Foundation of China (NSFC) (11474156, 11534006, 61605080). Acknowledgment. We acknowledge the support from the Collaborative Innovation Center of Advanced Microstructures and the Collaborative Innovation Center of Solid-State Lighting and Energy-Saving Electronics. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Fig. 5. Experimental results of the generated hybrid vector beams consisting of 2D ring curve and 3D Archimedean curve with continuously varying polarization state. The 2D ring is located at the focal plane. (Visualization 4, Visualization 5, Visualization 6.)

17. 18. 19. 20.

experiment exhibit a polarization gradient with a continuous linearly varying polarization state. This polarization gradient corresponds to the trajectory along the equator of the so-called “Poincare sphere,” wherein the local field is linearly polarized. It should be noted that the two QWPs used in our experiment can be replaced by two half-wave plates, where the two base

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