Orbital angular momentum generation via a spiral ... - OSA Publishing

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Letter

Vol. 43, No. 1 / January 1 2018 / Optics Letters

Orbital angular momentum generation via a spiral phase microsphere YAN ZHOU,1,2,† HUI GAO,1,3,4,† JINGHUA TENG,2 XIANGANG LUO,3,5

AND

MINGHUI HONG1,*

1

Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117576, Singapore Agency for Science, Technology and Research (A*STAR), 2 Fusionopolis Way, Singapore 138634, Singapore 3 State Key Laboratory of Optical Technologies for Nano-Fabrication and Micro-Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China 4 University of Chinese Academy of Sciences, Beijing 100049, China 5 e-mail: [email protected] *Corresponding author: [email protected] 2

Received 17 October 2017; revised 23 November 2017; accepted 23 November 2017; posted 27 November 2017 (Doc. ID 309301); published 20 December 2017

Vortex beam carrying orbital angular momentum (OAM) attracts much attention in many research fields for its special phase and intensity distributions. In this Letter, a novel design called the spiral phase microsphere (SPMS) is proposed for the first time, to the best of our knowledge, which can convert incident plane wave light into the focused vortex beam that carries OAM with different topological charges l
1 and 2. The vortex beam generation is verified by a self-interfered modification of the SPMS. The generation of the vortex beams by the SPMS irradiated by a single-wavelength incident light is studied using the CST MICROWAVE STUDIO simulation. The SPMS provides a new approach to achieve high-efficiency and high-integrated photonic applications related with OAM. © 2017 Optical Society of America OCIS codes: (050.4865) Optical vortices; (350.4990) Particles; (350.3950) Micro-optics; (070.7345) Wave propagation. https://doi.org/10.1364/OL.43.000034

Allen et al. discovered that the orbital angular momentum (OAM) modes of light initiate vortex beam formation with a helical wavefront and an azimuthal phase factor of exp
il θ, where l is the topological charge (or the azimuthal index) of a vortex, and θ is the azimuthal angle [1]. With their unique characteristics, vortex beams carrying OAM [1–3] are employed in a wide range of fields, including the superresolution stimulated emission depletion (STED) microscopy [4], manipulation of nano- and micro-particles as optical tweezers [5–7], communications in free space [8–10], and nanofabrication of three-dimensional (3D) [11,12] and surface structures [13,14]. Over the years, various methods and technologies have been employed to generate vortex beams carrying OAM, from spiral phase plates, Q-plates, and Dammann gratings to spatial light modulators [15–18]. However, the devices in aforementioned application fields are now expected to have 0146-9592/18/010034-04 Journal © 2018 Optical Society of America

a small volume for integration purposes, and nanophotonic applications are always based on nano-scale designs. To meet the stringent requirements, many current studies seek to use metasurfaces as micro- to nano-scale OAM generators [19–21]. However, the lossy metasurfaces often encounter the problem of low efficiency [19–21]. Therefore, an integrated OAM generator with focusing effect at high transmission efficiency is required. As one of the low-loss optical elements, transparent dielectric microspheres have been attracting much attention on exploring its applications in a variety of fields. With its strong capability to focus light to form a photonic nanojet, it has been studied for improving the performance of optical lithography [22]. Since 2011, transparent microspheres started to show their great potential in nano-imaging, as many works demonstrated that they can be used to observe nanoscale objects and features beyond the observation limitations [23–26]. Moreover, Wu et al. worked on engineering a transparent microsphere to enhance the focusing capability or to create the longitudinally polarized photonic nanojet [27,28]. The engineered microsphere is another field yet to be fully explored. Recently, one work also tried to use a monolayer of self-assembled micro-particles to generate a lattice of photonic vortex beams [29]. However, this design faces the drawback of limited choice on topological charge values of the OAM generated. Meanwhile, it requires a circular polarization condition for the incident irradiance. In this Letter, based on a low-loss transparent dielectric microsphere, we propose a novel approach to generate the highly focused vortex beam carrying OAM with different topological charges for the spiral phase microsphere (SPMS), which can be fabricated by focused ion beam milling. Analogous to the conventional planar spiral phase plate, under the laser irradiation, a continuous optical path difference is initiated on the engineered spherical surface and, thus, creates an azimuthal phase distribution of the light passing through it. The transmitted light carrying OAM is focused by the SPMS and, hence, forms a highly focused vortex beam. The theoretical analysis presented in this Letter demonstrates the method to convert

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Fig. 2. Geometric formulation for the SPMS structure. (a) 3D view of the dielectric microsphere in a Cartesian coordination space with spherical coordinate variables φ, θ, and r labeled; (b) cross-sectional view of half of the milled SPMS to show the fundamental of the spiral milling design.

Fig. 1. Schematic of concept. (a) Plane wave is incident onto the structured hemisphere of the SPMS, and a vortex beam carrying OAM is generated as a result. (b) Intensity of incident light. (c) Designed spiral phase generating structure etched on the upper half of a fused silica microsphere (diameter D  10 μm). (d) Vortex beam generated by the SPMS.

the incident plane waves into an optical vortex beam carrying OAM with different topological charges with the help of the SPMS. Such a micro-/nano-structured microsphere can be regarded as a highly compact and integrated light manipulation element that can be used in a large variety of fields. A 3D time-domain simulation solver (CST Microwave Studio, CST MWS) and analytical calculations by MATLAB are used for theoretical analyses. Figure 1 schematically shows the conceptual design of the SPMS, with only the upper hemisphere structured. As seen in Fig. 1(a), a plane wave is incident perpendicular onto the SPMS; after passing through this optical element, a focused vortex beam carrying OAM is generated underneath the SPMS. Figure 1(c) depicts a more detailed view of the 3D-modelled SPMS, which is made from a 10 μm diameter fused silica microsphere. The largest zenith height difference Δmax for topological charge l  1 condition is calculated to be around 865 nm. Figure 2 demonstrates how the SPMS structure is formulated. Figure 2(a) shows the position function of a point on the SPMS surface, P
φ; θ; r, governed by the three spherical coordinates. The sphere displayed is in its original condition before structuring. In Fig. 2(b), the structured SPMS is viewed at the cut-plane along the Z axis which has a polar angle of φ. From a pre-defined zero position, at each non-zero polar angle φ
0 < φ ≤ 2π, the original microsphere centered at O is milled with respect to the reference sphere centered at O 0 (with the same radius R as the original sphere), and Δ is the height difference of the zenith positions for the two spheres,

OO 0  Δ. The zenith height difference changes with respect to φ, such that 8 φ λ l ≥0 < l · 2π ·
n−1 ;   Δ ; (1) :l · φ − 1 · λ ; l < 0
n−1 2π where λ is the designed incident light wavelength, n is the refractive index of the dielectric microsphere, and l  0, 1, 2 is the vortex topological charge. We define anticlockwise as the positive rotation direction, represented by the upper row of Eq. (1), and clockwise rotation as the negative direction, as described by the lower row of Eq. (1). Δ has a maximum value at φ  2π when l is positive, or at φ  0 when l is negative. Because only the upper hemisphere is structured, by considering the relationship α  β  θ  π∕2, the radial distance r of each surface position with θ > 0 on the structured SPMS can be calculated from the equations below: R Δ r   : (2) sin
θ  π∕2 sin β sin α 3D models of the SPMS at five l values are analytically calculated and plotted in Figs. 3(a)–3(e) using MATLAB. Throughout this Letter, a fused silica microsphere with a diameter D of 10 μm is assumed for all calculations and simulations. Taking the tiny size of the microsphere, the incident light at the wavelength of 405 nm is assumed to be in the plane wave form and is set to be x-polarized. The color scale bar indicates the z coordinate (from −5 to 5 μm) of all points in the 3D model plotted. Different l values are achieved by changing the depth of the spiral pattern, Δmax . Specifically, Δmax  0 nm for l  0; Δmax  865 nm for l  1; and Δmax  1.73 μm for l  2. The case l  0 implies the original microsphere without any structuring. In simulations employing CST MWS, all boundaries are set to be open, while the simulation region has X Y Z dimensions of 12 μm × 12 μm × 20 μm, and the center of the SPMS is positioned at the origin. The typical dimension of meshes varies from 45 to 67 nm. The background has a refractive index of 1.0. Figures 3(f )–3(j) are the simulated phase distributions of the five SPMS models. By simulating the light field distribution in the 3D space, cross-sectional cut-plane results for power intensity distributions are obtained

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Fig. 3. Effect of different topological values l on the vortex beam generated. (a)–(e) 3D model designs for generating vortex beams carrying OAM at l  −2, −1, 0, 1, and 2, respectively. (f )–(j) Simulated phase distributions positioning 800 nm away from the bottom of the SPMS. (k)–(o) Corresponding xy-plane light field distribution results at the same positions as in (f )–(j); the dotted graph at the left half-side of each resulting image shows the normalized intensity distribution along the cut line vertically through the center.

and presented in Figs. 3(k)–3(o). The results in the middle and the bottom rows are captured at the xy-plane positioning 800 nm (around 2λ) away from the bottom of the SPMS. In addition, for Figs. 3(k)–3(o), the vertical dotted line at the center of each resulting image marks the cross-sectional line to view its power intensity distribution, which is depicted by the dotted graph at the left half-side of the image. Such graphs verify the center-hollowed vortex beam formation by the SPMSs. The gray scales at the right sides of Figs. 3(k)–3(o) indicate the range of optical field power intensity at the same arbitrary unit. It is noted that the absolute optical field power intensities decrease, while the sizes of vortex beams become larger as jl j increases from 0 to 2. More specifically, the fullwidth at half-maximum of the focused spot in Fig. 3(m) is 352 nm, while the peak-to-peak distances of the donuts generated by their corresponding SPMSs are 601, 434, 418, and 601 nm for l  −2, −1, 1, and 2, respectively. From the phase distribution graphs and the simulated optical field results, it is clearly seen that vortex beams carrying OAM with the five different topological charges can be successfully generated by the designed SPMSs. OAM with topological charge values beyond the presented range can also be achieved. However, the maximum magnitude of l is limited by the size of the microsphere used. The simulation results demonstrate that this method can be used in highly integrated optical systems related with OAM, such as the STED system, optical tweezers, optical communications, and photolithography. The SPMS is a basic design for the high-integrated OAM generation. It can be developed to achieve more novel functions. As an example, to further verify the generation of a vortex

beam carrying OAM of the proposed topological charge values and explore more possible applications based on this method, we modified the 3D model design of the SPMS, as demonstrated in Fig. 4; and it is called the self-interfered SPMS (SI-SPMS). This modified design is composed of two flattened surfaces at the top and bottom of the microsphere, while the spirally milled structure is carried out on the peripheral regions around one of the flattened surfaces. The two flattened circular surfaces are of the same radius R 0 , where R 0  R∕2. Similar to

Fig. 4. Verification of a vortex beam carrying OAM with l  1 and 2 through the SI-SPMS design. Center, cut-plane view of how incident light passing through different regions interferes in the bottom space of the SI-SPMS; the green-colored region marks the self-interference. Left, 3D model and resulting image of the l  1 design. Right, 3D model and resulting image of the l  2 design.

Letter the SPMS model design, the topological charge value of the SI-SPMS depends on the depth of the spiral pattern. Under laser incidence, the part of light passing through the central flattened region remains as a plane wave and should not carry any OAM. Light that passes through the structured peripheral region undergoes OAM generation and is focused towards the central region after exiting the SPMS element. At the center of Fig. 4, from the xz cut-plane view, the focused OAM-carrying light interferes with the central region plane wave within the green-colored space. As a conventional way to verify the generation of a OAM-carrying vortex beam [19], such interference forms a single-arm spiral beam (l  1) or dual-arm spiral beam (l  2) as viewed from a horizontal cut-plane in their light field distributions, as displayed at the left and right bottom corners in Fig. 4. The resulting images are obtained from the xy-plane 3.6 μm away from the bottom of the SI-SPMS. The spiral distribution achieved by interference can be used to fabricate chiral structure by photolithography. In conclusion, the focused vortex beams that can be produced by the SPMS under single-wavelength irradiation are studied analytically. With numerical calculations and simulations, the capability of OAM vortex beam formation for different topological charges is demonstrated. As a highly integrable element, the novel SPMS design would have many promising applications, for example, high-precision optical tweezers, optical lithography to form complex spiral structures, optical communications, and STED microscopy. Funding. National Research Foundation Singapore (NRF) (1421200080); National Program on Key Basic Research Project of China (973 Program) (2013CBA01700). Acknowledgment. Yan Zhou acknowledges the support from NUS Graduate School for Integrative Sciences and Engineering (NGS). Hui Gao acknowledges the support from China Scholarship Council (CSC). †

These authors contributed equally to this Letter.

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