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Abstract
Focal position control of vortex beams has tremendous applications in optical field. Herein, non-classical Archimedean arrays were proposed for optical devices with bifocal length and polarization-switchable focal length. The Archimedean arrays were constructed by rotational elliptical holes in a silver film, which were followed by two one-turned Archimedean trajectories. The elliptical holes in this Archimedean array provide the freedom of polarization control for the optical performance by their rotation status. The rotation of elliptical hole can provide additional phase to affect the shape of vortex beam (converged or diverged) under the illumination of circular polarization. The geometric phase of Archimedes trajectory will also determine the focal position of vortex beam. This Archimedean array can produce a converged vortex beam at the specific focal plane according to the handedness of the incident circular polarization and geometrical arrangement of array. The Archimedean array was also demonstrated by experiment and numerical simulation for its exotic optical performance.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Vortex beams possess unique optical properties, and their propagating wave can carry orbital angular momentum (OAM) to form a helical phase. The optical vortex in a vortex beam rotates around the optical axis to generate phase singularities. The vortex beams promised great potential applications for the optical tweezer [1–3], quantum information [4–6], optical communication [7–10], and biomedical applications [11–13]. Traditionally, vortex beam is generated using a combination of lens and spiral plates [14,15]. However, the traditional method in the generation of vortex beam presents the bulky volume. Recently, the concept of metasurfaces [16–21] can provide another route to generate vortex beam with desired optical performance. With the advancement of nanofabrication, the arrangement and shape of nanoapertures can realize the specific order of topological charge for the vortex beam. The metasurface can be an ultrathin optical element for the generation of vortex beam.
The Archimedean spiral slit [22] can generate a transmitted vortex beam with a specific topological charge according to the handedness of incident circular polarization. The geometric phase of Archimedean spiral slit can determine the topological charge of the optical vortex, and it can engineer a helical phase. The one one-turned Archimedean spiral slit only can provide a single focal length for the vortex beam. The complex mode of optical vortex is difficult to obtain because of the geometry of Archimedean spiral slit. The spatial arrangement of nanoslits can provide an artificial phase to convert the polarization of the transmitted light. The nanoslits with modified Archimedean trajectory can alter the order of topological charge for the plasmonic vortex [23], but the scattering light has no characterization with polarization-controlled focal length. Most studies in the generation of optical vortex only were concentrated on the mode of the optical vortex (topological charge and phase singularity). Our study proposed the integration of nanoslits and two one-turned Archimedean trajectories to provide the polarization-controlled optical vortex with the desired focal position and bifocal length in the far-field region. The metalens with polarization-controlled focal length was previously explored by the polysilicon nanoantenna [24], but the transmitted beam did not have the feature of optical vortex. The bifocal vortex beam with coaxial shape [25,26] was also studied by the Pancharatnam–Berry geometric phase. However, the propagation behavior of focused vortex beam with a circular shape in free space is rarely explored with polarization control. In this study, a two-armed Archimedean array was proposed to generate the polarization-controlled focal length for the focused vortex beam. This Archimedean array can also generate a bifocal vortex beam with proper geometrical parameters.
Herein, the non-classical Archimedean lens was developed to generate the focused optical vortex with different positions of optical axis according to the handedness of incident circular polarization. This Archimedean lens was constructed with several pairs of orthogonal nano-elliptical holes, which were placed in the two one-turned Archimedean arrays. Several pairs of nano-elliptical holes were arranged and rotated with the Archimedean trajectory. The rotating direction of nano-elliptical hole influence the phase and emitting direction of transmitted light. Hence, the alterations of rotating direction of nano-elliptical hole and the geometrical parameters of Archimedean trajectory can provide the capability of manipulating topological charge in the transmitted vortex beam. The manipulation of topological charge can enable the control of propagation distance for the focused vortex beam. Hence, the Archimedean array can be developed to control focal position of vortex beam according to the handedness of incident circular polarization.
2. Archimedean array design
The Archimedean trajectory in a metal film can be described by the following equation:
where r(θ) and θ are the polar and azimuthal coordinates of the Archimedean trajectory, r0 is the initial radius, λsp is the wavelength of surface plasmon (SP) at the metal surface, and m is the integer number, which presents the geometrical charge of Archimedean trajectory. The rotation elliptical hole can provide another geometric phase to the incident light for varying the optical properties of transmitted beam. Hence, we incorporate nano-elliptical holes in the Archimedean trajectory. The rotational nano-elliptical holes with the Archimedean trajectory can provide a flexible design for the manipulation of vortex beam. Figure 1 shows the illustrations of the one-turned Archimedean array. This Archimedean array contained the two arrays of nano-elliptical holes, which are followed by the two parallel Archimedean trajectories. In the two arrays of nano-elliptical holes, adjacent elliptical holes are perpendicular to each other. The nano-elliptical hole rotates around the center of the hole, and it also revolves with the Archimedean trajectory. The rotational angle of nano-elliptical holes is synchronous with the azimuthal angle of Archimedean trajectory. The inner and outer arrays in the Archimedean array have the same opening width as the Archimedean trajectory. The two Archimedean arrays are shown in Fig. 1. Figure 1(a) is the Archimedean array, in which rotational direction of nano-elliptical holes is the same as the azimuthal direction of Archimedean trajectory (RSA). Figure 1(b) is another Archimedean array, in which the rotational direction of nano-elliptical holes is opposite to the azimuthal direction of Archimedean trajectory (ROA).
Fig. 1. Illustrations of the one-turned Archimedean array. The r0 and |m|λsp are the initial radius and opening width of Archimedean trajectory, respectively. (a) RSA. (b) ROA.
Figure 2 shows the optical simulations of RSA and ROA under the illumination (wavelength: 532 nm) with right-handed circular polarization (RCP) and left-handed circular polarization (LCP) light. Finite-difference time-domain (FDTD) method (Lumerical software) is used. All simulation parameters are listed in Supplement 1 (Fig. S1, Tables S1, and S2). The RSA and ROA are in the silver film, supported on glass substrate. The silver thickness is 180 nm. The lengths of major and minor axes are 350 and 150 nm for each elliptical hole, respectively. Two adjacent elliptical holes are separated by a distance of SP wavelength (∼505 nm). The order of topological charge of optical vortex which is located at the metal surface are shown in Fig. 2. The optical vortex which is located at the metal surface can be described by the plasmonic vortex. The order of topological charge of plasmonic vortex will be m + l, where m is the geometrical charge of Archimedean trajectory, l is the handedness of incident circular polarization (l is -1 and 1 for RCP and LCP incidence, respectively). The order of topological charge of plasmonic vortex highly depends on the geometrical charge of Archimedean trajectory and handedness of circular polarization regardless of the rotational direction of the nano-elliptical holes. However, the transmitted beams of RSA and ROA are different for RCP and LCP incidences. The RSA can provide diverged and converged vortex beams under RCP and LCP incidences, respectively. Compared with the RSA, the ROA provides the opposite results under RCP and LCP incidences. The transmitted phase of Archimedean array is related to lφ, where φ is the rotational angle of nano-elliptical holes. The φ have opposite values in the RSA and ROA of Fig. 2. It reveals that the RSA with LCP (l is 1, and φ is negative) will be similar to the ROA with RCP (l is -1, and φ is positive) for their optical performance. The propagation process of vortex beam has also been described in Supplement 1 (Fig. S2). Both RSA and ROA possess the capability of polarization-selective beam shaping, and the opposite polarization response is also exhibited in them.
Fig. 2. Optical simulations of RSA and ROA with different incident circular polarizations. The incident wavelength is 532 nm. The upper parts are optical transmission. The right parts are the phase distributions on the metal surface. (a) RSA. (b) ROA.
Furthermore, the focal length of Archimedean array is investigated. According to Fresnel diffraction integral with the geometry of Archimedean trajectory, the propagation behavior of vortex beam can be described by the following equation:
The focal length of RSA with different geometric parameters is shown in Fig. 3 by FDTD. The focal length of the Archimedean array gets longer as the initial radius and order of geometrical charge increase. The vortex beam will carry specific OAM according to the m value. In the Archimedean array, the large initial radius and high order of geometrical charge represents a larger effective aperture of Archimedean lens. The scattering behavior is determined by the configuration of each pair nano-elliptical hole. In the same configuration of each pair nano-elliptical hole, the focal length can be enlarged by the Archimedean lens with larger aperture size. Hence, the rotational direction of elliptical holes and geometrical parameters of Archimedean trajectory can be varied to obtain the desired optical transmission. The excellent optical performance with bifocal length and polarization-controlled focal length can be facilitated by combinations of the two different Archimedean arrays.
Fig. 3. Focal length of RSA with different initial radii and geometrical charges. The incident wavelength and incident polarization are 532 nm and LCP, respectively.
3. Bifocal length and polarization-controlled focal length
Figure 4 shows the schematic of two Archimedean arrays, each having two arms, which were arranged by two one-turned Archimedean arrays in the same optical axis. The depiction of the focusing beam is also shown in Fig. 4. Figure 4(a) is the Archimedean array comprised of the two RSAs, which have same azimuthal direction of Archimedean trajectory (clockwise) with different geometrical parameters (r0 and m). The inner and outer RSAs provide the converged vortex beam under the illumination of LCP. The outer RSA has larger initial radius and opening width of Archimedean trajectory. Hence, the high and low focal positions of the Archimedean array can be attributed to the outer and inner RSAs, respectively. In Fig. 4(a), the design of Archimedean array can be an optical device, which provides vortex beam with bifocal length. The Archimedean array comprises of ROA and RSA in Fig. 4(b). These have different optical responses with the handedness of incident circular polarization according to the results of Fig. 2. The polarization-controlled focal length can be achieved by integrating the RSA and ROA.
Fig. 4. Configurations of two-armed Archimedean array and corresponding focusing statuses. (a) Two RSAs. (b) ROA and RSA.
Figure 5 shows the optical transmission of two-armed Archimedean array under the illuminations with RCP and LCP. The geometrical parameters of the two two-armed Archimedean arrays are listed in Supplement 1 (Tables S3 and S4). Figure 5(a) shows that the two-armed Archimedean array can provide the vortex beam with bifocal length under the illumination of LCP. The long and short focal lengths are dedicated by the outer and inner RSAs, respectively. Furthermore, the two-armed Archimedean array of Fig. 5(a) generate the diverged vortex beam under the illumination of RCP. The vortex beam with the polarization-controlled focal length is shown in Fig. 5(b). The combination of ROA and RSA can offer a focal length with flexible design according to the handedness of incident circular polarization. The ROA and RSA can provide the converged vortex beam under the illuminations of RCP and LCP, respectively. In Fig. 5(b), some stray light is present around the focal point. The two-armed Archimedean array with ROA and RSA will produce both the converged and diverged vortex beams, whether the illumination is LCP or RCP. The side lobe near the focal point is caused by the diverged vortex beam. The proposed two-armed Archimedean arrays have novel optical properties by the appropriate spatial arrangements of nano-elliptical holes.
Fig. 5. Optical simulations of two-armed Archimedean array. The incident wavelength is 532 nm. (a) Two RSAs. (b) ROA and RSA.
4. Experimental results and discussion
In the experimental method, the focused ion beam (FIB, FEI Helios G3CX) and E-beam evaporator (ULVAC) were adopted to fabricate the Archimedean array. The silver film is deposited on the glass substrate by the E-beam evaporator (thickness: 180 nm). The Archimedean arrays were etched by the FIB. The fabricated samples of Fig. 6 had the same geometrical parameters as the simulated two-armed Archimedean arrays of Fig. 5. A solid-state laser (wavelength: 532 nm) is used as the light source (Shanghai Dream Lasers). The circular polarization of incident light is constructed by the polarizer, half-wave plate, and quarter-wave plate. An inverted microscopy (Olympus, IX73) is utilized to characterize the focusing properties of Archimedean arrays, which was equipped with charge-coupled device (Olympus DP22) and motorized stage (Marzhauser MFD). Figure 6 shows the experimental results of optical transmission for the Archimedean arrays. The propagating behaviors of vortex beam were recorded at the different image planes (acquired one image per 0.5 µm). The experimental results in Fig. 6 are consistent with the simulation results of Fig. 5. The optical beams with the bifocal length and polarization-controlled focal length can be realized from the experimental results. The versatile polarization-controlled vortex beam can be obtained at the desired focal position by the two-armed Archimedean array. The deviations between experimental and simulation results can be attributed to the fabrication variations of sample and out-of-focusing light of image plane. The fabrication error induces the bias ellipticity of nano-elliptical holes to affect the scattering properties. The out-of-focusing light and positioning accuracy of motorized stage influence the beam shape and focal position, respectively.
Fig. 6. Scanning electron microscopy images and optical transmissions of two-armed Archimedean array by experimental methods. The incident wavelength is 532 nm. (a) Two RSAs. (b) ROA and RSA.
5. Conclusion
In conclusion, the two types of Archimedean arrays demonstrated the capabilities of the bifocal length and polarization-controlled focal length by experiment and numerical simulation. The two-armed Archimedean array can generate the bifocal length under the specific circular polarization incidence, which was constructed by the two RSAs. The bifocal length can be tailored by the proper geometrical parameters of Archimedean trajectory. In the two-armed Archimedean array, the polarization-controlled focal length can also be achieved by reversing rotating direction of nano-elliptical holes between the inner and outer Archimedean arrays. The proposed RSA and ROA are complementary in the optical response. In Figs. 5(b) and 6(b), the inner ROA and outer RSA generate the focused vortex beam under the different incident circular polarizations. The switchable focal length results in prominent optical properties by the handedness of incident circular-polarization. The two-armed Archimedean array is a potential candidate of optical applications, such as optical manipulation, communication technology, and optical imaging.
Funding
Ministry of Science and Technology, Taiwan (111-2221-E-006-199).
Acknowledgments
The authors would like to thank the technical services (focused ion beam and e-beam evaporation system) provided by the “Core Facility Center of National Cheng”.
Disclosures
The authors declare no conflicts of interest.
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.
Supplemental document
See Supplement 1 for supporting content.
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