Janus vortex beams realized via liquid crystal Pancharatnam–Berry phase elements

Abstract. Emerging as a family of waves, Janus waves are known to have “real” and “virtual” components under inversion of the propagation direction. Although tremendous interest has been evoked in vortex beams featuring spiral wavefronts, little research has been devoted to the vortex beam embedded Janus waves, i.e., Janus vortex beams. We propose a liquid crystal (LC) Pancharatnam–Berry (PB) phase element to demonstrate the realization of the Janus vortex beams and the modulation of the associated orbit angular momentum (OAM) and spin angular momentum (SAM). The generated Janus vortex beams show opposite OAM and SAM states at two distinct foci, revealing a spin-orbit interaction during propagation enabled by the LC PB phase element, which may play special roles in applications such as optical encryption and decryption. Other merits like reconfigurability and flexible switching between Janus vortex beams and autofocusing or autodefocusing vortex beams additionally increase the degree of freedom of manipulating vortex beams. This work provides a platform for tailoring complex structured light and may enrich the applications of vortex beams in classical and quantum optics.


Introduction
Manipulating light fields to generate spatially structured light beams 1 has been attracting significant interest. In particular, vortex beams have gained a great deal of attention owing to their outstanding performances. Different from ordinary Gaussian beams, vortex beams feature helical wavefronts described by expðimθÞ, where m is the topological charge and θ is the angular coordinate. 2,3 The spiral phase of the vortex beam, on the one hand, leads to the phase singularity at the center, resulting in the annular intensity distribution with a central dark region. On the other hand, it gives rise to the existence of the orbital angular momentum (OAM) with each photon carrying the value of mℏ (the quantum unit ℏ is the reduced Planck's constant), which distinguishes it from the spin angular momentum (SAM) related to the circular polarization state of light. These special characteristics endow vortex beams with a broad range of applications, such as optical trapping with controlled rotation, [4][5][6] laser processing with chiral microstructures, 7 stimulatedemission-depletion fluorescence microscopy with super high resolution, 8,9 quantum informatics with a large capacity, 10,11 optical encryption with high security, 12 and even vortex coronagraphs for the extrasolar planet detection. 13 It is conceivable that enriching the ways to modulate vortex beams will increase the degree of freedom of their applications in multidisciplinary fields.
Another interesting type of structured light beams that has emerged in recent years is Janus waves. First proposed by Papazoglou et al., Janus waves are a family of waves having "real" and "virtual" components located at two symmetrical foci. 14 The "real" wave and the "virtual" wave are conjugate to each other under inversion of the propagation direction, just like the Roman mythological deity Janus to some extent (as shown in the upper left illustration of Fig. 3). A typical example of Janus waves is the circular Airy beam (CAB), 14 which is widely known for its abruptly autofocusing property, i.e., it will keep a relatively low intensity before it abruptly autofocuses with a peak intensity increased by orders of magnitude. Owing to this distinctive property, CABs have played an important role in laser ablation, 15 microparticles manipulation, 16 generations of intense light bullets, 17 and stimulated Raman scattering, 18 and are excellent candidates for biomedicine and nanosurgery. 15 Besides the well-known real focus in the forward propagation direction, identified as the Janus wave, the CAB also has a virtual focus in the inverse direction. Generally, a lens is utilized to make the "virtual" component of the Janus wave accessible. 14 In addition to the aforementioned applications of the CAB, the Janus wave with two distinct foci could also be applied for phase memory preserving harmonics in strong field science. 19 Taking advantage of Janus waves to drive the spiral phase is a novel way to modulate vortex beams. However, few works have been devoted to this topic so far, one of which is the study of the propagation dynamics of the proposed Janus vortex waves only theoretically. 20 Therein, the spiral phases on the two focal planes have the same rotation direction, which means that the topological charge or the OAM remains unchanged, lacking a certain degree of modulation flexibility. In this work, specially designed liquid crystal (LC) Pancharatnam-Berry (PB) phase optical elements, Janus-q-plates, are proposed to experimentally realize the generation of Janus vortex beams. LCs have been widely used in display and nondisplay fields due to their outstanding electro-optical property, and the LC-mediated PB phase optical elements are remarkable candidates for multidimensional manipulation of light waves with high quality and high efficiency. [21][22][23][24][25][26][27][28] More importantly, the generated Janus vortex beams show opposite OAM states and SAM states at the two foci, exhibiting a spin-orbit interaction during propagation enabled by the LC PB phase element Janus-q-plate. In addition, the switch of the Janus vortex beam to a single circular Airy vortex beam 29,30 can be flexibly realized by just altering the input polarization state or removing a lens from the optical path. Our work provides an effective way to generate Janus vortex beams and a flexible approach for multidimensional manipulation of vortex beams, which will increase the degree of freedom for the applications of vortex beams in quantum informatics, optical communication, particle manipulation, laser processing, and so on.

Results and Discussion
As has been mentioned above, the CAB belongs to the Janus waves family. 14 The generation of CABs is commonly based on the Fourier transform (FT) patterns 15,31 or the holograms calculated by the interference of a reference plane wave and the target CABs. 16 Recently, a relatively easier way was proposed based on the superimposition of specific phase structures, 32 realizing the efficient generation 32 and modulation 33 of CABs. The superimposed phase structure is derived from the asymptotic analysis of the FT of a CAB whose characteristic phase term is composed of a cubic term and a linear term. Herein, on the basis of the radial cubic phase [as shown in Fig. 1(a)] and the radial linear phase [as shown in Fig. 1(b)], we purposefully add the spiral phase [as shown in Figs. 1(c), 1(e), and 1(g)] to introduce the vortex beam component. Figure 1 shows the design principle of the newly superimposed phase structure, which can be expressed as where β is the cubic phase modulation coefficient, , d is the period of the radial linear phase, m is the topological charge of the vortex beam, and θ ¼ arctanðy∕xÞ. Figures 1(d), 1(f), and 1(h) are superimposed phase structures with m ¼ 1, 2, and 6, respectively, in which the color variation from black to white represents the phase value from 0 to 2π. To improve the generation efficiency, PB phase modulation is further introduced. PB phase elements are usually composed of anisotropic media with optical axes distributing spatially, 23 and the amount of the PB phase modulation is proportional to the orientation angle of the optical axis. Herein, we choose nematic LC material E7 as the anisotropic medium, and the orientation of the LC director (i.e., optical axis of LC) is set to follow: where q ¼ m∕2 is half of the topological charge. Therefore, taking the geometrical characteristic of the PB phase into consideration, Fig. 1 can also be regarded as the simulated LC director distributions in which black to white corresponds to the orientation varying from 0 to π. As the first two terms of Eq.
(2) are related to the generation of CAB-based Janus waves, and the third term also describes the structure of the q-plate, 34 we define the designed LC PB phase element as Janus-q-plate. The diffraction property of the Janus-q-plate can be analyzed through Jones calculus. For the PB phase optical element under the half-wave condition with the optical axis orientation following α, the Jones matrix can be expressed as 34 For an incident Gaussian beam with a linear polarization state, its electric field can be expressed as where θ in is the angle between the incident polarization direction and x axis, and jLi ¼ 1∕ ffiffi ffi 2 p ½1; i T and jRi ¼ 1∕ ffiffi ffi 2 p ½1; −i T represent left circular polarization (LCP) and right circular polarization (RCP), respectively. After passing through the LC Janus-q-plate, the Gaussian beam will be transformed into From Eq. (5), we can see that the desired PB phase modulation component consists of two parts with orthogonal SAM states: RCP component and LCP component. For the RCP component, it exhibits a helical phase front expðimθÞ and a linear phase enhanced cubic phase modulation exp½iðβr 3 þ 2πr∕dÞ, revealing the generation of autofocusing CAB carrying mℏ OAM. However, for the LCP component, the phase modulation is conjugate, revealing an opposite OAM carried by the conjugate CAB. A lens can be used to bring the virtual focus of the conjugate CAB into real space 14 together with the OAM and SAM. In this case, the positions of the focused beams z 1 and z 2 can be calculated via the lens equation where z CAB and z CABÃ are the autofocus positions of the CAB and conjugate CAB before being modulated by the lens, respectively, and f L is the focal length of the autofocusing CAB, conjugate CAB, and the lens. As a result, two-foci vortex beams with opposite OAM carried by the autofocusing CAB will be formed, realizing the generation of the Janus vortex beam.
To implement the space-variant LC director distribution following Eq. (2), the rewritable sulfonic azo-dye SD1-based photoalignment technology and the digital micromirror device (DMD) based dynamic microlithography system are employed. 35 Under linearly polarized UV light irradiation, the SD1 molecules tend to orient their absorption oscillators perpendicularly to the incident polarization direction. As the alignment layer, SD1 molecules will spread their orientation to adjacent LC molecules by intermolecular interaction. Patterned UV light with a specific polarization direction can be created through the DMD (resolution: 1920 × 1080; pitch: 10.8 μm) exposure system. After an 18-step, five-time partly overlapping dynamic exposure process, 36 the Janus-q-plate pattern in Fig. 1 can be precisely duplicated onto the LC cell, which is composed of two SD1-coated indium-tin-oxide glass substrates separated by 6 μm spacers. After capillary filling of nematic LC E7, a director-variant PB phase element LC Janus-q-plate can thus be formed.
Figures 2(a)-2(c) exhibit the micrographs of the Janus-qplates with q ¼ 0.5, 1, and 3, respectively, which are observed under a polarized optical microscope. The greenish color of the micrographs is due to the corresponding wavelength meeting with the LC sample's half-wave condition. 37 The bright regions correspond to the LC directors at about π∕4 or 3π∕4 to one of the polarizers of the microscope, and the dark regions correspond to the angles of about 0 or π∕2. As the orientation of the LC directors gradually changes from 0 to π, the brightto-dark regions alternate twice, leading to the number of spiral fringes in the micrographs being twice that of the phase patterns shown in Figs. 1(d), 1(f), and 1(h). Therefore, 4q times, i.e., 2, 4, and 12 times bright-to-dark alternations are observed in Figs. 2(a)-2(c), respectively. As a comparison, it can be found that with the increase of q, the spiral distortion at the central part of the phase structure or the Janus-q-plate sample becomes larger. However, the consecutive variation of the brightness distributed in designed structures also confirms the continuous and periodic orientation of LC molecules, ensuring the high quality of the PB phase Janus-q-plates.
To further verify the performance of the Janus-q-plate and carry out the generation and detection of Janus vortex beams, an optical setup illustrated in Fig. 3 is adopted. A polarizer and a λ∕4 plate are placed in sequence to adjust the polarization state of the incident laser beam (wavelength λ ¼ 632.8 nm). A lens (Lens 1) with focal length f L1 ¼ 10 cm is placed at a distance f L1 from the LC Janus-q-plate to perform the FT. Another lens (Lens 2) with focal length f L2 ¼ 15 cm is placed at a distance f L1 from Lens 1 to bring the conjugate virtual vortex beam to the second focus, whose OAM is opposite to that of the vortex beam at the first focus, just as the inset shows. A CCD is used to capture the intensity distribution of the generated Janus vortex beams at different propagation distances, and the backplane of Lens 2 is defined as the initial observation point (z ¼ 0). Since for conventional vortex beams the radius increases with the topological charge jmj, 38-40 the radius of the generated Janus vortex beams will also increase with the q value of the Janus-q-plate. However, for a vortex beam driven by the CAB whose intensity will increase by orders of magnitude at the focus, the smaller the topological charge is, the less visibility of the dark region of the focused vortex beam will show. Therefore, in order to better demonstrate the generation and characterization of Janus vortex beams and simultaneously reflect the robustness of the LC sample, we mainly take the Janus-q-plate of q ¼ 3 with more complex structure for the following optical test. Figure 4 exhibits the experimental and simulated propagation dynamics of the proposed Janus vortex beam with Janus-q-plate of q ¼ 3.   The intensity distribution shown in Fig. 4(e) is captured at the focal plane or the FT plane of Lens 2, which is a result of the interference of the conjugate waves 14 and will be analyzed in detail below. After the second focal plane of the Janus vortex beam, the light intensity becomes weaker and weaker, and the radius keeps expanding [Figs. 4(h)-4(i)]. In order to better describe the propagation trajectory of the Janus vortex beam, the dependency of the size of the inner main ring or dark core on the propagation distance is plotted in Fig. 4(j). The fitted curves intuitively reveal that the Janus vortex beam first accelerates inward and parabolically, leading to the appearance of the first focal point at z ¼ 12 cm. Regardless of the size of the dark core at z ¼ 15 cm, the Janus vortex beam focuses again at z ¼ 18 cm and then diverges. The experimental trajectory and the above analyses are consistent with the theoretical trajectory shown in Fig. 4(j) and simulated side-view propagation shown in Fig. 4(k). Fortuitously, the simulated propagation dynamics better display the formation of the interference field of the conjugated waves at about z ¼ 15 cm. As analyzed above, the measured positions of the two foci are at z ¼ 12 cm and z ¼ 18 cm, respectively, and the theoretical values can be calculated according to Eq. (6). Herein, f L ¼ f L2 ¼ 15 cm, and the autofocus position of the CAB can be obtained via the focal length calculation, 32 where w is the width of the LC Janus-q-plate. With the designparameters, f CAB is calculated to be 78 cm. Considering the imaging formulas for the lens, z CABÃ ¼ −f CAB ¼ −78 cm, and z CAB ¼ f CAB ¼ 78 cm. Therefore, the theoretical values of the two-foci positions are z 1 ¼ 12.58 cm and z 2 ¼ 18.57 cm, respectively. The experimental results (12 cm, 18 cm) match well with the theoretical calculations considering the experimental errors. It is worth mentioning that for only LCP or RCP incidence, which can be carried out by tuning the angle between the c axis of the λ∕4 plate and the transmission direction of the polarizer to be 45 deg or −45 deg, a single circular Airy vortex beam focused at z 1 or z 2 can be individually obtained. Furthermore, by removing Lens 2 in the optical setup, autofocusing or an autofocusing Airy vortex beam can also be acquired, as shown in the simulation result in Fig. 4(l). Therefore, the switch between the Janus vortex beam and a single circular Airy vortex beam in the autofocusing or autodefocusing state can be flexibly realized by adjusting the input polarization or the number of the lens in the optical path, increasing the degree of freedom for the manipulation of vortex beam-based light waves.
To further verify the OAM property of the Janus vortex beam, the digital holographic imaging system 41,42 is applied to measure the phase distributions during propagation. The phase distributions at three featured positions, namely the two focal planes of the Janus vortex beam and the FT plane of Lens 2, are selected to be shown in Fig. 5. Figures 5(d)-5(f) are the experimental results, which are basically consistent with the corresponding simulations shown in Figs. 5(a)-5(c). From the images in the first and third columns, we can see that spiral phases are obviously presented. At the first focal plane shown in Figs. 5(a) and 5(d), the spiral phase varies counterclockwise from 0 − 2π six times, revealing a þ6 topological charge or a þ6ℏ OAM carried by the vortex beam. On the contrary, at the third focal plane shown in Figs. 5(c) and 5(f), the spiral phase varies clockwise from 0 − 2π also six times, revealing a −6ℏ OAM carried by the vortex beam. Between these two positions at the FT plane of Lens 2 shown in Fig. 3, the inner phase distribution exhibits divergent phase values with 12 times π leaps, revealing a vector beam characteristic with polarization order 6. 41,43 In addition, the SAM property is also analyzed via the digital holographic imaging system and the Stokes method. Figures 6(a) and 6(b), 6(d) and 6(e) are simulated/measured polarization distributions of the Janus vortex beam at the first and second focal planes, respectively, with corresponding intensity distributions shown in the background. The red or green circles in the central region of the image vividly show the RCP or LCP state at the corresponding focus. For the simulation shown in Figs. 6(a) and 6(d), the ellipses in the outer ring region result from the superposition of the focused RCP/LCP component and the unfocused LCP/RCP component [the LCP component in the outer ring region before converging at the second focal plane and the RCP component that diverges after the first focal plane, which can be seen from the simulated side-view intensity distribution in Fig. 4(k)]. The overall experimental results are coincident with the simulations except for some measurement errors and can be further optimized by improving the quality of the incident laser beam and blocking the stray light in the experimental environment to enhance the signalto-noise ratio received by the CCD. Moreover, the normalized Stokes parameter S 3 is calculated to better demonstrate the SAM feature. Herein, S 3 is defined as S 3 ¼ sinð2εÞ, where ε is the ellipticity. For RCP, ε ¼ π∕4 and S 3 ¼ 1; for LCP, ε ¼ −π∕4 and S 3 ¼ −1. For linear polarization, ε ¼ 0 and S 3 ¼ 0. From the detected S 3 distributions shown in Figs. 6(c) and 6(f), we can see that S 3 mainly presents þ1 value at the first focal plane while −1 value at the second plane, revealing the RCP and LCP light carrying an SAM of þℏ and −ℏ per photon, respectively. However, for the light field detected at the FT plane of Lens 2, as shown in Fig. 6(i), S 3 is basically 0, reflecting a linear polarization distribution. As analyzed above, the light field herein is a result of the interference of the two conjugate waves, i.e., the vortex beams with opposite SAM and OAM, leading to the production of the vector beam. The detected 12 arms displayed in the intensity distribution analyzed by a polarizer shown in Fig. 6(h), which match well with the simulation shown in Fig. 6(g), further verify the polarization character of the six-order vector beam. Despite this middle vector beam state, during the propagation process of the Janus vortex beam, the change of SAM together with OAM can be written as jRij þ 6i → jLij − 6i, where the phase factor expðimθÞ is recorded as jmi for brevity. 44 In other words, the auto-transition between different SAM and OAM values is realized during the propagation of the Janus vortex beam, exhibiting a spin-orbit interaction enabled by the Janus-q-plate. Additionally, the rewritability of SD1 35 and LC Janus-q-plate can endow the Janus vortex beam with the ability to refocus at other required positions, providing a multidimensional way to manipulate vortex beams. Such functionalities may enrich the applications of the vortex beams in quantum informatics, particle manipulation, laser processing, and even optical encryption and decryption.

Conclusions
We have proposed an LC PB phase optical element Janus-qplate and experimentally demonstrated the generation of Janus vortex beams. The design of the Janus-q-plate is based on the superimposition of specific phase structures, and the fabrication is complemented by the SD1-based photoaglignment technology and the DMD-based dynamic exposure process. The Janus vortex beam with two foci located at designed positions is realized, and the experimental propagation dynamics match well with the simulations. Both the OAM and SAM properties at featured positions are characterized. Our study reveals that the vortex beams at the two foci exhibit opposite OAM and SAM values, and the interference of these two conjugate waves leads to the appearance of a vector beam located between the two focal planes. The spin-orbit interaction during the propagation of the Janus vortex beam enabled by the Janusq-plate may play a special role in applications like optical encryption and decryption. In addition, the switch between the Janus vortex beam and a single autofocusing or autodefocusing circular Airy vortex beam can be flexibly realized by adjusting the input polarization or the number of lenses in the optical path, and the structure parameters of the LC Janus-qplate and the focus property of the Janus vortex beam can be reconfigured and customized as required. Based on the regime of the Janus waves, our work increases the degree of freedom of manipulating vortex beams associated with OAM and SAM, which we believe will promote multidimensional applications of vortex beams in quantum informatics, optical communication, particles manipulation, laser processing, and so on.
interests include light field manipulation, liquid crystals, and photoalignment technology.
Yuan Zhang is currently working at the Chinese Flight Test Establishment, Xi'an, China. She received her MS degree from NPU in 2021. She is the author of two journal papers and holds one issued patent. Her research interests include liquid crystals, vortex beams, circular Airy beams, etc.
Haozhe Xiong received his BS degree from NPU in 2020. He is currently a graduate student of NPU. His research interests include liquid crystals, vortex beams, circular Airy beams, etc.
Sheng Liu is currently an associate professor at the School of Physical Science and Technology, NPU, Xi'an, China. He received his BS and PhD degrees from NPU in 2005 and 2011, respectively. He is the author of more than 90 journal papers. His current research interests include the physical properties and propagation modulation of structured light fields, such as vortex beam, vector beam, non-diffracting beam, etc.
Peng Li is currently an associate professor at the School of Physical Science and Technology, NPU, Xi'an, China. He received his BS and PhD degrees from NPU in 2008 and 2013, respectively. He is the author of more than 80 journal papers. His current research interests include multi-dimensional modulation of light field, information processing based on metasurface, and its application.
Dandan Wen is currently a professor at the School of Physical Science and Technology, NPU, Xi'an, China. He received his BS and MS degrees from NPU in 2009 and 2012, respectively, and his PhD from Heriot-Watt University (UK) in 2017. He worked as a research fellow at the Department of Electrical and Electronic Engineering, University of Melbourne (Australia) between 2017 and 2021. He is the author of more than 20 journal papers and conference papers including nature photonics, nature communications, advanced materials, nano letters, and so on. His current research interests include optical metasurfaces, nanooptics, and optoelectronics.
Jianlin Zhao is a professor at the School of Physical Science and Technology, NPU, Xi'an, China. He received his BS and MS degrees from NPU in 1981 and 1987, respectively. He received his PhD in optics from Xi'an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi'an, in 1998. He is the director of the Shaanxi Key Laboratory of Optical Information Technology, and the MOE Key Laboratory of Material Physics and Chemistry under Extraordinary Conditions. His research interests include light field manipulation, imaging, information processing, and applications.