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Abstract
Optical vortex array has drawn widespread attention since the boom of special applications such as molecular selecting and optical communication. Here, we propose an integrated phase-only scheme to generate multiple multiplexed vortex beams simultaneously, constituting a multiplexed vortex state array, where the spatial position, as well as the corresponding orbital angular momentum (OAM) spectrum, can be manipulated flexibly as desired. Proof-of-concept experiments are carried out and show a few different multiplexed vortex state arrays that fit well with the simulation. Moreover, regarding the array as a data-carrier, a one-to-many multicasting link through multi-state OAM shift keying, a high-dimensional data coding, is also available in free space. In the experiment, four various OAM states are employed and achieve four bits binary symbols, and finally distribute three different images to three separate receivers independently from the same transmitter, showing great potential in the future high-dimensional optical networks.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recently, structured beams have attracted more and more attention due to their unique physical properties, novel physical effects and potential application scenarios, developing into the hotspot in the field of optics and optoelectronics [1–3]. Optical vortex beam with helical wave front is a kind of two-dimensional transverse structured light due to its inhomogeneous phase distribution at the plane perpendicular to the propagating axis [2,3]. In 1992, Allen et al. connected OAM with optical vortex beams, proving that the vortex beam characterized by a phase term exp(ilφ) can carry lħ OAM, where l refers to the topological charge (also called OAM state), φ is the azimuthal angle and ħ is the reduced Planck constant [4,5]. Moreover, the topological charge l is the eigenvalue of vortex beams, which can be taken as infinite integers, recording both of the direction and number of intertwined helices. Driven by a phase singularity at the beam center, vortex beams feature a doughnut intensity distribution, helical phase fronts and OAM, giving rise to tremendous applications in various domains, such as quantum entanglement [6–9], rotation detection [10,11], optical trapping [12–14], large-capacity data transmission and so on [15–17].
Plenty of approaches to generating optical vortex beams have been put forward and demonstrated, for instance, mode selection inside a resonator [18,19] and mode conversion outside a resonator [20–22]. On the basis of optical vortex beams generation, optical vortex array is further proposed by tailoring the spatial dimension, allowing the simultaneous generation of multiple various optical vortex beams. In other words, optical vortex array is the result of manipulating two degrees of freedoms, the OAM and spatial position, which consists of multiple vortex beams carrying identical or different OAM states in distinct spatial positions, and thus has multiple phase singularities [23]. The unique properties inspire optical vortex array exploited in large-capacity optical communication [24–27], high-resolution far-filed microscopy [28], laser multi-point machining [29,30], molecular selecting [31–33] and object displacement measurement [34]. Creating an optical vortex array with favorable performance lays vital basis for all of the above applications. At present, there are three commonly used methods to generate optical vortex arrays, which are multi-beam interference [35–45], special microstructure material method [46–51] and specially designed diffraction gratings [25,52–57]. Multi-beam interference can produce connected optical vortex arrays with various structures, such like circular optical vortex array [36], flower-shaped optical vortex array [40], high-order optical vortex lattice [45] and so on, which are usually employed to manipulate the movement of particles. The special microstructure material method relies on the material structure like defects in liquid crystals [46–48], which brings complexity and difficulty of modulation once they are fabricated. As for the specially designed diffraction gratings, it is undoubtedly the most flexible and convenient way to obtain the vortex array, which not only simplifies the setup of system but also avoids the cumbersome manufacturing process. The gratings modulate the phase and amplitude of Gaussian beam directly and convert it into a vortex array, the most classic of which is the Dammann vortex grating [52]. Based on this idea, various optical vortex arrays have been proposed, including vortex array with selective mode position, mode states and relative power [25], vector vortex array [53–55] and perfect vortex arrays [56,57]. Other generation methods include mode conversion [58], forming spontaneously within a resonator [59], and metasurfaces [60–65], etc.
However, previous optical vortex arrays mainly concentrate on generating a single OAM state in different spatial positions. In fact, taking advantage of inherent orthogonality between different OAM states, generating multiple OAM states in various spatial positions simultaneously may bring a lot of benefits. For example, a multiplexed vortex beam containing multiple OAM states can be employed to realize high-dimensional data distribution based on multi-state OAM shift keying [66], where only M OAM states are required to encode a M-bit symbol. Expanding it in spatial dimension, generating multiple multiplexed vortex beams simultaneously, namely a multiplexed vortex state array, single-channel data transmission can be extended to multicasting, which enhances coding efficiency of existing OAM encoding-based multicasting link [26] significantly as well where M OAM states can merely encode a symbol carrying log2M bits. Additionally, above encoding systems usually adopt iterative algorithms to obtain relevant holograms, whose calculation process is complicated and time-consuming. Specificly, when iterative algorithms are chosen to calculate the desired hologram, the whole process needs several iterations to converge, and each iteration requires measuring the OAM spectrum of the light field in each diffraction order.
In this paper, a multiplexed vortex state array composed of multiple multiplexed vortex beams with selective spatial positions and OAM spectrum in each channel is presented, whose concept is sketched in Fig. 1. That is, three degrees of freedom, the spatial position, the OAM state, and amplitude of each OAM component, are selectively tailored on demand. The key to create such array is a specially designed phase-only hologram, which consists of two parts, one is a checkerboard phase for amplitude modulation, the other is an ordinary phase for phase modulation. Based on complex amplitude modulation, all kinds of multiplexed vortex state arrays as desired can be achieved simply and rapidly as long as the transmission functions of the target arrays are determined in advance. In the experiment, a few multiplexed vortex state arrays are shown and the mean square error (MSE) is evaluated as 1.06 × 10−3 among eleven OAM channels, conforming to the trend of simulation. Furthermore, employing an array as data-carrier, we demonstrate a one-to-many multicasting link through multi-state OAM shift keying, where four various states are utilized to encode a four-bit symbol, achieving a high-dimensional encoding. As a result, three grayscale images are transferred simultaneously from one transmitter to 3 distinct receivers separately with a total bit error rate (BER) of 4.16 × 10−4.
2. Hologram designing
Generally speaking, to generate a vortex beam with a single OAM state, the most straightforward way is passing a Gaussian beam through a coaxially placed element with a transmission function of exp(ilφ). Moreover, a multiplexed vortex beam with multiple OAM states can be created via an element with a transmission function of ΣkCkexp(ilkφ), with k = 1, 2, …, K [67]. For the sake of generating various multiplexed vortex beams in spatial positions simultaneously, namely the multiplexed vortex state array, the desired transmission function in Cartesian coordinates can be expressed as
Notably, the desired transmission function Td(x,y) appears in a complex form, implying both amplitude and phase must be modulated. To simplify it, lots of iterative algorithms [67–70] are proposed to search for an approximate phase-only function to replace it. Nevertheless, these approaches usually cost long time for lots of iterations. Considering the speed and complexity of grating generation, we divide the expected transmittance function into two parts, one is amplitude modulation and the other is phase modulation. Specifically, we adopt checkerboard phase to simulate amplitude modulation and then overlay the phase modulation, realizing the complex amplitude modulation by a phase-only grating within a tolerable error range.
The standard checkerboard phase, consisting of phase unit [0, π; π, 0] arranged horizontally and vertically, can redistribute the energy in the zero order into higher orders [71]. Hence, it is able to mimic amplitude modulation. Changing the phase value in the phase unit of original chekerboard phase, 0 remains, while π is replaced by Δϕ. In this way, the average modulation effect of the improved checkerboard phase can be approximately expressed as [72]
where cos(Δϕ/2) is the amplitude modulation achieved by the checkerboard phase under this distribution law, and exp(iΔϕ/2) is the redundant phase introduced by the checkerboard phase. Therefore, we can obtain the phase value of the non-zero phase unit in the checkerboard phase by taking modulo of our desired transmittance function after normalization:The phase distribution Pa(x,y) of checkerboard can be obtained by alternate arrangement of phase value 0 and Δϕ(x,y) into two-dimensional pattern. By phase distribution Pa(x,y), the amplitude modulation of the desired transmittance function described in Eq. (1) can be accomplished.
Meanwhile, taking argument of the expected transmittance function, the phase distribution Pp(x,y) for phase modulation can be easily gotten,
in which arg(ς) represents taking argument of ς. Ultimately, by removing unwanted phase modulation Δϕ(x,y)/2, integrating Pa(x,y) and Pp(x,y), a multiplexed vortex state array can be acquired when a Gaussian beam illuminates. Figures 2(a)–2(e) illustrate the integrating progress. Figure 2(d) displays one of the described holographic gratings, and its corresponding far-field diffraction pattern [Fig. 2(f)] when a Gaussian beam is incident.
Fig. 2. Integrating process of hologram to produce the multiplexed vortex state array. (a) Checkerboard phase. (b) Redundant phase. (c) Phase modulation part. (e) Details of the checkerboard phase. (f) Simulated far-filed diffraction pattern when a Gaussian beam passes through the hologram in (d).
3. Experiment verification
3.1 Experimental setup
Figure 3 shows the experimental setup for generating the multiplexed vortex state array. A distributed feedback laser diode is employed as the source to produce a fundamental Gaussian beam at 1617 nm. Then the Gaussian beam is coupled into a single-mode fiber and emerged from a collimator with a diameter of 3 mm. After passing through a half wave plate (HWP) and a polarized beam splitter (PBS) successively, the beam is transformed into horizontally linear polarization to match the demand of phase-only liquid-crystal spatial light modulators (SLMs) (Holoeye, PLUTO-TELCO-013-C) used here. By encoding a specially designed hologram into SLM1, a multiplexed vortex state array is obtained. The 4-f system (L2 & L3) along with an aperture stop (AS) and SLM2 are utilized to filter each multiplexed vortex beams in the array and analyze their OAM spectra. Fourier transformation of optical fields is finished by Fourier lens (L1 & L4). Both of infrared CCDs cameras (Xenics, Bobcat-320-star) are placed to observe far-field diffraction patterns.
Fig. 3. Experimental setup. DFB, distributed feedback laser diode; SMF, single mode fiber; Col., collimator; HWP, half wave pate; PBS, polarized beam splitter; SLM1&SLM2, liquid-crystal spatial light modulator; BS, beam splitter; L1∼L4, lenses; CCD1&CCD2, infrared CCD camera; AS, aperture stop. (a) Hologram to generate the various multiplexed vortex state arrays. (b) Anti-spiral phase holograms.
3.2 Vortex array generation
In the experiment, we test the generation of vortex arrays with different OAM states multiplexed in various diffraction orders firstly. The results and corresponding phase profiles of hologramss are displayed in Fig. 4. Figure 4(e) shows a multiplexed vortex state array with the far-field pattern like a smiley face, which is made up of three 2-fold multiplexed vortex beam(for two, l1=-3, l2 = 3; for one, l1=−1, l2 = 1) and four single-state vortex beams(half l = 1, half l=−1). Meanwhile, a snowflake shaped pattern is given in Fig. 4(f). It contains seven beams, which are two single-state vortex beam(l=−1, l = 2), two 2-fold (l1=−2, l2 = 2; l1=−1, l2 = 1), two 3-fold (l1 = 1, l2=−1, l3 = 2; l1=−1, l2 = 2, l3 = 5) and a 4-fold multiplexed vortex beams(l1=−1, l2 = 1, l3 = 2, l4=−2), respectively. Besides, many other multiplexed vortex state arrays could be found in Visualization 1.
Fig. 4. Generated multiplexed vortex state arrays. (a),(b) Holographic gratings. (c),(d) Simulated far-field diffraction pattern. (e),(f) The corresponding experimental far-field diffraction pattern.
What’s more, to verify whether the multiplexed vortex beam in the array comprises desired OAM states, we take a hologram for example and analyze the OAM spectrum of its far-field diffraction patterns. The grating periods in x and y directions are both 0.4 mm. When a Gaussian beam passes through above grating, it will be separated into four channels, which are Gaussian beam (l = 0), 2-fold multiplexed vortex beams (l1 = 2, l2=−2), 3-fold multiplexed vortex beams (l1 = 2, l2 = 3, l3 = 4) and 5-fold multiplexed vortex beams (l1=−4, l2=−2, l3=−1, l4 = 3, l5 = 5) respectively. The OAM spectrum of each channel is measured with back-converted method. By loading a series of anti-spiral phases (l={−15,−14,…,14,15}) into SLM2, the projections under different substrates (exp(ilφ), l={−15,−14,…,14,15}) can be obtained in the form of a central bright spot whose intensity represents the intensity of each OAM state (l={15,14,…,−14,−15}). The experimentally recorded intensity pattern along with corresponding simulated and experimental OAM spectrum of each channel are given in Fig. 5. It can be clearly seen that every desired OAM state of each channel appears in experiment, fitting well with the simulation. And the total MSE of the four OAM spectra equals to 1.06 × 10−3.
Fig. 5. Experimentally captured far-field pattern of a multiplexed vortex state array and corresponding simulated and experimental OAM spectrum of each spatial position. (a) Far-field pattern of the multiplexed vortex state array. (b)-(e) OAM spectrum of one-fold, two-fold, three-fold and five-fold multiplexed vortex beams.
3.3 Multicasting link based on multi-state OAM encoding
Furthermore, by employing multiplexed vortex state arrays, we successfully demonstrate a one-to-three multicasting link based on multi-state OAM shift keying in free space experimentally. As exhibited in Fig. 6(a), the experimental facilities in the transmitter are identical to that of array generation. SLM1 is employed to carry out multicast OAM encoding. By switching holograms loaded on SLM1 continuously, a coded time-varying multiplexed vortex state array sequence is generated, which records the input signals. Here, the array contains three diffraction orders. Regarding every diffraction order as a channel, there are three channels to send information, which are labeled as channel 1 to channel 3. After propagating in free space for 1.5 meters, the three encoded multiplexed vortex beams reach three identical receivers, respectively. Each receiver is composed of three parts: a SLM, a lens and an infrared CCD camera. To decode the data transformed, we adopt a Dammann vortex grating (DVG) [52] assisted with the gray-scale algorithm [73]. Such DVG can produce a 3 × 3 optical vortex array with OAM state −4∼4, and the intensity of each OAM state is the same.
Fig. 6. One-to-three multicasting based on multi-state OAM shift keying. (a) Experimental setup. DFB, distributed feedback laser diode; SMF, single mode fiber; Col., collimator; HWP, half wave pate; PBS, polarized beam splitter; SLM1∼SLM4, liquid-crystal spatial light modulator; L1∼L3, lenses; CCD1∼CCD3, infrared CCD camera; DVG, Dammann vortex grating. (b) A group of hexadecimal codes and its corresponding decoding pattern captured by CCDs. (c) Three grayscale images to be transmitted and corresponding images recovered by each receiver separately.
Considering multi-state OAM shift keying [66], the presence or absence of any OAM state contained in a multiplexed vortex beams can represent a binary bit ‘1’ or ‘0’, which means that M different OAM states are able to encode a M-bit symbol, carrying M bits information. In the demonstration, we select 4 various OAM states (l1=−3, l2=−1, l3 = 1, l4 = 3) for each channel, thus 4 bits information, namely a hexadecimal symbol, for each of the three channels can be encoded by one grating. To vividly evaluate the performance of the one-to-three multicasting link, we send a different 50 × 50 grayscale image from one transmitter to each of the three receivers. The three images include a tiger, the emblem of the Olympic Winter Games Beijing 2022 and the badge of Beijing Institute of Technology. Since each pixel of a grayscale image contains 8 bits information, 50 × 50 × 50 = 20 kilobits data is transferred in each channel which can be mapped to a sequence of 20000/4 = 5000 holograms generated using the method we proposed above. By loading such holographs into SLM1, the multicasting is achieved. Figure 6(b) exhibits a group of hexadecimal symbols and their corresponding decoding pattern captured by the three receivers, respectively. Apparently, the OAM states in three channels are different, representing different hexadecimal symbols. Figure 6(c) shows the received three images recovered by each receiver separately, which are almost identical with the transmitted images, indicating the successful multicasting of various signals. Specifically, the total bit error rate (BER) of the three channels reaches 4.16 × 10−4. It is noteworthy that the bit error rate is affected by decoding performance of the receivers to a large extent. If a more accurate OAM measurement method like mode sorter [74,75] or deep learning [76] is used in the multicasting system, its bit error will drop significantly.
Besides, it is worth noting that this one-to-three multicasting link is demonstrated on the indoor condition and only transmitting 1.5 meters. As the transmission distance increases, multiplexed vortex beams will suffer from the turbulent atmosphere, resulting in spiral phase distortion and OAM spectrum broadening, which may lead to an inaccurate identification of the transmitted OAM state and the increase of BER. Under the circumstances, adaptive compensation methods [77–79] are expected to address such issue.
4. Conclusions
In summary, we propose an approach both theoretically and experimentally to generating a multiplexed vortex state array with selective spatial positions and customizable OAM spectrum in each position by a single phase-only hologram. Utilizing checkerboard phase to simulate amplitude modulation, the phase-only hologram can achieve complex amplitude modulation quickly and simply. When a Gaussian beam passing through the hologram from the center, multiple multiplexed vortex beams exit simultaneously. By setting different parameters of the hologram, we are qualified to tailor the number of multiplexed vortex beams as well as their corresponding OAM spectrum. As a proof of concept, different multiplexed vortex state arrays are successfully generated and the mean square error (MSE) of OAM spectrum of the generated arrays among eleven OAM channels is 1.06 × 10−3. Furthermore, taking the multiplexed vortex state array as a data-carrier, we demonstrate a multi-state OAM encoding multicasting in free space, where three various grayscale images are encoded as 4 bits per symbol and sent to three separate receivers independently. Compared with previous OAM encoding-based multicasting link, adopting the multiplexed vortex state array improves OAM-encoding efficiency along with the transmission rate of information to some extent. This work shows great potentials in large-capacity networking and the rate of data transmission can be further improved as more OAM states are chosen or more degrees of freedoms are introduced in the future. In addition to optical communication, the flexible way to create multiplexed vortex state array may path the way for domains where multiple multiplexed OAM beams are required, like high-dimensional quantum key distribution, high-resolution microscopy, multiple particle manipulation.
Funding
National Natural Science Foundation of China (11834001, 61905012); National Defense Basic Scientific Research Program of China (JCKY2020602C007); National Postdoctoral Program for Innovative Talents (BX20190036).
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.
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