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Abstract
Optical vortex beams, with phase singularity characterized by a topological charge (TC), introduces a new dimension for optical communication, quantum information, and optical light manipulation. However, the evaluation of TCs after beam propagation remains a substantial challenge, impeding practical applications. Here, we introduce vortices in lateral arrays (VOILA), a novel spatial multiplexing approach that enables simultaneous transmission of a lateral array of multiple vortices. Leveraging advanced learning techniques, VOILA effectively decodes TCs, even in the presence of strong optical nonlinearities simulated experimentally. Notably, our approach achieves substantial improvements in single-shot bandwidth, surpassing single-vortex scheme by several orders of magnitude. Furthermore, our system exhibits precise fractional TC recognition in both linear and nonlinear regimes, providing possibilities for high-bandwidth communication. The capabilities of VOILA promise transformative contributions to optical information processing and structured light research, with significant potential for advancements in diverse fields.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical communication has experienced tremendous growth in the past few decades, driven by various multiplexing techniques such as wavelength, amplitude, phase, and polarization multiplexing [1]. Despite all these approaches emerging, developing, and maturing, the capacity of optical communication is approaching its limit defined by intrinsic nonlinear effects in optical fibers [2,3]. A promising solution is to utilize vortex beams, which carry orbital angular momentum (OAM) and provide a new degree of freedom that can increase the transmission capacity [4–6]. Optical vortices maintain their topological properties upon propagation in optical fibers [7], making them attractive for a wide range of applications, including optical manipulation with optical tweezers [8–11] and quantum information processing [12–14]. One of the advantages of optical vortices is their theoretically unlimited transmission bandwidth due to the infinite-dimensional Hilbert space in which OAM states reside [2,15–17]. Vortex beams are characterized by their phase front, exp(iℓφ), where ℓ is the number of topological charges (TCs), defined as how many twists per wavelength, and φ is the azimuthal angle. Various methods, such as laser mode selection [18,19], spiral phase plates [20–23], metasurfaces [24,25], and spatial light modulators (SLM) [26,27], can be used to generate vortex beams. However, under realistic conditions, the TCs of the vortices are limited by the physical size and fabrication constraints, with the highest reported OAM being 10,010ℏ, where ℏ is the Dirac constant [28]. Fractional TCs provide an alternative way to increase the degrees of freedom [29], but their recognition is challenging due to the greatly increased complexity and difficulty.
Despite the various approaches developed for TC recognition in optical vortices, accurately extracting information from these beams remains a challenge. Some methods, such as optical interferometry, utilizes a reference beam to measure the phase for accurate TC determination [30–34]. Other techniques, like optical diffraction methods, read out TC directly using diffraction elements on light with OAM [35,36]. Transmission matrix-based TC recognition has also been developed, which yields integer TC readout after measuring the transmission matrix of the propagation medium [37]. Additionally, entangled photons have been used in optical quantum correlation detection [38,39] and quantum entanglement spiral imaging [40] to demonstrate integer or fractional TC recognition. However, these approaches often require additional complex measurement devices that need careful calibration and rely on complicated calculation and additional data augmenting and processing. As a result, accurately recognizing high-dimensional or fractional TCs can be challenging.
Deep learning [41,42] has emerged as a promising approach for recognizing fractional TCs, with successful applications in demultiplexing OAM states [43,44] and identifying OAM states in free space [45–50]. This approach has been shown to enable fractional TC recognition down to 1/100 TCs interval [46], in addition to integer TC recognition. However, the current designs have limited bit-depth scalability, restricting their potential for OAM-based high-bandwidth information transfer. Furthermore, these methods are limited to linear transmission systems and face challenges accurately recognizing fractional TCs in the presence of nonlinear optical processes.
To overcome this challenge, we propose a new strategy termed vortices in lateral arrays (VOILA) and experimentally validated it, which involves propagating lateral arrays of optical vortices through a transmission medium, followed by arbitrary TC recognition. In contrast to traditional deep learning-based recognition network that relies on a classifier to distinguish TCs (limited to 35,000 to date [51]), VOILA decodes, decouples, and identifies representations of TCs from complex patterns through regression-type algorithms that can assess continuous values, far beyond the current limit [52–55]. This arbitrary recognition of the topological charge network is termed ARTnet and provides several advantages (Tab. 1), including a significant in single-shot bandwidth by increasing the possible number of encoded TCs, M, in the form of MN-1, where N is the multiplexing number [56–58]. For example, a 256-time bandwidth increase would be achieved using nine binary vortices, which is beyond the capacity of current deep learning methods due to the significantly increased number of different TC values.
Table 1. Comparison of different modalities to characterize TCs
Furthermore, VOILA is capable of addressing the ubiquitous nonlinearity problem in fiber transmission and optical communications. Various nonlinear optical processes in optical fibers lead to changes in the light field, such as the wavefront, wavelength, or power, breaking the strict linear correspondence between the input and output light. Compared to self-phase modulation / cross-phase modulation and stimulated Raman scattering, we adopt a process with generalized nonlinearity, i.e., second harmonic generation (SHG), to mimic the nonlinear processes along the beam propagation. As demonstrated in the experiment, VOILA shows high-performance recognition of the 2 × 2 spatially multiplexed optical vortices after the SHG process, which brings new opportunities to break the limits of nonlinearity problems in fiber communications. On a side note, the SHG processes demonstrated in our setup have another inherent advantage, i.e., to distinguish chirality, which is not the main focus of this work but has the potential in chiral TC applications.
2. Principles
VOILA’s core, ARTnet [59], explores specific projections of complex physical processes involved in light propagation, offering valuable insights into changes resulting from media scattering or optical nonlinearities (Fig. 1). Despite the exceptional capability of traditional convolutional neural networks to extract local information in an image [46], their performance significantly deteriorates when confronted with non-local information, such as scattering (linear) or optical nonlinearities (nonlinear process). To address this limitation, we introduce a two-part structure for ARTnet, comprising the latent variable representation module and TC decoder. The latent variable representation module seamlessly transforms non-local intensity patterns into spaces solvable by the TC decoder. Meanwhile, the pre-processing module provides regularizations, such that the performance is improved by increasing the number of parameters without increasing the risk of overfitting. Moreover, we explore the assessment of vortices as a continuous TC value, providing remarkable accuracy and information space. This feature enables spatial multiplexing of optical vortices for information transmission, providing a new degree of freedom in optical communications. ARTnet takes a unique approach by performing a regression of TC values from the dataset, eliminating the need for prior knowledge about the transmission process to adjust network parameters. Our experimental results demonstrate successful transmission of high-fidelity, high-bandwidth information using 2 × 2 multiplexed optical vortices with 202 fractional TC values. In these experiments, lateral arrays of optical vortices are transmitted through regular and nonlinear media, respectively, with ARTnet efficiently recognizing TC values.
Fig. 1. Overview of VOILA. (a) Bandwidth advantages of lateral arrays of vortices as compared to a single vortex. (b) Workflow of VOILA. The original data is encoded on the TC and converted to the corresponding phase of vortex (array). The lateral arrays of the optical vortex are transmitted through a regular or nonlinear medium, followed by high-fidelity recognition of TC by ARTnet. Wavefront multiplexing of optical vortices exponentially increases the transmission bandwidth of single frames. In both linear and nonlinear medium, the camera captures light at the same wavelength. Free space and nonlinear optical patterns were used separately. SLM: spatial light modulator.
VOILA is, by design, capable of handling nonlinear effects in light propagation. The intensity patterns of vortex beams with varying TCs primarily exhibit distinct characteristics, such as broken lines arising from phase discontinuities and variations in the size of the hollow region. While the intensity discontinuity caused by phase dislocation in optical vortices has been observed [60–62], it is too subtle to directly discern chiral TCs. However, nonlinear processes offer a means to encode these features within the intensity distribution, going beyond just the discontinuous features. To illustrate this, we consider the example of SHG [63]; nevertheless, the methodology proposed here is extendable to address more complex nonlinear interactions typically encountered in optical fibers. Such complex nonlinear processes lead to complicated intensity structures involving the coupling of OAM in crystals and OAM doubling [64,65]. While nonlinear effect has been viewed unfavorable in previous studies [66,67], VOILA defies these notions by demonstrating its experimental capability to effectively tackle nonlinear challenges and achieve high-accuracy sorting of 2 × 2 vortex arrays.
The nonlinear optical encoding in VOILA is designed to be effectively deciphered by ARTnet. Although the SHG signals captured by the camera are fuzzy (Fig. 2) and lack a clear connection to the phase pattern, the TC information is still stored in the intensity patterns. However, existing methods, including state-of-the-art deep learning algorithms, have difficulties recognizing TCs (see Supplemental Materials Note). In contrast, ARTnet is designed to fully exploit the information from input images by learning the particular projection about physical processes of light propagation (Fig. 3). Particularly, the latent variable representation module (this module contain skip connection) performed a nonlinear mapping for pixel-level transformation that captures high-dimensional features of high-level semantic information, followed by the decoding process that extracts region-level features for TC recognition. Furthermore, the latent variable representation module and the decoder in the back end are trained simultaneously, making the neural network end-to-end with image input and TCs array output. We also conducted the ablation study to demonstrate the effectiveness of the algorithm design in the Supplemental Materials Note.
Fig. 2. Transmission characteristics of vortex beams with different TCs. First row: helical wavefront phase structures loaded to the SLM. Second row: intensity patterns of the vortex beams after free space propagation. Third row: intensity patterns of the vortex beams after free space propagation and SHG process. More experimental results are in the Supplemental Materials Note.
Fig. 3. Schematic of ARTnet. The input is the intensity image of the SHG signal captured by the camera, followed by an encoding and decoding structure performing pixel-wise data transformation. The output comprises four predicted TC values in sequence. The back-end of ARTnet is a decoding structure to extract regional features for recognizing the TCs array. The latent variable representation module, similar to a Unet with encoding-decoding structure, is directly connected to the decoder and maintains an end-to-end training process.
Restoring TCs arrays from the intensity images via ARTnet is realized by solving the optimization problem of the regression of TCs of optical vortices array from the intensity pattern (details in Supplemental Materials Note). In brief, the optimization problem can be written as follows,
3. Results and discussions
We evaluate the performance of ARTnet quantitatively, as shown in Fig. 4. To highlight the advantages, we make a quantitative comparison with a similar deep learning algorithm, ORNN, in Ref. [46]. Images in the test set are used as the input to test the ORNN and the ARTnet. The result shows that ARTnet prediction values are close to the corresponding ground truth, represented by the distribution near a 45° straight line. However, the prediction results of ORNN have higher uncertainty and many extreme deviation points. Figure 4(c) and (g) show the statistical accuracy of TCs predicted by ORRN and ARTnet versus the ground truth. The high accuracy is not spatial-dependent, as shown in Fig. 4(d) and (h), where TC arrays in different positions within the light beam yield similar predictions. In ARTnet, the accuracy for an exact match of the TCs is 70.4% (53.2%); the accuracy of TCs within the ±Δℓ, ± 2Δℓ, and ±3Δℓ range is 99.8% (86.7%), 100% (99.5%), and 100% (100%) in free space (SHG), respectively, showing extraordinary accuracy of the system.
Fig. 4. Quantitative comparison of ARTnet performance. (a-b) Predicted TCs (red dots) versus ground truth (dashed line) in the test set by ORNN for negative (a) and positive (b) TC values. (c) Error histogram of the prediction by ORNN. (d) Prediction accuracy of TCs at four different spatial positions by ORNN. (e)-(h) Performance of ARTnet by the same evaluations as in (a)-(d). A total number of 8080 TCs was included in the test set.
Compared to existing methods, ARTnet shows superior performance. Standard regression algorithms adopt feature extraction layers, de-redundancy pooling, and fully connected layers to achieve continuous value prediction. In contrast, ARTnet extracts the hidden features from the input images into a feature space via a latent variable representation module. Particularly, our neural network fits the intrinsic relationship of the data, which is the generation and transmission process of vortex beams. As a result, ARTnet can predict enormous TCs modes with only a tiny fractional amount of training data instead of having multiple samples for each class in standard classifiers. It is essential for real-world data transmission because it is unrealistic to acquire all possible combinations of information to train a classifier at a scale of millions or even billions. Specifically, we demonstrate wavefront multiplexing techniques by randomly combining 202 classes of TCs. As a result, the number of classes reaches 2024≈1 600 000 000, a previously unimaginable number.
ARTnet enables an OAM-based wavefront multiplexed communication (OAM-WMC) scheme (Fig. 5). The OAM-WMC encodes information of an image into multiplexed wavefronts of vortex beam, which propagates in free space and through a nonlinear crystal, and generates SHG signals captured by the camera. TCs are restored by ARTnet using the images captured, and the information encoded by TCs is then decoded. Experimental results show that VOILA-based OAM-WMC with high recognition accuracy is potentially ground-breaking in optical communication (Fig. 5). By using wavefront multiplexing, the capacity of the OAM transmission system can be significantly improved. For example, compared with the existing 8-bit binary bits transmission scheme with perfect transmission (100% accuracy) [46], our method can transmit 20 bits simultaneously with 100% accuracy in SHG and 22 bits in free space per single shot. This capability is unimaginable for the existing classification scheme because a total of 220, i.e., approximately 1 million classes, have to be distinguished, far beyond the ability of the contemporary classification neural network. Furthermore, if a tiny bit error ratio is allowed by advanced encoding protocols, the capacity would be even higher. For example, with ±2Δℓ and ±1Δℓ recognition error, the scheme can reach the bandwidth of 22 bits and 26 bits per shot, which significantly boosts the bandwidth for transmission. For more experimental results and details on the information encoding and decoding process, see the Supplemental Materials Note.
Fig. 5. Demonstration of information transmission using OAM-WMC. (a) Ground truth image of a photograph of Alan Turing (quarter-plate glass negative on March 29, 1951 by Elliott & Fry, adapted with permission from the © National Portrait Gallery, London). (b) Transmitted image. (c) Transmission scheme. Four pixels of the Turing picture were parallelly transmitted via vortex arrays. (d) and (e) correspond to the transmission of RGB color images. The Pearson correlation coefficient of the original image and the image retrieved is 0.9999, while the structural similarity factor is 0.9963 for the gray-scale image, where one is perfect. The nonlinear signals are directly used for data transmission.
Nonlinear effects are generally more susceptible to noise and considered parasitic for information transmission. Therefore, to assess ARTnet's robustness against noise, we added to the dataset every type of noise we could think of (Fig. 6), corresponding to the possible instrument and transmission channel noises. The results show an average signal-to-noise-ratio larger than 14 dB within the ±1Δℓ recognition error, in which ARTnet still provides meaningful results, with slightly reduced precision, without retraining. Thus, data retraining is not needed unless for precision-critical tasks. Notably, because the noises were randomly added to each pattern, which can be considered as a disturbance from the environmental vibrations, it is reasonable to assume that VOILA is robust against turbulence in transmission. Moreover, with the capability of amplifying the wavefront information of the structured beam via second-order nonlinear effect, for small Δℓ, utilization of nonlinear effects of the fiber may provide an effective means for amplifying the signal. Collectively, the results show potential in expanding the optical transmission bandwidth by the nonlinear effects in the fiber.
Fig. 6. VOILA's robustness against various noise. (a1-a8): nonlinear intensity patterns. (b1-b8): enlarged images of the area indicated by a square in the first row. (c1-c8): pixel intensity profiles at the center of the corresponding images of the second row. (d1-d7) and (e1-e7): prediction error histograms for retrained and not retrained, respectively. The error for each bar is 0, ± 0.01, ± 0.02, and ±0.03. The y-axes of (d) and (e) are the same, shown first.
It is worth discussing the figures of merit of VOILA. (1) ARTnet demonstrated that it learns the particular projection about physical propagation processes, including the nonlinear transformation of array vortex beams. Even with a training dataset that only covers 0.0011% of all possible classes, ARTnet still performed reasonably well. It would thus allow VOILA to address a key topic of OAM communications, i.e., transmission over long distances [44]. (2) VOILA readily applies to recognize an extensive range of TCs, such as vortices with huge topological charges or TC values out of the training range. For TC values far beyond the training range, VOILA can handle the task through quick training over the broader range. (3) VOILA is translational to other physical processes by design, such as using composite vortex beams for encoding, and can be used in a turbulent environment. The unique combination of algorithms and optical elements in VOILA makes it highly expandable and compatible. When utilizing pure light of high beam quality, the transmission system becomes more robust, making the acquired data with a higher signal-to-noise ratio that further improves the performance of VOILA.
Also, with transfer learning, VOILA can be extended to more application scenarios. For example, when massive datasets are difficult to obtain, simulated data can be used to optimize the parameters of ARTnet, followed by fine-tuning the parameters by a small amount of actual data. Currently, the system we demonstrated has not reached its full potential due to limitations of the response rate of the SLM (60 Hz), the detector (camera) acquisition speed (57 frames per second), and the limits of computing power (70 ms/frame on Nvidia Geforce RTX 2080Ti). However, such limitations in instrumentation can be removed by advanced techniques such as high-speed modulators like digital micromirror devices (DMD; > 20 kHz), mode selection directly from optical fiber or laser cavity to obtain high quality vortex beam, cameras at a high acquisition rate (>10 kHz), and higher performance computing units. In addition, TC is robust against noise in optical communication and can use optical fiber channels for transmission. Without considering these factors, our method applies to any orthogonal two-dimensional patterns, e. g., to the analysis of monochromatic [69] and polychromatic [70] Laguerre-Gaussian beams.
4. Conclusions
In this work, we present a new wavefront multiplexing approach, VOILA, for precise recognition of arbitrary TC in optical vortices, extending the applications of OAM. By using a regression-type neural network architecture, we demonstrated high-speed and high-fidelity TC recognition, which can be applied in high-bandwidth communication. VOILA shows unparalleled performance compared to conventional methods, including robustness to nonlinearity and turbulence, modulation-free and reference-free fractional vortices recognition, and multiple orders of magnitude higher single-shot bandwidth in optical communication. This new approach expands the current communication bandwidth greatly by exploring a new optical degree of freedom. VOILA provides a universal platform for automated design to solve complex real physical problems without introducing additional system control steps such as reference light and secondary modulation decoding. The robustness of VOILA enables nonlinear optical interactions to be mitigated in optical communications while deploying a novel deep learning algorithm, creating a simple yet powerful approach to exploiting the potential of structured light.
Furthermore, the impact of our work extends beyond its direct application to optical communications. By developing regression-type deep learning algorithms, we demonstrate the potential for utilizing complex structured light to achieve high-bandwidth information transfer. Our approach also provides a unique platform for data encryption/decryption applications through optical vortex array transmissions [71]. In addition, our strategy is expected to benefit a variety of optical tasks involving structured light, such as creating novel light structures and high-fidelity biomedical endoscopy applications [72]. The versatility and practicality of our approach make it applicable to various fields, opening up new opportunities for optical information processing and structured light research.
5. Method
Experimental Setup. In our experiments, an infrared pulse laser is used as the fundamental light (1064 nm, 330 fs; pump: SPIRIT 1040−16_30-HE; OPA: Orpheus-HP, Spectra-Physics), and the power was adjusted by a half waveplate and polarizing beam splitter. The laser passes through a 1064 nm bandpass filter of 10 mm bandwidth and through a laser collimation system formed by a pair of lenses. A SLM (X13139−09, Hamamatsu) was used to generate a lateral vortices array. The modulated light was then collected and focused into a nonlinear crystal (periodically poled potassium titanyl phosphate, Raicol; length: 2 cm), in which the SHG signal was generated. The SHG signal was filtered by a 532 nm bandpass filter and dichroic mirror (T/R @532/1064), and the intensity information was recorded by a CCD (Prosilica GT1910, AVT) with 5 frame/s. More details are provided in Supplemental Materials Note.
Data Acquirement. ARTnet uses a large number of samples to optimize the parameters of each H such that the objective function is minimized. The performance of ARTnet is evaluated with images unseen by the neural network. We first design the TCs array. We take the TCs interval Δℓ=0.01 within the range [−2, −1] ∪ [1,2], and 202 values of TCs are obtained. Then TCs with different values are placed at four spatial positions within a 2 × 2 grid. In detail, each of the 202 TC values repeats 100 times to generate a sequence, which is randomly scrambled four times to generate four new sequences. Each value of these four sequences was taken out for the vortex array—such arrangement results in 20,200 combinations of the TCs arrays. The arrays and the corresponding images captured by the camera from the dataset were used for training and testing ARTnet, in which 90% were used for the training set and 10% for the testing set. Since the trick of early stopping is not used in the training process, such that the validation set is unneeded. The training set is fed directly to ARTnet to optimize parameters without data augmentation. The resolution of the TCs at each spatial location of the SLM is 200 × 200 pixels (400 × 400 pixels for 2 × 2 TCs), while the camera images are composed of 512 × 512 pixels.
Funding
National Natural Science Foundation of China (11874322, 11934011, 12074339); National Key Research and Development Program of China (2018YFA0307200, 2019YFA0308100); Zhejiang Province Key Research and Development Program (2020C01019); Basic Research Funding of Zhejiang University; Information Technology Center of Zhejiang University; Fundamental Research Funds for the Central Universities; Major Scientific Project of Zhejiang Laboratory (20190057, 2019MB0AD01); Fund of State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiaotong University (2023GZKKF024); National Institutes of Health (1R01GM127696, 1R21CA269099, 1R21GM142107); Cancer Prevention and Research Institute of Texas (RP180588); National Science Foundation (CMMI-1826078); Air Force Office of Scientific Research (FA9550-20-1-0366).
Disclosures
The authors declare no conflicts of interest.
Data availability
The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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