1. Introduction

Vortex beam carrying orbital angular momentum (OAM) mode provides a promising physical dimension for expanding the capacity of optical communication due to its mode orthogonality [1–4]. Multiplexing and shift-keying (SK) are two primary mechanisms that leverage OAM modes to increase the communication capacity. OAM mode-division-multiplexing communication has achieved tremendous advancement, and various technologies including Damman grating [5–9], coordinate transformation [10–13], etc. have been proposed for (de)multiplexing OAM modes to support greater capacity [14–18]. However, the OAM-based SK is still severely hampered by the lack of effective modulation degree of freedoms (DoFs). OAM-based SK can be mainly divided into pure mode and superposed mode. The former use single integer or fractional OAM mode to represent a coded symbol, which can achieve hundreds of modulation orders [19–24]. Increasing the orders requires continuously escalating the integer OAM mode or further enhancing the fractional order resolution. However, the mode field mismatch caused by the increased radius and divergence of the OAM mode, and the finite phase resolution constrained by inherent accuracy of devices, set an upper limit on the codable mode order [20,25]. Hybrid mode schemes are able to obtain a greater number of codable modes through coherent superposition of multiple vortex beams with different radial and azimuthal index [26–32]. However, they usually require occupying many OAM modes, and their further development are still limited by the mode-field mismatch resulting from the rapidly increased divergence with the mode order [25,28,35]. Constraint by limited modulation DoFs, their modulation orders are approximately proportional or quadratic to the number of utilized OAM modes [20,28]. Therefore, exploring effective DoFs to construct more codable modes has become an urgent issue in developing high-order OAM-based SK communications.

Herein, we propose a phase-difference modulation strategy that introduces phase-difference DoF to provide more codable modes under limited OAM modes set and thus alleviate the mode-field mismatch problem. When multiple beams with different OAM modes propagate coaxially, the spatial transverse intensity distribution is determined not only by the topological charge of carried OAM modes but also by the phase differences. By utilizing the phase sensitivity of mutual interference between OAM modes, it becomes possible to provide a series of spatial modes with continuous variation via modulating the relative phase difference between different OAM modes. This continuous correspondence between phase differences and spatial modes enables obtaining as many distinct codable modes as possible through phase-difference modulation. By discretizing and quantizing the phase differences, discrete spatial modes that matches the discrete digital signal coding requirements can be obtained. Thus, the low order OAM modes with low divergence can be used to construct high-order shift-keying communication, alleviating the mode-field mismatch problem.

To validate its feasibility of achieving high-order SK communication, we simulated and experimentally demonstrated a shift-keying communication link using phase-difference modulation, and a neural network decoder was constructed to achieve phase difference inversion and signal demodulation. Results show that a modulation order of $4 \times {10^4}$ is achieved by using only 3 OAM modes (+1, + 2 and +3), and the signal decoding accuracy exceeds 99%. By increasing the OAM modes to 4, $8 \times {10^4}$ modulation order is achieved. Since the phase differences between different OAM modes can be modulated independently, the number of codable modes increases exponentially with the number of superposed OAM modes and the phase differences level. Therefore, it is expected to break the traditional linear or quadratic restraints on modulation order by leveraging the exponential scaling of order achievable through increasing superposed low-order modes, thus mitigating issues of mode divergence and mode-field mismatch, which may provide new insight for high-order OAM-based SK communication.

2. Principles and methods

In addition to the initial phase $\exp (i\phi )$, vortex beam also incorporates a spiral phase$\exp (il\theta )$ related to OAM mode, which makes the wavefront of vortex beam in free-space a helical structure [1]. In $\exp (il\theta )$, l is the topological charge, and θ is the azimuth angle. Since the light intensity distribution of vortex beam exhibits a rotationally symmetric ring shape, the initial phase is typically ignored in previous studies, as it does not affect the intensity distribution but only results in a global phase shift of the helical wavefront. For instance, considering the Laguerre-Gaussian beam as vortex beam [1,28], the initial phase $\exp (i{\pi / 2})$ only results in a 90° global rotation of the helical wavefront without altering the intensity distribution, as illustrated in Fig.S1(a) in Supplementary Materials (SM). However, such situation changes when multiple OAM modes are coaxially coherently superposition. Due to the mode interference, the transverse spatial intensity distribution of the superposition beam now depends not only on the carried OAM modes (l), but also on the relative phase differences ($\Delta \phi $) between the OAM mode components. As shown in Fig.S1(b) and (c), when the set of l is determined, changing the $\Delta \varphi $ will alter the interference condition at the cross-section perpendicular to the propagation direction, resulting in a series of transverse intensity distributions regarding $\varDelta \phi $.

According to the interference theory [36], the intensity distribution expression of the superposition beam with m OAM modes contains two terms: a constant term and a coherent term. The later is related to the topological charge differences$\varDelta {l_k}$ and phase differences$\varDelta {\phi _k}$, where $k = 1,2,..,m - 1$. For a given set of OAM modes, the $\varDelta {l_k}$ are determined. Therefore, the coherent term depends on the relative phase differences $\varDelta {\phi _k}$. Due to the periodicity and relative nature of phase, there are only m-1 independent phase differences for m OAM modes. Accodingly, denoting the k-th phase difference $\varDelta {\phi _k}$ as ${\varphi _k}$, the intensity distribution can be expressed as:

 

Fig. 1. The schematic diagram of phase-difference modulation with m modes (l₁, l₂, …, l_m) and m-1 phase differences (φ₁, φ₂, …, φ_m-1).

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Compared with the traditional hybrid modes scheme [28–30], where $T = {2^m}$, the number of codable modes is scaled by a factor of ${{(n} / 2}{)^m}/n$. As shown in Fig. 2, the number of codable modes increases exponentially with the number of superimposed OAM modes and the quantization level of phase differences. When the number of OAM modes is 10 and the quantization level is 200, there are more than 6 × 10²⁰ codable modes, and the bits of per symbol exceeds 60. Therefore, it facilitates establishing ultra-high-order OAM-SK within a limited number of OAM modes. By doing so, we can take advantage of lower-order OAM modes with low divergence to build higher-order shift-keying communications, alleviating problems such as mode field mismatch that are otherwise caused by the high divergence of higher-order OAM modes.

 

Fig. 2. The relationship between (a) the total number of codable modes and (b) bits per symbol with the number of OAM modes and the quantization level of phase differences.

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The following contents are organized as follows: Firstly, in Section 3, we validate the feasibility of constructing a neural network decoder for phase difference inversion. We utilize the trained decoder to elaborate on constructing and evaluating the performance of high-modulation-order shift-keying communication. Following that, in Section 4, we delve deeper into discussing potential approaches and feasibilities for extending the modulation orders based on the prior demonstrations. Finally, we summarize the key results.

3. Results and analysis

As a conceptual demonstration, we firstly illustrate the phase-difference modulation based 4 × 10⁴-order shift-keying communication link using three OAM modes (+1, + 2, and +3) and the phase difference quantization level of 200. The schematic diagram of shift-keying communication is shown in Fig. 3(a). A n-bit binary sequence is firstly mapped to the quantization level of the corresponding phase differences by numeral system conversion, and then encoded into the superimposed OAM modes by modulating the relative phase differences. After transmitting in free space, the light intensity distribution is collected at the receiving end and fed into the trained neural network decoder for phase differences inversion, which accepts intensity distribution as input and outputs the quantized phase difference level $({\varphi _1},{\varphi _2})$. Finally, the original binary sequence is demodulated by reverse numeral system conversion. For adequately exploiting the phase differences in constructing high-order OAM-based SK and considering the periodicity of phase, the phase differences are changed from 0 to 1.99π and sampled with an interval of 0.01π, which is close to the phase resolution of spatial light modulator (SLM) and the smallest variation that can be displayed between codable modes. It is important to note that this selection represents the largest set of phase differences. Any non-empty subset of this largest set can also be chosen and utilized to construct shift-keying communications, albeit with relatively lower modulation orders. Thus, for three superimposed OAM modes, we can obtain 4 × 10⁴ discrete phase differences, which correspond to 4 × 10⁴ codable modes. Details on phase difference quantization and encoding are given in the part II of SM.

 

Fig. 3. (a) The schematic diagram of the OAM-SK communication based on phase-difference modulation. The samples of (b) simulated and (c) experimentally collected codable modes with different ${\varphi _1}$ and ${\varphi _2}$. (d) and (e) are the accuracy curves of the decoder in simulation and experiment, respectively.

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Since the phase differences are quantified, the problem of phase differences inversion can be formulated as a classification problem rather than a regression problem. By exploiting the relativity of the phase differences, the decoder is designed as a single-input-multiple-output structure, which consists of a feature extraction module and a classification module. The feature extraction module is used to extract feature parameters from the light intensity, while the classification module, composed of multiple independent fully connected layers, is used to invert each phase difference from the same extracted feature parameters. The ResNet50 [37] is used as feature extraction module and the detailed structure of the decoder is shown in Fig.S3(a) in SM. In this work, the decoder was implemented using Python version 3.6.1 and Tensorflow framework version 2.4.0 on a desktop computer (GeForce RTX 3080 Ti GPU, and Intel Core i9-10900X CPU 3.70 GHz and 256 GB of RAM, running a Microsoft Windows 10 operating system).

Figure 3 shows the examples of discretely codable modes generated by three OAM modes superposition with phase-difference modulation. It is worth to emphasize that the concept of the phase difference here is essentially a relative definition. Specifically, we can specify arbitrary initial phase value as the reference phase (phase difference is 0), and then define other phase differences accordingly. Here, we take the initial phase of the OAM mode with l=+3 as the reference phase, and then determine the phases of the OAM modes with l=+1 and +2, which correspond to the phase differences ${\varphi _1}$ and ${\varphi _2}$, respective. As shown in Fig. 3(b), due to the relativity between ${\varphi _1}$ and ${\varphi _2}$, the generated modes exhibit some relativity as well. Specifically, the modes within each row (or column) can be obtained through periodic translation with some center rotation of the modes in other rows (or columns). For instance, cyclically shifting the modes in the first row to the right by two intervals and simultaneously rotating them by 2/5π yields the modes in the second row. It actually reveals that ${\varphi _1}$ and ${\varphi _2}$ have equal status to optical modes, and their relative nature.

Accurate signal demodulation is a prerequisite for utilizing these modes to constructing shift-keying communication links. As mentioned above, in this work we introduce a neural network decoder to assist signal decode. We first theoretically verify the performance of the constructed decoder. Here, decode accuracy is employed as a metric to evaluate the performance of the decoder. It is defined as the ratio between the number of accurately inverted codable modes and the total number of codable modes processed by the neural network decoder. To train the neural network decoder, we collected 4 × 10⁴ light intensity patterns corresponding to the 4 × 10⁴ codable modes, and fed them together with the labels (the quantization level of ${\varphi _1}$ and ${\varphi _2}$) into the neural network for supervised learning. Since we know all possible modes theoretically, we use the entire dataset to train the decoder without partitioning into separate training and validation sets. As mentioned above, what we are performing here is actually a classification task, so the cross-entropy error is chosen as the loss function accordingly [37]. Figure 3(d) plots the curve of decoding accuracy versus training epoch. It can be seen that when the epoch exceeds 80, the accuracy stabilizes at 1, which indicates the decoder has well fitted the mapping relationship between the input intensity patterns and the output phase differences. Therefore, the trained decoder can accurately identify the phase differences and achieve signal demodulation. Notably, the neural network has theoretically learned all possible modes during training. Therefore, the results of such accurate identification are predictable. This choice is reasonable, given that our goal is to develop an effective decoder, rather than a standard discriminator for unknowns, even though both employ the classification paradigm.

While we have theoretically demonstrated that the constructed neural network decoder can perform accurate phase differences inversion for all 4 × 10⁴ codable modes, in realistic application scenarios, signals will inevitably contain noise, so the encoder requires a certain degree of robustness to handle noisy inputs. To further verify the feasibility of the neural network decoder in practical application, we tested the performance of neural network decoder under experimental conditions. The corresponding experimental setup is shown in Fig.S5 in SM. A total of 80,000 data sets were collected experimentally and subsequently partitioned into training and testing sets using a ratio of 9:1, where each optical mode contained two samples. Figure 3(c) depicts the experimentally measured intensities of the codable modes corresponding to that in Fig. 3(b). Figure 3(e) shows the curve of the accuracy of the decoder on the training set during the training process. As the training progressed, the accuracies of the decoder for ${\varphi _1}$ and ${\varphi _2}$ are increasing and approaching 100%, demonstrating successful fitting. The trained decoder achieved 99.9% accuracy on the test set, which is consistent with the state of the art from previous related work [20,31]. Notably, the test set was independent of and not used during the training process. The decoder's ability to achieve high accuracy on unknown testing examples demonstrates its great generalization capabilities and robustness, allowing it to precisely decode new data despite potential noise or variations.

To verify the performance of constructed OAM-SK communication link using the trained neural network decoder, we transmitted four 100 × 100 gray and color images. Figure 4(a) shows the schematic diagram of the communication links. To match the modulation order of the communication link, the information of each pixel of the image is first converted into a series of 15-bit binary sequence. These binary sequences are then mapped to unique phase differences level via numeral system conversion, loading onto the codable modes. The series of codable modes is generated by continuously refreshing the SLM. After being transmitted 1 m in free space, the patterns are captured by the CCD camera. The received intensities are then sent to the pre-trained neural network decoder. The decoder performs phase difference inversion to reconstruct the images. As shown in Fig. 4(b), there are little difference between the received and the sent images. The bit error rates of both gray and color images are less than 10⁻³, indicating that the constructed OAM-SK communication link achieved reliable performance.

 

Fig. 4. The OAM-SK communication using three OAM modes (l = + 1, + 2 and +3). (a) The schematic diagram. (b) The communication results of gray images and color images.

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4. Discussion

In this article, we propose a phase-difference modulation strategy to construct ultra-high order OAM-SK. A neural network decoder is also developed to assist signal demodulation. Since the increase in the number of codable modes is exponentially related to the OAM mode numbers, it facilitates the implementation of high-order keying communication using a limited number of low-order OAM modes. However, since a neural network decoder is required to invert the phase differences from the light intensity, higher modulation order means more codable modes, which also means larger data sets. It poses a challenge for the training of the neural network decoder because it places extremely high demands on computing resources. Due to the symmetry between ${\varphi _1}$ and ${\varphi _2}$, the neural network decoder can implement their inversion through the same feature extraction module. Fortunately, such symmetry still exists when the number of superimposed OAM modes is greater than 3. Therefore, it provides a way to enable higher order modulation without significant increasing the complexity of decoder and sacrificing generalization performance.

To verify its feasibility, we increase the number of OAM modes to four (+1, + 2, + 3, and +4), and thus there are three independent phase differences $({\varphi _1},{\varphi _2},{\varphi _3})$. The generated codable modes are depending on the three phase differences, and the total number increases to $8 \times {10^6}$. Figure 5(a) and (b) show some samples of codable modes collected theoretically and experimentally, respectively. The slices of light intensity distribution corresponding to Fig. 5(a) and (b) are provided in Fig.S7. Since the codable modes have relative symmetry with respect to three phase differences, similar to the situation in Fig. 3(b) and (c), modes that continuously change about a certain phase difference can be obtained by performing certain rotations and translations on the modes on the other two axes. Therefore, this means that we can still directly construct a decoder by adding three classifiers corresponding to the phase differences on a shared feature extraction module, without needing to construct a more complex feature extraction module, thus reducing computational demands. The structure of the new decoder is shown in Fig.S3(b) in SM. Here, we sequentially selected the first 80000 modes of $8 \times {10^6}$ for verification purposes. This selection was primarily constrained by the computing capacity of our existing equipment, but it is still effective. To validate the approach, we first prove it in simulation. We collected 80000 intensity patterns and used them all to train the decoder. The changes in accuracy during the training process are shown in Fig. 5(c). It can be seen that after about 100 iterations, the accuracy stabilizes at 100%. This means that all modes can be accurately identified. The accuracy for identifying the three phase differences follows a similar trend, reflecting the intrinsic symmetry between the three phase differences. Note that compared to Fig. 5(c), the lines in Fig. 3(d) exhibit relatively serious jitter near 40 epochs. Given that the same hyperparameters (epochs = 200, batch size = 64, initial learning rate = 0.001) were used for training the decoders corresponding to Fig. 5(c) and Fig. 3(d), these fluctuations are essentially due to the scale of dataset used. Larger-scale datasets can mitigate the impact of overfitting and noise that may exist during the training processes, enabling the neural network decoder to better comprehend the overall sample features and enhance its robustness.

 

Fig. 5. The samples of (b) simulated and (c) experimentally collected codable modes with different ${\varphi _1}$, ${\varphi _2}$ and ${\varphi _3}$. (c) and (d) are the accuracy curves of the decoder in simulation and experiment, respectively.

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Furthermore, we experimentally collected 160,000 samples (2 samples per mode) to evaluate its practical performance. 9:1 splitting was applied for the training and test sets. As shown in Fig. 5(d), the accuracy on the training set approaches 100% after approximately 150 epochs. Note that a similar phenomenon mentioned above can also be observed in Fig. 3(e) and Fig. 5(d), where the lines in Fig. 3(e) (trained with 8 × 10⁴ samples) display slightly larger fluctuations compared to Fig. 5(d) (trained with 1.6 × 10⁵ samples). Despite varying degrees of fluctuations, these lines consistently exhibit an upward trend with increasing epochs, signifying the decoder’s ability to continue learning sample features and achieve accurate decoding. Remarkably, the trained decoder achieved 99.9% accuracy on the held-out test set, validating the simulation results. Therefore, even as the number of codable modes scales exponentially with increasing OAM modes, benefiting from the relative relationships between phase differences, accurate identification of all modes can be achieved through a linear expansion of classification heads, which is beneficial for establishing ultra-high-order shift-keying communication.

To further prove this point, we expanded the number of codable modes to $1.6 \times {10^5}$ and analyzed the performance of the decoder in simulation. Fig.S6 gives the corresponding loss and accuracy change curves. It can be found that using $1.6 \times {10^5}$ samples for training compared to using $8 \times {10^4}$ samples for training, the model converges faster, and the accuracy is stable at 100% after 120 epochs. This confirms that even as the codable modes grows exponentially with OAM modes, the complexity demands on the neural network decoder do not increase equivalently. Therefore, the approach enables the practical implementation of ultra-high-order OAM-based SK communications using decoders incorporating only modest additional complexity.

The above elucidation emphasizes the feasibility of enhancing the modulation order in the shift-keying communication links by augmenting the number of OAM modes. Moreover, achieving higher orders through finer quantization is plausible due to the continuous variation of phase difference values across the 0 to 2π. However, finer quantization necessitates higher phase modulation accuracy and smaller phase intervals. Although smaller phase intervals will result in smaller intensity differences between adjacent codable modes, developing and training more robust neural network decoders to accommodate such requirements is no longer a major challenge, thanks to the evolution of deep learning [38]. Hence, the quantization order primarily hinges on the phase modulation accuracy of prevailing spatial light modulators. Currently, a phase separation of 0.01π, corresponding to a quantization level of 200, is already close to the minimum achievable phase modulation accuracy for most commercially available spatial light modulators. Nevertheless, the ongoing advancement in phase modulation technologies, such as liquid crystal developments [39], offers promising prospects for further augmenting the modulation order through higher quantization orders.

5. Conclusion

In summary, we propose a phase-difference modulation strategy for achieving ultra-high-order OAM-SK communication within a limit number of OAM modes. Through exploiting the sensitivity of the coherent superposition modes to phase differences, a new DOF of phase difference can be obtained beyond OAM mode. By discretizing and quantizing the continuous phase differences, we can construct a series of codable optical modes whose number scales exponentially with the number of OAM modes. Therefore, a limited number of low-order modes with low-divergence can be used to construct high-order shift-keying communication to alleviate the problem of mode field mismatch caused by mode divergence. Furthermore, we demonstrate that using a neural-network decoder to perform end-to-end mapping between light intensity and phase differences allows for accurate signal demodulation. An OAM-SK communication link with $4 \times {10^4}$ order using only 3 modes (+1, + 2, and +3) has been achieved and the decode accuracy has reached 99.9%.