Abstract

Vortex beams have application potential in multiplexing communication because of their orthogonal orbital angular momentum (OAM) modes. OAM add–drop multiplexing remains a challenge owing to the lack of mode selective coupling and separation technologies. We proposed an OAM add–drop multiplexer (OADM) using an optical diffractive deep neural network (ODNN). By exploiting the effective data-fitting capability of deep neural networks and the complex light-field manipulation ability of multilayer diffraction screens, we constructed a five-layer ODNN to manipulate the spatial location of vortex beams, which can selectively couple and separate OAM modes. Both the diffraction efficiency and mode purity exceeded 95% in simulations and four OAM channels carrying 16-quadrature-amplitude-modulation signals were successfully downloaded and uploaded with optical signal-to-noise ratio penalties of ∼1 dB at a bit error rate of 3.8 × 10−3. This method can break through the constraints of conventional OADM, such as single function and poor flexibility, which may create new opportunities for OAM multiplexing and all-optical interconnection.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Vortex beams, circularly symmetric structured light, have been proved to have helical phase front and exhibit orbital angular momentum (OAM) modes [17]. Owing to their unique optical properties, they have been extensively applied in optical tweezers [810], high-resolution imaging [11,12], quantum information [13], and optical communication. In optical communication in particular, the orthogonal OAM modes provide infinite channels for signal multiplexing, which can greatly improve the communication capacity density [1421]. Moreover, OAM add–drop multiplexing, which enables the selective downloading and uploading of OAM channels, can significantly enhance the dynamic interactivity of communication networks [22]. Although various OAM multiplexing communications have been reported, and the transmission rate has increased Tbit/s, the OAM add–drop multiplexing remains elusive owing to the lack of effective mode selective coupling and separation techniques, which severely hinders its application in OAM communication and all-optical interconnections.

Several studies have been devoted to downloading and uploading OAM channels [22], and the OAM add–drop multiplexer (OADM) is primarily constructed using blazed phase gratings. However, the beam splitting characteristics and fixed structure of blazed phase gratings limit their diffraction efficiency, flexibility, and functionality. The major difficulty in selectively downloading and uploading OAM channels results from the independent manipulation of OAM modes at different spatial locations. Over the last decades, deep neural networks [2330] that can theoretically approximate functional relationships in arbitrary input–output domains seem to provide a potential method for OAM mode information processing [3134]. Traditional neural network models, such as feedforward and convolution neural networks, are computer-driven, which means that they can only process light field information offline. Recently, optical diffractive deep neural networks (ODNNs) using multilayer diffraction screens to manipulate the amplitude and phase distribution of light beam have been proposed and widely used in pattern recognition [3542,48], computational optical imaging [43], optical logic operation [44], etc. However, they focus only on controlling the energy distribution of light fields on the output plane, and the required light wave modulation accuracy is less stringent. The use of ODNNs to selectively couple and separate OAM modes remains a challenge for OAM add–drop multiplexing.

In this paper, we introduce an ODNN-based OADM and investigate its application in OAM add–drop multiplexing. Combining the effective data-fitting function of deep neural networks and the excellent light-field control ability of multi-layer diffraction, we designed a five-layer ODNN that can process complex vortex light fields. Based on the light-matter interaction and free space diffraction theory, the ODNN was trained to convert the spatial position of OAM modes by updating the amplitude and phase distribution of diffraction screens. Further, we cascaded two well-trained ODNNs to construct an OADM to selectively couple and separate OAM modes. The diffraction efficiency and mode purity were higher than 95% in simulations. Employing the OADM for OAM add–drop multiplexing, four OAM channels carrying 16-quadrature-amplitude-modulation (16-QAM) signals were successfully downloaded and uploaded, and the optical signal-to-noise ratio (OSNR) penalty was ∼1 dB with a forward-error-correction (FEC) threshold of 3.8 × 10−3. These results demonstrated that the ODNN can selectively couple and separate OAM modes, which may benefit OAM multiplexing communication and all-optical interconnection.

2. Results and analysis

2.1 Construction of OADM

Figure 1(a) depicts the concept of OADM for OAM add–drop multiplexing, where the colored arrows indicate OAM mode channels. The OADM includes four ports: (1) In port (inputs the multiplexed OAM channels); (2) Drop port (downloads the selected OAM channel from the multiplexed channels); (3) Add port (uploads an OAM channel carrying new data stream); (4) Through port (outputs re-multiplexed OAM channels). By exploiting the add and drop functions, the OADM can selectively upload and download OAM channels without affecting other transmitted OAM channels. The key to the selective downloading and uploading of OAM channels lies in the effective coupling and separation of OAM modes. Here, we trained ODNNs consisting of a multilayer diffraction screen and used them to construct OADM to selectively couple and separate OAM modes. The structural framework of the ODNN model is presented in Fig. 1(b), and the training steps were as follows: (1) Forward propagate the light field through the ODNN model; (2) calculate the error loss between the predicted and target output light fields; (3) backward propagate the error to previous diffraction layers; (4) update the amplitude and phase parameters of each diffraction layer. The principle of ODNNs for light field manipulation is provided in Appendix A.

 figure: Fig. 1.

Fig. 1. (a) Concept of OADM for OAM add–drop multiplexing. (b) Structural framework of the ODNN model. The training of ODNN includes four steps: information forward propagation, loss calculation, error backpropagation, and weight update.

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Since the coupling and separation of OAM modes are performed independently, OADM can be formed by cascading two ODNNs, one of which is used for mode separation and the other is used for mode coupling. Here, we divided the input and output planes of ODNN model into functional areas, making it both OAM mode drop and add functions (see Appendix B). This has two main advantages: first, the division of different modes into different regions can increase the number of controllable modes and reduce the crosstalk between OAM modes; second, the combination of mode dropping and adding functions can improve the functional integration and light field manipulation flexibility of the ODNN model. In simulations, the most common Laguerre-Gaussian beam was employed to generate the vortex beam, which was the special solution of Helmholtz equation in cylindrical coordinate system under paraxial approximation (radial index was set to 0, waist was 0.5 mm, and working wavelength was 1550 nm). Since we only trained an ODNN model with both mode add and drop functions, its training data only needed one input-output bases, as shown in Fig. 2(a) and (b). To obtain an ODNN model that can selectively couple and separate the four OAM modes, we employed twelve pairs of input and target output bases as training data. Figures 2(a1)–2(a4) show the four OAM modes input from different positions of the Add port, and the corresponding outputs are represented in Figs. 2(b1)–2(b4). Similarly, the OAM modes in the In port were directly output from the Through port (see Figs. 2(a5)–2(a8) and 2(b5)–2(b8)). In addition, the dropping of OAM modes was realized by inputting a Gaussian beam (waist was 0.1 mm, wavelength was 1550 nm) in the upper left corner of the mode position of the In port of ODNN1 (see Figs. 2(a9)–2(a12)). The dropping of different OAM modes was accompanied by the input of Gaussian beams at different positions, and the corresponding output is shown in Figs. 2(b9)–2(b12). The energy ratio of OAM and Gaussian modes was approximately 10:1. Here, we introduced Gaussian beams with different location inputs (they can be considered guide beams for dropping modes) because they carry different additional feature data, which enables the ODNN model to properly identify and drop OAM modes. The coupling and separation of the OAM mode (l=4) is shown in Fig. 2(c). Four multiplexed OAM modes (l=1, 2, 3, 4) were input from the In port of ODNN1, and the selected OAM mode (l=4) was dropped from the Drop port. The other three modes were output from the Through port of ODNN1 and fed directly into the In port of ODNN2. After adding a new OAM mode (l=4) from the Add port of the ODNN2, four modes were re-multiplexed and outputted from the Through port of ODNN2.

 figure: Fig. 2.

Fig. 2. (a)–(b) Ideal input–output bases for ODNN model training. (a1)–(a4) and (b1)–(b4) represent the OAM modes input from different positions of the Add port correspond to the OAM modes output by the Through port one at a time. (a5)–(a8) and (b5)–(b8) show the OAM modes of the In port correspond to the OAM modes output by the Through port one at a time. (a9)–(a12) and (b9)–(b12) show the OAM modes attaching Gaussian beams with different positions input from In port correspond to OAM modes dropped by the Drop port one at a tome. (c) Schematic diagram of OADM constructed using two ODNNs for OAM mode coupling and separation. The black arrows represents the Gaussian guide beams.

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After the data preparation, we trained a five-layer ODNN model with amplitude and phase hybrid modulation. Figure 3(a) depicts the training loss curve of the ODNN model. The loss curve decreased monotonously and finally became stable as the training epochs increased, indicating that the model became optimal and had an effective light-field processing ability. The obtained amplitude and phase screens after iterative training are shown in Fig. 3(b). With the amplitude and phase modulations of five-layer screens, the ODNN model made the input optical light as close as possible to the ideal output optical field, enabling selectively coupling and separating the four OAM modes. Figure 3(c) depicts the actual light fields predicted by the well-trained ODNN model. The actual test outputs were almost consistent with the desired output, and the phase planes of the OAM modes after being modulated exhibited a standard spiral shape. These indicated that the ODNN model can effectively manipulate light fields for selective coupling and separation of OAM modes.

 figure: Fig. 3.

Fig. 3. (a) Loss curves as functions of training epochs of the ODNN model. (b) Five diffraction screens obtained after the ODNN model training, each of which contained amplitude and phase screens. (c) Intensity and phase distributions of the actual output light fields obtained using the ODNN model training.

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To quantify the performance of the ODNN model, we calculated the diffraction efficiency ($\eta$) and OAM mode purity ($p$), which can be defined as: $\eta = {I_o}/{I_i}$, $p = {I_t}/{I_o}$, where ${I_i}$ and ${I_o}$ are the total energies of the input and output, respectively, and ${I_t}$ is the target mode energy obtained through spiral spectrum decomposition [45]. Figure 4 depicts the diffraction efficiencies and mode purities of four dropped and added modes modulated by the ODNN model. As the figure shows, the diffraction efficiencies of the OAM modes were as high as 99.96%, indicating that the ODNN model effectively converted the incident light intensity. Additionally, the dropping operation of the OAM mode was within the training range of the ODNN model, whereas the added modes required to be demultiplexed by the model at the receiver and were not within the training range, causing the mode addition operation to lose more energy; therefore, the diffraction efficiency of the dropped OAM modes was higher than that of the added modes. Nevertheless, the diffraction efficiencies of the added modes exceeded 95%, which confirmed the extrapolation ability of the ODNN model, that is, the mode space position conversion that was not in the training set could still be completed. In addition, the mode purities of all OAM modes were above 99%, indicating that this ODNN method has accurate mode conversion capabilities.

 figure: Fig. 4.

Fig. 4. Diffraction efficiencies and mode purities of OAM modes after the well-trained ODNN model modulation. (a) Dropped OAM modes, (b) added OAM modes.

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2.2 OAM add–drop multiplexing

To verify the mode selective coupling and separation properties of the ODNN-based OADM, we simulated a four-channel OAM add–drop multiplexing communication link (see Fig. 5). The transmitter of the system emitted a fundamental mode Gaussian beam with a working wavelength of 1550 nm. The 16-QAM signal generated by the signal generator was loaded into the Gaussian beam through the QAM modulator. The signal beam was divided into four sub-beams and converted into the four multiplexed OAM channels (l=1, 2, 3, 4) through vortex phase masks and multiplexer. Here, we used the drop and add functions of OADM consisting of two ODNNs to download and upload OAM channels. The OAM channels required by the local user were downloaded by the first ODNN, and the second ODNN was employed to re-multiplex and output the uploaded channels and other channels. By inputting different Gaussian guide beams, the OAM channels could be flexibly downloaded. After being transmitted 0.1 m in free space, the re-multiplexed OAM channels were de-multiplexed by the third ODNN at the receiver. Note that the demultiplexing of the OAM channels was also implemented by the well-trained ODNN model. Finally, the demodulated signal beam was detected by the photodetector and processed by the signal processor. In simulations, we generated 50,000 random binary data sequences to produce 16-QAM signals, and then converted every four binary values to decimal to get 12,500 signal data with values ranging from 0 to 15. These decimal data were mapped to coordinates (3,3), (1,3), (−3,3), (−1,3), (3,1), (1,1), (−3,1), (−1,1), (3,−3), (−3,−3), (−1,−3), (3,−1), (1, −1), (3, −1) and (−1, −1), respectively, where the x and y coordinates correspond to the real and imaginary parts of the complex plane, respectively, i.e., in-phase and quadrature signal components. In addition, a delay effect was introduced by randomly shuffling the signals for the 16-QAM signal loading of the other three OAM channels. At the receiver of the system, we performed envelope demodulation on the in-phase and quadrature signals, and obtained the demodulated binary signal sequence according to the hard decision method. Here, the bit error rates (BERs) of communication system can be defined as: $BER = {N_e}/{N_t}$, where ${N_e}$ represents the number of bit errors, and ${N_t}$ is the total number of bits.

 figure: Fig. 5.

Fig. 5. Schematic of simulated OAM add–drop multiplexing communication link using ODNN-based OADM. Col: coupler, FC: fiber collimator, VPM: vortex phase mask, MUX: multiplexer, GB: Gaussian beam, DEMUX: de-multiplexer, PD: photodetector.

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Figure 6(a) depicts the optical characterization of the downloaded and uploaded OAM channel (l=1) and the final de-multiplexed channels. After OADM modulation, the selective coupling and separation of OAM modes were accomplished. In addition, the demultiplexing of the four multiplexed OAM modes at the receiver was achieved by simultaneously inputting four Gaussian guide beams on the In port of the OADM (see the enlarged red box). To evaluate the performance of the communication system, we used the Gaussian white noise to interfere with the signals in the electrical domain and the OSNR to characterize the noise level. The BERs as functions of the OSNR of all OAM channels are shown in Fig. 6(b). When the OSNR was greater than 16 dB, the BERs were lower than the FEC threshold of 3.8 × 10−3, indicating excellent communication performance. The OSNR penalty of the channel (l=1) was less than 1 dB, and this penalty was primarily caused by crosstalk between OAM channels. Figures 6(c) and 6(d) show the error vector magnitude (EVM) curves and constellation diagrams under different OSNRs, respectively. As the OSNR increased from 15 to 25 dB, the EVM value decreased from 0.1777 to 0.0633, and the constellation diagram converged noticeably, which proved the effectiveness of the proposed ODNN-based OADM. Notably, in Fig. 6(a), the residual light energy at the Through port can be regarded as a power loss and be defined as: $P = 10\lg ({{E_r}/{E_t}} )$, where ${E_r}$ is residual energy, and ${E_t}$ represents the total energy. The results show that the final de-multiplexed power loss was −11.82 dB, which mainly came from the modulation error caused by the third ODNN model. Since the added mode needs to be dropped by the third ODNN, which was not within the training range of the model, a portion of the light after the mode conversion remained in the Through port, it eventually led to slightly worse BER and EVM of the uploaded (l = 1) and surplus OAM channels (l = 2, 3, 4) than that of the downloaded channel (l = 1).

 figure: Fig. 6.

Fig. 6. Add–drop multiplexing results of the selected OAM channel (l=1). (a) Optical characterization of the downloaded and uploaded channel and four final de-multiplexed channels. (b) Bit error rate and (c) error vector magnitude curves as the function of OSNR for different OAM channels. (d) Constellation diagrams and corresponding EVM values of the downloaded channel under different OSNRs (15–25 dB with steps of 2 dB).

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To demonstrate the reconfigurability of the OADM, we also studied the uploading and downloading of the other OAM channels in the multiplexing link. Figures 7(a), 7(b), and 7(c) show the BER curves of OAM channels (l=2, 3, 4), respectively. When the OSNR was greater than 17 dB, the BERs of all channels were lower than the FEC threshold, and a similar OSNR penalty was observed for the downloaded/uploaded channels and other channels. Moreover, the BERs of the system decreased by nearly three orders of magnitude as the OSNR increased from 15 to 25 dB. The final de-multiplexed power losses in these three scenarios were −12.26, −13.19, −13.24 dB, respectively. The constellation diagrams and EVM values of the back-to-back (B2B), downloaded and uploaded OAM channel of l=4, and the other three channels (OSNR=18 dB) are shown in Fig. 7(d). As the figure shows, the constellation diagrams converged successfully. Although the EVMs were slightly higher than B2B (0.1261) owing to the existence of inter-mode crosstalk, it was still within the acceptable range. These communication results indicated that this OADM can selectively couple and separate modes for the uploading and downloading of OAM channels in OAM multiplexing communication.

 figure: Fig. 7.

Fig. 7. Communication results for dropping and adding different OAM channels. BER curves as the function of OSNR for the OAM channel of (a) l=2, (b) l=3, and (c) l=4. (d) Constellation diagrams and EVM values of back to back, downloaded/uploaded OAM channel of l=4 and the others three channels under OSNR=18 dB (from top to bottom and left to right).

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In OAM multiplexing communication, it is important to download and upload multiple OAM channels simultaneously at particular nodes of all-optical networks to significantly improve the business processing performance and quality of service of multi-user networks. Therefore, we further investigated the simultaneous upload and download of two OAM channels (l=1, 2). As the light field distribution results in Fig. 8(a) show, the OADM successfully completed the simultaneous download and upload of the two channels, and the final de-multiplexed power loss was −11.79 dB. Figures 8(b) and 8(c) show the BER and EVM curves of different channels in the multiplexing link, respectively. The OSNR penalty of OAM channels was less than 2.5 dB with a BER of 3.8 × 10−3, which was worse than the download and upload of a single OAM channel. This can be attributed to the synchronous upload and download of two channels brings causing more mode crosstalk. Note that this ODNN-based OADM was constructed using four OAM modes; thus, up to four OAM channels can be downloaded and uploaded simultaneously. By further training the ODNN model with more mode space position conversion function, more channels can be manipulated. Here, a greater OSNR penalty may be observed owing to crosstalk from other channels.

 figure: Fig. 8.

Fig. 8. Add–drop multiplexing results of the selected OAM channels (l=1, 2). (a) Optical characterization of the downloaded and uploaded channels and the final de-multiplexed channels. (b) BER and (c) EVM curves as the function of OSNR of different channels.

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3. Discussion

Here, we created an ODNN-based OADM and numerically investigated its application in OAM add–drop multiplexing. The key to add–drop multiplexing is in the selective coupling and separation of OAM modes, and the effective manipulation of multiple OAM modes depends on the multi-layer diffraction screens of the ODNN. Based on the light–matter interaction and the Rayleigh–Sommerfeld theory of the beam between the diffraction layers, the ODNN in training automatically establishes the desired mapping relationship between the input and output light fields, which can be expressed as the amplitude and phase distribution on the diffractive layer [46]. After training, the obtained multi-layer diffraction screens are endowed with OAM mode information processing capability in real time and can be used for selective downloading and uploading of OAM channels. Thus, we can theoretically train and obtain a multilayer diffraction structure that can manage the desired light field information by assigninging different target light field tasks to the ODNN. Therefore, the ODNN can be considered a general-purpose optical-field-processing platform. In future research, the ODNN model can be trained for all-optical information processing (mode (de)multiplexing and conversion) of other structured light such as vector beams carrying non-uniform polarization distributions, since vector beams are more robust than vortex beams for transmission in free space.

In free-space OAM communication, the wavefront phase of the vortex beam is extremely sensitive to atmospheric turbulence, resulting in inter-mode crosstalk and signal distortion, which severely deteriorates the communication performance. To investigate the robustness of the well-trained ODNN model in the turbulence environment, the Hill-Andrew model [47] was introduced to simulate the turbulence phase screens. In the turbulence model, the inner and outer scales were 0.2 mm and 50 m, respectively, the equivalent transmission distance was 10 m. Here we employed turbulence with intensities of 1, 5, 9, 10, 30, 50, 70, 90, 100, 500, 900 (× 10−16 m−2/3) to perturb the OAM channel, and used the purity of the OAM mode (l = 1) as the metrics to evaluate the robustness of the trained ODNN model. As can be seen in Table 1, the mode purity of the three mode conversions exceeded 89.40% at the turbulence of < 1 × 10−15 m−2/3, indicating that this ODNN model has powerful robustness under weak turbulence. With the increase of turbulence strength, the mode purity of OAM mode decreased to 42.63% at the turbulence of 9 × 10−14 m−2/3. Moreover, from the Table, the mode purity of mode dropping and through was lower than that of mode addition. This can be attributed to the fact that these two mode conversions were realized by distinguishing the presence or absence of the Gaussian guide beam, and it was difficult for the ODNN model to judge whether the mode is dropped or passed through as the turbulence strength increased. However, the turbulence resistance of the ODNN model can be further improved by introducing turbulence distortion effects in the training set to increase the data diversity.

Tables Icon

Table 1. Robustness test results of the well-trained ODNN model under different turbulence strengths.

In practical applications, factors such as the input offsets and waist radius of the incident beam can affect the optical field modulation abilities of the ODNN. First, we introduced position offsets to the incident beam with OAM mode (l = 1) to test the trained model. These offsets from the center position of the OAM mode include 5 categories: 0.1, 0.2, 0.3, 0.4, 0.5 (mm). From Table 2, the mode purity of the converted vortex beam was higher than 94.24% for the displacements (< 0.2 mm) of the off-center position, indicating that this model can still effectively modulate the OAM mode at small displacements. As the offset continued to increase, the mode purity decreased gradually, which was mainly because the matching relationship between the originally trained diffraction screen and the input beam had been seriously destroyed. However, this decrease can be mitigated by introducing an offset factor in the training set to reduce the position sensitivity of the model to the incident beam. In addition, we also changed the waist radius of the OAM mode (l = 1) in the test set to analyze the modulation performance of the optimal ODNN model. It can be seen from Table 3 that when the waist radius was within the range of [0.3 mm, 0.6 mm], the mode purity of the original trained model (with a waist radius of 0.5 mm) exceeded 86.17%, which proves that the model has good robustness. To reduce the influence of the decrease of mode purity as the radius increased or decreased, it might be an effective way to increase the diversity of model data by introducing different waists in the training set, which will be further studied later.

Tables Icon

Table 2. Robustness test results of the well-trained ODNN model under different input offsets.

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Table 3. Robustness test results of the well-trained ODNN model under different waist radiuses.

During the training process of ODNN, the model structure parameters such as diffraction layers, interlayer spacings, and diffraction screen sizes can be manually adjusted according to the target task requirements. Figure 9(a) shows the training loss curves and time-consuming of the ODNN model under different diffractive layers. The more diffractive layers we had, the faster the training loss of the model converges, and the better the final convergence effect. Compared with a single diffractive layer, the multi-layer diffraction screens could operate together to increase the diffraction energy of the network output plane, which proved that the multi-layer diffraction method can effectively manage complex light fields. When the number of diffraction layers increased to seven, the final loss decreased from 24898.6 to 23.24, while the training time gradually increased to 18.2 min. This was because the training parameters of the model increased with the increase in the number of layers, resulting in increased computational complexity. Meanwhile, we explored the performance of the model with different layer spacings, and the results are shown in Fig. 9(b). As the layer spacing increased, the final loss of the models decreased, meaning that the ODNN must meet an appropriately long layer spacing to ensure that the incident beam can be effectively manipulated. In addition, we also trained models with different diffraction screen sizes (see Fig. 9(c)). The final loss of these ODNNs decreased as the screen size increased, which can be attributed to the fact that smaller diffraction screen size allows more diffraction neurons per unit size, improving the light field modulation capability of the model. However, larger layer spacing and diffraction screen size will result in large ODNN model size, making it difficult to achieve small system integration. Therefore, we traded off and employed an ODNN model (N = 5, D = 30 mm, S = 15 mm) to perform the OAM mode add–drop multiplexing task.

 figure: Fig. 9.

Fig. 9. Training loss curves and time-consuming of ODNN models with different (a) diffraction layers, (b) interlayer spacings, and (c) diffraction screen sizes. N represents the number of layers, D is the layer spacing, and S shows the size of the diffraction screen. FL: final loss value.

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In this study, we only verified the effectiveness of ODNN-based OADM in OAM add–drop multiplexing through simulations. The multiple diffraction layers obtained from the ODNN model training can be physically solidified for experimental implementation. Currently, the most common application is the classification of handwritten digits and clothing in the terahertz spectra using ODNNs based on three-dimensional printing technology. Since the terahertz bandwidth involves the limited inter-particle coupling and material loss, a recent study used a multi-step photolithography etching process to prepare a new visible-light ODNN on a SiO2 substrate [48], which resolved the contradiction between wavelength, neuron size, and manufacturing limitations and realized the recognition of handwritten digits at a wavelength of 632.8 nm. However, these classification tasks focus only on the amplitude distributions of the ODNN model on the output surface, which seem to be less stringent in terms of the accuracy of the optical wave modulation. In contrast to classification, we aim to utilize the ODNN model to accurately point-to-point output the amplitude and phase distributions of the vortex beams, which is more sensitive to light field modulation. As a regression task, it requires an ODNN to output continuous values rather than discrete categories. The physical realization of the ODNN can be achieved by loading multiple diffraction screens on optical devices that can simultaneously modulate the amplitude and phase of light waves, such as spatial light modulators or metasurfaces [4951]. The OAM modes can be controlled by cascading multiple spatial light modulators or multiple reflections on one spatial light modulator, but as the vortex beam is modulated layer by layer, problems such as misalignment and aberration will become increasingly severe, and the final target result will be less ideal. The few-layer metasurface containing multiple modulation layers can be used for the independent and flexible control of amplitude and phase owing to its multi-dimensional manipulation capabilities. However, its interlayer coupling effect may deteriorate the light field modulation ability owing to the absorption and reflection of nanomaterials. Therefore, the effective suppression of misalignments, wavefront aberrations, and inter-layer coupling effects, and realizing the physicalization of ODNNs for OAM multiplexing communication remain to be further studied.

4. Conclusion

We created an ODNN-based OADM and investigated its application in OAM add–drop multiplexing. For ideal input and output light fields, and configuring the amplitude and phase parameters of multiple diffractive layers, the ODNN had effective information-processing capabilities. After being trained, the five-layer ODNN model selectively coupled and separated OAM modes. Both the diffraction efficiency and mode purity reached 95% in the simulations. Moreover, we built a four-channel add–drop multiplexing communication link, and the downloading and uploading of OAM channels were successfully realized. The OSNR penalties were ∼1 dB at the BER of 3.8 × 10−3. These indicated that this OADM can selectively couple and separate OAM modes, providing an efficient method for signal processing in OAM multiplexing communication and all-optical interconnection.

Appendix A

Compared with computer-based deep neural networks with an insufficient computing speed and high energy consumption, the ODNN [52,53] can perform light field manipulation tasks at the speed of light with almost zero loss. An ODNN model includes an input plane, multiple diffraction layers, and an output plane (Fig. 1(b)). During the training process of the ODNN, the input light field is fed to the input plane, and the output of each diffraction layer is calculated via the forward information propagation. Based on the Rayleigh–Sommerfeld theory, the neurons on each diffractive layer can be considered the source of the secondary incident wave of the next diffractive screen, which can be expressed as

$$w_i^n(x,y,z) = \frac{{z - {z_i}}}{{{r^2}}}\left( {\frac{1}{{2\pi r}} + \frac{1}{{j\lambda }}} \right)\textrm{exp} \left( {\frac{{j2\pi r}}{\lambda }} \right),$$
where n is the number of diffractive layers, i is the neuron at $({x_i},{y_i},{z_i})$, and $\lambda $ is the wavelength. $r = {({(x - {x_i})^2} + {(y - {y_i})^2} + {(z - {z_i})^2})^{1/2}}$ expresses the propagation distance, and $j = {( - 1)^{1/2}}$ is the imaginary unit. The output light field distribution of the i-th neuron in the n-th layer can be expressed as
$$s_i^n({x,y,z} )= w_i^n({x,y,z} )t_i^n({{x_i},{y_i},{z_i}} )\sum\nolimits_k {s_k^{n - 1}} ({{x_i},{y_i},{z_i}} ),$$
where $\sum\nolimits_k {s_k^{n - 1}} ({{x_i},{y_i},{z_i}} )$ is the superimposed light field output by the n-1-th diffraction screen, $t_i^n({x_i},{y_i},{z_i}) = a_i^n({x_i},{y_i},{z_i})\textrm{exp} (j\phi _i^n({x_i},{y_i},{z_i}))$ is the light transmission coefficient, which contains two parts called the amplitude and phase. It is known that an initial ODNN framework can be constructed by fixing the number of diffraction layers, the size of neurons, and the distance between layers; thus, only the amplitude and phase parameters on each diffraction screen require to be updated to train the model. According to different transmission coefficients, the ODNN includes amplitude-only modulation, phase-only modulation, and amplitude–phase hybrid modulation. In the training process of the ODNN model, a Sigmoid function was employed to normalize the amplitude and phase distributions of multilayer diffraction screens to [0, 1] and [0, $2\pi$], respectively. For an ODNN with n-layer diffraction screens, the light intensity of the output plane can be represented as
$$S = {|{FF{T^{ - 1}}(FFT(S_{out}^n({x_n},{y_n})) \cdot H({f_x},{f_y}))} |^2},$$
where $FFT$ and $FF{T^{ - 1}}$ are the two-dimensional forward and inverse Fourier transforms, respectively, $S_{out}^n({x_n},{y_n}) = S_{in}^n({x_n},{y_n}) \cdot {t^n}({x_n},{y_n})$ is the output light field of n-layer, $H({f_x},{f_y})$ is a transmission matrix characterizing Fresnel diffraction, and $({f_x},{f_y})$ is the frequency domain coordinate. After obtaining the light intensity distribution of the output plane, the error can be backpropagated by calculating the network loss to optimize the amplitude and phase parameters of each diffraction layer. Here, the loss function was defined as the mean-square-error between the predicted output (O) and the ground-truth (G), which can be expressed as
$$L = \frac{1}{M}\sum\limits_i^M {{{|{{O_i} - {G_i}} |}^2}} .$$
where M denotes the pixel number of the diffractive layers. Here, the size of the input, diffraction, and output planes was 15 mm × 15 mm, and the distance between adjacent layers was 30 mm. The epoch of model training was set to 2000, and the learning rate was 0.01. The training of the ODNN model was implemented in Python 3.5 and the Google TensorFlow platform, and the Adaptive Moment Estimation algorithm was used to optimize the model. With an NVIDIA Ge Force GTX 1080 Ti graphics processing unit, the training process lasted approximately 15 min.

Appendix B

An ODNN model contains an input plane and output plane, representing the input and output light field in a three-dimensional space, respectively. Figure 10(a) shows the spatial position of different OAM modes in the light fields. The spatial pixel size of the light field was set to M × M = 256 × 256 and was divided into four parts. The top and bottom areas were used as Add and Drop ports for channel upload and download respectively, and four different OAM modes (l=1, 2, 3, 4) individually occupied a quarter of the ports. The In and Through ports divided the intermediate area into two, each accounting for half. The Add and In ports were on the input plane of the ODNN model, and the Through and Drop ports were located on the output plane (see Fig. 10(b)).

 figure: Fig. 10.

Fig. 10. (a) Spatial position arrangement of OAM modes in the input and output light fields. (b) Illustration of port settings on the input and output planes of the ODNN model.

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Funding

Shenzhen University Graduate Innovation and Development Foundation (315-0000470835); China Postdoctoral Science Foundation (2020M682867); Shenzhen Excellent Scientific and Technological Innovative Talent Training Program (RCBS20200714114818094); Shenzhen Universities Stabilization Support Program (SZWD2021013); Science and Technology Project of Shenzhen (GJHZ20180928160407303); Shenzhen Fundamental Research Program (JCYJ20180507182035270, JCYJ20200109144001800); Basic and Applied Basic Research Foundation of Guangdong Province (2019A1515111153, 2020A1515011392, 2020A1515110572, 2021A1515011762); National Natural Science Foundation of China (12047539, 61805149).

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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References

  • View by:

  1. A. M. Yao and M. J. Padgett, “Orbital angular momentum: Origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
    [Crossref]
  2. X. Yi, X. Ling, Z. Zhang, Y. Li, X. Zhou, Y. Liu, S. Chen, H. Luo, and S. Wen, “Generation of cylindrical vector vortex beams by two cascaded metasurfaces,” Opt. Express 22(14), 17207–17215 (2014).
    [Crossref]
  3. Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017).
    [Crossref]
  4. Y. He, Z. Xie, B. Yang, X. Chen, J. Liu, H. Ye, X. Zhou, Y. Li, S. Chen, and D. Fan, “Controllable photonic spin Hall effect with phase function construction,” Photonics Res. 8(6), 963–971 (2020).
    [Crossref]
  5. Y. He, P. Wang, C. Wang, J. Liu, H. Ye, X. Zhou, Y. Li, S. Chen, X. Zhang, and D. Fan, “All-optical signal processing in structured light multiplexing with dielectric meta-optics,” ACS Photon. 7(1), 135–146 (2020).
    [Crossref]
  6. S. Fu, T. Wang, and C. Gao, “Perfect optical vortex array with controllable diffraction order and topological charge,” J. Opt. Soc. Am. A 33(9), 1836–1842 (2016).
    [Crossref]
  7. S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4(5), B1–B4 (2016).
    [Crossref]
  8. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
    [Crossref]
  9. A. I. Yakimenko, Y. M. Bidasyuk, O. O. Prikhodko, S. I. Vilchinskii, E. A. Ostrovskaya, and Y. S. Kivshar, ““Optical tweezers for vortex rings in Bose-Einstein condensates,” Phys. Rev. A 88(4), 043637 (2013).
    [Crossref]
  10. M. Gecevičius, M. R. Drevinskas, M. Beresna M, and P. G. Kazansky, “Single beam optical vortex tweezers with tunable orbital angular momentum,” Appl. Phys. Lett. 104(23), 231110–299 (2014).
    [Crossref]
  11. K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: Optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5(6), 491–505 (2008).
    [Crossref]
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    [Crossref]
  14. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
    [Crossref]
  15. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
    [Crossref]
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    [Crossref]
  17. Z. Wang, N. Zhang, and X. C. Yuan, “High-volume optical vortex multiplexing and demultiplexing for free-space optical communication,” Opt. Express 19(2), 482–492 (2011).
    [Crossref]
  18. Y. Ren, L. Li, G. Xie, Y. Yan, Y. Cao, H. Huang, N. Ahmed, Z. Zhao, P. Liao, C. Zhang, G. Caire, A. F. Molisch, M. Tur, and A. E. Willner, “Line-of-sight millimeter-wave communications using orbital angular momentum multiplexing combined with conventional spatial multiplexing,” IEEE Trans. Wirel. Commun. 16(5), 3151–3161 (2017).
    [Crossref]
  19. H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39(2), 197–200 (2014).
    [Crossref]
  20. Y. Zhu, K. Zou, Z. Zheng, and F. Zhang, “1 λ × 1.44 Tbps free-space IM-DD transmission employing OAM multiplexing and PDM,” Opt. Express 24(4), 3967–3980 (2016).
    [Crossref]
  21. J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
    [Crossref]
  22. H. Huang, Y. Yue, Y. Yan, N. Ahmed, Y. Ren, M. Tur, and A. E. Willner, “Liquid-crystal-on-silicon-based optical add/drop multiplexer for orbital angular momentum multiplexed optical links,” Opt. Lett. 38(23), 5142–5145 (2013).
    [Crossref]
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  24. J. Schmidhuber, “Deep learning in neural networks: An overview,” Neural Netw. 61, 85–117 (2015).
    [Crossref]
  25. Q. Zhang, H. Yu, M. Barbiero, B. Wang, and M. Gu, “Artificial neural networks enabled by nanophotonics,” Light: Sci. Appl. 8(1), 1–14 (2019).
    [Crossref]
  26. W. Ma, Z. Liu, Z. A. Kudyshev, A. Boltasseva, W. Cai, and Y. Liu, “Deep learning for the design of photonic structures,” Nat. Photonics 15(2), 77–90 (2021).
    [Crossref]
  27. H. Ren, W. Shao, Y. Li, F. Salim, and M. Gu, “Three-dimensional vectorial holography based on machine learning inverse design,” Sci. Adv. 6(16), eaaz4261 (2020).
    [Crossref]
  28. Y. Rivenson, Y. Zhang, H. Gunaydin, T. Da, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” Light: Sci. Appl. 7(2), 17141 (2018).
    [Crossref]
  29. G. Barbastathis, A. Ozcan, and G. Situ, “On the use of deep learning for computational imaging,” Optica 6(8), 921–943 (2019).
    [Crossref]
  30. S. Feng, Q. Chen, G. Gu, T. Tao, L. Zhang, Y. Hu, and C. Zuo, “Fringe pattern analysis using deep learning,” Adv. Photonics 1(02), 1 (2019).
    [Crossref]
  31. Z. Liu, S. Yan, H. Liu, and X. Chen, “Superhigh-resolution recognition of optical vortex modes assisted by a deep-learning method,” Phys. Rev. Lett. 123(18), 183902 (2019).
    [Crossref]
  32. J. Liu, P. Wang, X. Zhang, Y. He, X. Zhou, H. Ye, Y. Li, S. Xu, S. Chen, and D. Fan, “Deep learning based atmospheric turbulence compensation for orbital angular momentum beam distortion and communication,” Opt. Express 27(12), 16671–16688 (2019).
    [Crossref]
  33. W. Xiong, P. Wang, M. Cheng, J. Liu, Y. He, X. Zhou, J. Xiao, Y. Li, S. Chen, and D. Fan, “Convolutional neural network based atmospheric turbulence compensation for optical orbital angular momentum multiplexing,” J. Lightwave Technol. 38(7), 1712–1721 (2020).
    [Crossref]
  34. T. Giordani, A. Suprano, E. Polino, F. Acanfora, L. Innocenti, A. Ferraro, M. Paternostro, N. Spagnolo, and F. Sciarrino, “Machine learning-based classification of vector vortex beams,” Phys. Rev. Lett. 124(16), 160401 (2020).
    [Crossref]
  35. X. Lin, Y. Rivenson, N. T. Yardimci, M. Veli, Y. Luo, M. Jarrahi, and A. Ozcan, “All-optical machine learning using diffractive deep neural networks,” Science 361(6406), 1004–1008 (2018).
    [Crossref]
  36. T. Yan, J. Wu, T. Zhou, H. Xie, F. Xu, J. Fan, L. Fang, X. Lin, and Q. Dai, “Fourier-space diffractive deep neural network,” Phys. Rev. Lett. 123(2), 023901 (2019).
    [Crossref]
  37. Y. Zuo, B. Li, Y. Zhao, Y. Jiang, Y. Chen, P. Chen, G. Jo, J. Liu, and S. Du, “All-optical neural network with nonlinear activation functions,” Optica 6(9), 1132–1137 (2019).
    [Crossref]
  38. S. Jiao, J. Feng, Y. Gao, T. Lei, Z. Xie, and X. Yuan, “Optical machine learning with incoherent light and a single-pixel detector,” Opt. Lett. 44(21), 5186–5189 (2019).
    [Crossref]
  39. J. Feldmann, N. Youngblood, C. D. Wright, H. Bhaskaran, and W. H. P. Pernice, “All-optical spiking neurosynaptic networks with self-learning capabilities,” Nature 569(7755), 208–214 (2019).
    [Crossref]
  40. T. Zhou, L. Fang, T. Yan, J. Wu, Y. Li, J. Fan, H. Wu, X. Lin, and Q. Dai, “In situ optical backpropagation training of diffractive optical neural networks,” Photonics Res. 8(6), 940 (2020).
    [Crossref]
  41. D. Mengu, Y. Rivenson, and A. Ozcan, “Scale-, shift-, and rotation-invariant diffractive optical networks,” ACS Photonics 8(1), 324–334 (2021).
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2022 (1)

P. Wang, W. Xiong, Z. Huang, Y. He, J. Liu, H. Ye, J. Xiao, Y. Li, D. Fan, and S. Chen, “Diffractive deep neural network for optical orbital angular momentum multiplexing and demultiplexing,” IEEE J. Select. Topics Quantum Electron. 28(4), 1–11 (2022).
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2021 (7)

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2020 (8)

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2019 (9)

T. Yan, J. Wu, T. Zhou, H. Xie, F. Xu, J. Fan, L. Fang, X. Lin, and Q. Dai, “Fourier-space diffractive deep neural network,” Phys. Rev. Lett. 123(2), 023901 (2019).
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2018 (2)

X. Lin, Y. Rivenson, N. T. Yardimci, M. Veli, Y. Luo, M. Jarrahi, and A. Ozcan, “All-optical machine learning using diffractive deep neural networks,” Science 361(6406), 1004–1008 (2018).
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Y. Rivenson, Y. Zhang, H. Gunaydin, T. Da, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” Light: Sci. Appl. 7(2), 17141 (2018).
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2017 (3)

Y. Ren, L. Li, G. Xie, Y. Yan, Y. Cao, H. Huang, N. Ahmed, Z. Zhao, P. Liao, C. Zhang, G. Caire, A. F. Molisch, M. Tur, and A. E. Willner, “Line-of-sight millimeter-wave communications using orbital angular momentum multiplexing combined with conventional spatial multiplexing,” IEEE Trans. Wirel. Commun. 16(5), 3151–3161 (2017).
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2014 (3)

2013 (3)

H. Huang, Y. Yue, Y. Yan, N. Ahmed, Y. Ren, M. Tur, and A. E. Willner, “Liquid-crystal-on-silicon-based optical add/drop multiplexer for orbital angular momentum multiplexed optical links,” Opt. Lett. 38(23), 5142–5145 (2013).
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2012 (1)

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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2011 (3)

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2008 (1)

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2005 (1)

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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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Figures (10)

Fig. 1.
Fig. 1. (a) Concept of OADM for OAM add–drop multiplexing. (b) Structural framework of the ODNN model. The training of ODNN includes four steps: information forward propagation, loss calculation, error backpropagation, and weight update.
Fig. 2.
Fig. 2. (a)–(b) Ideal input–output bases for ODNN model training. (a1)–(a4) and (b1)–(b4) represent the OAM modes input from different positions of the Add port correspond to the OAM modes output by the Through port one at a time. (a5)–(a8) and (b5)–(b8) show the OAM modes of the In port correspond to the OAM modes output by the Through port one at a time. (a9)–(a12) and (b9)–(b12) show the OAM modes attaching Gaussian beams with different positions input from In port correspond to OAM modes dropped by the Drop port one at a tome. (c) Schematic diagram of OADM constructed using two ODNNs for OAM mode coupling and separation. The black arrows represents the Gaussian guide beams.
Fig. 3.
Fig. 3. (a) Loss curves as functions of training epochs of the ODNN model. (b) Five diffraction screens obtained after the ODNN model training, each of which contained amplitude and phase screens. (c) Intensity and phase distributions of the actual output light fields obtained using the ODNN model training.
Fig. 4.
Fig. 4. Diffraction efficiencies and mode purities of OAM modes after the well-trained ODNN model modulation. (a) Dropped OAM modes, (b) added OAM modes.
Fig. 5.
Fig. 5. Schematic of simulated OAM add–drop multiplexing communication link using ODNN-based OADM. Col: coupler, FC: fiber collimator, VPM: vortex phase mask, MUX: multiplexer, GB: Gaussian beam, DEMUX: de-multiplexer, PD: photodetector.
Fig. 6.
Fig. 6. Add–drop multiplexing results of the selected OAM channel (l=1). (a) Optical characterization of the downloaded and uploaded channel and four final de-multiplexed channels. (b) Bit error rate and (c) error vector magnitude curves as the function of OSNR for different OAM channels. (d) Constellation diagrams and corresponding EVM values of the downloaded channel under different OSNRs (15–25 dB with steps of 2 dB).
Fig. 7.
Fig. 7. Communication results for dropping and adding different OAM channels. BER curves as the function of OSNR for the OAM channel of (a) l=2, (b) l=3, and (c) l=4. (d) Constellation diagrams and EVM values of back to back, downloaded/uploaded OAM channel of l=4 and the others three channels under OSNR=18 dB (from top to bottom and left to right).
Fig. 8.
Fig. 8. Add–drop multiplexing results of the selected OAM channels (l=1, 2). (a) Optical characterization of the downloaded and uploaded channels and the final de-multiplexed channels. (b) BER and (c) EVM curves as the function of OSNR of different channels.
Fig. 9.
Fig. 9. Training loss curves and time-consuming of ODNN models with different (a) diffraction layers, (b) interlayer spacings, and (c) diffraction screen sizes. N represents the number of layers, D is the layer spacing, and S shows the size of the diffraction screen. FL: final loss value.
Fig. 10.
Fig. 10. (a) Spatial position arrangement of OAM modes in the input and output light fields. (b) Illustration of port settings on the input and output planes of the ODNN model.

Tables (3)

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Table 1. Robustness test results of the well-trained ODNN model under different turbulence strengths.

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Table 2. Robustness test results of the well-trained ODNN model under different input offsets.

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Table 3. Robustness test results of the well-trained ODNN model under different waist radiuses.

Equations (4)

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w i n ( x , y , z ) = z z i r 2 ( 1 2 π r + 1 j λ ) exp ( j 2 π r λ ) ,
s i n ( x , y , z ) = w i n ( x , y , z ) t i n ( x i , y i , z i ) k s k n 1 ( x i , y i , z i ) ,
S = | F F T 1 ( F F T ( S o u t n ( x n , y n ) ) H ( f x , f y ) ) | 2 ,
L = 1 M i M | O i G i | 2 .

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