Open Access
Add to BibSonomy
Abstract
In this study, we theoretically and experimentally demonstrate that the convolutional neural network (CNN) in combination with the residual blocks and the regression methods can be used to precisely and quickly reconstruct the OAM spectrum of a hybrid OAM mode no matter how the consistent OAM modes have the same or different order indices in both the azimuthal and the radial direction. For cases of the simulation testing, the mean errors of all recognized parameters for hybrid OAM modes in a four-mode fiber (4MF) and a six-mode fiber (6MF) are smaller than 0.003 and 0.008, respectively. To the best of our knowledge, this is the first time that all the OAM modes, probably existing in the core of 4MFs or 6MFs, can be precisely and quickly recognized from intensity distribution of the hybrid OAM mode itself via the deep learning method.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical orbital-angular-momentum (OAM) beams featured with a helical phase front of exp(ilφ), where φ refers to the azimuthal angle and l represents the topological charge [1], have been extensively studied and have been found widespread applications in the fields of optical fine metrology, optical communication, optical manipulation, and quantum information processing, etc. [2–6]. To date, various methods enabling to realize the OAM beams have been developed, which includes the cylindrical lens, the q-plate, the integrated silicon device, J-plate, and the fiber-based converters etc. [7]. Among which, owing to their superior characteristics, e.g., the compact size, extremely low cost, low insertion-loss, high conversion-efficiency, and the inherent compatibility with any other fiber devices, etc., the fiber-based OAM mode generators [8–14], especially the chiral long-period fiber grating (CLPG)-based OAM generators [9–12], have recently attracted a significant research interest, which can be potentially used in optical vortex fiber laser [13], stimulated emission depletion (STED) microscopy [14] besides to the conventional OAM beams-based applications.
In accordance with the invention and the rapid development of the fiber-based OAM generators, the technique enabling to precisely yet quickly recognize the power distributions and the relative phases among the OAM modes, namely the complex OAM spectrum, becomes the significant importance, which is essential to the fiber-based OAM applications since the OAM modes existed in fibers may be a hybrid OAM one, i.e., the superposition of various OAM modes [8–14]. So far, various methods based on principles of either the interference [15], or the diffraction [16], or the geometric coordinate transformation [17] have been proposed and successfully used to recognize the OAM compositions of the hybrid OAM modes. However, some specific bulk apparatuses are generally required, which inevitably increase the complexity of the measurement system and make these methods unsuitable for some in-situ measurements. To overcome the above issues, the methods based on the de-synthesizing the intensity distribution of the hybrid OAM mode have been proposed and experimentally demonstrated, with which the power ratios and the relative phases among all the consisted OAM modes (OAM spectrum) can be precisely recognized [18–20]. However, only the azimuthal power distributions of the OAM modes are considered, the power distribution in radial direction has been neglected, i.e., all the considered OAM modes are strictly limited to the first order ones in radial direction. The OAM modes with identical azimuthal indexes but with the different radial indices, cannot be discerned at all, which certainly restrains OAM beams-based applications, where the OAM modes existed in fibers may have distinct indices not only in azimuthal but also in radial directions. To develop a simple and robust method which enables the well recognition of the OAM modes but with different radial indexes is strongly desired.
On the other hand, as a powerful and state-in-art technology, machine learning (ML) have been developed and widely used in the fields of the computer vision, natural language processing, and the imaging object recognition etc. [21]. Of which, the convolutional neural network (CNN)-based ML, also called the deep learning (DL) has attracted a significant interest and has brought many breakthroughs in processing the images, videos, speeches and audios etc. [22]. Recently, such technology has also been found potential in OAM beams-based applications. It has been proposed and experimentally demonstrated that with the aid of the DL technology, both the topological charges and power ratios of the utilized OAM modes can be extracted directly from the intensity distribution of the OAM hybrid modes [23–28]. However, DL-based recognition methods reported to date are limited to the cases where the considered OAM modes are either the mode with unitary topological charge [23,24] or the hybrid modes but all with the lowest radial index [25–28], which are not available to the more general cases that the OAM modes existed in fibers may have distinct indices not only in azimuthal but also in radial directions.
In this study, with the aid of the DL technology, we theoretically and experimentally demonstrate that the convolutional neural network (CNN) in combination with residual blocks and regression methods can be used to precisely and quickly reconstruct the OAM spectrum of a hybrid OAM mode existed in few-mode fiber. This is the first time that an arbitrary hybrid mode has been recognized no matter how the superimposed mode has the same or different order-indices in azimuthal- and the radial-directions. In addition, unlike the conventional CNN method, the residual block, called the residual neural network (ResNet) and the regression method, are particularly used and combined into the CNN, which would considerably decrease the training time of the CNN and meanwhile largely enhance the accuracy of the pattern recognition.
2. Theory and method
2.1 Modelling and characterizing hybrid OAM modes existed in a few-mode fiber
Under the orthogonal basis of the two circularly-polarized OAM modes, the electric-field distribution of a fiber mode in transverse direction U(r, φ) can be expressed as [29],
where r and φ represent the radial and the azimuthal coordinates, respectively. IL and IR represent the power weight of the left circular-polarization (LCP) and the right circular-polarization (RCP) OAM fields, respectively. UL and UR represent the field distribution of the LCP and RCP parts, respectively. Since the LCP light or the RCP light can be solely obtained just by using some bulk components like polarizer and the quarter wave plate (QWP) in experiments, the mode with LCP status is only considered in this study. The electric-field UL(r, φ) then can be expressed as,
Fig. 1. Intensity distributions of four hybrid modes with different power ratios existed in a four-mode fiber. The power ratios of the modes OAM-2,1, OAM-1,1, OAM0,1, OAM0,2, OAM + 1,1, and OAM + 2,1 are (a) 0.0596, 0.6413, 0.0126, 0.0204, 0.1387, 0.1272; (b) 0.0195, 0.3243, 0.2783, 0.1020, 0.0543, 0.2212; (c) 0.2894, 0.0485, 0.0562, 0.0413, 0.2312, 0.3330; and (d) 0.0136, 0.3689, 0.5696, 0.0260, 0.0102, 0.0113.
2.2 Convolutional neural network (CNN)-based deep-learning networks
Convolutional neural network (CNN) has been proved to be one highly-effective approach in image processing [22]. The CNN in combination with the residual blocks, generally called the residual neural network (ResNet) and the regression method, are specially used in this study, which has been thought to be able to considerably decrease the training time of the CNN and meanwhile largely enhance the accuracy of the pattern recognition [26,30]. The architecture of the CNN model is shown in Fig. 2, where Fig. 2(a) shows architecture of the CNN and Fig. 2(b) shows the residual block. As shown in Fig. 2(a), there exist six blocks in the adopted CNN, in which the first block (Block_1) contains four layers, i.e., the layers Conv, BN, ReLU and MaxPool, respectively. Specifically, the first layer Conv represents a convolution unit constructed by 64-feature maps, where the adopted convolutional kernel is 7 × 7 pixels (K7), and the sliding size and the filling amount are set as 2 (S2) and 3 (P3), respectively. Function of the convolution kernel is performed the convolution to the input image which is filled with the filling amount and slid with the sliding size. The second layer BN represent a batch normalization, the purpose to add this layer is help CNN to overcome the gradient-vanishing or gradient-explosion issues. The third layer ReLU represents a rectified linear unit, which is utilized as an activation function enabling to maintain a large change in gradient to further speed the learning procedures. The fourth layer MaxPool represents the max pooling function, which is used to considerably reduce the features and thus considerably decrease the processing time. The next four blocks (Block_2, Block_3, Block_4, and Block_5) are the residual blocks with a structure like the one shown in Fig. 2(b), where the convolution layers are constructed with different feature maps 64, 128, 256 and 512, respectively. In residual blocks as shown in Fig. 2(b), similarly, the first and forth layers represent the convolution unit, in which K3 represents the convolutional kernel 3 × 3 pixels, P1 represents the filling amount 1, and S2 and S1 represent the sliding size 2 and 1, respectively. The second and fifth layers both are of the batch normalization (BN). The third layer represents a rectified linear unit (ReLU). A shortcut connection in residual blocks is additionally introduced and used to perform the identity mapping, as a result the CNN becomes easier to be optimized and high accuracy in recognition of the imaging pattern can be expected to obtain [30]. In the last block (Block_6), there exist three layers, i.e., AvgPool, FC512, and Sigmoid, where the layer AvgPool represents an average pooling function, which is used to decrease the result fluctuations in iterations. The layer FC512 is a fully connected one with 512 units, which is used for resolving feature maps 512. The layer Sigmoid is particularly added for doing the sigmoid function, which is generally used to solve the regression task instead of the classification task used in Refs. [23,24]. As a result, a complex OAM spectrum can be expected to be reconstructed. In reality, the size and number of the pixel for the CCD used in the experiment are 20 × 20 µm2, and 320 × 256, respectively, the image size used for CNN is 224 × 224 in this study. Before the image is input into each block of the network, the image sizes are resized as 64@224 × 224, 64@112 × 112, 128@56 × 56, 256@28 × 28 and 512@28 × 28, respectively for block1 to block5, respectively, where 64, 64, 128, 256 and 512 are number of feature maps adopted in each individual block.
Fig. 2. Deep learning-based recognition system. (a) The architecture of the CNN model, and (b) the residual block.
To precisely and quickly recognize the parameters of ρn and θn in Eq. (2), namely the complex OAM spectrum, for the hybrid OAM mode, the mode intensity distribution like the one shown in Fig. 2, which is captured by the CCD camera is used as the input image of the ResNet-based CNN. In each iteration, as the output of the last layer Sigmoid, the recognized parameters ρpn and θpn could be automatically obtained. By comparing the recognized parameters ρpn and θpn with the preset parameters ρan and θan in the label, the loss function for the image samples with count S can be obtained, which is expressed as,
2.3 Evaluation factors for the CNN-based recognition system
By using the testing samples obtained from simulation by using Eq. (3), the trained CNN was then evaluated. For better evaluation, the factor Me, i.e., mean value of the deviation square for parameter ρpn and θpn is utilized, which can be defined as,
where yp(n) and ya(n) represent the recognized and the actual parameters of the testing sample. To further show the statistical accuracy of all the recognized parameters, the count percentage Cp for testing examples with count S is defined and given as, where tol denotes the target tolerance for the Me. num[] represents the count of the testing samples that the condition Me < tol is satisfied. S represents total count of the testing samples. One can easily calculate the intensity distributions of the hybrid modes by using Eq. (3), once the corresponding parameters ρn and θn are correctly recognized. To evaluate the quality of the reconstructed intensity distribution, the residual value of the intensity distribution of the hybrid mode is defined as, where Ip and Ia represent the reconstructed and the actual values of the intensity distribution of the hybrid mode. To further evaluate the accuracy of the reconstructed intensity distribution, the correlation coefficient Cr can be used and be defined by,3. Results and discussions
3.1 Testing conditions and evolution of the loss
Two kinds of fibers, i.e., the four-mode (4MF) one and the six-mode fiber (6MF) are considered and used for simulation. The fiber parameters of the 4MF, such as the radii of the core a1 and the cladding a2, the refractive indices of the core n1, the cladding n2, are assumed to be 9.25 µm, 62.5 µm, 1.4499, and 1.4440, respectively. The fiber parameters of the 6MF, such as the radii of the core a1 and the cladding a2, the refractive indices of the core n1, the cladding n2, are assumed to be 8 µm, 62.5 µm, 1.458, and 1.4466, respectively. Accordingly, two types of the hybrid OAM modes are particularly considered in the simulation below. The first is the one existed in a 4MF which consists of all the six OAM core modes, i.e., the modes OAM01, OAM02, and the first- and the second-order OAM modes (OAM ± 1,1 and OAM ± 2,1 modes). The second is the one existed in a 6MF which consists of all the ten OAM core modes, i.e., the modes OAM01 and OAM02, the first-order OAM modes (OAM ± 1,1 and OAM ± 1,2 modes) and the second- and third-order OAM modes (OAM ± 2,1 and OAM ± 3,1 modes). To train the CNN, large amounts of the samples, including the near-field beams (i.e., intensity distributions of the hybrid modes) and their corresponding labels are essential. In this study, the training samples include the simulation ones and the really-measured ones as well. For the simulation ones, the modal weights within the range of [0, 1] and the modal phases within the range of [-π, π] are randomly selected for generation of the near-field beam images. The pixel values of these images are set in the range of [0, 255] and resolution of the image is 224 × 224. All the simulations are run on a desktop computer with an Intel Xeon Silver 4214 CPU @ 2.20 GHz and GTX 2080Ti GPU. The weights of the modified layers are randomly initialized and the others are initialized by the pre-trained CNN. The learning rate is 0.001, and the batch size is 16. Within each epoch, 120000 (for 4MF) and 150000 (for 6MF) are generated, respectively with a resolution of 224 × 224.
Figure 3 shows the results for evolution of the loss defined by Eq. (4) via the number of the iterations (epochs), where the results for the cases of 4MF and 6MF are shown in Fig. 3(a) and (b), respectively. From these two figures, it can be seen that in both cases, within all the training period, the obtained losses turn to be rather small (the maximum values are less than 0.050 and 0.070, respectively) and finally get to convergence after 188 epochs and 181 epochs, respectively. In addition, the overall training times required for the cases of 4 MF and 6MF are approximately 72 hours and 96 hours based on the using the computer mentioned above.
Fig. 3. Evolution of the loss vs. the iteration number for the hybrid modes existed in (a) a 4MF, and (b) a 6MF.
3.2 Evaluating the system performances with the simulated beam patterns
To numerically validate the above results, we apply the proposed CNN-based method to the modes really existed in a 4MF. Figures 4(a1)–(a3) show the beam profiles of three typical samples (hybrid modes), respectively, whereas the actual and the recognized power-part of the OAM spectra are shown in Figs. 4(b1)–(b3), and the actual and the recognized phase-part of the OAM spectra are shown in Figs. 4(c1)–(c3), respectively. The evaluation factor Me obtained for the reconstructed three OAM spectra are 0.0008, 0.0012, and 0.00002, respectively. Whereas the obtained correlation coefficients Cr are 0.9991, 0.9983, and 0.9999, respectively. The above results obviously show that the proposed method works well for the hybrid modes shown in Figs. 4(a1)–(a3). The results shown in Fig. 4 also implicitly means that the modes with identical azimuthal indexes but with the different radial indexes, such as OAM0,1 and OAM0,2, can also be discerned by using the proposed method. To further validate the accuracy of the proposed method from the point-of-view of the statistics, much more samples whose testing numbers are 1 to 1000 (i.e., the parameter S in Eq. (5) is assumed to be 1000) were used as the testing subjects, which are obtained by changing the power ratios of the six OAM core modes differently in a 4MF.
Fig. 4. Simulation results for the reconstructed OAM spectra of hybrid modes in a 4MF where three samples with different mode-weights are considered, respectively. (a1)–(a3) The actual and the reconstructed beam profiles. (b1)–(b3) The actual and the recognized power spectra, and (c1)–(c3) the actual and the recognized phase spectra.
Figure 5(a) shows evolution of the evaluation factor Me of the 1000 samples with various testing number, where for comparison purpose, the input beam profiles (A), the correspondingly reconstructed beams (R), the residual values (ΔI) between the input and reconstructed beams, and the correlation coefficients Cr of five randomly selected samples labelled (1), (2), …, (5) are specially given in the inset of this figure. The position of theses labels indicate the specific quantity values of Me for the five samples. Figure 5(a) shows that no matter what the testing number is, the obtained factor Me remains a small value less than 0.003, and the obtained correlation coefficients Cr remains a large value larger than 0.99. The inset of Fig. 5(a) shows the results (A, R, ΔI and Cr) of five samples, indicating the intensity distribution of the hybrid modes can be well reconstructed with both the low residual values and high correlation coefficients. Figure 5(b) shows histogram of the factor Cp as a function of the tolerance while the sample count remained 1000. From Fig. 5(b), it can be seen that a large Cp of 91.5% can be obtained, once if the tolerance target is set to be larger than 0.0028. As the tolerance target is assumed to be further increased to 0.0036, the magnitude of Cp up to 100% can be expected to obtain.
Fig. 5. The statistical results for the case of 4MF. (a) Evolution of the factor Me with various testing numbers. (b) Histogram of the CP as a function of the tolerance while the sample count was remained as 1000.
The proposed CNN-based method was also exploited to the hybrid modes probably existed in a 6MF. Some samples whose testing numbers are 1 to 1000 were also used as the testing subjects, which are obtained by randomly changing the power ratios of the ten types of OAM core modes. Figure 6(a) shows evolution of the evaluation factor Me of the 1000 samples with various testing number and parameters (A, R, ΔI, and Cr) of five randomly selected samples labelled (1), (2), (3), (4), and (5), indicating that no matter what the sample count is, the obtained factor Me remains a small value less than 0.008, and the obtained correlation coefficients Cr remains a large value larger than 0.97. The inset of Fig. 6(a) further indicates the intensity distribution of hybrid modes can be well reconstructed. Figure 6(b) shows histogram of the factor Cp as a function of the tolerance while the sample count remained 1000, indicating that a large Cp of 95.7% can be obtained once if the tolerance target is set to be larger than 0.0055, and the magnitude of Cp up to 100% can be expected to obtain as the tolerance target is assumed to be further increased to 0.0075.
Fig. 6. Statistical results for the case of 6MF. (a) Evolution of the factor Me with various testing numbers. (b) Histogram of the Cp as a function of the tolerance while the sample count was remained a constant of 1000.
3.3 Performance evaluation with the really-measured beam profiles of the hybrid modes
In order to further validate the proposed method, a proof-of-concept test was conducted in the experiment. The experimental setup is shown in Fig. 7, which consists of a single-longitudinal-mode laser (operating at wavelength of 1550 nm), a piece of single-mode fiber (SMF), a piece of four-mode fiber (4MF), a three-dimensional displacement platform, a non-polarizing beam splitter (NPBS), a half-wave plate (HWP), a polarizer (Pol), a quarter-wave plate (QWP), and a CCD camera. A large amount of the samples for intensity distributions of the hybrid OAM modes with various power radios in 4MF are obtained by precisely controlling the relative positions of the SMF and the 4MF through the two nano-positioners. In addition, a home-made mode stripper (MS) which was realized by wounding the 4MF 6 turns on a small rod with a diameter of 40 mm, was specially used to ensure OAM modes launching only in the core region. More specifically, the polarization status of the light beam is adjusted by using the HWP, Pol and QWP. The intensity distribution of hybrid OAM modes output from the 4MF were captured by the CCD camera (Xenics, Bobcat-320-staris).
The 1000 samples for proof-of-concept tests were measured and recorded. Noted that in practical measurement, since the real magnitudes of ρpn and θpn are unknown, therefore the factor Me cannot be used for the real evaluation. As an alternative, the correlation coefficient Cr was used to evaluate the performance of the proposed method. Figure 8(a) shows evolution of the correlation coefficient Cr with various testing numbers. For all the cases adopting different number the samples, the minimum correlation coefficient Cr remain a value larger than greater than 0.83, and the average correlation coefficient Cr of the 1000 samples is 0.9359. The inset of Fig. 8(a) shows the five samples labelled (1), (2), (3), (4), and (5) with different correlation coefficients, among which the highest and the lowest correlation coefficient are 0.9923 and 0.8336, respectively. These results show that the intensity distribution of the hybrid OAM modes can be well reconstructed, indicating that the parameters, such as the power ratios and the relative phases of all the OAM modes have been precisely recognized. Moreover, from the results shown in the inset of Fig. 8(a) (the second column), it can be seen that the modes with identical azimuthal index but with the different radial indexes, e.g., OAM0,2 can also be well reconstructed. Whereas, Fig. 8(b) shows the histogram of the factor Cp as a function of the tolerance, while the sample count remained a constant of 1000. From this figure, it can be seen that as long as the tolerance target is assumed to be a magnitude less than 0.018, the magnitude of Cp can be a high magnitude up to 93.5%. As the tolerance target is further increased a little to 0.026, the magnitude of Cp can be increased up to 100%. To compare the above results with those shown in Fig. 5, one can find that the obtained correlation coefficients Cr are slightly lower than those of the simulation samples, which could be ascribed by the noises inevitably existed in the real beam patterns. In addition, the obtained average recognizing rate is 23 Hz under the general running environment, indicating that the proposed method may be potentially used for a real-time measurement.
Fig. 8. The statistical results for the concept-proof experiment. (a) Evolution of the factor Cr with various testing numbers. (b) Histogram of the factor Cp as a function of the tolerance while the sample count is remained a constant of 1000.
4. Conclusion
In conclusion, a CNN-based method enabling to precisely and quickly recognize the hybrid OAM modes exited in few-mode fibers is proposed and demonstrated in this study. Owing to the utilization of the residual blocks and the regression methods, the parameters, such as the power ratios and the relative phases of all the OAM modes probably-existed in cores of the 4MF and the 6MF can be recognized directly from intensity distribution of the hybrid mode itself with both a high accuracy and high speed, no matter what the hybrid modes have the same or different order indices in both the azimuthal and the radial direction in fibers. The concept-proof tests were also introduced in this study, all the experimental results agree well with the simulation ones. It is believed that the proposed CNN-based method may open a new way for analyzing and synthesizing the hybrid modes in fiber-based OAM mode application system.
Funding
Natural Science Foundation of Jiangsu Province (BK20201370); Yazaki Memorial Foundation for Science and Technology; Japan Society for the Promotion of Science (JP 22H01546); Postgraduate Practice and Innovation Program of Jiangsu Province (SJCX22_0548); Natural Science Research of Jiangsu Higher Education Institutions of China (22KJB510030); Certificate of Scientific Research Project of Nanjing Xiaozhuang University (2022NXY22).
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.
References
1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]
2. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013). [CrossRef]
3. 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]
4. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]
5. J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle–orbital angular momentum variables,” Science 329(5992), 662–665 (2010). [CrossRef]
6. D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013). [CrossRef]
7. Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019). [CrossRef]
8. M. Feng, W. Chang, B. Mao, P. Wang, Z. Wang, and Y. G. Liu, “108.96 nm broadband generation of second-order OAM beam using an angular modulated cascading LPFG,” J. Lightwave Technol. 41(7), 2152–2158 (2023). [CrossRef]
9. G. Wong, M. Kang, H. Lee, F. Biancalana, C. Conti, T. Weiss, and P. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337(6093), 446–449 (2012). [CrossRef]
10. C. Fu, S. Liu, Z. Bai, J. He, C. Liao, Y. Wang, Z. Li, Y. Zhang, K. Yang, B. Yu, and Y. Wang, “Orbital angular momentum mode converter based on helical long period fiber grating inscribed by Hydrogen-Oxygen flame,” J. Lightwave Technol. 36(9), 1683–1688 (2018). [CrossRef]
11. H. Zhao, P. Wang, T. Yamakawa, and H. Li, “All-fiber second-order orbital angular momentum generator based on a single-helix helical fiber grating,” Opt. Lett. 44(21), 5370–5373 (2019). [CrossRef]
12. C. Jiang, K. Zhou, K. Zhou, B. Sun, Y. Wan, Y. Ma, Z. Wang, Z. Zhang, C. Mou, and Y. Liu, “Multiple core modes conversion using helical long-period fiber gratings,” Opt. Lett. 48(11), 2965–2968 (2023). [CrossRef]
13. D. Mao, M. Li, Z. He, X. Cui, H. Lu, W. Zhang, H. Zhang, and J. Zhao, “Optical vortex fiber laser based on modulation of transverse modes in two mode fiber,” APL Photonics 4(6), 060801 (2019). [CrossRef]
14. L. Yan, P. Kristensen, and S. Ramachandran, “Vortex fibers for STED microscopy,” APL Photonics 4(2), 022903 (2019). [CrossRef]
15. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002). [CrossRef]
16. S. Fu, S. Zhang, T. Wang, and C. Gao, “Measurement of orbital angular momentum spectra of multiplexing optical vortices,” Opt. Express 24(6), 6240–6248 (2016). [CrossRef]
17. Y. Wen, I. Chremmos, Y. Chen, J. Zhu, Y. Zhang, and S. Yu, “Spiral transformation for high-resolution and efficient sorting of optical vortex modes,” Phys. Rev. Lett. 120(19), 193904 (2018). [CrossRef]
18. N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, “Control of orbital angular momentum of light with optical fibers,” Opt. Lett. 37(13), 2451–2453 (2012). [CrossRef]
19. Y. Jiang, G. Ren, H. Li, M. Tang, Y. Liu, Y. Wu, W. Jian, and S. Jian, “Linearly polarized orbital angular momentum mode purity measurement in optical fibers,” Appl. Opt. 56(7), 1990–1995 (2017). [CrossRef]
20. Y. Hao, C. Guo, X. Huang, J. Xu, H. Lu, H. Zhao, P. Wang, and H. Li, “Synthesizing the complex orbital-angular-momentum spectrum of hybrid modes existed in a few-mode fiber,” Opt. Express 30(15), 26286–26296 (2022). [CrossRef]
21. D. Michie, D. J. Spiegelhalter, and C. C. Taylor, Machine learning, neural and statistical classification, Ellis Horwood, New York, 1994.
22. Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015). [CrossRef]
23. 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]
24. L. L. Wang, Z. S. Ruan, H. Y. Wang, L. Shen, L. Zhang, J. Luo, and J. Wang, “Deep learning based recognition of different mode bases in ring-core fiber,” Laser & Photonics Reviews. 14(11), 2000249 (2020). [CrossRef]
25. T. Y. Lin, A. Liu, X. P. Zhang, H. Li, L. P. Wang, H. L. Han, Z. Chen, X. P. Liu, and H. B. Lü, “Analyzing OAM mode purity in optical fibers with CNN-based deep learning,” Chin. Opt. Lett. 17(10), 100603 (2019). [CrossRef]
26. B. Pinheiro da Silva, B. A. D. Marques, R. B. Rodrigues, P. H. Souto Ribeiro, and A. Z. Khoury, “Machine-learning recognition of light orbital-angular-momentum superpositions,” Phys. Rev. A 103(6), 063704 (2021). [CrossRef]
27. L. F. Zhang, Y. Y. Lin, Z. Y. She, Z. H. Huang, J. Z. Li, X. J. Luo, H. Yan, W. Huang, D. W. Zhang, and S. L. Zhu, “Recognition of orbital-angular-momentum modes with different topological charges and their unknown superpositions via machine learning,” Phys. Rev. A 104(5), 053525 (2021). [CrossRef]
28. M. Hou, M. Xu, J. Xu, J. Lu, Y. An, L. Huang, X. Zeng, F. Pang, J. Li, and L. Yi, “Deep learning–based vortex decomposition and switching based on fiber vector eigenmodes,” Nanophotonics 12(15), 3165 (2023). [CrossRef]
29. G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincare sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011). [CrossRef]
30. K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, New York, 2016), 770–778 (2016).
