Research on Enhancing Target Recognition Rate Based on Orbital Angular Momentum Spectrum with Assistance of Neural Network

Abstract

In this paper, the single-mode vortex beam is used to illuminate targets of different shapes, and the targets are recognized using machine learning algorithms based on the orbital angular momentum (OAM) spectral information of the echo signal. We innovatively utilize three neural networks—multilayer perceptron (MLP), convolutional neural network (CNN) and residual neural network (ResNet)—to train extensive echo OAM spectrum data. The trained models can rapidly and accurately classify the OAM spectrum data of different targets’ echo signals. The results show that the residual network (ResNet) performs best under all turbulence intensities and can achieve a high recognition rate when $C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$. In addition, even when the target size is $\eta =0.3$, the recognition rate of ResNet can reach 97%, while the robustness of MLP and CNN to the target size is lower; the recognition rates are 91.75% and 91%, respectively. However, although the recognition performance of CNN and MLP is slightly lower than that of ResNet, their training time is much lower than that of ResNet, which can achieve a good balance between recognition performance and training time cost. This research has a promising future in the fields of target recognition and intelligent navigation based on multi-dimensional information.

1. Introduction

In laser-active target recognition, the laser echo signal carries information about the target’s shape, size, material, reflectivity, and other properties. Target recognition can be achieved by processing characteristic information such as the echo signal’s amplitude, phase, frequency, and polarization. The traditional Gaussian beam has limited imaging resolution, and its propagation is greatly affected by the environment, with a limited range of action and insufficient adaptability, which limits its application in complex and long-distance target recognition. In recent years, due to their unique helical phase and polarization characteristic [1], vortex beams have been widely used in optical manipulation [2,3], quantum communication [4,5], imaging technology [6], underwater communication [7,8] and other fields. This paper improves the accuracy of target recognition by processing the orbital angular momentum (OAM) spectral information carried in the echo signals of the emitted vortex beam.

When vortex beams propagate through the atmosphere, they experience phase distortion, changes in intensity distribution, and dispersion of orbital angular momentum (OAM) modes. These effects degrade or distort the information carried by the originally pure OAM modes. To mitigate these issues, adaptive correction algorithms—such as the stochastic parallel gradient descent (SPGD) method [9] and the Shack-Hartmann method [10]—are required. Among the Gerchberg–Saxton (GS) algorithm, the GS iterative optimization phase recovery algorithm is the most widely used method, which works on the principle that generating pre-compensation phases can effectively counteract phase perturbations caused by the atmosphere. In 2017, Fu et al. [11] modified the GS algorithm tailored for OAM beam phase recovery in turbulent atmospheres, demonstrating mode purity restoration compared to traditional approaches. In 2018, Li et al. [12] systematically examine advancements in atmospheric turbulence compensation for OAM communication systems, highlighting techniques like phase retrieval algorithms, adaptive optics, and hybrid methods. Similarly, in 2022, Zhao et al. [13] proposed a modified GS algorithm-based wavefront sensorless adaptive optics system to compensate for atmospheric turbulence-induced phase distortions in OAM beams, achieving better compensation performance compared to traditional methods. In 2024, Cui et al. and Wang et al. [14] used orbital angular momentum spectra for target shape recognition and compensated for the distorted phase of different target echo light fields with the GS algorithm to restore the OAM spectrums of the echo light field and improve the target recognition rate. However, the GS algorithm has some limitations: its convergence depends on the initial guess and the choice of constraints, and it has a high computational complexity in the processing of complex wavefronts, which may result in failure to converge or require a large number of iterations, limiting its efficiency and applicability.

In recent years, the development of neural networks has offered new possibilities for the recognition of vortex light OAM [15,16,17,18,19]. First, the adaptive learning ability of neural networks enables models to be continuously optimized to handle complex problems without the need for precise physical modeling. In addition, neural networks support real-time and large-scale data analysis, which greatly improves the efficiency and accuracy of target recognition. This paper will use three different neural networks: multi-layer perceptron (MLP) [20], convolutional neural network (CNN), and residual network (ResNet) [21], trained with a large amount of OAM spectral data, which can quickly and accurately classify the OAM spectral characteristic of different target echo signals. Even when the turbulence intensity is high, the three networks can still maintain high recognition rates. This study has broad application prospects in areas such as target recognition based on multi-dimensional information and intelligent navigation.

2. Methods

2.1. Orbital Angular Momentum Spectrum Theory

The Laguerre–Gauss beam (LG) is the most typical and commonly used vortex beam. The expression for an LG propagating along the z-direction in free space in cylindrical coordinates is expressed as follows [22,23]:

$$\begin{array}{c}E\left(r,z\right)=\sqrt{{\displaystyle \frac{2p!}{\pi \left(p+\left|l\right|\right)!}}}{\displaystyle \frac{1}{\omega \left(z\right)}}{\left[{\displaystyle \frac{\sqrt{2}r}{\omega \left(z\right)}}\right]}^{\left|l\right|}\exp \left({\displaystyle \frac{-r^2}{{\omega }^2\left(z\right)}}\right)L_p^{\left|l\right|}\left({\displaystyle \frac{2r^2}{{\omega }^2\left(z\right)}}\right)\\ \hspace{1em}\hspace{1em} \times \exp \left[i\left(2p+\left|l\right|+1\right){\tan }^{-1}\left(z/z_R\right)-{\displaystyle \frac{ik{r}^{2}z}{2\left(z^2+z_R^2\right)}}\right]\exp \left(il\varphi \right)\end{array},$$

(1)

where p is the radial exponent, l is the topological charge number of the vortex beam, is the waist radius of the beam, r is the distance from the beam to the propagation axis, $L_p^{\left|l\right|}$ is the Laguerre polynomial, $z_R$ is the Rayleigh distance, k is the wave number of the light wave, is the azimuth angle, $\left(2p+\left|l\right|+1\right){\tan }^{-1}\left(z/z_R\right)$ is the Kuy phase shift, and z is the axial propagation distance. Since the OAM states with different topological charges are orthogonal to each other, it is possible to expand any beam with the base $\exp \left(il\phi \right)$, where $a_l$ is the expansion coefficient:

$$E\left(x,y,z\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle \sum _{l=-\infty }^{+\infty }{a}_l}\left(r,z\right)\exp \left(il\phi \right).$$

(2)

$$a_l\left(r,z\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_0^{2\pi }E\left(x,y,z\right)}\exp \left(-il\phi \right)d\phi .$$

(3)

Integrating the expansion coefficient over the entire region can yield the energy on this helical harmonic:

$$C_l={\displaystyle {\int }_0^{\infty }{\left|a_l\left(r,z\right)\right|}^2}rdr.$$

(4)

This value is independent of the z-coordinate, and thus the relative energy of the harmonic can be obtained.

$$R_l={\displaystyle \frac{C_l}{{\displaystyle \sum _{q=-\infty }^{+\infty }{C}_q}}}.$$

(5)

The formula represents the energy ratio of each OAM mode of the beam, which is the orbital angular momentum spectral data we measured.

The method for obtaining the OAM spectrum of the target signal is as shown in Figure 1. First, a laser with a wavelength of 532 nm is emitted using a laser, and a single-mode forked grating is loaded onto SLM1 through a spatial light modulator 1 (SLM1). A single-mode vortex beam is generated using the grating diffraction method [24,25,26,27,28,29]. When discussing the influence of atmospheric turbulence, the power spectrum inversion method is used to simulate the disturbance of atmospheric turbulence to the light field in this study. The superimposed phase of the turbulent screen of the Von Karman model [30,31,32] and the fork grating is loaded on SLM1, and the parameters of atmospheric turbulence are: The inner scale l₀ = 0.0003 m, the outer scale L₀ = 50 m and four kinds of atmospheric turbulence intensity ($C_n^2=1\times {10}^{-15} {\mathrm{m}}^{-2/3}$, $C_n^2=1\times {10}^{-14} {\mathrm{m}}^{-2/3}$, $C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$, $C_n^2=1\times {10}^{-12} {\mathrm{m}}^{-2/3}$) can be obtained by changing the refractive index constant of the atmosphere and taking successively.

With increasing of the refractive index constant $C_n^2$, the distortions caused by the turbulence become more severe, resulting in greater phase aberrations and significant changes in the OAM spectrum, as is shown in Figure A1 in Appendix A. It is worth noting that we just consider the influences of atmospheric turbulence in the distance of 100 m using a modified von Kármán spectrum, which enables us to isolate and analyze the effects of atmospheric turbulence without introducing additional variables related to different distances.

The first-order diffraction modulated signal is selected by using a 4F system with an aperture. After the beam is emitted to the target surface through a beam splitter (BS) and an echo signal is generated, a part of the scattered light is focused onto the CCD by the lens to capture the intensity distribution; the other part is demodulated through SLM2, which sequentially loads the demodulation phase from −10 to 10 orders, and the corresponding reverse-ordered beam is demodulated to the fundamental mode. For example, when there is an OAM component of l = 2 in the echo signal, loading a demodulation phase with l = −2 onto the SLM2 can demodulate the l = 2 component into a fundamental Gaussian beam with l = 2 − 2 = 0. The demodulated fundamental mode Gaussian beam is coupled into a single-mode fiber. The photon counter is used to record the number of photons received within a fixed time period. After normalization, it is used as an indicator of the intensity of each OAM component to obtain the OAM spectral information. The target shapes are equilateral triangles, squares, regular pentagons and circles. In this study, the target size $\eta$ is defined as the ratio of the distance between the geometric centers and vertices of each shape to the scale of the light field screen. The definition of target size $\eta$ as the ratio of the distance from the geometric center to the vertices of the shape relative to the optical field screen scale is intrinsically tied to the beam waist radius. This normalization ensures that different targets maintain consistent relative proportions with the beam waist, enabling fair comparisons across varying shapes and experimental conditions.

2.2. OAM Spectral Characteristic of the Vortex Beam After Being Reflected by Targets of Different Shapes

First, we investigated the effects of targets with different shapes (triangle, square, pentagon, circle) on the OAM spectrum of the echo light field, as shown in Figure 2. The parameters are set as follows: The beam waist radius is ω = 6 mm, the distance is z = 100 m, the target size is $\eta =0.2$, and the topological charge of the emitted signal is 3. Target reflection induces modal dispersion in the optical angular momentum, manifested by enhanced OAM spectral components at kn-mode separation from the incident beam—here defined as the target’s k-order modal dispersion characteristic. Here k is an integer, and n represents the number of symmetry axes of the target. For a circle, which can be regarded as having countless axes of symmetry, the degree of dispersion of its OAM spectrum is relatively low and can be omitted. For the same target size, the relative power of the characteristic order of the triangle is the largest, followed by the square, and the pentagon is the smallest. Because the target with the smaller interior angle has a larger area of unreflected light spot, which makes the target with the smaller interior angle reflect the OAM spectral dispersion characteristic more prominently.

The following formula can be used to describe the characteristic of OAM spectral dispersion for targets of different shapes:

$$R_n\left|l\right.〉={\displaystyle \sum _k{a}_k}\left|l+kn\right.〉.$$

(6)

$\left|l\right.〉$ represents the vortex beam with topological charge l, $a_k$ is the expansion coefficient, k is an integer, and n is the number of symmetry axes of the target.

In addition, we also considered using a superposition state of multi-mode vortex beams as the emitted beam, but it is worth noting that illuminating a target with multi-mode vortex beams may actually reduce the proportion of modes containing target feature information, as shown in Figure 3a,b. It can be seen that the ±4 superposition state diffuses its OAM component towards l = −7, −1, 1, 7 after being reflected by the triangular target. However, the relative power of the ±1-order dispersion characteristics of a single topological charge of 4 (l = 1, 7) is higher than that of the main dispersion characteristics of the ±4 superposition state (l = −7, −1, 1, 7). Moreover, if we employ a ±4 superposition state, the dispersion features of different targets are overly complex and more likely to be confused, making it more difficult for neural networks to learn and classify. For example, when we use a ±4 superposition state to illuminate a pentagon, the dispersion feature of the pentagon contains l = −4 + 5 = 1 and l = 4 − 5 = −1, which has a repetition of features with a triangle (l = −7, −1, 1, 7). Furthermore, we use 1, 3, and 5 vortex beam superposition states to illuminate a triangular target, as shown in Figure 3c; the target information contained in the echo signal is even less regular, with almost no significant target features. It is also difficult to maintain stability under atmospheric turbulence perturbations. Therefore, we mainly employ single-mode vortex beams for target recognition in this article; the recognition results of multi-mode vortex beam recognition can be used as references.

2.3. Neural Network Structure Analysis and OAM Spectrum Data Training Method

Three neural network architectures are used in this study to handle the recognition and classification tasks of angular spectral data: MLP, 1D-CNN [33], and 1D-ResNet [34]. These three networks take advantage of the strengths of fully connected layers, convolutional layers, and residual blocks, as is shown in Figure A2 of Appendix B, respectively, to identify and classify the OAM spectra of echo signals from four types of targets (triangle, square, pentagon, and circle). MLP passes through five fully connected layers (with 128, 256, 512, 256, and 128 neurons, respectively) and ultimately outputs four types of target categories, and the number of neurons and layers we employ has been carefully adjusted to achieve the highest recognition accuracy for MLP under strong turbulence conditions. MLP is simple to design and easy to implement, but it may need more datasets and the construction of more fully connected layers to achieve better recognition performance in strong turbulence. In the proposed CNN, two convolutional-pooling stages with increasing channel dimensions progressively extract spatial features, which are flattened and transformed by two fully connected layers to generate final predictions among four target classes. Owing to its parameter-sharing architecture and hierarchical feature extraction capability, the CNN demonstrates higher training efficiency compared to conventional models. Finally, we utilize a one-dimensional ResNet, which solves the vanishing gradient problem of deep networks through residual blocks, enabling ResNet18 to build deep networks and improve the accuracy and stability of the model. The simplified 1D-ResNet model is designed for 21-length input sequences, featuring an initial convolution layer followed by residual blocks with increasing channel dimensions. Despite the low dimension of the input data, we choose CNN and ResNet mainly for the following reasons. The OAM spectra of different targets have spatial characteristics. When dealing with one-dimensional sequence data with spatial characteristics, the convolution operation of 1D-CNN can utilize the local structure of the data effectively, thus having strong spatial feature extraction ability and maintaining high computational efficiency in strong turbulence. Meanwhile, ResNet, with its cross-layer characteristic fusion mechanism achieved through residual connections, can extract deep characteristics from complex data when strong turbulence interference causes degradation of OAM spectral characteristic; thus, it can show better recognition performance. We will further verify the feature extraction capability of ResNet in Section 3.

In this paper, the distance is set to 100 m. A total of 8000 sets of OAM spectral data collected from four targets exposed to four levels of atmospheric turbulence are divided into a training set and a test set in an 8:2 ratio. These datasets are then fed into the aforementioned three network models via batch processing for training. The models are trained with a batch size of 32, where the initial learning rate is set to 0.001, which has the best recognition performance after our adjustment and validation. All experiments are conducted on an NVIDIA RTX 2060 GPU,(Manufacturer: NVIDIA Corporation, Santa Clara, United States). To prevent overfitting, we implement early stopping when the loss plateaus for 10 consecutive epochs. These three models are trained using the cross-entropy loss function and the Adam optimizer. The expression of the cross-entropy loss L is

$$L=-{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{c=1}^C{y}_{i,c}}}\log (\widehat{y_{i,c}}),$$

(7)

where N is the number of samples, C is the number of categories (here C is 4), $y_{i.c}$ is the true label (0 or 1) that sample i belongs to category c, and $\widehat{y_{i.c}}$ is the model’s predicted probability that sample i belongs to category c.

First, we use a vortex beam with a topological charge of 3 as the emitted signal when $C_n^2=1\times {10}^{-14} {\mathrm{m}}^{-2/3}$. We take the beam waist radius ω = 6 mm, the distance z = 100 m, the target size $\eta =0.2$, and trained with the OAM spectral data of the echo signal. As can be seen from the loss curves shown in Figure 4, the training loss and test loss of both MLP and ResNet decrease rapidly at the beginning and then stabilize, showing good convergence; although CNN converges more epochs than MLP and ResNet, it also converges rapidly in the end.

3. Results and Discussion

3.1. The Impact of Atmospheric Turbulence Intensity on Recognition Rate

In this section, we will study the influence of the distortion of vortex light in atmospheric turbulence on the target recognition rate. The topological charge of the emitted light is 3, and all parameters of the emitted light, target size and distance are consistent with those in 2.3. We will also recognize 400 groups of OAM spectra of the four types of targets echo signals for each turbulence intensity using the three trained neural networks. As shown in Figure 5, the recognition rates of MLP, CNN and ResNet are all close to 100% under conditions of $C_n^2=1\times {10}^{-15} {\mathrm{m}}^{-2/3}$ and $C_n^2=1\times {10}^{-14} {\mathrm{m}}^{-2/3}$, and decrease to 92%, 90.75% and 95.5%, respectively, with the increase in turbulence intensity to $C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$. When the turbulence intensity increases further to $C_n^2=1\times {10}^{-12} {\mathrm{m}}^{-2/3}$, due to the excessively high atmospheric turbulence intensity at this time, the beam’s wavefront distortion becomes severe, leading to the failure of the OAM spectrum in the echo signal. Consequently, the recognition rates of all three types of networks experienced a sharp decline, dropping below 50%. Further, we construct confusion matrices for identifying different targets under four kinds of turbulence intensity, as shown in Figure 6. It can be seen that under conditions of $C_n^2=1\times {10}^{-15} {\mathrm{m}}^{-2/3}$ and $C_n^2=1\times {10}^{-14} {\mathrm{m}}^{-2/3}$, the three networks achieve 100% recognition rates for targets of all shapes, but when the turbulence intensity reaches $C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$, recognition rates of the three networks for pentagons and circles significantly reduce. However, the recognition rate for triangles remains at a relatively high order. This is because the relative power of the dispersion characteristic of the pentagon is too low, and the target characteristic is easily lost under strong turbulence, which causes all three models to easily confuse the pentagon with the circle. The relative power of the dispersion characteristic of triangular targets is relatively high, and they can still maintain high recognition rates under strong atmospheric turbulence disturbances. At the same time, we find that compared with the other two networks, the confusion matrix of ResNet still maintains a higher diagonal value when $C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$, showing stronger stability and generalization ability. When $C_n^2=1\times {10}^{-12} {\mathrm{m}}^{-2/3}$, the confusion matrices of the three networks deteriorate further, it is worth noting that all the characteristics of the targets are completely destroyed, at which point the OAM spectral information is almost ineffective, and the recognitions are severely confused, and none of the three neural networks are capable of extracting features effectively from the invalid OAM spectrum in the case of all topological charges, resulting in a sharp decline in recognition accuracy. Thus, our neural networks mainly reach a high recognition rate when atmospheric turbulence is within the range of $C_n^2=1\times {10}^{-15} {\mathrm{m}}^{-2/3}$ to $C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$.

It is worth noting that although CNN’s recognition rate is slightly lower in the case of strong atmospheric turbulence ($C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$), the training time of CNN is much lower than that of MLP and ResNet. Meanwhile, when the MLP employs fewer fully connected layers (128-256-128), its recognition rate is 86.25%, which is much lower than that of the CNN (90.75%), as is shown in

Table A1. The recognition rates of three neural networks under different layers of MLP and ResNet when l = 3 and turbulence intensity is $C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$.

(a) MLP
Layers	Accuracy (%)	Time (s)
21-128-4	77.5	22
21-128-256-128-4	86.25	126
21-128-256-512-256-128-4	92	332
(b) ResNet
The retained residual block number	Accuracy (%)	Time (s)
Block 1,2	92.5	152
Block 1,2,3	93.75	226
Block 1,2,3,4(ResNet18)	95.5	476

in Appendix A. Therefore, in cases where there is a constraint on training time cost, using a simple CNN to train OAM data and recognize targets is also a very good choice, as is shown in

Table A2. The recognition rates of three proposed neural networks under different layers of MLP and ResNet when l = 3 and turbulence intensity is $C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$.

Network	Accuracy (%)	Time (s)
MLP	92	332
CNN	90.75	96
ResNet	95.5	476

.

3.2. Impact of Emission Beam Topological Charge on Recognition Rate

In this section, we will discuss the influence of the topological charge of the emitted light on the recognition rate. Parameters such as the beam waist radius, target size and distance are all the same as those in 3.1. In this paper, the k-order dispersion characteristic of the target is defined as OAM spectral components at kn-mode separation from the incident beam. Since the relative power of each target’s −1-order dispersion characteristic is the largest, we use them to represent the degree of dispersion of different targets’ OAM spectrum. As shown in Figure 7a, with the increase in topological charge, the spiral phase is more complex, which is more susceptible to the geometric characteristics of the target, so the dispersion degree of the OAM spectrum of each target echo signal also increases. It is worth noting that this phenomenon is inherently determined by the field properties of vortex beams with different topological charges. Taking the pentagonal as an example, as shown in Figure 7b, the −1-order dispersion characteristics of emitted vortex beams with different topological charges increase first and then decrease as the target size increases. But when comparing the relative power of −1-order dispersion characteristics across various topological charges under identical target size $\eta$, we can conclude that vortex beams with higher topological charges exhibit more pronounced −1-order dispersion characteristics.

In order to achieve better training performance and to control variables for a clearer investigation of the ability of emitted light to reflect target characteristics, datasets with different topological charges are trained separately. As shown in Figure 8, when the topological charge of the emitted beam is increased from l = 1 to l = 5 under the condition of $C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$, we can observe the tendency that the recognition performance increases with the increase in topological charge. However, when l is further increased to 6 and 7, the recognition rates of the three networks at lower orders tend to decline compared to l = 5 when $C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$. As shown in Figure 9, we furthermore construct matrices of the recognition rates of the three networks for the four specific patterns at l = 1, 5, 7 under different atmospheric turbulence. When l = 1, under medium and strong turbulence, the recognition rates of the three networks for each target (especially the pentagon) are significantly lower compared to when l = 3. When the topological charge of the emitted beam increases to 5, ResNet and MLP achieve 100% recognition for all targets. Further, when the topological charge of the emitted light is increased to 7 as the turbulence intensity reaches $C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$, we find that due to the more complex light intensity and phase distribution of the vortex beam with l = 7, which is more susceptible to the disturbance of strong turbulence, the purity of the topological charge at the center of the beam decreases, resulting in a lower recognition rate for circular targets than other targets. From the above discussion, it can be seen that by appropriately increasing the topological charge of the emitted beam to improve the dispersion, the recognition rate of targets such as triangles, squares, and pentagons can be effectively enhanced. However, the topological charge of the emitted beam should not be increased too much either, as higher-order vortex beams are more susceptible to atmospheric turbulence. Meanwhile, when topological charge increases to 6, the impact of atmospheric turbulence disturbance on the recognition rate is greater than the effect of the increase in relative power when $C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$, which ultimately leads to a decrease in recognition accuracy.

In addition, we also compared the recognition rates of the vortex beam with topological charge of 4, the ±4 vortex beam superposition state, and the 1, 3, and 5 superposition state mentioned in Section 2.2 as the emitted beam when $C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$, as shown in

Table 1. Recognition rates of three kinds of emitted beams when $C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$.

Network	l = 4	l = ±4	l = 1, 3, 5
CNN	93.25%	88%	76%
MLP	96%	86.5%	71.75%
ResNet	96%	93%	80.5%

. We can see the single-mode vortex beam has better recognition performance than the superposition states, which is consistent with the conclusion in Section 2.2. Interestingly, the phenomenon is more evident from the performance of the MLP because of its lack of spatial feature extraction ability. However, we can also change the topological charge of the emitted single-mode vortex beam and integrate the recognition results of multiple topological charges (such as the result of the emitted topological charge being l = 3, l = 5, l = 7, respectively) and make a comprehensive analysis and judgment to ultimately identify the target more accurately.

In a complex atmospheric turbulence environment, different neural networks show differences in recognition performance closely related to their structural characteristics. ResNet, with its cross-layer characteristic fusion mechanism achieved through residual connections, can extract deep characteristic from complex data when turbulence interference causes degradation of OAM spectral characteristic, thus maintaining the highest recognition rate at all turbulence intensities; CNN excels at capturing the residual local spatial patterns after nonlinear distortion through the convolution features of local receptive fields and weight sharing; MLP relies on the global mapping advantage of the fully connected layer to quickly establish end-to-end classification decision boundaries when the features are not complex, but its lack of spatial feature extraction ability leads to a sudden drop in performance when the spatial features are deep and complex. Moreover, when the turbulence intensity is extremely high ($C_n^2=1\times {10}^{-12} {\mathrm{m}}^{-2/3}$), the echo signal wavefront distortion is severe, the OAM spectral data becomes extremely chaotic, and all three networks have difficulty extracting the chaotic and complex data characteristic, especially MLP, whose fully connected structure fails to learn the characteristic of circles, pentagons, and squares at l = 1 and l = 7, and thus recognizes all the targets as triangles.

3.3. The Impact of Target Size on Recognition Rate

This section examines the effect of target size on recognition rate, with the target size gradually increasing from 0.2 to 0.3 under the condition of $C_n^2=1\times {10}^{-14} {\mathrm{m}}^{-2/3}$ and l = 5. As shown in Figure 10a, we find that the CNN’s recognition rate drops sharply to 91% when $\eta$ reaches 0.3; MLP’s recognition rate drops from 100% to 91.75% when $\eta$ reaches 0.3; by contrast, ResNet is least affected by the target size and still maintains a recognition rate of 97% when $\eta =0.3$. All three networks tend to confuse circles with pentagons, which also confirms the conclusion that the OAM spectral dispersion characteristic is positively correlated with the recognition rate. As shown in Figure 10b, with the increase of $\eta$, the relative power of the −1-order dispersion characteristic of the targets decreases, especially for the pentagon. When $\eta$ increases to 0.3, its OAM spectral dispersion characteristic almost disappears, which leads to a sharp decline in the recognition ability of the three networks for pentagons and circles. From the above results, it can be seen that the target size should not be too large, because at this time the relative power of the −1-order dispersion characteristic is too low to clearly reflect the characteristic of the target, resulting in a decrease in recognition rate, especially for targets with large interior angles such as pentagons.

From the recognition performance of the three networks at medium turbulence intensity, when the actual value of the OAM spectral dispersion characteristic of the target echo signal is very small (such as a pentagon), ResNet, due to its internal residual block and skip connection design, can effectively enhance the transmission efficiency of the information flow, ensuring that even tiny dispersion characteristics can be fully utilized and transmitted. This maintains the effectiveness of learning in deep network structures, and the recognition rate is less affected by the size of the target.

4. Conclusions

In this paper, single-mode vortex beams are used to illuminate targets of different shapes, and the targets are recognized by the OAM spectral information of the echo signals. MLP, CNN and ResNet are used to train a large amount of echo OAM spectral data to achieve the recognition and classification of the targets. All three networks are able to quickly, accurately and stably classify the OAM spectral data of different target echo signals under the disturbance of strong atmospheric turbulence. Among them, ResNet, with its cross-layer characteristic fusion mechanism achieved through residual connections, can extract deep characteristics from complex data when turbulence interference causes degradation of OAM spectral characteristics and still maintain high recognition rates when $C_n^2=1\times {10}^{-13}{\mathrm{m}}^{-2/3}$. Secondly, since the relative power of the −1-order dispersion characteristic of the target decreases as the target size $\eta$ increases; the recognition rate also decreases accordingly. ResNet enhances gradient flow and feature reuse through residual blocks and skip connections, demonstrating the strongest adaptability to the increasing of $\eta$, with a recognition rate of 97% even when $\eta$ reaches 0.3. MLP and CNN are less robust to target size, with recognition rates of 91.75% and 91%, respectively. Although CNN and MLP have slightly lower recognition performance than ResNet in general, they take much less time to train than ResNet, achieving a good balance between recognition performance and training time cost. Our proposed models not only outperform traditional methods and simpler classifiers but also offer comparable or superior performance to the GS algorithm while requiring fewer computational resources, as is shown in

Table A3. The recognition performance comparison when $C_n^2=1\times {10}^{-13} {\mathrm{m}}^{-2/3}$ and l = 3.

Method	Accuracy (%)	Notes
Random Guessing	25	Baseline for random selection among four categories.
Simpler Classifier (21-10-4)	71.5	Average performance using simpler classifiers on the same dataset.
GS Algorithm in [14]	88	High recognition rate but suffers from high computational demands and real-time adaptation issues.
Proposed MLP	92	Optimal configuration with high recognition rate and medium computational cost.
Proposed CNN	90.75	Most efficient processing of spatially structured data like OAM spectra in three proposed models.
Proposed ResNet	95.5	Deep network structure with residual blocks, offering superior performance and stability under strong turbulence and increasing of target size.

.

This paper studies the target recognition rates of different neural networks based on the OAM spectrum of echo signals in an atmospheric turbulent environment. In this paper, we assume that the targets being detected are flat and free of undulations. However, we plan to expand our study to include the influence of surface roughness on recognition performance in future work. Furthermore, we may utilize perfect vortex beams or Bessel–Gaussian beams as the emitted beams and compare the results with those of LGs to investigate whether using perfect vortex beams as the emitted source can alleviate the disturbance caused by atmospheric turbulence and further enhance the recognition rate. Subsequently, our research can be used for the recognition and classification of asymmetric targets and has broad application prospects in areas such as multi-dimensional information-based target recognition and intelligent navigation.