High-resolution recognition of FOAM modes via an improved EfficientNet V2 based convolutional neural network

Abstract

Vortex beam with fractional orbital angular momentum (FOAM) is the excellent candidate for improving the capacity of free-space optical (FSO) communication system due to its infinite modes. Therefore, the recognition of FOAM modes with higher resolution is always of great concern. In this work, through an improved EfficientNetV2 based convolutional neural network (CNN), we experimentally achieve the implementation of the recognition of FOAM modes with a resolution as high as 0.001. To the best of our knowledge, it is the first time this high resolution has been achieved. Under the strong atmospheric turbulence (AT) (C_n^2={10}^{-15}{\mathrm{m}}^{-2/3}), the recognition accuracy of FOAM modes at 0.1 and 0.01 resolution with our model is up to 99.12% and 92.24% for a long transmission distance of 2000 m. Even for the resolution at 0.001, the recognition accuracy can still remain at 78.77%. This work provides an effective method for the recognition of FOAM modes, which may largely improve the channel capacity of the free-space optical communication.

Graphical abstract

Keywords

OAM / free-space optical communication / deep learning / convolutional neural network

1 Introduction

The vortex beams (VBs) can be characterized by a phase factor \mathrm{e} \mathrm{x}\mathrm{p}\left(\mathrm{i}l\phi \right) and the orbital angular momentum (OAM) l\hslash carried for each photon [1]. The VBs exhibit a spiral wave front and phase singularity at the center, where the l,\hslash ,\phi represent the topological charge, the Planck constant and the azimuthal angle respectively. Due to these special characteristics and more manipulation dimension, the VBs have highly potential applications in quantum entanglement [2], sensing [3], super-resolution microscopy [4], biomedicine [5], nonlinear optics [6], and optical communication [7]. In a free-space optical communication (FSO) system [8-10], the transmission capability can be greatly expanded with the introduction of OAM, especially the fractional orbital angular momentum (FOAM) due to the theoretically infinite number of available OAM modes [11]. However, the slight difference in the radius of the central circular ring between adjacent modes poses a great challenge to the recognition of FOAM modes compared to the integer modes [12, 13]. Furthermore, in an OAM-FSO system, the atmospheric turbulence (AT) is a common harmful factor that causes the optical intensity fluctuation, random phase perturbation, and position displacement on the VB signal [14, 15]. Therefore, the high-resolution recognition of FOAM modes under strong AT is still a problem demanding prompt solution. Various methods have been proposed to recognize integer OAM modes [16-21] and FOAM modes at 0.1 resolution [22, 23]. However, achieving higher-resolution recognition of FOAM modes remains a challenging endeavor.

In recent years, deep learning (DL) has been widely used in computer vision [24-27]. Compared to other neural networks, convolutional neural network (CNN) [28, 29] possesses superior capabilities in feature extraction and image data processing, which are the critical factors in image recognition. By building a CNN with an 8-layer structure, the recognition of FOAM modes at 0.01 resolution was firstly achieved with the recognition accuracy more than 99% [30]. However, this architecture is not suitable under strong AT because it lacks a module for extracting detailed features from images. Then, a 50-layer network structure based residual neural network (ResNet 50, a common CNN architecture) was introduced to recognize the FOAM modes at 0.1 resolution under strong AT (C_n^2= {5\times 10}^{-14} {\mathrm{m}}^{-2/3}) and long transmission distance (z = 1500 m). The recognition accuracy was as high as 85.30% [31]. By passing the FOAM through a column lens and then feeding it into the ResNet 50, the recognition resolution of FOAM modes was further improved to 0.01 even under strong AT (C_n^2={10}^{-15}{\mathrm{m}}^{-2/3}) and long transmission distance (z = 1500 m). The recognition accuracy reached 99.07% at 0.1 resolution and 73.68% at 0.01 resolution [32]. However, the ResNet 50 will suffer overfitting and gradient vanishing easily when training large-scale datasets, which is unfriendly to the image recognition task with a high similarity [33]. Recently, the EfficientNet V2 based CNN has been demonstrated that it can achieve higher recognition accuracy in image classification by employing incremental learning methods and architectural improvements such as the squeeze-and-excitation (SE) module and self-attention mechanism [34-36]. Therefore, the EfficientNet V2 may help further improve the recognition accuracy of FOAM modes at higher resolution (like 0.001) even under strong AT and long transmission distance.

In this work, we experimentally achieve the recognition of FOAM modes at 0.1, 0.01, and 0.001 resolution with much high recognition accuracy under strong AT and long transmission distance by an improved EfficientNet V2 based CNN. By modifying the layers of the SE module and optimizing other parameters (dropout, width factor, depth factor, activation function, etc.), the improved model can extract more detail features of FOAM modes. Based on this proposed model, we experimentally verify its great effectiveness for the recognition of FOAM modes at 0.1 and 0.01 resolution with much high recognition accuracy. Most importantly, for the FOAM modes at 0.001 resolution, the recognition accuracy can still reach 78.77% even under strong AT (C_n^2={10}^{-15}{\mathrm{m}}^{-2/3}) and long transmission distance (z = 2000 m). Our work exhibits the huge potential of CNN in the recognition of FOAM modes, which may promote the booming development of FSO communication.

2 Principles and concepts

2.1 Experimental setup

Fig.1 shows the experimental setup for generating and recognizing the FOAM modes. A highly stable He-Ne laser (wavelength: 632.8 nm; beam waist: 1.0 mm) is used as the Gaussian beam source. The beam is firstly expanded to 6 mm by the beam expansion system [formed by lens L₁ (50 mm) and L₂ (300 mm)]. A Glan−Teller (GL) prism is used to control the beam polarization. Then, the beam incidents on a spatial light modulator (SLM: Holoeye, ERIS-VIS-109), which is preloaded with the phase hologram of FOAM, AT, and AD. Finally, the diffraction pattern captured by the charge-coupled device (CCD: Thorlabs, BC106N-VIS/M) is input to the improved EfficientNet V2 model for recognizing the FOAM modes.

2.2 Intensity distribution of VBs after passing AD

FOAM modes (with topological charge l) can be easily obtained by loading the spiral wavefront phase on a Gaussian beam using SLM [37]

(1)E\left(r_0,{\theta }_0\right)=A\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{r_0^2}{w_0^2}}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{i}l{\theta }_0\right),

where \left(r_0,{\theta }_0\right) denotes the cylindrical coordinate, A is the complex amplitude, w₀ is the waist of the Gaussian beam, and l is the topological charge.

In order to obtain more detailed features of the FOAM, we design an aperture diaphragm (AD) consisting of a square hole with an internal circular screen shown in Fig.2(a). The transmittance function of AD can be expressed as

(2)T\left(r_1,{\theta }_1\right)=f_1\left(r_1,{\theta }_1\right)+f_2\left(r_1,{\theta }_1\right),

(3)

(4)

where f_1 and f_2 are the transmittance functions of the square hole and circular screen, respectively, a is the side length of the square hole, and R is the radius of circular screen. In this work, R=a/2=1.5\mathrm{m} \mathrm{m}.

According to the Collins diffraction integral equation [38, 39], the field distribution of the VB after passing through the AD can be expressed as

(5)\begin{array}{rl}U\left(r,\theta ,z\right)=& \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{i}kz\right)}{\mathrm{i}\lambda z}{\int }_0^{\mathrm{\infty }}{\int }_0^{2\pi }E(r_0,{\theta }_0)T\left(r_1,{\theta }_1\right)\\ & \times \mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{\mathrm{i}k}{2z}{[r}_0^2+r_1^2-2r_0{r}_1\cos \left({\theta }_0-{\theta }_1\right)]\right)r_0\mathrm{d}{r}_0\mathrm{d}{\theta }_0,\end{array}

where z is the transmission distance.

Finally, the beam intensity received on the CCD can be expressed as

(6)I\left(r,\theta ,z\right)=U\left(r,\theta ,z\right)\times U^{\ast }\left(r,\theta ,z\right).

To verify the effectiveness of our designed AD, we give out the simulation and experimental results of OAM modes in Fig.2. Fig.2(b) and (c) show the simulated intensity distributions of the original OAM modes (l = 1−5) and diffraction patterns after AD. In Fig.2(b), the radius of center ring increases gradually with the increase of l. After the AD, the diffraction patterns in Fig.2(c) present a four-fold symmetry [40], arising from the square diaphragm contained in AD. When l > 1, the number of diffracted spots in the outermost layer is (l + 1)/2 (when l is odd), or (l + 2)/2 (when l is even). In addition, whether there are intersection points in the center region can also determine the even or odd value of l (When l is odd, the intersection point appears). For example, when l = 3, the number of diffracted spots in the outermost layer is 2, and there are four intersection points in the center.

2.3 Construction of AT

During the beam transmission, the AT can seriously distort the wavefront phase and intensity distribution of the VB, which brings serious disturbance to the OAM communication [41, 42]. Here, we use the Hill Andrews model to simulate the AT, the equivalent phase can be shown as [43]

(7)\begin{array}{rl}{\mathrm{\Phi }}_n\left(k\right)=& 0.33C_n^2\left[1+1.802\sqrt{\frac{k^2}{k_l^2}}-0.254{\left(\frac{k^2}{k_l^2}\right)}^{\frac{7}{12}}\right]\\ & \times \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{k^2}{k_l^2}}\right)}{{\left(k_0^2+k^2\right)}^{\frac{11}{6}}},\end{array}

where k_0= 2\pi /L_0, L_0 is the outer scale of AT, k_l=2\pi /l_0, l_0 is the inner scale of AT, k^2=k_x^2+k_y^2, k_x and k_y denote the wavenumbers in the x and y directions, respectively, and C_n^2 is the structure constant of the refractive index of air which describes the intensity of AT [44].

The spatial variation in AT ({\sigma }^2\left(k\right) ) can be approximated by a number of phase screens, which modify the phase profile as the beam transmission. The relationship between {\sigma }^2\left(k\right) and {\mathrm{\Phi }}_n\left(k\right) can be expressed as [45]

(8){\sigma }^2\left(k\right)= {\left({\displaystyle \frac{2\pi }{N\mathrm{\Delta }x}}\right)}^{2}2 \pi k_z^2\mathrm{\Delta }z {\mathrm{\Phi }}_n\left(k\right),

where \mathrm{\Delta }x denotes the grid spacing of a random phase screen, N indicates the phase screen size, \mathrm{\Delta }z is the interval distance between adjacent phase screens, and k_z=2\pi /\lambda. We utilize a fast Fourier transform (FFT) operation to express the phase screen in the frequency domain,

(9)\xi \left(k\right)=FFT \left(M\sigma \left(k\right)\right),

where M is a complex Gaussian random matrix with mean 0 and variance 1 in the frequency domain, and FFT represents the fast Fourier transform operation.

During the beam transmission, the phase information of each layer can be designed and superimposed on the SLM. It should be noted that as the interval distance Δz between the adjacent phase screens decreases, the simulation accuracy increases rapidly. However, the computational resource requirement increases significantly, which is not conducive to the collection of large data sets. After several tests, we finally chose \mathrm{\Delta }z= 250 m. The inner scale and outer scale of AT are set to l₀ = 0.01 m and L₀ = 100 m. Fig.3 shows the intensity of FOAM modes under three different AT and transmission distance (z = 2000 m) experimentally. As C_n^2 increases, the intensity of FOAM modes undergoes increasingly obvious distortions which enlarges the challenge of recognizing.

2.4 Architecture of the improved EfficientNet V2 based CNN

Noting that some components of the EfficientNet V2 model like SE module is useful for analyzing the detailed features of images. Therefore, for higher recognition resolution and accuracy, we increase the layers of the SE module and optimize other parameters (dropout, width factor, depth factor, activation function, etc.) to extract more detail feature of the FOAM modes.

The architecture of the CNN is shown in Fig.4(a), where the input image size is adjusted to 224 × 224, Conv3 × 3 is the 3 × 3 convolution + activation function (SiLU) + Batch Normalization (BN). The Fused-MBConv module enhances the CNN performance by combining feature maps of different resolutions. When expansion is set to 1 [Fig.4(b)], it consists of a 3 × 3 convolutional layer, a BN layer with SiLU activation, and a dropout layer with 0.2 dropout rate. In the case of expansion ≠ 1 [Fig.4(c)], it includes an additional Project Conv module. By using these two different expansion ratios (1 and ≠ 1), the CNN can achieve faster training and manage complexity efficiently in image recognition. The structure of the MBConv module is shown in Fig.4(d), compared to Fused-MBConv module, it has an additional SE module, which means it can automatically learn the importance of different channel features. 0.25 represents that the node number of the first fully connected layer in the SE module occupies 25% of the input MBConv module feature matrix channels. The key distinction between the MBConv and Fused-MBConv modules lies in the incorporation of fused point-wise and depth-wise convolution, which enhances the computational efficiency and network performance in the latter.

In the end of the CNN model, a module consisting of a 1 × 1 convolutional layer + pooling layer + FC (fully connected layer, activation = Softmax) is used as the output for the classification of the FOAM modes. As for the loss function, we use the Sparse Categorical Cross Entropy [46], which can be expressed as

(10)loss\left(f\left(X,b\right),Y\right)=-{\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m{y}_i\mathrm{l}\mathrm{o}\mathrm{g}(y^{\ast }),}

where m is the number of data set input to the CNN, y^* is the predicted output, and y_i is the ideal output. The weight and bias parameters are updated using the Adam optimizer with a learning rate of 0.005, aiming to minimize the loss function.

3 Results and discussion

3.1 Recognition of FOAM modes at 0.1 and 0.01 resolution

For the FOAM modes at 0.1 resolution, we experimentally collect 10000 images (80% serving as the training sets and 20% as the test sets). Fig.5(a)−(d) show the recognition results under a medium AT (C_n^2={10}^{-16}{\mathrm{m}}^{-2/3}) and transmission distance (z = 500 m) of FOAM modes at 0.1 resolution. Fig.5(a) presents the intensity distribution of FOAM modes transmitting under AT with l = 4.1, 4.3, 4.5, 4.7, and 4.9, respectively. As l increases, the radius of central circular ring increases and the size of the notch increases first and then decreases. The corresponding intensity distributions diffracting after the ADs are shown in Fig.5(b). As l increases, the diffraction center gradually becomes a rectangular hole, and the rest of the diffracted spots gradually transition into the dart-shaped structures. It is worth noting that at this point, the topological charge l can still be recognized narrowly. After applying our proposed CNN, the recognition accuracy and loss values of the FOAM modes in the test set are shown in Fig.5(c) and (d) (red line). For comparison, we also show the situation under C_n^2={10}^{-15} {\mathrm{m}}^{-2/3} (blackline) and C_n^2={10}^{-17} {\mathrm{m}}^{-2/3} (bule line). As a result, the recognition accuracy in test set can quickly approach 100% after only 3 epochs. After 20 epochs, the accuracy and loss curves under three different AT all tend to be stable. When the epochs exceed 100, the recognition accuracy of FOAM modes finally reach 99.80%, 100%, and 100% under three different AT, while the losses are 1.25\times {10}^{-2}, 2.44\times {10}^{-7}, and 1.57\times {10}^{-7}. It is shown that our proposed CNN model exhibits perfect performance for the recognition of FOAM modes at 0.1 resolution under strong AT and long transmission distance.

For the FOAM modes at 0.01 resolution, we experimentally collect 25000 images (80% serving as the training set and 20% as the test set), results show in Fig.5(e)‒(h). From Fig.5(e), the radius of the center ring of adjacent FOAM modes increases at a much slower rate than the FOAM modes at 0.1 resolution and the appearance of notches is barely visible. Fig.5(f) shows the intensity distribution of FOAM modes diffracting after AD. Unfortunately, it is not possible to directly distinguish the FOAM modes at 0.01 resolution. Then, our proposed CNN model is applied for the recognition of FAOM modes. From Fig.5(g) and (h), after 40 epochs, the recognition accuracy curve and loss curve in test set all become stably, the recognition accuracy of FOAM modes finally reaches 94.76%, 96.34%, and 98.46%, while the losses are 0.3411, 0.2038, and 0.0513 under three different AT. The model converges more slowly than that at 0.1 resolution. However, our model can still possess high performance at 0.01 resolution.

3.2 Recognition of FOAM modes at 0.001 resolution

Furtherly, the proposed CNN model is utilized in recognition of the FOAM modes at 0.001 resolution. Due to the more difficulty for the recognition of FAOM modes at higher resolution, we expand the number of data set to improve the recognition accuracy. We eventually collect 60 000 images, in which 48 000 are used as the training set and others as the test set. Fig.6 shows the recognition results under a medium AT of C_n^2= {10}^{-16}{\mathrm{m}}^{-2/3} and transmission distance of z = 500 m at 0.001 resolution. Unfortunately, both from the radius of the center ring of the FOAM modes in Fig.6(a) and from the spot distribution of the FOAM modes after AD in Fig.6(b), the differences between adjacent FOAM modes are essentially invisible and can no longer be recognized. Nevertheless, our proposed CNN model can solve this problem. Fig.6(c) and (d) show the recognition accuracy and loss value in the test set under three different AT. Due to the increase in training difficulty, the recognition accuracy for the first few epochs dropped significantly comparing to the previous ones. However, since our proposed model has great advantages in performing feature extraction, it can still converge around 60 epochs. After 100 epochs, the recognition accuracy of FOAM modes under weak (C_n^2= {10}^{-17}{\mathrm{m}}^{-2/3}), medium (C_n^2= {10}^{-16}{\mathrm{m}}^{-2/3}), strong (C_n^2= {10}^{-15}{\mathrm{m}}^{-2/3}) AT are 94.88%, 91.48%, and 89.47%. The losses are 0.3102, 0.4271, and 0.4861. It can be concluded that our proposed model is still highly effective for the recognition of FOAM modes at 0.001 resolution, even under strong AT and long transmission distance.

Finally, we comprehensively investigate the effects of different transmission distance and AT on the recognition accuracy of FOAM modes. Fig.7(a) shows the recognition accuracy of FOAM modes at 0.1 resolution under different AT and transmission distances. As the AT increase, the recognition accuracy decreases slightly but still remains at a very high level. Even under strong AT {(C}_n^2= {10}^{-15}{\mathrm{m}}^{-2/3}) and long distance (z = 2000 m), the recognition accuracy is still up to 99.12%. From the above results, our model is extremely effective in recognizing FOAM modes at 0.1 resolution. Fig.7(b) shows the recognition accuracy of FOAM modes at 0.01 resolution. Obviously, even under strong AT (C_n^2={10}^{-15}{\mathrm{m}}^{-2/3}) and long transmission distance (z = 2000 m), the recognition accuracy reaches 92.24%. As anticipated, the recognition accuracy of FOAM modes at 0.01 resolution still remains at a high level, further confirming the great effectiveness of our model. Fig.7(c) shows the recognition accuracy of FOAM modes at 0.001 resolution. With the increase of AT and transmission distance, the recognition accuracy decreases significantly. Under the weak AT {(C}_n^2= {10}^{-17}{\mathrm{m}}^{-2/3}), the recognition accuracy decreases to 94.88%, 93.17%, 92.45% and 91.18% with the increase of transmission distance. Under the medium AT {(C}_n^2= {10}^{-16}{\mathrm{m}}^{-2/3}), the recognition accuracy decreases to 91.48%, 90.67%, 90.37% and 89.42%. Under the strong AT {(C}_n^2= {10}^{-15}{\mathrm{m}}^{-2/3}), the recognition accuracy decreases to 89.47%, 86.73%, 80.63% and 78.77%. The above results show that the accuracy is relate to AT and remain high value even under strong AT.

In addition, compared with the ResNet 50 [32], our model shows significant improvement, especially under strong AT (C_n^2={10}^{-15} {\mathrm{m}}^{-2/3}) and long transmission distance (z = 1500 m), the recognition accuracy is improved by 20.3% for the FOAM modes at 0.01 resolution. Furthermore, we also successfully realize the recognition of 0.001 FOAM modes. The specific comparative information for this improved EfficientNet V2 and ResNet 50 is shown in Tab.1. It can be seen that our network is superior to ResNet 50 in terms of network complexity and training time, which greatly reduces the training time and computational resources. Therefore, our model has an advantage over the ResNet 50 in handling the task of FOAM recognition.

4 Conclusion

In conclusion, we propose an improved EfficientNet V2 based CNN model for the recognition of FOAM modes experimentally. By modifying the layers of the SE module and optimizing other parameters, the model has a better ability to extract image features. By learning the mapping relationship between the topological charge l and the intensity distribution of the FOAM modes after the AD, the different FOAM modes at 0.1, 0.01 and 0.001 resolution under strong AT and long transmission distance can be precisely recognized based on the model. Specifically, for the FOAM modes at 0.1, 0.01 and 0.001 resolution, the recognition accuracy is 99.12%, 92.24% and 78.77%, respectively, even in the presence of C_n^2= {10}^{-15}{\mathrm{m}}^{-2/3} and z = 2000 m. These findings represent significant advancements in enhancing the channel capacity of FSO communication.

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