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Abstract
High-order Gaussian beams with multiple propagation modes have been studied for free-space optical communications. Fast classification of beams using a diffractive deep neural network (D2NN) has been proposed. D2NN optimization is important because it has numerous hyperparameters, such as interlayer distances and mode combinations. In this study, we classify Hermite–Gaussian beams, which are high-order Gaussian beams, using a D2NN, and automatically tune one of its hyperparameters known as the interlayer distance. We used the tree-structured Parzen estimator, a hyperparameter auto-tuning algorithm, to search for the best model. As a result, the proposed method improved the classification accuracy in a 16 mode classification from 98.3% in the case of equal spacing of layers to 98.8%. In a 36 mode classification, the proposed method significantly improved the classification accuracy from 84.9% to 94.9%. In addition, we confirmed that accuracy by auto-tuning improves as the number of classification modes increases.
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1. Introduction
Gaussian beams have high-order propagation modes. Examples include Laguerre–Gaussian (LG) beams and Hermite–Gaussian (HG) beams, which have cylindrical symmetry and rectangular symmetry, respectively. LG beams are light with a helical wavefront known as optical vortex. LG beams have a quantity called orbital angular momentum of $l\hbar $ ($l$ represents an integer and $\hbar $ represents Planck’s constant) per photon [1], which can increase the number of channels in free-space optical communications. An LG beam has orders p and l in the radial direction and azimuthal directions, respectively. p represents an integer greater than or equal to zero, and l represents a positive or negative integer; thus, theoretically an LG beam has an infinite number of propagation modes.
An HG beam is a rectangular beam with symmetry along the propagation axis and has an order m in the x-direction and an order n in the y-direction in the plane perpendicular to the direction of propagation z. Since both orders are integer values that are greater than or equal to zero, an HG beam also has an infinite number of propagation modes. Multiple-bit communication by assigning codes to each of these modes [2,3] along with conventional amplitude shift keying (ASK) [4] and multiplexing using orthogonality [5] have been studied. Terabit free-space optcical communication with high-order Gaussian beams has already been achieved by combining coding and multiplexing [6]. To generate multi-mode HG beams, phase patterns that produce a single-mode HG beam are displayed on spatial light modulators (SLMs), then lasers are switched to irradiate several phase patterns, and the obtained single HG beams are optically combined to produce the desired multi-mode HG beam [7]. In this method, multi-mode HG beams can be generated at the operating frequency of the laser sources.
A system for detecting the mode of a beam is required to realize multibit communications. Several methods for detecting a beam’s code have been investigated, including mode sorters [8], computer-generated holograms displayed on a spatial light modulator, and deep learning [6,9]. Imaging using a charge coupled device (CCD) camera and computational load in neural networks are bottlenecks of these methods.
Beam classification using electrical deep neural networks based on semiconductors can be expected to have high accuracy; however, the classification speed is limited by the semiconductor’s clock frequency, and power consumption is also high. Recently, an optical computer known as a diffractive deep neural network (D2NN), which uses passive optical modulators, has been proposed [10]. In a D2NN, training calculation is performed using a computer, and trained parameters are recorded in an optical modulator (photopolymer or SLM). Light modulators correspond to the layers of a neural network, and by arranging them in equal intervals, a deep neural network is mimicked. The input light is modulated in each layer, and a photodetector placed in the output layer detects the calculation result of D2NN. Since classification can be performed only by light propagation and diffraction, it is performed at the speed of light with a minute amount of power consumption.
Multimode classification of high-order Gaussian beams using a D2NN has been performed [11,12], as has multiplexed beam generation and separation [12,13,14]. In both cases, beam classification can be detected by a photodetector, which can speed up the detection speed by two to three orders of magnitude compared to a CCD camera.
However, in D2NN training, the number of layers, distance between layers, and choice of propagation mode are hyperparameters that should be optimized to improve classification accuracy. In general, hyperparameters are empirically determined.
In this study, we propose an automatic tuning of the hyperparameters of a D2NN for classifying HG beams of high-order Gaussian beams. Instead of manual tuning, automatic tuning algorithms using random and grid search and Bayesian optimization have been proposed to tune hyperparameters [15]. We use Bayesian optimization, tree-structured Parzen estimator (TPE), as an auto-tuning algorithm to optimize the interlayer distance and compare it with the case of equal spacing of layers. Phase disturbance due to the atmosphere was added to HG beams to assume free-space optical communications. The proposed method improved the classification accuracy in a 16 mode classification from 98.3% in the case of equal spacing of layers to 98.8%. In a 36 mode classification, the proposed method significantly improved the classification accuracy from 84.9% to 94.9%.
Section 2 describes HG beams and atmospheric turbulence, Section 3 explains the proposed method, Section 4 presents simulation results, and Section 5 concludes this study.
2. High-order Gaussian beam and atmospheric turbulence
In this session, we describe the HG beam used in this study and the disturbance model of the beam in the atmosphere.
2.1 Hermite–Gaussian beam
An HG beam is a beam having a rectangular symmetry [1], which has the following two degrees of freedom in the plane perpendicular to the optical axis: order m in the x-direction and order n in the y-direction. It is represented by
2.2 Atmospheric turbulence
When light propagates through the atmosphere, the phase of the beam is disturbed. In this study, we used the von Karman model to add atmospheric disturbance to an HG beam [16]. The fluctuation spectrum of the refractive index ${\phi _n}(\kappa )$ is expressed by
Fig. 1. Examples of Hermite–Gaussian (HG) beams. ($m,\textrm{n}$) represents the orders of m and n for an HG beam, respectively.
3. Proposed method
Figure 2 shows the D2NN model used in this study. The D2NN consists of multiple diffraction layers. In the conventional method shown in Fig. 2(a), each layer is equally spaced, whereas, in the proposed method shown in Fig. 2(b), each layer is unequally spaced. Pixels in each layer correspond to neurons, and the light diffracted from the previous layer is optically coupled to each neuron in the next layer. The angular spectrum method was used for light propagation calculation. The lightwave ${u^l}$ propagating from the $l - 1$ layer to l layer is calculated by
The D2NN has several diffraction layers, with lightwave as input. In the output layer, photodetectors are classified. The D2NN is trained to focus light on the detector corresponding to the input beam.
3.1 Black-box optimization
Hyperparameter optimization is essential to improve the performance of neural networks. The hyperparameters of the D2NN in this study include the number of layers, distance between each layer, and selection/combination of the beam pattern. Since finding the gradient calculation of these hyperparameters is difficult, we did not use stochastic gradient methods, such as Adam optimizer [17] used in deep learning. Thus, we need other optimization methods to find the optimal hyperparameters.
The relationship between hyperparameter values and classification accuracy is unknown. Therefore, the best hyperparameter settings are obtained by training with various values and comparing losses. For this, grid and random searches have been used. Grid search is intuitive and can be parallelized. However, it evaluates nondominant hyperparameters, which increase the computation time [18]. Random search has the advantage of the search being terminated in the middle of optimization, and the results of multiple searches executed independently can be combined [19]. However, methods such as random and grid searches are inefficient because they require several computationally expensive learning calculations.
Therfore, we used black-box optimization. Black-box optimizations observe some hyperparameters $x\; $ and the value of the loss function y and predict the x value that minimizes y from the results. Since the search proceeds while predicting the optimal hyperparameters, it is more efficient than random and grid searches.
TPE is a black-box optimization technique used as an automatic hyperparameter tuning algorithm for neural networks. TPE is a type of Bayesian optimization, consisting of a surrogate model to approximate the distribution of promising hyperparameters and an acquisition function to sample the next hyperparameters to be evaluated. TPE uses a proxy model and expected improvement as the acquisition function [20,21]. Figure 3 shows an example of a hyperparameter search $x = ({{z_0},{z_1},{z_2}} )$ of Fig. 2 by TPE. TPE first sets a random hyperparameter $x = ({{z_0},{z_1},{z_2}} )$ and evaluates it. In this study, we set five random hyperparameters and after the evaluations of the random hyperparameters, they are categorized into good and bad groups as in Fig. 3(a), and their distributions $l(x )$ and $g(x )$ are respectively approximated by kernel density estimation as in Fig. 3(b). Next, several points are sampled from the kernel density estimation, and a new hyperparameter $x = ({{z_0},{z_1},{z_2}} )\; $ is selected as the next observation point, maximizing $g(x )/l(x )$. By repeating this search, TPE can efficiently obtain excellent hyperparameters with few trials. After finishing the TPE optimization, we could obtain the optimized $x = ({{z_0},{z_1},{z_2}} )$.
4. Results
In this study, we set the wavelength of HG beams to 1550 [nm], the beam waist to 0.01 [m], and the propagation distance through the atmosphere to 250 [m]. For the atmospheric disturbance, the internal scale ${l_0} = 0.001$ [m], external scale ${L_0} = 20$ [m], and atmospheric structure constant $C_n^2 = 1 \times {10^{ - 14}}\; [{{\textrm{m}^{ - 2/3}}} ]$ are assumed for free-space optical communication during calm daytime. For the D2NN model, the number of pixels of the input beam plane, detector plane, and all layers is $256\; \times \; 256$ pixels, and the sampling pitch is 39 [µm]. We prepared 1000 images for each beam mode, 800 images for training data, and 200 images for validation data. We fixed the distance from the input plane to the output plane at 30 [cm] and searched for the best layer arrangement. The loss function E is the mean squared error:
where K represents the number of pixels in the output image and ground-truth data, ${s_k}$ represents the light intensity of each pixel in the output image, and ${g_k}$ represents the light intensity of each pixel in the ground-truth image. For the ground-truth image, we prepared an image, in which only the detector region corresponding to each label was brightened. An example of our dataset is shown in Fig. 4, and parameters used for HG beam classification are listed in Table 1. In this study, we set the number of layers to two to improve the flexibility of tuning the interlayer distance.Fig. 4. Example dataset: The top images are Hermite–Gaussian (HG) beams inputted to the diffractive deep neural network, and the bottom images are the corresponding ground-truth images. ($m,n$) represents the orders of m and n for an HG beam, respectively.
Table 1. Parameters of the diffractive deep neural network model.
Figure 5 shows the accuracy against TPE iterations and Table 2 shows the locations of layers and classification accuracy obtained using the proposed method. In the middle stage of the TPE optimization, the results are sometimes worse than the conventional method (equidistant placement). However, TPE can explores several parameters and thus is able to choose the best one. Table 2 also indicates the classification accuracy of the conventional method with equally spaced layers of 10 [cm], as shown in Fig. 2(a). Since classification accuracy depends on the randomness of the initial hyperparameter values of each layer, we trained the conventional method 40 times. We set the number of searches to 20 times in 36 classification modes because it takes a long time to train.
Table 2. Classification accuracy and the location of layers.
As shown in Table 2, comparing accuracy for 16 classification modes and auto-tuning the interlayer distance using TPE improved accuracy by 0.5% over equally spaced layer placement. Similarly, accuracy was improved by 3.4% for 25 classification modes and 10.1% for 36 classification modes over equally spaced layer placement. The difference in accuracy between TPE and equidistant placement is larger for 25 and 36 classification modes than for 16 classification modes. Note that the importance of auto-tuning the interlayer distance using TPE increases as the number of modes increases. Figure 6 illustrates the interlayer distances obtained by TPE optimization. It is difficult to search such interlayer distances manually.
Figure 7 shows examples of inferred images from the trained D2NN. The inference was performed using the D2NN that achieved the highest accuracies in classifying the 16, 25 and 36 modes of HG beams in Table 2, respectively. As shown in the figure, a specific detector region is brightened by the inference calculation. This light intensity can be used to determine original bits encoded in an HG beam using a photodetector. The surrounding detector areas become brighter along with the desired detector area. This results in a classification error. The error occurs in areas above, below, and to the left and right of the desired detector area. This error can be caused by the proximity of the detector areas or by the similarity of beam patterns due to the proximity of input beam modes. To confirm this, we randomly switched the detector’s positions and the mode of each beam; however, the same errors occurred for beams with similar modes. For HG beam classification using the D2NN, classification errors occurred between close modes and did not depend on the position of the detector.
The effect of the alignment error on the accuracy was examined. For the models with the highest accuracy for each classification (Fig. 6), we tested the accuracy when the alignment error in the propagation direction (z-axis) was applied to each layer and when the alignment error in the lateral direction (x-y direction) was applied to the input beam. The errors were generated by a normal distribution with mean 0 and standard deviation $\sigma $.
The results for the alignment error in the propagation direction (z-axis) are shown in Fig. 8. The accuracy in this figure is plotted as the average of 10 trials with different alignment errors. From the figure, it can be seen that the classification accuracy decreases as the alignment error increases, but the accuracy does not decrease significantly with an error of about 1 mm in standard deviation, so we consider that the error in the propagation direction is not significant in practical use.
Next, the results of the lateral (x-y axis) alignment error are shown in Fig. 8. The misalignment of the input beam is given independently in the x- and y-directions by a normal distribution with mean 0 and standard deviation σ. The minimum unit of the misalignment is 39 [µm], which is the pixel pitch of the modulation layers. The figure shows that the classification accuracy is strongly influenced by the input beam misalignment. We consider that an accuracy of a few tens of µm is necessary to adjust the optical axis.
Lastly, we describe the computational time required for training. We used Windows 10 Enterprise as the operating system, AMD Ryzen 5 2600 as the CPU, and NVIDIA GeForce RTX 2060 as the GPU. We used TensorFlow-GPU 2.2.0 and Keras 2.4.3 as the deep learning framework. In this study, the batch size was 64, and the number of epochs was 20. The computation time for training was 650 ms per batch; approximately 43 min were required to train 16 modes and approximately 29 h to search 40 D2NN models. For the 25-mode classification, approximately 45 h were required to search 40 D2NN models and for the 36-mode classification, approximately 32 h were required to search 20 D2NN models.
The challenge and limitation of the proposed method is the determination of the optimal number of layers and the distance between layers. In our other validation (not shown in this paper), it has been found that increasing the number of modulation layers improves the classification accuracy, but the disadvantage is that it increases the size of the classifier (from the input plane to the output plane) and the diffraction efficiency decreases with increasing the number of layers because of the energy loss due to the modulation layers. In addition, as the number of layers increases, the degree of freedom of the configuration increases, and the time required to find the optimal configuration also increases. Therefore, it is a challenge to find the optimal number of layers and the distance between the layers which are feasible and have the best classification accuracy.
The other limitation is the memory size of GPUs. In this study, the modulation layers were trained as $256 \times 256$ neurons. Increasing the number of parameters from $256 \times 256$ to $512 \times 512$ would improve the classification accuracy, but this was not done due to the time required for training and the memory size.
5. Conclusion
In this study, we investigated how hyperparameter tuning could improve classification accuracy in multibit optical free-space communications with HG beams, when using a D2NN as a classifier. Results indicate that, by hyperparameter tuning, we can search for interlayer distances that yield higher classification accuracy than that obtained by manually placing D2NN layers at equal intervals. Classification accuracy was improved by approximately 0.5%, 3.4%, and 10.1% for 16, 25, and 36 classification modes, respectively, over equally placed spaced layers. In general, as the number of classification modes increases, the classification accuracy decreases, making it more important to improve accuracy by tuning the interlayer distance. To increase communication capacity in the future, it is necessary to increase the number of beam modes used, such as 49 or 64 modes. Automatic hyperparameter tuning should be performed to cope with an increase in communication capacity.
The use of non-rewritable modulation layers can lead to a significant reduction in classification accuracy due to the wide range of atmospheric fluctuations in a real environment. In order to solve this problem, we need to use rewritable modulation layers such as SLMs. In addition, in recent years, a technique called the camera-in-the-loop-holography [22,23] has been developed to solve the problem in the holography field where neural networks trained only on simulated data do not perform well in real environments. This technique allows us to decrease the difference between the simulation and the real environment by incorporating the reproduced images captured by a camera into the loss function calculation of the neural network. By combining the idea of the camera-in-the-loop-holography with our proposed method, we may be able to achieve a more robust to real-world conditions. This method may be used in situations where mode sorters are difficult to use.
Funding
Japan Society for the Promotion of Science (19H04132, 19H01097).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available but may be obtained from the authors upon reasonable request.
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