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Abstract
In recent years, optical neural networks (ONNs) have received considerable attention for their intrinsic parallelism and low energy consumption, making them a vital area of research. However, the current passive diffractive ONNs lack dynamic tunability after fabrication for specific tasks. Here, we propose a dynamically reconfigurable diffractive deep neural network based on a hybrid graphene metasurface array, wherein the transmission and refractive index of each pixel can be finely adjusted via gate voltage. This capability enables the tailored modulation of the incident light’s amplitude and phase at each pixel, aligning with specific task requirements. The simulation results show the attainability of a dynamic modulation range of 7.97dB (ranging from −8.56dB to −0.591dB). Additionally, this proposed diffractive neural network platform incorporates an ultrathin structure comprising a one-atom-thick graphene layer and nanoscale metallic metastructures, rendering it compatible with complementary metal-oxide-semiconductor technology. Notably, a classification accuracy of 92.14% for a single-layer neural network operating in the terahertz spectrum is achieved based on the calculation result. This proposed platform presents compelling prospects for constructing various artificial neural network architectures with applications ranging from drug screening to automotive driving and vision sensing.
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1. Introduction
Artificial neural networks (ANNs) that inspired by the structure and functioning of a biological brain have been impulsing a various modern technologies, such as autonomous driving [1,2], natural language processing [3], and vision sensing [4]. They are trained on input data to perform intricate signal processing or ’inference’ [5], which relies on optical scattering theory as a foundational mathematical operation. Among all the ANNs, the ONNs [6–19] are one of the most promising optical computing platforms. In addition, they show several compelling advantages over their electronic counterparts. Firstly, ONNs rely on fixed matrix multiplications, and operations can be executed at the speed of light [7]. Moreover, the optical signals in ONNs can be detected at rates exceeding 100 GHz within photonic networks [20]. Secondly, they make effective use of inherent optical nonlinearities [19] to implement various operations. These nonlinearities serve as natural mechanisms for enhancing computational capabilities within the network. Lastly, there is no additional energy consumption for the computation on the optical signals once the ONNs are trained and fabricated. However, the diffractive ONNs [7,12,14] lack tunability due to their fixed network architectures and cannot be retrained for other tasks once fabricated. This inherent characteristic imposes constraints on their functional versatility and potential applications.
Recently, the tunable diffractive ONNs, which utilize components like spatial light modulator (SLM) [6,9], digital micromirror devices (DMD) [13] has been developed and investigated. While these approaches effectively tackle the issue of tunability, they manifest several additional drawbacks. These include constraints on the number of layers in the neural networks due to the reflection-type light path, and the introduction of significant time delays through the incorporation of electronic circuits. Moreover, these architectures are characterised by substantial physical dimensions, rendering their integration a challenging endeavour. Meanwhile the active and reprogrammable transmission-type deep neural networks that based on a metasurface array [21–24] address this problem by providing a solution with the weight-reprogrammable nodes.
In this study, we leverage the tunable optoelectronic characteristics inherent to two-dimensional materials [25–32], alongside the light resonant capability of a metasurface array, to propose a dynamically reconfigurable diffractive ONN based on a hybrid graphene metasurface (HGM) array. The designed neuron size is on the order of 20 micrometres, in contrast to the prior dimension of approximately 20 millimetres as reported in Ref. [23], which is accompanied by a corresponding shift in the operational frequency regime from gigahertz (GHz) to terahertz (THz). In addition, the prospective modulation speed of this platform is potentially limited only by the intrinsic carrier relaxation time of graphene, which operates at the picosecond scale [33,34], akin to other neural networks employing two-dimensional materials [29]. This reconfigurable ONN performs the multiplication of the project image with a complex-valued transmission matrix. Training of the network requires adjusting the transmission value of each pixel individually. At the resonant state of the HGM array, modifying the Fermi level of the graphene layer in a specific pixel induces alterations in its transmission. Consequently, the transmission matrix can be tailored to specific values post-training for a given task and subsequently fine-tuned for alternative tasks. This proposed work provides a new concept towards the designing of advanced reconfigurable neural network chips in the future.
2. Design and methods
As shown in Fig. 1(a), the proposed reconfigurable ONN consist of an array of HGM architecture that covered with 4 micrometres-thick PMMA layer. The HGM array, which is separated from the bottom electrodes by a 30-nanometer-thick $Al_2O_3$ spacer, is organized in a square lattice and positioned above the $SiO_2/Si$ substrate. Each HGM comprises a monolayer graphene and four ’L-shaped’ metal metasurface structures that located at the four corners of the unit cell. The period of HGM array, width of graphene layer, length and width of metasurface are denoted as p, a, b, c, respectively. The metal material of metasurface is selected as gold, which can be characterised as perfect electric conductor (PEC) in THz range [35,36]. The refractive index of PMMA layer is set to 1.56 at THz range [37]. The optical property of graphene layer used in this paper can be characterised by the frequency-dependent surface conductivity model [38,39], which can be quantitatively described as a sum of two parts:
Fig. 1. Schematic diagram of the proposed reconfigurable ONN serving as a classifier. (a) The illustration of the dynamically reconfigurable ONN made of a HGM array. (b) Top view of a single HGM unit cell, wherein P, a, b, c are the period of one unit cell, length of graphene layer in one unit cell, length and width of the ’L" shaped metasurface structure. (c) Side view offering a comprehensive view of the constituent material layers within a single HGM unit cell. (d) Schematic depicting the implementation of the reconfigurable ONN as a classifier.
The first term represents the interband transition, which can be written as the following form:
The second term corresponding to the intraband photon-electron transition that described as:
Here, $\Gamma$, T, $\mu _c$, e, $k_B$, and $\hbar$, are the scattering rate, temperature, chemical potential, electron charge, Boltzmann constant and reduced Plank constant respectively. and
In this study, the finite-difference time-domain (FDTD) method is used to investigate the photonic response of the proposed HGM array. In the FDTD simulations, we use the periodic boundary conditions in the x and y directions and perfectly matched layer boundary conditions in the z direction of a unit cell. A plane wave with electric field oscillating along the direction of x-axis is incident on the HGM. The geometrical parameters are originally set as p=8 $\mathrm{\mu}$m. a=5 $\mathrm{\mu}$m, b=2 $\mathrm{\mu}$m and c=0.5 $\mathrm{\mu}$m respectively. The thickness of gold metasurface, top and bottom metal gates are all specified as 50 nm. Subsequently, the absorption of the HGM layer is calculated as:
where $P_{In}$ and $P_{Out}$ are the powers traversing the top and bottom surfaces of the HGM layer respectively. The schematic of the implementation of the proposed reconfigurable ONN as a classifier in shown in Fig. 1(c). Here, the HGM array is operated as a single-layer perceptron, with pre-processed visual information serving as the input layer. The nonlinear activation functions are executed off-chip. Provided that the transmission ($T_{MN}$) and refractive index ($n_{MN}$) of each pixel of the proposed reconfigurable ONN can be tuned by the Fermi level of graphene layer via gate voltage, which modulates both the amplitude and phase of light incident on each pixel, with the synaptic weights encoded in the gate voltage matrix. This type of reconfigurable ONN configuration represents a supervised learning algorithm proficient in classifying images P into different categories y.For the classification process, training was initially conducted. The input digits (or fashion products) images were encoded into the amplitude of the input field to the neural network. The neural network was then trained to map input digits into 10 detector regions, one for each digit. After training, the optimized transmission of each unit is determined, which means the Fermi level of the graphene layer of each hybrid metasurface unit is fixed. During the classification process, the amplitude of the output field on the ten detector regions differs, and the detector with the maximum optical signal corresponds to the input digit or fashion product.
3. Results and discussion
3.1 Characteristics of the HGM
In order to comprehensively evaluate the performance and tunability of the proposed reconfigurable ONN tailored for image processing, we initiated our investigation by examining the electromagnetic characteristics of the HGM array structure. Figure 2(a) shows the variation of the Fermi level of graphene layer in the HGM structure at different applied voltages [40,41]. Notably, the inset within the figure illustrates a cross-sectional schematic of the back-gate HGM structure, where a 30 nm sapphire layer from the underlying electrical gate separates the graphene layer, as mentioned above. It shows that a readily adjustable Fermi level range from −0.5 eV to 0.5 eV has been achieved, even with a modest gate voltage of 10 volts, demonstrating its dynamic tunability through electrostatic gating. Subsequently, to explore the impact of the tunable Fermi level on the transmission of the HGM structure, we conducted rigorous calculations of the transmission spectra. The initial geometric parameters were set as p=8 um, a=5 um, b=2 um, and c=0.5 um. As shown in Fig. 2(b), an increase in the Fermi level of graphene resulted in a discernible blue-shift in the resonant wavelength, accompanied by a decrease in transmission at the resonant wavelength. This observed phenomenon can be attributed to the evolving metallic nature of graphene, where an increased number of electrons (or holes in the opposite case) penetrate the graphene layer. At a resonant wavelength of 6 $\mathrm{\mu}$m, the modulation range of transmission spectra is from 0.873 to 0.139 when Fermi level ranges from 0 eV to 0.5 eV.
Fig. 2. Optical characteristics of the HGM structure. (a) Fermi level variation of the HGM structure modulated via electrical field with a 30 nm Al2O3 spacer. (b) Calculated transmission spectra of HGM structure at different Fermi levels. (c) and (d) depict the absorption, reflection and transmission spectra of the HGM at Fermi level of 0 eV and 0.5 eV, respectively. The insets are the normalised electrical field distributions within the HGM structure with graphene’s Fermi level of 0 eV and 0.5 eV at a frequency of 6 THz.
To further elucidate the transmission difference for varying Fermi levels at an incident wavelength of 6 $\mathrm{\mu}$m, we calculated the absorption, reflection and transmission spectra of the HGM structure when the Fermi level are 0 eV and 0.5 eV respectively, as depicted in Fig. 2(c) and 2(d). At a reduced Fermi level, the interaction between light and-HGM structure exhibits diminished strength, resulting in low absorption and high transmission. Conversely, at an elevated Fermi level within the graphene material, the interaction between the HGM structure and incident light experiences substantial augmentation, thereby yielding heightened absorption and attenuated transmission at the resonant wavelength.
Moreover, we simulated and showed the normalized distribution of electric fields within the HGM structure, as depicted in the insets of Fig. 2(c) and Fig. 2(d), which aimed to enhance our comprehension of how variations in electromagnetic characteristics influence the transmission properties of the HGM structure. At low electric doping states, the electric field experiences strong confinement only within the narrow gaps between the top two and bottom two "L" shaped metal metastructures, resulting in a low absorption while a high transmission. Conversely, at high electric doping states, the confined electric field oscillates between the four "L" shaped metal metastructures between the surrounding adjacent unit cells, leading to the emergence of strong localized surface plasmon resonance in the separation area between the two adjacent unit cells. This, in turn, significantly reduces the transmission efficiency. Overall, these detailed investigations of the electromagnetic characteristics and transmission behaviours provide valuable insights into the tunability and functionality of the proposed reconfigurable ONN based on the HGM array structure for image processing applications.
3.2 Characteristic of the HGM array as a reconfigurable ONN
To assess the functionality of the HGM array as a reconfigurable ONN, we firstly trained it as a digit classifier for classifying the MNIST (modified National Institute of Standards and Technology) handwritten digits, as shown in Fig. 3. With deep training process, the transmission value of each neuron is iteratively altered to perform a specific function. The input digit information is encoded in the amplitude of the incident light. There are $80\times 80$ modulation array with $3\times 3$ HGM units per modulation pixel. Experimentally achieving such a neural network is conceptually straightforward and remains a mainly technological task. The reconfigurable ONN was trained to be some specific transmission values to map the input digits into 10 detection regions on the output plane, with one for each digit. Subsequently, the output intensity incident on each detector from the neural network was measured and the loss function defined as the mean square error (MSE) against the target image was used to feedback the training. Finding the detector with a maximum optical intensity is the training criteria. During training, Gaussian noise (with standard deviation of $\sigma$ =0.1, 0.2, and $0.3$) was added to expand the input data to test the stability of the neural networks’ functionality.
Fig. 3. Characterisation of the reconfigurable ONN as a handwritten digits classifier. (a) Accuracies and losses of the reconfigurable ONN during training for diverse artificial noise levels. (b) Confusion matrix of the numerical testing results for 10,000 different handwritten digits, with 1000 instances for each digit. (c) The image of input digit number "0" at a Gaussian noise level of 0.2. (d) The image of output plane for a digit input of "0". The red dashed squares represent 10 different detector regions. The scale bars in (c) and (d) are 120 $\mu$m.
Following the training phase, the reconfigurable ONN as a digit classifier was evaluated through an assessment using images sourced from the MNIST test dataset. It is noteworthy that these test images were different from the training image sets. The accuracy and loss are plotted over 20 training epochs in Fig. 3(a). The accuracy increases quickly for all the noise levels and reaches a maximum of 92.14%, 90.5%, 88.69%, 86.83% after 4, 6, 8 and 10 epochs for $\sigma =0,$ $\sigma =0.1,$ $\sigma =0.2,$ and $\sigma =0.3$, respectively. Meanwhile, the loss decreases drastically to a minimum for all noise levels, indicating an expedited convergence, particularly in the presence of lower levels of noise. Figure 3(b) illustrates the confusion matrix generated from testing ten thousands of input digits from "0" to "9" with Gaussian noise of 0.2, corresponding to an accuracy of 88.69%. Furthermore, the input plane and detection plane of digit "0" with Gaussian noise of 0.2 are shown in Fig. 3(c) and Fig. 3(d).
To illustrate the dynamic tunability of the HGM array as a reconfigurable ONN, we then trained and operated it as a classifier for fashion products. Analogous to the training regimen employed in previous classifiers for handwritten digits, the transmission matrix of the HGM array underwent retraining for this new task. The Fashion-MNIST dataset encompasses various categories of fashion products, including items such as t-shirts, trousers, pullovers, dresses, coats, sandals, shirts, sneakers, bags, and ankle boots, each of which is denoted by numerical labels ranging from 0 to 9. After the training process, an assortment of 10,000 distinct fashion product images, comprising 1000 images per category drawn from the MNIST test dataset, were utilized for accuracy assessment, as depicted in Fig. 4. The achieved accuracy rates were 80.3%, 79.39%, 77.39%, and 76.47% in the presence of Gaussian noise levels of 0, 0.1, 0.2, and 0.3, respectively. This moderate reduction in accuracy primarily stems from the heightened complexity of the image dataset as compared to that of handwritten digits. Notably, these accuracy metrics are commensurate with those reported for 3D-printed 5-layer diffractive deep neuron networks ($D^2NN$) operating in the terahertz (THz) frequency range, which achieved accuracies of 81.13% and 86.33%. The confusion matrix, an input image of "trousers" sized at 100 by 100 pixels, and the corresponding output image are presented in Fig. 4(b) to 4(d).
Fig. 4. Characterisation of the reconfigurable ONN as a fashion products classifier. (a) Accuracies and losses of the reconfigurable ONN during training for diverse artificial noise levels. (b) Confusion matrix of the numerical testing results for 10,000 different fashion products, with 1000 samples per category. (c) The image of the input fashion product "trousers" in the absence of Gaussian noise (d) Output image plane corresponding to the input fashion product "trousers". The red dashed squares delineate 10 different detector regions. The scale bars in (c) and (d) represent 120 $\mu$m.
3.3 Enhancing the accuracy of the reconfigurable ONN
While the accuracies computed previously align with those reported for optical neural networks (ONNs), they remain inferior to the performance of their electronic counterparts, specifically the state-of-the-art convolutional neural networks [42,43]. Consequently, we conducted an in-depth investigation into enhancing the accuracy of the proposed reconfigurable ONN model as a classifier. Here, we take the digit classifier as an example. This was achieved through an augmentation of both neuron numbers (N*N) (with neuron counts of N) and neural network layers, as depicted in Fig. 5(a) and 5(b). In the case of a single-layer neural network, we observed a positive correlation between classification accuracy and neuron count, reaching an approximate peak accuracy of 92.3% when the neuron counts reached 6400 ($80\times 80$). Consequently, all simulations were executed employing 6400 neurons per layer, unless specified otherwise. Furthermore, augmenting the neural network’s layer resulted in an additional improvement in classification accuracy. As depicted in Fig. 5(b), we computed the accuracy of the neural network across layer counts ranging from 1 to 5, under varying levels of Gaussian noise (characterized by standard deviations $\sigma = 0.1, 0.2,$ and $0.3$). The accuracy exhibited an upward trend with increasing layer count for all cases of Gaussian noise. Specifically, in instances featuring fewer layers, such as in the case of 1 and 2 layers, Gaussian noise exerted a discernible influence on accuracy, with substantial enhancement observed upon an increase in layer count. In contrast, for multi-layer neural networks, the influence of Gaussian noise on accuracy was notably diminished, achieving a peak of approximately 94.3% when the layer count was set at 4. This result is commensurate with the 93.39% accuracy attained in the proposed multilayer all-optical diffractive deep neural network. Alternatively, expanding the training dataset and optimizing hyperparameters such as learning rate and batch size, can also increase the accuracy of the reconfigurable ONN.
Fig. 5. Accuracy of the HGM array-based reconfigurable ONN as a classifier with varying neuron count (a) and neuron layer under different levels of Gaussian noise (b).
4. Conclusion
In summary, we proposed and investigated a dynamically reconfigurable ONN based on a HGM array as a tunable classifier. The matrix of transmission and refractive index can be modulated by manipulating the Fermi level of graphene layer through gate voltage. This allows for precise modulation of the amplitude and phase of incident light on each neuron. This device concept demonstrates great potential for machine vision applications.
Funding
National Key Research and Development Program of China (2021YFB2802000); Science and Technology Commission of Shanghai Municipality (21DZ1100500); Shanghai Municipal Science and Technology Major Project; Shanghai Frontiers Science Center Program (2021-2025 No.20); National Natural Science Foundation of China (61975123); China Postdoctoral Science Foundation (2022M712141).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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