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Abstract
Graphene material has excellent performance and unique variable carrier density characteristics, making it an excellent mid-infrared material. And deep learning makes it possible to quickly design mid-infrared band devices with good performance. A graphene nano-ring-symmetric sector-shaped disk array structure based on the PIT principle is proposed here for sensing. The influence of structural parameters and Fermi energy changes are studied. And its FOM (Figure Of Merit) can reach 28.7; the sensitivity is 574 cm-1 / RIU (Refractive Index Unit). At the same time, we designed a six-layer deep learning network that can predict structural parameters and curve predictions. When predicting structural parameters, its MAPE (Mean Absolute Percentage Error) converges to 0.5. In curve prediction, MSE (Mean Square Error) converges to 1.2. It shows that predictions can be made very well. This paper proposes a symmetrical sector disk array structure and a 6-layer deep learning network. And the deep neural network designed based on the device data has good prediction accuracy under the premise of ensuring the network is simple. This will lay a good foundation for future sensor design and device acceleration optimization design.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Electromagnetically induced transparency (EIT) was first observed in atomic systems. It is a phenomenon of enhanced absorption and transmission due to quantum interference. It can be applied to control the optical response of materials using electromagnetic fields [1]. Through quantum interference between different excitation orbits, a light transmitting area is formed in the opaque area, which ultimately causes EIT [2]. The EIT effect can produce a narrow transparent window while having a wide absorption spectrum range. This characteristic makes it have very broad application prospects in many aspects [3–5], such as low-loss slow-light equipment [6,7] and nonlinear optical processes [8,9]. However, the realization of the atomic EIT effect is very difficult and requires very harsh environmental and operating conditions, which greatly limits the application and development of traditional atomic EIT [10]. To overcome these problems, people have developed a new system which is similar to the atomic EIT system. Plasma-induced transparency (PIT) is an EIT-like effect that has aroused widespread concern. The realization of the PIT effect is usually the use of bright and dark mode coupling, which is the direct destructive interference between the bright state mode and the dark state mode to generate the PIT effect [11,12]. The PIT effect provides a good foundation for the application of EIT.
In recent years, as researchers have deepened their research on graphene, the plasma-induced transparency effect based on graphene has also attracted everyone's attention. Graphene is a two-dimensional material consisting of a single layer of carbon atoms, which is light, thin and hard. It has a thermal conductivity of $5000W{m^{ - 1}}{K^{ - 1}}$ at room temperature [13] and an electron mobility of $15000c{m^2}{V^{ - 1}}{s^{ - 1}}$[14]. In addition, by making graphene to make a plasma induced transparent device, the transparent window can be controlled by adjusting the Fermi level of graphene [15]. In this way, the problem that it is difficult to change the operating wavelength of the transparent window based on the plasma-induced transparent device excited by the traditional metal metamaterial can be overcome. This can make the device have better applicability.
At the same time, machine learning has begun to flourish as research into computers has intensified. In contrast to traditional computers that execute written instructions step-by-step, machine learning builds models based on data and then uses the models to analyze and predict the data. This is a way to improve performance by learning data [16]. It can be used in many tasks, including classification, regression, etc [17,18]. Deep learning, as a branch of machine learning, can automatically learn the relationship between input and output, and continuously optimize the loss function to obtain an optimal solution [19]. Traditional device design and optimization requires a lot of time and effort. Here, we combine the use of machine learning techniques to help design graphene-based plasma-induced transparent devices. This can speed up the design of the device and optimize it better.
In this paper, we propose a metamaterial structure that utilizes graphene ring-symmetrical sector disks to stimulate plasma-induced transparency. It explained in principle the reason why the proposed metamaterial structure excited plasma induced transparency, and then analyzed the influence of its tunability and other structural parameter changes. Then the performance of the proposed metamaterial structure for sensing is analyzed, and the results show that this metamaterial exhibits superior sensing performance. At the same time, based on the existing structure and collecting data, we designed a 6-layer deep neural network to predict the forward and reverse of the device, so that it can be better designed and optimized. The deep neural network we proposed can provide great potential for accelerating the design of metamaterials based on the PIT effect.
2. PIT device structure design and analysis
In this article, a graphene-based nano-ring-symmetric sector disk array structure is proposed. Figure 1 shows the proposed metamaterial structure. Figure 1(a) is a unit structure, which is a symmetrical fan shape of graphene nanorings. And it is placed on top of calcium fluoride (CaF2). The thickness of calcium fluoride is d = 20nm. The period of graphene nanostructures is px = py = 100 nm, r1 = 25 nm, r2 = 40 nm, w = 10 nm and θ = 150°. And as shown in Fig. 1(b), to adjust the Fermi level of graphene, we placed a gel and metal contacts above the structure, and a metallic wire gratings substrate below, which was controlled by applying a bias voltage on the metal contacts and metallic wire gratings substrate.
Fig. 1. (a) Schematic graph of the array of graphene nano-structure, which is fabricated on the CaF2 layer. The periodicity of unit cell is fixed as 100 nm, where r1 = 25 nm, r2 = 40 nm, w = 10nm and θ=150°. (b) Schematic diagram of a periodic array. The green part is the gel, covering the cell structure, and the golden color is the metal contact. Graphene adjusts the chemical potential of graphene by controlling the bias voltage between the metal contacts and the metallic wire gratings substrate.
In the mid-infrared region, theintraband conductivity term usually dominates over the interband term. Therefore, the interband transmission is negligible comparing to the intraband transmission in the mid-infrared range. Graphene's surface conductivity is well described by the Drude model and the equivalent relative permittivity is derived from the surface conductivity, as follows:
where e represents electron charge, ω is the angular frequency of the incident wave, ћ is the reduced Planck constant, Ef is the Fermi energy of the graphene layer, ɛ0 is the permittivity of free space and the graphene layer thickness is ${t_g} = 1nm$. The relaxation time is $\tau = \mu {E_f}/ev_f^2$, with Fermi velocity of ${v_f} = c/300$ and the carrier mobility of grapheme $\mu = 10000c{m^2}{V^{ - 1}}{s^{ - 1}}$.In this work, the room temperature (T = 300 K) is assumed. The simulation is performed using commercial software COMSOL Multiphysics, based on the finite-element method, and the periodic boundary conditions are used in the xy plane; the Floquet ports are applied in the z direction. In the simulation, the electric field is polarized in the x direction.
Through design and simulation, we obtain the transmission spectrum of the graphene ring and the graphene symmetric sector disk at normal incidence, as shown in Fig. 2 Using these transmission spectrums, the plasmon induced transparency effect in the graphene ring-symmetric sector disk structure can be analyzed. In the simulation, the polarization direction of the electric field is set to the x direction, and the Fermi level of the graphene is 0.7 eV. Under the same conditions, the graphene ring and the graphene symmetrical fan disk were simulated and numerically studied, as shown in Fig. 2(a) and Fig. 2(b). As shown in Fig. 2(a), the single graphene ring can excite dipole resonance at 19.4 THz. The resonance curve is a typical Lorentz line type, and the electric field diagram of the dipole resonance is shown in the illustration. Figure 2(b) shows the transmission spectrum of the graphene symmetric sector disk structure. It can be seen that the graphene symmetric sector disk can excite dipole resonance at 49.2 THz, and the electric field is shown in the illustration. Figure 2(c) shows the transmission spectrum of the combination of graphene ring-symmetric sector disk structure, it can be seen that plasmon-induced transparency resonance is excited at this time and the frequencies of the corresponding two dip and transmission windows are respectively 45.2 THz. 46.1 THz and 47.2 THz. In addition, we can also see that the graphene ring shows no response near the frequency of the PIT resonance excitation, because the octopole mode of the graphene ring cannot be excited by the normal incident wave, but it can be excited by the adjacent dipole mode structure. Therefore, by adjusting the geometric parameters of the graphene ring and the graphene symmetrical sector disk, when the resonance frequency of the dipole resonance excited by the symmetric sector disk is the same as the resonance frequency of the octopole mode of the graphene ring, the symmetrical sector disks can be used to excite the octopole mode of graphene rings. Eventually, the symmetrical sector disk acts as the bright state, and the excited graphene ring acts as the dark state, resulting in the PIT phenomenon.
Fig. 2. (a) The transmission spectrum and the z component electric field distribution of graphene ring; (b) The transmission spectrum of a symmetric sector disk structure, the illustration is the z-component electric field distribution of the symmetric sector disk structure; (c) Transmission spectrum of the metamaterial structure combined with graphene ring-symmetric sector disk.
To further explain the physical principles of PIT resonance in the proposed structure, Fig. 3 plots the electric field distribution of the graphene ring-symmetric sector disk in the x-y plane. Figures 3(a)-(c) correspond to the two dip and transparency window of the PIT resonance curve in Fig. 2(c). Since the graphene symmetrical sector disk can directly excite the dipole resonance from the incident light, it is a bright mode of the proposed structure. The graphene ring cannot be directly excited at this frequency, so it is used as the dark mode. However, the resonance of the dark graphene ring can be indirectly excited by the coupling between the bright state and the dark state. However, there is a π phase difference between the direct excitation and the indirect excitation. Currently, destructive interference occurs between the bright state and the dark state, and the electric field in the bright mode basically disappears, thus creating a transparency window in the PIT resonance. Figure 3(b) is the electric field diagram of the transparency window in the PIT resonance. The ring and the symmetrical sector disk are destructively interfered. The electric field of the symmetric sector disk as the bright mode disappears.
Fig. 3. The electrical near-field distribution of the graphene ring-symmetric sector disk structure (z component, x-polarized light normal incidence). (a) 45.2 THz, (b) 46.1 THz and (c) 47.2 THz correspond to two dip and transparency windows of plasmon induced transparency resonance. The Fermi level is 0.7 eV.
Next, we studied the effect of structural parameters on the transmission curve. Since PIT resonance is produced by the interference effect between the bright mode and the dark mode, the coupling strength between the two modes can be adjusted by changing the structural parameters of the graphene ring and the graphene symmetric sector disk. To better understand the physical principle of the PIT produced by the proposed metamaterial structure, the graphene ring radius r2 and the symmetrical sector disk radius r1 of the symmetric sector disc are changed under the same conditions, and then the variation of the simulated transmission spectrum is observed. Figure 4(a) shows the variation of the radius r2 of the ring. It can be seen that as the radius of the graphene ring increases from 80 nm to 90 nm, the intensity of the second dip angle becomes weak and the transparency window becomes wider. This is mainly due to the fact that as the radius of the ring increases, the coupling strength between the ring and the symmetrical sector disk gradually decreases. Similarly, when the graphene symmetric sector disk is increased from 40 nm to 50 nm, as shown in Fig. 4(b), the coupling strength between the ring and the symmetric sector disk is gradually increased, and it can be seen that with the increase of r1, the intensity of the first dip is enhanced, the transparency window is narrowed, and the resonance curve is red-shifted.
Fig. 4. Effect of (a) ring radius r2 and (b) symmetrical sector disk radius r1 on transmission spectrum, where Ef =0.7eV.
To better show the adjustability of graphene materials, we have drawn Fig. 5 to show the effect of Fermi energy level on PIT resonance. The adjustable surface conductivity is obtained by adjusting its Fermi level, which can dynamically adjust the metamaterial. As shown in the Fig. 5, when the Fermi level is increased from 0.5eV to 1.0eV, the corresponding amplitude is almost constant, and the resonance peak has a blue shift, realizing the modulation function in the target frequency range. Thus, by adjusting the Fermi level, the plasmon is dynamically controlled to induce the resonance frequency of the transparency window without rebuilding the geometry or integrating the metamaterial with other active components.
Fig. 5. Schematic diagram of the effect of Fermi level on PIT resonance, where the Fermi level ranges from 0.5-1.0 eV, r1 = 25 nm, r2 = 40 nm, w = 10 nm, θ = 150°.
Finally, we studied the sensing performance of the device. In applications based on graphene PIT sensors, since the PIT transparency window is very sensitive to changes in surrounding media, changes in measurement sensitivity are commonly used for sensing. The refractive index change sensitivity is analyzed from the shift of the transmission spectrum by changing the refractive index of the analyte above the metamaterial.
As shown in Fig. 6, we plotted the transmission spectra of graphene PIT metamaterial structures with different refractive indices from 1 to 1.5. The Fermi level is set to 0.7 eV, r1 = 25 nm, r2 = 40 nm, w = 10 nm, θ = 150°. When the refractive index is increased from 1 to 1.5, the transparency window peak frequency of the PIT resonance produces a red shift (from 1538 cm-1 to 1251 cm-1). Here, the wavenumber is defined as:$\tilde{v} = 1/\lambda $. Where $\lambda $ is the wavelength. The relationship between the transparency window frequency shift of the PIT resonance and the refractive index change is plotted, as shown in Fig. 6(b). There is a linear change in these frequency shifts, and the slopes of these lines are defined as the sensitivity (approximately 400 cm−1/RIU). The ratio of the sensitivity to the FWHM of the resonance [(cm−1 /RIU)/FWHM] determines the FOM of metamaterials. After calculation, the sensitivity of the graphene ring-symmetric sector disk metamaterial for sensing is about 574 cm-1 / RIU, and the FOM is about 28.7.
Fig. 6. (a) Transmission spectra of graphene PIT resonances with different refractive indices. (b)Frequency shift of the PIT resonance transparency window with different refractive indices.
3. Deep learning optimization device
In the past device design, it often takes a lot of time to optimize. However, with the development of current deep learning, a new door has been opened for device design. An artificial neural network is a powerful, but also very complex machine learning technology that can imitate the human brain, and then imitate the operation of the brain. Just like our human brain, there are millions of neurons on a level and in a network of neurons. These neurons are closely connected to each other through a structure called synapses. It can transmit electrical signals from one layer to another through Axons. This is how we humans learn things. Whenever we see, hear, feel and think, a synapse (electric pulse) is fired from one neuron in the hierarchy to another. In the artificial neural network, we use the activation function to introduce nonlinear features into our network, and convert the input signal of a node in the model into an output signal, which is used as the input of the next layer [18]. The most common activation functions include Sigmoid function, ReLu function and Leaky ReLU [20–22].The sigmoid function is a saturated activation function, there is a problem of gradient disappearance, and the output center is non-zero, making gradient update optimization difficult and slow convergence. Therefore, the ReLu function (Rectified linear units) is a better choice than the Sigmoid function. The ReLu function is shown in the formula below [20]:
However, ReLu has the problem that the neurons are too fragile and die during the training process. Therefore, we finally chose the optimized version of ReLu function called Leaky ReLu [21]. The Leaky ReLu function is shown in the formula below: Where a is a very small constantin this way, we can correct the data distribution and retain some negative axis values, so that the negative axis information will not be completely lost.In deep learning, Cost is often used to describe the degree of learning. Through the optimizer, Cost is continuously reduced and eventually converges to a value, thereby completing the learning. For the loss function, one way is to use MSE (Mean Square Error) to characterize, also use MAPE (Mean Absolute Percentage Error). The MSE function is shown in the formula below [23]:
Where n is the number of samples, $y(x )$ is the true value, and ${a^L}(x )$ is the estimated value. The MAPE function is shown in the formula below [24]:For the optimizer, we chose Adam Optimizer, which uses gradient first-order moment estimation and second-order moment estimation to dynamically adjust the learning rate of each parameter. The main advantage of Adam Optimizer is that after the offset correction, the learning rate of each iteration has a certain range, making the parameters relatively stable.
The function is shown in the formula below [25]:
To better optimize the structure and design the device faster according to the needs, a machine learning method for designing graphene-based plasma-induced transparent metamaterials is proposed here, which can predict the forward direction and direction so as to better optimize performance and quickly design according to demand. In previous studies, designing a structure based on graphene electromagnetically induced transparent metamaterials and optimizing it was extremely time-consuming, and it was difficult to reverse design the required structure size through demand. It is a very natural thing to combine deep learning and device design to solve these problems with the continuous development of deep learning.
First, we propose a 6-layer fully connected network. All data input to the network must first be standardized to make the statistical distribution of the samples more reasonable. Each layer uses Leaky ReLu as the activation function. For structural parameter prediction, we use MSE as the loss function, and for curve prediction, we choose MAPE as the loss function. Finally, use the Adam Optimizer to reduce the cost and complete the learning. The network structure diagram of forward and reverse prediction is shown in the Fig. 7 below. For predicting the structural parameters from the curve, after the curve data is normalized, it is input into a 5-layer fully connected network, and the input is a 201-dimensional vector. The number of neurons is 2048, 1024, 512, 256, 128. The output is a 5-dimensional vector, θ is the angle of the umbrella, r1 is the outer diameter of the ring, r2is the inner diameter of the ring, and r3 is the fan radius. For predicting a curve from structural parameters, the input is a 5-dimensional vector, and a 4-layer fully connected neural network is input. The number of neurons is 256, 512, 1024, and 2048. Finally, a 201-dimensional vector is output, which is 201 points of the curve.
Fig. 7. Schematic diagram of predictive neural network. The curve data is standardized, and then input to the 5-layer fully connected network, and output 5 structural parameters. The structural parameters are input to a 4-layer fully connected neural network, and 201 points are output to form a transmission curve.
We use Adam Optimizer to optimize the training of the loss function. For predicting curves from structural parameters, we use MAPE as the loss function. For curve prediction structural parameters, we use MSE as the loss function. The training situation is shown in Fig. 8(a) and Fig. 8(b), both of which can converge well. And the two can reach convergence after 400 trainings, which shows that the network is easier to converge. MAPE converges to 0.5, while MSE converges to 1.2. This shows that the network has good accuracy.
Fig. 8. (a) MAPE's convergence graph, which converges to 0.5.(b) Convergence graph of MSE, which converges to 1.2
By entering the size parameters of the structure, we can quickly obtain the curve of the corresponding structure. As shown in Fig. 9, the prediction point can have a good prediction effect for one peak, two peaks, three peaks, and four peaks. This shows that there can be good predictions for multiple peaks. The prediction point can show the frequency point, width, and the number of peaks well.
Fig. 9. Curve prediction diagram. (a) Prediction of one peak. (b) Prediction of two peaks. (c) Prediction of three peaks. (d) Prediction of four peaks.
At the same time, using curves to predict the direction of structural parameters is also our concern. Using reverse prediction, we can quickly set the structural parameters through demand to achieve our desired purpose. As shown in Table 1, for the required curve, the gap between our predicted value and the original true value is extremely small, indicating that the structure can be predicted better, with an extremely accuracy.
Table 1. Structure parameter prediction result
4. Conclusion
We propose a symmetrical disk array structure composed of graphene nanorings. The octupole mode of the graphene ring is excited by a symmetrical sector disk to produce the effect of PIT. We studied the effect of ring width, ring radius, fan radius and fermi level on the EIT effect. At the same time, the refractive index of the structure was studied. Its sensitivity is 574 cm-1 / RIU, and the FOM is about 28.7. This shows that the structure has good sensing performance. At the same time, we designed a deep learning network for the prediction of the forward and reverse of the device. By entering the structural parameters, the corresponding curve can be accurately obtained. When the curve is input, the required structural parameters can be well predicted. This can help better application of deep learning in device design in the future and speed up device design.
Funding
Natural Science Foundation of Zhejiang Province (No.LY16F010010, No. LY21F010012).
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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