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Abstract
In this work, we propose and implement a machine learning method of using a forward deep learning neural network and Fano function inverse matching to design and optimize hybrid metal-dielectric guided mode resonance narrow linewidth optical filters. First, a forward deep learning neural network is trained with a small design sample set generated with finite difference time domain physical simulations. The trained forward neural network is then used to generate a large sample set of three million designs. In inverse matching process, filter peak wavelength and spectral linewidth are two matching parameters first used for down selecting designs from the large sample set to a small sample set. Because of the asymmetric nature of the guided mode resonance filter spectral line-shape, Fano functions are used to match against the spectra in the small sample set to find the design with narrow filter linewidth. Optical transmission filters with linewidth between 6.8 nm and 8.7 nm are designed in visible spectrum.
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1. Introduction
Optical filters are critical optical elements widely used in many applications, such as imaging [1,2], displays [3,4], and optical communications [5–7]. With increasing demand for miniaturization and integration of optical filter devices, nanostructure optical filters based on guided mode resonance (GMR) [8–11] and localized surface plasmon resonance [12–15] have been extensively investigated in the past decade aimed to replace traditional thin-film optical filters. Traditional designs of optical filters, rely on trial-and-error and parameter sweeping approach to achieve design objectives [16]. While the method of parameter sweeping and trial-and-error approach is useful, it is tedious, inefficient, and computationally intensive. In the past decade, various optimization methods and algorithms such as the genetic algorithm [17], particle swarm optimization [18], and differential evolution [19] have been developed. However, these optimization techniques and algorithms require tremendous computing power for multiple-parameter design optimizations.
Deep learning neural networks (NNs) have been previously adopted to predict optical responses of nanostructure photonic devices [20–29]. To design functional photonic devices, two inverse design methods using neural networks have been reported [30–32]. One is the dictionary lookup inverse searching method [30], another is the two-way network prediction method [31,32]. Although both techniques can solve the issue of data inconsistency in inverse design problems, computing time and memory required by inverse design methods scale up rapidly with number of design parameters. In this work, we propose and implement a new machine learning inverse design method to design hybrid metal-dielectric guided mode resonance optical filters. Narrow linewidth hybrid metal-dielectric GMR filters are designed at different wavelengths in the visible spectral range.
2. Hybrid metal-dielectric guided mode resonance filter structure and FDTD numerical simulations
It has been previously reported that hybrid metal-dielectric guided mode resonance optical filters can have narrow filter linewidth and high peak transmittance [33–36]. The hybrid metal-dielectric GMR filter structure is schematically illustrated in Fig. 1(a). In the structure, an aluminum subwavelength grating sits on the top to couple incident light into the waveguide beneath the grating. Below the aluminum (Al) grating is a low index of refraction spacing layer (silicon oxide) and a high index of refraction Al2O3 waveguide layer. The substrate is a silicon oxide wafer. Figure 1(b) shows the cross-section of the hybrid filter device structure. The structure of hybrid GMR filter can be fully characterized by five geometric structure parameters: period of the aluminum subwavelength grating Λ, width of Al grating line w, thickness of the grating layer t1, thickness of the SiO2 spacing layer t2, and thickness of Al2O3 waveguide layer t3.
Fig. 1. (a) Schematic of the hybrid metal-dielectric guided mode resonance (GMR) optical filter. (b) Cross section of the hybrid GMR filter structure. Λ, w, t1, t2, and t3 denote the grating period, width of the metal line, metal grating thickness, SiO2 spacing layer thickness, and Al2O3 waveguide layer thickness, respectively. (c) A FDTD simulated transmittance spectrum of a hybrid metal-dielectric GMR filter.
Transmission spectrum of the hybrid GMR filter can be calculated by using finite difference time domain (FDTD) simulations. In this work, a FDTD simulation software (developed by Lumerical Solutions, Inc.) is used to calculate the transmittance spectrum. The FDTD simulation domain is a two-dimensional region with a 0.25 nm mesh resolution. Anti-symmetric boundaries are used in the lateral x-directions. Perfect matching layer boundaries are used in the y-directions to eliminate boundary reflection. The electric permittivity of Al, SiO2, and Al2O3 are taken from Ref. [37]. A plane wave is incident normally from air to the grating with polarization in x-direction, as shown in Fig. 1(a). The Al grating period Λ, width of the grating line w, the thickness of the Al grating t1, the thickness of SiO2 layer t2, and the Al2O3 layer thickness t3 are five structural parameters, as shown in Fig. 1 (b). The parameter f is the fill factor defined as the ratio of the grating metal line width w over the grating period Λ. The spectrum of a hybrid filter with a set of structural parameters {Λ, f, t1, t2, t3} = {0.4 µm, 0.7, 0.03 µm, 0.10 µm, 0.15 µm}, is calculated using FDTD simulations. The simulated transmission spectrum is shown in Fig. 1(c). In the spectrum, there is a single transmission peak at 610 nm wavelength.
3. Forward deep learning neural network
3.1 Deep learning neural network architecture
To design and optimize the hybrid GMR filter, a deep learning neural network (NN) and an inverse matching method are developed. The forward deep learning NN is shown in Fig. 2(a). The input of the network is a one-dimensional device structure data vector consisting of five structural parameters. The output is the transmittance spectrum with 501 numerical numbers at wavelengths between 400 nm and 900 nm with a step size of 1.0 nanometer. In the forward neural network, we use four fully connected layers with 200, 400, 800, and 600 neurons to provide sufficient network parameters to capture all spectral features. However, an overly complicated model can cause over fit of the network which weakens its generalization capability. To avoid the problem, a dropout layer with dropout rate of 0.2 is added after the last fully connected layer, which means that 20% of neurons are dropped out during the training process. The dropout layer randomly drops out network neurons and their connections in training process, which reduces the co-adapting of neurons to mitigate neural network overfitting [38]. The activation function used in each layer is the rectified linear unit [39], ‘Adam’ is the optimization algorithm [40], and the loss function is the mean square error (MSE).
Fig. 2. (a) Architecture of the forward neural network. (b) Transmittance spectra of FDTD simulation (red) and neural network calculation (blue). Each transmittance spectrum consists of 501 data points uniformly distributed in the wavelength range from 400 nm to 900 nm.
The output transmittance spectra are shown in Fig. 2(b). The red line curve is obtained using FDTD simulations, whereas the blue line curve is obtained using the forward neural network. The spectrum is discretized into 501 data points at equal intervals of 1.0 nm for the requirement of high spectral resolution narrow linewidth filter. The pairs of filter structural parameters and their transmittance spectra are used to train the forward NN. To collect training data set, we sweep the structural parameter space to identify the space boundary as shown in Table 1. The grating period Λ ranges from 300 nm to 550 nm to make the filter transmission peak wavelength in the range from 400 nm to 900 nm. The SiO2 layer thickness t2 is limited in the range from 20 nm to 200 nm to avoid the higher order guided resonance modes and multiple transmission peaks.
Table 1. Device structure parameter space of the training sample set
The neural network training dataset is obtained by FDTD physical simulations. An optimal training dataset must be dense enough to include as all spectral features to guarantee enough accuracy of neural network but as sparse as possible to reduce the data collection time. The forward NN input parameters used for obtaining the training data set is shown in Table 1. A total of 12000 spectra are calculated with an Intel Xeon Gold 6136 CPU in 8 hours. The training set is uniformly distributed in the parameter space to avoid uneven distribution of training samples caused by random sampling [25,41] and Monte Carlo sampling [28]. A total of 12,000 samples of filter structural parameters and their spectra are obtained by FDTD simulations for neural network training. The gradient descent method is used to train the neural network on a single NVIDIA Quadro P5000 GPU machine.
3.2 Neural network training and evaluation
The sample set of 12,000 designs is randomly shuffled. Then 70%, 20%, and 10% of the total design samples are used for training, validation, and testing, respectively. The training set of 8400 samples and validation set of 2400 samples are used to train the network in every training iteration for achieving low loss. After the training is completed, the trained forward network is evaluated using the testing set of 1200 samples. The network is trained for 1200 epochs, and the obtained training and validation set loss versus training epoch is shown in Fig. 3.
Fig. 3. Loss of the training and validation sets during 1200 training epochs. The black and red curves are the losses of the training and validation sets, respectively. The inset shows the loss curve after removing the first few data points with rapid decline.
The losses on the training and validation sets decreased rapidly in the initial epochs shown in Fig. 3, which demonstrates that the proposed forward NN has learned connections between geometrical parameters and the transmission spectrum, and reach convergence in a small number of epochs. The inset in Fig. 3 shows that there are local fluctuations and a general declining trend in the loss curve during training. The decreasing speed of loss in the training and validation sets approach zero after 1000 epochs. The training is stopped at 1200 epochs, and the loss values on the training and validation sets are 4.92×10−4 and 1.29×10−4, respectively.
The average loss value is 1.3×10−4 for the 1200 testing data, which demonstrates that the network can accurately predict the response spectra for a given structural parameter set. To evaluate the network prediction effect more intuitively, nine structures are randomly selected from the testing set and their FDTD simulated and neural network predicted spectral curves are plotted together and shown in Fig. 4. A sample of device structural parameter is represented by a vector {Λ, f, t1, t2, t3}. Results in Figs. 4(a) to 4(i) correspond to structure parameter set of {0.5 µm, 0.6, 0.02 µm, 0.16 µm, 0.14 µm}, {0.5 µm, 0.7, 0.05 µm, 0.12 µm, 0.08 µm}, {0.5 µm, 0.8, 0.06 µm, 0.08 µm, 0.12 µm}, {0.4 µm, 0.7, 0.03 µm, 0.06 µm, 0.08 µm}, {0.55 µm, 0.9, 0.03 µm, 0.12 µm, 0.16 µm}, {0.45 µm, 0.9, 0.06 µm, 0.04 µm, 0.02 µm}, {0.45 µm, 0.8, 0.02 µm, 0.06 µm, 0.16 µm}, {0.35 µm, 0.8, 0.05 µm, 0.12 µm, 0.1 µm}, and {0.35 µm, 0.8, 0.03 µm, 0.04 µm, 0.06 µm}, respectively. The black solid curves are FDTD simulation spectral curves. The red dashed curves are obtained by using the trained forward neural network. The MSE errors are listed in each figure. The FDTD simulation speed is 25 spectra per minute while the neural network prediction speed is 54000 spectra per minute. The calculation speed using the neural network is three orders of magnitude faster than the FDTD simulation speed.
Fig. 4. (a)-(i) Transmittance spectra of hybrid GMR structures with different structural parameters in the testing sample set. The black solid and red dashed spectra are obtained using FDTD simulation and the deep learning NN, respectively.
It can be seen in Fig. 4 that the neural network predicted transmittance spectra match well the FDTD simulated spectra. The low MSE loss value and well-matched curves indicate that the forward neural network can achieve the objective of calculating filter spectra with high accuracy.
4. Inverse design of hybrid guided mode resonance filters with Fano functions
4.1 Generation of a large design sample set with neural network
The trained forward neural network is used to calculate the transmittance spectra of a large sample set of three million filter structures for the inverse matching. The calculated parameter space for the inverse matching is listed in Table 2. The grating period Λ varies from 300 nm to 600 nm and is evenly distributed with a step of 5 nm. The grating ratio f ranges from 0.6 to 0.95 with an even step of 0.05. The grating thickness t1 ranges from 20 nm to 50 nm with a 5 nm increment step. The SiO2 layer thickness t2 and Al2O3 layer thickness t3 ranges from 20 nm to 200 nm with an increment step of 6 nm. The transmittance spectra of three million filter sample structures are calculated. It took one hour to obtain a large dataset of three million samples in the same machine. The design samples in the large data sample set have wide distributions of peak wavelengths and filter linewidths. Figure 5 shows transmittance spectra of four guided mode resonance filter structures in the large sample set generated by the neural network and transmittance spectra calculated by FDTD simulations. The spectra in Figs. 5(a) to 5(d) correspond to filter structure parameter set {Λ, f, t1, t2, t3} of {0.35 µm, 0.8, 0.03 µm, 0.05 µm, 0.062 µm}, {0.4 µm, 0.75, 0.03 µm, 0.068 µm, 0.08 µm}, {0.45 µm, 0.8, 0.02 µm, 0.064 µm, 0.148 µm}, {0.4 µm, 0.8, 0.05 µm, 0.108 µm, 0.098 µm}, respectively. It can be seen that the transmission spectra calculated using the deep-learning neural network agree well with the transmission spectra calculated with FDTD numerical simulations.
Fig. 5. Forwarded neural network generated transmittance spectra of guided mode resonance filter samples in the large sample set and rigorously calculated transmittance spectra by using FDTD simulations. Forward neural network designs agree well with FDTD numerical simulations.
Table 2. Parameter space of a large design sample set generated with trained neural network
4.2 Preliminary sample selection of narrow linewidth filter designs
Inverse matching design of optical filters requires matching a target filter spectrum with all spectra in the neural network generated large sample set. Since the target peak wavelength is predetermined, we can first down select from the large design sample set to a small sample set with the peak wavelength as a selecting criteria. Also, since our task is to design narrow linewidth filters, we can eliminate designs with large spectral linewidth in the sample set and only keep designs with small spectral linewidth. In the sample down-selection process, the filter peak wavelength and the spectral linewidth of three million design samples are obtained using a peak searching and linewidth calculation algorithm from SciPy open-source library [42]. The spectral linewidth is the full-width at the half-maximum (FWHM) linewidth.
To illustrate the design method, we design a narrow linewidth optical transmission filter at 600 nm peak wavelength. First, a preliminary sample selection is carried out to keep the design samples with peak wavelength in the range between 598 nm and 602 nm and spectral linewidth less than 15 nm. Design samples with peak wavelength and linewidth not meeting the requirements are eliminated. Preliminary sample selection reduces the number of design samples from three million to 4958 design samples. This reduces the computational time for follow-on inverse matching. Two filter property scatter plots of designs of the down-selected small sample set are shown in Fig. 6. Figure 6(a) is the sample distribution versus the filter linewidth and peak transmittance. The lack of distribution of data points in the lower right corner of the scatter plot indicates that the hybrid GMR filter cannot have a narrow linewidth and a high peak transmittance simultaneously. The sample distribution versus the sideband transmittance and peak transmittance is shown in Fig. 6(b), which indicates that the hybrid GMR filters cannot have low sideband transmittance and high peak transmittance simultaneously. Figure 6(a) shows that there is a trade-off between spectral linewidth and transmittance. It can be explained that a small energy leakage of GMR mode leads to a narrow filter linewidth and a small transmittance. The energy leakage is due to the coupling of surface plasmon resonance mode and GMR mode. When the spatial overlap of the two modes decreases, the coupling becomes weaker and energy leakage becomes smaller, resulting a narrow filter linewidth and small transmittance.
Fig. 6. Scattering plots of: (a) filter linewidth and peak transmittance sample distribution, and (b) maximal sideband transmittance and peak transmittance sample distribution. Each point represents a filter structure from preliminary selected data set containing 4958 structures.
4.3 Inverse matching design with Fano functions of different q-parameters
To design a filter at a predetermined peak wavelength with a narrow linewidth, a model based MSE matching method is proposed and implemented. Since the spectral line-shape of hybrid GMR filters is asymmetric and close to the Fano resonance function line-shape, Fano resonance functions [43] are used to match against the spectra in the small down-selected sample set. Fano resonance occurs when a continuum state interferes with a discrete state, which has been observed in various resonance systems [44,45]. With Fano resonance function as target matching function, smaller matching error can be expected. Indeed, we also conducted inverse matching with Gaussian and Lorentzian functions. Fano resonance function inverse matching produces the smallest mean square error. In our hybrid GMR filter design, the normalized Fano resonance function F (λ) used for inverse matching is,
In Eq. (1), λc is the peak wavelength of the target filter, Δλ1/2 is the FWHM linewidth of the filter, and q is the asymmetric parameter of the Fano function. The asymmetric parameter q influences the asymmetric line-shape of Fano function and thus has impact on the inverse matching results. To investigate the impact of the q-parameter, we use Fano functions with different asymmetric parameter q to match against the preliminary selected samples to design filters at peak wavelength of 600 nm with linewidth of 12 nm. The loss function of the matching is the mean square error (MSE) of the spectral data points,
In Eq. (2), yi is the is the sampled value of the FDTD simulated transmittance spectrum, ypi is the sampled value of the neural network predicted transmittance spectrum. N is the number of spectral data points used to calculate MSE. The wavelength range used to calculate MSE is from 500 nm to 700 nm.
Fano resonance functions of asymmetric parameter q=6, 7, 10, 20, 100 are plotted and shown in Fig. 7(a). All Fano functions have a same peak transmittance of 100% and a 12 nm FWHM linewidth. With increasing the parameter q, the asymmetry of Fano function lessens and Fano function gradually approaches a Lorentzian function. When parameter q is small, the transmittance of the left sideband is as high as 10% while the transmittance of the right sideband is close to zero. As the parameter q increases, the transmittance of the left sideband gradually decreases to be same as the transmittance of the right sideband of the left one. With the Fano resonance functions of parameter q=6, 7, 8, 9, 10, 15, 20, 50, 100, we match the preliminary selected samples and pick the filter design with smallest matching error. The matching MSE of Fano functions with different asymmetric factor q are plotted as Fig. 7(b) shown. The red data dot represents matching error of Fano functions while the black straight line represents the matching error of Lorentzian function which has same peak transmittance of 100% and linewidth of 12 nm. The matching MSE first diminishes and reach a minimum of 1.87×10−3 when the Fano parameter q = 20. When q exceeds 50, the Fano function approaches Lorentzian function and the matching error increases to 1.97×10−3. Because hybrid GMR filter spectral line shape is asymmetric, the Fano function with q of 20 matches well the filter spectrum and produces the smallest matching MSE. The MSEs of matching with q = 15, 20 and 50 are smaller than MSE of Lorentzian function matching of 2.02×10−3, which means that Fano resonance function better matches the filter spectrum than Lorentzian function does.
Fig. 7. (a) Fano functions of peak wavelength of 600 nm and linewidth of 12 nm with different asymmetric q parameter. The asymmetry of the Fano function curve increases with the decrease of parameter q. (b) Matching MSEs with Fano functions of different asymmetric parameter q. The straight dash line indicates the matching MSE error (2.02×10−3) using a Lorentzian function (q=∞) with the same peak wavelength and same linewidth.
The Fano function matched spectra with different asymmetric parameters of q = 6, 8, 10, 20 are shown in Figs. 8(a)-(d). The blue dotted curves are matched design spectra calculated by the trained forward network, and the red solid curves are calculated using the FDTD simulation. The inset shows the peak fitting degree, and the FWHMs of the matched filter design spectra are listed. In addition, the matching MSEs are shown in the images. The FDTD simulated curves and network predicted curves agree well. Designed filter structure parameters with Fano function matching {0.4 µm, 0.7, 0.03 µm, 0.098 µm, 0.116 µm} for q=6, {0.4 µm, 0.7, 0.02 µm, 0.086 µm, 0.122 µm} for q=8, {0.4 µm, 0.75, 0.025 µm, 0.08 µm, 0.116 µm} for q = 10 and {0.4 µm, 0.75, 0.02 µm, 0.086 µm, 0.116 µm} for q=20, respectively. The matched designs have high peak transmittance of 90%. When q = 20, the matched filter design spectrum has smallest filter linewidth of 12.6 nm which is close to target linewidth of 12 nm. Also it gives smaller sideband transmittance than designs with q=6, 8, and 10. The smallest linewidth and sideband transmittance account for the smallest matching MSE of Fano function with q=20. So, Fano function with q=20 is used as target function for following inverse designs in this work.
Fig. 8. (a)-(d) Fano function matched design spectra with asymmetric parameter q=6, 8, 10, 20. The insets show the close view of the spectra and the filter linewidths. The MSE errors of matching with Fano functions of different q parameters are indicated.
Since hybrid GMR filters have asymmetric line-shapes, Fano resonance function can match well the asymmetric spectral line shape. Fano resonance function with large asymmetric parameter q approaches the Lorentzian function. Fano resonance functions with small asymmetric parameter q have high sideband transmittance and large spectral asymmetry. As shown in Fig. 7(b) and Fig. 8, the Fano function matching with q=20 gives the smallest MSE error.
4.4 Inverse design of narrow linewidth hybrid guided mode resonance optical filters
Narrow linewidth optical transmission filters are designed by using inverse matching against the Fano functions at different wavelengths. In the design, Fano parameter of q = 20 is used to accommodate the asymmetry line-shape of the hybrid GMR filters. In the Fano function, the spectral linewidth is set at 5 nm which is smaller than linewidths of all filters in the design sample set. Because we use the MSE as a criterion in searching for narrow linewidth filters, this strategy enables to find the filter structure with the narrowest spectral linewidth. The peak transmittance of the Fano functions is set at 100%. Although none of the peak transmittance in the sample set reaches 100%, this strategy enables that higher peak transmittance filter has smaller MSE error. The sample with the smallest matching MSE error against the Fano function is the optimal design with high peak transmittance and narrow linewidth. The spectral range of Fano function matching is from the minus 100 nm to the plus 100 nm of the target filter peak wavelength.
In the Fano function matching process, MSEs between filter spectra of the pre-selected design samples and target Fano functions are calculated. Then, all filter designs are sorted based on the MSEs using a sort algorithm. The design having the smallest MSE is the optimal design. Then we verify the design with FDTD numerical simulations. The designed narrow linewidth hybrid GMR filter spectra centered at the wavelengths of 450 nm, 550 nm, 600 nm, and 700 nm are shown in Figs. 9(a)-(d) respectively. The blue dotted curves represent the transmittance spectra of the Fano function inversely matched design spectra. The red curves are transmittance curves calculated by using FDTD simulations with the optimal design structure parameters. The filter linewidths of the four designs inversely matched by Fano functions are smaller than 9 nm. The peak transmittances exceed 60% and the sideband transmittances are smaller than 6%. The filter at the peak wavelength of 450 nm achieves the narrow linewidth as small as 6.8 nm, as shown in Fig. 9(a). The inverse design results indicate that Fano function inverse matching method can be used to design narrow linewidth hybrid GMR optical filters at different peak wavelengths.
Fig. 9. (a)-(d) Transmittance spectra of the filter designs at peak wavelengths of 450 nm, 550 nm, 600 nm, and 700 nm, respectively. The black lines curves are target Fano functions, the blue dotted lines are transmission spectra obtained from the inverse matching designs with Fano functions at different wavelength, and the red line curves are obtained by using FDTD simulations with the optimally designed filter structures.
Previously, broadband optical filters have been designed with inverse matching method [30,32]. The inverse matching designed optical filters are wideband filters. The Fano model based inverse method shown here is to design and optimize narrow linewidth optical filters. Hybrid metal-dielectric GMR transmission filters with linewidth between 6.8 nm and 8.7 nm have been designed with this method. Compared with parameter sweeping method, the inverse matching with Fano functions avoids inefficient parameter sweeping and tuning. Fano function inverse matching also has advantage over tandem networks [26,46], in which both forward and inverse networks need to be trained. Since time and memory required by inverse searching scale up dramatically with number of design parameters [30], our inverse matching using a small selected sample set reduces the computational time. The model function based inverse matching method illustrated in this work is also applicable for design of other types of photonic devices such as optical filters, absorbers, emitters for biosensor [47], and on-chip spectrometer [34] to achieve desired spectral resonance in terms of transmission, reflection, or absorption.
5. Summary
In this work, we propose and implement a new inverse matching method to design hybrid metal-dielectric guided mode resonance optical filters. First, a four-layer forward neural network with a dropout layer is designed and trained for calculations of filter spectra of structures in the design space. With the trained deep learning neural network, a large sample set consisted of three million designs are generated. In the inverse matching design process, we first down select design samples from three million to about five thousand for efficient inverse matching. To match the asymmetric spectral line-shape of hybrid guided mode resonance optical filters, Fano resonance functions with different asymmetric parameters are tested to match against the spectral line-shape to determine the optimal Fano parameter q. Using Fano functions with optimized asymmetric q parameter, narrow linewidth hybrid guided mode resonance optical filters with spectral linewidth as narrow as 6.8 nm are designed in visible spectrum. The design method of using sample down selection and model-based Fano function inverse matching can be applied to design a variety of photonic devices with complex optical resonance for spectral engineering, sensing, and energy applications.
Funding
Fudan University Yanchang Petroleum Research Fund; Fudan University Changguang Research Fund; Yiwu Research Institute of Fudan University Research Fund.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper may be obtained from the author upon request.
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