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Abstract
Terahertz quantum cascade lasers (THz QCLs) are the most powerful solid-state THz sources so far and THz QCLs with various distributed feedback (DFB) gratings have demonstrated single-mode emission, collimated beam, frequency tunability and high output power. Resonant mode characteristics of THz QCLs with DFB, including frequency, loss and electric-field distributions, are important for waveguide analysis, fabrication and indication of THz QCLs’ radiative performance. Typically, predictions of these characteristics rely on numerical simulations. However, traditional numerical simulations demand a large amount of running time and computing resources, and have to deal with the trade-off between accuracy and efficiency. In this work, machine learning models are designed to predict resonant mode characteristics of THz QCLs with first-order, second-order, third-order DFB and antenna-feedback waveguides according to the four input structural parameters, i.e. grating period, total length of waveguide, duty cycle of grating and length of highly-doped contact layer. The machine learning models are composed of a multi-layer perceptron for predictions of frequency and loss, and an up-sampling convolutional neural network for predictions of electric-field distribution of the lowest-loss mode, respectively. A detailed study on more than 1000 samples shows high accuracy and efficiency of the proposed models, with Pearson correlation coefficients over 0.99 for predictions of lasing frequency and loss, median peak signal-to-noise ratios over 33.74dB for predictions of electric-field distribution, and the required time of prediction is within several seconds. Moreover, the designed models are widely applicable to various DFB structures for THz QCLs. Resonators with graded photonic heterostructures and novel phase-locked arrays are accurately predicted as examples.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Terahertz quantum cascade lasers (THz QCLs) are the electrically-pumped semiconductor lasers, which emit terahertz radiation due to inter-subband optical transitions in semiconductor superlattices [1,2]. The lasing frequency of THz QCLs can cover the range of ∼1-5THz and emit high output power at watt level [3,4]. And most recently, a record-high operating temperature of 250K and a THz-QCL based portable terahertz laser system haves been demonstrated [5]. In terms of beam quality and frequency tunability, full-width half-maximum (FWHM) of far-field radiation pattern from THz QCLs can be collimated within a few degrees [6–9] and a broad continuous frequency tuning up to ∼880GHz has been achieved [10]. High-performance THz QCLs with excellent beam quality and frequency tunability, high operating temperature and output power are highly desirable in a variety of applications, such as biomedical imaging, sensing and non-destructive identification.
Waveguide plays a vital role in the radiation characteristics and emission performance of THz QCLs. Such lasers with double-metal resonant cavities can achieve high operating temperatures and low lasing threshold. Recent progresses on THz QCLs with high temperature are based on double-metal waveguides [5,11]. However, such cavities with Fabry-Perot type ridge always suffer from multi-mode lasing and highly-divergent beam pattern due to the subwavelength dimension in vertical direction. To achieve single-mode emission, distributed-feedback (DFB) scheme has been introduced on the top cladding of Fabry-Perot metallic cavities or ridge cavities of THz QCLs. DFB structure builds up a one-dimensional interference grating and the grating provides optical feedback for the laser, while in Fabry-Perot lasers, facets of cavity form the two mirrors that provide the feedback. The DFB grating could be treated as a wavelength selective element to reflect only a narrow band of wavelength, and thus produce a single longitudinal lasing mode. THz QCLs with edge-emitting first-order DFB [12] and surface-emitting second-order DFB [13–15] have demonstrated robust single-mode operation, but they are incapable of narrowing down the far-field radiation pattern in both lateral and longitudinal directions. Further DFB waveguide engineering of THz QCLs with third-order DFB [7,16] and antenna-feedback scheme [6] achieved collimated beam pattern in both directions and significantly improved the beam quality, when maintaining single-mode output and enhancing the extraction of terahertz radiation.
Calculation of resonant modes for THz QCLs with DFB scheme [15–19] mainly relies on numerical simulations, such as finite-element methods (FEM) and finite difference time domain (FDTD). Figure 1 presents the geometry of two-dimensional finite-element simulations and the corresponding simulation results by COMSOL Multiphysics for THz QCLs with first-order, second-order, third-order DFB and antenna-feedback scheme, respectively. In COMSOL simulations, a complex eigenfrequency solver is used to solve the electromagnetic eigenvalues and the DFB grating is treated with finite length, height and infinite width. The absorbing boundary, implemented by adding a linearly-graded imaginary component to the dielectric constant of vacuum, serves as a boundary condition that absorbs radiation isotropically and prevents the disturbance from irrelevant reflection, similar to [13]. Loss and frequency of different eigenmodes located on both sides of photonic bandgap of the DFB waveguide are calculated, and the eigenmode with the lowest loss would be the lasing mode. As the core characteristics for THz QCLs with DFB gratings, eigenmode spectrum and electric-field distributions of the lasing mode are crucial in design, analysis and optimization of waveguide for THz QCLs. However, FEM simulation requires a lot of running time, typically from half an hour to a couple of hours, and a large amount of computational resources, resulting in lower productivity and higher requirement for hardware supports.
Fig. 1. (a) Geometry for 2D simulations of THz QCLs with DFB gratings, where Λ denotes the grating period and Wslit is the width of air slit. (b)-(i) Finite-element simulation results for THz QCLs with first-order, second-order, third-order DFB and antenna-feedback scheme, respectively. (b)-(e) Eigenmode spectrum, in which the lowest-loss modes are indicated by red circles, and (f)-(i) electric-field distribution in y and z directions of the resonant mode with the lowest loss. The total waveguide length is ∼1300µm; the duty cycle, defined as $\Gamma = 1 - {W_{slit}}/\Lambda $, is 83%; the absorber length, i.e. length of highly-doped contact layer, is 50µm; and the grating period is 15µm, 30µm, 45µm and 23µm for above DFB gratings, respectively.
Recently, machine learning-based, especially deep neural network-based methods have gained more and more attention for their powerful abilities to characterize the intrinsic relationships among large amounts of samples, and have been widely used in analysis, design and optimization of photonic structures [20–22]. For example, to reduce the full-width half-maximum (FWHM) as well as the side lobes of the focused beam, Turduev et al. [23] integrated additive updates of perceptron and state reward strategy of reinforcement learning into the design of subwavelength light focusing photonic structures. Asano and Noda [24] employed a four-layer neural network in cavity design for better learning the relationship between air holes’ displacements and Q factors to achieve a high Q factor in air-bridge-type 2D photonic crystal nanocavities. Statistical evaluations proved that cavities designed by the abovementioned neural network have higher Q factors and fabrication yields than those designed by conventional leaky-mode visualization [25]. Another interesting study was proposed by Tak et al. in [26], which developed a three-layer multi-layer perceptron for further optimization of 3D printed W-band slotted waveguide array antenna after a preliminary design procedure. Similarly, Chugh et al. [27] utilized a five-layer multi-layer perceptron for predictions of optical properties in photonic crystal fibers. Furthermore, Ferreira et al. [28] combined multi-layer perceptron with extreme learning machine algorithm for the fast estimations of dispersion relations and photonic band gaps in polarized photonic crystals. Long et al. [29] proposed an auto-encoding neural network for forward prediction and inversion design of photonics topological state of Zak phases in dielectric photonics crystal. And Khan et al. [30] designed and predicted transmission-spectral response of slab-waveguide based optical filters via a regression analysis model, which successfully relates design parameters, e.g. waveguide thickness and air hole radius, with the transmission wavelength. Salmela et al. [31] proposed to predict the peak power, duration and temporal walk-off of solitons in a supercontinuum spectrum by training a feed-forward neural network. And a recurrent neural network with long short-term memory (LSTM) has been modeled to predict the complex dynamics associated with the nonlinear propagation of short pulses in optical fibers in real time [32,33]. Meng and Dudley [34] optimized the femtosecond pulses of ultrafast mode-locked fiber laser by combining single-shot spectral measurements with genetic algorithms. Moreover, machine learning-based methods get extensive adoption in many other photonics and optics-related applications, such as simulations of plasmonic nanoparticle [35], design of metamaterials and metasurfaces [36–38], neuromorphic photonics [39], and control of photonic circuits [40].
In this work, we combine numerical simulations and machine learning models to predict resonant mode characteristics for THz QCLs with first-order DFB, second-order DFB, third-order DFB and antenna-feedback scheme. Firstly, we prepare a dataset using numerical simulations, in which the structural parameters of waveguides are varied to record different resonant mode characteristics, including frequency, loss and electric-field distributions. Secondly, we design the machine learning models and optimize them using the training dataset in the collected data. Thirdly, these machine learning models are utilized to predict the outputs for new input structural parameters. In contrast to direct numerical simulations, the running time and computational resource of our data-driven models are significantly decreased, with a comparable prediction accuracy. To the best of our knowledge, it is the first time to demonstrate both efficient and accurate predictions of resonant mode characteristics for THz QCLs with DFB waveguide based on machine learning models. The proposed machine learning models will be greatly beneficial to waveguide analysis, parameter optimization and experimental fabrication of THz QCLs.
2. Methods
Resonant mode characteristics of THz QCLs with DFB gratings are highly dependent on the four waveguide structural parameters, i.e. grating period, total waveguide length, duty cycle and absorber length. Therefore, machine learning models are proposed to predict resonant mode characteristics according to these specific structural parameters, which are defined and illustrated in Fig. 1. A flow path of implementing our machine learning models is given in Fig. 2. As an initial step, a dataset is prepared using a finite-element solver, COMSOL Multiphysics. Each sample in the dataset has an array of structural parameters, called inputs, and an array of resonant mode characteristics, called labels. The structural parameters are varied to generate different resonant mode characteristics. Then the dataset is split into two non-overlap sub-sets, training dataset and testing dataset. Training dataset contains 80% of the data, while testing dataset contains the rest 20%. Training dataset is utilized to train the proposed machine learning models. Model training is an iterative process that learns the complex mapping from the input to the output space by optimizing weights and biases of the model, and gradually drives the outputs of the model towards the desired labels for the given inputs. This iterative optimization continues until the deviations between the predicted values and the labels get minimized, which means the models have been well trained. Once the trained machine learning models are obtained, we utilize the trained models to predict resonant mode characteristics on new input structural parameters from testing dataset. Then we evaluate the performance of the trained models by comparing the corresponding outputs from FEM simulations and machine learning predictions.
Fig. 2. The flow chart of implementing machine learning models to predict resonant mode characteristics in THz QCLs with DFB gratings.
The proposed machine learning models consist of a multi-layer perceptron (MLP) for predictions of frequency and loss, and an up-sampling convolutional neural network (UCNN) for predictions of electric-field distribution. In these machine learning models, structural parameters are the inputs and resonant mode characteristics are the outputs. Detailed descriptions about the design of MLP and UCNN models are given in Section 2.1 and 2.2, respectively.
2.1 MLP for predictions of frequency and loss
We model the predictions of frequency and loss for resonant modes in THz QCLs with DFB as a regression problem, and therefore exploit a multi-layer perceptron (MLP) architecture, as shown in Fig. 3. There are five layers in the proposed MLP, which includes one input layer, three hidden layers and one output layer. The dots with red, blue and yellow colors in Fig. 3 represent the specific neurons, called nodes as well, in different layers. The input layer is used to receive the four structural parameters of DFB waveguide, i.e. grating period, waveguide length, duty cycle and absorber length, so that there are four neurons in the input layer. The output layer is used to present prediction results and contains two neurons, representing lasing frequency and loss, respectively. Between input and output layer are three hidden layers. There are 50 neurons in each hidden layer, which is determined by the trade-off between prediction capability and computational burdens. And in Fig. 3, symbols “I”, “H” and “O” inside the dots indicate input, hidden and output neurons, respectively. Each of the three hidden layers in the middle is fully connected with adjacent layers with a non-linear activation, which enables MLP model to learn the underlying relationship and build non-linear correlation between input and output data.
Fig. 3. The architecture of multi-layer perceptron (MLP) with one input layer (4 input neurons), three hidden layers (50 neurons in each layer), and one output layer (2 output neurons).
We utilize Adaptive moment estimation (Adam) algorithm [41] as the optimizer during back-propagation and set learning rate as 10−2 with full batch. It is notable that as a pre-processing, the collected data samples are normalized to the uniform scale, and therefore we utilize a bounded activation function, Tanh, in our MLP models. And the loss function is in the form of Huber loss [42]:
2.2 UCNN for predictions of electric-field distribution
Considering the huge difference between the input structural parameters (containing 4 values) and the output electric-field maps (containing at least 40 × 160 values), we develop an up-sampling convolutional neural network (UCNN) to perform multi-scale predictions of electric-field distribution. The proposed UCNN model contains three fully-connected layers, a re-arrange operator, three up-sampling layers, and two convolutional layers, as shown in Fig. 4, in which red and blue dots represent the specific neurons in fully-connected layers and cuboids denote feature maps in convolutional layers and up-sampling layers. The numbers at the top of Fig. 4 indicate the output data size for each layer. Specifically, the output of each fully-connected layer is in the form of one-dimensional array, and we characterize the corresponding data size with a single integer, while for up-sampling layers and convolutional layers, the output is in the form of multi-dimensional matrix, and therefore characterized by h × w×c, such as 10 × 40 × 64 for the first up-sampling layer and 20 × 80 × 64 for the second up-sampling layer, where h denotes the height, w the width, and c the channels.
Fig. 4. The architecture of up-sampling convolutional neural network (UCNN) with three fully-connected layers, a re-arrange operator, three up-sampling layers and two convolutional layers, where E-field denotes electric-field.
We can see from Fig. 4 that, after passing through three fully-connected layers and a re-arrange operation, the data points get enlarged preliminarily from 4 values to 100 values and reshaped into an image of 5 × 20 pixels. Then the image is fed into a dimension-expansion convolutional layer to expand the channels of feature maps from 1 to 64. Subsequently, three up-sampling layers are employed to perform map enlargement and resolution enhancement, with sampling rate set as 2-fold in each layer. And finally, the enlarged feature maps get compressed from 64 channels back to 1 channel by a dimension-reduction convolutional layer, which also serves as the output layer.
Kernel sizes of the two dimension-transformation (expansion/reduction) convolutional layers in our UCNN model are uniformly set as 1 × 1. Sigmoid is employed as the activation function in the output layer, while others adopt ReLU [43] for activation. Besides, we set learning rate to 10−3, and batch size to 64. And batch normalization [44] is introduced in each layer of UCNN to prevent overfitting.
There have been a number of ways to perform up-sampling for feature maps, such as bilinear interpolation, deconvolution [45], pixel shuffle [46], content-aware reassembly of features (CARAFE) [47] etc. Among all of them, CARAFE operator is exploited in this work, where the reassembly kernel size is set as 5 × 5 and the encoder kernel size is 3 × 3.
The proposed UCNN model may work with any differentiable loss. Here, we formulate structural similarity (SSIM) [48] as the loss function. SSIM is a popular subjective measurement of image quality and is calculated over a local window. Let X and Y represent the target and reference images, respectively, the SSIM index can be defined as:
3. Results and analysis
In this section, we evaluate and discuss our machine learning models from multiple perspectives. We first describe the details of dataset generation for THz QCLs with first-order, second-order, third-order DFB and antenna-feedback scheme. Next, we evaluate the performance of the proposed MLP model in predictions for frequency and loss of the lowest-loss mode. After that, both qualitative and quantitate results of our UCNN model in predictions for electric-field distribution of the lowest-loss mode are presented. We also analyze the influence of up-sampling layers on the whole network. Subsequently, we compare the designed machine learning models with numerical simulations and show the superiority of our models in terms of much shorter running time and less consumed computational resources, while maintaining high accuracy. Finally, generalization capability of the designed models is discussed.
3.1 Data preparation
The characteristics of resonant mode, i.e. frequency, loss, and electric-field distributions, for THz QCLs with first-order, second-order, third-order DFB and antenna-feedback scheme were calculated using a commercially-available FEM solver, COMSOL Multiphysics. The simulations are done in 2D with infinite width. The active region is considered as lossless with a thickness of 10µm. Metals are modeled as perfect electrical conductors with a thickness of 0.4µm. The thickness of highly doped contact layers at both longitudinal ends of the cavity is 0.1µm, which are implemented using a complex dielectric constant computed by Drude-model and mainly used to reduce the reflection from end facets and increase the discriminations of DFB modes [13]. The detailed geometry for simulations of THz QCLs with DFB is given in Fig. 1(a). Values of the four structural parameters are acquired by random sampling and combination within their respective empirical ranges, as shown in Table 1. After FEM simulations, 269, 322, 291 and 469 samples have been prepared for THz QCLs with first-order, second-order, third-order DFB and antenna-feedback scheme, respectively.
Table 1. Empirical ranges of structural parameters and the corresponding E-map size for THz QCLs with first-order, second-order, third-order DFB and antenna-feedback scheme, in which GP denotes grating period, WL waveguide length, DC duty cycle, AL absorber length, and E-map denotes the map showing electric-field distributions in y direction (Ey) or z direction (Ez).
For THz QCLs with first-order, second-order and third-order DFB waveguides, the vertical and horizontal sampling ranges for both y-direction electric-field (Ey) and z-direction electric-field (Ez) are set as 0∼20µm and -Λ∼Λ at equal intervals, respectively, with the center of bottom metal in double-metal metallic cavity as the origin. The extracted electric-field maps are able to cover the entire 10µm thickness of active region and 10µm thick air region in the vertical direction, while in the horizontal direction, the length covers two grating periods around longitudinal center of the cavity. Consequently, the extracted electric-field maps are capable of capturing the relation between the grating period and guided wavelength, clearly demonstrating spatial distribution characteristics of electric-field both inside and outside the laser cavity and other important features for THz QCLs with corresponding DFB structures. Then the Ey and Ez maps are resampled to 40 × 160 pixels. A larger sampling area is required for THz QCLs with antenna-feedback scheme because a strong and coherent single-sided surface plasmon polariton (SPP) mode with large spatial extent is excited in surrounding medium on top of the metallic gratings. Therefore, to fully demonstrate this unique feature of antenna-feedback scheme, the electric-field distributions are sampled within -2Λ∼2Λ horizontally, and the vertical sampling range is increased to 0∼120µm for Ey while still 0∼20µm for Ez. Correspondingly, we obtain Ey maps of 240 × 160 pixels and Ez of 40 × 160 pixels for THz QCLs with antenna-feedback scheme, which are also summarized in Table 1, where E-map denotes the map showing distributions of Ey or Ez. To eliminate the impact of different physical units to the proposed MLP and UCNN models, normalization is performed for the above-mentioned structural parameters and resonant mode characteristics before training and testing procedures. Once training and testing are done, we reverse the normalization operation back for a quantitative evaluation of the designed models.
3.2 Predictions for frequency and loss
To predict frequency and loss of the lowest-loss resonant mode for THz QCLs with first-order, second-order, third-order DFB and antenna-feedback waveguides with MLP, we employ five-fold cross validation mechanism to make a more accurate and robust evaluation, which means the entire datasets are randomly partitioned into five equal-size subsets, and consequently, five iterations of training and testing procedures will be performed. For each iteration, four subsets are used for training, and the rest subset for testing. By this way, each sample in the datasets will get validated exactly once. Figure 5 illustrates the loss curves during training and testing process of MLP. Both training and testing loss decrease rapidly within dozens of epochs, which implies the fast convergence and strong reliability of the designed MLP model.
Fig. 5. Loss curves of multi-layer perceptron (MLP) in (a) training and (b) testing stage for THz QCLs with first-order, second-order, third-order DFB, and antenna-feedback scheme, respectively.
To further validate the effectiveness and robustness of the designed MLP model, we introduce another two popular machine learning (ML) algorithms, linear regression (LR) [49] and random forest (RF) [50], as comparisons. Figure 6 presents the scatter plots of predictions for frequency and loss using MLP, LR and RF models, respectively, in which the values predicted by machine learning models are plotted against values from FEM simulations. These machine learning models predict the outputs using structural parameters from testing dataset as the inputs, and the results of FEM simulations with the same structural parameters serve as the references. And prior to the predictions, machine learning models have been well-trained using training dataset. Each circle in Fig. 6 evaluates the prediction result of a single data sample. Red circles refer to the predictions by MLP model, blue circles by LR model, and yellow ones by RF model.
Fig. 6. Scatter plots of frequency and loss predictions by three machine learning (ML) algorithms, i.e. multi-layer perceptron (MLP), linear regression (LR), and random forest (RF), against finite element methods (FEM) for THz QCLs with (a)-(b) first-order DFB, (c)-(d) second-order DFB, (e)-(f) third-order DFB, and (g)-(h) antenna-feedback scheme, along with the ideal linear model (y = x). Predictions by MLP are indicated with red circles, LR with blue circles, and RF with yellow circles. Insets show their quantitative comparisons measured with Pearson correlation coefficient (PCC).
Frequency of the lowest-loss mode is closely related to grating period, one of the four input structural parameters, and relatively easy to be predicted. Therefore, lasing frequency of various DFB gratings can be successfully predicted by either of LR, RF and MLP models, while the accuracy of our MLP is slightly higher than that of LR and RF methods, as shown by Figs. 6(a), (c), (e), and (g). In contrast, the prediction for loss is a little more difficult, because the values of loss typically cover several orders of magnitude, from negligible to more than 10cm-1, and suffer from a non-uniform distribution. In this case, small variations of loss would bring great prediction errors after normalization, especially for the modes whose loss is in lower order of magnitude. To solve this issue, logarithmic transformation is applied additionally to stretch the data points from skewed distribution to approximate normal distribution. Prediction results of loss for THz QCLs with first-order, second-order, third-order DFB and antenna-feedback scheme are presented in Figs. 6(b), (d), (f), and (h), respectively, from which we can see that compared with blue circles by LR and yellow circles by RF, the red circles by MLP are much more close to the ideal linear model, y = x, indicating that MLP model owns a high accuracy in predictions for loss, and is better than existing LR and RF methods. Quantitative comparison also reveals the excellent performance of our MLP models, with the Pearson correlation coefficients (PCCs) over 0.9999 in prediction for lasing frequency, and over 0.9921 in prediction for loss, as shown by the insets of Figs. 6(a)-(h).
3.3 Predictions for electric-field distributions
As a composite framework, the proposed UCNN model can be flexibly modified and adjusted. Variations of this framework are summarized in Fig. 7, where “C”, “D”, “B” in the first column denotes CARAFE operator [47], Deconvolution [45], and Bilinear interpolation serving as the up-sampling layers in UCNN, and the following two integers indicate the number of fully connected layers and up-sampling layers, respectively. For instance, D_3_3_UCNN refers to UCNN with three fully-connected layers and three deconvolution-based up-sampling layers, and C_4_2_UCNN refers to UCNN with four fully-connected layers and two CARAFE-based up-sampling layers. Specially, a convolutional layer is followed by each bilinear interpolation operation to have a better representation for features. In each up-sampling layer of D-based and B-based UCNN model, the kernel size is uniformly set as 3 × 3 and the channels of feature map are 64. It is worth mentioning that large spatial extent of SPP mode is excited in the surrounding medium of THz QCLs with antenna-feedback scheme [6], which does not exist in conventional DFB structures, such as first-order, second-order and third-order DFB. To present the entire landscape of electric-field distributions for antenna-feedback scheme, we sample the Ey-field data in a wider spatial range and resample it to 240 × 160 pixels, much larger than that of 40 × 160 pixels in first-order, second-order, and third-order DFB. To tackle with this difference of map size, a new architecture with more up-sampling layers is employed for Ey prediction in antenna-feedback scheme, as shown by Architecture II in Fig. 7(b), while Architecture I in Fig. 7(a) is used for the predictions of Ez-field distribution of antenna-feedback scheme as well as Ey- and Ez-field distributions of first-order, second-order and third-order DFB.
Fig. 7. Variations of UCNN model, where (a) and (b) illustrate two different architectures, Architecture I and Architecture II, respectively. Architecture I is applicable to E-map of 40 × 160 pixels, i.e. Ey- and Ez-field distribution in THz QCLs with first-order, second-order, third-order DFB, and Ez in antenna-feedback scheme, and Architecture II for E-map of 240 × 160 pixels, i.e. Ey-field distribution in THz QCLs with antenna-feedback scheme. The background colors indicate operations in UCNN and the values denote the corresponding output data size in each layer.
To quantitatively evaluate the prediction accuracy for electric-field distribution of UCNN models, we adopt two objective image-quality evaluation metrics, i.e. mean absolute error (MAE) and peak signal-to-noise ratio (PSNR). They are respectively defined as:
where X and Y represent E-maps by UCNN prediction and FEM simulation, respectively, N is the number of pixels in E-maps, and Q denotes the maximal values of X and Y. Obviously, a small MAE and a large PSNR indicate a more accurate prediction.The performance of UCNN with different numbers and types of up-sampling layers (corresponding to models in Fig. 7) in terms of MAE and PSNR is presented in Fig. 8. Evaluations in Fig. 8 demonstrate the superiority of CARAFE operator over deconvolution and bilinear interpolation when serving as up-sampling layers, due to the large reception field and content-aware strategy. Moreover, when increasing the number of up-sampling layers, the prediction accuracy shows an ascending trend, which confirms the effectiveness of up-sampling layers in extracting spatial and shape features and their applicability for predictions of electric-field distribution. Therefore, it can be discovered that among UCNN variations, the models with the largest number of CARAFE-based up-sampling layers, i.e. C_3_3_UCNN in Architecture I (exactly shown in Fig. 4) and C_3_4_UCNN in Architecture II, achieve the highest accuracy for first-order, second-order, third-order DFB and antenna-feedback scheme, respectively, with the median MAEs of 0.0101, 0.0099, 0.0104 and 0.0137, median PSNRs of 35.35 dB, 36.10 dB, 33.91 dB and 33.74 dB for predictions of Ey-field distribution, and median MAEs of 0.0018, 0.0025, 0.0021 and 0.0049, median PSNRs of 45.84 dB, 45.88 dB, 45.78 dB and 41.11 dB for predictions of Ez-field distribution. The analysis above would be useful for predictions of electric-field distribution in any type of DFB scheme, and even for other tasks mapping from small-size values to large-size images. It can also be found in Fig. 8 that the predictions for Ez-field distribution are slightly better than those for Ey field, with lower MAEs and higher PSNRs. This is mainly because Ey-field is the dominant electric-field of THz QCLs and contains the most information of field distribution both inside and outside DFB waveguides, as shown in Fig. 1(f)-(i).
Fig. 8. Quantitative comparisons of electric-field predictions by different UCNN networks in Fig. 7, for THz QCLs with (a) first-order, (b) second-order, (c) third-order DFB, and (d) antenna-feedback scheme. The first and second columns indicate MAE and PSNR for Ey-field distribution, and the third and fourth columns refer to MAE and PSNR for Ez-field distribution, respectively. The horizontal labels denote the specific UCNN networks.
In Fig. 9, training and testing loss of UCNN models with the best performance among the above variations, i.e. C_3_3_UCNN and C_3_4_UCNN, are presented, respectively. The fast and sharp reduction in loss curves demonstrates the effective convergence and high reliability of the models. Figure 10 intuitively illustrates the prediction results for electric-field distribution of the lowest-loss mode by utilizing C_3_3_UCNN and C_3_4_UCNN. The first and second columns show the simulations by FEM and predictions by UCNN models, respectively. The third column gives the absolute error maps, which denote the absolute values for the difference between electric-field distributions by UCNN and FEM. In each of THz QCLs with first-order, second-order, third-order DFB, and antenna-feedback scheme, four representative samples are presented. To further demonstrate the robustness of UCNN models, various grating periods, waveguide total length, duty cycles and absorber length are included. It can be observed that the predicted Ey and Ez maps agree well with those from FEM simulations, even with different waveguide structures and parameters. And the absolute error maps are shown with negligible amplitude, which means the errors between FEM and UCNN are close to 0, suggesting a high similarity between UCNN predictions and numerical simulations for both Ey- and Ez-field distributions and indicating the high accuracy of our designed models.
Fig. 9. Loss curves of up-sampling convolutional neural network (UCNN) for THz QCLs with first-order, second-order, third-order DFB, and antenna-feedback scheme, respectively, where (a) training loss for Ey prediction, (b) testing loss for Ey prediction, (c) training loss for Ez prediction, and (d) testing loss for Ez prediction.
Fig. 10. Prediction of electric-field distributions by UCNN for THz QCLs with (a) first-order DFB, (b) second-order DFB, (c) third-order DFB, and (d) antenna-feedback scheme with various grating periods, waveguide length, duty cycles and absorber length. The first column indicates numerical simulations by FEM, the second column is the predictions by UCNN model, and the last column is the absolute error maps between FEM and UCNN.
3.4 Running time
The numerical simulations via COMSOL Multiphysics are performed on Dell Precision T5820 workstation with an Intel Xeon W-2123 3.6GHz CPU and 32G RAM. The proposed MLP and UCNN methods are evaluated on a laptop with Intel Core i5-9300H 2.4GHz CPU and 4G RAM, once the training was accomplished. With the hardware supports above, the running time of a THz QCL with DFB grating (total length ∼2000µm, 80 eigenmodes) in a single COMSOL simulation is around 1hour. Longer total length of waveguide, finer mesh size and a larger number of calculated modes will take much longer simulation time and even up to several hours. In contrast, MLP-based predictions take less than a second, and for UCNN-based predictions, only several seconds are required. This enormous difference proved that the proposed MLP and UCNN models are highly efficient and computational resource-saving. Note that once training procedure was done, the predictions of resonant mode characteristics were performed completely by machine learning models, without any participation of FEM simulations.
3.5 Generalization capabilities
The proposed models are generally applicable to any type of waveguide structures, irrespective of whether they are periodic or non-periodic, DFB gratings or photonic crystal structures, etc. In order to verify the generalization ability of our machine learning models in non-periodic systems, prediction of THz QCLs with graded photonic heterostructure (GPH) resonators [15] is characterized as an example, in which the top metallic grating is not periodic and the grating period (ai) gradually decreases from center to each end of the laser ridge, with ai+1 = ai×0.99 (i = 0, 1, 2,…), where ai is the periodicity of i-th grating from the center and a0 is the periodicity of grating in the center of waveguide. Periodicity of grating in the center of waveguide (PGCW), number of slits (NS) and width of slits (WIS) are taken as the inputs, and the predictions by MLP and UCNN are shown in Table 2 and Fig. 11, respectively. The high similarities between the predicted and labelled lasing frequency, loss, and electric-field distributions reveal that the proposed machine learning models maintain high accuracy even for THz QCLs with grating structures that are not in periodic situations.
Fig. 11. Predictions of electric-field distribution for the lowest-loss resonant mode in THz QCLs with GPH resonators by UCNN, in which the first column indicates numerical simulations by FEM, the middle one is the predictions by UCNN model, and the last one is the absolute error maps between FEM and UCNN.
Table 2. Prediction results for frequency and loss of the lowest-loss resonant mode in THz QCLs with GPH resonators by MLP, where PGCW denotes the periodicity of grating in the center of waveguide, NS number of slits and WIS width of slits.
Recently, THz QCLs with novel phase-locked array were demonstrated and a record-high power output of ∼2W was detected from such single-mode terahertz lasers [3]. To achieve phase-locked scheme, a long ridge cavity is split into several shorter microcavities with slit-like apertures in the top metal layer, and the gain medium between two microcavities gets removed. For this case, our designed models maintain a good performance and accurately predict the lasing frequency, loss and electric-field distributions of lowest-loss mode. Structural parameters of the phase-locked array, including the number of microcavities (NMC), width of apertures in the top metal layer of each microcavity (AW) and wavelength of the single-sided SPPs (WS), are considered as the input. Representative results with different values of NMC, AW and WS are shown in Table 3 and Fig. 12, respectively.
Fig. 12. Predictions of electric-field distribution for the lowest-loss resonant mode in phase-locked THz QCLs by UCNN, in which the first column indicates numerical simulations by FEM, the middle one is the predictions by UCNN model, and the last one is the absolute error maps between FEM and UCNN.
Table 3. Prediction results for frequency and loss of the lowest-loss resonant mode in phase-locked THz QCLs by MLP, where NMC denotes the number of microcavities, AW width of apertures and WS wavelength of the single-sided SPPs.
4. Conclusions
In this paper, two machine learning models, MLP and UCNN, are developed to predict resonant mode characteristics including frequency, loss and electric-field distributions for THz QCLs with various types of DFB structures, i.e. first-order, second-order, third-order DFB and antenna-feedback scheme, using structural parameters as the inputs. Specifically, MLP is utilized for predictions of frequency and loss, and UCNN for predictions of electric-field distribution. Both intuitive and quantitative results show that the proposed MLP and UCNN models are able to perform accurate and fast predictions for resonant mode characteristics in THz QCLs with DFB waveguides, without extensive cost of running time and computational resources. As a new approach to calculate resonant mode characteristics for THz QCLs with DFB gratings, the designed machine learning models would provide a more efficient and computational resource-saving solution to the analysis and optimization, not only for DFB gratings of THz QCLs, but also in numerous photonic structures. A combination of numerical simulations and machine learning has the potential to significantly impact a broader field related to optics and photonics.
Funding
Natural Science Foundation of Shanghai (20ZR1428300); Shanghai Sailing Program (19YF1425100); Shanghai Jiao Tong University Innovation Program (TMSK-2020-105); Science and Technology Commission of Shanghai Municipality (20DZ2220400).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data Availability
The proposed MLP and UCNN models are implemented in Pytorch [51]. Relevant datasets and codes in our paper are available upon request from the corresponding author.
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