1. Introduction

Classical simulations, usually based on assumptions and simplifications, are employed in order to map complex systems to an analytical model. Real world problems are rarely completely mapped or only with great computational effort. In a digital experiment, where the physical component is replaced by a predictive model, no assumptions are required, which enables it to map the real world behavior, including effects that would be neglected in simulations. In contrast to classical simulations, which require details about the experiment geometry, boundary conditions and a sound mathematical description of the involved physics, but not a single measurement, the machine-learning based digital experiment only requires a measured data set of the parameter space.

Machine learning based solutions in photonics are rapidly gaining momentum. They are successfully used for controlling and modelling lasers [1,2], all-optical object recognition [3,4] and neuromorphic photonics [5,6]. Machine learning is successfully employed for designing photonic structures [7]. In conjunction with the development of Terahertz (THz) Quantum Cascade Lasers (QCLs) it was for example recently shown that machine learning methods can complement and perhaps even replace the finite-elements method for certain problems [8]. Another area of photonics, where machine learning is already successfully used in ultra-fast photonics [9].

An example of a complex system, which is difficult to map by using classical simulations are THz QCLs, which are efficient and powerful sources of coherent terahertz light [10]. Recently thermo-electrically cooled terahertz quantum cascade lasers [11] with operating temperatures up to 250 K [12] have been presented, allowing for portable systems. A sub-class of these lasers are the Terahertz Quantum Cascade Random Lasers (QCRLs) [13]. State-of-the-art QCRLs exhibit broadband, multi-mode terahertz emission with a collimated far field, making them ideal candidates for spectroscopic applications [14–16]. Molecular absorption lines result from transitions between vibrational and rotational states of molecules and lie within the THz spectral region. Molecules such as HCN or NO$_{2}$ exhibit their rotational transitions at 2.30 THz and 2.32 THz [17] and hence can be identified by THz spectroscopy.

Contrary to conventional Quantum Cascade (QC) resonator structures, such as ridge- or micro-disk resonators, which feature periodic Fabry-Perot and whispering gallery modes, the random laser is based on feedback by multiple scattering at scattering centers that are randomly distributed among the active region [18]. Considering the passive resonator only, such a cavity is broadband and not frequency selective, since the optical feedback is quasi-continuously broadband [19].

However, emission spectra of the THz QCRL exhibit discrete laser lines only. This is due to non-linear interaction between individual lasing modes that arise from the multiple scattering. The modes compete for the available gain in the spatial ("spatial hole burning") and the spectral ("mode competition") domains, which selects a set of modes.

The tuning of QCLs has previously been achieved by tuning the cavity refractive index via temperature [10], employing external cavities [20–23], and using micro-mechanical systems [24,25]. Recently published results show, that QCRLs may also be tuned by illumination with near infrared light (NIR) [26]. The interplay between the modes can be influenced by spatially shaping NIR illumination of the QCRL and thus the scattering conditions of the resonator structure. By optical control of the QCRL, a specific laser mode can be selectively enhanced until a single-mode spectrum is achieved. However, this iterative procedure requires repeated feedback from a FTIR spectrometer, which is fed into an optimization algorithm to shape the NIR modulation pattern. A typical optimization routine requires measuring dozens of spectra, which is very time consuming. In fact, every single desired spectrum requires remeasuring a large amount of spectra to achieve the desired output, hence the field of practical applications is limited.

To overcome this vexing limitation of the truly broadband random resonator, we demonstrate predictive deep learning models, which once trained are able to precisely predict the QCRL’s behavior. The rapid and instantaneous feedback of the predictive models significantly accelerates possible applications, such as the generation of designed output spectra. To modulate the QCRL towards specific output spectra, tuning of the hyperparameter space is necessary. While this process is inconvenient and time consuming for real experiments, as it requires restarting the experiment for each set of parameters, the digital experiment easily enables the tuning of the optimization process due to its high speed. The digital experiment allows the modulation of any NIR modulation pattern, thus enabling remote adaption of the output spectra.

For the fast generation of the desired THz spectra, we make use of a direct predictive approach by applying a convolutional neural network (CNN) [27]. For the exploration of boundaries of the physical system, an iterative approach demonstrates the potential of the digital experiment.

These results pave the way for real-time emission spectrum modulation of QCRLs with high flexibility, which enables advanced spectroscopic applications.

2. Methods

2.1 General

The THz beam is emitted from a QCRL cryogenically cooled to 10 K. The active region design is based on a three-well phonon depopulation, bound-to-continuum $\mathrm {Al_{0.15}Ga_{0.85}As}$ heterostructure as introduced by Amanti et al. [28]. The epitaxial layer has a thickness of 13 µm and consists of 340 repetitions of the active region as well as a 100 nm highly doped contact layer on the bottom. The device is fabricated as double-metal waveguide structure with a disk diameter of 500 µm. Etched holes with diameters of 20 µm act as random scatterers. They are distributed such that the filling fraction is 18%. The QCRL is operated in pulsed mode with a repetition rate of 10 kHz and a pulse width of 4 µs. The bias current density at a voltage of 11.8 V is 350 A cm⁻².

The modulation is achieved by expanding a 810 nm NIR laser beam with a continuous wave power of 950 mW to a diameter of 1 cm. The beam is directed onto a spatial light modulator (SLM), which spatially modulates the wave front of the beam in such a way that the desired image is created at the focal plane, where the QCRL is located. The diameter of the pattern in the focal plane is slightly larger than the diameter of the QCRL (500 µm), which is compensated by the pattern generation. For the projection phase-masks are calculated with the Gerchberg-Saxton algorithm [29]. This operation is performed on a computer, which also controls the experiment. The THz beam, emitted by the QCRL is directed into a Bruker Vertex 80 Fourier Transform Infrared (FTIR) spectrometer, which records the emission spectrum of the QCRL. The resolution of the FTIR spectrometer is set to 3 GHz. An illustration of the setup is depicted in Fig. 1.

 

Fig. 1. Illustration of the measurement setup used for data generation and validation. A 950 mW near infrared laser beam with a 810 nm wavelength is expanded and directed onto a spatial light modulator (SLM), which controls the phase front of the beam. The light reflected from the SLM is projected on the THz Quantum Cascade Random Laser (QCRL) using a lens. As a result, the image, imposed by the SLM on the light beam, is projected on the surface of the QCRL. Simultaneously, the emission spectrum of the QCRL is measured by using a Fourier-Transform Infrared (FTIR) spectrometer. The experiment is controlled by a computer, which continuously generates patterns and stores the corresponding spectra.

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During the data generation phase the computer continuously generates random patterns and displays them with the SLM. The pattern is projected onto the QCRL. As a consequence the interaction of the NIR, which is coupled through the holes in the device, with the lasing material significantly changes the local electrical permittivity of the laser device and thus the scattering conditions responsible for the mode formation, which leads to strong changes in the occurring modes [26,30]. The resulting spectrum is recorded by the FTIR spectrometer and stored on the computer. This procedure is repeated $N$ times, where $N$ is the number of samples and the data set consisting of $N$ random patterns and their corresponding QCRL emission spectra is stored.

Two different approaches are compared. The first approach is an iterative approach, which in combination with an optimizer is capable of intrinsically capturing the limits of the system but is limited to a one-dimensional column-wise modulation due to the complexity of the system. The complexity of this one-dimensional approach is given by the 16 utilized columns and the gray-scale-depth of 256 values, which results in more than $10^{38}$ possible patterns. Increasing the number of degrees of freedom by e.g. moving to a two-dimensional scheme leads to a poor performance of the optimization algorithm [31]. The second approach is a direct predictive approach, which is capable of handling the complexity of two-dimensional modulation, but does not provide limits, which is why we complement it with an auto-encoder based anomaly detection [32,33]. The complexity of the two-dimensional approach is given by $16\times 16=256$ pixels with a gray-scale-depth of 256 values, resulting in an astoundingly high number of more than $10^{616}$ possible patterns. As in this approach no optimization is employed, the amount of possible degrees of freedom can be significantly increased. Obviously for neither approach the whole set can be measured, which fortunately is not required, as for both approaches a sub-sample, which significantly describes the whole space is sufficient.

2.2 Iterative approach

In the training phase of the iterative approach an artificial neural network (ANN) is presented with the column weight vectors, which are a compressed representation of the NIR illumination patterns, and with the corresponding emission spectra. The column weight vector contains 16 values, each of which is a 8-bit weighting factor for a corresponding column of the $16\times 16$ Hadamard matrix. The artificial neural network learns to predict the QCRL behavior for different applied patterns by running through hundreds of training steps, which are commonly called epochs. During training the weight and bias values of the ANN are optimized by using the back propagation algorithm [34], such that the network improves its predictions over the course of the training cycle. Thereby, an Adam optimizer [35] with a learning rate of $10^{-3}$ and first and second momentum decay rates of $0.9$ and $0.999$ respectively is employed. After training the ANN is capable of predicting the emission spectrum for a given column weight vector. An optimizer is used to iteratively approximate the desired spectrum. This approach requires hundreds of neural network predictions. The main advantage of this approach is that it intrinsically captures the limits of the real device.

The ANN consists of three layers (see Fig. 2(b)). The first layer, which acts as the input layer has 16 neurons, one neuron for each column weight vector value. The hidden layer consists of 128 neurons with leaky rectified linear activation functions [36], which help to reduce the vanishing gradient problem and introduce non-linearities into the ANN. The output layer has 60 neurons with linear activation functions. Each of the output layer neurons corresponds to a unique frequency in the predicted spectrum.

 

Fig. 2. Iterative approach: (a) Schematic illustration of the pattern generation algorithm. The desired spectrum is translated into a cost function and fed into the optimizer, which iteratively creates column weight vectors, presents them to the ANN, and returns the predicted spectra. After convergence the resulting vector is returned by the optimizer, and is converted into an illumination pattern. This pattern is projected onto the THz Quantum Cascade Random Laser (QCRL), which changes its emission spectrum accordingly. (b) Sketch of the ANN. The input layer, which accepts the column weight vector has 16 neurons, one for each column weight vector entry. The hidden layer consists of 128 neurons with leaky rectified linear activation functions. The output layer has 60 neurons with linear activation functions. Each output neuron corresponds to a unique frequency in the predicted spectrum.

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A data set consisting of 1325 pairs of illumination patterns, generated from column weight vectors and the corresponding spectra is recorded. For each sample the column weight vector is converted to an illumination pattern, which is subsequently converted to a phase-mask by using the Gerchberg-Saxton algorithm. The phase-mask is displayed by the SLM, which modulates the phase-front of the NIR accordingly. As the pattern is projected on the QCRL, the emission THz spectrum is modulated, and measured by the FTIR spectrometer. Thereby the FTIR measurement is the slowest step taking between 20 and 30 seconds per measurement. 10% of the data set, equaling 133 pairs, is split off for testing of the model. Another 10% of the remaining 1192 samples, resulting in 119 samples are used for evaluation during the training of the ANN. The remaining 1073 samples are used for training. The ANN is trained for 200 epochs. The input column weight vectors do not have to be normalized as they already only consist of values between 0 and 1. The spectra on the other hand are Z-score normalized for each frequency. The loss function used for training is mean squared error. The ANN training converges already after 50 epochs. Overfitting is successfully prevented by dropout layers and L2 kernel regularization [37,38]. As a result, the validation loss and training loss are very similar. After training the ANN is evaluated on the 10% test set.

The trained ANN can be used to quickly generate the desired output spectra. A scheme of the spectrum generation process is depicted in Fig. 2(a). To generate a desired spectrum, a cost function is derived from it (e.g. maximize the intensity at a certain frequency) and fed into a Nelder-Mead downhill simplex optimizer [39]. The optimizer gets feedback from the ANN, which predicts the QCRL behavior and subsequently adapts the modulation pattern. After convergence the pattern and the predicted spectrum are returned.

The behavior of the ANN is defined by its weight and bias values. These values are fit during the training process and are finite. Our ANN is bounded-input, bounded-output stable, which means that for a bounded input the output is also bounded. The output bounds are defined by the weight and bias values and are implicitly learned by the neural network. This is the reason that the neural network is able to model the limitations of the real system. For the iterative approach, where the ANN is a direct model of the QCRL behavior under NIR modulation, no additional steps have to be taken to ensure realistic outputs.

2.3 Direct predictive approach

For an even faster generation of lasing spectra, that is also capable of leveraging the full two-dimensional surface of the QCRL the neural network is trained to return the NIR illumination pattern directly. This approach scales much better than the iterative approach at the cost of not capturing the limitations of the device. The neural network returns a NIR illumination pattern for any desired spectrum and is validated by using an auto-encoder based anomaly detection.

For this approach no optimizer is required. Instead, the NIR illumination pattern is directly generated by the ANN. This is especially beneficial when dealing with larger parameter spaces such as in two-dimensional patterns. Here a Convolutional Neural Network (CNN) [27] is used, which directly generates modulation patterns for any given spectrum (see Fig. 3(a)). CNNs are commonly used for object recognition tasks, as they are very efficient in learning patterns with a spatial structure, such as our two-dimensional NIR illumination patterns. To further enhance the effect of the random modulation patterns Perlin noise [40], a pseudo-randomly generated noise is used. Perlin noise is the ideal candidate for this task, as it resembles the spatial distribution of the modes in the QCRL cavity, leading to a higher impact of the NIR illumination on the output spectra. Additionally, Perlin noise has a lower information entropy than white noise, which enables the CNN to learn its structure, effectively reducing the parameter space, while preserving the two-dimensional nature of the data. Perlin noise leads to a $2.7$ times higher variance in the spectral data when doing random modulation compared to uniform noise due to the higher spatial coherence of Perlin noise.

 

Fig. 3. Direct predictive approach: (a) Schematic of the pattern generation algorithm. After checking the validity of the desired spectrum, the spectrum is fed into the convolutional neural network (CNN), which directly returns the corresponding NIR illumination pattern. The pattern is projected onto the QCRL, which emits the desired spectrum. (b) Illustration of the convolutional neural network used for pattern generation. The numbers above the layers indicate their shape. The CNN accepts a desired spectrum and predicts the corresponding NIR illumination pattern. The kernel size of the hidden convolutional layer is $2\times 2$, whereas the kernel size of the output layer is $4\times 4$.

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The CNN consists of 60 input nodes, one for each frequency value in the spectrum. The second layer is a 64 node dense layer with rectified linear activation functions. The third layer is an $8\times 8$ 2D convolutional layer with a kernel size of $2\times 2$ pixels, $16$ filters and likewise rectified linear activation functions. Symmetric zero-padding is used in order to retain the image size. After the third layer an up-sampling layer is added to scale the image to the desired format of $16 \times 16$ pixels. The output layer is another 2D convolutional layer with a $4\times 4$ pixel kernel, a total size of $16\times 16$ pixels, symmetric zero padding and only one filter (gray-scale). The stride size for all employed convolutional layers is $1\times 1$. The topology of the CNN is depicted in Fig. 3(b).

For training, an Adam optimizer [35] with a learning rate of $5\cdot 10^{-4}$ and first and second momentum decay rates of $0.9$ and $0.999$ respectively is employed. The batch-size for the training is 32.

A data set containing 696 pairs of Perlin noise patterns and corresponding QCRL spectra is recorded. Again 10% are split-off for after training evaluation and another 10% are used for during training validation of the model. The CNN is trained for 200 epochs, but also already shows convergence after 50 epochs. No critical overfitting is observed, eliminating the need for dropout layers and kernel regularization.

In contrast to the iterative approach, the direct prediction with a CNN only requires one evaluation. The desired spectrum (see dashed line in Fig. 5) is fed into the ANN. The network is able to return a NIR pattern that modulates the QCL accordingly. The main advantage of this method is its speed. Predictions are performed within one evaluation only, which takes less than 10 ms on state of the art computer systems. By using this approach a desired spectrum can be realized within a second, which includes the calculation of the phase-mask and displaying it on the SLM. Note that this approach does not intrinsically capture the limits of the QCRL. The CNN will always provide a NIR illumination pattern, even if the desired spectrum is not achievable. This can be negated by adding an auto-encoder based anomaly detection to the system. Therefore an auto-encoder neural network [41] with one hidden layer (10 neurons) is trained to perform a lossy compression of pre-processed, measured spectra and to recreate the spectra as closely as possible. The recreation error is small for spectra that are similar to the real spectra and larger for spectra, which are very different from the measured spectra. A T-Test performed on a set of generated spectra, based on real spectra and similar looking, but randomly generated spectra, yields a statistical significance ($p \ll 0.01$) of the results produced by this approach.

3. Results

For the iterative approach the calculated mean absolute error for the 133 tested pairs of patterns and spectra is $6.3$ intensity points per frequency. Considering that a typical mode peak in the spectrum has a height between 100 and 300 intensity points this is fairly low. This error is mostly accumulated at the edges of the spectra and small portions of the spectrum that are not properly predicted. The major peaks and their behaviors are very well predicted by the ANN, which is confirmed by a mean Pearson correlation between the real and predicted spectra of 99.1%.

In addition to comparing the performance of the ANN to previously recorded data it is even more interesting to perform a full optimization cycle. Therefore optimization routine is used to enhance a specific mode of the original QCRL spectrum by choosing a cost function $cost_{max}(k_0) = 1/(\mathrm {max}\ S(k_0))$, where $S(k)$ is the predicted spectrum and $k_0$ is the selected mode. Starting from an initial condition, the selected mode at 2.302 THz is subsequently enhanced in its intensity for 300 iterations. Figure 4(a) shows the predicted column weight vector for a maximization of the mode at 2.302 THz. Figure 4(b) shows the corresponding predicted spectrum (dashed line). To confirm the predicted output spectra under illumination with the simulated NIR illumination pattern, the pattern is projected onto the QCRL and the spectrum is measured. The experimental data (solid line in Fig. 4(b)) is in excellent agreement with the predicted spectrum. The neural network is capable of predicting the absolute intensities and position of the occurring modes in the emission spectrum. Thus, one can conclude that the ANN correctly learns the mode dynamics as well as the out-coupling efficiency of the respective modes. Note that the ANN is evaluated 300 times and each significant error would lead to an accumulation as each optimization step depends on the previous one. It can therefore be concluded that the presented method is very accurate and stable.

 

Fig. 4. Column weight vectors generated by using the iterative approach and spectra after 500 optimizer iterations. (a) Column wise predicted normalized NIR illumination of the QCRL to maximize the mode at 2.302 THz. (b) The dotted line shows the predicted spectrum, maximizing the mode at 2.302 THz as indicated by the red arrow. The solid green line shows the spectrum, which is measured when projecting the predicted pattern on the real device. (c) Column wise predicted normalized NIR illumination of the QCRL to minimize the mode at 2.302 THz. (d) The dotted line shows the predicted spectrum, minimizing the mode at 2.302 THz, indicated by the red arrow. The solid green line shows the spectrum, which is measured when projecting the predicted pattern on the real device.

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The optimization routine is not limited to enhancing modal lines in the spectrum. The suppression of lasing lines can be advantageous for self-referenced spectroscopy. The minimization is realized by choosing an appropriate cost function $cost_{min}(k_0) = \mathrm {max}\ S(k_0)$.

The results of the mode minimization and maximization are depicted in Fig. 4. The optimization routine returns the column weight vector (c), which leads to a modulated spectrum as depicted in (dö). The optimization routine works so well that the mode completely vanishes. The comparison of the predicted and measured spectra demonstrates that the ANN is capable of predicting both intensities and frequencies of the occurring modes. While the cost function favors a real single mode spectrum, the ANN predicts that the actual emission spectrum consists of at least two modes, which cannot be suppressed independently. The complex nonlinear interactions between the lasing modes hinders the single mode operation at 2.302 THz. Gain competition and spatial hole burning lead to the fact that only certain configurations are possible under the prevailing conditions. To achieve a single mode spectrum in this case a (free) parameter must be changed: either the current is reduced, so that all existing modes are weaker, or the magnitude of the NIR modulation is increased.

Evaluating the direct predictive approach is more difficult as it does not return spectra that can be directly graded by a metric but merely illumination patterns. During the training and by testing on the separate testing data set a significant reduction in mean-absolute error between the predicted and original NIR illumination patterns is observed. The mean-absolute error is reduced from 64 to 16. From the presented data the ANN learns which regions impact which modes and therefore only illuminates the necessary regions, which is in strong contrast to the random patterns. As a result the predicted illumination patterns look different than the corresponding ground-truth in many cases. The Pearson correlation between the pattern pairs is only about 0.5. Nonetheless the illumination patterns returned by the CNN yield the desired results as can be seen by two examples shown in Fig. 5. Thereby the dashed lines represent the desired spectra and the solid lines the actually measured ones for the returned illumination patterns, which are shown as an inset.

 

Fig. 5. Desired spectra (dashed lines) and their measured counterparts (solid lines) for two-dimensional NIR illumination. The insets show the NIR illumination patterns, generated by using the direct predictive approach. The QCRL outline is indicated by the black circle.

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4. Conclusion

We find that deep learning models are well suited to predict the emission spectra of QCRLs. A spatially modulated NIR laser with a continuous wave output power of 950 mW is used to modify a THz QCRL with an emission spectrum around 2.3 THz. Pairs of randomly generated NIR patterns and THz spectra are recorded and subsequently deep learning models are trained on the stored data. An iterative approach, which intrinsically captures the limits of the experiment, allows the generation of modulation patterns within less than 10 seconds. A direct predictive approach, which generates predicted NIR patterns for a desired spectrum within less than 100 ms is trained on 696 pairs and is complemented by an auto-encoder based anomaly detection for spectrum verification. We show that by using these methods the complete elimination or maximization of a mode can be achieved. The mean correlation between predicted and the corresponding real spectra is above 99%. As these methods are both fast and highly flexible, they may be useful for applications in spectroscopy and sensing.