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Abstract
The inverse design of photonic devices through the training of artificial neural networks (ANNs) has been proven as an invaluable tool for researchers to uncover interesting structures and designs that produce optical devices with enhanced performance. Here, we demonstrate the inverse design of ultra-narrowband selective thermal emitters that operate in the wavelength regime of 2-8 µm using ANNs. By training the network on a dataset of around 200,000 samples, wavelength-selective thermal emitters are designed with an average mean squared error of less than 0.006. Q-factors as high as 109.2 are achieved, proving the ultra-narrowband properties of the thermal emitters. We further investigate the physical mechanisms of the designed emitters and characterize their angular responses to verify their use as thermal emitters for practical applications such as thermophotovoltaics, IR sensing and imaging, and infrared heating.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As the quest for sustainable sources of energy is ever present, methods of effectively utilizing the power of the sun are paramount. Photovoltaic techniques use sunlight to excite electron-hole pairs in semiconductors to produce a usable flow of electrons [1]. This is then generally used to power electrical equipment or to charge a battery to store the energy to use later. A second well-known method is to use the heat from the sunlight to drive mechanical heat engines [2]. Photonic devices can be integrated into both strategies in order to increase the efficiency and generated power by engineering the optical properties in a way to enhance the absorption at the desired wavelengths [3–5]. Furthermore, according to Kirchhoff’s law of thermal radiation, the emissivity of an arbitrary body in thermodynamic equilibrium is equal to the absorptivity. Therefore, the design of broad and narrowband absorbers/emitters is of great interest for use in energy applications such as thermophotovoltaics [6,7] and radiative cooling [8–12], as well as in sensing [13–16], infrared heating [17,18], and thermal imaging [19,20].
Metamaterials have been proven as particularly exciting candidates to produce tailored optical responses through the careful design of multiple subwavelength sized elements to produce optical properties that are not found in naturally occurring materials [21,22]. Metamaterials have already been proven in applications such as negative refraction [23,24], super-resolution imaging [25–27], sensing [28,29], optical displays [30–33], and artificial chirality [34,35]. Work on absorbers/emitters based on metamaterials has been plentiful [36–39]. The desired response is achieved through the manipulation of the permittivity and permeability by the careful engineering of the constituent materials and their relative geometric parameters. Metals have been used to produce plasmonic absorbers [40–44], while devices based on all-dielectric materials have also been demonstrated [45,46]. This type of design usually depends on some form of subwavelength structure in order to induce the required impedance matching conditions. Although these devices have proven their applicability as absorbers/emitters, the reliance on subwavelength structures means that the fabrication process generally includes etching or lithography which limits the overall footprint and scalability [22,47]. On the other hand, simple multilayer structures only require the control of the thickness of the individual layers in order to produce the desired optical properties [48]. The important properties of reflection (R), transmission (T), and absorption (A) are related by the equation: 1 = R + T + A. By using a metallic mirror, transmission is completely suppressed to 0, therefore, A can be manipulated through the control of R. By designing the interference due to the multilayer structure, R can be completely suppressed at specific wavelengths, leading to the design of narrow and broadband absorbers/emitters.
Although the interference of multilayer structures is well understood, the design of the thickness and material of each layer to produce a desired response at a desired wavelength is still an arduous task. Recently, computational techniques have been used to inversely design metamaterials [49], with deep learning proving a valuable tool for the inverse design of photonic devices for various applications [50–60]. Among them, artificial neural networks (ANNs) have provided an extremely powerful way to generalize the relationships between the physical dimensions of meta-atoms and their optical properties, allowing for the efficient inverse design of photonic devices [58–60]. Here, we report an ANN that is trained for the inverse design of multilayer structures for narrowband thermal emitters. After training, the network is able to almost instantaneously produce high quality designs that closely match arbitrarily drawn Lorentzian-like functions to represent narrowband absorption spectra for wavelengths between 2 and 8 $\mu$m.
2. Method
2.1 Preparation of the dataset
To generate the dataset required to train the network, rigorous coupled-wave analysis (RCWA) simulations were performed using a home-built code. Several multilayer structures have been previously reported for narrowband thermal emitters, such as Fabry-Perot resonators [61], one-dimensional photonic crystals [62], and Tamm plasmon polaritons (TPPs) [48,63]. TPPs are formed at the interface between a metal and a distributed Bragg reflector (DBR). Therefore, we have set up the following basic structures of interest to excite TPPs to produce ultra-narrowband thermal emitter, each structure consists of a functional layer made up of silicon (Si) and three pairs of alternating multilayers of the chosen material and silicon dioxide ($\textrm {SiO}_2$), all on top of a 500 nm thick silver (Ag) metallic mirror (Fig. 1(a)). The top three pairs of high- and low- refractive index materials are designed as a DBR. To this end, three candidate materials of Si, germanium (Ge), and silicon nitride ($\textrm {Si}_{3}\textrm {N}_{4}$) are chosen due to their high refractive indices. This structure considers not only the general TPPs of the DBR, but also the function of the spacer between the DBR and the metal mirror [64,65]. The variables of the design of the ultra-narrowband emitters are the selection of the material (Mat) for use in the DBR and its thickness (t$_{\textrm {Mat}}$), the thickness of the $\textrm {SiO}_{2}$ (t$_{\textrm {SiO}_{2}}$) in the DBR, and finally the thickness of the functional Si layer (t$_{\textrm {Si}}$). Table 1 shows the order of the layers, the material candidates, and the range of thickness used to create the dataset. For the thin film thickness less than 100 nm, a smaller step size is desirable for more accurate prediction, whereas for larger thicknesses, smaller differences have little effect on the emissivity. Therefore, we only collected relatively dense data for thicknesses less than 100 nm in order to allow for a higher accuracy over large design space, while managing the size of the dataset to be within reasonable levels. The refractive indices of the materials were obtained from the following literature [66–69]. A total of 243,675 simulations were performed to prepare the dataset, with 300 equidistant spectral points over the wavelengths of interest (2-8 $\mu$m) were obtained for the input emissivity spectra.
Fig. 1. (a) Schematic illustration of the multilayer structure. The design parameters of the material type (Mat) and layer thicknesses ($\textrm {t}_{\textrm {Mat}}$, $\textrm {t}_{\textrm {SiO}_{2}}$, $\textrm {t}_{\textrm {Si}}$) are indicated. (b) The architecture of the ANN. The ANN is composed of an input layer that corresponds to 300 spectral points, four hidden layers, and an output layer that corresponds to the design parameters. The number of neurons in the hidden layers are shown.
Table 1. Details of multilayer structures used to create the dataset
2.2 Training the artificial neural network
The ANN takes the 300 emissivity spectral points as the input ($X$) and provides 6 parameters as the output ($Y$ in Eq. (1)). The ANN is completed with four hidden layers that use Rectified Linear Unit (ReLU) activation functions (Fig. 1(b)).
Table 2. Hyperparameters used for training the network
3. Results and discussion
The training of the ANN converged to a loss of around 0.06 after 5,000 epochs (Fig. 2(a)). After training, the ANN was used to design structures for the simulated emissivity spectra in the test dataset. For the given inputs, the ANN provided design parameters which were then used for RCWA simulations to obtain the emissivity of the designed structures. To quantitatively evaluate the discrepancy between the designed and target emissivity spectra, the spectral MSE between the two was calculated. For the 31,838 samples in the test dataset, an average spectral MSE of 0.001 was obtained. A histogram of this data is shown in Fig. 2(b), for 6 bins with an error interval of 0.001. 93% of the total test data have a spectral MSE of less than 0.005. Figure 2(c) shows eight random examples from the test dataset. When target spectra were input to the trained ANN, the design parameters were retrieved, and then used in RCWA and finite-difference time-domain (FDTD) simulations to obtain the emissivity spectra. RCWA is widely known for its fast simulation speed, but it can be less accurate when dealing with high refractive index contrasts or complex structures. Therefore, the commercially available FDTD solver from Lumerical Inc., Ansys. was used to perform additional validation of the RCWA results. As shown in Fig. 2(c), all examples show an excellent agreement between the target and designed emissivity spectra obtained from both RCWA and FDTD simulations. These results prove that the ANN has successfully learned to design structures that produce the desired emissivity spectra over target wavelengths.
Fig. 2. (a) Learning curves for the training (black) and validation (red) losses over 5,000 epochs. A logarithmic scale is used on the x-axis. (b) Histogram of the spectral MSE for the test set. The values above each bar represent the percentage of the data that falls within that loss range. (c) Examples of test results. The solid black line shows the target emissivity spectra, and dotted red lines and blue dots show the designed emissivity spectra obtained from RCWA and FDTD simulation, respectively.
3.1 Inverse design of ultra-narrowband thermal emitters
To demonstrate the inverse design of ultra-narrowband selective thermal emitters using the trained ANN, Lorentzian-like functions were used as the target emissivity spectra ($\epsilon _{\textrm {target}}$) as given by
where $w=0.05$ was used to produce functions with ultra-narrow bandwidths. Some examples of the inversely designed structures are shown in Fig. 3 for target emissivity spectra with $\lambda _{\textrm {peak}}$ gradually changing from $3.5~\mu \textrm {m} - 6.0 ~\mu \textrm {m}$. In particular, the designed parameters of $\textrm {t}_{\textrm {Mat}}$ (Fig. 3(a)) and the $\textrm {t}_{\textrm {Si}}$ (Fig. 3(b)) clearly show the tendency of an increase of the thickness with increasing $\lambda _{\textrm {peak}}$. Figure 3(c-f) show the target emissivity spectra and the corresponding designed spectra with $\lambda _{\textrm {peak}}$ of (c) 3.5 $\mu$m, (d) 4 $\mu$m, (e) 4.5 $\mu$m, (f) 5 $\mu$m, (g) 5.5 $\mu$m, and (h) 6 $\mu$m, respectively. For all examples, the target and the designed emissivity spectra show good agreements with a spectral MSE of less than 0.006. To further quantify spectral accuracy, we also calculated the spectral percentage error of the peak wavelengths asFig. 3. Designed parameters of (a) $\textrm {t}_{\textrm {Mat}}$ and (b) $\textrm {t}_{\textrm {Si}}$ for given input spectra. Design results of ultra-narrowband thermal emitter aimed at $\lambda _{\textrm {peak}}$ of (c) 3.5 $\mu$m, (d) 4 $\mu$m, (e) 4.5 $\mu$m, (f) 5 $\mu$m, (g) 5.5 $\mu$m, and (h) 6 $\mu$m.
The calculated error of the peak wavelengths are (c) 0.6%, (d) 3.5%, (e) 1.3%, (f) 0.8%, (g) 5%, and (h) 0%, respectively, which all represent reasonable peak wavelength percentage errors within 5%. The Q-factors of the designed structures are 93.2, 104.1, 87.4, 104.1, 109.2, and 103.9 for $\lambda _{\textrm {peak}}$ = 3.5 $\mu$m, 4 $\mu$m, 4.5 $\mu$m, 5 $\mu$m, 5.5 $\mu$m, and 6 $\mu$m, respectively, proving the ability of the trained ANN to inversely design arbitrary spectrally selective ultra-narrowband thermal emitters in the designated wavelength regime.
The designed parameters for these examples are summarized in Table 3. For arbitrarily drawn Lorentzian-like functions, the all design results of the materials used for DBR were Ge, which has the highest refractive index among the all three candidates. However, we observe that the two other materials are also able to produce high-Q emitters with similar target emissivity spectra (Fig. 4). This is a typical challenge that arises due to the non-uniqueness of solutions in photonic inverse design problems, where several different designs can have similar optical properties. The non-uniqueness in inverse design problem is known to cause networks to be difficult to converge or to converge between ground truth solutions [59,60,70,71]. In our case, the network returned a single solution somewhere in the ground truth design spaces among the several possible solutions, and accordingly, only materials of Ge were chosen for high-Q emitters drawn with Lorentzian-like functions. This problem could be alleviated by utilizing probabilistic models, where multiple possible candidates can be designed for a single target optical property [56,71].
Fig. 4. Design result of ultra-narrowband thermal emitters with three different materials of (a) Si, (b) Ge, and (c) $\textrm {Si}_{3}\textrm {N}_{4}$. The text boxes indicate the design parameters of [Mat, $\textrm {t}_{\textrm {Si}}$(nm), $\textrm {t}_{\textrm {SiO}_{2}}$(nm), $\textrm {t}_{\textrm {Mat}}$(nm)]
Table 3. Design parameters of structures in Fig. 3
To understand the mechanism behind the ultra-narrowband thermal emitters produced by the ANN, we conducted full-wave simulations using the FDTD method. Figure 5(a-c) show the calculated electric field profiles that are normalized by the incident fields for $\lambda _{\textrm {peak}}$ = 4 $\mu$m, 5 $\mu$m, and 6 $\mu$m, respectively. It can be clearly seen that the electric field is strongly localized in the bottom Ge-SiO2 layers, where the fields decay exponentially at the interfaces of the Si and metal layer due to the excitation of Tamm plasmon polaritons.
Fig. 5. Normalized electric field distribution at the peak wavelengths of the structures designed to target (a) 4 $\mu$m, (b) 5 $\mu$m, and (c) 6 $\mu$m, respectively.
For the practical application of thermal emitters, angle-independent thermal emission is preferred. Therefore, we analyzed the angular dependence of one of the inversely designed structures. Figure 6 shows the spectral directional emissivity of the structure targeting $\lambda _{\textrm {peak}}$ = 6 $\mu$m. It can be seen that the designed structure maintains the ultra-narrowband thermal emissivity properties in both TE and TM modes. However, the structure has an angular dependence due to the localized mode, where $\lambda _{\textrm {peak}}$ changes depending on the incident angle. This is especially obvious in TM mode. However, for small angles of incidence up to around $25^\circ$ in both modes, the peak wavelength change is minimal, proving the applicability of the inversely designed ultra-narrowband thermal emitters for practical uses.
Fig. 6. Directional and spectral emissivity of the designed structure targeting 6 $\mu$m for (a) TE mode and (B) TM mode.
4. Conclusion
In this study, we demonstrated the inverse design of multilayer structures for ultra-narrowband thermal emitters using an ANN. The trained ANN was able to successfully design ultra-narrowband thermal emitters that operate at target wavelengths between 2-8 $\mu$m with an average spectral MSE of less than 0.006. Using the ANN, we designed a thermal emitter with a high Q-factor of 109.2 at the wavelength of $\lambda _{\textrm {peak}}$ = 5 $\mu$m. The calculated electric fields inside the thermal emitters show strong light fields in the multilayer structure. We also calculated the spectral directional emissivity which proved the angular independence of the designed thermal emitters within a small angle of incidence of around $25^\circ$. In this study, only four design parameters of the thermal emitter were designed, but this could be easily expanded to a larger parameter space. The designed thermal emitters can be realized in a lithography-free manner, making them promising candidates for many practical applications, such as infrared heaters, sensing, imaging, and thermophotovoltaics.
Funding
National Research Foundation of Korea (CAMM2019M3A6B3030637, NRF-2018M3D1A1058997, NRF-2019R1A2C3003129, NRF-2019R1A5A8080290).
Acknowledgments
S.S. acknowledges the NRF Global Ph.D. fellowship (NRF-2017H1A2A1043322) funded by the Ministry of Education of the Korean government. D.L. acknowledges the PIURI fellowship funded by POSTECH.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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