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Abstract
6 G communication technology using terahertz (THz) frequencies has increased the demand for components such as terahertz band filters. A high Q-factor filter capable of transmitting only narrowband frequencies, in particular, has gained significant research attention due to its wide range of applications. Here we obtained an optimal high Q-factor THz filter using an inverse design method that combines a double deep Q-learning model with an analytical solution within eight hours. Further, we confirmed the results of the inverse design using a numerical simulation, experimentally realized the high Q-factor THz filter, and discussed limitations of the spectral resolution of THz spectroscopy.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The terahertz (THz) range, which represents frequencies between 0.1 and 10 THz, is applied in various fields such as solid-state physics, optics, biological and chemical sciences due to its label-free, non-contact, and non-destructive advantages [1–9]. In order to utilize the physical characteristics of the terahertz range, metasurfaces, nanogap loops, and slot antennas have been recently studied for enhancing light-matter interactions [10,11] as well as for sensing applications [12–15]. Furthermore, the THz optical devices have been studied as next-generation wire-less communication devices, in response to the increasing attention towards the THz range as a future communication frequency [16,17]. A high Q-factor filter, capable of transmitting only narrowband frequencies, have attracted significant attention in various applications such as optical communication systems [18,19] and optical mode trapping [20,21]. Despite the extensive researches on filter designs with various structures [22–29], it is possible to achieve a high Q-factor by manipulating the positions of peaks and dips based on the aspect ratio and periodicity in a simple rectangular hole array [23,30,31].
In order to design such optical devices, three dimensional (3D) numerical simulation methods are primarily used instead of costly sample fabrication and measurements, which saves a considerable amount of time and effort. A recent advancement in inverse design methods involves the incorporation of artificial intelligence in order to reduce the need to manually adjust individual parameters [32–37]. The inverse design method involves the agent exploring and adjusting parameters to find the optimal state, which can require tens of thousands iterative calculations. When combined with 3D numerical simulation methods that can take several minutes to complete for a single state, the entire process may span over a month to complete [33]. Analytical solutions can be introduced as alternatives to numerical simulations to overcome this challenge [34,38]. Using physics-informed machine learning in combination with an analytical model, the inverse design method has been successfully applied to the terahertz nanophotonic structure, which had been difficult to achieve previously due to a significant scale mismatch between the wavelength and the structure [38].
A noble metal, such as gold, has a large conductivity at the THz frequencies, making it a perfect electric conductor (PEC) in this region. The THz transmission spectrum in a rectangular hole antenna array can be theoretically determined using the analytical solution starting from treating metal as a PEC [23,39–42]. In this study, we demonstrated the high Q-factor THz filter designed by the inverse design method that combines double deep Q-learning and the analytical solutions within eight hours on a middle-level personal computer (CPU: 3.40 GHz, 6 cores, 12 threads, RAM: 16 GB, GPU: NVIDIA GeForce GTX 1050). In addition to numerical simulations, the optimal THz filters are experimentally verified by fabricating samples and utilizing terahertz time-domain spectroscopy (THz-TDS). Lastly, we discussed the limitations of experimental realization for high Q-factor filters using the THz-TDS method in Section 2.2.
2. Results and discussions
2.1 Inverse design method and structure optimization
As shown in Fig. 1, the state consists of four parameters: ${a_x}$ and ${a_y}$ representing the width and height of the pattern, and ${l_x}$ and ${l_y}$ representing the horizontal and vertical periods. The rectangular hole array consists of rectangular holes perforated in a 100 nm thick silver film on an undoped silicon substrate. An Action determines the next state by selecting one of nine options: eight correspond to increasing or decreasing each parameter, and one represents no change. By using the analytical calculation, the Environment provides a spectrum of the far-field transmitted electric field amplitude, called transmitted amplitude, and the Rewards are determined based upon the data in the spectrum.
Fig. 1. Schematic of the inverse design method for rectangular hole array structure. The structure consists of the parameters ${{\boldsymbol a}_{\boldsymbol x}}$, ${{\boldsymbol a}_{\boldsymbol y}}$, ${{\boldsymbol l}_{\boldsymbol x}}$, and ${{\boldsymbol l}_{\boldsymbol y}}$, where ${{\boldsymbol a}_{\boldsymbol x}}$ and ${{\boldsymbol a}_{\boldsymbol y}}$ represent the width and height of the rectangular hole, and ${{\boldsymbol l}_{\boldsymbol x}}$ and ${{\boldsymbol l}_{\boldsymbol y}}$ represent the horizontal and vertical periods, respectively. Thickness of the metal film is fixed as 100 nm. Based on the results of the analytical calculation, the Environment consists of an analytical calculation and a reward calculation. The Rewards are determined from the far-field transmitted electric field amplitude spectrum obtained by the analytical calculation. The double deep Q-learning network is applied to the agent to determine the optimal actions. A new state is determined by the action, and this process is repeated until the optimal design is achieved.
In the Environment, the transmitted amplitude spectra are calculated by modal expansion method for rectangular hole arrays [23,39,42]. Based on the modal expansion method, the analytical transmittance (T) of a rectangular hole array structure with an x-polarized electric field can be calculated as follows [23,39,42]:
As shown in Section S1 of the Supplement 1, the spectral feature of the high Q-factor can be obtained by overlapping peaks and dips in asymmetric resonances. Here, the peak and dip can be explained by extraordinary optical transmission (EOT) attributed to leaky wave modes [43–45]. In the visible spectrum, EOT is typically explained through a combination of surface plasmon polaritons and diffraction modes. However, in the terahertz range, surface plasmons cannot be supported, and all coupling occurs through diffraction. Leaky wave modes are generated due to the interference between the direct beam and the leakage beam from metal mesh. For a high Q-factor, the peak and dip positions are key parameters, and the Q-factor is also affected by the slopes surrounding the resonance peak. Based on these factors, the functions for the Rewards can be constructed as follows:
where Q is the Q-factor of the transmitted amplitude spectrum, ${R_{dist}}$ is a Reward function for the peak position, ${x_{target}}$ is the target peak position, ${x_{peak}}$ is a peak position from the calculated spectrum, ${R_{interdist}}$ is a Reward function for a distance between peak and dip, called an inter-distance, ${x_{dip}}$ is a dip position from the calculated spectrum, and $\Delta {x_{target}}$ is the target inter-distance.The Q-factor is obtained by dividing transmitted amplitude at the first peak (tmax) by the full width at half-maximum (FWHM). The Reward function of Rdist is a function based on distance from the target resonance frequency of 1 THz; the closer to the target resonance frequency, the greater the Reward of Rdist. In the same manner, another Reward function of Rinterdist is a function based on inter-distance between the positions of peak and dip. Thus, the Total Reward is composed of the three key functions of Q, Rdist, and Rinterdist in the appropriate proportions in order to achieve the highest Q-factor THz filter. Here are the ranges of each parameter of the rectangular hole array: ${a_x}$: 10 ∼ 200 µm with 5 µm interval, ${a_y}$: 10 ∼ 200 µm with 5 µm interval, ${l_x}$: 20 ∼ 400 µm with 1 µm interval, and ${l_y}$: 20 ∼ 400 µm with 1 µm interval. The thickness h is fixed as 100 nm. This results in a total of 220,789,881 cases. For each case of design, we are able to get acceptable results after around 60,000 iterative calculations within about eight hours. The calculations are conducted by a typical middle-level personal computer (CPU: 3.40 GHz, 6 cores, 12 threads, RAM: 16 GB, GPU: NVIDIA GeForce GTX 1050). The reinforcement learning toolbox in MATLAB is used for the Agent configuration and the application of double deep Q-learning method.
Reinforcement learning methods are known to be more efficient compared to classic optimization algorithms, which have the drawback of easily getting stuck in local minima and depending on the choice of the initial approximation. While there are more recent reinforcement learning algorithms available, such as the Asynchronous Advantage Actor-Critic (A3C) algorithm, we expect that in a situation with a relatively small number of states, such as ours, there will be little difference in performance [46,47]. We also adopted the relatively intuitive algorithm, double deep Q-learning, due to its simplicity and the advantage of using only one deep learning network. It should be noted that the analytical solution based on the modal expansion method takes approximately 0.8 seconds for a single state. For numerical simulations that include real metal properties and thickness, the computation time takes approximately ten hours using the wave optics module in COMSOL Multiphysics. To reduce the simulation time, a 1/4 geometry based on structural symmetry and adaptive meshing were applied. Furthermore, by implementing a PEC plane instead of 100 nm-thick real metal, the simulation time can potentially be reduced to as short as 3 minutes. Nevertheless, it is still evident that numerical simulations take more than 200 times longer than modal expansions. Accordingly, if the inverse design method with a conventional numerical simulation was employed, it would take about two months using the same computer.
Several inter-distances have been considered using our inverse design method, and the results are presented in Table 1. The inter-distance for samples 1-9 is 0.03 THz, 0.06 THz, 0.09 THz, 0.12 THz, 0.15 THz, 0.18 THz, 0.21 THz, 0.27 THz, and 0.30 THz, respectively. In addition, the Q-factors obtained from the analytical solution and numerical simulation are displayed in Table 1. For the purpose of comparison with the experimental results, numerical simulations were conducted concerning the optical properties of real metal and the perforated film of 100 nm thickness. The corresponding optimal design parameters ${a_x}$, ${a_y}$, ${l_x}$ and ${l_y}$ were determined by the inverse design method. The highest Q-factor is achieved at the smallest inter-distance of 0.03 THz, as expected. It is noted that the modal expansion method includes considerations for PEC. In numerical simulations that account for real metals, there can be a slight blue shift in the peak position [48]. This leads to a slight difference in the Q-factor between the analytical model and numerical simulation. Accordingly, there may be differences in the Q-factor and peak position. Detailed values and analysis of peak position, dip position, Q-factor, and peak transmitted amplitude are presented in the following section along with the experimental results.
Table 1. The optimal THz filters are calculated using our inverse design method for each case of the inter-distance. The peak positions are set at 1 THz for all cases. For samples 1-9, the inter-distance indicates how far the dip is located behind the peak: 0.03, 0.06, 0.09, 0.12, 0.15, 0.18, 0.21, 0.27, and 0.30 THz, respectively.
2.2 Experimental results and discussions
In order to demonstrate the feasibility of our inverse design method, we fabricate rectangular hole array samples using conventional photolithography and analyze them using THz-TDS [49,50]. The details of the fabrication process and THz-TDS are shown in Section S2 in the Supplement 1. In Fig. 2, we compare the first peak position, the first dip position, the Q-factor, and the transmitted amplitude for different inter-distances using analytical calculation, numerical simulation, and THz-TDS measurements. The full transmitted amplitude spectrum of each sample can be found in Section S3 of the Supplement 1. As shown in Figs. 2(a) and 2(b), the first peak and dip positions are nearly identical for all the three methods. However, the experimental results in the Q-factor and the transmitted amplitude, especially for the inter-distances less than 0.15 THz, differ from those of the analytical calculations and numerical simulations shown in Figs. 2(c) and 2(d). Indeed, as a result of the limited frequency resolution of THz-TDS, it might be difficult to obtain accurate experimental values when the peak and dip positions are too close together.
Fig. 2. Comparison of the (black squares) analytical, (red circles) numerical, and (blue triangles) experimental results for (a) the first peak position, (b) first dip position, (c) quality factor (Q-factor), and (d) transmitted amplitudes at the first peak on the inter-distance range between 0.03 and 0.30 THz.
One important point should be noted when obtaining the time-domain signal and frequency-domain spectrum after performing a fast Fourier transformation (FFT) of the time signal. The optical path length (OPL) is defined as the product of the real part of the refractive index and the thickness of the material [7], denoted as n and d, respectively, as follows:
Our sample is fabricated on the silicon substrate with n = 3.4 and d = $500 \pm 25$ µm. By measuring the optical path length, we can quantify the time delay $\tau $ caused by the light with the speed of c passing through the material, as follows:
The first multiple reflection signal in the time-domain signal is located $2\tau $ away from the first signal. The second multiple reflection signal occurs due to the detection crystal of ZnTe. Thus, the frequency-domain signal can be distorted when the multiple reflection signals are contained in the Fourier transform. To remove unwanted oscillations in the frequency domain spectrum, the “effective time window” of $11.34 \pm 0.56\; ps$, as illustrated by the blue dashed line in Figs. 3(a-b), should be considered. The minimum distinguishable frequency resolution $\Delta f$ in the frequency-domain spectrum with the given time window is as follows:
Fig. 3. The time-domain signals for (a) the sample 1 and (b) the sample 7 measured by the THz-TDS experiment. (c) and (d) show the Fourier-transformed spectra in frequency domain for the samples 1 and 7, respectively. To remove unwanted oscillations in the frequency-domain signal when the multiple reflections inside the substrate, only the “effective time window” area, indicated by the blue dashed line box, is used for the Fourier-transform of the time signal.
Accordingly, we can consider the minimum distinguishable frequency interval is about 0.09 THz. As a result of interpolation with the zero-padding method [51] during the Fourier transformation process, the frequency-domain signal may contain more data points than 0.09 THz spacing. Furthermore, the Nyquist theorem [52] states that a minimum resolution frequency, which is half the distance between the peak and dip, is required to distinguish between the peak and dip in the frequency domain. For example, the sample 1 (Fig. 3(c) and Figure S3 (a) of the Supplement 1) with a small inter-distance of 0.03 THz shows significant differences in the spectral features compared to the analytical calculation and numerical simulation, whereas the sample 7 (Fig. 3(d) and Figure S3 (g) of the Supplement 1) with an inter-distance of 0.21 THz close to twice the minimum resolution shows a good quantitative agreement with both of the calculation and simulation results.
A time-domain simulation was performed in order to account for the scenario without the effect of multiple reflections. To eliminate the multiple reflections, the electric field is absorbed completely underneath the substrate in the simulation. A seed pulse for the simulation was obtained from the THz signal through N2 purged air in the THz-TDS experiment. The simulation results for the transmitted THz signals through the sample 1 and 7 are shown in Fig. 4(a) and (b), respectively. The effective time window of 13 ps, indicated by blue dashed lines in Fig. 4(a) and (b), corresponds to the same range as Fig. 3(a) and (b). In Fig. 4(a), the oscillation caused by the sample 1 does not terminate within the entire time window of 90 ps, while in Fig. 4(b), the oscillation caused by the sample 7 terminates already within 20 ps, resulting in a higher Q-factor for the sample 1 when compared to the sample 7. Fourier-transformed frequency-domain spectra are shown in Fig. 4(c) and (d), respectively, for the time windows of 13 and 90 ps. As expected, in the sample 1, the transmitted amplitude spectrum with the time window of 90 ps shows a significant difference from that with the time window of 13 ps, whereas both the transmitted amplitude spectra in the sample 7 are in agreement. In order to experimentally measure a high Q-factor over 28 of the THz filter, thicker substrates and thicker detection crystals beyond 3 mm are required.
Fig. 4. The time-domain signals obtained by the numerical simulation for (a) the sample 1 and (b) sample 7. To eliminate multiple reflections, the electric field is absorbed completely underneath the substrate in the simulation. (c) and (d) show the Fourier transformed frequency-domain spectra of the sample 1 and 7, respectively. The black lines represent the Fourier-transformed frequency domain spectra with an effective time window of 13 ps, indicated by the blue dashed line in (a) and (b), in accordance with Fig. 3. The red lines in (c) and (d) represent the Fourier-transformed frequency-domain spectra for the entire time window of 90 ps.
3. Conclusion
In summary, we demonstrated optimal THz filters with the best Q-factor using the inverse design, significantly reducing the design time to only 8 hours by combining the analytical solution based on the modal expansion. Our inverse design method is applied to find the best structure that can transmit the desired frequency with a sharp peak for a subwavelength rectangular hole array. A sharp peak with a high Q-factor was obtained in the asymmetric resonance from EOT, which was formed by overlapping a dip from x-periodicity and a peak from shape. In addition, the analytical calculations contain considerations for PEC, so there may be slight discrepancies in peak position and amplitude when compared to numerical simulations for real metal, resulting in slightly different Q-factors. Interestingly, the experimental results were consistent with the simulations with inter-distances of 0.18 THz or higher due to constraints imposed by the frequency resolution of our THz-TDS system and the Nyquist theorem. Indeed, we confirmed that the sample 1 with the minimum inter-distance of 0.03 THz achieved the best Q-factor of over 28 with the time window of 90 ps after eliminating unwanted oscillations caused by the multiple reflections using time-domain simulations. With our inverse design method, we are able to design not only microstructures, but also nanostructures that are hundreds of thousands of times smaller than wavelengths. Despite the fact that nano slot antennas can broaden the peaks, making them less suitable for high Q-factor THz filters, we can rapidly develop THz nanophotonic structures tailored to ultra-low density sensing applications and high performance THz detectors by incorporating new rewards such as field enhancement into the inverse design method. Our inverse design method is a promising technique for designing terahertz filters for next-generation communication technologies, and it can be an excellent alternative to simulation-based inverse design methods.
Funding
Institute for Information and Communications Technology Promotion (IITP-2023-RS-2023-00259676, 2021-0-01580); Ulsan National Institute of Science and Technology (1.230022.01); National Research Foundation of Korea (NRF-2021R1A2C1008452, NRF-2022M3H4A1A04096465).
Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (NRF-2021R1A2C1008452 and NRF-2022M3H4A1A04096465), the Republic of Korea’s MSIT (Ministry of Science and ICT) under the High-Potential Individuals Global Training Program (Task No. 2021-0-01580) and the ITRC (Information Technology Research Center) support program (IITP-2023-RS-2023-00259676) supervised by the IITP (Institute of Information and Communications Technology Planning & Evaluation), and 2023 Research Fund (1.230022.01) of Ulsan National Institute of Science and Technology (UNIST).
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.
Supplemental document
See Supplement 1 for supporting content.
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