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Abstract
We designed and simulated an ultra-compact 1 × 2 power splitter operating in the terahertz region. A machine learning approach was implemented to design the photonic device. The designed power splitter has a footprint of 500 µm × 500 µm. We calculated the insertion loss using a three-dimensional finite difference time domain method. The calculated insertion loss was less than 4 dB over the operating wavelength range of 275–325 µm. The machine learning algorithm implemented in this work can be applied to the inverse design of various photonic devices.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Wireless data traffic has exponentially increased over recent years, with increasing demand from mobile traffic and new applications driving the need for transmission technologies with higher data rates across a broader spectrum [1,2,3]. Edholm’s law states that the “Required data rates double every 18 months” [4]. If the trend continues, wireless data rates of 100 Gbit/s will soon be required. Wireless communications will need to use the terahertz (THz) band for ‘beyond 5G’ [5,6].
The THz band is loosely defined as the frequency range from 0.1 to 10 THz. It has potential for large bandwidth and data rates above 100 Gbit/s [7,8]. Several photonic devices that operate in THz band were developed [9,10]. However, to make THz wireless communication a reality, many new photonic devices must still be developed including power splitters. Power splitters are an essential component for optical communications systems, channeling specific fractions of input power into different output channels [11]. Subwavelength graphene waveguides [12] and multi-output splitters without a grating structure [13] have been developed. Due to the long wavelength of THz radiation, most THz photonic devices have large footprints. For compact THz communication systems, compact THz power splitter must be designed.
When light encounters a sub-wavelength material with a different refractive index, scattering or refraction occurs. By engineering the refractive index distribution at a sub-wavelength scale, photonic devices with high efficiencies and small footprints become feasible. Conventionally, photonic device designers work with direct binary searches (DBS) [14], evolutionary algorithms [15], adjoint methods [16], or sweep parameters based on specific structures [17] to optimize the refractive index distribution and to design the structure. However, these methods depend on the initial conditions, making the optimization process time-consuming. Recently, machine learning (ML) approaches, including artificial neural networks, have been used in the inverse design of photonic devices [18]. However, training an artificial neural network is still very challenging with long learning times required.
We designed and simulated an ultra-compact THz 50:50 power splitter, using a ML approach based on a perceptron algorithm, to maximize the THz transmission. As the ML algorithm uses independent training data, the result does not depend on the initial conditions.
The device has an ultra-compact size of 500 µm × 500 µm. To the best of our knowledge, our power splitter has the smallest footprint per wavelength. The device can be fabricated using conventional semiconductor patterning techniques. The simulated insertion loss of each port is less than 4 dB over an operating bandwidth of 50 µm centered at 300 µm.
2. Problem statement and design approach
We aim to design an ultra-compact THz 50:50 power splitter. We chose a 1 × 2 power splitter design with one input waveguide, two output waveguides, and a square slab. We chose silicon (Si) as the device material to make the design compatible with current fabrication technology. Each 100 µm wide Si waveguide is connected to the 500 µm × 500 µm Si slab where device design begins. The distance between the output ports is 150 µm. Figure 1 shows the Si slab design region discretized into 20 × 20 pixels. Each pixel is a 25 µm × 25 µm square and can either have a circular etched air hole (radius 9 µm) at the center, or not. The circular void is much easier to fabricate than a square void which has sharp corners. The two possible pixel states, unetched (n = nsi = 3.415) [19] and etched (n = nair = 1), can be represented as “0” and “1”, respectively. The Si slab design region can then be described as a 20 × 20 binary matrix. In addition, the minimum feature size of the device (hole diameter of 18 µm) is easily fabricated using conventional processes.
Fig. 1. Schematic of a 1 × 2 power splitter with randomly generated etch holes (white). The footprint of 500 µm × 500 µm is discretized into 20 × 20 pixels.
We generated training data for the ML algorithm using the finite difference time domain (FDTD) method. The input mode was the fundamental TM mode and the simulation was conducted over the spectral range from 275 µm to 325 µm. The training data were 40,000 randomly generated photonic structures with different hole positions and their transmission responses in the wavelength range 275 µm - 325 µm with 100 spectral points. As our proposed device is a 50:50 power splitter, the structure must be symmetric about the y-axis with equal spectra at port 1 (T1) and port 2 (T2). Therefore, each training data is reduced to a 20 × 10 binary matrix and its corresponding transmission spectrum at port 1 (T1). In the simulation, we set the environment to 2D FDTD and symmetry boundary for time-saving. We used commercially-available software Lumerical FDTD on a 2 Intel Xeon CPU with 2.5-GHz clock speed and 383 GB RAM. The simulations could process 10,000 data/days, so the complete FDTD simulations took 4d.
We used a perceptron-like algorithm to maximize the broadband transmission [20]. The algorithm has two properties [21]: additive update features of the perceptron [22] and a reward system to reinforce learning [23]. The algorithm consists of two phases: training and inference. In the training phase, ML calculates the reward of the training data. To maximize the transmission of broadband light, the reward is defined as follows:
where, min(T1) is the minimum transmission of one random structure across the wavelength band, and Tmin and Tmax are the minimum and maximum of all min(T1) across the training dataset (Fig. 2 (a)). If the reward is set to the average transmission, rather than the minimum transmission, the uniformity of the transmission decreases and a peak may occur. Therefore, we defined the minimum value of transmission as the reward and normalized using the overall minimum and maximum values. We didn’t consider the reflection term as maximizing transmission also reduces reflection. This also decreases the complexity of the problem by reducing the number of features the algorithm considers.Fig. 2. (a) Transmission spectrum of random photonic structure in training data and it’s minimum transmission value. (b) Histogram of calculated min(T) from all 40,000 training data generated by numerical method.
Next, the binary matrices of training data and their corresponding reward are combined by multiplying them and the resulting matrix is accumulated in a summation matrix. In the inference phase, the accumulated rewards for each pixel in the summation matrix are normalized by subtracting the minimum value and activated with an average unit step function to determine the final design that maximizes the reward. If the element of the summation matrix is above average, the algorithm infers “1”, and if the element is below average “0” is inferred. In this way, ML determines the final 20 × 10 binary matrix photonic structure.
3. Results and discussion
The ML algorithm produced a binary matrix that described the hole positions. We then ran a 3D FDTD simulation to check the transmission response of our 1 × 2 power splitter (Fig. 3). During the 3D FDTD simulations, we used SiO2 material as the supporting substrate and a surrounding air medium around the waveguide. The refractive index of SiO2 substrate was 1.956 [19]. A fundamental TM mode was launched into the input Si waveguide with a wavelength range 275–325 µm. In addition, perfectly matched boundary conditions are used to absorb outgoing waves from the computational region.
Fig. 3. Final designed photonic structure. (a) 3D and (b) top view of the designed power splitter. (c) The fundamental TM mode launched into the input waveguide of the power splitter.
The performance of the power splitter is defined by its insertion loss and the distribution of input power across the two output waveguides. The calculated insertion loss of our device is shown in Fig. 4(a). The measured spectra of T1 and T2 are identical as expected from the symmetry. The insertion loss of each port is less than 4 dB over the operating bandwidth. As we designed an all-dielectric power splitter instead of using metal, there is some insertion loss. One of the biggest parts of its loss is the loss to the SiO2 substrate because in the THz region the refractive index of SiO2 is much bigger than in the infrared region. So, insertion loss can be further improved by using low refractive index material such as PDMS [24,25,26]. The back reflection at the input port is -16 dB at the central wavelength and the average back reflection is about -19 dB across the full range (Fig. 4(b)). Very low back reflection was achieved despite not including the reflection term in the reward. The calculated electric field intensity distribution for the device is shown in Fig. 4(c). In this work, as our proposed device is a 50:50 power splitter, we used a 20 × 10 binary matrix which describes only half of the design area and their corresponding transmission spectrum (T1) as training data. However, the algorithm we used in this work can be used for power splitters with other ratios if we use the training data with a 20 × 20 binary matrix which describes the entire design area and their corresponding transmission spectrum at output port 1 (T1) and port 2 (T2). In this case, the reward function has to be changed to consider both output spectrums and their split ratio [27].
Fig. 4. (a) Insertion loss of the power splitter at the output ports. (b) Reflection spectrum at the input port. (c) Electric field intensity distribution of 300 µm at z =0 plane. The photonic structure region is indicated by the white line.
Due to fabrication imperfection, fabricating the micro-size holes with exactly the designed size is challenging. So, to analysis about the fabrication tolerance of the designed power splitter, we run simulations with ±500 nm errors (up to 5%) in each hole diameter (Fig. 5). The insertion losses and back reflections are slightly changed with errors. However, averaged differences between non-error state are only 0.16 and 0.09 dB for +500 nm and -500 nm, respectively. Also, fluctuations of insertion losses are only within 0.5 dB, and the averaged back reflections are still lower than -17.7 dB. So, the simulation results suggest an acceptable fabrication tolerance of the designed power splitter in ±500 nm variation of diameter [28].
Fig. 5. Simulation results with fabrication errors (a) Insertion losses of the power splitters at the output ports. (b) Reflection spectrums at the input ports.
Unlike conventional optimization algorithms such as evolutionary algorithms, the algorithm we used in this work uses independent data in the training phase instead of iterating in the initial conditions. All processes can be easily parallelized when gathering training data as the result does not vary depending on the initial conditions. Hence, designers can utilize multiple computers which can be avoided because of diversity issues that are mainly encountered in other optimization algorithms for time-saving [21]. Also, updating the designed result using other optimization algorithms such as DBS after running ML algorithm with fewer data can be a good way to save more time and increase device performance [29]. Moreover, unlike artificial neural networks, our algorithm doesn’t need time to learn the model, because it learns with a reward.
4. Conclusion
We design an ultra-compact THz 50:50 power splitter using a ML algorithm and numerically calculate the transmission response using a 3D FDTD method. Our device has a small footprint of 500 µm × 500 µm. The calculated results show high performance, with an insertion loss less than 4 dB over an operating bandwidth of 50 µm centered at 300 µm and an average back reflection of -19 dB. We believe that the proposed power splitter has great potential for practical applications in compact THz wireless communication systems. Furthermore, the ML algorithm we implemented can be used for the inverse design of various photonic devices based on digitalized subwavelength patterns.
Funding
Samsung (IO201210-08035-01, SRFC-IT1802-01).
Acknowledgments
This work was supported by Samsung Electronics Co., Ltd. (IO201210-08035-01) and Samsung Research Funding and Incubation Center of Samsung Electronics, Korea (SRFC-IT1802-01) and LG innotek Co. Ltd.
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.
References
1. I. F. Akyildiz, J. M. Jornet, and C. Han, “Terahertz band: next frontier for wireless communications,” Phys. Commun. 12, 16–32 (2014). [CrossRef]
2. A. J. Seeds, H. Shams, M. J. Fice, and C. C. Renaud, “TeraHertz Photonics for Wireless Communications,” J. Lightwave Technol. 33(3), 579–587 (2015). [CrossRef]
3. T. Kürner and S. Priebe, “Towards THz communications - Status in research, standardization and regulation,” J. Infrared, Millimeter, Terahertz Waves 35(1), 53–62 (2014). [CrossRef]
4. S. Cherry, “Edholm’s law of bandwidth,” IEEE Spectrum 41(7), 58–60 (2004). [CrossRef]
5. J. Federici and L. Moeller, “Review of terahertz and subterahertz wireless communications,” J. Appl. Phys. 107(11), 111101 (2010). [CrossRef]
6. K. C. Huang and Z. Wang, “Terahertz terabit wireless communication,” IEEE Microwave 12(4), 108–116 (2011). [CrossRef]
7. H. Elayan, O. Amin, R. M. Shubair, and M. Alouini, “Terahertz communication: The opportunities of wireless technology beyond 5G,” in International Conference on Advanced Communication Technologies and Networking (CommNet) (2018), pp. 1–5
8. I. F. Akyildiz, J. M. Jornet, and C. Han, “TeraNets: ultra-broadband communication networks in the terahertz band,” IEEE Wireless Commun. 21(4), 130–135 (2014). [CrossRef]
9. J. Xie, X. Zhu, X. Zang, Q. Cheng, L. Chen, and Y. Zhu, “Terahertz integrated device: high-Q silicon dielectric resonators,” Opt. Mater. Express 8(1), 50–58 (2018). [CrossRef]
10. C. Gui and J. Wang, “Wedge hybrid plasmonic THz waveguide with long propagation length and ultra-small deep-subwavelength mode area,” Sci. Rep. 5(1), 11457 (2015). [CrossRef]
11. William S. C. Chang, Fundamentals of guided-wave optoelectronic devices (Cambridge University Press, 2010), Chap. 6.
12. Z. Guo, X. Nie, F. Shen, H. Zhou, Q. Zhou, J. Gao, and K. Guo, “Actively Tunable Terahertz Switches Based on Subwavelength Graphene Waveguide,” Nanomaterials 8(9), 665 (2018). [CrossRef]
13. F. Zhang, K. Song, and Y. Fan, “New 2D diffraction model and its applications to terahertz parallel-plate waveguide power splitters,” Sci. Rep. 7(1), 41726 (2017). [CrossRef]
14. L. Lu, M. Zhang, F. Zhou, and D. Liu, “An ultra-compact colorless 50: 50 coupler based on PhC-like metamaterial structure,” in Proceedings of 2016 Optical Fiber Communications Conference and Exhibition (OFC). IEEE, 1–3 (2016).
15. J. C. C. Mak, C. Sideris, J. Jeong, A. Hajimiri, and J. K. S. Poon, “Binary particle swarm optimized 2 × 2 power splitters in a standard foundry silicon photonic platform,” Opt. Lett. 41(16), 3868–3871 (2016). [CrossRef]
16. K. Wang, X. Ren, W. Chang, L. Lu, D. Liu, and M. Zhang, “Inverse design of digital nanophotonic devices using the adjoint method,” Photonics Res. 8(4), 528–533 (2020). [CrossRef]
17. P. A. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightwave Technol. 14(10), 2286–2293 (1996). [CrossRef]
18. M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9(1), 1368 (2019). [CrossRef]
19. D. Smith, E. Shiles, and M. Inokuti, Handbook of Optical Constants of Solids, D. Palik, ed. (Academic, 1988), pp. 568–763.
20. C. Latifoglu, “Binary matrix guessing problem,” arXiv preprint arXiv:1701.06167 (2017).
21. M. Turduev, E. Bor, C. Latifoglu, I. H. Giden, Y. S. Hanay, and H. Kurt, “Ultra-compact photonic structure design for strong light confinement and coupling into nano-waveguide,” J. Lightwave Technol. 36(14), 2812–2819 (2018). [CrossRef]
22. F. Rosenblatt, “The perceptron: A probabilistic model for information storage and organization in the brain,” Psychological Rev. 65(6), 386–408 (1958). [CrossRef]
23. M. V. Otterlo and M. Wiering, Reinforcement Learning and Markov Decision Processes (Springer, 2012), pp. 3–42.
24. K. Fan, J. Y. Suen, X. Liu, and W. J. Padilla, “All-dielectric metasurface absorbers for uncooled terahertz imaging,” Optica 4(6), 601–604 (2017). [CrossRef]
25. C. C. Nadell, B. Huang, J. M. Malof, and W. J. Padilla, “Deep learning for accelerated all-dielectric metasurface design,” Opt. Express 27(20), 27523–27535 (2019). [CrossRef]
26. T. Ma, Q. Huang, H. He, Y. Zhao, X. Lin, and Y. Lu, “All-dielectric metamaterial analogue of electromagnetically induced transparency and its sensing application in terahertz range,” Opt. Express 27(12), 16624–16634 (2019). [CrossRef]
27. Y. Xie, T. Huang, Q. Ji, M. Yang, J. Wang, X. Tu, Z. Cheng, G. Xu, Q. Wei, Y. Wu, and P. P. Shum, “Design of an arbitrary ratio optical power splitter based on a discrete differential multiobjective evolutionary algorithm,” Appl. Opt. 59(6), 1780–1785 (2020). [CrossRef]
28. H. Ma, J. Huang, K. Zhang, and J. Yang, “Inverse-designed arbitrary-input and ultra-compact 1× N power splitters based on high symmetric structure,” Sci. Rep. 10(1), 1–7 (2020). [CrossRef]
29. H. X. Chen, H. Jia, T. Wang, and J. H. Yang, “A Gradient-Oriented Binary Search Method for Photonic Device Design,” J. Lightwave Technol. 39(8), 2407–2412 (2021). [CrossRef]
