## Abstract

This article presents design methods for a transmissive metasurface antenna composed of four layers of meta-structures based on the deep neural network (DNN). Owing to the structural complexity as well as side effects such as couplings among the adjacent meta-structures, the conventional design of metasurface unit cell strongly relies on the researcher’s intuition as well as time-consuming iterative simulations. A design method for a metasurface antenna unit cell with a size of a quarter wavelength operating at a frequency of 5.8GHz is presented. We describe two unique implementations for designing the target metasurfaces: 1) utilizing the inverse network 2) data augmentation by the forward network and a random search algorithm. With the usage of the two DNNs, the average transmittance of the unit cells is improved by about 0.024 than that of the unit cells designed by the conventional approach. This research invokes the application of DNN in designing antennas and other structures operating at radio frequency.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Metasurface antennas are emerging devices that can reradiate electromagnetic waves toward the desired directions [1,2], focus beam on a point [3,4], or split single beam into multi-beams [5,6] by reflecting or transmitting input beam from feeding antennas. The former and later mechanism can be categorized into reflective metasurface antennas (RMAs) [1,3,5] and transmissive metasurface antennas (TMAs) [2,4,6], respectively. The metasurface is a two-dimensional version of metamaterials that can control the magnitude and phase of the electromagnetic wave utilizing engineered artificial structures [1–9]. Based on the merits of the metasurface, e.g., ultra-thin thickness, light weight, and simply controllable reflection or transmission phases using varactor or PIN diodes [10,11], both of RMA and TMA attracts great attention.

Between two schemes, TMA has a great advantage of high antenna gain because it does not suffer from the blockage loss by locating the feeding antenna at the opposite side of the propagating direction relative to the metasurface [2,4,6,12]. Typically, the unit cells of the conventional TMA are composed with multi-layer metasurfaces to achieve the phase modulation from 0° to 360° [2,4,6,12]. In Lee, et al. [12], modified four-layer-Jerusalem-cross-shaped patches were proposed and designed by changing one of the geometric parameters. Owing to the limited control of the parameters considering design complexity, it is found that there remained incomplete unit cells that present slightly low transmittance. Therefore, there is still room for improvement of TMA in terms of optimizing the unit cells by tuning all of the geometric parameters. However, it is obvious that it requires a time-consuming iterative process to optimize the unit cell with various variables. To reduce the time and effort for the design, deep learning (DL) is adopted to find the optimal solution in an efficient way.

DL is a class of machine learning that uses DNNs [13]. DNNs are layers of computational neurons packed together to search the correlation between input and output. The neurons are connected to each other by parameters called weights. By going through a training process, the weights of the DNN is adjusted for the task at hand. Finding the appropriate set of hyper-parameters (activation function, layer number, etc $\cdots$) is critical in DL. The purpose of designing a metasurface is mostly focused on the structural information of the sub-wavelength structure that displays a certain optical property such as reflection / transmission spectrum [14,15], polarization conversion [16,17], and dual modes [18]. The process of tracing back from optical phenomena to structural variables is called inverse design in nanophotonics [19]. Several topics such as color generation [20–22], meta hologram [23,24], transmissive nano structures [25–28], and core-shell nano particles [29,30] have applied DL to their cases. As the nature of finding optimal solution is by iteratively running simulations relying on the researcher’s intuitions, the employment has reduced time and effort on the search [31–33]. In this paper, two fully connected networks; forward network (FN) and inverse network (IN) are adopted to design meta surfaces composing a transmissive metasurface antenna. FN predicts transmittance from the design parameters, like most simulation does. IN does the exact opposite and draw structural variables from transmittance. By the usage of IN, metasurface design for wanted transmittance can be deducted, and FN is used to compensate IN [25,26,29]. Both of FN and IN, accompanied by a random search algorithm and data augmentation technique, are used to inversely design unit cells that compose a transmissive metasurface antenna.

## 2. Results and discussions

#### 2.1 Design goal and setups

The unit cell structure of Lee, et al. [12] is shown in Fig. 1. Twelve types of unit cells composed with four layers of metasurfaces are designed to realize a transmissive metasurface antenna including 17 $\times$ 17 unit cell array. Both of the unit-cell size ($p$) and the gap ($h$) between the metasurfaces (Fig. 1) are 12.9 mm that is approximately a quarter of the wavelength at a frequency 5.8 GHz. The thickness of the metasurface ($t$) is 3.175 mm. In this paper, to improve the transmission coefficient ($S_{21}$) of the metasurface antenna unit cell with the same size of it, geometric parameters of the meta-structure $\alpha , w, L$ and $R$ indicated in Fig. 1 were controlled.

Four thousand data sets for FN and IN were generated using ANSYS HFSS software that is a commercial finite-element-method solver. A ratio of 8:1 was taken to divide the training data sets and the validation data sets. While training the DNNs with training sets, validation sets were used simultaneously to check the training process. This analysis was done by examining and comparing the loss graphs. To guarantee the function of the metasurface antenna, e.g., splitting a single beam into quad beams, it was found that twelve types of unit cells are needed for discretizing a continuous phase profile with a sufficient interval. The phase of transmission coefficient ($\angle S_{21}$) of each unit cell has to satisfy the value in the range from interval of 30°, i.e., −150°, −120°,…, 180°. In addition, the magnitude of the transmittance ($|S_{21}|^2$) has to be close to one. To obtain the geometric parameters of the unit cells satisfying the performance criteria above, we used IN that can predict them with the data set of $S_{21}$ values. To gain final security of the network, the twelve designs are tested with the HFSS software.

#### 2.2 Inverse network

A fully connected layer model was selected to build FN and IN. Various combinations of hyper-parameters were tested and the best combinations are listed on Table 1.

The loss function of IN is denoted by $loss_{IN}$. The loss is expressed as an L1 norm, which can be expressed as

where $N$ is the number of data, $\hat {y}$ the target label, and $y_{pred}$ the DNN’s predicted value. Without FN, IN cannot function well because of its input-output relationship. There can be multiple design choices for a certain set of $S_{21}$ values, which can confuse IN at the training stage. Therefore, a pre-trained FN is connected to the rear of IN to form a new network as shown in Fig. 2(a). The network sends the output of IN to FN as an input. Since the input and the output have to be the same, giving restrains to IN by comparing those will prevent IN from making false choices. The regulation is expressed as a loss function given to IN.By adding the new loss to the original loss of IN, IN double checks the labels at training stage and the problems of multiple design choices are resolved.

Parameter $\rho$ was set to 1/3 to regard that both of $loss_{IN}$ and $loss_{new}$ are on the same level of magnitude and to make the total loss below 1.

#### 2.3 Data augmentation with the forward network and random search algorithm

Although both of FN and IN were well established, the prediction using the networks was poor at $\angle S_{21}$ = −150°. By checking the data set, it was found that only 0.18% of the total data was assigned for the training of the feature $\angle S_{21}$ = −150°, which might cause under fitting for it. Therefore, it was needed to supply more data for the sparse data set to build a concrete network. Data augmentation using FN integrated with a random search algorithm successfully enhanced IN’s performance. With a unique iterational process, reproduction of data are done.First with the original FN (black dashed arrow in Fig. 2) and ten thousand randomly distributed design parameters ($\alpha , w, L, R$), the transmittance values ($S_{21}$) are derived (Fig. 2(b)). From the newly generated data set, twenty data that are closest to the objective ‘$\angle S_{21}$ = −150° , $|S_{21}|^2$ = 1’ are drawn and added to the training data set (red dashed arrow in Fig. 2). Lastly, a new set of FN and IN is trained from the enlarged training set. This is a one cycle of random search based data augmentation that we have come up with. The cycle is repeated until FN is ensured from the under-fitting issue, i.e. when its loss graph shows convergence to $\angle S_{21}$ = −150° data.

Figure 3(a) represents the minimal value of FNs’ validation loss graph by each iteration. The loss increases at first because the networks are not familiar with $\angle S_{21}$ = −150° data. After $4^{th}$ iteration, the loss decreases, showing evidence of adaptation to the new data. The $5^{th}$ FN model (Fig. 3(b)) is chosen as the best fit and is used to derive IN. The result is plotted in Fig. 4. Overall IN outputs are better than the results from the reference [12], especially at $\angle S_{21}$ = −150°. $|S_{21}|^2$ at $\angle S_{21}$ = −150° has increased 11.6%. With the collected data sets, we were able to realize a transmissive metasurface antenna with a high transmittance efficiency.

#### 2.4 Discussion

A unit cell analysis using ANSYS HFSS simulator takes three minutes on average using Intel Xeon E5-2698 v4 CPU. The search for better transmittance may take a long time because repetitive simulations have to be done until the optimization is completed based on the researcher’s intuitions and predictions. Also, it is unsure whether the conventional unit cell design process, which relies on intuition, found the optimal solution or not. On the other hand, the process of utilizing DL could circumvent those problems because DL is an end-to-end procedure that can neglect all mid-processes by directly finding the correlation between the input and the output. In addition, it is found that the performances of the unit cells are improved compared with those reported in the previous paper [12]. To verify the improvement of the transmittance of the twelve unit cells of which the phases of the transmission coefficients are ranging from −150° to 180° with an interval of 30° the average of the transmittance was evaluated. As a result, the averaged transmittances were calculated as 0.902 and 0.926 for the unit cells of the reference paper [12] and those designed by DNNs, respectively. From the calculation, it is verified that the averaged transmittance is increased about 0.024 by applying the proposed DNNs. In Table 2, the design variable results of our DNN and the results from the reference are displayed.

In the Ref. [12], it is studied that the mutual capacitances induced by the coupling among the adjacent unit cells affect the phase of the transmission coefficient. Therefore, the phase error may be non-negligible if different types of unit cells are assembled into a metasurface antenna. Owing to the side effect, the increment of the transmittance of the unit cells did not lead to abrupt improvement of the aperture efficiency of the metasurface antenna assembled with the individual unit cells. The development of DNNs to find an optimal arrangement of metasurface antenna unit cells is remained as a future work.

## 3. Conclusion

We successfully demonstrated the accuracy and efficiency of two DNNs, FN and IN, developed for inverse design of transmissive metasurface antenna unit cells. Since scarcity of training data at $\angle S_{21}$ = −150° was hindering IN, data augmentation with FN connected to a random search algorithm was done to provide more data. After adding the augmented data to the training data set, the best FN was selected by observing the loss of each FN. The fifth FN showed converged results with the additional data, therefore, it was used to build IN. Compared with the previous result [12], the averaged transmittance of the unit cells was increased by 0.024. We believe that our study on the inverse design of the transmissive metasurface antenna will lead to the solution of other complicated antenna design problems and beyond.

## Funding

Agency for Defense Development (Defense Challengeable Future Technology Program); National Research Foundation of Korea (NRF-2017H1A2A1043322).

## Acknowledgments

This research has been supported by a grant-in-aid of HANWHA SYSTEMS based on the Defense Challengeable Future Technology Program of ADD. S.S. acknowledges the NRF Global Ph.D. fellowship (NRF-2017H1A2A1043322) funded by the Ministry of Education of the Korean government.

## 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. **B. H. Fong, J. S. Colburn, J. J. Ottusch, J. L. Visher, and D. F. Sievenpiper, “Scalar and tensor holographic artificial impedance surfaces,” IEEE Trans. Antennas Propag. **58**(10), 3212–3221 (2010). [CrossRef]

**2. **T. Cai, G.-M. Wang, J.-G. Liang, Y.-Q. Zhuang, and T.-J. Li, “High-performance transmissive meta-surface for *c*-/*x*-band lens antenna application,” IEEE Trans. Antennas Propag. **65**(7), 3598–3606 (2017). [CrossRef]

**3. **C. Saeidi and D. van der Weide, “A figure of merit for focusing metasurfaces,” Appl. Phys. Lett. **106**(11), 113110 (2015). [CrossRef]

**4. **J.-G. Lee, T.-S. Kwon, and J.-H. Lee, “Beam pattern reconfigurable circularly polarized transmitarray antenna by rearrangement of sources,” Microw Opt Technol Lett **61**(4), 999–1003 (2019). [CrossRef]

**5. **M. Karimipour and N. Komjani, “Holographic-inspired multibeam reflectarray with linear polarization,” IEEE Trans. Antennas Propag. **66**(6), 2870–2882 (2018). [CrossRef]

**6. **G. Liu, M. R. D. Kodnoeih, K. T. Pham, E. M. Cruz, D. Gonzalez-Ovejero, and R. Sauleau, “A millimeter-wave multibeam transparent transmitarray antenna at ka-band,” Antennas Wirel. Propag. Lett. **18**(4), 631–635 (2019). [CrossRef]

**7. **N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science **334**(6054), 333–337 (2011). [CrossRef]

**8. **M. Abdelsalam, A. M. Mahmoud, and M. A. Swillam, “Polarization independent dielectric metasurface for infrared beam steering applications,” Sci. Rep. **9**(1), 10824–7 (2019). [CrossRef]

**9. **K. K. Katare, A. Biswas, and M. J. Akhtar, “Microwave beam steering of planar antennas by hybrid phase gradient metasurface structure under spherical wave illumination,” J. Appl. Phys. **122**(23), 234901 (2017). [CrossRef]

**10. **K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, and S. Zhang, “A reconfigurable active huygens’ metalens,” Adv. Mater. **29**(17), 1606422 (2017). [CrossRef]

**11. **A. Clemente, L. Dussopt, R. Sauleau, P. Potier, and P. Pouliguen, “Wideband 400-element electronically reconfigurable transmitarray in x band,” IEEE Trans. Antennas Propag. **61**(10), 5017–5027 (2013). [CrossRef]

**12. **C.-H. Lee, S. W. Chi, S.-G. Lee, and J.-H. Lee, “Low profile quad-beam circularly polarised antenna using transmissive metasurface,” IET Microwaves, Antennas & Propagation **13**(10), 1690–1698 (2019). [CrossRef]

**13. **I. Goodfellow, Y. Bengio, and A. Courville, *Deep Learning* (MIT Press, 2016). http://www.deeplearningbook.org.

**14. **B. Yang, W. Liu, Z. Li, H. Cheng, D.-Y. Choi, S. Chen, and J. Tian, “Ultrahighly saturated structural colors enhanced by multipolar-modulated metasurfaces,” Nano Lett. **19**(7), 4221–4228 (2019). [CrossRef]

**15. **H.-E. Lee, H.-Y. Ahn, J. Mun, Y. Y. Lee, M. Kim, N. H. Cho, K. Chang, W. S. Kim, J. Rho, and K. T. Nam, “Amino-acid-and peptide-directed synthesis of chiral plasmonic gold nanoparticles,” Nature **556**(7701), 360–365 (2018). [CrossRef]

**16. **M. A. Ansari, I. Kim, D. Lee, M. H. Waseem, M. Zubair, N. Mahmood, T. Badloe, S. Yerci, T. Tauqeer, and M. Q. Mehmood, “A spin-encoded all-dielectric metahologram for visible light,” Laser & Photonics Reviews **13**(5), 1900065 (2019). [CrossRef]

**17. **M. Kim, K. Yao, G. Yoon, I. Kim, Y. Liu, and J. Rho, “A broadband optical diode for linearly polarized light using symmetry-breaking metamaterials,” Adv. Opt. Materials **5**(19), 1700600 (2017). [CrossRef]

**18. **G. Yoon, D. Lee, K. T. Nam, and J. Rho, “crypto-display in dual-mode metasurfaces by simultaneous control of phase and spectral responses,” ACS Nano **12**(7), 6421–6428 (2018). [CrossRef]

**19. **S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics **12**(11), 659–670 (2018). [CrossRef]

**20. **J. Baxter, A. C. Lesina, J.-M. Guay, A. Weck, P. Berini, and L. Ramunno, “Plasmonic colours predicted by deep learning,” Sci. Rep. **9**(1), 8074 (2019). [CrossRef]

**21. **I. Sajedian, T. Badloe, and J. Rho, “Finding the best design parameters for optical nanostructures using reinforcement learning,” *arXiv preprint arXiv:1810.10964* (2018).

**22. **Z. Huang, X. Liu, and J. Zang, “The inverse design of structural color using machine learning,” Nanoscale **11**(45), 21748–21758 (2019). [CrossRef]

**23. **I. Sajedian, H. Lee, and J. Rho, “Double-deep q-learning to increase the efficiency of metasurface holograms,” Sci. Rep. **9**(1), 10899–8 (2019). [CrossRef]

**24. **H. Ren, W. Shao, Y. Li, F. Salim, and M. Gu, “Three-dimensional vectorial holography based on machine learning inverse design,” Sci. Adv. **6**(16), eaaz4261 (2020). [CrossRef]

**25. **I. Malkiel, M. Mrejen, A. Nagler, U. Arieli, L. Wolf, and H. Suchowski, “Plasmonic nanostructure design and characterization via deep learning,” Light: Sci. Appl. **7**(1), 60 (2018). [CrossRef]

**26. **D. Liu, Y. Tan, E. Khoram, and Z. Yu, “Training deep neural networks for the inverse design of nanophotonic structures,” ACS Photonics **5**(4), 1365–1369 (2018). [CrossRef]

**27. **S. So and J. Rho, “Designing nanophotonic structures using conditional deep convolutional generative adversarial networks,” Nanophotonics **8**(7), 1255–1261 (2019). [CrossRef]

**28. **Z. Liu, D. Zhu, S. P. Rodrigues, K.-T. Lee, and W. Cai, “Generative model for the inverse design of metasurfaces,” Nano Lett. **18**(10), 6570–6576 (2018). [CrossRef]

**29. **S. So, J. Mun, and J. Rho, “Simultaneous inverse design of materials and structures via deep learning: demonstration of dipole resonance engineering using core–shell nanoparticles,” ACS Appl. Mater. Interfaces **11**(27), 24264–24268 (2019). [CrossRef]

**30. **J. Peurifoy, Y. Shen, L. Jing, Y. Yang, F. Cano-Renteria, B. G. DeLacy, J. D. Joannopoulos, M. Tegmark, and M. Soljačić, “Nanophotonic particle simulation and inverse design using artificial neural networks,” Sci. Adv. **4**(6), eaar4206 (2018). [CrossRef]

**31. **S. So, T. Badloe, J. Noh, J. Rho, and J. Bravo-Abad, “Deep learning enabled inverse design in nanophotonics,” Nanophotonics **9**(5), 1041–1057 (2020). [CrossRef]

**32. **W. Ma, Z. Liu, Z. A. Kudyshev, A. Boltasseva, W. Cai, and Y. Liu, “Deep learning for the design of photonic structures,” Nat. Photonics **15**(2), 77–90 (2021). [CrossRef]

**33. **J. Jiang, M. Chen, and J. A. Fan, “Deep neural networks for the evaluation and design of photonic devices,” Nat. Rev. Mater **2020**, 1–22 (2020). [CrossRef]