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.

 

Fig. 1. Schematics of metasurface antenna and its unit cell. Left hand-side inset: the magnitude of transmission coefficient $|S_{21}|$ and phase of transmission coefficient $\angle S_{21}$ determined by geometric parameters $\alpha , w, L$ and $R$ of the unit cell. The unit cell size $p$ and gap between metasurface layers $h$ are 12.9mm, and the thickness of the metasurface layer $t$ is 3.175 mm.

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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.

Table 1. List of best hyper-parameter sets of FN and IN. Learning rate of the network is expressed as lr in the table.

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

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.

 

Fig. 2. Schematics of proposed DNN (a) IN and FN connected together. (b) FN with a random search algorithm producing training data for sparse region. Data for $\angle S_{21}$ = −150 ° are made and sent to the training data set.

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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.

 

Fig. 3. (a) Loss values of FNs. Original model and other models state FN trained by original training data set and FN trained by original added by augmented training data set, respectively. (b) Loss curve of model 5 on training iterations by log scale. The reticle point on the loss graph indicates the blue box in (a)

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Fig. 4. Comparison of inverse & forward network results with reference results [12]. Verification is done by ANSYS HFSS software.

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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.

Table 2. Design variables predicted by the DNN and reference results

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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.