1. Introduction

The concept of achromatism is originated from optics, and in a diffractive optical element, the beam deflection angle and the focal length varied with the variation of wavelength [1]. With the development of all kinds of materials, the achromatic problems have been extended to frequency bands beyond optics [2–4]. Frequency dispersion is a key characteristic of electromagnetic materials, which always plays an important role in communication, detection, imaging, displaying etc. [5–7] Frequency dispersion will narrow working bands, therefore, a great many researches have been conducted to decrease the frequency dispersion and expand working band. Many previous demonstrations of dispersion-engineered meta-lenses focused on reducing the focal-length shift across continuous bandwidths, including unique chromatic phase shift behavior of polymer nanostructure [8–10], judicious design of nanofins on a surface [11,12], and dielectric nano-posts [13–15].

According to the above researches, metasurfaces are of great significance in the dispersion engineering. Dispersion engineering is of great value not only in optics but also in microwave frequency band. Metasurface is the 2D counterpart of metamaterial, and it can manipulate electromagnetic waves accurately [16,17]. The inhomogeneous arrays composed of different phase units can achieve the phase gradient metasurface (PGM) which can be analyzed by the Generalized Snell's law [17]. The PGMs exhibited a flexible manipulation ability of the electromagnetic waves and had wide application in optical vortexes, light bending, anomalous reflection and focusing [18–22]. Chromatism is commonly exists in metamaterials/metasurfaces, especially in PGMs. In general, resonant phase and Pancharatnam−Berry (PB) phase are widely used in the design of PGMs. However, the strong dispersion makes the working bandwidth narrow [23,24]. In previous work, the broadband dispersion engineering has been achieved by spoof surface plasmon polaritons (SSPP) [25]. By feat of the peculiarity of weak dispersion region, SSPP employed the accumulation of propagation phases to obtain the phase compensation and achieve the dispersion engineering further. However, a drawback of this method is the thickness of the SSPP structure is too thick to be miniaturized. Fortunately, metasurface can modulate the electromagnetic wave with the abrupt phase change produced by strong resonances or spatial orientations rather than the accumulation of propagation phases, which provides some novel properties to control phase, amplitude, polarization of electromagnetic waves in space [26–30]. However, strong resonance properties make the phase changed suddenly, which may cause the variation of phase gradient. And with the phase changed along the frequency, the phase gradient at different frequency also will be dramatically changed while frequency changed. Therefore, by means of phase compensation, the achromatic abnormal reflection can be realized by designing the meta-atoms with specified phase gradient. However, the customization of functions needs to design the exquisite structure, which will greatly increase the complexity of design.

 

Fig. 1. The sketch map of this work.

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In recent years, the rapid development of machine learning provides new ideas for solving problems in all walks of life. In the metasurface field, some advanced works about designing metasurface with machine learning have been proposed. The design of metasurface with machine learning can be roughly divided into two categories: supervised learning and unsupervised learning. In the supervised learning design, the maps of metasurface structures and spectrums were collected for training artificial neural network or deep learning network [31–40]. In the unsupervised learning design, generative adversarial network (GAN) is applied in inverse design [41–43]. All of these works provide the novel ideas of metasurface design. Similarly, machine learning also can be applied in dispersion engineering.

In this paper, we proposed a Self-Organizing-Map-Neural-Network-prepositioned Greedy Algorithm to design an ultrathin broadband achromatic metasurface deflectors (AMD). The sketch map is shown as the Fig. 1, including neural network design, greedy algorithm search and functional realization. We firstly collected 1,000 sample data to establish the Self-Organizing Map (SOM) Neural Network as a prepositive model to cluster the meta-atoms. With the trained SOM, the dataset can be divided into 100 class. The samples in 100 class are inhomogeneous distribution, meaning that there are many similar samples in the dataset. Reducing similar units in sample collection can improve search efficiency and diversity of data. The trained SOM can classify the samples, therefore the meta-atoms that have already gathered a large number of similar samples will be effectively filtered out by the prepositive SOM network. And then additional 1,000 sample data are generated randomly with SOM similarity discrimination to ensure the samples uniform distribution. On this basis, we used the 2,000 sample data to establish a searching map whose points and sides represent the meta-atom and difference of reflection phase respectively. In the searching map, the Greedy Algorithm is applied to find the meta-atoms with specified phase difference. Finally, we found 4 units with linear variation phase gradient to assemble an AMD for validation by simulation and measurement. The theory, simulation and measurement results are well consistent, in which the angle of reflection is constant 22° in 9.5GHz – 10.5GHz. More importantly, the design method we proposed can be extended in other functional design, and the ultrathin AMD we designed has potential application value in mobile network base station communications, satellite communications and others.

2. Theory and design

2.1 Achromatic abnormal reflection condition

The abnormal reflection can be analysed based on generalized Snell’s law [17,44–46]:

According to the Eq. (2), the influencing factors of the reflection angle is ΔΦ, Δx and f. The achromatic reflection demands that θr keeps constant despite frequency variation. In Eq. (2), c, sin(θr) and △x are constants, therefore the Eq. (2) can be transformed as Eq. (3):

2.2 SOM network design

 

Fig. 2. The preparation and construction of SOM network: (a) The geometrical parameters of meta-atoms. (b) The structure of SOM network. (c) SOM neighbor weight distances. (d) The distribution of samples in different clusters. (e) The collection of searching map via SOM. (f) The distribution of samples in different categories.

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To design AMD, we firstly customized meta-atom which can be represented as a coding sequence. The geometrical parameters of meta-atom are shown in Fig. 2(a), in which L = 10 mm, d = 1 mm, u = 0.5 mm, h = 2 mm and t = 0.017 mm. This structure contains 3 layers: the bottom layer is copper reflection sheet (with a conductivity of 5.8×10⁷ S/m); the middle layer is F4B dielectric substrate (with a dielectric constant 2.65 and a loss tangent 0.001); and the top layer is the meta-atom pattern (‘1’ means copper and ‘0’ means vacuum). The pattern can be simplified as a 64 bit coding sequence. The coding sequence is generated randomly, and the meta-atom is fabricated and simulated by CST microwave studio. The meta-atoms and simulated reflection phases are recorded, and a total of 1,000 sample data are collected as dataset.

As an unsupervised learning algorithms, SOM neural network is an effective tool for analysis of multidimensional data [47,48]. The SOM can be applied for clustering while preserving topology data structure [49]. It's worth noting that the meta-atom we designed can be represented as a 64-bit coding sequence and the neurons in the SOM can adjust their weights by competing with input vector. The coding sequence only contains 0 and 1, which can be quickly compared by adjusting the weights. Therefore, SOM neural network provides a reasonable solution for the clustering of meta-atom coding sequences.

A SOM neural network was established in the Fig. 2(b) including 10×10 neurons in the SOM layer. All of 1,000 sample data are clustered in 100 classes through SOM. The ‘trainbu’ function is applied to train this network with weight and bias learning rules with batch updates. The learning algorithm is ‘learnsomb’ that each neuron’s new weight vector is the weighted average of the input vectors [50]. The performance of training is illustrated in Fig. 2. Figure 2(c) shows the SOM neighbor weight distances and Fig. 2(d) exhibits the amount of samples in the 100 classes, where a large number of units are clustered in a few classes. From the training results, we can conclude that the sample is not evenly distributed which will cause that many similar structures and similar electromagnetic parameters are collected. These 1,000 random sample data are lack of low diversity. However, the trained SOM is appropriate to solve this problem. The diagram of subsequent meta-atom collection is illustrated in Fig. 2(e). The coding sequences are generated randomly and the subordination of their clusters is identified. If the coding sequence is identified to be subordinated to a cluster with many samples, a new sequence will be generated for the data collection. In this way, we can get more different samples to enrich the diversity of collected data, which provides convenience and diversity for subsequent searches. On this basis, we compare different distributions of initial samples, random samples and SOM samples, which is shown in Fig. 2(f). It is obvious from Fig. 2(f) that the sample distribution is more uniform than the others.

2.3 Searching map via Greedy Algorithm

 

Fig. 3. The exploration of meta-atoms via Greedy Algorithm in searching map: (a) Schematic diagram of searching map. (b) Greedy Algorithm searching strategy. (c) Reflection phase of the found units. (d) Phase different of the found units.

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Base on the trained SOM network, a total of 2,000 meta-atoms were collected as the original data to establish a searching map whose points and sides are defined as coding sequence and phase difference respectively. The schematic diagram of searching map is shown in Fig. 3(a). And then, the required meta-atoms can be explored in this searching map by Greedy Algorithm. Figure 3(b) illustrates the progress of Greedy Algorithm. According to the AMD condition, the reflection phase gradient and frequency are correlated, therefore we should search the meta-atoms with phase gradients that vary linearly. In the first iteration, two units with specified phase gradient are found in the searching map. Subsequently, the searching for a next unit with specified phase difference is performed based on the found unit. The iteration is repeated until all the units are found. In this work, we found four meta-atoms whose phase gradients vary linearly with frequency. Figure 3(c) shows the reflection phase curves of the found units, where the phase varied with the frequency. In order to observe the phase difference more clearly, the phase difference between the adjacent units is calculated in Fig. 3(d). From the Fig. 3(d), we can conclude that the phase difference varies linearly with frequency. Since greedy algorithm is used, the error increases gradually with the iteration of the unit. The mean error between the unit-1 and unit-2 is about 0°, the mean error between the unit-2 and unit-3 is about 5°, and the mean error between the unit-3 and unit-4 is about 10°. The overall mean error is about 5°, which means that we have found a comparatively excellent solution set via the greedy algorithm. And then the found units are assembled for a metasurface to verify our design. Greedy algorithm is a locally optimal algorithm which has the advantage of simplicity and rapidity. In this work, we try to approximate the global optimal solution by using greedy algorithm, and find a feasible solution in limited time. Although the feasible solution is not necessarily the global optimal solution, it also provides a better feasible solution for the next design.

3. Simulation and experiment

 

Fig. 4. Metasurface design and simulation. (a) Metasurface design. (b) The 3D far-field results of AMD. (c) The cross profile of far-field result in θ = 0°. (d) The reflection angle and main lobe magnitude with frequency variation.

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To further validate our design, we fabricated an abnormal reflection metasurface consisting of the four meta-atoms to achieve achromatism. The design of metasurface is shown in Fig. 4(a), in which the side length of the metasurface is 240 mm, the unit length is 20 mm, and it contains 3 phase gradient cycles. In order to verify the achromatic reflection of the metasurface, full-wave EM simulations were carried out using the time-domain solver in CST Microwave Studio, and the far-field monitors were set at intervals of 0.1 GHz. The simulation setups are as follows. The metasurface lies on XOY plane, X-polarized plane waves are normally incident from the -Z direction and the six boundary conditions in the x, y and z directions are all set as ‘open add space’. The 3D snapshot of reflected far-field results of this metasurface at 9.5GHz, 10.0GHz and 10.5GHz are shown in Fig. 4(b), in which the beams are reflected in the same direction. To observe the reflection angle more intuitively, the directivity diagrams at 9.5GHz, 10.0GHz and 10.5GHz are merged in the Fig. 4(c). The central reflected angles at different frequencies are recorded at Fig. 4(d), where the reflection angles are almost always at 22°. And the variation of main lobe magnitude is also recorded at Fig. 4(d), where the reflected magnitude is basically the same.

 

Fig. 5. Experimental measurement: (a) Photograph of fabricated achromatic metasurface deflector. (b) Far-field test environment in microwave anechoic chamber. (c) The measured directivity diagram of achromatic metasurface deflector. (d) The reflection angles and main lobe level at different frequencies.

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In order to further verify the performance of the designed AMD, we fabricated a prototype of the AMD using conventional Printed Circuit Board (PCB) techniques. The photograph of the sample was shown in Fig. 5(a). The size of this metasurface is 240×240 mm including 24×24 meta-atoms. Figure 5(b) illustrated the experiment measuring systems and the measurement was carried out in the microwave anechoic chamber. The metasurface is placed at the turntable to test the directivity diagrams and the measured results are recorded at Fig. 5(c). When 22 degrees is within the half-beam width range, it is recorded as 22 degrees, and the reflection angles at different frequencies are also neatened in Fig. 5(d). The main lobe levels that measured by far-field measurement system are also recorded in Fig. 5(d). From the measured results, we can conclude that the reflection angles are constant 22° in 9.5GHz – 10.5GHz, which are consisted well with theory and simulation. In practical measurement, it has edge scattering phenomenon on the metasurface and loss of antenna, therefore, the directional pattern is tested with a wider main lobe. However, the center point of all the angles of reflection is basically 22°, which proves the validity of our design. It also convincingly proved that the design method can be generalized to other functional designs.

4. Conclusion

In this paper, we proposed an ultrathin broadband metasurface that can achieve achromatic metasurface deflectors (AMD) in broadband by meta-atom resonance. By analyzing achromatic conditions, we ratiocinated the relationship between the phase gradient and the frequency. The achromatic condition was set to design a searching algorithm. And the Greedy Algorithm Empowered by SOM Neural Network was applied to design the meta-atoms that can satisfy achromatic condition. We used the SOM neural network to realize the clustering of training samples, and used the trained network to identify the similarity of the subsequently collected samples, so as to achieve the diversity of meta-atoms collection. The trained SOM as prepositive model provided the basis for searching map. On the basis of prepositive neural networks, 2,000 diverse sample data were collected to establish searching map. And then, the greedy algorithm was employed to find the meta-atoms whose phase differences satisfy achromatic condition. The found meta-atoms were assembled to an AMD, and the fabricated metasurface was simulated and measured. The results indicated that the reflected angle was constant 22° in 9.5GHz–10.5GHz, which convincingly testified our design and proposed method. More importantly, our designed metasurface has potential application value in satellite communications, mobile wireless communications and others.