Improved Huygens’ Principle for Metamaterial

Abstract

In this paper, we propose a new method based on Huygens’ principle for calculations of transmission spectra with weak coupling and we call this method an improved Huygens’ principle. The original Huygens’ principle for metamaterial can only deal with transmission spectra without coupling between metamaterial structures. Our improved Huygens’ principle can give the approximate calculations of transmission spectra while considering coupling by employing the original Huygens’ principle. We demonstrate our method by employing full-wave simulations and experimental results.

1. Introduction

Huygens’ principle [1,2,3,4] can calculate transmission spectra by considering every metamaterial structure as a wave source and only requires the interference between every two metamaterial structures. By using Huygens’ principle, it is very easy to calculate the transmission spectra of very complex and large-sized metamaterial arrays by knowing the transmission and phase spectra of every single metamaterial structure [5,6,7]. However, the original Huygens’ principle implies that the calculations do not consider coupling between metamaterial structures. Thus, the original Huygens’ principle cannot deal with metamaterial with coupling [5]. Normally, calculating transmission spectra with coupling between metamaterial structures exploits coupled mode theory (CMT) [8,9,10,11], which explains the coupling of many metamaterial phenomena, such as electromagnetic-induced transparency (EIT) [12,13,14], bound states in the continuum (BIC) [15,16,17] and parity–time (PT) symmetry [18,19,20,21,22]. However, the shortcoming of CMT is that it is very complex to calculate transmission spectra for complex and large-sized metamaterial arrays and it is also necessary to introduce some fitting numbers to obtain the transmission spectra.

A metasurface [23,24,25] is a two-dimensional planar array composed of superatomic structures with subwavelength size, which has efficient interaction with incident light and can detect, manipulate and modulate electromagnetic waves in a small size range. Metasurfaces have realized many exotic physical phenomena and engineering applications, such as hyper-holograms [26,27,28,29], anomalous refraction or reflection [30,31], polarization control and multi-functional device design [32,33], and devices based on metamaterials including filters [34,35,36], stealth materials [37], perfect absorption [8,38], patch antennas [39], etc. Huygens’ metasurfaces can excite both electrical and magnetic responses, can more effectively control the transmission direction of a beam, and have high transmittance, which is widely used in antenna engineering [40,41,42,43].

In our previous research, we have already demonstrated the equivalence of the original Huygens’ principle and CMT with coupling strength $g=0$ [5]. Therefore, the original Huygens’ principle is invalid for considering coupling between metamaterial structures. In this paper, we propose a brand-new method to improve the original Huygens’ principle, which can calculate the transmission spectra of metamaterial with weak coupling. Metamaterial arrays with weak coupling can be separated into several sub cells and the transmission spectra can be given by determining the transmission and phase spectra of those sub cells based on Huygens’ principle. In order to consider the coupling, we should separate the metamaterial into sub cells with the overlapping of one structure. The transmission and phase spectra of sub cells can be obtained by using CMT and we consider the coupling effect within the sub cells. Therefore, our improved Huygens’ principle is combined with the original Huygens’ principle and CMT. Our improved Huygens’ principle does not require full information of CMT for the whole metamaterial array and we only need to know the sub cells’ coupled information to approximately obtain the transmission spectra of the whole metamaterial array. Therefore, we do not need to do very complex calculations of CMT and can obtain accurate enough transmission spectra of the whole metamaterial array.

In order to demonstrate our idea, we start by using a one-dimensional random layout of metamaterial string with different periods of $P_x$ metamaterial, varying the coupling strengths between the metamaterial structures. We provide three different metamaterial structures as the examples, as shown in Figure 1. The three structures we selected are all toroidal and, in recent years, the practical application of toroidal structures has received considerable research attention [44]. Note that there is no restriction for selecting metamaterial structures or limitations for the layout of metamaterial structures. Our improved Huygens’ principle can also work for two-dimensional random metamaterial arrays. We employ two different metamaterial structures to demonstrate our findings. In this paper, we provide full-wave simulations and experimental results to illustrate our improved Huygens’ principle.

2. Model

The original Huygens’ principle can consider every metamaterial structure as the wave source and only consider the interference of every two metamaterial structures. The original Huygens’ principle can be given by [3,4,5]

$$T=\sqrt{{\displaystyle \sum _i{P}_i^2{T}_i^2}+{\displaystyle \sum _{i\ne j}2P_i{P}_j{T}_i{T}_j\cos ({\varphi }_i-{\varphi }_j)}}$$

(1)

where $P_{i,j}$ is the probability of metamaterial structures $\left\{i,j\right\}$ and $T_{i,j}$, ${\varphi }_{i,j}$ are the transmission and phase spectra with corresponding metamaterial structures $\left\{i,j\right\}$.

In order to combine CMT and Huygens’ principle, we utilize Huygens’ principle to solve the transmission spectra of larger metamaterial arrays with a coupling effect. The idea of our improved Huygens’ principle is that we take the sub cells of metamaterial by fully considering the coupling effect and then we consider Huygens’ principle by using every sub cell. We designed three different metamaterial structures (named structure ‘A’, ‘B’ and ‘C’ as the examples) to demonstrate our idea, as shown in Figure 1. At the beginning, we randomly arranged three metamaterial structures in a one-dimensional string, forming a supercell with a total of 15 metamaterial structures, as shown in Figure 2. In order to distinguish the supercell, we added one more period of the substrate, resulting in a total of 16 periods $P$. Subsequently, we selected the sub cells while considering CMT theory, allowing the number of structures in the sub cells to be positive integers. The configuration of sub cells should be a one-by-one closed arrangement. After that, we obtained the transmission and phase spectra for every sub cell and we were able to count the probabilities of different types of sub cells. Finally, we were able to utilize the sub cells as the basic elements for Huygens’ principle (Equation (1)) to calculate the transmission spectra of complex and large-sized metamaterial arrays.

Therefore, it is very easy to ascertain that when the number of sub cells is one, it is equivalent to the original Huygens’ principle. The number of sub cells can be selected as any positive integer. For example, when the number of sub cells is two and three, the schematic figures of configurations of sub cells are shown in Figure 2a,b, respectively. When the number of sub cells is equal to the number of the metamaterial’s supercells, our sub cell is equal to the supercell and the probability of the sub cell is one. Therefore, we can obtain the transmission spectrum of the sub cells which coincides with the supercells. Therefore, our improved Huygens’ principle combines the original Huygens’ principle with CMT and the precision of our method is between the transmission spectrum of the original Huygens’ principle and the transmission spectrum of the whole metamaterial array. To sum up, our method only requires the transmission spectra of the sub cells to approximately obtain the transmission spectrum of the whole metamaterial array, which means a small amount of information can give full information on the whole system.

3. Full-Wave Simulations

In this section, we demonstrate the functional operations of our improved Huygens’ principle. We take an example of a one-dimensional metamaterial array, as shown in Figure 2, which is ‘CCACBAABCBBACAB’ constituted by structures ‘A’, ‘B’ and ‘C’, as shown in Figure 1. Note that there is no limitation in selecting a combination of the one-dimensional metamaterial array and this specific configuration is only an example. In order to distinguish the metamaterial array, we added one additional period $P_x$ in the full-wave simulations and experimental verifications. When the number of sub cells is one, our method is exactly equivalent to the original Huygens’ principle and the probabilities of structures ‘A’, ‘B’ and ‘C’ are $P_A=1/3$, $P_B=1/3$ and $P_C=1/3$. Subsequently, we can obtain the transmission and phase spectra of structures ‘A’, ‘B’ and ‘C’, respectively, and then we can calculate the transmission spectrum of the whole system based on Huygens’ principle (Equation (1)).

The calculations of our improved Huygens’ principle are as follows: when the number of sub cells is two, the probabilities of the sub cells are $P_{AA}=1/14$, $P_B{}_B=1/14$, $P_{CC}=1/14$, $P_{AB/BA}=4/14$, $P_{AC/CA}=4/14$ and $P_{BC/CB}=3/14$. There are a few transmissions and phase spectra that are the same due to the period of sub cells. Thus, we can take those same sub cells as one sub cell. After that, we can obtain the transmission and phase spectra of the structures ‘AA’, ‘BB’, ‘CC’, ‘AB’, ‘AC’ and ‘BC’ from the full-wave simulations and experimental results. Finally, we can obtain the transmission spectrum of the whole system based on the ‘two’ sub cells. When the number of sub cells is three, the probabilities of the sub cells are $P_{CCA/CAC/ACC}=2/13$, $P_{ACB/CBA/BAC}=3/13$, $P_{BAA/AAB/ABA}=2/13$, $P_{BCB/CBB/BBC}=2/13$, $P_{BBA/BAB/ABB}=1/13$, $P_{ACA/CAA/AAC}=1/13$, $P_{ABC/CAB/BCA}=2/13$. According to those combinations of structures, we can obtain the transmission and phase spectra of the structures from the full-wave simulations and experimental results and then substitute Equation (1) to obtain the transmission spectrum of the whole system based on ‘three’ sub cells. Finally, we should consider the filling factor effect and we only have 15 structures but 16 periods of substrate. Therefore, the final transmission spectrum should be multiplied by the factor as ($16/15=1.06667$).

Based on the calculations of our method, the full-wave simulations of a full one-dimensional metamaterial array are shown by the red lines in Figure 3. The black lines in Figure 3 show the original Huygens’ principle, where the number of sub cells is one. The blue and green lines show the calculations of ‘two’ and ‘three’ sub cells based on our improved Huygens’ principle, respectively. The differences between Figure 3a and d are the differences of the period $P_x$ of the structures with (a) $P_x$ = 60 μm, (b) $P_x$ = 80 μm, (c) $P_x$ = 100 μm and (d) $P_x$ = 140 μm. With a varying period $P_x$ of the structures, we investigated our method using varying coupling strengths between structures. It is very easy to ascertain that the original Huygens’ principle cannot predict the transmission spectrum of the whole system due to the coupling shown by the black lines in Figure 3.

As we can see, when the period $P_x$ is increasing, which means the coupling is increasingly smaller, the original Huygens’ principle is more and more accurate. For instance, when the period $P_x$ = 60 μm, the black line in Figure 3a is totally different from the red line of the simulation, and when the period $P_x$ = 140 μm, the black line in Figure 3d has the same tendency as the red line of simulation. It is remarkable to find that our method of the improved Huygens’ principle (for both ‘two’ and ‘three’ sub cell configuration) is much better than original Huygens’ principle. The blue and green lines in Figure 3 are much better than the black lines in Figure 3 in any coupling configurations, which are functional for Figure 3a–d. In order words, our improved Huygens’ principle can provide a good approximation of a full one-dimensional metamaterial array with a coupling effect. In addition, the green lines (’three’ sub cell of the improved Huygens’ principle) are better than the blue lines (‘two’ sub cell configuration) from Figure 3a,b. However, the blue lines (’two’ sub cell configuration) and green lines (’three’ sub cell configuration) are almost the same in Figure 3c,d. It is very straightforward to explain that the ‘three’ sub cell configuration in our method is better than the ‘two’ sub cell configuration when the coupling is larger, as shown in Figure 3a,b, because ‘three’ sub cell configuration has more information than ‘two’ sub cell configuration. When the coupling is not strong enough, the two lines (blue and green lines) become the same and both are good enough for full-wave simulations, which means the ‘two’ sub cell configuration is good enough and we do not need as much information as the ‘three’ sub cell configuration.

4. Two-Dimensional Case and Experiments

Our improved Huygens’ principle can easily extend our calculations from one-dimensional to two-dimensional metamaterial arrays. The selection of sub cells of two-dimensional metamaterial arrays should be ‘2 × 2’ sub cell, ‘3 × 3’ sub cell and so on. In this section, we give an example of a two-dimensional metamaterial array (‘5 × 5’) and the selection of ‘2 × 2’ sub cells is shown in Figure 4. Note that the selection of structures (’B’ and ‘C’) and two-dimensional array are universal, which means that our method can work for any structures and configurations of a two-dimensional array. In order to distinguish the two-dimensional array, we added one more period $P_x$ along with an x and y axis. Therefore, the final transmission spectrum should be multiplied by the factor as ($36/25=1.44$), due to the filling factor effect. The rest of the calculations for the two-dimensional array are the same as the one-dimensional array. In this special example of the two-dimensional array, the full-wave simulations of two-dimensional array, the original Huygens’ principle and the ‘2 × 2’ sub cells of our improved Huygens’ principle are shown in the red lines, black lines and green lines in Figure 5. Figure 5a,b present the different coupling between the structures with varying period $P_x$ ((a) $P_x$ = 80 μm and (b) $P_x$ = 120 μm).

As we can see from Figure 5, our improved Huygens’ principle method (green lines) can perform well for a two-dimensional metamaterial array, which is much better than the original Huygens’ principle (black lines) and we can ascertain that our method can give a very good approximation compared to full-wave simulation (red lines) for both strong coupling (Figure 5a) and weak coupling (Figure 5b) configurations. In addition, the green line in Figure 5b is closer to the red line in Figure 5b, compared to Figure 5a. It is not surprising that our method is more suitable for weak coupling compared to a strong coupling configuration because the system requires more information to predict the transmission spectrum of the whole system in the stronger coupling configuration. Therefore, if we want to obtain a better prediction, it is very easy to calculate our method by using more information from more sub cells, for example ‘3 × 3’ sub cells.

Furthermore, we provide experiments to verify our idea, which use a two-dimensional case with $P_x=120 \mathsf{\mu }\mathrm{m}$, as shown in Figure 6. We employed the THz time-domain spectrometer to obtain the transmission spectra in the time domain. Subsequently, we applied Fourier transform to change the transmission spectra from time domain to frequency domain. The transmission spectra of our experimental results and simulations (as shown in Figure 6) can be given as $T(\omega )=\left.\left|E_S(\omega )/E_R(\omega )\right.\right|$, where $E_S(\omega )$ is the transmission spectra obtained by measuring our designed samples and $E_R(\omega )$ is the transmission spectra of pure substrate (Si). The transmission spectra of experiments and simulations are shown in black and red lines, respectively, in Figure 6. Figure 6a–e show the ‘2 × 2’ sub cell configurations which are ‘BCCB’, ‘BBCC’, ‘BBBC’, ‘BCBC’ and ‘BCCC’, respectively, and Figure 6f shows the transmission spectrum full structure of Figure 4. The blue, red and black lines are the simulations, experiments and calculations by Huygens’ principle. As we can see from Figure 6, comparing simulations with experiments, it is easy to ascertain that there is the same tendency between the experiments and simulations. The frequency of simulations and experiments are nearly equivalent. The amplitude of transmission spectra of experiments is higher than our simulations due to the larger intrinsic loss of metal in experiments and defects of fabrications of our metamaterials, which are quite common issues in experiments. Therefore, we can easily conclude that our simulations are reliable to lead to the correct findings. Thus, we can conclude that our experimental results well support our simulations, and we believe that our full-wave simulations in Figure 3 and Figure 5 are correct in this paper. In addition, we provide calculations based on Huygens’ principle, as shown in black lines in Figure 6. The transmission spectra of calculations based on Huygens’ principle are totally different compared to experiments and simulations due to the coupling between metamaterial structures.

To sum up, we propose a new approximate calculation method of transmission spectrum based on Huygens’ principle and full-wave simulations of sub cells. The shortcoming of the original Huygens’ principle is that it cannot deal with transmission spectra with coupling between the structures. The full-wave simulations can provide the transmission spectra. However, when we deal with large unit cells, full-wave simulations consume a lot of time and large calculation resources. In our new method, we only require the full-wave simulations of finite sub cells. Subsequently, we can employ our method to count the number of sub cells, as demonstrated in Figure 2 and Figure 4. Finally, we can calculate the probability of every sub cell and then substitute it into Huygens’ principle. Our method considers some sort of coupling due to the sub cell selection. Thus, our method is an approximate calculation method of transmission spectrum with weak coupling. Therefore, we can obtain the corollary which is equivalent to the original Huygens’ principle when we choose one structure as the sub cell. When we choose the whole structure as one sub cell, our method is equivalent to full-wave simulation of the whole structure. Therefore, the selection of the number of structures as the sub cell depends on the accuracy of the approximation. This conclusion is also consistent with Figure 3 and Figure 5, which demonstrate that the transmission spectra are increasingly close to full-wave simulations when we transition from the original Huygens’ principle to ‘three’ sub cells (or ‘3 × 3’ sub cell).

It is worth discussing Figure 3 and Figure 5 more. It seems that using ‘three’ sub cells in the improved Huygens’ principle is much better than using ‘two’ sub cells and the original Huygens’ principle with strong coupling in Figure 3a. However, using ‘three’ sub cells in the improved Huygens’ principle also works well for weak coupling cases ($P_x$ = 80 µm, $P_x$ = 100 µm, $P_x$ = 140 µm). In addition, the ‘two’ sub cells of the improved Huygens’ perform well when there is weak coupling and the original Huygens’ principle provides better and better performance when the coupling decreases. Therefore, we conclude that the ‘three’ sub cells of the improved Huygens’ are more accurate than the ‘two’ sub cells of the improved Huygens’ principle, which is better than the original Huygens’ principle. As we can see from the results, the accuracy of the approximation also depends on the coupling strength. Thus, when the coupling of structures is stronger, we should select a greater number of structures in the sub cells. For the same number of structures in the sub cells, our improved Huygens’ works better for weak coupling between structures. This finding is not contradictory to 2D cases. In Figure 5, our 2D case does not have much strong coupling ($P_x$ = 80 µm and $P_x$ = 120 µm), thus we chose the ‘2 × 2’ sub cell which is good enough for our approximation.

5. Conclusions

The original Huygens’ principle for metamaterial can only deal with transmission spectra without considering coupling. In this paper, we proposed an improved Huygens’ principle for calculating the approximate transmission spectra with weak coupling by applying Huygens’ principle of sub cells. Based on our improved Huygens’ principle, we can simply obtain the approximate transmission spectra with weak coupling between structures.