Metasurfaces have attracted significant attention due to their novel designs for flat optics. However, the approach usually used to engineer metasurface devices assumes that neighboring elements are identical, by extracting the phase information from simulations with periodic boundaries, or that near-field coupling between particles is negligible, by extracting the phase from single particle simulations. This is not the case most of the time and the approach thus prevents the optimization of devices that operate away from their optimum. Here, we propose a versatile numerical method to obtain the phase of each element within the metasurface (meta-atoms) while accounting for near-field coupling. Quantifying the phase error of each element of the metasurfaces with the proposed local phase method paves the way to the design of highly efficient metasurface devices including, but not limited to, deflectors, high numerical aperture metasurface concentrators, lenses, cloaks, and modulators.
© 2017 Optical Society of America
Optical metasurfaces are artificial planar structures, composed of dielectric or plasmonic nano-resonators [1–3]. Metasurfaces offer a wide control over the reflection and refraction by tuning the amplitude, the phase, and the polarization states of light incident on a surface . In addition to having strong potentials in applied research such as optical components (lenses, optical wave-plates, modulators, etc.) [5–9] cloaking devices [10,11], solar applications [12,13], biomedical imaging , sensing [15,16], hologram , metasurfaces are of great scientific interest to explore fundamental questions in optics and wave physics in complex environments. In general, there are two primary methods to design efficient metasurfaces. Both methods involve the engineering of the phase and amplitude information. The first approach is the single meta-atom method  and it consists in calculating the reflection or transmission phase of a single particle by assuming that near-field coupling between particles is negligible. The second method is a widely used approach that we term the unit cell method (UCM). In this approach, unit cell boundary conditions are used and the phase is thus obtained with identical neighbors. This method is only accurate if coupling differences between identical and different neighbors are negligible, which we called near-field coupling difference throughout the paper. However, in physical metasurface devices, the elements are not single particles or identical particles and devices constructed using these methods are thus not optimized. To the best of our knowledge, no numerical approach has been reported in the literature to address this problem. Here, we propose a method, that we term the local phase method (LPM). The LPM accounts for near-field coupling between adjacent neighbors. It is a powerful and versatile tool to quantify the phase error and amplitude of each element of the metasurface. These quantifications can be used to design high efficiency metasurface devices for various applications. The paper briefly presents the LPM and illustrates its capabilities and advantages for various applications such as deflectors (for different designs) and high numerical aperture concentrators. Furthermore, we compare the LPM to the UCM with the same examples.
2. Local phase method
For applications based on metasurfaces without strong near-field coupling, the unit cell method is sufficient. However, when strong near-field coupling is present, the unit cell method limits the performance of devices. To address these limitations, we propose the LPM that takes into account near-field coupling differences and quantifies the phase of meta-atoms in presence of non-identical neighbors. This quantification based on the LPM allows an efficient optimization of metasurfaces for a given application.
We first consider the standard UCM for a metasurface deflector that relies on the simulation and analysis of individual elements in a periodic array. The metasurface elements provide a linear phase shift as a function of their position with a constant phase difference of 40° between adjacent elements, which correspond to 9 (360° / 40°) elements. Figure 1(a) depicts the super-cell illuminated by a plane wave incident at an angle (10°) on the metasurface and polarized along the Y direction. The incident field interacts with the TiO2 dielectric resonator (in brown), and is reflected by a metallic ground plane modelled as a perfect electric conductor (PEC) in grey. The light blue color represents the SiO2 spacer. The boundary conditions in other directions (x and y) are set to periodic boundary conditions.
To obtain the phase of each element, we start by computing the field of the whole structure using the commercial software CST. From this total field (E, H), we obtain the equivalent sources (Js, Ms) of each element (blue boxes in Fig. 1(a)) using the equivalence principle . According to the equivalence principle, the field outside a virtual closed surface can be obtained by suitable electric- and magnetic-current densities over the closed surface. The equivalent current densities are given by the following equations:Figure 1(b) presents the schematic of the new simulation. The distance between the probe and the new equivalent source is 2 μm, and, it is large enough to be considered as far field for a wavelength of 800nm. Besides, the boundary conditions in x and y directions [Fig. 1(b)] are set to periodic boundary conditions. The boundary conditions in ± z direction are set to open. Using this method, the field radiated by the previous neighbors is not taken into account, and the phase shift produced by the considered element can be measured. Therefore, we refer to it as the Local Phase Method (LPM).
To compare the UCM and the LPM, we use metasurface deflectors as a first example. A plane wave comes from the top z plane with a 10° incident angle. We design deflectors based on three types of elements. The first is a simple rectangular element based reflector [Fig. 2(a)] with weak coupling between its elements. The second deflector is made of rectangular elements embedded in a low refractive index material, SiO2 [Fig. 2(b)] with nSiO2 = 1.45. In this configuration, the smaller refractive index contrast decreases the field confinement inside the TiO2 particles and the coupling between particles increases . The third design consists in tilted parallelepiped elements with a tilted angle of 20° from the normal [Fig. 2(c)] . The blue-green color in Figs. 2(a)-2(c) represents the low refractive index material (SiO2). The brown color represents the TiO2 material with a refractive index of 2.52 and the grey-bold represents the metallic ground plane. We chose to work in the visible spectrum at 800 nm to illustrate the LPM. Figures 2(d)-2(f) presents the phase shift obtained in simulation with the UCM. In all cases, we take the phase provided by the longest length (L) as phase reference. The other dimensions are listed as insets. The theoretical deflection angles for the three cases are respectively 0.5°, 1.3° and 0.5°, given by , with θi = 10°, ΔΦ = 40° and λ0 = 800 nm.
We first start with the rectangular elements design and compare it with the standard UCM design. The initial elements lengths of the optimized deflector are chosen with the UCM. The corresponding required lengths are shown in blue in Fig. 3(a). The radiated energy in the far-field, or equivalently the radar cross section (RCS), is given by the blue curve in Fig. 3(b). The RCS exhibits a maximum value of 187 λ2 at 0°. Subsequently, we use the LPM to obtain the real phase of the elements. Then, we tune the length of each element to optimize the phase difference between adjacent elements to be 40°. To do so, we use a derivative free approach based on the Matlab fminsearch algorithm which is well suited to optimize discontinuous functions, and we constrain the lengths to be between 100 and 540 nm . The cost function to minimize is given by the sum of the square difference of the phases: where is the phase error of the mth element, in other words, the phase difference between the target and the considered method (LPM and UCM).
The green circles in Fig. 3(a) correspond to the optimized dimensions by LPM. These optimized dimensions are used to provide the RCS plotted in Fig. 3(b). The RCS has been increased to 200 λ2, corresponding to an enhancement of 7%. Figure 3(c) represents the comparison between the phase difference obtained from the LPM and the UCM. Figure 3(d) presents the absolute value of the phase for both methods with the UCM in blue and the LPM in green. The maximum phase error is about 40° for the UCM and is decreased to 20° for the LPM. The final RCS has not increased a lot as the near-field coupling difference of rectangular elements is initially minor.
To further investigate our method, we now consider a new design with strong near-field coupling as is the case for rectangle elements embedded in a SiO2 layer shown in Fig. 2(b). Figure 4(a) presents the lengths obtained with the UCM (blue) and LPM (green). Unlike the first situation, the dimensions of the LPM are quite different from the ones of the UCM. This is due to the strong near-field coupling effect. Figure 4(b) shows the RCS obtained with the two methods. We observe that the RCS exhibits two low peaks at −3° with a value of 120 λ2 and at 8° with a value of 90 λ2. It is worth noting that the period, in this case, is a little larger than in the previous case [Fig. 2(a)] and that the theoretical deflection angle is 1.3°. By using the same approach as in the case of the rectangle elements, the RCS for the LPM increases to 269 λ2, i.e. more than a 124% enhancement compared to the UCM design. We also note that the maximum phase error (given by the phase obtained with the method minus the required theoretical phase) for the LPM is only about 30° while the maximum phase error of the UCM is more than 100°.
The third design consists of slanted parallelepiped elements with a tilted angle of 20° as shown in Fig. 2(e). The concept figure is presented in Fig. 1(c) and the results are presented in Fig. 5. Unlike the rectangular ones [Fig. 3], the RCS at 0° is as small as 85 λ2. Moreover, there is another beam at −10° which corresponds to specular reflection. On the other hand, the RCS for the LPM is shown in Fig. 5(b) and the RCS at 0° drastically increases to 180 λ2, corresponding to an enhancement of 111%, and the secondary lobes are now negligible. Similarly, Figs. 5(c)-5(d) present the comparison of the phase difference between the two methods (UCM and LPM). As we can see, the LPM provides a phase much closer to the target. These results clearly show that, even if the near-field coupling difference is small, the LPM improves the efficiency of the metasurface devices. More importantly, when the near-field coupling difference becomes strong, our LPM significantly improves the results.
4. High Numerical Aperture Concentrator
We now investigate a second class of devices by implementing a high numerical aperture concentrator where the advantage of the LPM over the usual UCM becomes evident. Figure 6(a) shows the unit cell dimensions (length) used to design the concentrator. The height is large enough to provide waveguide modes to the elements . Figure 6(b) presents the phase shift obtained with the UCM. Again, we take the phase obtained with the largest dimension as the phase reference. The red curve in Fig. 6(c) represents the required theoretical phase (target) for the concentrator. We can observe that the first 8 elements have a smooth phase variation and the average of the absolute value of the phase difference between UCM and the target is 35°. However, when the slope or the derivative of φ(x) gets steeper, or discontinuous, the imposed phase shift is very different from the targeted one. About half of the elements are almost 150° away from their targets. This is a consequence of the fact that the dimensions of the elements calculated by the UCM are extremely different from their neighbors and the coupling difference cannot be neglected anymore [Fig. 6(c)]. Figure 7(a) shows the real part of the reflected electric field. We can observe that the field is reflected towards the focal spot merely by the first eight elements. Figure 7(a) still shows a large discontinuity of the wave front around the 10th element because of its large phase error. On the other hand, in the LPM design, we optimize the dimensions of the elements from number eight to the end to obtain the desired phases. Overall, the phases obtained with our LPM optimization (in green in Fig. 6(c)) are much closer to their targets (in red) than with the UCM (in blue), except for some elements. This is due to the choice of our cost function which optimize the sum of the square difference and not all individual phase differences. For instance, to minimize the large error at element 18, we sacrifice a bit of the phase error at element 19. Interestingly, this may also indicate that the near-field coupling may impose some limits on the optimization approach and that it is not always possible to find a length distribution that fits exactly the theoretical phase profile of a perfect concentrator . This may also be attributed to our optimization algorithm that is known to converge towards local maxima for high dimensional problems . However, the optimization is still very efficient. Figure 8 presents the total power density produced by the metasurface concentrator with the UCM [Fig. 8(a)] and with the optimization based on the LPM [Fig. 8(b)]. We can see that the energy at the focal spot is increased by 15%, with a larger longitudinal length for a slightly reduced width.
In summary, we have proposed the local phase method, a versatile approach to quantify the phase error of each element within a metasurface accounting for the near-field coupling. The method improves the performance of devices based on metasurfaces. We have investigated two different applications, namely deflectors and high numerical aperture concentrators, to illustrate the impact of the LPM on the efficiency of metasurface devices. Even for devices with small near-field coupling, for which the UCM performs quite well, quantifying the local phase can improve the efficiency. In more advanced devices were near-field coupling is more important, optimizing the phase error using our LPM drastically improves the radar cross-section. The simplicity and the versatility of the LPM will lead to the design of highly-efficient metasurfaces and complex electromagnetic devices that rely on the discretization of theoretical profiles such as in transformation optics.
This material is based upon work supported by the U.S. Department of Energy under Award Number DE-EE0007341.
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