Inverse design of perimeter-controlled InAs-assisted metasurface for two-dimensional dynamic beam steering

Abstract The current commercially viable light detection and ranging systems demand continuous, full-scene, and dynamic two-dimensional point scanning, while featuring large aperture size to ensure long distance operation. However, the biasing architecture of large-area arrays with numerous individually controlled tunable elements is substantially complicated. Herein, inverse design of a perimeter-controlled active metasurface for two-dimensional dynamic beam steering at mid-infrared regime is theoretically presented. The perimeter-control approach simplifies biasing architecture by allowing column-row addressing of the elements. The metasurface consists of a periodic array of plasmonic patch nanoantennas in a metal-insulator-metal configuration, wherein two active layers of indium arsenide are incorporated into its building block. The metasurface profile facilitates wide phase modulation of ≈355° on the reflected light at the individual element level through applying independent voltages to its respective columns and rows. The multi-objective genetic algorithm (GA) for optimizing user-defined metrics toward shaping desired far-zone radiation pattern is implemented. It is demonstrated that multi-objective GA yields better results for directivity and spatial resolution of perimeter-controlled metasurface by identifying the design tradeoffs inherent to the system, compared to the single-objective optimizer. A high directivity and continuous beam scanning with full and wide field-of-view along the azimuth and elevation angles are respectively maintained.


S1. DEVICE SIMULATIONS AND CARRIER DYNAMICS
The carrier dynamic simulation of our proposed InAs-integrated plasmonic metasurface is carried out by using Lumerical device solver. In the device simulations, the spatial distributions of the charge carriers are modeled under the application of the bias voltages through the Poisson and drift-diffusion equations. The carriers (electrons and holes) move under the influence of drift due to the applied bias voltages and random thermal diffusion due to the gradient in the density. In the main manuscript, we remarked that the upper boundaries of the bias voltages are limited by the breakdown field strength of the alumina gate-dielectric that is reported as 7.4 MV/cm 1 . On the other hand, the lower threshold of the bias voltages is chosen in order to avoid the accumulation of the holes at the InAs/gate-dielectric interfaces. Figure S1 Fig. S1(B). The hole densities attenuate exponentially for larger distances from the interface. The accumulation of holes at InAs/alumina interface for the applied bias voltages V < V T leads to a decrement in the real part of the permittivity of InAs, while giving rise to its imaginary part. As a result, the reflection amplitude and phase of the metasurface are reduced.
To avoid the destructive impact of hole accumulation in the optical response of the metasurface, we have set the lower boundaries of the bias voltage into −2.5 V.
It should be noted that in order to generate the dataset demonstrated in Fig. 4(B) of the main manuscript, the charge carrier distributions for all possible combinations of the bias voltages V and U, that are continuously varying in the range of −2.5 V to +13.8 V, are maintained. However, for the sake of brevity, only the typical case in which U is adjusted to 0 V, is depicted. After generating the charge carrier distribution within the InAs active layer, its permittivity is calculated  of the main manuscript, we note that the reflection spectrum is characterized by one reflection dip for the bias voltage of V < 4 V. As soon as the bias voltage is increased beyond 4 V, the permittivity of the InAs layer approaches to ≈ 0, and the second resonant mode at the longer wavelengths is excited. This resonant mode is associated to the epsilon-near-zero (ENZ) material and is referred to as the ENZ resonance. To investigate this behavior further, we performed FDTD simulations and calculated the metasurface response to the variations of the parameters namely bias voltage, collision frequency, thickness of the ENZ layer, and the width of patch nanoantenna and the results are summarized in Figs. S2(A)-(E). We note that when the electrode voltages of the unit cell are set to (0, 0) V, there is only one resonance at the reflection spectrum that is attributed to the gap plasmon mode. By increasing the level of the bias voltage assigned to the top electrode into 4 V and 4.5 V, respectively, the permittivity of the accumulation layer within the top InAs approaches to the ENZ regime (−1 < ε InAs < 1), which results in the excitation of ENZ mode (Figs. S2(A)-(B)). This mode is heavily damped by the material loss. To verify, we calculated the optical response of the metasurface by increasing the collision frequency within the accumulation layer from 2 × 10 13 to 1 × 10 14 . While the gap plasmon resonance remains almost unchanged by injecting more dissipative loss into the ENZ thin film, the ENZ mode is substantially damped, as shown in Fig. S2(C). In addition, the ENZ wavelength is strongly dependent to thickness of the accumulation layer, while having a weak dependence on the antenna dimension. These are within all the ENZ layer, InAs layers and the gate-dielectrics. Since the ENZ wavelength is independent of the antenna dimension, the light interaction is weak with the patch antenna at the second resonance wavelength, while it is strongly enhanced within the top accumulation layer and the intensity of the electric field is at least three times larger than the gap plasmon mode, corresponding to an enhancement of approximately 50 (See Figs. S2(F)-(G)). Therefore, our results confirm that the second mode is associated to the ENZ resonance of the InAs accumulation layer.
Our results regarding the nature of ENZ mode are consistent with the observations in Ref. 2 , where the dependence of optical response of the dipole antenna on an ENZ substrate is illustrated.

S3. THE CROSS COUPLING EFFECTS OF THE ADJACENT UNIT CELLS
To study the mutual effects of the under-biased adjacent unit cells, we conducted FDTD simulations for a super cell that is composed of two InAs-assisted MIM unit cells, whose building block is demonstrated in Fig. 1(B) of the main manuscript. We have considered three cases cells, respectively. Figure S3(A) demonstrates the phase of the electric field reflected from the metasurface super cell upon applying the bias voltage configurations in Cases I-III into the elements that is obtained immediately above the super cell. It is observed that the reflection phase over the left element in Case III overlaps with Case I, however there is a small deviation between the phase profiles of Case II and Case III over the right element that is denoted by the dashed black box in Fig. 3(A). This deviation which is smaller than 4 • is attributed to the destructive effect of its adjacent unit cell. This implies that the crosstalk effects of the metasurface neighboring unit cells are negligible. It is worth mentioning that to further increase the device durability, the elements can be coated by a capping insulator layer that can further isolate the adjacent elements and minimize their side effects on each others performance as proposed in 3 .
In addition, we have calculated the near-field distribution within the metasurface super cell to

PERIMETER-CONTROLLED REFLECTARRAYS
The implementation of two-dimensional beam steering metasurface requires simultaneous and independent control over its constituent unit cells. Such controlling technique can be obtained by either individual biasing of the elements or the perimeter-control architecture. For the longrange communication, the large-aperture arrays are required, where individual biasing of their constituent elements presents a significant challenge. The perimeter-controlled architecture simplifies the biasing mechanism of the large-aperture arrays by reducing the control signals to 2N through addressing the corresponding rows and columns. In this section, we have performed a comparative study on the spatial distribution of the amplitude and phase of the perimeter-controlled and individually-controlled reflectarrays and their resultant radiation patterns. For this purpose, we have employed multi-objective GA for generating the desired far-zone radiation patterns and the amplitude and phase profiles of the perimeter-controlled reflectarray, while the forward approach is utilized for designing the individually-controlled metasurface. The reflectarray is composed of an ensemble of 9 × 9 unit cells that are arranged along the x and y directions, where the spacing between the elements is adjusted to Λ. Figure S5 shows two examples of beam steering that are obtained by the two biasing configuration (individually-and perimeter-controlled architectures). For the forward design of the individually-controlled reflectarray, we have used the data that provides an almost 2π phase span with the constant reflection amplitude level of ≈ 0.15 for the metasurface unit cell. Such data for the reflection is achievable and can be confirmed from the reflectivity result at the complex r-plane demonstrated in Fig. 4(B)   can be still maintained. In addition, due to the covarying amplitude and phase response of unit cells, uniform amplitude for all the constituent elements of the reflectarray cannot be obtained.
However, in our optimization problem, we optimize toward the maximum achievable directivity, therefore the algorithm aims to minimize the amplitude modulation over the entire array. This results in reduced sidelobe levels, as well. Moreover, the full FOV along elevation direction cannot be attained. The DC voltage profiles for addressing the columns and rows of the beam steering perimeter-controlled reflectarray are illustrated across their corresponding amplitude profiles, whose values are changing within the range −2.5 V to 13.8V. From Fig. S6 it can be concluded that the perimeter-controlled reflectarray allows for simplified biasing architecture and enables two dimensional dynamic beam steering with wide FOV along the azimuth and elevation angles, respectively. The inverse design technique based on multi-objective GA outputs nonintuitive amplitude and phase profiles for the perimeter-controlled array that succeeds in generating high resolution radiation patterns with enhanced directivity and reduced sidelobe levels.