0018-926X (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2021.3083806, IEEE Transactions on Antennas and Propagation 1 Angular Memory of Photonic Metasurfaces Constantinos Valagiannopoulos, Senior Member, IEEE, Adilkhan Sarsen, and Andrea Al` u, Fellow, IEEE Abstract—Nonlinear materials can exhibit bistability, namely support different possible outputs for the same input, depending on the excitation history. This property can find direct application to photonic memory devices, the vast majority of which operate by changing the amplitude of the incoming signals. In this work, we study impedance metasurfaces producing hysteresis responses with respect to impinging beam angle of fixed intensity profiles; accordingly, the considered setups remember previous history of illumination incidence. The reported findings may open new unexplored avenues towards the design of photonic systems supporting a wide range of fixed-power memory functionalities from switching and sensing to information storage and data writing. Index Terms—Angular memory, directional bistability, Kerr nonlinearity, nonlinear metasurface. I. I NTRODUCTION Nonlinearity is present in most materials as long as the input excitation is high enough to activate their natural saturation mechanisms. Therefore, the textural properties of the photonic materials are inevitably dependent on the intensity of the incoming electromagnetic fields. Bloembergen (Nobel Prize in Physics, 1981) was among the first to systematically formulate the driving principles behind nonlinear optics, revealing har- monic wave generation [1] and studying the interaction of light with a nonlinear half-space [2], back in the early 1960’s. Since then, optical nonlinearities have been extensively scrutinized; accordingly, their properties and application potential have become common knowledge through multiple well-established textbooks focusing on the physical concepts [3], on the device setups enabled by nonlinear optics [4] or even on the quantum mechanical origin of the nonlinear coefficients [5]. A large part of the related bibliography is devoted to the propagation of light into nonlinear media giving rise to interesting effects like tailored dispersion and ultra-fast solitons [6]. There are also several techniques available to study the nonlinear properties of optical materials as incorporated in the wave equation and solving it by imposing the dictated boundary conditions. The boundary value problem of a single nonlinear slab has been considered either as a Fabry-Perot interferometer with multi-valued response [7], as a part of bilayer configurations [8] or in terms of the supported lo- calized resonance modes [9], [10], typically with some help from elliptic integral functions. The same story holds for the detection of surface polaritons propagating along the interfaces of optically nonlinear materials [11], where Maxwell’s equa- tions are solved in intensity-dependent media. Furthermore, C. Valagiannopoulos and A. Sarsen are with Nazarbayev University, Nur- Sultan KZ-10000, Kazakhstan. A. Al` u is with the Photonics Initiative, Advanced Science Research Center of the City University of New York and Physics Program, Graduate Center, City University of New York, New York, NY 10016, USA. C. Valagiannopoulos is the corresponding author (e-mail: konstantinos.valagiannopoulos@nu.edu.kz). electromagnetic scattering by nonlinear cylinders has been investigated via efficient numerical discretization [12] and semi-analytical iterative approximations [13]. Structures that can be rigorously solved by considering similar boundary value problem formulations and, at the same time, offer realistic opportunities for applications, are nonlin- ear metasurfaces [14]. They literally constitute a paradigm shift in photonic design since they allow, at a theoretically infinitesimal thickness, for functionalities like wavelength con- version [15], giant second-harmonic generation [16], hologra- phy [17] and extreme wavevector selectivity [18]. Complete platforms based on nonlinear metasurfaces for a wide range of applications have been analytically developed [19], [20] and experimentally tested in photonic crystal configurations [21] to demonstrate controllable intensive focusing and ultra- wide angular scanning. The simplest type of nonlinearity considered in metasurface configurations is the so-called Kerr effect, based on which the surface impedance or conductivity is proportional to the squared amplitude of the local electric field. However, even this simplified model works well and captures successfully the basic wave interaction mechanisms in widely utilized setups like graphene nanoribbons [22] and gold quantum wells [23]. It is apparent that linear Maxwell’s equations satisfy the uniqueness theorem, but when nonlinearities are involved, the solutions can routinely be multi-stable. That property makes such Kerr-nonlinear metasurfaces operating effectively as memory elements. The phase change appearing in com- pound media has long been proposed for optical data stor- age [24] and it is still extensively employed in ultra-fast on-chip nanophotonic memory cells [25], [26]. Therefore, metasurfaces incorporating nonlinear materials can enable reconfigurable memory based on various control mechanisms. Importantly, the backbone of most optical memory elements is the bistability effect, which is associated with the ability of the structure to modify its output not only according to its current input but also based on the previous input values [27]. This hysteresis can occur in various metasurfaces characterized by Kerr nonlinearity, including acoustic [28] or thermal [29] com- ponents for information processing but, primarily, in photonic structures [30], [31] deployed for memory applications. In this work, we consider a free-standing impedance meta- surface with Kerr-nonlinear admittance; the setup is illumi- nated by an obliquely incident plane wave. The formulated boundary value problem is rigorously treated; accordingly, the transmissivity is calculated and found to exhibit bistability with respect to the input power. Since the structure supports an interaction that depends on the angle of incidence, we discov- ered that the metasurface remembers not only the amplitude of the incoming wave but also its direction; such a functionality is demonstrated in realistic nonlinear media excited by strong Authorized licensed use limited to: Constantinos Valagiannopoulos. Downloaded on June 07,2021 at 12:27:56 UTC from IEEE Xplore. Restrictions apply.