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Multipolar Modeling of Spatially Dispersive Metasurfaces Karim Achouri and Olivier J

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Multipolar Modeling of Spatially Dispersive Metasurfaces Karim Achouri and Olivier J 1 Multipolar Modeling of Spatially Dispersive Metasurfaces Karim Achouri and Olivier J. F. Martin Abstract—There is today a growing need to accurately model metasurface operations requiring large angular spectrum, as the angular scattering response of metasurfaces for optical we shall demonstrate thereafter. analog processing applications. However, the current metasurface In order to overcome this limitation, we propose to improve modeling techniques are not well suited for such a task since they are limited to small angular spectrum transformations, as the angular modeling accuracy of these techniques by deriving shall be demonstrated. The goal of this work is to overcome a model that includes higher-order multipolar moments and this limitation by improving the modeling accuracy of these their associated spatially dispersive components [20]–[22]. techniques and, specifically, to provide a better description of the Indeed, the main limiting factor of the current models is angular response of metasurfaces. This is achieved by extending that they only take into account dipolar responses and only the current methods, which are restricted to dipolar responses and weak spatially dispersive effects, so as to include quadrupolar incorporate weak spatially dispersive effects. This is unfor- responses and higher-order spatially dispersive components. The tunately not well suited for designing operations that deal accuracy of the newly derived multipolar model is demonstrated with large angular spectrum since it is precisely in such cases by predicting the angular scattering of a dielectric metasurface. that spatial dispersion starts to play a significant role [23]– This results in a modeling accuracy that is at least two times [32]. Therefore, our goal is to derive a model that includes better than the standard dipolar model. both dipolar and quadrupolar moments and all the associated Index Terms—Metasurface, angular scattering, GSTCs, spatial hypersusceptibility components, many of which are related to dispersion, multipoles. spatial dispersion [33]. For self-consistency, we first review the conventional meta- I. INTRODUCTION surface modeling approach based on electric and magnetic Over the past decade, the research field of metasurfaces – dipoles in Sec. II-A. Then, in Sec. II-B, we show its limitations the two-dimensional counterparts of three-dimensional meta- when it comes to modeling the angular scattering response of materials – has spectacularly flourished and has led to a a metasurface. The extended model that includes both dipolar plethora of metasurface concepts and applications [1]–[6]. To and quadrupolar moments is then derived in Sec. III-A and help designing metasurfaces, one of the most common model- the associated concept of spatial dispersion is discussed in ing approach has been to treat them as discontinuous electro- Sec. III-B. An example is then presented in Sec. III-C to magnetic sheets harboring effective dipolar responses [7]–[9]. illustrate the method and demonstrate its performance. Finally, Over the past few years, these metasurface modeling tech- we conclude in Sec. IV. niques have shown to be particularly effective at describing the complex electromagnetic responses of these structures and II. DIPOLAR MODELING OF METASURFACES have enabled the implementation of optimized transformations A. General Concepts such as perfect anomalous reflection and refraction [10], [11]. A common method for modeling an electromagnetic meta- Today, there is a growing interest towards the design of surface is to assume that it may be reduced to a discontin- metasurface-based spatial processors for optical analog signal uous interface consisting of a zero-thickness fictitious sheet processing, as evidenced by the various examples of spatial supporting electric and magnetic polarization currents [7]–[9], differentiators and integrators that have already been reported [34]–[39]. The interactions of the metasurface with an exciting in the literature [12]–[18]. From these studies, it is apparent electromagnetic field may then be modeled using appropriate that the successful implementation of more advanced signal boundary conditions – the Generalized Sheet Transition Con- arXiv:2103.10345v1 [physics.optics] 18 Mar 2021 processing operations clearly requires a very accurate control ditions (GSTCs) – that relate incident and scattered fields to of the angular scattering response of a metasurface. Naturally, the dipolar moments induced on the metasurface [40], [41]. the metasurface modeling techniques mentioned above may For a metasurface lying in the xy-plane at z = 0, the time- be used as powerful design tools for realizing such operations domain GSTCs read efficiently. @ However, a major drawback of these techniques is that, z^ ∆H = P z^ Mz; (1a) × @t k − × r although they work well in the paraxial regime or for illu- @ 1 minations with a fixed direction of propagation [19], they are z^ ∆E = µ0 Mk z^ Pz; (1b) × − @t − × r currently rather limited when it comes to properly modeling 0 where ∆H = H+ H− and ∆E = E+ E− are respec- K. Achouri, and O. J. F. Martin are with the Nanophotonics and − − ´ tively the differences of the fields between both sides of the Metrology Laboratory, Institute of Microengineering, Ecole Polytechnique + F´ed´erale de Lausanne, Route Cantonale, 1015 Lausanne, Switzerland (e-mail: metasurface with ‘+’ and ‘-’ indicating the fields at z = 0 − karim.achouri@epfl.ch). and z = 0 , and Pk and Mk are the tangential electric and 2 magnetic polarization densities induced on the metasurface, z E respectively. Under such a dipolar approximation, P and t M may generally be expressed in terms of the metasurface Ht effective bianisotropic surface susceptibility tensors χee, χmm, D P χem and χme as [7]–[9], [36], [38], [39] 1 H P 0χ χ E x = ee c0 em av ; (2) n M 1 χ χ H η0 me mm · av + − + − E where Eav = (E + E )=2 and Hav = (H + H )=2 are Hr i the average electric and magnetic fields on the metasurface, θ respectively. Hi Combining (2) with (1) leads to a system of equations Er that may either be used to predict the scattering response Fig. 1: Cross-section of a uniform metasurface composed of a metasurface with known susceptibilities or to compute of a periodic array of dielectric cylinders illuminated by an the metasurface susceptibilities required to achieve a desired obliquely propagating TM-polarized plane wave. scattering behavior specified in terms of incident and scattered fields. Since most metasurfaces are designed to operate within the paraxial limit and are ultimately composed of scattering metasurface period is subwavelength, all waves share the same particles that induce only negligible normal polarizations, it tangential wavenumber kx by phase matching. Finally, the 2 2 2 has been common practice to remove the spatial derivatives dispersion relation is k = kx + kz with kz = k cos θ and in (1) and thus transform the GSTCs into a set of linear equa- kx = k sin θ. The top and bottom signs in (4) correspond to an illumination propagating in the +z or the z directions, tions [19]. Substituting (2) into (1) and converting the resulting − expressions into the frequency-domain1 for convenience yields respectively. Due to the many structural symmetries that the scatter- ing particle (dielectric cylinder) composing this metasurface z^ ∆H = +j!0χ E + jkχ H ; (3a) × ee · av em · av exhibits, we know that χem = χme = 0 and that χee and z^ ∆E = j!kχme Eav j!µ0χmm Hav: (3b) χmm are diagonal matrices [19], [42]. Moreover, since we × − · − · have purposefully decided to ignore the presence of normal While these equations have proven to be very effective at polarizations, when transforming (1) into (3), and since we are modeling and synthesizing the vast majority of metasurfaces, only considering TM-polarized waves propagating in the xz- they still suffer from an important limitation, which is their plane, the only susceptibility components that remain relevant limited capability to accurately model the angular scattering xx yy to this problem are χ and χmm. This reduces (2) to response of a metasurface [19], as we shall now demonstrate. ee xx Px = 0χee Ex;av; (5a) yy B. Limitations of the model My = χmmHy;av: (5b) To illustrate the limitations of (3), we consider the following Substituting (5) and (4) into (3) and solving the resulting simple example: model the angular scattering response of a xx yy system of equations for χee and χmm in the case of an uniform metasurface composed of a subwavelength periodic illumination propagating in the +z direction, as in Fig. 1, array of lossless dielectric cylinders. The metasurface is illu- yields minated by a TM-polarized plane wave propagating in the +z direction, as illustrated in Fig. 1, where the plane of incidence xx 2j R 1 + T χee = − ; (6a) is limited to the xz-plane, for simplicity. The media on both kz R + 1 + T sides of the metasurface is vacuum. 2jkz R + 1 T χyy = − : (6b) To model the interaction depicted in Fig. 1, we define the mm k2 R 1 T fields that take part in it as − − In order to compute these susceptibilities, the metasurface k z;a ∓j(kxx+kz;az) reflection and transmission coefficients are now required. We Ex;a = +Aa e ; (4a) k obtain them from full-wave simulations for a metasurface kx ∓j(kxx+kz;az) whose physical parameters are provided in the caption of Ez;a = Aa e ; (4b) − k Fig. 4. These simulations are performed for an incidence A ◦ ◦ a ∓j(kxx+kz;az) angle ranging from θ = 0 to θ = 85 within the frequency Hy;a = e ; (4c) ± η0 range 200 660 THz (corresponding to a wavelength range − 2 where a = i; r; t to differentiate between the incident, of 500 1500 nm). The resulting transmitted power ( T ) is f g − j j reflected and transmitted waves, A = 1; R; T with R and plotted in Fig. 4b. a f − g T being the complex reflection and transmission coefficients, The metasurface susceptibilities are now computed with (6) respectively, and kz;a = kz; kz; kz .
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