Multipolar Modeling of Spatially Dispersive Metasurfaces Karim Achouri and Olivier J

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 MODELINGOF 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, zˆ ∆H = P zˆ Mz, (1a) × ∂t k − × ∇ although they work well in the paraxial regime or for illu- ∂ 1 minations with a fixed direction of propagation [19], they are zˆ ∆E = µ0 Mk zˆ Pz, (1b) × − ∂t − × ∇ 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 + Fédérale 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 scattering particle (dielectric cylinder) composing this metasurface zˆ ∆H = +jω0χ E + jkχ H , (3a) × ee · av em · av exhibits, we know that χem = χme = 0 and that χee and zˆ ∆E = jωkχme Eav jωµ0χmm 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 { } − | | reflected and transmitted waves, A = 1, R,T with R and plotted in Fig. 4b. a { − } T being the complex reflection and transmission coefficients, The metasurface susceptibilities are now computed with (6) respectively, and kz,a = kz, kz, kz . Note that since the using the full-wave simulated reflection and transmission coef- { − } ficients arbitrarily chosen to be those at normal incidence only, ◦ 1The time dependence ejωt is assumed and omitted throughout. i.e., θ = 0 . The computed real part of these susceptibilities 3

(a) (b) (c) (d) 10 -5 2 1 1 1 80 80 80 0.8 0.8 0.8 1 60 60 60 0.6 0.6 0.6 } χ

{ 0 40 40 40 0.4 0.4 0.4 Re

-1 20 0.2 20 0.2 20 0.2 Incidence angle (°) Incidence angle (°) Incidence angle (°)

-2 0 0 0 0 0 0 200 300 400 500 600 200 300 400 500 600 200 300 400 500 600 200 300 400 500 600 r qec Tz Feuny(H)rqec (THz) (THz)Frequency Frequency (THz)Frequency Frequency (THz) Fig. 2: Response of the metasurface in Fig. 1 with P = 225 nm, D = 200 nm, H = 400 nm and n = 2.55. (a) Retrieved real part of the metasurface electric (solid line) and magnetic (dashed line) susceptibilities using (6). (b) Full-wave simulated transmission coefficient, T 2. (c) Predicted transmission coefficient, T 2, using (7). (d) Relative transmission error defined | | | pred| as 1 T /T 2 . | − | pred | | are plotted in Fig. 4a. Note that if the metasurface could quadrupolar moments. We purposefully ignore higher-order xx yy really be only described by χee and χmm, then the angle multipolar moments for convenience but emphasize that the used to retrieve them would not matter. However, this is not derivation that we next provide is easily extendable to any the case here as modeling the full angular scattering response multipole moment. of this metasurface requires the introduction of higher-order Originally, the GSTCs were derived using a distribution- susceptibility components, as shall be discussed in Sec. III. based approach where all quantities in Maxwell equations are Now that we known the metasurface susceptibilities, we expanded in terms of series of derivatives of the Dirac delta can use them to predict the angular scattering response of the function [40]. For instance, the expressions in (1) correspond metasurface and see if it indeed corresponds to the simulated to such a series expansion truncated at the 0th Dirac delta result shown in Fig. 4b. To do so, we solve (6) for the reflection derivative order [8], [40]. It is obviously possible to derive and transmission coefficients, which yields higher-order GSTCs by truncating these series at higher Dirac 2 yy 2 xx delta derivative orders, an example of which is provided in [43] 2j k χmm kz χee Rpred = xx − 2 yy , (7a) where the series are truncated at the 1st derivative order. (2 + jkzχ ) (2kz + jk χ ) ee mm However, the exact physical meaning of the higher-order terms 4k + k2k χxxχyy z z ee mm that appear in these series and their potential relationships with Tpred = xx 2 yy , (7b) (2 + jkzχee ) (2kz + jk χmm) quadrupolar moments remain unclear, as explained in [43]. where we use the subscript ‘pred’, which stands for ‘pre- This suggests that this distribution-based approach may not be dicted’, to avoid confusion with the simulated coefficients R the best suited to extend the GSTCs to quadrupolar moments. and T used previously. Substituting the susceptibilities plotted An alternative derivation approach has been proposed in Fig. 4a into (7) and varying the incidence angle from in [44], which consists in splitting Maxwell equations in ◦ ◦ 0 to 85 (which changes the value of kz in (7)) results tangential and normal components and then using conven- in the predicted transmission coefficient plotted in Fig. 4c. tional pillbox integration techniques to arrive at the GSTCs. For comparison, we also plot the relative error between T 2 However, this approach is also not well suited because it is 2 | | and Tpred in Fig. 4d. As can be seen, the error between not obvious how should the electric and magnetic quadrupolar 2 | | 2 T and Tpred remains acceptable for low frequencies (large tensors be split into tangential and normal parts. | | | | wavelength-to-period ratios) and small incidence angles, which The approach that we shall rather employ is based on the confirms that the modeling approach provided by (3) is effec- vector potential and was proposed in [29] to derive boundary tive at least in the paraxial limit. However, the error becomes conditions that apply at the interface between media with substantial for higher frequencies (small wavelength-to-period quadrupolar moments. For our case, the method consists in ratios) as well as greater incidence angles, which clearly computing the fields radiated by a metasurface using the vector demonstrates the limitations of this method for modeling the potential, which may be related to a multipolar decomposition general angular response of a metasurface. of the induced surface current on the metasurface. The GSTCs In order to improve the angular scattering modeling of meta- are then directly obtained by subtracting the fields on both surfaces, we derive, in the next section, an extended version sides of the metasurface. of the boundary conditions in (1) that includes higher-order Let us consider the vector potential A, combined with the multipolar components and spatially dispersive susceptibility Lorenz gauge, from which the electric and magnetic fields are tensors. given by [45]

III.MULTIPOLAR MODELING 1 h i E = + k2I A, (8a) A. Derivation of Higher-Order GSTCs jωµ ∇∇ · 1 We are now interested in deriving an expression of the H = A. (8b) GSTCs in (1) that include electric and magnetic dipolar and µ∇ × 4

In this approach, the metasurface is mathematically described by a ﬁctitious sheet of electric current density J, which forms, 2 along with the vector potential A, the inhomogeneous wave k zˆ ∆H = jωPk + ˆz S ˆz equation × 2 × · 2 2 1 A + k A = µJ. (9) zˆ Mz ( ˆz + ˆz ): S (16b) ∇ − − × ∇ − 2 ∇k ∇k For a zero-thickness metasurface in the xy-plane at z = 0, the jω Q QzzI . current density may generally be expressed as − 2 − · ∇k

J = δ(z)J (x, y), (10) In these expressions, Qzz = ˆz Q ˆz, Szz = ˆz S ˆz, the dyadic s · · · · kˆz + ˆz k is defined as where δ(z) is the Dirac delta function and J (x, y) is the ∇ ∇ s   metasurface spatially varying surface current density. Noting 0 0 ∂x that Js may be expressed in the spatial Fourier domain as kˆz + ˆz k =  0 0 ∂y , (17) ∇ ∇ ∂ ∂ 0 Z +∞ Z +∞ x y −j(kxx+ky y) Js = J˜se dkxdky, (11) −∞ −∞ and the double dot product between the arbitrary dyadics A and B, as we may solve (9) for A, which yields2 A : B = AijBij = A11B11 + A12B12 + A13B13 + ... (18) jµ −jkz |z| A = Jse . (12) −2kz Now that we have derived GSTCs that include both dipolar The fields radiated by a metasurface with an arbitrary surface and quadrupole moment densities, we have to properly express current distribution Js are now obtained by substituting (12) these moments in terms of the fields at the metasurface. In the into (4), which leads to case of relations (1), which include only dipolar moments, the latter could be fully described in terms of the 4 bianisotropic 1 h i 2 −jkz |z| susceptibility tensors in (2). However, by adding the electric E = + k I Jse , (13a) −2kzω ∇∇ · and magnetic quadrupolar tensors in (16), we now have the j −jkz |z| capability to include many more susceptibility tensors by H = Jse . (13b) −2kz ∇ × leveraging spatial dispersion, as shall be discussed in the next Since we are interested in expressing the GSTCs in terms section. of a multipolar expansion, we can replace Js in (13) by its multipolar expanded counterpart, which, truncated at the B. Spatial Dispersion quadrupolar moments order, reads [22], [33], [46] Before providing the expressions of the moment densities jω 1 that may be used in (16), we shall first briefly review the J = jωP + M Q (S ), (14) s ∇ × − 2 · ∇ − 2∇ × · ∇ concept of spatial dispersion. Fundamentally, spatial dispersion corresponds to a non-local response of a medium due to where Q and S are the electric and magnetic quadrupolar an exciting field. For instance, this implies that the induced moment densities, respectively. The extended GSTCs are now current J in a medium due to the presence of an exciting obtained by substituting (14) into (13) and computing the electric field E may be expressed as the convolution [22], differences of the fields between both sides of the metasurface [33], [47] with Z J(r) = K(r r0) E(r0) dV 0, (19) ∆E = E |z|=z E |z|=−z, (15a) − · | − | ∆H = H |z|=z H |z|=−z. (15b) | − | where K represents the current response of the medium. By After simplifying and rearranging the resulting expressions considering the three first terms of the Taylor expansion of 0 and setting z = 0, we finally obtain the tangential components E(r ) around r, we may transform (19) into [22], [33], [47] of the extended GSTCs as Ji = bijEj + bijk kEj + bijkl l kEj, (20) k2 ∇ ∇ ∇ zˆ ∆E = jωµM + ˆz Q ˆz × − k 2 × · where the tensor bij represents a local response of the medium 1 1 to the exciting electric field, while the tensors bijk and zˆ Pz ( ˆz + ˆz ): Q (16a) − × ∇ − 2 ∇k ∇k bijkl represent its first- and second-order nonlocal responses, jωµ respectively. + S SzzI k, The tensors in (20) are conventionally split into symmetric 2 − · ∇ and antisymmetric parts so as to associate them to electric and 2This solution may be easily verified by substituting (12) into (9) and magnetic excitations, respectively. For instance, by splitting 2 −jkz |z| 2 −jkz |z| sym asym considering that ∂z e = −kz e − 2jkzδ(z). bijk into its symmetric (bijk ) and antisymmetric (bijk ) parts, 5 the second term on the right-hand side of (20) may be means that in modeling a reciprocal metasurface, it would not 0 expressed as make sense to include χee,ijk without also including Qee,ijk sym asym and thus using boundary conditions that include Q. This is bijk kEj = b + b kEj, ∇ ijk ijk ∇ why in (2) we can safely consider only the 4 bianisotropic sym j (21) susceptibility tensors without including higher-order terms = bijk + εljkgil kEj, ωµ0 ∇ since these equations are limited to the dipolar moments. sym By the same token, the symmetric part of bijkl in (20), = bijk kEj + gijHj, ∇ which is related to l kEj, would be reciprocally connected ∇ ∇ where we have used the fact that the antisymmetric third-rank to components of the electric octupole moment. Since the asym tensor bijk may be equivalently expressed as the second-rank octupole moments are not taken into account in (16), we do not tensor gil using the Levi-Civita symbol εijk [48]. In the last include the components related to l kEj in (22). However, ∇ ∇ part of (21), we have also used the fact that the Maxwell the antisymmetric part of bijkl, which may be expressed equation E = jωµ0H may be written as εijk kEj = ∇ × − ∇ in terms of kHj, is reciprocally related to the magnetic jωµ0Hi. The decomposition in (21) explains, for instance, ∇ − quadrupolar moment S, hence its presence in (22). the dependence of P on H via the susceptibility tensor χem in (2). Noting that the steps from (19) to (21) may be repeated for C. Illustrative Example P, M, Q and S, we may now express the extension of (2) as We shall now come back to the modeling problem discussed     in Sec. II-B and apply the newly derived extended GSTCs (16) Pi Eav,j and the associated spatially-dispersive moments (22). Mi  Hav,j    = χ   , (22) In this problem, we are only interested in modeling the Qil ·  kEav,j  ∇ interactions of TM-polarized waves propagating in the xz- Sil kHav,j ∇ plane, so we can reduce the GSTCs in (16) to where the hypersusceptibility tensor χ is given by jωµ 1 2 0 0 ∆E = (∂ S jM ) + ∂ (Q + Q )  ij 1 ij 0 ijk 1 ijk  x x yx y x zx xz 0χ χ χ χ 2 − 2 ee c0 em 2k0 ee 2c0k0 em 0 0 2 (24a)  1 χij χij 1 χ ijk 1 χ ijk  ωk 1 η0 me mm 2η0k0 me 2k0 mm + Qxz ∂xPz,  0 0  χ =  0 ilj 1 ilj 0 iljk 1 iljk . 2 −  k Qee c k Qem 2k2 Qee 2c k2 Qem   0 0 0 0 0 0  1 2 1 ilj 1 ilj 1 0iljk 1 0iljk ∆Hy = 2jωPx k Syz j∂xω(Qxx Qzz) . (24b) Sme Smm 2 Sme 2 Smm −2 − − − η0k0 k0 2η0k0 2k0 (23) Similarly, we reduce (22) so that the multipolar moments In these expressions, we retrieve the 4 bianisotropic sus- in (24) are only expressed in terms of the field components ceptibility tensors that were already present in (2) and gain Ex, Ez and Hy as well as their derivatives along x and z. several additional tensors that appear due to spatial dispersion The resulting expression is provided in (25), where we have and the presence of the quadrupolar moment densities. The not included all the prefactors in (23) for convenience. Note constants present in (23) are used to ensure that all tensors are that the symmetry relations (28) are already taken into account dimensionless. We emphasize that the tensorial components in (25). in (23) are not all independent from each. Indeed, one must To model the metasurface in Fig. 1, we now follow the consider the symmetry relations (28), which always apply, and same procedure as in Sec. II-B, i.e., we aim at finding an the reciprocity conditions (29) and (30), which apply if the expression equivalent to (6) using (25) instead of (5) and (24) metasurface is reciprocal. instead of (3). The problem is obviously more complicated A striking feature that results from the reciprocity conditions now since there are 64 components in (25), whereas there was is that, for instance, the dependence of P on the gradient only 2 components in (5). However, since we are interested 0 of the electric field (via χee,ijk) is reciprocally connected to in developing a model that is physically sound, and thus con- the dependence of Q on the electric field (via Qee,ijk). This sistent with the electromagnetic response of the metasurface

0 0  xx xz xy 0xxx 0xxz 0xzz xyz xyx    χee χee χem χee χee χee χem χem   0 0 Px  zx zz zy 0zxx 0zxz 0zzz zyz zyx  Eav,x  χee χee χem χee χee χee χem χem   Pz  0 0 0 0 0  Eav,z     yx yz yy yxx yxz yzz yyz yyx     My   χme χme χmm χme χme χme χmm χmm   Hav,y   0 0 0 0 0     xxx xxz xxy xxxx xxxz xxzz xxyz xxyx   Qxx Qee Qee Qem Qee Qee Qee Qem Qem   ∂xEav,x    xzy 0 0 0 0xzyz 0xzyx   (25) Qxz  xzx xzz xzxx xzxz xzzz  ∂xE ,z/∂zE ,x   ∝ Qee Qee Qem Qee Qee Qee Qem Qem  ·  av av    0 0 0 0 0   Qzz   zzx zzz zzy zzxx zzxz zzzz zzyz zzyx   ∂zEav,z    Qee Qee Qem Qee Qee Qee Qem Qem     0 0 0 0 0  Syz   yzx yzz yzy yzxx yzxz yzzz yzyz yzyx   ∂zHav,y  Sme Sme Smm Sme Sme Sme Smm Smm  Syx yxx yxz yxy 0yxxx 0yxxz 0yxzz 0yxyz 0yxyx ∂xHav,y Sme Sme Smm Sme Sme Sme Smm Smm 6

(a) (b) Transmission symmetry Reﬂection symmetry

Tf/b(θ)= Tf/b( θ) Rf/b(θ)= Rb/f(θ) z − z θ T (θ) T ( θ) Rb(θ) f f −

x x

Rf(θ) θ θ (c)

Fig. 3: Scattering symmetries and their impacts on the metasurface modeling. (a) Transmission symmetry. (b) Reflection symmetry. (c) Hypersusceptibilities playing a role in the scattering of TM-polarized waves propagating in the xz-plane. The blue and red rectangles indicate hypersusceptibilities that lead to asymmetric transmission and reflection, respectively, whereas the black rectangles correspond to hypersusceptibilities that self-cancel due to reciprocity. scattering particles, we will see that most of these components in (25) lead to asymmetric angular scattering. Therefore, these either do not contribute or are in fact dependent on each other. components must not be taken into account in our case To simplify (25), the first fundamental concept that must because they would lead to an unphysical model. To find be considered is reciprocity. We know that the metasurface out which components lead to asymmetric angular scattering, in Fig. 1 is reciprocal since we do not bias it with a time- we substitute (4) into (24) and investigate the scattering odd external quantity [49]–[51]. Therefore, the conditions (29) response of each of the remaining 28 independent components and (30) must be satisfied implying that the lower triangular in (25) individually. Those who do not satisfy the transmission part of (25) depends on its upper triangular part. This reduces symmetry condition in Fig. 3a are highlighted by a blue the number of independent unknowns from 64 to 36. More- rectangle in Fig. 3c, whereas those who do not satisfy the over, reciprocity also implies that some components in (25) reflection symmetry condition in Fig. 3b are highlighted by a cancel each other when substituted in (24). Indeed, this is red rectangle. We are now left with only 12 independent and zy yz for instance the case of χem and χme, which are connected relevant components in (25). to each other by reciprocity and that end up canceling each We now further simplify (24) by using (4) along with the other when substituted in (24a) via Pz and My, respectively. 12 independent components that are left in (25), which leads All the terms that cancel each other like this are highlighted by to a black rectangle in Fig. 3c. This further reduces the number of independent unknowns in (25) to 28. 2 (1 + R T ) − = sec(θ)˜χyy + sin(θ) tan(θ)˜χzz The second concept that must be considered is the angular jk (1 R + T ) mm ee − (26a) response of the metasurface scattering particle. The one that is 1 0 + cos2(2θ) sec(θ)Q xzxz, used in our case is a simple cylinder that exhibits 2 particularly 4 ee important symmetries. Among others, it is mirror symmetric 2 (1 R T ) xx 1 2 0zzzz through the yz and the xy planes. The mirror symmetry − − = cos(θ)˜χ + cos(θ) sin (θ)Q˜ , jk (1 + R + T ) ee 4 ee through the yz-plane implies that its angular transmission (26b) coefficient in the xz-plane is symmetric with respect to θ [19], i.e., T (θ) = T ( θ) where ‘f’ and ‘b’ stand for forward f/b f/b − (+z) and backward ( z) illumination directions, respectively. where kx and kz have been replaced by kx = k sin θ and − Similarly, the mirror symmetry through the xy-plane implies kz = k cos θ for convenience. In simplifying these expressions, that its angular reflection coefficient is the same for illumi- we have grouped together several of the remaining components nations impinging on either sides of the metasurface [19], in (25) because they exhibit the same angular scattering 0xzxz i.e., Rf/b(θ) = Rb/f(θ). These two cases of symmetric angular response. At the exception of Qee , which presents a unique transmission and reflection are depicted in Figs. 3a and 3b, angular scattering response, all the other terms have been respectively. Now, it turns out that some of the components replaced by arbitrarily named variables indicated by a tilde 7 in (26) and that are defined as To provide the system of equations with enough information

0 about the angular response of the metasurface, we select the yy yy yxz ◦ ◦ ◦ χ˜mm = χmm jχme , (27a) 3 angles for (26a) to be θ = [28 , 74 , 86 ], and the 2 angles − ◦ ◦ zz zz 0zyx 1 0yxyx 0yxz for (26b) to be θ = [6 , 76 ]. Once the 5 unknowns in (26) χ˜ = χ + jχ + S + 2jχ , (27b) ee ee em 4 mm me are obtained, we then reverse these equations, as done in (7), 0 1 0 χ˜xx = χxx jχ xyz + S yzyz, (27c) to predict the angular reflection and transmission coefficients ee ee − em 4 mm for all other angles. The resulting predicted transmission ˜0zzzz 0xxzz 1 0xxxx 1 0zzzz coefficient is plotted in Fig. 4a, and the relative error between Qee = Qee Qee Qee . (27d) − 2 − 2 the simulated data shown in Fig. 2a and the predicted one is This means that even though there are 12 independent un- plotted in Fig. 4b. knowns that could, in principle, be solved for in (25), we Beside the appearance of some undesired sharp features, cannot compute them all since some of them have identical the overall predicted transmission coefficient is in much better effects on the metasurface scattering response and would thus agreement with the simulated data in Fig. 4a than it is in lead to an ill-conditioned system of equations3. We shall Fig. 2c, especially for large angles of incidence. In fact, the instead restrict our attention to the 5 unknowns that are left total error, defined as the sum of the difference between simu- in (26). To obtain them, we note that the system (26) is in lated and predicted data for each combination of incidence an- fact made of two independent equations that can thus be gle and frequency point, is about 2.1 times smaller when using solved individually. Accordingly, Eq. (26a), which contains the multipolar modeling approach than the purely dipolar one. 3 unknowns, is solved by using the simulated metasurface For higher frequencies (small wavelength-to-period ratios), the reflection and transmission coefficients at 3 different angles multipolar model also provides a better agreement with the of incidence, whereas Eq. (26b), which contains 2 unknowns, simulated data although it remains unable to fully capture is solved by using only 2 different angles of incidence. the complex angular scattering response of the metasurface. It is thus hypothesized that a model with multipolar moments beyond the quadrupolar ones should provide an even better 1 agreement. 80 0.8

60 IV. CONCLUSIONS 0.6 We have shown that the conventional GSTCs, which only 40 0.4 include dipolar moments, are rather limited when it comes to modeling the angular scattering response of a metasurface, 20 0.2

Incidence angle (°) especially in the case of high frequencies (small wavelength- to-period ratios) and/or large incidence angles. In order to im- 0 0 200 300 400 500 600 prove this model, we have extended it by adding quadrupolar Frequency (THz) moments and a plethora of associated susceptibility tensors (a) that result from the presence of spatially dispersive effects. We have demonstrated that the derived multipolar GSTCs provide 1 an improvement by at least a factor of 2 in the modeling accu- 80 racy of the angular response of metasurfaces. Note that, in the 0.8 proposed example, this accuracy improvement was achieved 60 by only considering 5 effective (hyper)susceptibilities, which 0.6 is only 3 more than the number of susceptibilities used in 40 0.4 the dipolar model. Taking into account more terms or higher- order multipolar components should most like help providing 20 0.2

Incidence angle (°) an even better accuracy. In addition, the presence of the large number of hypersus- 0 0 200 300 400 500 600 ceptibility components in the multipolar GSTCs may not be Frequency (THz) only attractive for improving its modeling accuracy but also (b) for providing many new degrees of freedom for controlling electromagnetic waves. The ability to engineer not only the Fig. 4: Multipolar modeling of the metasurface in Fig. 1 with dipolar but also the quadrupolar responses of a metasurface P = 225 nm, D = 200 nm, H = 400 nm and n = 2.55. (a) is thus expected to be instrumental in the design of multi- Predicted transmission coefﬁcient. (b) Relative transmission functional angular processing metasurfaces. error.

3While the method that we propose here is unable to retrieve all the APPENDIX A independent components in (25), it may be possible to compute them using the SYMMETRY RELATIONS alternative technique proposed in [30], which consists in selectively canceling the fields or their derivatives at the metasurface to excite only the desired Due to the way the quadrupolar tensor Q is defined and due component. to the symmetric-antisymmetric splitting in (21), the following 8 symmetry relations apply [22], [33], [50], [52] [17] A. Momeni, H. Rajabalipanah, A. Abdolali, and K. Achouri, “Gener- alized optical signal processing based on multioperator metasurfaces Qee,ijk = Qee,jik,Qem,ijk = Qem,jik, synthesized by susceptibility tensors,” Physical Review Applied, vol. 11, 0 0 0 0 no. 6, p. 064042, 2019. χ ,ijk = χ ,ikj, χ ,ijk = χ ,ikj, ee ee me me (28) [18] X. Zhang, Q. Li, F. Liu, M. Qiu, S. Sun, Q. He, and L. Zhou, 0 0 0 0 “Controlling angular dispersions in optical metasurfaces,” Light: Science Q ,ijkl = Q ,jikl,S ,ijkl = S ,ijlk, em em me me & Applications 0 0 0 0 , vol. 9, no. 1, p. 76, Dec. 2020. Qee,ijkl = Qee,jikl = Qee,ijlk = Qee,jilk. [19] K. Achouri and O. J. F. Martin, “Angular scattering properties of metasurfaces,” IEEE Transactions on Antennas and Propagation, vol. 68, APPENDIX B no. 1, pp. 432–442, 2020. [20] R. E. Raab and O. L. De Lange, Multipole Theory in Electromagnetism: RECIPROCITY RELATIONS Classical, Quantum, and Symmetry Aspects, with Applications, ser. The reciprocity relations for a bianisotropic medium Oxford Science Publications. Oxford : New York: Clarendon Press ; Oxford University Press, 2005, no. 128. are [49], [51], [53] [21] V. M. Agranovich and V. Ginzburg, Crystal optics with spatial dispersion, and excitons. Springer Science & Business Media, 2013, vol. 42. χ ,ij = χ ,ji, χ ,ij = χ ,ji, χ ,ij = χ ,ji, (29) ee ee mm mm em − me [22] C. Simovski, Composite Media with Weak Spatial Dispersion. CRC Press, 2018. while those for higher-order components are given by [22], [23] D. J. Cho, F. Wang, X. Zhang, and Y. R. Shen, “Contribution of the [33], [52] electric quadrupole resonance in optical metamaterials,” Phys. Rev. B, 0 0 vol. 78, no. 12, p. 121101, Sep. 2008. χee,kji = Qee,ijk, χmm,kij = Smm,ijk, [24] J. Petschulat, J. Yang, C. Menzel, C. Rockstuhl, A. Chipouline, 0 0 P. Lalanne, A. Tuennermann,¨ F. Lederer, and T. Pertsch, “Understanding χ ,kij = Sme,ijk, χ ,kji = Qem,ijk, em − me − (30) the electric and magnetic response of isolated metaatoms by means of 0 0 0 0 a multipolar field decomposition,” Optics Express, vol. 18, no. 14, p. Qee,klij = Qee,ijkl,Qem,klij = Sme,ijkl, 0 0 − 14454, Jul. 2010. Smm,klij = Smm,ijkl. [25]S.M uhlig,¨ C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials, vol. 5, no. 2-3, pp. 64–73, Jun. 2011. REFERENCES [26] P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New Journal of Physics, vol. 14, no. 9, [1] N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, p. 093033, Sep. 2012. “Perfect metamaterial absorber,” Physical review letters, vol. 100, no. 20, [27] A. B. Evlyukhin, C. Reinhardt, U. Zywietz, and B. N. Chichkov, “Col- p. 207402, 2008. lective resonances in metal nanoparticle arrays with dipole-quadrupole [2] D. Schurig, J. J. Mock, B. Justice, S. A. Cummer, J. B. Pendry, interactions,” Phys. Rev. B, vol. 85, no. 24, p. 245411, Jun. 2012. A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at [28] A. D. Yaghjian, A. Alu,` and M. G. Silveirinha, “Homogenization microwave frequencies,” Science, vol. 314, no. 5801, pp. 977–980, 2006. of spatially dispersive metamaterial arrays in terms of generalized [3] A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics electric and magnetic polarizations,” Photonics and Nanostructures - with metasurfaces,” Science, vol. 339, no. 6125, 2013. Fundamentals and Applications, vol. 11, no. 4, pp. 374–396, Nov. 2013. [4] N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nature [29] M. G. Silveirinha, “Boundary conditions for quadrupolar metamaterials,” Mater., vol. 13, no. 2, pp. 139–150, 2014. New Journal of Physics, vol. 16, no. 8, p. 083042, Aug. 2014. [5] M. Chen, M. Kim, A. M. Wong, and G. V. Eleftheriades, “Huygens’ [30] F. Bernal Arango, T. Coenen, and A. F. Koenderink, “Underpinning metasurfaces from microwaves to optics: A review,” Nanophotonics, hybridization intuition for complex nanoantennas by magnetoelectric vol. 7, no. 6, pp. 1207–1231, Jun. 2018. quadrupolar polarizability retrieval,” ACS Photonics, vol. 1, no. 5, pp. [6] K. Achouri and C. Caloz, Electromagnetic Metasurfaces: Theory and 444–453, 2014. Applications. Wiley-IEEE Press, 2021. [31] E. Poutrina and A. Urbas, “Multipole analysis of unidirectional light [7] T. Niemi, A. Karilainen, and S. Tretyakov, “Synthesis of polarization scattering from plasmonic dimers,” Journal of Optics, vol. 16, no. 11, transformers,” IEEE Trans. Antennas Propag., vol. 61, no. 6, pp. 3102– p. 114005, 2014. 3111, June 2013. [32] V. E. Babicheva and A. B. Evlyukhin, “Metasurfaces with Electric [8] K. Achouri, M. A. Salem, and C. Caloz, “General metasurface synthesis Quadrupole and Magnetic Dipole Resonant Coupling,” ACS Photonics, based on susceptibility tensors,” IEEE Trans. Antennas Propag., vol. 63, vol. 5, no. 5, pp. 2022–2033, May 2018. no. 7, pp. 2977–2991, Jul. 2015. [33] K. Achouri and O. J. Martin, “Extension of lorentz reciprocity and [9] A. Epstein and G. Eleftheriades, “Passive lossless Huygens metasurfaces poynting theorems for spatially dispersive media with quadrupolar for conversion of arbitrary source field to directive radiation,” IEEE responses,” arXiv preprint arXiv:2102.08197, 2021. Trans. Antennas Propag., vol. 62, no. 11, pp. 5680–5695, Nov 2014. [34] C. Holloway, E. F. Kuester, J. Gordon, J. O’Hara, J. Booth, and D. Smith, [10]A.D ´ıaz-Rubio, V. S. Asadchy, A. Elsakka, and S. A. Tretyakov, “From “An overview of the theory and applications of metasurfaces: the two- the generalized reflection law to the realization of perfect anomalous dimensional equivalents of metamaterials,” IEEE Antennas Propag. reflectors,” SCIENCE ADVANCES, p. 11, 2017. Mag., vol. 54, no. 2, pp. 10–35, April 2012. [11] G. Lavigne, K. Achouri, V. S. Asadchy, S. A. Tretyakov, and C. Caloz, [35] C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoring “Susceptibility derivation and experimental demonstration of refract- wave fronts with reflectionless sheets,” Phys. Rev. Lett., vol. 110, p. ing metasurfaces without spurious diffraction,” IEEE Trans. Antennas 197401, May 2013. Propag., vol. 66, no. 3, pp. 1321–1330, March 2018. [36] C. Pfeiffer and A. Grbic, “Bianisotropic metasurfaces for optimal [12] A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog Computing polarization control: Analysis and synthesis,” Phys. Rev. Applied, vol. 2, Using Reflective Plasmonic Metasurfaces,” Nano Letters, vol. 15, no. 1, p. 044011, Oct 2014. pp. 791–797, Jan. 2015. [37] M. Selvanayagam and G. Eleftheriades, “Polarization control using [13] A. Chizari, S. Abdollahramezani, M. V. Jamali, and J. A. Salehi, “Analog tensor Huygens surfaces,” IEEE Trans. Antennas Propag., vol. 62, optical computing based on a dielectric meta-reflect array,” Optics no. 12, pp. 6155–6168, Dec 2014. Letters, vol. 41, no. 15, p. 3451, Aug. 2016. [38] K. Achouri, B. A. Khan, S. Gupta, G. Lavigne, M. A. Salem, and [14] S. Abdollahramezani, A. Chizari, A. E. Dorche, M. V. Jamali, and C. Caloz, “Synthesis of electromagnetic metasurfaces: principles and J. A. Salehi, “Dielectric metasurfaces solve differential and integro- illustrations,” EPJ Applied Metamaterials, vol. 2, p. 12, 2015. differential equations,” Optics Letters, vol. 42, no. 7, p. 1197, Apr. 2017. [39] K. Achouri and C. Caloz, “Design, concepts, and applications of [15] H. Babashah, Z. Kavehvash, S. Koohi, and A. Khavasi, “Integration electromagnetic metasurfaces,” Nanophotonics, vol. 7, no. 6, pp. 1095– in analog optical computing using metasurfaces revisited: Toward ideal 1116, 2018. optical integration,” Journal of the Optical Society of America B, vol. 34, [40] M. M. Idemen, Discontinuities in the Electromagnetic Field. John no. 6, p. 1270, Jun. 2017. Wiley & Sons, 2011. [16] T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic [41] E. F. Kuester, M. Mohamed, M. Piket-May, and C. Holloway, “Averaged computing of spatial differentiation,” Nature Communications, vol. 8, transition conditions for electromagnetic fields at a metafilm,” IEEE no. 1, p. 15391, Aug. 2017. Trans. Antennas Propag., vol. 51, no. 10, pp. 2641–2651, Oct 2003. 9

[42] K. Achouri and O. J. Martin, “Fundamental properties and classiﬁcation of polarization converting bianisotropic metasurfaces,” arXiv preprint arXiv:2008.05776, 2020. [43] K. Achouri, M. A. Salem, and C. Caloz, “Improvement of metasurface continuity conditions,” in 2015 International Symposium on Antennas and Propagation (ISAP). IEEE, 2015, pp. 1–3. [44] M. Albooyeh, S. Tretyakov, and C. Simovski, “Electromagnetic charac- terization of bianisotropic metasurfaces on refractive substrates: General theoretical framework,” Ann. Phys., vol. 528, no. 9-10, pp. 721–737, 2016. [45] J. D. Jackson, Classical electrodynamics, 3rd ed. New York, NY: Wiley, 1999. [46] C. H. Papas, Theory of electromagnetic wave propagation. Courier Corporation, 2014. [47] A. Serdiukov, I. Semchenko, S. Tretyakov, and A. Sihvola, Electromag- netics of bi-anisotropic materials-Theory and Application. Gordon and Breach science publishers, 2001, vol. 11. [48] G. B. Arfken and H. J. Weber, “Mathematical methods for physicists,” 1999. [49] J. Kong, Electromagnetic wave theory, ser. A Wiley-Interscience publi- cation. John Wiley & Sons, 1986. [50] J. D. Jackson, Classical Electrodynamics, 3rd ed. Wiley, 1998. [51] C. Caloz, A. Alu,` S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Leger,´ “Electromagnetic Nonreciprocity,” Physical Review Ap- plied, vol. 10, no. 4, Oct. 2018. [52] “Light propagation in cubic and other anisotropic crystals,” Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, vol. 430, no. 1880, pp. 593–614, Sep. 1990. [53] E. J. Rothwell and M. J. Cloud, Electromagnetics. CRC press, 2018.