# Impedance Operator Description of a Metasurface with Electric and Magnetic Dipoles

Authors: Didier Felbacq
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A metasurface made of a collection of nanoresonators characterized by an electric dipole and a magnetic dipole was studied in the regime where the wavelength is large with respect to the size of the resonators. An effective description in terms of an impedance operator was derived.


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## Body

## 1. Introduction

Metasurfaces are the bidimensional analogue of metamaterials [1]. They are made of basic cells containing resonant elements disposed periodically or not on a surface. The cells can contain one or several elements (nanoparticles, nanoantennas, etc.) whose aim is to produce a dephasing of some incident field, whose wavelength is large with respect to the size of the cells. The collection of cells is then able to produce a collective effect resulting in refracting and transmitting properties far from that of a homogeneous material [2]. For instance, it was shown in [3] that they could be used to design flat achromatic lenses. Metasurfaces are thus a wide generalization of subwavelength diffraction gratings, where the basic cell can be quite complicated. The periodic structure of gratings can be relaxed so as to obtain interesting effect on the polarization of the incident field [4]. In this context, we study the field diffracted by a periodic set of linear nanoresonators, electromagnetically characterized by their scattering matrix S. We are interested in the regime where the wavelength is much larger than the size of the nanoresonators. We proceed to an asymptotic analysis related to homogenization theory [5, 6]. The field diffracted by the structure is derived and it is shown that it is characterized by an impedance operator. Our results extend to electric and magnetic dipoles results in [7] where only electric dipoles were considered.


## 2. Setting of the Problem

The structure under study is made of an infinite number of resonators invariant alongz, periodically disposed at points Mp=(p×d,0), where d is the period and p∈Z. Each scatterer at position Mp is characterized in the frequency domain by a scattering matrix S(ω) as well as by electromagnetic parameters γs and δs. We assume that the wavelength in vacuum λ=2πc/ω is much larger than the size of the resonators, which are assumed to be contained in a cylinder of diameter a. We therefore consider a linearly polarized incident field Uie-iωtez, where Ui(x,y)=ei(αx±βy) and α=k0sin⁡θ, θ is an angle of incidence, and k0=ω/c. According to the polarization γs and δs can be either the relative permittivity or relative permeability.The stationary scattering problem considered therefore reads the following: find a fieldU∈Hloc⁡1(R2) (i.e., the space of locally square integrable functions with a locally square integrable weak derivative) satisfying div⁡(γ∇U)+k02δU=0 in the sense of Schwartz distribution in R2 and such that Us=U-Ui satisfies the outgoing wave conditions:(i)
Fory>0:Im⁡(Us∇Us)>0.(ii)
Fory<0:Im⁡(Us∇Us)<0.The functionsγ and δ are equal, respectively, to γs and δs inside the scatterers and to 1 outside. The following holds; see [8].Proposition 1.
Apart possibly from a discrete set of wavenumbersk1,k2,…, the field U exists and is unique.Our point is to provide a simplified expression of the scattering problem by means of an impedance operator.LetTd denote the translation along x of amplitude d; that is, Td(f)(x)=f(x-d). For later purpose, we define Y∗=-π/d,π/d, K=2π/d, αn=α+nK, and βn=k02-αn2, for n∈Z. For αn2>k02, we impose iβn<0. In the following, we denote T=R/dZ.LetH=-div⁡(γ∇·)-k02δ and let us define for α∈Y∗ the field of Hilbert spaces Lα2(T)=u;ue-iαx∈L2(T). It obviously holds the following.Lemma 2.
The commutator ofTd and H vanishes: [Td,H]=0.Applying Floquet-Bloch analysis [9], we obtain the following.Proposition 3.
The operatorH has a direct integral decomposition H=∫Y∗⊕Hαdα/K where Hα=-div⁡(γ∇·)-k02δ with domain D(Hα)=Lloc2(Ry;Lα2(T))∩Hloc2(Y×R).


## 3. Multiple Scattering Approach

The incident field has the expansion [10] Ui(x,y)=∑nanJn(k0r)einθ. For one scatterer alone, the incident field gives rise to a field Ups(x,y)=∑nsnpφn(x,y) where φn(x,y)=Hn(1)(k0r)einθ. Jn (resp. Hn(1)) is a Bessel (resp., Hankel) function. For the infinite set of scatterers, this gives a diffracted field that reads (1)Usx,y=∑p,nsnpφnx-pd,y.Multiple scattering theory [10] allows to write that for p=0(2)b0^=1-SΣ-1Sa^,where b0^=(…,b-n0,…,bn0,…)T and a^=(…,a-n,…,an,…)T. The matrix Σ is given by (3)Σk0,α0=∑m≠0eiα0mT0m.Here Proposition 3 was used through the introduction of a Bloch phase eiα0m. In this expression T0mpq=eip-qθ0mHp-q1(k0md); that is,(4)T0m=⋱⋮⋮⋮⋯⋯H0k0md-ϵmH1k0mdH2k0md⋯⋯ϵmH1k0mdH0k0md-ϵmH1k0md⋯⋯H2k0mdϵmH1k0mdH0k0md⋯⋯⋮⋮⋮⋱,where ϵm=sign⁡(m) (note that eiθ0m=-sign⁡(m)). The following series [11] indexed by p appear:(5)Σp=∑m≠0eimα0ϵmpHpk0md.And the entries of the matrix Σ(k0,α0) are Σ(k0,α0)pq=Σp-q.In the regime wherek0a≪1, the cylinder can be described by a 3×3 scattering matrix (this corresponds to an electric dipole and a magnetic dipole) and the field by 3 coefficients b-1, b0, and b1 [10]. Therefore, only 3 series are involved: Σ0, Σ1, and Σ2. It holds that(6)Σk0,α0=Σ0-Σ1Σ2Σ1Σ0-Σ1Σ2Σ1Σ0.In the extreme limit (a≪d) where the scatterers are very small as compared to the wavelength and the period, the scattering matrix S(ω) reduces to a scalar matrix s0(ω): the scatterers are thus dipoles with a dipole moment along ez and the only involved series is Σ0; this situation was addressed in [7]. The multiple scattering relation (2) then becomes(7)b00k0,α0=1-S0Σ0-1S0,where the series Σ0 can be written [7, 12, 13]:(8)Σ0k0,α0=∑m≠0eikmdH0k0md.For the more general case of an electric dipole and a magnetic dipole the following asymptotic expressions hold in the limit k0d≪1 [14].Proposition 4.
Consider the following:(9)Σ0k0,α0~-1-2iπγ+2iπln⁡2Kk0+Kπβ0,Σ1k0,α0~α0πk0-2+iKβ0,Σ2k0,α0~Kπk02β02-α02β0-iπk02K23-β02+α02.


## 4. Scattering Properties of the Meta Surface

DefinePz=b00 the electric moment and M=(Mx,My)=(b10+b-10),i(b10-b-10) the magnetic moment. We write m=Mx+iMy, m∗=Mx-iMy, and κn+=(αn,βn), κn-=(αn,-βn).We can now state the following.Theorem 5.
The total field has the following expression:(10)y>a:Ux,y=eiα0x-β0y+∑nrneiαnx+βny,y<a:Ux,y=∑ntneiαnx-βny,where (11)rn=KπβnPz+iM·κn+,tn=δn0+KπβnPz+iM·κn-.Proof.
We start with the following relation, obtained from Poisson formula:(12)∑nH0k0r-ndexeiknd=2d∑n1βneiαnx+βny.Upon applying the operator ∂=∂x+i∂y, using the fact that the series on the right-hand side is normally convergent (thanks to the term eiβny) and using the relation ∂H0(r)=-H1(r)eiθ, we obtain (13)-k0∑nφ1x-nd,yeiknd=2d∑niαn-βnϵβneiαnx+βny,where ϵ=sign⁡(y). Therefore we get (14)Usx,y=2d∑nb00+iαnb10+b-10+b-10-b10βnϵβneiαnx+βny.The result follows after some simple algebra.A simple, but interesting corollary is as follows.Corollary 6.
Asn~+∞,(15)rn~Kπm=2Kb-10π,tn~Kπm∗=2Kb10π.We are nota priori in the homogenization regime where k0d≪1 and hence there can be several reflected and transmitted orders. In expression (10), the propagative waves (i.e., the diffractive orders of the grating) correspond to the βn’s that are real. They are labelled by the finite set U=n∈Z,βn∈R+. The evanescent waves are labelled by the infinite set U+=n∈Z,iβn∈R-.


## 5. Impedance Operator Formulation

Our point is now to replace the set of nanoresonators by a metasurfaceS which is simply the line y=0. This requires to specify the boundary conditions there in terms of an impedance operator.So, consider the continuation of the fieldU obtained by making a=0 in (10). The continued field is still denoted as U. It is a singular distribution. To handle this situation, let us introduce the following fields of Sobolev spaces (16)Hαs/2Y=u=∑nuneiαnx;∑n1+kn2s/2un2<+∞and the dual spaces (17)Hα-s/2Y=u=∑nuneiαnx;∑n1+kn2-s/2un2<+∞.Let Z be the pseudodifferential operator defined by Z[u]=v, where(18)ux=∑nuneiαnx,vx=∑niβnuneiαnx.It is straightforward to show the following.Proposition 7.
Z is continuous and invertible from Hα-s/2(Y) to Hα-s/2-1(Y), for s>1.The inverse ofZ is the admittance operator Y defined by v=Yu.Let us denoteF as an element of Hα-s/2(Y)×Hα-s/2-1(Y), representing the discontinuity of the field and its derivative through S. The traces of the field and its derivative are (19)F+=Ux,0+∂U∂yx,0+,F-=-Ux,0-∂U∂yx,0-.By definition, it holds that F++F-=F. The Calderòn projectors [15] P+ and P- are defined by F+=P+F and F-=P-F. The preceding shows the following.Proposition 8.
Consider the following:(20)P+=121YZ1,P-=121-Y-Z1.Obviously, it holds thatP++P-=I, (P+)2=P+, P-2=P-, and P+P-=P-P+=0, as it should.The transmission conditions onS can be written as follows: (21)Ux,0+-Ux,0-=2iKMyπ∑neiαnx=2iKMyπ∑n1iβnrniβnrneiαnx,∂U∂yx,0+-∂U∂yx,0-=∑niβnδn0+rn+tneiαnx=∑niβnδn0+rn+tnδn0+rnrneiαnx.This suggests to define the following pseudodifferential operators, acting on u=∑nuneiαnx: (22)Xux=∑n1+tnδn0+rnuneiαnx,Wux=2iKMyπ∑n1rnuneiαnx.Both rn and tn are bounded with respect to n (see (15)); hence see the following.Proposition 9.
The pseudodifferential operatorsX and W are isomorphism of Lα2(T).These conditions can be rewritten conveniently in the operator form.Theorem 10.
The traces of the fieldU(x,y) diffracted by the metasurface under the illumination of a plane wave Ui satisfy the impedance conditions:(23)TF+=F-,where (24)T=1-YW-ZX1.The operatorT is the transfer matrix of the meta surface. The discontinuity of the (effective) field U at y=0 is due to the existence of a magnetic dipole moment. In the homogenization limit of very large wavelengths, that is, larger than the wavelengths corresponding to magnetic resonances and larger than twice the period, there are only one transmitted and one reflected wave; the evanescent waves can be discarded and the magnetic resonances have no effect; we then have the following.Proposition 11.
For a wavelengthλ≫2d and larger than the largest magnetic resonance, the propagative part of the field is given by (25)y>0:Ux,y=eiα0x-β0y+r0eiα0x+β0y,y<0:Ux,y=t0eiα0x-β0y,where r0=2b00/dβ0 and t0=1+r0. The transfer matrix becomes (26)T=10-2iβ0r1+r1.The form found for the transfer matrix matches that obtained in another context in [12].


## 6. Conclusion

The scattering of a plane wave by a grating of nanorods was described in the framework of metasurfaces by exhibiting an impedance condition. This takes into account both the electric and the magnetic dipoles characterizing each nanorod. This study can be generalized to higher multipoles but also to nonperiodic, for instance, quasiperiodic, structures [16] and to elementary scatterers deposited on an arbitrary smooth surface as well. A similar approach was used in [17] to study the coupling of a quantum emitter with the modes supported by the metasurface; see also [18]. The proposed formalism can be used in this context to analyze the role of resonances.



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