Design, concepts, and applications of electromagnetic metasurfaces

Auteurs:

Authors

: Karim Achouri et Christophe Caloz

Date: 2018

Type:

Référence:

Citation

:

Achouri, K. & Caloz, C. (2018). Design, concepts, and applications of electromagnetic metasurfaces. Nanophotonics, 7(6), p. 1095-1116. doi:10.1515/nanoph-2017-0119

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Journal Title: Nanophotonics (vol. 7, no 6)

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Rev

iew

art

ic

le

Kar

im

Achour

i*

and

Chr

is

tophe

Ca

loz*

Des

ign

,

concepts

,

and

app

l

icat

ions

o

f

 e

lectromagnet

ic

metasur

faces

https://doi.org/10.1515/nanoph-2017-0119

Received November 30, 2017;revised April 4, 2018; accepted April 6, 2018

Abstract: The paper overviews ourrecent work onthe synthesis of metasurfaces and related concepts and appli -cations. The synthesisis based on generalized sheet tran -sition conditions (GSTCs) with a bianisotropic surface susceptibility tensor model of the metasurface structure. We first place metasurfaces in a proper historical context and describe the GSTC technique with some fundamental susceptibilitytensor considerations. Onthis basis, we next provide anin-depth development of our susceptibility -GSTC synthesis technique. Finally, we present five recent metasurface concepts and applications, which cover thetopics of birefringenttransformations, bianisotropic refraction,light emission enhancement,remote spatial processing, and nonlinear second-harmonic generation. Keywords: bianisotropy; electromagnetics; metasurfaces.

1

Introduct

ion

Metamaterials reached a peak of interest in the first decade of the 21st century. Then, due to their fabrication complexity, bulkiness, and weight andtheirlimitations in terms of losses, frequency range, and scalability, they becameless attractive and were progressively superseded by their two-dimensional counterparts, the metasurfaces [1–5].

Theidea of controlling electromagnetic waves with electromagneticallythinstructuresis clearly not a new concept. The first example is probably that of Lamb who studiedthe reflection andtransmissionfrom an array of metallicstrips, already backin1897[6]. Later,inthe

1910s, Marconi used arrays of straight wirestorealize polarization reflectors [7]. These first two-dimensional electromagnetic structures were later followed by a great diversity of systemsthat emerged mainly withthe deve lop-ments of the radar technology during World War II. Many of these systems date back to the 1960s. The Fresnel zone plate reflectors,illustratedin Figure 1A, were based onthe concept ofthe Fresnellens demonstrated almost 150 years earlier and used in radio transmitters [8]. The frequency -selective surfaces (FSS), illustrated in Figure  1B, were developed as spatial filters [9, 10]. The reflectarray anten -nas [11] were developed as the flat counterparts of para -bolic reflectors and were initially formed by short-ended waveguides[12]. They werelater progressivelyimproved and the short-ended waveguides were replaced with microstrip printable scattering elements in the late 1970s [13, 14], as shown in Figure 1C. The transmissive counter -parts ofthereflectarrays arethetransmitarrays, which were used as arraylens systems and date backtothe 1960s [15–17]. They werefirstimplementedintheform oftwo interconnected planar arrays of dipole antennas, one for receiving and onefortransmitting, where each antenna on the receiver side was connected via a delay line to an antenna onthetransmitside, as depictedin Figure1D. Throughthe 1990s,thetransmitarrays evolvedfrominter -connected antenna arraystolayered metallicstructures that were essentiallythefunctional extensions of FSS [18–20] with efficiency limited by the difficulty to control thetransmission phase over a 2π range while maintaining a high enough amplitude. Finally, compact quasi-trans -parenttransmitarrays or phase-shiftingsurfaces, ableto cover a 2π-phase range, were demonstratedin 2010[21].

The aforementioned Fresnel lenses, FSS, reflectar -rays, and transmitarrays are the precursors of today’s “metasurfaces”.1

*Corresponding authors: Karim Achouri and Christophe Caloz, Department of Electrical Engineering, École Polytechnique de Montréal, Montréal, GC H3T 1J4, Canada,

e-mail: karim.achouri@polymtl.ca(K. Achouri);

christophe.caloz@polymtl.ca(C. Caloz). http://orcid.org/0000 -0002-6444-5215(K. Achouri)

1 Thus far, and throughout this paper, we essentially consider meta -surfacesilluminated by wavesincident onthethem under a nonzero angle with respect to their plane, i.e. space waves, which represent the metasurfaces leading to the main applications. However, meta -surface may also be excited within their plane,i.e. by surface waves orleaky waves, asin Refs.[22–26].

From a general perspective, metasurfaces can be used to manipulate the polarization, the phase, and the ampli -tude of electromagneticfields. Arich diversity of metas -urface applications has been reported in the literature to date and many more are expected to emerge. These appli -cations are too numerous to be exhaustively cited. Some ofthe mostsignificant ones arereportedin Refs.[27–35] (polarization transformations), Refs. [36–42] (absorp-tion), and Refs. [43–56] (wavefront manipulations). More sophisticated metasurfaces,transforming both phase and polarization, have beenrecentlyrealized. Thisincludes metasurfaces producing beams possessing angular orbital momentum[57] or vortex waves[58–63], holograms[64, 65], and stable beamtraction[66]. Additionally, nonrec ip-rocal transformations [67–71], nonlinear interactions [72– 74], analog computing[75, 76], and spatialfiltering[77–79] have also been reported.

To deploytheirfull potential, metasurfaces must be designed efficiently. Thisrequires asolid “model”that bothsimplifiesthe actual problem and providesinsight intoits physics. Metasurfaces are best modeled, accord -ing to Huygens principle, as “surface polarization current sheets” via continuous(locally homogeneous) bian iso-tropic surface susceptibilitytensorialfunctions.Insert -ing the corresponding surface polarization densities into Maxwell equations results in electromagnetic sheet tran -sition conditions, which consistinthe key equationsto solveinthe design of metasurfaces.

The objective ofthis paperistwofold. First,it will present a generalframeworkforthe synthesis ofthe afore -mentioned metasurface surface susceptibilityfunctions for arbitrary(amplitude, phase, polarization, propaga -tion direction, and waveform) specified fields. From this point,the physicalstructure(material and geometry of thescattering particles,substrate parameters, andlayer configuration, thickness, and size) is tediously but straightforwardly determined, afterthe discretization of thesusceptibilityfunctions, usingscattering parameter mapping. The synthesis of metasurfaces has beenthe objective of many researches in recent years [42, 80–90]. Second,the paper willshow howthissynthesisframe -work provides a general perspective ofthe electromag -netic transformations achievable by metasurfaces and then present subsequent concepts and applications.

2

Sheet

trans

it

ion

cond

it

ions

The generalsynthesis problem of a metasurfaceis rep-resented in Figure 2. As mentioned in Section 1, the metasurfaceis modeled as an electromagnetic sheet (zero-thicknessfilm).2_{Inthe most general case, a metasurfaceis}

x z y z = 0 Ly L_x ψt(r) χ () =? t << λ ψi (r) ψr (r) ρ

Figure 2: Metasurface synthesis problem.

The metasurface to be synthesizedliesin the xy-plane atz = 0. The synthesis procedure consists offinding the susceptibility tensors characterizing the metasurface, χρ(),in terms of specified arbitrary incident[ψi(r)], reflected[ψr(r)], and transmitted[ψt(r)] waves.

Source A B Source C φ1 φ2 φ3 φ4 φ5 φ6 Source D

Figure 1: Examples of two-dimensional wave manipulating struc -tures:(A) Fresnel zone plate reflector,(B) reflectarray,(C)intercon -nected arraylens, and(D) FSS.

2 This approximationisjustified by thefact that a physical metasur -faceis electromagnetically verythin, sothatit cannot support signifi -cant phase shifts and related effects, such as Fabry-Perot resonances.

made of an array of polarizablescattering particlesthat induce both electric and magneticfield discontinuities. Itistherefore necessaryto expressthe discontinuities ofthesefields asfunctions ofthe electric and magnetic surface polarization densities(P and M). Therigorous boundary conditions that apply to such aninterface have been originally derived by Idemen[91].

For a metasurface lyingin the xy-plane at z = 0, these transition conditionsfollowfromtheideathat allthe quantities in Maxwell equations can be expressed in the followingform:

() =0 () {()} N k(), k k fz fz= +

∑

where the function f(z) is discontinuous at z = 0. The first term of the right-hand side of Eq. (1) is the “regular part” of f, which corresponds to the value of the function eve -rywhere except at z = 0, whereasthesecondtermisthe “singular part” of f, whichis an expansion overthe kth derivatives of the Dirac delta distribution (corresponding tothe discontinuity off andthekth derivatives off).

Most often,theseriesin Eq.(1) may betruncated at N = 0, sothat onlythe discontinuities ofthefields are takeninto account, whereasthe discontinuities ofthe derivatives ofthefields are neglected. Withthistrunca -tion, the metasurface transition conditions, known as the generalizedsheettransition conditions(GSTCs), are found as3 ˆ×∆ =jω − ×∇ˆ M_z, z H P z (2) 0 0 ˆ=_j Pz ˆ, ∆ ωµ ε   × −∇_{ }×   E z M z (3) ˆ⋅ =−∇⋅∆ , z D P (4) 0 ˆ⋅∆ =− ∇⋅µ , z B M (5)

wheretheterms ontheleft-handsides ofthe equations correspondtothe differences ofthefields on both sides of the metasurface, which may be expressed as

0 ,t ,i ,r 0 ˆ ( ), , , , u z u u u u x yz ∆Ψ=⋅u Ψ∆ +₌−=Ψ −Ψ +Ψ = (6) where Ψ represents any ofthefields E, H, D, or B,the subscriptsi,r, andt denotetheincident,reflected, and

transmitted fields, and P and M are the electric and magnetic surface polarization densities, respectively.

In the general case of a linear bianisotropic metasur -face, these polarization densities are related to the acting (orlocal)fields, E_act andH_act, by[93–95]:

0 ee act em act 0 1 _, N N c α α = ⋅ + ⋅ P ε E H ₍₇₎ mm act mm act 0 1 _, Nα Nα η = ⋅ + ⋅ M H E (8)

wheretheα abtensorsrepresentthe polarizabilities of a given

scatterer, N isthe number ofscatterers per unit area,c₀is the speed of light in vacuum, and η0 is the vacuum imped

-ance.4 _{Thisis a “microscopic” description ofthe metasur}

-face response that requires an appropriate definition of thecoupling between adjacentscattering particles.Inthis work, we usethe concept of susceptibilitiesratherthan the polarizabilities to provide a “macroscopic” description of the metasurface, which allows a direct connection with material parameterssuch asε _rand .µ _r To bring aboutthe susceptibilities,relations(7) and(8) can betransformed by notingthatthe actingfields, atthe position of asca t-tering particle,can be defined asthe averagetotalfields minusthefield scattered bythe considered particle[97],i.e.

part act= av−scat.

E E E Thecontributions ofthe particle may be expressed byconsideringthe particle as acombination of electric and magnetic dipoles contained within a small disk, andthefield scatteredfromthis disk can be relatedto P and M bytakinginto accountthe coupling with adjacent scatter -ing particles. Therefore,the actingfields arefunctions ofthe averagefields andthe polarization densities. Upon substi tu-tion of this definition of the acting fieldsin Eqs. (7) and (8), the expressions ofthe polarization densities become

0 ee av em av 0 1 _, c χ χ = ⋅ + ⋅ P ε E H (9) mm av me av 0 1 _, χ _ηχ = ⋅ + ⋅ M H E (10)

wherethe averagefields are defined as

,t ,i ,r ,av av ( ) ˆ , , , , 2 u u u u u x yz Ψ Ψ Ψ Ψ =⋅uΨ = + + = (11) where Ψ correspondstoE orH.

3 Note that these relations can also be obtained following the more traditionaltechnique of boxintegration, as demonstratedin Ref.[92].

4 Despite beingindeed quite general,these relations are still restrict -edto “linear” and “time-invariant” metasurfaces. Thesynthesis of nonlinear metasurfaces has been approached using extended GSTCs in Ref.[96].

3

Suscept

ib

i

l

ity

tensor

cons

iderat

ions

Before delvinginto the metasurface synthesis,itisimpor -tant to examine the susceptibility tensors in Eqs. (9) and (10)inthelight offundamental electromagnetic consid -erations pertainingto reciprocity, passivity, andloss.

The “reciprocity” conditionsfor a bianisotropic meta -surface, resultingfromthe Lorentztheorem[93], read

T T T

ee ee,mm mm ,me em,

χ χ χ= =χ χ =− χ (12) where the superscript T denotes the matrix transpose operation.5

Addingthe property oflosslessness, resultingfromthe bianisotropic Poyntingtheorem[93], restricts Eq. (12)to

T T T

ee ee,mm mm ,me em,

χ χ χ₌∗ ₌χ∗ χ ₌χ∗ ₍₁₃₎

which characterize a simultaneously passive, lossless, and reciprocal metasurface.

Conditions(12) and(13) establishrelations between different susceptibility components of the constitutive tensors. Therefore, requiringthe metasurfaceto be rec ipro-cal or reciprocal and lossless/gainless, as often practically desirable, reduces the number ofindependent susceptibi l-ity components[80, 98, 99] and hence reducesthe diversity of achievablefieldtransformations, as will be shown next.

4

Metasur

face

synthes

is

4

.1

Genera

l

concepts

We follow here the metasurface synthesis procedure6

introduced in Ref. [80], which seems to be the most

general approach reported to date. This procedure consists ofsolvingthe GSTC equations(2)–(5)to deter -mine the unknown susceptibilitiesin (9) and (10) required for the metasurface to perform the electromagnetic trans -formation specifiedinterms oftheincident,reflected, and transmittedfields. Notethat Eqs.(4) and(5) areredundant in system(2)–(5) duetothe absence ofimpressed sources, so that Eqs. (2) and (3) are sufficient to fully describe the metasurface and synthesizeit. Consequently, only the transverse(tangentialtothe metasurface) components of the specified fields, explicitly apparent in (6) and (11), areinvolvedinthesynthesis, althoughthesefields may generallyincludelongitudinal(normaltothe metasur -face) components as well. Accordingtothe uniqueness theorem,thelongitudinal components ofthefields are automatically determinedfromthetransversefields.

GSTC equations(2) and(3)form aset of( inhomo-geneous) coupled partial differential equations due tothe partial derivatives ofthe normal components of the polarization densities, P_z andM_z. The resolution of the corresponding inverse problem is nontrivial and requiresinvolved numerical processing.In contrast,if P_z = M_z = 0,the differentialsystemreducesto asimple algebraic system of equations, most conveniently adm it-ting closed-formsolutionsforthesynthesized suscep-tibilities. For this reason, we will focus on this case in this section, whereas atransformation example with nonzero normal susceptibilities will be discussed in Section 4.4.

Enforcingthat P_z = M_z = 0  may a priori seemto rep-resent animportantrestriction, particularly, as weshall see,inthesensethatitreducesthe number of degrees offreedom ofthe metasurface. However,thisis not a major restriction as a metasurface with normal polariza -tion currents can generally bereducedto an equivalent metasurface with purely tangential polarization currents, accordingto Huygenstheorem. This restriction mostly affectstherealization ofthescattering particlesthat are thenforbiddento exhibit normal polarizations, which ultimatelylimitstheir practicalimplementation.7

5 These conditions areidenticaltothosefor a bianisotropic medium [93, 94], except that the metasurfaces in Eq. (12) are surface instead of volume susceptibilities.

6 The “synthesis” procedure consists of determiningthe physical metasurfacestructureforspecifiedfields. Theinverse procedureis the “analysis”, which consists of determining the fields scattered by a given physical metasurface structuresfor a givenincidentfield and is generally coupled (typically iteratively) with the synthesis for the efficient design of a metasurface[100]. The overall design procedure thus consists ofthe combination ofthe synthesis and analysis opera -tions. This paper focuses on the direct synthesis of the susceptibility functions, asthisisthe mostimportant aspectforthe understanding of the physical properties of metasurfaces, the elaboration of related concepts, andthe development of resulting applications.

7 Moreover, in the particular case where all the specified waves are normaltothe metasurface,the excitation of normal polarization densities does not induce any discontinuity in the fields. This is be -causethe correspondingfields, and hencetherelatedsusceptibili -ties, are not functions of the x andy coordinates, so that the spatial derivatives of Pz andMzin Eqs. (2) and (3) are zero,i.e. do notinduce

any discontinuityin thefields across the metasurface. Thus, suscep-tibilities producing normal polarizations can beignored, and only tangential susceptibility components must be considered, when the metasurfaceis synthesizedfor normal waves.

Substitutingthe constitutiverelations(9) and(10) intothe GSTCs (2) and (3) withM_z = P_z = 0leadsto

0 ee av 0 em av ˆ×∆ =jω χ⋅ +jkχ ⋅ , z H ε E H (14) 0 mm av 0 me av ˆ j jk , ∆E z× =ωµχ ⋅ +H χ ⋅E (15) where k₀ = ω/c₀ is the free-space wavenumber and where the susceptibility tensors only contain the tangential sus -ceptibility components. This system can also be writtenin matrixform ,av ee ee em em ,av ee ee em em ,av me me mm mm ,av me me mm mm , xx xy xx xy x y yx yy yx yy y x xx xy xx xy x y yx yy yx yy y x E H E H H E H E ∆ χ χ χ χ ∆ χ χ χ χ ∆ χ χ χ χ ∆ χ χ χ χ               _{= ⋅}                     (16)

where the tilde symbolindicates normalized susceptibili -ties, related to the nonnormalized susceptibilities in (14) and (15) by ee ee em em ee ee em em me me mm mm me me mm mm ee ee em em 0 0 0 0 ee ee em em 0 0 0 0 me me 0 0 xx xy xx xy yx yy yx yy xx xy xx xy yx yy yx yy xx xy xx xy yx yy yx yy xx xy j j j j k k j j j j k k j j j k k χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ ω ω χ χ χ χ ω ω χ χ       =         − − − − = − − − ε ε ε ε mm mm 0 0 me me mm mm 0 0 0 0 . xx xy yx yy yx yy j j j j j k k χ χ ωµ ωµ χ χ χ χ ωµ ωµ                −            (17)

System (16) contains four equations for 16 unknown susceptibilities.Itistherefore heavily underdetermined and cannot be solved directly.8 _{Thisleaves us withtwo dis}

-tinct resolution possibilities.

Thefirst possibility would beto reducethe number of susceptibilities from 16 to 4 to obtain a fully determined (full-rank)system. Becausethere exists many combina -tions of susceptibility quadruplets,9 _{different sets can be}

chosen, each ofthem naturally correspondingto different field transformations. This approach thus requires an educated selection of the susceptibility quadruplet thatis the most likely to enable the specified operation, within existing constraints.10

These considerations immediately suggest that a second possibility would beto augmentthe number of fieldtransformation specifications,i.e. allowthe metasur -faceto perform moreindependenttransformations, which may be of great practicalinterestin some applications. We would havethus ultimatelythree possibilitiestoresolve (16): (a) reducing the number of independent unknowns, (b)increasingthe number oftransformations, and(c) a combination of (a) and (b).

As we shall see in the forthcoming sections, the number N of physically or practically achievabletrans -formationsfor a metasurface with P susceptibility para -meters, N(P),is not trivial; specifically, N(P) = P/4, which may be expected from a purely mathematical viewpoint, is not alwaystrue.

4

.2

Four

-parameter

trans

format

ion

We now provide an example for the approach where the number of susceptibility parameters has been reduced to 4, or P = 4,sothatsystem(16)is offull-rank nature. We thus haveto selectfour susceptibility parameters and set all the others to zeroin Eq. (17). We decide to consider the simplest case of a monoanisotropic (eight parameters

em,me 0, uv

χ = u, v = x, y) axial(four parameters_χeeuv_,mm= 0_for u ≠ v, u, v = x, y) metasurface, which is thus characterized by the four parameters_{χ ee}xx,

ee, yy χ mm xx χ andmm, yy χ so that Eq. (16) reducestothe diagonal system:

,av ee ,av ee ,av mm ,av mm 0 0 0 0 0 0 . 0 0 0 0 0 0 xx x y yy y x xx x y yy y x E H E H H E H E ∆ χ ∆ χ ∆ χ ∆ χ                = ⋅                   (18)

This metasurfaceis a “birefringent”structure[101], with decoupled x- andy-polarized susceptibility pairs

ee mm 0 ,av 0 ,av , y xx yy x x y j H j E E H ∆ ∆ χ χ ω ωµ = = ε (19) and

8 Evenifit would besolved,this would probablyresultin anin -efficient metasurface, asit would use more susceptibility terms than requiredto accomplishthe specifiedtask.

9 Mathematically, the number of combinations would be 16!/ [(16 − 4)!4!] = 1820, but only asubset ofthese combinationsrepre -sents physically meaningful combinations.

10 Forinstance,the specification of areciprocaltransformation, corresponding to the metasurface propertiesin Eq. (12), would au to-matically precludethe selection of off-diagonal pairsforχee,mm.

ee mm 0 ,av 0 ,av , y , yy x xx y x j E j H E H ∆ ∆ χ χ ω ωµ − − = = ε (20)

respectively.11 _{In these relations, according to Eqs. (6) and}

(11), ∆H_y = H_y,t – (H_y,i + H_y,r), ∆E_x,av =(E_x,t + E_x,i + E_x,r)/2, and so on. By synthesis,the metasurface withthe susceptibilities (19) and (20) will exactly transform the specifiedincident fieldinto the specified reflected and transmittedfields,in an arbitrary fashion, except for the constraint of reciproc -ity as the susceptibility tensorin Eq. (18)inherently satis -fies Eq. (12).

It should be noted that the example of Eq. (18), with four distinct susceptibility parameters,is a very particular case of afour-parametertransformation asthe compo-nentsin Eqs. (19) and (20) are decoupledfrom each other, whichisthe origin of birefringence. Now, birefringence may be considered as a “pair” of distinct and independ -enttransformations(onefor x-polarization and onefor y-polarization),i.e. N(4) = 2 > 4/4. Thus,thespecification offoursusceptibility parameters mayleadto morethan onetransformation, which, by extension, already sug -geststhat P susceptibilities mayleadto morethan P/4 transformations, as announced in Section 4.1 and will be further discussedin Section 4.3.

Thus far, the fields have not be explicitly specifiedin the metasurface described by Eq. (18). Because the meta -surface can perform arbitrary transformations under the reservation of reciprocity,it mayforinstance by used for polarizationrotation, which willturnto be a most instructive example here. Consider the reflectionless metasurface, depictedin Figure 3, whichtransformsthe polarization of a normallyincident plane wave. Thefields correspondingtothistransformation are

i(, ) cosxy=ˆ ( /8) sinπ +ˆ ( /8π ), E x y (21) i 0 1 _{ˆ ˆ} (, ) [ sxy in( /8π ) cos( /8)π ], η = − + H x y ₍₂₂₎ r(, )0xy=, E (23) r(, )0xy=, H (24) and t(, ) cosxy=ˆ (11 /24π ) s+ˆin(11 /24π ), E x y (25) t 0 1 _{ˆ ˆ} (, )= [ sxy in(11 /24π ) cos(11 /24π )]. η− + H x y ₍₂₆₎

Insertingthesefieldsinto Eqs. (6) and (11) and substi -tutingthe resultin (19) and (20) yieldsthe susceptibilities

ee mm 0 1_.5048, xx yy _j k χ χ= =− (27) ee mm 0 0_.88063. yy xx _j k χ χ= = ₍₂₈₎

Note that, in this example,12 _{the aforementioned}

doubletransformation reducesto a singletransformation, N(4) = 1 = 4/4, because the specified fields possess both x-and y-polarizations. The susceptibilities do not depend on the position as the specified transformation, being purely normal, only rotates the polarization angle and does not affectthe direction of wave propagation.

The negative and positive imaginary natures of

ee mm xx yy

χ χ= andχ χeeyy= mmxx in (27) and (28) correspond to

absorption and gain, respectively. These features may be understood by noting, withthe help of Figure 3,that polarization rotationis accomplished here by attenuation

x x y y E E H H z π 8 11π 24

Figure 3: Polarization reflectionless rotating metasurface. The metasurface rotates the polarization of alinearly polarized normallyincident plane wavefrom the angle π/8 to the angle 11π/24 with respect to the x-axis(rotation ofπ/3). The metasurface is surrounded on both sides by vacuum,i.e.η1 = η2 = η0.

11 Ifthetwo electric andthetwo magneticsusceptibilitiesin(19) and (20) are equalto each other (χ χee xx = andyyee χmm xx= ),χmmyy the

mono-anisotropic metasurfacein Eq. (18) reduces to the simplest possible case of a monoisotropic metasurface and hence performs the same operationfor x- andy-polarized waves.

12 Incidentally, the equality between the electric and magnetic sus -ceptibilities results from the specification of zero reflection in addi -tionto normalincidence. Thereader may easily verifythat,inthe presence of reflection,the equalities do not hold.

and amplification of (E_x,i, H_y,i) and (E_y,i, H_x,i), respectively. Moreover,this metasurface can rotatethe polarization “only” by the angle π/3 when the incident wave is polar -ized at aπ/8 angle.13 _{This example certainly represents an}

awkward approachto rotatethefield polarization. A morereasonable approachisto consider a “gyro-tropic” metasurface, wherethe only nonzero suscepti -bilities areχ eexy,χ eeyx,χ mmxy andχ Thmmyx. is correspondsto a

different quadruplet oftensor parametersthanin Eq.(18), which illustrates the aforementioned multiplicity of pos -sible parameter set selection. With these susceptibilities, system (16) yieldsthefollowing relations:

ee 0 ,av , y xy y j H E ∆ χ ω = ε (29) ee 0 ,av , yx x x j H E ∆ χ ω − = ε (30) mm 0 ,av , y xy y j E H ∆ χ ωµ − = (31) mm 0 ,av , yx x x j E H ∆ χ ωµ = (32)

which, upon substitution of the fieldsin (21) to (26), become ee mm 0 1_.1547, xy xy _j k χ χ= =− ₍₃₃₎ ee mm 0 1_.1547. yx yx _j k χ χ= = (34)

Contrary to the susceptibilitiesin (27) and (28), those in(33) and(34) performthe specified π/3 polarization rotation “irrespectively” oftheinitial polarization ofthe incident wave duetothe gyrotropic nature ofthe meta -surface.It appearsthatthesesusceptibilities violatethe reciprocity conditionsin Eq.(12), andthe metasurface isthus “nonreciprocal”, whichis a necessary condition for polarizationrotation withthis choice ofsusceptibili -ties. Thus, the metasurface is a Faraday rotation surface,

whose direction of polarizationrotationisindependent ofthe direction of wave propagation[70,102]. However, contrary to conventional Faraday rotators [93], this meta -surface is also reflectionless due to the presence of both electric and magnetic gyrotropic susceptibility compo-nents (Huygens matching). The positive and negative imaginarysusceptibilitiesindicatethatthe metasurface is simultaneously active and lossy, respectively. It is this combination of gain and loss that allows perfect rotation inthislossless design. This designis naturally appropriate if Faraday rotation is required. However, it is not optimal in applications not requiring nonreciprocity, i.e. rec ipro-cal gyrotropy, wheretherequiredloss and gain would clearly represent a drawback.

Reciprocal gyrotropy may be achieved using biani -sotropic chirality,i.e. whichinvolvesthe parameterset

em em, ,me xx yy xx

χ χ χ and_{χ me}yy._{Followingthesamesynthesis} pro-cedure as before, wefind

em em 0 2 , 3 xx yy _j k χ =χ =− ₍₃₅₎ me me 0 2 . 3 xx yy _j k χ =χ = ₍₃₆₎

The corresponding metasurfaceisreadily verifiedto bereciprocal, passive, andlossless, asthe susceptibil -ity (35) and (36) satisfies condition (13). Therefore, if the purpose of the metasurfaceis to simply perform polariza -tion rotationin a given direction, without specificationfor the opposite direction,this designisthe most appropriate ofthethree discussed, asitis purely passive,lossless, and workingfor allincident polarizations.

Note that metasurfaces (33)–(36) correspond to N(4) = 1 = 4/4.

4

.3

More-than-

four

-parameter

trans

format

ion

Inthe previous section, we have seen how system(16) can besolved byreducingthe number ofsusceptibilitiesto P = 4 parametersto matchthe number of GSTCs equations and we have also seen some of the resulting single-trans -formation (N = 1, e.g. monoisotropic structure) or double -transformation (N = 2, e.g. birefringence) metasurface possibilities.

However, as mentionedin Section 4.1,the general system of equation(16), givenits16 degrees offreedom (16  susceptibility components), corresponds to a

13 If,forinstance,theincident was polarized along x only,then only thesusceptibilitiesin Eq.(27) would be excited andtheresulting transmittedfield would still be polarized alongx,just with a reduced amplitude with respect to that of the incident wave due to the loss induced bythese susceptibilities.

metasurface withthe potential capabilityto perform “more transformations” than a metasurface with 4 parameters, or generallyless than 16 parameters, N(16) > N(P < 16). In whatfollows, we will see how system(16) can be solvedfor several “independent”transformations, whichincludes the possibility of differently processing wavesincident from either sides. To accommodate for the additional degrees of freedom, a total of four wave transformations are consideredinstead of only one as donein Section 4.2, so that (16) becomes a full-rank system. The correspond -ing equations relatedto system(16) maythen be writtenin the compactform:

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 ee ee em em ee ee em em me me mm mm me me mm mm y y y y x x x x y y y y x x x x xx xy xx xy yx yy yx yy xx xy xx xy yx yy yx yy H H H H H H H H E E E E E E E E ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ      ₌                      

1,av 2,av 3,av 4,av 1,av 2,av 3,av 4,av 1,av 2,av 3,av 4,av 1,av 2,av 3,av 4,av

, x x x x y y y y x x x x y y y y E E E E E E E E H H H H H H H H       ⋅        (37)

wheresubscripts1–4indicatethe electromagneticfields corresponding to four distinct and “independent” sets of waves.14 _{Thesusceptibilities can be obtained by matrix}

inversion conjointly with the normalization (17). The resulting susceptibilities will, in general, be all different from each other. This meansthatthe corresponding meta -surfaceis both active/lossy and nonreciprocal.

Consider, for example, a metasurface with P = 8 parameters.In such a case, system(16)is underdeter -mined asitfeaturesfour equationsin eight unknowns. This suggeststhe possibilityto specify morethan one transformation, N > 1. Let us thus consider, for instance, a monoanisotropic (eight-parameter) metasurface and see whether such a metasurface can indeed perform

twotransformations. The correspondingsystemfortwo transformation reads

1 2 1 2 1 2 1 2 1,av 2,av ee ee 1,av 2,av ee ee 1,av 2,av mm mm 1,av 2,av mm mm 0 0 0 0 . 0 0 0 0 y y x x y y x x xx xy x x yx yy y y xx xy x x yx yy y y H H H H E E E E E E E E H H H H ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ χ χ χ χ χ χ χ χ      ₌                  ⋅              (38)

This system (38), being full-rank, automatically admits a solution for the eight susceptibilities, i.e. N = 2. The only question is whether this solution complies with practical design constraints. For instance, the electric and magnetic susceptibility submatrices are nondiagonal and may therefore violate the reciprocity condition (12). If nonreciprocity is undesirable or unrealizable in a practi -calsituation,then one would havetotry anotherset of eight parameters.

Ifthis eight-parameter metasurface performs onlytwo transformations, then one may wonder whatis the differ -ence with the four-parameter birefringent metasurface in Eq.(18), which can also providetwotransformations with just four parameters. The difference is that the two-transformation property of the metasurface in Eq. (18) is restricted to the case where the fields of the two transfor -mations are orthogonally polarized,15 _whereasthe

two-transformation property of the metasurface in Eq. (38) is completely general.

As an illustration of the latter metasurface, consider thetwotransformations depictedin Figure 4. Thefirst transformation, shownin Figure 4A, consists of reflecting at 45° a normally incident plane wave. The second trans -formation, shownin Figure 4B, consists offully absorbing anincident waveimpinging onthe metasurface under 45°. In both cases, the transmitted fieldis specified to be zero forthefirst andsecondtransformations. Thetransverse components of the electric fields for the two transforma -tions are, atz = 0, given by

i,1 r,1 r 2_{(ˆ ˆ),} 2_{( cos ˆ ˆ) ,} 2 2 x jkx e θ − = + = − + E x y E x y (39)

14 Itis also possibleto solve a system of equationsthat containsless than these 16 susceptibility components. In that case,less than four wavetransformations should be specified sothatthe system remains fully determined. Forinstance,twoindependent wavetransforma -tions (possessing both x- and y-polarizations) could be solved with eight susceptibilities. Similarly,three wavetransformations could be solved with 12 susceptibilities.

15 Forinstance,ifthefields ofthefirsttransformation are only x- polarized, the fields of the second transformation are only y-polarized.

i,2 ₂2(cosiˆ ˆ) x, jkx e θ − = + E x y (40) respectively.

The synthesis is then performed by inserting the electricfield(39) and(40), andthe corresponding mag -neticfields,into Eq.(38). The susceptibilities arethen straightforwardly obtained by matrixinversionin Eq.(38). Forthe sake of conciseness, we do not givethem here, but we point outthattheyinclude nonreciprocity,loss and gain, and complex spatial variations.

This double-transformationresponseis verified by full-wavesimulation andtheresultingsimulations are plottedin Figure 5. Thetwosimulationsinthisfigure have been realized in the commercial FEM software

COMSOL, wherethe metasurfaceisimplemented as a thin material slab of thickness d = λ₀/100.16 _{The simula}

-tion correspondingtothetransformation of Figure 4A isshownin Figure  5A, whereasthesimulation corre -sponding to the transformation of Figure 4Bis shownin Figure 5B. The simulated results are in agreement with thespecification[Eqs.(39) and(40)], exceptforsome scattering due to the nonzero thickness of the full-wave slab approximation.

The examplejust presented, where both transfor-mations 1 and 2include allthe components ofthefields, correspondsto N(8) = 2 = 8/4,i.e. N(P) = P/4. However, in the same manner as the birefringent metasurface of (19) and (20), featuring N(4) = 2 > 4/4, i.e. specifi -cally N(P) = P/2,the metasurfacein Eq.(38) maylead to N(P) > P/4. This depends essentially on whetherthe specifiedtransformations are composed offieldsthat are either only x- or y-polarized or both x- and y-polar -ized. Thetwotransformations given bythefields(39) and (40) are both x- and y-polarized, which thus limits the number of transformations to N(P) = P/4. If the transformation given by Eq.(40) was specified suchthat E_{iy, 2} = 0(i.e. no polarization along y),thenthis would release degrees offreedom and hence allow atriple transformation,i.e.N(8) = 3 > 8/4.In addition,ifthefirst transformation, given by Eq.(39), also hadtransverse components of the electric field polarized only along x or y,then we could achieve N(8) = 4 > 8/4transforma -tions. These considerationsillustratethe necessityto perform educated selectionsin the metasurface synthe-sis procedure, as announcedin Section 4.1.

4

.4

Metasur

face

w

ith

nonzero

norma

l

po

lar

izat

ions

Thusfar, we have discardedthe possibility of normal polarizations by enforcing P_z = M_z = 0 in Eqs. (2)–(5). This is not only synthesis-wise convenient, as this suppresses thespatial derivativesin Eqs.(2)–(5), but alsotypically justified by the fact that any electromagnetic field can be produced by purely tangential surface currents/polariza -tions accordingto Huygenstheorem.It was accordingly claimedin Ref.[103] that these normal polarizations, and corresponding susceptibility components, do not bring

x z θr = 45° θi = 45° λ0 λ0 100 A x z 100 B

Figure 4: Example of double-transformation metasurface:(A) first transformation[corresponding to subscript 1in Eq.(38)]: the normallyincident plane waveisfully reflected at a 45° angle and(B) second transformation[corresponding to subscript 2in Eq.(38)]: the obliquelyincident plane waveisfully absorbed.

0.5 0 1 0 5 10 15 20 25 –5 –10 0 5 10 –5 –10 –15 –20 –25 x/λ0 z/ λ0 B A 0.5 0 1 0 5 10 15 20 25 –5 –10 0 5 10 –5 –10 –15 –20 –25 x/λ0 z/ λ0

Figure 5: Eight-parameter metasurface simulations: COMSOL simulated normalized absolute value of the total electricfield corresponding to(A) the transformationin Figure 4A and(B) the transformationin Figure 4B.

16 Thesynthesistechnique yieldsthesusceptibilitiesfor anideal zero-thickness metasurface. However,the metasurface sheet may be “approximated” by an electrically thin slab of thickness d (d  λ) with volume susceptibility corresponding to a diluted version of the surface susceptibility,i.e. χvol = χ/d [80].

about any additional degrees of freedom and can thus be completelyignored.Itturns outthatthis claimis gener -ally nottrue:infact, P_z and M_z provide extra degrees of freedom that allow a metasurface to perform alarger number of distinct operations for “differentincident field configurations” and “at differenttimes”.

Huygenstheorem “exclusively” appliesto a “single” (arbitrarily complex) combination ofincident,reflected, and transmitted waves. This means that any metasurface, possibly involving normal polarizations, which performs the specified operation for such a single combination offields, can bereducedto an equivalent metasurface with purelytransverse polarizations. However, Huygens theorem does not applyto case of wavesimpinging onthe metasurface at “different times”. Indeed, it is in this case impossibleto superimposethe differentincident wavesto form a total incident field as they are not simultaneously illuminatingthe metasurface. Consequently, a purelytan -gential description of the metasurface is incomplete, and normal polarizationsthus become necessaryto perform the synthesis.

Infact,the presence ofthese normalsusceptibility components greatly increases the number of degrees of freedom asthe susceptibilitytensors are now 3 × 3 matri -cesinstead of 2 × 2 asin Eq.(16). This meansthat,for the four relevant GSTCs equations, we have now access to 36 unknown susceptibilities,instead of only 16, which increasesthe potential number of electromagnetictrans -formationsfrom 4to 9, providedthatthesetransfor -mationsincludefieldsthat areindependentfrom each other.

Thesynthesis of metasurfaces with nonzero normal polarization densities may be performedfollowing similar procedures asthose already discussed. As before, one needsto balancethe number of unknownsusceptibili -ties to the number of available equations provided by the GSTCs. Depending onthe specifications,this may become difficult as many transformations may be required to obtain afull-ranksystem. Additionally,ifthespecified transformationsinvolve changingthe direction of wave propagation, then system (2)–(5) becomes a coupled system of partial differential equationsinterms ofthe sus -ceptibilities as the latter would now depend on the posi -tion. This generally preventsthe derivation of closed-form solutions ofthesusceptibilities, whichshouldrather be obtained numerically. However, we will now provide an example of asynthesis problem, wherethesusceptibili -ties are obtainablein closedform.

More specifically, we discuss the synthesis and analysis of a reciprocal metasurface with controllable angle-dependent scattering [104–106]. To synthesize this

metasurface, we considerthethree “independent”17_trans

-formations depictedin Figure 6.

Specifying these three transformations allows one to achieve a relatively smooth control of the scatter -ing response of the metasurface for any nonspecified incidence angles.

For simplicity, we specifythatthe metasurface does not change the direction of wave propagation, which implies that it is uniform, i.e. susceptibilities are not functions of position. Moreover, we specify thatitis also reflectionless and only affectsthetransmission phase of p-polarized incident waves asfunction oftheirincidence angle.

To designthis metasurface, we considerthatit may be composed of a total number of 36 susceptibility com -ponents. However, as allthe wavesinteracting withthe metasurface are p-polarized, most ofthese susceptibilities will not be excited by these fields and thus will not play a role in the electromagnetic transformations. Accordingly, the only susceptibilitiesthat are excited bythefields are

ee ee em ee em ee ee em 0 0 0 0 0 0 , 0 0 0, 0 0 0 xx xz xy zx zz zy χ χ χ χ χ χ χ χ         =_{ } =_{ }         (41) me me me mm mm 0 0 0 0 0 0 0 , 0 0, 0 0 0 0 0 0 yx yz yy χ χ χ χ χ         =_{ } =_{ }         (42) Ei,1 Ei,3 θt,1 θi,1 θi,3 Et,1 Ei,2 θt,3 Et,3 Et,2 x z

Figure 6: Multiple scatteringfrom a uniform bianisotropic reflec -tionless metasurface.

17 It is essential to understand that these three sets of incident and transmitted waves “cannot” be combined, bysuperposition,into a single incident and a single transmitted wave because these waves are not necessarily impinging on the metasurface at the same time. This meansthat Huygenstheorem cannot be usedtofind purelytan -gential equivalent surface currents correspondingtothesefields.

wherethe susceptibilities not excited have been setto zero for simplicity. To satisfy the aforementioned specifi -cation of reciprocity, condition (12) must be satisfied. This implies thatχ χxzee = zxee,χemxy=− andχyxme χemzy=− Asχmeyz. a con

-sequence, the total number ofindependent susceptibility componentsin (41) and (42) reducesfrom 9to 6.

Upon insertion of (41) and (42), the GSTCs in Eqs. (2) and (3) become 0 ee ,av (xx ee ,axz v) 0 emxy ,av, y x z y H j E E jk H ∆ =−ω χε +χ − χ (43) 0 mm ,av 0 em ,av em ,av ee ,av ee ,av 0 em ,av ( ) , yy xy zy x y x z xz zz zy x x x z x y E j H jk E E E E H ∆ ωµχ χ χ χ χ ηχ =− + + − ∂ − ∂ − ∂ (44)

where the spatial derivatives only apply to the fields and nottothesusceptibilities asthelatter are notfunctions space duetothe uniformity ofthe metasurface.

System(43) and(44) containstwo equationsinsix unknownsusceptibilities andisthus underdetermined. To solveit, we apply the multiple transformation concept discussedin Section 4.3, which consists of specifying threeindependentsets ofincident,reflected, andtrans -mitted waves. These fields can be simply defined by their respectivereflection(R)18 _{andtransmission(T) coefficients}

as well astheirincidence angle(θ_i).In our case,the metas -urface exhibits atransmission phase shift, φ, whichisthe function oftheincidence angle,i.e.T e=jφθ( )i.

Let us consider,forinstance,thatthethreeincident plane wavesimpinge onthe metasurface at θ_i,1 = − 45°, θ_i,2 = 0°, and θ_i,3 = + 45° and aretransmitted at θ_t = θ_i with transmission coefficientsT₁ = e−jα_,T

2 = 1, andT3 = ejα, where

α is a given phaseshift. Solvingrelation(43) and(44) with these specifications yields the following nonzero susceptibilities: ee ee 0 2 2tan . 2 xz zx k α χ χ= =   _{ } (45) It can be easily verified that these susceptibilities satisfythereciprocity, passivity, andlosslessness cond i-tions (13).

As susceptibility(45) correspondstothe only solution of system (43) and (44)for our specifications and asthese susceptibilities correspondtothe excitation of normal polarization densities, the normal polarizations are indeed useful and provide additional degrees of freedom. This provesthe claiminthefirst paragraph ofthis section that normal polarizationsleadto metasurfacefunctionali -tiesthat are unattainable withoutthem.

Nowthatthe metasurface has beensynthesized, we analyzeits scattering response for all (including non -specified)incidence angles. Forthis purpose, wesubsti -tute susceptibility (45) into (43) and (44) and consider an incident wave, impinging on the metasurface at an angle θ_i, being reflected and transmitted with unknown scatter -ing parameters. System(43) and(44) canthen be solvedto obtainthese unknown scattering parametersfor any value of θ_i. In our case, the analysisis simple because the meta -surfaceis uniform, which meansthatthereflected and transmitted waves obey Snelllaws. The resulting angular -dependenttransmission coefficientis

i i 2 ( ) 1 , 1 2sin( )tan 2 T j θ α θ =−+   −  _{ } (46)

whilethe reflection coefficientis R(θ_i) = 0.

Toillustratethe angular behavior ofthetransmis -sion coefficientin Eq.(46),itis plottedin Figure 7for a

18 Here, R = 0 asthe metasurfaceis reflectionless by specification.

–90 –45 0 45 90 0 0.2 0.4 0.6 0.8 1 A mp lit ud e A –90 –135 –90 –45 0 45 90 135 ∠T = –90° ∠T = 90° ∠T = 0° Ph as e ( °) B θi(°) θ_i(°) |T| =1 |T| =1 |T| = 1 90 –45 0 45

Figure 7: Transmission amplitude(A) and phase(B) asfunctions of theincidence anglefor a metasurface synthesizefor the trans -mission coefficients T = {e−j90°_;1;ej90°_{}(and R = 0) at the respective}

specified phaseshift of α = 90°. As expected,thetrans -mission amplitude remains unityfor allincidence angles, whereasthetransmission phaseis asymmetric around broadside and covers about a 220°-phase range.

4

.5

Re

lat

ions

w

ith

scatter

ing

parameters

and

imp

lementat

ion

We haveseen how a metasurface can besynthesizedto obtain its susceptibilities in terms of specified fields. We shall nowinvestigate howthe synthesized susceptibilities may be relatedtothe shape ofthe scattering particlesthat will constitutethe metasurfacesto berealized. Here, we will only presentthe mathematical expressionsthat relate the susceptibilities to the scattering particles. The reader is referredto[42, 83–90, 98, 107, 108]for moreinformation onthe practical realization ofthese structures.

The conventional method to relate the scattering par -ticle shapeto equivalent susceptibilities(or material para -meters)is based on homogenizationtechniques.Inthe case of metamaterials,thesetechniques may be usedto relate homogenized material parameters to the scattering parameters of the scatterers. From a general perspective, a single isolated scatterer is not sufficient to describe an homogenized medium. Instead, we shall rather consider a periodic array ofscatterers, whichtakesinto account the interactions and coupling between adjacent scat -terers, henceleadingto a more accurate description of a “medium” comparedto asinglescatterer. The suscep-tibilities, which describethe macroscopicresponses of a medium, are thus naturally well-suited to describe the homogenized material parameters of metasurfaces. It follows that the equivalent susceptibilities of a scattering particle may berelatedtothe correspondingscattering parameters, conventionally obtained viafull-wavesimu -lations, of a periodic array made of an infinite repetition ofthat scattering particle[83, 84, 109, 110].

Becausethe periodic array ofscatterersis uniform with subwavelength periodicity, the scattered fields obey Snell laws. More specifically, if the incident wave propa -gates normally with respecttothe array,thenthe reflected and transmitted waves also propagate normally. In most cases, the periodic array of scattering particles is excited with normally propagating waves. This allows one to obtainthe16 “tangential”susceptibility componentsin Eq.(37). However,it does not provide anyinformation aboutthe normalsusceptibility components ofthescat -tering particles. Thisis because,inthe case of normally propagating waves, the normal susceptibilities do not induce any discontinuity ofthefields, as explainedin

Section 4.1. Nevertheless,this method allows oneto match the tangential susceptibilities of the scattering particle to the susceptibilities found from the metasurface synthesis procedure andthat precisely yieldstheidealtangential susceptibility components.

Itis clearthatthescattering particles may,in addi -tiontotheirtangentialsusceptibilities, possess nonzero normal susceptibility components. In that case, the scat -tering response of the metasurface, when illuminated with obliquely propagating waves, will differfromthe expectedideal behavior prescribedin the synthesis. Con -sequently,the homogenizationtechniqueserves only as an initial guess to describe the scattering behavior of the metasurface.19

We will now derivethe explicit expressionsrelating thetangential susceptibilitiestothe scattering parameters inthe general case of afully bianisotropic uniform metas -urface surrounded by different media and excited by nor -mally incident plane waves. Let us first write system (37) inthefollowing compactform:

,

v

A

∆ χ=⋅ (47)

where matrices ,∆ χ andA correspond_v to the field differ -ences, the normalized susceptibilities, and the field aver -ages, respectively.

To obtainthe 16tangential susceptibility components in Eq.(37), we will now definefourtransformations by specifying the fields on both sides of the metasurface. Let us consider that the metasurface is illuminated from the left with an x-polarized normallyincident plane wave. The correspondingincident,reflected, andtransmitted electricfields read

i=ˆ, r=S11xxˆ+S 11yxˆ, =t Sxx21ˆ+S21yxˆ,

E x E x y E x y (48)

where the terms ,uv ab

S witha, b = {1, 2} andu, v = {x, y}, are the scattering parameters with ports 1 and 2 correspond -ing to the left and right sides of the metasurface, respec -tively, as shownin Figure 8.

The medium oftheleft ofthe metasurface hasthe intrinsic impedance η₁, whereas the medium on the right hastheintrinsicimpedance η₂.In additionto Eq.(48),three other cases have to be considered, i.e. y-polarized excita -tion incident from the left (port 1) and x- and y-polarized

19 Notethatis possibleto obtain all 36 susceptibility components of a scattering particle provided that the four GSTC relations are solved for nineindependent sets ofincident,reflected, andtransmitted waves. In practice, such an operation is particularity tedious and is thus generally avoided.

excitationsincidentfromthe right(port 2).Insertingthese fields into Eq. (37) leads, after simplification, to matrices ∆ andA g_v iven below:

2 1 2 11 1 2 21 2 2 2 2 12 1 2 22 2 11 21 1 2 1 2 1 2 1 2 1 2 12 1 2 22 / / / / / / , N N S N S N N S N S N N N N S N N S N N N N S N N S ∆ η η η η η η = − + ⋅ + ⋅ − + ⋅ + ⋅    _{− ⋅ − ⋅ ⋅ + ⋅ ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅}    (49) 11 21 12 22 1 1 1 11 1 1 21 2 1 2 1 12 1 1 22 2 1 _. 2 / / / / / / v A I S S I S S N η N S η N S η N η N S η N S η =  _{+ + + +}     _{− ⋅ + ⋅ − − ⋅ + ⋅}    (50) wherethe matrices ,S ,_abI N and₁ N are₂ defined by

1 2 1 0 , , 0 1 0 1 1 0 , . 1 0 0 1 xx xy ab ab ab yx yy ab ab S S S I S S N N     =_ _ = _{ }      −   =_ _ =_ _ −     (51)

Now, the procedure to obtain the susceptibilities of a given scattering particleis as follows: first, the scattering particleissimulated with periodic boundary conditions (PBCs) and normal excitation. Second, the resulting scat -tering parameters obtainedfromthe simulations are used to definethe matricesin Eqs.(49) and(50). Finally,the susceptibilities correspondingtothe particle are obtained by matrixinversion of Eq. (47).

Alternatively,itis possibleto obtainthe scattering parameters of a normallyincident plane beingscattered by a uniform metasurface with known susceptibilities [111]. This can be achieved by solving Eq.(47)forthe scattering parameters. This leads to the following matrix equation:

1 1 2,

S M M_{= ⋅} − ₍₅₂₎

wherethe scattering parameter matrix, ,S is defined as

11 12 21 22 , S S S S S   =    (53)

and matricesM and₁ M are₂ obtained from Eqs. (47), (49), and (50) by expressing the scattering parametersin terms of the normalized susceptibility tensors. The resulting matricesM and₁ M are₂ given below:

1 2 1 ee em 1 1 2 2 ee em 1 2 1 2 me mm 1 1 1 2 me mm 1 2 / /2 /(2 ) / /2 /(2 ) , /2 /(2 ) /2 /(2 ) M N N N N N N N N N N η χ χ η η χ χ η χ χ η χ χ η =  _{− + ⋅ − − ⋅}    − ⋅ − + ⋅ ⋅ − − ⋅    (54) 2 ee 2 1 em 1 1 ee 2 2 em 1 2 me 1 2 mm 1 1 me 1 2 mm 1 2 /2 / /(2 ) /2 / /(2 ) . /2 /(2 ) /2 /(2 ) M N N N N N N N N N N χ η χ η χ η χ η χ χ η χ χ η =  + + ⋅ + − ⋅     _{+ ⋅ + ⋅ − ⋅ − ⋅}    (55) Thus, the final metasurface physical structure is obtained by mapping the scattering parameters (53) obtainedfromthe discretized synthesized susceptibili -ties by Eq. (52) via Eqs. (54) and (55) to those obtained by full-wave simulating metasurface unit cells with tunable parameters, in an approximate periodic environment, as illustratedin Figure 8. Unfortunately,thereis currently nostraightforward methodtorelatethesusceptibilities foundfromthesynthesistothespecific geometry ofthe scattering particles and one therefore has to strongly rely on numericalsimulations and optimizationsinthat step ofthesynthesis. Notethat grouptheory has been proposed as a potential mathematicaltooltorelatethe geometry of the scattering particles to their effective sus -ceptibilities [112–114]. However, this approach has not been widely used thus far, and less elegant but still quite efficient empirical approaches are preferred atthis point.

A typical unit cell structure allowing arbitrary designs, atleast at microwavefrequencies,is astack of three metalliclayersseparated by dielectricspacers[86, 115–117]. In this structure, the metalliclayers may for instancetaketheshape ofJerusalem crossessimilarto that depictedin Figure 8. The advantage of this geometry is that the x- and y-polarizations may be almostindepen -dently controlled by varying the dimensions of the arms ofthe crosses.

The unit cell may be longitudinally symmetric (same outer layers) or asymmetric. In the symmetric case, thetwolowestresonant modes ofthe entire structure may be decomposedinto even and odd modes, which

x y Port 1 Port 2 z PBC

Figure 8: Full-wave simulation setupfor the scattering parameter techniqueleading to the metasurface physical structurefrom the metasurface model based on Eq.(47).

respectively correspondto electric and magnetic dipolar resonances. Controllingthe overallresponse ofthe unit cellisthus achieved bytuningits electric and magnetic responses, whichleads to different values of the sus -ceptibilities. Duetosymmetry,the magneticresponseis controlled by changing onlythetwo outerlayers as,for the odd mode, there is no current flowing on the middle layer. In contrast, the geometry of the three layers affects the electric response. Accordingly, the general procedure toimplement a unit cellsothatit exhibitsthesuscepti -bilities obtainedfromthesynthesisis asfollows:first, the magneticsusceptibilityis obtained by changingthe geometry ofthe unit cell outerlayers. Then,the electric susceptibilityis obtained by changingthe geometry of the remaining middlelayer. If the unit cellis asymmetric, thenthetwolowest resonances cannot splitinto even and an odd modes as they are now coupled to each other. Due to this asymmetry, and resulting mode coupling, the unit cell exhibits more degrees of freedom, whichincidentally allows onetoimplement bianisotropic metasurfaces.

Implementing unit cell structures made of three metalliclayersis also possible at optical frequencies[83]. However, such structures are practically difficultto realize using current fabrication processes. A more practical optical unit cell structure would be one based on dielec -tricresonators[118–120]. Suchresonators, which natu -rally possess electric and magnetic resonances, typically have a cylindrical or elliptical cross-sectionforisotropic or anisotropicfield control,respectively. Their designis muchsimplerthanthat ofthethree-layerstructures, as theirresponses are generally controlled bytuning only the radius for the cylindrical ones and the ellipticity and orientationforthe elliptical ones.

Duetothesubwavelength unit cell period of meta-surfaces, coupling between the scattering particles is unavoidable. However, strong couplingis not necessarily detrimental.Indeed,it may help reducingthe overall unit cell size viaincreasedfieldinteractions andit may also be leveragedtoincreasethe bandwidth ofthe structure[121]. In cases where coupling would be detrimental,itis practically difficult to address it otherwise than resorting to brute-force numerical optimizationtools offull-wave simulation software. Nevertheless,in the case of spatially varying metasurfaces,itis possibletoreducethe detri -mental effects of coupling by ensuring slow spatial vari -ations compared to the unit cell size. Then, the scattering particles of adjacent unit cells are almostidenticalto each other, meaningthatthe coupling betweenthemis also almostidenticaltothat whenthey wereinitially simulated in a perfectly periodic environment. This greatlysimpli -fiestherealization ofspatially varying metasurfaces as theirimplementation becomes muchless cumbersome.

5

Concepts

and

app

l

icat

ions

Inthe previoussection, we haveshownseveral metas -urface examples as “illustrations” ofthe proposed syn-thesis technique. These examples did not necessarily correspond to practical designs but, in addition to il lus-trating the proposed synthesis technique, they did set up the stageforthe development of useful and practical con-cepts and applications, whichis the object of the present section.

Weshall present herefive of our mostrecent works representing novel concepts and applications of metas -urfaces. In the order of appearance, we present our work on birefringenttransformations[98, 122], bianisotropic refraction[123],light emission enhancement[124], remote spatial processing [125], and nonlinear second-harmonic generation(SHG)[96]. Thereaderis alsoreferredto our related works on nonreciprocal nongyrotropicisolators [126], dielectric metasurfacesfor dispersion engineering [127], and radiation pressure control[128].

5

.1

B

ire

fr

ingent

operat

ions

A direct application of the synthesis procedure discussed in Section 4, and more specifically of the susceptibilities in(19) and(20),isthe design of birefringent metasurfaces. These susceptibilities are split into two independent sets that allow to individually control the scattering of s- and p-polarized waves. In particular, the manipulation of the respectivetransmission phases ofthese orthogonal waves allows severalinteresting operations.

In Ref.[122], we have usedthis approachtorealize half-wave plates, which rotate the polarization oflinearly polarized waves by 90° orinvert the handedness of circu -larly polarized waves, quarter-wave platesthat convert linear polarizationinto circular polarization, a polariza -tion beamsplitterthatspatiallyseparates orthogonally polarized waves, and an orbital angular momentum gen -erator that generates topological charges that depend ontheincident wave polarization. These operations are depictedin Figure 9.

5

.2

“Per

fect”

re

fract

ion

Most refractive operations realized thus far with a metas -urface have been based on the concept of the generalized law of refraction [43], which requires theimplementation of a phase gradient structure. However, such structures are plagued by undesired diffraction orders and are thus not

fully efficient.Itturns outthatthefundamental reasonfor this efficiencylimitationisthe symmetric nature of simple (early asin Ref. [43]) refractive metasurfaces with respect tothez-direction. This can be demonstrated bythefollow -ingad absurdum argument.

Let us consider a passive metasurface surrounded by a given reciprocal20 _{medium and denote the two sides of}

the structure by indices 1 and 2. Assume that this meta-surface “perfectly”refracts(withoutreflection and spu-rious diffraction) a waveincident underthe angle θ₁in side 1tothe angle θ₂in side 2, and assume,ad absurdum, thatthis metasurfaceis “symmetric” withrespecttoits normal. Asitis reciprocally perfectly refracting,itis per-fectly matchedfor both propagation directions:1–2 and 2–1. Consider first wave propagation from side 1 to side 2. Due to perfect matching, the wave experiences no ref lec-tion and, due to perfect refraction, it is fully transmitted to the angle θ₂ in side 2. Consider now wave propagation inthe opposite direction alongthereciprocal(ortime -reversed) path. Now, the wave incident in side 2 has di f-ferent tangential field components than that incident in side 1, assuming θ₂≠ θ₁; therefore, it will see a different impedance, which means that the metasurface is neces -sarily mismatchedinthe direction 2–1. However,thisis in contradiction with the assumption of perfect (recipro -cal) refraction. Consequently,the symmetric metasurface does not produce perfect refraction. Part ofthe waveinc i-dent from side 2 is reflected back; therefore, by rec iproc-ity, matching also did not actually exist in the direction 1–2, so all of the energy of the wave incident under θ₁ in side 1 cannot completely refractinto θ₂; part ofit hasto be

transmitted to other directions in side 2, which typically representsspurious diffraction orders assuming a per i-odic-gradient metasurface. These diffraction orders are consistently visibleinreportedsimulations and exper i-ments of symmetric metasurfacesintendedto perform refraction.

It was demonstratedin Refs.[116,123]that “biani -sotropy” wasthesolutiontorealize perfect(reciprocal) refraction(100% powertransmission efficiencyfrom θ₁ to θ₂). In what follows, we summarize the main synthesis stepsfor such a metasurface.

Let us consider the bianisotropic GSTC relations in Eq. (16). For a refractive metasurface, the rotation of polarizationis notrequired and usually undesired. Therefore, the relevant nonzero susceptibility compo-nents reduce to the diagonal components ofχ andee χ mm

andthe off-diagonal components ofχ and_em χ Th_me. is cor -respondsto 4 × 2 = 8 susceptibility parameters,leading, accordingto Section 4.3,tothe double-transformation full-rank system 1 2 1 2 1 2 1 2 1,av 2,av ee em 1,av 2,av ee em 1,av 2,av me mm 1,av 2,av me mm 0 0 0 0 , 0 0 0 0 y y x x y y x x xx xy x x yy yx y y xy xx x x yx yy y y H H H H E E E E E E E E H H H H ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ χ χ χ χ χ χ χ χ       =                  ⋅              (56)

where we naturally specify the second transformation as thereciprocal ofthefirst one. Assumingthattherefrac -tiontakes placesinthexz-plane andthatthe waves are all p-polarized, system (56) reducesto

1 2 ee em 1,av 2,av 1,av 2,av 1 2 me mm , xx xy y y x x yx yy y y x x H H E E H H E E ∆ ∆ χ χ ∆ ∆ χ χ       =_{ }⋅          (57)

whichstrictly correspondsto asystemthatis N(4) = 2, although the initial goal might have been to perform refraction in one propagation direction only. An illustra -tion ofthefirst and secondtransformationsis presentedin Figure 10A and B, respectively. Note that subscripts i and t, respectively, refer to the incident and transmit sides of the metasurface rather than theincident andtransmitted waves.

The electromagnetic fields on theincident and trans -mit sides of the metasurface, assuming that the media on

Half-waveplate Quarter-waveplate

Polarization beam splitter OAM generation Figure 9: Birefringent metasurface transformations presentedin Ref.[122].

20 The quasi-totality of the refracting metasurfaces discussedin the literature thus far has been reciprocal. The following argument does not holdforthe nonreciprocal case, where perfect refraction couldin principle be achieved by a symmetric metasurface structure.