homogenization and scattering analysis of second-harmonic generation in nonlinear ... · 2019. 1....

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 66, NO. 11, NOVEMBER 2018 6061

Homogenization and Scattering Analysisof Second-Harmonic Generation in

Nonlinear MetasurfacesKarim Achouri , Gabriel D. Bernasconi, Jérémy Butet, and Olivier J. F. Martin

Abstract— We propose an extensive discussion on the homog-enization and scattering analysis of the second-order nonlinearmetasurfaces. Our developments are based on the generalizedsheet transition conditions (GSTCs) that are used to modelthe electromagnetic responses of nonlinear metasurfaces. TheGSTCs are solved both in the frequency domain, assuming anundepleted pump regime, and in the time domain, assumingdispersionless material properties but a possible depleted pumpregime. Based on these two modeling approaches, we derivethe general second-harmonic reflectionless and transmissionlessconditions as well as the conditions of asymmetric reflectionand transmission. We also discuss and clarify the concept ofnonreciprocal scattering pertaining to nonlinear metasurfaces.

Index Terms— Generalized sheet transition conditions(GSTCs), metasurface, nonlinear optics, susceptibility tensor.

I. INTRODUCTION

OVER the past decade, metasurfaces have attractedtremendous attention due to their incredible capabil-ities to control electromagnetic waves together with theirlow weight, reduced thickness, and relative ease of fabrica-tion [1]–[5]. More recently, nonlinear metasurfaces made theirapparition and were rapidly considered as a new paradigmshift in nonlinear optics [6]–[8]. Indeed, the capability ofengineering the shape and the composition of the nonlinearscattering particles composing the metasurfaces allows one tostrongly confine the electromagnetic fields and thus achievevery high nonlinear effects, which may be several ordersof magnitude stronger than those obtainable in conventionalnonlinear crystals [7], [9], [10]. In addition, nonlinear meta-surfaces may be used to manipulate the second- or third-harmonic scattered fields in a very elaborate fashion and withmuch more degrees of freedom compared to our current abilityof controlling linear fields in conventional linear metasur-faces [11]–[15]. For instance, it was suggested that nonlinearmetasurfaces may be used to realize functionalities such asoptical switches, diodes, and transistors [16] and also generate

Manuscript received March 7, 2018; revised July 16, 2018; acceptedJuly 20, 2018. Date of publication August 3, 2018; date of current versionOctober 29, 2018. This work was supported by the European Research Councilunder Grant ERC-2015-AdG-695206 Nanofactory. (Corresponding author:Karim Achouri.)

The authors are with the Department of Microengineering, École Poly-technique Fédérale de Lausanne, 1015 Lausanne, Switzerland (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAP.2018.2863116

multiple holographic images based on the Pancharatnam–Berry phase effect [17], [18]. We also note the particularlyinteresting potential of metasurfaces to exhibit magnetic non-linear effects [19]–[21]. This ability to control both electricand magnetic nonlinear responses is of utmost importance fora complete control of the nonlinear scattered fields, as will bediscussed thereafter.

Even though nonlinear metasurfaces are already quite attrac-tive, there still needs to be major theoretical and practicalworks to be done before they achieve their full potential. Thispaper tackles the theoretical aspect of the second-harmonicgeneration in the second-order nonlinear metasurfaces. Themathematical basis of this paper is a direct extension of ourprevious works on linear [22]–[24] and nonlinear metasur-faces [25]. It also extends other similar theoretical works per-taining to the second-order nonlinear metasurfaces [26]–[31].The theory presented in these papers mostly refers to relativelyspecific situations, while we try here to be as general aspossible in our description of the second-harmonic scattering.

The goal of this paper is to provide the reader with a com-prehensive discussion on the homogenization and scatteredfield analysis of the second-order nonlinear metasurfaces andmore specifically on their properties in terms of the second-harmonic generation. Accordingly, we propose a frequency-and time-domain analysis of nonlinear metasurfaces, uponwhich we notably derive the general reflectionless and trans-missionless second-harmonic conditions. The mathematicalmodel that we use is based on rigorous zero-thickness transi-tion conditions, which have been extensively used in the caseof linear metasurfaces [22]–[24], [32]–[34]. This model relatesthe incident, reflected, and transmitted fields to the metasur-face linear and now also nonlinear susceptibilities. Finally,we also propose a discussion on the nonreciprocal behavior ofnonlinear metasurfaces and, more generally, nonlinear opticalstructures [35]–[38]. This is to clarify certain misconceptionsabout the definition of nonreciprocity in nonlinear optics.

This paper is organized as follows. In Section II, we startby discussing the electromagnetic modeling of nonlinearmetasurfaces. More specifically, we explain how thesemetasurfaces may be mathematically modeled by zero-thickness transition conditions. Then, in Sections III and IV,we present a frequency-domain and a time-domain approach,respectively, to perform the homogenization and the scatteringanalysis of these metasurfaces. In Section V, we brieflyconclude our discussion. In addition, we also provide several

0018-926X © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

https://orcid.org/0000-0002-6444-5215

6062 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 66, NO. 11, NOVEMBER 2018

Fig. 1. Pump field at frequency ω being scattered at frequencies ω and 2ωdue to its interactions with the second-order periodically uniform nonlinearmetasurfaces. (a) Physical metasurface built from nonlinear scattering parti-cles. (b) Homogenized metasurface with effective susceptibilities that producethe same scattering as that in (a).

useful considerations in the appendices. These includea discussion on the homogenization and scattering analysis oflinear metasurfaces in Appendix A and a discussion that aimsat clarifying the nonreciprocal behavior of linear and nonlinearmetasurfaces in Appendix B. Finally, we provide the generalreflectionless conditions of linear metasurfaces in Appendix D.

II. GSTCS’ MODELING OF THE SECOND-ORDERNONLINEAR METASURFACES

Let us consider a periodically uniform metasurface madeof the second-order nonlinear scattering particles lying in thexy plane at z = 0, as shown in Fig. 1(a). The period ofthe scattering particles within the plane of the metasurfaceis also much smaller than the wavelength with a typical unitcell lateral dimension of λ/3–λ/5. From the perspective of an

incident pump field at frequency ω, the metasurface appears asa perfectly uniform, homogeneous, and time-invariant mediumwith effective susceptibilities independent of the coordinates(x, y, t), as suggested in Fig. 1(b). Note that, in these figures,we only represent the scattered fields at frequencies ω and 2ωfor convenience, while there may be higher order harmonicsgenerated by the metasurface.

Metasurfaces may, in most cases, be conveniently modeled,assuming that they consist of zero-thickness sheets of excitableelectric and magnetic polarizations [32], [39]. This assumptionof zero thickness is a very good approximation, since metasur-faces are electrically thin structures with a typical thickness, d ,much smaller than the operation wavelength (d � λ). When ametasurface is modeled as a zero-thickness sheet, it is possibleto relate its interactions with electromagnetic fields in a generaland complete fashion using the generalized sheet transitionconditions (GSTCs) [22], [32], [40].

From a general perspective, the relationship between theincident, reflected, and transmitted fields and the linear andnonlinear susceptibilities of the metasurface may be expressedthrough the GSTCs. In the time domain, assuming a timedependence e jωt , the GSTCs are given by [40]

z × �H = ∂∂ t

P − z × ∇Mz (1a)

z × �E = −μ0 ∂∂ t

M − 1�0

z × ∇ Pz (1b)

where P and M are the metasurface electric and magneticpolarization densities, proportional to the susceptibilities, and�E and �H are the differences of the electric and magneticfields on both sides of the metasurface, respectively [22]. Notethat the spatial derivatives on the right-hand sides of (1) onlyapply to the xy plane.

The GSTCs may be used to perform three distinct oper-ations [22]–[24]. They may be used to synthesize—find thesusceptibility functions—of a metasurface so that it scatterslight in a specified fashion. In this case, the susceptibilitiesare mathematical functions and the corresponding scatteringparticles need to be practically designed and realized. TheGSTCs may also be used to homogenize—find the effectivelinear and nonlinear susceptibilities of—a metasurface made ofscattering particles with specific shapes and materials. Finally,they may be used to analyze—find the fields scattered by—a metasurface with known (effective) susceptibilities. Theseoperations may be realized by solving (1) to either express thesusceptibilities in terms of known electromagnetic fields (spec-ified for the synthesis, and simulated or measured for thehomogenization) or to express the scattered fields in terms ofknown susceptibilities to analyze the metasurface scattering.

In this paper, we will mostly discuss the last two operations,i.e., the homogenization and scattering analysis of the second-order nonlinear metasurfaces. This is because the synthesis ofsuch structures, although possible, is generally difficult due tothe very large number of susceptibility components, as will bediscussed shortly. Moreover, we will concentrate our attentionon the generation of the second-harmonic light since the studyof linear scattering is already very well documented in theliterature. We will also neglect the presence of all higher order

ACHOURI et al.: HOMOGENIZATION AND SCATTERING ANALYSIS OF SECOND-HARMONIC GENERATION 6063

harmonics that may emerge from the interactions of waves atfrequencies ω and 2ω (and so on) with the nonlinear medium.

In order to solve the system (1), we will consider twodifferent approaches: a frequency-domain approach, for whichwe will assume that the pump at frequency ω is undepletedand that the susceptibilities are dispersive so that in generalχ(ω) �= χ(2ω), and a time-domain approach, for whichthe pump may be depleted but the susceptibilities are notdispersive so that χ(ω) = χ(2ω). As we will see, thesetwo approaches present their own benefits and drawbacks.Therefore, the most adequate approach mostly depends on theneeds of the user.

Note that the fact that the metasurface is spatially uniformimplies that no diffraction orders will be produced upon lightscattering. Therefore, an incident plane wave, impinging on themetasurface at arbitrary angles (θi, φi), will be reflected andtransmitted with reflection and transmission angles obeyingSnell’s law. This is of particular importance since, for anyincidence angles, we know how the metasurface will scatterlight, which greatly simplifies the operations of homogeniza-tion and scattered field analysis.

III. FREQUENCY-DOMAIN APPROACH

In this frequency-domain approach, we consider by approx-imation that the interactions of an incident wave at frequencyω with the metasurface linear and nonlinear susceptibili-ties produce scattered fields only at frequencies ω and 2ω,respectively. This assumption is almost always valid since theamplitudes of the higher order (>3) harmonics generated bya second-order nonlinear material are negligible [41]. We alsoconsider that the pump is not depleted. This assumptionviolates the conservation of energy, but it greatly simplifiesthe modeling of nonlinear metasurfaces and still provides theexcellent results as long as the power of the second-harmonicis much smaller than that of the pump, which is usually thecase with current nonlinear metasurfaces [31].

With this assumption of undepleted pump, it is trivialto homogenize and analyze the response of metasurfaces atfrequency ω. It may be easily done by following the well-known techniques applied to linear bianisotropic metasurfaces.In Appendix A, we briefly summarize the main steps to obtainthe effective linear susceptibilities of a metasurface and tocompute the fields scattered by a metasurface with the knowneffective susceptibilities.

We are now interested in homogenizing the nonlinearsusceptibilities and analyzing the second-harmonic scatteredwaves of a second-order nonlinear metasurface. To do so,we consider the GSTCs in (1) and write them in the frequencydomain so that they relate the scattered fields and the polar-ization densities both at frequency 2ω. They thus read1

z × �H2ω = j2ωP2ω − z × ∇M2ωz (2a)z × �E2ω = − j2ωμ0M2ω − 1

�0z × ∇ P2ωz (2b)

1Note that the time derivatives in (1) reduce to j2ω in this frequency-domainrepresentation.

where the electric and magnetic polarization densities are splitinto linear and nonlinear terms as

P2ω = P2ωlin + P2ωnl (3a)M2ω = M2ωlin + M2ωnl . (3b)

The linear polarizations correspond to the interactions of thefields at 2ω with the linear metasurface susceptibilities at thatfrequency and may be generally expressed as

P2ωlin = �0χee · E2ωav + �0η0χem · H2ωav (4a)M2ωlin = χmm · H2ωav +

1

η0χme · E2ωav (4b)

where η0 is the vacuum impedance, E2ωav and H2ωav are the

arithmetic averages of the fields on both sides of the metasur-face, and χee, χmm, χme, and χem are the electric, magnetic,magnetoelectric, and electromagnetic linear susceptibilitytensors, respectively [42].

The nonlinear polarizations correspond to the interactionsof the pump fields at ω with the nonlinear metasurfacesusceptibilities. These polarizations can thus be considered asthe sources of the second-harmonic generation. Taking intoaccount all possible interactions of the fields, they may begenerally expressed as

P2ωnl =1

2�0

�χeee : Eωav Eωav + η0χ eem : Eωav Hωav+ η20χemm : Hωav Hωav

�(5a)

M2ωnl =1

2

�η0χmmm : Hωav Hωav + χmem : Eωav Hωav+ 1

η0χmee : Eωav Eωav

�(5b)

where we have used our own convention for the dimension ofthe susceptibilities, by introducing the impedance parameter,such that they all have the same dimension. Note that the factor(1/2) in front of relations (5) comes from the time derivativesin (1). This factor appears in (5) and not in (4) is because

∂

∂ tE2ωav ∝

∂

∂ tcos (2ωt) = −2ω sin (2ωt) (6)

while

∂

∂ tEωav E

ωav ∝

∂

∂ tcos2 (ωt) = −ω sin (2ωt). (7)

Therefore, a factor of (1/2) must be considered for thenonlinear polarization densities.

A. Homogenization Technique

The condition that must be met to be able to homogenizea metasurface is that the latter does not produce diffractionorders (besides the zeroth orders) when illuminated. Thiscondition is satisfied when the periodicity of the scatteringparticles is less than the smallest wavelength considered mul-tiplied by the background refractive index. In order to homog-enize and thus find the effective nonlinear susceptibilities of ametasurface, we have to solve (2) along with relations (3)–(5).Considering plane wave illumination, the general nonlinearGSTCs thus form a set of four equations that are given in


−�H 2ωy = j2ω�0�

χ xiee E2ωi + η0χ xiem H 2ωi +

1

2χ xi jeee E

ωi E

ωj +

1

2η0χ

xi jeem E

ωi H

ωj +

1

2η20χ

xi jemm H

ωi H

ωj

�

− jk2ωy�

χ zimm H2ωi +

1

η0χ zime E

2ωi +

1

2η0χ

zi jmmm H

ωi H

ωj +

1

2χ zi jmem E

ωi H

ωj +

1

2η0χ zi jmee E

ωi E

ωj

�(8a)

�H 2ωx = j2ω�0�

χ yiee E2ωi + η0χ yiem H 2ωi +

1

2χ yi jeee E

ωi E

ωj +

1

2η0χ

yi jeem E

ωi H

ωj +

1

2η20χ

yi jemm H

ωi H

ωj

�

+ jk2ωx�

χ zimm H2ωi +

1

η0χ zime E

2ωi +

1

2η0χ

zi jmmm H

ωi H

ωj +

1

2χ zi jmem E

ωi H

ωj +

1

2η0χ zi jmee E

ωi E

ωj

�(8b)

−�E2ωy = − j2ωμ0�

χ ximm H2ωi +

1

η0χ xime E

2ωi +

1

2η0χ

xi jmmm H

ωi H

ωj +

1

2χ xi jmem E

ωi H

ωj +

1

2η0χ xi jmee E

ωi E

ωj

�

− jk2ωy�

χ ziee E2ωi + η0χ ziem H 2ωi +

1

2χ zi jeee E

ωi E

ωj +

1

2η0χ

zi jeem E

ωi H

ωj +

1

2η20χ

zi jemm H

ωi H

ωj

�(8c)

�E2ωx = − j2ωμ0�

χ yimm H2ωi +

1

η0χ yime E

2ωi +

1

2η0χ

yi jmmm H

ωi H

ωj +

1

2χ yi jmem E

ωi H

ωj +

1

2η0χ yi jmee E

ωi E

ωj

�

+ jk2ωx�

χ ziee E2ωi + η0χ ziem H 2ωi +

1

2χ zi jeee E

ωi E

ωj +

1

2η0χ

zi jeem E

ωi H

ωj +

1

2η20χ

zi jemm H

ωi H

ωj

�(8d)

reduced tensor notation for convenience2 in (8), as shown atthe top of this page, where i, j = {x, y, z} and where wehave transformed the gradients in (2) into ∇ → − jk2ωu withu = {x, y} since we are considering plane waves and wherej = √−1 is not to be confused with the tensor notation. Notethat the (1/2) factor discussed in (6) and (7) must also existin the case of the spatial derivatives in (2).

As discussed in Appendix A, the linear susceptibilities in (8)can be obtained using the retrieval method applied to linearmetasurfaces [31]. For the nonlinear susceptibilities, the sys-tem (8) must be solved so as to express the susceptibilitiesin terms of the incident, reflected, and transmitted fields atfrequency 2ω that may be found from numerical simula-tions. However, it is clear that this system of equations is,in most cases, largely underdetermined. Indeed, assuming thatthe nonlinear susceptibilities only have intrinsic permutationsymmetry (χ i j k = χ ikj as is the case of most nonlinearsystems [41]), there is a total number of 108 unknownsusceptibility components in (8) for only 4 equations. Notethat, in the case of linear susceptibilities, some susceptibilitycomponents may be related to each other through the reci-procity conditions (41) (see Appendix B), which reduce thetotal number of independent unknowns. Such a considerationis not (yet) possible for the case of the second-order nonlinearsusceptibility tensors, since there are currently no reciprocityrelations available that would apply to these nonlinear suscep-tibility tensors, as discussed in Appendix B.

However, it must be emphasized that a scattering particlegenerally only exhibits a few relevant nonlinear susceptibilitycomponents, while most of the other terms are zero or atleast may be neglected [31]. The main reason for this isthe structural symmetries of the particles, which preventsthe generation of the second-harmonic light for some inci-dent/scattered field polarization combinations. Therefore, onemay reduce the 108 nonlinear susceptibility components down

2Note that in these expressions, we have dropped the summation terms suchthat Ai j Bi j = �i, j Ai j Bi j . We have also dropped the “av” term for the fieldsmultiplying the susceptibilities for conciseness.

to just a few relevant terms. In addition, when the scatteringparticle is symmetric in the metasurface plane, and then, itsresponse to x- or y-polarized waves will be the same. In thiscase, the susceptibility tensors will exhibit further fundamentalsymmetries in addition to the intrinsic permutation symmetryconsidered earlier, hence greatly reducing the number ofindependent susceptibility unknowns.

Unfortunately, even if the number of unknowns may bereduced, the system (8) generally remains underdetermined.To overcome this issue, we may exploit the same approachas that conventionally used for linear metasurfaces: performseveral simulations, each time for different pump incidenceangles. Indeed, for each set of incidence angles (θi, φi),the susceptibilities remain the same, but the incident andscattered fields differ, thus increasing the number of linearlyindependent equations, while keeping the same number ofunknowns. For example, if the total number of independentnonzero susceptibility components is 8, then the system (8)becomes fully determined when only two sets of pump inci-dence angles are used.

Finally, we would like to mention that besides the homog-enization technique discussed here, which is essentially thenonlinear extension of the conventional method used tohomogenize linear metasurfaces, there is another approachto retrieve the nonlinear susceptibilities. This alternativeapproach is discussed in [43]. It consists in exciting thestructure with contrapropagating waves so that their mag-netic (electric) fields cancel at the position of the struc-ture, while their electric (magnetic) constructively interfere.It is thus possible to selectively excite specific nonlinearsusceptibilities, for instance, only χeee. Then, by measuringthe scattered fields and by changing the parameters of thecontrapropagating waves, one may completely characterize themetasurface susceptibilities.

B. Scattering Analysis

When the metasurface effective linear and nonlinear sus-ceptibilities are known, it is possible to predict how the


metasurface will scatter light for any pump incidence angles.The general approach to compute the fields scattered bythe metasurface is to solve the system (8) for the unknownreflected and transmitted fields.

In what follows and for simplicity, we concentrate ourattention on the particular case where the pump is normallyincident and the metasurface is spatially uniform. In this case,the spatial derivatives in (2) are zero, since both the fields andthe susceptibilities are spatially uniform in the metasurfaceplane, which greatly simplifies the scattered field analysis.Indeed, since the spatial derivatives are zero, the presenceof normal polarizations (and hence susceptibilities) may beignored, since they do not contribute to the scattering. Thismeans that the only relevant components of the incident andscattered electromagnetic fields, Ei and Hi in (8), as well asthe linear (χ i j ) and nonlinear (χ i j k) susceptibility tensors aresuch that i, j, k = {x, y}.

This approach allows us to rigorously compute the fieldsscattered by the metasurface. However, in many cases, the inci-dent wave is not normally impinging on the metasurfaceand/or the latter may be spatially varying. In these scenarios,the spatial derivatives in (2) do not vanish. Nevertheless,if the incidence angle is small and/or the spatial variationsof the susceptibilities are slow compared to the wavelength at2ω, then it is possible to neglect the presence of the spatialderivatives (small angle approximation). Accordingly, it ispossible to obtain an approximate evaluation of the scatteringfrom the metasurface, which gets worse as the incidence angleand/or spatial variations increase [44].

To compute the second-harmonic scattering from nonlinearmetasurfaces, we simplify the GSTCs and write them in amore compact and convenient form. To do so, we start byexpressing the magnetic field in terms of the correspondingelectric field in the case of normally propagating plane waves.We thus only consider their transverse components. For planewaves propagating in the forward (+z) and backward (−z)directions, we, respectively, have that

H fw = 1η0

J · Efw, Hbw = − 1η0

J · Ebw (9)

where J is the rotation matrix defined as

J =�

0 −11 0

�. (10)

Then, by making use of (9), we express the average of thepump electromagnetic field that is used in (5). When the pumppropagates normally in the forward direction, this average fieldis given by

Eωav,fw =1

2(I + S11 + S21) · Eω0 (11a)

Hωav,fw =1

2η0J · (I − S11 + S21) · Eω0 (11b)

where Eω0 is the amplitude of the pump electric field,

the matrices S are the linear scattering matrices obtained inAppendix A, and I is the 2-D identity matrix. Note that ports1 and 2 refer to the left (z < 0) and right (z > 0) sides of themetasurface, respectively.

Similarly, if the pump is propagating in the backwarddirection, then the average pump fields are

Eωav,bw =1

2(I + S22 + S12) · Eω0 (12a)

Hωav,bw = −1

2η0J · (I − S22 + S12) · Eω0 . (12b)

We now express the averages of the second-harmonic fieldsthat are used in (4). Since there is no wave impinging on themetasurface at 2ω, these average fields are simply given by

E2ωav =1

2

�E2ωfw + E2ωbw

�(13a)

H2ωav =1

2η0J · �E2ωfw − E2ωbw

�. (13b)

By the same token, the difference of the fields in (2) read

�E2ω = E2ωfw − E2ωbw (14a)�H2ω = 1

η0J · �E2ωfw + E2ωbw

�. (14b)

The GSTCs may now be simplified by substitut-ing (13) and (14), along with (3) and (4), into (2).We thus get

E2ωfw + E2ωbw = − jω

c0

�χee ·

�E2ωfw + E2ωbw

� + χem · J× �E2ωfw − E2ωbw

�� − jωη0 P2ωnl(15a)

J · �E2ωfw − E2ωbw� = − j ω

c0

�χmm · J ·

�E2ωfw − E2ωbw

� + χme× �E2ωfw + E2ωbw

�� − jωμ0 M2ωnl(15b)

where c0 is the speed of light in vacuum.These two vectorial equations may now be solved so as

to express the backward and forward second-harmonic wavesas functions of the linear susceptibilities and the nonlinearpolarizations. They thus, respectively, read

E2ωbw =1

�0C1 · P2ωnl + η0C2 · J · M2ωnl (16a)

E2ωfw =1

�0C3 · P2ωnl − η0C4 · J · M2ωnl (16b)

where the matrices C, which contain the linear susceptibilities,are explicitly provided in Appendix C. Relations (16) are thegeneral expressions for the second-harmonic fields scatteredby a nonlinear metasurface. In these expressions, the nonlinearpolarizations, P2ωnl and M

2ωnl , play the role of nonlinear sources

excited by the pump at ω. These polarizations are given in (5)and are different for different directions of pump propagationaccording to (9).

C. Numerical Demonstration

In order to demonstrate the validity of the proposed homog-enization and analysis method, we will now apply it to aconcrete example. First, we will use relations (16) to retrievethe susceptibilities of a nonlinear metasurface and, second, use


Fig. 2. Nonlinear metasurface unit cell. The dimensions of the T-shapedparticles are Lx = 165 nm and L y = 125 nm with a diameter of D = 40 nm,and the separation between the two particles is Lz = 60 nm. The positionz = 0 is in-between the two T-shaped particles.

these susceptibilities to predict the amplitude of the nonlinearscattered fields.

The chosen metasurface unit cells are composed of T-shapedgold particles [45], which may be considered as simple build-ing blocs to generate in-plane nonlinear dipolar moments. Eachunit cell consists of two T-shaped particles placed on top ofeach other and with one being rotated 180◦ with respect tothe other, as shown in Fig. 2. This specific arrangement allowsone to generate a nonlinear electric dipole, px , when excitedwith a y-polarized normally incident wave [45]. As will bedemonstrated in the following, an additional nonlinear mag-netic moment, my , is created by the two parallel px dipoles ineach T-shaped particle. The metasurface consists of a periodicarray, with a subwavelength lattice period of 200 nm, formedby the unit cell shown in Fig. 2. The background mediumis SiO2 with an assumed nondispersive relative permittivityof �r = 2.22.

The homogenization of this metasurface is made difficultby the complex geometry of its unit cells. Indeed, it is partic-ularly difficult to know a priori which nonlinear susceptibilitycomponents will play a major role and which ones may beneglected. Nevertheless, one may still simplify the homoge-nization procedure by taking into account the two followingconsiderations: 1) we know from the xz plane symmetry ofthe structure that when excited with a y-polarized illumination,it only produces an x-polarized second-harmonic light [45] and2) we are here only interested in normally propagating wavesand thus decide to ignore the presence of z-oriented dipolarresponses and their corresponding susceptibility components.This allows us to select, out of the six nonlinear susceptibilitytensors in (5), only six susceptibility components (one in eachtensor) that are χ xyyeee , χ

xyxeem , χ

x x xemm, χ

yx xmmm, χ

yyxmem, and χ

xyymee.

All the other nonlinear susceptibility components may then beneglected, as they do not play a role in the scattering responseof the metasurface.

Now, the homogenization procedure consists in numeri-cally simulating the linear and nonlinear fields scattered bythe metasurface and then solving relations (16) in order toobtain these six susceptibilities. However, the system (16)

Fig. 3. Retrieved normalized nonlinear susceptibilities for the metasurfaceof Fig. 2. The susceptibility components χ xxxemm and χ

yyxmem are found to be

negligible and are thus not plotted.

is undertermined since it contains only two equations, whilethere are six unknown susceptibilities. To overcome thisissue, we have to increase the number of equations so asto match the number of unknowns, which may be achievedby considering several independent illuminations, as explainedin Section III-A. Specifically, we consider a total of threeindependent illuminations, which allows us to solve the sys-tem (16) three times, one for each illumination, leading to atotal number of six equations in six unknowns.

In our case, we specify that these illuminations only prop-agate normally with respect to the metasurface as we are notinterested in oblique illuminations or in retrieving suscepti-bility components with z-oriented contributions. Accordingly,we consider the three following pump illuminations:

Einc,1 = ye− j kz (17a)Einc,2 = y(e− j kz + e jkz) (17b)Einc,3 = y(e− j kz − e jkz) (17c)

where Einc,1 is a forward propagation plane wave, andEinc,2 and Einc,3 are superpositions of forward and backwardpropagating plane waves, which are in-phase and out-of-phaseat z = 0 (corresponding to the position of the metasurface),respectively.

The nonlinear numerical simulations are performed with ahomemade surface integral equation (SIE) method [46], [47].Note that this numerical method assumes an undepleted pumpregime. The pump illuminations’ wavelength covers the rangefrom 700 to 900 nm. Using the simulated linear and nonlinearscattered fields, relations (16) are solved for the six unknownsusceptibilities, whose absolute values are plotted in Fig. 3.We see that at the resonance peak (λ = 825 nm), the only sig-nificant susceptibility components are χ yyymee, χ

xyxeem , and χ

xyyeee ,

while all the other are essentially negligible. As expected,the metasurface exhibits both electric and magnetic dipolarresponses.

Now that we know the effective nonlinear susceptibilities ofthis metasurface, we use them to predict how the latter scattersthe second-harmonic light under a pump illumination different


Fig. 4. Comparison of forward and backward normalized nonlinear scatteredfields computed using SIE (solid lines) and the GSTCs’ method (dashed lines).

than those in (17). Specifically, we consider a backward planewave illumination Einc,4 = ye jkz . Using (16), it is nowstraightforward to find the nonlinear fields scattered by themetasurface. The resulting forward and backward scatteredfields are plotted in Fig. 4 along with their SIE simulatedcounterparts. This reveals an overall good agreement betweenthe predicted scattered fields and those obtained via numericalsimulations. The discrepancies are mostly due to the fact thatthe GSTCs’ method assumes a perfectly zero-thickness sheet,while the metasurface has an actual thickness of 140 nmcorresponding to about λNL/3. Nevertheless, this exampledemonstrates the validity of the GSTCs’ method even formetasurfaces that are not deeply subwavelength.

D. Simplification for Linearly Reflectionless Metasurfaces

From (16), we see that as it is the case for linear meta-surfaces, the forward or backward scattering can be com-pletely suppressed since the terms on the right-hand sides maycancel each other if the linear and nonlinear susceptibilitiesare properly engineered. It should be therefore possible toobtain the second-order nonlinear counterparts of the Kerkerconditions [48]. However, in the very general formulation (16),it is difficult to obtain the nonlinear reflectionless conditionssolely in terms of the susceptibilities, because the nonlinearpolarizations densities depend upon the scattering of the pumpfields, as can be seen in (11) and (12). Indeed, it is, in gen-eral, particularly cumbersome to factor the linear scatteringmatrices out of the double dot products in (5), since (11b)[or (12b)] cannot be expressed in terms of (11a) [or (12a)].

Nevertheless, this issue may be overcome by consideringa specific but particularly interesting and insightful situa-tion. The case where the metasurface is reflectionless at thefrequency of the pump. This means that at frequency ω,the reflectionless conditions (48) in Appendix D are satisfiedand, hence, that S11 = S22 = 0 in (11) and (12). In that case,the average forward and backward magnetic fields [see (11b)and (12b)] are, using (9), simply given by

Hωav,fw =1

η0J · Eωav,fw, Hωav,bw = −

1

η0J · Eωav,bw. (18)

Based on the assumption that the pump excitation is notreflected, the nonlinear polarization densities in (5) may besimplified using (18). For a forward propagating pump, theybecome3

P2ωnl,fw =�0

2

�χeee : Eωav,fw Eωav,fw − χeem : Eωav,fw Eωav,fw · J

− χemm : J · Eωav,fw Eωav,fw · J�

(19a)

M2ωnl,fw =1

2η0

� − χmmm : J · Eωav,fw Eωav,fw · J − χmem : Eωav,fw

× Eωav,fw · J+χmee : Eωav,fw Eωav,fw�

(19b)

and for a backward propagating pump, they become

P2ωnl,bw =�0

2

�χeee : Eωav,bw Eωav,bw + χeem : Eωav,bw Eωav,bw · J

−χemm : J · Eωav,bw Eωav,bw · J�

(20a)

M2ωnl,bw =1

2η0

� − χmmm : J · Eωav,bw Eωav,bw · J+χmem : Eωav,bw

× Eωav,bw · J+χmee : Eωav,bw Eωav,bw�. (20b)

We may now further simplify relations (19) and (20) byapplying the rotation matrices, J, directly on the nonlinearsusceptibility tensors instead of the electric field matrices,Eωav E

ωav. These rotations of the susceptibility tensors are

presented in Appendix C. With this simplification, it is possibleto factor the Eωav E

ωav terms out of the polarization densities.

Consequently, the second-harmonic scattered fields in (16)may thus be expressed in the following compact form:

E2ωbw,fw = Sω→2ω11 : Eωav Eωav (21a)

E2ωbw,bw = Sω→2ω22 : Eωav Eωav (21b)

E2ωfw,fw = Sω→2ω21 : Eωav Eωav (21c)

E2ωfw,bw = Sω→2ω12 : Eωav Eωav (21d)

where the first subscripts of the terms on the left-hand sidescorrespond to the direction of propagation of the scatteredfields, while the second subscripts correspond to the directionof propagation of the pump. In (21), we have introducedthe notion of nonlinear scattering tensors which are providedin (22), as shown at the bottom of the next page, where theprimed tensors are those that have been rotated accordingto the operations provided in Appendix C. Note that theconcept of nonlinear scattering parameters is only valid in theundepleted pump approximation.

At this point, it is important to realize that the nonlinearscattering tensors defined in (22) are the third-order tensorsand not just simple matrices such as the conventional linearscattering tensors used in Appendix A and in (11) and (12).Accordingly, there is a total number of 32 scattering parame-ters in (22), while there is only 16 linear scattering parametersin (39) in Appendix A.

3We have used the fact that E H = E · HT ∝ E · (J · E)T = E · ET ·JT = −E E · J, where T is the transpose operation and JT = −J. Similarly,

H H ∝ − J · E E · J.


In the absence of external time-odd bias, the metasurface islinearly reciprocal (refer to Appendix B), which is the case ofthe vast majority of metasurfaces. In that case, relations (42)apply. Therefore, the 16 linear scattering parameters in (39)are reduced to only 10 independent parameters.

According to the discussion in Appendix B, the nonlin-ear scattering tensors (22) are not subjected to any recip-rocal condition. However, they are affected by the intrinsicpermutation symmetries of the structure. This means thatSi jkab = Sikjab , where i, j, k = {x, y} and a, b = {1, 2}.Consequently, the 32 scattering parameters in (22) are reducedto 24 independent parameters. Hence, a second-order nonlinearmetasurface exhibits much more degrees of freedom availableto control the scattered fields compared with conventionallinear metasurfaces. More specifically, a nonlinear metasur-face has the particularly interesting property of exhibitingasymmetric second-harmonic generation. For instance, a non-linear metasurface may be perfectly reciprocal and have4

Sω→2ω,x x x21 �= Sω→2ω,x x x12 , while, by reciprocity, in the linearregime, the equality Sx x21 = Sx x12 must be satisfied accord-ing to (42).

In order to understand why, for a reciprocal metasurface,Sω→2ω,x x x21 can be different from S

ω→2ω,x x x12 , let us consider

the two following idealized situations. In the first situation,a pump at frequency ω illuminates a nonlinear metasurfacewhich only presents χ x x xeee as a nonzero susceptibility, whileall other susceptibility terms are assumed to be zero or negli-gible. In the second situation, the nonlinear metasurface onlypresents χ yyymmm as a nonzero susceptibility. For simplicity,we consider that5 χ x x xeee = χ yyymmm = 1 m2/V.

These two metasurfaces are successively excited with apump propagating once in the forward direction and then in thebackward direction. In both cases, the pump excites the meta-surface susceptibilities and, hence, the nonlinear polarizationsthat are responsible for the second-harmonic generation.

The scattering (here, for simplicity, the second-harmonictransmitted field only) from the electrically nonlinear meta-surface is shown in Fig. 5(a) and (b) for the two excitation

4In [25], [28], [29], and [49], this inequality is presented as a nonreciprocaloperation. However, according to the upcoming discussion in this section aswell as that in Appendix B, we shall rather refer to it as an asymmetric ratherthan a nonreciprocal operation.

5Note that we are using surface susceptibilities, and hence, the dimensionof linear susceptibilities is [m] instead of being dimensionless like it is thecase for bulk susceptibilities. Moreover, the nonlinear electric and magneticsusceptibilities have the same dimension thanks to the convention that wehave used in (5).

Fig. 5. Comparison of the second-harmonic scattering from two differentnonlinear metasurfaces. In (a) and (b), the metasurface is electrically nonlinearand nonlinear electric dipolar moments are excited within the metasurface.In (c) and (d), it is magnetically nonlinear and nonlinear magnetic dipolarmoments are excited within the metasurface. In (a) and (c), the pumppropagates forward, while in (b) and (d), it propagates backward. In all figures,the blue arrows around the dipolar moments represent the correspondingscattered electric fields.

directions. Similarly, the scattering from the magneticallynonlinear metasurface is shown in Fig. 5(c) and (d).

From these representations, it is clear that the electri-cally nonlinear metasurface produces the same transmittedfield irrespective of the direction of pump propagation sinceP2ωx = χ x x xeee E2x is symmetric with respect to Ex and,thus, Sω→2ω21 = Sω→2ω12 . However, the opposite occurs inthe case of the magnetically nonlinear metasurface sinceM2ωy = χ yyymmm H 2y is antisymmetric with respect to Hy whenthe direction of propagation is reversed and thus Sω→2ω21 �=Sω→2ω12 . This simple example shows that the presence of thismagnetic nonlinear susceptibility introduces an asymmetricsecond-harmonic generation in transmission.

In what follows, we will consider the nonlinear scatteringtensors in (22) and generalize the concept of asymmetricnonlinear scattering. We will also look into other importantaspects of nonlinear metasurfaces, which notably includesthe conditions that enable one to completely suppress theforward or backward second-harmonic generation.

Sω→2ω11 =

1

2C1 ·

�χeee − χ eem − χ emm

� + 12

C2 ·� − χ mmm − χ mem + χ mee

�(22a)

Sω→2ω22 =

1

2C3 ·

�χeee + χ eem − χ emm

� + 12

C4 ·�χ

mmm − χ mem − χ mee

�(22b)

Sω→2ω21 =

1

2C3 ·

�χeee − χ eem − χ emm

� + 12

C4 ·�χ

mmm + χ mem − χ mee

�(22c)

Sω→2ω12 =

1

2C1 ·

�χeee + χ eem − χ emm

� + 12

C2 ·� − χ mmm + χ mem + χ mee

�(22d)


TABLE I

NONLINEAR REFLECTIONLESS AND TRANSMISSIONLESS CONDITIONS AS WELL AS THE SUSCEPTIBILITIES RESPONSIBLE FOR ASYMMETRICREFLECTION AND TRANSMISSION FOR THREE DIFFERENT SCENARIOS: 1) IN THE ABSENCE OF BIANISOTROPY; 2) WHEN

THE LINEAR REFLECTIONLESS CONDITIONS ARE SATISFIED; AND 3) WHEN BOTHCONDITIONS ARE SIMULTANEOUSLY SATISFIED

Table I summarizes the upcoming results. We considerthree different scenarios that are reported in the columnsof the table. In order of appearance, we consider: 1) the

absence of bianisotropy such that χem = χme = 0 atall frequencies, which is usually the case if the surface issymmetric in the longitudinal (z) direction and if it exhibits nochirality (the scattered fields, at ω, have the same polarizationas the excitation, also at ω); 2) the linear reflectionlessconditions in (48) are satisfied at both ω and 2ω; and 3)both conditions 1) and 2) are simultaneously satisfied. In thecorresponding rows of the table, we start by providing the

updated C tensors for the three respective scenarios. Then,we present the general reflectionless and transmissionlessconditions for both forward and backward pump propagations.Finally, we provide the properties of asymmetric reflection andtransmission.

Being able to suppress either the reflected or transmittedfields is achieved by the superposition of the fields scatteredby both electric and magnetic dipolar moments. By controllingthe phase shift between these two dipoles, it is thus possibleto completely cancel the field scattered either in the back-ward or the forward direction. In the case of linear structures,this effect is referred to as the Kerker condition [48] and hasbeen extensively used to realize reflectionless (linear) meta-surfaces. The case of nonlinear metasurfaces is fundamentallyidentical when nonlinear electric and magnetic dipoles can be

excited. For instance, we see that an electrically (χeee �= 0)

and magnetically (χmmm �= 0) nonlinear metasurface, whichsatisfies the linear reflectionless conditions, is nonlinearly

reflectionless for a forward propagating pump (Sω→2ω11 = 0)

when χeee = χ mmm, which corresponds to the nonlinearcounterpart of (48a). However, for a backward propagating

pump, the corresponding reflectionless condition (Sω→2ω22 = 0)

is χeee = −χ mmm. The fact that the reflectionless conditionsare not the same from both sides is another evidence ofthe asymmetric second-harmonic scattering behavior of thesetypes of nonlinear metasurfaces. In fact, the two last rows ofthe table provide the expressions of the susceptibility compo-nents, respectively, responsible for the metasurface asymmetric

reflection (Sω→2ω11 �= S

ω→2ω22 ) and transmission (S

ω→2ω21 �=

Sω→2ω12 ). In the absence of bianisotropy, we see that the

susceptibility tensors χeem, χ

mmm, and χ

mee are inducing

some sorts of scattering asymmetry, which is a generalizationof the concept already shown in Fig. 5. However, in the pres-ence of bianisotropy, all nonlinear susceptibilities are naturallyinducing asymmetric scattering since, to achieve bianisotropy,the metasurface has to be spatially asymmetric, as discussedearlier.

It is also interesting to note that the reflectionless andtransmissionless conditions provided in Table I are satisfiedwhen the terms on the left-hand sides are equal to the termson the right-hand sides of the equalities, since electric andmagnetic terms should cancel each other. Therefore, in the


absence of magnetic nonlinear susceptibilities, if the electricsusceptibilities χeee, χ

eem, and χ

emm cancel out, then the

nonlinear scattering is completely suppressed both in reflectionand transmission. Note that such an effect, while theoret-ically possible, is practically difficult to implement, sincemany conditions on the susceptibilities must be simultaneouslysatisfied.

IV. TIME-DOMAIN APPROACH

We shall now discuss the approach that consists in solv-ing (1) in the time domain. This method is of practical interestwhen the depletion of the pump must be considered. Thisapproach is generally mathematically more involved than thefrequency-domain technique. Indeed, for the latter, it is rela-tively simple to obtain the second-order scattering response ofthe metasurface, since we assume that the pump is undepleted.In contrast, the time-domain approach naturally considers thepump depletion as well as the generation of higher orderharmonic due to the interactions of the signals at ω and 2ω(and so on) with the nonlinear metasurface. Moreover, in thetime-domain formulation, the dispersive nature of the suscep-tibilities is conventionally expressed as time convolutions withthe acting field such that the polarization densities in (1) takethe following general form [50]:

P(r, t) = �0 t

−∞dt χee(r, t − t ) · E(r, t) + · · · . (23)

In order to overcome these additional difficulties, we nextassume that the susceptibilities are dispersionless. Althoughthis seems a rather stringent assumption, we shall rememberthat the field interacting with the metasurface may generallybe expressed as E(r, ω) = �∞n=1 En(r)e jnωt . Moreover,the most important terms of this sum are typically thosefor which n = 1 (linear) and n = 2 (second harmonic).Considering (23), this means that in reality, the suscepti-bilities must satisfy the condition χ(ω) = χ(2ω), whichcorresponds to the conventional nonlinear phase match-ing condition [41], rather than being dispersionless at anyfrequency.

In what follows, we will solve (1) in the time domain anddiscuss both the operations of homogenization and scatteredfield analysis. However, we will not do it in a general fashion,as we did for the frequency-domain method, because of thecomplexity of the time-domain approach. We shall ratherillustrate the method to solve (1) with an example similarto that used in [25]. Accordingly, let us consider the caseof an electrically and magnetically nonlinear metasurface thatonly exhibits the following nonzero susceptibility components:χ x xee , χ

yymm, χ

x x xeee , and χ

yyymmm. In that case, the time-domain

GSTCs reduce to

−�H = �0χ x xee∂

∂ tEav + �0χ x x xeee

∂

∂ tE2av (24a)

−�E = μ0χ yymm∂

∂ tHav + μ0η0χ yyymmm

∂

∂ tH 2av (24b)

where we assume that the waves are x-polarized and normallypropagating. For a forward propagating pump and by making

use of (9), we have that

−Efw − Ebw + Epump= η0�0

2χ x xee

∂

∂ t(Efw + Ebw + Epump)

+η0�04

χ x x xeee∂

∂ t(Efw + Ebw + Epump)2 (25a)

−Efw + Ebw + Epump= μ0

2η0χ yymm

∂

∂ t(Efw − Ebw + Epump)

+ μ02η0

χ yyymmm∂

∂ t(Efw − Ebw + Epump)2. (25b)

This system of equation may now be solved either to obtainthe susceptibilities in terms of known fields or to get thescattered fields in terms of known susceptibilities. In [25],we already provide the susceptibilities in terms of knownfields, and therefore, we do not present them here again.However, the scattering from such a metasurface was onlypresented for the particular case where the linear and non-linear reflectionless conditions are satisfied, and thus, only thetransmission coefficients were provided. We shall now addressthe more general situation where the linear and nonlinearreflectionless are not necessarily satisfied leading to bothreflected and transmitted fields.

As it is, the system (25) forms a set of two first-order inho-mogeneous coupled nonlinear differential equations, whichdoes not possess analytical solutions. It is, however, possible toobtain approximate expressions of the scattered fields using theperturbation theory. We assume that the forward and backwardscattered fields may be, respectively, expressed as

Efw ≈ E0,fw + γ E1,fw + γ 2 E2,fw + · · · + γ n En,fw (26a)Ebw ≈ E0,bw + γ E1,bw + γ 2 E2,bw + · · · + γ n En,bw (26b)

where γ is a small quantity. We also consider the followingconditions on the susceptibilities:

χ x xee � χ x x xeee ∼ γ, and χ yymm � χ yyymmm ∼ γ. (27)These conditions reflect the difference in terms of amplitude

between the linear and nonlinear susceptibilities. For instance,in conventional optical systems, the value of gamma is aboutγ ∼ 10−12 [41].

It is now possible to obtain the approximate expressionof the backward, Ebw, and forward, Efw, scatteredfields by inserting (26) and (27) into (25) and usingEpump = E0 cos (ωt).

Now, for a given value of n, the nth term of expressions (26)can be solved for by removing all terms proportional to γ m

with m > n, which greatly simplifies the complexity ofthe system (25). By doing so, we derive the first 4 termsof the backward and forward scattered fields’ expansions.It turns out that each of these terms contains a certain numberof harmonics such that the nth term is proportional to thefollowing harmonic(s):

n = 0 → ω (28a)n = 1 → 2ω (28b)n = 2 → ω, 3ω (28c)n = 3 → 2ω, 4ω. (28d)


Finally, the fields scattered at frequency ω may be found bycombining the contributions from the n = 0 and n = 2 terms,while the fields scattered at frequency 2ω may be found usingthe n = 1 and n = 3 terms, and so on. The resulting fieldsscattered at ω are6

Eωfw = E04 + χ x xee χ yymmk2�

2 + jkχ x xee��

2 + jkχ yymm� − 4E30k2

×

�

χ x x xeee�2

�1 + jkχ x xee

��2−jkχ x xee

��2 + jkχ x xee

�3

+�χ

yyymmm

�2�1+ jkχ yymm

��2−jkχ yymm

��2+ jkχ yymm

�3

�

(29a)

Eωbw = E02 jk

�χ

yymm − χ x xee

�

�2 + jkχ x xee

��2 + jkχ yymm

� − 4E30k2

×

�

χ x x xeee�2

�1 + jkχ x xee

��2 − jkχ x xee


�3

−�χ

yyymmm

�2�1+ jkχ yymm

��2− jkχ yymm

��2+ jkχ yymm

�3

�

(29b)

where the first terms (proportional to E0) on the right-handsides correspond to the undepleted pump approximations.In fact, these two terms are the solutions that are found for thescattering of a conventional linear metasurface [22] and maybe derived directly from (38). The second terms (proportionalto E30) on the right-hand sides are correction terms that modelthe depletion of the pump. Note that these correction terms aredirectly proportional to the square of the wavenumber and thenonlinear susceptibilities, this means that they are generallyweak compared with the linear contributions except for verylarge values of E0.

We next present the solutions corresponding to the scatteredfields at 2ω. However, due to the length of these expressions,we only provide the n = 1 terms, which read

E2ωfw = −2 j E20k

χ x x xeee�1 + jkχ x xee


�2

+ χyyymmm�

1 + jkχ yymm�(2 + jkχ yymm)2

�

(30a)

E2ωbw = −2 j E20k

χ x x xeee�1 + jkχ x xee


�2

− χyyymmm

�1 + jkχ yymm

��2 + jkχ yymm

�2

�

. (30b)

Again, these terms correspond to the undepleted pump approx-imation. Adding the 2ω contributions from the n = 3terms would provide a correction which takes into accountthe depletion of the pump. As a consequence, the resultsin (30) are exactly those that would be obtained using thegeneral frequency-domain scattering relations (16) with theassumption that χ(ω) = χ(2ω).

6In these expressions, as well as in (30), the wavenumber is at frequencyω, i.e., k = ω/c0.

The time-domain approach presented here is thus morecomplicated to use than the frequency-domain one discussedearlier. However, the main advantage of this time-domainapproach is that it allows one to consider the depletion of thepump, which may be useful when the second-harmonic con-version efficiency of nonlinear metasurfaces will become sig-nificant. Finally, we also mention the fact that the time-domaintechnique may also be converted into a finite-difference time-domain technique, as discussed in [25].

V. CONCLUSION

In this paper, we have presented an elaborate discussion onthe electromagnetic theory of the second-harmonic generationin nonlinear metasurfaces. We have focused our attention onthe homogenization and second-harmonic scattering analysisof such structures. Both a frequency-domain and a time-domain approach have been presented.

It is clear that with the current conversion efficiency of non-linear metasurfaces, the frequency-domain approach, whichassumes an undepleted pump regime, is the most convenientof the two techniques.

Moreover, we have tried to remain as general as possibleso as to cover all possible scenarios. Accordingly, we havederived the general reflectionless and transmissionless condi-tions and highlighted the fundamental reasons of asymmetricreflection and transmission in these structures. We have alsoclarified the concept of asymmetric scattering versus nonrecip-rocal scattering in nonlinear media when changes in frequencyare considered.

APPENDIX AHOMOGENIZATION AND SCATTERING

OF LINEAR METASURFACES

In this appendix, we briefly present the main steps requiredto homogenize and obtain the scattering parameters of linearbianisotropic metasurfaces [24]. We will here assume thatthese metasurfaces are uniform and the incident wave prop-agates normally so that the spatial derivatives in (1) may bedropped.7 In the case of a bianisotropic linear metasurface,the GSTCs read [22]

ẑ × �H = jω�0χee · Eav + jk0χem · Hav (31a)�E × ẑ = jωμ0χmm · Hav + jk0χme · Eav. (31b)

It is often particularly convenient to cast this system ofequations into a matrix form to simplify the upcoming com-putations. Accordingly, the system becomes

⎛

⎜⎜⎝

�Hy�Hx�Ey�Ex

⎞

⎟⎟⎠ =

⎛

⎜⎜⎜⎝

�χ x xee �χxyee �χ x xem �χ

xyem

�χ yxee �χyyee �χ

yxem �χ

yyem

�χ x xme �χxyme �χ x xmm �χ

xymm

�χ yxme �χyyme �χ

yxmm �χ

yymm

⎞

⎟⎟⎟⎠

·

⎛

⎜⎜⎝

Ex,avEy,avHx,avHy,av

⎞

⎟⎟⎠ (32)

where the relationship between the susceptibilitiesin (31) and the normalized susceptibilities in (32) is

7The very general case of oblique incidence may be treated following theexact same procedure as that described in Section III-A


given by⎛

⎜⎜⎜⎝

χ x xee χxyee χ

x xem χ

xyem

χyxee χ

yyee χ

yxem χ

yyem

χ x xme χxyme χ

x xmm χ

xymm

χyxme χ

yyme χ

yxmm χ

yymm

⎞

⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

j

ω�0�χ x xee

j

ω�0�χ xyee

j

k0�χ x xem

j

k0�χ xyem

− jω�0

�χ yxee − jω�0

�χ yyee − jk0

�χ yxem − jk0

�χ yyem

− jk0

�χ x xme −j

k0�χ xyme − j

ωμ0�χ x xmm −

j

ωμ0�χ xymm

j

k0�χ yxme

j

k0�χ yyme

j

ωμ0�χ yxmm

j

ωμ0�χ yymm

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(33)

We now write the matrix system (32) in the followingcompact form:

� = �χ · Av (34)where �, �χ , and Av refer to the field differences, the normal-ized susceptibilities, and the field averages, respectively.

From this system of equations, we can now easily homog-enize the metasurface. This may be done by illuminating themetasurface with a normally incident plane wave and com-puting the electric and magnetic scattered fields. From theseknown fields, we can define the components of the field dif-ferences, �, and averages, Av , and ultimately obtain the sus-ceptibilities, �χ , by the matrix inversion of (34). However, thissystem of equations contains 16 unknown susceptibilities foronly 4 equations and is thus underdetermined. In order to solveit, we consider four different illuminations instead of just 1.As a consequence, the system now contains 16 equations (4 foreach illumination) for the same 16 unknown susceptibilitiesand is now fully determined. The four illuminations that weconsider are forward x-polarization, forward y-polarization,backward x-polarization, and backward y-polarization.

For each of these illuminations and resulting scattered fields,we express the corresponding electric and magnetic fieldsin terms of scattering parameters. For instance, the incident,

reflected, and transmitted electric fields, in the case of aforward x-polarized excitation, are, respectively, given by

Ei = x̂, Er = Sx x11 x̂ + Syx11 ŷ, Et = Sx x21 x̂ + Syx21 ŷ (35)where Suvab , with a, b = {1, 2} and u, v = {x, y}, are thescattering parameters. We consider that port 1 is on the left(z < 0) of the metasurface, while port 2 is on its right (z > 0).We also consider that the metasurface is surrounded on bothsides by different media with intrinsic impedances η1 and η2,respectively.

Expressing the electromagnetic fields of the four illumina-tions in the same fashion as in (35), leads, after simplification,to the matrices � and Av given in the following [see (37)],as shown at the bottom of this page, where the matrices Saband N are defined by:

Sab =�

Sx xab Sxyab

Syxab Syyab

�, N =

�1 00 −1

�. (36)

Instead of homogenizing a metasurface, we may use thesystem (34) to find to fields scattered by a metasurface withthe known susceptibilities. To do this, we insert (37) into (34)and solve for the scattering parameters. We thus obtain thefollowing relation:

S = M−11 · M2 (38)where S is a 4 × 4 matrix defined as

S =

S11 S12S21 S22

�

(39)

and the resulting matrices M1 and M2 are given [see (40)],as shown at the bottom of this page.

APPENDIX BDISCUSSION ON THE NONRECIPROCITY

OF METASURFACES

Nonreciprocity is a theoretically and practically importantconcept that is often misunderstood [51]. In what follows,we provide a brief discussion on the concept of nonreciprocitywhich applies to linear and nonlinear media so as to clarifyseveral points brought up in this paper.

� =

−N/η1 + N · S11/η1 + N · S21/η2 −N/η2 + N · S12/η1 + N · S22/η2−J · N − J · N · S11 + J · N · S21 J · N − J · N · S12 + J · N · S22

�

(37a)

Av = 12

I + S11 + S21 I + S12 + S22

J/η1 − J · S11/η1 + J · S21/η2 −J/η2 − J · S12/η1 + J · S22/η2

�

(37b)

M1 =⎛

⎝N/η1 − �χee/2 + �χem · J/(2η1) N/η2 − �χee/2 − �χem · J/(2η2)

−J · N − �χme/2 + �χmm · J/(2η1) J · N − �χme/2 − �χmm · J/(2η2)

⎞

⎠ (40a)

M2 =⎛

⎝�χee/2 + N/η1 + �χem · J/(2η1) �χ ee/2 + N/η2 − �χem · J/(2η2)�χme/2 + J · N + �χmm · J/(2η1) �χme/2 − J · N − �χmm · J/(2η2)

⎞

⎠ (40b)


In the case of a linear and time-invariant (LTI) system(a metasurface in our case), nonreciprocity may be achieved bybreaking the time-reversal symmetry of the system. Accordingto the Onsager–Casimir principle [52], [53], which providesthe time-symmetry relations of tensorial constitutive parame-ters, the action of breaking time-reversal symmetry may berealized by externally biasing the system with a time-oddquantity [54]. For instance, Faraday isolators are well-knownsystems that achieve nonreciprocity by biasing a ferrite witha static magnetic field [55].

In the absence of external bias, the Onsager–Casimir sym-metry relations reduce to the conventional reciprocity condi-tions provided by the Lorentz reciprocity theorem [50]. In thecase of a bianisotropic medium, they are given by

χTee = χee, χTmm = χmm, χTme = −χem. (41)

An LTI metasurface is, thus, reciprocal if these conditions aresatisfied.

In order to assess the nonreciprocal response of a system,it is often particularly convenient to analyze its scatteringparameters. In the case of a two-port system, the followingreciprocity conditions apply:

ST

21 = S12, ST

11 = S11, ST

22 = S22 (42)where the scattering matrices have the form of (36). Theseconditions may naturally be extended to the case of an N-portnetwork [56]. An important consideration is that an LTI system

may exhibit asymmetric scattering such that ST

21 �= S21,while still being perfectly reciprocal, S

T

21 = S12. In fact,a spatially asymmetric LTI system is always reciprocal expectif it is externally biased with a time-odd quantity as mentionedearlier.

The case of nonlinear media is more complicated. First ofall, there is no nonlinear counterpart to the Lorentz reciprocitytheorem [35]. Furthermore, spatial asymmetry is sufficientto achieve nonreciprocity [57]. Finally, we should considertwo different situations: when the frequency of the excitationis changed by the nonlinear process, it is for instance thecase with the second-harmonic generation, and when it is notchanged, it is for instance the case in the Kerr media [41]. Forthat latter situation, it should be noted that in some particularcases of four-wave mixing, where the input and output photonshave the same frequency, it is possible to obtain nonlinearOnsager relations [35], [36]. However, this does not applyto the case of second-order media that notably change thefrequency of the excitation and are the topic of this paper.

In what follows, we shall discuss the nonreciprocal behaviorof the second-order nonlinear media. We illustrate this dis-cussion with a simplified example. Consider the following1-D (with no spatial variations in the x- and y-directions,i.e., ∂/∂y = ∂/∂x = 0) two-port problem consist-ing of a second-order nonlinear metasurface surrounded byports 1 and 2 placed on its left- and right-hand sides, respec-tively. The metasurface is illuminated from port 1 with apump field, E1, at frequency ω. The resulting transmitted field,E2, is then measured at port 2. In a very general scenario,

the field E2 is proportional to all multiples of the fundamentalharmonic, ω. Assuming that all waves are x-polarized, the fieldE2 can thus be expressed as

E2 = Sω→ω21 E1 + Sω→2ω21 E21 + Sω→3ω21 E31 + · · · (43)where the scattering parameters correspond to the complexamplitude of the corresponding harmonic received at port 2.

The time-reversed (−t) operation of the process describedin (43) would consist in reversing the direction of wave prop-agation, such that the field E2 (and all its harmonics) emergesfrom port 2 and transmits back through the metasurface to thenbe received at port 1 [36]. By symmetry, the field received atport 1 is exactly the same as the original pump field E1 of thedirect time (t) scenario. Only in this case, would the systembe considered as reciprocal and we would have that

Sω→ω21 = Sω→ω12 , Sω→2ω21 = S2ω→ω12 , Sω→3ω21 = S3ω→ω12 , . . .(44)

It is important to realize that this time-reversed operation isa purely mathematical concept, which is practically impossibleto implement. Indeed, it would be impossible to generate allthe harmonics constituting E2 and reproduce the exact phaseshift between them so that the field received at port 1 is thesame as the original field E1 [36]. Therefore, the equalitiesin (44) are generally not satisfied and the reversed operationis thus nonreciprocal. Accordingly, we generally have that

Sω→ω21 �= Sω→ω12 , Sω→2ω21 �= S2ω→ω12 , Sω→3ω21 �= S3ω→ω12 , . . .(45)

Note that if the system is spatially symmetric (in thez-direction) and the field E2 is generated at port 2 such thatit contains only the fundamental harmonic, then the equalitySω→ω21 = Sω→ω12 is respected, provided that the conditions (41)are satisfied. However, if the field E2 contains only thefrequency 2ω, then the equality Sω→2ω21 = S2ω→ω12 is generallynot satisfied [even if the conditions (41) are satisfied]. Sucha scenario is discussed in [58] for the case of time-varyingmetasurfaces that behave in a very similar fashion as nonlinearmetasurfaces.

Finally, we point out that a nonlinear metasurface mayexhibit asymmetric nonlinear scattering, which has to beclearly differentiated from the nonreciprocal nonlinear scatter-ing described by relations (45). Asymmetric second-hamonicscattering is defined as Sω→2ω21 �= Sω→2ω12 , which essentiallymeans that the second-harmonic field scattered in transmissionby the metasurface is not the same as that when the latteris spatially flipped on itself. While this effect has oftenbeen referred to as a nonreciprocal process in the litera-ture [25], [28], [29], [49], the fact that Sω→2ω21 �= Sω→2ω12is not due to time-reversal symmetry breaking, such as theinequalities in (45), but rather due to the asymmetric scatteringof electric and magnetic nonlinear dipolar moments, as shownin Fig. 5. Accordingly, this effect should not be referred to asnonreciprocal but rather as asymmetric scattering, which doesnot make it less interesting or potentially useful.


C1 = − j2A ·�c0

ωJ + jχmm · J + jχme

�·�c0

ωI + jχee + jχem · J

�−1(46a)

C2 = − j2A · J (46b)C3 = j2

�c0ω

I+ jχee+ jχem ·J�−1 ·

��c0ω

I+ jχee− jχem · J�

· A ·�c0

ωJ + jχmm · J+ jχme

�·�c0

ωI+ jχee + jχem · J

�−1−I�

(46c)

C4 = − j2�c0


�−1 ·�c0

ωI + jχee − jχem · J

�· A · J (46d)

A =��c0

ωJ + jχmm · J − jχme

�+

�c0ω

J + jχmm · J + jχme�

·�c0


�−1 ·�c0

ωI + jχee − jχem · J

��−1

(46e)

APPENDIX CREDUCED TENSOR PARAMETERS

In this appendix, we first provide the reduced linear sus-ceptibility matrix parameters that are used to define generalsecond-harmonic scattered field relations (16) as well as thenonlinear scattering tensors in (21). These reduced matricesare provided in (46), as shown at the top of this page.

Next, we provide the relationships between the rotatednonlinear susceptibility tensors and their original form. Theserotations affect the inner matrices of these third-order tensorssuch that

�χ

eem

�x = −(χeem)x · J

�χ

eem

�y = −(χeem)y · J (47a)

�χ

emm

�x = J · (χ emm)x · J

�χ

emm

�y = J · (χemm)y · J

(47b)�χ

mmm

�x = −J · (χmmm)y · J

�χ

mmm

�y = J · (χmmm)x · J

(47c)�χ

mem

�x = (χmem)y · J

�χ

mem

�y = −(χmem)x · J (47d)

�χ

mee

�x = −(χmee)y

�χ

mee

�y = (χmee)x . (47e)

APPENDIX DREFLECTIONLESS CONDITIONS FOR LINEAR

BIANISOTROPIC METASURFACES

As discussed in Section III-B, the nonlinear reflectionlessconditions directly depend on both the linear and nonlinearmetasurface susceptibility tensors. Interestingly, it was shownthat these nonlinear reflectionless conditions are greatly sim-plified when the linear reflectionless conditions are satisfied.We thus provide here the general reflectionless conditions forlinear metasurfaces in the case of a normally incident planewave.

These linear reflectionless conditions are obtained in asimilar fashion as their nonlinear counter parts. The procedureto obtain them is to set S11 = 0 and S22 = 0 in (38) and solvefor the susceptibilities. Accordingly, we get

χee = −J · χmm · J (48a)χem = J · χme · J. (48b)

A particularly interesting scenario is when the metasurfaceis (linearly) reciprocal, which is most often the case. In this

situation, the only way to simultaneously satisfy the condi-tion (48b) and the reciprocity conditions (41) is when

χem = χme = κI (49)where κ is a chiral coefficient. This equality implies that areciprocal bianisotropic metasurface can only be reflectionlesswhen it corresponds to a chiral bi-isotropic structure [42].

REFERENCES

[1] N. Yu et al., “Light propagation with phase discontinuities: General-ized laws of reflection and refraction,” Science, vol. 334, no. 6054,pp. 333–337, Oct. 2011.

[2] N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” NatureMater., vol. 13, pp. 139–150, Jan. 2014.

[3] A. E. Minovich, A. E. Miroshnichenko, A. Y. Bykov, T. V. Murzina,D. N. Neshev, and Y. S. Kivshar, “Functional and nonlinear opticalmetasurfaces,” Laser Photon. Rev., vol. 9, no. 2, pp. 195–213, 2015.

[4] S. B. Glybovski, S. A. Tretyakov, P. A. Belov, Y. S. Kivshar, andC. R. Simovski, “Metasurfaces: From microwaves to visible,” Phys. Rep.,vol. 634, pp. 1–72, May 2016.

[5] A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonicswith metasurfaces,” Science, vol. 339, no. 6125, p. 1232009, 2013.

[6] A. Krasnok, M. Tymchenko, and A. Alù, “Nonlinear metasurfaces:A paradigm shift in nonlinear optics,” Mater. Today, vol. 21, no. 1,pp. 8–21, 2017.

[7] J. Lee et al., “Giant nonlinear response from plasmonic metasurfacescoupled to intersubband transitions,” Nature, vol. 511, pp. 65–69,Jul. 2014.

[8] M. Ren, E. Plum, J. Xu, and N. I. Zheludev, “Giant nonlinear opti-cal activity in a plasmonic metamaterial,” Nature Commun., vol. 3,May 2012, Art. no. 833.

[9] Y. Yang et al., “Nonlinear fano-resonant dielectric metasurfaces,” NanoLett., vol. 15, no. 11, pp. 7388–7393, 2015.

[10] M. Tymchenko, J. Gomez-Diaz, J. Lee, M. A. Belkin, and A. Alù,“Highly-efficient THz generation using nonlinear plasmonic metasur-faces,” J. Opt., vol. 19, no. 10, p. 104001, 2017.

[11] V. K. Valev et al., “Asymmetric optical second-harmonic generationfrom chiral G-shaped gold nanostructures,” Phys. Rev. Lett., vol. 104,p. 127401, Mar. 2010.

[12] R. Czaplicki et al., “Enhancement of second-harmonic generation frommetal nanoparticles by passive elements,” Phys. Rev. Lett., vol. 110,p. 093902, Feb. 2013.

[13] M. Celebrano et al., “Mode matching in multiresonant plasmonicnanoantennas for enhanced second harmonic generation,” NatureNanotechnol., vol. 10, pp. 412–417, Apr. 2015.

[14] K.-Y. Yang et al., “Wavevector-selective nonlinear plasmonic metasur-faces,” Nano Lett., vol. 17, no. 9, pp. 5258–5263, 2017.

[15] F. Walter, G. Li, C. Meier, S. Zhang, and T. Zentgraf, “Ultrathinnonlinear metasurface for optical image encoding,” Nano Lett., vol. 17,no. 5, pp. 3171–3175, 2017.

[16] P.-Y. Chen and A. Alù, “Optical nanoantenna arrays loaded withnonlinear materials,” Phys. Rev. B, Condens. Matter, vol. 82, no. 23,p. 235405, 2010.


[17] W. Ye et al., “Spin and wavelength multiplexed nonlinear metasurfaceholography,” Nature Commun., vol. 7, Jun. 2016, Art. no. 11930.

[18] M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin,and A. Alù, “Advanced control of nonlinear beams with pancharatnam-berry metasurfaces,” Phys. Rev. B, Condens. Matter, vol. 94, p. 214303,Dec. 2016.

[19] S. S. Kruk et al., “Nonlinear optical magnetism revealed by second-harmonic generation in nanoantennas,” Nano Lett., vol. 17, no. 6,pp. 3914–3918, 2017.

[20] R. Camacho-Morales et al., “Nonlinear generation of vector beams fromalgaas nanoantennas,” Nano Lett., vol. 16, no. 11, pp. 7191–7197, 2016.

[21] S. Kruk et al., “Enhanced magnetic second-harmonic generation fromresonant metasurfaces,” ACS Photon., vol. 2, no. 8, pp. 1007–1012, 2015.

[22] K. Achouri, M. A. Salem, and C. Caloz, “General metasurface synthesisbased on susceptibility tensors,” IEEE Trans. Antennas Propag., vol. 63,no. 7, pp. 2977–2991, Jul. 2015.

[23] K. Achouri, B. A. Khan, S. Gupta, G. Lavigne, M. A. Salem, andC. Caloz, “Synthesis of electromagnetic metasurfaces: Principles andillustrations,” EPJ Appl. Metamater., vol. 2, p. 12, Oct. 2015.

[24] K. Achouri and C. Caloz. (Dec. 2017). “Design, concepts and applica-tions of electromagnetic metasurfaces.” [Online]. Available: https://arxiv.org/abs/1712.00618

[25] K. Achouri, Y. Vahabzadeh, and C. Caloz, “Mathematical synthesis andanalysis of a second-order magneto-electrically nonlinear metasurface,”Opt. Express, vol. 25, no. 16, pp. 19013–19022, Aug. 2017.

[26] A. Rose, S. Larouche, E. Poutrina, and D. R. Smith, “Nonlinear magne-toelectric metamaterials: Analysis and homogenization via a microscopiccoupled-mode theory,” Phys. Rev. A, Gen. Phys., vol. 86, p. 033816,Sep. 2012.

[27] A. Rose, D. Huang, and D. R. Smith, “Nonlinear interference andunidirectional wave mixing in metamaterials,” Phys. Rev. Lett., vol. 110,p. 063901, Feb. 2013.

[28] E. Poutrina and A. Urbas, “Multipole analysis of unidirectional lightscattering from plasmonic dimers,” J. Opt., vol. 16, no. 11, p. 114005,2014.

[29] E. Poutrina and A. Urbas, “Multipolar interference for non-reciprocalnonlinear generation,” Sci. Rep., vol. 6, Apr. 2016, Art. no. 25113.

[30] M. Merano, “Nonlinear optical response of a two-dimensional atomiccrystal,” Opt. Lett., vol. 41, no. 1, pp. 187–190, Jan. 2016.

[31] X. Liu, S. Larouche, and D. R. Smith, “Homogenized description andretrieval method of nonlinear metasurfaces,” Opt. Commun., vol. 410,pp. 53–69, Mar. 2018.

[32] E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway,“Averaged transition conditions for electromagnetic fields at a metafilm,”IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2641–2651,Oct. 2003.

[33] C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth,and D. R. Smith, “An overview of the theory and applications ofmetasurfaces: The two-dimensional equivalents of metamaterials,” IEEEAntennas Propag. Mag., vol. 54, no. 2, pp. 10–35, Apr. 2012.

[34] M. Merano, “Role of the radiation-reaction electric field in the opticalresponse of two-dimensional crystals,” Ann. Phys., vol. 529, no. 12,p. 1700062, 2017.

[35] R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys., vol. 67, no. 5,p. 717, 2004.

[36] M. Trzeciecki and W. Hübner, “Time-reversal symmetry in nonlinearoptics,” Phys. Rev. B, Condens. Matter, vol. 62, pp. 13888–13891,Dec. 2000, doi: 10.1103/PhysRevB.62.13888.

[37] Y. Zheng, H. Ren, W. Wan, and X. Chen, “Time-reversed wave mixingin nonlinear optics,” Sci. Rep., vol. 3, Nov. 2013, Art. no. 3245.

[38] S. Naguleswaran and G. E. Stedman, “Onsager relations and time-reversal symmetry in nonlinear optics,” J. Phys. B, At., Mol. Opt. Phys.,vol. 31, no. 4, p. 935, 1998.

[39] C. L. Holloway, A. Dienstfrey, E. F. Kuester, J. F. O’Hara, A. K. Azad,and A. J. Taylor, “A discussion on the interpretation and characterizationof metafilms/metasurfaces: The two-dimensional equivalent of metama-terials,” Metamaterials, vol. 3, no. 2, pp. 100–112, Oct. 2009.

[40] M. M. Idemen, Discontinuities in the Electromagnetic Field. Hoboken,NJ, USA: Wiley, 2011.

[41] R. W. Boyd, “Nonlinear optics,” in Handbook of Laser Technologyand Applications (Three-Volume Set). New York, NY, USA: Taylor &Francis, 2003, pp. 161–183.

[42] A. H. Sihvola, A. J. Viitanen, I. V. Lindell, and S. A. Tretyakov,Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech HouseAntenna Library). Norwood, MA, USA: Artech House, 1994.

[43] F. B. Arango, T. Coenen, and A. F. Koenderink, “Underpinninghybridization intuition for complex nanoantennas by magnetoelec-tric quadrupolar polarizability retrieval,” ACS Photon., vol. 1, no. 5,pp. 444–453, 2014.

[44] K. Achouri, G. Lavigne, and C. Caloz, “Comparison of two synthesismethods for birefringent metasurfaces,” J. Appl. Phys., vol. 120, no. 23,p. 235305, 2016.

[45] R. Czaplicki et al., “Second-harmonic generation from metal nanopar-ticles: Resonance enhancement versus particle geometry,” Nano Lett.,vol. 15, no. 1, pp. 530–534, 2015.

[46] B. Gallinet, A. M. Kern, and O. J. F. Martin, “Accurate and versatilemodeling of electromagnetic scattering on periodic nanostructures witha surface integral approach,” J. Opt. Soc. Amer. A, Opt. Image Sci.,vol. 27, no. 10, pp. 2261–2271, Oct. 2010.

[47] J. Butet, B. Gallinet, K. Thyagarajan, and O. J. F. Martin, “Second-harmonic generation from periodic arrays of arbitrary shape plasmonicnanostructures: A surface integral approach,” J. Opt. Soc. Amer. B, Opt.Phys., vol. 30, no. 11, pp. 2970–2979, Nov. 2013.

[48] M. Kerker, D.-S. Wang, and C. L. Giles, “Electromagnetic scatteringby magnetic spheres,” J. Opt. Soc. Amer., vol. 73, no. 6, pp. 765–767,Jun. 1983.

[49] V. K. Valev et al., “Nonlinear superchiral meta-surfaces: Tuning chiralityand disentangling non-reciprocity at the nanoscale,” Adv. Mater., vol. 26,no. 24, pp. 4074–4081, 2014.

[50] E. J. Rothwell and M. J. Cloud, Electromagnetics. Boca Raton, FL,USA: CRC Press, 2008.

[51] D. Jalas et al., “What is—And what is not—An optical isolator,” NaturePhoton., vol. 7, pp. 579–582, Jul. 2013.

[52] L. Onsager, “Reciprocal relations in irreversible processes. I,” Phys. Rev.,vol. 37, pp. 405–426, Feb. 1931.

[53] H. B. G. Casimir, “On onsager’s principle of microscopic reversibility,”Rev. Mod. Phys., vol. 17, pp. 343–350, Apr. 1945.

[54] J. D. Jackson, Classical Electrodynamics, 3rd ed. New York, NY, USA:Wiley, 1999.

[55] J. F. Nye, Physical Properties of Crystals: Their Representation byTensors and Matrices. London, U.K.: Oxford Univ. Press, 1985.

[56] D. M. Pozar, Microwave Engineering, 4th ed. Hoboken, NJ, USA: Wiley,2011.

[57] D. Roy, “Few-photon optical diode,” Phys. Rev. B, Condens. Matter,vol. 81, p. 155117, Apr. 2010.

[58] A. Shaltout, A. Kildishev, and V. Shalaev, “Time-varying metasurfacesand Lorentz non-reciprocity,” Opt. Mater. Express, vol. 5, no. 11,pp. 2459–2467, 2015.

Karim Achouri, photograph and biography not available at the time ofpublication.

Gabriel D. Bernasconi, photograph and biography not available at the timeof publication.

Jérémy Butet, photograph and biography not available at the time ofpublication.

Olivier J. F. Martin, photograph and biography not available at the time ofpublication.

http://dx.doi.org/10.1103/PhysRevB.62.13888

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 150 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 600 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages false /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 400 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/CreateJDFFile false /Description >>> setdistillerparams> setpagedevice

homogenization and scattering analysis of second-harmonic generation in nonlinear ... · 2019. 1....

Documents