1 Design, Concepts and Applications of Electromagnetic Metasurfaces Karim Achouri and Christophe Caloz Abstract—he paper overviews our recent work on the synthesis of metasurfaces and related concepts and applications. The syn- thesis is based on generalized sheet transition 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 susceptibility tensor considerations. Upon this basis, we next provide an in-depth development of our susceptibility- GSTC synthesis technique. Finally, we present five recent meta- surface concepts and applications, which cover the topics of bire- fringent transformations, bianisotropic refraction, light emission enhancement, remote spatial processing and nonlinear second- harmonic generation.he paper overviews our recent work on the synthesis of metasurfaces and related concepts and applications. The synthesis is based on generalized sheet transition 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 susceptibility tensor considerations. Upon this basis, we next provide an in-depth development of our susceptibility-GSTC synthesis technique. Finally, we present five recent metasurface concepts and applications, which cover the topics of birefringent transformations, bianisotropic refraction, light emission enhancement, remote spatial processing and non- linear second-harmonic generation.T I. I NTRODUCTION Metamaterials reached a peak of interest in the first decade of the 21st century. Then, due to their fabrication complexity, bulkiness and weight, and their limitations in terms of losses, frequency range and scalability, they became less attractive, and were progressively superseded by their two-dimensional counterparts, the metasurfaces [1]–[4]. The idea of controlling electromagnetic waves with elec- tromagnetically thin structures is clearly not a new concept. The first example is probably that of Lamb, who studied the reflection and transmission from an array of metallic strips, already back in 1897 [5]. Later, in the 1910s, Marconi used arrays of straight wires to realize polarization reflectors [6]. These first two-dimensional electromagnetic structures were later followed by a great diversity of systems that emerged mainly with the developments of the radar technology during World War II. Many of these systems date back to the 1960s. The Fresnel zone plate reflectors, illustrated in Fig. 1a, were based on the concept of the Fresnel lens demonstrated almost 150 years earlier and used in radio transmitters [7]. The frequency selective surfaces (FSS), illustrated in Fig. 1b, were developed as spatial filters [8], [9]. The reflectarray K. Achouri and C. Caloz are with the Department of Electrical Engineering, Polytechnique Montreal, Montreal, Quebec, Canada email: [email protected], [email protected]antennas [10], were developed as the flat counterparts of parabolic reflectors, and were initially formed by short-ended waveguides [11]. They were later progressively improved and Source (a) e (b) Source (c) φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 Source (d) Fig. 1: Examples of two-dimensional wave manipulating struc- tures: (a) Fresnel zone plate reflector, (b) reflectarray, (c) in- terconnected array lens and (d) frequency-selective surface. the short-ended waveguides were replaced with mircrostrip printable scattering elements in the late 1970s [12], [13], as shown in Fig. 1c. The transmissive counterparts of the reflectarrays are the transmitarrays, which were used as array lens systems and date back to the 1960s [14]–[16]. They were first implemented in the form of two interconnected planar arrays of dipole antennas, one for receiving and one for transmitting, where each antenna on the receiver side was connected via a delay line to an antenna on the transmit side, as depicted in Fig. 1d. Through the 1990ies, the transmitarrays evolved from interconnected antenna arrays to layered metallic structures that were essentially the functional extensions of FSS [17]–[19] with efficiency limited by the difficulty to con- trol the transmission phase over a 2π range while maintaining arXiv:1712.00618v1 [physics.optics] 2 Dec 2017
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1
Design, Concepts and Applications ofElectromagnetic Metasurfaces
Karim Achouri and Christophe Caloz
Abstract—he paper overviews our recent work on the synthesisof metasurfaces and related concepts and applications. The syn-thesis is based on generalized sheet transition conditions (GSTCs)with a bianisotropic surface susceptibility tensor model of themetasurface structure. We first place metasurfaces in a properhistorical context and describe the GSTC technique with somefundamental susceptibility tensor considerations. Upon this basis,we next provide an in-depth development of our susceptibility-GSTC synthesis technique. Finally, we present five recent meta-surface concepts and applications, which cover the topics of bire-fringent transformations, bianisotropic refraction, light emissionenhancement, remote spatial processing and nonlinear second-harmonic generation.he paper overviews our recent work on thesynthesis of metasurfaces and related concepts and applications.The synthesis is based on generalized sheet transition conditions(GSTCs) with a bianisotropic surface susceptibility tensor modelof the metasurface structure. We first place metasurfaces ina proper historical context and describe the GSTC techniquewith some fundamental susceptibility tensor considerations. Uponthis basis, we next provide an in-depth development of oursusceptibility-GSTC synthesis technique. Finally, we present fiverecent metasurface concepts and applications, which cover thetopics of birefringent transformations, bianisotropic refraction,light emission enhancement, remote spatial processing and non-linear second-harmonic generation.T
I. INTRODUCTION
Metamaterials reached a peak of interest in the first decadeof the 21st century. Then, due to their fabrication complexity,bulkiness and weight, and their limitations in terms of losses,frequency range and scalability, they became less attractive,and were progressively superseded by their two-dimensionalcounterparts, the metasurfaces [1]–[4].
The idea of controlling electromagnetic waves with elec-tromagnetically thin structures is clearly not a new concept.The first example is probably that of Lamb, who studied thereflection and transmission from an array of metallic strips,already back in 1897 [5]. Later, in the 1910s, Marconi usedarrays of straight wires to realize polarization reflectors [6].These first two-dimensional electromagnetic structures werelater followed by a great diversity of systems that emergedmainly with the developments of the radar technology duringWorld War II. Many of these systems date back to the1960s. The Fresnel zone plate reflectors, illustrated in Fig. 1a,were based on the concept of the Fresnel lens demonstratedalmost 150 years earlier and used in radio transmitters [7].The frequency selective surfaces (FSS), illustrated in Fig. 1b,were developed as spatial filters [8], [9]. The reflectarray
K. Achouri and C. Caloz are with the Department of ElectricalEngineering, Polytechnique Montreal, Montreal, Quebec, Canada email:[email protected], [email protected]
antennas [10], were developed as the flat counterparts ofparabolic reflectors, and were initially formed by short-endedwaveguides [11]. They were later progressively improved and
Source
(a)Source
(b)
Source
(c)
φ1
φ2
φ3
φ4
φ5
φ6
Source
(d)
Fig. 1: Examples of two-dimensional wave manipulating struc-tures: (a) Fresnel zone plate reflector, (b) reflectarray, (c) in-terconnected array lens and (d) frequency-selective surface.
the short-ended waveguides were replaced with mircrostripprintable scattering elements in the late 1970s [12], [13],as shown in Fig. 1c. The transmissive counterparts of thereflectarrays are the transmitarrays, which were used as arraylens systems and date back to the 1960s [14]–[16]. Theywere first implemented in the form of two interconnectedplanar arrays of dipole antennas, one for receiving and onefor transmitting, where each antenna on the receiver side wasconnected via a delay line to an antenna on the transmit side,as depicted in Fig. 1d. Through the 1990ies, the transmitarraysevolved from interconnected antenna arrays to layered metallicstructures that were essentially the functional extensions ofFSS [17]–[19] with efficiency limited by the difficulty to con-trol the transmission phase over a 2π range while maintaining
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a high enough amplitude. Finally, compact quasi-transparenttransmitarrays or phase-shifting surfaces, able to cover a 2π-phase range, were demonstrated in 2010 [20].
The aforementioned Fresnel lenses, FSS, reflectarrays andtransmitarrays are the precursors of today’s metasurfaces1.
From a general perspective, metasurfaces can be used to ma-nipulate the polarization, the phase and the amplitude of elec-tromagnetic fields. A rich diversity of metasurface applicationshave been reported in the literature to date and many moreare expected to emerge. These applications are too numerousto be exhaustively cited. Some of the most significant onesare reported in [26]–[34] (polarization tranformations), [35]–[41] (absorption) and [42]–[55] (wavefront manipulations).More sophisticated metasurfaces, transforming both phaseand polarization, have been recently realized. This includesmetasurfaces producing beams possessing angular orbital mo-mentum [56] or vortex waves [57]–[62], holograms [63], [64]and stable beam traction [65]. Additionally, nonreciprocaltransformations [66]–[70], nonlinear interactions [71]–[73],analog computing [74], [75] and spatial filtering [76]–[78]have also been reported.
To deploy their full potential, metasurfaces must be de-signed efficiently. This requires a solid model, that bothsimplifies the actual problem and provides insight into itsphysics. Metasurfaces are best modelled, according to Huy-gens’ principle, as surface polarization current sheets viacontinuous (locally homogeneous) bianisotropic surface sus-ceptibility tensorial functions. Inserting the corresponding sur-face polarization densities into Maxwell equations results inelectromagnetic sheet transition conditions, which consist inthe key equations to solve in the design of metasurfaces.
The objective of this paper is twofold. First, it will presenta general framework for the synthesis of the aforementionedmetasurface surface susceptibility functions for arbitrary (am-plitude, phase, polarization, propagation direction and wave-form) specified fields. From this point, the physical structure(material and geometry of the scattering particles, substrateparameters and layer configuration, thickness and size) istediously but straightforwardly determined, after discretizationof the susceptibility functions, using scattering parametermapping. The synthesis of metasurfaces has been the objectiveof many researches in recent years [79]–[90]. Second, thepaper will show how this synthesis framework provides ageneral perspective of the electromagnetic transformationsachievable by metasurfaces, and present subsequent conceptsand applications.
II. SHEET TRANSITION CONDITIONS
The general synthesis problem of a metasurface is repre-sented in Fig. 2. As mentioned in Sec. I, the metasurface ismodeled as an electromagnetic sheet (zero-thickness film)2.
1So far, and throughout this paper, we essentially consider metasurfacesilluminated by waves incident on the them under a non-zero angle with respectto their plane, i.e. space waves, which represent the metasurfaces leading tothe main applications. However, metasurface may also be excited within theirplane, i.e. by surface waves or leaky-waves, as in [21]–[25].
2This approximation is justified by the fact that a physical metasurface iselectromagnetically very thin, so that it cannot support significant phase shiftsand related effects, such as Fabry-Perot resonances.
In the most general case, a metasurface is made of an arrayof polarizable scattering particles that induce both electricand magnetic field discontinuities. It is therefore necessary toexpress the discontinuities of these fields as functions of theelectric and magnetic surface polarization densities (P andM ). The rigorous boundary conditions that apply to such aninterface have been originally derived by Idemen [91].
x
z
y
z “ 0
Ly
Lx
ψtprq
χpρq “? t ! λ
ψiprq
ψrprq
Fig. 2: Metasurface synthesis problem. The metasurface tobe synthesized lies in the xy-plane at z = 0. The synthe-sis procedure consists in finding the susceptibility tensorscharacterizing the metasurface, χ(ρ), in terms of specifiedarbitrary incident, ψi(r), reflected, ψr(r), and transmitted,ψt(r), waves.
For a metasurface lying in the xy-plane at z = 0, thesetransition conditions follow from the idea that all the quantitiesin Maxwell equations can be expressed in the following form
f(z) = {f(z)}+
N∑k=0
fkδ(k)(z), (1)
where the function f(z) is discontinuous at z = 0. The firstterm of the right-hand side of (1) is the regular part of f ,which corresponds to the value of the function everywhereexcept at z = 0, while the second term is the singular partof f , which is an expansion over the k-th derivatives of theDirac delta distribution (corresponding to the discontinuity off and the k-th derivatives of f ).
Most often, the series in (1) may be truncated at N = 0,so that only the discontinuities of the fields are taken intoaccount while the discontinuities of the derivatives of thefields are neglected. With this truncation, the metasurfacetransition conditions, known as the generalized sheet transitionconditions (GSTCs), are found as3
3Note that these relations can also be obtained following the more tradi-tional technique of box integration, as demonstrated in [92].
3
z ×∆H = jωP‖ − z ×∇‖Mz, (2a)
∆E × z = jωµ0M‖ −∇‖(Pzε0
)× z, (2b)
z ·∆D = −∇ · P‖, (2c)z ·∆B = −µ0∇ ·M‖, (2d)
where the terms on the left-hand sides of the equationscorrespond to the differences of the fields on both sides ofthe metasurface, which may be expressed as
∆Ψu = u·∆Ψ∣∣∣0+z=0−
= Ψu,t−(Ψu,i+Ψu,r), u = x, y, z, (3)
where Ψ represents any of the fields E, H , D or B, andwhere the subscripts i, r, and t denote the incident, reflectedand transmitted fields, and P and M are the electric andmagnetic surface polarization densities, respectively.
In the general case of a linear bianisotropic metasurface,these polarization densities are related to the acting (or local)fields, Eact and Hact, by [93], [94]
P = ε0Nαee ·Eact +1
c0Nαem ·Hact, (4a)
M = Nαmm ·Hact +1
η0Nαmm ·Eact, (4b)
where the αab tensors represent the polarizabilities of a givenscatterer, N is the number of scatterers per unit area, c0 is thespeed of light in vacuum and η0 is the vacuum impedance4.This is a microscopic description of the metasurface responsewhich requires an appropriate definition of the coupling be-tween adjacent scattering particles. In this work, we use theconcept of susceptibilities rather than the polarizabilities toprovide a macroscopic description of the metasurface, whichallows a direct connection with material parameters such as εrand µr. To bring about the susceptibilities, relations (4) can betransformed by noting that the acting fields, at the position ofa scattering particle, can be defined as the average total fieldsminus the field scattered by the considered particle [96], i.e.Eact = Eav − Epart
scat . The contributions of the particle maybe expressed by considering the particle as a combination ofelectric and magnetic dipoles contained within a small disk,and the field scattered from this disk can be related to Pand M by taking into account the coupling with adjacentscattering particles. Therefore, the acting fields are functionsof the average fields and the polarization densities. Uponsubstitution of this definition of the acting fields in (4), theexpressions of the polarization densities become
P = ε0χee ·Eav +1
c0χem ·Hav, (5a)
M = χmm ·Hav +1
η0χme ·Eav, (5b)
where the average fields are defined as
Ψu,av = u ·Ψav =Ψu,t + (Ψu,i + Ψu,r)
2, u = x, y, z, (6)
4Despite being indeed quite general, these relations are still restricted to lin-ear and time-invariant metasurfaces. The synthesis of nonlinear metasurfaceshas been approached using extended GSTCs in [95].
where Ψ corresponds to E or H .
III. SUSCEPTIBILITY TENSOR CONSIDERATIONS
Before delving into the metasurface synthesis, it is impor-tant to examine the susceptibility tensors in (5) in the lightof fundamental electromagnetic considerations pertaining toreciprocity, passivity and loss.
The reciprocity conditions for a bianisotropic metasurface,resulting from the Lorentz theorem [93], read
χTee = χee, χ
Tmm = χmm, χ
Tme = −χem, (7)
where the superscripts T denotes the matrix transpose opera-tion5.
Adding the property of losslessness, resulting from thebianisotropic Poynting theorem [93], restricts (7) to
χTee = χ
∗ee, χ
Tmm = χ
∗mm, χ
Tme = χ
∗em, (8)
which characterize a simultaneously passive, lossless andreciprocal metasurface.
The conditions (7) and (8) establish relations betweendifferent susceptibility components of the constitutive tensors.Therefore, requiring the metasurface to be reciprocal or re-ciprocal and lossless/gainless, as often practically desirable,reduces the number of independent susceptibility compo-nents [79], [97], [98], and hence reduces the diversity ofachievable field transformations, as will be shown next.
IV. METASURFACE SYNTHESIS
A. General Concepts
We follow here the metasurface synthesis procedure6 in-troduced in [79], which seems the most general approachreported to date. This procedure consists in solving the GSTCsequations (2) to determine the unknown susceptibilities in (5)required for the metasurface to perform the electromagnetictransformation specified in terms of the incident, reflected andtransmitted fields. Note that Eqs. (2c) and (2d) are redundantin the system (2), due to the absence of impressed sources,so that Eqs. (2a) and (2b) are sufficient to fully describethe metasurface and synthesize it. Consequently, only thetransverse (tangential to the metasurface) components of thespecified fields, explicitly apparent in (3) and in (6), areinvolved in the the synthesis, even though these fields maygenerally include longitudinal (normal to the metasurface)components as well. According to the uniqueness theorem,the longitudinal components of the fields are automaticallydetermined from the transverse fields.
5These conditions are identical to those for a bianisotropic medium [93],[94], except that the metasurfaces in (7) are surface instead of volumesusceptibilities
6The synthesis procedure consists in determining the physical metasurfacestructure for specified fields. The inverse procedure is the analysis, whichconsists in determining the fields scattered by a given physical metasurfacestructures for a given incident field, and is generally coupled (typicallyiteratively) with the synthesis for the efficient design of a metasurface [99].The overall design procedure thus consists of the combination of the synthesisand analysis operations. This paper focuses on the direct synthesis of the sus-ceptibility functions, as this is the most important aspect for the understandingof the physical properties of metasurfaces, the elaboration of related concepts,and the development of resulting applications.
4
The GSTC equations (2a) and (2b) form a set of (inhomoge-neous) coupled partial differential equations, due to the partialderivatives of the normal components of the polarizationdensities, Pz and Mz . The resolution of the correspondinginverse problem is nontrivial and requires involved numericalprocessing. In contrast, if Pz = Mz = 0, the differentialsystem reduces to a simple algebraic system of equations,most conveniently admitting closed-form solutions for thesynthesized susceptibilities. For this reason, we will focus onthis case in this section, while a transformation example withnonzero normal susceptibilities will be discussed in Sec. IV-D.
Enforcing that Pz = Mz = 0 may a priori seem to representan important restriction, particularly, as we shall see, in thesense that it reduces the number of degrees of freedom ofthe metasurface. However, this is not a major restriction sincea metasurface with normal polarization currents can generallybe reduced to an equivalent metasurface with purely tangentialpolarization currents, according to Huygens’ theorem. Thisrestriction mostly affects the realization of the scattering par-ticles, that are then forbidden to exhibit normal polarizations,which ultimately limits their practical implementation7.
Substituting the constitutive relations (5) into theGSTCs (2a) and (2b) with Mz = Pz = 0 leads to
z ×∆H = jωε0χee ·Eav + jk0χem ·Hav, (9a)∆E × z = jωµ0χmm ·Hav + jk0χme ·Eav, (9b)
where k0 = ω/c0 is the free-space wavenumber and where thesusceptibility tensors only contain the tangential susceptibilitycomponents. This system can also be written in matrix form
∆Hy
∆Hx
∆Ey∆Ex
=
χxxee χxyee χxxem χxyem
χyxee χyyee χyxem χyyem
χxxme χxyme χxxmm χxymm
χyxme χyyme χyxmm χyymm
·Ex,avEy,avHx,avHy,av
, (10)
where the tilde symbol indicates normalized susceptibilities,related to the non-normalized susceptibilities in (9) by
χxxee χxyee χxxem χxyem
χyxee χyyee χyxem χyyem
χxxme χxyme χxxmm χxymm
χyxme χyyme χyxmm χyymm
=
=
jωε0
χxxeejωε0
χxyeejk0χxxem
jk0χxyem
− jωε0
χyxee − jωε0
χyyee − jk0χyxem − j
k0χyyem
− jk0χxxme − j
k0χxyme − j
ωµ0χxxmm − j
ωµ0χxymm
jk0χyxme
jk0χyyme
jωµ0
χyxmmjωµ0
χyymm
.
(11)
The system (10) contains 4 equations for 16 unknown suscep-tibilities. It is therefore heavily under-determined and cannot
7Moreover, in the particular case where all the specified waves are normalto the metasurface, the excitation of normal polarization densities do notinduce any discontinuity in the fields. This is because the correspondingfields, and hence the related susceptibilities, are not functions of the x andy coordinates, so that the spatial derivatives of Pz and Mz in Eqs. (2a)and (2b) are zero, i.e. do not induce any discontinuity in the fields acrossthe metasurface. Thus, susceptibilities producing normal polarizations can beignored, and only tangential susceptibility components must be considered,when the metasurface is synthesized for normal waves.
be solved directly8. This leaves us with two distinct resolutionpossibilities.
The first possibility would be to reduce the number of sus-ceptibilities from 16 to 4 in order to obtain a fully determined(full-rank) system. Since there exists many combinations ofsusceptibility quadruplets9, different sets can be chosen, eachof them naturally corresponding to different field transforma-tions. This approach thus requires an educated selection of thesusceptibility quadruplet that is the most likely to enable thespecified operation, within existing constraints10.
These considerations immediately suggest that a secondpossibility would be to augment the number of field transfor-mation specifications, i.e. allow the metasurface to performmore independent transformations, which may be of greatpractical interest in some applications. We would have thusultimately three possibilities to resolve (10): a) reducing thenumber of independent unknowns, b) increasing the numberof transformations and c) a combination of a) and b).
As we shall see in the forthcoming sections, the numberN of physically or practically achievable transformations fora metasurface with P susceptibility parameters, N (P ), is nottrivial; specifically, N (P ) = P/4, that may be expected froma purely mathematical viewpoint, is not always true!
B. Four-Parameter Transformation
We now provide an example for the approach where thenumber of susceptibility parameters has been reduced to 4, orP = 4, so that the system (10) is of full-rank nature. We thushave to select 4 susceptibility parameters and set all the othersto zero in (11). We decide to consider the simplest case of amonoanisotropic (8 parameters χuvem,me = 0, u, v = x, y) axial(4 parameters χuvee,mm = 0 for u 6= v, u, v = x, y) metasurface,which is thus characterized by the four parameters χxxee , χyyee ,χxxmm and χyymm, so that Eq. (10) reduces to the diagonal system
∆Hy
∆Hx
∆Ey∆Ex
=
χxxee 0 0 00 χyyee 0 00 0 χxxmm 00 0 0 χyymm
·Ex,avEy,avHx,avHy,av
. (12)
This metasurface is a biregringent structure [100], with de-coupled x-polarized and y-polarized susceptibility pairs
χxxee =j∆Hy
ωε0Ex,av, χyymm =
j∆Exωµ0Hy,av
(13a)
and
χyyee =−j∆Hx
ωε0Ey,av, χxxmm =
−j∆Eyωµ0Hx,av
, (13b)
8Even if it would be solved, this would probably result in an inefficientmetasurface, as it would use more susceptibility terms than required toaccomplish the specified task.
9Mathematically, the number of combinations would be 16!/[(16−4)!4!] =1, 820, but only a subset of these combinations represent physically mean-ingful combinations.
10For instance, the specification of a reciprocal transformation, correspond-ing to the metasurface properties in Eq. (7), would automatically preclude theselection of off-diagonal pairs for χee,mm.
5
respectively11. In these relations, according to (3) and (6),∆Hy = Hy,t− (Hy,i +Hy,r), Ex,av = (Ex,t +Ex,i +Ex,r)/2,and so on. By synthesis, the metasurface with the suscep-tibilities (13) will exactly transform the specified incidentfield into the specified reflected and transmitted fields, in anarbitrary fashion, except for the constraint of reciprocity sincethe susceptibility tensor in (12) inherently satisfies (7).
It should be noted that the example of (12), with 4 dis-tinct susceptibility parameters, is a very particular case of afour-parameter transformation since the components in (13a)and (13b) are decoupled from each other, which is the originof birefringence. Now, birefringence may be considered as apair of distinct and independent transformations (one for x-polarization one for y-polarization), i.e. N (4) = 2 > 4/4.Thus, the specification of 4 susceptibility parameters may leadto more than 1 transformation, which, by extension, alreadysuggests that P susceptibilities may lead to more than P/4transformations, as announced in Sec. IV-A and will be furtherdiscussed in Sec. IV-C.
So far, the fields have not be explicitly specified in the meta-surface described by (12). Since the metasurface can performarbitrary transformations under the reservation of reciprocity,it may for instance by used for polarization rotation, whichwill turn to be a most instructive example here. Consider thereflectionless metasurface, depicted in Fig. 3, which transformsthe polarization of a normally incident plane wave. The fieldscorresponding to this transformation are
Ei(x, y) = x cos(π/8) + y sin(π/8), (14a)
Hi(x, y) =1
η0[−x sin(π/8) + y cos(π/8)] , (14b)
Er(x, y) = 0, (15a)Hr(x, y) = 0, (15b)
and
Et(x, y) = x cos(11π/24) + y sin(11π/24), (16a)
Ht(x, y) =1
η0[−x sin(11π/24) + y cos(11π/24)] . (16b)
Inserting these fields into (3) and (6), and substituting theresult in (13) yields the susceptibilities
χxxee = χyymm = −1.5048
k0j, (17a)
χyyee = χxxmm =0.88063
k0j. (17b)
Note that in this example12, the aforementioned doubletransformation reduces to a single transformation, N (4) =1 = 4/4, because the specified fields possess both x and ypolarizations. The susceptibilities do not depend on position
11If the two electric and the two magnetic susceptibilities in (13) are equal toeach other (χxx
ee = χyyee and χxx
mm = χyymm), the monoanisotropic metasurface
in (12) reduces to the simplest possible case of a monoisotropic metasurface,and hence performs the same operation for x- and y-polarized waves.
12Incidently, the equality between the electric and magnetic susceptibilitiesresults from the specification of zero reflection in addition to normal incidence.The reader may easily verify that in the presence of reflection, the equalitiesdo not hold.
x
xy
y
E
E
H
H
z
π8
11π24
Fig. 3: Polarization reflectionless rotating metasurface. Themetasurface rotates the polarization of a linearly polarizednormally incident plane wave from the angle π/8 to theangle 11π/24 with respect to the x-axis (rotation of π/3).The metasurface is surrounded on both sides by vacuum, i.e.η1 = η2 = η0.
since the specified transformation, being purely normal, onlyrotates the polarization angle and does not affect the directionof wave propagation.
The negative and positive imaginary natures of χxxee = χyymm
and χyyee = χxxmm in (17) correspond to absorption and gain,respectively. These features may be understood by noting,with the help of Fig. 3, that polarization rotation is accom-plished here by attenuation and amplification of (Ex,i, Hy,i)and (Ey,i, Hx,i), respectively. Moreover, this metasurface canrotate the polarization only by the angle π/3 when the incidentwave is polarized at a π/8 angle13. This example certainly rep-resents an awkward approach to rotate the field polarization!
A more reasonable approach is to consider a gyrotropicmetasurface, where the only nonzero susceptibilities are χxyee ,χyxee , χxymm and χyxmm. This corresponds to a different quadrupletof tensor parameters than in (12), which illustrates the afore-mentioned multiplicity of possible parameter set selection.With these susceptibilities, the system (10) yields the followingrelations:
χxyee =j∆Hy
ωε0Ey,av, (18a)
χyxee =−j∆Hx
ωε0Ex,av, (18b)
χxymm =−j∆Eyωµ0Hy,av
, (18c)
χyxmm =j∆Ex
ωµ0Hx,av, (18d)
which, upon substitution of the fields in (14) to (16), become
13If, for instance, the incident was polarized along x only, then only thesusceptibilities in (17a) would be excited and the resulting transmitted fieldwould still be polarized along x, just with a reduced amplitude with respectto that of the incident wave due to the loss induced by these susceptibilities.
6
χxyee = χxymm = −1.1547
k0j, (19a)
χyxee = χyxmm =1.1547
k0j. (19b)
Contrary to the susceptibilities in (17), those in (19) performthe specified π/3 polarization rotation irrespectively of theinitial polarization of the incident wave, due to the gyrotropicnature of the metasurface. It appears that these susceptibilitiesviolate the reciprocity conditions in (7), and the metasurfaceis thus nonreciprocal, which is a necessary condition forpolarization rotation with this choice of susceptibilities. Thus,the metasurface is a Faraday rotation surface, whose directionof polarization rotation is independent of the direction of wavepropagation [101], [102]. However, contrary to conventionalFaraday rotators [93], this metasurface is also reflectionlessdue to the presence of both electric and magnetic gyrotropicsusceptibility components (Huygens matching). The positiveand negative imaginary susceptibilities indicate that the meta-surface is simultaneously active and lossy, respectively. It isthis combination of gain and loss that allows perfect rotationin this lossless design. This design is naturally appropriateif Faraday rotation is required. However, it is not optimalin applications not requiring non-reciprocity, i.e. reciprocalgyrotropy, where the required loss and gain would clearlyrepresent a drawback.
Reciprocal gyrotropy may be achieved using bianisotropicchirality, i.e which involves the parameter set χxxem , χ
yyem, χxxme
and χyyme. Following the same synthesis procedure as before,we find
χxxem = χyyem = − 2√3k0
j, (20a)
χxxme = χyyme =2√3k0
j. (20b)
The corresponding metasurface is readily verified to be re-ciprocal, passive and lossless, since the susceptibilities (20)satisfy the conditions (8). So, if the purpose of the metasurfaceis to simply perform polarization rotation in a given direction,without specification for the opposite direction, this design isthe most appropriate of the three discussed, as it is purelypassive, lossless and working for all incident polarizations.
Note that the metasurfaces (19) and (20) both correspondto N (4) = 1 = 4/4.
C. More-Than-Four-Parameter Transformation
In the previous section, we have seen how the system (10)can be solved by reducing the number of susceptibilitiesto P = 4 parameters so as to match the number ofGSTCs equations, and seen some of the resulting single-transformation (N = 1, e.g. monoisotropic structure) ordouble-transformation (N = 2, e.g. birefringence) metasurfacepossibilities.
However, as mentioned in Sec. IV-A, the general systemof equations (10), given its 16 degrees of freedom (16 sus-ceptibility components), corresponds to a metasurface withthe potential capability to perform more transformations than
a metasurface with 4 parameters, or generally less than 16parameters, N (16) > N (P < 16). In what follows, we willsee how the system (10) can be solved for several independenttransformations, which includes the possibility of differentlyprocessing waves incident from either sides. To accommodatefor the additional degrees of freedom, a total of 4 wavetransformations are considered, instead of only one as donein Sec. IV-B, so that (10) becomes a full-rank system. Thecorresponding equations related to the system (10) may thenbe written in the compact form
where the subscripts 1, 2, 3 and 4 indicate the electromag-netic fields corresponding to 4 distinct and independent setsof waves14. The susceptibilities can be obtained by matrixinversion conjointly with the normalization (11). The resultingsusceptibilities will, in general, be all different from eachother. This means that the corresponding metasurface is bothactive/lossy and nonreciprocal.
Consider for example a metasurface with P = 8 parameters.In such a case, the system (10) is under-determined, sinceit features 4 equations in 8 unknowns. This suggests thepossibility to specify more than 1 transformations, N > 1. Letus thus consider for instance a monoanisotropic (8-parameter)metasurface, and let us see whether such a metasurface canindeed perform 2 transformations. The corresponding systemfor 2 transformation reads
This system (22), being full-rank, automatically admits a so-lution for the 8 susceptibilities, i.e. N = 2. The only questionis whether this solution complies with practical design con-straints. For instance, the electric and magnetic susceptibilitysubmatrices are non-diagonal, and may therefore violate the
14It is also possible to solve a system of equations that contains less thanthese 16 susceptibility components. In that case, less than 4 wave transfor-mations should be specified so that the system remains fully determined.For instance, two independent wave transformations (possessing both x andy polarizations) could be solved with 8 susceptibilities. Similarly, 3 wavetransformations could be solved with 12 susceptibilities.
7
reciprocity condition (7). If nonreciprocity is undesirable orunrealizable in a practical situation, then one would have totry another set of 8 parameters.
If this 8-parameter metasurface performs only 2 transfor-mations, then one may wonder what is the difference withthe 4-parameter birefringent metasurface in (12) which canalso provide 2 transformations with just 4 parameters. Thedifference is that the 2-transformation property of the meta-surface in (12) is restricted to the case where the fields ofthe two transformations are orthogonally polarized15, whereasthe 2-transformation property of the metasurface in (22) iscompletely general.
As an illustration of the latter metasurface, consider the twotransformations depicted in Figs. 4. The first transformation,shown in Fig. 4a, consists in reflecting at 45◦ a normally inci-dent plane wave. The second transformation, shown in Fig. 4b,consists in fully absorbing an incident wave impinging on themetasurface under 45◦. In both cases, the transmitted field isspecified to be zero for the first and second transformations.The transverse components of the electric fields for the twotransformations are, at z = 0, given by
Ei,1 =
√2
2(x+ y), Er,1 =
√2
2(− cos θrx+ y)e−jkxx, (23a)
Ei,2 =
√2
2(cos θix+ y)e−jkxx, (23b)
respectively.
x
z
θr “ 45˝
λ0100
(a)
x
z
θi “ 45˝
λ0100
(b)
Fig. 4: Example of double-transformation metasurface.(a) First transformation [corresponding to the subscript 1in (22)]: the normally incident plane wave is fully reflectedat a 45◦ angle. (b) Second transformation [corresponding tothe subscript 2 in (22)]: the obliquely incident plane wave isfully absorbed.
The synthesis is then performed by inserting the electricfields (23), and the corresponding magnetic fields, into (22).The susceptibilities are then straightforwardly obtained bymatrix inversion in (22). For the sake of conciseness, wedo not give them here, but we point out that they includenonreciprocity, loss and gain, and complex spatial variations.
This double-transformation response is verified by full-wave simulation and the resulting simulations are plottedin Figs. 5. The two simulations in this figure have beenrealized in the commercial FEM software COMSOL, wherethe metasurface is implemented as a thin material slab of
15For instance, if the fields of the first transformation are only x-polarized,while the fields of the second transformation are only y-polarized.
thickness d = λ0/10016. The simulation corresponding tothe transformation of Fig. 4a is shown in Fig. 5a, while thesimulation corresponding to the transformation of Fig. 4b isshown in Fig. 5b. The simulated results are in agreement withthe specification [Eq. (23)], except for some scattering due tothe non-zero thickness of the full-wave slab approximation.
0.5
0
1
0 5 10 15 15 20-5-10
0
5
10
-5
-10
-15-20-25
x{λ0
z{λ
0
(a)
0.5
0
1
0 5 10 15 15 20-5-10
0
5
10
-5
-10
-15-20-25
x{λ0
z{λ
0
(b)
Fig. 5: 8-parameter metasurface simulations. COMSOL sim-ulated normalized absolute value of the total electric fieldcorresponding to: (a) the transformation in Fig. 4a, and (b)the transformation in Fig. 4b.
The example just presented, where both the transforma-tions 1 and 2 include all the components of the fields,corresponds to N (8) = 2 = 8/4, i.e. N (P ) = P/4. However,in the same manner as the birefringent metasurface of (13),featuring N (4) = 2 > 4/4, i.e. specifically N (P ) = P/2,the metasurface in (22) may lead to N (P ) > P/4. This de-pends essentially on whether the specified transformations arecomposed of fields that are either: a) only x- or y-polarized,or b) both x- and y-polarized. The two transformations givenby the fields (23) are both x- and y-polarized, which thuslimits the number of transformations to N (P ) = P/4. Ifthe transformation given by (23b) was specified such thatEiy,2 = 0 (i.e. no polarization along y), then this would releasedegrees of freedom, and hence allow a triple transformation,i.e. N (8) = 3 > 8/4. In addition, if the first transformation,given by (23a), also had transverse components of the electricfield polarized only along x or y, then we could achieveN (8) = 4 > 8/4 transformations. These considerations
16The synthesis technique yields the susceptibilities for an ideal zero-thickness metasurface. However, the metasurface sheet may be approximatedby an electrically thin slab of thickness d (d � λ) with volume suscep-tibility corresponding to a diluted version of the surface susceptibility, i.e.χvol = χ/d [79].
8
illustrate the necessity to perform educated selections in themetasurface synthesis procedure, as announced in Sec. (IV-A).
D. Metasurface with Nonzero Normal Polarizations
So far, we have discarded the possibility of normal polar-izations by enforcing Pz = Mz = 0 in (2). This is not onlysynthesis-wise convenient, since this suppresses the spatialderivatives in (2), but also typically justified by the fact thatany electromagnetic field can be produced by purely tangentialsurface currents/polarizations according to the Huygens the-orem. It was accordingly claimed in [103] that these normalpolarizations, and corresponding susceptibility components, donot bring about any additional degrees of freedom and canthus be completely ignored. It turns out that this claim isgenerally not true: in fact Pz and Mz provide extra degrees offreedom that allow a metasurface to perform a larger numberof distinct operations for different incident field configurationsand at different times.
The Huygens theorem exclusively applies to a single (arbi-trarily complex) combination of incident, reflected and trans-mitted waves. This means that any metasurface, possiblyinvolving normal polarizations, that performs the specifiedoperation for such a single combination of fields can bereduced to an equivalent metasurface with purely transversepolarizations. However, the Huygens theorem does not applyto case of waves impinging on the metasurface at differenttimes. Indeed, it is in this case impossible to superimpose thedifferent incident waves to form a total incident field sincethey are not simultaneously illuminating the metasurface. Con-sequently, a purely tangential description of the metasurface isincomplete, and normal polarizations thus become necessaryto perform the synthesis.
In fact, the presence of these normal susceptibility compo-nents greatly increases the number of degrees of freedom sincethe susceptibility tensors are now 3 × 3 matrices, instead of2 × 2 as in (10). This means that, for the 4 relevant GSTCsequations, we have now access to 36 unknown susceptibilities,instead of only 16, which increases the potential number ofelectromagnetic transformations from 4 to 9, provided thatthese transformations include fields that are independent fromeach other.
The synthesis of metasurfaces with nonzero normal polar-ization densities may be performed following similar proce-dures as those already discussed. As before, one needs tobalance the number of unknown susceptibilities to the numberof available equations provided by the GSTCs. Dependingon the specifications, this may become difficult since manytransformations may be required to obtain a full-rank system.Additionally, if the specified transformations involve changingthe direction of wave propagation, then the system (2) becomesa coupled system of partial differential equations in termsof the susceptibilities since the latter would now depend onthe position. This generally prevents the derivation of closed-form solutions of the susceptibilities, which should ratherbe obtained numerically. However, we will now provide anexample of a synthesis problem, where the susceptibilities areobtainable in closed form.
More specifically, we discuss the synthesis and analysisof a reciprocal metasurface with controllable angle-dependentscattering [104]–[106]. To synthesize this metasurface, weconsider the three independent17 transformations depicted inFig. 6. Specifying these three transformations allows one to
Ei,1
Ei,3
θt,1
θi,1
θi,3
Et,1
Ei,2
θt,3
Et,3
Et,2
x
z
Fig. 6: Multiple scattering from a uniform bianisotropic re-flectionless metasurface.
achieve a relatively smooth control of the scattering responseof the metasurface for any non-specified incidence angles.
For simplicity, we specify that the metasurface does notchange the direction of wave propagation, which implies thatit is uniform, i.e. susceptibilities are not functions of position.Moreover, we specify that it is also reflectionless and onlyaffects the transmission phase of p-polarized incident wavesas function of their incidence angle.
To design this metasurface, we consider that it may becomposed of a total number of 36 susceptibility components.However, since all the waves interacting with the metasurfaceare p-polarized, most of these susceptibilities will not beexcited by these fields and, thus, will not play a role inthe electromagnetic transformations. Accordingly, the onlysusceptibilities that are excited by the fields are
χee =
χxxee 0 χxzee0 0 0χzxee 0 χzzee
, χem =
0 χxyem 00 0 00 χzyem 0
, (24a)
χme =
0 0 0χyxme 0 χyzme
0 0 0
, χmm =
0 0 00 χyymm 00 0 0
, (24b)
where the susceptibilities not excited have been set to zero forsimplicity. In order to satisfy the aforementioned specificationof reciprocity, the conditions (7) must be satisfied. This impliesthat χxzee = χzxee , χxyem = −χyxme and χzyem = −χyzme. As aconsequence, the total number of independent susceptibilitycomponents in (24) reduces from 9 to 6.
Upon insertion of (24), the GSTCs in (2a) and (2b) become
17It is essential to understand that these three sets of incident and transmit-ted waves cannot be combined, by superposition, into a single incident and asingle transmitted wave because these waves are not necessarily impinging onthe metasurface at the same time. This means that the Huygens theorem cannotbe used to find purely tangential equivalent surface currents corresponding tothese fields.
where the spatial derivatives only apply to the fields and notto the susceptibilities since the latter are not functions spacedue to the uniformity of the metasurface.
The system (25) contains 2 equations in 6 unknown sus-ceptibilities and is thus under-determined. In order to solveit, we apply the multiple transformation concept discussedin Sec. IV-C, which consists in specifying three independentsets of incident, reflected and transmitted waves. These fieldscan be simply defined by their respective reflection (R)18 andtransmission (T ) coefficients as well as their incidence angle(θi). In our case, the metasurface exhibits a transmission phaseshift, φ, that is function of the incidence angle, i.e. T = ejφ(θi).
Let us consider, for instance, that the 3 incident plane wavesimpinge on the metasurface at θi,1 = −45◦, θi,2 = 0◦ andθi,3 = +45◦, and are transmitted at θt = θi with transmissioncoefficients T1 = e−jα, T2 = 1 and T3 = ejα, whereα is a given phase shift. Solving relations (25) with thesespecifications yields the following nonzero susceptibilities:
χxzee = χzxee =2√
2
k0tan
(α2
). (26)
It can be easily verified that these susceptibilities satisfy thereciprocity, passivity and losslessness conditions (8).
Since the susceptibilities (26) correspond to the only solu-tion of the system (25) for our specifications and since thesesusceptibilities correspond to the excitation of normal polariza-tion densities, the normal polarizations are indeed useful andprovide additional degrees of freedom. This proves the claim inthe first paragraph of this section that normal polarizations leadto metasurface functionalities that are unattainable withoutthem.
Now that the metasurface has been synthesized, we analyzeits scattering response for all (including non-specified) inci-dence angles. For this purpose, we substitute the susceptibili-ties (26) into (25) and consider an incident wave, impinging onthe metasurface at an angle θi, being reflected and transmittedwith unknown scattering parameters. The system (25) can thenbe solved to obtain these unknown scattering parameters forany value of θi. In our case, the analysis is simple becausethe metasurface is uniform, which means that the reflectedand transmitted waves obey Snell laws. The resulting angulardependent transmission coefficient is
T (θi) = −1 +2
1− j√
2 sin(θi) tan(α2
) , (27)
while the reflection coefficient is R(θi) = 0.In order to illustrate the angular behavior of the transmission
coefficient in (27), it is plotted in Figs. 7 for a specified phaseshift of α = 90◦. As expected, the transmission amplituderemains unity for all incidence angles while the transmissionphase is asymmetric around broadside and covers about a220◦-phase range.
18Here R = 0 since the metasurface is reflectionless by specification.
-90 -45 0 45 90
0
0.2
0.4
0.6
0.8
1
|T | = 1 |T | = 1 |T | = 1
θi (◦)
Am
pli
tude
(a)
-90 -45 0 45 90
-135
-90
-45
0
45
90
135
6 T = −90◦
6 T = 0◦
6 T = 90◦
θi (◦)
Phase(◦)
(b)
Fig. 7: Transmission amplitude (a) and phase (b) as functionsof the incidence angle for a metasurface synthesize for thetransmission coefficients T = {e−j90◦ ; 1; ej90
◦} (and R = 0)at the respective incidence angles θi = {−45◦; 0◦; +45◦}.
E. Relations with Scattering Parameters
We have seen how a metasurface can be synthesized so as toobtain its susceptibilities in terms of specified fields. We shallnow investigate how the synthesized susceptibilities may berelated to the shape of the scattering particles that will consti-tute the metasurfaces to be realized. Here, we will only presentthe mathematical expressions that relate the susceptibilities tothe scattering particles. The reader is referred to [82]–[90],[97] for more information on the practical realization of thesestructures.
The conventional method to relate the scattering particleshape to equivalent susceptibilities (or material parameters) isbased on homogenization techniques. In the case of metama-terials, these techniques may be used to relate homogenizedmaterial parameters to the scattering parameters of the scatter-ers. From a general perspective, a single isolated scatterer isnot sufficient to describe an homogenized medium. Instead, weshall rather consider a periodic array of scatterers, which takesinto account the interactions and coupling between adjacentscatterers hence leading to a more accurate description of a“medium” compared to a single scatterer. The susceptibilities,which describe the macroscopic responses of a medium, arethus naturally well-suited to describe the homogenized mate-rial parameters of metasurfaces. It follows that the equivalentsusceptibilities of a scattering particle may be related to the
10
corresponding scattering parameters, conventionally obtainedvia full-wave simulations, of a periodic array made of aninfinite repetition of that scattering particle [83], [107]–[109].
Because the periodic array of scatterers is uniform withsubwavelength periodicity, the scattered fields obey Snell laws.More specifically, if the incident wave propagates normallywith respect to the array, then the reflected and transmittedwaves also propagate normally. In most cases, the periodic ar-ray of scattering particles is excited with normally propagatingwaves. This allows one to obtain the 16 tangential suscepti-bility components in (21). However, it does not provide anyinformation about the normal susceptibility components of thescattering particles. This is because, in the case of normallypropagating waves, the normal susceptibilities do not induceany discontinuity of the fields, as explained in Sec. IV-A.Nevertheless, this method allows one to match the tangentialsusceptibilities of the scattering particle to the susceptibilitiesfound from the metasurface synthesis procedure and thatprecisely yields the ideal tangential susceptibility components.
It is clear that the scattering particles may, in addition totheir tangential susceptibilities, possess nonzero normal sus-ceptibility components. In that case, the scattering response ofthe metasurface, when illuminated with obliquely propagatingwaves, will differ from the expected ideal behavior prescribedin the synthesis. Consequently, the homogenization techniqueonly serve as an initial guess to describe the scattering behav-ior of the metasurface19.
We will now derive the explicit expressions relating thetangential susceptibilities to the scattering parameters in thegeneral case of a fully bianisotropic uniform metasurface sur-rounded by different media and excited by normally incidentplane waves. Let us first write the system (21) in the followingcompact form:
∆ = χ ·Av, (28)
where the matrices ∆, χ and Av correspond to the field differ-ences, the normalized susceptibilities and the field averages,respectively.
In order to obtain the 16 tangential susceptibility com-ponents in (21), we will now define four transformationsby specifying the fields on both sides of the metasurface.Let us consider that the metasurface is illuminated from theleft with an x-polarized normally incident plane wave. Thecorresponding incident, reflected and transmitted electric fieldsread
Ei = x, Er = Sxx11 x+ Syx11 y, Et = Sxx21 x+ Syx21 y, (29)
where the terms Suvab , with a, b = {1, 2} and u, v = {x, y},are the scattering parameters with ports 1 and 2 correspondingto the left and right sides of the metasurface, respectively, asshown in Fig. 8. The medium of the left of the metasurfacehas the intrinsic impedance η1, while the medium on the righthas the intrinsic impedance η2. In addition to (29), three othercases have to be considered, i.e. y-polarized excitation incident
19Note that is possible to obtain all 36 susceptibility components of ascattering particle provided that the 4 GSTCs relations are solved for 9independent sets of incident, reflected and transmitted waves. In practice,such an operation is particularity tedious and is thus generally avoided.
x
y
Port 1
Port 2
z
PBC
Fig. 8: Full-wave simulation setup for the scattering parametertechnique leading to the metasurface physical structure fromthe metasurface model based on (28). The unit cell is sur-rounded by periodic boundary conditions (PBC) and excitedfrom port 1 and 2.
from the left (port 1), and x- and y-polarized excitationsincident from the right (port 2). Inserting these fields into (21),leads, after simplification, to the matrices ∆ and Av givenbelow, where the matrices Sab, I , N1 and N2 are defined by
Sab =
(Sxxab SxyabSyxab Syyab
), I =
(1 00 1
),
N1 =
(0 −11 0
), N2 =
(1 00 −1
).
(31)
Now, the procedure to obtain the susceptibilities of a givenscattering particle is as follows: firstly, the scattering particleis simulated with periodic boundary conditions and normalexcitation. Secondly, the resulting scattering parameters ob-tained from the simulations are used to define the matricesin (30a) and (30b). Finally, the susceptibilities correspondingto the particle are obtained by matrix inversion of (28).
Alternatively, it is possible to obtain the scattering parame-ters of a normally incident plane being scattered by a uniformmetasurface with known susceptibilities. This can be achievedby solving (28) for the scattering parameters. This leads to thefollowing matrix equation:
S = M−11 ·M2, (32)
where the scattering parameter matrix, S, is defined as
S =
(S11 S12
S21 S22
), (33)
and the matrices M1 and M2 are obtained from (28), (30a)and (30b) by expressing the scattering parameters in terms ofthe normalized susceptibility tensors. The resulting matricesM1 and M2 are given below.
Thus, the final metasurface physical structure is obtainedby mapping the scattering parameters (33) obtained fromthe discretized synthesized susceptibilities by (32) via (34a)and (34b) to those obtained by full-wave simulating meta-surface unit cells with tunable parameters, in an approximateperiodic environment, as illustrated in Fig. 8.
In the previous section, we have shown several metasurfaceexamples as illustrations of the proposed synthesis technique.These examples did not necessarily correspond to practicaldesigns but, in addition to illustrating the proposed synthesistechnique, they did set up the stage for the development ofuseful and practical concepts and applications, which is theobject of the present section.
We shall present here 5 of our most recent works represent-ing novel concepts and applications of metasurfaces. In theorder of appearance, we present our work on birefringent trans-formations [97], [110], bianisotropic refraction [111], lightemission enhancement [112], remote spatial processing [113]and nonlinear second-harmonic generation [95]. The reader isalso referred to our related works on nonreciprocal nongy-rotropic isolators [114], dielectric metasurfaces for dispersionengineering [115] and radiation pressure control [116].
A. Birefringent Operations
A direct application of the synthesis procedure discussed inSec. IV, and more specifically of the susceptibilities in (13), isthe design of birefringent metasurfaces. These susceptibilitiesare split into two independent sets that allow to individuallycontrol the scattering of s- and p-polarized waves. In particular,the manipulation of the respective transmission phases of theseorthogonal waves allows several interesting operations.
In [110], we have used this approach to realize half-waveplates, which rotate the polarization of linearly polarizedwaves by 90◦ or invert the handedness of circularly polarizedwaves, quarter-wave plates, which convert linear polarizationinto circular polarization, a polarization beam splitter, whichspatially separates orthogonally polarized waves, and an orbitalangular momentum generator, which generates topologicalcharges that depend on the incident wave polarization. Theseoperations are depicted in Fig. 9.
B. “Perfect” Refraction
Most refractive operations realized so far with a metasurfacehave been based on the concept of the generalized law ofrefraction [42], which requires the implementation of a phasegradient structure. However, such structures are plagued by
undesired diffraction orders and are thus not fully efficient.It turns out that the fundamental reason for this efficiencylimitation is the symmetric nature of simple (early as in [42])refractive metasurfaces with respect to the z-direction. Thiscan be demonstrated by the following ad absurdum argument.
Let us consider a passive metasurface surrounded by a givenreciprocal20 medium and denote the two sides of the structureby the indices 1 and 2. Assume that this metasurface perfectlyrefracts (without reflection and spurious diffraction) a wave in-cident under the angle θ1 in side 1 to the angle θ2 in side 2, andassume, ad absurdum, that this metasurface is symmetric withrespect to its normal. Since it is reciprocally perfectly refract-ing, it is perfectly matched for both propagation directions,1 to 2 and 2 to 1. Consider first wave propagation from side 1to side 2. Due to perfect matching, the wave experiences noreflection and, due to perfect refraction, it is fully transmittedto the angle θ2 in side 2. Consider now wave propagation in theopposite direction, along the reciprocal (or time-reversed) path.Now, the wave incident in side 2 has different tangential fieldcomponents than that incident in side 1, assuming θ2 6= θ1,and, therefore, it will see a different impedance, which meansthat the metasurface is necessarily mismatched in the direction
20The quasi-totality of the refracting metasurfaces discussed in the literatureso far have been reciprocal. The following argument does not hold for thenonreciprocal case, where perfect refraction could in principle be achieved bya symmetric metasurface structure.
12
2 to 1. But this is in contradiction with the assumption ofperfect (reciprocal) refraction! Consequently, the symmetricmetasurface does not produce perfect refraction. Part of thewave incident from side 2 is reflected back and therefore, byreciprocity, matching also did not actually exist in the direction1 to 2, so all of the energy of the wave incident under θ1in side 1 cannot completely refract into θ2; part of it has tobe transmitted to other directions in side 2, which typicallyrepresents spurious diffraction orders assuming a periodic-gradient metasurface. These diffraction orders are consistentlyvisible in reported simulations and experiments of symmetricmetasurfaces intended to perform refraction.
It was demonstrated in [111], [117] that bianisotropy wasthe solution to realize perfect (reciprocal) refraction (100%power transmission efficiency from θ1 to θ2). In what follows,we summarize the main synthesis steps for such a metasurface.
Let us consider the bianisotropic GSTCs relations in (10).For a refractive metasurface, rotation of polarization is not re-quired and usually undesired. Therefore, the relevant nonzerosusceptibility components reduce to the diagonal componentsof χee and χmm and the off-diagonal components of χem andχme. This corresponds to 4× 2 = 8 susceptibility parameters,leading, according to Sec. IV-C, to the double-transformationfull-rank system
where we naturally specify the second transformation as thereciprocal of the first one. Assuming that the refraction takesplaces in the xz-plane and that the waves are all p-polarized,the system (35) reduces to(
∆Hy1 ∆Hy2
∆Ex1 ∆Ex2
)=
(χxxee χxyem
χyxme χyymm
)·(Ex1,av Ex2,avHy1,av Hy2,av
),
(36)which strictly corresponds to a system that is N (4) = 2although the initial goal might have been to perform refractionin one propagation direction only. An illustration of the firstand second transformations is presented in Figs. 10a and 10b,respectively. Note that the subscripts i and t respectively referto the incident and transmit sides of the metasurface ratherthan the incident and transmitted waves. The electromagneticfields on the incident and transmit sides of the metasurface,assuming that the media on both sides are vacuum, and thatcorrespond to the first transformation read
Ex1,i =kz,ik0
e−jkx,ix, Ex1,t = Atkz,tk0
e−jkx,tx, (37a)
Hy1,i = e−jkx,ix/η0, Hy1,t = Ate−jkx,tx/η0, (37b)
metasurface
x
zy
θi
θt
Ψ1,i
Ψ1,t
(a)
metasurface
x
zy
θi
θt
Ψ1,i
Ψ1,t
(b)
Fig. 10: Representation of the two transformations specifiedin the system (36). (a) First transformation, corresponding tothe fields in (37), (b) Second transformation, corresponding tothe fields in (38).
where At is the amplitude of the wave on the transmit side.The fields corresponding to the second transformation read
Ex2,i = −kz,ik0
ejkx,ix, Ex2,t = −Atkz,tk0
ejkx,tx, (38a)
Hy2,i = ejkx,ix/η0, Hy2,t = Atejkx,tx/η0. (38b)
In order to ensure power conservation between the incident andtransmitted waves, the amplitude of the transmitted wave mustbe At =
√kz,i/kz,t =
√cos θi/ cos θt, as shown in [111].
Under this condition, the metasurface susceptibilities, obtainedby substituting (37) and (38) into (36) and considering thenormalization (11), read
χxxee =4 sin(αx)
β cos(αx) +√β2 − γ2
, (39a)
χyymm =β2 − γ2
4k20
4 sin(αx)
β cos(αx) +√β2 − γ2
, (39b)
χxyem = −χyxme =2j
k0
γ cos(αx)
β cos(αx) +√β2 − γ2
, (39c)
where α = kx,t−kx,i, β = kz,i +kz,t and γ = kz,i−kz,t. It canbe easily verified, using (8), that the bianisotropic refractivemetasurface with the susceptibilities (39) corresponds to areciprocal, passive and lossless structure, in addition to beingimmune to reflection and spurious diffraction, and is hence aperfectly refractive metasurface.
To demonstrate the performance of the synthesis method,we have built two bianisotropic refractive metasurfaces [111].They respectively transform an incident wave impinging atθi = 20◦ into a transmitted wave refracted at θi = −28◦, anda normally incident wave into a transmitted wave refracted atθi = −70◦. The full-wave simulations corresponding to thesetransformations are respectively plotted in Figs. 11a and. 11b.The simulated power transmission of these two structures isrespectively 86.7% and 83.2%. These efficiencies are mostlylimited to the inherent dielectric and metallic losses of thescattering particles and, to a lesser extent, to the undesireddiffraction orders due to the imperfection of these particles.A corresponding metasurface was demonstrated in [111] withan efficiency (79 %) that is around 4 % superior to the
13
theoretical limit of a lossless monoanisotropic metasurface,hence unquestionably demonstrating the superiority of thebianisotropic design!
Metasurface remote spatial processing, introduced in [113],consists in controlling the transmission of a signal beamthrough a metasurface by remotely sending a control beam,which properly interferes with the signal beam. This interfer-ence is thus used to shape the metasurface transmission patternby varying the phase and/or amplitude of the control beam.
Figure 12 presents an example of such remote spatialprocessing. Initially, the signal beam (in blue) in Fig. 12ais refracted by the metasurface according to some initialspecification. When the control beam (in red) is next added tothe signal beam on the metasurface, as in Fig. 12b, it changesthe overall radiation pattern of the metasurface.
We have used this concept to implement remote spatialswitch/modulators. The operation principle of such a mod-ulator is presented in Fig. 13. To avoid the collocation of thecontrol and signal beam sources, the control beam impinges onthe metasurface at an angle while the signal beam is normallyincident. In order to independently control the transmissionof both beams, they must be orthogonally polarized on theincident side of the metasurface. However, they must exhibitthe same polarization on the transmit side so as to interfere.In [113], we show that such a transformation can only beachieved using a bianisotropic metasurface, which must alsobe chiral so as to rotate the polarization of the controlbeam. On the transmit side, the two beams interfere and thecorresponding amplitude thus depends of the phase differencebetween them.
The fabricated metasurface performing the operation de-picted in Fig. 13 has been experimentally measured, and the
(a) (b)
Fig. 12: Example of a remote spatial processing operation.(a) Signal beam being refracted by the metasurface. (b) Su-perposition of signal and control beams interacting with eachother, which leads to a different transmitted wave.
z
Signal
Control
s s
sp
x
y
Fig. 13: Coherent modulator metasurface. The signal andcontrol beams are impinging on the metasurface at differentangles to avoid collocation of their source. The amplitude ofthe transmitted wave depends on the phase difference betweenthe two beams by interference.
corresponding results are plotted in Fig. 14 for an operatingfrequency of 16 GHz.
D. Light Emission Enhancement
In the perspective of enhancing the efficiency of light-emitting diodes (LEDs), we have reported in [112] a partiallyreflecting metasurface cavity (PRMC) increasing the emissionof photon sources in layered semiconductor structures, usingthe susceptibility-GSTC technique presented in this paper. ThisPRMC simultaneously enhances the light extraction efficiency(LEE), spontaneous emission rate (SER) and far-field direc-tivity of the photon source.
The LEE is enhanced by enforcing the emitted light tooptimally refract/radiate perpendicularly to the device. Suchrefraction suppresses the wave trapping loss, represented inFig. 15a. The requirement of total normal refraction, rep-resented in Fig. 15b, is excessively stringent, leading tosusceptibilities with prohibitive spatial variations, and is notrequired in this application. A better strategy consists, asillustrated in Fig. 15c, in allowing partial local reflection, andultimately collecting the reflected part of the energy by Fabry-
14
14 15 16 17 18−35
−30
−25
−20
−15
−10
−5
0
5
10
ON
OFF
Gain
Tra
nsm
issi
on
(dB
)
Frequency (GHz)
Fig. 14: Measured transmission coefficients for the metasur-face in Fig. 13. The blue curve is the transmission of the signalbeam only, while the black and green curves are the destructiveand constructive interferences of the signal and control beams,respectively.
Perot resonance in the PRMC formed with a mirror plane atthe bottom of the slab. The double-metasurface cavity, depictedin Fig. (15d) is an even more sophisticated design, leading todramatic LEE enhancement.
(a) (b)
(c) (d)
Fig. 15: Radiation of a light source (quantum well) embeddedin a semiconductor (e.g. GaN) substrate. (a) Bare structure.(b) Reflectionless metasurface, placed on top of the slab,that collimates the dipole fields. (c) Introduction of perfectlyrefractive metasurface cavity (PRMC). (d) Double metasur-face cavity, with partially reflective top metasuface and fullyreflective bottom metasuraface.
The SER is enhanced by maximizing the confinement ofcoherent electromagnetic energy in the vicinity of the sourceand leveraging the Purcell effect, which is particularly wellachieved in the double-metasurface PRMC (Fig. 15d). Finally,the far-field directivity is maximized as an optimization trade-off for maximal overall power conversion ratio.
Figure 16 shows full-wave simulated flux densities for thedesigns of Figs. 15a and 15d, where the latter features LEEand SER enhancements by factors of 4.0 and 1.9, respectively,with half-power beam width of 22.5◦. The case of a realLED is more complex due to the incoherence and distributionemission of the quantum well emitters. Different metasurface
3
2
1
-1
-2
-3
0
2 4-2-4 0
3
2
1
-1
-2
-3
0
2 4-2-4 0
(a) (b)
0 1 (W/m2)
z (μm)z (μm)
x(μ
m)
x(μ
m)
Fig. 16: Full-wave (COMSOL) simulated energy flux densitiesfor a dipole emitter embedded in a GaN slab. (a) Configurationof Fig. 15a. (b) Configuration of Fig. 15d. Original imagesfrom [112].
strategies are currently being investigated to maximize thepower conversion efficiency of a complete LED.
E. Second-Order Nonlinearity
So far, we have only discussed linear metasurfaces, i.e.metasurfaces whose polarization densities are linear functionsof the electric and magnetic fields. Given the wealth ofpotential applications of nonlinear metasurfaces, it is highlydesirable to develop tools for the design of such metasurfaces.Therefore, we extended our susceptibility-GSTC technique tothe case of a second-order nonlinear metasurface in [95].
In this case, the polarization densities can be written as
P = ε0χ(1)ee ·Eav + ε0χ
(2)ee : EavEav, (40a)
M = χ(1)mm ·Hav + χ
(2)mm : HavHav, (40b)
where χ(1) and χ(2) are to the linear and nonlinear (second-order) susceptibilities of the metasurface. For the sake of sim-plicity, we assume that these susceptibility tensors are scalar.Being nonlinear, the metasurface will generate harmonics ofthe excitation frequency ω0. Consequently, we have to expressthe GSTCs in (2) in the time-domain to properly take intoaccount the generation of these new frequencies. The relevantGSTCs are then, in the case of x-polarized waves, given by21
−∆H = ε0χ(1)ee
∂
∂tEav + ε0χ
(2)ee
∂
∂tE2
av, (41a)
−∆E = µ0χ(1)mm
∂
∂tHav + µ0χ
(2)mm
∂
∂tH2
av, (41b)
where E and H are, respectively, the x-component of theelectric field and the y-component of the magnetic field.From these relations, we can either perform a synthesis, i.e.expressing the susceptibilities as functions of the fields, or ananalysis, i.e. computing the fields scattered from a metasurfacewith known susceptibilities. Here, for the sake of briefness, wewill not elaborate on the synthesis and analysis operations butshall rather present one of the main results obtained in [95],
21In these expressions, the susceptibilities are dispersion-less. Meaning thatχ(ω0) = χ(2ω0) = χ(3ω0) = ..., as discussed in [95], which is essentiallyequivalent to the conventional condition of phase-matching in nonlinear optics.
15
which are the reflectionless conditions for the metasurface.The metasurface with susceptibilities (41) exhibit differentreflectionless conditions for the two propagation directionssince, due to the presence of the square of both the electric andmagnetic fields, the relations (41) are asymmetric with respectto the z-direction. It follows that the reflectionless conditionsfor waves propagating in the forward (+z) direction are
χ(1)ee = χ(1)
mm, (42a)
η0χ(2)ee = χ(2)
mm, (42b)
while for backward (-z) propagation they are
χ(1)ee = χ(1)
mm, (43a)
−η0χ(2)ee = χ(2)
mm. (43b)
An important consequences of the fact that the metasurfacecannot be matched from both sides is that its second-harmonicgeneration (SHG) is inherently nonreciprocal.
VI. CONCLUSIONS
We have presented an overview of electromagnetic meta-surface designs, concepts and applications based on a bian-isotropic surface susceptibility tensors model. This overviewprobably represents only a small fraction of this approach,which nevertheless already represents a solid foundation forfuture metasurface technology.
ACKNOWLEDGMENT
This work was accomplished in the framework of the Col-laborative Research and Development Project CRDPJ 478303-14 of the Natural Sciences and Engineering Research Councilof Canada (NSERC) in partnership with the company Meta-material Technology Inc.
REFERENCES
[1] C. Holloway, A. Dienstfrey, E. F. Kuester, J. F. O’Hara, A. K. Azad,and A. J. Taylor, “A discussion on the interpretation and character-ization of metafilms/metasurfaces: the two-dimensional equivalent ofmetamaterials,” Metamaterials, vol. 3, no. 2, pp. 100–112, Oct. 2009.
[2] C. Holloway, E. F. Kuester, J. Gordon, J. O’Hara, J. Booth, andD. Smith, “An overview of the theory and applications of metasurfaces:the two-dimensional equivalents of metamaterials,” IEEE AntennasPropag. Mag., vol. 54, no. 2, pp. 10–35, April 2012.
[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 & Photonics Reviews, vol. 9, no. 2, pp. 195–213,Mar. 2015.
[4] S. B. Glybovski, S. A. Tretyakov, P. A. Belov, Y. S. Kivshar, and C. R.Simovski, “Metasurfaces: from microwaves to visible,” Phys. Rep., vol.634, pp. 1 – 72, 2016.
[5] H. Lamb, “On the reflection and transmission of electric waves by ametallic grating,” Proceedings of the London Mathematical Society,vol. 1, no. 1, pp. 523–546, 1897.
[6] G. Marconi and C. Franklin, “Reflector for use in wireless telegraphyand telephony.” Apr. 22 1919, US Patent 1,301,473.
[7] L. V. Buskirk and C. Hendrix, “The zone plate as a radio-frequencyfocusing element,” IRE Trans. Antennas Propag., vol. 9, no. 3, pp.319–320, May 1961.
[8] B. A. Munk, Frequency Selective Surfaces: theory and design. JohnWiley & Sons, 2000.
[9] A. K. Rashid, B. Li, and Z. Shen, “An overview of three-dimensionalfrequency-selective structures,” IEEE Antennas Propag. Mag., vol. 56,no. 3, pp. 43–67, June 2014.
[10] J. Huang and J. A. Encinar, Reflectarray antenna. Wiley OnlineLibrary, 2007.
[11] D. Berry, R. Malech, and W. Kennedy, “The reflectarray antenna,” IEEETrans. Antennas Propag., vol. 11, no. 6, pp. 645–651, Nov 1963.
[12] C. Malagisi, “Microstrip disc element reflect array,” in EASCON’78:Electronics and Aerospace Systems Convention, vol. 1, 1978, pp. 186–192.
[13] J. Montgomery, “A microstrip reflectarray antenna element,” in Annalesd’Astrophysique, 1978.
[14] J. Stangel, “The thinned lens approach to phased array design,” in 1965Antennas and Propagation Society International Symposium, vol. 3,Aug 1965, pp. 10–13.
[15] P. J. Kahrilas, “Hapdar 8212: An operational phased array radar,”Proceedings of the IEEE, vol. 56, no. 11, pp. 1967–1975, Nov 1968.
[16] D. McGrath, “Planar three-dimensional constrained lenses,” IEEETrans. Antennas Propag., vol. 34, no. 1, pp. 46–50, January 1986.
[17] D. M. Pozar, “Flat lens antenna concept using aperture coupledmicrostrip patches,” Electron. Lett., vol. 32, no. 23, pp. 2109–2111,Nov 1996.
[18] K.-W. Lam, S.-W. Kwok, Y. Hwang, and T. K. Lo, “Implementationof transmitarray antenna concept by using aperture-coupled microstrippatches,” in Proceedings of 1997 Asia-Pacific Microwave Conference,vol. 1, Dec 1997, pp. 433–436 vol.1.
[19] S. Datthanasombat, A. Prata, L. R. Arnaro, J. A. Harrell, S. Spitz,and J. Perret, “Layered lens antennas,” in IEEE Antennas and Propa-gation Society International Symposium. 2001 Digest. Held in con-junction with: USNC/URSI National Radio Science Meeting (Cat.No.01CH37229), vol. 2, Jul. 2001, pp. 777–780 vol.2.
[20] N. Gagnon, A. Petosa, and D. McNamara, “Thin microwave quasi-transparent phase-shifting surface (PSS),” IEEE Trans. AntennasPropag., vol. 58, no. 4, pp. 1193–1201, April 2010.
[21] M. F. Imani and A. Grbic, “An analytical investigation of near-fieldplates,” Metamaterials, vol. 4, no. 2, pp. 104–111, 2010.
[22] A. Grbic, L. Jiang, and R. Merlin, “Near-field plates: subdiffractionfocusing with patterned surfaces,” Science, vol. 320, no. 5875, pp. 511–513, 2008.
[23] M. Imani and A. Grbic, “Planar near-field plates,” IEEE Trans. Anten-nas Propag., vol. 61, no. 11, pp. 5425–5434, Nov 2013.
[24] B. Fong, J. Colburn, J. Ottusch, J. Visher, and D. Sievenpiper, “Scalarand tensor holographic artificial impedance surfaces,” IEEE Trans.Antennas Propag., vol. 58, no. 10, pp. 3212–3221, Oct 2010.
[25] S. Maci, G. Minatti, M. Casaletti, and M. Bosiljevac, “Metasurfing: ad-dressing waves on impenetrable metasurfaces,” IEEE Antennas WirelessPropag. Lett., vol. 10, pp. 1499–1502, 2011.
[26] K. Song, X. Zhao, Y. Liu, Q. Fu, and C. Luo, “A frequency-tunable90◦-polarization rotation device using composite chiral metamaterials,”Appl. Phys. Lett., vol. 103, no. 10, p. 101908, 2013.
[27] L. Martinez-Lopez, J. Rodriguez-Cuevas, J. Martinez-Lopez, andA. Martynyuk, “A multilayer circular polarizer based on bisected split-ring frequency selective surfaces,” IEEE Antennas Wireless Propag.Lett., vol. 13, pp. 153–156, 2014.
[28] J. Gao, K. Zhang, G. Yang, and Q. Wu, “A novel four-face polarizationtwister based on three-dimensional magnetic toroidal dipoles,” IEEETrans. Magn., vol. 50, no. 1, pp. 1–4, Jan 2014.
[29] L. Wu, Z. Yang, Y. Cheng, R. Gong, M. Zhao, Y. Zheng, J. Duan,and X. Yuan, “Circular polarization converters based on bi-layeredasymmetrical split ring metamaterials,” Appl. Phys. A, vol. 116, no. 2,pp. 643–648, 2014.
[30] S.-C. Jiang, X. Xiong, Y.-S. Hu, Y.-H. Hu, G.-B. Ma, R.-W. Peng,C. Sun, and M. Wang, “Controlling the polarization state of light witha dispersion-free metastructure,” Phys. Rev. X, vol. 4, p. 021026, May2014.
[31] C. Huang, X. Ma, M. Pu, G. Yi, Y. Wang, and X. Luo, “Dual-band90◦ polarization rotator using twisted split ring resonators array,” Opt.Commun., vol. 291, no. 0, pp. 345 – 348, 2013.
[32] J.-X. Zhao, B.-X. Xiao, X.-J. Huang, and H.-L. Yang, “Multiple-band reflective polarization converter based on complementary l-shapedmetamaterial,” Microw. Opt. Technol. Lett., vol. 57, no. 4, pp. 978–983,2015.
[33] Q. Levesque, M. Makhsiyan, P. Bouchon, F. Pardo, J. Jaeck, N. Bardou,C. Dupuis, R. Hadar, and J.-L. Pelouard, “Plasmonic planar antennafor wideband and efficient linear polarization conversion,” Appl. Phys.Lett., vol. 104, no. 11, pp. –, 2014.
[34] G. Kenanakis, R. Zhao, A. Stavrinidis, G. Konstantinidis, N. Kat-sarakis, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Flexiblechiral metamaterials in the terahertz regime: a comparative study of
16
various designs,” Opt. Mater. Express, vol. 2, no. 12, pp. 1702–1712,Dec 2012.
[35] M. Li, L. Guo, J. Dong, and H. Yang, “An ultra-thin chiral metamaterialabsorber with high selectivity for LCP and RCP waves,” J. Phys. D,vol. 47, no. 18, p. 185102, 2014.
[36] B. Bian, S. Liu, S. Wang, X. Kong, H. Zhang, B. Ma, and H. Yang,“Novel triple-band polarization-insensitive wide-angle ultra-thin mi-crowave metamaterial absorber,” J. Appl. Phys., vol. 114, no. 19, pp. –,2013.
[37] C. A. Valagiannopoulos, A. Tukiainen, T. Aho, T. Niemi, M. Guina,S. A. Tretyakov, and C. R. Simovski, “Perfect magnetic mirror andsimple perfect absorber in the visible spectrum,” Phys. Rev. B, vol. 91,p. 115305, Mar. 2015.
[38] Y. Wen, W. Ma, J. Bailey, G. Matmon, X. Yu, and G. Aeppli, “Pla-nar broadband and high absorption metamaterial using single nestedresonator at terahertz frequencies,” Opt. Lett., vol. 39, no. 6, pp. 1589–1592, Mar. 2014.
[39] F. Dincer, O. Akgol, M. Karaaslan, E. Unal, and C. Sabah, “Polariza-tion angle independent perfect metamaterial absorbers for solar cellapplications in the microwave, infrared, and visible regime,” Prog.Electromagn. Res., vol. 144, pp. 93–101, 2014.
[40] M. Yoo and S. Lim, “Polarization-independent and ultrawidebandmetamaterial absorber using a hexagonal artificial impedance surfaceand a resistor-capacitor layer,” IEEE Trans. Antennas Propag., vol. 62,no. 5, pp. 2652–2658, May 2014.
[41] Y. Ra’di, V. Asadchy, and S. Tretyakov, “Total absorption of elec-tromagnetic waves in ultimately thin layers,” IEEE Trans. AntennasPropag., vol. 61, no. 9, pp. 4606–4614, Sept 2013.
[42] N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, andZ. Gaburro, “Light propagation with phase discontinuities: generalizedlaws of reflection and refraction,” Science, vol. 334, no. 6054, pp. 333–337, 2011.
[43] N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” NatureMater., vol. 13, no. 2, pp. 139–150, 2014.
[44] P. Genevet and F. Capasso, “Breakthroughs in photonics 2013: flat op-tics: wavefronts control with Huygens’ interfaces,” Photonics Journal,IEEE, vol. 6, no. 2, pp. 1–4, April 2014.
[45] J. Shi, X. Fang, E. T. F. Rogers, E. Plum, K. F. MacDonald, andN. I. Zheludev, “Coherent control of snell’s law at metasurfaces,” Opt.Express, vol. 22, no. 17, pp. 21 051–21 060, Aug 2014.
[46] Q. Yang, J. Gu, D. Wang, X. Zhang, Z. Tian, C. Ouyang, R. Singh,J. Han, and W. Zhang, “Efficient flat metasurface lens for terahertzimaging,” Opt. Express, vol. 22, no. 21, pp. 25 931–25 939, Oct 2014.
[47] S. Asadchy, V. Y. Ra’di, J. Vehmas, and A. Tretyakov, S. “Functionalmetamirrors using bianisotropic elements,” Phys. Rev. Lett., vol. 114,p. 095503, Mar. 2015.
[48] G. Lipworth, A. Mrozack, J. Hunt, D. L. Marks, T. Driscoll, D. Brady,and D. R. Smith, “Metamaterial apertures for coherent computationalimaging on the physical layer,” J. Opt. Soc. Am. A, vol. 30, no. 8, pp.1603–1612, Aug 2013.
[49] J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady,and D. R. Smith, “Metamaterial apertures for computational imaging,”Science, vol. 339, no. 6117, pp. 310–313, 2013.
[50] F. Liu, S. Xiao, A. Sihvola, and J. Li, “Perfect co-circular polarizationreflector: a class of reciprocal perfect conductors with total co-circularpolarization reflection,” IEEE Trans. Antennas Propag., vol. 62, no. 12,pp. 6274–6281, Dec 2014.
[51] D. Germain, D. Seetharamdoo, S. Nawaz Burokur, and A. de Lustrac,“Phase-compensated metasurface for a conformal microwave antenna,”Appl. Phys. Lett., vol. 103, no. 12, p. 124102, 2013.
[52] C. Holloway, “Sub-wavelength resonators: on the use of metafilmsto overcome the λ/2 size limit,” IET Microwaves, Antennas andPropagation, vol. 2, pp. 120–129(9), Mar. 2008.
[53] Y. Ra’di, V. Asadchy, and S. Tretyakov, “Tailoring reflections from thincomposite metamirrors,” IEEE Trans. Antennas Propag., vol. 62, no. 7,pp. 3749–3760, Jul. 2014.
[54] A. Abdelrahman, A. Elsherbeni, and F. Yang, “Transmitarray antennadesign using cross-slot elements with no dielectric substrate,” IEEEAntennas Wireless Propag. Lett., vol. 13, pp. 177–180, 2014.
[55] H. Shi, J. Li, A. Zhang, Y. Jiang, J. Wang, Z. Xu, and S. Xia, “Gradientmetasurface with both polarization-controlled directional surface wavecoupling and anomalous reflection,” IEEE Antennas Wireless Propag.Lett., vol. 14, pp. 104–107, 2015.
[56] J. Nye and M. Berry, “Dislocations in wave trains,” in Proc. R. Soc.A A: Mathematical, Physical and Engineering Sciences, vol. 336, no.1605, 1974, pp. 165–190.
[57] E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W.Boyd, “Generating optical orbital angular momentum at visible wave-lengths using a plasmonic metasurface,” Light: Science & Applications,vol. 3, no. 5, p. e167, 2014.
[58] C. Pfeiffer and A. Grbic, “Controlling vector bessel beams withmetasurfaces,” Phys. Rev. Applied, vol. 2, p. 044012, Oct 2014.
[59] X. Yi, X. Ling, Z. Zhang, Y. Li, X. Zhou, Y. Liu, S. Chen, H. Luo,and S. Wen, “Generation of cylindrical vector vortex beams by twocascaded metasurfaces,” Opt. Express, vol. 22, no. 14, pp. 17 207–17 215, Jul 2014.
[60] M. Veysi, C. Guclu, and F. Capolino, “Vortex beams with stronglongitudinally polarized magnetic field and their generation by usingmetasurfaces,” J. Opt. Soc. Am. B, vol. 32, no. 2, pp. 345–354, Feb2015.
[61] J. Sun, X. Wang, T. Xu, Z. A. Kudyshev, A. N. Cartwright, and N. M.Litchinitser, “Spinning light on the nanoscale,” Nano Lett., vol. 14,no. 5, pp. 2726–2729, 2014.
[62] J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanos-tructured holograms for broadband manipulation of vector beams,”Nano Lett., vol. 13, no. 9, pp. 4269–4274, 2013.
[63] J. Hunt, J. Gollub, T. Driscoll, G. Lipworth, A. Mrozack, M. S.Reynolds, D. J. Brady, and D. R. Smith, “Metamaterial microwaveholographic imaging system,” J. Opt. Soc. Am. A, vol. 31, no. 10, pp.2109–2119, Oct 2014.
[64] G. Zheng, H. Muhlenbernd, M. Kenney, G. Li, T. Zentgraf, andS. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat.Nanotechnol., 2015.
[65] C. Pfeiffer and A. Grbic, “Generating stable tractor beams withdielectric metasurfaces,” Phys. Rev. B, vol. 91, p. 115408, Mar. 2015.
[66] H. Da, Q. Bao, R. Sanaei, J. Teng, K. P. Loh, F. J. Garcia-Vidal, and C.-W. Qiu, “Monolayer graphene photonic metastructures: giant Faradayrotation and nearly perfect transmission,” Phys. Rev. B, vol. 88, p.205405, Nov 2013.
[67] A. Degiron and D. R. Smith, “One-way glass for microwaves usingnonreciprocal metamaterials,” Phys. Rev. E, vol. 89, p. 053203, May2014.
[68] Y. Ra’di, V. S. Asadchy, and S. A. Tretyakov, “One-way transparentsheets,” Phys. Rev. B, vol. 89, p. 075109, Feb 2014.
[69] T. Kodera, D. L. Sounas, and C. Caloz, “Artificial Faraday rotationusing a ring metamaterial structure without static magnetic field,” Appl.Phys. Lett., vol. 99, no. 3, p. 031114, 2011.
[70] D. Sounas, T. Kodera, and C. Caloz, “Electromagnetic modelingof a magnetless nonreciprocal gyrotropic metasurface,” IEEE Trans.Antennas Propag., vol. 61, no. 1, pp. 221–231, Jan 2013.
[71] Y. Zhu, X. Hu, Z. Chai, H. Yang, and Q. Gong, “Active control ofchirality in nonlinear metamaterials,” Appl. Phys. Lett., vol. 106, no. 9,p. 091109, 2015.
[72] V. K. Valev, J. Baumberg, B. De Clercq, N. Braz, X. Zheng, E. Osley,S. Vandendriessche, M. Hojeij, C. Blejean, J. Mertens et al., “Nonlin-ear superchiral meta-surfaces: tuning chirality and disentangling non-reciprocity at the nanoscale,” Adv. Mater., vol. 26, no. 24, pp. 4074–4081, 2014.
[73] J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Dem-merle, G. Boehm, M.-C. Amann, A. Alu, and M. A. Belkin, “Giantnonlinear response from plasmonic metasurfaces coupled to intersub-band transitions,” Nature, vol. 511, no. 7507, pp. 65–69, 2014.
[74] A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computingusing reflective plasmonic metasurfaces,” Nano Lett., vol. 15, no. 1,pp. 791–797, 2015.
[75] A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Al, and N. Engheta,“Performing mathematical operations with metamaterials,” Science,vol. 343, no. 6167, pp. 160–163, 2014.
[76] J. Ortiz, J. Baena, V. Losada, F. Medina, and J. Araque, “Spatial angularfiltering by fsss made of chains of interconnected SRRs and CSRRs,”IEEE Microw. Compon. Lett., vol. 23, no. 9, pp. 477–479, 2013.
[77] Y. Shen, D. Ye, L. Wang, I. Celanovic, L. Ran, J. D. Joannopoulos,and M. Soljacic, “Metamaterial broadband angular selectivity,” Phys.Rev. B, vol. 90, p. 125422, Sep 2014.
[78] Y. Shen, D. Ye, I. Celanovic, S. G. Johnson, J. D. Joannopoulos, andM. Soljai, “Optical broadband angular selectivity,” Science, vol. 343,no. 6178, pp. 1499–1501, 2014.
[79] K. Achouri, M. A. Salem, and C. Caloz, “General metasurface syn-thesis based on susceptibility tensors,” IEEE Trans. Antennas Propag.,vol. 63, no. 7, pp. 2977–2991, Jul. 2015.
[80] M. A. Salem, K. Achouri, and C. Caloz, “Metasurface synthesisfor time-harmonic waves: Exact spectral and spatial methods,” Prog.Electromagn. Res., vol. 149, pp. 205–216, 2014.
17
[81] M. A. Salem and C. Caloz, “Manipulating light at distance by ametasurface using momentum transformation,” Opt. Express, vol. 22,no. 12, pp. 14 530–14 543, Jun 2014.
[82] C. Pfeiffer, N. K. Emani, A. M. Shaltout, A. Boltasseva, V. M. Shalaev,and A. Grbic, “Efficient light bending with isotropic metamaterialHuygens surfaces,” Nano Lett., vol. 14, no. 5, pp. 2491–2497, 2014.
[83] C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoringwave fronts with reflectionless sheets,” Phys. Rev. Lett., vol. 110, p.197401, May 2013.
[84] C. Pfeiffer and A. Grbic, “Millimeter-wave transmitarrays for wave-front and polarization control,” IEEE Trans. Microwave Theory Tech.,vol. 61, no. 12, pp. 4407–4417, Dec 2013.
[85] C. Pfeiffer and A. Grbic, “Bianisotropic metasurfaces for optimalpolarization control: Analysis and synthesis,” Phys. Rev. Applied, vol. 2,p. 044011, Oct 2014.
[86] J. P. Wong, M. Selvanayagam, and G. V. Eleftheriades, “Design ofunit cells and demonstration of methods for synthesizing Huygensmetasurfaces,” Photonics and Nanostructures - Fundamentals andApplications, vol. 12, no. 4, pp. 360 – 375, 2014, metamaterials-2013Congress.
[87] M. Selvanayagam and G. Eleftheriades, “Polarization control usingtensor Huygens surfaces,” IEEE Trans. Antennas Propag., vol. 62,no. 12, pp. 6155–6168, Dec 2014.
[88] A. Epstein and G. Eleftheriades, “Passive lossless Huygens metasur-faces for conversion of arbitrary source field to directive radiation,”IEEE Trans. Antennas Propag., vol. 62, no. 11, pp. 5680–5695, Nov2014.
[89] T. Niemi, A. Karilainen, and S. Tretyakov, “Synthesis of polarizationtransformers,” IEEE Trans. Antennas Propag., vol. 61, no. 6, pp. 3102–3111, June 2013.
[90] Y. Ra’di, V. Asadchy, and S. Tretyakov, “Total absorption of elec-tromagnetic waves in ultimately thin layers,” IEEE Trans. AntennasPropag., vol. 61, no. 9, pp. 4606–4614, Sept 2013.
[91] M. M. Idemen, Discontinuities in the Electromagnetic Field. JohnWiley & Sons, 2011.
[92] M. Albooyeh, S. Tretyakov, and C. Simovski, “Electromagnetic char-acterization of bianisotropic metasurfaces on refractive substrates:General theoretical framework,” Ann. Phys., vol. 528, no. 9-10, pp.721–737, 2016.
[93] J. Kong, Electromagnetic wave theory, ser. A Wiley-Interscience pub-lication. John Wiley & Sons, 1986.
[94] A. H. Sihvola, A. J. Viitanen, I. V. Lindell, and S. A. Tretyakov,Electromagnetic waves in chiral and bi-isotropic media, ser. The ArtechHouse Antenna Library. Artech House, 1994.
[95] K. Achouri, Y. Vahabzadeh, and C. Caloz, “Mathematical synthesis andanalysis of a second-order magneto-electrically nonlinear metasurface,”Optics Express, vol. 25, no. 16, pp. 19 013–19 022, 2017.
[96] E. F. Kuester, M. Mohamed, M. Piket-May, and C. Holloway, “Av-eraged transition conditions for electromagnetic fields at a metafilm,”IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2641–2651, Oct2003.
[97] K. Achouri, B. A. Khan, S. Gupta, G. Lavigne, M. A. Salem, andC. Caloz, “Synthesis of electromagnetic metasurfaces: principles andillustrations,” EPJ Applied Metamaterials, vol. 2, p. 12, 2015.
[98] K. Achouri, M. A. Salem, and C. Caloz, “Electromagnetic metasurfaceperforming up to four independent wave transformations,” in 2015IEEE Conference on Antenna Measurements Applications (CAMA),Nov 2015, pp. 1–3.
[99] Y. Vahabzadeh, N. Chamanara, K. Achouri, and C. Caloz, “Compu-tational analysis of metasurfaces,” arXiv preprint arXiv:1710.11264,2017.
[100] B. Saleh and M. Teich, Fundamentals of Photonics, ser. Wiley Seriesin Pure and Applied Optics. Wiley, 2007.
[101] T. Kodera, D. L. Sounas, and C. Caloz, “Artificial Faraday rotationusing a ring metamaterial structure without static magnetic field,” Appl.Phys. Lett., vol. 99, no. 3, pp. 031 114:1–3, Jul. 2011.
[102] D. L. Sounas and C. Caloz, “Electromagnetic non-reciprocity andgyrotropy of graphene,” Appl. Phys. Lett., vol. 98, no. 2, pp. 021 911:1–3, Jan. 2011.
[103] M. Albooyeh, H. Kazemi, F. Capolino, D. H. Kwon, and S. A.Tretyakov, “Normal vs tangential polarizations in metasurfaces,” in2017 IEEE International Symposium on Antennas and PropagationUSNC/URSI National Radio Science Meeting, July 2017, pp. 1707–1708.
[104] J. A. Gordon, C. L. Holloway, and A. Dienstfrey, “A physical expla-nation of angle-independent reflection and transmission properties of
[105] A. Di Falco, Y. Zhao, and A. Alu, “Optical metasurfaces with robustangular response on flexible substrates,” Appl. Phys. Lett., vol. 99,no. 16, p. 163110, 2011.
[106] Y. Ra’di and S. Tretyakov, “Angularly-independent Huygens’ meta-surfaces,” in 2015 IEEE International Symposium on Antennas andPropagation & USNC/URSI National Radio Science Meeting. IEEE,2015, pp. 874–875.
[107] C. Pfeiffer, N. K. Emani, A. M. Shaltout, A. Boltasseva, V. M. Shalaev,and A. Grbic, “Efficient light bending with isotropic metamaterialHuygens surfaces,” Nano Lett., vol. 14, no. 5, pp. 2491–2497, 2014.
[108] V. S. Asadchy and I. A. Fanyaev, “Simulation of the electromagneticproperties of helices with optimal shape, which provides radiation of acircularly polarized wave,” Journal of Advanced Research in Physics,vol. 2, no. 1, 2011.
[109] V. S. Asadchy, I. A. Faniayeu, Y. Radi, and S. A. Tretyakov, “De-termining polarizability tensors for an arbitrary small electromagneticscatterer,” Photonics and Nanostructures-Fundamentals and Applica-tions, vol. 12, no. 4, pp. 298–304, 2014.
[110] K. Achouri, G. Lavigne, and C. Caloz, “Comparison of two synthesismethods for birefringent metasurfaces,” J. Appl. Phys., vol. 120, no. 23,p. 235305, 2016.
[111] G. Lavigne, K. Achouri, V. Asadchy, S. Tretyakov, and C. Caloz,“Refracting metasurfaces without spurious diffraction,” arXiv preprintarXiv:1705.09286v2, 2017.
[112] L. Chen, K. Achouri, E. Kallos, and C. Caloz, “Simultaneous enhance-ment of light extraction and spontaneous emission using a partiallyreflecting metasurface cavity,” Phys. Rev. A, vol. 95, p. 053808, May2017.
[113] K. Achouri, G. Lavigne, M. A. Salem, and C. Caloz, “Metasurfacespatial processor for electromagnetic remote control,” IEEE Trans.Antennas Propag., vol. 64, no. 5, pp. 1759–1767, 2016.
[114] S. Taravati, B. A. Khan, S. Gupta, K. Achouri, and C. Caloz, “Nonre-ciprocal nongyrotropic magnetless metasurface,” IEEE Trans. AntennasPropag., vol. PP, no. 99, pp. 1–1, 2017.
[115] K. Achouri, A. Yahyaoui, S. Gupta, H. Rmili, and C. Caloz, “Dielectricresonator metasurface for dispersion engineering,” IEEE Trans. Anten-nas Propag., vol. 65, no. 2, pp. 673–680, Feb 2017.
[116] K. Achouri and C. Caloz, “Metasurface solar sail for flexible radiationpressure control,” arXiv preprint arXiv:1710.02837, 2017.
[117] A. Epstein and G. V. Eleftheriades, “Arbitrary power-conserving fieldtransformations with passive lossless omega-type bianisotropic meta-surfaces,” IEEE Trans. Antennas Propag., vol. 64, no. 9, pp. 3880–3895, Sept 2016.
