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"Ra'di", Younes; Simovski, Constantin R.; Tretyakov, Sergei A.Thin perfect absorbers for electromagnetic waves: Theory, design, and realizations

Published in:Physical Review Applied

DOI:10.1103/PhysRevApplied.3.037001

Published: 01/01/2015

Document VersionPublisher's PDF, also known as Version of record

Please cite the original version:"Ra'di", Y., Simovski, C. R., & Tretyakov, S. A. (2015). Thin perfect absorbers for electromagnetic waves:Theory, design, and realizations. Physical Review Applied, 3(3).https://doi.org/10.1103/PhysRevApplied.3.037001

https://doi.org/10.1103/PhysRevApplied.3.037001

https://doi.org/10.1103/PhysRevApplied.3.037001

Thin Perfect Absorbers for Electromagnetic Waves: Theory, Design, and Realizations

Y. Ra’di,* C. R. Simovski, and S. A. TretyakovDepartment of Radio Science and Engineering, School of Electrical Engineering, Aalto University,

P.O. Box 13000, FI-00076 Aalto, Finland(Received 24 October 2014; published 17 March 2015)

With recent advances in nanophotonics and nanofabrication, considerable progress has been achieved inrealizations of thin composite layers designed for full absorption of incident electromagnetic radiation,from microwaves to the visible. If the layer is structured at a subwavelength scale, thin perfect absorbersare usually called “metamaterial absorbers,” because these composite structures are designed to emulatesome material responses not reachable with any natural material. On the other hand, many thin absorbingcomposite layers were designed and used already in the time of the introduction of radar technology,predominantly as a means to reduce radar visibility of targets. In view of a wide variety of classical and newtopologies of optically thin metamaterial absorbers and plurality of applications, there is a need for ageneral, conceptual overview of the fundamental mechanisms of full absorption of light or microwaveradiation in thin layers. Here, we present such an overview in the form of a general theory of thin perfectlyabsorbing layers. Possible topologies of perfect metamaterial absorbers are classified based on theirfundamental operational principles. For each of the identified classes, we provide design equations and giveexamples of particular realizations. The concluding section provides a summary and gives an outlook onfuture developments in this field.

DOI: 10.1103/PhysRevApplied.3.037001

I. INTRODUCTION

In numerous applications of electromagnetic waves,from radio to optical frequencies, there is a need to designand realize layers which effectively absorb the power ofincident electromagnetic waves. Dallenbach [1] andSalisbury [2] absorbers are classical examples of micro-wave absorbers, developed along the work on the firstradars. The Dallenbach absorber uses a lossy dielectriclayer backed by a metal ground plane. The layer thicknessis close to a quarter wavelength in the dielectric, so that theinput impedance is real. The loss tangent of the dielectric ischosen so that the input resistance is matched with the free-space impedance. The thickness of Salisbury screens is alsoequal to a quarter wavelength: A single resistive sheet islocated at quarter-wavelength distance from the metalground, where the electric field resulting from the inter-ference of the incident and mirror-reflected waves has themaximum value. Information on classical microwaveabsorbers based on the use of conventional and artificialmaterials can be found in Refs. [3–5]. Analysis of singe-and multilayered Salisbury absorbers can be found inRef. [6]. Because of their resonant sizes, most of theseabsorbers are considerably thick. The absorption frequencyband of optically thick absorbers can be quite wide, but theconsiderable thickness of these structures makes them notpractical for many applications. To absorb visible light,

layers of various black substances have been used fromprehistorical times. Also, in this case the absorbers are verythick in terms of the wavelength. Naturally, there is a needfor electrically thin structures (in radio-engineering lan-guage, it means the same as optically thin, i.e., muchthinner than the wavelength) as microwave absorbers. Atpresent, the research focus is on optically thin terahertz andoptical absorbers. This research is motivated by the currentdevelopment of nanotechnologies and new arising appli-cations in optical [7] and terahertz [8] sensing of individualquantum objects and energy harvesting for solar-photovoltaic [9] and thermophotovoltaic [10] systems.These new applications require the perfect absorption ofa certain spectrum of incident light in submicron layers.The recent literature has witnessed great activity in design-ing thin absorbers for infrared and optical frequencies.Figure 1 illustrates typical topologies of nearly perfectabsorbers for microwave, terahertz, and infrared frequen-cies. Interestingly, all of these devices are similar in whatconcerns the operational principle (Sec. V B 4).As we shall see, there are many possibilities to realize

full absorption of normally incident plane waves inextremely thin layers, but perfect absorption takes placeat just one frequency (or possibly at a set of discretefrequencies). For thin absorbers, the most practicallyrelevant problem is the fundamental limitation on theabsorption bandwidth for absorbers of a given thickness[15,16]. Another important issue is the sensitivity ofabsorption on the incidence angle and polarization of theincident waves. Considerable work has been done in

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finding designs providing angular- and polarization-stableoperation of thin absorbers (see, e.g., Refs. [13,17–25]).In many recent papers, the name perfect absorber is

adopted for a structure which has zero-reflection andtransmission coefficients for normally incident plane wavesat one given frequency. This terminology appears unfortu-nate, because from the engineering point of view a perfectabsorber would work at all frequencies and for all incidentangles and polarizations. Moreover, the same word perfectalready appears in the name of the perfectly matched layer[26], which is indeed perfect in the sense that it fullyabsorbs waves of any polarization coming from anydirection. Of course, perfect absorbers in the engineeringsense do not exist. In traditional terminology, a boundarywhich fully absorbs a normally incident plane wave at onefrequency is called the matched boundary. However, sincethe term perfect absorber appears to be commonly acceptedin the current literature, we use it also in this review.In a sense, the design of perfect absorbers (stressing

again, perfect at one frequency and for one incidence angleonly) is rather trivial. Just like a load characterized by

an arbitrary impedance with a nonzero real part can bematched to a cable with any characteristic impedance usingonly one reactive element, any reflector-backed materiallayer of any nonzero thickness (except totally losslesslayers) can be forced to fully absorb normally incidentplane waves at one frequency using only one reactivefrequency-selective surface (the frequency-selective sur-face is a sheet characterized by some reactive averagedsurface impedance, for example, an array of thin metalpatches [4]). However, if we impose conditions on theallowed thickness and materials, and, most importantly, onthe absorption bandwidth and angular stability, design taskswill be far from trivial and simple.Recently, the work on perfect absorbers has been

focused on the use of metamaterials and metasurfaces,two-dimensional analogues of metamaterials. Probably thefirst known realization of thin metamaterial absorbers isbased on the use of artificial impedance surfaces [27]. On asurface with a very high reactive surface impedance, themagnetic field is close to zero and the electric field equalstwice the incident electric field. Positioning a resistive sheetat such a high-impedance surface (HIS), one can effectivelydissipate the incident power and decrease reflectivity tozero. Later, it was shown that there is no need for a separateresistive sheet, and full absorption in high-impedancestructures can be realized by using a lossy dielectric as asubstrate [19]. Another implementation of the same idea isthe use of an array of split-ring resonators (SRRs) to realizean artificial magnetic conductor and position a resistivesheet on top of that surface [28]. More recently, after thepublication of Ref. [29], where the name metamaterial-based perfect absorber is introduced, designs of thincomposite absorbing layers (now commonly called meta-material-based absorbers) developed earlier for microwavefrequencies have been extended to higher frequencies, up tothe visible range. There have been perfect absorbersreported in terahertz [12,30], near-infrared [31], and visiblelight [23] frequencies. A recent review of metamaterial-based electromagnetic wave absorbers can be found inRef. [32]. These absorbers can be potentially used in filtersand sensors, for solar energy harvesting, and for many otherapplications [7,9,25,33–45].For the design and optimization of these structures, it is

really critical to understand the physical phenomena whichensure the absorber operation. One needs conceptuallysimple and clear models which should give the conditionson the effective parameters of the composite layers astargets for the design and optimization. Earlier approachesare based on the description of thin absorbing layers aseffectively homogeneous bulk media or on using themethods based on impedance matching [29,46–49]. Themain idea in this approach is to tune the effectivepermittivity ϵ and permeability μ of a metamaterialabsorber to match the effective wave impedance of theabsorber η ¼ ffiffiffiffiffiffiffiffi

μ=ϵp

to the free-space wave impedance

FIG. 1. (a) Experimental realizations of thin nearly perfectabsorbers at microwaves. This figure is reproduced with permis-sion [11]. ©The Institution of Engineering and Technology.(b) Experimental realizations of thin nearly perfect absorbers atterahertz. This figure is reproduced with permission [12]. ©SPIEpublication. (c) Experimental realizations of thin nearly perfectabsorbers at infrared [13]. ©IOP Publishing. This figure isreproduced by permission of IOP Publishing. All rights reserved.(d) Experimental realizations of thin nearly perfect absorbers atnear-infrared frequencies. This figure is reprinted with permis-sion from Ref. [14]. Copyright (2010), AIP Publishing LLC.

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η0 ¼ffiffiffiffiffiffiffiffiffiffiffiμ0=ϵ0

p. However, the use of this method is limited

by the problem of the applicability of the effective mediummodel to (usually single) arrays of resonant cells. There arealso methods based on the reflection interference model[50–53]. Another popular method is using circuit- andtransmission-line-based models [54–59]. Also, models andexplanations based on Fabry-Pérot and other cavity reso-nances [60–62] have been introduced as alternative inter-pretations of the absorption mechanism in metamaterialabsorbers.Although all these models explain the same physical

phenomena, they are not always ideal in clarifying differentphysical aspects of the processes in various absorbers andfor use as design tools. In some of these models interactionsbetween inclusions in the absorbing array are not takeninto account, and in some others near-field interactionsbetween two layers on the two sides of a spacer areneglected. The majority of the known models are mostlyqualitative and cannot be used as effective design tools, sothat the design is based only on extensive numericalsimulations and optimizations. Moreover, the analysis ofrecent literature shows that there is still some confusion inthe explanations of the absorption mechanisms. For exam-ple, in Refs. [51,63], it is claimed that a magnetic responseis not necessary to realize metamaterial-based absorbers.However, opposite statements are found in numerouspublications, e.g., Refs. [46,64–66]. Another physicallyimportant aspect of the perfect absorption in thin layers isthe symmetry or asymmetry of absorption for illuminationof the two opposite sides of the absorbing structure. In fact,the possible asymmetry of absorption is linked to bianiso-tropic properties of the layers (excitation of both electricand magnetic polarizations by both electric and magneticexcitations), and it is possible to find very general andsimple design requirements for all kinds of electricallythin perfect absorbers, both symmetric and asymmetric atilluminations from the opposite sides [67].The goal of this tutorial overview is to provide a general

view on the phenomenon of perfect absorption in opticallythin layers, give the most general design equations for therequired induced currents, and provide classification ofperfect absorbers based on their operational principles.Explanations of the general concepts are complementedby a broad overview of particular realizations of perfectabsorbers of each fundamental class, which illustrate andexemplify the general design principles. The concludingsection summarizes the state of the art and provides anoutlook for further developments.

II. PERFECT ABSORBERS:OPERATIONAL PRINCIPLE

Here we will show that the operational principles of anythin perfect absorber can be well understood in terms ofeffective electric- and magnetic-current sheets which modelthe polarization and conduction currents induced in the

respective composite structures. Figure 2 illustrates ageneric composite layer illuminated by a normally incidentplane wave. The layer thickness is assumed to be opticallysmall, so that the layer can be considered as a metasurface:a microscopically structured sheet (usually periodic) withthe period of the structure smaller than the wavelength inthe surrounding media. This restriction on the period isnecessary to ensure that the absorbing layer does notgenerate diffraction lobes, and for an observer in the farzone the response is that of an effectively homogeneoussheet. We do not impose any other restrictions on thegeometry of the layer. It can be asymmetric with respectto the two sides; it can be asymmetric with respect tomirror inversion (chiral) or have nonreciprocal elements(magnetized ferrite or plasma inclusions).Under the assumption of an electrically small thickness

and the period smaller than the wavelength, the reflectedand transmitted waves are plane waves created by thesurface-averaged electric- and magnetic-current sheets Jeand Jm, respectively, as illustrated in Fig. 2. In compositesheets, the layer has a complicated microstructure, usuallycontaining some electrically small but resonant inclusions(such as complex-shape patches or split rings or smallhelices). The surface-averaged current densities can berelated to the electric and magnetic dipole moments p andm, respectively, induced in each unit cell as

Je ¼jωpS

; Jm ¼ jωmS

: ð1Þ

Here S is the unit-cell area, and we use the time-harmonicconvention expðjωtÞ. The higher-order multipoles inducedin the unit cells do not contribute to the radiated plane-wavefields of the infinite array (higher-order modes are impor-tant in calculations of the dependence of the inducedaveraged currents on the incident fields).In this overview, we describe thin composite layers

acting as perfect absorbers, that is, layers which havezero-reflection and transmission coefficients: R ¼ T ¼ 0.We formulate the necessary requirements on the induced

Y

Z X

I

R

T

I

R

T

z0

,Je Jm

d<<λ

FIG. 2. Geometry of a generic optically thin absorbing layer(left) and its equivalent model as a set of two current sheets (right).

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surface-current densities and explain how these require-ments can be realized in various absorbing structures.

A. General considerations

To ensure zero reflection and zero transmission, theabsorber device must create a forward-directed plane wavewhich cancels the incident plane wave. At the same time,the absorber device should not create any radiation in thedirection towards the source (no reflection). First of all,we note that an infinitely thin sheet of any nonmagneticmaterial cannot act as a perfect absorber. It is clear from thefact that, in this case, conduction or polarization currentsinduced on the sheet by the incident plane wave areconfined to one plane in space. This current sheet willradiate symmetrically in the forward and backward direc-tions, while a perfect absorber should radiate only in theforward direction. In fact, it is easy to show that aninfinitely thin sheet can absorb maximum 50% of theincident power (e.g., Ref. [68]). This simple considerationbrings us to a conclusion that any perfect absorber musthave a nonzero physical thickness (the use of negligiblythin magnetic-material sheets implies nonphysical infinitevalues of the permeability).The simplest volumetric distribution of current in a

thin finite-thickness layer is a set of two parallel infinitelythin sheets of electric current. Thus, let us next consideran absorber device formed by two parallel material orcomposite sheets separated by distance d. The incident fieldwill induce some electric currents on the two sheets(conduction or displacement currents), and we denote thecorresponding surface-current densities as Je1 and Je2. Letthe incident planewave come from the side of sheet 1, and itsamplitude at the position of the second sheet isEinc. The twosheets will act as a perfect absorber if the sum of the electricfields in two plane waves created by the two sheets cancelsthe incident field behind the second sheet:

− η02Je1e−jk0d − η0

2Je2 ¼ −Einc; ð2Þ

while the electric fields of the two reflected plane wavescancel in the opposite direction:

− η02Je1 − η0

2Je2e−jk0d ¼ 0: ð3Þ

Here, η0 is the free-space impedance and k0 is thepropagation constant. The above formulas for the ampli-tudes of electric fields created by electric-current sheetssimply express the classical boundary conditions onsurfaces which maintain surface currents [a more generalform valid for oblique incidence can be found, e.g., inEq. (4.34) of Ref. [69]]. The solution of this system ofequations reads

Je1 ¼j

η0 sinðk0dÞEinc; ð4Þ

Je2 ¼ − jejk0d

η0 sinðk0dÞEinc: ð5Þ

In this simple case of currents concentrated on two sheets,we have a unique solution for the surface-current densitiesof a perfect absorber (for a fixed thickness d). Examples ofabsorbers of this type include the Salisbury screen [2]and lossy capacitive grids (for example, arrays of patches)over a ground plane [11,19,27], which we will callhigh-impedance surface absorbers. While the realizationscan be quite different, in all of these absorbersthe surface-current densities on the two layers obey (4)and (5).Let us next find the required boundary conditions on the

two surfaces, relating the electric surface-current densitieswith the total tangential fields on these surfaces. Actually,Eq. (2) implies that in absorbers formed by only twoelectric-current sheets the second surface must be perfectlyelectrically conducting (PEC boundary). Indeed, Eq. (2)directly states that the total tangential electric field at thesecond surface

Etot ¼ Einc − η02Je1e−jk0d − η0

2Je2 ¼ 0: ð6Þ

As a check, one can use the last relation to expressthe incident field in terms of the total field and currentdensities and substitute in Eq. (5). As a result, it is seen thatthe surface impedance of the second surface is zero(Je2 × 0 ¼ Etot). Because also the tangential magnetic fieldbehind a perfect absorber is zero, the theory can bereformulated for the dual case of a perfect magneticconductor (PMC) as a second surface or, more generally,for an arbitrary boundary. From the practical point of view,this result tells us that within this realization scenario weneed at least three effective current sheets to realizesymmetric absorbers (equally absorbing from both sides).Of course, the boundary condition Eq. (6) bounds only

the surface-averaged fields and currents, which means thatthe conclusion that the second sheet is a PEC or PMCboundary holds only for homogeneous-material sheets.In structured layers, the evanescent fields formed closeto the inhomogeneities do not equal zero everywhere at andbehind the second sheet (they exponentially decay behindand in front of the absorber). We will see examples ofabsorbers formed by arrays of particles without any groundplane in the following sections.Let us write the required sheet impedance of the first

sheet, which relates the total electric field and the surface-current density Je1 at that plane. The total field is the sumof the incident field Eincejk0d at that plane and the fieldscreated by electric currents Je1;2 induced on the twosurfaces, but, since the reflection coefficient is zero, thesum of the fields created by the currents is zero [Eq. (3)].Thus, we can write the total field at the first sheet as

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Etot ¼ Eincejk0d ¼ ZgJe1 ¼ −jη0 sinðk0dÞejk0dJe1: ð7Þ

Here we have expressed Einc in terms of Je1 using Eq. (4).We see that the required sheet (grid) impedance [69] of thefirst layer reads

Zg ¼ −jη0 sinðk0dÞejk0d; ð8Þ

and the corresponding admittance equals

Yg ¼1

Zg¼ j

η0

e−jk0dsinðk0dÞ

¼ jη0 tan k0d

þ 1

η0: ð9Þ

As an example, for the Salisbury absorber (d ¼ λ0=4,k0d ¼ π=2), Eq. (8) shows that the surface impedance ofthe first layer should be purely resistive and equal toZg ¼ η0. In the modern optical literature, the effect of fullabsorption in a thin sheet when the matching conditionEq. (9) is satisfied is usually called critical coupling[70–74]. Because of the complexity of actual geometriesused in optical devices, the effect is usually revealed byusing numerical optimizations. However, the physics of thiseffect is the same as of the Salisbury absorber described inRef. [75]. If the distance between the two sheets is arbitrary(usually considerably smaller than λ0=4), Eqs. (8) and (9)give the general design requirements for high-impedancesurface absorbers [see Eq. (90), discussions in Sec. V B 4,and detailed derivations in Appendix A].More generally, we can consider perfect absorbers where

induced currents are filling the whole volume of a slab ofthe thickness d instead of being concentrated only on twosheets. In this case, the requirements for perfect absorptiontake the form of two integral equations [the sums in Eqs. (2)and (3) replaced by integrals over the slab thickness], andthe solution for the required current densities is no longerunique.An important conclusion at this stage is that for perfect

absorption of the power of a plane wave we need to ensurethat the absorber creates at least two coherent and interfer-ing plane waves, to enable asymmetric actions in theforward and backward directions. These waves can becreated by current sheets induced on two infinitesimallythin layers (one of them can be a mirror), by currents inthe whole volume of a slab, or by other means. With thisrespect, it is interesting to note recent papers on so-called“coherent absorbers,” where the second coherent wave iscreated by splitting the incident beam into two anddirecting both beams to the same thin lossy sheet via asystem of mirrors [76–79] or a prism [80]. The namecoherent absorber was introduced in Ref. [81] as specific tothat particular device, but, as we see, the operation of anyperfect absorber is based on interference of mutuallycoherent waves. Actually, in any perfect absorber wherethe energy is dissipated in an optically thin layer, that layeris illuminated by two (or more) coherent plane waves. The

operational principle of the two-beam device [76] is verysimilar to the Salisbury screen [2] or the circuit-analogabsorber [4], where the second incident coherent beam,which illuminates a thin absorbing sheet, is created simplyby a mirror behind a lossy sheet.Let us note in passing that Eqs. (4) and (5) suggest a

possibility to realize an active absorber if the incident fieldis known. Indeed, if we know Einc in advance, we canrealize the required surface-current densities by construct-ing, for example, two arrays of electric dipole antennas fedby appropriately chosen sources. Although the system isactive, for the incident plane wave it will work as a perfectabsorber. This idea is similar to the recently introducedconcept of the active cloak [82,83], where arrays ofantennas are used to generate fields which cancel fieldsscattered from an object which one wants to cloak.After this brief discussion which concerns resonant

perfect absorbers of any thickness, let us next considerthe fundamentals of optically thin absorbing layers.

B. Optically thin perfect absorbers

The operational principle of a generic optically thinabsorber is convenient to explain by using the sameexample of two polarizable sheets tuned to work as aperfect absorber. Let us rewrite Eqs. (2) and (3) in terms ofthe symmetric and antisymmetric surface-current sheets,defining

Js ¼1

2ðJe1 þ Je2Þ; Ja ¼

1

2ðJe2 − Je1Þ: ð10Þ

Equations (2) and (3) take the form

− η02Jsð1þ e−jk0dÞ − η0

2Jað1 − e−jk0dÞ ¼ −Einc; ð11Þ

− η02Jsð1þ e−jk0dÞ − η0

2Jað−1þ e−jk0dÞ ¼ 0: ð12Þ

For optically thin (meaning that jk0jd ≪ 1) absorbers, wecan replace the exponents with the first terms of their Taylorexpansions. The first term in Eqs. (11) and (12) reduces to

− η02Jsð1þ e−jk0dÞ ≈ − η0

2ð2JsÞ ¼ − η0

2ðJe1 þ Je2Þ:

ð13ÞWe recognize this quantity as the electric field of the planewave generated by the electric-current sheet

Je ¼ Je1 þ Je2 ð14Þat the position of this sheet. Obviously, the surface-currentdensity Je is simply the total electric-current densityinduced in the thin absorber.The second term in Eq. (11) for thin absorbers

simplifies to

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− η02Jað1 − e−jk0dÞ ≈ − η0

2Jajk0d ¼ − 1

2ðjωμ0dÞJa:

ð15Þ

We recognize this quantity as the electric field of the planewave generated by a sheet of magnetic current at theabsorber plane. Indeed, the magnetic surface-current den-sity Jm is related to the magnetic moment per unit areaM asJm ¼ jωM, and the magnetic moment is related to theantisymmetric part of the electric-current distribution asM ¼ −μ0dz0 × Ja (z0 is the unit vector normal to theabsorber plane and pointing towards the source of theincident plane wave). Thus, jωμ0dJa ¼ z0 × Jm.Finally, Eqs. (11) and (12) take the physically clear,

simple, and general form

− η02Je − 1

2z0 × Jm ¼ −Einc; ð16Þ

− η02Je þ

1

2z0 × Jm ¼ 0: ð17Þ

Solving these equations, we find the required electric andmagnetic surface-current densities:

Je ¼1

η0Einc; ð18Þ

z0 × Jm ¼ Einc; Jm ¼ −z0 ×Einc ¼ η0Hinc: ð19Þ

As a check, we can see that the same follows directly fromEqs. (4) and (5) in the assumption k0d ≪ 1:

Je ¼ Je1 þ Je2 ¼1

η0

jk0d

ð1 − ejk0dÞEinc ≈1

η0Einc; ð20Þ

Jm ¼ −jωμ0 d2z0 × ðJe2 − Je1Þ ≈ −ωμ0 d

2

2

η0k0dz0 ×Einc

¼ −z0 ×Einc ¼ η0Hinc: ð21Þ

Although we have derived these formulas considering amodel of two polarizable sheets, it is clear that the resultin fact holds for any optically thin absorber, because theplane-wave response of any optically thin structure reducesto plane-wave fields created by surface-averaged electric-and magnetic-current sheets, as illustrated in Fig. 2. Wecan use this result for thin homogeneous layers, periodicalstructures, and arrays of discontinuous cells, with the onlylimitation that the structural period in the absorber planeis smaller than the wavelength in free space (ensuring thatno diffracted plane waves are generated). For absorbersrealized in the form of arrays of particles (or any dis-connected unit cells), it is more convenient to rewriteEqs. (16) and (17) in terms of the electric (p) and magnetic(m) dipole moment of a single unit cell. As already

discussed, these dipole moments relate to the respectivesurface-current densities as in Eq. (1).

C. Thin absorbers as Huygens sheets

Two important conclusions follow from Eqs. (16) and(17). First of all, and as was already noted before, a thinperfect absorber must support both electric and magneticsurface currents (both Je and Jm must be nonzero). Indeed,if one of them is zero, then the other one must be also zeroto ensure zero reflection, and in that case the layer is fullytransparent. Second, we observe that any optically thinperfect absorber is a Huygens surface, because the inducedsurface-current densities obey the relation (17):

η0Je ¼ z0 × Jm; ð22Þwhich is the same as the relation between the electric andmagnetic fields in the incident plane wave. Physically, thisresult means that the plane-wave fields created by theinduced electric and magnetic currents cancel each other inthe backward direction (again we see interference of twomutually coherent plane waves). Let us stress also here thatthis conclusion holds for any optically thin layer which hasa zero-reflection coefficient. In the case of a perfectabsorber, the two waves sum up in the forward directionand cancel the incident field [Eq. (16)].Actual realizations can contain some homogeneous

layers (e.g., a thin slab of magnetic or dielectric materialon a PEC ground plane) or periodical arrays with thetransverse period smaller than the wavelength in free space(e.g., arrays of resonant dielectric spheres, double arrays ofmetal patches, or arrays of metal patches over a PEC plane).In the case of a periodical structure, it can be moreconvenient to write the equations in terms of electric andmagnetic dipole moments induced in each unit cell, p andm; then Eq. (22) takes the form

η0p ¼ z0 ×m: ð23ÞBecause the dipole moments of cells and the surface-averaged current densities are simply proportional to eachother [Eq. (1)], both continuous and periodical absorberscan be understood in terms of induced electric andmagnetic moments, either per unit cell or per unit area.

III. THIN PERFECT ABSORBERS: GENERALDESIGN EQUATIONS

In the previous section, we found the required valuesof induced surface-current densities which ensure perfectabsorption of a normally incident plane wave. Next, weaddress the design question: What should be the electro-magnetic parameters of the layer, such that the inducedcurrents indeed take the desired values? As discussedabove, the electromagnetic response of thin material orcomposite sheets can be described in terms of either

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relations for induced surface-current densities or inducedelectric and magnetic moments per unit cell or per unit area.We will first use the formalism of induced dipole moments.The material of this section is based on Ref. [67].

A. Induced polarization densities and currents

In the most general case of a thin linear sheet, compositeor homogeneous, we can write the linear relations betweenthe induced moments in each unit cell (or in a unit area ofthe sheet) and the incident fields as

�p

m

�¼

"¯̂̄αee

¯̂̄αem¯̂̄αme

¯̂̄αmm

#·

�Einc

Hinc

�: ð24Þ

Here the dual bars denote dyadic (tensor) quantities, and thehat marks the collective polarizabilities, which measurethe response of the unit cells to external fields incident onthe infinite array. For composite sheets, these polarizabil-ities include the effects of particle interactions and dependnot only on the individual cell topology and dimensions butalso on the distances to the other inclusions in the lattice. Inview of Eq. (1), this dependence is equivalent to relating theinduced surface-current densities to the incident fields.Indeed, Eq. (24) is equivalent to

�JeJm

�¼ jω

S

"¯̂̄αee


¯̂̄αmm

#·

�Einc

Hinc

�ð25Þ

(S is the unit-cell area). We know that in perfect absorbersthe induced current densities relate to the incident fields asin Eqs. (18) and (19). Thus, we can determine the requiredunit-cell polarizabilities of perfect absorbers as solutions ofthe following system of equations:

jωS

"¯̂̄αee


¯̂̄αmm

#¼

�1η0¯̄It 0

0 η0¯̄It

�: ð26Þ

Here, ¯̄It is the two-dimensional unit dyadic (matrix) definedin the absorber plane. Obviously, there are infinitely manysolutions for the polarizabilities, since we have more freedesign parameters than the number of equations. To clarifythe possible choices and different absorption and matchingmechanisms, it is more convenient to first find the reflec-tion and transmission coefficients in terms of the collectivepolarizabilities and then study what combinations of theseparameters realize perfect absorbers.To do that, first we note that the dyadics in Eqs. (24)

and (26) are not the most general two-dimensional dyadics,but they possess rotational symmetry, because perfectabsorbers should absorb normally incident plane wavesof any polarization. This property means that the absorberstructure should have no preferred direction along theabsorber plane. In terms of the electromagnetic parameters,

the layer must be a uniaxial structure, where the onlypreferred direction is given by the unit vector normal to thesurface, z0. The uniaxial symmetry ensures the isotropicresponse for normally incident plane waves of arbitrarypolarizations.Thus, for the most general thin perfect absorber we can

write all the polarizabilities in (24) in the form

¯̂̄αee ¼ α̂coee¯̄It þ α̂cree

¯̄Jt;¯̂̄αmm ¼ α̂comm

¯̄It þ α̂crmm¯̄Jt;

¯̂̄αem ¼ α̂coem¯̄It þ α̂crem

¯̄Jt;¯̂̄αme ¼ α̂come

¯̄It þ α̂crme¯̄Jt; ð27Þ

where indices co and cr refer to the symmetric andantisymmetric parts of the corresponding dyadics, respec-tively. Here, ¯̄It ¼ ¯̄I − z0z0 is the two-dimensional unitdyadic, and ¯̄Jt ¼ z0 ×

¯̄It is the vector-product operator.In the last set of relations, it is convenient to separatethe coupling coefficients responsible for reciprocal andnonreciprocal coupling processes:

¯̂̄αem ¼ ðχ̂ þ jκ̂Þ¯̄It þ ðV̂ þ jΩ̂Þ ¯̄Jt;¯̂̄αme ¼ ðχ̂ − jκ̂Þ¯̄It þ ð−V̂ þ jΩ̂Þ ¯̄Jt: ð28Þ

In the following, we use the classification of magnetoelec-tric coupling effects in terms of reciprocity and thesymmetry of their magnetoelectric coupling dyadics.There are two reciprocal classes (chiral and omega,measured by the parameters κ̂ and Ω̂, respectively) andtwo nonreciprocal classes (“moving” and Tellegen, mea-sured by V̂ and χ̂, respectively) [5]. The four main types ofmagnetoelectric coupling are summarized in Table I. Notealso that for reciprocal particles the electric and magneticpolarizabilities ¯̂̄αee and

¯̂̄αmm are always symmetric dyadics;that is, α̂cree ¼ 0 and α̂crmm ¼ 0 in all reciprocal structures.In the rest of the review, we use double signs to

distinguish the illuminations of the thin absorbing layerfrom its two sides, where the top and bottom signscorrespond to the incident plane wave propagating alongthe −z0 and z0 directions, respectively (Fig. 2). In theincident plane wave, the electric and magnetic fields satisfy

Hinc ¼∓ 1

η0z0 ×Einc: ð29Þ

Radiation from infinite sheets of electric and magneticcurrents can be easily solved (e.g., Ref. [84]), and theelectric field amplitudes in the reflected (Eref ) and trans-mitted (Etr) plane waves are written as

Eref ¼ − jω2S

½η0p ∓ z0 ×m�;

Etr ¼ Einc − jω2S

½η0p� z0 ×m�: ð30Þ

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Expressing the induced dipole moments in terms of theincident electric field and the polarizabilities in the form ofEq. (28), we get

Eref ¼ − jω2S

��η0α̂

coee � 2jΩ̂ − 1

η0α̂comm

�¯̄It

þ�η0α̂

cree ∓ 2χ̂ − 1

η0α̂crmm

�¯̄Jt

�· Einc; ð31Þ

Etr ¼��

1 − jω2S

�η0α̂

coee � 2V̂ þ 1

η0α̂comm

��¯̄It

− jω2S

�η0α̂

cree ∓ 2jκ̂ þ 1

η0α̂crmm

�¯̄Jt

�· Einc: ð32Þ

General expressions (31) and (32) for the reflectedand transmitted fields from planar uniaxial bianisotropiclayers allow the derivation of required conditions forperfect absorption in terms of the polarizabilities of theunit cells. By definition, a perfect absorber does not reflectpower and creates perfect shadow behind the sheet; that is,

Eref ¼ 0; Etr ¼ 0: ð33Þ

Let us consider the two requirements of perfect absorptionEq. (33) separately.

B. Zero-reflection condition forcollective polarizabilities

As discussed above, the zero-reflection property requiresthat the layer behaves as a Huygens sheet, with the induceddipole moments satisfying Eq. (23). From Eq. (31), we seethat the zero-reflection condition requires the followingrelation between the two polarizabilities (electric andmagnetic) and the omega coupling coefficient:

η0α̂coee � 2jΩ̂ − 1

η0α̂comm ¼ 0 ð34Þ

(under this condition the copolarized reflection vanishes).Let us recall that the � signs in this relation refer toilluminations from the two sides of the thin layer. Thus, ifthere is no magnetoelectric coupling in the layer (Ω̂ ¼ 0),the reflection coefficient is the same for illumination fromboth sides. In particular, it is clear that any thin absorber ona metal surface is a bianisotropic layer with omega couplingin the microstructure. This result is obvious from the factthat the reflection coefficient from the side of the ground

plane is always −1, while it is zero from the side of theabsorbing layer.Continuing the analysis of the zero-reflection require-

ment, we note that the cross-polarized reflection vanishes if

η0α̂cree ∓ 2χ̂ − 1

η0α̂crmm ¼ 0: ð35Þ

All the polarizabilities which enter this relation can havenonzero values only if the layer is nonreciprocal. In viewof practical realizations, the simplest way to ensure thatEq. (35) is valid is to use only reciprocal materials in theabsorber design. If nonreciprocity is required (for example,to realize different transmission coefficients for wavescoming from the opposite directions), the nonreciprocalresponses due to electric, magnetic, and magnetoelectricpolarization processes must be balanced as is dictated byEq. (35) in order to ensure zero reflection from theabsorber. In the following, we concentrate on reciprocalabsorbers. Nonreciprocal absorbing sheets are discussedin Ref. [67].

C. Zero transmission condition forcollective polarizabilities

The transmitted field is zero when the sum of the plane-wave fields created by induced electric- and magnetic-current sheets compensates the incident field:

jω2S

½η0p� z0 ×m� ¼ Einc: ð36Þ

Recall that the amplitudes of waves created by the electricand magnetic dipole moments are equal (as they cancel outin the reflected field). Again, we note that magnetoelectriccoupling (nonreciprocal parameter V̂) is responsible forthe possible difference of transmission coefficients forillumination from the two sides of the sheet. Naturally,for reciprocal sheets (V̂ ¼ 0) the transmission coefficient isthe same as seen from both sides, and the requirement offull absorption reads

η0α̂coee þ

1

η0α̂comm ¼ 2S

jω: ð37Þ

If the sheet is reciprocal, then the electric and magneticpolarizabilities are symmetric dyadics (α̂cree ¼ 0, α̂crmm ¼ 0),which brings us to the conclusion that all reciprocalpolarization-insensitive thin perfect absorbers are nonchiral(κ̂ ¼ 0). With this respect, we note that this conclusion does

TABLE I. Magnetoelectric coupling effects (parameters Ω, κ, V, and χ for lossless structures are real).

Omega Chiral Moving Tellegen

¯̄αem ¼ ¯̄αme ¼ jΩ ¯̄Jt ¯̄αem ¼ − ¯̄αme ¼ jκ ¯̄It ¯̄αem ¼ − ¯̄αme ¼ V ¯̄Jt ¯̄αem ¼ ¯̄αme ¼ χ ¯̄It

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not exclude the use of chiral elements (like metal spirals) inthe design of thin absorbers. However, for proper operation,chiral elements should be arranged so that the overallstructure of the layer is nonchiral. The most commonknown topologies contain random mixtures or regulararrays of chiral particles over a perfectly conducting groundplane [85–87]. Because the mirror images of spirals havethe opposite handedness, the overall response is nonchiral(the reflected wave has the same polarization as the incidentwave, if the arrangement of spirals is isotropic, andthe transmitted field is zero behind the PEC plane). Adiscussion about advantages of microscopically chiralabsorbers can be found in Ref. [87] and references therein.In summary, thin perfect reciprocal absorbers which

absorb waves incident on both sides of the sheet are simpleelectrically and magnetically polarizable layers (no biani-sotropy). Thin perfect reciprocal absorbers which absorbwaves incident only on one side of the sheet (variousreflection properties are possible if one illuminates theother side) are bianisotropic layers with the antisymmetriccoupling dyadic (omega structures). Equations (34)and (37) give the necessary conditions on the collectivepolarizabilities of the unit cells of perfect absorbers.

D. Required values of collective polarizabilities

For the most general case of thin reciprocal perfectabsorbers, we can assume that the response can beasymmetric, meaning that only one side of the layer isperfectly absorbing, while the other side is characterized bythe reflection coefficient R. From Eqs. (31) and (32), wefind the requirements on the collective polarizabilities interms of the reflection coefficient R of the “imperfect” side:

η0α̂coee þ 2jΩ̂ − 1

η0α̂comm ¼ 0; ð38Þ

− jω2S

�η0α̂

coee − 2jΩ̂ − 1

η0α̂comm

�¼ R; ð39Þ

η0α̂coee þ

1

η0α̂comm ¼ 2S

jω: ð40Þ

Solving for the required polarizabilities, we find

η0α̂coee ¼

�1 − R

2

�Sjω

; ð41Þ

jΩ̂ ¼ R2

Sjω

; ð42Þ

and

1

η0α̂comm ¼

�1þ R

2

�Sjω

: ð43Þ

Let us write the induced electric and magnetic currents for asheet with such polarizabilities. Suppose that the perfectlyabsorbing side is illuminated (the incident wave propagatesalong−z0). In this bianisotropic layer, the electric current isinduced by both the electric and magnetic fields:

Je ¼jωSp ¼ jω

Sðα̂coeeEinc þ jΩ̂z0 ×HincÞ: ð44Þ

Using the plane-wave relation between the fields in theincident wave, z0 ×Hinc ¼ Einc=η0, and substituting theparameters from Eqs. (41) and (42), we find that

Je ¼jωS

�α̂coee þ

1

η0jΩ̂

�Einc ¼

1

η0Einc; ð45Þ

for an arbitrary reflection coefficient R of the oppositeside of the layer. Thus, the necessary relation between theincident field and induced current Eq. (18) is indeedsatisfied. Similarly, one can check that the induced mag-netic current satisfies Eq. (19) for any R.Let us specialize the above results for the case when a

thin absorbing structure is covering a perfectly conductingsurface (the most interesting scenario for microwaveabsorbers). In this situation, the reflection coefficient fromthe PEC side equals −1, while we demand perfectabsorption in the cover. If the structure is perfectlyabsorbing plane waves traveling along −z0 and fullyreflects (R ¼ −1) waves propagating along z0, the polar-izabilities read

η0α̂coee ¼

3

2

Sjω

; ð46Þ

jΩ̂ ¼ − 1

2

Sjω

; ð47Þ

and

1

η0α̂comm ¼ 1

2

Sjω

: ð48Þ

It is important to note that, due to the omega coupling, theinduced electric and magnetic surface-current densities arebalanced, although the normalized magnetic polarizabilityis 3 times as small as the electric polarizability. From thepoint of view of realizations at very high frequencies(especially in the infrared and visible ranges), the mainproblem is to provide enough strong magnetic response,because naturally magnetic media are not available, andartificial magnetism is a weak second-order effect of spatialdispersion (e.g., Ref. [5]). The possibility to use stronger,first-order effects of bianisotropic omega coupling isclearly an advantageous route. If it is enough to ensurefull absorption only from one side of the layer and the

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reflection coefficient from the other side can be arbitrary, itis reasonable to choose the realization with the smallestpossible absolute value of the required magnetic polar-izability α̂comm. Inspecting Eq. (43), we see that the bestchoice is the case of full mirror reflection from the backside, R ¼ −1.To realize symmetric perfect absorption from both sides

of a thin layer, we must ensure that

η0α̂coee ¼

1

η0α̂comm ¼ S

jω: ð49Þ

In this case there is no magnetoelectric coupling, and theelectric and magnetic responses are balanced (equallystrong).It is important to note that the results of this section are

not restricted by any assumption on the explicit topologyof bianisotropic sheets. The structure can contain one orseveral composite or homogeneous layers, and the requiredconditions for induced electric and magnetic polarizationsmay be realized in structures which mimic the requiredbianisotropic response, not containing any bianisotropicparticles with dipolar response.

IV. REQUIRED SURFACE SUSCEPTIBILITIESAND UNIT-CELL POLARIZABILITIES

A. Symmetric absorbers: Impedance or admittanceboundary conditions

While the above relations for the required inducedelectric- and magnetic-current densities and collectivepolarizabilities of unit cells provide good insight into theoperational principle of perfect absorbers, for the design ofabsorbing layers it is more convenient to express the currentdensities as functions of the total tangential electric andmagnetic fields, so that the coefficients would have theusual meaning of surface impedance or admittance. Forsymmetric structures (both sides perfectly absorptive), wecan find the total surface-averaged fields right on theabsorbing sheet plane as the averaged values of thetangential fields on the two opposite surfaces of the layer:

Etot ¼1

2ðEþ þ E−Þ; Htot ¼

1

2ðHþ þH−Þ: ð50Þ

The total fields can be related to the induced surface-currentdensities simply by noticing that the total fields on thesurface are the sums of the incident fields and the fieldscreated by the currents on the surface:

Etot ¼ Einc − η02Je; ð51Þ

Htot ¼ Hinc − 1

2η0Jm: ð52Þ

Note that, with the current values given by Eqs. (18)and (21), relations (50) are equivalent to

Etot ¼1

2Einc; Htot ¼

1

2Hinc ð53Þ

(this is expected, because on the illuminated side the fieldequals to the incident field, and on the other side it is zero).Substituting the incident fields into Eqs. (18) and (19), wefind that on the surface of a symmetric perfectly absorbingsheet the following sheet conditions are satisfied:

Je ¼2

η0Etot; Jm ¼ 2η0Htot: ð54Þ

Thus, the layer is a set of two purely resistive (thecoefficients in the above relations are purely real) sheets:one electric, with the surface resistance equal to η0=2, andthe dual “magnetic resistor” sheet, both positioned at thesame plane of the sheet absorber.

B. Asymmetric absorbers

1. Impedance or admittance matrices

For asymmetric absorbers, the impedance or admittancematrix (e.g., Ref. [69]) is an appropriate model. In thismodel, the structure response is described in terms of linearrelations between the total fields at the two sides of thelayer:

�EþE−

�¼

�Z11 Z12

Z21 Z22

�·

�z0 ×Hþz0 ×H−

�: ð55Þ

Here the indices � mark the tangential fields at the twoopposite sides of the layer. Let us find the impedanceparameters for a layer acting as a perfect absorber from the“þ” side (incident waves along the −z0 direction) andhaving reflection coefficient R for the illumination of theopposite side (the same as considered in Sec. III D). To dothat, we write the expression (55) first for illuminations ofthe absorbing side and then for the reflecting side:

�Einc

0

�¼

�Z11 Z12

Z21 Z22

�·

�z0 ×Hinc

0

�; ð56Þ

�0

ð1þRÞEinc

�¼�Z11 Z12

Z21 Z22

�·

�0

−ð1−RÞz0×Hinc

�:

ð57ÞObviously, these relations are satisfied if

Z11 ¼ η0; Z22 ¼ −η0 1þ R1 − R

; Z12 ¼ Z21 ¼ 0:

ð58Þ

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Knowing the Z matrix, we can find the parameters of theequivalent T circuit (e.g., Sec. 2.4 in Ref. [69]). The resultis shown in Fig. 3. Obviously, the symmetric absorberwhich is modeled as a set of resistive sheets in the sameplane [Eq. (54)] corresponds to a symmetric T circuit withboth impedances equal to η0.Similarly, to determine the structure of the equivalent

Π circuit, we can find the parameters of the admittancematrix ¯̄Y:

�z0 ×Hþz0 ×H−

�¼

�Y11 Y12

Y21 Y22

�·

�EþE−

�; ð59Þ

which is the inverse of ¯̄Z. Its components read

Y11 ¼1

η0; Y22 ¼ − 1

η0

1 − R1þ R

; Y12 ¼ Y21 ¼ 0:

ð60Þ

This matrix corresponds to the equivalent Π circuitshown in Fig. 4. For the purposes of the topology design,it may be convenient to make use of the equivalentscattering or transmission matrices of the layer. Theycan be found by using standard formulas of microwaveengineering, e.g., Ref. [88].The presence of either short- or open-circuit components

in the equivalent networks does not imply that the actualrealizations of perfect absorbers must include an impen-etrable wall. In fact, these two equivalent T and Π circuitscan model one and the same absorber, which does notcontain neither electric nor magnetic wall. For symmetric

absorbers (R ¼ 0 in the above relations), the equivalentcircuits are symmetric and contain only resistors equalto η0. Such absorbers can be modeled both as a symmetrictwo-port, described in terms of matrix ¯̄Z or ¯̄Y, or as a setof two resistive sheets (one electric and one magnetic,in the same plane), described by relations (54). If theabsorber is asymmetric (R ≠ 0), the two-port matrixmodel (55) is appropriate. The relations between theasymmetry of the equivalent circuit and the omega-typebianisotropy of the physical structure are analyzed in detailin Ref. [89].

2. Required parameters of discrete unit cells(individual polarizabilities)

In the previous section, we derived the required effectiveparameters of asymmetric perfect absorbers in terms oftheir impedance or admittance matrices. However, there is auseful alternative model for asymmetric perfect absorberswhich consist of an array of electrically small particles. Ifthe interaction of particles in the array can be considered asthe interaction of corresponding electric and magneticdipoles, it is possible to find the required polarizabilitiesof a single individual particle in free space, such that, whenarranged in a lattice, a perfect absorber would be realized.Knowing the requirements on single particles allowsdesigning individual unit cells.Finding relations between collective and individual

polarizabilities is one of the fundamental problems ofapplied electromagnetics, and it has been discussed indozens of papers and (for simple dipole arrays) in books,e.g., Ref. [69]. For understanding and designing perfectabsorbers, we need a rather general solution presenting therelation between the collective and individual polariz-abilities in closed form which would be applicable beyondthe quasistatic approximation (which is used in mostpublications). Moreover, here we need the solution forbianisotropic arrays. Recently, in Ref. [67], the neededexpressions were found for uniaxial square arrays ofbianisotropic particles whose electric and magneticmoments are induced in the array plane. By using theseresults, for arrays of bianisotropic omega particles theindividual polarizabilities as functions of the collectiveones can be written as

αcoee ¼α̂coee þ βmðα̂coeeα̂comm − Ω̂2Þ

1þ ðβeα̂coee þ βmα̂commÞ þ βeβmðα̂coeeα̂comm − Ω̂2Þ ;

αcomm ¼ α̂comm þ βeðα̂coeeα̂comm − Ω̂2Þ1þ ðβeα̂coee þ βmα̂

commÞ þ βeβmðα̂coeeα̂comm − Ω̂2Þ ;

Ω ¼ Ω̂1þ ðβeα̂coee þ βmα̂

commÞ þ βeβmðα̂coeeα̂comm − Ω̂2Þ :

ð61Þ

FIG. 3. T-circuit model of a perfectly absorbing layer.

FIG. 4. Π-circuit model of a perfectly absorbing layer (theshown values are resistances).

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Explicit analytical expressions for the interactionconstants βe and βm can be found, e.g., in Ref. [69].As an example, let us give explicit design formulasfor an asymmetric absorber which is fully reflective(R ¼ −1) from the nonabsorbing side. The requiredcollective polarizabilities are given by Eqs. (46)–(48).Substituting into the above relations, we find that therequired individual polarizabilities of unit cells read

αcoee ¼32þ βe

jωη0

ð1þ βejωη0

Þ21

jωη0;

αcomm ¼12þ βe

jωη0

ð1þ βejωη0

Þ2η0jω

;

Ω ¼12

ð1þ βejωη0

Þ21

ω: ð62Þ

By knowing these required polarizabilities of one singleunit cell in free space, the shape and dimensions of theparticle can be found from analytical models or by usingnumerical simulation and optimization techniques.

V. POSSIBLE REALIZATIONS OF PERFECTLYABSORBING SHEETS

Next, we consider particular realizations of thin absorb-ers and explain how they realize the required conditions oninduced electric- and magnetic-current sheets to formabsorptive Huygens pairs. We start from the simpler caseof symmetric structures, equally absorbing power fromboth sides of the layer.

A. Symmetric perfect absorbers

1. Impedance-matched layers

Let us consider a material layer with μr ¼ ϵr. Becausethe impedance is equal to that of free space, at an interfacewith such a medium there is no reflection (consideringonly normal incidence). If we in addition assume that thematerial is very lossy, the wave will quickly decay, and wecan use an electrically thin (compared to the free-spacewavelength) layer as a perfect absorber. In practice, one canposition this layer on a PEC ground plane, if the thickness dis such that the wave decays before reaching the PEC plane.It is possible if k0dj ffiffiffiffiffiffiffiffi

ϵrμrp j ≫ 1.

To find the equivalent electric surface-current density inthe skin layer, we first write the volumetric-current density

je ¼ jωP ¼ jωðD − ϵ0EÞ ¼ jωϵ0ðϵr − 1ÞE; ð63Þ

where E is the electric field inside the absorbing layer.The electric field at the input interface equals the incidentfield (because there is no reflected wave), and in thelayer it exponentially decays: EðzÞ ¼ Eince−jk0

ffiffiffiffiffiffiϵrμr

pz. The

equivalent surface-current density is found as an integral ofthe volumetric-current density over the skin depth:

Je ¼Z

∞

0

jedz ¼ jωϵ0ðϵr − 1Þ 1

jk0ffiffiffiffiffiffiffiffiϵrμr

p Einc ≈1

η0Einc;

ð64Þ

in the assumption jϵrj ¼ jμrj ≫ 1. Because of duality, wecan do the same derivation for the magnetic current toobtain

Jm ¼Z

∞

0

jmdz ¼ jωμ0ðμr − 1Þ 1

jk0ffiffiffiffiffiffiffiffiϵrμr

p Hinc ≈ η0Hinc:

ð65Þ

As we see, the induced surface current densities are indeedthe same as required for perfect absorption in a thin sheet[Eqs. (20) and (21)].Now we can find the effective impedances and admit-

tances, relating the induced surface current densities to thetotal tangential electric and magnetic fields in the sheetplane, instead of the incident fields as in Eqs. (64) and (65).To do that, we add to the left- and right-hand sides ofEq. (64) the plane-wave fields created by Je (the electricfields generated by Jm differ by sign on the two sides of thesheet and, thus, do not contribute to the averaged electricfield):

η0Je − η02Je ¼ Einc − η0

2Je ¼ Etot: ð66Þ

Thus,

Je ¼2

η0Etot: ð67Þ

Likewise, we obtain

Jm ¼ 2η0Htot: ð68Þ

As expected, these are the required resistive-sheet con-ditions for symmetric absorbers (54). In this case, themagnetoelectric coupling is absent, because essentially weassume that the wave decays to zero over the slab thickness.The same isotropic slab can be illuminated from theopposite direction, and the power will be equally totallyabsorbed. As we have shown before, in this case thinabsorbers have no bianisotropy. Practical realizations ofsheets implementing the condition k0dj ffiffiffiffiffiffiffiffi

ϵrμrp j ≫ 1 with

μr ¼ ϵr are challenging due to the lack of materials with theneeded permittivity and permeability, especially for broad-band and high-frequency applications.

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2. Arrays of small resonant particles

The required electromagnetic properties of matchedabsorbing layers can be emulated by periodical arrays ofsmall resonant particles. The electromagnetic properties ofdense arrays (the period is smaller than the wavelength inthe surrounding medium) can be modeled by the gridimpedance Zg, which connects the averaged electric sur-face-current density Je and the averaged tangential electricfield in the array plane:

Etot ¼ ZgJe: ð69Þ

For perfect absorption, the required value of this parameteris Zg ¼ η0=2 [Eq. (54)]. In terms of the polarizability ofeach individual particle in the array, the grid impedancereads [69]

Zg ¼ −j Sω

�Re

�1

αee− βe

�þ j

�Im

1

αee− k306πϵ0

��:

ð70Þ

Here, αee is the electric polarizability of one inclusion infree space, βe is the interaction constant in the infinite array,and, as before, S is the unit-cell area. Apparently, in order toachieve perfect absorption, the imaginary part of Zg shouldvanish. Physically, this requirement means that we shouldwork at the resonance frequency of the particles in the array[the reactive coupling with the other inclusions is modeledby ReðβeÞ]. Equating the real part of the grid impedance toη0=2, we find the required loss parameter of the inclusions.Obviously, the imaginary part of the inverse polarizabilityshould satisfy

Im

�1

αee

�¼ η0

2

ω

Sþ k306πϵ0

: ð71Þ

The last term (proportional to k30) is the measure ofscattering loss of any dipole particle, and the first termtells how strong dissipation we need in each particle. Notethat the required absorption parameter is inversely propor-tional to the unit-cell area S. This result means that there isno need to tune the particle losses to a prescribed value, asthe overall losses can be adjusted varying the array period.Dual relations hold for the magnetic-current density andmagnetic fields, from which we find the required conditionon magnetic losses in each particle:

Im�

1

αmm

�¼ 1

2η0

ω

Sþ k306πμ0

: ð72Þ

Similarly to the case of electric dipoles, the last termmodels magnetic dipole scattering loss, and the first termtells how much absorption loss we need in each magneticdipole particle.

For example, assuming the Lorentz resonant model forthe particle polarizability αee

1

αee¼ ω2

0e − ω2 þ jωΓe

Aeþ j

k306πϵ0

ð73Þ

(the last term models the radiation damping; see, e.g.,Ref. [69]), we find the perfect absorption condition

Γe

Ae¼ η0

2

1

S: ð74Þ

From duality, we find the condition on the loss factor ofresonant magnetic dipoles sitting in the same unit cells asthe electric dipoles:

Γm

Am¼ 1

2η0

1

S: ð75Þ

The realization of a single array of resonant particlesworking as a perfect absorber is reported in Refs. [90,91].In that work, a periodic planar array of right- and left-handed chiral particles is used [see Fig. 5(a)]. An equalnumber of right- and left-handed helices makes the effec-tive chirality negligible, as is required for symmetric perfectabsorbers, and the shape of the spirals is chosen so thatthe electric and magnetic polarizabilities are balanced.Figures 5(b) and 5(c) show an illustration of the responseof this symmetric absorber (the design parameters are takenfrom Ref. [90]).Another design for symmetric absorbers realized as

an array of resonant particles is presented in Ref. [92](the authors of Ref. [92] actually used two-layer arrays,but using the same concept it can be done also with a

(a)

3 3.05 3.1 3.15 3.20

0.2

0.4

0.6

0.8

1

Frequency (GHz)

Abs

orpt

ion

θ=0°

θ=35°

θ=70°

(b)

3 3.05 3.1 3.15 3.20

0.2

0.4

0.6

0.8

1

Frequency (GHz)

Abs

orpt

ion

θ=0°

θ=35

θ=70

(c)

FIG. 5. (a) Topology of a symmetric absorber. Absorption atdifferent incidence angles θ for (b) TE- and (c) TM-polarizedplane-wave incidences.

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single layer of resonant inclusions). Yet another proposedtopology uses an array of dielectric-coated metalcylinders [93].In the following sections, we give examples of thin

asymmetric absorbers, with different response to wavesincident on the two sides of the layer. As shown above, thesestructures realize balanced bianisotropic (omega) layers.

B. Asymmetric perfect absorbers

Asymmetric perfectly absorbing structures exhibit per-fect absorption properties only for illumination from oneof the two sides of the layer. The reflection coefficient ofthe opposite side is different from zero, and, as explainedabove, the layer is a bianisotropic omega structure.Although the physical operational principle of all thinasymmetric absorbers is the same, and the induced electricand magnetic surface-current densities are the same, actualtopologies of the layer and the materials from which theabsorber is made can vary a lot. In the following, weclassify asymmetric perfect absorbers first into singlearrays of electrically small dipolar inclusions in a homo-geneous host (usually free space) and structures whichcontain a mirror or an interface (from ideally conductingsurfaces to moderately lossy dielectric half-spaces). Stillthere is a great variety of structures which contain a groundplane, and we classify them into structures which containan ideal or nearly ideal boundary (PEC or PMC plane) andstructures which have a finite-conductivity (or moderatelylossy) ground plane.

1. Arrays of resonant bianisotropic omega particles

As explained in Sec. III D, the polarizabilities (per unitcell or per unit area) of asymmetric perfectly absorbingsheets are the same as those of bianisotropic omegaparticles. Thus, it appears that the most natural realizationof such absorbers would be in the form of periodicaltwo-dimensional arrays of omega particles, whose collec-tive polarizabilities are tuned to the values given byEqs. (41)–(43). This scenario has been studied in detailin Ref. [67]. In that paper, Ω-shaped pieces of thinconducting wires are considered as possible bianisotropicomega inclusions, but it is shown that particles formed by asingle Ω-shaped piece of wire are not suitable for thisapplication. The reason for this limitation is that thedifferent polarizabilities of such particles cannot be con-trolled independently, as they are related to each other (allthe polarizabilities are defined by the same resonant currentmode; see details in Ref. [94]). As shown in Ref. [94], thislimitation can be lifted by using asymmetric particles orunit cells containing more than one conducting element. Anatural choice is to use a set of two metal patches ofdifferent dimensions or shapes located close to each other,effectively forming one bianisotropic inclusion.As an example, Fig. 6(a) shows an absorbing layer

possessing omega coupling. This layer is composed by two

very closely positioned arrays of different metal patches.The calculated absorptions for waves hitting different sidesof the layer are shown in Fig. 6(b). The results demonstratecompletely different behavior for waves coming fromdifferent sides.Topologically, the structures described in Refs. [22,46,

95,96] function as omega unit cells (two arrays of asym-metric metal patches), which means that these structurescan fully absorb power from one side but can havecontrollable reflection for the wave illuminating the oppo-site side (transmission must be symmetric due to reciproc-ity). On the other side, the structure described in Ref. [29]looks like an omega unit cell (a complex-shaped capacitivepatch next to a metal strip), but in fact the dimensions of thestrip array are chosen so that it behaves as a nearly perfectlyconducting plane (our simulations show that within theoperational frequency band the reflection coefficient of thestrip array is close to 0.9, and the reflection phase is about−170°). Thus, it is one of the many realizations of high-impedance surface absorbers, described in Sec. V B 4.There are other bianisotropic structures which were

proposed to work as absorbing layers. For example, puttinga symmetric dual-patch structure on top of a substrate, onecan break the symmetry and introduce bianisotropic cou-pling in the structure [31] (see Fig. 7). However, the layersdescribed in that paper are electrically thick. More aboutthe use of substrate-induced bianisotropy can be found inSec. V B 11. For most applications, illumination of onlyone side of the absorbing structure is relevant. Apparentlyfor this reason, the role of bianisotropic coupling andconsequently the possibilities to realize and control differ-ent responses for different sides have not been realizedearlier. In many papers on asymmetric bianisotropicabsorbers, e.g., Refs. [20,22,29,31,46,95,96], only absorp-tion of waves coming from one of the two sides of the layeris considered.Most of the known realizations of asymmetric perfect

absorbers contain a ground plane or an interface betweentwo different media, possessing high optical contrast, andwe describe them next.

Y

Z

X

Top and bottom views of the unit cell

(a) (b)

65 70 75 800

0.2

0.4

0.6

0.8

1

Frequency (THz)

Abs

orpt

ion

Absorption for − z0−directed wave

Absorption for + z0−directed wave

FIG. 6. A typical topology of an omega layer. (a) An exampleof an array of resonant particles possessing omega coupling.(b) Absorption for −z- and þz-directed waves.

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2. Asymmetric absorbers containing a ground plane:General considerations

Possible strategies of the realization of perfect absorberspositioned at an interface with a homogeneous medium canbe conveniently understood with the use of an equivalentcircuit. The input impedance and admittance of a half-spacefilled with an isotropic medium with the relative parametersϵs and μs read

Zs ¼ η0

ffiffiffiffiffiμsϵs

r¼ Rs þ jωLs; Ys ¼

1

Zs¼ Gs − jBs

ð76Þ

(writing in this form, we emphasize that the imaginary partof the surface impedance of metals or lossy dielectrics isinductive: ωLs > 0, B > 0). Asymmetric absorbers can berealized by positioning an electrically thin layer of ahomogeneous or composite material on the interface sur-face. The electromagnetic properties of such a layer can bealways modeled by a set of linear relations between thesurface-averaged tangential field components on its twosurfaces, in the form of a transmission matrix or impedance(admittance) matrix; see Sec. IV B 1. This modeling ispossible even for composite layers (for example, arraysof complex-shaped particles) as long as the thicknessof the homogeneous substrate is much larger than theperiod of the arrays inside the covering layer, because theeffects of evanescent fields in the vicinity of the layer canbe incorporated into linear relations between the funda-mental propagating plane-wave fields (e.g., Ref. [69]).Furthermore, for any reciprocal layer, isotropic in theabsorber plane, the impedance or admittance matrix canbe presented in the form of an equivalent T circuit (or Πcircuit) with effective scalar impedance and (or) admittancecomponents. Thus, any absorber formed by a material or

composite layer on an interface with another material canbe modeled by the equivalent circuit shown in Fig. 8. Thesymmetric T circuit shown in the picture corresponds tolayers without intrinsic bianisotropy (e.g., an isotropic layeron an interface). In the most general case, the two seriesimpedances may be different.For the special case when the absorbing layer is filled

with a homogeneous isotropic material (permittivity ϵ ¼ϵrϵ0 and permeability μ ¼ μrμ0), the equivalent parametersread [69]

Z ¼ j2η tan

�kd2

�≈ jωμd ¼ jη0μrk0d; ð77Þ

Y ¼ jηsinðkdÞ ≈ jωϵd ¼ j

η0ϵrk0d: ð78Þ

Here k0 ¼ ωffiffiffiffiffiffiffiffiffiϵ0μ0

p, k ¼ ffiffiffiffiffiffiffiffi

ϵrμrp

k0, η0 ¼ffiffiffiffiffiffiffiffiffiffiffiμ0=ϵ0

p, and the

approximation refers to the case of electrically thin layers(jkjd ≪ 1). The perfect-absorption conditions can be for-mulated in terms of matching: The parameters of the covershould be chosen so that the input impedance of the circuitwould be equal to the free-space wave impedance η0.Analyzing the equivalent circuit of Fig. 8, we see thatthe choice of an appropriate absorbing layer is criticallydependent on the value of the surface impedance of theunderlying half-space.If the real part of the surface impedance Zs is smaller than

η0 and the absorbing layer is electrically thin, it is obviousthat we need a lossy magnetic layer, because the total inputresistance must be higher than the real part of Zs. Thissituation is realized in the case of highly conductingmaterials (such as metals at microwave frequencies) usedto form the ground plane or mirror. In this case, the absorberbehaves as a series resonant circuit, formed byZs andZ. It isdifficult to find natural magnetic materials with the desiredproperties, and usually the way to realize this response is touse artificial magnetic layers. In the rest of the review,different ways to realize such a response are discussed. Theuse of dielectric layers as perfectly absorbing layers backed

FIG. 8. Circuit model of a material layer on an arbitrarysubstrate. The substrate is modeled by the surface impedanceZs, and the T circuit models a layer of an arbitrary thickness. Forperfect absorption, the input impedance (ratio of the “inputvoltage” Eþ and “input current” z0 ×Hþ) must be equal tothe free-space impedance η0.

Y

Z

X

FIG. 7. Realizing omega layers positioning a symmetric patcharray on top of a dielectric layer.

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by good conductors implies the use of layers of resonantthickness (the Dallenbach absorber, Sec. V B 8).If the real part of the surface impedance is larger than η0,

adding a series resistor Z will only make it larger. It isobvious that in this case perfect absorption is possible toachieve with a thin lossy dielectric layer, acting as a shuntimpedance with respect to Zs. In this case, the absorbingstructure forms a parallel resonant circuit, formed by Zsand Y. This situation is typically realized in the case ofoptical frequencies and metals as materials for the groundplane, when the negative-permittivity response of metalscan provide the necessary inductive response. This internalinductance is due to reactive magnetic-field energy storedinside the metal substrate (sometimes called kinetic induct-ance). The resonance is similar to the plasmonic resonanceat an interface of metal (negative real part of the permit-tivity) and a dielectric or composite layer (positive per-mittivity), although in this case there is no surface-waveexcitation. As discussed above, the resonance is necessaryto make the input impedance real, and the amount of lossesin the dielectric layer is chosen so that the value of the totalinput impedance is equal to η0. For a detailed description ofthese absorbers, see Sec. V B 9.The internal-inductance absorbers have a clear advantage

of simplicity in design and realizations: In principle, theabsorber can be formed just by a dielectric (semiconductor)sheet on a metal surface. It may be necessary to use somecomposites to realize the needed electric response, though.However, realizations of this type of absorbers are possibleonly when two conditions on the ground-plane surfaceadmittance Ys are satisfied. The first condition we have justdiscussed: The real part of the wave admittance of theground-plane material should be smaller than the free-space wave admittance: Gs < 1=η0. The second conditionrequires that the fields inside the ground plane decay fast(the imaginary part of

ffiffiffiffiϵs

pis sufficiently large). If this

condition is not met, the waves in the substrate decayslowly, and, although matched, this structure is not workingas a thin absorber. In practice, these restrictions limit theconcept of the internal-inductance absorber to infrared andoptical frequencies.Furthermore, if the imaginary part of the permittivity ϵs

is very large (metals at microwave frequencies, for exam-ple), the reactive part of the input impedance (the surfaceinductance Ls) is very small. In this case, the internal-impedance absorber scenario does not allow one to ensurepractically sufficient frequency bandwidth of the absorber.Indeed, this absorber is a parallel-type resonant structurewhere the inductance is due to magnetic field energystored in the ground plane (positive reactance) and thecapacitance is due to the electric field energy stored in thedielectric covering layer. As is known from the theory ofhigh-impedance surfaces [97], the relative bandwidth isproportional to

ffiffiffiffiffiffiffiffiffiffiffiLs=C

p, where C is the equivalent sheet

capacitance of the cover [ImðYÞ ¼ ωC]. Thus, if the

surface inductance is low (too large conductivity of thesubstrate), there is no practical possibility to create aninternal-inductance absorber using a thin high-capacitancecover (a high-permittivity dielectric layer or a capacitivearray). On the other hand, note that one can get bothrequired inductive and capacitive responses by using alossy high-permittivity material as an absorbing layer ofresonant thickness. Such absorbers can be also very thin ifthe refractive index of the lossy layer is sufficiently high.These devices are known as Dallenbach absorbers; seeSec. V B 8.Only in the two extreme cases of electric or magnetic

sheets, when the thickness is negligible with respect tothe free-space wavelength and small with respect to thewavelength inside the layer, can we model the layer as apurely electric or magnetic (capacitive or inductive) sheet.Generally, a finite-thickness material layer or a composite(metamaterial) layer stores both electric and magneticenergies, and both tangential electric and magnetic fieldschange across the layer, as is evident from the generalequivalent circuit model of any thin absorber, shown inFig. 8. In other words, both electric and magnetic responsesin the covering layer are generally present.In the following sections, we consider various realiza-

tions of asymmetric absorbers as homogeneous orcomposite layers on perfect and imperfect ground planes.We start from perfect absorbers realized as electrically thinlayers of conventional lossy materials on top of a PEC orPMC ground plane.

3. Thin layer of a homogeneous magneticmaterial on a PEC plane

One of the classical thin microwave absorbers is a layerof a lossy magnetic material on a PEC surface. The mainidea of this device comes from the fact that in the vicinity ofa PEC surface the tangential magnetic field is strong (rightat the PEC wall, it is twice as strong as the incidentmagnetic field). Thus, it appears natural to position here aslab of a lossy magnetic material to absorb the incidentpower. To find the required value of the permeability, wespecialize the theory of Sec. V B 2 for the case whenZs ¼ 0 (PEC ground plane) and Y ¼ 0 (magnetic materialwith jϵrj ≪ jμrj) and write the input impedance in theapproximation of an electrically thin layer (thin also ascompared with the wavelength inside the slab):

Zinp ¼ jη0

ffiffiffiffiffiμrϵr

rtanðk0d

ffiffiffiffiffiffiffiffiμrϵr

p Þ ≈ jη0μrk0d ð79Þ

(here d is the layer thickness). Equating the input imped-ance (79) to the free-space impedance η0, we find that thisstructure works as a perfect absorber if μr is imaginary and

μr ¼ − jk0d

: ð80Þ

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This result appears, for example, in Ref. [6]. To check ifthe assumption of a small electrical thickness holdswith this value of the permeability, we calculate jkjd ¼k0dj ffiffiffiffiffiffiffiffi

μrϵrp j ¼ ffiffiffiffiffiffiffi

k0dp

. Thus, this analysis can be used underthe condition

ffiffiffiffiffiffiffik0d

p≪ 1.

The volumetric magnetic-current density in the magneticlayer is proportional to the magnetic field inside the layerH: jm ¼ jωðμ − μ0ÞH. The magnetic field inside the slabvaries as cosðkdÞ, which we should approximate as aconstant, because we have kept only the first-term expan-sion term in (79). Furthermore, since there is no reflection,this field equals the incident magnetic field: H ¼ Hinc.Thus, the equivalent magnetic surface-current density reads

Jm ¼ jmd ¼ jωμ0ðμr − 1ÞHincd ≈ η0Hinc: ð81Þ

Here we have substituted the value of μr from Eq. (80) andneglected the real part inside the brackets. Actually, this isnot an approximation: For the reference plane at theabsorbing layer surface, the equivalent boundary conditionreads Jm ¼ jωμ0μrdHinc (Sec. 2.5 in Ref. [69]). Asexpected, this value is as required for perfect absorptionin thin layers [Eq. (21)].The electric current is excited on the PEC sheet, and the

exciting electric field is the sum of the incident field and theplane-wave electric field created by the magnetic-currentsheet [Eq. (81)]:

E ¼ Einc − Jm2

¼ Einc − η0Hinc

2¼ 1

2Einc: ð82Þ

This field is compensated by the field created by the electriccurrent on the PEC surface, to fulfill the PEC boundarycondition:

− η02Je ¼ − 1

2Einc; ð83Þ

which determines the value of the electric surface-currentdensity

Je ¼1

η0Einc: ð84Þ

This result is the same value as required by Eq. (20), and wesee that the two current sheets indeed form a Huygens pair.As it was explained in Sec. III, all thin asymmetric

absorbers are bianisotropic omega layers. Let us calculatethe polarizabilities (per unit area) of this structure and showthat it is a bianisotropic omega layer with the parametersrequired by the theory of Sec. III D. If we excite thisabsorber by a tangential uniform external electric field(while the external magnetic field is zero), the electric fieldwill directly excite the electric current on the PEC surface,but then this current sheet will generate a plane wave,whose magnetic field will induce magnetic currents in the

magnetic layer. To find the effective polarizabilities ofthe structure, we will utilize the same approach as inAppendix A, calculating the currents induced by uniformelectric and magnetic external fields. The result of thecalculations (given in Appendix B) shows that the inducedelectric (px) and magnetic (my) moments per unit areasatisfy Eq. (24) with the coefficients

¯̂̄αee ¼ α̂ee¯̄It ¼

1

η0

3

2jω¯̄It;

¯̂̄αmm ¼ α̂mm¯̄It ¼ η0

1

2jω¯̄It;

ð85Þ

¯̂̄αme ¼ ¯̂̄αem ¼ jΩ̂ ¯̄Jt ¼ − 1

2jω¯̄Jt: ð86Þ

This result fully agrees with the requirements given inSec. III D and with the result for high-impedance structuresin Eqs. (46)–(48) and Appendix A. This result shows thatthe balance conditions between the polarizabilities hold, asshould be in any thin asymmetric perfect absorber.The simplest equivalent circuit of this absorber is of T

type, as shown in Fig. 8, where the shunt admittance can beneglected (an open circuit). The value of the seriesimpedance is equal to the sheet impedance of a thinmagnetic layer (e.g., Sec. 2.5 in Ref. [69]):

Z ¼ jωμ0μrd: ð87Þ

Obviously, with the permeability value required for theperfect absorption operation (80), we have Z ¼ η0, and thematching condition is satisfied. If the ground plane is notideally conducting, the total absorption condition reads

Zinp ¼ Z þ η0

ffiffiffiffiffiμsϵs

r≈ jωμ0μrdþ η0

1þ jffiffiffi2

pffiffiffiffiffiffiffiffiωϵ0σ

r¼ η0:

ð88Þ

Here ϵs and μs are the relative parameters of the ground-plane material. The last approximate expression is for thecase of a good conductor with ϵs ≈ −jσ=ðωϵ0Þ, where σ isthe substrate conductivity.As any thin absorber, this is a resonant structure (the

input impedance is real in the perfect absorption regime).For a PEC ground plane, the resonance parameters aredetermined by the dependence of the permeability of themagnetic substrate μr on the frequency. In the general caseof an arbitrary substrate, we have a series resonance of thesurface impedance of the substrate (half-space or a morecomplex layered structure) and the sheet impedance of themagnetic covering layer.Practical implementations of this design are limited by

the properties of available materials, and there has beenconsiderable effort in the synthesis of magnetic materialswith high losses at microwave frequencies but with limitedsuccess, especially above 1 GHz. As an example, by using

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nanostructured multilayers described in Refs. [98] or [99],it appears possible to realize absorbers of the normalizedthickness k0d of the order of 1=100 at the frequencies up to3–4 GHz. An extensive recent review of various magneticmaterials for absorber applications can be found inRef. [100]. Realizations of this principle at more elevatedfrequencies require the use of artificial magnetic materials(metamaterials and metasurfaces), which we consider next.In fact, even for applications in microwave absorbersartificial magnetics may have advantages over naturalmagnetics; see a review paper [101]. However, the band-width of absorbers [15] and material-filled small antennas[102] is determined by the static (zero frequency) value ofthe permeability. For example, for an absorber formed byan arbitrary isotropic material slab on a PEC plane, thefundamental limitation on the absorption bandwidth can beformulated in terms of the minimum allowed absolute valueof the reflection coefficient R0 within an interval ofwavelengths from λmin to λmax as

j lnR0jðλmax − λminÞ ≤ 2π2μsd; ð89Þ

where d is the absorber thickness and μs is the static valueof the relative permeability of the absorbing material.This result clearly calls for the use of natural magneticmaterials in contrast to artificial magnetics (like split-ringcomposites).

4. Thin layer of an artificial magnetic at a PEC plane(high-impedance surface absorbers)

As we have just noted, it is difficult or even not possibleto find natural magnetic materials with the propertiesneeded for the realization of simple absorbers describedin the previous section, especially at elevated frequencies.This problem can be solved by using artificial magneticmaterials or structures. Usually, these layers are thin also interms of the number of magnetically polarizable inclusions(“metaatoms”) across the layer thickness. Most commonly,only a single layer of some particles is used. Because ofthis, the theory of thin homogeneous magnetic layers(Sec. V B 3) cannot be applied, because the model of anequivalent bulk medium characterized by its effectivepermeability implies volume averaging, and, thus, it cannotbe introduced for single layers of particles. The notions ofthe effective grid or sheet impedance, which imply onlysurface averaging, will be used instead.One possible solution to realize artificial magnetic

response is to use a high-impedance surface. The mostcommon topology of high-impedance surfaces is oftencalled the mushroom structure [103]. In this structure, ametal patch array is located on a thin dielectric substrate,backed by a metal ground plane; see Fig. 9. Most of thesedesigns are for the microwave frequency range where theground plane can be assumed to be perfectly conducting(PEC). The name “mushroom structure” comes from the

original topology [103], where each patch is connected tothe ground plane with a thin conducting via, forming a“mushroom.” However, if this structure is excited bynormally incident plane waves, there are no currents inthin vias, because the electric field is orthogonal to thewires. Thus, the vias are not needed and can be removed.The presence of vias is important for the operation atoblique incidence, where they become useful also inabsorber applications; see, e.g., Refs. [19,21]. Differentshapes of patches can be chosen for unit cells of mushroomarrays, but all realizations use basically the same principle.In this absorber, the necessary resonant response (recall

that the values of the Z-matrix parameters of perfectabsorbers are purely real) is organized as a parallel-circuitresonance of the capacitive grid impedance of the array ofdisconnected metal patches and the inductive impedance ofa short-circuited transmission line between the patch arrayand the PEC ground plane. At the position of the grid, theimpedance matching (perfect absorption) condition can bewritten as

Yg þ1

jη tan kd¼ 1

η0; ð90Þ

where Yg is the grid admittance, and k and η are the wavenumber and wave impedance, respectively, of the substratematerial. Analytical formulas to design arrays with theneeded grid admittance can be found in Refs. [104,105].The total electric currents on the PEC plane and the

averaged electric currents on the patch arrays and thedifferential current on these two surfaces form a Huygenspair thanks to bianisotropic omega coupling in the struc-ture. This property can be shown by deriving polarizabil-ities per unit area in incident electric and magnetic fields,which can be assumed to be uniform over the layerthickness. To do that, one can assume that the structure

Y

Z

X d

PEC

Capacitive grid

Spacer

FIG. 9. A typical mushroom structure forming a high-impedance surface. No vias connectors are used in view ofapplications for absorbing normally incident plane waves.

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is in the field of a standing plane wave [69] and find theinduced currents satisfying the boundary conditions onthe PEC plane and on the patch array. As a result, thepolarizabilities per unit area in the regime of perfectabsorption are found to be equal to

α̂coee¼3

j2ωη0; α̂crme¼ jΩ̂¼ α̂crem¼− 1

j2ω; α̂comm¼ η0

j2ω

ð91Þ

(details of the derivation are given in Appendix A). Itshould be noted that the polarizabilities derived for thethin metamaterial-based absorbers are defined in terms ofinduced moments per unit area, which are related to thepolarizabilities of particles in Eq. (24) as αij → αij=S.Considering this, one can simply see that the polarizabil-ities satisfy the required conditions (34) and (37) forabsorbing arrays of reciprocal inclusions. For a plane-waveexcitation at normal incidence, we can write for the induceddipole moments per unit area

p ¼ 3

j2ωη0E0 − 1

j2ωH0 ¼

1

jωη0E0;

m ¼ 1

j2ωE0 þ

η0j2ω

H0 ¼1

jωE0: ð92Þ

Obviously, this is a Huygens pair which satisfies the perfectabsorption requirements (23) and (36).Numerous particular realizations of thin absorbers of this

type have been reported, e.g., Refs. [19,21,29,30,53,63,66,106]. The majority of the known realizations are for themicrowave frequencies, and they use metals, most com-monly, copper, as the material for the ground plane and thecapacitive patch array. It is interesting to note that thecircuit-analogue absorber [4] has the same topology: apatch array over a metal ground plane, but in that case thethickness of the spacer is close to λ=4, so the circuit-analogue absorber is a modified Salisbury screen. Here, thereactive response of the array is used to enhance thefrequency bandwidth through double-resonance propertiesof the device. The same is true for an early suggestion(published in 1999) to position a patch array on a lossy,metal-backed dielectric substrate [107]. As an example ofasymmetric absorbers, Fig. 10 shows the absorption spec-trum for the mushroom absorber shown in Fig. 9 with thedesign parameters given in Ref. [21].Alternatively to the mushroom high-impedance surface

topology, an artificial magnetic layer can be realized asan array of small magnetically polarizable particles madeof conductors or high-permittivity dielectrics. For instance,small resonant spirals [108] or split rings (SRRs) [109] orresonant ferroelectric inclusions [110] can be used. Anexample of such absorber design, where an SRR array islocated on top of a ground plane covered by a thin lossydielectric, can be found in Refs. [111,112]. Actually, in this

design one can simply make the SRR array from a lossymetal, without any need for a lossy dielectric spacer.

5. External-inductance andinternal-inductance absorbers

Up to now, we assumed that in high-impedance surfaceabsorbers the ground plane behaves as a PEC surface. Letus next discuss the influence of finite conductivity (andother finite electromagnetic parameters) of the groundplane in these absorbers. This discussion is especiallyrelevant for applications at terahertz and higher frequencies.Referring to Fig. 8, imagine a situation where the groundplane is not ideally conductive (Zs ≠ 0). For metals, thisproperty means that it has a considerable inductivesurface impedance in addition to some finite resistance.Fields penetrate into the ground-plane medium, and thetotal inductive response of the absorber structure is bothdue to the magnetic flux in the space between the groundplane and the patch array and due to the flux inside theground plane itself. We can state that if the inductiveresponse due to the magnetic fields inside the groundplane is smaller than that due to the magnetic fields inthe spacer volume (inductance is mainly external withrespect to the ground plane), the absorber can beconsidered as a high-impedance-surface absorber, whoseoperational principle is the same as of the microwavemushroom absorbers. In the opposite situation when mostof the reactive magnetic field energy is stored inside theground-plane medium (internal inductance), we classifythese absorbers as internal-inductance absorbers. In thecase of highly conducting ground planes at microwavefrequencies, when the inductance of high-impedancesurface absorbers is mainly due to the flux betweenthe ground plane and the patch array (Sec. V B 4), thename external-inductance absorber is appropriate.To understand what the physical nature of the resonance

is and to which class a particular absorber belongs, one canconsider the generalization of the perfect absorption con-dition (90) for the case of nonideal ground planes:

2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

Frequency (GHz)

Abs

orpt

ion

θ=0°

θ=35°

θ=70°

(a)

2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

Frequency (GHz)

Abs

orpt

ion

θ=0°

θ=35°

θ=70°

(b)

FIG. 10. An illustration of the response of an asymmetricabsorber (mushroom absorber). Absorption at different incidenceangles θ for (a) TE- and (b) TM-polarized plane-wave incidences.

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Yg þ1þ j Zs

η tan kd

Zs þ jη tan kd¼ 1

η0: ð93Þ

Here we have replaced the input admittance of a short-circuited transmission line by that of a transmission lineloaded by impedance Zs, e.g., Ref. [88]. Assuming thatthe spacer is electrically thin (that is, j tan kdj < 1), we seethat if

jZsj ≪ jη tan kdj; ð94Þ

the above equation transits to Eq. (90), and we deal witha high-impedance surface (mushroom) absorber. Theinternal-inductance regime is realized when, in contrast,jZsj ≫ jη tan kdj and, in addition, j Zs

η tan kdj ≪ 1. In thiscase, the perfect absorption condition becomes

Yg þ1

Zs¼ 1

η0; ð95Þ

and we deal with an internal-inductance absorber.Most of recently reported terahertz and infrared absorb-

ers (e.g., of Refs. [7,31,48,113,114]) are based on thehigh-impedance surface principle and work as external-impedance absorbers, meaning that the inductive reactanceof the ground plane gives only a small addition to the maininductance due to the flux between the ground plane andthe patch array. To illustrate this fact, we can consider theequivalent circuit in Fig. 8 and estimate the relevant valuesof the reactive impedances for some typical known designs.As examples, we compare the structures reported in two

of the papers mentioned above: Refs. [31,48]. Figures 11(a)and 11(b) compare the inductive impedance of the groundplane and the inductive impedance due to the magneticflux in the spacer. As can be seen, the inductance providedby the metallic ground plane is considerably smaller thanthe one provided by the spacer, and the condition (94) issatisfied. This result clearly indicates that for the perfor-mance of these absorbers the plasmonic properties of themetal used as a ground plane are not essential, even thoughthe metal has negative permittivity at the working frequen-cies and fields penetrate into the ground-plane volume.

According to the above classification, these structuresare high-impedance surface (mushroom) absorbers, astheir operation is based on the use of the parallel-circuitresonance. The operation is defined by the same Eq. (90),with a correction due to the final conductivity of thesubstrate, which can be estimated by using Eq. (93). Thephysics behind the operation of absorbers presented inRefs. [7,113,114] is basically the same.On the other hand, we find some examples of the

structures which are closer to the internal-inductanceabsorber in its operational principle [64,115]. Figure 12shows the relevant parameter values for the absorberdescribed in Ref. [115]. One can conclude that the metallicground plane, in the frequency region of absorption, playsthe main role in providing the required inductance incomparison to the spacer. This conclusion means thatthe structure is working as an internal-inductance absorber.In general, the internal-inductance absorber regime isrealized for substrates with Reð ffiffiffiffi

ϵsp Þ < 1 if the capacitive

array (artificial dielectric layer) is positioned right on themetal surface, so that j tanðkdÞj is very small.

6. Thin resistive sheet at an artificial PMC plane

The duality of electromagnetics suggests that it ispossible to replace the PEC ground plane by a PMCground plane and the thin layer of a lossy magnetic materialby a thin layer of a lossy dielectric, to achieve the sameperfect-absorption performance. To the best of our knowl-edge, the first absorber of this type was introduced bySchelkunoff in 1934 [116]. In that paper, he explained thatto eliminate reflections from an open end of a coaxial cable(which emulates a magnetic wall) one can put a resistivesheet right at the open end of the cable. In the well-knownbook by Ramo and Whinnery, published in 1944, a

50 52 54

(a) (b)

56 58 600.03

0.035

0.04

0.045

0.05

0.055

0.06

Frequency (THz)

|Zs|/|η|

|tan(kd)|

20 40 60 80 1000

0.1

0.2

0.3

0.4

Frequency (THz)

|Zs|/|η|

|tan(kd)|

FIG. 11. Comparison between inductive impedances providedby the metallic ground plane and the spacer.

500 600 700 800 900 10000

2

4

6

8

10

12

14

Frequency (THz)

|Zs|/|η|

|tan(kd)|1/|tan(kd)|

FIG. 12. Comparison between the inductive impedanceprovided by the metallic ground plane and the spacer forthe structure presented in Ref. [115]. The internal-inductanceabsorber regime corresponds to jZsj=jηj ≫ j tanðkdÞj and, atthe same time, jZsj=jηj ≪ j1= tanðkdÞj.

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topology to create this kind of absorber by positioning aresistive sheet at the λ=4 distance from a PEC ground planeis presented [75]. At this distance, the PEC ground plane isseen as a magnetic wall. In May 1943, Salisbury filed apatent application on this device, and a patent was grantedin 1952 [2]. Obviously, this absorber has a considerableelectrical thickness, and, since there are no naturallyavailable magnetic conductors, this scenario of realizingthin absorbers is possible only with the use of artificialmagnetic conductors.The most common compact artificial magnetic conduc-

tor for microwave applications (more generally, HIS) isthe mushroom layer: a dense array of conducting patchespositioned close to a conducting ground plane [103]. Wehave already discussed its use as an effective lossymagnetic layer in Sec. V B 4. In the absorber type whichwe consider now, the mushroom layer as such is nearlylossless and used to emulate the PMC boundary. Thenecessary resonant response (recall that the values of theZ-matrix parameters of perfect absorbers are purely real) isorganized as a parallel-circuit resonance of the capacitivegrid impedance of the array of disconnected metal patchesand the inductive impedance of a short-circuited trans-mission line between the patch array and the PEC groundplane (see Fig. 13). The PMC response takes place if thefollowing resonance condition is satisfied:

Yg þ1

jη0 tan k0d¼ 0; ð96Þ

where Yg is the grid admittance and for simplicity weassume that the volume between the patch array and thePEC plane is free space. At this frequency, the inputadmittance of the lossless mushroom layer is zero, andits input impedance is infinite. Analytical formulas todesign arrays with the needed grid admittance can befound in Refs. [104,105].

Now, if we position a thin resistive sheet with the surfaceresistance equal to η0 on the surface of this artificialmagnetic wall, the absorber is matched and the incidentpower is fully absorbed in the resistive sheet. As we see, theonly difference with the previously considered lossy mush-room structure, which emulates a lossy magnetic layer, isthat here the absorption is concentrated only in the thinresistive sheet, while in the previous case it was distributedover the whole substrate volume.To the best of our knowledge, the use of artificial

magnetic conductors in thin absorbers was proposed in2002 [27] and then experimentally verified in Ref. [11].Various patch shapes have been proposed by manyresearchers, including space-filling curves [117]. Anothertopology to realize an artificial magnetic wall is to usean array of small spirals, split rings, or high-permittivityparticles as resonant magnetic inclusions (see also theprevious section). Using the same principle, one canposition a resistive sheet close to such an array and achievefull absorption of electromagnetic waves [28,118]. Anillustration of this topology is shown in Fig. 14.

7. Using graphene in thin absorbers

At terahertz frequencies, graphene sheets can be used asthe material for patch arrays of mushroom absorbers (arraysof graphene nanodisks are described in Ref. [119], andgraphene nanoribbons are used in Ref. [120]). The electro-magnetic properties of graphene are usually modeled by itscomplex surface conductivity, which is the same as thesheet or grid admittance Yg used in this review. Typicalfrequency dependence of the graphene sheet impedance,calculated by using the Kubo approximation (e.g.,Ref. [121]), is shown in Fig. 15. In this example, thegraphene sheet parameters are the same as in Ref. [119]:The chemical potential is 0.4 eV, the carrier lifetime is4 × 10−13 s for both intraband and interband transitions,

Y

Z

X d

Capacitive gridSpacer

PEC

Resistive sheet

FIG. 13. A lossy sheet on top of a high-impedance surface(lossless mushroom layer).

Y

Z

X

Resistive sheet

PMC grid

FIG. 14. A lossy dielectric layer in the vicinity of an artificialPMC surface realized as an array of resonant split rings.

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and the temperature is 300 K. We can observe that underthese conditions the graphene sheet behaves as a moder-ately lossy conductor with the loss parameter (the real partof the sheet impedance) quite suitable for absorber appli-cations. However, the significant inductive impedance ofgraphene needs to be compensated by some sheet capaci-tance, which can be provided by breaking the graphenesheets into patches [119] or strips [120] (if only onepolarization is of interest). Two scenarios are, in principle,possible. The capacitance per unit area (due to gaps) can betuned to resonate with the graphene sheet inductance, sothat the total sheet impedance becomes resistive and closeto the free-space wave impedance. In this scenario, thesubstrate thickness should be close to a quarter wavelength,and the absorber is working as a Salisbury screen.Alternatively, the total sheet reactance can be tuned tobe strongly capacitive (very small gaps between graphenepatches). In this scenario, the substrate thickness can besignificantly smaller, and the operation is similar to thehigh-impedance-surface absorbers. In the example given inRef. [119], the thickness is 1.12 μm, which is smaller thanthe quarter wavelength (1.6 μm) but not significantlysmaller. Unfortunately, in both scenarios, the large induc-tive reactance of the graphene sheet reduces the absorberbandwidth (because some reactive energy is stored in thisinductor; see Ref. [97]), and, thus, the use of metals interahertz absorbers appears to be preferable to graphene.Graphene layers can be also used to achieve tunability

of absorbers. For example, in Ref. [122], a double layerof graphene strips was embedded into a lossy dielectricsubstrate of a high-impedance-surface terahertz-frequencyabsorber (a capacitive array of golden patches on a metal-backed lossy dielectric substrate, Sec. V B 4). Modulatingthe graphene surface impedance by using voltage bias, theresonant frequency of the device can be changed.

8. Dallenbach absorbers

As we saw from the general considerations of absorbinglayers on PEC ground planes (Sec. V B 2), to realize a thinabsorber we need to use magnetic layers (natural orartificial). An absorbing layer made of a purely dielectricmaterial must have a considerable (in fact, resonant)electrical thickness. However, if the permittivity of thematerial is high, the layer thickness can be still small ascompared with the free-space wavelength. This absorber isone of the classical designs (see Fig. 16), and it is called theDallenbach absorber [1,5,123]. The requirements on thematerial parameters and thickness can be found by equatingthe input impedance of the layer (thickness d and relativepermittivity ϵr) on a PEC ground plane to the free-spacewave impedance:

ZD ¼ jη01ffiffiffiffiϵr

p tanðk0dffiffiffiffiϵr

p Þ ¼ η0: ð97Þ

The input impedance takes real values at the thicknessresonances (the lowest one, at the quarter-wave thickness).The loss parameter should be chosen so as to ensurematching with free space. In terms of the refractive index ofthe dielectric material n ¼ n0 − jn00, the conditions read

n0 ≈π

2k0d; n00 ≈

2

πð98Þ

(e.g., Sec. 12.3 in Ref. [5]).A detailed analysis of the physical mechanism behind

the operation of the Dallenbach absorber is presented inAppendix C, where it is shown that close to the resonantfrequency the expressions for the input impedance of aDallenbach absorber have the same form as those for alossy high-impedance (mushroom) layer. Furthermore, inthe same Appendix, it is shown that the induced electric andmagnetic moments in the Dallenbach type absorbers read

px ¼2jE0

ξωη0

½Fð2 − jk0dÞ − 1þ jk0d2�

ð1 − jk0dÞ;

my ¼2jE0

ξω


ð1 − jk0dÞð99Þ

(the notations are defined in the Appendix). These equa-tions show that the condition of the Huygens layermy ¼ η0px is fulfilled. The induced electric and magneticmoments are balanced, as is required by the general theoryof thin absorbers presented above. In practice, the mainchallenge is to find (or realize as a metamaterial) a dielectricwith the refraction index approximately satisfying Eq. (98)at the design frequency and in as wide a frequency range aspossible.

10 20 30 40 500

2000

4000

6000

8000

Frequency (THz)

Shee

t im

peda

nce

Z g

(O

hm)

Real part

Imaginary part

FIG. 15. Typical dependence of the graphene sheet surfaceimpedance on the frequency in the terahertz frequency band. TheMATLAB code implementing the Kubo model is used courtesy ofDr. I. Nefedov.

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9. Thin layers on imperfectly conducting surfaces:Internal-inductance perfect absorber

Let us next apply the general theory of absorbing layerson imperfect ground planes (Sec. V B 2) to the case whenthe ground-plane material is a plasmonic metal (thinabsorbers for infrared and visible frequencies). The equiv-alent circuits of these structures can be considered as dualwith respect to the equivalent circuit of thin magnetic-layerabsorbers on good conductors: Instead of a series reso-nance, here we will have a parallel-circuit resonance.Indeed, the surface impedance of a conducting half-spaceis inductive [the imaginary part of the surface admittanceYs is negative (76)]. Thus, it appears to be possible tocancel its reactance by capacitive reactance of a thindielectric layer positioned on the conductor surface. Theequivalent sheet admittance of a thin dielectric layer reads(Sec. 2.5 in Ref. [69])

Yg ¼jη0

ðϵr − 1Þk0d: ð100Þ

The appropriate equivalent circuit of this absorber is likeshown in Fig. 8, where the series impedance Z can beneglected (approximated by zero). The total admittance ofthe parallel connection is

Y inp ¼ Ys þ Yg; ð101Þ

and the perfect-absorption condition reads

ffiffiffiffiϵs

p þ jðϵr − 1Þk0d ¼ 1 ð102Þ

[the branch of the square root is defined by Reð ffiffi·

p Þ ≥ 0].Unfortunately, from here we see that for microwave

absorbers this scenario requires active covering layers.For good conductors and microwave frequencies ϵs≈−jσ=ðωϵ0Þ, where σ is the metal conductivity. Becausein this case σ ≫ ωϵ0, the real part of

ffiffiffiffiϵs

p ≫ 1, and, thus, itis not possible to reduce it to the free-space level by addingsome positive equivalent resistance of the covering layer.However, this scenario is realizable in the visible range,

where metals no longer behave as nearly perfect conduc-tors. Assuming the Drude model of plasmonic metals

ϵs ¼ 1 − ω2p

ωðω − jΓÞ ; ð103Þ

the perfect-absorption condition Eq. (102) can beexpressed as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2 − ω2

p − jωΓω2 − jωΓ

sþ jðϵr − 1Þk0d ¼ 1: ð104Þ

It appears that if jω2 − ω2pj < ω2, the real part of

ffiffiffiffiϵs

pis

smaller than unity, and it is possible to realize a perfectabsorber as a thin dielectric (semiconductor) cover. Asimilar design is presented in Ref. [124] without realizingthe possibility of perfect absorption using this design, andan experimental demonstration of nearly perfect absorp-tion according to this scenario was recently published inRef. [125].Using the experimental data on the permittivity of metals

found in Ref. [126], one can see that for silver groundplanes Reð ffiffiffiffi

ϵsp Þ < 1 holds in the frequency range from 113

to 945 THz and for gold from 130 to 612 THz. As anexample, Fig. 17 illustrates theoretical performance ofperfect internal-inductance absorbers realized by position-ing dielectric layers with the thickness equal to λ0=20 on agold substrate. The design was made for a number offrequencies within the allowed range by selecting thedielectric layer permittivity in accord with Eq. (102). Tomodel gold properties, the data from Ref. [126] are used.Notice that the bandwidth of these absorbers becomes

smaller at lower frequencies, as expected (surface induct-ance decreases, which leads to a smaller bandwidth of the

Metal Metal

Dielectric |>>1|

d<< d

R R

(a) (b)

Z

x

FIG. 16. (a) Geometry of a dielectric Dallenbach absorber.(b) An equivalent model as a lossy capacitive grid over the sameground plane.

0 500 1000 15000

0.2

0.4

0.6

0.8

1

Frequency (THz)

|R|2

Design−A, fr=139.1 THz

Design−B, fr=291.7 THz

Design−C, fr=449.4 THz

Design−D, fr=599.4 THz

FIG. 17. Reflection coefficients from perfect internal-inductance absorbers formed by thin dielectric layers on a goldsubstrate. Different curves correspond to different requiredpermittivities of the dielectric cover. Design A, ϵr ¼ 17.1368−j4.8309; design B, ϵr ¼ 12.2021 − j4.2609; design C,ϵr ¼ 8.5919 − j3.8204; design D, ϵr ¼ 5.6498 − j1.8296. Thedielectric layer thickness equals λ0=20 at the resonant frequencyof the corresponding design.

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parallel-circuit resonance). Note also that, while the elec-trical thickness of the dielectric cover is small with respectto the free-space wavelength, it is not necessarily smallcompared to the wavelength inside the dielectric. In fact, forthe designs at the lower end of the allowed frequency range,the optical thickness of the layer is close to the quarter-waveresonance (for design A the thickness is approximately0.21λ), which means that the operation mechanism in thiscase is closer to that of the Dallenbach absorber than to theinternal-inductance one. Furthermore, internal-inductanceabsorbers can be classified as thin absorbers only if the skindepth of the metal ground plane is small enough, so that thefields in the substrate decay at a subwavelength depth.Finally, since it is difficult or even not possible to finddielectric materials with the required permittivities (theabove illustrative designs assume dispersion-free dielectricproperties), some artificial dielectrics, usually in the form ofarrays of electrically polarizable inclusions, need to bedesigned. Note that large values of the permittivity in ourexamples correspond to semiconductors, but real semi-conductors are dispersive media and their dispersion maybe not favorable for satisfying Eq. (102) with so small athickness as d ¼ λ0=20.

10. Plasmonic absorbers

Perfect absorption at infrared and possibly optical rangecan be realized also by using arrays of resonant plasmonicnanoparticles, separated by a thin dielectric substrate from ametal ground plane. As discussed above, if the real part ofthe refractive index of the ground-plane metal is larger thanunity (which is the case of most metals up to the infraredrange), the structure which is put on top of a metal surfacemust possess some effective magnetic response (referringto Fig. 8, the series resistance Z is significant for absorberoperation). Because the surface impedance of the groundplane Zs is inductive, this structure should also providesome electric-polarization (capacitive) response in order tocompensate the inductive reactance of the interface with theground plane. Two distinct regimes are possible: The spacerand the plasmonic-particle array are far from the resonance,so that the reactive energy stored inside and in the nearvicinity of plasmonic nanoparticles is predominantly elec-tric. In this regime (realized in Refs. [7,37,113]), most ofthe inductance is due to the magnetic flux in the dielectricspacer, and the absorber’s operation principle is similar tothat of a high-impedance (mushroom-type) absorber. In theother possible regime, the array of plasmonic particles isclose to the resonance, in which case the reactive magneticenergy is mainly in the particle array. In these structures,most of the power is lost in the resonating nanoparticles,and the operational principle is more similar to theDallenbach absorber. This similarity means that the struc-ture on top of a reflecting ground plane provides the phaseshift equivalent to a quarter-wave gap between the interfaceand the ground plane. Depending on the operational

frequency and used materials, intermediate mixed regimesare possible. As discussed in the previous subsection, somemetals in the visible range have the real part of the surfaceadmittance smaller than the inverse of free-space imped-ance. In this case, the needed dielectric-sheet response ofthe absorbing cover given by Eq. (102) corresponds to asemitransparent material with realistic complex permittiv-ity. If a natural material with the needed properties cannotbe found, this response can be achieved for a dielectric-semiconductor composite or even emulated by a planararray of metal particles. The topology of such absorbers(realized in Refs. [114,115,127]) is the same as for the caseof resonant plasmonic arrays.

11. Metamaterial absorber based on the effect ofsubstrate-induced bianisotropy

We see that the presence of magnetic polarization inthin layers is a necessary condition for perfect absorption.In the examples above, we see a number of techniques used toensure the required magnetic response: the use of naturalmagneticmaterials, artificial magnetic conductors in the formof mushroom layers, resonant split rings, small spirals, anddielectric nanospheres in the magnetic Mie resonance. Herewe discuss another possibility, offered by the bianisotropicresponse of arrays of small particles on a transparentsubstrate (see Fig. 18). Clearly, bianisotropy results ineffective magnetic surface polarization generated by theincident electric fields. The physical mechanism under-lying this effect is called substrate-induced bianisotropy(SIB) [128].Consider a regular planar array of plasmonic nano-

particles located at a small distance d over a transparentsubstrate with the relative permittivity ϵr [128]. Let thebottom parts of these particles be curved (e.g., as spheres orspheroids). In the plasmon resonance band, the electricfield is concentrated within a hot spot which is partiallylocated inside the nanoparticle but partially in the substrate.Local polarization currents in these two parts of the hotspot have opposite directions, because the real part of thecomplex permittivity of plasmonic metals is negative andthat of the substrate is positive. If the absolute values ofthese permittivities are comparable to each other, thepolarization of the hot spot (which is locally dominatingover the polarization of the remainder of a unit cell) can beadequately presented as a pair of electric dipoles withcomparable amplitudes and opposite directions. The effec-tive distance between these two dipoles is close to the hotspot size. This phenomenon is nothing but generation of alocal magnetic moment linked to the hot spot. In otherwords, the array acquires bianisotropic omega coupling.In the presence of SIB the array of plasmonic nanoparticlesis fully equivalent to an array of omega particles [129].In Ref. [130], it was shown that the SIB effect may allowsatisfaction of the zero reflection condition (36), togetherwith the zero transmission condition (37). Since the

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magnetic polarizability of nanoparticles is negligible, formetamaterial absorbers operating due to the SIB effectone should assume α̂comm ¼ 0 in both Eqs. (36) and (37).These conditions can be satisfied simultaneously within theplasmon resonance band of the original nanosphere array ifϵr ≫ ϵu (where ϵu ≥ 1 is the relative permittivity of theupper half-space) and 0 < d ≪ D, and the array period psatisfies p ≪ λ. Practically, the structure from Ref. [130]may be realized of silver or gold nanospheres floating in aliquid over the surface of a semiconductor.

VI. SUMMARY AND OUTLOOK

Based on the analysis of the possible physical mecha-nisms of perfect absorption in optically thin structures,we can classify all known perfect and nearly perfect thinabsorbers into the following categories:(1) Symmetric absorbers (electric and magnetic resistive

sheets at the same plane)(a) A lossy layer of a material with ϵr ¼ μr(b) Arrays of resonant electric and magnetic dipoles

(spirals, dipoles and split rings, etc.)(2) Asymmetric absorbers (bianisotropic omega layers)

(a) Arrays of resonant omega particles(b) Lossy layers on ideal boundaries(i) Thin lossy magnetic layer (natural or artificial

magnetic, commonly in the form of a mush-room structure) on a PEC ground plane

(ii) Resistive sheet on an artificial magnetic con-ductor surface (e.g., lossless high-impedancestructure)

(iii) Resonant lossy dielectric layer on a PECground plane (the Dallenbach absorber)

(c) Layers on imperfect boundaries(i) Thin lossy dielectric layer (natural or artificial

dielectric) on a conductor surface (internal-impedance absorber)

(ii) Lossy artificial magnetic conductor (e.g.,mushroom structure with a lossy dielectricsubstrate)

Symmetric absorbers as lossy layers of materials withequal relative permittivity and permeability (Sec. VA 1),being conceptually very simple, are problematic due to verystrict requirements on the material parameters. Althoughconsiderable progress has been achieved in synthesizinglossy magnetic materials (e.g., Ref. [100]), there are onlylimited options and mainly at microwave frequencies.Symmetric perfect absorbers in the form of arrays ofresonant particles (Sec. VA 2) appeared only recently,and the only known experimental realization [91] is inthe microwave frequency range. Realizations in the opticalrange need electrically small resonant particles with bal-anced resonant electric and magnetic moments. Similarrequirements hold for particles (metaatoms) needed forrealizations of double-negative metamaterials, and consid-erable experience of many groups working in that fieldcan be useful in designing perfect absorbers of this class.Various topologies, such as core-shell nanoparticles ordouble patches of various shapes, have been proposedand studied in the regime of as low a dissipation loss aspossible. To approach the problem of perfect absorptionrealization, dissipation losses should be significant andcontrollable in the design and manufacturing.Interestingly, while there is a huge literature on high-

impedance surface (mushroom-type, Sec. V B 4) absorbersin the form of various periodical structures on top of aconducting surface, some of the simplest possible realiza-tions of perfect absorbers have not yet been sufficientlystudied. In particular, we mean internal-impedance perfectabsorbers: thin lossy dielectric or artificial-dielectric sheetson a plasmonic metal or another conductor with lowconductivity and high surface inductance, and we expectinteresting new developments in these areas.In the design of thin broadband absorbers, the funda-

mental limitation Eq. (89) suggests the use of naturalmagnetic materials with high static values of permeability.However, such materials are not practical in terahertzand optical devices. In this respect, a largely unexploredpossibility is making use of active and parametricallypumped layers, as a route to overcome the fundamentallimitations on the performance of passive absorbers.Reference [131] discusses the general conditions of zerobackscattering from small particles, showing interestingnew possibilities offered by active inclusions. The activeroute is especially promising in view of a possibility toovercome the frequency bandwidth limitations for allpassive absorbers. Some initial work on active non-Fosterabsorbers can be found inRefs. [18,132–134]. Alternatively,frequency-bandwidth limitations can be overcome by mak-ing the absorber’s parameters tunable by some externalfields (most commonly, electric bias or light illumination).This route is being actively explored especially for micro-wave absorbers; see the review paper Ref. [135].In this review, we discuss only perfect absorption of

normally incident waves. The angular stability of thin

(a) (b)

FIG. 18. (a) Horizontally isotropic grid of silver nanosphereslocated at a very small distance d from a semiconductor substrate.(b) An equivalent array of scatterers with both electric andmagnetic responses which are balanced at a certain frequency.The action of the substrate at this frequency is modeled by themagnetic polarizability.

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absorbers presents another challenge which needs furtherstudies. Here, we expect that the key to success will belearning how to engineer spatial dispersion in metasurfaces.The use of active elements can offer an alternative route.It is important that for perfect absorption at a fixed butoblique incidence angle there are more open possibilitiesthan for the normal-incidence case. An interesting examplecan be found in Ref. [136], where in the extremely thinabsorbing layer the equivalent surface electric and magneticcurrents are not balanced, and the necessary asymmetry inradiation of the induced currents is ensured by an infini-tesimally thin layer of induced electric polarization which isdirected normally to the absorbing surface.Another possibility to improve performance of thin

absorbers is to make use of amorphous arrangements ofabsorbing elements. Nearly all known approaches exploitperiodical arrays of various topologies (possible deviationsfrom periodicity are commonly considered as unavoidableimperfections in manufacturing). However, it appears thatproperly engineered randomness of metasurfaces may offerpossibilities to control and potentially improve absorptionefficiency. Recently, some initial studies of absorption inresonant amorphous metasurfaces were published, e.g.,Refs. [64,127,137,138], but more studies and a betterunderstanding of such structures is needed to enable optimaluse of randomness in absorber design. Research in this areais motivated also by technological advantages, since amor-phous metasurfaces can be manufactured by using effectiveand cheap bottom-up methods [138]. On the other hand, weshould note that, while the use of random structures mayhelp in controlling the absorption level and increasingabsorption bandwidth, this way it is not possible to realizeperfectly nonreflecting absorbers. Amorphous absorbersinevitably create some diffuse scattering in all directions,which cannot be eliminated by any matching structures.

VII. CONCLUSIONS

The general consideration of physical mechanisms ofperfect absorption brings us to a number of importantconclusions. First, all optically thin absorbers are based onthe same principle of an absorbing Huygens sheet formedby induced electric and magnetic surface currents. Second,absorbers which equally absorb waves coming from thetwo sides of the sheet are simple combinations of an electricresistive sheet and a magnetic resistive sheet (no bianiso-tropic coupling in the layer). Third, reciprocal absorberswhich absorb waves only from one side of the sheet (e.g.,absorbers on a perfectly conducting ground plane) are lossyand resonant bianisotropic omega layers. As is clear fromthe above wide overview of different topologies which canbe used to create a thin absorber, all of these absorbers arebasically following the same physical concept.Knowing the required response of thin absorbing sheets

allows the use of metamaterial and metasurface synthesismethods to design particular devices. In order to realize a

thin perfect absorber, one must ensure that the structureresponds both electrically and magnetically and the strengthof these responses is balanced at the working frequency tohave zero reflection. At the same time, these inducedcurrents should be strong enough to cancel the incidentwave in the forward direction and ensure zero transmission.Having a magnetic response for an absorber structure is anecessary condition for perfect absorption. All passive thinperfect absorbers are resonant structures, which fundamen-tally limits their operational frequency band.Themajority of the knowndesigns of perfect absorbers are

based on only numerical simulations and optimizations. Wehope that the general framework theory and the wide over-view of known approaches to realizing perfect absorption inthin layers which we present here will be useful for futureresearch in this field and for practical device development.

ACKNOWLEDGMENTS

This work was supported in part by the NokiaFoundation.

APPENDIX A: POLARIZABILITIES OFHIGH-IMPEDANCE SURFACE

PERFECT ABSORBERS

Here we provide detailed derivations of polarizabilitiesfor a typical thin absorber made of a capacitive gridmounted close to a metal sheet. Deriving induced electricand magnetic moments, it is shown that these layers act asthe Huygens’ layer. To find the polarizabilities, we need toexcite the absorber layer by the electric field and magneticfield which are approximately constant inside the layer[69]. We choose the exciting field in the form of a standingwave as

E ¼ ½E0e−jk0z − E0eþjk0z�x0 ¼ −j2E0 sinðk0zÞx0;

H ¼ ½H0e−jk0z þH0eþjk0z�y0 ¼ 2H0 cosðk0zÞy0: ðA1Þ

To study the excitation by external electric fields, weposition the layer so that the maximum of the electricfield distribution of the standing wave is located at themiddle point between the grid and the ground plane(Fig. 19), where sinðk0zÞ ¼ −1, for example, assumingk0z ¼ −π=2 (the normalized distance between the gridand the ground plane is k0d ≪ 1). Next, we write therelations between the averaged surface-current density onthe ground plane Jmet and on the array of patches Jg andthe total surface-averaged electric fields at the respectivetwo planes:

j2E0 cos

�k0d2

�− η0

2Jg − η0

2Jmete−jk0d ¼ ZgJg;

j2E0 cos

�k0d2

�− η0

2Jge−jk0d − η0

2Jmet ¼ 0: ðA2Þ

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Here Zg is the grid impedance of the patch array. The valueof the grid impedance we choose so that the absorbermatches to the free space. Using the perfect-absorptioncondition (90), we find the required grid impedance

Zg ¼ jη0

�sin k0d

j sin k0d − cos k0d

�: ðA3Þ

Using Eq. (A2), the currents on the patch grid and on themetal surface can be found:

Jmet ¼ j4

η0E0 cos

k0d2

− Jge−jk0d;

Jg ¼−2E0e−jðk0d=2Þ sin k0djη0e−jk0d sin k0dþ Zg

: ðA4Þ

Substituting Zg from (A3), we can write the surface-currentdensity at the patch array as

Jg ¼−2E0e−jðk0d=2Þjη0ð−j2 sin k0dÞ ¼

−E0e−jðk0d=2Þη0 sin k0d

: ðA5Þ

The induced electric moment per unit area of the structure,by definition [139], reads

px ¼1

jωðJmet þ JgÞ: ðA6Þ

Substituting Eqs. (A4)–(A6), we find

px ¼1

jω

�j4

η0E0 − E0e−jðk0d=2Þð1 − e−jk0dÞ

η0 sin k0d

�: ðA7Þ

Assuming that the layer is optically thin (k0d ≪ 1), theabove result simplifies to

px ¼3

ωη0E0: ðA8Þ

Dividing the induced electric moment by the amplitudeof the external electric field j2E0 in Eq. (A1), we find theeffective electric polarizability per unit area:

α̂coee ¼px

j2E0

¼ 3

j2ωη0: ðA9Þ

By definition [139], the induced magnetic moment perunit area can also be written as

my ¼ − dμ02

ðJmet − JgÞ: ðA10Þ

Using Eqs. (A4), (A5), and (A10), we find the inducedmagnetic moment as

my ¼ −dμ02

�j4

η0E0 þ

E0ejðk0d=2Þð1þ e−jk0dÞη0 sin k0d

�: ðA11Þ

The assumption of an optically thin layer simplifies thisrelation to

my ¼ −E0

ω: ðA12Þ

Next, dividing the magnetic moment by the amplitudeof the exciting electric field in Eq. (A1), we cancalculate the magnetoelectric polarizability (omegacoupling coefficient):

α̂crme ¼my

j2E0

¼ −1j2ω

: ðA13Þ

In the second step, we derive the effective magnetic andelectromagnetic polarizabilities by considering excitationof the same layer by external magnetic fields. To do that, weposition the layer inside the standing wave Eq. (A1) so thatthe maximum of the magnetic field is in between the arrayof patches and the ground plane, as shown in Fig. 19 (e.g.,at k0z ¼ 0). Similarly to the above, writing the relationsbetween the induced averaged surface currents and theelectric fields in the plane of the patches and the groundplane, we get

−j2E0 sin

�k0d2

�− η0

2Jg − η0

2Jmete−jk0d ¼ ZgJg;

j2E0 sin

�k0d2

�− η0

2Jge−jk0d − η0

2Jmet ¼ 0: ðA14Þ

From these equations we find the induced current densitieson the metal surface and on the grid array:

Electric field

Magnetic field

Second case

First case

FIG. 19. Geometry of the problem. The gray plates show two testpositions of the absorbing structure in the field of a standing wave.

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Jmet ¼ j4

η0E0 sin

k0d2

− Jge−jk0d;

Jg ¼−j2E0e−jðk0d=2Þ sin k0djη0e−jk0d sin k0dþ Zg

: ðA15Þ

By using Eqs. (A3) and (A15), the patch-array current canbe written as

Jg ¼−j2E0e−jðk0d=2Þ sin k0d

jη0e−jk0d sin k0dþ jη0ð sin k0dj sin k0d−cos k0dÞ

; ðA16Þ

which for thin layers simplify to the following expression:

Jg ¼−jE0e−jðk0d=2Þη0 sin k0d

: ðA17Þ

The electric moment per unit area is calculated by usingEq. (A6):

px ¼1

jω

�j4

η0E0 sin

k0d2

− jE0e−jðk0d=2Þð1 − e−jk0dÞη0 sin k0d

�:

ðA18ÞAssuming an optically small thickness for the structure,we get

px ¼H0

jω: ðA19Þ

The coupling coefficient can be found by dividing theinduced electric moment by the amplitude of the externalmagnetic field in Eq. (A1):

α̂crem ¼ px

−2H0

¼ −1j2ω

: ðA20Þ

By using Eqs. (A4), (A5), and (A10), the magnetic momentcan be written as

my ¼ − dμ02

�j4

η0E0 sin

k0d2

þ jE0e−jðk0d=2Þð1þ e−jk0dÞη0 sin k0d

�;

ðA21Þ

and within the same assumption of optically smallthickness,

my ¼η0H0

jω: ðA22Þ

The magnetic polarizability can be determined by dividingthe induced magnetic moment by the external magneticfield in Eq. (A1):

α̂comm ¼ my

2H0

¼ η0j2ω

: ðA23Þ

Finally, the collective polarizabilities per unit area of thinhigh-impedance surface absorbers can be written as

α̂coee¼3

j2ωη0; α̂crme¼ jΩ̂¼ α̂crem¼− 1

j2ω; α̂comm¼ η0

j2ω:

ðA24Þ

These expressions show that there is bianisotropic omegacoupling in thin metasurface-based absorbers. It should benoted that the above polarizabilities are defined interms of induced moments per unit area, which are relatedto the polarizabilities of particles in Eq. (24) as αij → αij=S.Considering this, one can simply see that these polar-izabilities satisfy the required perfect-absorption conditions(34) and (37) written for absorbing using an array ofindividual reciprocal inclusions.This equivalence can be expressed also in terms of

induced electric and magnetic surface currents. Assuming aplane wave traveling in the z0 direction, the induced electricand magnetic moments in thin-layer absorbers can bewritten as

px ¼ α̂coeeEx − α̂crmeHy;

my ¼ α̂crmeEx þ α̂commHy; ðA25Þ

where Ex ¼ E0 and Hy ¼ E0=η0. Finally, substituting thepolarizabilities, we find

px ¼3

j2ωη0E0 − 1

j2ωH0 ¼

1

jωη0E0;

my ¼1

j2ωE0 þ

η0j2ω

H0 ¼1

jωE0: ðA26Þ

We can conclude that these form a Huygens’s pair and satisfythe required perfect-absorption conditions (23) and (36).

APPENDIX B: POLARIZABILITIES OF A THINMAGNETIC LAYER AT A PEC PLANE

Here we give details of derivations of polarizabilities(per unit area) of an asymmetric perfect absorber formed bya lossy magnetic-material layer at an ideally conductingmirror (Sec. V B 3). The geometry of the structure isillustrated by Fig. 20, and the configuration of the probefields is the same as shown in Fig. 19.Let us suppose that we excite this structure only by a

uniform electric field E ¼ E0x. In this scenario, theinduced electric (px) and magnetic (my) moments per unitarea can be written as

px ¼ α̂eeE0; my ¼ α̂meE0: ðB1Þ

The boundary condition on the PEC surface reads

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η02Jex þ

1

2Jmy ¼ E0: ðB2Þ

The magnetic surface-current density is related to the totalmagnetic field induced in the magnetic sheet as in Eq. (81).The total magnetic field is created by the induced currentsonly (the external magnetic field is zero), which allows usto write

Jmy ¼ η0Htot ¼ η0

�− 1

2Jex − 1

2η0Jmy

�: ðB3Þ

This result defines the relation between the induced electricand magnetic surface-current densities:

Jmy ¼ − η03Jex: ðB4Þ

Next, by using Eqs. (B2) and (B4), the induced averagedelectric- and magnetic-current densities per unit area of thestructure can be found:

Jex ¼3

2η0E0; Jmy ¼ − 1

2E0; ðB5Þ

and the respective induced surface-averaged electric andmagnetic moments read

px¼Jexjω

¼ 3

j2ωη0E0; my¼

Jmyjω

¼− 1

j2ωE0: ðB6Þ

Now we can find the electric and magnetic polarizabilitiessimply by dividing the moments by the amplitude of theincident field:

η0α̂ee ¼3

j2ω; α̂me ¼ − 1

j2ω: ðB7Þ

Similarly, we can excite the structure with a uniformmagnetic field while the external electric field is set to bezero at the position of the structure and find the respectivepolarizabilities, writing

px ¼ −α̂emH0; my ¼ α̂mmH0: ðB8Þ

To satisfy the boundary condition at the PEC surface, thefields created by the two induced current sheets must canceleach other:

Jmy ¼ η0Jex: ðB9ÞInstead of Eq. (B3), we have (the external magnetic field isnot zero in this excitation scenario)

Jmy ¼ η0Htot ¼ η0

�− 1

2Jex − 1

2η0Jmy þH0

�: ðB10Þ

By using Eqs. (B9) and (B10), the induced electric andmagnetic currents and the respective polarizabilities arefound:

Jex ¼1

2H0; Jmy ¼ η0

2H0; ðB11Þ

α̂em ¼ − 1

j2ω;

1

η0α̂mm ¼ 1

j2ω: ðB12Þ

Thus, the effective polarizabilities per unit area of thestructure read

η0α̂ee¼3

j2ω; α̂me¼ α̂em¼− 1

j2ω;

1

η0α̂mm¼ 1

j2ω;

ðB13Þwhich are the required polarizabilities for perfect absorp-tion in all thin asymmetric absorbers with a reflector at theback [Eqs. (46)–(48)].

APPENDIX C: DALLENBACH ABSORBERS ASBIANISOTROPIC HUYGENS LAYERS

Here we show that close to its operational frequency athin Dallenbach absorber response is equivalent to that of alossy capacitive grid separated by a free-space gap fromthe ground plane (high-impedance surface absorber). Thisequivalence is illustrated by Fig. 16. For the proof, it isimportant that the gap in Fig. 16(b) is optically thin,whereas the dielectric in Fig. 16(a) has the relativepermittivity ϵr of a large absolute value which makes thegap rather optically substantial in terms of the wavelengthin the dielectric medium (quarter-wavelength thicknesscorresponds to the thinnest Dallenbach absorber). We showthat the effect of the high complex permittivity slab canbe replaced by the action of a lossy grid whose reactance inthe vicinity of the operation frequency is capacitive. Inother words, we show the equivalence of the fundamentaloperation principles of the Dallenbach and the mushroom-type absorbers.First, let us show the equivalence of their surface

impedances (admittances). The surface impedance of thedielectric Dallenbach absorber is equal to (see, e.g., Ref. [4])

Y

Z

X

d Magnetic layer

PEC

FIG. 20. Magnetic layer on a PEC ground plane.

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ZD ¼ jη0tan kdffiffiffiffi

ϵrp ; ðC1Þ

where d is the dielectric layer thickness and k is the wavenumber in the dielectric k ¼ k0 − jα ¼ k0

ffiffiffiffiϵr

p. The com-

plex permittivity of the layer can be expressed throughthe loss tangent ϵr ¼ ϵ0r − jϵ00r ¼ ϵ0rð1 − j tan δÞ. The thin-nest Dallenbach absorber is described by the followingequations [123]:

ϵ0r ¼ cos δcoshðΘsÞ − coshðΘÞcoshðΘsÞ þ coshðΘÞ ; Θ ¼ π

1 − s2; ðC2Þ

where it is denoted Θ ¼ 2k0d and s ¼ tan ðδ=2Þ. Theseequations in the absence of magnetic properties of thedielectric have a solution within the region s < 0.25, whenthe second equation (C2) practically results in the well-known condition of the quarter-wavelength thickness2k0d ≈ π. Since jϵrj ≫ 1, the smallness of dielectric lossesand the quarter-wavelength restriction for d imply theinequality αd ≪ 1. By using the standard formula

tanðx − jyÞ ¼ sinð2xÞ − j sinhð2yÞcosð2xÞ þ coshð2yÞ ðC3Þ

and the quarter-wavelength condition for d, the surfaceadmittance of a thin dielectric Dallenbach absorber[Eq. (C1)] can be approximately written in the form:

YD ≡ 1

ZD≈½ðπ=2 − jαdÞðcoshð2αdÞ − 1�

η0k0d sinhð2αdÞ: ðC4Þ

Using the condition αd ≪ 1, we can write coshð2αdÞ −1 ≈ 2ðαdÞ2 and sinhð2αdÞ ≈ 2αd, reducing Eq. (C4) to

YD ≈πα

2k0η0− j

α2dk0η0

: ðC5Þ

Under the condition α ≈ ð2k0=πÞ, which can be writtenalso as ϵ00r ≈ ð4=πÞ or as δ ≈ ð4=πϵ0rÞ, we obtain the equationof the approximate impedance matching on top of theabsorbing surface:

YD ¼ 1

ZD≈

1

η0ð1 − jξÞ; ðC6Þ

where it is denoted ξ ¼ ð4k0d=πÞ < 1. This known resultcorresponds to the thinnest Dallenbach absorber andpractically requires ϵ0r > 15–16 so that the absorptioncoefficient may exceed 0.9 [123]. From Eq. (C6), it followsthat the frequency of the maximal absorption is redshiftedwith respect to the resonance of the surface impedance[at this resonance ZD ¼ ReðZDÞ ≫ η0].The surface admittance Yma of the effective mushroom

absorber depicted in Fig. 16(b) is the sum of the grid

admittance Yg of a presumably capacitive grid and the inputadmittance of the metal-backed gap:

Yma ¼ Yg þ1

jη0 tanðk0dÞ≈ Yg þ

1

jη0k0d: ðC7Þ

Equating admittances (C6) and (C7), we obtain

Yg ≈1

η0

�1þ j

k0d− jξ

�: ðC8Þ

In the reactive part of the grid admittance, one can neglectthe small term ξ and present Eq. (C8) in the form

Yg ¼ Gþ jωC; G ≈1

η0; C ≈

2

ωη0π: ðC9Þ

The frequency dispersion of the effective capacitance C inthe vicinity of the absorption frequency does not violateany physical requirements. What is really important is thepositive sign of the imaginary part of Yg that indicatesthe capacitive response of the equivalent grid, whereas thepositive sign of G corresponds to absorption.We see that the surface impedance of the Dallenbach

absorber is indeed equivalent to that of a mushroomabsorber with lossy and dispersive patches. The schemeof a mushroom absorber with lossy (and consequently moreor less dispersive) patches was first published probably inRef. [27] and later theoretically and experimentally devel-oped in Refs. [11,19,106,140], and many others. For theequivalent scheme of the Dallenbach absorber it is notessential that the gap in Fig. 16(b) is filled with free spaceand that it has the same thickness d as the actual absorber.In our model, we can fill the gap of the equivalent absorberwith a low-permittivity dielectric layer and simultaneouslyreduce its equivalent height that will slightly change theeffective grid capacitance compared to expressions (C9).However, the equivalence of the surface impedances

even within a nonzero range of frequencies around that ofthe maximal absorption does not guarantee the equivalenceof the electromagnetic responses of the Dallenbach andmushroom absorbers. To prove that the Dallenbachabsorber is fully within our general scheme of electromag-netic absorption in thin structures, we will derive thecondition of the total absorbance in terms of balancedelectric p and magnetic m moments per unit area, like itwas done in Appendixes A and B for other thin absorbers.To do that, we locate the coordinate origin at the central

plane of the absorber as is shown in Fig. 16(a). Then theground plane is at z ¼ −d=2 and the upper surface is atz ¼ d=2, and the effective moments p and m refer to theplane z ¼ 0. Let the electric field be polarized along thex axis and the magnetic field along y. Let us show thatthe Huygens’ scatterer condition is satisfied in the regimeof full absorption, i.e., that for the normal plane-wave

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incidence we have my ¼ η0px. Also, as was done inAppendix A, we combine this proof with the demonstrationof the effective bianisotropy of the absorber. In other words,we show that both mushroom and Dallenbach absorbersturn out to be equivalent to an array of omega particles,which means the equivalence of their electromagneticresponse to a plane wave.For this proof, we derive the effective polarizabilities

defined with respect to the local fields: αee ¼ px=Elocx ,

αmm ¼ my=Hlocy , αem ¼ px=Hloc

y ¼ −jΩ, and αme ¼my=Eloc

x ¼ jΩ. Here Ω is the magnetoelectric coefficientof an effective omega particle modeling the unit area of theDallenbach absorber. To model the local fields, we againuse the standing-wave approach of Appendix A, Fig. 19.For the electric excitation of the Dallenbach absorbershown in Fig. 16(a), the external fields in our model areequal to

Eextx ðzÞ ¼ 2E0 cosðk0zÞ; Hext

y ðzÞ ¼ 2E0 sinðk0zÞ=η0;ðC10Þ

i.e., the local fields are as follows:

Elocx ≡ Eext

x ð0Þ ¼ 2E0; Hlocy ≡Hext

y ð0Þ ¼ 0: ðC11Þ

For the magnetic excitation, the external fields are equal to

Hexty ðzÞ ¼ 2E0 cosðk0zÞ=η0; Eext

x ðzÞ ¼ 2E0 sinðk0zÞ;ðC12Þ

whereas the local fields are as follows:

Hlocy ≡Hext

y ð0Þ ¼ 2E0=η0; Elocx ≡ Eext

x ð0Þ ¼ 0:

ðC13Þ

First, consider the electric excitation described byEq. (C10). Let EPðzÞ ¼ EPðzÞx0 and HPðzÞ ¼ HPðzÞy0,respectively, denote the electric and magnetic fields pro-duced by the bulk polarization PðzÞ ¼ PðzÞx0 of thedielectric layer. Denoting Jmet ¼ Jmety0 the amplitude ofthe surface current induced on the metal plane, we canwrite the boundary condition at the upper interface of theabsorber:

2E0 cos

�k0d2

�þ EP

�d2

�− η0Jmet

2e−jk0d

¼ ZD2E0 sinðk0d2 Þ þHPðd2Þ − Jmet

2e−jk0d

η0: ðC14Þ

Here the left-hand part of Eq. (C14) is the total electric fieldE ¼ Ex0 on top of the structure, at z ¼ d=2. The right-hand side is the product of the surface impedance of theDallenbach absorber by the total magnetic field H ¼ Hy0

on top of it. The total electric field comprises even and oddparts and inside the structure can be searched in the form

EðzÞ ¼ A sin kzþ B cos kz: ðC15Þ

Coefficients A and B are related by the PEC boundarycondition at z ¼ −d=2:

−A sin

�kd2

�þ B cos

�kd2

�¼ 0: ðC16Þ

In the absorption regime kd ¼ π=2 − jαd, where αd ≪ 1.In these derivations we adopt an approximation kd ≈ π=2.Then Eq. (C16) results in B ¼ A.In Eq. (C14), the magnetic field HP is equal

HP ¼ EP=η0, and it is possible to express EPðd=2Þ throughthe unknown constant A. To do that, we split the dielectriclayer d to elementary layers dz and present the field EP asthe integral of partial fields created by elementary currentsheets dJPðzÞ ¼ jωPðzÞdz, where PðzÞ ¼ ϵ0ðϵr − 1ÞEðzÞ:

EP

�� d2

�¼ jk0ðϵr − 1Þ

2

Zd=2

−d=2EðzÞe−jk0½ðd=2Þ∓z�dz:

ðC17Þ

Substitution of Eq. (C15) with B ¼ A yields Eq. (C17) to

EP

�� d2

�¼ jk0Aðϵr − 1Þ

2

×Z

d=2

−d=2e−jk0½ðd=2Þ∓z�ðsin kzþ cos kzÞdz:

ðC18Þ

The value EPð�d=2Þ after integration of Eq. (C18) shouldbe substituted into Eq. (C14). This substitution gives us thefirst equation relating constants A and Jmet. The secondequation is the equivalence of Eðd=2Þ corresponding to theleft-hand side of Eq. (C14) and Eðd=2Þ corresponding tothe right-hand side of Eq. (C15):

2E0 cos

�k0d2

�þ EP

�d2

�− η0Jmet

2e−jk0d

¼ A

�sin

�kd2

�þ cos

�kd2

��: ðC19Þ

To calculate the integral in Eq. (C18), we used thecondition of the optically thin layer k0d ≪ 1:

e−jk0½ðd=2Þ∓z� ≈ 1 − jk0

�d2∓ z

�; if jzj ≤ d

2:

Then Eq. (C18) can be reduced to the following knownintegrals:

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Z �sin x

cos x

�dx ¼

�− cos x

sin x;

Zx

�sin x

cos x

�dx ¼

�sin x

cos x

�∓ x ×

�cos x

sin x: ðC20Þ

Finally, we obtain for the integral in Eq. (C18):

Zd=2

−d=2e−jk0½ðd=2Þ∓z�ðsin kzþ cos kzÞdz

≈2jk0dk

cos

�kd2

�þ 2ð1 ∓ jk0dÞ

ksin

�kd2

�: ðC21Þ

Substituting Eqs. (C18) and (C21) into the left-handside of Eq. (C14) and using approximations expð−jk0dÞ ≈1 − jk0d and cosðk0d=2Þ ≈ 1, we obtain

E

�d2

�¼ 2E0 þ

jk0Aðϵr − 1Þk

×

�jk0d sin

kd2þ ð1 − jk0dÞ cos

kd2

�

− Jmetη02

ð1 − jk0dÞ: ðC22Þ

The expression for Hðd=2Þ in the right-hand side ofEq. (C14) being multiplied by η0 differs from Eq. (C22)only by the first term, expressing the external magneticfield:

η0H

�d2

�¼ k0dE0 þ

jk0Aðϵr − 1Þk

×

�jk0d sin

kd2þ ð1 − jk0dÞ cos

kd2

�

þ Jmetη02

ð1 − jk0dÞ: ðC23Þ

Using Eqs. (C22) and (C23) and taking into account that inthe absorption regime cosðkd=2Þ ≈ sinðkd=2Þ ≈ 1=

ffiffiffi2

p, we

can rewrite Eq. (C14) after simple algebra in the form

Jmetη02

ð1 − jk0dÞ þffiffiffi2

pk0dðϵr − 1Þ

πA

¼�1 − k0dZD

η0

�2E0

1 − ZDη0

: ðC24Þ

The second condition—Eq. (C19)—after the samesubstitutions takes the form

2E0 − Jmetη02

ð1 − jk0dÞ

−ffiffiffi2

pk0dðϵr − 1Þ

πA½k0d − jð1 − jk0dÞ� ¼

ffiffiffi2

pA:

ðC25Þ

The solution for Jmet can be written in the form

Jmet ¼ΨA; Ψ¼ 2ffiffiffi2

p ðF− 1Þη0ð1− jk0dÞ

; F ¼ k0dðϵr − 1Þπ

;

ðC26Þ

and for A we obtain

A ¼ ΦE0; Φ ¼ffiffiffi2

p ð1 − k0dZDη0

Þð1 − ZD

η0Þ : ðC27Þ

Following to Eq. (C6), in the vicinity of the frequencyof total absorption 1 − ðZD=η0Þ ≈ −jξ, where ξ ≪ 1.Therefore,

Φ ≈ffiffiffi2

pj½1 − k0dð1 − jξÞ�

ξ: ðC28Þ

Notice that the calculation of EPð−d=2Þ using Eq. (C18)allowed a useful check of our derivations. We have checkedthat the equation

2E0 cos

�k0d2

�þ EP

�− d2

�− Jmet

η02¼ 0 ðC29Þ

is identically satisfied with Eqs. (C26) and (C28).After finding the two relevant constants A and Jmet, we

can determine the effective polarizabilities referring to theunit surface area. From the general definitions of electricand magnetic dipole moments, it follows that

px ¼Z

d=2

−d=2PðzÞdzþ Jmet

jω;

my ¼jωμ02

Zd=2

−d=2zPðzÞdz − μ0dJmet

2: ðC30Þ

Substituting formula PðzÞ ¼ ϵ0ðϵr − 1ÞAðsin kzþ cos kzÞinto Eq. (C30), we again come to known integrals (C20)and using the quarter-wavelength condition kd ≈ π=2obtain

px ≈2

ffiffiffi2

pE0A

ωη0

�F − 1 − F

1 − jk0d

�ðC31Þ

and

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my ≈−jk0d

ffiffiffi2

pE0A

ω

�F þ 1 − F

1 − jk0d

�: ðC32Þ

In accordance to the definitions of the effective polar-izabilities and using Eqs. (C11) and (C13), we can rewriteEqs. (C31) and (C32) as follows:

αee ≈2j½1 − k0dð1 − jξÞ�

ξωη0

�F − 1 − F

1 − jk0d

�; ðC33Þ

αme ≈k0d½1 − k0dð1 − jξÞ�

ξω

�F þ 1 − F

1 − jk0d

�: ðC34Þ

Since in our derivations we neglected the terms of the orderof ðk0dÞ2, Eqs. (C33) and (C34) should be simplifiedneglecting quadratic terms in them:

αee ≈2j

ξωη0

½Fð2 − jk0dÞ − 1�1 − jk0d

; αme ≈k0d

ξωð1 − jk0dÞ:

ðC35Þ

Formulas (C35) are self-consistent within our first-ordermodel.Next, let us consider the magnetic excitation given by

Eq. (C12). On the top surface of the absorber, we have theboundary condition:

2E0 sin

�k0d2

�þ EP

�d2

�− η0Jmet

2e−jk0d

¼ ZD2E0 cosðk0d2 Þ þHPðd2Þ − Jmet

2e−jk0d

η0: ðC36Þ

Again, the dielectric layer d is polarized electrically andits bulk polarization PðzÞ is still equal to ϵ0ðϵr − 1ÞEðzÞ,where the total electric field inside the layer has the formEðzÞ ¼ A0 sin kzþ B0 cos kz. The field produced by thepolarized dielectric layer is still expressed by Eq. (C17).Again, the PEC boundary condition at z ¼ −ðd=2Þ togetherwith the absorption condition kd ≈ π=2 results in theequivalence B0 ¼ A0. An interesting fact that the electricexcitation and magnetic excitation result in the samedistribution of the total field across the absorber is thepeculiarity of the thinnest Dallenbach absorption: Forany distribution of the external electromagnetic fieldacross it, the electric and magnetic fields are distributedin it in the same form: EðzÞ ∼ cos kzþ sin kz and HðzÞ∼coskz−sinkz.Further derivations practically repeat the same steps

as above, and we obtain for the magnetic excitation ofthe absorber the following formulas for the effectivepolarizabilities:

αmm≈2jη0½1−k0dð1−jξÞ�

ξω

�F

1−jk0d−ð1−FÞ

�; ðC37Þ

αem ≈ − k0d½1 − k0dð1 − jξÞ�ξω

�F

1 − jk0dþ ð1 − FÞ

�:

ðC38ÞFor self-consistency of the model, we should neglect thesecond-order terms in these relations, which gives

αmm ¼ 2jη0½Fð2 − jk0dÞ − 1þ jk0d�ξωð1 − jk0dÞ

;

αem ¼ −ame ¼ − k0dξωð1 − jk0dÞ

: ðC39Þ

Now let a plane wave be normally incident on aDallenbach absorber. Then we have for the inducedmagnetic and electric dipole moments per unit area

px ¼ αeeE0 þ αemH0 ¼ E0

�αee þ

αemη0

�; ðC40Þ

my ¼ αemE0 þ αmmH0 ¼ E0

�−αem þ αmm

η0

�: ðC41Þ

These equations after substitutions of Eqs. (C35) and (C39)become

px ¼2jE0

ξωη0


ð1 − jk0dÞ;

my ¼2jE0

ξω

½− jk0d2

þ Fð2 − jk0dÞ − 1þ jk0d�ð1 − jk0dÞ

: ðC42Þ

Equations (C42) show that the condition of the Huygensscatterer my ¼ η0px is fulfilled. The thin Dallenbachabsorber is fully within our general scheme.

[1] W. Dallenbach and W. Kleinsteuber, Reflection andabsorption of decimeter-waves by plane dielectric layers,Hochfrequenztechnik und Elektroakustik 51, 152 (1938).

[2] W.W. Salisbury, Absorbent body of electromagneticwaves, U.S. Patent No. 2,599,944, 10 June 1952.

[3] E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar CrossSection (Artech House, London, 1993).

[4] B. A. Munk, Frequency Selective Surfaces: Theory andDesign (Wiley, New York, 2000).

[5] A. Serdyukov, I. Semchenko, S. Tretyakov, and A.Sihvola, Electromagnetics of Bi-anisotropic Materials:Theory and Applications (Gordon and Breach, Amsterdam,2001).

[6] R. L. Fante and M. T. McCormack, Reflection propertiesof the Salisbury screen, IEEE Trans. Antennas Propag. 36,1443 (1988).

[7] N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen,Infrared perfect absorber and its application as plasmonicsensor, Nano Lett. 10, 2342 (2010).

REVIEW ARTICLE


037001-33

http://dx.doi.org/10.1109/8.8632

http://dx.doi.org/10.1109/8.8632

http://dx.doi.org/10.1021/nl9041033

[8] J. Grant, Y. Ma, S. Saha, L. B. Lok, A. Khalid, and D. R. S.Cumming, Polarization insensitive terahertz metamaterialabsorber, Opt. Lett. 36, 1524 (2011).

[9] H. A. Atwater and A. Polman, Plasmonics for improvedphotovoltaic devices, Nat. Mater. 9, 205 (2010).

[10] M. Laroche, R. Carminati, and J.-J. Greffet, Near-fieldthermophotovoltaic energy conversion, J. Appl. Phys. 100,063704 (2006).

[11] S. Simms and V. Fusco, Thin radar absorber using artificialmagnetic ground plane, Electron. Lett. 41, 1311 (2005).

[12] W. Padilla and X. Liu, Perfect electromagnetic absorb-ers from microwave to optical, SPIE Newsroom,doi:10.1117/2.1201009.003137, 2010.

[13] H. Cheng, S. Chen, H. Yang, J. Li, X. An, C. Gu, and J.Tian, A polarization insensitive and wide-angle dual-bandnearly perfect absorber in the infrared regime, J. Opt. 14,085102 (2012).

[14] J. Hao, J. Wang, X. Liu, W. J. Padilla, L. Zhou, andM. Qiu,High performance optical absorber based on a plasmonicmetamaterial, Appl. Phys. Lett. 96, 251104 (2010).

[15] K. N. Rozanov, Ultimate thickness to bandwidth ratio ofradar absorbers, IEEE Trans. Antennas Propag. 48, 1230(2000).

[16] C. Sohl, M. Gustafsson, and G. Kristensson, Physicallimitations on metamaterials: Restrictions on scatteringand absorption over a frequency interval, J. Phys. D 40,7146 (2007).

[17] S. A. Tretyakov, Uniaxial omega medium as a physicallyrealizable alternative for the perfectly matched layer(PML), J. Electromagn. Waves Appl. 12, 821 (1998).

[18] S. A. Tretyakov and T. G. Kharina, The perfectly matchedlayer as a synthetic material with active inclusions, Electro-magnetics 20, 155 (2000).

[19] S. A. Tretyakov and S. I. Maslovski, Thin absorbingstructure for all incident angles based on the use of ahigh-impedance surface, Microwave Opt. Technol. Lett.38, 175 (2003).

[20] N. I. Landy, C. M. Bingham, T. Tyler, N. Jokerst, D. R.Smith, and W. J. Padilla, Design, theory, and measurementof a polarization-insensitive absorber for terahertz imag-ing, Phys. Rev. B 79, 125104 (2009).

[21] O. Luukkonen, F. Costa, C. R. Simovski, A. Monorchio,and S. A. Tretyakov, A thin electromagnetic absorber forwide incidence angles and both polarizations, IEEE Trans.Antennas Propag. 57, 3119 (2009).

[22] B. Zhu, Z. Wang, C. Huang, Y. Feng, J. Zhao, and T.Jiang, Polarization insensitive metamaterial absorberwith wide incident angle, Prog. Electromagn. Res.101, 231 (2010).

[23] K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater,Broadband polarization-independent resonant light ab-sorption using ultrathin plasmonic super absorbers, Nat.Commun. 2, 517 (2011).

[24] L. Lu, S. Qu, H. Ma, F. Yu, S. Xia, Z. Xu, and P. Bai, Apolarization-independent wide-angle dual directional ab-sorption metamaterial absorber, Prog. Electromagn. Res.M 27, 91 (2012).

[25] M. Diem, T. Koschny, and C. M. Soukoulis, Wide-angleperfect absorber/thermal emitter in the terahertz regime,Phys. Rev. B 79, 033101 (2009).

[26] J.-P. Berenger, A perfectly matched layer for the absorptionof electromagnetic waves, J. Comput. Phys. 114, 185(1994).

[27] N. Engheta, in Proceedings of the IEEE InternationalSymposium on Antennas and Propagation, 2002 (IEEE,New York, 2002), Vol. 2, p. 392.

[28] F. Bilotti, A. Toscano, K. B. Alici, E. Ozbay, and L. Vegni,Design of miniaturized narrowband absorbers based onresonant-magnetic inclusions, IEEE Trans. Electromagn.Compat. 53, 63 (2011).

[29] N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, andW. J. Padilla, Perfect Metamaterial Absorber, Phys. Rev.Lett. 100, 207402 (2008).

[30] V. T. Pham, J. W. Park, D. L. Vu, H. Y. Zheng, J. Y. Rhee,K. W. Kim, and Y. P. Lee, THz-metamaterial absorbers,Adv. Nat. Sci. Nanosci. Nanotechnol. 4, 015001 (2013).

[31] G. Dayal and S. A. Ramakrishna, Design of highlyabsorbing metamaterials for infrared frequencies, Opt.Express 20, 17503 (2012).

[32] C. M. Watts, X. Liu, and W. J. Padilla, Metamaterialelectromagnetic wave absorbers, Adv. Mater. 24, OP98(2012).

[33] T. Maier and H. Brückl, Wavelength-tunable microbol-ometers with metamaterial absorbers, Opt. Lett. 34, 3012(2009).

[34] J. Rosenberg, R. V. Shenoi, T. E. Vandervelde, S. Krishna,and O. Painter, A multispectral and polarization-selectivesurface-plasmon resonant midinfrared detector, Appl.Phys. Lett. 95, 161101 (2009).

[35] E. Rephaeli and S. Fan, Absorber and emitter for solarthermo-photovoltaic systems to achieve efficiency exceed-ing the Shockley-Queisser limit, Opt. Express 17, 15145(2009).

[36] J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White,and M. L. Brongersma, Plasmonics for extreme lightconcentration and manipulation, Nat. Mater. 9, 193 (2010).

[37] X. Liu, T. Tyler, T. Starr, A. F. Starr, N. M. Jokerst, andW. J. Padilla, Taming the Blackbody with Infrared Meta-materials as Selective Thermal Emitters, Phys. Rev. Lett.107, 045901 (2011).

[38] A. Tittl, P. Mai, R. Taubert, D. Dregely, N. Liu, andH. Giessen, Palladium-based plasmonic perfect absorberin the visible wavelength range and its application toHydrogen sensing, Nano Lett. 11, 4366 (2011).

[39] J.-J. Greffet, Controlled incandescence, Nature (London)478, 191 (2011).

[40] J. N. Munday and H. A. Atwater, Large integrated absorp-tion enhancement in plasmonic solar cells by combiningmetallic gratings and antireflection coatings, Nano Lett.11, 2195 (2011).

[41] S. A. Kuznetsov, A. G. Paulish, A. V. Gelfand, P. A.Lazorskiy, and V. N. Fedorinin, Bolometric THz-to-IRconverter for terahertz imaging, Appl. Phys. Lett. 99,023501 (2011).

[42] C. Hägglund and S. P. Apell, Plasmonic near-field absorb-ers for ultrathin solar cells, J. Phys. Chem. Lett. 3, 1275(2012).

[43] Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X.Fang, Ultrabroadband light absorption by a sawtoothanisotropic metamaterial slab, Nano Lett. 12, 1443 (2012).

REVIEW ARTICLE


037001-34

http://dx.doi.org/10.1364/OL.36.001524

http://dx.doi.org/10.1038/nmat2629

http://dx.doi.org/10.1063/1.2234560

http://dx.doi.org/10.1063/1.2234560

http://dx.doi.org/10.1049/el:20053236

http://dx.doi.org/10.1117/2.1201009.003137

http://dx.doi.org/10.1088/2040-8978/14/8/085102

http://dx.doi.org/10.1088/2040-8978/14/8/085102

http://dx.doi.org/10.1063/1.3442904

http://dx.doi.org/10.1109/8.884491

http://dx.doi.org/10.1109/8.884491

http://dx.doi.org/10.1088/0022-3727/40/22/042

http://dx.doi.org/10.1088/0022-3727/40/22/042

http://dx.doi.org/10.1163/156939398X01060

http://dx.doi.org/10.1080/027263400308348

http://dx.doi.org/10.1080/027263400308348

http://dx.doi.org/10.1002/mop.11006


http://dx.doi.org/10.1103/PhysRevB.79.125104

http://dx.doi.org/10.1109/TAP.2009.2028601


http://dx.doi.org/10.2528/PIER10011110

http://dx.doi.org/10.2528/PIER10011110

http://dx.doi.org/10.1038/ncomms1528

http://dx.doi.org/10.1038/ncomms1528

http://dx.doi.org/10.2528/PIERM12102101

http://dx.doi.org/10.2528/PIERM12102101


http://dx.doi.org/10.1006/jcph.1994.1159

http://dx.doi.org/10.1006/jcph.1994.1159

http://dx.doi.org/10.1109/TEMC.2010.2051229

http://dx.doi.org/10.1109/TEMC.2010.2051229

http://dx.doi.org/10.1103/PhysRevLett.100.207402


http://dx.doi.org/10.1088/2043-6262/4/1/015001

http://dx.doi.org/10.1364/OE.20.017503

http://dx.doi.org/10.1364/OE.20.017503

http://dx.doi.org/10.1002/adma.201200674


http://dx.doi.org/10.1364/OL.34.003012

http://dx.doi.org/10.1364/OL.34.003012

http://dx.doi.org/10.1063/1.3244204

http://dx.doi.org/10.1063/1.3244204

http://dx.doi.org/10.1364/OE.17.015145

http://dx.doi.org/10.1364/OE.17.015145




http://dx.doi.org/10.1021/nl202489g

http://dx.doi.org/10.1038/478191a

http://dx.doi.org/10.1038/478191a

http://dx.doi.org/10.1021/nl101875t

http://dx.doi.org/10.1021/nl101875t

http://dx.doi.org/10.1063/1.3607474

http://dx.doi.org/10.1063/1.3607474

http://dx.doi.org/10.1021/jz300290d

http://dx.doi.org/10.1021/jz300290d

http://dx.doi.org/10.1021/nl204118h

[44] F. Alves, D. Grbovic, B. Kearney, N. V. Lavrik, and G.Karunasiri, Bi-material terahertz sensors using metamate-rial structures, Opt. Express 21, 13256 (2013).

[45] F. B. P. Niesler, J. K. Gansel, S. Fischbach, and M.Wegener, Metamaterial metal-based bolometers, Appl.Phys. Lett. 100, 203508 (2012).

[46] H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D.Averitt, and W. J. Padilla, A metamaterial absorber for theterahertz regime: Design, fabrication and characterization,Opt. Express 16, 7181 (2008).

[47] H. Tao, C. M. Bingham, A. C. Strikwerda, D. Pilon, D.Shrekenhamer, N. I. Landy, K. Fan, X. Zhang, W. J.Padilla, and R. D. Averitt, Highly flexible wide angle ofincidence terahertz metamaterial absorber: Design, fabrica-tion, and characterization, Phys. Rev. B 78, 241103 (2008).

[48] X. Liu, T. Starr, A. F. Starr, and W. J. Padilla, InfraredSpatial and Frequency Selective Metamaterial with Near-Unity Absorbance, Phys. Rev. Lett. 104, 207403 (2010).

[49] Y. Jin, S. Xiao, N. A. Mortensen, and S. He, Arbitrarilythin metamaterial structure for perfect absorption and giantmagnification, Opt. Express 19, 11114 (2011).

[50] H.-T. Chen, J. Zhou, J. F. O’Hara, F. Chen, A. K. Azad,and A. J. Taylor, Antireflection Coating using Metamate-rials and Identification of its Mechanism, Phys. Rev. Lett.105, 073901 (2010).

[51] H.-T. Chen, Interference theory of metamaterial perfectabsorbers, Opt. Express 20, 7165 (2012).

[52] X. Shen, Y. Yang, Y. Zang, J. Gu, J. Han, W. Zhang, andT. J. Cui, Triple-band terahertz metamaterial absorber:Design, experiment, and physical interpretation, Appl.Phys. Lett. 101, 154102 (2012).

[53] J. Sun, L. Liu, G. Dong, and J. Zhou, An extremely broadband metamaterial absorber based on destructive interfer-ence, Opt. Express 19, 21155 (2011).

[54] T. V. Teperik, V. V. Popov, and F. J. García de Abajo, Totallight absorption in plasmonic nanostructures, J. Opt. A 9,S458 (2007).

[55] Q.-Y. Wen, Y.-S. Xie, H.-W. Zhang, Q.-H. Yang, Y.-X. Li,and Y.-L. Liu, Transmission line model and fields analysisof metamaterial absorber in the terahertz band, Opt.Express 17, 20256 (2009).

[56] Y.-Q. Pang, Y.-J. Zhou, and J. Wang, Equivalent circuitmethod analysis of the influence of frequency selectivesurface resistance on the frequency response of metama-terial absorbers, J. Appl. Phys. 110, 023704 (2011).

[57] Y. Pang, H. Cheng, Y. Zhou, and J. Wang, Analysis anddesign of wire-based metamaterial absorbers using equiv-alent circuit approach, J. Appl. Phys. 113, 114902 (2013).

[58] M. P. Hokmabadi, D. S. Wilbert, P. Kung, and S. M. Kim,Design and analysis of perfect terahertz metamaterialabsorber by a novel dynamic circuit model, Opt. Express21, 16455 (2013).

[59] F. Costa, S. Genovesi, A. Monorchio, and G. Manara, Acircuit-based model for the interpretation of perfect meta-material absorbers, IEEE Trans. Antennas Propag. 61,1201 (2013).

[60] D. Y. Shchegolkov, A. K. Azad, J. F. O’Hara, and E. I.Simakov, Perfect subwavelength fishnetlike metamaterial-based film terahertz absorbers, Phys. Rev. B 82, 205117(2010).

[61] X.-Y. Peng, B. Wang, S. Lai, D. H. Zhang, and J.-H. Teng,Ultrathin multi-band planar metamaterial absorber basedon standing wave resonances, Opt. Express 20, 27756(2012).

[62] J. Zhou, H.-T. Chen, T. Koschny, A. K. Azad, A. J. Taylor,C. M. Soukoulis, and J. F. O’Hara, Application of metasur-face description for multilayered metamaterials andan alternative theory for metamaterial perfect absorber,arXiv:1111.0343.

[63] H. Y. Zheng, X. R. Jin, J. W. Park, Y. H. Lu, J. Y. Rhee,W. H. Jang, H. Cheong, and Y. P. Lee, Tunable dual-bandperfect absorbers based on extraordinary optical trans-mission and Fabry-Perot cavity resonance, Opt. Express20, 24002 (2012).

[64] M. K. Hedayati, M. Javaherirahim, B. Mozooni, R.Abdelaziz, A. Tavassolizadeh, V. S. K. Chakravadhanula,V. Zaporojtchenko, T. Strunkus, F. Faupel, and M. Elbahri,Design of a perfect black absorber at visible frequenciesusing plasmonic metamaterials, Adv. Mater. 23, 5410(2011).

[65] Y. Zeng, H.-T. Chen, and D. A. R. Dalvit, The role ofmagnetic dipoles and non-zero-order Bragg waves inmetamaterial perfect absorbers, Opt. Express 21, 3540(2013).

[66] J. W. Park, P. V. Tuong, J. Y. Rhee, K. W. Kim, W. H. Jang,E. H. Choi, L. Y. Chen, and Y. P. Lee, Multi-band meta-material absorber based on the arrangement of donut-typeresonators, Opt. Express 21, 9691 (2013).

[67] Y. Ra’di, V. S. Asadchy, and S. A. Tretyakov, Totalabsorption of electromagnetic waves in ultimately thinlayers, IEEE Trans. Antennas Propag. 61, 4606 (2013).

[68] D. Pozar, Scattered and absorbed powers in receivingantennas, IEEE Antennas Propag. Mag. 46, 144 (2004).

[69] S. Tretyakov, Analytical Modeling in Applied Electro-magnetics (Artech, Norwood, MA, 2003).

[70] J. R. Tischler, M. S. Bradley, and V. Bulović, Criticallycoupled resonators in vertical geometry using a planarmirror and a 5 nm thick absorbing film, Opt. Lett. 31, 2045(2006).

[71] J. R. Tischler, M. S. Bradley, Q. Zhang, T. Atay, A.Nurmikko, and V. Bulović, Solid state cavity QED:Strong coupling in organic thin films, Org. Electron. 8,94 (2007).

[72] S. D. Gupta, Strong-interaction-mediated critical cou-pling at two distinct frequencies, Opt. Lett. 32, 1483(2007).

[73] S. Deb, S. Dutta-Gupta, J. Banerji, and S. Dutta Gupta,Critical coupling at oblique incidence, J. Opt. A 9, 555(2007).

[74] Y. Xiang, X. Dai, J. Guo, H. Zhang, S. Wen, and D. Tang,Critical coupling with graphene-based hyperbolic meta-materials, Sci. Rep. 4, 5483 (2014).

[75] S. Ramo and J. R. Whinnery, Fields and Waves in ModernRadio (Wiley, New York, 1944).

[76] W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H.Cao, Time-reversed lasing and interferometric control ofabsorption, Science 331, 889 (2011).

[77] V. Klimov, S. Sun, and G.-Y. Guo, Coherent perfectnanoabsorbers based on negative refraction, Opt. Express20, 13071 (2012).

REVIEW ARTICLE


037001-35

http://dx.doi.org/10.1364/OE.21.013256

http://dx.doi.org/10.1063/1.4714741

http://dx.doi.org/10.1063/1.4714741

http://dx.doi.org/10.1364/OE.16.007181



http://dx.doi.org/10.1364/OE.19.011114



http://dx.doi.org/10.1364/OE.20.007165

http://dx.doi.org/10.1063/1.4757879

http://dx.doi.org/10.1063/1.4757879

http://dx.doi.org/10.1364/OE.19.021155

http://dx.doi.org/10.1088/1464-4258/9/9/S29

http://dx.doi.org/10.1088/1464-4258/9/9/S29

http://dx.doi.org/10.1364/OE.17.020256

http://dx.doi.org/10.1364/OE.17.020256

http://dx.doi.org/10.1063/1.3608169

http://dx.doi.org/10.1063/1.4795277

http://dx.doi.org/10.1364/OE.21.016455

http://dx.doi.org/10.1364/OE.21.016455





http://dx.doi.org/10.1364/OE.20.027756

http://dx.doi.org/10.1364/OE.20.027756

http://arXiv.org/abs/1111.0343

http://dx.doi.org/10.1364/OE.20.024002

http://dx.doi.org/10.1364/OE.20.024002



http://dx.doi.org/10.1364/OE.21.003540

http://dx.doi.org/10.1364/OE.21.003540

http://dx.doi.org/10.1364/OE.21.009691


http://dx.doi.org/10.1109/MAP.2004.1296172

http://dx.doi.org/10.1364/OL.31.002045

http://dx.doi.org/10.1364/OL.31.002045

http://dx.doi.org/10.1016/j.orgel.2007.01.008

http://dx.doi.org/10.1016/j.orgel.2007.01.008

http://dx.doi.org/10.1364/OL.32.001483

http://dx.doi.org/10.1364/OL.32.001483

http://dx.doi.org/10.1088/1464-4258/9/7/001

http://dx.doi.org/10.1088/1464-4258/9/7/001

http://dx.doi.org/10.1038/srep05483

http://dx.doi.org/10.1126/science.1200735

http://dx.doi.org/10.1364/OE.20.013071

http://dx.doi.org/10.1364/OE.20.013071

[78] S. Longhi, Coherent perfect absorption in a homo-geneously broadened two-level medium, Phys. Rev. A83, 055804 (2011).

[79] S. Longhi, PT -symmetric laser absorber, Phys. Rev. A 82,031801 (2010).

[80] G. Pirruccio, L. M. Moreno, G. Lozano, and J. G. Rivas,Coherent and broadband enhanced optical absorption ingraphene, ACS Nano 7, 4810 (2013).

[81] Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, CoherentPerfect Absorbers: Time-Reversed Lasers, Phys. Rev. Lett.105, 053901 (2010).

[82] D. A. B. Miller, On perfect cloaking, Opt. Express 14,12457 (2006).

[83] M. Selvanayagam and G. V. Eleftheriades, ExperimentalDemonstration of Active Electromagnetic Cloaking, Phys.Rev. X 3, 041011 (2013).

[84] T. Niemi, A. O. Karilainen, and S. A. Tretyakov, Synthesisof polarization transformers, IEEE Trans. AntennasPropag. 61, 3102 (2013).

[85] V. K. Varadan, V. V. Varadan, and A. Lakhtakia, On thepossibility of designing anti-reflection coating using chiralcomposites, J. Wave-Mater. Interact. 2, 71 (1987).

[86] D. L. Jaggard and N. Engheta, ChirosorbTM as an invisiblemedium, Electron. Lett. 25, 173 (1989).

[87] S. A. Tretyakov, A. A. Sochava, and C. R. Simovski,Influence of chiral shapes of individual inclusions onthe absorption in chiral composite coatings, Electromag-netics 16, 113 (1996).

[88] D. M. Pozar, Microwave Engineering (Wiley, New York,2012).

[89] J. Vehmas, S. Hrabar, and S. Tretyakov, Omega trans-mission lines with applications to effective medium modelsof metamaterials, J. Appl. Phys. 115, 134905 (2014).

[90] V. Asadchy, I. Faniayeu, Y. Ra’di, I. Semchenko, and S.Khakhomov, in Proceedings of the 7th InternationalCongress on Advanced Electromagnetic Materials inMicrowaves and Optics—Metamaterials, Bordeaux,France, 2013 (IEEE, New York, USA, 2013), p. 244.

[91] I. Faniayeu, V. Asadchy, T. Dziarzhauskaya, I. Semchenko,and S. Khakhomov, A single-layer meta-atom absorber,arXiv:1404.1816.

[92] S. Gu, J. P. Barrett, T. H. Hand, B.-I. Popa, and S. A.Cummer, A broadband low-reflection metamaterialabsorber, J. Appl. Phys. 108, 064913 (2010).

[93] C. A. Valagiannopoulos and S. A. Tretyakov, Symmetricabsorbers realized as gratings of PEC cylinders coveredby ordinary dielectrics, IEEE Trans. Antennas Propag. 62,5089 (2014).

[94] S. A. Tretyakov, C. R. Simovski, and A. A. Sochava, inAdvances in Complex Electromagnetic Materials, editedby A. Priou, A. Sihvola, S. Tretyakov, and A. Vinogradov(Kluwer Academic, Dordrecht, 1997), pp. 271–280.

[95] C. Hu, X. Li, Q. Feng, X. Chen, and X. Luo, Introducingdipole-like resonance into magnetic resonance to realizesimultaneous drop in transmission and reflection at tera-hertz frequency, J. Appl. Phys. 108, 053103 (2010).

[96] Y. Cheng, H. Yang, Z. Cheng, and B. Xiao, A planarpolarization-insensitive metamaterial absorber, PhotonicsNanostruct. Fundam. Appl. 9, 8 (2011).

[97] F. Costa, S. Genovesi, and A. Monorchio, On the band-width of high-impedance frequency selective surfaces,IEEE Antennas Wireless Propag. Lett. 8, 1341 (2009).

[98] H. Greve, C. Pochstein, H. Takele, V. Zaporojtchenko,F. Faupel, A. Gerber, M. Frommberger, and E. Quandt,Nanostructured magnetic Fe-Ni-Co/Teflon multilayers forhigh-frequency applications in the gigahertz range, Appl.Phys. Lett. 89, 242501 (2006).

[99] I. T. Iakubov, A. N. Lagarkov, S. A. Maklakov, A. V.Osipov, K. N. Rozanov, I. A. Ryzhikov, V. V. Samsonova,and A. O. Sboychakov, Microwave and static magneticproperties of multi-layered iron-based films, J. Magn.Magn. Mater. 321, 726 (2009).

[100] L. B. Kong, Z. W. Li, L. Liu, R. Huang, M. Abshinova,Z. H. Yang, C. B. Tang, P. K. Tan, C. R. Deng, and S.Matitsine, Recent progress in some composite materialsand structures for specific electromagnetic applications,Int. Mater. Rev. 58, 203 (2013).

[101] O. Acher, Copper vs. iron: Microwave magnetism in themetamaterial age, J. Magn. Magn. Mater. 321, 2093(2009).

[102] P. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A.Tretyakov, On artificial magnetodielectric loading forimproving the impedance bandwidth properties of micro-strip antennas, IEEE Trans. Antennas Propag. 54, 1654(2006).

[103] D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexópolous,and E. Yablonovitch, High-impedance electromagneticsurfaces with a forbidden frequency band, IEEE Trans.Microw. Theory Tech. 47, 2059 (1999).

[104] S. A. Tretyakov and C. R. Simovski, Dynamic model ofartificial reactive impedance surfaces, J. Electromagn.Waves Appl. 17, 131 (2003).

[105] O. Luukkonen, C. Simovski, G. Granet, G. Goussetis,D. Lioubtchenko, A. V. Räisänen, and S. A. Tretyakov,Simple and accurate analytical model of planar grids andhigh-impedance surfaces comprising metal strips orpatches, IEEE Trans. Antennas Propag. 56, 1624(2008); 58, 2162(E) (2010).

[106] Y. Okano, S. Ogino, and K. Ishikawa, Development ofoptically transparent ultrathin microwave absorber forultrahigh-frequency RF identification system, IEEE Trans.Microw. Theory Tech. 60, 2456 (2012).

[107] F. Terracher and G. Berginc, A broadband dielectricmicrowave absorber with periodic metallizations, J. Elec-tromagn. Waves Appl. 13, 1725 (1999).

[108] A. N. Lagarkov, V. N. Semenenko, V. A. Chistyaev, D. E.Ryabov, S. A. Tretyakov, and C. R. Simovski, Resonanceproperties of bi-helix media at microwaves, Electromag-netics 17, 213 (1997).

[109] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J.Stewart, Magnetism from conductors and enhanced non-linear phenomena, IEEE Trans. Microw. Theory Techn. 47,2075 (1999).

[110] J. Hao, V. Sadaune, L. Burgnies, and D. Lippens, Ferro-electrics based absorbing layers, J. Appl. Phys. 116,043520 (2014).

[111] C. R. Simovski, M. Kondratiev, and S. He, Anexplicit method for calculating the reflection from an

REVIEW ARTICLE


037001-36

http://dx.doi.org/10.1103/PhysRevA.83.055804




http://dx.doi.org/10.1021/nn4012253



http://dx.doi.org/10.1364/OE.14.012457

http://dx.doi.org/10.1364/OE.14.012457

http://dx.doi.org/10.1103/PhysRevX.3.041011

http://dx.doi.org/10.1103/PhysRevX.3.041011



http://dx.doi.org/10.1049/el:19890126

http://dx.doi.org/10.1080/02726349608908465

http://dx.doi.org/10.1080/02726349608908465

http://dx.doi.org/10.1063/1.4869655

http://arXiv.org/abs/1404.1816

http://dx.doi.org/10.1063/1.3485808



http://dx.doi.org/10.1063/1.3467528

http://dx.doi.org/10.1016/j.photonics.2010.07.002

http://dx.doi.org/10.1016/j.photonics.2010.07.002

http://dx.doi.org/10.1109/LAWP.2009.2038346

http://dx.doi.org/10.1063/1.2402877

http://dx.doi.org/10.1063/1.2402877

http://dx.doi.org/10.1016/j.jmmm.2008.11.036


http://dx.doi.org/10.1179/1743280412Y.0000000011





http://dx.doi.org/10.1109/22.798001

http://dx.doi.org/10.1109/22.798001

http://dx.doi.org/10.1163/156939303766975407

http://dx.doi.org/10.1163/156939303766975407




http://dx.doi.org/10.1109/TMTT.2012.2202680

http://dx.doi.org/10.1109/TMTT.2012.2202680

http://dx.doi.org/10.1163/156939399X00187

http://dx.doi.org/10.1163/156939399X00187

http://dx.doi.org/10.1080/02726349708908533

http://dx.doi.org/10.1080/02726349708908533

http://dx.doi.org/10.1109/22.798002

http://dx.doi.org/10.1109/22.798002

http://dx.doi.org/10.1063/1.4891728

http://dx.doi.org/10.1063/1.4891728

anti-reflection structure involving array of C-shaped wireelements, J. Electromagn. Waves Appl. 14, 1335 (2000).

[112] C. R. Simovski, M. Kondratiev, and S. He, Array ofC-shaped wire elements for the reduction of reflectionfrom a conducting plane, Microwave Opt. Technol. Lett.25, 302 (2000).

[113] M. Pu, C. Hu, M. Wang, C. Huang, Z. Zhao, C. Wang, Q.Feng, and X. Luo, Design principles for infrared wide-angle perfect absorber based on plasmonic structure, Opt.Express 19, 17413 (2011).

[114] N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H.Giessen, Plasmon hybridization in stacked cut-wire meta-materials, Adv. Mater. 19, 3628 (2007).

[115] J. Dai, F. Ye, Y. Chen, M. Muhammed, M. Qiu, and M.Yan, Light absorber based on nano-spheres on a substratereflector, Opt. Express 21, 6697 (2013).

[116] S. A. Schelkunoff, The electromagnetic theory of coaxialtransmission lines and cylindrical shields, Bell Syst. Tech.J. 13, 532 (1934).

[117] J. McVay, A. Hoorfar, and N. Engheta, Thin absorbersusing space-filling curve artificial magnetic conductors,Microwave Opt. Technol. Lett. 51, 785 (2009).

[118] F. Bilotti, L. Nucci, and L. Vegni, An SRR based micro-wave absorber, Microwave Opt. Technol. Lett. 48, 2171(2006).

[119] Sukosin Thongrattanasiri, Frank H. L. Koppens, and F.Javier Garcia de Abajo, Complete Optical Absorption inPeriodically Patterned Graphene, Phys. Rev. Lett. 108,047401 (2012).

[120] R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, Aperfect absorber made of a graphene micro-ribbon meta-material, Opt. Express 20, 28017 (2012).

[121] G.W. Hansen, Dyadic Green’s functions and guided sur-face waves for a surface conductivity model of graphene,J. Appl. Phys. 103, 064302 (2008).

[122] Y. Zhang, Y. Feng, B. Zhu, J. Zhao, and T. Jiang, Graphenebased tunable metamaterial absorber and polarizationmodulation in terahertz frequency, Opt. Express 22,22743 (2014).

[123] A. Fernandez and A. Valenzuela, General solution forsingle-layer electromagnetic-wave absorber, Electron.Lett. 21, 20 (1985).

[124] M. A. Kats, R. Blanchard, P. Genevet, and F. Capasso,Nanometre optical coatings based on strong interferenceeffects in highly absorbing media, Nat. Mater. 12, 20 (2013).

[125] J. Park, J.-H. Kang, A. P. Vasudev, D. T. Schoen, H. Kim,E. Hasman, and M. L. Brongersma, Omnidirectional near-unity absorption in an ultrathin planar semiconductor layeron a metal substrate, ACS Photonics 1, 812 (2014).

[126] A. D. Rakić, A. B. Djurišić, J. M. Elazar, and M. L.Majewski, Optical properties of metallic films forvertical-cavity optoelectronic devices, Appl. Opt. 37,5271 (1998).

[127] A. Moreau, C. Ciracì, J. J. Mock, R. T. Hill, Q. Wang,B. J. Wiley, A. Chilkoti, and D. R. Smith, Controlled-reflectance surfaces with film-coupled colloidal nanoan-tennas, Nature (London) 492, 86 (2012).

[128] M. Albooyeh and C. Simovski, Substrate-induced biani-sotropy in plasmonic grids, J. Opt. 13, 105102 (2011).

[129] M. Albooyeh, D. Morits, and C. R. Simovski, Electro-magnetic characterization of substrated metasurfaces,Metamaterials 5, 178 (2011).

[130] M. Albooyeh and C. R. Simovski, Huge local fieldenhancement in perfect plasmonic absorbers, Opt. Express20, 21888 (2012).

[131] J. Vehmas, Y. Ra’di, A. O. Karilainen, and S. A. Tretyakov,Eliminating electromagnetic scattering from small par-ticles, IEEE Trans. Antennas Propag. 61, 3747 (2013).

[132] M. Barbuto, A. Monti, F. Bilotti, and A. Toscano, inMetamaterials 2011: Proceedings of the FifthInternational Congress on Advanced ElectromagneticMaterials in Microwaves and Optics, Barcelona, Spain,2011 (Metamorphose VI, Brussels, Belgium, 2011), p. 86.

[133] Y. Ding and V. Fusco, Loading artificial magnetic con-ductor and artificial magnetic conductor absorber withnegative impedance converter elements, Microwave Opt.Technol. Lett. 54, 2111 (2012).

[134] K. L. Ford, in Proceedings of the Antennas and Propa-gation Conference (LAPC), Loughborough, UK, 2013(IEEE, New York, USA, 2013), p. 510.

[135] J. P. Turpin, J. A. Bossard, K. L. Morgan, D. H. Werner,and P. L. Werner, Reconfigurable and tunable metamate-rials: A review of the theory and applications, Int. J.Antennas Propag. 2014, 429837 (2014).

[136] I. S. Nefedov, C. A. Valagiannopoulos, S. M. Hashemi, andE. I. Nefedov, Total absorption in asymmetric hyperbolicmedia, Sci. Rep. 3, 2662 (2013).

[137] M. Albooyeh, D. Morits, and S. Tretyakov, Effectiveelectric and magnetic properties of metasurfaces in tran-sition from crystalline to amorphous state, Phys. Rev. B 85,205110 (2012).

[138] Amorphous Nanophotonics, edited by C. Rockstuhl and T.Scharf (Springer, New York, 2013).

[139] J. D. Jackson, in Classical Electrodynamics, 3rd ed.(Wiley, New York, 1999).

[140] A. Kazemzadeh and A. Karlsson, On the absorptionmechanism of ultra thin absorbers, IEEE Trans. AntennasPropag. 58, 3310 (2010).

REVIEW ARTICLE


037001-37

http://dx.doi.org/10.1163/156939300X00112

http://dx.doi.org/10.1002/(SICI)1098-2760(20000605)25:5%3C302::AID-MOP5%3E3.0.CO;2-J

http://dx.doi.org/10.1002/(SICI)1098-2760(20000605)25:5%3C302::AID-MOP5%3E3.0.CO;2-J

http://dx.doi.org/10.1364/OE.19.017413

http://dx.doi.org/10.1364/OE.19.017413


http://dx.doi.org/10.1364/OE.21.006697

http://dx.doi.org/10.1002/j.1538-7305.1934.tb00679.x

http://dx.doi.org/10.1002/j.1538-7305.1934.tb00679.x






http://dx.doi.org/10.1364/OE.20.028017

http://dx.doi.org/10.1063/1.2891452

http://dx.doi.org/10.1364/OE.22.022743

http://dx.doi.org/10.1364/OE.22.022743

http://dx.doi.org/10.1049/el:19850016

http://dx.doi.org/10.1049/el:19850016


http://dx.doi.org/10.1021/ph500093d

http://dx.doi.org/10.1364/AO.37.005271

http://dx.doi.org/10.1364/AO.37.005271

http://dx.doi.org/10.1038/nature11615

http://dx.doi.org/10.1088/2040-8978/13/10/105102

http://dx.doi.org/10.1016/j.metmat.2011.08.002

http://dx.doi.org/10.1364/OE.20.021888

http://dx.doi.org/10.1364/OE.20.021888




http://dx.doi.org/10.1155/2014/429837

http://dx.doi.org/10.1155/2014/429837

http://dx.doi.org/10.1038/srep02662





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