EPJ Appl. Metamat. 8, 3 (2021)© A.V. Marini et al., published by EDP Sciences, 2021https://doi.org/10.1051/epjam/2020014

Available online at:epjam.edp-open.org

RESEARCH ARTICLE

Metasurface virtual absorbers: unveiling operative conditionsthrough equivalent lumped circuit modelAngelica Viola Marini*, Davide Ramaccia, Alessandro Toscano, and Filiberto Bilotti

Department of Engineering, ROMA TRE University, Via Vito Volterra 62, Rome 00146, Italy

* e-mail: a

This is anO

Received: 29 October 2020 / Accepted: 20 December 2020

Abstract. Virtual absorption concept has been recently introduced as a new phenomenon observed inelectromagnetics and optics consisting of theoretically unlimited accumulation of energy within a finite volumeof material without dissipation. The anomalous behaviour is achieved by engaging the complex zero scatteringeigenmodes of the virtual absorbing system by illuminating it with a proper complex frequency v=vr+ jvi,whose value is strictly determined by the system characteristics. In this paper, we investigate on the position ofthe zero-pole scattering pairs in the complex frequency plane as a function of the input impedance of themetasurface-based lossless virtual absorber. We analytically derive the conditions under which a properlymodulated monochromatic plane wave can be virtually absorbed by the system and stored within its volume.The analysis is developed by modelling the propagation of a normally impinging plane wave through itsequivalent transmission line model terminated in an arbitrary reactive load, which in turn models the inputimpedance of themetasurface-based system under consideration. The study allows to determine a prioriwhetherthemetasurface-based system can support the virtual absorption or not by evaluating the time-constant from itsequivalent circuit.

Keywords: Virtual perfect matching / time-varying signal / complex frequency / virtual absorption

1 Introduction

The localization and confinement of electromagneticenergy in a finite volume without dissipation has beenextensively studied by the scientific community in the lastyears, by exploiting the anomalous interaction betweenlight and artificial metastructures [1–19]. In this frame-work, virtual absorption concept [20–24] represents one ofthe most appealing technique for storing and releasingelectromagnetic energy, due to passive and simplestructures that can support the desired energy accumula-tion. Indeed, originally the virtual absorption has beenachieved by illuminating a lossless dielectric slab of finitethickness with two perfectly synchronized signals at itsopposite sides, demonstrating the possibility to achieve zerototal scattering and energy accumulation when the illumi-nating signals have a proper complex temporal frequencyv=vr+ jvi [20]. This special temporal excitation isfundamental for engaging the complex zero scatteringeigenmodes of the system and enabling its anomalouselectromagnetic behaviour of zero scattering and energystoring.

ngelicaviola.marini@uniroma3.it

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More recently, to overcome the strictly requirementsof pure lossless materials and perfect coherent illumina-tion of the structure, we proposed a metasurface-basedvirtual absorber [25,26], able to store energy from animpinging plane wave under arbitrary illuminationcondition, and without the use of dielectric materials.As shown in Figure 1a, the proposed structure consists of a1D cavity filled by vacuum and bounded by a reflector anda metasurface. The metasurface is designed to bepenetrable by the external illuminating field, allowingthe cavity to interact with the external environment. In[25], exploiting the transmission line equivalence inFigure 1b, we analytically derived the required complexfrequency of the illuminating plane wave as a function ofthe surface reactance Xmts of the metasurface, theelectrical size of the cavity, polarization and incidenceangle of the illuminating wave. In [25] we also reported aneffective design guideline of a metasurface-based virtualabsorber for a given frequency, noticing that the metasur-face reactance value was strongly linked with the electricalsize of the cavity (Eq. (5) in [25]). This suggests that thevirtual absorption phenomenon does not occur for anypair of metasurface and cavity impedances.

The aim of this paper is to identify the conditions interms of equivalent lumped impedances modelling the

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Fig. 1. Schematic representation of (a) a virtual metasurfaceabsorber, (b) its transmission line model, and (c) circuitalequivalent transmission line with lumped elements.

2 A.V. Marini et al.: EPJ Appl. Metamat. 8, 3 (2021)

metasurface and cavity that enable virtual absorption.Therefore, the entire analysis is reduced to the circuitalresponse of the transmission line model shown in Figure 1b.The metasurface reactance Xmts and input reactance Xcavof the short-circuited stub modelling the metal-backedcavity can assume independently capacitive and inductivereactive values, according to the metasurface layout andelectrical dimension of the cavity. Their overall responsecan be synthesized through the reactive load XL shown inFigure 1c. Such a representation significantly simplifies theanalysis by reducing the problem of virtual absorption to avery well-established concept in microwave circuit, thatis perfect impedance matching [27]. Indeed, when thetransmission line in Figure 1c is matched under complexexcitation, the corresponding metasurface-bounded opencavity in Figure 1a is virtually absorbing the impingingelectromagnetic wave, allowing us to easily identify theoperative bounds for the proposed system. By exploring thescattering zero-pole pairs and their position in the complexfrequency plane as a function of such reactances, weanalytically derive the conditions under which the systemcan support virtual absorption and the correspondingoperative bounds.

The paper is organized as follows. In Section 2 weintroduce the analysis method by focusing our attention onthe response from the load in terms of reflection coefficientwhen an excitation signal with complex frequency ispropagating along the feeding transmission line. Here, weinvestigate the two possible configurations for the inputreactance XL given by the parallel connection of surface

reactance Xmts and input reactance Xcav of the short-circuited stub modelling the metal-backed cavity: (i) Xmtsand Xcav have the same equivalent electrical response, thatis, both capacitive or both inductive; and (ii) Xmts and Xcavhave opposite equivalent electrical responses, that is, onecapacitive and one inductive. In Section 3 we verify theperfect matching of the reactive load in the two config-urations identified in Section 2 by a proper set of numericalsimulations in ADS [28]. We demonstrate that if thelumped-element equivalent modelling the metasurfacevirtual absorber exhibits the proper values, the virtualabsorption can be enabled. Finally, in Section 4 someconclusions are drawn.

2 Operative conditions of metasurface virtualabsorber through TL model

Let us consider an arbitrary metasurface virtual absorberas shown in Figure 1a. Under plane wave excitation fromfree-space, the propagation and interaction with the virtualabsorber can be modelled as shown in Figure 1b: a losslesstransmission line with characteristic impedance modellingthe surrounding vacuum Z0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffim0=e0

p≃ 377V is loaded by

a parallel connection between the reactances Xmts(v) andXcav(v). The response of the system can be derived byevaluating the amplitude of the reflection coefficient at theterminals of the Thevenin impedance jXL(v)= jXmts(v) kjXcav(v), where “||” denotes parallel connection:

jG vð Þj ¼ Zin � Z0

Zin þ Z0

�������� ¼ jXLðvÞ � Z0

jXLðvÞ þ Z0

��������: ð1Þ

The overall input impedance Zin is a frequencydependent quantity and is purely imaginary. Being Z0 areal valued impedance, the system turns out to be alwaysmismatched, since jXL≠Z0 over the whole real frequencyspectrum. To achieve virtual absorption, that is, perfectimpedance matching from the circuital point-of-view, theimaginary quantity jXL (v) should be converted into a realone and equated to the real characteristic impedance Z0.This can be done by introducing a complex value of theexcitation frequency v, that is, v=vr+ jvi [25]. A complexfrequency excitation is a harmonic signal of frequency vr,whose amplitude temporally varies following an exponen-tial growing or decaying profile proportional to exp(vit).Under such particular excitation condition, the reflectioncoefficient in equation (1) presents some scattering singular-ities in the complex frequency plane. The singularitiesconsist of zero-pole pairs as complex conjugated values.Each pair occurs periodically over the real frequency axis, asshown in [20,21], due to the fact that the cavity exhibitsthe same input reactance for different cavity sizes [29].

On the contrary, using lumped-elements representation(Fig. 1c) where Xmts, Xcav are modelled through a lumpedinductor or capacitor, the equivalent reflection coefficientcomplex frequency plane will lose the periodic behaviour ofscattering singularities, but will preserve the informationon the feasibility of triggering the virtual absorptionphenomenon according to the values of reactancesXmts andXcav.

Fig. 2. Parallel connection of two reactances with the sameelectrical behaviour (a) both inductive and (b) both capacitive.(c) Amplitude of the reflection coefficient of the loaded transmissionline in the complex frequency plane.

A.V. Marini et al.: EPJ Appl. Metamat. 8, 3 (2021) 3

In the following subsections, we consider two relevantcases consisting of two different combinations for themetasurface and cavity reactances: in the former,the two reactances show the same electrical behaviour(Xmts ⋅Xcav> 0), whereas, in the latter, the opposite one(Xmts ⋅Xcav< 0).

2.1 Metasurface and cavity with same electricalbehaviour

Let us now consider the case where Xmts and Xcav have thesame equivalent electrical responses, that is, both capaci-tive or both inductive. In Figures 2a and 2b, we report thetransmission line when the load consists of a parallelconnection of two inductors or two capacitors, and realisesan input impedance Zin= jXL(v), where XL=vL andXL ¼ �1=vC, respectively, with L and C being the totalinductance and capacitance measured at the loadterminals.

The reflection coefficients for the two consideredscenarios can be easily written as:

GL vð Þ ¼ jvt � 1

jvt þ 1; GC vð Þ ¼ �GL vð Þ; ð2Þ

where t assumes the values tL ¼ L=Z0 and tC=Z0C incase of inductive and capacitive load, respectively.

Substituting the complex frequency v=vr+ jvi in equa-tion (2), the amplitude of the reflection coefficients can beplotted in the complex frequency plane, as shown inFigure 2c.

It is interesting to note that, regardless the reactiveload type, the amplitude of the reflection coefficientexhibits two singularities: a pole is in the positive half-space of the imaginary frequency, whereas a zero is in thenegative half-space. Both of them exist for the same realfrequency vr=0 and are complex conjugate quantities.From the physical point of view, the pole represents thecase when the load is reflecting more energy than theimpinging one. In passive systems, this is as a specialscattering condition for which the scattered field decaysslower than the excitation field, giving rise to a virtualgain effect, as shown in [30]. On the contrary, thezero corresponds to a zero-reflection condition, whichbehaves as an indefinite accumulator for the illuminatingsignal, without dissipating its energy but rather storingit within the reactive load. The zero-scattering conditionis kept if the complex frequency is such to approach thezero of the reflection coefficients. As soon as thefrequency of the signal changes, the reflection coefficientassumes a non-zero value and energy leaks out from thereactive load.

By forcing equation (2) to zero, we obtain thatthe reflection coefficient vanishes at the complexfrequencies:

vL;C0 ¼ vr þ jvi ¼ 0� jt�1; witht ¼ tL;C: ð3Þ

The complex frequencies vL;C0 are purely imaginary

(vr=0) in equation (3). Being the zero and the pole acomplex conjugate pair, they have the same imaginaryfrequency amplitude |vi| (Fig. 2c). The signal, then, is notoscillating in time, but simply exponentially growing. Thesteepness in time of the applied signal is given by theexponential factor exp(�vit). This is fully consistent withcircuit theory where the charging and dischargingbehaviours of RC and RL circuits are described by thesame exponential curve with a factor expð�t=tÞ for voltageand current, respectively.

To conclude, when metasurface and cavity exhibit thesame equivalent electrical behaviour, the metasurface-based virtual absorbing condition cannot be achieved, sincethe propagation of the exciting field requires a non-zero realfrequency to propagate.

2.2 Metasurface and cavity with opposite electricalbehaviour

Let us now consider the case where Xmts and Xcavhave opposite equivalent electrical responses, that is,Xmts ⋅Xcav< 0. Therefore, the lossless transmission line isloaded by a reactive network modelled as parallelconnection of an inductor L and capacitor C (Fig. 3a),leading to a load reactance:

jXL ¼ jvL

1� v2LC; ð4Þ

Fig. 3. Quality behaviour of the reflection coefficient amplitudeG in the complex plane showing zeros in lower half-plane and polesin higher half-plane. Respectively zeros are, for a (a) parallelconnection of an LC load case study, (b) purely imaginarywhen t2> 4LC, (c) coincident purely imaginary when t2= 4LC,(d) complex when t2< 4LC.

4 A.V. Marini et al.: EPJ Appl. Metamat. 8, 3 (2021)

Substituting equation (4) into equation (1), we obtain:

G v; tð Þ ¼ � v2LC � 1ð Þ þ jvt

v2LC � 1ð Þ � jvt; ð5Þ

where t ¼ L=Z0 is the circuit time constant definedaccording to circuit theory. By forcing equation (5) tovanish, we can obtain the following perfect matchedcomplex frequencies:

v0 ¼1

j

t±ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit2 � 4LC

p

2LC

!: ð6Þ

It is worth noticing that, at a first glance, equation (6)seems to identify pure imaginary frequencies, as in thecase reported in Section 2.1. However, this is true only ifthe argument of the square root in equation (6) ispositive. Conversely, for negative arguments, the squareroot returns an imaginary quantity, that allows achiev-ing a complex frequency with non-zero real part, that is,harmonic signals at frequency vr with temporalvarying envelope, growing or decaying, according tothe frequency vi. In this case, two distinct regions can beidentified, separated by the special condition for whichthe argument of the square root is zero, that is, t2 = 4LC.If t2≥ 4LC, the complex frequencies in equation (6) arepurely imaginary, as for the case in Section 2.1, whereas,if t2< 4LC, the angular frequency can assume complexvalues.

It is interesting now to plot the amplitude of thereflection coefficient in equation (5) for the three distinctcases: (i) t2> 4LC, (ii) t2= 4LC, and (iii) t2< 4LC, as afunction of the complex frequency. The plots are reportedin Figures 3b–3d. The reflection coefficient exhibits somesingularities in the spectrum, that are located differently ineach considered case:

(i) (t2> 4LC – Fig. 3b) the reflection coefficient has two

pairs of singularities, zeros and poles are symmetrical-ly distributed and lying along the imaginary axis. Allsingularities have zero real frequency.

(ii)

(t2= 4LC – Fig. 3c) the two poles and the two zerosdegenerate, allowing only one zero to be engaged.Again, the singularities are located along the axisvr=0, meaning that the excitation signal is notoscillating with a specific real frequency as desired forachieving matching in microwave networks.

(iii)

(t2< 4LC – Fig. 3d), the reflection coefficient exhibitstwo zeros for complex frequencies with a non-zero realpart.

This last case is of fundamental importance for ouranalysis because it demonstrates that it is theoreticallypossible to achieve zero reflection from a metasurface-based virtual absorber, when the metasurface and thecavity show an opposite electrical behaviour in the loadreactance (i.e., parallel connection of an inductor andcapacitor). This occurs under monochromatic excitationwith a signal at frequency vr, provided that the amplitudeof the wave follows the profile exp(�vit). It is worth

Fig. 4. Amplitude of the incident and released signals of thereactive load for parallel connection of Lmts=23 nH and (a)Lcav=11 nH and (b) Ccav=2.45 pF.

A.V. Marini et al.: EPJ Appl. Metamat. 8, 3 (2021) 5

noting that Figure 3d reports two possible zerossymmetrically distributed with respect to the imaginaryaxis, but only the zero for positive real frequency can beexcited. Hence the left pair of pole-zero can be neglected.

3 Results and verificationThis section is dedicated to the validation of operativeconditions for an arbitrary metasurface-based virtualabsorber as identified in Section 2. In the following,a metasurface with Xmts=100V at f0=692MHz is consid-ered, which will be modelled through an inductorLmts ¼ Xmts= 2pf0ð Þ ¼ 23 nH. The metasurface value iskept in the different configurations, whereas the cavity istime-by-time selected to fit with one of the specific casesdiscussed in Section 2. The simulations are performed byloading a scattering portwith an internal impedance of 377Vwith the different equivalent lumped elementsmodelling themetasurface and cavity. The incident and reflected signalsare recorded directly at the scattering port terminals.

Let us start considering an inductive cavity withelectrical thickness 0.02l0, showing an input reactanceXcav=47.6V, which can be modelled through an inductorLcav ¼ Xcav= 2pf0ð Þ ¼ 11 nH. According to equation (3) theinductance L=Lmts kLcav exhibits zero reflections at theimaginary frequency v1 ¼ 0� j5:1 � 1010

� �rad=s, that is,

a non-oscillating time-varying signal whose amplitudegrows in time, as shown in Figure 4a (orange line). Asexpected, as long as the voltage signal is applied to thecircuit, no reflections occur till the kick-off time t0. Beingthe entire circuit passive, energy cannot be dissipated,allowing only energy storing in the reactive load. As soon asthe signal stops, the zero-reflection condition is not satisfiedanymore, forcing the reactive load to release the storedenergy, as shown in Figure 4a (green line). From theelectrical point of view, this corresponds to the chargingand discharging behaviours of an RL circuit, where theresistive element controlling the damping factor is the free-space impedance Z0. However, this case cannot beimplemented in a realistic metasurface virtual absorber,due to the impossibility to excite it with a propagatingmonochromatic electromagnetic field.

Let us now consider a capacitive cavity with electricalthickness 0.46l0, showing an input reactance Xcav=� 94V,which can be modelled through a capacitorCcav ¼ 1= 2pf0Xcavð Þ ¼ 2:45 pF. The cavity size has beenselected in order to satisfy the condition t2< 4LC, whichenables perfect matching also when the excitation isoscillating with a real frequency vr≠ 0. Indeed, accordingto equation (6), the zero of the reflection coefficient isachieved at the frequency v2= (2pf0� j5.4� 108), that is, amonochromatic signal at frequency f0, with a time-varyinggrowing envelop following the profile exp(5.4� 108t), asshown in Figure 4b (orange line). Again, the voltage signal isvirtually absorbed by the reactive load as for the previouscase, storing the energy in the LC reactive load, andcontrolling the releasing through the kick-off instant of time.It is worth mentioning that, in this case, the system has itsrealistic free-space counterpart, being the exciting signaloscillating in time and supporting propagation as electro-magnetic wave [25].

4 ConclusionIn this paper, we have explored the analytical operationlimits to get virtual absorption for a metasurface-basedabsorber. By investigating on its lumped circuit equivalentloads, we have also found the conditions enabling the perfectmatching of purely reactive loads by exploiting time-varyingcomplex signals for metasurface and cavity reactancesshowing the same (Xmts ⋅Xcav> 0) and the opposite (Xmts ⋅Xcav< 0) electrical behaviour. For these cases, we havetheoretically derived the fundamental limits of theimpedance matching concept based on time-modulatedexcitation signals, in terms of load configurations, and rangesof values for load inductance and capacitance for which theperfect matching can be achieved.

6 A.V. Marini et al.: EPJ Appl. Metamat. 8, 3 (2021)

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Cite this article as: Angelica Viola Marini, Davide Ramaccia, Alessandro Toscano, Filiberto Bilotti, Metasurface virtualabsorbers: unveiling operative conditions through equivalent lumped circuit model, EPJ Appl. Metamat. 8, 3 (2021)