Shadow-free multimers as extreme-performance meta-atoms
SShadow-free multimers as extreme-performance meta-atoms
M. Safari , M. Albooyeh , ∗ , C. R. Simovski , and S. A. Tretyakov Department of Electrical Engineering,Iran University of Science and Technology,Narmak, Tehran, Iran Department of Electrical Engineering and Computer Science,University of California, Irvine, CA 92617, USA Department of Electronics and Nanoengineering,Aalto University, P. O. Box 15500,FI-00076 Aalto, Finland ∗ corresponding authorWe generalize the concept of parity-time symmetric structures with the goal to create meta-atomsexhibiting extraordinary abilities to overcome the presumed limitations in the scattering of overalllossless particles, such as non-zero forward scattering and the equality of scattering and extinctionpowers for all lossless particles. Although the forward scattering amplitude and the extinctioncross section of our proposed meta-atoms vanish, they scatter incident energy into other directions,with controllable directionality. These meta-atoms possess extreme electromagnetic properties notachievable for passive scatterers. As an example, we study meta-atoms consisting of two or threesmall dipole scatterers. We consider possible microwave realizations in the form of short dipoleantennas loaded by lumped elements. The proposed meta-atom empowers extraordinary responseof a shadow-free scatterer and theoretically enables most unusual material properties when used asa building block of an artificial medium. I. INTRODUCTION
The metamaterial paradigm is based on engineeringelectrically (optically) small particles called meta-atoms and exploiting them as optimized ingredients of com-posites with engineered electromagnetic properties (seee.g. [1–5]). The ultimate goal of the metamaterial tech-nology development would be to find means for realiza-tions of any arbitrary material properties, which wouldrequire creation of meta-atoms with any arbitrary elec-tromagnetic response. Basically, within the idealistic sce-nario we would like to be able to engineer and control thepolarizabilities, the scattering cross sections and absorp-tion cross sections of meta-atoms with full freedom. Foran arbitrary particle that is sufficiently small in order tobe described by a pair of the electric and magnetic dipolemoments p and m , the most general linear relations be-tween these moments and the local fields E and H read p = α ee E + α em H , m = α mm H + α me E . (1)Here, α ee , α mm , α em , and α me are, respectively, elec-tric, magnetic, magnetoelectric, and electromagnetic po-larizabilities of the meta-atom, which are scalar valuesfor an isotropic meta-atom and dyadics (tensors) in ananisotropic case [5–7]. Notice that the last two polar-izabilities i.e., α em and α me describe the bianisotropicresponse of the meta-atom which is a measure of cou-pling between the electric (magnetic) response of themeta-atom and the magnetic (electric) excitation field.The ultimate goal would be the full design control overthe values of the four dyadic polarizabilities of the meta-atom.However, the design freedom is limited by fundamentalphysics. For example, the conservation of energy imposes non-zero radiation losses for all passive particles (scatter-ers). Moreover, it dictates that four noted polarizabili-ties of a meta-atom cannot be tuned independently fromeach other [8]. By applying the energy conservation forthe simplest case of a lossless electric dipole polarizablemeta-atom (when α em = α me = α mm = 0) in an incidentelectromagnetic wave one obtains [8–11]1 α ee = Re (cid:18) α ee (cid:19) + j k π(cid:15) , (2)where k and (cid:15) are the ambient wavenumber and per-mittivity, respectively (e.g. [10]). Condition (2) meansthat the imaginary part of the inverse polarizability of alossless dipolar particle is fixed and there is no freedomto engineer it.Next, for a lossless isotropic scatterer both the cou-pling coefficients α em and α me vanish if α ee α mm = 0 [12].Thus, it appears that in order to create magnetic polar-ization in an applied electric field, one must obviouslypolarize the meta-atom electrically and vice versa. In-deed, hypothetical meta-atoms modeled by p = α em H , m = α me E (3)are forbidden if are lossless. The existence of suchmeta-atoms is not compatible with the classical limita-tions based on the energy conservation principle. How-ever, such meta-atoms called purely bianisotropic parti-cles would be extremely interesting and practically use-ful [12].Furthermore, energy conservation considerations leadto the optical theorem which defines a connection be-tween the forward scattering amplitude and the total ex-tinction cross section that is valid for all passive particles. a r X i v : . [ phy s i c s . op ti c s ] N ov In particular, if the particle is passive and absorptive,its forward scattering cross section and the extinctioncross section are not zero even for meta-atoms with ex-otic properties (see e.g. [13–16]). In other words, if aparticle receives some power from the incident waves, itmust create some shadow. This limitation does not allowus to realize an “invisible” meta-atom interacting withthe incident fields and extracting power from them whilecasting no shadow.Recently, a new concept of parity-time (PT) symmetricstructures gained a lot of attention in the literature (e.g.see [17, 18]). PT-symmetric structures possess proper-ties which are invariant with respect to the inversion ofboth spatial coordinates and time. For example, dielec-tric objects are said to be PT-symmetric if the followingsymmetry relation for the complex permittivity holds: (cid:15) ( r ) = (cid:15) ∗ ( − r ) , (4)where ∗ denotes complex conjugation and r is the posi-tion vector. For example, if we fill one half of a spherewith a dielectric with the permittivity (cid:15) = (cid:15) (cid:48) − j(cid:15) (cid:48)(cid:48) and theother half with the material modelled by (cid:15) = (cid:15) (cid:48) + j(cid:15) (cid:48)(cid:48) , thissphere will be a PT-symmetric structure. PT-symmetricobjects appear to be overall lossless, and they seem to beable to overcome the presumed limitations of non-zeroforward scattering and the equality of scattering and ex-tinction powers for passive and lossless scatterers.Basically, one can create a structure where the lossyhalf receives some power from the incident fields whilethe active half re-radiates the same amount of powerinto the forward direction, so that there is no shadowbehind. This property was demonstrated, both theoret-ically and experimentally, for acoustic waves using twospeakers [19]. Does it mean that we are able to realizean “invisible medium” formed by PT-symmetric meta-atoms? If yes, what would be a suitable topology formeta-atoms composing such media? Perhaps, the sim-plest overall lossless electromagnetic structure that wouldapparently be shadow-free is depicted in Fig. 1.Assuming that the polarizabilities of the constituentsof this dimer of electric dipoles are such that the twoinduced dipoles are equal in the amplitude and oscillatewith opposite phases, we see that the forward scatter-ing amplitude is exactly zero while the object scatterssome power in other directions. Realization of this regimeis not possible for any passive scatterer, and this struc-ture is not PT-symmetric either. Although the forwardscattering amplitude and the extinction cross section arezero, the loss is not compensated by gain since the scat-tering cross section is not zero. Clearly, the scatteringpattern depicted in Fig. 1(b) must be accompanied byscattering losses. Thus, without a detailed analysis it re-mains unclear if a PT symmetric structure in free spacecan be shadow-free.Next, the limitation on the values of bianisotropic pa-rameters, which forbids realization of particles obeying(3) also comes from the basic properties of usual loss-less particles. Since the example in Fig. 1 shows that the d l E k zxy P - P (a) | E| (V/m) E k 𝜑 𝜃 (b) FIG. 1: (a) Two oppositely directed dipoles withinduced moments p and − p separated by distance d = λ/
10 ( λ is the operational wavelength) which areexcited with a plane wave whose propagation vector k isnormal to the plane of two dipoles. (b) The scatteringpattern of the configuration in (a) which clearly showstwo nulls both in the forward and backward directions.The three-dimensional radiation pattern of the systemis calculated using full wave simulations (COMSOL).limitation on forward scattering can be overcome, can wefind meta-atoms which would be overall lossless but stillviolate the restrictive conditions [8] realizing constitutiverelations (3)? It would be interesting to see how to realizein practice a shadow-free lossless scatterer and a purelybianisotropic lossless scatterer. These are the questionswe address in this work. II. PARITY-TIME SYMMETRY AND LOSSCOMPENSATION FOR FINITE-SIZE OBJECTSIN UNBOUNDED SPACE
Since we are interested in engineered properties ofsmall particles in open space, we start with a generaldiscussion of the means to overcome the presumed lim-itations of zero forward scattering and the equality ofextinction and scattering by passive and lossless scat-terers using PT-symmetric objects. The known theo-retical and experimental work on PT-symmetric struc-tures which produce no shadow deal with scatterers in awaveguide environment [19]. Recently, similar scenariofor small scatterers in free space has been considered [20]with the conclusion that PT-symmetric dimers enableunusual scattering phenomena, including zero extinctionand large scattering. Here we discuss the concept of PT-symmetry of objects in open space in general, and showthat they cannot be PT-symmetric in the strict sense,leading to definition of shadow-free and loss-compensated scatterers.The top three panels of Fig. 2 illustrate the conven-tional scenario of a PT-symmetric object in a closedwaveguide environment. We denote the characteristicimpedance of the waveguide by η . A PT-symmetric ob-stacle (for instance, a double dielectric layer whose per- (a) (b) (c)(d) (e) (f) 𝑅 rad 𝑅 rad − 𝑅 rad 𝜂 𝜂 − 𝑅 rad rad 𝜂 𝜂 Original structure Time-reversed structure Space-inversed structure G a i nLo ss Lo ss G a i n Lo ss G a i n 𝜂 𝜂 −𝑅 rad FIG. 2: Conceptual illustration of the PT-symmetryconcept in a closed waveguide (a-c) and for a compactscatterer in free space (d-f).mittivity obeys relation (4)) is illuminated by a wave cre-ated by an ideal voltage source. For simplicity we assumethat there are no reflections from the scatterer towardsthe source. Panel (b) shows this structure after the oper-ation of time reversal. The power propagation directionis reversed, the active source becomes power sink and thesource is replaced by a matched load. Next, we apply thespatial coordinate inversion [see panel (c)]. This opera-tion brings the system to its initial state [i.e., panel (a)]and obviously demonstrates that the system is symmet-ric under the two successive operations of time reversaland space inversion (i.e., it is a PT-symmetric system).Let us now consider the same scenario for a compactobject (again, as the same example, a dielectric objectwhose permittivity obeys relation (4)). This case is il-lustrated in panels (d-f). The fundamental differencewith the previous case is that polarizable objects scat-ter power into various directions, which physically meansthat there is dissipation loss at infinity (where the scat-tered power is eventually dissipated), measured by theradiation resistance of the scatterer [denoted by R rad inFig. 2]. In the illustration in panel (d) we assume thatthe scattering is symmetric with respect to left and righthalf-spaces (it does not have to be symmetric, in gen-eral), and conceptually indicate this dissipation loss atinfinity as two absorbing hemispheres with the respectivedistributed surface resistance. The remaining two panelsillustrate the results of time inversion followed by spaceinversion. Obviously, PT symmetry cannot be ensuredbecause the symmetry relation (4) must hold globally,that is, the environment at infinity must have also sym-metrically distributed and balanced gain and loss prop-erties.On the other hand, the numerical example of Fig.1shows that pairs of properly engineered active and lossyscatterers in open space can show properties which arevery similar to those of PT-symmetric objects in waveg-uides, in particular, zero forward scattering amplitudewhile the total scattering cross section is not zero. Wecall these interesting objects shadow-free dimers, and in the next section we consider their properties in some de-tail.The impossibility to achieve true PT-symmetric scat-tering response using complex conjugate permittivitieswas noted in the conference abstract [20]. In that work,the authors added gain to both elements of the dimerto offset the radiation loss, however, we note that evenwith this added gain, the structure does not become PT-symmetric. Scattering by cylinders obeying (4) was stud-ied in [21], where it was assumed that these scattererswere PT-symmetric. III. SHADOW-FREE DIMERSA. Balance of loss and gain and zero forwardscattering
As a simple conceptual example of a small meta-atomwith both lossy and active components we consider a pairof two closely positioned electrically polarizable scatter-ers, similarly to the scenario studied in [20]. Each of thescatterers is approximated as a Hertzian dipole: an elec-tric dipole antenna with the electrically negligible length l and a uniform current along the antenna. We assumethat the two antennas are parallel and the distance be-tween them d is very small compared to the wavelength λ (but still much larger than the negligibly small lengthof each antenna l , i.e., l (cid:28) d (cid:28) λ . A uniform currentdistribution in a short conducting wire can be approxi-mately realized using capacitive caps at the two ends ofthe wire. Alternatively, we can work with ordinary shortwire antennas, replacing in all the following formulas theHertzian dipole length l by the length of one antenna arm l/ Z and Z . In particu-lar, we will be interested in the situations when this pairforms a loss-compensated structure. Thus, we allow thereal parts of the load impedances be positive or negative,so that the absorption and scattering can be balancedwith gain.This system can be analyzed using the antenna theoryand the corresponding equivalent circuit. Let us assumethat the dimer is excited by an external electromagneticplane wave, propagating in the direction normal to thedimer plane. In this case the amplitudes of the externalelectric fields at the positions of the two antennas areequal (we denote the complex amplitude of the incidentelectric fields as E inc ). The currents I , induced on thetwo dipoles obey linear relations I ( Z inp + Z ) + I Z m = E inc l, (5) I ( Z inp + Z ) + I Z m = E inc l. (6)The source voltages are the products of the incident elec-tric fields at the positions of the two dipole antennas andthe effective length of the antennas. The impedances arethe sums of the input impedances of the dipole anten-nas Z inp and the corresponding load impedances Z , .Moreover, the mutual impedance between the two dipolesis denoted by Z m . Knowing all the parameters, we caneasily solve for the induced currents I , and find the in-duced electric and magnetic dipole moments in the dimer(induced dipole moments are proportional to the currentsflowing in the two wire antennas).Let us study the most interesting scenario when theloads are selected so that I = − I , realizing the regimeillustrated in Fig. 1. In this situation, the total inducedelectric moment is zero, but the induced magnetic mo-ment is not zero. Apparently, this dimer would also re-alize a purely bianisotropic particle, as it obeys relations(3). The presumed limitation of non-zero forward scat-tering is violated, at least in the dipole approximation,because the induced magnetic moment is directed alongthe incidence direction and does not radiate in the for-ward direction.From (5) and (6) we can find the two currents I = Z inp + Z − Z m ( Z inp + Z )( Z inp + Z ) − Z E inc l, (7) I = Z inp + Z − Z m ( Z inp + Z )( Z inp + Z ) − Z E inc l, (8)and it is easy to derive the condition on the impedancesunder which I = − I :2 Z inp + Z + Z − Z m = 0 . (9)If this condition is satisfied, the induced currents areequal to I = − I = 2 lZ − Z E inc = lZ − ( Z m − Z inp ) E inc . (10)As noted above, the induced electric dipole of the pairis zero, while the induced magnetic dipole equals m = (cid:82) r × J d r = (cid:0) I ld (cid:1) n where n is the unit vector nor-mal to the plane of the dipole pair [23].Since all the involved impedances are complex num-bers, (9) is in fact a set of two conditions for the re-spective real and imaginary parts. The condition for theimaginary parts (reactances) is always possible to sat-isfy for any dipole antennas and any distance betweenthem by properly choosing the reactances of the loads.This is possible because reactances of passive circuits canbe either positive (inductance) or negative (capacitance),and there is no fundamental limitation on how small orlarge these reactances can be. However, in order to sat-isfy the condition on the real parts of the impedances,we have to allow for negative values of the load resis-tance in at least one of the dipoles. This is obvious fromthe fact that the real part of Z inp is always positive (it is the radiation resistance of the corresponding dipole),and Re( Z m ) < Re( Z inp ), as long as d >
0. This is an ex-pected conclusion, because otherwise we would be able toobtain zero extinction cross section, and hence, acquirethe unattainable regime of zero forward scattering for apassive scatterer.An exciting conclusion at this point is that we can over-come this limitation in an overall lossless dimer, becausethe equality of the total resistance to zero (9) means thatall losses are exactly compensated by gain.To estimate the required dimensions and loadimpedances, we can analytically calculate the inputimpedance and the mutual impedance. Again we stressthat we only need to estimate the corresponding realparts (resistances). The input impedance of a shortdipole is well known, and it reads Z inp = 1 jωC + R loss + R rad , (11)where C is the input capacitance of one of the antennas, R loss is the dissipation loss resistance due to the final con-ductivity of the antenna wires, and R rad is the radiationresistance of the dipole, which reads [24, 25] R rad = η ( kl ) π , (12)where η = (cid:113) µ (cid:15) the free-space wave impedance and l is the effective length of the dipole i.e., for the caseof Hertzian dipole the effective and physical lengths areequal while for the case of short dipole the effective lengthis half of that of the total physical length. This expres-sion for R rad is valid for electrically short dipoles, when l (cid:28) λ . The mutual impedance is, by definition, the ratioof the voltage induced in one of the two antennas if thecurrent in the other one is fixed: Z m = E | r = d lI , (13)where E | r = d is the electric field (the component alongthe dipole axis) created by the first antenna maintain-ing current I at the position of the second antenna (atdistance d ).To calculate this value, we make use of the standardexpression of the electric field of a Hertzian dipole in thedirection θ = π/ E θ = I l πd jkη (cid:20) jkd − kd ) (cid:21) e − jkd , (14)and calculate its real part at distance d :Re( E θ ) = I l πd kη (cid:20) cos( kd ) kd + (cid:18) − kd ) (cid:19) sin( kd ) (cid:21) . (15)While the reactive field (which determines the imaginarypart of the mutual impedance) is very high in the nearfield, the real part of the field at small distances givesa finite and small value (15). Multiplying (15) by l anddividing by I we findRe( Z m ) = η ( kl ) π (cid:20) cos( kd )( kd ) + 1 kd (cid:18) − kd ) (cid:19) sin( kd ) (cid:21) ≈ η ( kl ) π (cid:20) −
15 ( kd ) (cid:21) . (16)The approximate relation is obtained by expanding inthe Taylor series with respect to kd (valid for kd (cid:28) d → Z inp ) = η ( kl ) / (6 π ), and for small finite values of kd it is alwayssmaller than that. Formula (16) agrees with the resultsof Ref.[26], derived using a different approach.Now we are ready to calculate the required dipole loadresistances which correspond to the regime of zero totalinduced electric dipole and zero forward scattering am-plitude ( I = − I ). Defining Z , = R , + jX , , theresult reads: η ( k ld ) π + R + R + 2 R loss = 0 . (17)Clearly, the total negative resistance of the two loadsmust compensate the total dissipation and radiation lossin the system. Note again that the compensation of totalloss does not correspond to a PT-symmetric system in theconventional definition: the two load resistance in the twodipoles are not negative to each other. This is explainedby the fact that even in the absence of dissipation inthe antenna wires, both dipoles always exhibit radiationloss, which also needs to be compensated by an activeload. This is an example of shadow-free scatterers definedabove.In this regime, incident plane waves excite two equaldipole moments with the opposite directions, i.e., p = − p = p = − j I lω z ( z is the unit vector in z -direction).As is seen in Fig. 1(b), the radiation pattern has a nullin both forward and backward directions. Apparently,the particle has zero radar cross section as well as zeroforward cross section. However, it obviously scatters indirections other than the axial ones (here φ = 0 and 180 ◦ according to Fig. 1). The scattered power density can beanalytically solved (see Appendix A) as P scatt = η (cid:18) kI l πr kd sin θ sin ϕ (cid:19) . (18)We see that the scattered power density is zero only inthe directions along the incidence axis ( φ = 0 and π ) andwhen θ = 0 or π .From Eq. (18) we can find the total scattered power ofthe scatterer, which reads P totscatt = η π ( k ld ) I . (19) Normalizing to the amplitude of the incident Poyntingvector P inc = η | E inc | , we find the total scattering crosssection σ sc = k π(cid:15) | α ee | d (20)in term of the electric polarizability α ee of one of thedipoles, or equivalently σ sc = d ( kl ) π η | Z − Z | (21)in term of the load impedances on the dipoles [seeEq. (10)]. It is interesting to compare the scattered powerto the total scattered power from a single dipole with thesame length l and the same current I . The total scat-tering power for such dipole reads (see e.g. [27]) P totdipole = η π ( kl ) I . (22)The ratio of the total scattered powers of the two closelyspaced dipoles with opposite currents and from onedipole equals P normscatt = P totscatt P totdipole = 25 k d . (23)Next, the absorption cross section is found by nor-malizing the total power dissipated in the resistive partsof the loads and lost in the conducting dipole arms i.e, P abs = ( R | I | + R | I | + R loss | I | + R loss | I | ) tothe incident power density. Since at least one of the loadsis active (negative resistance), this power can be zero ornegative. Substituting the current amplitudes from (10),we get σ abs = η (2 l ) R + R + 2 R loss | Z − Z | . (24)Finally, the extinction cross section σ ext = σ sc + σ abs reads σ ext = η l | Z − Z | (cid:18) η k l π d + R + R + 2 R loss (cid:19) . (25)Comparing the value in the brackets of Eq. (25) withEq. (17), one clearly observes that the extinction crosssection equals zero for our proposed shadow-free meta-atom. It means that although the optical theorem is notviolated (i.e., σ Totalext = σ Forwardsc = 0), the total scatteringcross section is not zero and the meta-atom scatters inthe lateral directions rather than the forward one (seee.g. Figs. 1 and 3).Let us next study the resonant dependence of the scat-tering cross sections (21) on the load parameters. Equa-tion (10) tells that, counter-intuitively, the currents inthe two out-of-phase dipoles tend to infinity when thetwo loads become identical : i.e., when Z − Z → the same incident field are opposite inphase. Actually, from equations (7) and (8) we see thatif Z − Z → Z L = Z + a and Z L = Z − a , where Z = Z m − Z inp (so that for any complex value of param-eter a we satisfy Eq. (9), ensuring that the current modeis antisymmetric). A simple calculation shows that for a → I , ∼ ± a (26)which results from the ratio a/a . The argument of thecomplex parameter a determines the phase of the inducedcurrents in relation to the phase of the incident field.Thus, we see that in the vicinity of this resonant pointthe current amplitudes can take arbitrary high values andthe particle cross sections have no upper bound.This shadow-free particle can be classified as a recip-rocal bianisotropic particle with the omega type of mag-netoelectric coupling [6]. More specifically, the propertyof having zero co-polarizabilities is similar to the prop-erty of omega nihility composite materials [28], althoughhere we study single meta-atoms while in [28], effectivelyhomogeneous materials are considered. It is interestingto compare these results to the conclusions of paper 29,where passive or active particles have been studied. Inthat paper it is shown that zero forward and back scat-tering from a single omega particle is possible only if theparticle is active. Here we see that it is possible also foroverall lossless particles, provided that the loss is bal-anced with gain. On the other hand, it is important tostress that the gain compensates the total loss, includingscattering loss (radiation damping). Thus, as discussedabove, the loads are not exactly symmetric: the load re-sistances of the two dipoles are not exactly negative withrespect to each other, as is usually required in the defi-nition of PT-symmetric systems (4). We expect that thesymmetry of loads will be exact in the case of a periodicalsubwavelength arrays of such dimers, where the scatter-ing loss is compensated by particle interactions [10, 30],or in waveguide set-ups, where scattering is preventedby waveguide walls (as in the acoustical experiments de-scribed in [19]). B. Numerical example: Finite-size stronglycoupled shadow-free dimers
Next we study a particular system of two electricallysmall but finite-length loaded dipole antennas and dropthe assumption that the distance between the two anten-nas is large compared with the dipole length (Fig. 3).In this case we can bring the two antennas very close to each other, so that the pair can be treated as a sin-gle composite meta-atom, whose transverse size is of thesame order as the height. First we use the approximateformulas derived in the previous section to find a suit-able pair of loading resistances for a given pair of shortwire dipoles of length l at distance d . Note that for ashort dipole antenna the effective length is equal to thehalf-length of the antenna, and in our definition, we candirectly use equation (17) where l is the half-length. Toestimate the needed reactive loads, we use the known ap-proximate formula for the input reactance capacitance ofa short wire dipole [27, 31] X inp = − η ln( l/r ) − π tan πlλ (27)( r is the wire radius and l is the effective length of theantenna). The mutual reactance can be approximatelyfound in the model of Hertzian dipole interactions, as inthe previous section. Namely, we calculate the imagi-nary part of (14) at a small distance d and find the mu-tual reactance in the same way as we found the mutualresistance. The result readsIm( Z m ) = η ( kl ) π (cid:34)(cid:32) kd − kd ) (cid:33) cos( kd ) − kd ) sin( kd ) (cid:35) ≈ η ( kl ) π ( kd ) (cid:34) − kd ) − kd ) (cid:35) . (28)In order to realize the zero forward scattering regimewe need to satisfy condition (9). It is a complex-valuedequation, therefore, we need to solve it for both realand imaginary parts. We can freely choose the lengthand radius of each dipole, staying within the assump-tions of a small scatterer. To stay within the short-dipole approximation, we choose l = λ/
20 and r = l/
50 for both dipoles. Then, by using (12), (16), (27),and (28), we find the input and mutual impedances Z inp = (1 . − j . Z m = (0 . − j . Z inp = (1 . − j . Z m =(0 . − j . Z and Z . We may freely choose one of the impedancesand calculate the other one. It may seem to be enoughto set one load impedance to zero and find the other one,however, it is not a wise selection because to bring thedipole to resonance (without a reactive load) we oughtto go beyond the short-dipole regime. Thus, to find therequired values for load impedances we calculate the in-duced currents and scattering patterns numerically andfine-tune the loads to realize the regime with zero forwardscattering. The results for the required impedances aresummarized in Table I before and after numerical tuning. | E| (V/m) Z Z d E k(a) (b) E k zxy 𝜑 𝜃 (c) FIG. 3: (a) Schematic of two short dipole antennas with length l = λ/
20 ( λ is the operational wavelength) andradius r = l/
50, separated by distance d = 6 l and loaded by impedances Z and Z . The plane of the two dipolesis normal to that of the exciting plane wave incidence direction and the dipoles’ material is considered to be perfectelectric conductor (PEC). (b) Far-field scattering pattern of the configuration in (a) which clearly shows nulls bothin the forward and backward directions. (c) Electric near-field distribution of configuration (a).TABLE I: The required impedances for negligibleforward scattering in the case of a finite-size stronglycoupled loss-compensated dimer. The distance betweenthe dipoles is chosen to be three times of the dipolelength. Impedance initially chosen values after fine-tuning Z (Ω) − .
816 + j . − . j . Z (Ω) 0 . j . . j . The radiation pattern and the local electric field dis-tribution for the chosen load impedance of Table I areshown in Fig. 3. Note that in the exact numerical solu-tion the regime of zero forward scattering does not im-ply that the total electric dipole moment of the pair iszero, because higher-order modes also contribute to scat-tering in all directions. From Fig. 3(b), one clearly ob-serves the nulls in both forward and backward directions:the pair of a lossy (Re( Z ) >
0) and an active dipole(Re( Z ) <
0) obviously allows us to realize a shadow-free meta-atom. Figure 3(c) presents the local electricfield distribution in the vicinity of the two dipoles. As itis clear from this figure, the electric-field vectors aroundthe two dipoles are equal but have opposite directionswhich leads to a non-zero curl of the electric fields that es-sentially demonstrates the presence of an equivalent mag-netic dipole moment. That is, with the proper design, wehave suppressed the electric dipole moment of the meta-atom while have kept its magnetic dipole. This leads tothe extraordinary scattering in the lateral direction froman overall lossless meta-atom.Next, we illustrate the resonant enhancement of scat-tering close to the point where Z − Z →
0. We in-troduce deviations Z L = Z + a and Z L = Z − a , where Z = Z m − Z inp from the optimized values of the loadimpedances and plot the dependence of the total scatter-ing cross section on the absolute value of a . The resultsare shown in Fig. 4. Numerical simulations do not show |𝑎 | (Ω) 𝜎 s c ( m ) FIG. 4: Total scattering cross section obtained fromEq. (21), showing the resonant response of ashadow-free dimer when Z − Z →
0. The values forthe impedances are Z − a = Z + a = − . j . a is considered to be areal-valued variable.unbounded growth of the scattering amplitude: comingvery close to the resonant point, the numerical solutionbecomes not accurate. IV. SHADOW-FREE DIMER IN THEINCIDENCE PLANE
Next, let us study the same dimer as in Sec. III butexcited by a plane wave travelling in the dimer plane,orthogonal to the dipole axes. We are interested in theregime where the forward scattering is absent, while thecurrents in the dipoles are different from zero. Now theexternal fields exciting the two dipole are different inphase, and the equations for the induced currents takethe form I ( Z inp + Z ) + I Z m = E inc l, (29) I ( Z inp + Z ) + I Z m = E inc e − jkd l. (30)Here we assume that the incident plane wave propagatesalong the line from dipole 1 to dipole 2. The conditionfor zero forward scattering reads in this case I = − I e − jkd , (31) I = − I e jkd = 2 lZ − Z + 2 jZ m sin kd E inc , (32)which corresponds to the following relation between theimpedances2 Z inp + Z + Z − Z m cos kd = 0 . (33)Next, by using Eq. (15), we find the condition for therequired load resistances: η π ( k ld ) + R + R + 2 R loss = 0 . (34)To analyze this structure, we consider the same exampleas in the previous case, only assuming that the wave vec-tor k is in the plane of the two dipoles and the polariza-tion of the incident electric field matches the orientationof the two antennas. The results for the required loadimpedances are summarized in Table II.TABLE II: The required impedances for negligibleforward scattering when the loss-compensated dimerand the incidence direction are in the same plane. Impedance analytically found values after fine-tuning Z (Ω) − .
952 + j . − .
339 + j . Z (Ω) 0 . j . . j . The far-field radiation pattern and the electric fielddistribution in the vicinity of the considered scattererare shown in Fig. 5(b) and (c). As it is clear, we againobserve the regime of zero forward scattering while thescattering cross section is not zero. It can be shown thatthe scattered power of the proposed system of coupleddipoles when I = − I e jkd reads (see Appendix B) P scat = η (cid:20) kI l πr kd sin θ (1 − sin θ sin ϕ ) (cid:21) . (35)This is exactly what is observed from the full-wave sim-ulations in Fig. 5(b) i.e., a null of the scattered powerin forward direction φ = 90 ◦ and θ = 90 ◦ (a shadow-free meta-atom) and a maximum in the backward direc-tion φ = − ◦ and θ = 90 ◦ . Moreover, when θ = 0 or θ = 180 ◦ the radiation pattern (35) experiences two nullswhich again are clearly observable from the full-wave sim-ulations in Fig. 5(b) which are due to the presence of aquadrupole in the coupled system of dipoles. This systemis analogous to the one studied in [19] in the acoustical case, but the difference is that in [19] the passive andactive parts were in a closed waveguide, while in our casewe consider an isolated dimer scatterer in free space.From Eq. (35), the total scattered power of the pro-posed structure in this case reads P totscatt = η π ( k dl ) I (36)and the normalized total scattered power with respect tothe total power of a dipole [see (22)] reads P normscatt = P totscatt P totdipole = 75 k d . (37)The total scattering and extinction cross sections read σ sc = 730 π ( k ld ) (cid:12)(cid:12)(cid:12)(cid:12) η I E inc (cid:12)(cid:12)(cid:12)(cid:12) (38)and σ ext = η (cid:12)(cid:12)(cid:12)(cid:12) I E inc (cid:12)(cid:12)(cid:12)(cid:12) (cid:20) η π ( k ld ) + R + R + 2 R loss (cid:21) , (39)respectively, and I is defined in (32). Similar to the pre-vious scenario in Sec. III B, the extinction cross sectionis zero in this case [compare Eqs. (39) and (34)]. Thismeans that the scattering losses is fully compensated bythe introduction of the resistive (active and passive) loadimpedances. Although the present dimer still possessesa shadow-free characteristic, it mainly backscatters un-like the previous scenario which was scattering laterally(with respect to the incidence direction). V. SHADOW-FREE TRIMERS:SUBWAVELENGTH SUPERDIRECTIVESCATTERERS
In Section III we have symmetrically redirected the in-cident power into the lateral directions (with respect tothe incident wave direction) by using shadow-free dimers[see Figs. 1 and 3]. In those cases, as we show next, al-though the field amplitude was symmetrically redirectedto the two opposite lateral directions, the phases of thefields were opposite in sign. In this section, we go be-yond the limit of symmetrical power distribution andshow that the power amplitude can be tuned asymmet-rically in the opposite lateral directions [the y -direction,see e.g. Fig. 1]. In particular, we present the extremecase of a subwavelength superdirective scatterer whereall the received power is redirected into one side only. A. Symmetrical pattern of shadow-free dimers
We first consider the dimer example and prove thatit is impossible to asymmetrically tune the pattern in | E | (V/m) Z Z d2l Ek (a) (b) E k zx y 𝜑 𝜃 (c) FIG. 5: (a) Schematic of two short dipole antennas with the length l = λ/
20 ( λ is the operational wavelength) andthe radius r = l/
50, separated by the distance d = 6 l and loaded by impedances Z and Z . The dimer is locatedin the incidence plane and the dipoles are normal to the wave vector k . (b) Far-field scattering pattern of theconfiguration in (a) which clearly shows nulls both in the forward and backward directions. (c) Electric near-fielddistribution of the arrangement shown in (a).the lateral directions while the forward and backwardscatterings are canceled. That is, when one imposes thesimultaneous zero forward and backward scattering con-dition, then the lateral distribution of power is uncondi-tionally symmetric. To demonstrate that, let us considerthe vector potential of a dimer [see Fig. 8] as is discussedin Appendix A, i.e., A z = µ l π e − jkr r ( I e jkd sin θ sin ϕ + I e jkd sin θ sin ϕ ) . (40)Notice that d and d are the distances of the two dipolesfrom the origin of the coordinate system and, in our pre-vious examples, we had always considered d = d = d/ φ = 0) and backward ( φ = π )scattering, we require I = − I = I , as we discussed ear-lier, and therefore A z = µ lI π e − jkr r (e jkd sin θ sin ϕ − e jkd sin θ sin ϕ ) . (41)Our goal is to obtain asymmetric patterns in the oppo-site lateral directions φ = π/ φ = − π/ θ = π/ φ = π/ (cid:0) e jkd − e jkd (cid:1) , that of φ = − π/ (cid:0) e − jkd − e − jkd (cid:1) . Therefore, the amplitudesof the pattern in the opposite lateral directions φ = ± π/ (cid:12)(cid:12) e jkd − e jkd (cid:12)(cid:12) = (cid:12)(cid:12) e − jkd − e − jkd (cid:12)(cid:12) while the phases are not, i.e., ∠ (cid:0) e jkd − e jkd (cid:1) (cid:54) = ∠ (cid:0) e − jkd − e − jkd (cid:1) . Moreover, the phases only differby sign, i.e., ∠ (cid:0) e jkd − e − jkd (cid:1) = − ∠ (cid:0) e − jkd − e jkd (cid:1) , ifwe consider d = − d = d . Note that in Refs. 32 and33, asymmetric scattering patterns for dimers operatingclose to the PT-symmetry point were predicted, however,such asymmetry is only possible when the induced polar-izations in the two dimer elements are different (whichis the case in papers 32 and 33), and the loss and gainare not fully balanced. Next, we present an alternativeapproach to overcome this limitation. B. Asymmetrical scattering patterns ofloss-compensated trimers
We expect that one of the exciting possibilities for con-trolling scattering fields using loss-compensated scatter-ers will be a possibility to send the scattered power intoa specific lateral direction, orthogonal to the illumina-tion direction. The dimers considered above exhibit fullcontrol over scattering in the forward and backward di-rections, but in the regime of zero forward scattering theenergy is symmetrically scattered in the lateral directions(see Fig. 1).Here we show that it is possible to tune the amplitudesof the waves scattered in the opposite lateral directions.The shadow-free dimers have a symmetric scattering pat-tern in the lateral plane because the two dipoles were re-quired to have electric dipoles with equal amplitudes andopposite phases (i.e., p = − p = p ) to cancel out theoverall electric dipole moment while generating a puremagnetic dipole moment in the system, correspondingto simultaneously zero forward and backward scatteringamplitudes. In order to overcome this limitation, we addone more dipole to the system, creating possibilities totune the total dipole moment of the system to zero (i.e., p + p + p = 0) in an asymmetric system. Here weconsider the extreme case where the ratio of the scatter-ing amplitudes along the two opposite lateral directions( φ = π/ φ = − π/
2) is infinite or zero. This corre-sponds to total cancellation of scattering into one of theside half-spaces. In terms of the antenna theory, such anobject is a sub-wavelength superdirective scatterer.Without loss of generality and for simplicity, we con-sider three equispaced dipoles (see Fig. 6). As shown inAppendix B, the scattered power of this trimer systemreads P scatt = η (cid:18) kI l πr (cid:19) ( kd ) (cid:0) sin θ sin ϕ (cid:1) (1 − sin θ sin ϕ ) . (42)The total scattering power of the proposed structure in0 | E| (V/m) E k 𝜑 𝜃 z d x y P d P P FIG. 6: (a) Schematic of the trimer system of dipoleswith induced moments p , p , and p separated bydistances d = λ/
20 ( λ is the operational wavelength)which are excited with a plane wave whose propagationvector k is normal to the plane of two dipoles. Thescattering pattern of the configuration clearly showsthree nulls in the forward, backward, and one lateraldirection . this case reads P totscatt = η π ( k d l ) I , (43)and the normalized total scattered power with respectto the total power of a dipole with the same current[see (22)] reads P normscatt = P totscatt P totdipole = 4641155 ( kd ) ≈ . kd ) . (44)As is proven in Appendix C, to obtain such scatteredpower, we need to satisfy the conditions I = e − jkd I , I = − (cid:0) e − jkd (cid:1) I (45)for the trimer currents. The radiation pattern of thissystem is plotted in Fig. 6. As is clear from this figure,the pattern has three nulls in θ = π/ , φ = 0 , π/ , π withthe main beam directed along θ = π/ , φ = − π/
2. Thisbehavior is also inferred from Eq. (42).Next, similarly to what we have performed to ob-tain (9), we derive the condition required for an over-all lossless trimer system of loaded dipoles with bothactive and passive loads (loss-compensated, shadow-freetrimers) to generate such superdirective patterns. Weconsider mutual impedances of Z ij between the i -th and j -th ( i, j = 1 , ,
3) elements and equal input impedancesof equi-sized dipoles. Moreover, the loads are denotedas Z iL . The three equations for induced currents in theloaded trimer system of dipoles read I ( Z inp + Z ) + I Z + I Z = E inc l, (46) I Z + I ( Z inp + Z ) + I Z = E inc l, (47) I Z + I Z + I ( Z inp + Z ) = E inc l. (48) Next, by using the required currents of Eq. (45) andconsidering the mutual impedances of Z m , d and Z m , between the closer (with distance d ) and farther (withdistance 2 d ) dipoles, respectively, the above equationsreduce to( e − jkd − Z inp + Z ) + ( e jkd − e − jkd ) Z m , d +(1 − e jkd ) Z m , = E inc lI , (49)( e − jkd − Z m , d + ( e jkd − e − jkd )( Z inp + Z )+(1 − e jkd ) Z m , d = E inc lI , (50)( e − jkd − Z m , + ( e jkd − e − jkd ) Z m , d +(1 − e jkd )( Z inp + Z ) = E inc lI . (51)Notice that Z m , d and Z m , can be derived from (16)and (28). Finally, combining the above equations leadsto the conditions e − jkd Z − Z = (1 − e − jkd )( Z inp − Z m , ) (52) e − jkd Z + (1 + e − jkd ) Z = 2(1 + e − jkd ) Z m , d − (1 + 2 e − jkd ) Z inp − Z m , , (53)which imply simultaneous absence of forward and back-ward scattering and a unidirectional scattering patternin the lateral plane. Similarly to the analysis in Sec. IIIit is possible to derive the required active-passive loadsfor the trimer system to realize these properties.As a particular example, we consider three short dipoleantennas of equal length loaded by three bulk impedances Z , Z , and Z , as shown in Fig. 7. We assume similarphysical parameters for the antennas as in the previouscase, i.e., l = λ/
20 and r = l/
50 for all the dipoles, and d = 0 . λ . Next, by using (12), (16), (27), and (28), wefind the mutual impedances Z m , d = (1 . − j . Z m , = (0 . − j . Z m , d = (1 . − j . Z m , = (0 . − j . | E| (V/m) Z Z d E k(a) (b) E k zxy 𝜑 𝜃 Z d2l FIG. 7: (a) Schematic of three short dipole antennas with the length l = λ/
20 ( λ is the operational wavelength) andthe radius r = l/
50, separated by the distance d = 3 l and loaded by bulk impedances Z , Z , and Z . Thestructure is excited by a plane wave. (b) Far-field scattering pattern of the configuration in (a) which clearly showsunidirectional pattern in a lateral direction. (c) Electric near-field distribution of the scheme in (a).TABLE III: The required load impedances for thesimultaneous absence of forward and backwardscattering and unidirectional pattern in the lateraldirection in the case of a loss-compensated trimer. Impedance analytically estimated values after fine-tuning Z (Ω) 0 + j .
511 0 . j . Z (Ω) − .
665 + j . − . j . Z (Ω) 1 .
456 + j .
274 1 . j . Obviously, since we control the lateral scattering pat-tern of a shadow-free meta-atom, we need simultane-ous presence of both active (Re( Z ) <
0) and pas-sive (Re( Z , ) >
0) loads. The radiation pattern andthe local electric field distribution for the chosen loadimpedances of Table III are shown in Fig. 3. FromFig. 7(b), one clearly observes scattering nulls in theforward, backward, and one lateral direction. That is,our loss (Re( Z , ) >
0) and gain (Re( Z ) < VI. DISCUSSION AND CONCLUSION
We have introduced the concept of shadow-free or loss-compensated meta-atoms enabling extraordinary controlover scattering properties. We have demonstrated onnumerical examples that the scattering response controlfreedom of these meta-atoms is not limited by the com-monly adopted restrictions. We have benefited from thecombination of lossy and active impedances as the loadsfor two closely spaced dipole antennas to compensatethe loss of one scatterer with the gain of another and,hence, to suppress the forward scattering of the over-all lossless meta-atoms while preserving non-zero radia-tion towards other directions. Moreover, we have demon- strated that within this paradigm it becomes possible tocreate purely bianisotropic meta-atoms, where the onlyexisting polarization mechanism is the magnetoelectriccoupling. Furthermore, generalizing the proposed sce-nario to dipole trimers, it becomes possible to shape thescattering pattern in the lateral plane, pushing the scat-tered power aside from the propagation direction of theincident waves and providing end-fire superdirective ra-diation properties.The proposed meta-atoms can be employed in thedesign of engineered materials with extraordinary elec-tromagnetic and optical properties, where, for instance,magnetic response is created by external high-frequencyelectric fields. As another example, materials with unitypermittivity and permeability and non-zero and res-onant chirality coefficient or omega coupling parame-ter. The extreme values of optical parameters are re-alized by exploiting combinations of passive and activeimpedances which serve as the loads in our coupled-dipole systems. The introduced shadow-free meta-atomis hopefully defining a new paradigm in engineering ma-terials with extraordinary properties which are otherwiseimpossible to achieve. Indeed, due to the advent of newtechniques in the compact and efficient design of activenetworks, the realization of our proposed scheme is astraightforward task at radio and microwave frequencies,although special cares should be taken to ensure stabilityof the active components [34, 35].
Appendix A: Calculation of scattered power fromtwo oppositely oriented closely spaced dipoles
Here we derive the expression for scattered power den-sity of two closely spaced dipole antennas with equal am-plitudes and opposite phases of the antenna currents (i.e., I = − I ). The geometry of the problem is illustratedin Fig. 8. The vector potential for two dipoles of Fig. 8reads A z = µ I l π (cid:18) e − jkr r − e − jkr r (cid:19) , (A1)2 z yx 𝜑𝜃 𝜒 dl I I Observation point r r r z yx FIG. 8: Geometry of the problemwere r and r are the distances from each dipole, re-spectively, to the observation point. Next, for the phaseterm, we assume r (cid:39) r − d cos χ and r (cid:39) r + d cos χ (where cos χ = sin θ sin ϕ ) while for the amplitude termswe assume r (cid:39) r and r (cid:39) r . Therefore, the vectorpotential of the system of two dipole reads A z = µ I l π (cid:34) e − jk ( r − d cos χ ) r − e − jk ( r + d cos χ ) r (cid:35) (A2)or, equivalently, A z = j µ I l πr e − jkr sin (cid:18) k d θ sin ϕ (cid:19) . (A3)Since kd (cid:28)
1, we can approximate the vector potentialas A z (cid:39) j µ I l πr e − jkr kd sin θ sin ϕ. (A4)Next, by using relations H = µ ∇ × A and E = jω(cid:15) ∇ × H , we can find the scattered far-fields as E θ = η kI l πr e − jkr kd sin θ sin ϕ,H ϕ = kI l πr e − jkr kd sin θ sin ϕ. (A5)The scattered power density reads P scat = 12 Re( E θ H ∗ ϕ ) = η (cid:18) kI l πr (cid:19) ( kd ) sin θ sin ϕ, (A6)which is Eq. (18). Eq. (18) [or (A6)] demonstrates thatforward and radar cross section are zero and this resultis in agreement with the simulation since the total (inte-grated over all space) scattering cross section is not zero. Appendix B: Calculation of scattered power of twooppositely oriented closely spaced dipoles with aphase difference πλ d If we excite the system of coupled dipoles in theirplane, i.e., the electric field and the propagation vectorof the excitation field lie in the plane of dipoles, then weface a similar problem as in the previous section. How-ever, in this case the two dipoles have an extra phasedifference of πλ d rather than only a 180 ◦ phase differ-ence (i.e., I = − I e jkd ). In this case for kd (cid:28)
1, thevector potential approximately reads A z = j µ I l πr e − jkr kd (1 − sin θ sin ϕ ) (B1)and, therefore, we can find the scattered far-field as E θ = η kI l πr e − jkr kd sin θ (1 − sin θ sin ϕ ) ,H ϕ = kI l πr e − jkr kd sin θ (1 − sin θ sin ϕ ) , (B2)and the scattered power reads P scat = 12 Re( E scat H ∗ scat )= η (cid:18) kI l πr (cid:19) ( kd ) [sin θ (1 − sin θ sin ϕ )] , (B3)which is Eq. (35). In this case the scattered power densityequals zero only for φ = 90 ◦ (i.e., the forward direction)in the φ -plane (i.e., θ = 90 ◦ ). Moreover, the obtainedpower density is in full agreement with Fig. 5 since italso possesses two zeros for θ = 0 and 180 ◦ due to thepresence of a quadrupole. Appendix C: Calculation of scattered power fromthree closely spaced dipoles with unequal currents
Here we derive the expression for scattered power den-sity from three equispaced dipole antennas with unequalcurrents I , I , and I . The geometry of the problem isillustrated in Fig. 9. Similarly to Appendix A, the vectorpotential of the three dipoles read A z = µ l π (cid:18) I e − jkr r + I e − jkr r + I e − jkr r (cid:19) , (C1)where r i (cid:39) r − d i cos χ . Again, in the far-field zone wehave r (cid:39) r (cid:39) r (cid:39) r for the denominators. Next, sincecos χ = sin θ sin ϕ , we get A z = µ l π e − jkr r (cid:0) I e jkd sin θ sin ϕ + I e jkd sin θ sin ϕ + I e jkd sin θ sin ϕ (cid:1) . (C2)Therefore, from (C2), we have two parameters to tunein order to synthesize required patterns, i.e., the currents3 z yx 𝜑𝜃 𝜒 d l I I Observation point r r r z yx I d FIG. 9: Geometry of the problem I i and distances d i . We require zero forward ( φ = 0) andbackward ( φ = π ) scattering, which implies the cancella-tion of A z | φ =0 ,π = 0, that is, I + I + I = 0 . (C3)This condition simply means that the overall electricdipole moment of the trimer system is zero. For sim-plicity, we may position one dipole at the origin, and theother two dipoles symmetrically spaced ( d = d = d )along the y -axis, as is shown in Fig. 9, which reducesthe number of tuning parameters. The vector potentialreduces to A z = µ l π e − jkr r (cid:0) I e jkd sin θ sin ϕ + I + I e − jkd sin θ sin ϕ (cid:1) . (C4)Now, by using (C4) one is able to asymmetrically tunethe radiation pattern in the opposite directions φ = ± π/ θ = π/
2. As mentioned in the main text, we are in-terested in an extreme case when all the radiated poweris directed to one lateral direction which implies superdi-rectivity. To ensure that, we choose the currents in away that the radiation from these elements cancels outin either one of the lateral directions, e.g., in θ = π/ φ = π/
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