Cloaking of Arbitrarily-Shaped Objects with Homogeneous Coatings
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Cloaking of Arbitrarily-Shaped Objects with Homogeneous Coatings
Carlo Forestiere,
1, 2, ∗ Luca Dal Negro, and Giovanni Miano Department of Electrical and Computer Engineering & Photonics Center,Boston University, 8 Saint Mary’s Street, Boston, Massachusetts Department of Electrical Engineering and Information Technology,Universit`a degli Studi di Napoli Federico II, via Claudio 21, Napoli, 80125, Italy
We present a theory for the cloaking of arbitrarily-shaped objects and demonstrate electromag-netic scattering-cancellation through designed homogeneous coatings. First, in the small-particlelimit, we expand the dipole moment of a coated object in terms of its resonant modes. By zeroingthe numerator of the resulting rational function, we accurately predict the permittivity values of thecoating layer that abates the total scattered power. Then, we extend the applicability of the methodbeyond the small-particle limit, deriving the radiation corrections of the scattering-cancellation per-mittivity within a perturbation approach. Our method permits the design of “invisibility cloaks”for irregularly-shaped devices such as complex sensors and detectors.
Arguably the most studied inverse scattering prob-lem, the design of a cloaking environment that drasti-cally reduces or ideally cancels the electromagnetic scat-tering of a given object, fascinates scientists and engi-neers from a broad variety of disciplines. The most pop-ular cloaking approaches developed in the past decadeare based on transformation-optics and scattering-cancellation and have both been experimentally val-idated at microwaves . In particular, the latter tech-nique presents several practical advantages, since itonly makes use of homogeneous and isotropic materials,which are easier to fabricate than the inhomogeneousand anisotropic media required by the transformation-optics approach. In addition, the interior of the cloakedobject supports non-vanishing fields: this fact has in-spired the seminal idea of a cloaked-sensor which candrastically reduce the perturbation introduced by themeasuring apparatus on the physical quantity underinvestigation . However, a rigorous solution of thescattering-cancellation problem exists only for simpleshapes, such as cylinders or spheres due to the substantialdifficulties in tackling the inverse scattering problem inthe presence of non-canonical geometries. This fact mayprevent the application of the scattering-cancellation ap-proach to real-life problems since the shape of the objectto be cloaked is in general not under the control of thedesigner of the cloaking system. So far, this problemhas been circumvented by resorting to numerical opti-mization techniques . Unfortunately, besides the lackof physical understanding, numerical optimization usu-ally presents a high computational burden, requiring alarge number of iterations of the direct electromagneticproblem. In addition, it is also well-known that numericaloptimization techniques can be trapped in local minima.This fact is particularly relevant to the problem at handdue to the existence of multiple solution of different qual-ity. Deriving general design techniques for the electro-magnetic properties of the homogeneous coating of an ar-bitrarily shaped object to achieve scattering-cancellationremains a grand challenge.In this paper, we address this problem introducing ageneral theory of scattering-cancellation from an arbi- trarily shaped object with a homogeneous coating. Ourtheory enables the rigorous design of the permittivity ofthe cover of an arbitrarily shaped object to achieve cloak-ing or “invisibility” in the limit of low-losses. The prob-lem is tackled in two-steps. First, in the small-particlelimit (Rayleigh regime), we expand the dipole moment ofthe coated object in terms of its electrostatic modes, us-ing the theoretical framework developed in Refs. . Bynoting that the obtained expansion is a rational functionof the dielectric permittivity of the coating, we can deter-mine the scattering-cancellation condition by zeroing itsnumerator. Next, in order to extend our method beyondthe Rayleigh regime, we derive the radiation correctionsof the scattering-cancellation permittivity by using theperturbation approach introduced by Mayergoyz et al.in Ref. . In particular, the first- and second-orderradiation corrections of the scattering-cancellation per-mittivity are found by zeroing the corresponding-orderperturbation of the dipole moment. Then, we validateour method and estimate its accuracy by designing thesusceptibility of the cover of a sphere of several electricsizes. Finally, we design the invisibility cloaking of aC-shaped particle, showing that, by using the radiationcorrections, the scattering-cancellation permittivity canbe accurately predicted even for objects of size compara-ble to the incident wavelength. I. SMALL-PARTICLE LIMIT
We start by considering a core-shell object of arbi-trary shape sketched in Fig. 1 (a), and embedded infree-space. The core is assumed to be made of a linear,homogeneous, isotropic, lossless medium with relativepermittivity ε r, ∈ R and corresponding susceptibility χ = ε r, −
1, whereas the shell is composed by a linear,homogeneous, isotropic, and time-dispersive material ex-hibiting a complex permittivity ε r, ( ω ) = ε ′ r, − jε ′′ r, (atime-harmonic dependence e jωt has been assumed) and acorresponding susceptibility χ ( ω ) = ε r, −
1. We denotewith V and V the volumes occupied by the core and theshell, respectively, and with V the external space. Wealso denote with S and S the surfaces separating theshell with the core and with the external space, respec-tively. Both the outward-pointing normals to the twosurfaces S and S are indicated with n . Assuming thatthe investigated system is much smaller than the wave-length of operation we employ the quasi-electrostatic ap-proximation of the Maxwell’s equations. FIG. 1. Sketch of the studied core-shell geometry (a). In-vestigated spherical shell (b). Investigated coated C-shapedobject where only the portion of the boundary surfaces with z < x <
The source-free electric field that may exist in the pres-ence of a dielectric coating with ε ′ r, < σ ′ and σ ′′ distributed on S and S , re-spectively. They are solution of the following homoge-neous boundary integral equation : A (cid:20) σ ′ σ ′′ (cid:21) = β B (cid:20) σ ′ σ ′′ (cid:21) , (1)where A and B are endomorphisms of the vector space L ( S ) × L ( S ) defined as: A = (cid:20) ε r, ( L − I ) ε r, L −L ( −L − I ) (cid:21) , (2) B = (cid:20) ( L + I ) L −L ( −L + I ) (cid:21) , (3) L uv : L ( S v ) → L ( S u ) is defined ∀ u, v ∈ { , } as L uv { σ } ( Q ) = 12 π I S v σ ( M ) r MQ · n Q r MQ dS M , Q ∈ S u (4)and I is the identity operator. Equation 1 defines a gen-eralized eigenvalue problem in β for the operator pair( A , B ). As the operators A and B are not self-adjoint,the eigenmodes of the problem (1) are not orthogonal.Therefore, we consider the following problem: A † (cid:20) τ ′ τ ′′ (cid:21) = β B † (cid:20) τ ′ τ ′′ (cid:21) , (5) where A† and B† are the adjoint operators of A and B .Since the operators A and B are compact, problems 1and 5 support discrete spectra. Moreover, problems 1and 5 share the same eigenvalues { β k | k ∈ N } , whereastheir eigenmodes { σ k | k ∈ N } and { τ k | k ∈ N } form bi-orthonormal sets , namely: I S τ ′ h ( L + I ) σ ′ k dS + I S τ ′ h L σ ′′ k dS − I S τ ′′ h L σ ′ k dS − I S τ ′′ h ( L − I ) σ ′′ k dS = δ hk , (6)where δ h,k is the Kronecker delta, which is 1 if the k = h ,and 0 otherwise. It is fundamental to note that the eigen-values β k , and the associated eigenmodes depend solelyon the geometry of the shell and on the dielectric con-stant of the core. This fact is crucial for the design of thedielectric constant of the cover to achieve transparency.It can be also proved that the eigenvalue β k are real andnegative.We associate to each eigenmode k a dipole moment p k = ( p k,x , p k,y , p k,z ) p k = I S r σ ′ k dS + I S r σ ′′ k dS. (7)The kth plasmonic mode is dark if p k = 0, it is bright otherwise.When the coated object is excited by an external field E inc its dipole moment can be expressed as : p (0) ( χ ) = X k χ r ′ (0) inc,k + χ r ′′ (0) inc,k ψ k − χ p k , (8)where: r ′ (0) inc,k = − ε I S E inc · n τ ′ k dS,r ′′ (0) inc,k = 2 ε (cid:18)I S E inc · n τ ′ k dS − I S E inc · n τ ′′ k dS (cid:19) . (9)The real resonant frequency ω k of the mode k can beobtained by the equation:Re { ε r, ( ω k ) } = β k , (10)Equation 8 shows that the total dipole moment is a ratio-nal function of the susceptibility of the cover χ , since asalready noticed ψ k , r ′ (0) inc,k , r ′′ (0) inc,k , and p k are independentof χ . This fact allows a simple and elegant solution ofthe inverse scattering problem.The quantity c k ( χ ) = h χ r ′ (0) inc,k + χ r ′′ (0) inc,k i in the nu-merator of Eq. 8 represents the coupling coefficient ofthe mode k to the external excitation E inc . Moreover,we define the resonant radiative strength s k of the mode k s k = c k ( ψ k ) p k = h χ r ′ (0) inc,k + ψ k r ′′ (0) inc,k i p k . (11)which quantifies the contribution of the mode k to thetotal dipole moment at its resonance. It is immediateto notice that dark modes have vanishing magnitude of s k . As we will see later by examples, this synthetic pa-rameter is particularly useful to separate the modes thatmake a significant contribution to the dipole moment bythose who make a minor contribution and can thereforebe neglected.Once the total dipole moment is known, the totalpower scattered by the structure is given by: P rad = ω πε c | p | = ω πε c X t ∈{ x,y,z } | p t | , (12)being c the speed of light in free-space.The goal of our study is to find the values of suscep-tibility of the cover at which the scattered power P rad vanishes when the core-shell object is excited by the field E inc . We assume that the coated object exhibits n brightmodes, i.e. σ k = ( σ ′ k , σ ′′ k ) | k = 1 . . . n , with correspond-ing resonant susceptibilities ψ k . We also assume that thecomponent of the total dipole moment along a given di-rection ˆ α is strongly dominant in the frequency range ofinterest: | p · ˆ α |k p − ( p · ˆ α ) ˆ α k ≫ . (13)Thus, the problem reduces to finding the values of χ incorrespondence of which the ˆ α -component of the dipolemoment vanishes. In the presence of objects of mod-erate aspect ratio, the offset between the directions ofthe dipole moment and the incident polarization direc-tion is usually small, thus it is reasonable to assumeˆ α = E inc / k E inc k . Starting from Eq. 8 it is straight-forward to demonstrate that the zeros of the ˆ α compo-nent of the dipole moment are given by the roots of thefollowing polynomial of degree n P ˆ α { χ } = χ n n X k =1 r ′′ (0) inc,k p k, ˆ α − χ n − n X k =1 h r ′′ (0) inc,k e ( . . . ψ k − , ψ k +1 . . . ) − χ r ′ (0) inc,k i p k, ˆ α ++ χ n − n X k =1 h r ′′ (0) inc,k e ( . . . ψ k − , ψ k +1 . . . ) − χ r ′ (0) inc,k e ( . . . ψ k − , ψ k +1 . . . ) i p k, ˆ α + · · · = 0 , (14)where p k, ˆ α = p k · ˆ α and e i ( . . . ψ k − , ψ k +1 . . . ) is an ele-mentary symmetric polynomial of degree i in the n − ψ , . . . , ψ k − , ψ k +1 , . . . , ψ n , defined as : e ( . . . ψ k − , ψ k +1 . . . ) = 1 ,e ( . . . ψ k − , ψ k +1 . . . ) = X ≤ l ≤ nl = k ψ l ,e ( . . . ψ k − , ψ k +1 . . . ) = X ≤ l
As the diameter D of the minimum sphere circum-scribing the particle becomes comparable to the incidentwavelength λ , the quasi-static prediction made by Eq.14 becomes inaccurate. Nevertheless, our method can besignificantly extended by using a perturbation approach.We have adapted to the problem at hand the approachthat Mayergoyz et al. originally introduced to study theradiation correction of the plasmonic resonance . Byintroducing the perturbation parameter β = ω √ ǫ µ D ,we expand the excitation fields, the relative permittivity ε SC ,r at which the scattering-cancellation occurs, and thecorresponding electric and magnetic fields in powers of β . In particular, for ε SC ,r we have ε SC ,r ≈ ε (0) SC ,r + βε (1) SC ,r + β ε (2) SC ,r + · · · . (16)These expansions are then substituted into the Maxwell’sequations and first- and second-order boundary valueproblems are obtained by equating the terms of corre-sponding order. Eventually, the first- and second-orderradiation corrections of the scattering-cancellation per-mittivity, i.e. ε (1) SC ,r and ε (2) SC ,r , are found by zeroing thecorresponding perturbations of the ˆ α -component of thedipole moment of the coated object. The detailed deriva-tion is shown in the appendix. As we will see in thenext section, the quasi-electrostatic theory accompanied FIG. 2. (a) Electric charge density of the eigenmodes of thecoated sphere with non-vanishing radiative strength. Theouter shell is “opened” to allow the visualization of the sur-face charge density on the inner surface. (b) Power scatteredby the coated sphere calculated with the bi-orthogonal expan-sion and with the Mie-Theory. R is the external radius (seeFig 1 (b)). The power scattered by the uncloaked Si sphereis also shown (blue line). The zeros and the poles of the x-component of the total dipole moment are shown with redand black vertical lines. by the radiation corrections turns out to be the most nat-ural and predictive framework to describe the regime inwhich the dipole scattering is dominant. III. RESULTS AND DISCUSSIONA. Cloaking of a sphere
Aiming at the validation of our method, we design thesusceptibility of the cover of the silicon dioxide ( χ = 2 . ρ = r/R = 0 . x -polarized electric field.This problem admits an analytical solution (see forinstance ) that we use to estimate the error of our ap-proach. We first solve this problem in the small-particlelimit, then, by using the radiation corrections presentedin the appendix, we extend the solution to the case ofparticle’s sizes comparable to the wavelength.We start by numerically solving the electrostatic eigen-value problems 1 and 5, obtaining the set of resonantsusceptibilities ψ k and the corresponding eigenmodes( σ ′ k , σ ′′ k ) and ( τ ′ k , τ ′′ k ), normalized according to Eq. 6 .Then, the values of r ′ (0) inc,k and r ′′ (0) inc,k and of the resonantradiative strength s k are calculated for each mode usingEqs. 9 and 11. At this point, we notice that only twodegenerate eigenvalues (each of them with multiplicity 3)are associated to eigenmodes with non-vanishing radia-tive strength. Their surface charge density is shown inFig. 2 (a) and their resonant susceptibilities are listedin Tab. I. Furthermore, for symmetry considerations thetotal dipole moment has to be oriented along the x -axis.The coupling between these six eigenmodes, formally de-scribed by Eq. 14 with n = 6 and ˆ α = ˆ x , gives rise tozeros in the scattered power. The values of susceptibility ψ k -13.9 -1.3 χ (0) SC (theory) -0.6665 -6.8474 χ (0) SC (numeric) -0.6663 -6.8545TABLE I. Resonant susceptibilities of the modes of a spher-ical core-shell objects ( χ = 2 . , ρ = 0 .
8) with non-vanishingresonant radiative strength and zeros of the x-component ofthe overall dipole moment of the coated sphere. χ (0) SC satisfying the transparency condition, i.e. Eq. 14,are listed in Tab. I; they are real and in very good agree-ment with the analytical solution (error below 0.1%).To achieve transparency at a given wavelength λ , thesusceptibility of the cover should satisfy the constraint χ ( λ ) = χ (0) SC . It is worth noting that since actual mate-rials always exhibit losses, the transparency condition isnever exactly satisfied. If a Drude metal is the materialof choice, its plasma frequency ω p is given by: ω p = q − ( ω + γ ) χ (0) SC . (17)By assuming λ = 5 . cm and γ = 8 · s − andchoosing the zero χ (0) SC = − . ω p = 8 . · s − rad/s. For instance, a micro-structuremade of a regular array of thin wires can be properlydesigned to mimic a Drude metal with the prescribedplasma frequency .In Fig. 2 (b), we plot the corresponding scatteredpower as a function of the wavelength obtained usingEqs. 8 and 12 where the summation runs only over themodes 1 . . .
6. We also show with a blue line the scatteredpower from the uncloaked sphere and with red open cir-cles the scattered power of the cloaked sphere calculatedusing the Mie theory, which validates the bi-orthogonalexpansion.As soon as the diameter D of the sphere becomes com-parable to the incident wavelength, the quasi-static pre-diction of the scattering-cancellation permittivity madeby Eq. 14 becomes inaccurate and the use of the ra-diation corrections presented in the appendix is manda-tory. Thus, we determine the radiation corrections tothe zero ε (0) SC ,r = − . D/λ of the sphere and we compare the resultingvalue of ε SC ,r with the one obtained by the Mie theoryusing Ref. . By using Eqs. A.20 and A.34 we obtain ε (1) SC ,r = − . · − j and ε (2) SC ,r = − . ε (1) SC ,r is negligible, thus ε SC ,r canbe approximated by: ε SC ,r ≈ ε (0) SC ,r + ε (2) SC ,r (cid:18) π Dλ (cid:19) (18)In Fig. 3 we plot the scattering-cancellation permittivitycalculated by Eq. 18 and by the Mie theory . We notice Mie Theory sc , r D/ Radiation Correction
FIG. 3. Scattering-cancellation permittivity ε SC ,r of the coat-ing of a spherical object ( χ = 2 . , ρ = 0 .
8) computed byusing the radiation corrections (black line) and the full-waveMie theory (red line) as a function of the electric size of theobject
D/λ . very good agreement between the two approaches up tothe electric size of D/λ = 0 . D/λ ε SC ,r -6.5 -7.0 -7.7 - 8.5 ω p ( rad/s ) 9 . · . · . · . · Q ε SC ,r of the coating at which scattering-cancellation occurs, the corresponding value of plasma fre-quency ω p of the Drude cover and the achieved quality ofscattering-cancellation Q defined as the ratio between thepowers scattered by the uncloaked and the cloaked object at λ = 5 . Choosing λ = 5 . cm and four different electric sizes,namely D/λ = 0 . , . , . , . ε SC ,r that guarantee the dipolescattering-cancellation using Eq. 18. They are listed inTab. II together with the corresponding plasma frequen-cies of the Drude-metal coating obtained using Eq. 17.Thus, for each case, we plot in Fig. 4 the power scatteredby the coated sphere as a function of the wavelength. Inall the four cases the minimum of the scattering powercorresponds to the nominal wavelength λ = 550 nm (ver-tical red line) as expected. Nevertheless, we notice adegradation of the quality of scattering-cancellation asthe electric size of the object increases due to the on-set of higher-order scattering modes. In particular, incorrespondence to D/λ = 0 . λ , the scat-tered powers of the cloaked and the uncloaked structureare almost equal. For larger sizes, the cancellation of thedipole-scattering is not sufficient to reduce the total scat-tered power, since the scattering is dominated by higher -9 -7 -5 -10 -8 -6 -11 -9 -7 -5 -8 -7 -6 -5 cloacked uncloacked P r ad ( W ) P r ad ( W ) (d)(c) (b) wavelength (cm)wavelength (cm)(a) FIG. 4. Spectra of the power scattered by the investigatedcoated sphere ( χ = 2 . , ρ = 0 .
8) for different electric sizes,namely
D/λ = 0 .
15 (a) ,0 . . . λ = 5 . cm (red line) are also shown. order modes. B. Cloaking of a C-shaped object
In order to demonstrate the feasibility of the presentedapproach, we design the material of the cover of theC-shaped object sketched in Fig. 1 (c-d) to achievetransparency under a y -polarized excitation. The coreis made of silicon dioxide, i.e. χ = 2 .
9, with dimensions R = 0 . , d = 0 . , a = 0 .
5, while the thickness of thecover is δ = 0 .
05. We numerically solve the eigenvalueproblems 1 and 5, obtaining the set of resonant suscep-tibilities ψ k and the corresponding eigenmodes ( σ ′ k , σ ′′ k )and ( τ ′ k , τ ′′ k ). Then, the values of r ′ (0) inc,k and r ′′ (0) inc,k and ofthe resonant radiative strengths s k are calculated for eachmode using Eqs. 9 and 11. At this point, we notice thatonly nine eigenvalues are associated to eigenmodes withnon-negligible radiative strength. The remaining eigen-modes have resonant radiative strengths less than a pre-scribed limit k s k k / max k k s k k < . · − and have beendisregarded. The surface charge density of the eigen-modes is shown in Fig. 5 (a) and their resonant suscep-tibilities ψ k are reported in table III. Thus, we assumethat the component of the dipole moment along the in-cident polarization direction is dominant. The values ofsusceptibility satisfying the transparency condition, i.e.Eq. 14 with n = 9 and with ˆ α = ˆ y , are listed in Tab. III.To achieve transparency at a given wavelength λ , thesusceptibility of the cover should satisfy the constraint χ ( λ ) = χ (0) SC , where χ (0) SC is the generic root of Eq. 14. FIG. 5. (a) Electric charge density of the eigenmodes of thecoated C-shaped particle with appreciable radiative strengthswhen excited with a y -polarized electric field. The outer shellis “opened” to allow the visualization of the surface chargedensity on the inner surface. (b) Power scattered by a coatedC-shaped particle when excited by a y -polarized electric field. L is the horizontal length of the particle (see Fig. 1(c)). Thepower scattered by the uncloaked C-shaped particle is alsoshown (blue line). The zeros of the y-component of the totaldipole moment are also shown with red vertical lines. Since actual materials always exhibit losses, when a zeroof the dipole moment is in proximity of a pole, i.e. a plas-mon resonance, the pole-zero cancellation can deterioratethe quality of the designed transparency. Therefore, inthe presence of real materials the roots of Eq. 14 arenot equivalent in terms of the quality of the scattering-cancellation. When many solutions are allowed, as in thepresent scenario, the zeros far from the poles have to bepreferred. In this case, examining Tab. III, we select thezero χ (0) SC = − .
4. Considering a Drude metal,and assuming λ = 5 . cm and γ = ω p · − we obtain ω p = 9 . · rad/s. At this point, the condition 13has been verified a posteriori using Eq. 8. In Fig. 5(b) we plot the corresponding scattered power as a func-tion of the wavelength using Eqs. 8 and 12 where thesummation runs only over the modes 1 . . . . We alsoshow with a blue line the power scattered by the un-cloaked C-shaped object and with red vertical lines theposition of the zeros of the y-component of the overalldipole moment, which are directly obtained from the val-ues of susceptibility. At the wavelength λ = 5 . cm thescattered power is reduced of 24 . dB with respect to theuncloaked object. It is worth noting that in Fig. 5 (b) thezeros of the scattered power spectra give rise to asymmet-ric scattering line-shapes, usually referred to as Fano-likeresonances. As already shown for arrays of homogeneousplasmonic objects , also in strongly subwavelength plas-monic shells Fano-like resonances are originated by dipolescattering-cancellation of bright-modes .As the diameter D of the minimum sphere circum- ψ k -89.0 -27.3 -19.2 -16.5 -14.2 -12.4 -10.7 -10.1 -1.3 λ k (cm) 19.1 10.6 8.9 8.2 7.6 7.1 6.6 6.5 2.3 χ (0) SC -33.8 -27.3 -16.8 -14.5 -12.7 -10.8 -10.2 -7.4 -0.84 λ SC (cm) 11.8 10.6 8.3 7.7 7.2 6.7 6.5 5.5 1.9TABLE III. Resonant susceptibilities of the bright modes andzeros of the y − component of the total dipole moment of acoated C-shaped object excited by a y -polarized electric field.The corresponding wavelengths are also listed assuming ω p =9 . · rad/s and γ = ω p · − . scribing the C-shaped object become comparable to theincident wavelength λ , the scattering-cancellation per-mittivity calculated by Eq. 14 has to be corrected us-ing the perturbation approach. Thus, we determinethe radiation corrections to the zero ε (0) SC ,r = − .
4) as a function of the electric size
D/λ of the C-shaped object. By using Eqs. A.20 and A.34we obtain ε (1) SC ,r = − . · − j and ε (2) SC ,r = − . ε (1) SC ,r is negligible, also in this case ε SC ,r can be approximated by Eq. 18. In Fig. 3 we plot thescattering-cancellation permittivity calculated by Eq. 18as a function of the electric size of the object D/λ . D/ sc , r FIG. 6. Scattering-cancellation permittivity ε SC ,r of the coat-ing of a C-shaped object as a function of the electric size of theobject D/λ and computed using the radiation corrections. By choosing λ = 5 . cm and four different electricsizes, namely D/λ = 0 . , . , . , .
9, we obtain thevalues of ε SC ,r , listed in Tab. IV, that guarantee thedipole scattering-cancellation. Thus, for each value of ε SC ,r we show in Fig. 7 the power scattered by the coatedC-shaped object as a function of the wavelength. In allthe four cases the minimum of the scattering power cor-responds to the nominal wavelength λ = 5 . cm (verticalred line). This fact validates our design. As in the caseof the sphere, the onset of higher-order scattering modes,occurring for large electric sizes has a detrimental effecton quality of scattering-cancellation. Nevertheless, it isworth noting that a moderate reduction of the scatteredpower is achieved also in the case in which the dimensionof the C-shaped object is equal to the incident wavelength λ as shown Fig. 7. D/λ ε SC ,r -6.45 -7.23 -7.91 -8.79 ω p ( rad/s ) 9 . · . · . · . · Q ε SC ,r of the coating atwhich scattering-cancellation occurs, the corresponding valueof plasma frequency ω p of the Drude cover and the achievedquality of scattering cancellation Q defined as the ratio be-tween the powers scattered by the uncloaked and the cloakedobject at λ = 5 . P r ad ( W ) P r ad ( W ) wavelength (cm) wavelength (cm) -6 -4 -2 -6 -4 -2 (d) -6 -4 -2 -6 -4 -2 (c) -8 -6 -4 -8 -6 -4 (b) -10 -8 -6 -4 -10 -8 -6 -4 (a) FIG. 7. Spectra of the power scattered by the investigatedcoated C-shaped particle for different electric sizes, namely
D/λ = 0 . . . z . The spectra havebeen calculated by the full-retarded Method of Moments .The material of the coating has been designed using the quasi-electrostatic theory and the radiation corrections. The usedvalues of plasma frequency of the coating are listed in TabIV. The power scattered by the uncloaked C-shaped particle(blue line) and the wavelength λ = 5 . cm (red line) are alsoshown. IV. CONCLUSIONS
We have introduced a novel theory for the cloakingof arbitrarily-shaped objects through designed homoge-neous coatings. Our approach, which is valid beyond theRayleigh regime, permits the rigorous design of the per-mittivity values of the coating layer that abates the totalscattered power. It can be also easily extended to design the cloaking of objects lying on a substrate, multi-coatedobjects, and plasmonic cores with a dielectric shell.Nevertheless, it is important to point out thatthe achieved cloaking depends on the polarization ofthe incident light, being this limitation inherent tothe scattering-cancellation approach to cloaking whenapplied to arbitrary shapes using homogeneous andisotropic materials . Moreover, since this approach isbased on the cancellation of the dipole scattering, itis ineffective when the electric size of the particle islarge enough that high orders of scattering are domi-nant. Moreover, in the presence of objects of extremeaspect ratio it may not be possible to find a direction ˆ α satisfying the condition 13. In this case the scatteringcancellation is not limited by the losses but by a residualpolarization lying on the plane orthogonal to ˆ α with aconsequent degradation of the quality of the cloaking .Despite its limitations, the introduced frameworkpaves the way to the application of the scattering-cancellation to real-life problems where the shape of theobject to be cloaked, e.g. a complex sensor, is not underthe control of the designer. ACKNOWLEDGMENTS
This work was supported by the U.S. Army ResearchLaboratory through the Collaborative Research Alliance(CRA) for MultiScale multidisciplinary Modeling of Elec-tronic materials (MSME), and by the Italian Ministry ofEducation, University and Research through the projectPON01 02782.
Appendix: Derivation of the radiation correction ofthe scattering-cancellation permittivity
In the present section we derive the first- and second-order radiation correction of the permittivity at whichscattering-cancellation occurs. This is achieved by zero-ing the corresponding perturbations of the ˆ α -componentof the dipole moment of the coated object. In orderto accomplish this, we have adapted to the problem athand the approach that Mayergoyz et al. originally intro-duced to study the radiation correction of the plasmonicresonance . Thus, by introducing the normalized inci-dent and scattered fields e inc = √ ε E inc h inc = √ µ H inc , e t = √ ε E t h t = √ µ H t ∀ t ∈ { , , } , (A.1)and scaling the spatial coordinates by the diameter D ofthe smallest sphere circumscribing the object, we obtainthe following boundary value problem: ∇ × e = − jβ h ∇ × h = + jβε ,r e + jβ ( ε ,r − e inc ∇ · e = ∇ · h = in V , (A.2) ∇ × e = − jβ h ∇ × h = + jβε ,r e + jβ ( ε ,r − e inc ∇ · e = ∇ · h = in V , (A.3) ∇ × e = − jβ h ∇ × h = + jβ e ∇ · e = ∇ · h = in V , (A.4) n · ( ε ,r e − ε ,r e ) = − ( ε ,r − ε ,r ) n · e inc n × ( e − e ) = · ( h − h ) = × ( h − h ) = on S , (A.5) n · ( e − ε ,r e ) = − (1 − ε ,r ) n · e inc n × ( e − e ) = · ( h − h ) = × ( h − h ) = on S , (A.6)where we have defined the quantity β = ω √ ǫ µ D .When the dimension D is small compared to thefree-space wavelength, the forcing terms e inc , h inc , therelative permittivity ε SC ,r at which the scattering-cancellation occurs, and the corresponding fields e and h can be expanded in powers of β , namely e inc ≈ e (0) inc + β e (1) inc + β e (2) inc + · · · ε SC ,r ≈ ε (0) SC ,r + βε (1) SC ,r + β ε (2) SC ,r + · · · e ≈ e (0) + β e (1) + β e (2) + · · · h ≈ h (0) + β h (1) + β h (2) + · · · (A.7)In the particular case of a plane wave excitation of ex-pression: e inc = e exp ( − jβ i k · r ) , (A.8)we have: e (0) inc = e , e (1) inc = − j ( i k · r ) e , e (2) inc = − ( i k · r ) e . (A.9)
1. Zero-Order Boundary Value Problem
Substituting the expansion A.7 in Eqs. A.2-A.6 andequating the terms of zero-power we obtain the zero-orderboundary value problem for the electric field: ( ∇ × e (0) t = ∇ · e (0) t = 0 ∀ t ∈ { , , } , (A.10) n · (cid:16) ε (0) SC ,r e (0)2 − ε ,r e (0)1 (cid:17) = − (cid:16) ε (0) SC ,r − ε ,r (cid:17) n · e (0) inc , n · (cid:16) e (0)3 − ε (0) SC ,r e (0)2 (cid:17) = − (cid:16) − ε (0) SC ,r (cid:17) n · e (0) inc , n × (cid:16) e (0)2 − e (0)1 (cid:17) = , n × (cid:16) e (0)3 − e (0)2 (cid:17) = , (A.11)and the zero-order boundary value problem for the mag-netic field: ( ∇ × h (0) t = , ∇ · h (0) t = 0 , ∀ t ∈ { , , } (A.12) n · (cid:16) h (0)2 − h (0)1 (cid:17) = 0 , n · (cid:16) h (0)3 − h (0)2 (cid:17) = 0 , n × (cid:16) h (0)2 − h (0)1 (cid:17) = , n × (cid:16) h (0)3 − h (0)2 (cid:17) = . (A.13)First, from Eqs. A.12 and A.13 we can conclude that h (0) = 0 in R . Next, we notice that the set of Eqs.A.10-A.11 defines the electrostatic problem encounteredin the previous section. Thus, we have to use Eq. 14 inorder to find the zero-order values ε (0) SC ,r of the dielectricpermittivity zeroing the ˆ α -component of the zero-orderdipole moment.
2. First-Order Boundary Value Problem
Next, equating the terms of first-power in Eqs. A.2-A.6 we obtain the first-order boundary value problem forthe electric field: ∇ × e (1) t = ∇ · e (1) t = 0 ∀ t ∈ { , , } , (A.14) n · (cid:16) ε (0) SC ,r e (1)2 − ε ,r e (1)1 (cid:17) = − ε (1) SC ,r n · (cid:16) e (0)2 + e (0) inc (cid:17) − (cid:16) ε (0) SC ,r − ε ,r (cid:17) n · e (1) inc , n · (cid:16) e (1)3 − ε (0) SC ,r e (1)2 (cid:17) = + ε (1) SC ,r n · (cid:16) e (0)2 + e (0) inc (cid:17) − (cid:16) − ε (0) SC ,r (cid:17) n · e (1) inc , n × (cid:16) e (1)2 − e (1)1 (cid:17) = , n × (cid:16) e (1)3 − e (1)2 (cid:17) = . (A.15)and the first-order boundary value problem for the mag-netic field: ∇ × h (1)1 = jε ,r e (0)1 + j ( ε ,r − e (0) inc , ∇ × h (1)2 = jε (0) SC ,r e (0)2 + j (cid:16) ε (0) SC ,r − (cid:17) e (0) inc , ∇ × h (1)3 = j e (0)3 , ∇ · h (1) t = 0 ∀ t ∈ { , , } , (A.16) n · (cid:16) h (1)2 − h (1)1 (cid:17) = · (cid:16) h (1)3 − h (1)2 (cid:17) = , n × (cid:16) h (1)2 − h (1)1 (cid:17) = × (cid:16) h (1)3 − h (1)2 (cid:17) = . (A.17)From Eqs. A.14-A.15 we derive the expression of thefirst-order correction p (1) of the dipole moment of thecoated object in terms of its electrostatic modes p k : p (1) (cid:16) ε (1) SC ,r (cid:17) = X k ε (1) SC ,r (cid:16) r ′′ (0) s,k + r ′′ (0) inc,k (cid:17) + χ r ′ (1) inc,k + χ (0) SC r ′′ (1) inc,k ψ k − χ (0) SC p k , (A.18)where we have defined the quantities: r ′ (1) inc,k = − ε I S τ ′ k n · e (1) inc dS,r ′′ (1) inc,k = 2 ε (cid:18)I S τ ′ k n · e (1) inc dS − I S τ ′′ k n · e (1) inc dS (cid:19) ,r ′′ (0) s,k = 2 ε (cid:18)I S τ ′ k n · e (0)1 dS − I S τ ′′ k n · e (0)2 dS (cid:19) . (A.19)By zeroing the quantity p (1) in Eq. A.18 we obtain thefirst-order correction ε (1) SC ,r of the permittivity at whichscattering-cancellation occurs: ε (1) SC ,r = − X k χ r ′ (1) inc,k + χ (0) SC r ′′ (1) inc,k ψ k − χ (0) SC p k, ˆ α X k r ′′ (0) s,k + r ′′ (0) inc,k ψ k − χ (0) SC p k, ˆ α . (A.20)We now summarize the algorithm for the computationof ε (1) SC ,r . Assuming that the zero-order scattering can-cellation permittivity ε (0) SC ,r = χ (0) SC − e (0) , solution of the problem A.10-A.11, on S and S . Thus, we apply Eq. A.19 to computethe quantities r ′′ (0) s,k and r ′ (1) inc,k , r ′′ (1) inc,k using the fields e (0) , e (1) inc and the electrostatic eigenvectors ( τ ′ k , τ ′′ k ). Eventu-ally, we calculate ε (1) SC ,r by Eq. A.20.Let us now turn our attention to the first-order bound-ary problem for the magnetic field, shown in Eqs. A.16-A.17. Its solution that will be useful for the second-order problem, is given by: h (1) ( Q ) = j ε (0) SC ,r − ε ,r π I S n M × (cid:16) e (0) + e (0) inc (cid:17) r MQ dV M + j − ε (0) SC ,r π I S n M × (cid:16) e (0) + e (0) inc (cid:17) r MQ dV M . (A.21)
3. Second-Order Boundary Value Problem
Next, we discuss the second-order correction ε (2) SC ,r forthe scattering-cancellation permittivity. Equating theterms of second-power in Eqs. A.2-A.6 we obtain thesecond-order boundary value problem for the electricfield: ∇ × e (2) t = − j h (1) t ∇ · e (2) t = in V t ∀ i ∈ { , , } , (A.22) n · (cid:16) ε (0) SC ,r e (2)2 − ε ,r e (2)1 (cid:17) = − ε (2) SC ,r n · (cid:16) e (0)2 + e (0) inc (cid:17) − ε (1) SC ,r n · (cid:16) e (1)2 + e (1) inc (cid:17) − (cid:16) ε (0) SC ,r − ε ,r (cid:17) n · e (2) inc , n · (cid:16) e (2)3 − ε (0) SC ,r e (2)2 (cid:17) = + ε (2) SC ,r n · (cid:16) e (0)2 + e (0) inc (cid:17) + ε (1) SC ,r n · (cid:16) e (1)2 + e (1) inc (cid:17) − (cid:16) − ε (0) SC ,r (cid:17) n · e (2) inc , (A.23) n × (cid:16) e (2)2 − e (2)1 (cid:17) = , n × (cid:16) e (2)3 − e (2)2 (cid:17) = . (A.24)As suggested in Ref. , it is now convenient to split e (2) t into two components: e (2) t = ¯ e (2) t + ¯¯e (2) t ∀ t ∈ { , , } . (A.25)The first component ¯ e (2) t is the solution of the followingproblem: ( ∇ × ¯ e (2) t = − j h (1) t ∇ · ¯ e (2) t = 0 ∀ t ∈ { , , } , (A.26) n · (cid:16) ¯ e (2)2 − ¯ e (2)1 (cid:17) = 0 , n · (cid:16) ¯ e (2)3 − ¯ e (2)2 (cid:17) = 0 , n × (cid:16) ¯ e (2)2 − ¯ e (2)1 (cid:17) = , n × (cid:16) ¯ e (2)3 − ¯ e (2)2 (cid:17) = , (A.27)while the second component ¯¯e (2) t satisfies the followingboundary value problem: ( ∇ × ¯¯e (2) t = ∇ · ¯¯e (2) t = 0 ∀ t ∈ { , , } , (A.28)0 n · (cid:16) ε (0) SC ,r ¯¯e (2)2 − ε ,r ¯¯e (2)1 (cid:17) = − ε (2) SC ,r n · (cid:16) e (0)2 + e (0) inc (cid:17) − ε (1) SC ,r n · (cid:16) e (1)2 + e (1) inc (cid:17) − (cid:16) ε (0) SC ,r − ε ,r (cid:17) n · (cid:16) e (2) inc + ¯ e (2) (cid:17) , n · (cid:16) ¯¯e (2)3 − ε (0) SC ,r ¯¯e (2)2 (cid:17) = ε (2) SC ,r n · (cid:16) e (0)2 + e (0) inc (cid:17) + ε (1) SC ,r n · (cid:16) e (1)2 + e (1) inc (cid:17) + (cid:16) ε (0) SC ,r − (cid:17) n · (cid:16) e (2) inc + ¯ e (2) (cid:17) , (A.29) n × (cid:16) ¯¯e (2)2 − ¯¯e (2)1 (cid:17) = , n × (cid:16) ¯¯e (2)3 − ¯¯e (2)2 (cid:17) = . (A.30) By using the same line of reasoning of Ref. we obtainthe expression of ¯ e (2) that satisfies Eqs. A.26 and A.27:¯ e (2) ( P ) = (cid:16) ε (0) SC ,r − ε ,r (cid:17) π I S (cid:16) n M × (cid:16) e (0) ( M ) + e (0) inc (cid:17)(cid:17) × r MP r MP dS M + (cid:16) − ε (0) SC ,r (cid:17) π I S (cid:16) n M × (cid:16) e (0) ( M ) + e (0) inc (cid:17)(cid:17) × r MP r MP dS M , (A.31)Then, from Eqs. A.28-A.30 we obtain the expression ofthe second-order correction p (2) of the dipole moment ofthe coated-object in terms of its resonant modes p k : p (2) = X k ε (2) SC ,r (cid:16) r ′′ (0) s,k + r ′′ (0) inc,k (cid:17) + ε (1) SC ,r (cid:16) r ′′ (1) s,k + r ′′ (1) inc,k (cid:17) + χ (cid:16) r ′ (2) inc,k + ˜ r ′ (2) s,k (cid:17) + χ (0) SC (cid:16) r ′′ (2) inc,k + ˜ r ′′ (2) s,k (cid:17) ψ k − χ (0) SC p k , (A.32)where we have defined the quantities: r ′ (2) inc,k = − ε I S τ ′ k n · e (2) inc dS,r ′′ (2) inc,k = 2 ε (cid:18)I S τ ′ k n · e (2) inc dS − I S τ ′′ k n · e (2) inc dS (cid:19) ,r ′′ (1) s,k = 2 ε (cid:18)I S τ ′ k n · e (1)2 dS − I S τ ′′ k n · e (1)2 dS (cid:19) , ˜ r ′ (2) s,k = − ε I S τ ′ k n · ¯ e (2) dS, ˜ r ′′ (2) s,k = 2 ε (cid:18)I S τ ′ k n · ¯ e (2) dS − I S τ ′′ k n · ¯ e (2) dS (cid:19) . (A.33) By zeroing the quantity p (2) we obtain the second-ordercorrection of the permittivity ε (2) SC ,r at which scattering-cancellation occurs: ε (2) SC ,r = − X k ε (1) SC ,r (cid:16) r ′′ (1) s,k + r ′′ (1) inc,k (cid:17) + χ (cid:16) r ′ (2) inc,k + ˜ r ′ (2) s,k (cid:17) + χ (0) SC (cid:16) r ′′ (2) inc,k + ˜ r ′′ (2) s,k (cid:17) ψ k − χ (0) SC p k, ˆ α X k r ′′ (0) s,k + r ′′ (0) inc,k ψ k − χ (0) SC p k, ˆ α . (A.34)We now recapitulate the steps for the computation of thesecond-order radiation correction ε (2) SC ,r . Assuming thatthe calculation of ε (1) SC ,r has been already performed, wecalculate the corresponding scattered electric field e (1) ,solution of the problem A.14-A.15, on both the internal and the external surface. Moreover, using the electro-static field e (0) and the incident field e (0) inc we can cal-culate by Eq. A.31 the field ¯ e (2) on the both internaland the external surface. Thus, we compute the quan-tities r ′ (2) inc,k , r ′′ (2) inc,k , r ′′ (1) s,k , ˜ r ′ (2) s,k , ˜ r ′′ (2) s,k . Eventually, ε (2) SC ,r canbe calculated using Eq. A.34.1 ∗ Corresponding author: [email protected] J. B. Pendry, D. Schurig, and D. R. Smith,
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