Theoritical Analysis of Some Homogenized Metamaterials and Application of PML to Perform Cloaking and Back-scattering Invisibility

Transcription

1 PIERS ONLINE, VOL. 6, NO., Theoritical Analysis of Some Homogenized Metamaterials and Alication of PML to Perform Cloaking and Back-scattering Invisibility P. H. Cocquet, V. Mouysset, and P. A. Mazet Deartment of Information Treatment and Modelization DTIM, ONERA Toulouse, France Abstract A general Drude-Born-Fedorov DBF system, with materials like metamaterials, on a bounded domain is studied. Due to the resence, for some ulsation, of negative ermittivity and ermeability, the well-osedness of this kind of system remains a steady oint. In this aer, we treat examles dealing with homogenized metamaterials like a eriodic array of Slit- Ring-Resonators SRR, homogenized dielectric-chiral hotonic crystals and convex absorbing boundary conditions of PML tye have been treated. At last, we show both theoretically and numerically that PML can erform electromagnetic cloaking and back-scattering invisibility. 1. INTRODUCTION In 1968, V. G. Veselago [11] theoretically studied electromagnetic henomenons in materials with simultaneous negative ermittivity [ε] and ermeability [µ]. Due to the reversed Poynting vector, he named these materials Left-Handed-Materials LHM. He also remarks that these materials had exotics roerties reversed Doler effect for examle. The keen interest for the LHM was initiated by J. B. Pendry s in 000 [6]. He managed to build LHM using a eriodic array of Slit-Ring-Resonators SRR and suggested making a erfect lens with these structures. Futhermore, these exotic building were named metamaterials. However, as metamaterials can t be found in nature as negative indexes materials, they are usually the result of an homogenization see for examle [3, 4, 6, 7, 9]. The roblem arises from the choice for homogenization s method. Every method can lead to a different homogenized material and by the way to a new artial diffential equation. The question is then to know, in a first ste, which of these homogenized roblems are well-osed. A second ste will be linked with a suitable aroximation of the solution of these homogenized systems. Unfortunatly, the well-osedness of these homogenized roblems remains a steady oint. In fact, homogenized metamaterials have their ermitivitty and ermeability deending on the ulsation w and start to be negative definite for some w. This imlies that the well-osedness of the DBF system, which establishes electromagnetic henomenons in chiral materials, can t be a riori issued. So, we are going to study a generalized DBF system including some negative index henomenons. A well-osedness result for this generalised DBF system is given in the first section. Next, homogenized dielectric-chiral hotonic crystals [4], eriodical arrangement of Slit-Ring-Resonator SRR [7] and convex absorbing boundary conditions of PML tye for Maxwell s equations [5], seen as ideal metamaterial, are studied. At last, we show theoretically and check numerically that convex PML for Maxwell s equations can erform electromagnetic cloaking and back-scattering invisibility.. MATHEMATICAL FRAMEWORK Lots of electromagnetic henomenons in homogenized chiral hotonic crystals can be modelized by the following Drude-Born-Fedorov DBF system 1, e.g., [3]: Find e, h Hcurl, Ω such that : e K 0, x + K 1, xm = f, x Ω 1 h nx e + Λxnx h = 0, x Ω, 0 [ε, x] 0 M =, K 0 0, x = 0 [µ, x] I3 [β, x][ε, x] K 1, x =. [β, x][µ, x] I 3,

2 PIERS ONLINE, VOL. 6, NO., where e and h are the electric and the magnetic fields. The ermitivity [ε], the ermeability [µ] and the chirality [β] are the homogenized arameters of the media which are smoothly deending both on = iw, w is the ulsation, and x, f L Ω 6. At last Ω is a simly connected bounded domain in R 3 with C 1 boundary whose outward unitary normal is denoted n and Λ LiΩ, LC 6 such that ReΛx+Λ x is nonnegative. We also set H = e, h T L Ω 3, e, h T L Ω 3 and nx E + Λxnx H = 0, x Ω} for the functional sace to look solution wherein. In usual materials, [ε] and [µ] are bounded and ositively definite so the multilication oerators K 0 and K 1, in Lalace transform, are bounded and uniformly coercive with resect to x Ω for all = iw + γ C such that γ > 0. As the oerator M, H is closed maximal dissitive [8], taking the limit γ 0 leads to the well-osedness of the system 1, for all = iw. In some examles linked with the metamaterials, the ermitivity and/or the ermeability, which are functions of the ulsation w, become negative definite for some w thus the well-osedness of the system 1, can t be, a riori, issued. Note that system 1 can both modelize hotonic crystals with DBF equations, and electromagnetic henomenons with Maxwell s equations by setting K 1 = I 6. But, we have the following result []: Theorem 1. Let D be a domain of C such that D ir and let s assume: H1: K j,. L Ω, j = 0, 1 and K 1, x is invertible, x D Ω, H: K j., x is holomorhic for D. H3: There exists 0 D such that K j 0 for j = 0, 1 is coercive. Then, under assumtions H1 H H3, the roblem 1 is well-osed with comact resolvant for all in D excet for a discrete set of values in D. Theorem 1 gives an existence and uniqueness result for the generalized DBF system 1. H1 H are the descritive and henomenological hyotheses which describe the considered media. They ask for iecewise bounded indexes deending smoothly on the Lalace variable. Note that meromorhic K j can be taken into account by removing their oles from D. Hence, for examle, Debye s and Lorentz s materials satisfy H1 H. The hyothesis H3 have been stated in [11] and can be checked exerimentally see Figure 1 in [7]. It means that the considered media behaves as an usual one for some 0. Then, for this 0, the generalized DBF system is well-osed. And finally, theorem 1 extends the well-osedness of this system to almost all in the domain of study D. 3. STUDY OF A PERIODIC ARRAY OF SPLIT-RING-RESONATORS The Slit-Ring-Resonators SRR have been introduced by J. B. Pendry in 000 suggesting to make a erfect lens using negative index materials [4]. Some studies dealing with homogenization using a eriodic array of SRR follow [6, 9] but the well-osedness of this system remains unanswered at the best of our knowledge. The effective arameters of a eriodical array of intersaced conducting nonmagnetic slit-ringresonators and continuous wires calculated in [9] has the following exressions: [ε, x] = 1 + w I 3, [µ, x] = 1 + δ w 0 + Γ where w > 0 is the lasma ulsation of gold, w 0 = 3l π µ 0Cr 3 I 3, [β, x] = 0, 3 and Γ = lρ rµ 0 > 0 are constants. Here ρ is the resistance er unit length of the rings measured around the circumference, l is the distance between layers, a is the lattice arameter, r is defined on Figure 1 in [9] and C is the caacitance associated with the gas between the rings. First, remark that [ε] and [µ] defined by 3 are free from x. Futhermore, they deend on as rational fractions thus K 0 and K 1 defined by, 3 are holomorhic on D = C\Z where Z is the set of zeros of their denominators. Namely Z = Γ Γ 4w0, 0, Γ+ Γ 4w0 equal to i y if y < 0. Hyothese H1 H are then satisfied on D Ω. Now, let f : R\0} w 0 + Γ R. } where y is

3 PIERS ONLINE, VOL. 6, NO., The maximum of f is reached for 0 = Γ. As f 0 > 0, K 0 0 and K 1 0 defined by, 3 are coercive thus H3 is verified. Hence, homogenized arameters [ε], [µ] in 3 define an admissible roblem in the sence of theorem 1 which can be lately mathematically and numerically investigated. 4. STUDY OF A HOMOGENIZED DIELECTRIC-CHIRAL PHOTONIC CRYSTALS Let ε c, µ c, β c be the arameters of a chiral media. The hotonic crystal, hosted in the vacuum, formed by a eriodic array of chiral sheres can be described as a homogenised chiral medium noted Ω whose coefficients can be written as follow [7]: [ε, x] = + ε c δ1 ε c } + µ c δ1 µ c } + 4δ 1 ε c µ c βc + ε c + δ1 ε c } + µ c δ1 µ c } δ 1δ + ε c µ c βc I 3, [µ, x] = + ε c δ1 ε c } + µ c δ1 µ c } + 4δ 1 ε c µ c βc + ε c δ1 ε c } + µ c + δ1 µ c } δ 1δ + ε c µ c βc I 3, 9δε c µ c β c [β, x] = + ε c δ1 ε c } + µ c δ1 µ c } + 4δ 1 ε c µ c βc I 3, where 0 < δ < 1 is the fraction of the total volume occuied by the sheres. Notice first that [ε], [µ] and [β] defined by 4 are free from x. Thus coefficients K 0 and K 1 defined by 4 are bounded with resect to x Ω. Futhermore [ε], [µ], [µ][β] and [ε, x][β, x] deend on as rational fraction. Thus the holomorhy of K 0 and K 1 holds on D = C\Z where Z is the set of zeros of their denominators. This imlies that: Z= ± + ε c +δ1 ε c }+µ c δ1 µ c } δ 1δ + ε c µ c βc, ± +ε c δ1 ε c } + µ c + δ1 µ c } δ 1δ + ε c µ c β c with the same notation of. as reviously. So hyotheses H1 H are satisfied on D Ω where D = C\Z. We remark that K 0 1 and K 1 1 defined by 4 are coercive. Hence, the hysical modelling which gives the arameters defined by 4 leads to an admissible DBF system in the sence of theorem 1. This roblem can be lately numerically and mathematically studied. Remark 1. The fraction of the total volume occuied by the sheres, noted δ, can deend on x Ω. So [ε], [µ] and [β] defined in 4 will be able to deend on x too. The study of the wellosedness of the system 1,, 4 with δ = δx is not slighty different from the revious case. The main modification occurs on the definition of the set Z 5, now noted Z : Z x Ω, + εc δx1 ε c } + µ c + δx1 µ c } δx 1δx + ε c µ c β c = 0, + ε c + δx1 ε c } + µ c δx1 µ c } δx 1δx + ε c µ c β c = 0. Remark that Z is the disjoint union of four intervals located on the imaginary axis. Moreover, Z 0} = and, if δ is free from x, Z = Z. As for all x in Ω, 0 < δx < 1, hyotheses H1 H are satisfied on D Ω where D = C\Z. K 0 1 and K 1 1 still define coercive multilication oerators. Thus H3 is checked. Hence the same conclusion as reviously with δ free from x holds. 5. CLOAKING AND BACK-SCATTERING INVISIBILITY WITH PML At last we study some absorbing boundary conditions of PML tye. Using the result of the Section, we show that Maxwell s equations with convex PML are well-osed. We also show theoretically that PML can erform cloaking and back-scattering invisibility. Moreover we give some numerical results showing these roerties using finite volumes method for T. M. Maxwell s equations. Let x = x 1, x Ω R and consider the following T.M Maxwell s system: where B e1 e h 3 B = y 0 0 x y x 0 e1 e h 3 } 4, 5 = f, 6 eγ,x γ,x 0 0 γ,x 0 eγ,x 0, γ, x γ, x

4 PIERS ONLINE, VOL. 6, NO., acts as absorbing boundary condition of convex PML tye on a subset Θ of Ω. Two different tyes of PML will be used according to the geometry studied: circular PML and cartesian PML. Circular PML are defined on an annulus Γ Ω of radius r 1 and r by 7 with: R γ, x = 1 σρρ ρ ρ1, γ, x = 1 σρsds ρ, 8 where ρ = x, σ ρ ρ 0 for x in Γ and σ ρ ρ = 0 for x in Ω\Γ. Cartesian PML are defined on ] a, +a[ ] b, +b[ Ω, by 7 with: γ, x = σx+, γ, x = σ1x1+, 9 where σ x1 x 1 = 0 for x 1 in ] a, +a[ and σ x1 x 1 > 0 for x 1 in R\] a, +a[, σ x x = 0 for x in ] b, +b[ and σ x x > 0 for x in R\] b, +b[. We use Finite Volume method for T.M Maxwell s equation 6, 7 to investigate numerically the cloaking and back-scattering invisibility roerties. The cloaking effect is numerically shown on an annulus with circular PML 8. The back-scattering abilities has been comuted with cartesian PML 9 on a square. A oint source is introduced. It is described in Equation 6 with f = f[0, 0, 1] T where f is a mollification of δ x0 suorted by the oen disk of radius η = π 1.4 : 0, x x0 η, fx = exη ex η 10, x x 0 < η. x x 0 Figure 1a reresents the aroximation of the field Reh 3 solution of T. M. Maxwell s equations 6 8, 10 with x 0 = 0, 0 and σ ρ ρ = 5, 617 ex ρ r1+r. We can see that PML absorbs strongly the field Reh 3. Moreover it is reflectionless thus electromagnetic cloaking can be erformed with PML. The field outside the annulus, the concentric circles, is closed to zero ±0.. Inside the annulus, we have exactly the restriction of the solution of 6 8, 10 roagating on unbounded sace. Figure 1b reresents the aroximation of the field Reh 3 solution of T. M. Maxwell s equations 6, 7, 10, 9 with x 0 = 5, 0, σj kx j = x j ± d k. There are two layers of PML. One on the boundary of our comutational domain, for numerical urose, defined by 9 with σ 1 x j = x j ± 1. The other, which is lying in the middle of the aera, for invisibility aim, is 9 with σ x j = x j ± 3. We can see that a measurement of the field at x = x m 1, xm with x m 1 5 and x m = 0 is not modified by the resence of the internal PML box. Hence back scattering invisibility is erformed. The electromagnetic wave roagation in vacuum is aroached by the system 6, 7. The matrix B, x is bounded and holomohic on C\0} Ω thus satisfy H1 H. As the multilication oerator B1 is coercive, H3 is checked with 0 = 1. Again, the arameters in 7, 8 and 7, 9 define an admissible system in the sence of theorem 1. This result and the numerical simulations see Figures 1a and b lead us to investigate some roerties of the PML. Even if PML are not by nature, we investigate their roerties to roose a b Figure 1: a Cloaking. b Back-scattering invisibility.

5 PIERS ONLINE, VOL. 6, NO., a distribution of indexes erforming cloaking and back-scattering invisibility. It is well known that PML is reflectionless and absorbs strongly the scattered waves. Then, PML can erforms electromagnetic cloaking. To show that PML can erform back scattering invisibility, we assume, for simlicity, that Θ Ω is a circle. We use the following comlex formulation of PML [5]: x : R 3 C 3, x xx = ρcosθ, sinθ, where ρ x = ρ x + Kρ, θ [0, π] and K 0. Let y R, from [5], the solution of the Maxwell s equations 6, 7 with fx = δ y x is a radial function G x ỹ. Let f L Ω be comactly suorted in Ω. The solution of 6, 7 is given by ũx = G xx ỹy fỹydy. Let G x y be the Green s function of Ω the T. M. Maxwell s equations 6 in the vacuum B = I 3 thus the solution of 6 is given by ux = G x y fydy. A straightforward calculation of x ỹ leads to: Ω x ỹ = ρ x + ρ y cosθ x cosθ y + sinθ x sinθ y }. Thus x ỹ = ρ x + ρ y cosθ x θ y }, and, develoing ρ x, ρ y, and assuming θ x = θ y [π], it follows: x ỹ = ρ x ρ y. Then, for all x such that θ x = θ y [π], we have ũx = ux so the solution of 6, 7 and the solution of 6 in the vaccum match and we can conclude that PML erforms back-scattering invisibility. Remark that the roof of back-scattering invisibility erformed by PML can be done in R 3 with sherical PML for Maxwell s equations. 6. CONCLUSION In this aer we were interested in analyzing scattering by comlex materials either for Drude-Born- Fedorov system or Maxwell s equations. Using a general mathematical framework, we studied a Maxwell s equations with a eriodical array of intersaced Slit-Ring-Resonators. A DBF system in resence of homogenized dielectric-chiral hotonic crystals have been studied. The Maxwell s system with some absorbing boundary condition of PML tye, seen as ideal metamaterial, have also been considered. At last, we checked both numerically using Finite Volume method for T.M Maxwell s system and theoretically the cloaking and back-scattering invisibility abilities of the PML. The homogenized systems treated in this aer have been analyzed with a general mathematical framework. Futhermore, this tool works under comatible assumtions relevant for the hysical modelling thus more homogenized systems can be studied using it. It remains to find how we can aroximate the solution of these homogenized systems. The classical numerical methods Finite Volumes, Finite Elements, Finite Difference,... used for the aroximation of Maxwell s equations or DBF system in resence of usual materials are convergent mainly due to the ositiveness of the arameters [ε] and [µ]. The question is then to know if these numerical schemes are still convergent in resence of materials whose arameters can become negative definite for some ulsation w. An answer can be found in some works dealing with finite element methods for wave roagation in metamaterials which have been made in [1, 10] modifying the Finite Element scheme. However, these result can t be, at the best of our knowledge, straightforwardly used for systems like Maxwell s equations with a eriodical arrangement of slitring-resonators or for the DBF system with chiral hotonic crystals. REFERENCES 1. Bonnet-Bendhia, A. S., P. Ciarlet, and C. M. Zwölf, Time harmonique wave diffration roblems in materials with sign-shifting coefficients, J. Comut. Al. Math., Cocquet, P. H., P. A. Mazet, and V. Mouysset, On well-osedness of some homogenized Drude-Born-Fedorov systems on a bounded domain and alications to metamaterials, submitted. 3. Guenneau, S. and F. Zolla, Homogenization of 3D finite chiral hotonic crystals, Science Direct, Physica B, Vol. 394, , 007.

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