Two dimensional invisibility cloaking for Helmholtz equation and non-local boundary conditions
aa r X i v : . [ m a t h . A P ] J a n Two dimensional invisibility cloaking forHelmholtz equation and non-local boundaryconditions
Matti Lassas ∗ and Ting Zhou † Abstract
Transformation optics constructions have allowed the design ofcloaking devices that steer electromagnetic, acoustic and quantumwaves around a region without penetrating it, so that this region ishidden from external observations. The material parameters used todescribe these devices are anisotropic, and singular at the interfacebetween the cloaked and uncloaked regions, making physical realiza-tion a challenge. These singular material parameters correspond tosingular coefficient functions in the partial differential equations mod-eling these constructions and the presence of these singularities causesvarious mathematical problems and physical effects on the interfacesurface.In this paper, we analyze the two dimensional cloaking for Helmholtzequation when there are sources or sinks present inside the cloaked re-gion. In particular, we consider nonsingular approximate invisibilitycloaks based on the truncation of the singular transformations. Usingsuch truncation we analyze the limit when the approximate cloakingapproaches the ideal cloaking. We show that, surprisingly, a non-localboundary condition appears on the inner cloak interface. This effect ∗ Department of Mathematics and Statistics, P.O. Box 68, 00014 University of Helsinki,Finland. Partially supported by Academy of Finland CoE Project 213476 and Mathemat-ical Sciences Research Institute (MSRI). † Department of Mathematics, University of Washington, Seattle, WA 98195, US. Par-tially supported by MSRI n the two dimensional (or cylindrical) invisibility cloaks, which seemsto be caused by the infinite phase velocity near the interface betweenthe cloaked and uncloaked regions, is very different from the earlierstudied behavior of the solutions in the three dimensional cloaks. There has recently been much activity concerning cloaking , or rendering ob-jects invisible to detection by electromagnetic, acoustic, or other type ofwaves or physical fields. Many suggestions to implement cloaking has beenbased on transformation optics , that is, designs of electromagnetic or acous-tic devices with customized effects on wave propagation, made possible bytaking advantage of the transformation rules for the material properties ofoptics. All perfect cloaking devices based on transformation optics requireanisotropic and singular material parameters, whether the conductivity (elec-trostatic) [16, 17], index of refraction (Helmholtz) [22], [10], permittivity andpermeability (Maxwell) [29], [10], mass tensor (acoustic) [10], [6], [9], or ef-fective mass (Schr¨odinger) [13, 14, 35]. By singular material parameters,we mean that at least one of the eigenvalues or the values of the functionsdescribing the material properties goes to zero or infinity at some pointswhen the material parameters are represented in Euclidean coordinates, typ-ically on the interface between the cloaked and uncloaked regions. Both theanisotropy and singularity present serious challenges in trying to physicallyrealize such theoretical plans using metamaterials. Analogous difficulties areencountered in the study of invisibility cloaks base on ray-theory [23] andplasmonic resonances [1, 26].To justify the invisibility cloaking constructions, one needs to study phys-ically meaningful solutions of the resulting partial differential equations onthe whole domain, including the region where material parameters becomesingular. In [10], the finite energy solutions are defined to be at least measur-able functions with finite energy in (degenerate) singular weighted Sobolevspaces, and satisfy the equations in distributional sense.Due to the presence of singular material parameters, or mathematicallyspeaking, partial differential equations with singular coefficient functions, thequestion how the waves behave in cloaking devices near the surface wherethe material parameters are singular is complicated. Indeed, very different2ind of behaviors of solutions are possible: In the three dimensional case,it is proved in [10] that the transformation optics construction based ona blow up map allows cloaking with respect to time-harmonic solutions ofthe Helmholtz equation or Maxwell’s equations as long as the object beingcloaked is passive. In fact, for the Helmholtz equation, the object can bean active source or sink. Moreover, in [10] it is shown that the finite energysolutions for the Helmholtz equation in the three dimensional case satisfya hidden boundary condition, namely waves inside the cloaked region sat-isfy the Neumann boundary conditions. For Maxwell’s equations, the finiteenergy solutions inside the cloaked region need to have vanishing Cauchydata i.e., the hidden boundary conditions are over-determined. This leadsto non-existence of finite solutions for Maxwell’s equations with generic in-ternal currents [10]. Physically, this non-existence results is related to theso-called extraordinary boundary effects on the interface between the cloakedand uncloaked regions [36].Another point of view in dealing with the singular anisotropic design forcloaking devices is to approximate the ideal cloaking parameters by nonsin-gular, or even nonsingular and isotropic, parameters [12, 13, 14, 18, 19, 25],which has its advantages in practical fabrication. In the truncation basednonsingular approximate cloaking for three dimensional Helmholtz equation[14], when it approaches the ideal cloaking, one can obtain above Neumannhidden boundary condition for the finite energy solution. In the nonsingularand isotropic approximate cloaking, one can obtain different types of Robinboundary conditions by varying slightly the way how the approximative cloakin constructed, see [12, 13, 14].Similarly, for Maxwell’s equations it has been studied how the approximatecloak behave on the limit when the approximate cloaks approach the idealone [25]. We note that for Maxwell’s equations there are various suggestionswhat kind of limiting cloaks are possible in three dimensions. These sug-gestions are based on constructions where additional layers (e.g. perfectlyconduction layer) is attached inside the cloak [10] or where the ideal cloakcorresponds to some of the possible self-adjoint extensions of Maxwell’s equa-tions [33, 34]. In the two dimensional or cylindrical cloaking construction forMaxwell’s equations, the eigenvalues of permittivity and the permeability ofthe cloaking medium do not only contain eigenvalues approaching to zero(as in 3D) but also some of the eigenvalues approach infinity at the cloakinginterface. Then, the electric flux density D and magnetic flux density B may3low up even when there are no sources inside the cloak and an incident planewave is scatters from the cloak, see [11]. However, if a soft-hard (SH)-surfaceis included inside the cloak, the solutions behave well.These above examples show how different behavior the solutions may have,in different type of cloaking devices, near the interface between the cloakedand uncloaked regions.In this paper, we analyze the two dimensional cloaking for Helmholtz equa-tion when there is a point source inside the cloaked region. We start withthe nonsingular approximate cloaking based on the truncation of the singu-lar transformation. Taking the limit when the approximate cloaks approachthe ideal cloak, we show that a non-local boundary condition appears on theinner cloak interface. This type of boundary behavior is very different fromthat the solutions have in three dimensional case discussed in [14, 36]. Themain result is formulated as Theorem 3.2. Physically speaking, such non-local boundary condition is possible due to the fact that the phase velocityof the waves in the invisibility cloak approaches infinity near the interfacebetween the cloaked and uncloaked regions, even though the group veloc-ity stays finite, see [7]. We note that as the most important experimentalimplementations of invisibility cloaks [30] have been based on cylindricalcloaks, the appearance of such boundary condition could also be studied inthe present experimental configurations, at least on micro-wave frequencies.We also study the eigenvalues, i.e., resonances inside the ideal cloak corre-sponding to the non-local boundary condition. As these eigenvalues play anessential role in the study of almost trapped states [13] and in the develop-ment of the invisible sensors [2, 15], such resonances can be used to studyanalogous constructions in the cylindrical geometry.The rest of the paper is organized as following. In Section 2, we considerthe basics on transformation optics in the electrostatic setting and the idealacoustic cloak for the two dimensional Helmholtz equation. Section 3 isdevoted to the nonsingular approximate acoustic cloaking construction andanalysis of behaviors of acoustic waves as it approaches the ideal cloaking.4 Perfect acoustic cloaking
Our analysis is closely related to the inverse problem for electrostatics, orCalder´on’s conductivity problem [3, 5, 27, 28, 32]. Let Ω ⊂ R d be a domain,at the boundary of which electrostatic measurements are to be made, anddenote by σ ( x ) the anisotropic conductivity within. In the absence of sources,an electrostatic potential u satisfies a divergence form equation, ∇ · σ ∇ u = 0 (1)on Ω. To uniquely fix the solution u it is enough to give its value, f , on theboundary. In the idealized case, one measures, for all voltage distributions u | ∂ Ω = f on the boundary the corresponding current fluxes, ν · σ ∇ u , where ν is the exterior unit normal to ∂ Ω. Mathematically this amounts to theknowledge of the Dirichlet–Neumann (DN) map, Λ σ . corresponding to σ ,i.e., the map taking the Dirichlet boundary values of the solution to (1) tothe corresponding Neumann boundary values,Λ σ : u | ∂ Ω ν · σ ∇ u | ∂ Ω . If F : Ω → Ω , F = ( F , . . . , F d ), is a diffeomorphism with F | ∂ Ω = Identity,then by making the change of variables y = F ( x ) and setting u = v ◦ F − ,we obtain ∇ · e σ ∇ v = 0 , (2)where e σ = F ∗ σ is the push forward of σ in F ,( F ∗ σ ) jk ( x ) = 1det[ ∂F j ∂y k ( y )] d X p,q =1 ∂F j ∂y p ( y ) ∂F k ∂y q ( y ) σ pq ( y ) (cid:12)(cid:12)(cid:12) y = F − ( x ) . (3)This can be used to show thatΛ F ∗ σ = Λ σ . Thus, there is a large (infinite-dimensional) family of conductivities whichall give rise to the same electrostatic measurements at the boundary. Thisobservation is due to Luc Tartar (see [20] for an account.) Calder´on’s inverse5roblem for anisotropic conductivities is then the question of whether twoconductivities with the same DN operator must be push-forwards of eachother. There are a number of positive results in this direction in two dimen-sions [4, 21, 24, 31] , but it was shown in [16, 17] in three dimensions and in[19] two dimensions that, if one allows singular maps, then in fact there arecounterexamples, i.e., conductivities that are undetectable to electrostaticmeasurements at the boundary.From now on, we will restrict ourselves to the two dimensional case. Foreach
R >
0, let B R = {| x | ≤ R } and Σ R = {| x | = R } be the central balland sphere of radius R , resp., in R , and let O = (0 , ,
0) denote the origin.To construct an invisibility cloak, for simplicity we use the specific singularcoordinate transformation F : R \{ O } → R \ B , given by x = F ( y ) := ( y, for | y | > , (cid:16) | y | (cid:17) y | y | , for 0 < | y | ≤ . (4)Letting σ = 1 be the homogeneous isotropic conductivity on R , F thendefines a conductivity σ on R \ B by the formula σ jk ( x ) := ( F ∗ σ ) jk ( x ) , (5)cf. (3). More explicitly, the matrix σ = [ σ jk ] j,k =1 is of the form σ ( x ) = | x | − | x | Π( x ) + | x || x | − I − Π( x )) , < | x | < , where Π( x ) : R → R is the projection to the radial direction, defined byΠ( x ) v = (cid:18) v · x | x | (cid:19) x | x | , (6)i.e., Π( x ) is represented by the matrix | x | − xx t , cf. [19].One sees that σ ( x ) is singular at Σ , that is, in interface between the cloakedand uncloaked regions, as one of its eigenvalues, namely the one correspond-ing to the radial direction, tends to 0, and the other tends to ∞ as | x | approaching 1 + . We extend σ to B as an arbitrary smooth, nonsingu-lar (bounded from above and below) conductivity there. Let Ω = B ; theconductivity σ is then a cloaking conductivity on Ω, as it is indistinguish-able from σ , vis-a-vis electrostatic boundary measurements of electrostatic6otentials (treated rigorously as bounded, distributional solutions of the de-generate elliptic boundary value problem corresponding to σ [16, 17]).A transformation optics construction based on the above blow up map F was proposed in Pendry, Schurig and Smith [29] to cloak the region B in R from observation by electromagnetic waves at a positive frequency; seealso Leonhardt [22] for a related proposition for the Helmholtz equation in R based on the use of several leaves of an Riemannian sufrace. We consider the Helmholtz equation, with source term p , of the form λ ∇ · σ ∇ u + ω u = p ( x ) on Ω (7)corresponding cloaking medium with the mass tensor and the bulk modulusgiven by σ jk = σ jk for | x | > F ∗ σ ) jk for 1 < | x | ≤ σ jka for | x | ≤ , λ = λ for | x | > F ∗ λ for 1 < | x | ≤ λ a for | x | ≤ σ , λ ) corresponds to homogeneous vacuum space and ( σ a , λ a ) arearbitrary smooth, nondegenerate medium in cloaked region B . The push-forward of tensor F ∗ σ is defined by (3) and F ∗ λ by( F ∗ λ )( x ) := [det( DF ) λ ] ◦ F − ( x ) . More specifically, if ( σ , λ ) = ( I, < | x | < σ ( x ) = | x | − | x | Π( x ) + | x || x | − I − Π( x )) , λ ( x ) = | x | | x | − = ∂B .In the next section, we study the regularized approximate cloaking and obtainthe behavior of acoustic waves in a singular medium by taking the limit ofwaves propagating in the nonsingular medium.7 Nonsingular approximate cloaking for theHelmholtz equation with interior sources
At the present time, we assume that σ and λ be homogeneous isotropic inside B , i.e., ( σ, λ ) = ( σ a δ jk , λ a ) with σ a and λ a arbitrary positive constants.To start, let 1 < R < ρ = 2( R −
1) and introduce the coordinate transfor-mation F R : R \ B ρ → R \ B R , x := F R ( y ) = ( y, for | y | > , (cid:16) | y | (cid:17) y | y | , for ρ < | y | ≤ , cf. [12, 13, 14, 18, 19, 25]. We define the corresponding approximate masstensor σ R and bulk modulus λ R as σ jkR ( x ) = (cid:26) σ jk ( x ) for | x | > R,σ a δ jk for | x | ≤ R. λ R ( x ) = (cid:26) λ ( x ) for | x | > R,λ a for | x | ≤ R, (9)where σ jk and λ are as in (8). Note that then σ jk ( x ) = (( F R ) ∗ σ ) jk ( x )and λ ( x ) = (( F R ) ∗ λ ) ( x ) for | x | > R . Observe that, for each R >
1, themedium is nonsingular, i.e., is bounded from above and below with, however,the lower bound going to 0, and the upper bound going to ∞ as R → + .Consider the solutions of( λ R ∇· σ R ∇ + ω ) u R = p in Ω u R | ∂ Ω = f, (10)As σ R and λ R are now non-singular everywhere on Ω, we have the standardtransmission conditions on Σ R := { x : | x | = R } , u R | Σ R + = u R | Σ R − ,e r · σ R ∇ u R | Σ R + = e r · σ R ∇ u R | Σ R − , (11)where e r is the radial unit vector and ± indicates when the trace on Σ R iscomputed as the limit r → R ± .Let Ω = B . Then u R defines two functions v ± R such that u R ( x ) = (cid:26) v + R ( F − R ( x )) , for R < | x | < ,v − R ( x ) , for | x | ≤ R, v ± R satisfy ( ∇ + ω ) v + R ( y ) = p ( F R ( y )) in ρ < | y | < ,v + R | ∂B = f, (12)and ( ∇ + κ ω ) v − R ( x ) = κ p ( x ) , in | x | < R. (13)where κ = ( σ a λ a ) − is a constant. Moreover, if we assume ω is not aneigenvalue of the transmission problem, then by the transformation law wehave e r · σ R ∇ u R (cid:12)(cid:12) ∂ Ω = e r · ∇ v + R (cid:12)(cid:12) ∂ Ω . This implies that the DN-map Λ σ R ,λ R at ∂ Ω for the approximate cloakingmedium (9) is the same as the DN-map at ∂ Ω, denoted by Λ ρ , of a nearlyvacuum domain with a small inclusion present in B ρ .Next, using polar coordinates ( r, θ ), r = | y | , and ( e r, θ ), e r = | x | , the trans-mission conditions (11) on the surface Σ R yield v + R ( ρ, θ ) = v − R ( R, θ ) ,ρ ∂ r v + R ( ρ, θ ) = κR ∂ e r v − R ( R, θ ) . (14)For simplicity, we analyze the case of when the source is supported at origininside the cloak and has the form p ( x ) = κ − X | α |≤ N q α ∂ αx δ ( x ) , (15)where δ is the Dirac delta function at origin and q α ∈ C , i.e., there is a(possibly quite strong) point source in the cloaked region. The Helmholtzequation (13) on the entire space R , with the above point source and thestandard radiation condition, would give rise to a radiating wave w ∈ D ′ ( R ), w | R \{ } ∈ C ∞ ( R \ { } ), given by w ( e r, θ ) = N X n = − N p n H (1) | n | ( κω e r ) e inθ . (16)where H (1) | n | ( z ) and J | n | ( z ) denote the Hankel and Bessel functions, see [8],and p n ∈ C . The analysis below can be generalized for more general sources,but this generalization will be considered elsewhere.9n B R \ B R the function v − R ( x ) differs from w by a solution to the homoge-neous equation of (13), and thus for e r ∈ ( R , R ) v − R ( e r, θ ) = N X n = − N ( a n J | n | ( κω e r ) + p n H (1) | n | ( κω e r )) e inθ , with yet undefined a n = a n ( κ, ω ; R ). Similarly, for ρ < r < v + R ( r, θ ) = N X n = − N ( c n H (1) | n | ( ωr ) + b n J | n | ( ωr )) e inθ , with as yet unspecified b n = b n ( κ, ω ; R ) and c n = c n ( κ, ω ; R ).Rewriting the boundary value f on ∂ Ω as f ( θ ) = N X n = − N f n e inθ , we obtain, together with transmission conditions (14), the following equationsfor a n , b n and c n : f n = b n J | n | (3 ω ) + c n H (1) | n | (3 ω ) , (17) a n J | n | ( κωR ) + p n H (1) | n | ( κωR ) = b n J | n | ( ωρ ) + c n H (1) | n | ( ωρ ) , (18) κR ( κωa n ( J | n | ) ′ ( κωR ) + κωp n ( H (1) | n | ) ′ ( κωR ))= ρ ( b n ω ( J | n | ) ′ ( ωρ ) + ωc n ( H (1) | n | ) ′ ( ωρ )) . (19)Solve for a n and c n from (18)-(19) in terms of p n and b n , and use the solutionsobtained and the equation (17) to solve for b n in terms of f n and p n . Thisyields b n = 1 J | n | (3 ω ) + s n H (1) | n | (3 ω ) ( f n + e s n H (1) | n | (3 ω ) p n ) ,c n = s n b n − e s n p n ,a n = t n b n − e t n p n (20)10here s n = 1 D n (cid:8) ρJ | n | ( κωR ) J ′| n | ( ωρ ) − κ RJ ′| n | ( κωR ) J | n | ( ωρ ) (cid:9) ,t n = 1 D n n ρH (1) | n | ( ωρ ) J ′| n | ( ωρ ) − ρH (1) | n | ′ ( ωρ ) J | n | ( ωρ ) o , e s n = 1 D n n κ RH (1) | n | ′ ( κωR ) J | n | ( κωR ) − κ RJ ′| n | ( κωR ) H (1) | n | ( κωR ) o , e t n = 1 D n n κ RH (1) | n | ( ωρ ) H (1) | n | ′ ( κωR ) − ρH (1) | n | ′ ( ωρ ) H (1) | n | ( κωR ) o with D n the common denominator given by D n = κ RJ ′| n | ( κωR ) H (1) | n | ( ωρ ) − ρJ | n | ( κωR )( H (1) | n | ) ′ ( ωρ ) . Suppose that the boundary data vanishes, i.e., f ≡
0. Then by (20), we have b n = e s n H (1) | n | (3 ω ) J | n | (3 ω ) + s n H (1) | n | (3 ω ) p n . Therefore, one can show a n = ( t n e s n − e t n s n ) H (1) | n | (3 ω ) − e t n J | n | (3 ω ) J | n | (3 ω ) + s n H (1) | n | (3 ω ) p n = κ RH (1) | n | ′ ( κωR ) l − ρH (1) | n | ( κωR ) l ρJ | n | ( κωR ) l − κ RJ ′| n | ( κωR ) l p n := A n B n p n (21)where l = J | n | ( ωρ ) H (1) | n | (3 ω ) − H (1) | n | ( ωρ ) J | n | (3 ω ) ,l = J ′| n | ( ωρ ) H (1) | n | (3 ω ) − H (1) | n | ′ ( ωρ ) J | n | (3 ω ) . (22)11ince for small arguments x ≪ J | n | ( x ) ∼ n ! (cid:16) x (cid:17) n n ≥ , J ′| n | ( x ) ∼ (cid:26) − x n = 0 , n ( n − x n − n ≥ ,H (1) | n | ( x ) ∼ (cid:26) iπ ln ( x/ n = 0 , − i ( n − π n x − n n ≥ , H (1) | n | ′ ( x ) ∼ (cid:26) ix − /π n = 0 ,in !2 n x − n − /π n ≥ , (23)when n ≥
1, where we denote f ∼ g if f − g = o ( g ) as x →
0, and A n ∼ i n ω − n − ( n − π J | n | (3 ω ) h ωκ RH (1) | n | ′ ( κωR ) + nH (1) | n | ( κωR ) i ρ − n ,B n ∼ − i n ω − n − ( n − π J | n | (3 ω ) (cid:2) ωκ RJ ′| n | ( κωR ) + nJ | n | ( κωR ) (cid:3) ρ − n . (24)Now we observe that | a n | → ∞ as R → + if (cid:2) ωκ R ( J | n | ) ′ ( κωR ) + nJ | n | ( κωR ) (cid:3) (cid:12)(cid:12)(cid:12)(cid:12) R =1 = 0 . (25)Note that if (25) holds, then h ωκ R ( H (1) | n | ) ′ ( κωR ) + nH (1) | n | ( κωR ) i (cid:12)(cid:12)(cid:12)(cid:12) R =1 = 0 . (26)This implies that if ω is such that (25) is satisfied by functions J | n | and H (1) | n | for some n , then there are sources p for which for any with 0 < R < R < H ( B R \ B R )-norm of the solution u R restricted to B R \ B R goes toinfinity (i.e., a resonance happens in the cloaked region) as R → + (i.e., ρ → V ± n ( r, θ ) := J | n | ( κωr ) e ± inθ (27)satisfies the boundary value problem(∆ + κ ω ) V = 0 in B , (cid:2) κr∂ r V + ( − ∂ θ ) / V (cid:3) (cid:12)(cid:12)(cid:12) r =1 + = 0 . (28)12oughly speaking, this means that the ideal cloak has resonance solutionsinside the cloaked region which satisfy a non-local boundary condition.Next we consider the frequencies ω for which (cid:2) ωκ R ( J | n | ) ′ ( κωR ) + nJ | n | ( κωR ) (cid:3) (cid:12)(cid:12)(cid:12)(cid:12) R =1 = 0 ,J | n | ( ωr ) (cid:12)(cid:12) r =3 = 0 , for all n ∈ Z . (29) In the following, we show that when we have in B \ B R , 1 < R < ω , the boundary condition in (28) holds as well as R → + . Westart with the following lemma. Lemma 3.1.
Assume that in
Ω = B we have the material parameters ( σ R , λ R ) . Moreover, suppose ω is such that (29) . When R > is suffi-ciently close to , then for any source p of the form (15) with N ≥ andfor f ∈ H / ( ∂ Ω) the Helmholtz equation (10) has a unique solution u R .Moreover, as R → + , Fourier coefficients of the solution v − R := u R | B R in thecloaked region, v − R,n ( r ) = Z π e − inθ v − R ( r, θ ) dθ satisfy lim R → ( κr∂ r v − R,n ( r ) + n v − R,n ( r )) (cid:12)(cid:12) r = R = 0 (30) for all | n | ≤ N .Proof. We have v − R ( r, θ ) = N X n = − N ( a n J | n | ( κωr ) + p n H (1) | n | ( κωr )) e inθ = N X n = − N (cid:20) A n B n J | n | ( κωr ) + H (1) | n | ( κωr ) (cid:21) p n e inθ A n = κ RH (1) | n | ′ ( κωR ) l − ρH (1) | n | ( κωR ) l B n = ρJ | n | ( κωR ) l − κ RJ ′| n | ( κωR ) l . Denote Φ n ( r ) := A n B n J | n | ( κωr ) + H (1) | n | ( κωr ) . Apparently e Φ( r, θ ) := Φ n ( r ) e inθ satisfies the Helmholtz equation(∆ + κ ω ) e Φ = 0 in B R . To check (30), it is sufficient to show rκ∂ r Φ n ( r ) + n Φ n ( r ) (cid:12)(cid:12) r = R → R → + . (31)Indeed, by ∂ r Φ n ( r ) = A n B n κωJ ′| n | ( κωr ) + κωH (1) | n | ′ ( κωr ) , and B n = 0, we have rκ∂ r Φ n ( r ) + n Φ n ( r ) (cid:12)(cid:12) r = R = κ RB n ( nl + ωρl ) (cid:16) H (1) | n | ′ ( κωR ) J | n | ( κωR ) − J ′| n | ( κωR ) H (1) | n | ( κωR ) (cid:17) (32)where l and l are given by (22).For n ≥
1, as ρ → + , nl + ωρl = ωρ (cid:16) J | n |− ( ωρ ) H (1) | n | (3 ω ) − H (1) | n |− ( ωρ ) J | n | (3 ω ) (cid:17) = O ( ρ − n +2 ) . (33)Combining (32), (33) and (24), one has rκ∂ r Φ n ( r ) + n Φ n ( r ) (cid:12)(cid:12) r = R = O ( ρ ) as R → + (i.e. ρ → + ) , (34)which proves (31). 14or n = 0, from (23), one has A = − iκ Rπ H (1)0 ′ ( κωR ) J (3 ω ) ln (cid:16) ωρ (cid:17) + κ RH (1)0 ′ ( κωR ) H (1)0 (3 ω )+ 2 iπω H (1)0 ( κωR ) J (3 ω ) + O ( ρ ) ,B = 2 iκ Rπ J ′ ( κωR ) J (3 ω ) ln (cid:16) ωρ (cid:17) − κ RJ ′ ( κωR ) H (1)0 (3 ω )+ 2 iπω J ( κωR ) J (3 ω ) + O ( ρ ) . Therefore, ∂ r Φ ( R ) = A B κωJ ′ ( κωR ) + κωH (1)0 ′ ( κωR )has denominator B and numerator κω [ A J ′ ( κωR ) + B H (1)0 ′ ( κωR )] = 2 κiπ W n ( κωR ) + O ( ρ ) , where W n ( x ) = H (1)0 ( x ) J ′ ( x ) − H (1)0 ′ ( x ) J ( x ) This implies ∂ r Φ ( R ) ∼ W n ( κωR ) κRJ ′ ( κωR ) ln (cid:0) ωρ (cid:1) → ρ → + , i.e., the boundary condition (31) is satisfied for n = 0 and moreover, rκ∂ r Φ ( r ) (cid:12)(cid:12) r = R = O (cid:0) ωρ (cid:1) ! as R → + . (35)Now we are ready to prove our main result for the limit of the waves u R cor-responding to the approximate cloaking medium as R → + . The obtainedlimit can be considered as a model for solutions in an ideal (i.e. perfect)cylindrical invisibility cloak. Theorem 3.2.
Let ω be such that (29) is satisfied. Suppose u R is the solutionof (10) with f = 0 and source p is of the form (15) with N ≥ Then as R → + , u R converges almost everywhere in B to the limit u satisfying (cid:26) (∆ + κ ω ) u = κ p in B ,κr∂ r u + ( − ∂ θ ) / u (cid:12)(cid:12) ∂B = 0 , (36)15 nd u (cid:12)(cid:12) B \ B = 0 . (37) Proof.
Let 0 < R < R < R . Using Lemma 3.1 we see that the wave u R andits derivatives of any order converge uniformly in compact subsets of B \ B R to u = N X n = − N e a n J | n | ( κω e r ) e inθ + w, as R →
1. Moreover, we see u ∈ C ∞ ( B \ B R ) and that u satisfies (38).As u R satisfy Dirichlet problems ( ( ∇ + κ ω ) u R = κ p in B R ,u R (cid:12)(cid:12) ∂B R = h R , (38)where h R converge in C k ( ∂B R ) to u | ∂B R for any k as R →
1, we see that u R converges pointwisely to u in B R \ { } .We also see that u R converge pointwisely to zero in B \ B using the factthat c n ( ρ ) = O ( ρ n +1 ) , b n ( ρ ) = O ( ρ n +1 ) as ρ → + . This proves that (37) holds.We conclude our discussion by remarking that our analysis also explainsthe limit of approximate electromagnetic cloaks in the cylindrical case withTE/TM polarized incoming waves, as the solutions of Maxwell’s equationsin this case satisfy the Helmholtz equation.
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Phys.Rev. Lett. , (2008), 063904. 19 r X i v : . [ m a t h . A P ] J a n Two dimensional invisibility cloaking forHelmholtz equation and non-local boundaryconditions
Matti Lassas ∗ and Ting Zhou † Abstract
Transformation optics constructions have allowed the design ofcloaking devices that steer electromagnetic, acoustic and quantumparameters waves around a region without penetrating it, so thatthis region is hidden from external observations. The material pa-rameters used to describe these devices are anisotropic, and singularat the interface between the cloaked and uncloaked regions, makingphysical realization a challenge. These singular material parameterscorrespond to singular coefficient functions in the partial differentialequations modeling these constructions and the presence of these sin-gularities causes various mathematical problems and physical effectson the interface surface.In this paper, we analyze the two dimensional cloaking for Helmholtzequation when there are sources or sinks present inside the cloaked re-gion. In particular, we consider nonsingular approximate invisibilitycloaks based on the truncation of the singular transformations. Usingsuch truncation we analyze the limit when the approximate cloakingapproaches the ideal cloaking. We show that, surprisingly, a non-localboundary condition appears on the inner cloak interface. This effect inthe two dimensional (or cylindrical) invisibility cloaks, which seems tobe caused by the infinite phase velocity near the interface between thecloaked and uncloaked regions, is very different to the earlier studiedbehavior of the solutions in the three dimensional cloaks. ∗ Institute of Mathematics, Helsinki University of Technology, FIN-02015, Finland. † Department of Mathematics, University of Washington, Seattle, WA 98195, US. Introduction
There has recently been much activity concerning cloaking , or rendering ob-jects invisible to detection by electromagnetic, acoustic, or other type ofwaves or physical fields. Many suggestions to implement cloaking has beenbased on transformation optics , that is, designs of electromagnetic or acous-tic devices with customized effects on wave propagation, made possible bytaking advantage of the transformation rules for the material properties ofoptics. All perfect cloaking devices based on transformation optics requireanisotropic and singular material parameters, whether the conductivity (elec-trostatic) [16, 17], index of refraction (Helmholtz) [23], [10], permittivity andpermeability (Maxwell) [32], [10], mass tensor (acoustic) [10], [6], [9], or ef-fective mass (Schr¨odinger) [13, 14, 38]. By singular material parameters,we mean that at least one of the eigenvalues or the values of the functionsdescribing the material properties goes to zero or infinity at some pointswhen the material parameters are represented in Euclidean coordinates, typ-ically on the interface between the cloaked and uncloaked regions. Both theanisotropy and singularity present serious challenges in trying to physicallyrealize such theoretical plans using metamaterials. Analogous difficulties areencountered in the study of invisibility cloaks base on ray-theory [24] andplasmonic resonances [1, 28].To justify the invisibility cloaking constructions, one needs to study phys-ically meaningful solutions of the resulting partial differential equations onthe whole domain, including the region where material parameters becomesingular. In [10], the finite energy solutions are defined to be at least measur-able functions with finite energy in (degenerate) singular weighted Sobolevspaces, and satisfy the equations in distributional sense.Due to the presence of singular material parameters, or mathematicallyspeaking, partial differential equations with singular coefficient functions, thequestion how the waves behave in cloaking devices near the surface wherethe material parameters are singular is complicated. Indeed, very differentkind of behaviors of solutions are possible: In the three dimensional case,it is proved in [10] that the transformation optics construction based ona blow up map allows cloaking with respect to time-harmonic solutions ofthe Helmholtz equation or Maxwell’s equations as long as the object beingcloaked is passive. In fact, for the Helmholtz equation, the object can be2n active source or sink. Moreover, in [10] it is shown that the finite energysolutions for the Helmholtz equation in the three dimensional case satisfya hidden boundary condition, namely waves inside the cloaked region sat-isfy the Neumann boundary conditions. For Maxwell’s equations, the finiteenergy solutions inside the cloaked region need to have vanishing Cauchydata i.e., the hidden boundary conditions are over-determined. This leadsto non-existence of finite solutions for Maxwell’s equations with generic in-ternal currents [10]. Physically, this non-existence results is related to theso-called extraordinary boundary effects on the interface between the cloakedand uncloaked regions [39].Another point of view in dealing with the singular anisotropic design forcloaking devices is to approximate the ideal cloaking parameters by nonsin-gular, or even nonsingular and isotropic, parameters [12, 13, 14, 19, 20, 27],which has its advantages in practical fabrication. In the truncation basednonsingular approximate cloaking for three dimensional Helmholtz equation[14], when it approaches the ideal cloaking, one can obtain above Neumannhidden boundary condition for the finite energy solution. In the nonsingularand isotropic approximate cloaking, one can obtain different types of Robinboundary conditions by varying slightly the way how the approximative cloakin constructed, see [12, 13, 14].Similarly, for Maxwell’s equations it has been studied how the approximatecloak behave on the limit when the approximate cloaks approach the idealone [27]. We note that for Maxwell’s equations there are various suggestionswhat kind of limiting cloaks are possible in three dimensions. These sug-gestions are based on constructions where additional layers (e.g. perfectlyconduction layer) is attached inside the cloak [10] or where the ideal cloakcorresponds to some of the possible self-adjoint extensions of Maxwell’s equa-tions [36, 37]. In the two dimensional or cylindrical cloaking construction forMaxwell’s equations, the eigenvalues of permittivity and the permeability ofthe cloaking medium do not only contain eigenvalues approaching to zero(as in 3D) but also some of the eigenvalues approach infinity at the cloakinginterface. Then, the electric flux density D and magnetic flux density B mayblow up even when there are no sources inside the cloak and an incident planewave is scatters from the cloak, see [11]. However, if a soft-hard (SH)-surfaceis included inside the cloak, the solutions behave well.These above examples show how different behavior the solutions may have,3n different type of cloaking devices, near the interface between the cloakedand uncloaked regions.In this paper, we analyze the two dimensional cloaking for Helmholtz equa-tion when there are sources present inside the cloaked region. We start withthe nonsingular approximate cloaking based on the truncation of the singu-lar transformation. Taking the limit when the approximate cloaks approachthe ideal cloak, we show that a non-local boundary condition appears onthe inner cloak interface. This type of boundary behavior is very differentfrom that the solutions have in three dimensional case discussed in [14, 39].The main result is formulated as Theorem 3.2. We note that in the recentpreprint [31] of Hoai-Minh Nguyen a different type of formulation, based ona transmission problem, is given for the non-local boundary condition ap-pearing in two-dimensional cloaking. In [31], also cloaking for more generalsecond order equations and quantitative convergence properties of the threeand two dimensional approximative cloaks are analyzed.Physically speaking, the non-local boundary condition is possible due to thefact that the phase velocity of the waves in the invisibility cloak approachesinfinity near the interface between the cloaked and uncloaked regions, eventhough the group velocity stays finite, see [7]. We note that as the mostimportant experimental implementations of invisibility cloaks [33] have beenbased on cylindrical cloaks, the appearance of such boundary condition couldalso be studied in the present experimental configurations, at least on micro-wave frequencies. We also study the eigenvalues, i.e., resonances inside theideal cloak corresponding to the non-local boundary condition. As theseeigenvalues play an essential role in the study of almost trapped states [13]and in the development of the invisible sensors [2, 15], such resonances canbe used to study analogous constructions in cylindrical geometry.The rest of the paper is organized as following. In Section 2, we presentthe basics on transformation optics in the electrostatics setting and applythem to the construction of acoustic ideal cloak for the two dimensionalHelmholtz equation. Section 3 is devoted to the nonsingular approximateacoustic cloaking construction and analysis of behaviors of acoustic waves asit approaches the ideal cloaking. 4 Perfect acoustic cloaking
Our analysis is closely related to the inverse problem for electrostatics, orCalder´on’s conductivity problem [3, 5, 29, 30, 35]. Let Ω ⊂ R d be a domain,at the boundary of which electrostatic measurements are to be made, anddenote by σ ( x ) the anisotropic conductivity within. In the absence of sources,an electrostatic potential u satisfies a divergence form equation, ∇ · σ ∇ u = 0 (1)on Ω. To uniquely fix the solution u it is enough to give its value, f , on theboundary. In the idealized case, one measures, for all voltage distributions u | ∂ Ω = f on the boundary the corresponding current fluxes, ν · σ ∇ u , where ν is the exterior unit normal to ∂ Ω. Mathematically this amounts to theknowledge of the Dirichlet–Neumann (DN) map, Λ σ . corresponding to σ ,i.e., the map taking the Dirichlet boundary values of the solution to (1) tothe corresponding Neumann boundary values,Λ σ : u | ∂ Ω ν · σ ∇ u | ∂ Ω . If F : Ω → Ω , F = ( F , . . . , F d ), is a diffeomorphism with F | ∂ Ω = Identity,then by making the change of variables y = F ( x ) and setting u = v ◦ F − ,we obtain ∇ · e σ ∇ v = 0 , (2)where e σ = F ∗ σ is the push forward of σ in F ,( F ∗ σ ) jk ( x ) = 1det[ ∂F j ∂y k ( y )] d X p,q =1 ∂F j ∂y p ( y ) ∂F k ∂y q ( y ) σ pq ( y ) (cid:12)(cid:12)(cid:12) y = F − ( x ) . (3)This can be used to show thatΛ F ∗ σ = Λ σ . Thus, there is a large (infinite-dimensional) family of conductivities whichall give rise to the same electrostatic measurements at the boundary. Thisobservation is due to Luc Tartar (see [21] for an account.) Calder´on’s inverse5roblem for anisotropic conductivities is then the question of whether twoconductivities with the same DN operator must be push-forwards of eachother. There are a number of positive results in this direction in two dimen-sions [4, 22, 25, 34] , but it was shown in [16, 17] in three dimensions and in[20] two dimensions that, if one allows singular maps, then in fact there arecounterexamples, i.e., conductivities that are undetectable to electrostaticmeasurements at the boundary.From now on, we will restrict ourselves to the two dimensional case. Foreach
R >
0, let B R = {| x | ≤ R } and Σ R = {| x | = R } be the central balland sphere of radius R , resp., in R , and let O = (0 , ,
0) denote the origin.To construct an invisibility cloak, for simplicity we use the specific singularcoordinate transformation F : R \{ O } → R \ B , given by x = F ( y ) := ( y, for | y | > , (cid:16) | y | (cid:17) y | y | , for 0 < | y | ≤ . (4)Letting σ = 1 be the homogeneous isotropic conductivity on R , F thendefines a conductivity σ on R \ B by the formula σ jk ( x ) := ( F ∗ σ ) jk ( x ) , (5)cf. (3). More explicitly, the matrix σ = [ σ jk ] j,k =1 is of the form σ ( x ) = | x | − | x | Π( x ) + | x || x | − I − Π( x )) , < | x | < , where Π( x ) : R → R is the projection to the radial direction, defined byΠ( x ) v = (cid:18) v · x | x | (cid:19) x | x | , (6)i.e., Π( x ) is represented by the matrix | x | − xx t , cf. [20].One sees that σ ( x ) is singular at Σ , that is, in interface between the cloakedand uncloaked regions, as one of its eigenvalues, namely the one correspond-ing to the radial direction, tends to 0, and the other tends to ∞ as | x | approaching 1 + . We extend σ to B as an arbitrary smooth, nonsingu-lar (bounded from above and below) conductivity there. Let Ω = B ; theconductivity σ is then a cloaking conductivity on Ω, as it is indistinguish-able from σ , vis-a-vis electrostatic boundary measurements of electrostatic6otentials (treated rigorously as bounded, distributional solutions of the de-generate elliptic boundary value problem corresponding to σ [16, 17]).A similar construction based on a blow up map F was proposed in Pendry,Schurig and Smith [32] to cloak the region B in R from observation byelectromagnetic waves at a positive frequency; see also Leonhardt [23] fora related proposition for the Helmholtz equation in R based on the use ofseveral leaves of an Riemannian sufrace. We consider the Helmholtz equation, with source term p , of the form λ ∇ · σ ∇ u + ω u = p ( x ) on Ω (7)corresponding cloaking medium with the inverse of the anisotropic mass den-sity and the bulk modulus given by σ jk = σ jk for | x | > F ∗ σ ) jk for 1 < | x | ≤ σ jka for | x | ≤ , λ = λ for | x | > F ∗ λ for 1 < | x | ≤ λ a for | x | ≤ σ , λ ) corresponds to homogeneous vacuum space and ( σ a , λ a ) arearbitrary smooth, nondegenerate medium in cloaked region B . The push-forward of tensor F ∗ σ is defined by (3) and F ∗ λ by( F ∗ λ )( x ) := [det( DF ) λ ] ◦ F − ( x ) . More specifically, if ( σ , λ ) = ( I, < | x | < σ ( x ) = | x | − | x | Π( x ) + | x || x | − I − Π( x )) , λ ( x ) = | x | | x | − = ∂B .In the next section, we study the regularized approximate cloaking and obtainthe behavior of acoustic waves in a singular medium by taking the limit ofwaves propagating in the nonsingular medium.7 Nonsingular approximate cloaking for theHelmholtz equation with interior sources
At the present time, we assume that σ and λ be homogeneous isotropic inside B , i.e., ( σ, λ ) = ( σ a δ jk , λ a ) with σ a and λ a arbitrary positive constants.To start, let 1 < R < ρ = 2( R −
1) and introduce the coordinate transfor-mation F R : R \ B ρ → R \ B R , x := F R ( y ) = ( y, for | y | > , (cid:16) | y | (cid:17) y | y | , for ρ < | y | ≤ . We define the corresponding approximate mass tensor σ R and bulk modulus λ R as σ jkR ( x ) = (cid:26) σ jk ( x ) for | x | > R,σ a δ jk for | x | ≤ R. λ R ( x ) = (cid:26) λ ( x ) for | x | > R,λ a for | x | ≤ R, (9)where σ jk and λ are as in (8). Note that then σ jk ( x ) = (( F R ) ∗ σ ) jk ( x )and λ ( x ) = (( F R ) ∗ λ ) ( x ) for | x | > R . Observe that, for each R >
1, themedium is nonsingular, i.e., is bounded from above and below with, however,the lower bound going to 0, and the upper bound going to ∞ as R → + .Consider the solutions of( λ R ∇· σ R ∇ + ω ) u R = p in Ω u R | ∂ Ω = f, (10)As σ R and λ R are now non-singular everywhere on Ω, we have the standardtransmission conditions on Σ R := { x : | x | = R } , u R | Σ R + = u R | Σ R − ,e r · σ R ∇ u R | Σ R + = e r · σ R ∇ u R | Σ R − , (11)where e r is the radial unit vector and ± indicates when the trace on Σ R iscomputed as the limit r → R ± .Let Ω = B . Then u R defines two functions v ± R such that u R ( x ) = (cid:26) v + R ( F − R ( x )) , for R < | x | < ,v − R ( x ) , for | x | ≤ R, v ± R satisfy ( ∇ + ω ) v + R ( y ) = p ( F R ( y )) in ρ < | y | < ,v + R | ∂B = f, (12)and ( ∇ + κ ω ) v − R ( x ) = κ p ( x ) , in | x | < R. (13)where κ = ( σ a λ a ) − is a constant. Moreover, if we assume ω is not aneigenvalue of the transmission problem, then by the transformation law wehave e r · σ R ∇ u R (cid:12)(cid:12) ∂ Ω = e r · ∇ v + R (cid:12)(cid:12) ∂ Ω . This implies that the DN-map Λ σ R ,λ R at ∂ Ω for the approximate cloakingmedium (9) is the same as the DN-map at ∂ Ω, denoted by Λ ρ , of a nearlyvacuum domain with a small inclusion present in B ρ .Next, using polar coordinates ( r, θ ), r = | y | , and ( e r, θ ), e r = | x | , the trans-mission conditions (11) on the surface Σ R yield v + R ( ρ, θ ) = v − R ( R, θ ) ,ρ ∂ r v + R ( ρ, θ ) = κR ∂ e r v − R ( R, θ ) . (14)We consider the source term κ p ( x ) where p ( x ) ∈ C ∞ ( R ) with supp p ⊂ B R (0 < R < w ( x ) ∈ C ∞ ( R ), namely thesolution of ( ∇ + κ ω ) w = κ p in R (15)satisfying the Sommerfeld radiation condition. Moreover, one can write w as w ( e r, θ ) = ∞ X n = −∞ p n H (1) | n | ( κω e r ) e inθ , for e r > R (16)where H (1) | n | ( z ) and J | n | ( z ) denote the Hankel and Bessel functions, see [8].In B R \ B R the function v − R ( x ) differs from w by an entire solution to thehomogeneous equation of (13), and thus for e r ∈ ( R , R ) v − R ( e r, θ ) = ∞ X n = −∞ ( a n J | n | ( κω e r ) + p n H (1) | n | ( κω e r )) e inθ , a n = a n ( κ, ω ; R ). Similarly, for ρ < r < v + R ( r, θ ) = ∞ X n = −∞ (cid:0) c n H (1) | n | ( ωr ) + b n J | n | ( ωr ) (cid:1) e inθ , (17)with as yet unspecified b n = b n ( κ, ω ; R ) and c n = c n ( κ, ω ; R ).Rewriting the boundary value f on ∂ Ω as f ( θ ) = ∞ X n = −∞ f n e inθ , we obtain, together with transmission conditions (14), the following equationsfor a n , b n and c n : f n = b n J | n | (3 ω ) + c n H (1) | n | (3 ω ) , (18) a n J | n | ( κωR ) + p n H (1) | n | ( κωR ) = b n J | n | ( ωρ ) + c n H (1) | n | ( ωρ ) , (19) κR ( κωa n ( J | n | ) ′ ( κωR ) + κωp n ( H (1) | n | ) ′ ( κωR ))= ρ ( b n ω ( J | n | ) ′ ( ωρ ) + ωc n ( H (1) | n | ) ′ ( ωρ )) . (20)Solve for a n and c n from (19)-(20) in terms of p n and b n , and use the solutionsobtained and the equation (18) to solve for b n in terms of f n and p n . Thisyields b n = 1 J | n | (3 ω ) + s n H (1) | n | (3 ω ) ( f n + e s n H (1) | n | (3 ω ) p n ) ,c n = s n b n − e s n p n ,a n = t n b n − e t n p n (21)where s n = 1 D n (cid:8) ρJ | n | ( κωR ) J ′| n | ( ωρ ) − κ RJ ′| n | ( κωR ) J | n | ( ωρ ) (cid:9) ,t n = 1 D n n ρH (1) | n | ( ωρ ) J ′| n | ( ωρ ) − ρ ( H (1) | n | ) ′ ( ωρ ) J | n | ( ωρ ) o , e s n = 1 D n n κ R ( H (1) | n | ) ′ ( κωR ) J | n | ( κωR ) − κ RJ ′| n | ( κωR ) H (1) | n | ( κωR ) o , e t n = 1 D n n κ RH (1) | n | ( ωρ )( H (1) | n | ) ′ ( κωR ) − ρ ( H (1) | n | ) ′ ( ωρ ) H (1) | n | ( κωR ) o D n the common denominator given by D n = κ RJ ′| n | ( κωR ) H (1) | n | ( ωρ ) − ρJ | n | ( κωR )( H (1) | n | ) ′ ( ωρ ) . (22) Suppose that the boundary data vanishes, i.e., f ≡
0. Then by (21), we have b n = e s n H (1) | n | (3 ω ) J | n | (3 ω ) + s n H (1) | n | (3 ω ) p n . Therefore, one can show a n = ( t n e s n − e t n s n ) H (1) | n | (3 ω ) − e t n J | n | (3 ω ) J | n | (3 ω ) + s n H (1) | n | (3 ω ) p n = κ R ( H (1) | n | ) ′ ( κωR ) l − ρH (1) | n | ( κωR ) l ρJ | n | ( κωR ) l − κ RJ ′| n | ( κωR ) l p n := A n B n p n (23)where l = J | n | ( ωρ ) H (1) | n | (3 ω ) − H (1) | n | ( ωρ ) J | n | (3 ω ) ,l = J ′| n | ( ωρ ) H (1) | n | (3 ω ) − ( H (1) | n | ) ′ ( ωρ ) J | n | (3 ω ) . (24)Since for small arguments 0 < x ≪ J | n | ( x ) ∼ n ! (cid:16) x (cid:17) n n ≥ , J ′| n | ( x ) ∼ ( − x n = 0 , n − (cid:0) x (cid:1) n − n ≥ ,H (1) | n | ( x ) ∼ (cid:26) iπ ln (cid:0) x (cid:1) n = 0 , − i ( n − π (cid:0) x (cid:1) n n ≥ , ( H (1) | n | ) ′ ( x ) ∼ (cid:26) ix − /π n = 0 ,in !2 n x − n − /π n ≥ , (25)when n ≥
1, where we denote f ∼ g if f − g = o ( g ) as x →
0, and A n ∼ i n ω − n − ( n − π J n (3 ω ) (cid:2) ωκ R ( H (1) n ) ′ ( κωR ) + nH (1) n ( κωR ) (cid:3) ρ − n ,B n ∼ − i n ω − n − ( n − π J n (3 ω ) (cid:2) ωκ RJ ′ n ( κωR ) + nJ n ( κωR ) (cid:3) ρ − n , (26)11or n ≥
1. Now we observe that | a n | → ∞ as R → + if (cid:2) ωκ R ( J | n | ) ′ ( κωR ) + | n | J | n | ( κωR ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) R =1 = 0 . (27)Note that then (cid:2) ωκ R ( H (1) | n | ) ′ ( κωR ) + | n | H (1) | n | ( κωR ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) R =1 = 0 . (28)This implies that if κ is outside a discrete set and if ω is such that (27) and(28) are satisfied by functions J | n | and H (1) | n | for some n , then there are sources p for which the H -norm of the solution u R goes to infinity in the cloakedregion (i.e., when resonance happens) as R → + (i.e., ρ → V ± n ( e r, θ ) := J | n | ( κω e r ) e ± inθ (29)satisfies the boundary value problem(∆ + κ ω ) V = 0 in B , (cid:2) κ e r∂ e r V + ( − ∂ θ ) / V (cid:3) (cid:12)(cid:12)(cid:12) e r =1 + = 0 . (30)Next we consider the frequencies ω for which (cid:2) ωκ RJ ′| n | ( κωR ) + | n | J | n | ( κωR ) (cid:3) (cid:12)(cid:12)(cid:12)(cid:12) R =1 = 0 ,J | n | (3 ω ) = 0 , for any n ∈ Z . (31) In the following, we show that when we have in B \ B R , 1 < R < ω , the boundary condition in (30) holds for all solutions when R → + . Lemma 3.1.
Assume that in
Ω = B we have the material parameters ( σ R , λ R ) . Moreover, suppose ω is such that (31) holds. When R > is ufficiently close to , then for any source p ∈ L (Ω) supported compactly in B and for f ∈ H / ( ∂ Ω) the Helmholtz equation (10) has a unique solution u R . Moreover, as R → + , Fourier coefficients of the solution v − R := u R | B R in the cloaked region, v − R,n ( e r ) = Z π e − inθ v − R ( e r, θ ) dθ the limits lim R → v − R,n ( e r ) exists and we have for all n ∈ Z lim R → ( κ e r∂ e r v − R,n ( e r ) + | n | v − R,n ( e r )) (cid:12)(cid:12) e r = R = 0 , (32)lim R → ( κ e r∂ e r v − R,n ( e r ) + | n | v − R,n ( e r )) (cid:12)(cid:12) e r =1 = 0 . (33) Proof.
We suppose that ω is not an eigenvalue of (30). Then v − R ( e r, θ ) = ∞ X n = −∞ ( a n J | n | ( κω e r ) + p n H (1) | n | ( κω e r )) e inθ = ∞ X n = −∞ (cid:20) A n B n J | n | ( κω e r ) + H (1) | n | ( κω e r ) (cid:21) p n e inθ where as in (23) A n ( R ) = κ R ( H (1) | n | ) ′ ( κωR ) l − ρH (1) | n | ( κωR ) l B n ( R ) = ρJ | n | ( κωR ) l − κ RJ ′| n | ( κωR ) l . In following, we sometimes denote shortly A n ( R ) = A n and B n ( R ) = B n .Denote Φ n ( e r ) := A n B n J | n | ( κω e r ) + H (1) | n | ( κω e r ) . Apparently e Φ( e r, θ ) := Φ n ( e r ) e inθ satisfies the Helmholtz equation(∆ + κ ω ) e Φ = 0 in B R . To prove the existence of the limits lim R → v − R,n ( e r ) and (32), it is sufficientto show e rκ∂ e r Φ n ( e r ) + n Φ n ( e r ) (cid:12)(cid:12) e r = R → R → + . (34)13ndeed, by ∂ e r Φ n ( e r ) = A n B n κωJ ′| n | ( κω e r ) + κω ( H (1) | n | ) ′ ( κω e r ) , and lim R → B n ( R ) = 0 (corresponding to the non-resonant case), we have e rκ∂ e r Φ n ( e r ) + n Φ n ( e r ) (cid:12)(cid:12) e r = R = κ RB n ( nl + ωρl ) (cid:16) ( H (1) | n | ) ′ ( κωR ) J | n | ( κωR ) − J ′| n | ( κωR ) H (1) | n | ( κωR ) (cid:17) (35)where l and l are given by (24).For n ≥
1, as ρ → + , nl + ωρl = ωρ (cid:16) J | n |− ( ωρ ) H (1) | n | (3 ω ) − H (1) | n |− ( ωρ ) J | n | (3 ω ) (cid:17) = O ( ρ − n +2 ) . (36)Combining (35), (36) and (26), one has e rκ∂ e r Φ n ( e r ) + | n | Φ n ( e r ) (cid:12)(cid:12) e r = R = O ( ρ ) as R → + (i.e. ρ → + ) , (37)which proves (34).For n = 0, from (25), one has A = − iκ Rπ ( H (1)0 ) ′ ( κωR ) J (3 ω ) ln (cid:16) ωρ (cid:17) + κ R ( H (1)0 ) ′ ( κωR ) H (1)0 (3 ω )+ 2 iπω H (1)0 ( κωR ) J (3 ω ) + O ( ρ ) ,B = 2 iκ Rπ J ′ ( κωR ) J (3 ω ) ln (cid:16) ωρ (cid:17) − κ RJ ′ ( κωR ) H (1)0 (3 ω )+ 2 iπω J ( κωR ) J (3 ω ) + O ( ρ ) . Therefore, ∂ e r Φ ( R ) = A B κωJ ′ ( κωR ) + κω ( H (1)0 ) ′ ( κωR ) (38)has denominator B and numerator κω [ A J ′ ( κωR ) + B ( H (1)0 ) ′ ( κωR )] = 2 κiπ W n ( κωR ) + O ( ρ ) , W n ( x ) = H (1)0 ( x ) J ′ ( x ) − ( H (1)0 ) ′ ( x ) J ( x ). This implies ∂ r Φ ( R ) ∼ W n ( κωR ) κRJ ′ ( κωR ) ln (cid:0) ωρ (cid:1) → ρ → + , i.e., the boundary condition (34) is satisfied for n = 0 and moreover, rκ∂ r Φ ( r ) (cid:12)(cid:12) r = R = O (cid:0) ωρ (cid:1) ! as R → + . (39)This proves (32). The equation (33) follows similarly by evaluating (35) and(38) at e r = 1 instead of e r = R .Now we are ready to prove our main result of the limit of the waves u R of thephysical approximate cloaking medium as R → + . We recall that Σ = ∂B . Theorem 3.2.
Let ω be such that (31) is satisfied. Assume that u R is thesolution of (10) with f = 0 and p ∈ C ∞ ( B R ) with R < . Then as R → + , u R converges uniformly in compact subsets of B \ Σ to the limit u satisfying ( ∇ + κ ω ) u = κ p in B , (40) κ∂ r u + ( − ∂ θ ) / u (cid:12)(cid:12) ∂B = 0 , (41) and u (cid:12)(cid:12) B \ B = 0 . (42)We note that solutions of (40)-(42) with f = 0 and p = 0 have been analyzedin [11]. Proof.
Let R < R < R . Recall that solution w ∈ C ∞ ( R ) of (15) is theradiating solution produced by source κ p in R and it has the expansion(16) for e r > R . Consider the Fourier coefficients w n ( e r ) = Z π e − inθ w ( e r, θ ) dθ and denote P n ( x ) := − ( ∇ + κ − n ) w n ( | x | ). As w ∈ C ∞ ( R ), we see usingintegration by parts that k P n ( | x | ) k L ( B ) ≤ C M (1 + | n | ) − M for arbitrary M > . (43)15e consider the Fourier coefficients v − R,n ( e r ) = Z π e − inθ v − R ( e r, θ ) dθ, v + R,n ( r ) = Z π e − inθ v + R ( r, θ ) dθ. They satisfy the following problem( −∇ − κ ω + n ) v − R,n ( | x | ) = P n ( | x | ) for 0 ≤ | x | ≤ R, (44)( −∇ − ω + n ) v + R,n ( | y | ) = 0 for ρ ≤ | y | ≤ , (45) v + R,n | ∂B = 0 , (46) v − R,n | ∂B − R = v + R,n | ∂B + ρ , κR ( v − R,n ) ′ ( | x | ) | ∂B − R = ρ ( v + R,n ) ′ ( | y | ) | ∂B + ρ . (47)By the transmission conditions (47), we see for V ± R,n ( x ) = v ± R,n ( | x | ) Z ∂B R ∂ e r V − R,n V − R,n dS x = Rρ Z ∂B ρ ρκR ∂ r V + R,n V + R,n dS y = Z ∂B ρ κ ∂ r V + R,n V + R,n dS y . Thus, using integration by parts, we obtain I := 1 κ Z B R P n V − R,n dy = Z B R ( −∇ − κ ω + n ) V − R,n V − R,n dx + 1 κ Z B \ B ρ ( −∇ − ω + n ) V + R,n V + R,n dy = − Z ∂B R ∂ e r V − R,n V − R,n dS x − Z ∂B − Z ∂B ρ ! κ ∂ r V + R,n V + R,n dS y + Z B R ( |∇ V − R,n | + ( − κ ω + n ) | V − R,n | ) dx + 1 κ Z B \ B ρ ( |∇ V + R,n | + ( − ω + n ) | V + R,n | ) dy, and then I = Z B R ( |∇ V − R,n | + ( − κ ω + n ) | V − R,n | ) dx + 1 κ Z B \ B ρ ( |∇ V + R,n | + ( − ω + n ) | V + R,n | ) dy ≥ Z B R ( − κ ω + n ) | V − R,n | dx + 1 κ Z B \ B ρ ( − ω + n ) | V + R,n | dy. For | n | ≥ N > max { κ ω , ω } , the above and I ≤ k V − R,n k L k P n k L implyfirst that (cid:16) k V − R,n k L ( B R ) + k V + R,n k L ( B \ B ρ ) (cid:17) ≤ C N k P n k L ( B ) (48)16nd second that (cid:16) k V − R,n k H ( B R ) + k V + R,n k H ( B \ B ρ ) (cid:17) ≤ C ′ N k P n k L ( B ) (49)where C N and C ′ N are independent of R and n .By Lemma 3.1, for each n ∈ Z and e r ∈ [0 ,
1) and r ∈ (0 ,
2) there exists limits v − n ( e r ) = lim R → + v − R,n ( e r ) , v + n ( r ) = lim R → + v + R,n ( r ) (50)and we denote V ± n ( x ) = v ± n ( | x | ). Let now 0 < r < R <
1. Then by (49)the restrictions v − R,n | ( r ,R ) , R > R are uniformly bounded in H ( r , R ). BySobolev embedding theorem, the set { v − R,n | [ r ,R ] ; R < R < } is relativelycompact in C ([ r , R ]) ⊂ H s ( r , R ), 1 / < s <
1. Thus any sequence( v − R j ,n | [ r ,R ] ) ∞ j =1 with R j → C ([ r , R ])which limit has to coincide with v − n by (50). Thus v − R,n have to converge to v − n in C ([ r , R ]) and hence V − R,n have to converge to V − n in C ( B R \ B r )as R →
1. Similarly, for all ρ > V + R,n have toconverge to V + n in C ( B \ B ρ ) as R →
1. Now, as the Sobolev norm u u k H ( B R \ B r ) is a lower semi-continuous function in L ( B R \ B r ) and (49)holds, we see that k V − n k H ( B R \ B r ) ≤ C ′ N k P n k L ( B ) for all 0 < r < R < k V − n k H ( B ) = k V − n k H ( B \ = lim r → ,R → k V − n k H ( B R \ B r ) ≤ C ′ N k P n k L ( B ) . (51)By (43) we see that u ( e r, θ ) = P n ∈ Z v − n ( e r ) e inθ is a well defined function in H ( B ) satisfying (40). By (49), k V − R,n k C ( B R \ B r ) = k v − R,n k C ([ r ,R ]) ≤ C N ,r ,R k v − R,n k H ( r ,R ) ≤ C ′ N ,r ,R k P n k L ( B ) , where C ′ N ,r ,R does not depend on R or n > N . Thus, using (43) and theconvergence of V − R,n to V − n in C ( B R \ B r ) we see that u R converge to u uniformly in compact subsets of B \ { } . Using equation (44), we see theuniform convergence in a neighborhood of zero, too. Similarly, we see theuniform convergence in compact subsets of B \ B . Moreover, by [26] and(40) the boundary values κ∂ r u | ∂B ∈ H − / ( ∂B ) and u | ∂B ∈ H / ( ∂B )are well defined. Now, by (44) and [26] we have k κ∂ r V − n k H − / ( ∂B ) ≤ C (1 + n ) k V − n k H ( B \ B R ) where C > n . Thus using (43), (33),17nd (51) we see that the boundary value κ∂ r u +( − ∂ θ ) / u , vanishes. Hence,the boundary condition (41) is satisfied.Equation (42) for u | B \ B follows using (43), (49) and the fact that by (21),(22) for any fixed n ∈ Z we have in (17) c n ( ρ ) = O ( ρ | n | ) , b n ( ρ ) = O ( ρ | n | ) as ρ → + , | n | ≥ ,c ( ρ ) = O ((log ρ ) − ) , b ( ρ ) = O ((log ρ ) − ) as ρ → + . We conclude our discussion by remarking that our analysis also explainsthe limit of approximate electromagnetic cloaks in the cylindrical case withTE/TM polarized incoming waves, as the solutions of Maxwell’s equationsin this case satisfy the Helmholtz equation.
Acknowledgements.
ML was partly supported by Finnish Centre of Excel-lence in Inverse Problems Research, Academy of Finland COE 213476 and byMathematical Sciences Research Institute (MSRI). TZ was partly supportedby MSRI.
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