Superlensing using hyperbolic metamaterials: the scalar case

Superlensing using hyperbolic metamaterials: the scalar case Tome 4 (2017), p. 973-1003.

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SUPERLENSING USING HYPERBOLIC METAMATERIALS:

THE SCALAR CASE

by Eric Bonnetier & Hoai-Minh Nguyen

Abstract. — This paper is devoted to superlensing using hyperbolic metamaterials: the pos- sibility to image an arbitrary object using hyperbolic metamaterials without imposing any conditions on the size of the object and the wave length. To this end, two types of schemes are suggested and their analysis are given. The superlensing devices proposed are independent of the object. It is worth noting that the study of hyperbolic metamaterials is challenging due to the change of type of the modeling equations, elliptic in some regions, hyperbolic in some others.

Résumé(Propriété de superlensing de dispositifs constitués de méta-matériaux hyperboliques:

le cas scalaire)

Dans cet article, on s’intéresse à la propriété de superlensing des méta-matériaux, c’est-à-dire à la possibilité d’imager un objet arbitraire, sans condition sur le rapport entre sa taille et la longueur d’onde de la lumière incidente. Nous proposons et analysons deux types de dispositifs constitués de méta-matériaux hyperboliques, qui possèdent cette propriété. L’étude de tels milieux est délicate, car les EDP qui les modélisent changent de type : elles sont elliptiques dans certaines régions de l’espace et hyperboliques dans les autres.

Contents

1. Introduction. . . 973

2. Tuned superlensing using HMMs. . . 980

3. Superlenses using HMMs via complementary property. . . 993

4. Constructing hyperbolic metamaterials. . . 996

5. Stability of HMMs. . . 999

References. . . .1001

1. Introduction

Metamaterials are smart materials engineered to have properties that have not yet been found in nature. They have recently attracted a lot of attention from the scientific community, not only because of potentially interesting applications, but also because of challenges in understanding their peculiar properties.

Mathematical subject classification (2010). — 35B30, 35B40, 35J05, 35J70, 35M10, 35L53, 78A25.

Keywords. — Negative index materials, hyperbolic meta-materials, superlensing, degenerate elliptic equations.

The support of the Labex PERSYVAL-Lab 5ANR-11- LABX-0025 is gratefully acknowledged.

Negative index materials (NIMs) is an important class of metamaterials. Their study was initiated a few decades ago in the seminal paper of Veselago [30], in which he postulated the existence of such materials. New fabrication techniques now allow the construction of NIMs at scales that are interesting for applications, and have made them a very active topic of investigation. One of the interesting properties of NIMs is superlensing, i.e., the possibility to beat the Rayleigh diffraction limit:⁽¹⁾no constraint between the size of the object and the wavelength is imposed.

Based on the theory of optical rays, Veselago discovered that a slab lens of index−1 could exhibit an unexpected superlensing property with no constraint on the size of the object to be imaged [30]. Later studies by Nicorovici, McPhedran, and Milton [24], Pendry [25, 26], Ramakrishna and Pendry in [29], for constant isotropic objects and dipole sources, showed similar properties for cylindrical lenses in the two dimensional quasistatic regime, for the Veselago slab and cylindrical lenses in the finite frequency regime, and for spherical lenses in the finite frequency regime. Superlensing of arbi- trary inhomogeneous objects using NIMs in the acoustic and electromagnetic settings was established in [15, 20] for related lens designs. Other interesting properties of NIMs include cloaking using complementary media [11, 18, 23], cloaking a source via anomalous localized resonance [1, 2, 10, 13, 16, 21, 17], and cloaking an arbitrary object via anomalous localized resonance [22].

In this paper, we are concerned with another type of metamaterials: hyperbolic metamaterials (HMMs). These materials have quite promising potential applications to subwavelength imaging and focusing; see [27] for a recent interesting survey on hyperbolic materials and their applications. We focus here on their superlensing prop- erties. The peculiar properties and the difficulties in the study of NIMs come from (can be explained by) the fact that the equations modelling their behaviors have sign changing coefficients. In contrast, the modeling of HMMs involve equations of changing type, elliptic in some regions, hyperbolic in others.

We first describe a general setting concerning HMMs and point out some of their general properties. Consider a standard medium that occupies a region Ω of R^d (d= 2,3) with standard (elliptic) material constantA, except for a subsetDin which the material is hyperbolic with material constantA^H in the quasistatic regime (the finite frequency regime is also considered in this paper and is discussed later). Thus, A^H is a symmetric hyperbolic matrix-valued function defined in D andA is a sym- metric uniformly elliptic matrix-valued function defined inΩrD. Since metamaterials usually contain damping (metallic) elements, it is also relevant to assume that the medium in D is lossy (some of its electromagnetic energy is dissipated as heat) and study the situation as the loss goes to 0. The loss can be taken into account by adding

−iδI toA^H, whereI denotes the identity matrix, wherei²=−1, and whereδ >0is a parameter meant to be small. With the loss, the medium in the whole ofΩis thus

(1)The Rayleigh diffraction limit is on the resolution of lenses made of a standard dielectric material: the size of the smallest features in the images they produce is about a half of the wavelength of the incident light.

characterized by the matrix-valued functionAδ defined by

(1.1) Aδ =

(A inΩrD, A^H−iδI inD.

For a given (source) function f ∈L²(Ω), the propagation of light/sound is modeled in the quasistatic regime by the equation

(1.2) div(Aδ∇uδ) =f in Ω,

with an appropriate boundary condition on∂Ω.

Understanding the behaviour ofu_δ asδ→0₊ is a difficult question in general due to two facts. Firstly, equation (1.2) has both elliptic (inΩrD) and hyperbolic (inD) characters. It is hence out of the range of the standard theory of elliptic and hyperbolic equations. Secondly, even if (1.2) is of hyperbolic character inD, the situation is far from standard since the problem inD is not an initial boundary problem. There are constraints on both the Dirichlet and Neumann boundary conditions (the transmission conditions). As a consequence, equation (1.2) is very unstable (see Section 5).

In this paper, we study superlensing using HMMs. The use of hyperbolic media in the construction of lenses was suggested by Jacob et al. in [8] and was experimentally verified by Liu et al. in [12]. The proposal of [8] concerns cylindrical lenses in which the hyperbolic material is given in standard polar coordinates by

(1.3) A^H =aθeθ×eθ−arer×er,

where aθ and ar are positive constants.⁽²⁾ Denoting the inner radius and the outer radius of the cylinder respectively byr1andr2, Jacob et al. argued that

(1.4) the resolution is r1

r₂λ,

where λis the wave number. They supported their prediction by numerical simula- tions.

The goal of our paper is to go beyond the resolution problem to achieve superlensing using HMMs as discussed in [15, 20] in the context of NIMs, i.e., to be able to image an object without imposing restrictions on the ratio between its size and the wavelength of the incident light. We propose two constructions for superlensing, which are based on two different mechanisms, inspired by two basic properties of the one dimensional wave equation.

The first mechanism is based on the following simple observation. Letube a smooth solution of the system

(1.5)

(∂²_ttu(t, x)−∂_xx² u(t, x) = 0 inR+×[0,2π], u(t,·)is2π-periodic.

(2)It seems to us that in their proposal these constants can be chosen quite freely.

Thenucan be written in the form u(t, x) =a0+b0t+

∞

X

n=−∞

n6=0

X

±

a_n,±e^i(±nt+nx) inR+×[0,2π],

for some constants a₀, b₀, a_n,± ∈C. For the class of Cauchy data satisfying the con- ditionR2π

0 ∂_tu(0, x)dx= 0, we have

b0= 0.

This implies

(1.6) u(t,·) =u(t+ 2π,·) and ∂tu(t,·) =∂tu(t+ 2π,·) for allt>0, and thus the values ofuand∂_tuare transported without alteration over time inter- vals of length 2π. We speak of tuned superlensing to describe devices that achieve superlensing using this property.

In particular, we propose the following two dimensional superlensing device in the annulusBr₂rBr₁:

(1.7) A^H= 1

rer×er−reθ×eθ inBr₂rBr₁, under the requirement that

(1.8) r2−r1∈2πN⁺.

Throughout the paper, Br denotes the open ball in R^d centered at the origin and of radius r. We also use the standard notations for the polar coordinates in two dimensions and the spherical coordinates in three dimensions. With the choice ofA^H in (1.7), we have

div(A^H∇u) = 1

r(∂²_rru−∂_θθ² u) inBr2rBr1.

Hence, ifuis a solution to the equationdiv(A^H∇u) = 0inBr₂rBr₁ then (1.9) ∂²_rru−∂_θθ² u= 0 in Br₂rBr₁.

For the special choice of boundary conditions considered in its statement, Theorem 1 below shows that (1.8) and (1.9) imply that

(1.10) u(r2x/|x|) =u(r1x/|x|) and ∂ru(r2x/|x|) =∂ru(r1x/|x|).

This in turn implies the magnification of the medium contained insideB_r1by a factor r2/r1(the precise meaning of this is given in the statement of Theorem 1).

Our second class of superlensing devices is inspired by another observation con- cerning the one dimensional wave equation. Given T > 0, let u be a solution with

appropriate regularity to the system

(1.11)















∂_tt²u−∂_xx² u= 0 in(−T,0)×[0,2π],

−∂_tt²u+∂_xx² u= 0 in (0, T)×[0,2π], uis2π-periodic w.r.t.x,

u(0+,·) =u(0₋,·), ∂tu(0+,·) =−∂tu(0₋,·) in[0,2π].

Then

(1.12) u(t, x) =u(−t, x) for(t, x)∈(0, T)×[0,2π].

Indeed, set

v(t, x) =u(−t, x) and w(t, x) =v(t, x)−u(t, x) for(t, x)∈(0, T)×(0,2π).

Then















∂²_ttw−∂_xx² w= 0 in (0, T)×[0,2π], w(·,0) =w(·,2π) = 0 in(0, T), wis2π-periodic w.r.t. x,

w(0₊,·) =∂_tw(0₊,·) = 0 in[0,2π].

Therefore, w = 0 in (0, T)×(0,2π) by the uniqueness of the Cauchy problem for the wave equation. This implies thatu(t, x) =u(−t, x)for(t, x)∈(0, T)×(0,2π)as claimed. In this direction, we propose the following superlensing device inBr₂rBr₁

in both two and three dimensions, withrm= (r1+r2)/2:

A^H =





 1

re_r⊗e_r−re_θ⊗e_θ inB_r2rB_rm,

−1

re_r⊗e_r+re_θ⊗e_θ inB_rmrB_r1,

ford= 2 and

A^H =





 1

r²e_r⊗e_r−(e_θ⊗e_θ+e_ϕ⊗e_ϕ) inB_r2rB_rm,

−1

r²e_r⊗e_r+ (e_θ⊗e_θ+e_ϕ⊗e_ϕ) inB_rmrB_r1,

ford= 3.

In a compact form, one has (1.13) A^H =





 1

r^d−1e_r⊗e_r−r^3−d(I−e_r⊗e_r) in B_r2rB_rm,

− 1

r^d−1er⊗er+r^3−d(I−er⊗er) in Br_mrBr₁. The choice ofA^H in (1.13) implies that

div(A^H∇u) = 1 r^d−1

∂_rr² u−∆_∂B1u

in B_r2rB_rm, and

div(A^H∇u) =− 1 r^d−1

∂_rr² u−∆_∂B1u

inB_rmrB_r1,

where ∆∂B₁ denotes the Laplace-Beltrami operator on the unit sphere of R^d. Note also that

∂_ru(r₂⁻, θ) =∂_ru(r₂⁺, θ), ∂_ru(r⁻_m, θ) =−∂_ru(r⁺_m, θ), ∂_ru(r₁⁻, θ) =−∂_ru(r⁺₁, θ).

Hence, ifuis an appropriate solution to the equationdiv(A^H∇u) = 0in Br₂rBr₁, then, by taking into account the transmission conditions on∂B_rm, one has

(1.14)











∂_rr² u−∆∂B₁u= 0 inBr₂rBr_m,

−∂_rr²u+ ∆∂B₁u= 0 in Br_mrBr₁, u

_B

r2rB_rm =u _B

rmrB_r1, ∂_ru _B

r2rB_rm =−∂_ru _B

rmrB_r1 on∂B_rm. As in (1.12), one derives that

u (s+rm)bx

=u (rm−s)xb

forbx∈∂B1, s∈(0, r2−rm), which yields

(1.15) u(r⁻₂x) =b u(r⁺₁x)b and ∂ru(r⁻₂bx) =−∂ru(r₁⁺x)b forbx∈∂B1. This in turn implies the magnification of the medium contained insideB_r1by a factor r2/r1(the precise meaning is given in Theorem 2). In contrast with the first proposal (1.7) where (1.8) is required, no condition is imposed on r1 and r2 for the second scheme (1.13). We call this method superlensing using HHMs via complementary property. The idea of using reflection takes roots in the work of the second author [14].

Similar ideas were used in the study properties of NIMs such as superlensing [15, 20], cloaking [18, 23], cloaking via anomalous localized resonance in [16, 21, 22, 17], and the stability of NIMs in [19]. Nevertheless, the superlensing properties of NIMs and HMMs are based on two different phenomena: the unique continuation principle for NIMs, and the uniqueness of the Cauchy problem for the wave equation for HMMs.

We now state two results that illustrate tuned superlensing and superlensing via complementary property. Suppose that an object to-be-magnified in B_r1 is charac- terized by a symmetric uniformly elliptic matrix-valued function a. Throughout the paper, to deal with sufficiently regular solutions of the wave equation, we assume that (1.16) ais of classC¹ in a neighborhood of∂B_r1.

Suppose that outsideBr₂ the medium is homogeneous and the lens is characterized by a matrix-valued function A^H in B_r2 rB_r1. The whole system (taking loss into account) is then given by

(1.17) Aδ =











I inΩrBR₂, A^H−iδI inBr₂rBr₁, a inB_r1. Set

(1.18) H_m¹(Ω) :=n

u∈H¹(Ω);

Z

∂Ω

u= 0o . Concerning the scheme whereA^H is defined by (1.7), we have

Theorem1. — Let d = 2, 0 < δ < 1, 0 < r1 < r2 with r2−r1 ∈ 2πN+, let Ω be a smooth bounded connected open subset of R², and let f ∈ L²(Ω) with R

Ωf = 0.

Assume that B_r2bΩand suppf ⊂ΩrB_r2. Let u_δ ∈H_m¹(Ω) be the unique solution to the system

(1.19)

(div(A_δ∇u_δ) =f inΩ,

∂_νu_δ = 0 on∂Ω, whereAδ is given by (1.17)with A^H defined by(1.7). We have

(1.20) kuδk_H1(Ω)6Ckfk_L2(Ω) and uδ−→u0 strongly inH¹(Ω) asδ−→0, where u0 ∈ H_m¹(Ω) is the unique solution to (1.19) with δ = 0 and C is a positive constant independent of f and δ. Moreover, u₀ =buin ΩrB_r2 where bu∈H_m¹(Ω) is the unique solution to the system

(1.21)

(div(A∇b u) =b f inΩ,

∂_νbu= 0 on∂Ω, where A(x) =b

(I inΩrB_r2, a r₁x/r₂

inB_r2. Concerning the scheme whereA^H is defined by (1.13), we establish

Theorem 2. — Let d = 2,3, 0 < δ < 1, 0 < r1 < r2, Ω be a smooth bounded connected open subset of R^d, and letf ∈L²(Ω) withR

Ωf = 0. Assume thatBr₂ bΩ andsuppf ⊂ΩrB_r2. Letu_δ ∈H_m¹(Ω)be the unique solution to the system

(1.22)

(div(A_δ∇u_δ) =f inΩ,

∂_νu_δ = 0 on∂Ω,

whereAδ is given by (1.17)with A^H defined by(1.13). We have

(1.23) kuδk_H1(Ω)6Ckfk_L2(Ω) and uδ −→u0 strongly inH¹(Ω),

where u0 ∈ H_m¹(Ω) is the unique solution to (1.22) with δ = 0 and C is a positive constant independent of f and δ. Moreover, u₀ =buin ΩrB_r2 where bu∈H_m¹(Ω) is the unique solution to the system

(1.24)

(div(A∇b u) =b f inΩ,

∂_νbu= 0 on∂Ω, where A(x) =b







I inΩrB_r2, r^d−2₁

r^d−2₂ ar1

r2

x

inBr₂.

Some comments on Theorems 1 and 2 are in order. The well-posedness and the stability of (1.19) and (1.22) are established in Lemma 1. The existence and the uniqueness ofu0 are part of the statements of Theorems 1 and 2. Sincef is arbitrary with support inΩrBr₂, it follows from the definition ofAbthat the object inBr₁ is magnified by a factorr2/r1. It is worth noting that the matrixacan be an arbitrary function inside B_r1, provided it is uniformly elliptic and smooth near ∂B_r1. The lensing properties of the proposed devices inBr₂rBr₁ are independent of the object.

The paper is organized as follows. Section 2 is devoted to tuned superlensing via HMMs. There, besides the proof of Theorem 1, we also discuss variants in two and

three dimensions in the finite frequency regime (Theorems 3 and 4). Section 3 concerns superlensing using HMMs via the complementary property. In this section, we prove Theorem 2 and establish its finite frequency variant (Theorem 5). Finally, in Section 4, we construct HMMs with the required properties, as limits asδ→0of effective media obtained from the homogenization of composite structures, mixtures of a dielectric and a metal. The final section concerns the stability of HMMs. We show there on a simple example, that the properties of an inclusion of hyperbolic metamaterial, embedded in a matrix of dielectric material, are strongly dependent on the geometry of the inclusion. Numerical simulations of some of the results of our paper are presented in [6]. It would be interesting to analyse the corresponding problems for the full Maxwell system and to investigate other possible applications of HMMs. We plan to address these questions in future work.

2. Tuned superlensing using HMMs

In this section, we first present two lemmas on the stability of (1.2), (1.21), and (1.24) and their variants in the finite frequency regime. In the second part, we discuss a toy model which illustrates tuned superlensing with hyperbolic media. In the third part, we give the proof of Theorem 1. In the last part, we discuss its variants in the finite frequency regime.

2.1. Two useful lemmas. — We first establish the following lemma which implies the well-posedness of (1.2). In what follows, for a subset D of R^d, 1^D denotes its characteristic function. For a function u∈ L²(Ω) and D bΩ, we set u⁺ =u_|Ω

rD, u⁻=u_|D. Whenuhas a well-defined trace on∂D, we set[u] =u⁺−u⁻ on∂D. We also use similar notations for A∇u·ν. We have

Lemma1. — Let d = 2,3, k > 0, δ₀ > 0, 0 < δ < δ₀. Let D b Ω be two smooth bounded connected open subsets of R^d. Let A be a bounded matrix-valued function defined in Ωsuch thatA is uniformly elliptic inΩrD,Ais piecewiseC¹ inΩ, and letΣbe a complex bounded function such that=(Σ)>0. Set

(2.1) A_δ(x) =A(x)−iδ1D(x)I and Σ_δ(x) = Σ(x) +iδ1D(x) inΩ.

Let g_δ ∈ [H¹(Ω)]^∗, the dual space of H¹(Ω), be such that g_δ is square integrable near ∂Ω and in the case k = 0, assume in addition that R

Ωgδ = 0. There exists a unique solutionvδ ∈H¹(Ω)ifk >0 (respectivelyvδ ∈H_m¹(Ω)ifk= 0) to the system (2.2)

(div(Aδ∇vδ) +k²Σδvδ=gδ inΩ, A∇vδ·ν−ikvδ = 0 on∂Ω.

Moreover,

(2.3) kv_δk²_H1(Ω)6 C δ Z

Ω

g_δv_δ

+kg_δk²_[H1(Ω)]^∗,

for some positive constant C depending only onΩ,D, andk. Consequently,

(2.4) kvδk_H1(Ω)6 C

δ kgδk_[H1(Ω)]^∗.

Proof. — We only prove the result fork >0. The casek= 0 follows similarly and is left to the reader. The proof is in the spirit of that of [21, Lem. 2.1]. The existence ofvδ

can be derived from the uniqueness ofvδ by using the limiting absorption principle, see, e.g., [19]. We now establish the uniqueness ofv_δ by showing thatv_δ = 0ifg_δ= 0.

Multiplying the equation of vδ byvδ (the conjugate of vδ) and integrating by parts, we obtain

− Z

Ω

hAδ∇vδ,∇vδi+k² Z

Ω

Σδ|vδ|²+ Z

∂Ω

ik|vδ|²= 0.

Considering the imaginary part and using the definition (2.1) ofAδ andΣδ, we have

(2.5) vδ = 0 in D.

This impliesv⁻_δ =Aδ∇v⁻_δ ·ν= 0on∂D; which yields, by the transmission conditions on∂D,

v_δ⁺=A∇v⁺_δ ·ν= 0 on∂D.

It follows from the unique continuation (see, e.g., [28]) thatv_δ= 0also inΩrD. The proof of uniqueness is complete.

We next establish (2.3) by contradiction. Assume that there exists(gδ)⊂[H¹(Ω)]^∗ such thatg_δ is square integrable near∂Ω,

(2.6) kvδkH¹(Ω)= 1 and 1 δ Z

Ω

g_δv_δ

+kgδk²_[H1(Ω)]^∗−→0,

as δ→bδ∈[0, δ0]. In fact, we may assume that these properties hold for a sequence (δn) → δ. However, for the simplicity of notation, we still useb δ instead of δn to denote an element of such a sequence. We only consider the caseδb= 0; the caseδ >b 0 follows similarly. Without loss of generality, one may assume that(v_δ)converges tov₀ strongly inL²(Ω) and weakly inH¹(Ω)for some v0∈H¹(Ω). Then, by (2.6), (2.7)

(div(A0∇v0) +k²Σ0v0= 0 in Ω, A∇v₀·ν−ikv₀= 0 on∂Ω.

Multiplying the equation ofv_δ byv_δ and integrating by parts, we obtain

(2.8) −

Z

Ω

hA_δ∇v_δ,∇v_δi+k² Z

Ω

Σ_δ|v_δ|²+ Z

∂Ω

ik|v_δ|²= Z

Ω

g_δv_δ. Considering the imaginary part of (2.8) and using (2.6), we have

(2.9) lim

δ→0

k∇vδkL²(D)+kvδkL²(D)+kvδkL²(∂Ω)

= 0,

from which it follows by semi-continuity of the norm thatv0= 0inD. Invoking again the unique continuation principle shows that v0 = 0 in ΩrD as well. The weak convergence ofv_δ tov₀ then implies

(2.10) lim

δ→0kv_δk_L2(Ω)= 0.

Further, the real part of (2.8) together with (2.6), (2.10), and (2.9), yields

(2.11) lim

δ→0k∇vδkL²(ΩrD)= 0.

Combining (2.10), (2.9), and (2.11) yields

δ→0limkvδk_H1(Ω)= 0 :

which contradicts (2.6). The proof is complete.

Remark1. — In the casek= 0, the result in Lemma 1 also holds for zero Dirichlet boundary condition in which g ∈H⁻¹(Ω), the dual space of H₀¹(Ω). Moreover, the constant C depends only on the ellipticity of A, and on δ0, D, and Ω. The proof follows the same lines.

The following standard result is repeatedly used in this paper:

Lemma2. — Let d = 2,3, k > 0. Let D, V,Ω be smooth bounded connected open subsets of R^d such that D b Ω, ∂D ⊂ V ⊂ Ω. Let A be a matrix-valued function andΣbe a complex function, both defined in Ω, such that

A isuniformly elliptic inΩandΣ∈L^∞(Ω) with =(Σ)>0and<(Σ)>c >0, for some constantc. Assume thatA∈C¹(ΩrD)andA∈C¹(V∩D). Letg∈L²(Ω) and in the casek= 0assume in addition thatR

Ωg= 0. There exists a unique solution v∈H¹(Ω) if k >0 (respectivelyv∈H_m¹(Ω) ifk= 0) to the system

(div(A∇v) +k²Σv=g inΩ, A∇v·ν−ikv= 0 on ∂Ω.

Moreover,

(2.12) kvkH¹(Ω)6CkgkL²(Ω) and kvkH²(VrD)6CkgkL²(Ω), for some positive constant C independent off.

Proof. — The existence, the uniqueness, and the first inequality of (2.12) follow from the Fredholm theory by the uniform ellipticity ofAinΩand the boundary condition used. The second inequality of (2.12) can be obtained by Nirenberg’s method of difference quotients (see, e.g., [4]) using the smoothness assumption of A and the

boundedness ofΣ. The details are left to the reader.

2.2. A toy problem. — In this section, we consider a toy problem for tuned superlens- ing using HMMs, in which the geometry is rectangular. Given three positive constants l,LandT, we define⁽³⁾

R= [−l, L]×[0,2π], Rl= [−l,0]×[0,2π], Rc = [0, T]×[0,2π], Rr= [T, L]×[0,2π].

Denote

Γ :=∂R, Γc,0={0} ×[0,2π], and Γc,T ={T} ×[0,2π].

(3)Lettersc,l,rstand for center, left, and right.

Letabe a uniformly elliptic matrix-valued function defined inRl∪Rr. We set aδ = 1−iδ 0

0 −1−iδ

! ,

and define

Aδ =

(a inRl∪Rr, aδ inRc,

so that the superlensing device occupies the region Rc. For f ∈ L²(R) with suppf ∩Rc=∅, letu_δ∈H₀¹(R)be the unique solution to the equation

(2.13) div(Aδ∇uδ) =f inR.

Assume thatkuδk_H1(R)is bounded asδ→0. Then, up to a subsequence,uδ converges weakly to someu0∈H₀¹(R). It is clear that u0 is a solution to

(2.14) div(A0∇u0) =f inR.

More precisely, u0 ∈ H₀¹(R) satisfies (2.14) if and only if u0 satisfies the elliptic- hyperbolic system

div(a∇u0) =f in Rl∪Rr and ∂_x²_1x1u0−∂_x²_2x2u0= 0 inRc, and the transmission conditions





 u0

_R

l=u0

_R

c

∂x₁u0

_R

l=∂x₁u0

_R

c,

onΓc,0 and





 u0

_R

r =u0

_R

c

∂x₁u0

_R

r =∂x₁u0

_R

c,

onΓc,T. This problem is ill-posed: in general, there is no solution in H₀¹(R), and so, kuδk_H1(R) → +∞, as δ → 0 (see Section 5 for an example of such a situation).

Nevertheless, for some special choices ofT, discussed below, the problem is well-posed and its solutions have peculiar properties.

To describe them, we introduce an “effective domain”RT = [−l, L−T]×[0,2π]

and

A(xb 1, x2),fb(x1, x2) =

(a(x₁, x₂), f(x₁, x₂) in Rl

a(x₁+T, x₂), f(x₁+T, x₂) in RT rRl. In what follows, we assume thatAb∈C²(RT).

Proposition1. — Let0< δ <1,f ∈L²(R), anduδ∈H₀¹(R)be the unique solution of (2.13). Assume that T ∈2πN+ andsuppf∩Rc=∅. Then

(2.15) kuδk_H1 6Ckfk_L2(R) and uδ−→u0 strongly in H¹(R),

where u0 ∈ H₀¹(R) is the unique solution of (2.13) with δ = 0 and C is a positive constant independent ofδ andf. We also have

u0(x1, x2) = (

u(xb ₁, x₂) inRl, u(xb ₁−T, x₂) inRr,

whereub∈H₀¹(RT) is the unique solution to the equation (2.16) div(Ab∇u) =b fb inRT.

Remark2. — It follows from Proposition 1 thatu0can be computed as if the structure in Rc had disappeared. This phenomenon is similar to that in the Veselago setting:

superlensing occurs.

Proof. — The proof of Proposition 1 is in the spirit of the approach used by the second author in [14] to deal with negative index materials. The key point is to construct the unique solutionu0to the limiting problem appropriately and then estimateuδ by studying the differenceu_δ−u₀.

We first construct a solutionu0∈H₀¹(R)to (2.13) withδ= 0. SinceAb∈C²(RT) and sincef ∈L²(R), the regularity theory for elliptic equations (see, e.g., [7, 3.2.1.2]) implies thatub∈H²(RT)and

(2.17) kbukH²(RT)6CkfkL²(R).

Here and in what follows in this proof, C denotes a positive constant independent off andδ. It follows thatbu(0, x₂)∈H¹(Γ_c,0)and∂₁bu(0, x₂)∈L²(Γ_c,0). Interpreting x1 and x2 as respectively time and space variables in the rectangle Rc, we seek a solutionv∈C [0, T];H₀¹(0,2π)

∩C¹([0, T];L²(0,2π))of the wave equation (2.18) ∂_x²_1x1v−∂_x²_2x2v= 0 inRc,

with zero boundary condition, i.e., v = 0 onΓ∩∂Ωc, and with the following initial conditions

v(0, x₂) =u(0, xb ₂) and ∂_x1v(0, x₂) =∂_x1ub

Rl(0, x₂).

Existence and uniqueness ofvfollow from the standard theory of the wave equation by taking into account the regularity information in (2.17). We also have, for06x16T, Z 2π

0

|∂x₁v(x1, x2)|²+|∂x₂v(x1, x2)|²dx2= Z 2π

0

|∂x₁v(0, x2)|²+|∂x₂v(0, x2)|²dx2

= Z 2π

0

|∂x1ub

Rl(0, x₂)|²+|∂x2bu(0, x₂)|²dx₂. (2.19)

Furthermore, asv vanishes onΓ∩∂Ω_c, it can be represented inRc as (2.20) v(x1, x2) =

∞

X

n=1

sin(nx2)

ancos(nx1) +bnsin(nx1) ,

where an, bn ∈ R are determined by the initial conditions satisfied by v at x1 = 0.

SinceT ∈2πN, it follows from this representation that

(2.21) v(0,·) =v(T,·) and ∂x1v(0,·) =∂x1v(T,·) in[0,2π],

for any initial conditions, and hence for anyf withsuppf∩Rc=∅. Define

(2.22) u0(x1, x2) =











u(xb 1, x2) inRl, v(x1, x2) inRc, u(xb ₁−T, x₂) inRr.

It follows from (2.16), (2.18), and (2.21) thatu0∈H₀¹(Ω)and that it is a solution to (2.13) withδ= 0. Moreover, by (2.17) and (2.19),

(2.23) ku₀k_H1(R)6Ckfk_L2(R).

We next establish the uniqueness of u0. Let w0 ∈ H₀¹(Ω) be a solution to (2.13) withδ= 0. Sincew₀can be represented as in (2.20) inRc, we obtain

w0(0,·) =w0(T,·) and ∂x₁w0(0,·) =∂x₁w0(T,·) in[0,2π].

We can thus define for(x1, x2)inRT

w(xb 1, x2) =

(w₀(x₁, x₂) in Rl, w0(x1−T, x2) otherwise,

which is a solution to (2.16). By uniqueness for this elliptic equation, it follows that wb≡ubinRT, and (2.22) shows thatw0≡u0in R.

Finally, we establish (2.15). Define

(2.24) v_δ =u_δ−u₀ in R.

We have

div(Aδ∇vδ) = div(Aδ∇uδ)−div(Aδ∇u0)

= div(A_δ∇u_δ)−div(A₀∇u₀) + div(A₀∇u₀)−div(A_δ∇u₀) in R. It follows thatv_δ ∈H₀¹(Ω) is the solution to

(2.25) div(Aδ∇vδ) = div(iδ1Rc∇u0) inR. As in (2.4) in Lemma 1, we obtain from (2.23) that

(2.26) kvδk_H1(R)6C

δ kδ∇u0k_L2(Rc)6Ckfk_L2(R);

which implies the first inequality of (2.15). It follows that a subsequence of vδ con- verges weakly to somev0, solution to (2.14) withf = 0. Uniqueness shows thatv0= 0, and that the whole sequence v_δ converges weakly to0. As in (2.3) in Lemma 1, we deduce from (2.23), (2.25), and (2.26) that

(2.27) kuδ−u0k²_H1(R)=kvδk²_H1(R)6C Z

Rc

i∇u0∇vδ

−→0,

asvδ converges weakly to0in H¹(R). The proof is complete.

2.3. Tuned superlensing using HMMs in the quasistatic regime. Proof of Theorem1 The proof is in the spirit of Proposition 1: the main idea is to construct u0 and then estimateuδ−u0. We have

(2.28) kukb _H1(Ω)6Ckfk_L2(Ω).

Using (1.16) and applying Lemma 2, we derive thatub∈H²(ΩrBr₂)and (2.29) kbuk_H2(ΩrBr2)6Ckfk_L2(Ω).

Define a functionv in Br₂rBr₁ by

(2.30) ∂_rr² v−∂_θθ² v= 0, v is periodic with respect toθ, and

(2.31) v(r2, θ) =u(rb 2, θ) and ∂rv(r2, θ) =r2∂rub⁺(r2, θ) forθ∈[0,2π].

By considering (2.30) as a Cauchy problem for the wave equation with periodic boundary conditions, in which r and θ are seen as a time and a space variable respectively, the standard theory shows that there exists a unique such v(r, θ) ∈ C [r1, r2];H_per¹ (0,2π)

∩C¹([r1, r2];L²(0,2π)). We also have, for r16r6r2, Z 2π

0

|∂rv(r, θ)|²+|∂θv(r, θ)|²dθ= Z 2π

0

|∂rv(r2, θ)|²+|∂θv(r2, θ)|²dθ

= Z 2π

0

r²₂|∂rbu⁺(r₂, θ)|²+|∂θu(rb ₂, θ)|²dθ;

(2.32)

which yields, by (2.29),

(2.33) kvk_H1(Br2rBr1)6Ckfk_L2(Ω). Moreover,v can be represented in the form

(2.34) v(r, θ) =a₀+b₀r+

∞

X

n=−∞

n6=0

X

±

a_n,±e^i(nr±nθ) in B_r2rB_r1,

wherea0, b0, an,±∈C. Sinceub⁺ is harmonic inΩrBr₂, we have b0= 1

2π Z 2π

0

∂rv(r2, θ)dθ= 1 2π

Z 2π 0

r2∂rub⁺(r2, θ)dθ

= 1 2π

Z

∂B_r2

∂rub⁺(x)dx= 1 2π

Z

∂Ω

∂νu(x)b dx= 0.

Sincer2−r1∈2πN+, it follows that

(2.35) v(r1, θ) =v(r2, θ) and ∂rv(r1, θ) =∂rv(r2, θ) forθ∈[0,2π].

Set

(2.36) u₀=











ub inΩrBr₂, v inB_r2rB_r1, u rb 2·/r1

inBr₁.

It follows from (2.28), (2.31), (2.33), and (2.35) thatu0∈H¹(Ω) and (2.37) ku₀k_H1(Ω)6Ckfk_L2(Ω).

We also have

(2.38) div(A0∇u0) =f inΩr(∂Br₁∪∂Br₂).

On the other hand, from (1.7), (2.31) and the definition ofA0, we have (2.39) [A₀∇u0·e_r] =∂_ru⁺₀ − 1

r2

∂_ru⁻₀ =∂_ru⁺₀ − 1 r2

∂_rv= 0 on∂B_r2 and from (2.35) and the definition ofA, we obtainb

[A0∇u0·er](x) = 1 r1

∂ru⁺₀(x)−A0∇u⁻₀ ·er(x)

= 1

r₁∂rv(x)−r2

r₁a(x)∇bu⁻(r2x/r1)·er

= 1 r1

∂_rv(r₂x/r₁)−r₂ r1

A(rb ₂x/r₁)∇bu⁺(r₂x/r₁)·e_r= 0 on∂B_r1. (2.40)

A combination of (2.38), (2.39), and (2.40) yields that div(A₀∇u₀) =f inΩ;

which implies thatu₀ is a solution to (1.19) withδ= 0.

We next establish the uniqueness of u₀. Let w₀ ∈ H¹(Ω) be a solution to (1.19) withδ= 0. Sincew0can be represented as in (2.34) inBr₂rBr₁, we have

(2.41) w0(r1, θ) =w0(r2, θ) and ∂rw0(r1, θ) =∂rw0(r2, θ) forθ∈[0,2π].

Define

w(x) =b

(w0(x) inΩrBr₂, w0 r1x/r2

inBr₂.

It follows from (2.41) thatwb∈H¹(Ω). One can verify that wb is a solution of (1.21).

Hencewb=bu; which yieldsw0=u0.

We next establish the inequality in (1.20). Set

(2.42) vδ =uδ−u0 inΩ.

Thenvδ ∈H¹(Ω)and satisfies

div(Aδ∇vδ) = div(iδ1^Br2rB_r1∇u0) in Ω and

∂_νv_δ = 0 on∂Ω.

Applying (2.4) of Lemma 1, we obtain from (2.37) that kvδkH¹(Ω)6Ck∇u0kL²(Ω),

which implies the inequality in (1.20). The same argument as that in Proposition 1 shows that vδ → 0 weakly in H¹(Ω). Applying (2.3) of Lemma 1, we derive from (2.37) that

kuδ−u0k²_H1(Ω)=kvδk²_H1(Ω)6C Z

Br2rBr1

i∇u0∇vδ

−→0 asδ−→0,

which completes the proof.

2.4. Tuned superlensing using HMMs in the finite frequency regime

In this section we consider variants of Theorem 2 in the finite frequency regime.

Assume that the region B_r1 to be magnified is characterized by a pair (a, σ) of a matrix-valued functionaand a complex functionσsuch thatasatisfies the standard conditions mentioned in the introduction (a is uniformly elliptic inBr₁ and (1.16) holds) andσsatisfies the following standard conditions

(2.43) σ∈L^∞(Br₁), with=(σ)>0 and<(σ)>c >0,

for some constant c. Assume that the lens without loss is characterized by a pair (A^H,Σ^H) in Br₂rBr₁. Taking loss into account, the overall medium is character- ized by

(2.44) Aδ,Σδ =











I,1 in ΩrBr₂, A^H−iδI,Σ^H+iδ in Br₂rBr₁,

a, σ in B_r1,

Given a (source) functionf ∈L²(Ω)and given a frequencyk >0, standard arguments show that there is a unique solutionu_δ ∈H¹(Ω)to the system

(2.45)

(div(Aδ∇uδ) +k²Σδuδ=f in Ω,

∂νuδ−ikuδ = 0 on∂Ω.

We first consider the three dimensional finite frequency case. The superlens in Br₂rBr₁ is defined by

(2.46) (A^H,Σ^H) = 1

r²e_r⊗e_r−(e_θ⊗e_θ+e_ϕ⊗e_ϕ), 1 4k²r²

inB_r2rB_r1. Note thatΣ^H also depends onk. We have

Theorem3. — Letd= 3,k >0,0< δ <1, and letΩbe a smooth bounded connected open subset ofR³ and let0< r1< r2 be such thatr2−r1∈4πN+ andBr₂ bΩ. Let f ∈L²(Ω)withsuppf ⊂ΩrBr2 and letuδ ∈H¹(Ω)be the unique solution of (2.45) where(A^H,Σ^H) is given by (2.46). We have

(2.47) kuδkH¹(Ω)6CkfkL²(Ω) and uδ −→u0 strongly inH¹(Ω),

where u0 ∈ H¹(Ω) is the unique solution to (2.45) with δ = 0 and C is a positive constant independent of f andδ. Moreover,u0=buinΩrBr2 wherebuis the unique

solution to the system

(2.48)

(div(A∇b bu) +k²Σbbu=f inΩ,

∂νbu−ikub= 0 on∂Ω,

whereA,b Σ =b







I,1 inΩrBr2,

r₁ r2

ar₁ r2

· ,r³₁

r³₂σr₁ r2

·

inB_r2.

From the definition of(A^H,Σ^H)in (2.46), one derives that ifuis a solution to the equationdiv(A^H∇u) +k²Σ^Hu= 0in Br2rBr1 then

∂_rr² u−∆_∂B1u+1

4u= 0 inB_r2rB_r1. This equation plays a similar role as the wave equation in (1.5).

Proof. — We have, by Lemma 2, that

kukb _H1(Ω)6Ckfk_L2(Ω) and kbuk_H2(ΩrBr2)6Ckfk_L2(Ω). Set

(2.49) u₀=











ub inΩrBr₂, v inB_r2rB_r1, u rb ₂·/r₁

inB_r1, wherev∈H¹(Br₂rBr₁)is the unique solution of

∂_rr² v−∆∂B₁v+1

4v= 0 inBr₂rBr₁, (2.50)

v=ub on∂Br₂ and ∂rv=r²₂∂rbu⁺ on∂Br₂. (2.51)

For n > 0 and −n 6 m 6n, let Y_n^m denote the spherical harmonic function of degreenand of orderm, which satisfies

∆∂B1Y_n^m+n(n+ 1)Y_n^m= 0 on∂B1. Since the family Y_n^m

is dense inL²(∂B1),v can be represented in the form (2.52) v(x) =

∞

X

n=0 n

X

m=−n

X

±

a_nm,±e^±iλnrY_mⁿ(bx), x∈B_r2rB_r1,

where λn = (n+ 1/2),r =|x| andxb=x/|x|. Note that the 0-order term in (2.50) has been chosen inB_r2rB_r1 so that the dispersion relation writes

λ²_n=n(n+ 1) + 1/4 = (n+ 1/2)²,

which implies that all the termse^±iλnrin (2.52), and thusv, are4π-periodic functions ofr. Sincer2−r1∈4πN+, it follows that

(2.53) v(r1x) =b v(r2bx) and ∂rv(r1bx) =∂rv(r2bx) forbx∈∂B1.

We have, by (2.51),

(2.54) [A₀∇u0·e_r] =∂_rbu⁺− 1

r₂²∂_rv= 0 on∂B_r2 and, by (2.51) and (2.53),

[A0∇u0·er](x) = 1

r₁²∂rv(x)−r2

r₁a(x)∇ub⁻(r2x/r1)·er

= 1

r₁²∂rv(r2x/r1)−r₂²

r₁²A(rb 2x/r1)∇bu⁻(r2x/r1)·er

= 1

r₁²∂rv(r2x/r1)−r₂²

r₁²∂rub⁺(r2x/r1) = 0 on∂Br₁. (2.55)

As in the proof of Theorem 1, one can check thatu₀ is the unique solution of (2.45) withδ= 0 where(A^H,Σ^H)is given by (2.46). Moreover,

ku_δ−u₀k_H1(Ω)6Ckfk_L2(Ω)

and

uδ −→u0 in H¹(Ω).

The proof of these facts is the same as in the proof of Theorem 1.

We next deal with a variant of Theorem 1 in the two dimensional finite frequency regime. Set

(2.56) (A^H,Σ^H) =1

rer⊗er−reθ⊗eθ,0

inBr₂rBr₁.

The following theorem describes the superlensing property of the device defined by (2.56).

Theorem4. — Letd= 2,k >0,0< δ <1, and letΩbe a smooth bounded connected open subset of R². Let 0 < r₁ < r₂ be such that r₂−r₁ ∈ 2πN+ and B_r2 bΩ. Let f ∈L²(Ω)withsuppf ⊂ΩrBr₂ and letuδ ∈H¹(Ω)be the unique solution of (2.45) where(A^H,Σ^H) is given by (2.56). We have

(2.57) kuδk_H1(Ω)6Ckfk_L2(Ω) and uδ−→u0 strongly inH¹(Ω) asδ−→0, where u0 ∈ H¹(Ω) is the unique solution to (2.45) with δ = 0 and C is a positive constant independent off andδ. Moreover,u0=ubinΩrBr₂, whereub∈H¹(ΩrBr₂) is the unique solution to the system

(2.58)



















div(A∇b u) +b k²Σbub=f inΩ,

∂νub−ikbu= 0 on∂Ω, [A∇b bu·ν] = 0 on∂B_r2, [u] =b c

Z

∂Br2

A∇b bu·ν on∂Br₂,