Superlensing using complementary media and reflecting complementary media for electromagnetic waves

In this paper, we present the proof of superlensing an arbitrary object using complementary media and we study reflecting complementary media for electromagnetic waves. The analysis is based on the reflecting technique and new results on the compactness, existence, and stability for the Maxwell equations with low regularity data.


Introduction
Negative index materials (NIMs) were first investigated theoretically by Veselago in [36]. The existence of NIMs was confirmed experimentally by Shelby, Smith and Schultz in [35]. The study of NIMs has attracted a lot of attention in the scientific community thanks to their interesting properties and many possible applications, such as superlensing using complementary media, see [13,19,28,30,31,33,34], cloaking using complementary media, see [10,21,23,25], cloaking a source via anomalous localized resonance, see [2,4,9,12,[16][17][18]26] and references therein, and cloaking an object via anomalous localized resonance, see [20]. A survey for recent mathematics progress on these applications can be found in [24]. In this paper, we present the proof of superlensing using complementary media for electromagnetic waves.
Superlensing using complementary media was suggested by Veselago in [36] for a slab lens (a slab of index −1) using the ray theory. Later, cylindrical lenses in the two-dimensional quasistatic regime, the Veselago slab and cylindrical lenses in the finite frequency regime, and spherical lenses in the finite frequency regime were studied by Nicorovici, McPhedran and Milton in [28], Pendry in [30,31], and Pendry and Ramakrishna in [34] respectively for constant isotropic objects. Superlensing arbitrary inhomogeneous objects using complementary media in the acoustic setting was established in [19] for schemes inspired from [28,31,34] and guided by the concept of reflecting complementary media in [15]. The proof of superlensing arbitrary inhomogeneous objects using complementary media for electromagnetic waves presented in this paper therefore represents the natural completion of this line of work.
Let us describe how to magnify m times (m is a given real number greater than 1) the region B r 0 for some r 0 > 0 in which the medium is characterized by a pair of two uniformly elliptic matrix-valued functions (ε O , μ O ) using complementary media. The idea suggested by Pendry and Ramakrishna in [34] is to put a lens in B r 2 \ B r 0 whose medium is characterized by (−(r 2 2 /|x| 2 )I, −(r 2 2 /|x| 2 )I); the loss is ignored. Our lens construction is as follows. Let α, β > 1 be such that αβ − α − β = 0.
(1.1) Set r 1 = m 1− 1 α r 0 , r 2 = mr 0 , and r 3 = m 2− 1 α r 0 , (1.2) and define F : B r 2 \ {0} → ℝ 3 \B r 2 and G : Our lens contains two parts (see Figure 1). The first one of NIMs is given by (see (1.6) below for the explicit formula) and the second one is Given a diffeomorphism T from D onto D , the following standard notations are used: , (1.5) with x = T(x), for a matrix-valued function a and a vector-valued function j defined in D.
As showed later in Section 3, we have Letting α = β = 2, one rediscovers the construction suggested by Pendry and Ramakrishna in B r 2 \ B r 1 . Note that even in this case, our lens construction contains two layers and is different from theirs where one layer is used. We emphasize here that the lens-construction is independent of the object. Taking into account the loss, the medium is characterized by (ε δ , μ δ ), where Some comments on the construction are necessary. The media (ε 0 , μ 0 ) in B r 2 \ B r 1 and (I, I) in B r 3 \ B r 2 are complementary or more precisely reflecting complementary (see Section 2). For a given r 2 , we choose r 1 and r 3 such that r 3 r 1 = m and F(∂B r 1 ) = ∂B r 3 , since a superlens of m times magnification is considered. The choice of (ε δ , and  Recall that a solution (E, is said to satisfy the outgoing condition (or the Silver-Müller radiation condition) if Our result on superlensing is the following theorem.
be the unique outgoing solutions to (1.8) and (1.9), respectively. We have, for R > 0, for some positive constant C R independent of δ and j. In particular, For an observer outside B r by (1.10): one has a superlens whose magnification is m.
The proof of Theorem 1 given in Section 3 is derived from Theorem 2 in Section 2. Section 2 is devoted to the concept of reflecting complementary media (Definition 1) and their properties (Theorem 2). This concept appears naturally in the study of superlensing mentioned above and is inspired from [15]. The analysis of Theorem 2 is based on the reflecting technique which has root from [15] and a number of new results on the compactness, existence, and stability for the Maxwell equations with low regularity data.
The paper is organized as follows. In Section 2, we discuss reflecting complementary media. Proof of Theorem 1 is given in Section 3.

Reflecting complementary media
Let Ω 1 ⊂⊂ Ω 2 be smooth simply connected bounded open subsets of ℝ 3 . Let ε and μ be two real measurable matrix-valued functions defined in ℝ 3 . We assume that ε, μ are bounded in ℝ 3 and uniformly elliptic and Here and in what follows, ⟨ ⋅ , ⋅ ⟩ denotes the Euclidean scalar product. We also assume that ¹ (ε, μ) is piecewise C 1 . otherwise.
. Note that we do not impose the ellipticity of ε and μ in ℝ 3 . In fact, as seen later, in the setting of reflecting complementary media, they are negative in Ω 2 \ Ω 1 (see Remark 2). Fix k > 0. Given j ∈ L 2 c (ℝ 3 ), we are interested in the behavior of the unique outgoing solution as δ → 0 in the case (ε, μ) satisfies the reflecting complementary property, a concept introduced in Definition 1 below. For an open subset Ω of ℝ 3 , the following standard notations are used: We are ready to introduce: Definition 1 (Reflecting complementary media). Let Ω 1 ⊂⊂ Ω 2 ⊂⊂ Ω 3 be smooth simply connected bounded open subsets of ℝ 3 . The media (ε, μ) in Ω 2 \ Ω 1 and (ε, μ) in Ω 3 \ Ω 2 are reflecting complementary if there exists a diffeomorphism F : and the following two conditions hold: (1) There exists a diffeomorphism extension of F, which is still denoted by F, from Here and in what follows, when we mention a diffeomorphism F : Ω → Ω for two open smooth subsets Ω, Ω of ℝ d , we mean that F is a diffeomorphism, F ∈ C 1 (Ω), and F −1 ∈ C 1 (Ω ). The illustration of reflecting complementary media is given in Figure 2. Note that the superlensing setting in Theorem 1 has this property. Theorem 1 will be derived from Theorem 2-below, on properties of the reflecting complementary media. Remark 1. We emphasize here that in (1.5), det DT(x) is used and not |det DT(x)|, and in (2.6), one requires that (ε, μ) = (F * ε, F * μ) not (ε, μ) = (−F * ε, −F * μ). These conventions are different from the ones in the acoustic setting, see, e.g., [15], and are more convenient in the study of Maxwell equations.
We next make some comments on the definition. Condition (2.6) implies that (ε, μ) in Ω 2 \ Ω 1 and (ε, μ) in Ω 3 \ Ω 2 are complementary in the "usual" sense.² The term "reflecting" in the definition comes from (2.7) and the assumption Ω 1 ⊂ Ω 2 ⊂ Ω 3 . Conditions (2.6) and (2.7) are the main assumptions in the definition. They are motivated by the following observation. Assume that (ε, μ) in Ω 2 \ Ω 1 and (ε, μ) in Ω 3 \ Ω 2 are reflecting complementary and suppose that there exists a solution . Conditions (2.6) and (2.7) imply that, by the rule of change of variables (see, e.g., Lemma 7 in Section 2.1), is uniformly elliptic in Ω 3 \ Ω 2 by the unique continuation principle; this is the main motivation for conditions (2.6) and (2.7). Conditions (1) and (2) are mild assumptions. Introducing G in the definition makes the analysis more accessible; see Sections 2.2.
Here and in what follows, we denote The following definition is used in the statement of Theorem 2 below.
is the unique outgoing solution to the system

Remark 3.
It is important to note thatε andμ are uniformly elliptic in ℝ 3 by (2.8) since det ∇F and det ∇G are both negative. The existence and uniqueness of (Ê,Ĥ) then follow from Lemma 4 in Section 2.1. The uniqueness of (E, H) is a consequence of the unique continuation principle (see [3,27]). Remark 4. Note that (2.9) is a Cauchy problem: the uniqueness is ensured by the unique continuation principle but the existence is not; hence the resonance might appear.
Our main result on the reflecting complementary media for electromagnetic waves is: for some positive constant C R independent of j and δ.
The implication of Theorem 1 from Theorem 2 is given in Section 3. The rest of this section containing two subsections is devoted to the proof of Theorem 2. In the first one, we presents some lemmas used in the proof of Theorem 2. The proof of Theorem 2 is given in the second subsection.

Some useful lemmas
In this subsection, we present some technical lemmas which are used in the proof of Theorem 2.
The following compactness result plays an important role in our analysis. (2.11) There exists a subsequence of (u (n) ) which converges in [L 2 (D)] 3 .
be the unique solution with zero mean, i.e., ∫ B\D φ n = 0, to Set Here and in what follows, C denotes a positive constant depending only on B and D.
We derive from (2.12) that (w (n) ) is bounded in [H 1 (D)] 3 . Without loss of generality, one may assume that (2.14) Using the fact that Since D is simply connected, one has A combination of (2.11) and (2.14) yields On the other hand, 3 and converges in [L 2 (D)] 3 , it follows from (2.11) that (p (n) ) converges in H 1 2 (∂D). Combining this, (2.11), and (2.15), we derive that (p (n) ) converges in H 1 (D). Since u (n) = w (n) + ∇p (n) and The proof is complete in the case D is simply connected. The proof in the general case follows by using local charts. Remark 5. Lemma 1 is known if instead of (2.11) one assumes that (see [37]). It is clear that Lemma 1 implies the known compactness result. The case ε = I was established in [8, Lemma A5] under the additional assumption (∇ ⋅ (εu (n) )) is bounded in L 2 . The proof presented here is in the same spirit of the one given in [8], which has roots from [7]. Condition (2.11) appears naturally when one studies the existence and the stability for Maxwell equations (see Lemmas 3,4, and 5).
The second lemma is a known result on the trace of H(curl, D) (see [1,5,29]). Lemma 2. Let D be a smooth open bounded subset of ℝ 3 and set Γ = ∂D. The tangential trace operator for some positive constant C independent of ϕ. Here .
Using Lemmas 1 and 2, we can easily reach the following result, which is used in the proof of Lemma 6 to establish the stability of (2.5).
) (2.16) for some positive constant C depending on D, ε, μ, and k but independent of f , g, h 1 , and h 2 . Proof. Using Lemma 2, without loss of generality, one may assume that h 1 = h 2 = 0. We prove (2.16) by contradiction. Assume that there exist f n , g n ∈ L 2 (D) such that , H (n) )‖ H(curl,D) = 1 and lim n→+∞ ‖(f n , g n )‖ L 2 (D) = 0. (2.17) Here (E (n) , H (n) ) is the unique solution to We next deal with the existence, uniqueness, and stability of outgoing solutions defined in the whole space.
) (2.20) for some positive constant C R depending on R, R 0 , D, ε, μ, and k, but independent of f, g, h 1 , and h 2 .
The well-posedness of (2.19) is known for h 1 = h 2 = 0 and f, g ∈ H(div, ℝ 3 ) (in this case, ‖(f, g)‖ L 2 is replaced by ‖(f, g)‖ H(div) in (2.20) since the standard compactness criterion was used).⁶ To our knowledge, Lemma 4 is new and the proof requires the new compactness criterion in Lemma 1. Proof. Using Lemma 2, without loss of generality, one may assume that h 1 = h 2 = 0. The uniqueness is a consequence of Rellich's lemma (see, e.g., [6, Theoren 6.1]) and the unique continuation principle [27, Theorem 1.1]. The details are left to the reader. The existence and the stability can be derived from the uniqueness using the limiting absorption principle in the spirit of [11] and the compactness result in Lemma 1 as follows.
Multiplying the equation byĒ τ (the conjugate of E τ ), integrating on ℝ 3 , and considering the imaginary part, we have Here and in what follows in this proof, C denotes a positive constant independent of f , g, and τ. We claim that We prove this by contradiction. To this end, assume that there exist τ n → 0 + and f n , g n ∈ L 2 (ℝ 3 ) with supp f n , supp g n ⊂ B R 0 such that ‖(E (n) , H (n) )‖ H(curl,B R 0 +2 ) = 1 and lim n→+∞ ‖(f n , g n )‖ L 2 = 0.
Here (E (n) , H (n) ) ∈ [H(curl, ℝ 3 )] 2 is the unique outgoing solution to (2.21) with f = f n , g = g n , and τ = τ n . The Stratton-Chu formula (see, e.g., [6, Theorem 6.6]), gives, for |x| > R 0 + 1, and Letting n → ∞ in (2.23) and (2.24), we derive that (E, H) satisfies the Stratton-Chu formula and Applying Lemma 1, without loss of generality, one may assume that We also have (2.25) and (2.26) for (E, H). Therefore, (E, H) satisfies the outgoing condition. The estimate of (E, H) follows from the estimate of (E τ , H τ ). The proof is complete. Remark 6. The unique continuation of the Maxwell equations has a long story, see, e.g., [3,11,27,32] and the references therein. It has been known from [11] that the principle holds for ε, μ in C 2 . However, under the assumption ε, μ in C 1 , it was proved recently in [27] (see also [3] for a more general setting) using the fact the Maxwell equations can be reduced to a weakly coupled second order elliptic equations see, e.g., [11, p. 168].
Similarly, we obtain the following result on the exterior Dirichlet boundary problem. Then ) for some positive constant C R depends on R, R 0 , D, ε, μ, and k, but independent of f , g, and h. We are ready to state and prove the stability result for (2.5). Lemma 6. Let 0 < δ < 1, f, g ∈ [L 2 (ℝ 3 )] 3 , and let (ε δ , μ δ ) be defined in (2.4). Assume that ε and μ are bounded in ℝ d and satisfy (2.1), (2.2) and (2.3), and supp f, supp g,Ω 2 ⊂ B R 0 . There exists a unique outgoing solution Assume in addition that supp f ⊂D, supp g ⊂D, andD ∩ Ω 2 = 0 for some smooth open subset D of ℝ 3 . Then

28)
Here C R denotes a positive constant depending on R, R 0 , ε, μ, and D but independent of f , g, and δ. Remark 8. Lemma 6 does not require any assumptions on the reflecting complementary property. In the proof of Theorem 2, we apply Lemma 6 with D = B R \ Ω 2 for some R > 0. Proof. For δ > 0 fixed, the existence and uniqueness of (E δ , H δ ) can be obtained as in the proof of Lemma 4.
The details are omitted. We only give the proof of (2.27) and (2.28). We have, in ℝ 3 , Multiplying the equation byĒ δ , integrating in B R , and using the fact supp f ⊂ B R 0 , we have, for R > R 0 , Since μ δ = I, f = 0, and ∇ × E δ = ikH δ in ℝ 3 \ B R 0 , we derive that, for R > R 0 , Letting R → +∞, using the outgoing condition, and considering the imaginary part, we obtain This implies, by Lemma 2, ‖E δ × ν‖ 2 Applying Lemma 5, we have where sign := det ∇T(x)/|det ∇T(x)| for some x ∈ D. In particular, if D ∩ D = 0 and T(x) = x on ∂D ∩ ∂D , then Assertion (2.37) immediately follows from (2.36) by noting that sign = −1, ν = −ν, and T = I on ∂D ∩ ∂D in this case. This assertion is used several times in the proof of Theorem 2.

Remark 9.
Note that the definition of T * is different from T * for a field in ℝ 3 . It is helpful to remember that for electromagnetic fields (2.34) is used whereas for sources (1.5) is involved. Remark 10. System (2.35) is known for a smooth pair (E, H). Statement (2.36) might be known; however, we cannot find a reference for it. For the convenience of the reader, we give the details of the proof in Appendix A for the form stated here.

Proof of Theorem 2
The proof is divided into three steps.
(2.48) This is the key observation for the setting of Theorem 2. Remark 12. In this paper, we confine ourself to the case supp j ∩ Ω 3 = 0 for simple presentation. In fact, the proof of Theorem 2 can be extended to cover the case where no condition on supp j is required. The details are left to the reader (see [15] for a complete account in the acoustic setting). Remark 13. In [22, Theorems 2 and 3 and Proposition 2] we showed that in the acoustic setting the complementary property is "necessary" to the appearance of resonance in the sense that the field can blow up in L 2 -norm even in the region away from the interface of sign changing coefficients. This property would hold in the electromagnetic setting and will be considered elsewhere.

Proof of Theorem 1
Theorem 1 is a consequence of Theorem 2. In fact, we can derive from Theorem 2 the following more general result: 2 ‖j‖ L 2 for some positive constant C R independent of j and δ.
It follows from (3.1) that The conclusion now follows from Theorem 2. We are now ready to give the Proof of Theorem 1. Applying Proposition 1 with Ω j = B r j and noting that (ε ,μ ) = (ε, μ) in B r 3 \ B r 2 by (1.7), one obtains the conclusion of Theorem 1. Remark 14. It follows from (1.2) that r 3 → mr 0 as α → 1 + . Therefore, for any ε > 0, there exists a lensconstruction such that m times magnification for an object in B r 0 takes place for any f with supp f ∩ B mr 0 +ε = 0.
Remark 15. The construction of lenses is not restricted to the symmetric geometry considered here: the geometry of lenses can be quite arbitrary (see Proposition 1). This is one of the motivations of the study reflecting complementary media in a general form given in Section 2. Remark 16. In comparison with the lens construction in [19] for the acoustic setting, we assume here that mr 0 = r 2 instead of the condition mr 0 < √r 2 r 3 . A rate of the convergence which is δ 1 2 is obtained in this case and the proof does not involve the removing localized singularity technique.
We next prove (2.36). Fix φ ∈ [C 1 (D)] 3 . We have This implies, by an integration by parts, Since (A.7) also holds for φ ∈ C 1 (D), we derive that where φ 1 (x ) = φ(x). Integration by parts gives We obtain ∫ Similarly, we obtain E × ν = T * g on ∂D . The proof is complete.