Shielding at a distance due to anomalous resonance in superlens with eccentric core

A cylindrical plasmonic structure with a concentric core exhibits an anomalous localized resonance which results in cloaking effects. Here we show that, if the structure has an eccentric core, a new kind of shielding effect can happen. In contrast to the conventional shielding device, our proposed structure can block the effect of external electrical sources even on a region which is not enclosed by any conducting materials. In fact, the shielded region is located at a distance from the device. We analytically investigate this phenomenon by using the m\"{o}bius transformation via which an eccentric annulus is transformed into a concentric one. We also present several numerical examples.


Introduction
A cylindrical superlens with a shell having the relative permittivity ǫ δ = −1 + iδ exhibits the anomalous resonant behavior as the loss parameter δ → 0. The magnitude of the quasi-static electric field diverges throughout a region localized within a specific distance from the superlens, but it converges to a smooth field outside that region. And the boundary of the resonance region does not coincide with any discontinuity in permittivity distribution [2,4]. A superlens acts as a cloaking device since the anomalous localized resonance cancels the effect of a polarizable dipole source or certain sources that lie within a specific distance from the superlens as shown in [3,1].
In this paper, we show that a superlens act also as an electrostatic shielding device. The electrostatic shielding phenomena has been discovered by Michael Faraday in 1836. A region coated with an conducting material is not affected by an external electric field due to the polarization of electric charges, and it is the conventional method to achieve the shielding effect. What is remarkable of the proposed sheilding due to the anomalous localized resonace is that it can shield a region (without coating on it) located away from the device. Moreover, the size of the shielding region can be arbitrarily large while the size of the device is fixed. We call this phenomenon as the shielding at a distance. The aim of this paper is to investigate the condition for the shielding at a distance and the geometric features such as location and size of the shielding region.
Let us now describe a superlens with eccentric core in details. As shown in Fig. 1.1, the superlens consists of a cylindrical shell, a disk with radius r s , and a eccentric core, a disk with radius r c . We denote d the distance between the center of the shell and the core. The permittivity distribution ε δ is assumed to be ε δ = 1 in the core, ε 0 = 1 in the background and δ s = −1 + iδ in the shell. Here, δ is the loss parameter related to the dissipation of energy. We assume that the electromagnetic field is the transverse magnetic(TM) mode. We also assume extremely low frequency external field is applied so that it is valid to consider the quasistatic approximation, i.e., with the decay condition V δ (z) → 0 as |z| → ∞. The key element to study the superlens with eccentric core is the möbius transformation Φ. Since the möbius transformation is conformal mapping and (with well-chosen coefficients) maps an eccentric annulus to an concentric annulus, V • Φ −1 is a solution to (1.1) with the modified source term. Hence, the previously known knowledge on the anomalous resonance of the superlens with concentric core gives the understanding on that of the superlens with eccentric core.
Let us give a brief explanation of anomalous resonance for the concentric case. Let Ω * be a circular region of radius r 3/2 s r −1/2 c with the same center as that of concentric annulus, see Fig. 1.2. As shown in [3], the dissipation power W δ = δ Ωs\Ωc |E| 2 into heat in the shell may blow up as δ goes to zero while the electric field E outside Ω * remains bounded regardless of δ. For such case happens, the electric field should be normalized as E ′ = E/W 1/2 δ since the divergence of the power W δ is unphysical. Then the normalized electric field E ′ goes to zero outside the region Ω * . It was also shown in [3] that if a point dipole source is located in Ω * , then blow-up of the power W δ does occur. In other words, the point dipole source is almost cloaked. Now we turn to anomalous resonance for the eccentric case. Differently from the concentric case, the resonance region Ω * for the eccentric case are one of following three cases: (a) a circular region, (b) a half plane, (c) the exterior of a circular region. The type of the resonance region is determined by the radii r e , r c and the distance between the center d. The right figure in Fig. 1.1 is illustration of the case (c), where the resonance region Ω * is the outside the circular cylinder Ω 0 . Thus, the resonance region is not localized while the electric field is bounded the bounded region Ω 0 . If the dissipation power W δ blows up as δ tends to zero (for example, one has this blow-up when an uniform electric field applied), then in Ω 0 the normalized electric field E ′ = E/W 1/2 δ becomes very small. Therefore, the resonance (c) in fact means the shielding at a distance when the dissipation power blows up. Surprisingly, the shielding of the electric field takes place in the region Ω 0 even though it is not enclosed by conducting material. We will also show that the size of the shielding region can be arbitrarily large by changing the location of the core.
2 Transformation of the superlense to an annulus 2.1 Superlens with eccentric core Here and after we identify R 2 with C in the usual way, i.e, (x, y) ∈ R 2 is identified with z = x+iy. We consider the superlens consists of the circular core Ω c and the circular shell Ω s with radii r c and r s . The distance between the two centres are denoted as τ . After translating and rotating, the core and the shell can be written as We consider the following two problems for the electric potential V δ : where H is an external field given by an entire harmonic function. Here and throughout C means some constant. We assume that f is compactly supported outside Ω s and R 2 f dxdy = 0. We let F be the Newtonian potential of f , i.e., and W δ be the dissipation power into heat in the shell, that is,

The Möbius transformation
For the sake of simplification in the investigation of the properties for the superlens with eccentric core, we consider the so-called Möbius transformation Φ which maps the superlens composed of Ω c and Ω s to the concentric double ring-shaped structure, see Fig. 2.1. More precisely, we denote Φ the bilinear transformation given by with a τ given as (2.1). Then Φ is a conformal mapping from For |ζ| = ρ ∈ (0, 1) ∪ (1, ∞), let It is easy to show that Φ maps D ρ to the origin-centered disk D ρ . As shown in Fig. 2.1, the region Ω ρ for ρ < 1 is a circular cylinder containing the shell Ω s . If we let ρ → 1−, then the region Ω ρ converges to the left half plane {x < 0}. For ρ > 1, which is the most interesting case and is related to the shielding at a distance that will be explained in the next section, C \ Ω ρ is now a circular cylinder contained in the right half-plane {x > 0}. Using the definition of a τ , one can easily show Ω c = D ρc and Ω s = D ρs with with ρ c = 1 + (a τ /r c ) 2 − a τ /r c , ρ s = 1 + (a τ /r s ) 2 − a τ /r s . Therefore, as shown in Fig. 2.1, the superlense with eccentric core is transformed by the Möbius transformation to the concentric circles of center 0 in ζ-plane. Note also that ∞ in the (extended complex) z-plane is corresponding to the point 1 in ζ-plane and, conversely, ∞ in ζ-plane to a τ in z-plane.

Potential equation in ζ-plane
We now consider the problem (P1) and (P2) in ζ-plane. For Let us denote V : A straightforward application of the chain rule implies that Since Φ is a conformal mapping, it preserves harmonicity and the interface condition of the potential V . Therefore, we can consider V as the electric potential in the presence of the concentric cylindrical superlens with the source and the external field transformed via Φ.
Owing to the inversion formula of Φ, the entire electric potential in z-plane, of which the source is at ∞, is a potential generated by a point multipole source located at 1 in ζ-plane. More precisely, we have with complex constants b n 's and N ∈ N that And the transformed function V of the solution V to (P1) with H = H ∞ is the solution to If f is compactly supported away from a τ , then the solution V to (P2) is continuous at a τ and, thus, V converges at infinity. Therefore we have Assuming f being compactly supported away from a τ , the source term 4a 2 τ |ζ−1| 4f is compactly supported in C \ ({1} ∪ Ω s ).
Let us now assume f a point multipole source located at a τ , then F is a meromorphic function with poles at a τ and F is a polynomial with zero at 1. More precisely, we have for complex constants b n 's and N ∈ N that And V δ for the solution V δ to (P2) with f = f aτ satisfies ( V δ − F ∞ (ζ) → C as |ζ| → ∞ thanks to the fact that (V δ − F aτ )(z) is continuous at z = a τ and, hence, is the solution to 3 Anomalous resonance and Shielding at a distance
This value ρ * is known to be the critical radius for the cloaking due to the annulus plasmonic structure ε δ . Milton and Nicorovici [3] discovered that any finite collection of dipole sources is cloaked as the loss parameter δ goes to zero if they are located within the critical radius (but outside of Ω s ). The cloaking occurs due to the blow-up of the dissipation power, i.e., while the potential | V δ (ζ)| < C for |ζ| > a and some constant C and a independent of δ, i.e., the so called the cloaking due to the anomalous localized resonance (CALR) occurs. In fact, the electric field also satisfies the decaying property: a n ρ |n| e inθ , ρ < ρ e with the polar coordinate (ρ, θ) of ξ and the coefficients {a n } such that there exists {n k } with |n 1 | < |n 2 | < · · · satisfying lim k→∞ (ρ i ρ −1 e ) |n k+1 −|n k | |n k ||a n k | 2 ρ Applying the results in [1,3] to the superlense ε δ in ζ-plane, we have the followings with a = r 2 e r i −1 : (a ′ ) For the solution V δ to (P ′ 1), CALR takes place.
(b ′ ) For the solution V δ to (P ′ 2), CALR takes place if f is a finite collection of dipole sources located in Ω ρ * \ Ω s or if f is a source supported in Ω ρ * \ Ω s such that the the Newtonian potential of 4a 2 τ |ζ−1| 4f satisfies the GP condition (3.4).
(c ′ ) For the solution V δ to (P ′′ 2), then E δ is bounded independently of δ, i.e., CALR does not take place.
In z-plane, we are interested in the following feature: and We have the followings with a = r 2 e r i −1 owing to W δ = W δ : (a) For the solution V δ to (P1), we have (3.5) and (3.6).
(b) For the solution V δ to (P2), we have (3.5) and (3.6) if f is a finite collection of dipole sources located in Ω ρ * \ Ω s or if f is a source supported in Ω ρ * \ Ω s such that the the Newtonian potential of 4a 2 τ |ζ−1| 4f satisfies the GP condition (3.4).
(c) For the solution V δ to (P2) with f = f aτ , then E δ is bounded independently of δ.

Shielding at a distance
Now we characterize the condition for shielding at a distance. Suppose ρ * > 1. Remind that we assume an uniform external field is applied(in z-plane). Also, recall that the uniform electric field in z-plane is transformed to the point dipole source at ζ = 1 in ζ-plane. Since ρ * > 1, the resonance region {|ζ| < ρ * } in ζ-plane contains the dipole source at ζ = 1. So the anomalous resonance does happen. Therefore, the normalized electric field E ′ goes to zero in the region Ω * even though we apply an uniform electric field. Also, the region Ω * is located at a distance from the superlens, we can conclude that shielding at a distance does occur. In similar way, we can see that shielding at distance does not occur if ρ ≤ 1. So we can summarize as follows: Shielding at a distance does happen in the region Ω * if and only if ρ * > 1, where Ω * is given as (3.1).