## Abstract

If a body of dielectric material is coated by a plasmonic structure of negative dielectric material with non-zero loss parameter, then cloaking by anomalous localized resonance (CALR) may occur as the loss parameter tends to zero. If the coated structure is circular (two dimensions) and the dielectric constant of the shell is a negative constant (with loss parameter), then CALR occurs, and if the coated structure is spherical (three dimensions), then CALR does not occur. The aim of this paper is to show that CALR takes place if the spherical coated structure has a specially designed anisotropic dielectric tensor. The anisotropic dielectric tensor is designed by unfolding a folded geometry.

## 1. Introduction

If a body of dielectric material (core) is coated by a plasmonic structure of negative dielectric constant with non-zero loss parameter (shell), then anomalous localized resonance may occur as the loss parameter tends to zero. To be precise, let *Ω* be a bounded domain in , *d*=2,3, and *D* be a domain whose closure is contained in *Ω*. In other words, *D* is the core and is the shell. For a given loss parameter *δ*>0, the permittivity distribution in is given by
1.1Here *ϵ*_{c} is a positive constant, but *ϵ*_{s} is a negative constant representing the negative dielectric constant of the shell. For a given function *f* compactly supported in satisfying
1.2(which is required by conservation of charge), we consider the following dielectric problem:
1.3with the decay condition *V* _{δ}(*x*)→0 as . Equation (1.3) is known as the quasi-static equation and the real part of −∇*V* _{δ}(*x*) e^{−iωt}, where *ω* is the frequency and *t* is the time, represents an approximation for the physical electric field in the vicinity of *Ω*, when the wavelength of the electromagnetic radiation is large compared with *Ω*.

Let
1.4( for the imaginary part), which, within a factor proportional to the frequency, approximately represents the time-averaged electromagnetic power produced by the source dissipated into heat. (Note that, for the quasi-static approximation to be valid, it is not necessary for the frequency to be small, only that *Ω* is sufficiently small compared with the wavelength.) Also for any region *Υ*, let
1.5where, when *Υ* is outside, *Ω* approximately represents, within a proportionality constant, the time-averaged electrical energy stored in the region *Υ*. Anomalous localized resonance is the phenomenon of field blow-up in a localized region. It may (and may not) occur depending upon the structure and the location of the source. Quantitatively, it is characterized by as *δ*→0 for all regions *Υ* that overlap the region of anomalous resonance, and this defines that region. Cloaking by anomalous localized resonance (CALR) may occur when the support of the source, or part of it, lies in the anomalously resonant region. Physically the enormous fields in the anomalously resonant region interact with the source to create a sort of optical molasses, against which the source has to do a tremendous amount of work to maintain its amplitude, and this work tends to infinity as *δ*→0. Quantitatively it is characterized by as .

This phenomenon of anomalous resonance was first discovered by Nicorovici *et al.* [1] and is related to invisibility cloaking [2]: the localized resonant fields created by a source can act back on the source and mask it (assuming the source is normalized to produce fixed power). It is also related to superlenses [3,4] because, as shown by Nicorovici *et al*. [1], the anomalous resonance can create apparent point sources. For these connections and further developments tied to this form of invisibility cloaking, we refer to [5–9] and references therein. Anomalous resonance is also presumably responsible for cloaking owing to complementary media [10–12], although we do not study this here.

The problem of CALR can be formulated as the problem of identifying the sources *f* such that, first,
1.6and, second, goes to zero outside some radius *a*, as *δ*→0,
1.7Because the quantity *E*_{δ} is proportional to the electromagnetic power dissipated into heat by the time-harmonic electric field averaged over time, (1.6) implies an infinite amount of energy dissipated per unit time in the limit *δ*→0 that is unphysical. If we rescale the source *f* by a factor of , then the source will produce the same power independently of *δ* and the new associated potential will, by (1.7), approach zero outside the radius, *a*. Hence, CALR occurs. The normalized source is essentially invisible from the outside, yet the fields inside are very large. We also say that the weak CALR occurs if
1.8which is weaker than (1.6), and the limit in (1.7) is replaced by lim sup.

In recent papers [5,6], the authors developed a spectral approach to analyse the CALR phenomenon. In particular, they showed that if *D* and *Ω* are concentric discs in of radii *r*_{i} and *r*_{e}, respectively, and *ϵ*_{s}=−1, then there is a critical radius *r*_{*} such that for any source *f* supported outside *r*_{*} CALR does not occur, and for sources *f* satisfying a mild (gap) condition CALR takes place. The critical radius *r*_{*} is given by , if *ϵ*_{c}=1, and by , if *ϵ*_{c}≠1. It is also proved that if *ϵ*_{s}≠−1, then CALR does not occur: *E*_{δ} is bounded regardless of *δ* and the location of the source. It is worth mentioning that these results (when *ϵ*_{c}=−*ϵ*_{s}=1) were extended in Kohn *et al.* [13] to the case when the core *D* is not radial by a different method based on a variational approach. There the source *f* is assumed to be supported on circles.

The situation in three dimensions is completely different. If *D* and *Ω* are concentric balls in , CALR does not occur whatever *ϵ*_{s} and *ϵ*_{c} are, as long as they are constants. We emphasize that this discrepancy comes from the convergence rate of the singular values of the Neumann–Poincaré-type operator associated with the structure. In two dimensions, they converge to 0 exponentially fast, but in three dimensions they converge only at the rate of 1/*n* [6]. The absence of CALR in such coated sphere geometries is also linked with the absence of perfect plasmon waves: see the appendix in Kohn *et al.* [13]. On the other hand, in a slab geometry CALR is known to occur in three dimensions with a single dipolar source [2]. (CALR is also known to occur for the full time-harmonic Maxwell equations with a single dipolar source outside the slab superlens [2,14,15].)

The purpose of this paper is to show that we are able to make CALR occur in three dimensions by using a shell with a specially designed anisotropic dielectric constant. In fact, let *D* and *Ω* be concentric balls in of radii *r*_{i} and *r*_{e}, and choose *r*_{0} so that *r*_{0}>*r*_{e}. For a given loss parameter *δ*>0, define the dielectric constant *ϵ*_{δ} by
1.9where **I** is the 3×3 identity matrix, *ϵ*_{s} and *ϵ*_{c} are constants, , and
1.10Note that *ϵ*_{δ} is anisotropic and variable in the shell. This dielectric constant is obtained by push-forwarding (unfolding) that of a folded geometry, as in figure 1. (See the next section for details.) It is worth mentioning that this idea of a folded geometry has been used in Milton *et al.* [16] to prove CALR in the analogous two-dimensional cylinder structure for a finite set of dipolar sources. Folded geometries were first introduced in Leonhardt & Philbin [17] to explain the properties of superlenses, and their unfolding map was generalized in Milton *et al*. [16] to allow for three different fields, rather than a single one, in the overlapping regions. Folded cylinder structures were studied as superlenses in Yan *et al.* [18] and folded geometries using bipolar coordinates were introduced in Chen & Chan [19] to obtain new complementary media cloaking structures. More general folded geometries were rigorously investigated in Nguyen [12].

For a given source *f* supported outside , let *V* _{δ} be the solution to
1.11and define
1.12where is the complex conjugate of *V* _{δ}. Let *F* be the Newtonian potential of the source *f*, i.e.
1.13with *G*(**x**−**y**)=−1/4*π*|**x**−**y**|. Because *f* is supported in , *F* is harmonic in |**x**|<*R* for some *R*>*r*_{e}, and can be expressed there as
1.14where is the (real) spherical harmonic of degree *n* and order *k*. The coefficients can be calculated by
1.15for any *r*<*R*. The following is the main result of this paper.

### Theorem 1.1

*Let* **ϵ**_{δ} *be the permittivity profile in* *given by* (1.9).

(

*i*)*If*ϵ_{c}=−ϵ_{s}=1,*then weak CALR occurs and the critical radius is**i.e. if the source function f is supported inside the sphere of radius r*_{*}(*and the series in*(1.14)*does not extend harmonically to*|x|<r_{*}),*then the weak CALR occurs, i.e*. 1.16*and there exists a constant C such that*1.17*for all***x***with*.*If, in addition, the Fourier coefficients**of F satisfy the following gap condition*:[GC1]:

*There exists a sequence*{*n*_{j}}*with n*_{1}<*n*_{2}<⋯*such that**where*ρ:=*r*_{e}/*r*_{0},*then CALR occurs, i.e*. 1.18*and**goes to zero outside the radius*,*as implied by*(1.17).(

*ii*)*If*ϵ_{c}≠−ϵ_{s}=1,*then weak CALR occurs and the critical radius is r*_{**}=*r*_{0}.*If, in addition, the Fourier coefficients**of F satisfy*[GC2]:

*There exists a sequence*{*n*_{j}}*with n*_{1}<*n*_{2}<⋯*such that**then CALR occurs*.(

*iii*)*If*−ϵ_{s}≠1,*then CALR does not occur*.

We remark that, even if the source *f* is located inside in |*x*|<*r*_{*}, the corresponding series (1.14) may be harmonic in |*x*|<*r*_{*}. For example, the Newtonian potential of the form *f*=*c*_{1}*χ*_{r1<|x|<r2}−*c*_{2}*χ*_{r3<|x|<r4} with *r*_{e}<*r*_{j}<*r*_{*}, 1≤*j*≤4, is quadratic in |*x*|<*r*_{e}. We also emphasize that [GC1] and [GC2] are mild conditions on the Fourier coefficients of the Newtonian potential of the source function. For example, if the source function is a dipole in , i.e. *f*(**x**)=**a**⋅∇*δ*_{y}(**x**) for a vector **a** and , where *δ*_{y} is the Dirac delta function at **y**, [GC1] and [GC2] hold, and CALR takes place. A proof of this fact is provided in appendix A. Similarly one can show that if *f* is a quadrupole, for a 3×3 matrix *A*=(*a*_{ij}) and , then [GC1] and [GC2] hold.

## 2. Proof of theorem 1.1

Let *r*_{i}, *r*_{e} and *r*_{0} be positive constants satisfying *r*_{i}<*r*_{e}<*r*_{0}, as before. In terms of spherical coordinates (*r*,*θ*,*ϕ*), we define a mapping *Φ*={*Φ*_{c},*Φ*_{s},*Φ*_{m}}, called the unfolding map, by
2.1where *a* and *b* are constants defined in (1.10). Then, the folding map can be written (with an abuse of notation) as
2.2Let ** κ**(

**x**) be a permittivity profile (in the folded geometry) given by 2.3and let

**be the push-forward of**

*ϵ***by the unfolding map**

*κ**Φ*, namely 2.4where

**x**=

*Φ*(

**y**). By straight-forward computations one can see 2.5and

**=**

*ϵ*

*ϵ*_{δ}in (1.9) if we set 2.6

For a source *f* supported outside and the solution *V* _{δ} to (1.11), we define
2.7Then *u*_{c}, *u*_{s} and *u*_{m} satisfy
2.8We emphasize that the domains of *u*_{c}, *u*_{s} and *u*_{m} are overlapping on *r*_{e}≤|**x**|≤*r*_{0}, so that the solutions combined may be considered as the solution of the transmission problem with dielectric constants *κ*_{c}, *κ*_{s} and *κ*_{m} in the folded geometry, as shown in figure 1. We obtain *V* _{δ} by unfolding the solution (*u*_{m},*u*_{s},*u*_{c}) into one whose domain is not overlapping, following the idea in Milton *et al.* [16].

By the change of variables **x**=*Φ*_{s}(**y**) and (2.4), we have
2.9

Suppose that the source *f* is supported in |**x**|>*R* for some *R*>*r*_{e}. Then, the solution *u* to (2.8) can be expressed in |**x**|<*R* as follows:
2.10where the coefficients satisfy the following relations resulting from the interface conditions:
By solving this system of linear equations one can see that
where
2.11
2.12
2.13
and
2.14Here *ρ* is defined to be *r*_{e}/*r*_{0}.

Let *F* be the Newtonian potential of *f*, as before. Because *u*−*F* is harmonic in |**x**|>*r*_{e} and tends to 0 as , we have
2.15So *u*_{m} (the solution in the matrix) is given by
2.16Because , we have
2.17if . This proves (1.17).

The solution in the shell *u*_{s} is given by
2.18Using Green's identity and the orthogonality of , we obtain that
The estimate (2.9) yields
2.19Here and afterwards, *a*≈*b* means that there exist constants *C*_{1} and *C*_{2} independent of *n* and *δ* such that
(i) Suppose that *ϵ*_{c}=−*ϵ*_{s}=1. With the notation in (2.6), we have
and, hence,
2.20It then follows from (2.19) that
2.21

Let
2.22If *n*≤*N*_{δ}, then we know that *δ*≤*ρ*^{n} and . Hence,
for any *m*≤*N*_{δ}. By taking *δ* to be *ρ*^{n}, *n*=1,2,…, we see that if the following holds
2.23then there is a sequence {*n*_{k}} such that
2.24i.e. weak CALR occurs.

Suppose that the source function *f* is supported inside the critical radius (and outside *r*_{e}) and its Newtonian potential *F* cannot be extended harmonically in |*x*|<*r*_{*}. Then we have
2.25because, otherwise, *F* given by (1.14) converges in |*x*|<*r*_{*} because . Consequently, (2.23) holds. This proves that if the source function *f* is supported inside the sphere of critical radius *r*_{*}, then weak CALR occurs.

If the source function *f* is supported outside the sphere of critical radius , then its Newtonian potential *F* can be extended harmonically in |*x*|<*r*_{*}+2*η* for *η*>0 and
2.26So *E*_{δ} is bounded regardless of *δ* and CALR does not occur.

Suppose that *f* is supported inside *r*_{*} and [GC1] holds. Let {*n*_{j}} be the sequence satisfying
If *δ*=*ρ*^{α} for some *α*, let *j*(*α*) be the number in the sequence such that
Then, we have
as . So CALR takes place.

To prove (ii) assume that *ϵ*_{c}≠−*ϵ*_{s}=1. In this case, we have
and
The rest of proof of (ii) is the same as that for (i).

Suppose now that −*ϵ*_{s}≠1. If *ϵ*_{c}=1, then we have
and
Because the source function *f* is supported outside the radius *r*_{e}, we have
and *E*_{δ} is bounded independently of *δ*. The case when *ϵ*_{c}≠1 can be treated similarly.

## Acknowledgements

The authors thank the referees for their valuable comments on this paper. This work was supported by the ERC advanced grant project MULTIMOD–267184, by the Korean Ministry of Education, Sciences and Technology through NRF grant nos 2010-0004091 and 2010-0017532, and by the NSF through grant nos DMS-0707978 and DMS-1211359.

## Appendix A. Gap property of dipoles

In this appendix, we show that the Newtonian potentials of dipole source functions satisfy the gap conditions [GC1] and [GC2]. We only prove [GC1], because the other can be proved similarly.

Let *f* be a dipole in , i.e. *f*(**x**)=**a**⋅∇*δ*_{y}(**x**) for a vector **a** and . Then its Newtonian potential is given by *F*(**x**)=−**a**⋅∇_{y}*G*(**x**−**y**). It is well known (see [20]) that the fundamental solution *G*(**x**−**y**) admits the following expansion if |**y**|>|**x**|:
So we have
and, hence,
A1

We show that A2and hence [GC1] holds. The following lemma is needed.

### Lemma A.1

*For any* *a**and* *on S*^{2} *and for any positive integer n there is a homogeneous harmonic polynomial h of degree n such that*
A3 *and*
A4

### Proof.

After rotation, if necessary, we may assume that . We introduce three homogeneous harmonic polynomials of degree *n*,
Then one can easily see that
So if we define
then (A3) holds.

Since we obtain (A4) using the Cauchy–Schwartz inequality. This completes the proof. □

Let **a** and be two unit vectors, and let *h* be a homogeneous harmonic polynomial of degree *n* satisfying (A3) and (A4). Then *h* can be expressed as
where
A5Because of (A3), we have
So there is *k*, say *k*_{n}, between −*n* and *n* such that
A6On the other hand, from (A4) and (A5), it follows by using Jensen's inequality that
Thus, we have
A7for some *C* independent of *n*.

Now one can see from (A1) that
A8for some *C* independent of *n*. Because |**y**|<*r*_{*}, we obtain that
as desired. It is worth mentioning that the constants *C* in the above may be different at each occurrence, but are independent of *n*.

- Received January 24, 2013.
- Accepted March 12, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.