## Abstract

We present a simple design of electromagnetic shields for both field expelling and field confinement. Motivated by the concept of neutral inclusions in the theory of composites, we introduce two concepts of neutral shells, and use neutral shells to construct our designs of electromagnetic shields. We also discuss the relationships between electromagnetic shields and cloaking structures and argue that the designed shields are capable of ‘cloaking’ for plane waves in the long-wavelength limit.

## 1. Introduction

Passive electromagnetic shields for field expelling or confinement are necessary for the reliable operation of many electronic devices. Examples of shields for field expelling include superconducting quantum interface devices for precision measurements of magnetic fields (Polushkin *et al.* 1994). Examples of shields for field confinement include magnetic resonance imaging (MRI) machines and tokamaks in magnetic confinement fusion (Tavernier *et al.* 1979; Crozier *et al.* 1998). In addition to these examples, electromagnetic shields are commonly used in advanced nanotechnology research facilities, biomedical research laboratories, continuous beam accelerators and various facilities such as transformer vault and switchgear in electrical-power industry (Gustavson *et al.* 1997; Ohta & Matsui 2000). The use of electromagnetic shields reduces the interference between devices and devices or devices and environments.

For frequencies above 100 kHz, satisfactory shielding performance can be easily achieved by a conductive shell (Faraday shield), which uses the eddy current to absorb the electromagnetic waves (Okazaki & Ueno 1992). For a low-frequency electromagnetic wave or static electric or magnetic field, passive shields use, for example, high-permeability materials, to channel flux lines around the shielded region (Mager 1968, 1970; Baum & Bork 1991; Calvo *et al.* 2009). There is, however, a limitation in using a passive shield of a high-permeability material. Further, a careful design of the shield can greatly improve the shielding performance without using extra expensive high-permeability materials, as Mager (1970) has shown that a shield of double shells of high–low–high permeability materials performs much better than a single shell of the same material, weight and exterior boundary. Since then, various authors (Gubster *et al.* 1979; Dubbers 1986; Sumner *et al.* 1987; Burt & Ekstrom 2002; Paperno *et al.* 2004) have considered multiple cylindrical or spherical shells and achieved a good understanding of such shielding structures.

We present a simple design of electromagnetic shields for both field expelling and field confinement. This design is motivated by the concept of *neutral inclusions* in the theory of composite, based on which we introduce two concepts of *neutral shells*. A second motivation arises from recent waves of theoretical proposals and experimental efforts to realize cloaking by manipulating various materials (Leonhardt 2006; Milton *et al.* 2006; Pendry *et al.* 2006; Nicorovici *et al.* 2007; Norris 2008). Perfect cloaking, by definition, requires that the shield and the objects we desire to hide do not disturb the wave field on the exterior domain for an incident wave of any source. That is, the solution to the Maxwell equations remains exactly the same on the exterior domain as if we set the permittivity *ϵ*(**x**)=*ϵ*_{0} and permeability *μ*(**x**)=*μ*_{0} everywhere in space. This requirement is rather stringent but, nevertheless, could be satisfied by using materials including *singular* materials—materials with their physical moduli equal to negative numbers, zero or infinite, see e.g. Leonhardt (2006), Milton *et al.* (2006), Pendry *et al.* (2006), Schurig *et al.* (2006), Nicorovici *et al.* (2007) and Norris (2008). Though in theory, but not without debates (Valanju *et al.* 2002; Kildal *et al.* 2007; Paul *et al.* 2009), such singular materials may be realized by *metamaterials* (Walser *et al.* 2003). Metamaterials, exhibiting singular properties by resonance, are intrinsically lossy and strongly frequency dependent. These issues can be addressed by considering normal materials while relaxing the stringent requirements of perfect cloaking. The resulting cloaks, however, must be interpreted with caution.

We propose a relaxed concept of cloaking which requires that (i) the shield and what we desire to hide negligibly disturb the exterior wave field when a plane wave passes them and (ii) if there is a wave source inside the shield, the wave negligibly penetrates the shield into the exterior domain. In the long-wavelength limit, requirement (i) amounts to an electromagnetic shield that excludes electromagnetic fields from the interior domain and does not disturb the exterior fields; requirement (ii) amounts to an electromagnetic shield that confines electromagnetic fields inside the interior domain. We find that some of the ideas in the theory of composites and the design of electromagnetic shields are surprisingly useful for the design of cloaking structures. In particular, we demonstrate that, in the long-wavelength limit and for plane waves, the designed electromagnetic shields are capable of ‘cloaking’ in the relaxed sense discussed above.

We remark that the idea behind our designs is different from those of Qiu *et al.* (2009) and Popa & Cummer (2009), where the distribution of materials in the radian direction is optimized by numerical methods, though the final layout of materials appear similar, i.e. a sphere of multiple shells. Further, we can generalize our designs to geometries other than coated spheres. The idea is by regarding the requirements on a neutral shell as *overdetermined* conditions. The method presented in Liu *et al.* (in press) can be used to construct neutral shells of various shapes.

The paper is organized as follows. In §2, we formulate the design problems concerning electromagnetic shields for field expelling and field confinement. In §3, we introduce the concepts of neutral shells, discuss the methods of constructing neutral shells and present various examples of neutral shells. Using neutral shells as building blocks, we then proceed to the solutions of the design problems of electromagnetic shields in §4. In §5, we show that the designed electromagnetic shields indeed have the cloaking effects in the long-wavelength limit. We conclude in §6, providing an outlook. In the appendix, we derive bounds on the shielding factor of the first kind of neutral shells in terms of the threshold exponents (Milton 1986).

## 2. Formulation of the design problems

Let be the design region (*n*=2 or 3), *E* be the exterior domain, *Ω* be the region we aim to exclude or confine the fields, and *μ*(**x**) (*ϵ*(**x**)) be the permeability (permittivity) of the medium, which is equal to *μ*_{0} (*ϵ*_{0}) on and *μ*_{D}(**x**) (*ϵ*_{D}(**x**)) on the design region *D*, see figure 1*a*,*b*. We consider the Maxwell equations. At the long-wavelength limit, the magnetic and electric fields are decoupled Jackson (1999), and the magnetic and electric fields can be expressed as and , where *φ*(**x**) and *ξ*(**x**) are the static electric and magnetic potentials satisfying
2.1
and
2.2
respectively. Since equation (2.1) behaves similarly to equation (2.2), below we focus on the design of magnetic medium, i.e. *μ*_{D}(**x**) and assume *ϵ*(**x**)=*ϵ*_{0} everywhere.

We consider two types of shields as illustrated in figure 1*a*,*b*. In the first scenario, a magnetic field is applied externally and we aim to minimize the field inside the shield *D*, i.e. on *Ω*; in the second scenario, a magnetic-field source is placed inside *Ω* and we aim to minimize the field outside the shield *D*, i.e. on *E*. We shall achieve either goal or both by designing the materials profiles *μ*_{D}(**x**). Since natural materials have a finite range of permeability, we enforce the constraint
2.3
where *K*>1 is a design constraint.

Mathematically, the design problems are posed as follows. For the first scenario of field expelling, we consider the min–max problem
2.4
where the magnetostatic potential *ξ* is determined by the boundary-value problem
2.5
Here, with |**h**_{0}|=1 is interpreted as the polarization of the incident wave or simply the applied magnetic field in the static situation. For the second scenario of field confinement, we consider the min–max problem
2.6
where the magnetostatic potential *ξ* is determined by the boundary-value problem
2.7
Here, **n** is the unit normal on the interface ∂*Ω*, with |**h**_{0}|=1 is given. Note that the differences between min–max problems (2.4) and (2.6) lie on the shielded regions (*Ω* versus *E*) and the boundary-value problems that determine the field (boundary-value problems (2.5) versus (2.7)).

## 3. Neutral shells

Our solutions to the design problems (2.4) and (2.6) are motivated by the concept of neutral inclusion (Manfield 1953). In the theory of composites, an inclusion inside a homogenized medium is neutral if it does not perturb the effective property of the medium. In the context of a magnetic medium and in terms of the boundary-value problem (2.5), the inclusion sketched in figure 1*a* is a neutral inclusion if the solution to the boundary-value problem (2.5) satisfies
3.1
That is, the exterior field is undisturbed in the presence of the inhomogeneous structure . From solution (3.1), it is not hard to see that a composite medium with any number and any size of such neutral inclusions distributed in a matrix of permeability *μ*_{0} has its effective permeability equal to *μ*_{0}. The existence of neutral inclusions is well known, in particular, they include Hashin & Shtrikman’s (1962) construction of coated spheres and Milton’s (2002) construction of coated ellipsoids.

Below we present two concepts of neutral shells that require the following:

— the interior medium is the same as the exterior medium, i.e.

*μ*(**x**)=*μ*_{0}if ,— the solution to the boundary-value problem (2.5) satisfies solution (3.1) and 3.2 We call such a structure

*D*a neutral shell of the first kind, which has been introduced by Milgrom & Shtrikman (1989) for calculating the effective properties of composites, or— the solution to the boundary-value problem (2.7) satisfies 3.3 where

*r*=|**x**|,**e**_{r}=|**x**|/*r*and is a constant. We call such a structure*D*a neutral shell of the second kind. The physical meaning of equation (3.3) is that the field on ∂*Ω*coincides exactly with a point dipole**h**_{0}at the origin.

The motivation for the above definitions of neutral shells is that the solutions to the boundary-value problems (2.5) and (2.7) are exceptionally easy for neutral shells, and neutral shells have the property that a nested neutral shell remains to be a neutral shell, see discussions in §4.

### (a) Neutral shells of double layers

We now give various examples of neutral shells. In the simplest situation, we consider domains of spherical symmetry. Dividing our design region *D* into double spherical shells with radius *R*_{1}, *R*_{2} and *R*_{3}, as sketched in figure 2*a*, we denote by *χ*_{V} the characteristic function of the domain *V* , i.e. *χ*_{V} is equal to 1 on *V* and 0 otherwise. Then, the permeability on the entire space is given by
3.4
where *μ*_{3}=*μ*_{0} and
Below we show that for appropriate *μ*_{1},*μ*_{2},*R*_{1},*R*_{2} and *R*_{3}, the double shell is a neutral shell of the first or second kind.

By symmetry, we write the solutions to boundary-value problem (2.5) or (2.7) as
3.5
where *u* is given by (*r*=|**x**|)
3.6
and the constants are to be determined. We require that *u*′(*r*) be continuous for *r*>0, which implies that, for *i*=1,2,3,
3.7

By direct calculations, we find that
3.8
where *δ*(*r*) is the Dirac function, *σ*_{n}=2*π* if *n*=2 and =4*π* if *n*=3. Further, we verify that the function *v*_{h0}=−**h**_{0}⋅∇*u* satisfies
3.9
By equation (3.6), we have
which implies that, for any **x**∈*S*_{i}:={**x**:|**x**|=*R*_{i}} and *i*=1,2,3,
3.10
Here, **e**_{r}=**x**/|**x**| and **x**+ (**x**−) denotes the boundary points outside (inside) the sphere *S*_{i}. Moreover, we write equation (2.2) in a different form as
3.11
Comparing equation (3.11) with equations (3.9) and (3.10), we are motivated to require that, for any *i*=1,2,3,
3.12
which, together with equation (3.7), can be rewritten as
3.13
Note that . It will be useful to define a matrix
3.14
By equation (3.13), we immediately have
3.15

We call the matrix **T** the *transfer matrix*. The boundary-value problems (2.5) and (2.7) can be conveniently solved using this transfer matrix **T**. To see this, let us first consider the boundary-value problem (2.5), where the boundary conditions require that ∇*ξ*(*r*) is non-singular at *r*=0 and approaches −**h**_{0} as . These are satisfied by solution (3.5) if
3.16
Meanwhile, the boundary conditions in the boundary-value problem (2.7) are satisfied by equation (3.5) if
3.17
Therefore, by equations (3.15) and (3.16) or equation (3.17), we can solve for all *a*_{i},*b*_{i} for *i*=0,…,3 and obtain the solution to the boundary-value problem (2.5) or (2.7), as given by solution (3.5).

By equations (3.15) and (3.16), we see that if the matrix element *T*_{21}=0, then *b*_{3}=0, i.e. the solution to the boundary-value problem (2.5) satisfies solution (3.1). The converse is also true. Further, from equations (3.5), (3.6) and (3.16), we see that the solution to the boundary-value problem (2.5) automatically satisfies equation (3.2). Therefore, a double spherical shell is a neutral shell of the first kind if and only if the matrix element *T*_{21}=0. Moreover, by the divergence theorem and equation (3.8), we find that *T*_{21}=*b*_{3}=0 if
where |*D*_{i}| denotes the volume of the domain *D*_{i}. We define the shielding factor
3.18
which measures the effectiveness of the shield for field expelling.

Parallel to our discussions about the neutral shell of the first kind, we see that if the matrix element *T*_{12}=0, then, by equations (3.15) and (3.17), *a*_{0}=0, i.e. the solution to the boundary-value problem (2.7) satisfies equation (3.3). The converse is also true. Therefore, a double spherical shell is a neutral shell of the second kind if and only if the matrix element *T*_{12}=0. Moreover, by the divergence theorem and equation (3.8), we find that *T*_{12}=*a*_{0}=0 if
We define the shielding factor
3.19
which measures the effectiveness of the shield for field confinement.

In figure 2, we show examples of neutral shells in three dimensions (*n*=3) and their shielding effects, where we specify
3.20
and assume so that the transfer matrix **T** depends only on and *R*_{2}: . Note that the thickness of the structure *D* is only 1 per cent of the radius. For given , we solve for () such that (), and find the corresponding shielding factor (*s*^{2}_{f}=1/|*T*_{22}|). In figure 2*b*, we show the curves and : the solid line of corresponds to neutral shells of the first kind; the dashed line of corresponds to neutral shells of the second kind. In figure 2*c*, we show the curves of the shielding factors and of the neutral shells. Note that the two shielding factors *s*^{1}_{f} and *s*^{2}_{f} have no noticeable difference within the numerical resolution. Further, it is interesting to note that when *R*_{2}=1.0050 and , the structure *D* is simultaneously a neutral shell of the first kind and of the second kind. In this case, the transfer matrix defined in equation (3.15) is a diagonal matrix
3.21

### (b) Neutral shells of continuous gradings

We now generalize our constructions of neutral shells to allow continuous gradings. As in the last section, we assume *Ω*={**x**:|**x**|<*R*_{1}}, *E*={**x**:|**x**|>*R*_{3}} and *μ*=*μ*(*r*) is continuous for *r*∈(*R*_{1},*R*_{3}). Plugging equation (3.5) into equation (2.2), we find that
3.22
Note that *μ*(*r*) may be discontinuous across *r*=*R*_{1} and *r*=*R*_{3}. In this case, equation (2.2) shall be interpreted as [[*μ*(*r*)∇*ξ*]]⋅**e**_{r}=0. Therefore, across a discontinuous interface of *μ*(*r*) at *r*=*R*, we have
3.23
where *R*=*R*_{1} or *R*_{3}. Further, we require that *u*(*r*) is continuously differentiable (*C*^{1}) for *r*>0 and *u*′(*r*) satisfies
3.24

It is interesting to note that equations (3.22)–(3.24) may be solved from two directions. In one direction, we extend *u*(*r*) by interpolation such that *u*′(*r*) is at least continuous for all *r*>0. For example, let us assume
3.25
Requiring *u*′ and *u*′′ to be continuous at *r*=*R*_{1} and *R*_{3}, we obtain
3.26
Further, equation (3.23) implies
3.27
Plugging equation (3.25) into equation (3.22), we obtain a first-order ordinary differential equation (ODE) for *μ*(*r*). Specifying *μ*_{0}, *R*_{1} and *R*_{3} as in equation (3.20), we are left with nine unknowns: *a*_{0},*b*_{0},*a*_{3},*b*_{3},*α*_{0},…,*α*_{3}, and an integration constants associated with the solution of the first-order ODE for *μ* in equation (3.22). There are six equations in (3.26) and (3.27). Therefore, presumably we can specify three of the unknowns, e.g. (*a*_{0},*b*_{0}) and *α*_{0}, and solve for all others. In particular, analogous to equation (3.15), we can write the relation between (*a*_{0},*b*_{0}) and (*a*_{3},*b*_{3}) as
3.28
where the linear dependence of (*a*_{3},*b*_{3}) on (*a*_{0},*b*_{0}) follows from the ODE (3.22) is linear for *u*(*r*). By adjusting the parameter *α*_{0} such that the matrix element *T*_{21} (*T*_{12}) vanishes, we obtain a neutral shell of the first (second) kind.

In the opposite direction, we specify the functional dependence of *μ* on *r* and determine the associated parameters so that the solution to equations (3.22) and (3.23) can indeed be extended continuously differentiable to satisfy equation (3.24). In figure 2, we show such examples where *μ*(*r*) is a piecewise constant function. Below we assume *μ*(*r*) is linearly graded as
3.29
where is the gradient of *μ*(*r*) in the **e**_{r}-direction. Then, equations (3.22), (3.23) and (3.24) imply
3.30
An analytical solution to the above problem is desirable but not obvious; we turn to numerical solutions. Specifying *μ*_{0}, *R*_{1} and *R*_{3} as in equation (3.20), we are left with eight unknowns: and two integration constants associated with the solution to the second-order ODE for *u*′ in equation (3.30). Note that there are four boundary conditions in equation (3.30). Therefore, if four of the unknowns, e.g. (*a*_{0},*b*_{0}) and are specified, we can solve for all others. In particular, analogous to equation (3.15), we can write the relation between (*a*_{0},*b*_{0}) and (*a*_{3},*b*_{3}) as
3.31
where the linear dependence of (*a*_{3},*b*_{3}) on (*a*_{0},*b*_{0}) follows from the ODE (3.22) is linear for *u*(*r*). By adjusting the parameter such that the matrix element *T*_{21} (*T*_{12}) vanishes, we obtain a neutral shell of the first (second) kind.

In figure 3, we show examples of linearly graded neutral shells and their shielding factors, where *μ*_{0}, *R*_{1} and *R*_{3} are specified by equation (3.20), and so if and *R*_{2} are given, we can calculate the transfer matrix by equation (3.30). For given , we solve for () such that (), and find the corresponding shielding factor *s*^{1}_{f}=|*T*_{11}| (*s*^{2}_{f}=1/|*T*_{22}|). Figure 3*a* shows the curves and : the solid line of corresponds to neutral shells of the first kind; the dashed line of corresponds to neutral shells of the second kind. Figure 3*b* shows the curves of the shielding factors and of the neutral shells. Note that the two shielding factors *s*^{1}_{f} and *s*^{2}_{f} are slightly different.

## 4. Designs of electromagnetic shields

In this section, based on the concepts of neutral shells, we construct solutions to the design problems (2.4) and (2.6). First, we verify that the solutions to the boundary-value problems (2.5) and (2.7) have the transformation property that

Since ∇*ξ*′_{x}(**x**)=∇_{x}*ξ*(*λ***x**), we infer that a uniformly shrunk neutral shell remains to be a neutral shell and the shielding factor remains unchanged. Therefore, if we construct a shield by shrinking and nesting *N* neutral shells, as illustrated in figure 4, the transfer matrix of the overall shield, denoted by **T**_{N}, is given by
4.1
where **T** is the transfer matrix of the prototype neutral shell. Since products of lower (upper) triangular matrices remain to be lower (upper) triangular matrices, the overall shield is again a neutral shell of the first (second) kind if the prototype shell is a neutral shell of the first (second) kind. The key observation is that the shielding factor of the overall shield grows *exponentially*, whereas the growth of the thickness of the shield decreases *exponentially* as the nesting number increases. For example, let us begin with a double-layer neutral shell specified by equation (3.4). If and *R*_{2}=1.0064, the double-layer shell is a neutral shell of the first kind with shielding factor *s*^{1}_{f}=3.48, as shown in figure 2*b*,*c*. Upon shrinking this prototype neutral shell and nesting *N* such neutral shells, the permeability on the entire space is given by
4.2
Note that the thickness *H* and shielding factor of this shield is given by
4.3
Thus, if *N*=20, then *H*=0.18, the interior radius *R*_{0}=*R*_{3}−*H*=0.83 and *S*^{1}_{f}=6.79×10^{10}. For this shield, the solution to the boundary-value problem (2.5) has the property that the strength of the field inside the shield is *S*^{1}_{f} times smaller than the external field. On the other hand, if *R*_{2}=1.0036, the prototype double-layer shell is a neutral shell of the second kind with shielding factor , see figure 2*b*,*c*. Then for the shield of 20 such nested neutral shells, the solution to the boundary-value problem (2.7) has the property that the strength of the field outside the shield is times smaller than the interior field.

There are applications that require simultaneously expelling the external field and confining the interior field. In these applications, we can use neutral shells that are both the first and second kind, e.g. the double-layer shell with and *R*_{2}=1.0050, see figure 2*b*,*c* and discussions in §3 and equation (3.1). Then, the shield of 20 such nested neutral shells is a neutral shell of both the first and second kind with both shielding factors , see equation (3.21). Therefore, for this shield, the solution to the boundary-value problem (2.5) (the boundary-value problem (2.7)) has the property that the strength of the field inside (outside) the shield is 6.32×10^{5} times smaller than the external (interior) field.

## 5. Cloaking effect

In this section, we verify that the constructed electromagnetic shields have the following cloaking effects at the long-wavelength limit: (i) when a plane wave passes through the shield, the scattered wave field is negligible compared with the scattered wave field at the absence of the shield and (ii) a magnetic dipole inside the shield gives rise to a negligible radiation field compared with the radiation field at the absence of the shield.

To verify (i), we consider an incident plane wave passes through the shield, and for simplicity, assume the permittivity *ϵ*(**x**)=*ϵ*_{0} everywhere, but the permeability is given by equation (4.2) with *N*=20, and *R*_{2}=1.0050, i.e. the prototype shell is simultaneously a neutral shell of the first and second kinds with its transfer matrix given by equation (3.21).

If the wavelength of the incident wave in free space is much larger than the diameter (*R*_{3}) of the shield, then besides a harmonic factor , the scattered wave field on the exterior domain *E* is given by **h**_{sc}=−∇*ξ*−**h**_{0}, where *ξ* is the solution to the boundary-value problem (2.5). Since the shield is a neutral shell of the first kind, by solution (3.1) we see that **h**_{sc}=0 on *E*, i.e. the shield does not disturb the incident wave on the exterior domain. Moreover, we consider a spherical particle hidden inside the shield, and for simplicity, assume that the particle has permittivity *ϵ*_{0}, permeability *μ*_{*} and radius *R*_{*}<*R*_{0}, see figure 5. For the particle and the shield, the solution to the boundary-value problem (2.5) is given by equation (3.5) with
5.1
where, by discussions in §4 and equation (3.1) and equation (4.1), we have
Further, by equation (3.13), we have (*n*=3)
5.2
Note that the boundary condition in equation (2.5) and the non-singularity at *r*=0 imply *a*_{3}=1 and *b*_{*}=0. Additionally, the prototype shell is neutral with transfer matrix given by equation (3.21), by which we find
5.3
In the absence of the shield, the scattered wave field is given by
5.4
Comparing equation (5.3) with equation (5.4), we see that the scattered wave field at the presence of the shield is 1.95^{40}=3.99×10^{11} times smaller than the scattered wave field at the absence of the shield.

To verify (ii), we assume there is a magnetic dipole **m**^{i} at the origin. In the presence of the shield, the radiation field is determined by
5.5
Again, assuming the solution to the above problem is given by equations (3.5) and (5.1) we verify that
In the absence of the shield, the radiation field is directly given by −1/(4*π*){∇[**m**^{i}⋅∇(1/*r*)]}, which is 1.95^{20}=6.32×10^{5} times larger than the wave field at the presence of the shield.

From the energetic viewpoint, the shield of nested neutral shells essentially cuts off the magnetic interactions between bodies inside the shield and outside the shield. To see this, in addition to the magnetic dipole **m**^{i} inside the shield, we assume there is a second magnetic dipoles **m**^{e} at an exterior point **x**_{e}, see figure 4. In the absence of the shield, the interaction energy between the two dipoles are given by
where , see Jackson (1999). The interaction energy between the two dipoles at the presence of the shield is given by
which is 1.95^{20}=6.32×10^{5} times smaller than the interaction energy at the absence of the shield.

## 6. Summary and discussion

We consider the design of passive electromagnetic shields in the long-wavelength limit. By introducing the concepts of neutral shells, we construct our shields simply by shrinking and nesting a number of prototype neutral shells. The key observation is that the resulting shield remains as a neutral shell and the shielding factor increases exponentially as the nesting number increases. We also show that the designed shield is capable of cloaking in a relaxed sense discussed in the introduction.

The method of solving the governing boundary-value problems (2.5) and (2.7) follows from the observation that the potential of the boundary-value problems (2.5) and (2.7) is in fact given by a gradient field, see equation (3.5). This observation greatly simplifies the procedure of solving the boundary-value problems (2.5) and (2.7) and facilitates the definition of the transfer matrix, see equation (3.15). Further, we note that this is a special property of the structure, i.e. multiple spherical shells, but not restricted to structures with spherical symmetries. It can be shown that multiple ellipsoidal shells have a similar property and so neutral shells of confocal ellipsoidal surfaces can be defined as well. More generally, we can construct neutral shells of a variety of geometries by regarding the requirements on neutral shells as *overdetermined* conditions. A generalization of the work Liu *et al.* (in press) yields a method of constructing neutral shells of various shapes.

Two remarks are in order regarding the realizable range of the shielding factors and the relative permeability of materials. First, though it appears in figures 2*c* and 3*b* that the shielding factors are greater than or equal to 1, there are situations where the shielding factors are less than 1. Further, for the given design constraint (2.3), the shielding factors may be bounded from above and below in terms of the design constraint, the number *K*. These bounds are closely related to the threshold exponents defined by Milton (1986), see details in appendix A. Second, high-permeability materials, e.g. Mu-metals, are easily available with relative permeability *μ*/*μ*_{0} up to 2.0×10^{4} (Jiles 1998). Most natural diamagnetic materials such as water and bismuth have relative permeability close to 1, and the difference is of the order of 10^{−4}. Superconductors, however, ideally have relative permeability equal to 0. Therefore, materials with relative permeability between 0.005 and 200 are physically realizable, at least by composite materials with a superconductor phase.

As discussed in this paper, the design of cloaking structures is closely related to the design of electromagnetic shields in the long-wavelength limit. It will be interesting to generalize the concepts of neutral shells and the current designs of electromagnetic shields and cloaking structures to waves with finite wavelengths. The interested reader is referred to Dolin (1961) and Alù & Engheta (2005) for work in this direction.

## Acknowledgements

This work is supported by the startup funds from the University of Houston. The author thanks Professor Graeme Milton for pointing out the connection between the shielding factors and the threshold exponents, which motivates the calculations in the appendix, and the references Dolin (1961); Milgrom & Shtrikman (1989); Leonetti & Nesi (1997); Faraco (2003); Alù & Engheta (2005).

## Appendix A. Bounds on the shielding factors

Milton (1986) defined the threshold exponents to measure the degree of field concentration in a composite. Below we show that the threshold exponents imply bounds on the shielding factors of neutral shells. To see this, we recall the following definition of the threshold exponents, also see Leonetti & Nesi (1997) and Faraco (2003). Let be an open bounded domain and *ξ*∈*W*^{1,2}(*U*) be a weak solution to
where *μ*(**x**) satisfies
A1
It is known that |∇*ξ*| is in *L*^{p}(*U*) for some *p*>2 and the reciprocal 1/|∇*ξ*| is in *L*^{q}(*U*) for some *q*≥0, i.e.
A2
The supremum of such *p*(*q*) that the first (second) inequality holds for *any* *μ*(**x**) satisfying constraint (A1), denoted by *p*_{M} (*q*_{M}), is called the threshold exponent.

Let *D*={**x**:*R*_{1}<|**x**|<*R*_{3}} be a spherical neutral shell of the first kind with shielding factor *s*^{1}_{f}, *μ*(**x**) restricted on *D* satisfies constraint (A1), and *ξ*_{0} be the solution to min–max problem (2.4). Consider a shield of *N*-nested neutral shells of the scaled copies of *D*, for which the solution to min–max problem (2.4) is denoted by *ξ*_{N}. Let *V*_{k}={**x**:*R*_{1}(*R*_{1}/*R*_{3})^{k}<|**x**|<*R*_{3}} and and, without loss of generality, assume . Then, from the discussions in §4, we have that, for any *k*≥0,
Thus,
and
where and *ϱ*′(*q*):=(*R*_{1}/*R*_{3})^{n}(*s*^{1}_{f})^{q}. Sending , by the definition of *p*_{M} and *q*_{M}, we infer that *ϱ*(*p*_{M})≤1 and *ϱ*′(*q*_{M})≤1, which implies
A3
In two dimensions (*n*=2), *p*_{M}=2*K*/*K*−1 and hence the above inequality implies a lower bound on the shielding factor
On the other hand, if a neutral shell of the first kind with could be constructed, by equation (A3), we obtain the following non-trivial upper bound for the threshold exponents
A4

- Received March 24, 2010.
- Accepted May 5, 2010.

- © 2010 The Royal Society