1. Introduction

The advent of metamaterials [1, 2] over the past decade has brought about interesting electromagnetic phenomena, including the electromagnetic cloak [3]. The cloak represents a spherical shell where the permittivity and permeability tensors vary across the shell according to a precisely given formula. Although the idea has been confirmed by several recent contributions, both theoretically [4, 5, 6, 7, 8] and experimentally [9, 10], the requirements for the material comprising the cloaking shell are such that a considerable amount of skepticism is present regarding the feasibility of such a device. Not only that a precise form of spatial variation of material parameters is to be met, but the values of those parameters imply overlapping electric and magnetic resonances raising several issues such as extremely hard implementation in the visible range, strong dispersion (and hence limited bandwidth) and increasing losses. Additionally, the material parameters have singular values (zero or infinite) on the inner surface of the cloak, which we believe to be the ultimate limiting factor in the performance of a cloak. These arise from the singularity of the mapping used in the cloak design which is necessary if the concealed object is to be made invisible, i. e. mapped to a zero volume counterpart in the transformation domain.

Since the parameters of any real material can only approximate these values, we suggest an analysis of a device referred to as the imperfect cloak and investigate its performance as the limit of the ideal cloak is approached. A simple and elegant way to do this is to use the transformation media method [3, 11, 12] by considering a modified mapping without any singularities asymptotically approaching the original mapping. In this approach, the imperfect cloak does not completely hide the concealed object but merely makes it effectively smaller, which is further elaborated in the next section. We derive formulae predicting both scattering and radiation from within the concealed region for the cylindrical cloak as it is presently more interesting for practical implementation and easier to simulate numerically than the spherical cloak. Our analytical results are confirmed by finite element (FE) simulations using the COMSOL Multiphysics commercial software package. In a numerical simulation all the material parameters are essentially nonsingular so an ideal cloak cannot be simulated. The fact that our numerical results agree with analytical results, means that we have correctly described the deviations due to finite and nonzero material parameters. We point out that the cloaking performance with respect to this kind of imperfection is determined by an effective cross-section (or width in the 2D case) of the object in the transformation domain and does not depend on the particular type of mapping used for designing the cloaking shell, which leaves room for further optimization when a particular implementation is considered. An example of this type of optimization, but for nonmagnetic [9, 13] cloaks, is [14]. For previously considered piecewise linear mapping, we show that within reasonable assumptions, the cloaking shell merely reduces the radiation power and scattering width by an order of magnitude.

2. The imperfect cylindrical cloak

A cylindrical cloak, spanning the region R ₁<r<R ₂, of effectively infinite height and excited by electromagnetic waves that fall perpendicularly upon it is considered. We restrict our discussion to this two-dimensional problem noting that the three-dimensional generalization is straightforward. The present case allows for simpler formulae and is readily simulated numerically while retaining most of the interesting physical phenomena. Also, this is the only configuration considered so far in experimental realization [9, 10]. The cloaking shell is made of an inhomogeneous anisotropic material having diagonal permitivitty and permeability tensors when represented in cylindrical coordinates (r,ϕ,z). The concealed region, r<R ₁ is assumed to be homogeneous and isotropic with relative permittivity ε_c and permeability μ_c and containing current sources that are infinite along the vertical z-axis. Under these circumstances, transverse electric (TE) and transverse magnetic (TM) waves may be considered separately. TE waves have their electric field E polarized along the z-axis while for TM waves the magnetic field H is along the z-axis. All the results for one polarization are easily transformed to corresponding results for the other by making the following exchanges

 

Fig. 1. Mapping between the ‘transformation’ and ‘physical’ domain and the difference between a nonsingular (one-to-one) mapping corresponding to a imperfect cloak and the singular mapping of a ideal cloak. R ₂ and R ₁ are the outer and inner radius of the cloak, respectively, while ρ ₁ is the effective size of the concealed object in the ‘transformation’ domain.

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where j ^(e) and j ^(m) are the electric and magnetic current densities, respectively. Hence, we focus on TE waves and electric line current sources.

Throughout this paper, we use the form-invariance of the Maxwell equations under the change of coordinates which is elaborated in [3, 11, 12]. For convenience, we refer to the ‘physical’ and the ‘transformation’ domain as the r - and r′-domain, respectively. In the former we have the usual orthogonal metrics and the metamaterial cloak, while in the latter the metrics is curved so that the cloak becomes invisible. In both domains we use cylindrical coordinates and since we consider mappings changing only the radial coordinate

same symbols are used for the angle ϕ and height z. Quantities represented in the r′-domain are denoted by primed symbols while unprimed ones are used for representation in the r-domain. This is depicted in Fig. 1.

In the setting described above, the form-invariance of Maxwell equations in the two domains is attained provided that following relations are satisfied

where j_z and j′_z are the z components of the electric current density. As shown in [3, 11, 12], one might solve for electromagnetic fields in either domain if the conditions given in (3) are met. The connection between the fields, E and H, in the r-domain and the fields in the r′-domain, E′ and H′ is linear whereas it is particularly simple in the considered case

If the ideal cloak is to be transparent to any electromagnetic wave, in the r′-domain we need to have only vacuum, irrespectively on what is contained in r<R ₁. This is possible only if the region r<R ₁ has no counterpart in the r′-domain, meaning that this whole volume from the r-domain is mapped onto a single zero volume line in the r′-domain. In terms of g(r′), this means g(0)=R ₁ so that at least one of the permittivity and permeability tensor components of the cloaking shell, given in (3), will be zero or infinity, depending on the particular type of mapping used. Such values can never be fully achieved in a real device. To avoid singular parameters we assume that g(0)=0 and g(ρ ₁)=R ₁ where ρ ₁ is taken to be small (certainly smaller than R ₁) but nonzero. The cloak designed using this type of mapping g(r′) we call the imperfect cloak but it is still perfect in sense that we assume lossless media and the prescribed (3) electric (ε) and magnetic (μ) response which is still a very optimistic assumption. Obviously, in this case the region r<R ₁ is not completely decoupled from the surrounding medium (vacuum), but by decreasing ρ ₁ we may asymptotically approach the ideal case with r<R ₁ being completely invisible. Recent work [6, 7, 8] on how a cloak responds to an incoming plane wave avoids the problem of material parameter singularities on the r=R ₁ surface by a method which can be described as removing a thin layer, R ₁<r<R ₁+δ with δ>0, from the inner surface of the ideal cloak. Hence, in that case r<R ₁+δ is what is concealed and the ideal cloak is reached in the limit δ→0. This approach then allows a discussion of nonsingular cloaks without any reference to transformation media. However, we chose to analyze cloaks by taking the opposite point of view - instead of solving for the fields in the r-domain, as in [6, 7, 8], we do it in the r′-domain. Certainly, jumping from r-domain to r′-domain back and forth might be confusing. However we feel more than compensated for that because that is where the cloaking idea originally came from [3] suggesting it to be a good way to look for more insight into the physics of a cloak. Also, as shown below, we are able to solve for the fields more easily in the r′-domain because the cloaking shell is mapped to vacuum in it.

The concealed region r<R ₁ is taken to be homogeneous, so it is convenient to choose g(r′)=r′R ₁/ρ ₁ for r′<ρ ₁, though this is also arbitrary. In what follows, for r′>ρ ₁ the mapping may be arbitrary provided that some basic requirements, like being continuous, monotonously increasing and identical, g(r′)=r, for r′>R ₂, are satisfied. To each choice of g(r′) corresponds an implementation of a cloak, so we can make a cloak with a given effect, i.e. effectively shrinking from R ₁ to ρ ₁, in various ways. It means that there is a lot of room left to optimize and adapt for a particular realization. If the parameters of the cloaking shell, R ₁<r<R ₂ are given by

which is just (3) for the case when ε′=μ′=1 and expressed explicitly in terms of r, in the r′-domain there is only vacuum for r′>ρ ₁ and the cylinder, r′<ρ ₁ with anisotropic parameters and current sources given by

The TE waves, E′_z, in the r′-domain for this case are governed by the following wave equation

where k ₀=2π/λ ₀ is the free-space wavevector and ε=μ=1 for r′>ρ ₁, ε=α ² ε_c, μ=μ_c for r′<ρ ₁.

Let us consider first the sourceless case of a TE plane wave E_i,z=E ₀ exp(i(kr-ωt)), k=k ₀(x̂cosϕ_i+ŷsinϕ_i) being scattered on the imperfect cloak. In the r′-domain this is just the scattering of a TE plane wave on an infinite cylinder so the solutions for the internal, E′_int,z, and scattered, E′_scatt,z, field are obtained in the form [15]

where J_m is the mth order Bessel function and H_m is the mth order Hankel function of the first kind with the usual superscript ‘(1)’ left out for clarity. The coefficients a_m and b_m are given by

with J′_m and H′_m denoting the derivatives of the Bessel and Hankel functions with respect to their argument.

As mentioned above, the scattering from nonideal cylindrical cloaks has been already discussed in [6, 7]. We explained that the difference is that we are solving a scattering problem in the r′-domain with less unknown quantities (only a_m and b_m) and then just use (4) to obtain the fields in the r-domain. Now only the relation between paths along which the ideal cloak is asymptotically approached in [6, 7, 8] and here remains to be resolved. If we impose for our one-to-one mapping r=g(r′) the condition g(ρ ₁)=R ₁+δ instead of just g(ρ ₁)=R ₁, put ρ ₁=δ and do everything else the same as described above, we are obviously following the same asymptotic path. Therefore, our approach is very general and we are not missing anything when compared to that approach. However, δ>0 is of no use for our cause so we prefer to fix the geometry in the r-domain by putting δ=0.

When ρ ₁ becomes very small compared to the free-space wavelength λ ₀, more precisely for k ₀ ρ ₁≪1, the asymptotic forms of Bessel and Hankel functions might be used to obtain

while the expressions for the coefficients with negative m are found using a_m=a _-m and b_m=b _-m.

These results explicitly confirm that both the fields inside the concealed region r<R ₁ and the scattered field r>R ₁ drop to zero as the ideal cloak is approached. Two additional remarks are in place at this point. Firstly, as discussed in [7], the scattered field E′_scatt,z will go to zero as ρ ₁→0 everywhere except at the point r′=0 (i.e. r=R ₁) since the product of the zeroth order Hankel function, H ₀, and coefficient b ₀ stays finite in that limit (it is easy to check that it is equal to -E ₀). The other way of seeing this is just to note that since the field inside the cloak goes to zero, the scattered field has to cancel out the nonzero value of the incident field E_i,z on the surface r=R ₁ so it must be nonzero. The physical explanation of this effect given in [7] is that a surface displacement current (magnetic for the TE wave case) is excited by the scattering wave at the inner boundary of the cloak. We fully agree with this explanation and put forward that it is just another way of saying that the material parameters are singular at the cloak’s inner boundary because to say that there is a magnetic surface displacement current on a given surface means just the same as to say that the tangential component of the electric field is discontinuous on it, and in this case it is not the total electric field that has a discontinuity, but the electric field describing the response of the cloaking shell, i.e. the scattered field. Also, this type of behavior is not limited to the case of piecewise linear mapping considered in [7] where ε_z→0 and μ_ϕ→∞ for r=R ₁ in the limit ρ ₁→0, but will appear in other interesting cases including cloaks where one of the quantities ε_z and μ_ϕ stays finite and nonzero in the limit ρ ₁→0, all depending on what kind of mapping g(r′) is used.

The second remark to results given in (12)–(15), regards the case

From (14) it might appear that this means that b_m become infinite and that the scattered field diverges. However, it is not the case and that is easily confirmed by replacing (16) in (11) so the scattered field is always finite or zero. On the other hand, this does not apply to the field inside the concealed region which may be resonantly enhanced if (16) is satisfied. This is demonstrated by FE simulation results shown in Fig. 2. The occurrence of these cloak-induced resonances inside the concealed region is very interesting but has not been discussed in the literature so far. An analogous resonant field enhancement for the case of radiation is also possible. Note that the condition (16) needs to be accompanied by k ₀ ρ ₁≪1 for the resonant behavior to become pronounced, so an infinite resonant response can occur only in the limit of the ideal cloak. The choice of the order of magnitude of ρ ₁ for the simulation in Fig. 2 is explained somewhat below and is in connection of what can be reasonably expected from a realistic cloak. A small enough value is used to obtain a strong resonance. For g(r′) we used the piecewise linear mapping, as the one shown in Fig. 1. Details of cloak’s parameters are given in the caption. We emphasize that the cloak from Fig. 2 is very effective in usual, non-resonant, cases. For example, had a very good conductor been placed inside it, there would be hardly any scattering, more precisely, it would be the same as scattering from a perfectly conducting cylinder of radius ρ ₁. In this case, the scattering is still very low but there is a high field inside the cloak.

We now turn to the problem of radiation from within the concealed region r<R ₁. Assume that the source of radiation is a very thin, infinite along z, wire carrying the total current I and positioned at r=R_s<R ₁ and ϕ=ϕ_s. However small the cross section of this wire, it is reduced by the factor α ² when mapped into the r′-domain, so using (3) we obtain that the total current is preserved when switching to the r′-domain, just as it should be. Therefore, in order to account for the total field, E′_z, radiated by this source in the r′-domain, we just have to displace the source from R_s to ρ_s=R_s/α and replace j′_z in (7) with

 

Fig. 2. FE simulation results for the scattering of a TE plane wave with $\mid E_0\mid =1\frac{V}{m}$ and ϕ_i=0 on a imperfect cloak with ρ ₁=0.01m, R ₂=2R ₁=0.5m, λ ₀=1.2R ₁=0.3m, ε_c=3.5 and μ_c=1. a) shows the distribution of E_z (i.e. it’s real part). b) is the distribution of the amplitude of the electric field, |E_z|. This is a near-resonance case with particularly chosen values of ε_c and μ_c aiming to support the conclusions from the text. Outside the cloak, the field is perturbed only slightly since ρ ₁ is small compared to λ ₀. Notice how the wavefronts in a) are kept straight and parallel. However, the field in r<R ₁ is very high because of the resonance. Also, notice that this is clearly not the zeroth angular mode which usually dominates if the cloaking is effective Ref. [6].

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Following [16], the solution for E′_z reads

The coefficients A_m, B_m, C_m and D_m are given by

If k ₀ ρ ₁≪1 we may use the asymptotic forms for J_m and H_m to simplify these results to

while the formula for A ₀ is found by combining (21) with (28) and C_m is independent on ρ ₁. For the case of negative m, X_m=-X _-m, X=A,B,C,D is used.

 

Fig. 3. FE simulation results for the case R ₂=2R ₁=0.5m, λ ₀=1.2R ₁=0.3m, ε_c=μ_c=1, for the case of a very thin electric current source line located at R_s=0.1m, ϕ_s=0 and carrying a total amount of current I=1A. Electric field is given in V/m and magnetic field in A/m.

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Fig. 4. a) Radiated power, per unit length along z, for a thin electric current line located at R_s=0.1m, ϕ_s=0 carrying a total amount of current I=1A. b) Scattering width. λ ₀=0.3m, R ₁=0.25m. ρ ₁=R ₁ means that there is no cloak and ρ ₁=0 represents the case of an ideal cloak. The radiation peak around ρ ₁=0.05m for ε_c=1, μ_c=4, is due to the structure resonance mentioned in the text - here it’s for the m=4 angular mode.

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Equations (27) and (29) explicitly assert that the field outside the concealed region, r>R ₁ in the r-domain, or r>ρ ₁ in the r′-domain, tends to zero with ρ ₁→0. This is valid for every point except the very boundary r′=ρ ₁ on which the field is finite (nonzero) because Hankel functions in (20) diverge as ρ ₁→0. In other words, the requirement of continuity of E′_z and the fact that the fields inside r′<ρ ₁ remain nonzero as ρ ₁→0, implies that the field on the inner surface of the cloaking shell has a nonzero limit. This means that an ideal cloak would necessarily have a discontinuity of E_z (an infinitely steep drop from E_z(R ₁)=E_z(R ^- ₁) to E_z(R ⁺ ₁)=0) on its inner surface and, therefore, the corresponding magnetic field would be infinite, which is clearly unphysical so the picture has to break down at some point. The important difference between this and the scattering case is that now the total electric field has a discontinuity.

The ultimate aim is to design a cloak which would yield a very small ρ ₁ for the given value of R ₁. If the permittivity and permeability of the concealed region, ε_c and μ_c, were not changed in the r′-domain, the cloak would literally resize the concealed object decreasing its size from R ₁ to ρ ₁. For concealed objects with very big values of |ε_c| or |μ_c|, such as a perfect electric or magnetic conductor, the effect of the cloaking shell is very nearly the same as if the object was really shrunk. What value of ρ ₁ is feasible is still unclear since it depends both on the cloak’s thickness, R ₂-R ₁, and the particular mapping g(r′) used for designing it.

To get a meaningful estimate on a realistic cloak performance, assume R ₂=2R ₁ and consider the mapping shown in Fig. 1: r=ar′+b, a=(R ₂-R ₁)/(R ₂-ρ ₁), b=R ₂(R ₁-ρ ₁)/(R ₂-ρ ₁), which was previously discussed in [4, 5, 6, 7], though only for ρ ₁=0. In this case, to get ρ ₁=R ₁/10 the values of ε and μ tensors on the cloak’s inner surface should be ε_r=μ_r≈0.05, ε_ϕ=μ_ϕ≈20, and ε_z=μ_z≈0.2, which is not easily obtained especially considering the inevitably present dispersion with any increase in r and z or decrease of ϕ tensor components significantly hindering the cloaking effect. Namely, it can be said that in this situation a cloak yielding ρ ₁=R ₁/10 is a clearly optimistic result so we can now better appreciate the quality of the cloak used for simulation in Fig. 2. Figure 3 shows the electric and magnetic field norm, |E_z| and |H|, distribution for the case of a current source inside the cloak calculated using FE simulations. The logarithmic scale is used because the fields vary over several orders of magnitude. These simulation results fully agree with our analytical results. Note how the electric field is localized within the r<R ₁ region and how the magnetic field reaches very large values at the cloaks inner surface r=R ₁.

Using the above presented formulae, we have calculated the dependence of scattering width, $w=\frac{4}{k}{\sum }_m{\mid b_m\mid }^2$ , and radiation power per unit height, $P_l=\frac{2}{\omega {\mu }_0}{\sum }_m{\mid D_m\mid }^2$ , for the case R ₂=2R ₁ with λ ₀=1.2R ₁ and several cases for ε_c and μ_c. As the results shown in Fig. 4 indicate, the effect of an imperfect cloak is not always beneficial in terms of reducing P_l and w because the coupling between the shell and object has its resonances which may enhance radiation or scattering.

Figure 4 shows that reducing the value of ρ ₁ does not necessarily decrease the structure’s response to electromagnetic fields - it depends on what is hidden and how big ρ ₁ is. However, the fields eventually start decaying monotonously with ρ ₁ if it is small enough. We have discussed that reaching a value of ρ ₁=R ₁/10 should be hard if a piecewise linear mapping is used. Even if this could be exceeded, it is easy to check that bigger values of R ₁/ρ ₁ place increasingly stringent conditions on the parameter values at r=R ₁. This is a strong enough argument to assert that using the term ‘invisibility’ is not really appropriate for an imperfect cloak, not to mention a realistic cloak having strong dispersion and losses.

3. Conclusion

We have investigated one-to-one mappings between the ‘transformation’ and ‘physical’ domain to account for nonsingular material parameters of a physically meaningful cloak. It was shown how this approach might be used to give a consistent overview of the most important phenomena occurring in this type of ‘invisibility’, including radiation and resonant field enhancement which have not been studied before. For the piecewise linear mapping originally proposed for the cloak’s design, we have shown how it would operate within reasonable assumptions for the material parameters. In the proposed picture, the cloaking efficiency is determined by an effective length in the ‘transformation’ domain and depends on the medium that fills the concealed region. In cases where internal resonances are excited, the cloaking shell might even amplify scattering or radiation thus serving opposite of its purpose. We have also pointed out that a way to optimize the design of the cloak would be by considering different mappings and finding the most appropriate one, which depends on details how the metamaterial shell is realized. The ultimate performance of the cloak is shown to be determined by the cloaking shell material parameters on the cloaks inner surface.