Electromagnetic Wave Scattering on Imperfect Cloaking Devices

Cloaking devices based on the coordinate transform approach enable, in principle, a perfect concealment of a region in space provided that the material composing the cloaking shell meets certain criteria. To achieve ideal cloaking it is necessary that the shell material parameters have singular values on the surface bounding the cloaked region which is unphysical. In this paper we assume finite values of cloak parameters and apply the scattering theory formalism to give an estimate of the overall performance of an 'imperfect' cloak. We perform full-wave numerical calculations and use our theoretical results to discuss them.


Introduction
The traditional macroscopic Maxwell equations including the dielectric permittivity, ε, and magnetic permeability, μ, describing any inhomogenities (i.e. the response of the medium to fields) in the space are valid provided that the characteristic length (the wavelength) of the electromagnetic fields is much larger than the size of the structural unit cell of the material representing the inhomogenity [1].Therefore, if means to control the material structure on the subwavelength level are provided, it is possible to design the medium parameters, ε and μ, in detail and, hence, to control in the beahaviour of the fields in a given region of space.However, the question remains what values of ε and μ are physically meaningful.
In a, at that time, fantastic paper, Veselago [2] discussed the "upside-down" behavior of the fields in cases when both ε and μ are negative.It took several decades before such a material was created [3], with the associated phenomena bringing about lively discussions [4,5,6].These "new materials", called metamaterials, soon focused the attention on unusual effects (e. g. subwavelength focusing) showing that the limits of operation of optical devices may be pushed well beyond the extent previously deemed possible.
The electromagnetic cloak is, by definition, a invisible device having a hole inside in which we may put arbitrary objects so that the whole structure remains invisible.Alternately, it could be called the 'invisibility box', meaning that it does not perturb the exterior fields.In this paper, we discuss the cloak proposed by Pendry [7] which, in theory, provides invisibility for the full set of Maxwell equations (i.e. without any approximations, such as ray optics).The cloak is conceived using the clever idea of coordinate transform media, proposed in the same paper, but already discussed previously [8].Very soon after the seminal paper [7], it has been shown by Leonhardt [9] that the invisibility may be explained in terms of Maxwell equations written in the generally covariant form.Namely, in that case it becomes obvious that the material parameters and the metrics of the space are "clumped" together so instead of solving for the fields in a rectangular space with inhomogenities, we may solve for the fields in homogeneous media but with curved metrics.Then the cloak effectively creates a hole in space and this is where we hide the object.
In order to create such a cloak, the theoretically prescribed values of ε and μ (the cloak is anisotropic so these are tensor quantities now) have to be met.This is not easily done, but does fall within the domain of possible and there are already some experimental results by leading experts in metamaterial design [10], however for simplified material parameters.So far, most of the attention is given to 2D cloaks, since it is much easier to make 2D metamaterials than 3D ones.
The theoretical prescription [7,8,9] for a 2D cloak gives singular values of the material parameters for its inner surface.Our aim in this paper is to discuss to what extent is the cloaking effect hindered if finite parameter values are used.Here we focus on qualitative discussion and on the simulation, as more detailed analytical expressions are reported elsewhere [11].

Simulation results and discussion
When an "odd" theoretical prediction is made, we usually want to verify it somehow.Problably the most straightforward way is to use computer simulations, and this is what we have done.In addition to a possible confirmation of the effect, by using simulations we can get a very good idea of how sensitive the effect itself is to parameter variations.The first report on cloak simulations is given in [12] and it has been concluded that the cloaking effect is by far less sensitive to material parameter deviations than perfect lensing [4].
In the case of the 2D cylindrical cloak, some of the material parameter components [12], ε ij and μ ij , have values below unity, thus implying a superluminal phase velocity and, hence, a strong dispersion.Because of that, the cloaking effect is present only within a narrow bandwidth centered around the angular frequency ω 0 for which the values of ε and μ are closest to ideal ones.When looking for the effect, we simulate for the steady-state solution where only fields with exp(-iω 0 t) time variation are present.Consequently, we simulate the solutions to the Helmholtz equation, and this we do using the finite element method (FEM).
In what follows, we discuss transverse electric (TE) fields, with the electric field polarized along the axis perpendicular to the pictures (z-axis).Very similar conclusions apply to transverse magnetic (TM) fields, as they are obtained by exchanging E↔H and ε↔(-μ).The FEM simulation setup is shown in Fig. 1.This is a usual setup when plane wave scattering is investigated.Shaded regions represent perfectly matched layers (PMLs), with appropriate position dependant parameters, whose purpose is to effectively extend the boundary conditions on boundaries 1-4 to infinity.The relevant parameters of PMLs are as follows: h) Further information on PMLs is available in [13] and [14].Boundary 1 has the prescribed value of the electric field thus simulating the incoming plane wave (this is the "driving force" in the simulation).Boundary 2 has a prescribed value of electric field equal to 0, since all realistic fields eventually decay to zero at infinity.The perfect magnetic boundary condition on boundaries 3 and 4 has the purpose of ensuring the translational symmetry along the vertical (y-axis) direction (otherwise, the excitation from boundary 1 would appear to be finite and we would not obtain results corresponding to a plane wave).The described setup is somewhat different than the one used in [12] and an interested reader will easily verify that our's is approapriate for the plane wave scattering simulation.The annulus on the picture represents the cloak and the circle inside it is taken to be a perfect electric conductor (PEC).The medium surrounding the cloak is vacuum.
Boundary conditions 1-4 are explained in the text..
As in reality, any parameter value in a numerical simulation has to be finite thus implying that every simulation will yield some scattering on the cloak.Fig. 2. shows the overall and the scattered electric field distribution for the case we term "ideal cloak", meaning that we use the ideal parameters for the simulation, but the implementation of those has deviations due to the finite mesh element size (approximately λ 0 /100 in the vicinity of inner cloak surface and λ 0 /10 elsewhere, with λ 0 being the wavelength of the wave).The scattering is small, but still present due to finite numerical values and mesh size used in the simulation.Using the concept of modified mapping and calculating the scattered power from b), we obtain that the "ideal cloak" shown in the picture has ρ 1 ~0.001=a/250,where a is the size of the hidden mettalic object.Had we used a finer mesh at the cloak's inner surface, the scattering would have been smaller.In order to theoretically investigate the influence of finite values of material parameters, we devised a modified mapping with a parameter ρ 1 which restores the original one [12] in the limit ρ 1 →0.This mapping is plotted in Fig. 3. (solid) together with the original mapping (dashed).The dashed line represents the modified mapping mentioned in the text, while the solid line is the ideal mapping, attained when ρ 1 becomes zero.These lines correspond to the case of ideal and imperfect cloak, respectively.There are two line-breaks: first one at r=a=0.25 and the second one at r=b=0.5, since outside the plotted region, both mappings are identical, i.e. r=ρ.Our mapping, with nonzero ρ 1 , yields finite (and nonzero) values for all the material parameter components ε ij and μ ij and can be writtean as while the cloak's parameters are given by In the above formulae, a and b are the inner and outer radius of the cloak, respectively.The price paid for having nonsingular material parameters is that our "imperfect cloak" does not create a "hole" in space but merely shrinks the object (creating a "hole" in space then means that the object is shrinked to a point), thereby inducing some scattering which will depend both on how small ρ 1 is and on what is hidden.Now, imagine that we want to make the cloak and estimate in advance the amount of scattering it will induce when illuminated with a plane wave.The key parameter we would have to find in order to make this estimate, is how big (for ε ϕ and μ ϕ ) and how close to zero (ε r , ε z and μ r , μ z ) are the parameters of the material we can make.Based on that, we would assign an appropriate value of ρ 1 [11] for our structure and then proceed to calculate the scattering.We may continue in a similar manner to make an estimate on the operating bandwidth of the cloak (if a dispersion formula is given, we can find the magnitude of parameter deviations within a given spectral region).Fig. 3 Diagram showing the mapping between the "transformation" (horizontal axis) and the "physical" (vertical axis) space.
The analytic results for the scattering problem on the imperfect cylindrical cloak (to simplify, we assume that the hidden object is a perfect electric conductor, e. g. a metallic wire), are obtained in the "transformation space" (this is where we have the shrinked object).
Here, the problem is reduced to the familiar problem of plane wave scattering on a infinite cylinder [14] (since we work with 2D geometry, the physical picture is that the plane wave falls perpendicularly on the scattering cylinder).The scattered field is obtained in terms of Hankel functions of the first kind, usually labeled H n (1) (k 0 r), where k 0 is the propagation constant in the exterior medium, taken to be vacuum, and r is the radial coordinate.Our main analytic result concerns the scattering width, d s , (scattered power divided by the incoming power flux per unit width) in the limit ρ 1 →0:

Conclusion
We have discussed the 2D electromagnetic cloak recently proposed in literature [7,12] and its practical limitations stemming from the fact that the device which would lead to ideal invisibility should have singular material parameters on the inner surface of the cloaking cylinder.The FEM simulations are described and results presented when an ideal cloak is simulated.We have, briefly, described the procedure of analytically solving the problem by introducing a modified mapping and we gave the asymptotic formula for the scattering width of the structure (cloak and the perfectly conducting cylinder inside) in terms of ρ 1 , as the ideal cloak is approached.The formula not only proves the convergence towards the ideal cloaking (zero scattering width), but also gives a simple and useful quantitative measure on how well an object can be concealed using the cylindrical cloak.

Fig. 2
Fig. 2 FEM simulation results for the case of "ideal cloak" -a plane wave is incident on the cloak from left to right.. a) Total electric field, b) scattered field.