Waves in composites and metamaterials/Transformation-based cloaking in electromagnetism

The content of these notes is based on the lectures by Prof. Graeme W. Milton (University of Utah) given in a course on metamaterials in Spring 2007.

Introduction edit

In this lecture we will give a brief description of cloaking in the context of conductivity. It is useful to start off with a desciption of some variational principles for electrical conductivity at this stage.

Variational principle edit

Suppose that the electrical conductivity   is real and symmetric. Also assume that


Consider the body ( ) with boundary ( ) shown in Figure 1.

Figure 1. Body   with boundary   with a specified potential   on the boundary.

We would like to minimize the power dissipation into heat inside the body. This statement can be expressed as




Now consider a variation   where   on   and let   be a small parameter. Then


Using the identity


in the middle term on the right hand side leads to


From the divergence theorem, we have


where   is the outward unit normal to the surface   and  . Since   on  , we have




For   to be positive for all  , it is sufficient to have


If this is to be true for all  , then


If we define the flux as


then we have


Coordinate transformation equations for currents edit

Let us take new curvilinear coordinates   as shown in Figure 2. The new coordinates are material coordinates.

Figure 2. Transformation from spatial coordinates to material coordinates.

The Jacobian of the transformation   is given by


Then an infinitesimal volume   of the body transforms as


Recall that


Then, using the chain rule, we get






Hence, in the transformed coordinates, the functional   takes the form


where   denotes a gradient with respect to the   coordinates and the conductivity transforms as


Interpretation edit

We can now interpret the minimization problem in the transformed coordinates as follows:

  • The function   minimizes   in a body   filled with material with conductivity   with   as Cartesian coordinates in   space.

Therefore, for   to remain positive, we must have








This is the transformation law for currents. Using the same arguments as before, we can show that


Let the electric field   be derived from the potential  . Then the fields


are related via


Therefore, there are two transformations which are equivalent. However, an isotropic material transforms to an anisotropic material via the transformation equation for conductivity.

Electrical tomography edit

Consider the situation shown in Figure 1. Let the conductivity of the body be   and let us require that   inside the body. In electrical tomography one measures the current flux   at the surface for all choices of the potential  .

Suppose one knows the Dirchlet to Neumann map ( )


Can one find  ? No, not generally. Figure 3 illustrates why that is the case. For the body in the figure, the transformation is   outside the blue region while inside the blue region  . Also, outside the blue region,  ,  , and  . Inside the blue region   and   is obtained via the transformation rule.

Figure 3. Illustration of why the Dirchlet to Neumann map on the surface may not, in general, be used to determine the conductivity inside a body.

From the figure we can see that the Dirichlet-Neumman map will remain unchanged on  . Hence, the body appears to be exactly the same in  -space but has a different conductivity.

Even though this fact has been known for a while, there was still hope that you could determine   uniquely, modulo a coordinate transformation. However, such hopes were dashed when Greenleaf, Lassas, and Uhlmann provided a counterexample in 2003 (Greenleaf03).

First transformation based example of cloaking edit

Greenleaf et al. (Greenleaf03) provided the first example of transformation based cloaking. They considered a singular transformation


The effect of this mapping is shown in the schematic in Figure 4. An epsilon ball at the center of   is mapped into a sphere of radius 1 in  . The value of   is singular at the boundary of this sphere. Inside the sphere of radius 1, the transformed conductivity has the form  .

Figure 4. Transformation cloaking using the Greenleaf-Lassas-Uhlmann map.

Therefore we can put a small body inside and the potential outside will be undisturbed by the presence of the body in the cloaking region.

Cloaking for Electromagnetism edit

Pendry, Schurig, and Smith (Pendry06) showed in 2006 that cloaking could be achieved for electromagnetic waves. The concept of cloaking follows from the observation that Maxwell's equations keep their form under coordinate transformations. The Maxwell's equations at fixed frequency   are


A coordinate transformation ( ) gives us the equivalent relations








To see that this invariance of form under coordinate transformations does indeed hold, observe that


We want to show that this equals  .

In index notation, (1) can be written as


On the other hand,


The first term above evaluates to zero because of   if   is skew and   is symmetric.

So we now need to show that


or that,


Multiply both sides of (2) by   and sum over  , (i.e., multiply by   which is non-singular). Then we get




Both sides are completely antisymmetric with respect o  . So it suffices to take  ,  ,   and we can write


The right hand side above is the well known formula for the determinant of the Jacobian. Hence the first of the transformed Maxwell equations holds. We can follow the same procedure to show that the second Maxwell's equation also maintains its form under coordinate transformations. Hence Maxwell's equations are invariant with respect to coordinate transformations.

References edit

  • [Greenleaf03]     A. Greenleaf, M. Lassas, and G. Uhlmann. On non-uniqueness for Calderon's inverse problem. Mathematical Research Letters, 10:685--693, 2003.
  • [Pendry06]     J. B. Pendry, D. Schurig, and D. R. Smith. Controlling electromegnetic fields. Science, 312:1780--1782, 2006.