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

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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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

  • [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.