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

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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

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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

 

where

 

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

 

Therefore,

 

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

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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

 

or,

 

where

 

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

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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

 

Now,

 

Hence,

 

or,

 

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

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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

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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

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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

 

with

 

where

 

and

 

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

 

or,

 

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

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  • [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.