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.
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.
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 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
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
Then, using the chain rule, we get
Hence, in the transformed coordinates, the functional takes the
where denotes a gradient with respect to the
coordinates and the conductivity transforms as
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
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.
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
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.
- [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.