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

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.

Recap edit

In the previous lecture we showed that for a material with a real, symmetric, and positive definite conductivity tensor ( ), we can set up a variational principle for minimal power dissipation. We also showed that this variational principle is satisfied if


Next we showed that under a coordinate transformation from  , the conductivity tensor transforms as




We also found that the variational principle in transformed coordinates has the alternative interpretation that { the function   minimizes   in a body   filled with material with conductivity   with   as Cartesian coordinates in   space.}

Next we derived the transformation rule for currents:


We also saw that the transformation rule for the electric field could be written as


Next, based on insights obtained from electric tomography, we found that a transformation-based cloaking effect could be obtained using the Greenleaf-Lassas-Uhlmann mapping (Greenleaf03). This mapping is singular and of the form


The effect of this mapping is shown in Figure 1.

Figure 1. Transformation cloaking using the Greenleaf-Lassas-Uhlmann map.

Some other unusual transformations edit

One unusual mapping that can be used to achieve cloaking is to fold back space upon itself (Leonhardt06,Pendry06). An example of such a mapping is


The effect of this transformation is shown in Figure~2. Note that there is a sharp (discontinuous) fold and the separation shown in the thickness direction is simply for the purpose of illustration. In reality, space is folded upon itself and the determination of the Jacobian inside the fold is -1.

Figure 2. Transformation that causes folding over of space.

Transformations of Maxwell's equations edit

We also showed in the previous lecture that Maxwell's equations at fixed frequency are invariant with respect to coordinate transformations. Thus the equations in  -space


transform, in  -space, to


In the transformed equations,




Let us consider the effect of the fold-back transformation shown in Figure~2 on Maxwell's equations.

Recall that this transformation has the form


The perfect lens edit

Now, let us suppose that   everywhere in the region. Since the Jacobian of the transformation is


we have


This implies that in the region   we have


Since the materials in the region are isotropic, i.e.,   and  , then from equation (1) we see that in the region  ,


Therefore the fold back transformation is realized in a geometry that is equivalent to the perfect lens that we discussed earlier. Figure 3 shows the geometry involved.

Figure 3. The "perfect" lens corresponds to a transformation that causes folding over of space.

For a source that is less than a distance   from the first interface, the fields blow up to infinity and there is no solution. For a solution to exist, we need to regularize the problem and add a small loss  , i.e.,   in the lens. In that case, the fields blow up to infinity in two strips of length   where   is the distance of the source from the first interface. Outside this region, the fields converge to that expected by the Pendry solution (see Figure 4 for a schematic.)

Figure 4. Fields in the vicinity of a slightly lossy "perfect" lens.

If  , i.e.,  , then the sources will be in a region of enormous fields. In fact, the source produces infinite energy per unit time in such regions as the loss  . This is clearly unphysical. So any realistic point or line source with finite energy such as a polarizable particle must have an amplitude which goes to zero as  . This means that the particle will have become cloaked!

Figures 5(a) and (b) show the cloaking caused by a cylindrical perfect lens with a small loss (Milton06). When a polarizable diople is located close to the lens, the field is barely perturbed. However, when the dipole is at a distance from the lens, the field shows significant perturbations.

Figure 5. Cloaking due to a cylindrical perfect lens (with a slight loss). Source: Milton and Nicorovici (Milton06).

Magnification edit

So far we have not dealt with the issue of magnification. Is there a coordinate transformation that leads to magnification? One such possible transformation is illustrated in Figure 6.

Figure 6. Magnification due to a folding coordinate transformation.

In this case, in the region of dilation, the transformation is


Therefore, the Jacobian of the transformation is


Hence the material tensors in the region of dilation transform as


However, in the folded region, the   and   tensors are anisotropic and negative. Such a transformation therefore acts like a magnifying lens.

Transformation-based cloaking in elasticity edit

It turns out the Willis equations in elastodynamics also transform in a manner that is very similar to the Maxwell equation in electromagnetism. Before we describe the Willis equations, let us get into a brief description of ensemble averaging (a opposed to volume averaging). The hope is that the ensemble average is a good descriptor of behavior in individual realizations.

Examples of ensembles edit

Some examples of ensembles are:

  • Periodic media with a period   where the fields are not necessarily periodic (see Figure 7(a)). The ensemble is the material and all translations of it. Of course, a translation that is equal to the period gives back the same material.
  • Media generated by some translation invariant statistical process. This means that a particular realization and its translations are equally likely to occur (roughly speaking). An example is a medium generated by a Poisson process. We can represent the ensemble by constructing a Voronoi tessellation and assigning constants to each cell at random (see Figure 7(b)).
  • Media generated by some statistical process where the statistics depend slowly on position.
Figure 7(a). Periodic media.
Figure 7(b). Poisson distribution based media.

Willis' equations edit

Recall that the equations governing the motion of a linear elastic body are


where   is the momentum,   is the stress, and   is the body force. We assume that the body force is independent of the realization. The microscopic constitutive relations are assumed to be




By ensemble averaging (2) we get


where   is the ensemble average over realizations and not a volume average.

However, we cannot ensemble average (3) since


We therefore need some effective constitutive relation. Willis (Willis81,Willis81a,Willis83,Willis97,Milton07) found that


where all the operators are nonlocal in time (and in general also nonlocal in space).

By the adjoint operator (represented by the superscript  ), we mean


for all fields   and   and at time  .

In the next lecture we will show how the Willis' equations are derived.

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.
  • [Leonhardt06]     U. Leonhardt. Optical conformal mapping. Science, 23:1777--1780, 2006.
  • [Milton06]     G. W. Milton and N-A. P. Nicorovici. On the cloaking effects associated with anomalous localized resonance. Proc. R. Soc. London A, 462:3027--3059, 2006.
  • [Milton07]     G. W. Milton and J. R. Willis. On modifications of newton's second law and linear continuum elastodynamics. Proc. R. Soc. London A, 463:855--880, 2007.
  • [Pendry06]     J. B. Pendry, D. Schurig, and D. R. Smith. Controlling electromagnetic fields. Science, 312:1780--1782, 2006.
  • [Willis81]     J. R. Willis. Variational and related methods for the overall properties of composites. Advanced in Applied Mechanics, 21:1--78, 1981.
  • [Willis81a]     J. R. Willis. Variational principles for dynamics problems in inhomogenous elastic media. Wave Motion, 3:1--11, 1981.
  • [Willis83]     J. R. Willis. The overall elastic response of composite materials. J. Appl. Mech., 50:1202--1209, 1983.
  • [Willis97]     J. R. Willis. Dynamics of composites. In Suquet P., editor, Continuum Micromechanics: CISM Courses and Lectures No. 377, pages 265--290. Springer-Verlag-Wien, New York, 1997.