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Transformation Optics

Metamaterials give enormous choice of material parameters for electromagnetic applications. So much so that we might ask if there is a new way to design electromagnetic systems exploiting this new flexibility. In an ideal world magnetic and electrical field lines can be placed anywhere that the laws of physics allow and a suitable metamaterial found to accommodate the desired configuration of fields. It was to answer the question of what parameters to choose for the metamaterial that we developed transformation optics.

A

The idea is quite straightforward: start with a field pattern that obeys Maxwell’s equations for a system that is topologically similar to the desired configuration but confined either to free space or a simple configuration of permittivity and permeability, then distort the system until the fields are in the desired configuration. If we imagine that the original system was embedded in an elastic matrix in which Cartesian coordinate lines were drawn, then after distortion the deformed coordinates could be described by a coordinate transformation. Next rewrite Maxwell’s equations using the new coordinate system. Some time ago it was shown that Maxwell’s equation are of the same form in any coordinate system but the precise values of permittivity and permeability will change. These new values of permittivity and permeability are the ones we must give to our metamaterial if we want the fields to take up the distorted configuration.

 

 

 

 

 

Ward and Pendry used transformations to adapt computer programs to different coordinate systems: transformation from a a Cartesian to a cylindrical system simply modifies the values of e and m used in the codes. Shyroki simplified the formulation using modern techniques an approach we subsequently followed. If the distorted system is described by a coordinate transform

 we define,

Then in the new coordinate system we must use modified values of the permittivity and permeability to ensure that Maxwell’s equations are satisfied,

       

 

Although the prescription is simple enough it is very powerful in what can be achieved. For example: the perfect lens suffers from the limitation that the image is exactly the same size as the object. Our new design methodology easily remedies this limitation: imagine a coordinate system that is stretched in the region of the image thus stretching the image itself. Transformation optics then prescribes the metamaterials with which this can be achieved.

Take figure B, a periodic array of radiation sources embedded in a periodic structure comprising a repeated structure of three layers of positively refracting material and one layer of material having an equal and opposite refractive index.

B

A transformation to cylindrical coordinates,

maps figure B to a new geometry shown in figure C below.

C

The periodic sources combine into one source and two images are produced: one inside at B and the other outside the material at C. The process of refraction reverses the direction of each ray exactly a a corner mirror would do.

In this manner a single device such a a lens can generate a whole family of new devices simply by applying new transformations.

As an extreme challenge for transformation optics we set ourselves the problem of constructing a cloak of invisibility. Two problems confronted us: first we must eliminate scattered radiation and hence no radiation must reach the hidden object; second the hidden object must cast no shadow.

 

 

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