Focusing light using negative refraction

A slab of negatively refracting material, thickness d, can focus an image at a distance 2d from the object. The negative slab cancels an equal thickness of positive space. This result is a special case of a much wider class of focusing: any medium can be optically cancelled by an equal thickness of material constructed to be an inverted mirror image of the medium, with epsilon, mu reversed in sign. We introduce the powerful technique of coordinate transformation, mapping a known system into an equivalent system, to extend the result to a much wider class of structures including cylinders, spheres and intersecting planes, and hence show how to produce magnified images. All the images are 'perfect' in the sense that both the near and far fields are brought to a focus and hence reveal sub-wavelength details.