Refraction and geometry in Maxwell's equations

Computational studies of Maxwell's equations in complex geometries encountered in photonic band structure calculations run into difficulties when several length scales occur, such as the wavelength of light in free space and the skin depth in metal. These problems are remedied by using an adaptive co-ordinate system which expands or contracts length scales as necessary. Here we show that moving to a general co-ordinate transformation is equivalent to renormalizing ε and μ. This is an huge simplification because now we need only write one computer code in a Cartesian system, and we can use this same code to handle any co-ordinate system by adjusting the ε and μ we feed into the calculation.