A fast and simple design methodology for transformation optics (TO) is described for devices having completely arbitrary geometries. An intuitive approach to the design of arbitrary devices is presented which enables possibilities not available through analytical coordinate transformations. Laplace's equation is solved using the finite-difference method to generate the arbitrary spatial transforms. Simple techniques are presented for enforcing boundary conditions and for isolating the solution of Laplace's equation to just the device itself. It is then described how to calculate the permittivity and permeability functions via TO from the numerical spatial transforms. Last, a modification is made to the standard anisotropic finite-difference frequency-domain (AFDFD) method for much faster and more efficient simulations. Several examples are given at the end to benchmark and to demonstrate the versatility of the approach. This work provides the basis for a complete set of tools to design and simulate transformation electromagnetic devices of any shape and size.
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