The use of Z transform theory for numerical simulation of dispersive and non-linear materials with the FDTD method

The finite-difference time-domain (FDTD) method has been proven capable of modeling electromagnetic (EM) interactions in a wide variety of applications . Among these have been biological applications, such as EM dosimetry and hyperthermia cancer therapy, and optical applications, such as the simulation of soliton propagation. Progress in biological applications proceeded with a method to simulate the frequency dependence of biological tissues while using a time domain method. However, the formulation for even a relatively straightforward problem proved to be extremely complicated. Progress in optical simulation proceeded when the ability to simulate non-linear materials was developed. However, these methods usually involved inserting an iterative procedure to allow for the non-linearity, making the formulation cumbersome, and substantially adding to the computation time. It has been found that both of these obstacles, the simulation of dispersive materials for biological application, and the simulation of non-linear materials for optical applications, can be addressed more easily by formulating the Maxwell´s equations as a digital filtering problem, and then using concepts from Z transform theory to implement the solution into an FDTD algorithm.