This high level monograph for the optics research market explores a large number of novel interactive methods and algorithms for calculating the transmission function of phase diffractive optical elements. The text includes accounts of well-established methods and algorithms for calculating DOEs, but its major contribution is to include current methods and examine the theoretical and practical aspects of synthesising optical components. All the methods discussed in this book have been verified by their numerical simulation. A fast fourier transform algorithm presents computational basis of all the methods considered. A portion of the algorithms have received a comparative study in terms of their suitability for solving the same problem. For a number of the interactive algorithms a rigorous proof to their convergence is given.