The purpose of this paper is twofold. First, a simple but comprehensive and powerful arrayed-waveguide grating (AWG) field model is presented which,based on Fourier optics, borrows some principles of that developed by Takeouchi and coworkers  for the analysis of reflective-type AWGs for optical signal processing, but at the same time adds more features, such as the calculation of device losses and the refinement of the mathematical model to obtain a simple expression for the output field for any input-output waveguide configuration where the meaning of the different high-level parameters of the AWG becomes very clear to the reader. Second, we elaborate on the model developed to present an useful design procedure of the AWG based on two steps illustrated by design flowcharts.
Pascual Munoz, Daniel Pastor, and Jose Capmany, "Modeling and Design of Arrayed Waveguide Gratings," J. Lightwave Technol. 20, 661- (2002)