Abstract

The finite-difference beam propagation method using the paraxial approximation¹ has been applied successfully to photonics integrated circuits with shallow angles. Recent extensions to these techniques based on Pade approximants of the square root operator in the one way wave equation² have made wide-angle propagation possible with a significant improvement in accuracy. However, in order to effectively apply these new techniques to real-world problems, it is necessary to have a thorough understanding of their fundamental limits and how to achieve them. Much of the previous work in this area has focused on implementation and comparison of algorithms using one or two benchmark examples³, rather than considering detailed parametric studies. Here we theoretically examine the Pade-based BPM scheme and derive general expressions for angular and truncation error as a function of various waveguiding and numerical parameters. These results are then validated with numerical experiments. From the general expression for the error, the significance of the choice of reference wavenumber is clearly evident. For waveguiding circuits containing variable angles, the optimal reference wave number, which is normally assumed constant, will change with position, thus we explore the possibility of varying the reference number for efficient solution of variable angle photonic integrated circuits.

© 1995 Optical Society of America