Abstract

An important challenge in computer-aided design and simulation of advanced waveguiding circuits is to perform efficient calculations of arbitrarily shaped, wide-angled structures using effective computational numerical schemes. The finite-difference beam propagation method using the paraxial approximation¹ has been applied successfully to treat photonics integrated circuits with shallow angles. Recent extension of this technique based on the Padé approximants^2,3 of the square root operator in the scalar one-way wave equation, specifically the multistep method of implementation by Hadley⁴ have made wide-angle propagation possible with a significant improvement in accuracy. To derive the full benefit of this technique requires a thorough understanding of its limitations. We previously⁵ derived analytic expressions which measure the accuracy that can be achieved for different Padé orders and showed that the error is composed of a fundamental part and a part dependent on the reference wavenumber. In this paper, we perform an exhaustive study to test these scaling laws. We show that there is an additional error which fundamentally limits the accuracy of the solution and which is not reduced with higher Padé approximants. From these results we have now obtained useful recommendations on the best application of higher order Padé approximants to achieve an acceptable absolute error for wide-angle waveguides with step-index profile.

© 1996 Optical Society of America