 
Summary: A construction of Beam Propagation Methods for
optical waveguides
Xavier Antoine
, Pierre Dreyfuss
, Karim Ramdani
Abstract. This paper presents a systematic method to derive Beam Propagation Models
for optical waveguides. The technique is based on the use of the symbolic calculus rules
for pseudodifferential operators. The cases of straight and bent optical waveguides are
successively considered.
Keywords. Bent Waveguides; Beam Propagation Method; Optics
1 Introduction
Complex optical waveguides play a key role in the design of optical communications systems
and integrated optical circuits [13]. In many applications, the waveguides considered are not
uniform in the propagating direction, called the zdirection in this paper (inhomogeneous
structures, bent waveguides e.g.). In order to simulate numerically such optical devices,
one can truncate the structure in the transverse xvariable by using for example a Perfectly
Matched Layer (see e.g. [11]). Since the length of the waveguide (of the order of the
millimeter) is much larger than the free space wavelength 0 (of the order of the micrometer),
a numerical simulation remains extremely costly. This is the reason why approximate efficient
models like Beam Propagation Methods (BPMs) have been introduced [13]. The idea is to
