M odeling the propagation of coherent fields, from the macroscopic (such as bulk optics) to the microscopic (such as guided optics) has typically involved two methods. For free- space propagation in bulk optics, Fourier methods have been applied to Gaussian- beam decomposition (GBD). For the guid- ed-wave domain, methods based on finite-difference methods, either in time or space, can be employed. For example, the finite-difference beam- propagation method (FD-BPM) is used to approximate the behavior of optical wave fields in media with microscopic variations in refractive index and in optical fibers, and it can be used to model devices that utilize evanescent coupling. Entire optical systems, from bulk optics to optical fibers and other integrated optical components, can be modeled with both GBD and FD-BPM. The major drawback of FD-BPM is that backward propagating optical fields are not calculated. Modeling of the electric wave-field propagation in inhomogeneous optical sys- tems using FD-BPM can be performed with Advanced Systems Analysis Program (ASAP) of Breault Research Organization. FD-BPM models the propagation of optical wave fields in media that are inhomoge- neous and isotropic—that is, media with microscopic variations in refractive index where the refractive index is fully repre- sented by a three-dimensional scalar func- tion. The FD-BPM tool in the ASAP complements the GBD 1 used to model the propagation of scalar wave fields through optical systems immersed in free space (such as bulk optics). So a complete coher- ent optical system can be modeled in both the free-space and guided-wave domains. In FD-BPM, the wave field is propagat- ed along the axial direction by dividing the three-dimensional volume into a sequence of planes each separated by a small and finite distance ∆z. The wave field is defined in an initial plane and only the forward-propagating wave field is chosen. Hence, the FD-BPM technique cannot be reliably used for modeling systems that require the formation of the backward propagating wave field. FD-BPM is more reliable than the Fourier BPM methods used, 3 because the Fourier com- ponents of wave propagation require the mean refractive index in a given plane to be defined, so that the wavelength of the field in that plane can also be defined. FD-BPM uses a cor- rection factor, similar to the Wentzel, Kramers, and Brillouin (WKB) approximation used in quantum theory, so the average refractive index is included to account for the inho- mogeneity of the medium. Other formalisms, such as the finite difference time-domain element Modeling coherent propagation aids accurate coupling Robert S. Upton and R. John Koshel ools and techniques for modeling the propagation of coherent fields— Fourier methods applied to Gaussian-beam decomposition and the finite-difference beam-propagation method—can help ensure accurate and efficient coupling. But all perspectives, from the macroscopic to the microscopic, must be considered. DESIGN SOFTWARE FIGURE 1. A ball lens fiber-coupling geometry shows the light focused by the ball lens into a single-mode fiber. Reprinted with revisions, from the June 2001 edition of WDM SOLUTIONS Copyright 2001 by PennWell Corporation T