Transparent boundary condition for the beam propagation method

Abstract

A new boundary condition algorithm is presented that passes outgoing radiation freely with a minimum reflection coefficient (typically 10-5) while inhibiting the flux of incoming radiation. In contrast to the commonly used absorber method, this algorithm contains no adjustable parameters and is thus problem independent. It adapts naturally to a standard Crank-Nicholson difference scheme and is shown to be accurate and robust for both twoand three-dimensional problems.The beam-propagation method is currently the most widely used tool for the investigation of complex optoelectronic structures such as tapered or bent waveguides and Y junctions. 1 -3 Unfortunately this method is notoriously weak in modeling structures that permit radiation loss, since that radiation tends to reflect from the problem boundaries back into the solution region where it causes unwanted interference. The seriousness of this problem follows from the fact that virtually all the structures of interest do in fact result in scattered radiation. The most common way of preventing boundary reflection has been the insertion of artificial absorbing regions adjacent to the pertinent boundaries. 4 This procedure is accurate, provided that the absorbing region is carefully tailored, i.e., by using a small enough absorption gradient so that the absorber itself does not generate reflections and a thickness sufficient to absorb all radiation impinging upon the region. Unfortunately, ensuring that these conditions are properly met for each new problem is often a difficult and time-consuming process. Even when one is successful, the addition of extra problem zones results in computational penalties of run time and storage space. Recently, a different beam-propagation algorithm was reported (the socalled method of lines) that properly treats radiation loss through the problem boundaries. 5 This method is quite effective for longitudinally uniform structures. Because it is essentially an eigenmode expansion technique, however, the treatment of problems containing longitudinally varying dielectric constants is hindered by the necessity of recalculating the eigenmode spectrum whenever the dielectric constant changes.Thus, this method is cumbersome at best for simulating a number of important structures such as tapered waveguides or Y junctions.In this paper I describe a new boundary condition algorithm that allows radiation to escape the problem freely without appreciable reflection while prohibiting the flux of radiation back into the problem region. This transparent boundary condition (TBC) employs no adjustable parameters and is thus problem independent. In addition, it is easily incorporated into a standard Crank-Nicholson differencing scheme and is applicable to longitudinally varying structures of interest for optoelectronics research.A technical description of this new technique begins by considering the scalar paraxial beam-propagation equation. Since only the boundary region is of interest, we further restrict ourselves to the diffraction terms,where ...

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