We present new, stabilized shape-perturbation methods for calculations of scattering from rough surfaces. For practical purposes, we present new algorithms for both low- (first- and second-) and high-order implementations. The new schemes are designed with guidance from our previous results that uncovered the basic mechanism behind the instabilities that can arise in methods based on shape perturbations [D. P. Nicholls and F. Reitich, J. Opt. Soc. Am. A <b>21</b>, 590 (2004)]. As was shown there, these instabilities stem from significant cancellations that are inevitably present in the recursions underlying these methods. This clear identification of the source of instabilities resulted also in a collection of guiding principles, which we now test and confirm. As predicted, improved low-order algorithms can be attained from an <i>explicit</i> consideration of the recurrence. At high orders, on the other hand, the complexity of the formulas precludes an explicit account of cancellations. In this case, however, the theory suggests a number of alternatives to <i>implicitly</i> mollify them. We show that two such alternatives, based on a change of independent variables and on Dirichlet-to-interior-derivative operators, respectively, successfully resolve the cancellations and thus allow for very-high-order calculations that can significantly expand the domain of applicability of shape-perturbation approaches.
© 2004 Optical Society of America
David P. Nicholls and Fernando Reitich, "Shape deformations in rough-surface scattering: improved algorithms," J. Opt. Soc. Am. A 21, 606-621 (2004)