## Shape deformations in rough-surface scattering: improved algorithms

JOSA A, Vol. 21, Issue 4, pp. 606-621 (2004)

http://dx.doi.org/10.1364/JOSAA.21.000606

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### Abstract

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 [NichollsD. P.ReitichF., J. Opt. Soc. Am. A 21, 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 *explicit* 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 *implicitly* 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

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(050.2770) Diffraction and gratings : Gratings

(290.0290) Scattering : Scattering

**History**

Original Manuscript: February 21, 2003

Revised Manuscript: August 27, 2003

Manuscript Accepted: November 14, 2003

Published: April 1, 2004

**Citation**

David P. Nicholls and Fernando Reitich, "Shape deformations in rough-surface scattering: improved algorithms," J. Opt. Soc. Am. A **21**, 606-621 (2004)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-21-4-606

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