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Shape deformations in rough-surface scattering: cancellations, conditioning, and convergence.

David P Nicholls1, Fernando Reitich

  • 1Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556, USA. Nicholls.2@nd.edu

Journal of the Optical Society of America. A, Optics, Image Science, and Vision
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Summary
This summary is machine-generated.

We analyzed shape-perturbation methods for scattering from rough surfaces. Cancellations in the series limit performance, but a variable change enables convergence proof and improved methods.

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Area of Science:

  • Electromagnetics
  • Computational physics
  • Applied mathematics

Background:

  • Classical shape-perturbation methods are used for scattering predictions from rough surfaces.
  • These methods often face performance limitations with decreasing surface regularity.

Purpose of the Study:

  • To analyze the conditioning properties of shape-perturbation methods.
  • To understand the role of cancellations in perturbation series for scattering predictions.
  • To develop principles for improved, stabilized methods.

Main Methods:

  • Analysis of recurrence relations in perturbation series.
  • Identification and characterization of cancellations.
  • Mathematical manipulation involving a change of independent variables.

Main Results:

  • Significant cancellations in recurrence relations were identified as the cause of performance deterioration.
  • These cancellations prevent straightforward recursive estimation of series term sizes and direct convergence proofs.
  • A direct proof of convergence is achievable through a simple change of independent variables prior to series derivation.

Conclusions:

  • The observed cancellations are fundamental to the limitations of current shape-deformation methods.
  • A modified approach using a change of variables can lead to a direct convergence proof.
  • These findings offer guiding principles for designing more stable and effective scattering prediction methods.