Journal of the European Optical Society - Rapid publications, Vol 5 (2010)

Application of the method of auxiliary sources to a defect-detection inverse problem of optical diffraction microscopy

M. Karamehmedovic, M.-P. Soerensen, P.-E. Hansen, A. V. Lavrinenko


We propose a method of numerical solution of a type of inverse scattering problem that arises in the optical characterisation/quality control of nanostructures. The underlying global, ill-posed, nonlinear optimisation problem is first localised by best-fit matching of library and measured diffraction efficiency patterns. The inverse problem is then solved using piecewise linear interpolation between the best far-field matches. Finally, the results are refined, on average, by solving an additional local optimisation problem formulated in terms of the Method of Auxiliary Sources. To illustrate the proposed method, we apply it in a concrete quantitative characterisation of a non-periodic, nano-scale grating defect, with numerically simulated measurements. It is shown that the presented procedure can solve the inverse problem with an accuracy usually thought to require rigorous electromagnetic theories.

© The Authors. All rights reserved. [DOI: 10.2971/jeos.2010.10021]

Full Text: PDF

Citation Details

Cite this article


N. Agersnap, P.-E. Hansen, J. C. Petersen, J. Garnaes, N. Destouches, and O. Parriaux, "Critical dimension metrology using optical diffraction microscopy" Proc. SPIE 5965, 68-78 (2005).

F. Borsetto, K. Carneiro, I. Davi, J. Garnaes, J. C. Petersen, N. Agersnap, P.-E. Hansen, J. Holm, and L. H. Christensen, "Profile characterization with combined atomic force (AFM) and optical diffraction microscopy (ODM)" in Proceedings of the euspen 7th International Conference and 8th General Meeting of the European Society for Precision Engineering and Nanotechnology (Baaden, 2006).

J. Garnaes, P.-E. Hansen, N. Agersnap, I. Davi, J. C. Petersen, A. Kühle, J. Holm, and L. H. Christensen, "Determination of submicrometer high aspect ratio grating profiles" Proc. SPIE 5878, 1-9 (2005).

J. Garnaes, P.-E. Hansen, N. Agersnap, J. Holm, F. Borsetto, and A. Kühle, "Profiles of high aspect ratio grating determined by optical diffraction microscopy and atomic force microscopy" Appl. Opt. 45, 3201-3212 (2006).

P.-E. Hansen, and L. Nielsen, "Combined optimization and hybrid scalar-vector diffraction method for grating topography parameters determination" Mater. Sci. Eng. B 165, 165-168 (2009).

V. D. Kupradze, "On the approximate solution of problems of mathematical physics" Russ. Math. Surv.+ 22, 59-107 (1967).

G. Fairweather, and A. Karageorghis, "The method of fundamental solutions for elliptic boundary value problems" Adv. Comput. Math. 9, 69-95 (1998).

G. Fairweather, A. Karageorghis, and P. A. Martin, "The method of fundamental solutions for scattering and radiation problems" Eng. Anal. Bound. Elem. 27, 759-769 (2003).

A. Doicu, Y. A. Eremin, and T. Wriedt, Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources (Academic Press, 2000).

T. Wriedt (ed.), Generalized Multipole Techniques for Electromagnetic and Light Scattering (Elsevier, 1999).

D. I. Kaklamani, and H. T. Anastassiu, "Aspects of the method of auxiliary sources (MAS) in computational electromagnetics" IEEE Antenn. Propag. Mag. 44, 48-64 (2002).

H. T. Anastassiu, D. I. Kaklamani, D. P. Economou, and O. Breinbjerg, "Electromagnetic scattering analysis of coated conductors with edges using the method of auxiliary sources (MAS) in conjunction with the standard impedance boundary condition (SIBC)" IEEE Trans. Antenn. Propag. 50, 59-66 (2002).

Y. Leviatan, A. Boag, and A. Boag, "Analysis of TE scattering from dielectric cylinders using a multifilament magnetic current model" IEEE Trans. Antenn. Propag. 36, 1026-1031 (1988).

M. Karamehmedovic, and O. Brienbjerg, "Application of the method of auxiliary sources for the analysis of plane-wave scattering by impedance spheres" in Proceedings of the 2002 International Conference on Mathematical Methods in Electromagnetic Theory MMET02, 499-501 (Kiev, 2002).

M. Karamehmedovic, and O. Brienbjerg, "A convergent method of auxiliary sources for two-dimensional impedance scatterers with edges" in Proceedings of the 16th International Conference on Applied Electromagnetics and Communications, 46-49 (Dubrovnik, 2001).

B. Sternin, and V. Shatalov, Differential Equations on Complex Manifolds (Kluwer Academic Publishers, 1994).

D. Colton, and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer, 1998).

COMSOL Multiphysics demonstration CD-ROM can be requested at

W. B. J. Zimmerman, Multiphysics Modelling with Finite Element Methods (World Scientific, 2006).

A. Wächter, and L. T. Biegler, "On the implementation of a primaldual interior point filter line search algorithm for large-scale nonlinear programming" Math. Program. 106, 25-57 (2006).