DFG-Forschungszentrum Matheon Weierstrass Institute for Applied Analysis and Stochastics Technische Universität Berlin

Matheon Project D25: Computation of shape derivatives for conical diffraction by polygonal gratings

Factual information: duration, research team, collaboration, software, funding

back to top   

Duration July 2010 -- December 2010
Project leaders Andreas Rathsfeld
Weierstrass Institute for Applied Analysis and Stochastics
Mohrenstrasse 39
10117 Berlin
phone: +49 (0)30 - 20372 457 (office)

email: andreas.rathsfeld[at]wias-berlin.de

Dietmar Hömberg
Weierstrass Institute for Applied Analysis and Stochastics
Mohrenstrasse 39
10117 Berlin
phone: +49 (0)30 - 20372 491 (office)
+49 (0)30 - 20372 555 (secretary)

email: dietmar.hoemberg[at]wias-berlin.de
Technical University of Berlin
Department of Mathematics
Strasse des 17. Juni 136
10623 Berlin
phone: +49 (0)30 - 31428 034 (office)
+49 (0)30 - 31479 687 (secretary)

Support Project D25 is funded by DFG Research Center MATHEON "Mathematics for Key Technologies"


back to top   

Diffraction of elecromagnetic waves by grating structures plays an important role in the micro-optics industry, especially in the design of optical devices and in scatterometry, where grating structures have to be determined by measurements of the diffracted wave [8,1]. Both problems require the solution of an inverse problem for a Maxwell transmission problem. A particularly interesting case regarding applications is the conical illumination of periodic interfaces by plane waves. It has been shown that in this setting the Maxwell problem can be reduced to a Helmholtz system coupled by transmission conditions [3].

The inverse problem can be solved using iterative Newton-type methods. One approach, which has been investigated by Elschner and Schmidt [7], uses the Eulerian derivative of a shape functional. For this method, a direct and an adjoint problem have to be solved in each iteration step. Another promising strategy [14,9,11] suggests to use the shape derivative of the solution to the direct problem. However, up to now only transmission by smooth structures has been investigated in this context. In [12], a priori estimates, existence and uniqueness results for shape derivatives of solutions of conical diffraction problems with non-smooth gratings have been proven for the first time, using weighted Sobolev spaces of Kondratiev type [13]. Moreover, the shape derivative has been characterized as a solution of a diffraction problem with different right-hand sides, but with the same operator. Using this result together with the above mentioned shape derivative method should lead to a more efficient numerical approach for shape reconstruction. However, the presence of singularities that occur due to the non-smoothness of the interface proposes challenging questions regarding the development of a numerical method and error estimates.

Current research

back to top   

The aim of the project is the numerical computation of shape derivatives for conical diffraction problems with non-smooth gratings. The shape derivative should be used for shape reconstruction in combination with iterative method, such as Gauss-Newton or Levenberg-Marquardt. Each iteration step requires the solution of two diffraction problems, i.e. one direct problem and one problem for the shape derivative, which have the same operator, as stated above. Since inverse problems of this type are typically ill-posed, a shape reconstruction method will require regularization.

The algorithmic basis for our approach is a BEM solver for conical diffraction problems which already exists at WIAS [16]. Up to now, this program handles incoming plane waves as right-hand sides. It has to be modified in such a way that also the modified right-hand sides characterizing the shape derivative can be treated. Note that boundary element methods are known to be very efficient in the case of interface problems with homogeneous materials and high frequencies.

As theoretical results have shown [12], the solution of the diffraction problem characterizing the shape derivative fails to be of H-regularity. Moreover, the singularities of the shape derivative at the corners of the grating structure are not explicitly known. This makes it difficult to use the canonical ansatz of splitting the solution into a singular and a regular part and computing the stress intensity factors by the (dual) singular function method [2].

However, the results obtained in [12] suggest a new approach that consists in dealing with the singularities of the right-hand side of the problem by choosing suitable cut-off functions. Moreover, the a priori estimate shown in [12] paves the way for proving convergence of this method in the case of domains with corners. Furthermore, an error analysis for the described cut-off process needs to be done. In summary, we have the following goals:

  • to compute shape derivatives for conical diffraction problems with non-smooth interfaces by extending an existing BEM code
  • to use the shape derivative for an iterative shape-reconstruction method
  • to derive error estimates for the numerical approach

  1. Y. Achdou, O. Pironneau, Optimization of a Photocell Optimal Control Application and Methods, 12(4):221-246, 1991

  2. H. Blum, M. Dobrowolski, On finite element methods for elliptic equations on domains with corners, Computing 28:53-63, 1982

  3. J. Elschner, R. Hinder, F. Penzel, G. Schmidt, Existence, uniqueness and regularity for solutions of the conical diffraction problem, Mathematical Models and Methods in Applied Sciences, 10:317-341, 2000

  4. J. Elschner, R. Hinder, G. Schmidt, Finite element solution of conical diffraction problem Advances in Computational Mathematics, 16(2-3):139-156, 2002

  5. J. Elschner, G. Schmidt, Diffraction in periodic structures and optimal design of binary gratings I: Direct problems and gradient formulas, Mathematical Methods in the Applied Sciences, 21(14): 1297-1342

  6. J. Elschner, G. Schmidt, A rigorous numerical method for the optimal design of binary gratings Journal of computational Physics, 146:603-626, 1998

  7. J. Elschner, G. Schmidt, Conical diffraction by periodic structures: Variation of interfaces and gradient formulas, Mathematische Nachrichten, 252:24-42, 2003

  8. H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, Mathematical modelling of indirect measurement in scatterometry Measurement, 39:782-794, 2006

  9. F. Hettlich, Fréchet derivatives in inverse obstacle scattering, Inverse Problems, 11:371-382, 1995

  10. D. Hömberg, J. Sokolowski, Optimal shape design of inductor coils for surface hardening SIAM Journal on Control and Optimization, 42(3): 1087-1117, 2003

  11. A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems, 9:81-96, 1993

  12. N. Kleemann, Shape derivatives in Kondratiev spaces for conical diffraction, Preprint No. 1489, WIAS, 2010

  13. V. A. Kondratiev, Boundary problems for elliptic equations in domains with conical or angular points, Transactions of the Moscow Mathematical Society, 16:209-292, 1967

  14. R. Potthast, Fréchet differentiability of boundary integral operators in inverse acoustic scattering, Inverse Problems, 10:431-447, 1994

  15. A. Rathsfeld, G. Schmidt, B. Kleemann, On a fast integral equation method for diffraction gratings, Communications on Computational Physics, 1:934-1009, 2006

  16. G, Schmidt, L. Goray, Solving conical diffraction grating problems with integral equations Journal of the Optical Society of America A, 27:585-597, 2010

last modified June 15, 2010