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Optimization of diffractive structures and reconstruction methods for severely ill-posed problems

Collaborator: G. Bruckner, J. Elschner, A. Rathsfeld, G. Schmidt

Cooperation with: B. Kleemann (Carl Zeiss Oberkochen), R. Güther (Ferdinand-Braun-Institut für Höchstfrequenztechnik Berlin), G. Bao (Michigan State University, East Lansing, USA), G.C. Hsiao (University of Delaware, Newark, USA), M. Yamamoto (University of Tokyo, Japan), CiS Institut für Mikrosensorik gGmbH (Erfurt)

Supported by: BMBF: ``Modellierung und Optimierung mikrooptischer Oberflächenstrukturen'' (Modeling and optimization of microoptical surface structures ), DFG: ``Scientific cooperation with Japan: Inverse problems in electromagnetics and optics''

Description:1. Accurate FEM and BEM simulation of diffraction by binary and polygonal gratings (A. Rathsfeld, G. Schmidt).

The diffraction of light waves by optical gratings can be reduced to boundary value problems for the Helmholtz equation on a rectangular domain which contains one period of the surface structure. In case of complicated surface geometries, the FEM is the natural method for the numerical solution. We continued to develop our FEM program package DiPoG, versions 1.4 and 2.0 (cf. [16], [17]), which treats the outgoing wave conditions at infinity by coupling with BEM and which includes a generalized FEM approach for high frequency solutions.

DiPoG-1.4, which is used to simulate and optimize binary and multilevel structures, now includes all modifications which were developed due to requirements of the industrial partners from Carl Zeiss and CiS Erfurt. This concerns the possibility to model homogeneous layers of any thickness also within the grating structure, to choose different output formats for the computed diffracted fields and related values, and the improved GMRES-based iterative solver. Most of these extensions have been included into the programs for solving optimal design problems. In particular, in version 1.4 we implemented new functionals, which take into account prescribed polarizations of the incoming waves, and we introduced a new solver for the direct and dual problems based on the PARDISO library. Both of the implemented optimization algorithms, a conjugate gradient and an interior point method, can now be controlled by the same set of parameters.

In DiPoG-2.0 we implemented in accordance with the requirements of our cooperation partners from Carl Zeiss a new presentation of the computed results, a graphical display of the far-field solution, and an improved output of efficiency data for the conical diffraction. Since a simple handling of geometric data is essential for a user-friendly operation of DiPoG, we increased the number of standard gratings which can be generated by simple code words and a few parameters. In particular, two types of echelle gratings, of sine-shaped, lamellar, and coated trapezoidal gratings have been realized and a stack of such profiles can now easily be assembled. Finally, based on our FEM code and on simulated annealing, we developed a first version of a global optimization algorithm to design polygonal grating profiles. This work is to be continued next year.

For gratings with a single transition profile and, possibly, a single coated layer, a boundary integral equation method like the IESMP (owned by Carl Zeiss, cf. [13]) turns out to be more efficient. However, in order to treat thin coatings and corner profiles, the simple combination of trigonometric collocation for the main part operator and of a Nyström quadrature for the remainder within IESMP is not sufficient. We changed the basic discretization scheme of the integral equation method to spline collocation over graded meshes and developed an adapted quadrature algorithm. Further we implemented an extension of Ewald's method for computing the kernel functions of the integral operators. The acceleration method for these infinite sums implemented in IESMP breaks down if the kernel is nearly singular, whereas Ewald's method allows to extract the main singularity and to handle it separately. For special profile curves, the resulting method converges and the numerical error diminishes in accordance with theoretically predicted orders. This work is to be continued next year.

2. Inverse problems for diffraction gratings: Uniqueness results and reconstruction methods (G. Bruckner, J. Elschner, A. Rathsfeld, G. Schmidt).

The reconstruction of the shape of periodic structures from measurements of scattered electromagnetic waves is a problem of great practical importance in modern diffractive optics, [2]. We studied the scattering of monochromatic plane waves by a two-dimensional diffraction grating, i.e. a periodic curve (the grating profile) which separates two regions with different optical materials. Let k⁺ > 0 be the refractive index (or wave number) above the grating, whereas the refractive index below the interface satisfies $\Re$k^- > 0 ,  $\Im$k^-$\ge$ 0. The direct diffraction problem is modeled by a transmission problem for the periodic Helmholtz equation.

Let the profile of the diffraction grating be given by the curve $\Lambda_{f}^{}$ : = {(x₁, x₂) $\in$ $\mathbb {R}$² : x₂ = f (x₁)} where f is a 2$\pi$-periodic Lipschitz function. Suppose that a plane wave given by

uⁱⁿ : = exp(i$$\alpha$$x₁ - i$$\beta$$x₂),($$\alpha$$,$$\beta$$) = k⁺(sin$$\theta$$, cos$$\theta$$)

is incident on $\Lambda_{f}^{}$ from the top, where $\theta$ $\in$ (- $\pi$/2,$\pi$/2) is the incident angle. The inverse problem or the profile reconstruction problem can be formulated as follows.

(IP): Consider a fixed refractive index k^- below $\Lambda_{f}^{}$, and let $\theta$ be a fixed incident angle. Determine the profile function f from incident waves uⁱⁿ, given for several wave numbers k⁺, and the knowledge of the corresponding scattered fields on two straight lines {x $\in$ $\mathbb {R}$² : x₂ = b^$\pm$} above and below the structure.

Note that this problem also involves near-field measurements since the evanescent modes cannot be measured far away from the grating profile. The uniqueness with a single arbitrary wave number k^- in problem (IP) is presently only known for reflection gratings, i.e. for $\Im$k^- > 0 ([8]). General uniqueness results with a single wave number are also available for perfectly reflecting diffraction gratings with polygonal profiles ([10]).

In the practically important case of transparent gratings (k^- > 0), we were recently able to show uniqueness for a finite number of refractive indices, where this number only depends on the maximal value k_max of k^-, k⁺, $\theta$, and the amplitude h of the profile function f ([9]). In particular, if h$\sqrt{{(k^+)^2+(k^-)^2}}$ < $\pi$, i.e. the refractive indices are sufficiently small, then uniqueness with a single wave number holds. The proof is based on the Courant-Weyl min-max principle for the eigenvalues of a fourth-order elliptic problem.

The efficient numerical solution of the profile reconstruction problem is challenging due to the fact that it is both nonlinear and severely ill-posed. For the reconstruction of perfectly reflecting periodic interfaces leading to the inverse Dirichlet problem, several inversion algorithms based on iterative regularization ([12]), linear sampling ([1]), and the Kirsch-Kress optimization method ([4], [6]) became recently available. The latter approach was originally developed for acoustic obstacle scattering and avoids the solution of direct diffraction problems. In [5] a corresponding reconstruction method for the inverse transmission problem (IP) was developed and analyzed for the first time.

As in [4], [6], this method splits the inverse problem into a linear ill-posed part to reconstruct the scattered field and a nonlinear well-posed part to find the profile curve. The minimization of the Tikhonov functional for the linear problem and the defect minimization of the transmission conditions are then combined into one cost functional. We obtained convergence results in the general case of Lipschitz grating profiles, extending the variational approach for the perfectly reflecting case. However, in the transmission case, it is harder to establish convergence of the cost functional. The proof of this is based on nontrivial continuity and solvability properties of layer potentials on periodic Lipschitz graphs.

The reconstruction algorithm was implemented as a two-step method. Two unknown density functions are first computed from near-field data measured above and below the grating structure, which allows us to represent the scattered and transmitted fields as single layer potentials. Then these density functions are used as inputs to a nonlinear least squares problem, which determines the unknown profile as a curve where the associated transmission conditions are fulfilled. After discretization, the least squares problem is solved iteratively by the Gauss-Newton method. Numerical results with exact and noisy data demonstrated the efficiency of the inversion algorithm, [5].

Fig. 1: Given and reconstructed profile using 100 and 800 Gauss-Newton iterations

3. Optimal design of nonlinear diffraction gratings (G. Schmidt).

If a plane wave of frequency $\omega_{1}^{}$ illuminates on a grating or periodic structure ruled on some nonlinear optical material, then the nonlinear optical interaction gives rise to diffracted waves at frequencies $\omega_{1}^{}$ and $\omega_{2}^{}$ = 2$\omega_{1}^{}$. This process represents the simplest situation in nonlinear optics, the so-called second harmonic generation (SHG). An exciting application of SHG is to obtain coherent radiation at a wavelength shorter than that of the available lasers. Unfortunately, it is well known that nonlinear optical effects from SHG are generally so weak that their observation requires an extremely high intensity of laser beams. The effective enhancement of nonlinear optical effects presents one of the most challenging tasks in nonlinear optics. It has been announced recently that SHG can be greatly enhanced by using diffraction gratings or periodic structures, and the PDE model can predict the field propagation accurately.

The joint paper [3] with Gang Bao and Kai Huang (Michigan State University) is concerned with some aspects of the systematic design of surface (grating) enhanced nonlinear optical effects. We give the mathematical foundation of optimization methods for solving the optimal design problem of nonlinear periodic gratings. By conducting a perturbation analysis of the grating problems that arise from smooth variations of the interfaces, we derive explicit formulas for the partial derivatives of the reflection and transmission coefficients. Such derivatives allow us to compute the gradients for a general class of functionals involving the Rayleigh coefficients.

4. Reconstruction of curve source profiles from boundary measurements in a 2D wave equation model (G. Bruckner).

Generalizing earlier investigations, [7], where point sources in a one-dimensional vibrating string were identified from dislocations at one fixed point, here a corresponding 2D problem is investigated: the identification of curves in a plane domain from measurements at the boundary, where excitations are governed by the 2D wave equation. This can be considered as a first step towards a reduced earthquake model. A second step could be replacing the wave equation by the Lamé system. In [11] the Lamé system has been considered with an L² source function, while here H^-1 sources are asked for. In our case a main difficulty consists in finding an adequate distance for curves in a 2D domain. So far, this problem could be solved by the authors only for pieces of straight lines. The stability estimate is of logarithmic type, and in the proofs, Duhamel's principle and Carleman estimates are essential.

A paper is in preparation.

5. New solution method for volume integral equations of scattering theory (G. Schmidt).

Scattering of incoming plane waves by inhomogeneous media can be described by Lippmann-Schwinger-type integral equations, which contain the diffraction potential over the volume of the scatterer. Especially in the case of high wave numbers, the approximation of the diffraction potential is very expensive. In [14] the cubature approach which had been developed in [15] was extended to the numerical solution of Lippmann-Schwinger equations. We considered a collocation method where the unknown is sought as a linear combination of scaled and shifted Gaussians. Then the discrete system could be obtained from the semi-analytic representation. We proved spectral convergence rates of the method, which were confirmed in one-dimensional numerical tests.

References:

1. T. ARENS, A. KIRSCH, The factorization method in inverse scattering from periodic structures, Inverse Probl., 19 (2003), pp. 1195-1211.

2. G. BAO, L. COWSAR, W. MASTERS (EDS.), Mathematical Modeling in Optical Science, SIAM, Philadelphia, 2001.

3. G. BAO, K. HUANG, G. SCHMIDT, Optimal design of nonlinear diffraction gratings, J. Comput. Phys., 184 (2003), pp. 106-121.

4. G. BRUCKNER, J. ELSCHNER, A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles, Inverse Probl., 19 (2003), pp. 315-329.

5.          , The numerical solution of an inverse periodic transmission problem, WIAS Preprint no. 888, 2003.

6. G. BRUCKNER, J. ELSCHNER, M. YAMAMOTO, An optimization method for grating profile reconstruction, in: Progress in Analysis, Proc. 3rd ISAAC congress, World Scientific, Singapore, 2003, pp. 1391-1404.

7. G. BRUCKNER, M. YAMAMOTO, Determination of point wave sources by pointwise observations: Stability and reconstruction, Inverse Probl., 16 (2000), pp. 723-748.

8. J. ELSCHNER, G.C. HSIAO, A. RATHSFELD, Grating profile reconstruction based on finite elements and optimization techniques, SIAM J. Appl. Math., 64 (2003), pp. 525-545.

9. J. ELSCHNER, M. YAMAMOTO, Uniqueness results for an inverse periodic transmission problem, in preparation.

10. J. ELSCHNER, M. YAMAMOTO, G. SCHMIDT, An inverse problem in periodic diffractive optics: Global uniqueness with a single wave number, Inverse Probl., 19 (2003), pp. 779-787.

11. M. GRASSELLI, M. YAMAMOTO, Identifying a spatial body force in linear elastodynamics via fraction measurements, SIAM J. Control Optimization, 36 (1998), pp. 1190-1206.

12. F. HETTLICH, Iterative regularization schemes in inverse scattering by periodic structures, Inverse Probl., 18 (2002), pp. 701-714.

13. B. KLEEMANN, Elektromagnetische Analyse von Oberflächengittern von IR bis XUV mittels einer parametrisierten Randelementmethode: Theorie, Vergleich und Anwendungen, Ph.D. Thesis, TU Ilmenau, Forschungsberichte aus den Ingenieurwissenschaften, Mensch & Buch Verlag, Berlin, 2003.

14. F. LANZARA, V. MAZ'YA, G. SCHMIDT, Numerical solution of the Lippmann-Schwinger equation by approximate approximations, Preprint no. 1412, The Erwin Schrödinger International Institute for Mathematical Physics, Vienna, Austria, 2003; to appear in: J. Fourier Anal. Appl.

15. V. MAZ'YA, G. SCHMIDT, ``Approximate approximations'' and the cubature of potentials, Rend. Mat. Acc. Lincei, Ser. 9, 6 (1995), pp. 161-184.

16. A. RATHSFELD, DIPOG-2.0, User guide, Direct problems for optical gratings over triangular grids, http://www.wias-berlin.de/software/DIPOG.

17. G. SCHMIDT, Handbuch DIPOG-1.4, http://www.wias-berlin.de/software/DIPOG.

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