Diffractive optical elements

The operation of most passive optical elements is based on effects that occur at an interface between two homogeneous media. In traditional refractive optics the interface does not vary on the scale of the optical wavelength. Diffractive optical elements employ the controlled use of light propagation on microstructered interfaces. The recent emergence of microlithographic fabrication technology paved the way for fabricating optical devices with complicated structural features within the length-scale of optical waves. Such diffractive elements have many technological advantages and can be designed to perform functions unattainable with traditional optical devices. They provide the potential to significantly reduce the size, weight, and cost of a variety of optical systems.

objective with diffractive lens (red) designed by Carl Zeiss AG
The only classic optical devices that employ diffraction rather than refraction and reflection are diffraction gratings, which possess periodically corrugated interfaces and have been indispensable spectroscopic instruments for a long time. Modern semiconductor technologies enable to realize diffraction gratings, which have applications in many domains in physics and astronomy; e.g., spectroscopy, acoustics, solid-state physics, X-ray instrumentations, laser technology, but also applications in optical communications, information processing, together with many everyday employments ranging from home electronics to photography equipment and safety features in credit cards and banknotes. It should be noted that the properties of diffraction gratings are fundamental for the understanding of the properties of all diffractive elements since an elemental area of those elements can always be considered as a small grating.

The practical application of diffractive optics technology has driven the need for mathematical models and numerical codes both to provide rigorous solutions of the Maxwell equations for complicated grating structures, thus predicting performance given the structure, and to carry out optimal design of new structures (cf. our program package WIAS-DiPoG). Moreover, the same numerical algorithms can be applied to the reconstruction of grating structures from their diffraction properties, i.e., to scatterometric measurements which will be described next.

Scatterometric measurement

The production of computer chips with increasingly small details requires a more and more accurate measurement technique. In particular, the development and evaluation of photolithographic manufacturing techniques necessitate a high precision measurement of special test geometries, which can be generated on the surface of lithographic masks and wafers. Besides well-established measurement techniques including AFM and SEM microscopy, the fast and non-destructive scatterometric methods are promising. In contrast to the optical microscopy, scatterometry does not produce a true image of the target. The geometry data is hidden in the measurement data and needs to be extracted by a mathematical optimization algorithm. In other words, an inverse problem for the reconstruction of the geometry data is to be solved.

Measurement equipment at PTB/BESSY

The test geometries to be checked are simple periodic line-space structures (resp. biperiodic block structures). They are illuminated by light rays, and the distribution of the scattered wave is measured (cf. the measurement equipment of our cooperation partners at the PTB operating in the extreme UV range). Using the energy distribution and/or phase shifts of the scattered light, the geometry is to be reconstructed by mathematical optimization algorithms.

For real line-space structures, the desired rectangular cross section of the multi-layered lines (cf. Figure below) is changed to a trapezoidal shape, and the design values of the layer thicknesses are modified. Thus the measured side-wall angles and the deviations of thicknesses characterize the quality of the manufacturing process. In spite of working beyond the limits of diffraction (resp. in spite of the high degree of ill-posedness of the inverse problem), these geometry parameters can be determined with high precision. However, the impact of inaccurate material constants, the edge roundings, and stochastic perturbations of lines and interfaces are to be taken into account.

Cross section of one line in line-space grating
Similar scatterometric problems are in the center of the ZIM project with the title "Development of in-situ measurement technique to control and determine the structures of plasma-etch prosesses". In collaboration with the Leibniz institute IHP (Frankfurt O.) and the company LayTec (Berlin) we have to measure structures during etch processes, i.e. the inverse problem is to be solved within seconds.



  • D. Hömberg, G. Hu, eds., Issue on the workshop ``Electromagnetics -- Modelling, Simulation, Control and Industrial Applications'', 8, no. 3 of Discrete Contin. Dyn. Syst. Ser. S, American Institute of Mathematical Sciences, Springfield, 2015, 259 pages, (Collection Published).

  • L.I. Goray, G. Schmidt, Chapter 12: Boundary Integral Equation Methods for Conical Diffraction and Short Waves, in: Gratings: Theory and Numerical Applications, Second Revisited Edition, E. Popov, ed., Université d'Aix-Marseille, Institut Fresnel, Marseille, 2014, pp. 12.1--12.86, (Chapter Published).

  • P. Deuflhard, M. Grötschel, D. Hömberg, U. Horst, J. Kramer, V. Mehrmann, K. Polthier, F. Schmidt, Ch. Schütte, M. Skutella, J. Sprekels, eds., MATHEON -- Mathematics for Key Technologies, 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, 453 pages, (Collection Published).

  Articles in Refereed Journals

  • C. Brée, D. Gailevičius, V. Purlys, G.G. Werner, K. Staliunas, A. Rathsfeld, G. Schmidt, M. Radziunas, Chirped photonic crystal for spatially filtered optical feedback to a broad-area laser, Journal of Optics, 20 (2018), pp. 095804/1--095804/7, DOI 10.1088/2040-8986/aada98 .
    We derive and analyze an efficient model for reinjection of spatially filtered optical feedback from an external resonator to a broad area, edge emitting semiconductor laser diode. Spatial filtering is achieved by a chirped photonic crystal, with variable periodicity along the optical axis and negligible resonant backscattering. The optimal chirp is obtained from a genetic algorithm, which yields solutions that are robust against perturbations. Extensive numerical simulations of the composite system with our optoelectronic solver indicate that spatially filtered reinjection enhances lower-order transversal optical modes in the laser diode and, consequently, improves the spatial beam quality.

  • T. Abbas, H. Ammari, G. Hu, A. Wahab, J.Ch. Ye, Elastic scattering coefficients and enhancement of nearly elastic cloaking, Journal of Elasticity, 128 (2017), pp. 203--243, DOI 10.1007/s10659-017-9624-7 .
    The concept of scattering coefficients has played a pivotal role in a broad range of inverse scattering and imaging problems in acoustic and electromagnetic media. In view of their promising applications, we introduce the notion of scattering coefficients of an elastic inclusion in this article. First, we define elastic scattering coefficients and substantiate that they naturally appear in the expansions of elastic scattered field and far field scattering amplitudes corresponding to a plane wave incidence. Then an algorithm is developed and analyzed for extracting the elastic scattering coefficients from multi-static response measurements of the scattered field. Moreover, the estimate of the maximal resolving order is provided in terms of the signal-to-noise ratio. The decay rate and symmetry of the elastic scattering coefficients are also discussed. Finally, we design scattering-coefficients-vanishing structures and elucidate their utility for enhancement of nearly elastic cloaking.

  • G. Hu, A. Kirsch, T. Yin, Factorization method in inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves, Inverse Problems and Imaging, 10 (2016), pp. 103--129.
    Consider a time-harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with homogeneous compressible inviscid fluid with a constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by the Lamé constants. This paper is concerned with direct (or forward) and inverse fluid-solid interaction (FSI) problems with unbounded bi-periodic interfaces between acoustic and elastic waves. We present a variational approach to the forward interaction problem with Lipschitz interfaces. Existence of quasi-periodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. Concerning the inverse problem, we show that the factorization method by Kirsch (1998) is applicable to the FSI problem in periodic structures. A computational criterion and a uniqueness result are justified for precisely characterizing the elastic body by utilizing the scattered acoustic near field measured in the fluid.

  • Y. Guo, D. Hömberg, G. Hu, J. Li, H. Liu, A time domain sampling method for inverse acoustic scattering problems, Journal of Computational Physics, 314 (2016), pp. 647--660.
    This work concerns the inverse scattering problems of imaging unknown/inaccessible scatterers by transient acoustic near-field measurements. Based on the analysis of the migration method, we propose efficient and effective sampling schemes for imaging small and extended scatterers from knowledge of time-dependent scattered data due to incident impulsive point sources. Though the inverse scattering problems are known to be nonlinear and ill-posed, the proposed imaging algorithms are totally “direct” involving only integral calculations on the measurement surface. Theoretical justifications are presented and numerical experiments are conducted to demonstrate the effectiveness and robustness of our methods. In particular, the proposed static imaging functionals enhance the performance of the total focusing method (TFM) and the dynamic imaging functionals show analogous behavior to the time reversal inversion but without solving time-dependent wave equations.

  • G. Hu, A. Rathsfeld, T. Yin, Finite element method to fluid-solid interaction problems with unbounded periodic interfaces, Numerical Methods for Partial Differential Equations. An International Journal, 32 (2016), pp. 5--35.
    Consider a time-harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid fluid of constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by its Lamé constants. This paper is concerned with a variational approach to the fluid-solid interaction problems with unbounded biperiodic Lipschitz interfaces between the domains of the acoustic and elastic waves. The existence of quasi-periodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. A finite element scheme coupled with Dirichlet-to-Neumann mappings is proposed. The Dirichlet-to-Neumann mappings are approximated by truncated Rayleigh series expansions, and, finally, numerical tests in 2D are performed.

  • G. Hu, H. Liu, Nearly cloaking the elastic wave fields, Journal de Mathématiques Pures et Appliquées, 104 (2015), pp. 1045--1074.
    In this work, we develop a general mathematical framework on regularized approximate cloaking of elastic waves governed by the Lamé system via the approach of transformation elastodynamics. Our study is rather comprehensive. We first provide a rigorous justification of the transformation elastodynamics. Based on the blow-up-a-point construction, elastic material tensors for a perfect cloak are derived and shown to possess singularities. In order to avoid the singular structure, we propose to regularize the blow-up-a-point construction to be the blow-up-a-small-region construction. However, it is shown that without incorporating a suitable lossy layer, the regularized construction would fail due to resonant inclusions. In order to defeat the failure of the lossless construction, a properly designed lossy layer is introduced into the regularized cloaking construction . We derive sharp asymptotic estimates in assessing the cloaking performance. The proposed cloaking scheme is capable of nearly cloaking an arbitrary content with a high accuracy.

  • H. Gross, S. Heidenreich, M.-A. Henn, M. Bär, A. Rathsfeld, Modeling aspects to improve the solution of the inverse problem in scatterometry, Discrete and Continuous Dynamical Systems -- Series S, 8 (2015), pp. 497--519.

  • G. Hu, X. Liu, F. Qu, B. Zhang, Variational approach to rough surface scattering problems with Neumann and generalized impedance boundary conditions, Communications in Mathematical Sciences, 13 (2015), pp. 511--537.

  • G. Hu, A. Rathsfeld, Scattering of time-harmonic electromagnetic plane waves by perfectly conducting diffraction gratings, IMA Journal of Applied Mathematics, 80 (2015), pp. 508--532.
    Consider scattering of time-harmonic electromagnetic plane waves by a doubly periodic surface in $R^3$. The medium above the surface is supposed to be homogeneous and isotropic with a constant dielectric coefficient, while below is a perfectly conducting material. This paper is concerned with the existence of quasiperiodic solutions for any frequency of incidence. Based on an equivalent variational formulation established by the mortar technique of Nitsche, we verify the existence of solutions for a broad class of incident waves including plane waves, under the assumption that the grating profile is a Lipschitz biperiodic surface. Our solvability result covers the resonance case where a Rayleigh frequency is allowed. Non-uniqueness examples are also presented in the resonance case and the TE or TM polarization case for classical gratings.

  • TH. Arnold, A. Rathsfeld, Reflection of plane waves by rough surfaces in the sense of Born approximation, Mathematical Methods in the Applied Sciences, 37 (2014), pp. 2091--2111.
    The topic of the present paper is the reflection of electromagnetic plane waves by rough surfaces, i.e., by smooth and bounded perturbations of planar faces. Moreover, the contrast between the cover material and the substrate beneath the rough surface is supposed to be low. In this case, a modification of Stearns' formula based on Born approximation and Fourier techniques is derived for a special class of surfaces. This class contains the graphs of functions if the interface function is a radially modulated almost periodic function. For the Born formula to converge, a sufficient and almost necessary condition is given. A further technical condition is defined, which guarantees the existence of the corresponding far field of the Born approximation. This far field contains plane waves, far-field terms like those for bounded scatterers, and, additionally, a new type of terms. The derived formulas can be used for the fast numerical computations of far fields and for the statistics of random rough surfaces.

  • G. Hu, A. Rathsfeld, Convergence analysis of the FEM coupled with Fourier-mode expansion for the electromagnetic scattering by biperiodic structures, Electronic Transactions on Numerical Analysis, 41 (2014), pp. 350--375.

  • G. Schmidt, Conical diffraction by multilayer gratings: A recursive integral equations approach, Applications of Mathematics, 58 (2013), pp. 279--307.
    In this paper we consider an integral equation algorithm to study the scattering of plane waves by multilayer diffraction gratings under oblique incidence. The scattering problem is described by a system of Helmholtz equations with piecewise constant coefficients in $R^2$ coupled by special transmission conditions at the interfaces between different layers. Boundary integral methods lead to a system of singular integral equations, containing at least two equations for each interface. To deal with an arbitrary number of material layers we present the extension of a recursive procedure developed by Maystre for normal incidence, which transforms the problem to a sequence of equations with $2 times 2$ operator matrices on each interface. Necessary and sufficient conditions for the applicability of the algorithm are derived.

  • G. Hu, F. Qu, B. Zhang, A linear sampling method for inverse problems of diffraction gratings of mixed type, Mathematical Methods in the Applied Sciences, 35 (2012), pp. 1047--1066.

  • G. Hu, Inverse wave scattering by unbounded obstacles: Uniqueness for the two-dimensional Helmholtz equation, Applicable Analysis. An International Journal, 91 (2012), pp. 703--717.
    In this paper we present some uniqueness results on inverse wave scattering by unbounded obstacles for the two-dimensional Helmholtz equation. We prove that an impenetrable one-dimensional rough surface can be uniquely determined by the values of the scattered field taken on a line segment above the surface that correspond to the incident waves generated by a countable number of point sources. For penetrable rough layers in a piecewise constant medium, the refractive indices together with the rough interfaces (on which the TM transmission conditions are imposed) can be uniquely identified using the same measurements and the same incident point source waves. Moreover, a Dirichlet polygonal rough surface can be uniquely determined by a single incident point source wave provided a certain condition is imposed on it.

  • L. Goray, G. Schmidt, Analysis of two-dimensional photonic band gaps of any rod shape and conductivity using a conical-integral-equation method, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 85 (2012), pp. 036701/1--036701/12.
    The conical boundary integral equation method has been proposed to calculate the sensitive optical response of 2D photonic band gaps (PBGs), including dielectric, absorbing, and high-conductive rods of various shapes working in any wavelength range. It is possible to determine the diffracted field by computing the scattering matrices separately for any grating boundary profile. The computation of the matrices is based on the solution of a 2 x 2 system of singular integral equations at each interface between two different materials. The advantage of our integral formulation is that the discretization of the integral equations system and the factorization of the discrete matrices, which takes the major computing time, are carried out only once for a boundary. It turned out that a small number of collocation points per boundary combined with a high convergence rate can provide adequate description of the dependence on diffracted energy of very different PBGs illuminated at arbitrary incident and polarization angles. The numerical results presented describe the significant impact of rod shape on diffraction in PBGs supporting polariton-plasmon excitation, particularly in the vicinity of resonances and at high filling ratios. The diffracted energy response calculated vs. array cell geometry parameters was found to vary from a few percent up to a few hundred percent. The influence of other types of anomalies (i.e. waveguide anomalies, cavity modes, Fabry-Perot and Bragg resonances, Rayleigh orders, etc), conductivity, and polarization states on the optical response has been demonstrated.

  • H. Gross, M.-A. Henn, S. Heidenreich, A. Rathsfeld, M. Bär, Modeling of line roughness and its impact on the diffraction intensities and the reconstructed critical dimensions in scatterometry, Applied Optics, 51 (2012), pp. 7384--7394.
    We investigate the impact of line edge and line width roughness (LER, LWR) on the measured diffraction intensities in angular resolved extreme ultraviolet (EUV) scatterometry for a periodic line-space structure designed for EUV lithography. LER and LWR with typical amplitudes of a few nanometers were previously neglected in the course of the profile reconstruction. The 2D rigorous numerical simulations of the diffraction process for periodic structures are carried out with the finite element method (FEM) providing a numerical solution of the two-dimensional Helmholtz equation. To model roughness, multiple calculations are performed for domains with large periods, containing many pairs of line and space with stochastically chosen line and space widths. A systematic decrease of the mean efficiencies for higher diffraction orders along with increasing variances is observed and established for different degrees of roughness. In particular, we obtain simple analytical expressions for the bias in the mean efficiencies and the additional uncertainty contribution stemming from the presence of LER and/or LWR. As a consequence this bias can easily be included into the reconstruction model to provide accurate values for the evaluated profile parameters. We resolve the sensitivity of the reconstruction from this bias by using the LER/LWR perturbed efficiency datasets for multiple reconstructions. If the scattering efficiencies are bias-corrected, significant improvements are found in the reconstructed bottom and top widths toward the nominal values.

  • M.-A. Henn, S. Heidenreich, H. Gross, A. Rathsfeld, F. Scholze, M. Bär, Improved grating reconstruction by determination of line roughness in extreme ultraviolet scatterometry, Optics Letters, 37 (2012), pp. 5229--5231.

  • C. Pfüller, M. Ramsteiner, O. Brandt, F. Grosse, A. Rathsfeld, G. Schmidt, L. Geelhaar, H. Riechert, Raman spectroscopy as a probe for the coupling of light into ensembles of sub-wavelength-sized nanowires, Applied Physics Letters, 101 (2012), pp. 083104/1--083104/4.

  • G. Hu, B. Zhang, The linear sampling method for the inverse electromagnetic scattering by a partially coated bi-periodic structure, Mathematical Methods in the Applied Sciences, 34 (2011), pp. 509--519.

  • F. Lanzara, V. Maz'ya, G. Schmidt, On the fast computation of high dimensional volume potentials, Mathematics of Computation, 80 (2011), pp. 887--904.

  • F. Lanzara, V.G. Maz'ya, G. Schmidt, Accurate cubature of volume potentials over high-dimensional half-spaces, Journal of Mathematical Sciences (New York), 173 (2011), pp. 683--700.

  • J. Elschner, G. Hu, Inverse scattering of elastic waves by periodic structures: Uniqueness under the third or fourth kind boundary conditions, Methods and Applications of Analysis, 18 (2011), pp. 215--244.
    The inverse scattering of a time-harmonic elastic wave by a two-dimensional periodic structure in $R^2$ is investigated. The grating profile is assumed to be a graph given by a piecewise linear function on which the third or fourth kind boundary conditions are satisfied. Via an equivalent variational formulation, existence of quasi-periodic solutions for general Lipschitz grating profiles is proved by applying the Fredholm alternative. However, uniqueness of solution to the direct problem does not hold in general. For the inverse problem, we determine and classify all the unidentifiable grating profiles corresponding to a given incident elastic field, relying on the reflection principle for the Navier equation and the rotational invariance of propagating directions of the total field. Moreover, global uniqueness for the inverse problem is established with a minimal number of incident pressure or shear waves, including the resonance case where a Rayleigh frequency is allowed. The gratings that are unidentifiable by one incident elastic wave provide non-uniqueness examples for appropriately chosen wave number and incident angles.

  • J. Elschner, G. Hu, Uniqueness in inverse scattering of elastic waves by three-dimensional polyhedral diffraction gratings, Journal of Inverse and Ill-Posed Problems, 19 (2011), pp. 717--768.
    We consider the inverse elastic scattering problem of determining a three-dimensional diffraction grating profile from scattered waves measured above the structure. In general, a grating profile cannot be uniquely determined by a single incoming plane wave. We completely characterize and classify the bi-periodic polyhedral structures under the boundary conditions of the third and fourth kinds that cannot be uniquely recovered by only one incident plane wave. Thus we have global uniqueness for a polyhedral grating profile by one incident elastic plane wave if and only if the profile belongs to neither of the unidentifiable classes, which can be explicitly described depending on the incident field and the type of boundary conditions. Our approach is based on the reflection principle for the Navier equation and the reflectional and rotational invariance of the total field.

  • J. Elschner, G. Hu, Uniqueness in inverse transmission scattering problems for multilayered obstacles, Inverse Problems and Imaging, 5 (2011), pp. 793--813.
    Assume a time-harmonic electromagnetic wave is scattered by an infinitely long cylindrical conductor surrounded by an unknown piecewise homogenous medium remaining invariant along the cylinder axis. We prove that, in TM mode, the far field patterns for all observation directions at a fixed frequency uniquely determine the unknown surrounding medium as well as the shape of the cylindrical conductor. A similar uniqueness result is obtained for the scattering by multilayered penetrable periodic structures in a piecewise homogenous medium. The periodic interfaces and refractive indices can be uniquely identified from the near field data measured only above (or below) the structure for all quasi-periodic incident waves with a fixed phase-shift. The proofs are based on the singularity of the Green function to a two dimensional elliptic equation with piecewise constant leading coefficients.

  • G. Schmidt, B.H. Kleemann, Integral equation methods from grating theory to photonics: An overview and new approaches for conical diffraction, Journal of Modern Optics, 58 (2011), pp. 407--423.

  • G. Schmidt, Integral equations for conical diffraction by coated gratings, Journal of Integral Equations and Applications, 23 (2011), pp. 71--112.
    The paper is devoted to integral formulations for the scattering of plane waves by diffraction gratings under oblique incidence. For the case of coated gratings Maxwell's equations can be reduced to a system of four singular integral equations on the piecewise smooth interfaces between different materials. We study analytic properties of the integral operators for periodic diffraction problems and obtain existence and uniqueness results for solutions of the systems corresponding to electromagnetic fields with locally finite energy.

  • G. Hu, F. Qu, B. Zhang, Direct and inverse problems for electromagnetic scattering by a doubly periodic structure with a partially coated dielectric, Mathematical Methods in the Applied Sciences, 33 (2010), pp. 147--156.

  • S. Chandler-Wilde, J. Elschner, Variational approach in weighted Sobolev spaces to scattering by unbounded rough surface, SIAM Journal on Mathematical Analysis, 42 (2010), pp. 2554--2580.

  • L.I. Goray, G. Schmidt, Solving conical diffraction grating problems with integral equations, Journal of the Optical Society of America A. Optics, Image Science, and Vision, 27 (2010), pp. 585--597.
    Off-plane scattering of time-harmonic plane waves by a diffraction grating with arbitrary conductivity and general border profile is considered in a rigorous electromagnetic formulation. The integral equations for conical diffraction were obtained using the boundary integrals of the single and double layer potentials including the tangential derivative of single layer potentials interpreted as singular integrals. We derive an important formula for the calculation of the absorption in conical diffraction. Some rules which are expedient for the numerical implementation of the theory are presented. The efficiencies and polarization angles compared with those obtained by Lifeng Li for transmission and reflection gratings are in a good agreement. The code developed and tested is found to be accurate and efficient for solving off-plane diffraction problems including high-conductive surfaces, borders with edges, real border profiles, and gratings working at short wavelengths.

  • H. Gross, J. Richter, A. Rathsfeld, M. Bär, Investigations on a robust profile model for the reconstruction of 2D periodic absorber lines in scatterometry, Journal of the European Optical Society - Rapid Publications, 5 (2010), pp. 10053/1--10053/7.

  • J. Elschner, G. Hu, Global uniqueness in determining polygonal periodic structures with a minimal number of incident plane waves, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 26 (2010), pp. 115002/1--115002/23.
    In this paper, we investigate the inverse problem of recovering a two-dimensional perfectly reflecting diffraction grating from the scattered waves measured above the structure. Inspired by a novel idea developed by Bao, Zhang and Zou [to appear in Trans. Amer. Math. Soc.], we present a complete characterization of the global uniqueness in determining polygonal periodic structures using a minimal number of incident plane waves. The idea in this paper combines the reflection principle for the Helmholtz equation and the dihedral group theory. We characterize all periodic polygonal structures that cannot be identified by one incident plane wave, including the resonance case where a Rayleigh frequency is allowed. Furthermore, we show that those unidentifiable gratings provide non-uniqueness examples for appropriately chosen wave number and incident angles. We also indicate and fix a gap in the proof of the main theorem of Elschner and Yamamoto [Z. Anal. Anwend., 26 (2007), 165-177], and generalize the uniqueness results of that paper.

  • H. Gross, A. Rathsfeld, F. Scholze, M. Bär, Profile reconstruction in extreme ultraviolet (EUV) scatterometry: Modeling and uncertainty estimates, Measurement Science and Technology, 20 (2009), pp. 105102/1--105102/11.
    Scatterometry as a non-imaging indirect optical method in wafer metrology is also relevant to lithography masks designed for Extreme Ultraviolet Lithography, where light with wavelengths in the range of 13 nm is applied. The solution of the inverse problem, i.e. the determination of periodic surface structures regarding critical dimensions (CD) and other profile properties from light diffraction patterns, is incomplete without knowledge of the uncertainties associated with the reconstructed parameters. With decreasing feature sizes of lithography masks, increasing demands on metrology techniques and their uncertainties arise. The numerical simulation of the diffraction process for periodic 2D structures can be realized by the finite element solution of the two-dimensional Helmholtz equation. For typical EUV masks the ratio period over wave length is so large, that a generalized finite element method has to be used to ensure reliable results with reasonable computational costs. The inverse problem can be formulated as a non-linear operator equation in Euclidean spaces. The operator maps the sought mask parameters to the efficiencies of diffracted plane wave modes. We employ a Gauß-Newton type iterative method to solve this operator equation and end up minimizing the deviation of the measured efficiency or phase shift values from the calculated ones. We apply our reconstruction algorithm for the measurement of a typical EUV mask composed of TaN absorber lines of about 80 nm height, a period of 420 nm resp. 720 nm, and with an underlying MoSi-multilayer stack of 300 nm thickness. Clearly, the uncertainties of the reconstructed geometric parameters essentially depend on the uncertainties of the input data and can be estimated by various methods. We apply a Monte Carlo procedure and an approximative covariance method to evaluate the reconstruction algorithm. Finally, we analyze the influence of uncertainties in the widths of the multilayer stack by the Monte Carlo method.

  • H. Gross, A. Rathsfeld, Sensitivity analysis for indirect measurement in scatterometry and the reconstruction of periodic grating structures, Waves in Random and Complex Media. Propagation, Scattering and Imaging, 18 (2008), pp. 129--149.

  • J. Elschner, M. Yamamoto, Uniqueness in determining polygonal periodic structures, Zeitschrift fur Analysis und ihre Anwendungen. Journal for Analysis and its Applications, 26 (2007), pp. 165--177.

  • H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, Mathematical modelling of indirect measurements in scatterometry, Measurement, 39 (2006), pp. 782--794.

  • A. Rathsfeld, G. Schmidt, B.H. Kleemann, On a fast integral equation method for diffraction gratings, Communications in Computational Physics, 1 (2006), pp. 984-1009.

  • G. Bruckner, J. Elschner, The numerical solution of an inverse periodic transmission problem, Mathematical Methods in the Applied Sciences, 28 (2005), pp. 757--778.

  • J. Elschner, M. Yamamoto, Uniqueness results for an inverse periodic transmission problem, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 20 (2004), pp. 1841--1852.

  • G. Bao, K. Huang, G. Schmidt, Optimal design of nonlinear diffraction gratings, Journal of Computational Physics, 184 (2003), pp. 106--121.

  • G. Bruckner, J. Elschner, A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 19 (2003), pp. 315--329.

  • J. Elschner, G.C. Hsiao, A. Rathsfeld, Grating profile reconstruction based on finite elements and optimization techniques, SIAM Journal on Applied Mathematics, 64 (2003), pp. 525--545.

  • J. Elschner, G. Schmidt, M. Yamamoto, An inverse problem in periodic diffractive optics: Global uniqueness with a single wavenumber, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 19 (2003), pp. 779--787.

  • J. Elschner, G. Schmidt, M. Yamamoto, Global uniqueness in determining rectangular periodic structures by scattering data with a single wave number, Journal of Inverse and Ill-Posed Problems, 11 (2003), pp. 235--244.

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

  Contributions to Collected Editions

  • C. Brée, V. Raab, D. Gailevičius, V. Purlys, J. Montiel, G.G. Werner, K. Staliunas, A. Rathsfeld, U. Bandelow, M. Radziunas, Genetically optimized photonic crystal for spatial filtering of reinjection into broad-area diode lasers, in: 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, OSA Technical Digest, IEEE, Piscataway, 2019, DOI 10.1109/CLEO-EQEC.2019.8871622 .
    Modern high-power broad-area semiconductor laser diodes (BASLDs) deliver optical output powers of several ten Watts at high electro-optical conversion efficiencies, which makes them highly relevant for numerous industrial, medical and scientific applications. However, lateral multimode behavior in BASLDs due to thermal lensing turns out highly detrimental, as it results in poor focusability and decreased laser beam brightnesss. Approaches to overcome this issue include improved epitaxial layer design, the optimization of evanescent spatial filtering by tailoring the emitter geometry and facet reflectivity, or Fourier spatially filtered reinjection from an external resonator [1].

  • G. Schmidt, Boundary integral methods for periodic scattering problems, in: Around the Research of Vladimir Maz'ya II. Partial Differential Equations, A. Laptev, ed., 12 of International Mathematical Series, Springer Science+Business Media, New York [et al.], 2010, pp. 337--363.

  • H. Gross, F. Scholze, A. Rathsfeld, M. Bär, Evaluation of measurement uncertainties in EUV scatterometry, in: Modeling Aspects in Optical Metrology II, H. Bosse, B. Bodermann, R.M. Silver, eds., 7390 of Proceedings of SPIE, SPIE, 2009, pp. 7390OT/1--7390OT/11.

  • H. Gross, A. Rathsfeld, M. Bär, Modelling and uncertainty estimates for numerically reconstructed profiles in scatterometry, in: Advanced Mathematical and Computational Tools in Metrology and Testing VIII, F. Pavese, M. Bär, A.B. Forbes, J.M. Linares, C. Perruchet, N.F. Zhang, eds., 78 of Series on Advances in Mathematics for Applied Sciences, World Scientific, Singapore, 2009, pp. 142--147.

  • M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, H. Gross, On numerical reconstructions of lithographic masks in DUV scatterometry, in: Modeling Aspects in Optical Metrology II, H. Bosse, B. Bodermann, R.M. Silver, eds., 7390 of Proceedings of SPIE, SPIE, 2009, pp. 7390OQ/1--7390OQ/11.

  • H. Gross, R. Model, A. Rathsfeld, F. Scholze, M. Wurm, B. Bodermann, M. Bär, Modellbildung, Bestimmung der Messunsicherheit und Validierung für diskrete inverse Probleme am Beispiel der Scatterometrie, in: Sensoren und Messsysteme, 14. Fachtagung Ludwigsburg, 11./12. März 2008, 2011 of VDI-Berichte, VDI, 2008, pp. 337--346.

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