Fast online inverse scattering with reduced basis method (RBM) for a 3D phase grating with specific line roughness

Transcription

1 Fast online inverse scattering with reduced basis method (RBM) for a 3D phase grating with specific line roughness Bernd H. Kleemann a, Julian Kurz b, Jochen Hetzler c, Jan Pomplun de, Sven Burger de, Lin Zschiedrich d, and Frank Schmidt de a Carl Zeiss AG, D Oberkochen, Germany; b Karlsruhe Institute of Technology (KIT), D Karlsruhe, Germany; c Carl Zeiss SMT AG, D Oberkochen, Germany; d JCMwave GmbH, D Berlin, Germany; e Zuse Institute Berlin (ZIB), D Berlin, Germany ABSTRACT Finite element methods (FEM) for the rigorous electromagnetic solution of Maxwell s equations are known to be very accurate. They possess a high convergence rate for the determination of near field and far field quantities of scattering and diffraction processes of light with structures having feature sizes in the range of the light wavelength. We are using FEM software for 3D scatterometric diffraction calculations allowing the application of a brilliant and extremely fast solution method: the reduced basis method (RBM). The RBM constructs a reduced model of the scattering problem from precalculated snapshot solutions, guided self-adaptively by an error estimator. Using RBM, we achieve an efficiency accuracy of about 4 compared to the direct problem with only 35 precalculated snapshots being the reduced basis dimension. This speeds up the calculation of diffraction amplitudes by a factor of about 00 compared to the conventional solution of Maxwell s equations by FEM. This allows us to reconstruct the three geometrical parameters of our phase grating from measured scattering data in a 3D parameter manifold online in a minute having the full FEM accuracy available. Additionally, also a sensitivity analysis or the choice of robust measuring strategies, for example, can be done online in a few minutes. Keywords: Inverse scattering, 3D phase grating, line roughness, FEM Maxwell solver, reduced basis method (RBM) 1. INTRODUCTION Optical measurement for the characterization of micro- and nanostructures 1 especially model based dimensional metrology 2 is an increasing field. Meanwhile, inverse scatterometry is an established method as a non-destructive measuring tool for parameter identification of known lithographic structures. It becomes more important with shrinking feature sizes along the lithography road map. 3,4 Finite element methods (FEM) for the rigorous solution of Maxwell s equations are known to be very accurate. They possess a high convergence rate for the determination of near field and far field quantities of scattering and diffraction processes of light with structures having feature sizes in the dimension of the light wavelength. Therefore, FEM are used for inverse scattering of 1D periodic gratings and lithographic masks. 3,4 However, FEM are actually still rarely used as numerical method for the solution of Maxwell s equations in 3D scatterometric applications. 5 This may be due to the assumption that 3D FEM is supposed to be too slow for that purpose. In this paper we show that this assumption is totally wrong as until now only FEM allow the application of a brilliant and extremely fast solution method: the so-called reduced basis method (RBM) preserving all good properties of FEM like stability, excellent accuracy, and high convergence rate. The RBM package consists of mathematically founded methods for (1) a FEM for the solution of a differential equation, (2) an approach for the construction of a reduced basis, and (3) a corresponding error estimator. All these methods form a unit and are based on the mathematics of finite elements. Send correspondence to B.H.K.: [email protected], J.P.: [email protected]

2 Figure 1. 2D-periodic phase grating with specific line roughness and incidence angle θ with the incidence plane perpendicular to the grating lines. Periods in x- and y-directions p x, p y are p x = 20λ, p y = λ and the height of the lines is h = 1λ. The sinusoidal boundaries have different periods on both sides of 2λ and 5λ with amplitudes r = 1λ. Since the popularization to a broader audience by MIT Ford Prof. Anthony Patera, the reduced basis method allows the FEM solution of large physical problems on a simple smartphone, 6 originally intended only for supercomputers. One dimensional scattering and diffraction methods such as RCWA, C-method, Differential, Boundary Integral-, Rayleigh-Fourier-method, and others can be used for grating like structures. Really 3D structures and 2D-periodic structures often need adequate 3D methods such as FDTD or FEM leading to significantly increased computation times with FDTD having a too weak accuracy and a too slow convergence. The reduced basis method 7 with a posteriori error estimation and together with FEM provides a fast and accurate direct and inverse scattering parameter identification method fully based on Maxwell s equations. In the following, we start with a detailed description of the scattering structure in Sect. 2, describe some details of the theory and discuss the results of the direct scattering problem by FEM in Sect. 3. A brief instruction to the RBM is given in Sect. 4 showing the way for achieving a fast and reliable RBM for our 3D diffractive structure. The really convincing results and the potential of the method are discussed in Sect INVERSE SCATTERING STRUCTURE The problem of inverse scattering of a 3D phase grating is studied. It is well known that the difficulty of electromagnetic solutions of Maxwell s equations for scattering and diffraction problems mainly depends on the ratio of scatterer s dimension to incident wavelength. Hence, we can define the 2D periodic phase diffraction grating in dimensions of the wavelength (cf. Fig. 1). The periods in x- and y-direction p x,p y are p x = 20λ and p y = λ fixing the xy-plane as the plane parallel to the geometry with the height of the lines h = 1λ perpendicular to this plane. Together with a filling factor of the lines of 0.5 in x-direction the FEM triangulation volume with 200λ 3 is quite large for FEM calculations of Maxwell s equations and hence, tests the capability and suitability of the FEM approach together with the applicability of the reduced basis method for this specific inverse scattering parameter identification application. The unit cell is periodically continued in x- and y- direction forming lines with sinusoidally modulated boundaries with differing periods 2λ and 5λ, respectively, having the same amplitude r = 1λ. Then the filling factor has to be defined as the ratio of the area of the grating line to the area of the unit cell defining the line width w. The grating material and the substrate are assumed to have a refractive index of n = 1.5 while the superstrate material is air. The light as a plane wave is incident from above as defined in Fig. 1 by the inclination angle θ with sin(θ) = 0.1. For reasons of clarity the incidence plane is chosen to be perpendicular to the grating lines and hence, defines the classical case of incidence so that

3 the two polarization cases TE- and TM-polarization are considered. Clearly, this is not a restriction and also the conical diffraction case can be treated and arbitrary polarization directions can be composed from the two fundamental cases mentioned above. The inverse scattering task consists in determining the three grating parameters line width w, line height h, and roughness amplitude r of a 3D manifold being the parameter domain with w [9λ,11λ], h [0.9λ,1.1λ], and r [0.9λ, 1.1λ] from efficiency measurements of three diffraction orders in transmission for TE- and TMpolarization. Hence, we have six efficiency values for the determination of three structure parameters which should work well if parameter changes cause polarization dependent efficiency modifications. Instead of real measurement data we are using exactly calculated efficiencies by solving Maxwell s equations for the direct diffraction problem with FEM solver JCMsuite. For demonstrating the practicability of the approach it is sufficient to provide these efficiencies with a certain random noise level. 3. DIRECT FEM CALCULATIONS: THEORY AND PROPERTIES The direct problem consists in calculating the reflection and transmission efficiencies (= relative intensities in the far field compared to the incident one) for TE- and TM-polarized plane waves incident with angle θ on the 3D phase grating of Fig. 1. It is determined by Fourier Transformation from 2D near-field data of the electromagnetic field in the vicinity of the grating. Due to the relations of wavelength λ, main period in direction perpendicular to the grating lines p x = 20λ, incidence direction sinθ = 0.1, and refractive indices of substrate n substrate = 1.5 and superstrate n air = 1.0, the diffraction of the plane wave on the structure of Fig. 1 generates 41 reflection and 61 transmission orders. We are interested in only three transmission orders: 1, 0, 1. Nevertheless, if the reconstruction would fail with these three orders, we have the opportunity to add some more diffraction orders or to modify the incidence conditions to improve the accuracy of the reconstruction process. 3.1 Theory of direct FEM Calculation Diffraction at the given grating in Fig. 1 generates many diffraction orders as we have seen above. This in turn implies high oscillations of the electromagnetic field. Additionally, the sinusoidal boundaries of the grating introduce high field intensities as we can observe in Fig. 2. Figure 2. Intensity plot of the electric near-field in the xy-plane at z = 0.5λ of the 2D-grating with specific line roughness. Red spots represent higher intensity. These circumstances require a numerical method that can sufficiently resolve the sinusoidal fine structures and the oscillating electromagnetic field to produce accurate results. We are using the most elegant, universal, accurate, and fastest numerical method for solving Maxwell s equations and for the calculation of the electromagnetic field generated by scattering of light at the structure of Fig. 1: a finite element method (FEM). None of the other methods known as RCWA (Rigorous Coupled Wave Analysis), FDTD (Finite Difference Time Domain), or others would adequately be able to deliver similar accurate results and, more important, do not allow the application of the reduced basis method for fast solution of the direct and/or inverse scattering problem. The commercially available FE package JCMsuite numerically solves Maxwell s equations in our case. In the following, let us consider a free geometrical parameter set denoted by a ˆp-tupel ν in a bounded parameter space TE = transverse electric or s-polarized; TM = transverse magnetic or p-polarized.

4 D Rˆp. This parameter set describes the geometrical properties of our grating structure given in Fig. 1 by width w, height h, and amplitude r. Then ν = (w,h,r) D and the dimension of the parameter space is ˆp = 3 in our example. For every possible configuration ν D and a given incoming field the resulting electromagnetic field can accurately be calculated employing the FEM. In general terms the electromagnetic scattering problem can hence be stated in a so-called variational formulation: Find u N X N such that: a(u N,v;ν) = f(v), v X N. (1) The geometrical parameter dependent sesquilinear form a(, ; ν) denotes Maxwell s equations while the linear functional f( ) represents the incoming field, e.g. a plane wave. The electromagnetic field solution u N (ν) is numerically given in the finite element space X N. The solution of equation (1) on X N leads to a large sparse matrix equation of dimension N which is typically 5 up to several millions. This corresponds to the number of unknowns determined by the applied mesh and the polynomial order of the basis functions defining the finite elements. Hence, the computation time maybe large, especially when it is necessary to repeat similar calculations with different parameter sets ν D very often as it is the case in inverse scattering or optimization applications, for example. Interestingly, our final output of interest, e.g. the calculated diffraction efficiencies of the 3D grating, are nothing else but a linear functional called output functional 8 l o (X N ) applied to the discrete electromagnetic field u N : l o : X N C; u N (ν) l o ( u N (ν) ) = s N (ν). (2) This equation describes the so-called input-output relationship of the scattering problem, for example. It fully abstracts from the underlying physical context and mathematically describes what one is interested in when solving partial differential equations, in our case, Maxwell s equations. Function s N (ν) calculates the diffraction efficiencies for arbitrary parameters (w,h,r) but fully based on the accurate numerical solution of the electromagnetic field u N for these parameters. In Sect. 4 we will see that the calculation of function s N (ν) can speeded up significantly using RBM. The discretized version of Maxwell s equations (1) is referred to as truth approximation in the context of the reduced basis approach below. For inverse problems in scatterometry, design, and optimization applications the truth approximation has to be solved many times for different parameter values ν. In the considered case of the 3D parameter domain (w,h,r), the usage of only sampling values in each direction would result in the calculation of 3 direct FEM solutions of dimension N. And although FEM is quite fast, it would take a long time to produce all these solutions. Hence, a fast solution of inverse scattering and other inverse problems is prohibited. Therefore, the reduced basis method is used to overcome this serious problem as described in Sect Properties and accuracy of direct FEM Calculation At first the FEM grid triangulation has to be noticed. Usually, a tetrahedral grid is used for triangulation of the structures. However, the given scattering structure in Fig. 1 needs variation of the height in z-direction, among others. This variation is cylindrical in z-direction and hence, the grid should be cylindrical (also called tensorial) in z-direction, too. A prism grid fulfills this condition as one can see in Fig. 4. Then, the height of the given structure can easily be changed without changing the grid s nodes (i.e. the grid topology) as is visualized in the left part of Fig. 3. Figure 3. FEM prism grid. Left: Variation of line height h. Right: Variation of line roughness amplitude r.

5 Figure 4. Triangulation of half of the symmetric unit cell of the considered 3D grating structure by JCMgeo using a prism grid. Additionally, the sinusoidal profile of the line roughness has to be approximated accordingly to be able to reconstruct the amplitude from efficiency measurement. Figure Fig. 4 shows the approximation of a sine subperiod with 13 nodes corresponding to 12 polygons. Seven nodes have been used for the calculation of the results presented later and the approximation is yet quite good as can be seen from Fig. 3. The amplitude change of the line roughness without changing the grid s nodes is now also possible as visualized in the right part of Fig. 3. The RBM can now be applied to all grid types used for FEM, e.g. tetrahedral as well as prism grids. Next the accuracy and the convergence rate of the direct electromagnetic FEM field solver has been analysed to verify the applicability of the FEM to the specified structure. Fig. 4 shows the triangulation of the unit cell using a prism grid generated by JCMgeo. Due to a symmetry of the grating structure only half of the unit cell has to be discretized. The direct problem was simulated with different polynomial degrees p = of the ansatz functions and different triangulation refinement levels (and mesh cells) determining the computational cost. The dependency of the number of unknowns on the number of mesh cells is linear and the number of unknowns is proportional to pd in d space dimensions. The computational cost in turn depends almost linearly on the number of unknowns, which is due to sparsity of the finite element matrix and usage of highly efficient sparse matrix solvers. This is however only the case as long as all computations can be fully done in a sufficiently large random access memory. The most important figures in this section, Fig. 5, Fig. 6 show the already mentioned relative error of the FEM simulation results with respect to the number of unknowns N and to computation time (which is proportional to the number of unknowns). Obviously, the relative errors converge with increasing numbers of unknowns (corresponding to linearly increasing computation time) in the graphs of all considered cases. All calculations Relative error Relative error 4 n = 1 n=0 n= n = 1 n=0 n=1 6 2 Computation time [sec] Computation time [sec] 4 Figure 5. Convergence of the three diffraction orders of interest with respect to computation time (which is proportional TM R Opteron Dual Core 2.2 GHz). Datapoints to the number of unknowns N ) using 8 CPU cores (4 AMD of the same symbol belong to different refinement levels and different polynomial degrees p = The relative error of the calculated FEM efficiencies is compared to a quasi exact numerical solution with a one rank higher refinement level and highest polynomial degree p = 6. Left: TE polarization. Right: TM polarization.

6 2 Convergence: 1st diffraction order Relative error P=2 P=3 P=4 P=5 P= N Figure 6. Convergence of the 1 st diffraction order (TE polarization) for polynomial degrees p = of the ansatz functions of the finite elements with respect to the number of unknowns N. Relative error of the calculated FEM efficiencies compared to a quasi exact numerical solution with a one rank higher refinement level. of the relative error are based on the comparison with a solution of the same problem with one rank higher refinement level and highest order p = 6. The two graphs of Fig. 5 show the efficiency convergence of the three interesting diffraction orders 1,0,1 for s- and p-polarization corresponding to TE- and TM-polarization with respect to computation time using 8 CPU cores. This also corresponds to the number of unknowns N since computing time is proportional to N. Datapoints of the same symbol belong to different refinement levels and different polynomial degrees p = The computational cost of the forward FEM solution for the scattering problem of Sect. 2 for a very satisfying accuracy with a relative error in the order of 5 to 6 in all considered diffraction orders range from some minutes up to an hour using 8 CPU cores (cf. Fig. 5). Obviously, the relative errors of the zero diffraction order are slightly larger than for the ±1 st orders. This is due to the fact that the zero order is significantly smaller by some magnitudes than the ±1 st orders. The reason is the following: if width w is half the period and height h is about 1λ then the zero order is nearly cancelled for normal incidence. Hence, the accuracy of the zero order is quite important as especially this efficiency responds very sensitive on height changes. Fig. 6 shows the convergence of the 1 st diffraction order efficiency for different basis polynomial degrees p. When the number of unknowns is equal for two different degrees that means, that in the case of the lower degree the triangulation was chosen finer which again increased the number of unknowns. For small numbers of unknowns the basis polynomials with lower degree have the advantage, that their mesh is a lot finer than the one for higher degree basis functions. This fact is useful if many coarse calculations of a problem shall be done for orientation purposes. Another interesting illustration is the near-field of the considered 2D periodic phase grating shown in Fig. 2. One recognizes intensity spots in red caused by the sinusoidal line roughness for the smaller period highlighted in the intensity plot of the electric field in the vicinity of the scattering structure. It shows a significant interaction of the electromagnetic field with the roughness structures and indicates the possibility of reconstruction also for this rather difficult parameter. 4. REDUCED BASIS METHOD A BRIEF INSTRUCTION This section summarizes and explains general concepts of the reduced basis method 7 which is applicable to any FEM solution of (typically linear) differential equations. The PhD thesis 8 deals with RBM in the field of electromagnetic scattering problems. Further reading and more examples can be found in. 5,9 11

7 4.1 RBM approximation and error estimator The purpose of the reduced basis method is to construct an approximative so-called input-output relationship which can be solved very fast. As mentioned in equation (2), the input is our parameter vector ν i with values (w i,h i,r i ) and our aim is to find accurate efficiencies s N (ν i ) D (= output) to the parameters as quickly as possible. At first, one computes a set of snapshot solutions u N (ν) one-time in the so-called offline phase. This process takes some time, but afterwards the result can be used to make fast quasi real-time computation possible. Also this offline phase is self adaptive and optimized using an a posteriori error estimator (cf. next paragraph) to minimize the number of full FEM snapshot computations. The manifold of all possible solutions of the truth approximation (1) is denoted by: M N = { u N (ν) is a solution to (1) ν D }. (3) Now we approximate M N by a low dimensional space X N M N with dim(x N ) = N N. This means, we do not compute our diffraction efficiencies (= output of interest) employing the FEM, rather we use the truth approximations to construct the desired result. We just pick a comparatively small number of snapshot parameters Figure 7. 2D parameter space example with three snapshots and error map calculated by the error estimator. ν i D, i = 1,...,N max out of the parameter space and use the respective fields u N (ν i ) to approximate the field for any other given parameter set within the original parameter space ν D. By this approximation an error is introduced in our output of interest. Amazingly, due to the properties of Maxwell s equations together with the used FE space for the numerical solution, it is possible to construct an a posteriori error estimation. The estimator guides us constructing the reduced basis to minimize the error for given maximum error or maximum dimension N. The snapshot parameters are thereby chosen with the goal to minimize the overall deviation:! {ν i } i=1,...,n = min s s, (4) where s is the output of interest calculated in (2) and s its approximation. Such an error estimator is extremely useful during the process of snapshot selection for the ν i D. This can schematically be studied in Fig. 7 for the example of a 2D parameter space with the parameters parameter1 and parameter2. After having calculated three snapshots, the error map is determined by the error estimator in the whole parameter space. Our goal should be to minimize the maximal error in the given parameter space, for example. Hence, we select locations for next snapshots in the vicinity of large errors to decrease the maximal error. This process can be repeated after the next calculated snapshot and so forth leading to a fast reduction of maximal error in the parameter space D. The reliability of the a posteriori error estimator for establishing the reduced basis for the grating from Fig. 1 can be studied in Fig. 8. The figure shows the relation of the estimated and the true H(curl) error of the near field. The error estimator is best if there is a clear linear relation between both quantities. The relation in Fig. 8 is sufficiently linear so that the error estimations are sufficiently good. 4.2 Computation of the reduced basis solution In order to calculate the electric field for arbitrary parameters ν D a variational formulation similar to (1) is stated in the reduced basis space. Hence, RBM solves Maxwell s equations, in contrast to a pure parameter ν D

8 estimated H(curl) error true H(curl) error Figure 8. Reliability of the a posteriori error estimator for establishing the reduced basis for the grating from Fig. 1. interpolation. For construction of the reduced basis system, first an orthonormal basis of the reduced basis space X N is computed from the snapshots: B N N = { ζ N q q = 1,...,N }. Then we can expand the reduced basis solution into this basis: N u N (ν) = α q (ν)ζ N q. (5) q=1 By choosing an appropriate basis (from snapshots as explained in the last subsection) the deviation between truth approximation and the result computed within the reduced basis space is very small. In order to calculate the output of interest we only have to calculate the parameter dependent coefficients α q (ν). This is what actually has to be done when solving the problem for given parameters in the online phase. By inserting ansatz (5) into the reduced basis system (equivalent to (1)) a(u N,v;ν) = f(v), v X N we receive the linear system of equations for the coefficients α q (ν) which has to be solved, N α q (ν)a(ζ N q,ζ N n ;ν) = f(ζ N n ), n = 1,...,N. (6) q=1 The sesquilinear form a(, ;ν) can be affinely decomposed into parameter dependent functions Θ m (ν) and parameter independent sesquilinear forms a m (, ). The latter can be computed in the offline phase such that the costs of the online phase do not depend on the size of the finite element truth approximation. For the computation of the output of interest s N (ν) we get after some rearrangements: ( N ) N s N (ν) = l o (u N (ν)) = l o α q (ν)ζ N q = α q (ν)l o (ζ N q ). (7) q=1 The cost for this online operation is significantly smaller than calculating the standard forward FEM problem since l o (ζ N q ) can be computed offline and α q (ν) can be calculated online fast. Generally speaking, after having assembled the reduced basis, solution of the parameter dependent problem is up to several orders of magnitude faster than regularly by direct FEM calculations. On the other hand, the accuracy is nearly the same and decreases with the number of snapshots taken into account. Therefore, it is well suited for the approximative forward solution of inverse scattering problems and optimization issues. 5. REDUCED BASIS RESULTS FOR DIRECT AND INVERSE SCATTERING The reduced basis was constructed as explained above. The step of basis construction together with the according solutions of FEM truth approximations for several parameter sets ν i is the so-called offline stage. It may take q=1

9 a long time but is only computed once, while later, the fast RBM can respond to new geometry parameter sets. In our case the offline stage took the time to calculate the N = 35 snapshots (= 35 direct FEM truth approximations) and to construct the reduced basis, altogether this took about (35 + 1) 0.5 hours. Guided by the ingenious a posteriori error estimator we need only about sampling points per space dimension of our 3D parameter space D reaching a sufficiently high accuracy as will be shown below. This has to be compared with a naive approach by sampling the 3D space with e.g. samples per space dimension leading to 00 FEM calculations which would lead to times longer computing times than for the RBM offline stage without having the reliability of a sufficiently good accuracy. At first we compare the reduced basis results to standard FEM forward calculations in terms of accuracy and computational cost. For an extensive testing of the RBM and to get a sufficiently large variety of solutions for different geometry parameter sets, 250 exact FEM calculations were carried out for comparison, 125 for TE and 125 for TM polarization for relative variations { 0.1, 0.05, 0, +0.05, +0.1} from the default values (0.5, 1λ, 1λ) of the three geometry parameters ( wi p x,h i,r i ) =: ν i D. This operation took a long time with an average for a single calculation of about 30 minutes with the standard FEM solver while a single computation took only 2 seconds using the RBM with basis dimension N = 35. Therefore, the reduced basis calculations for the solution of the direct scattering task were faster with a factor of about 00. absolute error of intensity order -1 order 0 order reduced basis dimension Figure 9. Convergence of the reduced basis method with respect to the reduced basis dimension N. Mean error of 35 randomly chosen geometry parameters ν i, i = 1,..., 35. For the calculation of the efficiency results (= output of interest) using the reduced basis method the number of snapshots taken into account is specified by the user (N = 35 with our basis). As expected, an increasing accuracy with increasing number of used snapshots is obviously verified in Fig. 9. Certainly, the deviations of the efficiencies in the diffraction orders 1,0 and 1 to the standard FEM results were used for comparing the accuracy of the RBM. For N = 35 the absolute error ranges between 1 4 and 3 4 for all three diffraction orders. This is a very impressive result considering the fact, that the computational cost was reduced by a factor of 00! Also the relative error for the larger intensity orders 1 and 1 is around 0.1% very impressive, too. Altogether we can conclude that the efficiency determination using the reduced basis method works impressively good. After having a reduced basis of small dimension N = 35 at hand for our 3D grating structure we are now able and can apply this new technique to perform inverse scattering operations very fast and online in a few seconds. Inverse scatterometry is the task to find the parameters of a diffractive geometry that created a measured diffraction pattern under the assumption that the general form of the diffractive structure is given and only the values of some parameters have to be determined. Hence, compared to a real inverse problem, inverse scatterometry is only a parameter identification task. But sometimes, already in this simpler case of parameter identification, we are considering here, the parameters cannot be reconstructed uniquely since different geometry

10 11 reconstructed line width true line width.5 11 Figure. Reconstruction accuracy for line width w from noisy data with 0.1% noise. sets can yield the same output of interest. And this may be especially then the case if we do not use all appearing diffracted orders for reconstruction, but just a small number with the highest intensities as we do it in this application. Nevertheless, weak variational formulations as given in (1) together with FEM sometimes allow the iterative solution of inverse scattering problems by evaluating gradients of the objective function, directly depending on the given structure. 12 On the other hand, it can make sense to translate the inverse problem into a forward optimization problem and defining an objective function as we do it in this publication. Thus, we simulate the direct diffraction problem using the determined reduced basis as a model for the diffraction process. Using the input-output relation of equation (7) we can calculate our output of interest s N (ν), the diffraction efficiencies, for many geometry parameter sets ν and try to find the largest correlation, corresponding to e.g. the smallest value of the objective function, between the calculated output of interest depending on the geometry and the given diffraction efficiencies. A possible simple objective function can be the root mean square value between the calculated and given three diffraction efficiencies. Together with the two polarizations, only 6 values are taken into account in our example. Here are some of the results of the parameter reconstruction process for the three parameters (w, h, r) of the 3D grating. Measuring data has been simulated by randomly adding 0.1% noise to the 6 diffraction efficiencies determined by the RBM. The results are noise level dependent and become worse with increasing noise level. Figures and 11 are results from the reconstruction of 40 randomly chosen ν i D, the 3D parameter domain with w [9λ,11λ], h [0.9λ,1.1λ], and r [0.9λ,1.1λ], computed with global and local minimization of the objective function. The determination of line width w in Fig. is best and can be explained by the main influence of the larger intensity orders 1 and 1 which additionally have the smallest RBM error. The deviation of the only outlier is about 5% to the real value. Also the height h determination in the left part of Fig. 11 is quite good. Some reconstructed values have deviations up to 9%. An explanation may be the influence of noise and the fact that height h is sensitive mainly to the zero order which is quite small and the relative error of the RBM results is large for this order. Also the reconstruction of amplitude r of the line roughness in the right part of Fig. 11 works well. The number of visual deviations is larger than for the other two parameters but the values are sufficiently good with maximal deviations of ±5%. Obviously, the three considered diffraction orders are less sensitive to roughness amplitude variations than to the two other parameters. A sensitivity analysis like that presented in Figs., 11 using other noise levels and objective functions, including dozens of solutions of Maxwell s equations for ever new geometry sets during optimization for each of the 40 randomly chosen geometries ν i D, runs only a few minutes using RBM. This is really great as it is even

11 reconstructed height reconstructed amplitude true height true amplitude Figure 11. Reconstruction accuracy from noisy data with 0.1% noise. Left: Line height h. Right: Line roughess amplitude r. faster than a single direct FEM forward calculation! Hence, RBM should significantly improve the reconstruction results in inverse scattering and other time consuming applications. It should be remarked that deviations in the reconstruction of all three grating parameters also appear in the case of no noise. This may be astonishing but it is not. At first one assumes RBM may be too inaccurate using N = 35 basis functions but similar deviations also appear in the case of a reconstruction process solving the full Maxwell equations by direct FEM calculations. Hence, one has to realize the real reason for the deviations: Maxwell s equations, the underlying physics. We should remember what we have mentioned above: parameter reconstruction for inverse scattering is an inverse operation and as such it may be not unique. There may exist different geometry sets producing the same output of interest (= diffraction efficiencies). A possible solution improving the reconstruction results could be to increase the number of measured diffraction efficiencies included in the reconstruction process or to change the incidence condition to find a better one producing unique reconstruction results CONCLUSION The problem of inverse scattering of a 3D phase grating has been studied. It looks like a one-dimensional phase grating (cf. Fig. 1). Due to the specific line roughnesses at both sides of the grating lines, it necessarily must be treated fully three-dimensional. The discretized FEM area of the 2D elementary cell is large and takes 20 times wavelengths. The structure is parametrized and the three parameters to be reconstructed from diffraction efficiency measurement are line width w, line height h, and amplitude r of the line roughness. Calculated efficiencies with added statistical errors are used instead of measured efficiencies. Due to the grating structure the direct Maxwell computations need specific FEM prism grid triangulations capable of the necessary height variations and use 6 polygons per sub-period approximating a sine sub-period of the line roughness. Both actions result in a grid which does not change the number of nodes and the topology with modified geometry parameters (w,h,r). The reduced basis method is based on the full decoupling of the truth solutions of Maxwell s equations by FEM and the RB results through offline-online procedures: the complexity of the offline stage depends on the huge dimension of the truth FE solution; but the complexity of the online stage - in which RBM responds to new values of the input parameter set - depends only on the small dimension of the reduced basis space and our three dimensional parameter space. 7 In essence, one is guaranteed the accuracy of a high-fidelity finite element Maxwell solver but at the very low cost of a reduced-order model. Maxwell solver software firm JCMwave is the first one developing a RBM for commercial applications in 3D electromagnetic scattering and diffraction applications. Using this FEM software for 3D scatterometric diffraction calculations together with fast RBM we achieve an efficiency accuracy of about 4 for the direct problem with only 35 snapshots being the reduced basis dimension, guided by the unique error estimator. This speeds

12 up the calculation of diffraction amplitudes by a factor of about 00 compared to the conventional solution of Maxwell s equations by FEM. Having this in mind we were able to solve the inverse problem to reconstruct the three parameters of our phase grating from measured scattering data in a 3D manifold online in less than a minute having the full FEM accuracy available. Additionally, also a sensitivity analysis or the choice of robust measuring strategies, for example, which usually needs dozens or hundreds of very time consuming FEM runs, can now be done online in a few minutes. Clearly, similar investigations would also be possible by library or data based methods with precalculated results for samples of the whole parameter space. This procedure would have at least two significant drawbacks compared to the RBM approach: (1) the computational amount would be tremendous as the whole parameter space has to be sampled which is not the case for RBM construction, (2) the accuracy of results for parameters between the sampling points is very weak because of the interpolation of data points while RBM uses a real basis having inherently the full Maxwell equations incorporated, and (3) the reduced basis approximation is built self-adaptively. Hence, no user has to waste time finding an interpolation grid which guarantees sufficiently accurate results. The reduced basis method with a posteriori error estimation and together with FEM provides an impressively fast and very accurate direct and inverse scattering parameter identification method fully based on Maxwell s equations. REFERENCES [1] Bodermann, B. and Wurm, M., Numerical investigations of prospects, challenges and limitations of nonimaging optical metrology of structured surfaces, in [Modeling Aspects in Optical Metrology II], Bosse, H., Bodermann, B., and Silver, R., eds., Proc. SPIE 7390, 73900R (2009). [2] Bodermann, B. and Bosse, H., Model based reference metrology for dimensional characterization of microand nanostructures, Optoelectr. Lett. 4(2), (2008). [3] Gross, H. and Rathsfeld, A., Sensitivity analysis for indirect measurement in scatterometry and the reconstruction of periodic grating structures, Waves Random Complex Media 18, (2008). [4] Model, R., Rathsfeld, A., Gross, H., Wurm, M., and Bodermann, B., A scatterometry inverse problem in optical mask technology, J. Phys. Conference Series 135, (2008). [5] Pomplun, J., Zschiedrich, L., Burger, S., Schmidt, F., Tyminski, J., Flagello, D., and Toshiharu, N., Reduced basis method for source mask optimization, in [Photomask Technology], Montgomery, M. W. and Maurer, W., eds., Proc. SPIE 7823, 78230E (2009). [6] Patera, A. T., Smartphone as Supercomputer Surrogate: In-Situ Scientific Simulation. Lecture at University of Ulm, Germany (07 December 20). Anthony Patera is MIT Ford Professor of Engineering. [7] Patera, A. and Rozza, G., Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations (version 1.0), to appear in: MIT Pappalardo Graduate Monographs in Mechanical Engineering, Massachusetts (2006). Copyright MIT. [8] Pomplun, J., Reduced Basis Method for Electromagnetic Scattering Problems, PhD thesis, Free University, Berlin (20). [9] Pomplun, J., Zschiedrich, L., Burger, S., and Schmidt, F., Reduced basis method for computational lithography, in [Photomask Technology], Zurbrick, L. S. and Montgomery, M. W., eds., Proc. SPIE 7488, 74882B (2009). [] Pomplun, J. and Schmidt, F., Accelerated a posteriori error estimation for the Reduced Basis Method with application to 3D electromagnetic scattering problems, SIAM J. Scientific Computing 32(2), (20). [11] Pomplun, J. and Schmidt, F., Reduced basis method for fast and robust simulation of electromagnetic scattering problems, in [Modeling Aspects in Optical Metrology II], Bosse, H., Bodermann, B., and Silver, R., eds., Proc. SPIE 7390, 73900I (2009). [12] Elschner, J. and Schmidt, G., Analysis and numerics for the optimal design of binary diffractive gratings, Math. Meth. Appl. Sci. 21, (1998).