## A maximum likelihood approach to the inverse problem of scatterometry |

Optics Express, Vol. 20, Issue 12, pp. 12771-12786 (2012)

http://dx.doi.org/10.1364/OE.20.012771

Acrobat PDF (1006 KB)

### Abstract

Scatterometry is frequently used as a non-imaging indirect optical method to reconstruct the critical dimensions (CD) of periodic nanostructures. A particular promising direction is EUV scatterometry with wavelengths in the range of 13 – 14 nm. The conventional approach to determine CDs is the minimization of a least squares function (LSQ). In this paper, we introduce an alternative method based on the maximum likelihood estimation (MLE) that determines the statistical error model parameters directly from measurement data. By using simulation data, we show that the MLE method is able to correct the systematic errors present in LSQ results and improves the accuracy of scatterometry. In a second step, the MLE approach is applied to measurement data from both extreme ultraviolet (EUV) and deep ultraviolet (DUV) scatterometry. Using MLE removes the systematic disagreement of EUV with other methods such as scanning electron microscopy and gives consistent results for DUV.

© 2012 OSA

## 1. Introduction

1. H. Huang and F. Terry Jr., “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films **455**, 828–836 (2004). [CrossRef]

2. C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B **15**, 361–368 (1997). [CrossRef]

3. J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B **22**, 3059 (2004). [CrossRef]

5. M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE **7390**, 73900Q (2009). [CrossRef]

6. M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol. **22**, 094024 (2011). [CrossRef]

7. X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE **3677**, 159–168 (1999). [CrossRef]

12. P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A **13**, 779–784 (1996). [CrossRef]

13. R. Petit and L. Botten, *Electromagnetic theory of gratings* (Springer, 1980). [CrossRef]

15. J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. **16**, 139–156 (2002). [CrossRef]

19. J. Melenk and I. Babuška, “The partition of unity finite element method: basic theory and applications,” Comput. Meth. Appl. Mech. Eng. **139**, 289–314 (1996). [CrossRef]

22. S. Coulombe, B. Minhas, C. Raymond, S. Naqvi, and J. McNeil, “Scatterometry measurement of sub-0.1 *μ*m linewidth gratings,” J. Vac. Sci. Technol. B **16**, 80 (1998). [CrossRef]

23. H. Gross, A. Rathsfeld, F. Scholze, and M. Bär, “Profile reconstruction in extreme ultraviolet (EUV) scatterometry: modeling and uncertainty estimates,” Meas. Sci. Technol. **20**, 105102 (2009). [CrossRef]

24. R. Al-Assaad and D. Byrne, “Error analysis in inverse scatterometry. I. Modeling,” J. Opt. Soc. Am. A **24**, 326–338 (2007). [CrossRef]

7. X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE **3677**, 159–168 (1999). [CrossRef]

8. H. Patrick, T. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE **7271**, 727128 (2009). [CrossRef]

23. H. Gross, A. Rathsfeld, F. Scholze, and M. Bär, “Profile reconstruction in extreme ultraviolet (EUV) scatterometry: modeling and uncertainty estimates,” Meas. Sci. Technol. **20**, 105102 (2009). [CrossRef]

23. H. Gross, A. Rathsfeld, F. Scholze, and M. Bär, “Profile reconstruction in extreme ultraviolet (EUV) scatterometry: modeling and uncertainty estimates,” Meas. Sci. Technol. **20**, 105102 (2009). [CrossRef]

28. H. Gross, M.-A. Henn, A. Rathsfeld, and M. Bär, “Stochastic modelling aspects for an improved solution of the inverse problem in scatterometry,” in “Advanced mathematical and computational tools in metrology and testing: AMCTM IX,” F. Pavese, M. Bär, J.-R. Filtz, A. B. Forbes, L. Pendrill, and H. Shirono, eds. (World Scientific Pub. Co. Inc., 2012), 202–209.

29. J. Ruanaidh and W. Fitzgerald, *Numerical Bayesian Methods Applied to Signal Processing* (Springer, 1996). [CrossRef]

30. C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE **6151**, 61510I (2006). [CrossRef]

## 2. Methods

### 2.1. Scatterometric measuring setup for the EUV mask

*TaO*,

*TaN*, and

*SiO*

_{2}). These trapezoids are defined by the heights of the three layers

*p*,

_{i}*i*= 1, 6, 11 and by coordinates

*p*,

_{i}*i*= 2, 3, 7, 8, 12, 13 describing the corner positions. Beneath the line-space structure, there are two capping layers of

*SiO*

_{2}and of

*Si*on top of a Molybdenum silicide (

*MoSi*) multilayer stack (MLS). The latter stack consists of a periodically repeated structure composed of a

*Mo*layer and a

*Si*layer separated by two intermediate layers. Note, that the MLS is added to enable the reflection of EUV waves. It acts as a Bragg mirror at the design wavelength of about 13 nm. Important geometric profile parameters are the height

*p*

_{6}of the

*TaN*layer (55 – 60 nm) and the

*x*-coordinates

*p*

_{2}and

*p*

_{7}of the right corners of the

*TaN*layer. The complex indices of refraction for the involved materials are given in Table 1 for one of the three wavelengths in the range of 13 nm.

*x*-coordinates of the corresponding left corners depend on those of the right corners such as

*p*

_{3}=

*d*–

*p*

_{2}or

*p*

_{8}=

*d*–

*p*

_{7}, where

*d*is the period of the EUV mask. Furthermore the sidewall angle (SWA) for the

*TaO*layer was fixed. The cross-section area of this trapezoidal layer is equal to a corresponding

*TaO*layer having curved upper edges with a radius of about 6 nm. Additionally, we have assumed that the SWA of the

*SiO*

_{2}layer is always equal to the SWA of the

*TaN*layer above. The latter angle depends on the corners and the height of the

*TaN*layer:

*p*

_{6},

*p*

_{7}and

*p*

_{2}(cf. Fig. 1). In the following we will refer to these parameters as height, top CD and bottom CD. An important quantity that derives from these three parameters is the sidewall angle (SWA) of the

*TaN*absorber layer with a design value of 90°. In our evaluations all remaining parameters were set to the values given in Table 1 that represent the manufacturer’s design values [31

31. J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV Masks by EUV-Scatterometry and Finite Element Analysis,” Proc. SPIE **7028**, 70280P (2008). [CrossRef]

30. C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE **6151**, 61510I (2006). [CrossRef]

### 2.2. Scatterometric measuring setup for the MoSi mask

*MoSi*) photomask. Its cross section profile for one period and a scheme of the measurement set-up are shown in Fig. 2 below, its optical constants can be found in Table 2.

*MoSi*based on a glass substrate. The trapezium is completely defined by its height

*h*and by the

*x*-coordinates

*p*,

_{i}*i*= 1, 2, 3, 4 of its corners. Again a symmetric profile is imposed and the sidewall angle (SWA) of the

*MoSi*absorber layer has a design value of 90°.

*MoSi*mask derives from measurements with a DUV goniometric scatterometer. It can be operated in a reflectometric, ellipsometric, and diffractometric measurement mode in a spectral range from 190 nm to 840 nm. A measurement data set in the present case includes reflected and transmitted diffraction orders at seven different incident angles for a laser beam with a wavelength of 193 nm and consists typically of 43 data points, see [6

6. M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol. **22**, 094024 (2011). [CrossRef]

### 2.3. Mathematical model of scatterometry

13. R. Petit and L. Botten, *Electromagnetic theory of gratings* (Springer, 1980). [CrossRef]

*u*is the transversal field component that oscillates in the groove direction, resp. along the

*z*-axis (cf. Fig. 2) and

*k*is the wave number that is assumed to be constant for areas filled with the same material. The boundary conditions that are imposed on this partial differential equation are the common transmission conditions on the interfaces between domains, quasi-periodic boundary conditions on the lateral boundaries due to the periodic nature of the structure and the usual outgoing wave conditions in the infinite regions [16

16. H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, and A. Rathsfeld, “Mathematical modelling of indirect measurements in scatterometry,” Measurement **39**, 782–794 (2006). [CrossRef]

33. J. Elschner, R. Hinder, A. Rathsfeld, and G. Schmidt, http://www.wias-berlin.de/software/DIPOG.

**p**= (

*p*

_{1}, . . . ,

*p*,

_{n}*λ*,

_{k}*θ*) characterizing the surface structure by its profile and optical indices

_{l}*p*

_{1}, . . . ,

*p*and the incidental light by its wavelength

_{n}*λ*and angle of incidence

_{k}*θ*is called the solution of the forward problem. It can be regarded as a nonlinear operator mapping

_{l}**p**to the corresponding diffraction pattern:

*f*(

**p**) = (

*f*

_{1}(

**p**), . . . ,

*f*(

_{m}**p**)). The model function

*f*is calculated by solving the PDE in Eq. (1).

### 2.4. Measurement error model

**y**= (

*y*

_{1}, . . . ,

*y*), consisting of efficiencies or phase shift differences for different wavelengths, incident angles or polarization states perturbed by measurement noise. The

_{m}*j*th data point is a sum of the value of the model function and a noise contribution: Assuming no correlation between the measurements and the absence of systematic errors, the errors for the

*j*th data point are modelled as normally distributed with zero mean, i.e. where the variances are composed of the variances of two independent terms: The first term (

*a*·

*f*(

_{j}**p**))

^{2}indicates the contribution of a linearly dependent noise. From an experimental point of view, power fluctuations of the incidental beam during the recording of the diffraction patterns are the main determinant for the values of

*a*. The second term

*b*

^{2}is the contribution of the background noise independent of the measured light intensities.

### 2.5. The solution of the inverse problem

#### 2.5.1. Least squares method

**20**, 105102 (2009). [CrossRef]

24. R. Al-Assaad and D. Byrne, “Error analysis in inverse scatterometry. I. Modeling,” J. Opt. Soc. Am. A **24**, 326–338 (2007). [CrossRef]

*ω*are usually chosen to be the reciprocal variances of the measurement values (cf. Eq. (5)). The solution to the inverse problem is then found by minimizing the resulting weighted least squares function. Below, this is achieved with the DIPOG software that employs a Gauß-Newton type iterative algorithm. Note that this method requires knowledge of the noise level of the measurement process. This information is, of course, not always available, and the question arises how an inappropriate choice of the weights influences the reconstruction.

_{j}#### 2.5.2. Maximum likelihood estimation

*a*and

*b*can not be determined by LSQ. We employ the maximum likelihood estimation (MLE) [29

29. J. Ruanaidh and W. Fitzgerald, *Numerical Bayesian Methods Applied to Signal Processing* (Springer, 1996). [CrossRef]

*a*,

*b*and the geometry parameters

**p**for given measurement data

**y**[34

34. R. Millar, *Maximum Likelihood Estimation and Inference* (Wiley, 2011). [CrossRef]

^{®}, DIPOG [33

33. J. Elschner, R. Hinder, A. Rathsfeld, and G. Schmidt, http://www.wias-berlin.de/software/DIPOG.

*f*(

**p**), i.e. to solve the forward problem and also to calculate the partial derivatives of

*f*w.r.t. the parameters

*p*.

_{i}### 2.6. Variance estimation

#### 2.6.1. Least squares method

24. R. Al-Assaad and D. Byrne, “Error analysis in inverse scatterometry. I. Modeling,” J. Opt. Soc. Am. A **24**, 326–338 (2007). [CrossRef]

*f*is approximately linear in the relevant regions of the parameter values

*p*, then the errors of the reconstructed parameters are, again, normally distributed random numbers with zero mean. The standard deviations

_{i}*u*of the quantities

_{i}*p*are given by the square root of the main diagonal entries of the covariance matrix Σ of the parameters. The matrix Σ can be approximated as hence Note that in earlier work [23

_{i}**20**, 105102 (2009). [CrossRef]

**24**, 326–338 (2007). [CrossRef]

*σ*in Eq. (10) were chosen according to the predefined error model. A modified approach to the variance estimation for LSQ is to chose the scaling factors according to the resulting residuals of the optimal solution, based on the following reasoning. A consistent solution

_{j}**p̂**to the optimization problem (cf. Eq. (7)) should pass the

*χ*

^{2}-test, namely where

*ν*=

*m*–

*n*denotes the degrees of freedom and

*α*= 5 ·10

^{−2}e.g.. If this is not the case we can fulfill the criterium of Eq. (12) by rescaling the variances of the input data resp. the weights with some scaling factor

*κ*: chosen such that the rescaled

*ν*. Note, that this rescaling of the weights does not affect the result of the optimization given by the parameter values describing the minimum of the function in Eq. (7). In contrast, it increases only the variances of the reconstructed parameters. In this work LSQ refers to the minimization of the function in Eq. 6 followed by a rescaling of the variances according to Eq. 13. In [23

**20**, 105102 (2009). [CrossRef]

35. H. Gross, A. Rathsfeld, and M. Bär, “Modelling and uncertainty estimates for numerically reconstructed profiles in scatterometry,” in “Advanced mathematical and computational tools in metrology and testing: AMCTM VIII,” F. Pavese, M. Bär, A. B. Forbes, J.-M. Linares, C. Perruchet, and N.-F. Zhang, eds. (World Scientific Pub. Co. Inc., 2009), 142–147.

*a*and

*b*and sufficient linearity of the model function

*f*.

#### 2.6.2. Maximum likelihood estimation

34. R. Millar, *Maximum Likelihood Estimation and Inference* (Wiley, 2011). [CrossRef]

*θ*̂ the maximum likelihood estimator, then its standard error can be calculated as

## 3. Results

### 3.1. Dependency of the reconstruction on the chosen weight factors for the EUV measurement setup

*a*and

*b*. Different methods to solve the inverse problems are used and compared. We start with a least-squares approach assuming fixed uncertainty parameters

*a*and

*b*. Figure 3 shows the values of the

*χ*

^{2}-function defined in Eq. (7) for a simulated data set perturbed by a noise contribution with

*a*= 10% and

*b*= 10

^{−3}(cf. Eq. (5)) in dependence on the bottom CD and top CD for two different noise models, i.e. two different weightings in the least squares function from Eq. (7). Figure 4 shows the dependency of the reconstructed CDs on the ratio

*b*/

*a*for the same analysis. Note, that the reconstruction results depend only on the ratio of

*b*to

*a*and not on the absolute values of the two parameters.

*TaN*and the

*SiO*

_{2}layer and 82.6° for the

*TaO*layer on top of the absorber line (cf. Fig. 1). In Fig. 3 one clearly sees how the minimum (the blue dot) shifts upon change of the ratio

*b*/

*a*resulting in two different solutions to the inverse problem. This dependency of course also has an impact on the reconstructed sidewall angle (see Fig. 4). One sees that the reconstructed value of the top CD (bottom CD) decreases (increases) by several nanometers as the value of

*b*/

*a*in the reconstruction is chosen by one order of magnitude bigger than the true value. It is this sensitivity that suggested to us to treat the noise parameters

*a*and

*b*as additional variables that need to be reconstructed as well, since a wrong or incomplete assessment regarding the variances of the input parameters (scattering efficiencies) will not only lead to an under- or overestimation of the variances of the output parameters (reconstructed geometry parameters) but also causes significant systematic errors in the results. This needs to be considered especially in the case of a newly set-up experiment when the knowledge of the variances of the measurement errors is usually incomplete.

### 3.2. Application to simulated EUV data

*a*= 10% and

*b*= 10

^{−2}(cf. Eq. (5)) resulting in a total of 50 “noisy” datasets. The weights for the LSQ in Eq. (7) were set to

*a*= 1% and

*b*= 10

^{−3}representing a typical estimate of the variances in the real measurement processes used in earlier publications [23

**20**, 105102 (2009). [CrossRef]

*b*/

*a*along with the approximate 95% confidence intervals based on the variance estimation described in Section 2.6 are presented in Fig. 5. MLE is found to be capable to reconstruct the noise parameter

*a*from a data set of limited size with a typical relative error of 10 – 20 %, while errors in

*b*can be substantially larger. Figure 6 below presents the reconstructed sidewall angles along with the approximate 95% confidence intervals for the solutions of the two methods. Note that there is a systematic shift of about 1.5° between the actual value of 90° and the mean estimated sidewall angle obtained by LSQ, while the mean estimated sidewall angle obtained by MLE is almost identical to the actual value. This bias is observed because of the nonlinearity of the mathematical model. It is found to vanish for the present model if the “correct” weights are chosen.

### 3.3. Application to measured EUV data

*a*and the ratios

*b*/

*a*for all measured data sets are plotted in Fig. 9. The mean estimated value for

*b*/

*a*lies around 1.2·10

^{−2}, a value that differs significantly from the value of ca. 3 ·10

^{−2}– 10

^{−1}(

*a*= 1 − 3%,

*b*= 10

^{−3}) used for the LSQ method in previous publications [23

**20**, 105102 (2009). [CrossRef]

*b*of about 2 ·10

^{−3}agrees quite well with the value given by the experimenters, the relative error

*a*lies around 16%, which is about a magnitude larger than expected. Similar to the observations in the case of simulated input data we observe a systematic shift of the sidewall angles reconstructed using MLE towards the design value of 90° compared to the LSQ solutions with fixed weights, see Fig. 10. Note that the mean sidewall angle obtained by SEM in [31

31. J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV Masks by EUV-Scatterometry and Finite Element Analysis,” Proc. SPIE **7028**, 70280P (2008). [CrossRef]

### 3.4. Application to measured DUV data

37. J. Richter, J. Rudolf, B. Bodermann, and J. C. Lam, “Comparative scatterometric CD measurements on a MoSi photo mask using different metrology tools,” Proc. SPIE **7122**, 71222U (2008). [CrossRef]

*a*and

*b*characterizing the uncertainty of the input data with their approximate 95% confidence intervals are presented in Fig. 11. The parameter

*b*representing the background noise of the measurement setup lies around 3 ·10

^{−2}, which is in a fair agreement with the value of 4 ·10

^{−2}given by the experimenters in [38

38. M. Wurm, S. Bonifer, B. Bodermann, and M. Gerhard, “Comparison of far field characterisation of DOEs with a goniometric DUV-scatterometer and a CCD-based system,” J. Eur. Opt. Soc. Rapid Publ. **6**, 11015s (2011). [CrossRef]

*b*/

*a*(cf. Fig. 4 and 8), hence the geometrical features obtained by LSQ with fixed weights

*a*= 2.5% and

*b*= 4 ·10

^{−2}(

*b*/

*a*= 1.6) are almost identical to those obtained by MLE (

*b*/

*a*≈ 1), both in terms of consistency and variance, as shown in Fig. 12.

## 4. Discussion and conclusions

8. H. Patrick, T. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE **7271**, 727128 (2009). [CrossRef]

**20**, 105102 (2009). [CrossRef]

**24**, 326–338 (2007). [CrossRef]

^{−3}for the measurement data. The resulting variances of the CDs were found to be in the range of 2 – 3 nm, whereas the height variances are ca. 0.5 nm. The sidewall angles were systematicly larger than with application of the LSQ method and showed better agreement with the design values as well as with independent measurement with scanning microscopy [31

31. J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV Masks by EUV-Scatterometry and Finite Element Analysis,” Proc. SPIE **7028**, 70280P (2008). [CrossRef]

*MoSi*mask with identical periods of 560 nm, CDs around 250 nm and profile height around 70 nm. Application of MLE yielded relative variances of 3 % superposed by absolute background noises in the range of 3·10

^{−2}. The variances of the CDS were found to be in the range of 1 – 2 nm and the height variances were determined to be 0.5 nm.

*MoSi*mask in a DUV scatterometer. A reliable treatment of EUV masks in an EUV scatterometer will require further improvement of the mathematical modelling and inverse methods.

27. A. Kato and F. Scholze, “Effect of line roughness on the diffraction intensities in angular resolved scatterometry,” Appl. Opt. **49**, 6102–6110 (2010). [CrossRef]

**20**, 105102 (2009). [CrossRef]

29. J. Ruanaidh and W. Fitzgerald, *Numerical Bayesian Methods Applied to Signal Processing* (Springer, 1996). [CrossRef]

## Acknowledgments

## References and links

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2. | C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B |

3. | J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B |

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6. | M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol. |

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33. | J. Elschner, R. Hinder, A. Rathsfeld, and G. Schmidt, http://www.wias-berlin.de/software/DIPOG. |

34. | R. Millar, |

35. | H. Gross, A. Rathsfeld, and M. Bär, “Modelling and uncertainty estimates for numerically reconstructed profiles in scatterometry,” in “Advanced mathematical and computational tools in metrology and testing: AMCTM VIII,” F. Pavese, M. Bär, A. B. Forbes, J.-M. Linares, C. Perruchet, and N.-F. Zhang, eds. (World Scientific Pub. Co. Inc., 2009), 142–147. |

36. | H. Gross, R. Model, F. Scholze, M. Wurm, B. Bodermann, M. Bär, and A. Rathsfeld, “Modellbildung, Bestimmung der Messunsicherheit und Validierung für diskrete inverse Probleme am Beispiel der Scatterometrie,” VDI-Berichte |

37. | J. Richter, J. Rudolf, B. Bodermann, and J. C. Lam, “Comparative scatterometric CD measurements on a MoSi photo mask using different metrology tools,” Proc. SPIE |

38. | M. Wurm, S. Bonifer, B. Bodermann, and M. Gerhard, “Comparison of far field characterisation of DOEs with a goniometric DUV-scatterometer and a CCD-based system,” J. Eur. Opt. Soc. Rapid Publ. |

39. | J. Kaipio and E. Somersalo, |

40. | J. Berger, |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(120.3940) Instrumentation, measurement, and metrology : Metrology

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: February 8, 2012

Revised Manuscript: March 16, 2012

Manuscript Accepted: April 1, 2012

Published: May 23, 2012

**Citation**

Mark-Alexander Henn, Hermann Gross, Frank Scholze, Matthias Wurm, Clemens Elster, and Markus Bär, "A maximum likelihood approach to the inverse problem of scatterometry," Opt. Express **20**, 12771-12786 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-12771

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