Model Uniqueness and Maximum Likelihood Estimation

Reconstruction Based on Specular EUV Reflectance 4.1

Maximum likelihood

A solution of the aforementioned problem requires to determine the value of ˜χ² in vicinity of the PSO solution or possibly the whole parameter space. This is approached by numerically sampling the functional based on a MCMC method [56]. An application of this technique to the design process of multilayer mirrors has been demonstrated by Hobson and Baldwin [64]. In our case, the match of model and experimental result is evaluated based on a non-centeredχ²distribution assuming independent measurements.

It was further assumed that any measured point is distributed around the actual reflectiv-ity curve following a Gaussian distribution, i.e. Gaussian uncertainties for the experiment are assumed. The corresponding probability density function for a measurement result matching with the actual reflectivity curve, which is assumed to be obtainable exactly through the theoretical calculation, is then of Gaussian form [1]. Thus, the likelihood that the measured values match with the theoretical curve under the assumption that the model is correct is proportional to

L(E|^M(~x))∝exp −^χ˜2(~x)/2

, (4.2)

whereEdenotes the experiment, i.e. the measured data andM(~x)represents the model given through parameter set~x, e.g. the parameters of the model in table4.1. In our case however, we seek to evaluate the likelihoodL(M(~x)|^E)that the model M(~_x)with a given set of parameters~xis valid assuming the experiment Eyields the correct curve (the so called “posterior distribution”). Those two quantities are linked through the Bayesian theorem [16,97] stating

L(M(~x)|^E)∝L(E|^M(~x))L(M(~x)), (4.3) where L(M(~x)) denotes the likelihood for the model to be valid for a specific set of parameters~x(the so called “prior distribution”). The prior distribution does contain any prior knowledge about the model and allowed parameters. For the example of the model parameters in table 4.1, the prior distribution is L(M(~x)) → −^∞ for any parameter set outside the listed boundaries and L(M(~x)) = 1 everywhere else. In addition, the maximum total period thickness is limited, i.e. the sum of all layers in one period to only allow the appearance of the first Bragg peak within the measured spectral range through the same condition. Combining Eq. (4.2) and Eq. (4.3) then yields the likelihood functional

L(~x) = L(M(~x)|^E)∝exp −^χ˜2(~x)/2

L(M(~x)). (4.4) Solving the optimization problem posed in the previous section within this context is then, equivalently to the minimization of ˜χ², the maximization of the likelihood L(~x). The MCMC method poses a statistical approach on evaluating (mapping) the likelihood across the parameter space within the previously defined limits as in the PSO approach.

It was proven that after a theoretical number of infinite iterations, the distribution of the individual samples within the MCMC algorithm, corresponds to the likelihood functional in Eq. (4.4) [34,92]. With a limited number of iterations, a numerical approximation of that distribution is obtained after reaching an equilibrium state in the algorithm [51]. It thus yields an alternative method on solving the optimization problem by extracting the maximum likelihood from the final result. However, in addition to the maximum value,

the likelihood distribution in parameter space is obtained allowing to extract confidence intervals for each of the parameters. Thereby, the aforementioned ambiguity of solutions can be quantified within the defined model and the available experimental data. The confidence intervals are defined as the one- or two-sigma standard deviations of the respective distributions for each parameter.

Confidence intervals for the Mo/B₄C/Si/C sample

An existing implementation of the MCMC algorithm by Foreman-Mackey et al. [51] was applied to the EUV measurement of the Mo/B₄C/Si/C sample in Fig.4.1with the model in Fig.4.2. The likelihood, as defined in Eq. (4.4) with the ˜χ²functional from Eq. (4.1), is sampled in a high-dimensional space depending on the number of parameters in the model. We therefore need to project the distribution for each parameter by marginalizing over all other parameters. Alternatively, two-parameter correlations can be visualized by projecting on a two-dimensional area, again marginalizing across all other parameters.

The projection for the Si and Mo layer thicknesses are shown in Fig. 4.5b and4.5c. In both cases, a well defined distribution is obtained. In the two-dimensional projection in Fig.4.5a, no correlations are apparent and a two-dimensional Gaussian-like shape results.

In all cases, the one-sigma standard deviations for Gaussian distributions are shown

1 2 3 4 5

d_Si/ nm 1

2 3 4 5

dMo/nm

1σ 2σ

a) ^{PSO fit}

1 2 3 4 5

d_Si/ nm b)

1 2 3 4 5

d_Mo/ nm

0.0 0.2 0.4 0.6 0.8 1.0

likelihood/arb.units

c) ±^1σ

50%

Figure 4.5 |Results of the maximum likelihood estimation obtained via the MCMC procedure. a) Two dimensional projection of the likelihood distribution for the parameter pairdSianddMo. The projection was obtained by marginalizing over all other parameters of the model. The black contours indicate the areas for one and two standard deviations (one and two sigma contours). The blue lines in all three sub-figures indicate the best parameter set found with the PSO method. b) One dimensional projection of the likelihood distribution for the silicon layer thicknessd^Si. The solid black line marks the center position (50%percentile) of the distribution. The dotted lines are the limits of one standard deviation. c) The one dimensional distribution similarly to b) for the molybdenum layer thickness.

together with the weighted center, i.e. the 50th percentile. The PSO result is also indicated, which is compatible with the one sigma standard deviation, but does not match the center of the likelihood result. The reason for that lies in higher order correlations of the parameters. In Fig.4.6, all one-dimensional projections of the likelihood distribution are shown for all remaining parameters. Clearly, while a reasonably small confidence interval (again, one standard deviation for all distributions) can be found for the thickness of the carbon and boroncarbite layers, the off-center value for the silicon thickness of the PSO result in Fig.4.5c is compensated by a larger than center value for the boroncarbite layer in Fig.4.6. Thus, the thicknesses are correlated and are no independent model parameters. Nevertheless, confidence intervals can be obtained within the given model

Reconstruction Based on Specular EUV Reflectance 4.1

0 1 2 3 4 5 6 7 d_SiO2(cap)/ nm

0 1 2 3 4 5 6 7 d_C/ nm

0 1 2 3 4 5 6 7

dB₄C/ nm 0.0 0.5 1.0 1.5 2.0 σ/ nm

likelihood/arb.units

0.5 0.8 1.1 1.4 1.7 2.0 ρSi

0.5 0.8 1.1 1.4 1.7 2.0 ρMo

0.5 0.8 1.1 1.4 1.7 2.0 ρC

0.5 0.8 1.1 1.4 1.7 2.0

ρ_B4C likelihood/arb.units

Figure 4.6 |In analogy to Fig. 4.5b and 4.5c the one dimensional projections of the likelihood distribution estimation for the Mo/B₄C/Si/C sample are shown for the remaining parameters of the model with the PSO result, the center value and one standard deviation.

and the given prior (the boundaries listed in table 4.1) and are listed accordingly in table4.3 for one and two standard deviations. Within the allowed boundaries, some parameters remain entirely undefined with similar likelihood for any parameter value, such as the SiO2 capping layer thickness, the silicon, carbon and boroncarbite relative densities. Their corresponding total confidence intervals thus cover almost exactly 68.2%

(one standard deviation) and 95.4% (two standard deviations) of the allowed respective parameter range. Hence, with respect to the model defined and the measured EUV reflectivity curve, no reliable value for those sample properties can be determined.

Table 4.3 |MCMC results obtained by the analysis of the EUV reflectivity for the Mo/B₄C/Si/C sample.

The center values (50%percentile) together with confidence intervals (c.i.) of one and two standard deviations are shown.

Parameter PSO result center value 1σc.i. 2σ c.i.

d_SiO2(cap) / nm 3.194 3.139 (−^1.077/+1.108) (−^2.704/+3.378) d_Mo / nm 2.460 1.998 (−^0.422/+0.429) (−^0.789/+0.945) dSi / nm 2.421 2.910 (−^0.529/+0.473) (−^1.162/+0.862) d_C / nm 0.811 1.190 (−^0.516/+0.459) (−^0.947/+0.968) d_B4C / nm 1.308 0.894 (−^0.531/+0.560) (−^0.825/+1.060) σ/ nm 0.322 0.456 (−^0.211/+0.206) (−^0.376/+0.399) ρ_Mo 0.989 1.086 (−^0.098/+0.147) (−^0.183/+0.340) ρSi 0.883 0.851 (−^0.219/+0.253) (−^0.330/+0.491) ρC 0.833 0.941 (−^0.297/+0.418) (−^0.421/+0.846) ρ_B4C 0.909 1.115 (−^0.435/+0.572) (−^0.588/+0.845) It should be noted, that the given center values here are not a good solution to the optimization problem. The reason for that is, that the parameters are highly correlated.

The center values of the one-dimensional projections may therefore not be suitable pa-rameters for the model based on the low amount of data available. A valid optimization result can therefore only be obtained by either applying the PSO routine or by iterative application of the MCMC procedure. The latter may be achieved by fixing single parame-ters according to their maximum likelihood value found in the previous iteration and obtaining the resulting likelihood distributions for the remaining parameters according to that restricted prior distribution.

The results listed in table4.3serve as the model parameters for the analysis of diffuse scattering from the Mo/B₄C/Si/C sample in chapter 5.

4.2 Molybdenum Thickness Variation in Mo/Si/C