6.2 Automatic Calibration and Feature Detection Algorithm
As already evaluated in Section 5.2, the best-suited flux modeling approach is the top-down pseudo-particle tracking from Section 2.3.3 using constant and effective sticking coefficient (\(\beta _\mathrm {eff}\)). The normalized flux convention from Eqs. (2.7) and (2.8) is applied, therefore, there are two key parameters per involved material: the plane-wafer etch
rate (PWR) and \(\beta _\mathrm {eff}\). As seen in Fig. 6.1, the materials involved in the etching process are Si and the photoresist (\(\mathrm {res}\)).
Additionally, as discussed in Section 5.3, the reactor loading effect might play a role after the removal of the photoresist, thus the \(\mathit {PWR}_\mathrm {Si}\) can take
different values for each etch step. In summary, there are five model parameters: \(\beta _\mathrm {Si}\), \(\beta _\mathrm {res}\), \(\mathit {PWR}_\mathrm {res}\), the first etch step \(\mathit {PWR}_\mathrm {Si}\),
and the second step \(\mathit {PWR}_\mathrm {Si}\).
Some of these parameters, most notably the second etch step \(\mathit {PWR}_\mathrm {Si}\) and \(\mathit {PWR}_\mathrm {res}\), are in principle experimentally accessible. However, their values were not recorded,
therefore, all five parameters must be calibrated to the experimental topographies. This number of free parameters is relatively high, as the only experimental data are the state of the cavities at the end of processing. Therefore, a
robust calibration procedure is required to extract physically meaningful parameters values. In particular, the challenge of comparing disparate geometrical data — the simulated topography and the profilometer data — must be
addressed. The algorithm which achieves this calibration is represented in Fig. 6.3.
The role of the topography simulator, which ingests the parameters and produces a simulated mirror microcavity, is shown in Fig. 6.3.a) through c). However, as discussed
in Section 6.1, the simulated microcavity is outputted as a triangle mesh, while the experimentally characterized microcavity is described by a coordinate list.
Furthermore, the coordinate systems used by both representations is not the same. Therefore, to meaningfully compare both surfaces and evaluate if the parameters are correct, feature detection is required. At the heart of the
custom developed feature detection algorithm is the circular Hough transform [200], implemented in the OpenCV library [201]. This method is used to determine the center of the cavity and its maximum opening
diameter \(O\), therefore, delimiting the area of interest. It is very robust, being able to operate on incomplete experimental data, shown in Fig. 6.4.
Having extracted not only the value \(O\) of the microcavity opening but also the position of its center, the indeterminacy of the coordinate systems is resolved, since they can be matched at the microcavity minima. Therefore, the
microcavity maximum depth \(h\) is also determined. To further simplify the structures in the presence of radial symmetry, all the data points are projected into their radial coordinate, thus reducing the data set from a
three-dimensional (3D) to a two-dimensional (2D) problem. This is done instead of a simple cross-section to maximize the use of the experimental data, as the tilted profilometry measurement is not available for the entire
microcavity. A functional description of the microcavity shape is then extracted via a sixth-order polynomial fit \(\mathrm {poly}(x)\) over this 2D data set. Figure 6.3.e)
summarizes the outputs of the feature detection procedure: The maximum opening \(O\), the maximum depth \(h\), and the polynomial shape representation \(\mathrm {poly}(x)\).
Figure 6.4: Microcavity opening extraction using the circular Hough transform on incomplete experimental data. The microcavity has been removed in a preprocessing step, therefore, the experimental data shows only the flat regions.
These outputs are by themselves not sufficient for the calibration algorithm. They must be combined into a cost function which is minimized during the calibration procedure, shown in Fig. 6.3.g). It is defined as the Euclidean norm of a vector of residuals which is itself constructed for each microcavity to be evaluated as
\(\seteqnumber{0}{6.}{0}\)
\begin{align}
\label {eq::cost_function_residuals} \left ( \begin{array}{c} |h_{\mathrm {exp}} - h_{\mathrm {sim}}| \\ \frac {1}{N_x}\sum _x |\mathrm {poly}_{\mathrm {exp}}(x) - \mathrm {poly}_{\mathrm
{sim}}(x)| \end {array} \right ) \, ,
\end{align}
where \(N_x\) is the number of samples of \(x\) taken homogeneously from the region delimited by the microcavity opening \(O\), and the subscripts \(\mathrm {exp}\) and \(\mathrm {sim}\) refer to experimental and
simulation data, respectively. The opening \(O\) is not considered directly since it is already implicitly taken into account by the entry involving \(\mathrm {poly}(x)\). As this calibration procedure can be simultaneously
performed on \(m\) multiple microcavities, the total vector of residuals has \(2m\) entries.
The determination of the minimum of the cost function and, thus, the calibrated parameters, is performed with a global optimizer from the SciPy library [202], shown in Fig. 6.3.h). From the multiple available global optimizers, the generalized simulated annealing (GSA) algorithm is used [203]. Since the top-down pseudo-particle tracking is a
Monte Carlo method, the simulated surfaces possess some noise which can propagate to the evaluated cost function. Therefore, local explorations of the parameter space should be avoided, as differences in Monte Carlo noise
between similar simulations might lead to a fruitless search. The GSA algorithm, being stochastic in nature, performing comparatively large jumps in parameters, and not conducting local searches, is the most suited algorithm from
those available within SciPy.
The calibration is performed over \(m=3\) experimental microcavities with initial photoresist opening diameters of \(\SI {12.4}{\micro \meter }\), \(\SI {34}{\micro \meter }\), and \(\SI {52}{\micro \meter }\). The
simulation domain is \(\SI {160}{\micro \meter } \times \SI {160}{\micro \meter }\) with a grid spacing \(\Delta x = \SI {0.5}{\micro \meter }\). Reflective boundary conditions are used. Approximately 6.1
million pseudo-particles are used per LS advection time step, that is, \(120\) pseudo-particles per source plane grid point. The calibration procedure is performed on a dual-socket compute node equipped with dual Intel Xeon Gold
6248 processors, for a total of 40 physical computing cores. Since each cavity can be simulated independently, each simulation is assigned \(13\) threads to homogeneously distribute the workload, since the topography simulation in
Fig. 6.3.b) is the most resource-intensive operation. The remaining steps are single-threaded. Although the precise runtime varies greatly depending on the parameter
bounds, a typical GSA run requires approximately \(\SI {60}{\hour }\) to converge to an optimized set of parameters.
After successfully applying the GSA algorithm, a comparison of the simulated surfaces using the calibrated parameters to the experimental data is shown in Fig. 6.5, showing
excellent agreement. The automatically calibrated parameters are shown in Tab. 6.1. The calibrated values of \(\mathit {PWR}_{\mathrm {Si}/\mathrm {res}}\) and
\(\beta _{\mathrm {Si}/\mathrm {res}}\) are consistent with those reported in the literature [45, 171, 185, 186]. The reduction in \(\mathit {PWR}_{\mathrm {Si}}\) of approximately \(70\,\%\) in the second etch
step is a consequence of the reactor loading effect, as discussed in Section 5.3, and is consistent with other reported experiments [46, 131, 204]. The ratio of the first step
\(\mathit {PWR}_{\mathrm {Si}}\) to \(\mathit {PWR}_{\mathrm {res}}\), that is, the observed photoresist selectivity of approximately 10 to 1, is in line with reported data from the literature [186, 205, 206]. This is
remarkable evidence that the automatic calibration procedure is very effective, as it can obtain physically meaningful parameters even without direct measurements.
