Ellipsometry is widely used to characterize the thickness and optical parameters of thin films deposited, for example, in industrial processes. It is based on the measurement of polarization change upon reflection of, for example, visible light at a material sample. Commercially available devices are designed for stationary applications and often rely on precise geometric adjustment of the optical setup to maximize the measurement precision. In this work, a simplified spectral ellipsometer is proposed and tested with the aim of flexible implementation in space-limited applications in thermonuclear fusion research: on the one hand, as a hand-held device for large thickness scans of coatings deposited on first-wall components inside the vacuum vessel of fusion experiments and, on the other hand, for in situ monitoring of plasma deposited coatings on diagnostic vacuum windows, reducing their transmission in the optical spectral range, which hampers spectroscopic diagnostics in long-pulse plasma experiments. The simplicity of the hardware setup is partially compensated by complex Bayesian inference of the coating parameters, which incorporates all uncertainties of the measurement and the model and provides a quantitative assessment of the final uncertainties of inferred coating parameters. The Bayesian inference based on synthetic observations is also used to optimize the diagnostic design, identifying the limiting parameters and quantifying their impact on final accuracy. For real-time analysis of layer thickness on first-wall components in fusion devices measured with the hand-held device, a neural network based analysis has been implemented, and promising test results are presented.

Electromagnetic waves reflected at an interface of materials with different optical parameters, the refractive index and extinction coefficient (n, k), experience under certain conditions not only intensity reduction but also a phase delay, both depending on the incident angle, optical parameters, and the polarization direction (s and p). Multiple reflections occur in the case of a thin not opaque material layer covering the reflecting surface. Interference of the reflected waves can significantly change the reflected intensities and phase delays, encoding information on the optical parameters and the layer thickness. These effects are utilized in ellipsometry, which measures the polarization state (ellipticity) of the reflected wave.1 Ellipsometers are used to monitor many industrial processes based on the deposition of thin material layers,2 to mention one example closely related to spectroscopic diagnostics in fusion devices, coating of optical components to reduce reflections or to produce band-pass transmission filters at wavelength ranges of interest. Specific to thermonuclear fusion research is the application of ellipsometry to study migration processes inside the vacuum vessel of materials sputtered from the first-wall components and deposited as thin layers elsewhere in the device.3,4 While for ex situ layer analysis,5 commercial ellipsometers using monochromatic (incl. lasers) or broad-band light sources1 are employed, for in situ thickness scans of coating at first-wall components in the breaks between experimental campaigns and with human access to the plasma vessel, a hand-held colorimeter has been developed6,7 and applied at the experiments Wendelstein 7-X8 and Large Helical Device.9,10 It is based on a reduced detection domain, including only the reflected light intensity at three colors, and results in a significantly smaller range of coating thicknesses, which can be unambiguously inferred. Another potential application of ellipsometry in fusion research is connected to long pulse operation of modern experiments, which poses a risk of gradual transmission losses of vacuum windows applied in spectroscopic diagnostics and coated by plasma deposition.11 Such transmission losses can be monitored using ellipsometry.

This work aims at proposing and testing a spectral ellipsometer for in situ thin layer measurements comprising a minimum set of hardware components to allow for their flexible adaptation in limited space in specific fusion-related applications while keeping the measurement accuracy at an acceptable level. The shortcomings of the simple hardware setup, e.g., light-weight mechanical support structure, shall be partially compensated by complex Bayesian data analysis, which significantly enhances the capabilities of the method (an example application of Bayesian analysis of ellipsometry is given in Ref. 12). Two specific applications for fusion experiments are discussed in Sec. II: (1) A hand-held ellipsometer for multiple measurements at large surfaces of first-wall components in devices with neutron production at a low enough level that allows man access to the vacuum vessel between the experimental campaigns. Such an ellipsometer assembled on a remote handling arm in experiments with high neutron production could be considered as well. However, the impact of the neutron fluxes on the mechanical and optical components (e.g., darkening of lenses and fibers) would need to be accounted for, which is not the subject of this work. (2) An in situ ellipsometer for online monitoring transmission losses in the visible light of vacuum windows in diagnostic ports, coated by plasma deposition. The layer thicknesses, which are expected to be diagnosed with both methods, lie in the range of ∼10–1000 nm.

The assessment of the final accuracy of the inferred coating parameters is not straightforward due to the complex patterns of recorded raw spectra of the reflected light, which strongly depend on the coating parameters and the geometric parameters of the diagnostic setup. In order to properly quantify the error propagation, a Bayesian model of the measurement has been implemented in the Bayesian modeling framework Minerva,13 which is described in Sec. III. The diagnostic setup and the Bayesian inference model were validated with test measurements at a transparent coating standard with known refractive index n(λ) and layer thickness. Example analysis results in terms of the probability distribution function (PDF) of the inferred parameters are shown in Sec. IV. These results demonstrate the capabilities of the diagnostic setup for the case of a single smooth transparent layer as well as the ability to precisely calibrate/characterize the geometric parameters of the assembled diagnostic setup. The Bayesian analysis approach based on the inference of synthetic observations is a very powerful tool for diagnostic design in the sense of finding and optimizing critical diagnostic parameters that limit the accuracy of the inferred physics parameters. In Sec. V, it is demonstrated to which accuracy the refractive index of a transparent coating can be inferred, given realistic model uncertainties, such as the detector noise and geometric assembly tolerances. Critical parameters of the diagnostic design have been identified, which are the drivers in limiting the inference accuracy. In Sec. VI, a neural network based analysis approach is presented, which is aiming at providing a real-time analysis tool for the layer thickness, important for large area thickness scans of coated first-wall components in fusion devices.

Ellipsometry of thin films utilizes the strong dependence of the amplitude and phase delay of light reflected at a substrate with layers on the incident angle, layer parameters, as well as the polarization and wavelength of the incident light. The complex reflection coefficients for both polarizations (s, p) in the case of reflection at a single interface with isotropic complex refractive indices n1 = n1ik1 and n2 = n2ik2 are given by the Fresnel equations,
(1)
(2)
Summation of an infinite number of interfering waves reflected at a single-layer film yields the following complex reflection coefficients:14 
(3)
with λ denoting the wavelength, d denoting the layer thickness, as well as r1s,p and r2s,p denoting the complex reflection coefficients derived from Eqs. (1) and (2) at interfaces 1 and 2 (see Fig. 1) for the s and p polarization, respectively.
FIG. 1.

Multiple reflection of light at a single-layer thin film.

FIG. 1.

Multiple reflection of light at a single-layer thin film.

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The proposed ellipsometer setup is based on detecting the change of the ellipticity, i.e., of the relative phase delay and the relative intensities, of both polarization components of incoming visible light reflected at a substrate with a thin layer. The incoming light at the sample is linearly polarized with equal intensity of the s and p polarization components. The relative phase delay between both components is measured by including a linear polarizer in front of the collection lens and rotating the polarizer in steps of, for example, 10° over the range of 180°.

The knowledge of the (absolute) intensity of the incident light provides additional information on the reflectivity of both polarization components (and not only their relative reflectivity), reducing the final measurement uncertainties. The intensity of the incident light can, in principle, be measured by coupling out its fraction and feeding it directly to the detector bypassing all components of the ellipsometer. However, in measurements of coatings on diffusively reflecting surfaces a large fraction of the light entering the ellipsometer will not reach the detector. Therefore, the method of redirecting a fraction of the incident light directly to the detector will merely provide monitoring of the spectral shape of the incident light, not its intensity. In the presented work, the incident light intensity was inferred from reference spectra recorded upon reflection at a pure substrate (of known optical parameters), which provides a way of in situ calibration, eliminating the potential impact of a misalignment of any component inside the ellipsometer path. Such reference spectra can also be obtained on a coated substrate if the optical parameters of the film are known as well (this would apply in the case of in situ monitoring of vacuum windows, which is described at the end of this section).

The principle setup of the device is shown in Fig. 2. The incoming light originates from a stabilized white light source coupled into a light fiber and imaged at the sampled film surface by a single uncoated lens. A simple multimode short (∼1 m) fiber with a core diameter of 0.5 mm was used. An incident angle of 70° was selected as often used in the thin film ellipsometry to enhance its sensitivity.15 The imaged spot size amounts to ∼2 × 2 mm2 and could be further reduced in case of a strong thickness inhomogeneity of the investigated coating. A Glan–Thompson polarizer is inserted between the lens and the sample adjusted at a fixed polarization direction of 45°, providing linearly polarized light of equal intensity of the s and p polarization components. The observation arm of the setup is a symmetric copy of the illumination arm with the only difference that the analyzer (the same type of Glan–Thompson polarizer) is mounted on a rotatory stage. In addition, a longer fiber (∼10 m) was chosen to depolarize the detected light in order to avoid the consideration of polarization dependent sensitivity of the spectrometer. For measurements on rough surfaces, an optional, potentially rotating compensator (typically a λ/4 delay plate) could be added in the illumination or observation arm to account for potential depolarization effects. This is deferred to future work with the aim to reduce inaccuracies of measurements at rough surfaces of first-wall components in fusion experiments. A one-channel universal serial bus (USB) type spectrometer with an uncooled CCD chip covering the spectral range of 400–900 nm was selected as a detector. The spectral resolution of 1 nm is fully sufficient. In fact, to speed up the Bayesian inference, most of the measured data points (the detector array contains 2048 pixels) were excluded from the analysis: it turned out that including the intensities at only every ∼15 nm was enough to minimize the final uncertainties of the inferred coating parameters (see Sec. V). With the exposure time of 10 ms, averaging over several tens of spectra and rotating the analyzer in 10–20 steps, the full measurement time amounts to a few seconds. In applications with diffusively reflecting surfaces, the amount of collected light is strongly reduced. For such a case, a USB spectrometer with a Peltier-cooled detector has been tested and provided a much better signal-to-noise ratio.

FIG. 2.

Principle hardware setup of the in situ ellipsometer.

FIG. 2.

Principle hardware setup of the in situ ellipsometer.

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The optional housing indicated in Fig. 2 encompasses the minimum set of the ellipsometer components, which require precise relative adjustment. The total weight and small size of the assembly make it possible to be used as a hand-held device for plasma deposition layer analysis at first-wall components of fusion experiments. Another application of the proposed ellipsometer setup is in situ monitoring of thin layers deposited by the plasma on the vacuum side of diagnostic observation windows in fusion experiments, which lead to gradual transmission losses in the UV, visible, and IR spectral range. The principle arrangement of the components is shown in Fig. 3. In the case of the usually smooth coatings on vacuum windows, the use of a compensator is not necessary. To save space, thin foil instead of Glan–Thompson polarizers could be used. It is noteworthy to mention that all these components, including piezo-crystal based rotatory stages, are nowadays available with a compatibility with ultrahigh vacuum and relatively high magnetic fields and could, in principle, be installed inside the vacuum vessel of a fusion device. The lens with the polarizer of the illuminating and detecting arm of the ellipsometer can be fixed on either side of the optical head of the primary spectroscopic diagnostic. Special attention needs to be paid to avoid collecting light from the reflection at the front (uncoated) side of the window, which would effectively lead to partial depolarization of the detected light.

FIG. 3.

Arrangement of the ellipsometer components in a diagnostic port for monitoring thin layers deposited on vacuum windows in fusion devices.

FIG. 3.

Arrangement of the ellipsometer components in a diagnostic port for monitoring thin layers deposited on vacuum windows in fusion devices.

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The probabilistic Bayesian data analysis is a powerful approach that allows quantitative error propagation in complex models.16 All uncertain model parameters (e.g., physics constants, uncertain results from additional measurements, hardware setup parameters, and thin layer parameters to be inferred) as well as actual observations are described in terms of probability distributions and are combined using Bayes’ theorem into a final so-called posterior probability distribution, a function of all uncertain parameters. Optimizers, for example, gradient descent or pattern search algorithms, can be used to find the most probable set of parameters, while Markov chain Monte Carlo (MCMC) sampling (e.g., adaptive Metropolis–Hastings algorithms17,18 used in this work) explores the multi-dimensional posterior and provides not only one-dimensional marginal PDFs for single parameters but also multi-dimensional correlations between them. The Bayesian modeling framework Minerva was employed for data analysis shown in this work. The model contains the following uncertain parameters, and the form of the a priori assumptions applied are given in brackets:

  • The thickness of the (single) layer (uniform PDF).

  • The refractive index n(λ) as a function of wavelength is modeled using a non-parametric Gaussian process19,20 with a squared exponential covariance function. The chosen scale length is 200 nm. In the same way, the extinction coefficient k(λ) is included in the model; however, in this work, only transparent layers have been analyzed, and hence, k(λ) was set to zero for all wavelengths.

  • The incident angle (normal PDF).

  • Common offset of rotation angle for all angles of the analyzer at which the measurements were done. It accounts for the imperfection of positioning in the space of the rotatory stage with the analyzer (normal PDF).

  • Additional imprecision of each analyzer angle used in the measurements (multivariate normal PDF).

  • Spectral shape of the incident light intensity (multivariate normal PDF). It accounts for the change in the spectral shape, for example, induced by filament temperature drifts of the light source (halogen lamp). The mean value is derived from a reference measurement at one selected analyzer angle.

  • Scaling factors for the entire spectrum of the incident light for measurements taken at further analyzer angles (i.e., in addition to the one from which the mean value of the incident light was derived; multivariate normal PDF).

  • Detector non-linearity correction: a (linear) intensity correction factor (normal PDF).

In order to validate the hardware setup as well as the Bayesian analysis model and the inference in Minerva, test measurements were done using standard layers of SiO2 with known thickness on a silicon wafer, which featured also a reference surface without any coating. The spectra were recorded at nine analyzer angles (between 0° and 160° relative to the incident plane, in 20° steps). At each setting of the analyzer angle, two spectra (background subtracted) were recorded: with the incident light reflected at the coated and the reference area to ensure exactly the same setting of the analyzer angle since in some cases, the measured spectra reacted very sensitive to small changes in the analyzer angle. One example of spectra recorded at an analyzer angle of 80° is shown in Fig. 4. For the analysis, only ∼30 data points (every ∼15 nm) in each spectrum have been selected to speed up the Bayesian inference (see the explanation in Sec. II and as well as the test result with a doubled number of data points included which is explained in detail in Sec. V and summarized in Table II).

FIG. 4.

Measured and fitted raw spectra recorded at the analyzer angle of 80° (precisely 81.6°) for the case of illuminating the coated and the reference area. The very good fits lead to overlaps of the lines representing the predicted and measured spectra. In light-blue color, the residuum for the case of the measurement with the coating is shown.

FIG. 4.

Measured and fitted raw spectra recorded at the analyzer angle of 80° (precisely 81.6°) for the case of illuminating the coated and the reference area. The very good fits lead to overlaps of the lines representing the predicted and measured spectra. In light-blue color, the residuum for the case of the measurement with the coating is shown.

Close modal
TABLE I.

Impact of increased model uncertainties on the accuracy of inferred layer thickness and refractive index based on synthetic data.

Model assumptionsResulting uncertainties
Assume that lowest expected uncertaintyof  ±0.05 nm 
geometric parameters of ±0.1° (n, k) are known  
  
Add n(λ) to fitted parameters Considerably increased ±0.8 nm [n(λ): ±0.001] 
  
Increased by factor 5 (up to ±0.5°), the Not increased ±0.8 nm [n(λ): ±0.001] 
uncertainty of incident angle n(λ) is fitted as well  
  
Increased by factor 5 (up to ±0.5°), uncertainty Increased ±1.07 nm [n(λ): ±0.0015] 
of all nine analyzer angles n(λ) is fitted as well  
Model assumptionsResulting uncertainties
Assume that lowest expected uncertaintyof  ±0.05 nm 
geometric parameters of ±0.1° (n, k) are known  
  
Add n(λ) to fitted parameters Considerably increased ±0.8 nm [n(λ): ±0.001] 
  
Increased by factor 5 (up to ±0.5°), the Not increased ±0.8 nm [n(λ): ±0.001] 
uncertainty of incident angle n(λ) is fitted as well  
  
Increased by factor 5 (up to ±0.5°), uncertainty Increased ±1.07 nm [n(λ): ±0.0015] 
of all nine analyzer angles n(λ) is fitted as well  

A total of 18 measured spectra were analyzed using MCMC exploration of the posterior distribution. In addition to the thickness (uniform PDF), the following uncertain model parameters (prior PDFs) were assumed in the analysis:

  • Incident angle: 70° ± 3°.

  • Polarizer angle: 45° ± 3°.

  • (Common) offset of analyzer rotation angle: 0° ± 3°.

  • Additional imprecision of each of nine analyzer rotation angles: 0° ± 0.2°.

  • Spectral shape of the incident light intensity: mean values from reference measurement at analyzer angle of 0° ± 105 counts.

  • Scaling factors for incident intensity at remaining analyzer angles: 1 ± 0.2.

  • Non-linearity correction factor: 1 ± 0.03, truncated to 0.98–1.02.

The refractive index n(λ) of SiO2 was assumed to be known. In Fig. 5, the histogram of converged MCMC samples is shown, constituting the marginal PDF for the layer thickness. Its mean value of 493.3 nm lies within the confidence interval provided by the manufacturer of the thickness standard of 492.4 ± 1 nm, confirming that the ellipsometer setup and the analysis model work properly. In Fig. 4, the mean values of the predicted raw spectra calculated from converged MCMC samples of the fitted parameters are plotted as well, showing very good fit of the observations. The light-blue curve in the same figure depicts the relative residuum of the fit of the measurement with the coating. The residuum shows remaining spectral features, which cannot be explained by detector non-linearity nor by other effects related to uncertain parameters assumed in the model. The features in the residuum are potentially caused by stray light in the spectrometer or some low level of light depolarization at imperfect surfaces of optical components used in the test ellipsometer setup. The existence of the features in the residuum could also explain the small difference of 0.9 nm between the inferred mean thickness and the nominal value provided by the manufacturer of the thickness standard.

FIG. 5.

Marginal PDF (histogram of MCMC samples) of the film thickness inferred from the Bayesian model using real measurements shown exemplarily in Fig. 4. The mean value of 493.3 nm lies within the confidence interval 492.4 ± 1 nm provided by the manufacturer of the thickness standard.

FIG. 5.

Marginal PDF (histogram of MCMC samples) of the film thickness inferred from the Bayesian model using real measurements shown exemplarily in Fig. 4. The mean value of 493.3 nm lies within the confidence interval 492.4 ± 1 nm provided by the manufacturer of the thickness standard.

Close modal

The standard deviation of the inferred thickness of 0.24 nm lies below 0.1% of the thickness itself, showing, in principle, a very high precision of the measurement even given the relatively large prior uncertainties (as listed above) assumed in the inference. In fact, all geometric parameters assumed uncertain in the model were inferred from the test measurement with much higher precision than the initial uncertainty inputted via the prior PDFs. One example is shown in Fig. 6: The inferred incident angle of 70.1° is close to the maximum prior PDF value at 70°, while the standard deviation of the posterior PDF of 0.09° is much lower than the assumed value of 3°. Furthermore, the inferred value of the polarizer angle was 46.9° ± 0.15°, the analyzer angle offset was 0° ± 0.12°, and the first four analyzer angles was [0.6° ± 0.15°, 21.2° ± 0.16°, 40.9° ± 0.17°, 60.9° ± 0.18°, …]. It is evident that the measurement at the one thickness standard provides a way for a relatively precise calibration (of ∼±0.15°) of the geometric parameters of the ellipsometer setup, at least in the case of a perfect diagnostic model, since other values are just not supported by the measured data points (observations). Given the fact of the remaining features in the residuum, the inferred angles may not perfectly reflect the reality but provide a good guess for the use in the employed imperfect model. Determining the geometric parameters using 3D scans of the assembled hardware components seems rather demanding and would need to be repeated periodically to monitor any changes introduced, for example, by mechanical vibrations, which cannot be excluded in the case of a hand-held device.

FIG. 6.

Marginal PDF (histogram of MCMC samples) of the incident angle inferred from the Bayesian model using real measurements shown exemplarily in Fig. 4.

FIG. 6.

Marginal PDF (histogram of MCMC samples) of the incident angle inferred from the Bayesian model using real measurements shown exemplarily in Fig. 4.

Close modal

In another test analysis using the same measurements and the adjusted geometric parameters as described above and assuming them as known, the inference of the refractive index n(λ) was investigated. The inferred spectrum of n(λ) is shown in Fig. 7 and demonstrates the principle ability to derive the refractive index with relatively high precision (the resulting thickness was 492.7 ± 0.98 nm).

FIG. 7.

Initial and inferred refractive index using real measurements shown exemplarily in Fig. 4.

FIG. 7.

Initial and inferred refractive index using real measurements shown exemplarily in Fig. 4.

Close modal

The quantified full error propagation accessible with the Bayesian inference makes it an ideal tool for optimization of a diagnostic design by finding its critical parameters that limit the final precision of the inferred physics quantities. In a case when real data are not available, synthetic observations with realistic noise level can be easily generated and used instead. In the following, two examples of Bayesian diagnostic design using synthetic data will be discussed: the impact of increased model uncertainties (mostly the mechanical precision of the diagnostic assembly) and extending the observation dataset on the final uncertainty of inferred layer thickness.

Table I summarizes the results of the first example (in this case, the non-linearity correction factor as well as the spectral shape of the incident intensity were assumed as known). The precision of inferred thickness in the case of the synthetic data (±0.05 nm) is significantly lower than in the case of real measurements (±0.24 nm, see Fig. 5), which is most likely caused by the perfect inference model (as the same model was used to generate the synthetic observations). The thickness uncertainty increases considerably to ±0.8 nm in case the refractive index n(λ) is fitted as well [the corresponding value of ±0.98 nm value was found when fitting real data (the inferred n(λ) is shown in Fig. 7)]. The thickness uncertainty does not change in the case of increasing by factor 5 the uncertainty of the incident angle, while it does (up to the value of ±1.1 nm) if increasing by the same factor the uncertainties of all analyzer angles. Noticeably, the uncertainty of the inferred refractive index is rather low (∼±0.001, increased to ±0.0015 in the last case). This demonstrates the sensitivity of the final inference accuracy on the precision of critical geometric parameters and the importance of (repeated) calibration measurements in order to reduce the uncertainties of, for example, the analyzer angles down to ±0.1°.

TABLE II.

Impact of the extended set of observations on the accuracy of inferred layer thickness and refractive index based on synthetic data.

Model assumptionsResulting uncertainties
Increased by factor 5 (up to ±0.5°), uncertainty n(λ±1.1 nm [n(λ): ±0.0017] 
is fitted as well of all nine analyzer angles   
  
Doubled number of data points ±1.1 nm [n(λ): ±0.0017] 
(wavelengths) in the spectra (to 60)  
  
Extended wavelength range (to 350–1200 nm) ±1.1 nm [n(λ): ±0.0017] 
  
Doubled number of analyzer angles (to 18) Considerably reduced 
 ±0.5 nm [n(λ): ±0.0008] 
Model assumptionsResulting uncertainties
Increased by factor 5 (up to ±0.5°), uncertainty n(λ±1.1 nm [n(λ): ±0.0017] 
is fitted as well of all nine analyzer angles   
  
Doubled number of data points ±1.1 nm [n(λ): ±0.0017] 
(wavelengths) in the spectra (to 60)  
  
Extended wavelength range (to 350–1200 nm) ±1.1 nm [n(λ): ±0.0017] 
  
Doubled number of analyzer angles (to 18) Considerably reduced 
 ±0.5 nm [n(λ): ±0.0008] 

In the second example, it is investigated to which degree the initial thickness uncertainty of ±1.1 nm as obtained in the case of increased uncertainties of all nine analyzer angles can be reduced by extending the set of observations (the results of these tests are summarized in Table II). Neither doubling the number of data points (from 30 to 60) included in the analysis nor extending the wavelength range from initially 400–900 to 350–1200 nm provides more information such that the final uncertainties of the thickness and refractive index remain unchanged. In contrast, doubling the number of analyzer angles at which the spectra are recorded and analyzed (reducing the steps between the angles from 20° to 10°) significantly reduced (by factor 2) both the uncertainty of the thickness and the refractive index. This example demonstrates the design optimization capabilities in the sense of selecting the hardware components, such as the spectral range of the light source and the spectrometer. Any other diagnostic design parameter can be analyzed in this way, and it is planned for the application as a hand-held device to study the performance improvement of the device when adding a second measurement channel at a significantly different incident angle of, for example, 40°–50°, which can likely be accommodated in the diagnostic head without jeopardizing convenient handling of the device during measurements.

The aim of the presented considerations was to point out the principle capabilities of the Bayesian diagnostic design. The discussed examples show results just in a small fraction of the parameter space. At higher layer thicknesses, the interference patterns observed in the spectra would likely significantly change, showing an increased value of extending the observed wavelength range in the UV. Therefore, it is necessary to conduct such an analysis for a specific application, assuming expected realistic types of coatings and substrates as well as their thicknesses. If n(λ) and/or k(λ) should be fitted as well, surely ambiguous solutions will be found, and in this case, it should be assessed to which degree this ambiguity can be avoided by limiting the parameter space, for example, to values expected on the basis of previous experiments and observations. These limitations can be provided as hard limits (truncation) and/or as prior probability distributions of the fitted parameters. Hilfikter et al.21 demonstrated additional methods to increase the information content of ellipsometry. For inference of k(λ), the relationship between n(λ) and k(λ) described by the Kramers–Kronig relations22 can be implemented in the Bayesian model to better constrain the available parameter space.

The complex Bayesian analysis as discussed in Secs. IIIV is a very powerful tool for the diagnostic design as well as for the final analysis of data taken at layers of limited knowledge of their optical parameters. In such cases, the Bayesian inference will provide reliable final uncertainties, given all uncertain model parameters. However, this complex analysis comes at a cost of computational time: one full MCMC based inference takes a few hours on a modern desktop PC. In the envisaged application of the ellipsometer as a hand-held device for multiple thickness measurements of large surface areas of first-wall components in a fusion experiment, a fast analysis will be helpful for a quick judgment of spatial gradients of the thicknesses for a proper choice of an optimum grid the measurements should be taken at.

For this purpose, a neural network based on the Tensor Flow keras library was trained and the first optimization was done, providing promising results. Between one half and one million samples of synthetic spectra for measurement with and without (reference) a coating have been used to train the network. Realistic signal noise was applied, 30 data points in the wavelength range of 400–900 nm were created for each spectrum, and the incident angle of 70° was assumed as well as 45° for the polarizer angle and nine settings of the analyzer angle (in 20° steps). The optical parameters n(λ) and k(λ) of SiO2 in the layer and Si in the substrate were used and assumed as known. In the top panel of Fig. 8, the layer thicknesses predicted by the network vs the assumed thicknesses are plotted for the case that the synthetic spectra were created by sampling the layer thickness (in the range of 1–1000 nm) and from one common incident intensity scaling factor for all spectra (in the range of 0.33–3.0). The layer thickness is predicted with relatively high precision for almost the entire input thickness range (the relative standard deviation of the predicted thicknesses lies below ∼27% for input thickness greater than ∼20 nm; the same is true for the scaling parameter, not shown in the figure). In the bottom panel of Fig. 8, training results for the thickness are plotted for a more realistic case that all geometric parameters of the diagnostic setup are assumed uncertain with the standard deviation of 0.1°. This assumption indeed increases the scattering of the predicted layer thickness especially for lower values (the relative standard deviation now exceeds 25% for thicknesses below ∼100 nm). Figure 9 shows the results of an optimized network: In this case, approximately one half of the samples were created by sampling from lower values of the layer thickness (below 200 nm) to enhance the network training in this thickness range. In addition, the spectra were created for 18 analyzer angle settings (in 10° steps), and a 2D convolution kernel was employed. The range of thickness with a relative standard deviation exceeding 25% reduces now to values below ∼50 nm. Further optimization of the network is likely possible and will be pursued with the aim of reducing the lower thickness limit to ∼10 nm.

FIG. 8.

Predicted vs assumed (input) layer thickness (blue dots) resulting from neural network based analysis. Top panel: geometric parameters uncertainties not included. Bottom panel: geometric parameters uncertainties included with ±0.1°. The green lines denote the relative standard deviation of thicknesses predicted by the network for a given input thickness. Yellow boxes mark the input thickness range for which the relative standard deviation is lower than 25%.

FIG. 8.

Predicted vs assumed (input) layer thickness (blue dots) resulting from neural network based analysis. Top panel: geometric parameters uncertainties not included. Bottom panel: geometric parameters uncertainties included with ±0.1°. The green lines denote the relative standard deviation of thicknesses predicted by the network for a given input thickness. Yellow boxes mark the input thickness range for which the relative standard deviation is lower than 25%.

Close modal
FIG. 9.

Predicted vs assumed (input) layer thickness (blue dots) resulting from an optimized neural network (uncertainties of geometric parameters are included with ±0.1°). For a more detailed explanation, see the caption of Fig. 8.

FIG. 9.

Predicted vs assumed (input) layer thickness (blue dots) resulting from an optimized neural network (uncertainties of geometric parameters are included with ±0.1°). For a more detailed explanation, see the caption of Fig. 8.

Close modal

Spectral ellipsometers are widely used to derive thickness, and in a limited range, the optical parameters n(λ) and k(λ) of thin layers are deposited, for example, in various manufacturing processes. In thermonuclear fusion experiments, in situ characterization of such layers supports the understanding of material migration with the plasma inside the vacuum vessel and can be used to monitor transmission losses of vacuum windows used to diagnose the plasma by optical spectroscopy. In this work, a simplified setup of an in situ spectral ellipsometer was investigated with the aim to provide an as simple as possible setup, which is flexible enough to be easily adapted in fusion applications with strongly limited space. The simplicity of the setup is partially compensated by application of a complex Bayesian inference with a full propagation of all measurement and model uncertainties.

The measurement is based on the detection of the change in relative phase delay and relative intensities of the s and p polarized light reflected at a coated substrate. A stabilized white light source and a USB type single-channel spectrometer are used enabling measurement in the full visible and potentially in the near UV and near IR spectral range. In the simplest configuration for monitoring smooth coatings on vacuum windows, only two polarizers (one of them on a rotatory stage) constitute the core of the setup. For measurements on rough surfaces of first-wall components (with a hand-held diagnostic head), an additional compensator is needed to detect the depolarization degree of the reflected light. The reflected spectra are recorded for multiple (10–20) settings of the analyzer angle. The incident light spectrum can be either monitored by coupling out a fraction of the incoming light to the spectrometer or derived from a reference measurement at the initial state of the probe (without the coating under investigation).

The probabilistic Bayesian inference was implemented in the Minerva framework. Test measurements on a coating standard (SiO2 film on Si substrate) with known parameters and analysis of the test datasets validated the hardware setup and the Bayesian model. The inferred coating thickness was lying in the range of the confidence interval provided by the manufacturer of the coating standard. In addition, the inference clearly showed a way of calibrating the geometric parameters (angles) of the hardware assembly with the resulting much narrower marginal PDFs of the angles (in the range of ±0.1°) than the assumed prior PDFs (normal distribution with a standard deviation of 0.2°–3°). The probabilistic approach was also used to generate and analyze synthetic observations with the aim to find critical diagnostic parameters limiting the final accuracy of inferred physics quantities. It was found that in the case of the SiO2 coating on the Si substrate, the refractive index n(λ) can be inferred simultaneously to the layer thickness; however, increased uncertainty of the light incident angle from ±0.1° to ±0.5° leads to an increase by 35% of the uncertainty of inferred layer thickness. Doubling the number of data points in the spectrum or extending the measured wavelength range in the near UV and near IR did not improve the accuracy in contrast to doubling the number of analyzer angles at which the spectra are recorded. Such a diagnostic design optimization cannot be generalized but rather needs to be performed in a limited parameter space depending on the specific application of the ellipsometry method.

For the application as a hand-held device for multiple measurements of layer thickness on large surfaces of first-wall components in fusion devices, a neural network has been trained and partially optimized to provide real-time thickness estimates of layers with known optical parameters. In the trained thickness domain of 1–1000 nm, the relative standard deviation of derived values exceeds 25% only for thicknesses below 50 nm. Further optimization of the network seems possible to reduce the lower limit of the accessible layer thicknesses.

Further research is necessary to achieve a final design of the two proposed applications of the ellipsometer. This includes tests with other substrates such as graphite, steel, tungsten, and fused silica and sapphire glass. Layers on rough substrates, for example, of first wall components in fusion devices will need compensation for the fact of partial light depolarization on such surfaces. The reflection model used in the Bayesian inference needs an extension to account for multiple layers and tests with not fully transparent coatings. Both proposed applications of the ellipsometer were primarily meant to be used in fusion devices with a moderate level of radioactivity, allowing, for example, man access to the plasma vessel between experimental campaigns for layer measurements with the hand-held device inside the vacuum vessel. Hardening the ellipsometer design for work at higher radiation levels is conceivable but would need a significant amount of additional development, which was not discussed in this work. For example, the hand-held design would need a mechanical adaptation to a remote handling arm, and the darkening of optical components caused by neutron fluxes would need to be considered.

The author would like to acknowledge the support and discussions with Dr. Ralf König, Dr. Sehyun Kwak, Dr. Jakob Svensson, Dr. Andrea Pavone, Dr. Daniel Böckenhoff, Dr. Dirk Naujoks, and Dr. Oliver Ford. Furthermore, the author would like to thank Seed eScience Research for providing the Minerva framework as well as Dr. Chandra-Prakash Dhard for support with hardware components.

The author has no conflicts to disclose.

Maciej Krychowiak: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available within the article and also from the author upon reasonable request.

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