1. Introduction

The measurement of transmittance spectra of a growing thin solid film under in situ conditions has manifested itself as a powerful source of information on both geometric and material parameters of single- and multilayer coatings [1–4]. In the past decades, the experimental techniques for optical monitoring techniques, including single wavelength monitoring, monochromatic monitoring as well as broadband optical monitoring have been refined, and the accessible spectral ranges have been extended. A detailed survey on relevant milestones of this development may be found in [5]. Today, spectrally broadband optical monitoring is accepted to represent state of the art among the optical in situ measurement techniques [6]. Also, novel technical solutions have been reported that allow combining transmittance measurements with reflectance measurements or other spectroscopic techniques [7–10].

In parallel, the spectra evaluation techniques have been refined. The time-sequential, as well as the full triangular algorithms [11], have been developed and tested; several studies have shown that the full triangular algorithm provides the most reliable coating information from the spectra [12–15].

Let us mention at this point that optical monitors may be used for two entirely different tasks. Firstly, the recorded signals may be used for reliable identification of layer deposition termination points (endpoint detection), thus producing coatings with an exorbitant level of specification adherence. In particular, it could be shown that layer thickness control by broadband optical monitoring techniques incorporates error self-compensation mechanisms [12,16–24]. Secondly, the archived transmittance (and/or reflectance) spectra represent an enormous treasure of information for re-engineering of the coating at a later time.

The purpose of this paper is to contribute to the second point. We will use in situ recorded transmittance spectra to gain information on thickness and in situ relevant optical constants of the individual layer of an optical coating. We make use of the OptiMon in situ spectrometer [9] integrated into a Bühler plasma ion-assisted deposition (PIAD) system.

The OptiMon evaluation software allows combining the full triangular spectra evaluation algorithm with different Kramers-Kronig consistent dispersion models [25,26], among them, the oscillator model [27]. Thus, even absorbing layers or rugate systems can be successfully analyzed [28–30]. In a combined transmittance / reflectance version, an OptiMon-based characterization strategy even allowed determining the effective thickness of growing metal island and closed metal films [31]. In this study, our goal is to demonstrate the simultaneous determination of thickness and in situ optical constants of PVD (physical vapor deposited) coatings with moderate porosity.

The present study has been performed in tight correlation to an already published repeatability study on the deposition of a titania-silica 5-layer model system [32]. The purpose of that study was to optimize PIAD processes to achieve better repeatability of the optical coating performance. Note that we use the terminus repeatability here instead of reproducibility to emphasize that all coating preparation experiments are performed at the same deposition setup. In order to investigate uncertainties in refractive index and layer thickness, a reliable characterization methodology had to be developed, and this methodology is in the focus of the present paper.

We will proceed with motivating and explaining the model we are going to apply. Then, the deposition experiments will be presented in brief, followed by the presentation of the characterization results. Finally, we will discuss the consistency of the obtained optical constants.

2. Materials and methods

2.1 In situ spectra evaluation aspects

2.1.1 Strategy

A particular problem of in situ optical monitoring techniques is in the possibly vague knowledge of the optical constants. Although ex situ determined refractive index and extinction coefficient values are usually available, their relation to in situ data is not a priori clear. This is particularly true for layers with moderate porosity, when the size and shape of the pores are unknown, as well as their water content after having been stored at atmosphere for a certain time. All in all, one may distinguish between 3 basic strategies in this context:

What we propose here is a spectra evaluation strategy that merges aspects of all the extreme strategies i. – iii. The idea is the following:

2.1.2 Model system

We demonstrate the method by re-engineering a five-layer stack built from titania (TiO₂) and silica (SiO₂), making use of in situ spectra recorded in the wavelength range from 400–1000 nm.

2.1.3 Dispersion approach

In order to keep the number of dispersion parameters low, the titania (H) ex situ spectra have been fitted in terms of a Lorentzian oscillator model according to:

For silica, ex situ spectra have been fitted (neglecting absorption) by:

Figures 1 and 2 show ex situ optical constants from ex situ spectra.

 

Fig. 1. Refractive index ${n}_{H}^{{atm}}$ (solid line) and corresponding extinction coefficient ${k}_{H}^{{atm}}$ (dashed line) of titania obtained from fitting spectra of a 199 nm thick sample in terms of Eq. (1).

Download Full Size | PDF

 

Fig. 2. Refractive index ${n}_{L}^{{atm}}$ of silica obtained from fitting spectra of a 258 nm thick sample in terms of Eq. (3).

Download Full Size | PDF

2.1.4 Consistency considerations

Of course, a dispersion approach with only one free parameter seems comfortable, but its consistency may be questioned. Therefore, let us compare this approach to other accepted treatments.

Provided that the difference between in situ (${n^{vac}}$) and ex situ (${n^{atm}}$) optical constants is caused by porosity, mixing model approaches may be used to estimate their differences. In terms of the Bragg Pippard approach (columnar film structure [38]), one could write for the resulting refractive index n (possibly complex):

In application to ex situ spectra, i.e., $n = {n^{atm}}$, while the pores should be filled with water, such that ${n_\nu }$ of the order 1.333-1.339 [39] provides a reasonable assumption. Then the packing density may be directly determined by inverting Eq. (5):

This allows calculating the in situ refractive index from Eq. (5) again, assuming now empty pores (${n_\nu } = 1$). This leads to an in situ refractive index shown in Fig. 3 in red.

 

Fig. 3. Comparison of different approaches for estimating vacuum refractive indices in the case of a porous titania coating.

Download Full Size | PDF

Another possible approach is to assume a frequency-independent constant offset between in situ and ex situ refractive indices. This is shown in Fig. 3 as the dashed line.

And finally, we can apply Eq. (2), which leads to the navy curve in Fig. 3. Note that the refractive index at 500 nm (20000 cm^-1) has been used as the isosbestic point in these calculations.

Note that all of these approaches use one free parameter (porosity, offset, $\mathrm{\Delta }J$) in order to quantify the difference in refractive indices. All results are similar, but nevertheless, the oscillator model approach falls closer to the Bragg Pippard approach than the constant offset does. We therefore conclude, that our one-parametric oscillator model approach describes physically reasonable dispersion behavior. It does, moreover, not rely on morphology assumptions such as columnar film structure, and does not need a pore-free reference.

2.2 Experiment

2.2.1 Thin film deposition

The experimental setup and schematics of the quarter-wave (QW) stack repeatability trial has been described in great detail in two successive publications [32,41]. Here we confine the explanation to a summary of the most relevant aspects of the deposition process.

2.2.1.1 Coating environment

All experiments have been carried out at INP in a Leybold Optics Syrus LC-III PIAD box coater, a midsize industrial production plant that relates to recent Bühler Syrus 1100 devices. Material evaporation is provided by means of one electron beam gun (EBG) where both layer materials are processed using crucibles mounted in a rotary carrier. An Advanced Plasma Source (APS^pro, Bühler) serves as assist source. An apparently trivial statement is to mention that a multiplicity of operating parameters regarding various components (vacuum system, heating system, substrate drive, EBG, APS^pro) affects coating conditions, i.e., deposition rate, substrate temperature, chamber pressure, as well as flux and energy of accelerated particles. While the first three quantities are determined by appropriate sensors, the latter two are merely set indirectly by variables like gas flux and discharge voltage on the dc plasma source. An overview of actual plasma parameters, in particular, ion beam properties, related to varying operating conditions, can be found in Ref. [42].

First of all, our experimental setup represents state of the art in PIAD with respect to operation of evaporation source and assist source, control of deposition rate by quartz crystal microbalance (QCM), and a possible process control and endpoint detection by broadband optical monitoring (OM) using the already mentioned OptiMon technology. Furthermore, the coating plant is augmented by a system for plasma monitoring and an extended control capability, in order to stabilize the level of plasma assistance.

2.2.1.2 Plasma monitoring diagnostics

The aforementioned goal of monitoring and control of the plasma state is accomplished by active plasma resonance spectroscopy (APRS) employing a multipole resonance probe (MRP [43,44]). The measurement principle is based on an analytical description of the interaction of probe head and plasma in its vicinity. A vector network analyzer serves as source of a broadband hf signal and detection unit of reflected power as a function of frequency. The quantity of interest is a resonance frequency as a function of electron density ${f_r}({{n_e}} )$. Outstanding quality of the MRP is its insensitivity against dielectric coating and its small footprint, which makes it a promising tool for permanent monitoring of deposition processes. In our case, the probe is mounted close to the substrate holder at the mid radius. It is operated with a repetition rate of ${f_{APRS}}$=2 Hz and the essential resonance frequency is provided by online analysis for further data processing, i.e., either for recording only or as an input variable for feedback control.

2.2.1.3 Process control schemes

In this study, two control modes are utilized for the operation of the assist source APS^pro. Mode A is of conventional type, while mode B takes into account the plasma state as monitored by APRS. The main goal of mode A is control of beam ion energy represented by the anode voltage. It is worth noting that in standard devices, there exists no instrumentation to characterize the actual plasma properties, and therefore potential drifts in the plasma assistance level are simply unknown. Monitoring by APRS has revealed that in mode A, strong variations of plasma density can occur. While the ion energy might change only slightly due to a constant anode voltage ${V_A}$, the variation in ion density will lead to an alteration of the ion flux [32].

In order to counteract variations in density, data from APRS has been included in an extended mode of operation of the APS^pro. In mode B the conventional mode A is complemented by ${f_r}$ as new process variable and ${V_A}$ becomes a dependent variable. The combination of the conventional control scheme and APRS is described in detail in [32].

The repeatability experiment was projected in order to analyze the impact of assist source control mode (A, B) on thin film properties. Specific working points are described in [32]. Deposition conditions have been chosen to provide a moderate level of plasma assistance by setting ${V_A}$=80 V (mode A). At the same time, it is a challenging regime regarding the stability of the operation of the APS^pro. Also, the process window chosen becomes relevant when low film stress is pursued.

2.2.1.4 Sample preparation

Optical constants of the layer materials have been obtained from single layer coatings on borosilicate glass substrates with a nominal thickness of $d$=200 nm or 250 nm at process conditions of mode A. Then, 5-layer QW stacks have been prepared designed for a central wavelength of 550 nm. In the QW stack series, fused silica substrates were used. Ex situ optical characterization was conducted at INP by photometry using a Shimadzu UV-2450 spectrophotometer. Film stress measurements have been performed at IOF in a standard manner [45] with selected samples deposited on silicon wafers.

2.2.2 In situ transmittance spectra

In order to demonstrate the in situ optical characterization process, we focus on the analysis of the first sample that has been prepared in terms of process control mode A using quartz crystal monitoring for layer thickness control. Figures 4–8 show in situ recorded transmittance spectra of such a 5-layer system after completion of each of the individual layers. While Fig. 4 shows a simple single layer spectrum that contains information about the first layer only, the spectra shown in Fig. 8 already contain information about thicknesses and optical constants of all 5 layers. Modeling is performed using the dispersion laws (2) and (4). The transmittance minimum in Fig. 8 corresponds to a wavelength of 542 nm and is thus slightly blue-shifted with respect to the design wavelength of 550 nm.

 

Fig. 4. Measured (gray cross) and modeled (solid line) in situ transmittance after completion of the 1^st layer (= subsystem 1).

Download Full Size | PDF

 

Fig. 5. Measured (gray cross) and modeled (solid line) in situ transmittance after completion of the 2^nd layer (= subsystem 2).

Download Full Size | PDF

 

Fig. 6. Measured (gray cross) and modeled (solid line) in situ transmittance after completion of the 3^rd layer (= subsystem 3).

Download Full Size | PDF

 

Fig. 7. Measured (gray cross) and modeled (solid line) in situ transmittance after completion of the 4^th layer (= subsystem 4).

Download Full Size | PDF

 

Fig. 8. Measured (gray cross) and modeled (solid line) in situ transmittance after completion of the 5^th layer (= subsystem 5).

Download Full Size | PDF

2.2.3 Spectra shift after venting

Once our purpose was to deposit stress-poor samples, we used deposition conditions, which result in certain layer porosity. Therefore, we expect that after venting, atmospheric water penetrates into the pores and leads to an increase of the effective refractive index of the layers. Consequently, the spectral features from Fig. 8 should shift along the wavenumber scale. In order to visualize that vacuum-air shift, transmittance spectra of the coatings have been recorded prior to venting by means of the OptiMon in situ spectrophotometer, as well as ex situ in the Shimadzu UV-2450. In atmosphere, water can penetrate into the pores, which gives rise to changes in the optical thickness of the sample, and consequently in the transmittance spectrum. This is shown in Fig. 9. The relative shift is around –2.9%. This rather significant shift should correspond to changes in the refractive index of the participating materials. This is verified by measuring the corresponding shift of the spectra of the rather thick samples already used in Figs. 1 and 2. The spectra are shown in Figs. 10 and 11. Generally, spectra recorded in vacuum conditions will further be called in situ spectra, while those recorded in atmospheric conditions are called ex situ spectra.

 

Fig. 9. In situ (dashed line) and ex situ (solid line) transmittance of the quarter-wave stack on fused silica substrate. The average shift corresponds to -2.9% on the wavenumber scale.

Download Full Size | PDF

 

Fig. 10. In situ (dashed line) and ex situ (solid line) transmittance of a titania single layer coating on a borosilicate substrate, arrows indicate the position of the corresponding halfwave points, corresponding shift: -1.1%. Values in brackets indicate the coordinates of the extrema in the diagram.

Download Full Size | PDF

 

Fig. 11. In situ (dashed line) and ex situ (solid line) transmittance of a silica single layer coating on a borosilicate substrate, arrows indicate the position of the corresponding transmittance extremum, corresponding shift: -3.3%/-4.4%. Values in brackets indicate the coordinates of the extrema in the diagram.

Download Full Size | PDF

As expected, in situ transmittance spectra show an interference pattern that appears to be blue-shifted with respect to the ex situ features. Thus, the in situ refractive index must be smaller than ex situ, which is a typical phenomenon observed in porous coatings [46] and may be used for estimating the porosity [47]. For single layers, we observe a shift of approximately -1.1% for titania, and -3.8% for silica. Then, the shift of -3% as obtained from the 5-layer coating appears as a reasonable compromise between these single layer shifts.

3. Results and discussion

The main result of this study is in the dispersion curves of the in situ measured refractive indices of porous titania and silica. Naturally, in terms of the triangular spectra evaluation algorithm, the most reliable information is obtained for the individual layers that have been deposited first. In this context, Fig. 12 shows the in situ refractive index of the first (i.e., titania) layer in the stack. Dispersion curves are displayed that are calculated from in situ spectra recorded after the first run (= subsystem1), after the second run (the input are spectra from subsystem 1 and subsystem 2), and so on. We recognize that from the first layer, the re-engineering result is quite stable, which is an indication of the reliability of the obtained dispersion curves. In Fig. 13, the same is shown for the second (i.e., silica) layer.

 

Fig. 12. Ex situ (black line) and in situ refractive indices (colored lines) of the 1^st layer (titania) from in situ transmittance of (sub–)systems with 1…5 layers. The color code corresponds to that introduced in Figs. 4–8.

Download Full Size | PDF

 

Fig. 13. Ex situ (black line) and in situ refractive index (colored lines) of the 2^nd layer (silica) from in situ transmittance of (sub–)system with 2…5 layers. The color code corresponds to that introduced in Figs. 4–8.

Download Full Size | PDF

In Tab. 1, obtained thicknesses and refractive indices at 550 nm wavelength are collected for all individual layers. Also, the obtained individual layer thicknesses appear to be stable for consecutive recordings.

Table 1. Obtained Thicknesses and Refractive Indices at 550 nm.

View Table | View all tables in this article

According to [48], the error in the obtained refractive indices and film thicknesses are mainly caused by systematic measurement errors of the transmittance. In order to estimate errors in n and d, systematically falsified transmittance spectra have been fitted in terms of (2) and (4), and the obtained results have been compared to the results obtained from the originally measured spectra. Thereby, the errors given in Tab.1 correspond to an assumed systematic transmittance error equal to±0.005.

In Figs. 14–16, the data from Tab. 1 are graphically visualized. Figure 14 shows the refractive index profile as designed for the 550 nm wavelength, together with the corresponding data obtained from in situ spectra. Naturally, as a result of porosity, all in situ refractive indices are smaller than the ex situ indices used for design calculations.

 

Fig. 14. Assumed (dotted line) and modeled (circle) refractive index at 550 nm for individual layers, the left symbol for each layer corresponds to the result from the initial subsystem with this layer. The color code corresponds to that introduced in Figs. 4–8.

Download Full Size | PDF

 

Fig. 15. Assumed (dotted line) and modeled (circle) physical thickness for individual layers, the left symbol for each layer corresponds to the result from the initial subsystem with this layer. The color code corresponds to that introduced in Figs. 4–8.

Download Full Size | PDF

 

Fig. 16. Assumed (dotted line) and modeled (circle) quarter-wave optical thickness (QWOT) at 550 nm for individual layers. The color code corresponds to that introduced in Figs. 4–8.

Download Full Size | PDF

In Fig. 15, the same is shown for the individual metric layer thicknesses. Note that the thicknesses are remarkably larger than designed. As a result, the error in the optical thicknesses is only 2% or smaller (Fig. 16).

This anticorrelation between in-situ refractive index and metric film thickness results from a specific error self-compensation mechanism inherent to quartz monitoring. Indeed, a larger porosity results in a smaller density. Therefore, a larger metric thickness will be deposited because the quartz monitor detects the mass coverage. On the other hand, in a dielectric film, a larger porosity is connected with a smaller refractive index. Consequently, the refractive index is smaller, and the metric thickness larger, such that the optical thickness is rather stable [46].

The mentioned compensation mechanism is, of course, not perfect, and therefore, the minimum position of the in- situ transmittance is 542 nm and, therefore, somewhat smaller than the design value of 550 nm (Fig. 9). After venting, the transmittance minimum shifts to 558 nm, so that in sum, we have a net overshoot of approximately 1.5% in the ex situ optical thickness.

Hence, when summarizing the immediate characterization results, we establish the following properties of the stack:

 

Fig. 17. Correlation between shift, stress and ${{n}^{{atm}}}$ from published PVD data (see [46]). Red symbols indicate data from Figs. 1, 2, 10, 11. Arrows indicate the guess of the mechanical stress. For more details, see text. EBE = electron beam evaporation without assistance; MS = magnetron sputtering. In our convention, negative stress is tensile and positive compressive.

Download Full Size | PDF

Table 2. In situ Refractive Indices at 400 nm.

View Table | View all tables in this article

It must be noted that the literature data collected in [46] concern ex situ determined refractive indices. Therefore, in order to compare our 400 nm data with the literature data, we must estimate their ex situ counterparts (${n^{atm}}$). This can easily be done using Eqs. (5) and (6), and the results are given in Tab. 3.

Table 3. Layer Thicknesses, Porosity, and Refractive Indices at 400 nm.

View Table | View all tables in this article

In the graphs of Fig. 17 (left side), we recognize the general trend that samples with a stronger shift tend to have smaller refractive indices, although there is considerable scatter in the data observed by different deposition techniques and conditions. The highest indices (and weakest shifts) are obtained from strongly assisted evaporated or from sputtered layers, while layers prepared by evaporation without assistance show the strongest shifts and smallest refractive indices. The mentioned trend is a trivial result when the shift is expected to arise from porosity, which is the largest for evaporated layers. Concerning titania, so the value obtained in this study (the large circle) falls well inside the data cloud formed by the PIAD samples, such that the expected moderate porosity of about 6 vol% seems to be reasonable. In silica, we observe a stronger shift, corresponding to an estimated porosity of 23 vol%.

The quarter-wave stack is expected to exhibit a shift that provides a compromise of the shift behavior of the individual layers (Fig. 9). Here we must keep in mind that the titania films in the stack seem to have different porosity. Thus, the first titania film in the stack has an estimated porosity of only 4 vol%, has a rather large refractive index (vertical arrow in Fig. 17 left on top) and might, therefore, be almost shift-free. The other titania films in Tab. 1 turned out to have a somewhat lower refractive index, again corresponding to a porosity of around 6 vol%, so that they should provide a stronger shift of about -1.1%. Taking the stronger shift of the silica layers into account, shifts of the whole stack in the range -2…-3% seem reasonable.

Once the correlation between shift and internal layer stress is also supported by numerous experimental data (Fig. 17 on the right), the mentioned refractive index and shift values may be used to derive a hypothesis on the expected individual layer stress (horizontal arrows between the images on the left and right in Fig. 17). When taking the data from Tab. 3, we arrive at moderate compressive stress in the silica layers, while the stress in the titania layers is also moderate, but tensile. This is consistent with the observation that the stacks are almost strain-free at atmosphere.

Let us finally return to the repeatability aspect. We have already mentioned that for the reported sample, the minimum transmittance wavelength is 542 nm in situ, and 558 nm ex situ. This does not coincide with the target wavelength of 550 nm, and in real deposition practice, one would have to modify the deposition process in order to hit the target central wavelength more accurately.

However, it would be dangerous to modify the process based on the relative success of the described single deposition run. It does not provide any information on whether the recorded deviation from the target wavelength is of stochastic or systematic nature. To discriminate between these two contributions, the deposition procedure has been repeated four times with the same deposition strategy (process control mode A and endpoint detection by QCM). The resulting four in situ spectra are published in [32] and result in a medium in situ position of the transmittance minimum of ${\lambda _{min}}$=548 nm with a standard deviation of $\sqrt {{{({{\lambda_{min}}} )}^2}} $=5 nm. It is therefore likely that stochastic contributions are essential so that in practice, improving repeatability of the deposition process should have the highest priority. As described in [32], much better repeatability could be achieved when changing to process control mode B and endpoint detection by broadband optical monitoring. In these conditions, we were able to achieve values of ${\lambda _{min}}$=544.7 nm and $\sqrt {{{({{\lambda_{min}}} )}^2}} $=0.5 nm, again from four deposition runs. Thus, after process stabilization, we are able to quantify the systematic deviation in the minimum wavelength more reliably. In deposition practice, this systematic deviation, combined with knowledge on the air-vacuum shift, would finally be used to re-calibrate the relevant endpoint detection strategy.

4. Conclusion

We have presented an in situ based optical characterization strategy for calculating thickness and optical constants of the individual layers in a layer stack. The differences between in situ and ex situ optical constants are approximated in terms of a single parameter in a simple Lorentzian oscillator model. Kramers-Kronig consistency of the obtained optical constants may be a priori guaranteed this way.

The methodology has been demonstrated by analyzing a single PIAD coating experiment of a somewhat porous five-layer stack. Correspondingly, the stack showed a significant vacuum-to-air shift, while the layer strain turned out to be small enough that it was below our detection limit. This result is consistent with otherwise reported data on the relationship between porosity, shift, and stress of PVD titania and silica layers. Thus, based on the obtained optical constants, the titania fraction should show moderate tensile stress, while the silica stress should be compressive. In the stack, these different contributions seem to cancel out each other such that a nearly strain-free stack was produced.

As a result of our analysis we come to the conclusion, that the indication of one set of in situ optical constants for titania and silica may be insufficient for the correct description of the optical behavior of the stack. Thus, it turned out that the first titania layer has a higher refractive index than the following layers. If this difference is not taken into account, it may be a source of systematic deposition errors.

Process optimization needs to minimize both stochastic and systematic deposition errors. Information on stochastic deposition errors may be obtained from repeatability studies and was topic of [32]. In particular, it turned out that the process control strategy may be optimized such that stochastic errors are minimized. Remaining differences are of systematic nature and need a recalibration of the deposition process, or maybe of underlying model parameters. In light of the goals of the cited repeatability study, the presented analysis methodology of single layer properties appears as a substantial part besides the characterization of key parameters of the deposition process itself.

Appendix: Dispersion parameters

Table 4 provides data on the dispersion parameters in Eqs. (1)-(4) as obtained from fits of single layer spectra:

Table 4. Dispersion Parameters as Obtained From the Spectra Fits in cm^-1.

View Table | View all tables in this article

Thereby, Eqs. (3) and (4) as used for modelling silica essentially represent Sellmeier-type dispersion expressions, while (1) and (2) as used for titania provide a merger of a Sellmeier expression with a single Lorentzian oscillator. The additional oscillator is essential for modelling weak absorption contributions in the vicinity of the band edge arising from optical excitations of valence electrons. Hence Eqs. (1) or (2) work in the transparency region of a material only, while the application of (3) or (4) is possible even in the vicinity of the absorption edge of the corresponding material. This is illustrated in Fig. 18:

 

Fig. 18. Typical dispersion behavior of n and k in a dielectric material and application ranges of Eqs. (1) – (4). MIR denotes the middle infrared spectral range, NIR the near infrared spectral range, VIS the visible spectral range, and UV the ultraviolet.

Download Full Size | PDF

It is now the relation between the spectral range used for the spectrophotometric measurement and the absorption onset frequency of the particular material that defines whether (1) or (3) should find application for modelling the ex situ optical constants: In our study, measurements were performed in the wavelength range from 400 nm to 1000 nm. Once silica is transparent down to vacuum ultraviolet wavelengths, absorption does not need to be taken into account in our particular case, and the application of (3) is absolutely sufficient for modelling the optical constants. In the case of titania, the absorption onset is observed at a wavelength around 400 nm. Thus it has to be taken into account in the modelling procedure. Correspondingly, it is Eq.(1) that should find application for modelling ex situ optical constants of titania. In this sense, the mentioned dispersion approach may be used for modelling layer stacks built from other materials as well, as long as measurement range and transparency ranges of the used materials are in proper relation.

Concerning the applicability of Eqs. (2) and (4), it is presumed here that changes in the refractive index caused by porosity dominate over those observed in extinction. This is certainly the case as long as both, pore and solid materials are weakly absorbing in the measurement range. Water as well as certain hydrocarbons which might penetrate into the pores are certainly transparent in the VIS, which is the reason for the applicability of (2) and (4) in our study. In the UV, however, water as well as hydrocarbons become absorbing, and therefore the mentioned approach will become insufficient at least for wavelengths smaller than 200 nm. In that case, more than one fitting parameter must be taken into account when modelling the difference between optical constants in vacuum and at atmosphere. Then, the simplest approach would be to include an additional correction $\mathrm{\Delta }{J_2}$ to the parameter ${J_2}$.

Note that in our situation (Tab. 4, titania), $\mathrm{\Delta }{J_1} \approx {J_2} \ll {J_1}$. Once water is practically non-absorbing in our measurement range, a small porosity would induce only a small correction to ${J_2}$ caused by changes in the density of absorbers. Therefore, $\mathrm{\;\ \Delta }{J_2} \ll {J_2} \approx \mathrm{\Delta }{J_1} \ll {J_1}$. The correction to ${J_2}$ would therefore be small to second order. For this reason it is not considered in our approach.

Finally, Eqs. (1)-(4) are not suitable for application in the MIR, because the response of oscillations of atomic cores should additionally be taken into account.

Funding

Bundesministerium für Bildung und Forschung (13N13213, 13N13214).

Acknowledgments

The authors are grateful to D. Köpp for the skillful operation and maintenance of the PIAD box coater at INP. Support for strain measurements at IOF by C. Franke is highly acknowledged.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. A. Tikhonravov, I. Kochikov, I. Matvienko, T. Isaev, and A. Yagola, “Strategies of Broadband Monitoring Aimed at Minimizing Deposition Errors,” Coatings 9(12), 809 (2019). [CrossRef]  

2. M. Vignaux, F. Lemarchand, T. Begou, C. Grezes-Besset, and J. Lumeau, “Semi-automated method for the determination of the all-optical monitoring strategy of complex thin-film filters,” Opt. Express 27(9), 12373–12390 (2019). [CrossRef]  

3. A. V. Tikhonravov, I. V. Kochikov, I. A. Matvienko, T. F. Isaev, D. V. Lukyanenko, S. A. Sharapova, and A. G. Yagola, “Correlation of errors in optical coating production with broad band monitoring,” Numerical methods and programming 19(4), 439–448 (2018). [CrossRef]  

4. A. Moreau, T. Begou, F. Lemarchand, F. Lemarquis, K. Mathieu, and J. Lumeau, “Comparative study of bandpass filters manufactured by different deposition technologies,” Proc. of Optical Interference Coatings Conference (OIC) (2019),Santa Ana Pueblo, New Mexico, 2019/06/02; p. WB.7.

5. R. R. Willey, Practical design and production of optical thin films (CRC press, 2002).

6. B. T. Sullivan and G. Carlow, “An Overview of Optical Monitoring Techniques,” Proc. of Optical Interference Coatings (OIC) (2010), Tucson, Arizona, 2010/06/06, p. TuC1.

7. S. Waldner, R. Benz, P. Biedermann, and A. Jaunzens, “Broadband Optical Monitoring Combined with Additional Rate Measurement for Accurate and Robust Coating Processes,” Proc. of Optical Interference Coatings (OIC) (2010), Tucson, Arizona, 2010/06/06, p. TuC10.

8. S. Wilbrandt, O. Stenzel, and N. Kaiser, “Verfahren zur Messung der optischen Eigenschaften einer bewegten Probe in einer Beschichtungsanlage,” Patent DE 102009012756 B4.

9. S. Wilbrandt, O. Stenzel, and N. Kaiser, “All-optical in-situ monitoring of PIAD deposition processes,” Proc. SPIE 7101, 71010D (2008). [CrossRef]  

10. S. Waldner, J. Buchholz, and R. Benz, “Hybrid Mode Optical Monitoring – Monochromatic and Broadband Algorithms in the same Coating Process,” Proc. of Optical Interference Coatings Conference (OIC) (2019), Santa Ana Pueblo, New Mexico, 2019/06/02, p. WA.9.

11. A. V. Tikhonravov and M. K. Trubetskov, “Online characterization and reoptimization of optical coatings,” Proc. SPIE 5250, 406 (2004). [CrossRef]  

12. T. V. Amotchkina, M. K. Trubetskov, V. Pervak, S. Schlichting, H. Ehlers, D. Ristau, and A. V. Tikhonravov, “Comparison of algorithms used for optical characterization of multilayer optical coatings,” Appl. Opt. 50(20), 3389–3395 (2011). [CrossRef]  

13. S. Wilbrandt, O. Stenzel, N. Kaiser, M. K. Trubetskov, and A. V. Tikhonravov, “In situ optical characterization and reengineering of interference coatings,” Appl. Opt. 47(13), C49–C54 (2008). [CrossRef]  

14. S. Wilbrandt, O. Stenzel, N. Kaiser, M. K. Trubetskov, and A. V. Tikhonravov, “On-line re-engineering of interference coatings,” Proc. of Optical Interference Coatings OIC (2007), Tucson, Arizona, 2007/06/03, p. WC10.

15. A. V. Tikhonravov, M. K. Trubetskov, I. V. Kochikov, I. B. Oliver, and D. J. Smith, “Real-Time Characterization and Optimization of E-Beam Evaporated Optical Coatings,” Proc. of Optical Interference Coatings (OIC) (2001), Banff, 2001/07/15, p. ME8.

16. X. Ji, J. Zhang, X. Cheng, Z. Wang, I. Matvienko, T. Isaev, and A. Tikhonravov, “The Error Self-Compensation Effect in the Broadband Monitoring of Multiband Filters,” Proc. of Optical Interference Coatings Conference (OIC) (2019), Santa Ana Pueblo, New Mexico, 2019/06/02, p. WA.8.

17. A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, Optical Thin Films and Coatings: Optical monitoring strategies for optical coating manufacturing, A Piegari and F. Flory, eds. (Woodhead Publishing2018), pp. 65–101.

18. A. Tikhonravov, M. Trubetskov, I. Kozlov, S. Alekseev, P. Konotopov, and V. Zhupanov, “Correlated Choice of Design and Monitoring Strategy,” Proc. of Optical Interference Coatings (OIC) (2013), Whistler, 2013/06/16, p. WB.3.

19. A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Investigation of the error self-compensation effect associated with broadband optical monitoring,” Appl. Opt. 50(9), C111–C116 (2011). [CrossRef]  

20. A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Does Broadband Optical Monitoring Provide an Error Self-compensation Mechanism?” Proc.of Optical Interference Coatings (OIC) (2010), Tucson, Arizona, 2010/06/06; p. TuC3.

21. A. V. Tikhonravov and M. K. Trubetskov, “Elimination of cumulative effect of thickness errors in monochromatic monitoring of optical coating production: theory,” Appl. Opt. 46(11), 2084–2090 (2007). [CrossRef]  

22. A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Computational experiments on optical coating production using monochromatic monitoring strategy aimed at eliminating a cumulative effect of thickness errors,” Appl. Opt. 46(28), 6936–6944 (2007). [CrossRef]  

23. B. T. Sullivan and J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings. I. Theoretical description,” Appl. Opt. 31(19), 3821–3835 (1992). [CrossRef]  

24. B. T. Sullivan and J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings. II. Experimental results—sputtering system,” Appl. Opt. 32(13), 2351–2360 (1993). [CrossRef]  

25. D. Franta, J. Vohánka, and M. Čermák, Optical Characterization of Thin Solid Films: Universal dispersion model for characterization of thin films over wide spectral range, O. Stenzel and M. Ohlídal, eds. (Springer2018), pp. 31–82.

26. O. Stenzel and S. Wilbrandt, “Beta-distributed oscillator model as an empirical extension to the Lorentzian oscillator model: physical interpretation of the β_do model parameters,” Appl. Opt. 58(33), 9318–9325 (2019). [CrossRef]  

27. J. A. Dobrowolski, F. C. Ho, and A. Waldorf, “Determination of optical constants of thin film coating materials based on inverse synthesis,” Appl. Opt. 22(20), 3191–3200 (1983). [CrossRef]  

28. S. Wilbrandt, N. Kaiser, and O. Stenzel, “In-situ broadband monitoring of heterogeneous optical coatings,” Thin Solid Films 502(1-2), 153–157 (2006). [CrossRef]  

29. S. Wilbrandt, O. Stenzel, and N. Kaiser, “Experimental determination of the refractive index profile of rugate filters based on in situ measurements of transmission spectra,” J. Phys. D: Appl. Phys. 40(5), 1435–1441 (2007). [CrossRef]  

30. S. Wilbrandt, O. Stenzel, M. Bischoff, and N. Kaiser, “Combined in situ and ex situ optical data analysis of magnesium fluoride coatings deposited by plasma ion assisted deposition,” Appl. Opt. 50(9), C5–C10 (2011). [CrossRef]  

31. J. Gäbler, O. Stenzel, S. Wilbrandt, and N. Kaiser, “Optische in-situ Prozessverfolgung und-steuerung: Messung des Transmissions-und Reflexionsvermögens wachsender Schichten,” Vak. Forsch. Prax. 25(6), 22–28 (2013). [CrossRef]  

32. J. Harhausen, R. Foest, O. Stenzel, S. Wilbrandt, C. Franke, and R. P. Brinkmann, “Concepts for in situ characterization and control of plasma ion assisted deposition processes,” Thin Solid Films 673, 94–103 (2019). [CrossRef]  

33. F. Carstens, H. Ehlers, S. Schlichting, L. Jensen, and D. Ristau, “High-Resolution Optical Broadband Monitoring for the Production of Miniaturized Thin-Film Filters,” Proc.of Optical Interference Coatings Conference (OIC) (2019), Santa Ana Pueblo, New Mexico, 2019/06/02; p. WA.7.

34. T. Amotchkina, M. Trubetskov, A. Tikhonravov, I. B. Angelov, and V. Pervak, “Reliable optical characterization of e-beam evaporated TiO₂ films deposited at different substrate temperatures,” Appl. Opt. 53(4), A8–A15 (2014). [CrossRef]  

35. S. Wilbrandt, O. Stenzel, and N. Kaiser, “All-oxide broadband antireflection coatings by plasma ion assisted deposition: design, simulation, manufacturing and re-optimization,” Opt. Express 18(19), 19732–19742 (2010). [CrossRef]  

36. S. L. Nadji, M. Lequime, T. Begou, C. Koc, C. Grézes-Besset, and J. Lumeau, “Use of a broadband monitoring system for the determination of the optical constants of a dielectric bilayer,” Appl. Opt. 57(4), 877–883 (2018). [CrossRef]  

37. L. Dubé-Riopel, B. Baloukas, O. Zabeida, and L. Martinu, “In Situ and Ex Situ Spectroscopic Ellipsometry of Electrochromic NiO Films,” Proc. of Optical Interference Coatings Conference (OIC) (2019), Santa Ana Pueblo, New Mexico, 2019/06/02; p. ThC.6.

38. O. Stenzel, The Physics of Thin Film Optical Spectra: An Introduction, 2nd ed. (Springer, 2016).

39. G. M. Hale and M. R. Querry, “Optical Constants of Water in the 200-nm to 200-µm Wavelength Region,” Appl. Opt. 12(3), 555–563 (1973). [CrossRef]  

40. M. Landmann, T. Köhler, S. Köppen, E. Rauls, T. Frauenheim, and W. G. Schmidt, “Fingerprints of order and disorder in the electronic and optical properties of crystalline and amorphous TiO₂,” Phys. Rev. B 86(6), 064201 (2012). [CrossRef]  

41. J. Harhausen, R. Foest, J. Wauer, O. Stenzel, S. Wilbrandt, C. Franke, M. Oberberg, and R. P. Brinkmann, “Prospects for the enhancement of PIAD processes by plasma diagnostics,” Proc. SPIE 10448, 69 (2017). [CrossRef]  

42. J. Harhausen, R. P. Brinkmann, R. Foest, M. Hannemann, A. Ohl, and B. B. Schröder, “On plasma ion beam formation in the Advanced Plasma Source,” Plasma Sources Sci. Technol. 21(3), 035012 (2012). [CrossRef]  

43. M. Lapke, T. Mussenbrock, and R. P. Brinkmann, “The multipole resonance probe: A concept for simultaneous determination of plasma density, electron temperature, and collision rate in low-pressure plasmas,” Appl. Phys. Lett. 93(5), 051502 (2008). [CrossRef]  

44. M. Lapke, J. Oberrath, C. Schulz, R. Storch, T. Styrnoll, C. Zietz, P. Awakowicz, R. P. Brinkmann, T. Musch, and T. Mussenbrock, “The multipole resonance probe: characterization of a prototype,” Plasma Sources Sci. Technol. 20(4), 042001 (2011). [CrossRef]  

45. O. Stenzel, M. Schürmann, S. Wilbrandt, N. Kaiser, A. Tünnermann, M. Mende, H. Ehlers, D. Ristau, S. Bruns, and M. Vergöhl, “Optical and mechanical properties of oxide UV coatings, prepared by PVD techniques,” Proc. SPIE 8168, 81681W (2011). [CrossRef]  

46. O. Stenzel, Optical Coatings: Material Aspects in Theory and Practice, 1st ed. (Springer, 2014).

47. O. Stenzel and M. Ohlídal, eds., Optical characterization of thin solid films, 1st ed. (Springer, 2018).

48. A. V. Tikhonravov, M. K. Trubetskov, M. A. Kokarev, T. V. Amotchkina, A. Duparré, E. Quesnel, D. Ristau, and S. Günster, “Effect of systematic errors in spectral photometric data on the accuracy of determination of optical parameters of dielectric thin films,” Appl. Opt. 41(13), 2555–2560 (2002). [CrossRef]