1. Introduction

1.1 Motivation

The performance of any optical multilayer coating is essentially determined by the accurate adjustment of two groups of parameters: geometrical parameters (e.g. film thicknesses), and material parameters (e.g. optical constants) [1]. In a real deposition process, the operator tries his best to tune these parameters to the desired values, defining stable process conditions in order to obtain reproducible optical constants, and making use of thickness monitoring techniques to terminate the layer deposition when the film has exactly the correct thickness. But all this can be realized only with a limited accuracy, so that stochastic fluctuations as well as systematic errors will occur in both geometrical and material parameters of a multilayer coating.

On the other hand, the development of sophisticated design methods and their implementation into commercially available software tools, has practically led to the situation that for every reasonably specified coating design problem, theoretical solutions may be found as the result of some affordable computational effort. But there is absolutely no guarantee that a theoretical design, and particularly that with the best theoretical performance, has a chance to be manufactured with the precision required to obtain the calculated performance in practice. As the result of a real deposition process, all construction parameters of the coating will be consistent with the theoretically designated parameters only within a certain error margin. The latter depends on a multiplicity of parameters, and may be influenced by the choice of the deposition process, deposition parameters, and the deposition strategy (which includes monitoring issues). Some deposition errors may appear to be mutually correlated, while others are uncorrelated. Moreover, the sensitivity of the coatings’ spectral performance to these deposition errors may be related to its particular design. A design with excellent theoretical performance may turn out to be very sensitive to deposition errors, and therefore not practicable to be manufactured with the given equipment. In order to comply with this problem, the attention of coating designers has shifted from the identification of designs with best spectral performance to the identification of designs that are more reliably manufactured in practice [2]. Of course, the result of such a design optimization process will depend on the available deposition and monitoring equipment. Thus, any numerical procedure which identifies the most practicable design must be specific for the given coating facility. In this context, in [3], the significance of computational manufacturing (CM) of optical coatings as a linking element between design and production is emphasized. Today, the CMs (or virtual deposition runs) have been identified as reliable tools to judge the practicability of a preselected design with respect to the available technology.

In this context, identification and tabulation of typical deposition errors such as fluctuations in thicknesses, refractive indices, and the like appear to be a matter of primary importance, because the relevance of assumed deposition errors defines the degree of relevance of the output of any CM. Hereby, much work has been done to quantify thickness errors and their accumulation for specific monitoring strategies [2,4,5 ]. Much less work has been conducted in the course of modelling and quantifying typical errors in thin film optical constants. Clearly, fluctuations in optical constants depend on the deposition process condition stability and reproducibility, and are thus at least dependent on the deposition technique chosen. In this context, the present paper is focused on the quantification of errors in optical constants observed in a plasma ion assisted electron beam evaporation (PIAD) deposition experiment.

In order to keep the CM procedure as realistic as possible, it is the opinion of the authors that only physically reasonable (particularly Kramers—Kronig (KK) consistent [6]) dispersion models should be considered for the corresponding modelling purposes. We have demonstrated corresponding CMs in ref [5], where deposition errors in a PIAD environment have been simulated in terms of fluctuations of the parameters of a multi-oscillator model. However, as a matter of fact, important commercial software packages for thin film design do not support performing CM procedures in terms of KK-consistent models such as the oscillator model, but rather operate in terms of independently determined fluctuations in refractive index n and extinction coefficient k [7,8 ]. In the case of interference coatings that are practically free of absorption in the spectral range of interest, k is often negligible, and in this case fluctuations in optical constants practically concern fluctuations in n only. In this particular case, a simple modelling of refractive index stochastic errors in terms of a stochastic additive and wavelength-independent contribution (constant offset) to n or to Sellmeier parameters [9] seems reasonable, as long as the specified wavelength range is not too broad.

It is the primary purpose of the present paper to provide a detailed study of deposition errors relevant for the refractive index of typical oxide coating materials in a PIAD process.

1.2 Basic concept of the experiment

In order to keep CMs realistic, a clear separation between systematic and stochastic deposition errors is of primary importance. Imagine the situation of a CM, where a coating deposition is simulated taking various deposition errors into account. In the simplest approach, any parameter f that is subject to stochastic fluctuations (further called random parameter) may be characterized by its mean value <f> and its standard variation δf.

In the practically relevant case that the knowledge about the random parameter f is restricted to a finite set of measured independent data f ₁, f ₂, ..., f_N, with each value having the same probability, mean value and standard deviation may be estimated by:

Then, the necessary effort for acquiring realistic and reliable data for the standard deviations of all relevant random parameters includes acquisition of the data sets f ₁, f ₂, ..., f_N. When associating the random parameter f with such film construction parameters like film thickness d and refractive index n (at a certain reference wavelength), data acquisition may be accomplished from repeated independent deposition experiments pursuing the same target thickness d _target and refractive index n _target while using, of course, the same deposition technique and monitoring strategy. After determination of all values n_j and d_j, corresponding to the j ^th deposition experiment, mean values (<d>, <n>) as well as standard deviations (δd, δn) may be calculated from (1) and (2), correspondingly.

The thus obtained values for the standard deviations are a measure for the reproducibility of the corresponding film construction parameter and are characteristic for the applied deposition technique and conditions. They may immediately find application in CMs to check the stability of a chosen design with respect to relevant stochastic deposition errors [3,5 ].

In principle, we can obtain additional information from these experiments. In the absence of any systematic errors, one would expect that for a sufficiently high N, the obtained mean values of f coincide with their target values. Hence, in the ideal case

should be fulfilled. In practice, a coincidence of the determined mean value with the target value is usually not obtained, because in reality N is always finite. Moreover, systematic errors Δ_syst f may result in a violation of (3). Hereby, effects caused by the finite value of N can be estimated through the expression for the standard error of the mean, the latter following from the Bienayme-formula [10], according to:

In the case that observed differences between established mean values of f and their target values are substantially larger (by absolute value) than$\frac{\delta f}{\sqrt{N}}$, it makes sense to associate this difference with a systematic deposition (or measurement) error. We thus have:

Therefore, in a CM experiment, random deposition errors can be considered in terms of the corresponding standard deviations, while systematic errors can be modelled by changes in the mean values. Having acquired reliable information about all these parameters, a CM experiment can be performed in terms of the procedure described in [5].

In the present paper, our focus is on the deposition errors relevant for the refractive index in a PIAD deposition experiment. We chose three oxide materials, namely TiO₂, SiO₂, and Al₂O₃. Hereby, we include the combination of TiO₂ and SiO₂ which might be relevant for near infrared and visible (NIR/VIS) coating specifications, and we will focus on the reference wavelength λ ₁ = 400nm. The combination of SiO₂ and Al₂O₃ may be of interest for deep ultraviolet (DUV) coatings, and we will consider a reference wavelength λ ₂ = 250nm.

For any set of deposition conditions chosen, the refractive index reproducibility will be checked by comparing the results of 6 identical but independent deposition runs. Hence, it is N = 6 throughout this study. This might seem to be a small number, but in practice, as it will become clear later, we had to perform a total of 54 deposition experiments, resulting in a number of 132 individual samples that had to be characterized. Nevertheless, even a number of only 6 independent attempts provides an estimation of random variations in n in terms of the standard deviation given in (2).

Considerations regarding systematic errors are more complex. Clearly, when speaking about the film thickness, it is common to define a target thickness, so that systematic errors may be estimated in terms of (5a). Nevertheless, once systematic errors are identified, it appears to be much more reasonable to eliminate them by a careful adjustment of the experiment than to include them into the CM. Thus, in the case of systematic thickness errors, recalibration of the monitoring device or a refinement of the evaporation mask [11] could be useful to eliminate systematic thickness errors.

The situation is different with the refractive index. In practice, there is usually no target value specified, but one rather relies on reproducibility. However, systematic deviations in the refractive index occur. Indeed, in a PIAD process, the growing film is subject to a bombardment with high energy inert gas atoms and ions, generated by the plasma source. As a result of the different angular distributions of high energy particles emitted from the plasma source [12] and layer forming species emitted from the evaporators [13], the net assistance effect is different at any point in the deposition chamber. Thus samples, placed at an outer position of the substrate holder, will therefore receive a net assistance that is different from that acting on a sample placed at an inner position. This will result in differences in the mean values of the refractive index, as obtained from samples placed in different conditions. Such effects form a specific mechanism for systematic deviations in the refractive index, and they cannot be eliminated by a certain recalibration procedure. They are quite relevant for the production yield, because in production conditions, one would naturally try to use the full substrate holder area in order to keep the production costs low.

We will not try to develop a manageable model to quantify these effects here. Instead, in each of our experiments, samples have been placed at different positions in order to obtain information about the scatter in mean values. Moreover, different control concepts for plasma assistance have been employed (see later Sect. 2.2 and compare [12,14 ]) for deposition experiments with each of the mentioned materials. As the result, mean values and standard deviations are obtained corresponding to a rather representative set of relevant but different sample positions and assistance strategies. In order to quantify systematic errors, the scatter in mean values obtained from these different experimental situations is analyzed. When the latter are numbered by the index l, a pragmatic recipe for realistic estimation of the magnitude of systematic deviations is given by (5b):

Thus, the essence of the present study is in performing repeated deposition experiments while considering different relevant sample positions and assistance concepts. The result of the study will be composed from quantitative data on systematic (according to (5b)) and stochastic (according to (2)) deviations in the refractive indices in a PIAD process.

2. Deposition process

2.1 PIAD environment

The deposition experiments were carried out in SYRUSpro 1100 (Bühler Alzenau) type batch coaters. In these devices the gridless plasma source APS [15] is placed at the central region of the bottom of the chamber pointing upwards and surrounded by electron beam evaporators (EBG) that are located at mid radius. Figure 1 shows the experimental setup including rough measures of the area between sources and substrates.

 

Fig. 1 Experimental setup.

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A large part of the deposition chamber as shown in Fig. 1 is occupied by components of the plasma source and diagnostics to characterize its ion emission. Actually, performance and stability of the APS have a fundamental impact on thin film properties and in particular on the issue of repeatability addressed in this work. The APS is a hot cathode dc discharge with an auxiliary magnetic field. The combination of high electron pressure (p _e≈10Pa) in the source and high vacuum in the chamber (p≈10⁻²Pa) generates an ion beam by plasma expansion. Results of characterization of the plasma plume [12,16 ] and an investigation on ion beam propagation in the chamber [17] have been published recently. Actuating variables to operate the APS include fluxes Γ ₁, Γ ₂ of the working gas argon, Γ ₃ of oxygen introduced at the source exit, heating power of the cathode P _H, coil current I _C of the solenoid encompassing the anode tube, discharge voltage V _D and current I _D, further anode voltage against ground V _A which is commonly termed the BIAS voltage of the arrangement. This set of eight parameters spans a large space of possible working points. The actual APS configurations employed for various deposition processes correspond to results of empirical procedures for the optimization of layer properties. In typical cases gas fluxes and cathode heating power are kept constant during a PIAD process phase which leaves the set of four parameters {V _A, V _D, I _D, I _C} for measures of source control. In turn, this set of four consists of two current-voltage pairs where a fixed setting for one implies dependent settings of the remaining pair. E.g. when I _C and V _A are to be kept constant, V _D and I _D need to be adapted. During operation values for V _D and I _D are subject to drifts due to changes of anode and cathode properties. Analogous behaviour is observed for all set/adapt voltage-current pairs. This means that continuous control of the plasma source during a deposition process is inherent to the PIAD system and a unique set of control parameters does not exist.

2.2 Considerations on assistance control concept and its relation to the stability of deposition process conditions

The setup of an optimum coating environment requires solutions regarding two fields of problems. Firstly, thorough understanding of the underlying physical mechanisms to obtain high quality thin films, and secondly, preparation of accurate tools and methods to technically implement such assistance effects are mandatory. In order to obtain dense films which tend to exhibit compressive stress, present understanding is that sufficient momentum flux generated by energetic particles needs to be applied during the layer growth process. Such reasoning is supported e.g. by comparison of different noble gas species in an ion assisted electron beam evaporation (IAD) process by Targove and Macleod [18]. A systematic approach was presented by Davis [19] who elucidated the significance of energy and momentum input by the assisting species per deposited particle and the trade-off between layer densification and generation of defects by energetic particles. Ideally, energy and flux density of the assisting species would be set separately to promote the desired layer properties at a given deposition rate.

When plasma sources are used, again two problem areas need to be addressed. On the one hand, fundamental properties of the plasma responsible for the assistance effect have to be identified and on the other hand, concepts for measurement and control of relevant parameters are required. Employing the APS as assist source, operators are focused on gas fluxes and the BIAS voltage V _A. The latter parameter is commonly interpreted as ion energy and has actually evolved to a highly rated parameter to describe the assistance effect. Investigations on the ion energy distribution [12] have confirmed that V _A is related to the ion beam component of the plasma. However, V _A cannot be considered unique concerning characterization of the plasma state. Figure 2 gives an impression on the spatial variation of the ion velocity distribution function f(v _i) in the coating chamber. The high energy part (E _i ∈ [60;120]eV) corresponds to the anisotropic ion beam, while the low energy part (E _i <20eV) is due to the cold isotropic background plasma. Not only is f(v _i) a function of the polar angle ϑ, but also it is strongly depending on APS parameters like P _H or I _C. We pass on details already presented in [12] and exemplify this issue by showing the polar profile of the ion kinetic energy flux density j _E(ϑ) which is the third velocity moment of f(v _i). A distinct variation of j _E(ϑ) is observed by variation of P _H and connected change in V _D, while V _A is the same for both. This kind of investigation has been carried out for many APS configurations as to provide a set of reference states reflecting the parameter space spanned by the drifting APS parameters during ordinary operation. In order to reduce complexity and quantity of f(v _i) data, an integral parameter to characterize the ion component of the plasma may be considered. The total ion energy flux or ion beam power J _E is defined as

 

Fig. 2 Ion velocity distribution functions for two polar angles.

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where R ₀ is the distance between source exit and measurement location and the functions j _E(ϑ) are of the kind j _E∼[cos(ϑ)]^β fitting to experimental data (Fig. 3 ). The ion beam power can be related to APS parameters by employing a nonlinear multidimensional scaling law (6b) [12]:

 

Fig. 3 Polar profiles of total ion kinetic energy flux density for two cathode states.

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The parameters a _j need to be determined by an appropriate optimization routine. Figure 4 shows the idea, where a ₀ = 2.8⋅10⁻²A²V⁻¹, a ₁ = 2.5, a ₂ = a ₃ = a ₄ = −0.5. The figure displays the actual values J _E,exp on the abscissa and the estimated values J _E,scl by the power law. The distribution of data points on the abscissa indicates the limitation of using V _A as a measure of ion beam properties.

 

Fig. 4 Basic scaling law for total ion kinetic energy flux.

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With the availability of scaling laws relating plasma properties to external control variables, a new class of control scheme can be implemented. For preparation, standard processes were run in V _A control mode (i.e. V _A = const.) and values of J _E,scl had been recorded. It was found that in particular in the deposition phase J _E is not constant, but subject to drifts and shifts from run to run. Hence, even when V _A is constant, the plasma properties may vary, which is at least one reason for the process instabilities and limits on layer reproducibility that are in the focus of the present study. Consequently, the alternative assistance concept (J _E control) is to adapt V _A such that J _E is constant during the PIAD process, while V _A is drifting. Notably, this kind of concept allows to counteract apparent variations in plasma properties without operating diagnostics during the process. We regard this approach as a bridge between conventional control of external parameters, like currents or voltages, and future concepts based on in situ diagnostics of plasma parameters.

2.3 Deposition experiments

Because the deposition technique as introduced in Sect.2.1 has often been described in the literature, we refrain from a further detailed explanation and restrict ourselves to listing main deposition parameters. Thus, a couple of relevant deposition parameters are given in Table 1 . The level of plasma assistance has been chosen to be moderate, i.e. the sensitivity of the deposited layers on plasma parameters is rather high. This behaviour corresponds to the assistance branch of knock-on densification [19]. Thickness control has been accomplished by means of quartz crystal monitors. Fused silica substrates have been used throughout this study to enable ex situ optical characterization from the NIR up to the DUV spectral regions. The target thickness was 200nm for TiO₂, 250nm for Al₂O₃, and 300nm for SiO₂.

Table 1. Overview on selected deposition parameters

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According to Sect. 2.2, different assistance control concepts have been adopted for each of the materials investigated:

In a)-c), the target values of V _A and J _E are corresponding to the same assistance effect, i.e. identical film properties. Hence, the goal was to achieve the same value <n> by means of both control principles. Remaining deviations between the <n>-data obtained from the different strategies are therefore interpreted as systematic deviations, according to (5b). Moreover, in order to account for systematic deviations resulting from spatial particle flux distributions during deposition, substrates were placed on various positions of the substrate holder. In particular, the phrase “in” denotes samples which have been located on an inner circle of the substrate holder, while “out” corresponds to the outermost position (Fig. 1). In some experiments, additional substrates have been placed at an intermediate position (“mid”). In practice, our substrate holder is supplied with numerous substrate mounting fixtures arranged in 6 circles, so that “in” corresponds to the second circle when counting from the centre of the substrate holder. The position “mid” corresponds to the fourth circle, and “out” to the sixth one.

In the case of TiO₂ deposition, in addition to the fused silica substrates, silicon wafers have also been coated for mechanical stress measurements. This allowed identification of possible correlations between the reproducibility in optical and mechanical properties.

Almost all experiments have been carried out at the same deposition system (deposition plant P1). Only in the case of SiO₂, one of the V _A series has been prepared in another system (deposition plant P2) for reference purposes. Table 1 summarizes main characteristics of the deposition series accomplished in our study. The total number of deposition runs thus turns out to be equal to 54, while the total number of samples characterized in this study is 132.

3. Layer characterization

Clearly, the focus in this study is on optical characterization. All samples have been characterized by spectrophotometry using dispersive double beam Perkin Elmer spectrophotometers, equipped with so-called VN-accessories for absolute measurements of transmittance T and reflectance R at an incidence angle of 6° [21]. The thickness and the optical constants have been obtained from spectra fits assuming homogeneous and isotropic coatings, in terms of a multi-oscillator dispersion model. Before fitting, all spectra have been converted to a wavenumber-equidistant grid of size M, with M ≈180 wavenumber points ν_l per spectrum. The fit quality has been quantified in terms of the discrepancy function DF:

The fitting routine has been manually interrupted when the DF was observed to be smaller than a pre-defined threshold value, which has been defined for each of the three materials specifically. In the case that the threshold value could not be achieved, the fit was interrupted when no further improvement of the fitting quality could be observed. All fits corresponding to the same material were started from the same initial approximation.

As an example, Fig. 5 visualizes the fits of two TiO₂ reflection spectra with different fit quality. The spectrum shown on the left corresponds to the best TiO₂ spectra fit within this study, with a value of the discrepancy function of DF = 1.23e-03. This corresponds to a rather excellent agreement between the theoretical spectrum and experimental data. On the contrary, the spectrum on the right shows the fit of a TiO₂ spectrum with the largest final DF, the latter being equal to 2.34e-03, slightly larger than the threshold DF specified for TiO₂ with DF _threshold = 2.0e-03. Nevertheless the fit looks qualitatively good, but in the extrema of the interference pattern, a slight misfit between theory and experiment is observed, most probably caused by a weak refractive index inhomogeneity in this case. Nevertheless, a glance on this “worst fit” compared to the “best fit” confirms, that our fits reproduce the main spectral features of the experimental data. The fit of the transmittance (not shown here) is of similar quality.

 

Fig. 5 Typical fit quality of TiO₂ reflection spectra. on left: DF = 1.23e-03; on right: DF = 2.34e-03. Symbols correspond to measured data, red lines to the theoretical fit.

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Clearly, when fitting the spectra from samples produced in (intentionally) identical conditions, we obtain direct information about the scatter in their thicknesses and optical constants, i.e. some information about the reproducibility of the optical properties from run to run. However, one has to bear in mind that the spectral measurements are subject to stochastic errors as well. In order to quantify this effect, one pre-selected reference sample (of each of the materials) has been measured repeatedly, and those spectra have been fitted by means of the same procedure as described before. The scatter in optical constants and thicknesses as obtained from these repeated measurements of the same reference sample defines a reference level of δn and δd, which represent threshold values for the detection of the reproducibility of optical film properties that are inherent to our method itself.

Just to get an impression, Fig. 6 provides a comparison between experimental spectra obtained from the TiO₂ reference sample (6 spectra measurements, left on top), 6 samples deposited with the V _A control concept (Centre, on the left), and 6 samples deposited with the J _E control concept (left on bottom). The nearly identical spectra obtained from independent measurements of the reference sample correspond to a reference level of the thickness standard deviation of only 0.06nm, and of 0.001 in the refractive index (compare histograms on the right). In the case of 6 different samples prepared by means of the V _A control concept and quartz monitoring at position “out,” the standard deviation in the thickness is close to 4nm, and that of the refractive index about 0.009. With the J _E control concept (again position “out”), we obtain remarkably reduced standard deviations when comparing with the V _A series. Anyway, the standard deviations obtained from different samples appear to be substantially larger than the detection threshold resulting from repeated measurements of the same sample. From here we conclude that our spectrophotometric characterization technique is sensitive enough to judge the reproducibility in optical constants obtained from different deposition runs.

 

Fig. 6 Left: sample sets of 6 experimental titanium dioxide reflection spectra; Right: scatter in corresponding film thicknesses and refractive indices. On top: same sample, 6 independent measurements; centre: 6 samples produced in V _A control mode; on bottom: 6 samples produced in J _E control mode.

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4. Results

The primary result of this study consists of a set of numerical data concerning the mean values and standard deviations of the refractive index at the mentioned reference wavelengths. These data are collected in Table 2 and Table 3 . Note that every row in any of these tables corresponds to the results obtained from a set of 6 (intentionally) identical samples, calculated according to Eqs. (1) and (2) .

Table 2. Mean values and standard deviations of refractive index, shift and mechanical stress (TiO₂)

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Table 3. Mean values and standard deviations of the refractive index (SiO₂ and Al₂O₃)

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Although the present study generally deals with deposition errors characteristic for three PIAD materials, TiO₂ was chosen as a model substance for a more detailed investigation of the observed scatter in optical properties. In particular, the TiO₂ samples have also been investigated with respect to their vacuum shift behavior, as well as their mechanical stress. The idea was to get additional information useful to judge their degree of their porosity. Shift and stress measurements have been performed in the way published elsewhere [22].

In Table 2, the results obtained from the TiO₂ films are summarized. Here we have the strongest experimental basis, because in addition to the scatter in the refractive indices at λ ₁ = 400nm, corresponding data are available for the shift as well as for the mechanical stress. In our convention, positive stress values correspond to compressive stress, and negative to tensile. Due to the rather weak assistance (BIAS voltage V _A around 80V), the stress is small by absolute value. The layers tend to exhibit an air-to-vacuum shift, which indicates some porosity. When comparing the standard deviations of the refractive index with those of the shift, we observe a rough correlation, i.e. a larger scatter in the refractive index appears to be accompanied by a larger scatter in the shift values. This indicates that the level in reproducibility of both these parameters is connected to the degree of reproducibility of the properties of the pore fraction.

In addition, from the data in Table 2 it appears that experimental TiO₂ series prepared by means of the J _E assistance control concept have lower standard deviations than the corresponding series prepared with V _A control. This concerns the scatter in refractive indices, as well as that in shift and in mechanical stress.

In Table 3, refractive index data (mean values and standard deviations) are collected as relevant for the other materials SiO₂ and Al₂O₃. As a main result, we recognize that the standard deviations are now generally lower than in the case of TiO₂. Furthermore, the differences in standard deviations resulting from the different assistance control concepts are much smaller than in the case of TiO₂.

5. Discussion

Comparison between materials

First of all, when comparing the data in Table 2 and Table 3 for the three materials, we recognize the rough trend that a higher average refractive index tends to be accompanied by a higher standard deviation. This finding is visualized in Fig. 7 .

 

Fig. 7 Standard deviations in refractive index vs. average refractive index. Navy symbols correspond to 250nm, and red symbols to 400nm. Open symbols denote J _E series, full symbols V _A series.

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Considerations on porosity

A further evaluation of the data requires postulating certain model assumptions on the physical origin of the observed scatter in optical constants. Keeping in mind the rather moderate assistance level and the presence of a certain vacuum shift (as verified here for TiO₂), it is likely that it is the limited reproducibility in the properties of the pore fraction that gives rise to the observed scatter in refractive indices. Fortunately, it is possible to perform a crude estimation of the porosity of the TiO₂ coatings from the shift values given in Table 2, making use of the so-called Wiener bounds (compare [23–25 ]). These are given by Eqs. (8a) and (8b) :

Here, to consider the effects of pores in a quantitative manner, the film material is regarded as a binary mixture of the “pure solid material” (with a refractive index n_b) and the pores (with a refractive index n_p) [25]. p is the packing density, and 1-p the porosity. Then, from the measured shift and the average refractive indices, the following constraints on the film porosity are obtained (Table 4 ) [24]:

Table 4. Estimation of porosity in terms of the Wiener bounds (TiO₂)

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Table 4 summarizes the span of principally allowed porosity values when assuming, that the shift measurements gather changes in optical film properties between the extremes of completely empty pores (in vacuum) and pores that are completely filled with water (in air). The thus obtained span in porosity appears to be rather large. Nevertheless, the upper Wiener bound corresponds to the rather unlike scenario of flat pancake-shaped pores parallel to the film surface. As result of a typical columnar film growth mechanism, pores are rather expected to be elongated and oriented perpendicular to the film surface. This morphology appears to be closer to what is predicted by the lower Wiener bound. It is therefore reasonable to assume that in most samples, the porosity is lower than 0.1 (or 10%). Hence, in our further model calculation the coatings will be regarded as almost dense, with the porosity (1-p) as a small parameter.

Simple modelling of refractive index fluctuations

In the case of dense coatings, all the typical mixing models known from the literature give similar results for the resulting effective refractive indices of a binary dielectric mixture coating, while their properties are again found within the span defined by the Wiener bounds (8a) and (8b). Therefore we further restrict on the mathematically simple discussion of what is obtained from these bounds.

In practice, all parameters entering into (8a) and (8b) are subject to stochastic fluctuations. Thus, pores can be partially filled with water, which may results in a relevant spread in possible refractive indices of the pore fraction dn_p. Also, the solid fraction of the film is not necessarily well-defined: Various solid phases (different amorphous and crystalline modifications, hydroxides and the like) may coexist in a real film depending on the film material, which may result in some spread of refractive indices of the solid fraction (dn_b) from charge to charge as well. Thus, the three different crystalline phases of TiO₂, in combination with relevant amorphous phases, give certain degrees of freedom for the variation of the properties of the solid fraction. The same is valid for alumina – the refractive index of corundum is entirely different from that of typical aluminum hydroxides, again giving rise for some freedom in the value of n_b. And finally, the properties of silicon oxide are also dependent on the degree of crystallinity and the particular stoichiometry.

Furthermore, the packing density itself will never be absolutely reproducible, giving rise to some variation dp. Then, by calculating the total differential of the upper bound (8a), we find:

For high packing densities (p→1), n_u ≈n_b, and in this case we find the simple approximation:

In analogy, for highest packing densities, for the lower bound it is obtained:

Of course, the fact that n is confined between n_u and n_l does not mean that derivatives like ∂n/∂p are generally confined between ∂n_u/∂p and ∂n_l/∂p. Nevertheless, in the case of dense coatings, any real differentiable n(p) dependence will have a derivative confined between the derivatives of n_u and n_l at least in a small environment of p = 1. Therefore, in the case of packing densities that are close to 1, (9a) and (9b) may be tackled as an estimation for the higher and lower bounds of the variation in n as caused by a small variation in p and n_b. Because of the underlying topological considerations, the behaviour of our coatings is expected to be somewhere in between the predictions of (9a) and (9b), but probably closer to (9b).

Both relations (9a) and (9b) have a rather transparent physical meaning. In the developed picture, variations in the refractive index of an almost dense dielectric film arise from variations in the index of the solid fraction itself, as well as from remaining variations in the packing density. The latter enters with a pre-factor which depends on the films refractive index. This dependence is visualized in Fig. 8 for two assumed values of the pore fractions refractive index n_p. It has further been assumed here that dn_b = 0.001 and dp = 0.002 for all materials. The data as presented in Fig. 8 may therefore be used for a straightforward quantitative estimation of variations in the refractive index, when variations in porosity are in the region of 0.002 or 0.2%. When associating the calculated values dn (Fig. 8) with the experimentally established standard deviations δn (Fig. 7), it turns out that within the presented model, our experimental data are consistent with the assumption that standard deviations in porosity are in the range of 0.1-0.3%.

 

Fig. 8 Model calculation: Variations in the refractive index as calculated from Wiener bounds.

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At lower packing densities, there is no reason to assume that dn is anymore confined between dn_u and dn_l. Therefore, the variations in the refractive index may be larger than what is predicted by the Wiener bounds. The conclusion is that the reproducibility in refractive index, as expressed in terms of the standard deviation, should be worse for moderately or strongly porous layers, when comparing with dense coatings.

Relation to experimentally established standard deviations

As it is evident from Fig. 8, the assumed small variations in packing density cause highest effects on the value of the titania refractive index, while the effect on the silica index is more than 5 times smaller. This means, that a comparable level in the reproducibility in packing density may cause significantly higher standard deviations in the refractive index of high-index coatings than it will be expected in low-index coatings. This straightforward result obviously explains major features in the arrangement of experimental points presented in Fig. 7. Indeed, when associating the calculated values dn (Fig. 8) with the experimentally established standard deviations δn (Fig. 7), we observe an astonishing qualitative agreement between the “allowed” dn-ranges (as confined between the black dotted and red solid lines) from Fig. 8 with the experimental results presented in Fig. 7. We even obtain a reasonable quantitative agreement, when taking the detection threshold into account (Table 5 ):

Table 5. Comparison between experimental refractive index standard deviations and the results of the model calculations

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Equations (9a) and (9b) thus predict highest variations in the film refractive index of TiO₂, and lowest for SiO₂, in best agreement with the experimental findings from Fig. 7. In terms of this model, the differences in the reproducibility are primarily defined by the second term in Eqs. (9a) and (9b) , i.e. caused by the limited reproducibility in packing density. On the other hand, it is one of the striking effects of the plasma assistance that it results in an enhancement of the packing density in a reproducible manner. When optimizing the assistance concept (for example by replacing the V _A controlling concept by the J _E controlling concept), we may primarily expect a decrease in dp. This is reflected, for example, in the standard deviations of the measured shift values tabulated in Table 2. However, a decrease in dp will diminish dn only in high index coatings, while in low index coatings, the effect will be negligible. This is at least one reason for the observed finding that the J _E control concept has led to an improvement in the TiO₂ refractive index reproducibility, while no comparable effect could be observed for the low-index materials.

Returning finally to the primary motivation of this paper, let us condense the data presented in Table 2 and Table 3 into some “rule of thumb” for typical deposition errors relevant in a carefully optimized PIAD experiment with deposition conditions similar to those used here. These data are finally summarized in Table 6 .

Table 6. Typical order of stochastic and systematic refractive index deviations

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Let us emphasize once again that the primary goal of the present study was to quantify the reproducibility in the refractive index of PIAD oxide coatings. We have provided a recipe for that problem, and it is rather this recipe which is the primary result of the study than the concrete numbers fixed in Table 6. In other deposition systems, and with other deposition conditions, other error levels will be found. But one should note that the success of the implementation of suchlike data into a computational manufacturing experiment depends on the correct assignment of stochastic and systematic errors. Thus, a scatter of data caused by systematic long-term drifts in deposition conditions should not be included into the standard deviations, but rather into time-dependent mean values. Otherwise there is a risk of confusing correlated and uncorrelated errors in practice, which will lead to a less reliable estimation of the production yield of a coating. The clean separation between systematic and stochastic effects is extremely challenging in practice. We have done our best to exclude systematic drifts in our experiments, but as it might be seen from some of the diagrams in Fig. 6, this was not always successful. Therefore, the stochastic error levels given in Table 6 may be regarded as an upper limit for estimating realistic standard deviations in the optical constants.

6. Summary and outlook

Essentially, we have addressed three topics in this paper, all of them being connected with the investigation of the refractive index reproducibility of different oxide coatings prepared in a series of intentionally independent PIAD deposition runs.

First of all, we presented a methodology for quantifying stochastic and systematic deviations in the refractive index. Stochastic deviations are described in terms of the usual standard deviations, while systematic deviations are obtained for samples located at different positions of the rotating substrate holder, as well as by using different technological assistance control concepts. As a trend, coatings with a higher refractive index also had a higher refractive index standard deviation. This methodology may find practical applications in the data acquisition for realistic computational manufacturing experiments.

Particularly with respect to assistance control concepts, we obtained experimental evidence that the proper choice of the control concept may result in an improvement of the refractive index reproducibility. In this study, we could show that the application of the J _E control concept has led to an improvement in the reproducibility of refractive index, shift and mechanical stress of titanium dioxide coatings when comparing with the widely used BIAS (V _A) control concept. From the topological point of view, we attribute the improved reproducibility in these macroscopic properties to an improved reproducibility in the porosity of the coatings.

A simple model has been presented for reproducing the recorded trends. When regarding the coatings as an almost dense binary mixture of solid material with a small amount of pores, variations in the resulting refractive index appear to be dominated by variations in the refractive index of the solid material as well as remaining variations in porosity. The latter become the more significant the higher the refractive index of the coating is. This result of the model calculation is in best agreement with the experimental fact, that titanium dioxide showed highest standard deviations in refractive index, and appeared to be most sensitive to changes in the assistance control concept.

Basing on the results of this study, we see certain potential for still improving the reproducibility of the already well-established PIAD deposition technique by optimizing the assistance control concept. The J _E control concept as applied in this study already allows to counteract apparent variation in plasma properties without operating diagnostics during the deposition process, but still in terms of control of external parameters, like currents or voltages. Future developments pursue innovative control concepts based on in situ diagnostics of plasma parameters.