Procedure to characterize microroughness ofoptical thin films: application to ion-beam-sputteredvacuum-ultraviolet coatings

A method for characterizing the microroughness of samples in optical coating technology is developed. Measurements over different spatial-frequency ranges are composed into a single power spectral density ~PSD! covering a large bandwidth. This is followed by the extraction of characteristic parameters through fitting of the PSD to a suitable combination of theoretical models. The method allows us to combine microroughness measurements performed with different techniques, and the fitting procedure can be adapted to any behavior of a combined PSD. The method has been applied to a set of ion-beamsputtered fluoride vacuum-UV coatings with increasing number of alternative lowand high-index layers. Conclusions about roughness development and microstructural growth are drawn. © 2001 Optical Society of America OCIS codes: 310.6870, 180.5810, 120.6660.


Introduction
With the ongoing trend of optical lithography toward ever-decreasing wavelengths, the interest in the study of the microroughness of optical coating components has taken a new perspective because of its crucial influence on the scattering losses of deep-UV and vacuum-UV ͑VUV͒ coatings. 1,2This makes residual roughnesses a dominant scatter source and a technological problem that has to be engineered if high-quality components are to be achieved.
Within this framework, new concepts for the optimization of coatings that take into account specific problems relevant only at these shorter wavelengths have to be developed.Among these concepts, the development of accurate modeling of microroughness of optical surfaces is of particular interest.Power spectral density ͑PSD͒ 3,4 functions offer a comprehensive description of the surface statistics and are a preferred tool for specifying surface roughness. 5owever, when different surfaces are to be compared, the PSD functions have to be represented together in the same graph.Hence only qualitative information can be obtained.To accomplish a quantitative microroughness description, first-order statistics such as the rms roughness or the correlation length are commonly used. 3Nevertheless, these statistics presume that the surfaces follow certain models, which in general are too simple to describe the actual properties of the surfaces.For instance, common models for representing optical surfaces are combinations of Gaussian and exponential cross-correlation functions. 6 -11These models, however, cannot appropriately describe the fractal behavior. 12Moreover, Jakobs 13 showed that the behavior of the PSD of a coated sample contains two components: a fractal one caused by the substrate roughness and a second component caused by pure film roughness.
In this study we provide a means of characterizing the microroughness of optical coatings with a reduced set of parameters, each with physical meaning ͑such as rms roughness or correlation length͒.The method is based on the PSD description of a surface with subsequent extraction of characteristic parameters through fitting of the PSD to theoretical models.These models can be chosen according to the actual behavior of the PSD's.
This method has been successfully applied to a set of coatings for the VUV range.The investigation of these samples was driven by the technological challenges at short wavelengths: the selection of materials with low absorption and scattering in the VUV and the choice and optimization of the deposition processes. 14,15It has been proven that for applications in the UV range fluorides are the preferred materials because of their larger energy gap.However, not every deposition technique is suitable to produce appropriate fluoride coatings: Conventional evaporation techniques tend to produce coatings with high tensile stress because substrate heating is required for improving the coating adhesion and because of the large differences in thermal expansion coefficient between substrates and coating materials.However, the ion-beam-sputtering ͑IBS͒ technique solves this problem, since it does not require substrate heating, hence producing coatings with higher density and compressive stress. 15,16he roughness properties of IBS fluoride coatings are studied with the methods proposed in the present paper.In Section 2 a brief overview of the concepts involved in surface microroughness characterization is given.Then a revision of the available models to represent surface roughness is provided, and finally the method for combining these models to analyze the roughness of thin-film coatings is described.In Section 3 experimental results obtained from atomic force microscope ͑AFM͒ measurements on IBS fluoride coatings are presented, and an initial qualitative study is performed.Then the results of applying the method to the same coatings are presented.Finally, conclusions are drawn in Section 4.

Measurement and Characterization of Surface Microroughness
In this study we propose a method for reducing the large amount of data produced by microtopography measurements.Although the measurements presented in this study were performed with an AFM, the method is suitable for any technique that results in the surface profile of the sample as a one-or twodimensional function of the position such as mechanical profilometry, white-light interferometry, or confocal microscopy.
The simplest reduction procedure consists of characterizing the surface by its rms roughness ͑ rms ͒. 3 This quantity is useful for comparing different surfaces.It is well known, however, that the rms roughness depends on the spatial-frequency bandwidth of the particular measurement technique.Therefore, when the results from different measurements are compared, the measurement conditions ͑scan length and sampling distance͒ need to be equivalent.Moreover, the rms roughness does not give information about the spatial distribution; this means that two surfaces with the same rms roughness could contain different lateral structures.
The PSD yields a more complete roughness description, since it takes into account the statistical correlation between the different surface points.It is defined as the Fourier transform of the surface autocovariance function. 17In optical coating tech-nology it is common to use the two-dimensional isotropic form of the PSD, since the majority of optical coatings and their substrates justify sufficiently the assumption of isotropic roughness.This PSD is then related to the surface autocovariance function through the Hankel transform. 6SD functions obtained from measurements are defined only in a limited range of spatial frequencies.This range depends on the scan length and the sampling distance, and it also can be additionally reduced by the effect of measurement artifacts. 13,18These limitations can be overcome when we combine measurements performed on different scales and͞or with different measuring methods. 19These measurements should fulfill at least two conditions: ͑i͒ The spatial-frequency ranges where the measurements are defined should partially overlap.This condition is easy to meet by adequate selection of the scan sizes and sampling distances.
͑ii͒ In the overlapping region the different PSD functions should be of the same order of magnitude.
Then the result of the PSD combination is a function whose value at a given frequency f is the geometrical average over all the PSD's that are defined at this frequency, where N is the number of PSD functions overlapping at that particular frequency.Figure 1 shows an example of this procedure: The original PSD functions correspond to AFM measurements of a sample corresponding to a three-layer fluoride coating on CaF 2 .A more complete description of all studied samples is given below.The measurements were performed at the same position and with three different scan sizes: 0.2 m ϫ 0.2 m, 1 m ϫ 1 and 10 m ϫ 10 m.The decrease observed at the low-frequency range in the original PSD functions is a result of the detrending procedure commonly applied to the raw data. 12These points are not considered for calculating the combined PSD.Furthermore, the influence of noise can be perceived in the high-spatial-frequency ranges of each individual measured PSD.These points have not been considered for the combination, either.With this example it is shown that this procedure enables the generation of a complete description of surface roughness on a large bandwidth of spatial frequencies.This procedure can be applied to combine any number of microtopography measurements, including measurements taken at different positions on the same sample or even measurements performed with different methods.
Although the PSD has a more complete description of the surface than the rms roughness, it is difficult to compare the roughness characteristics of different samples on the basis of a function.Consequently, it is useful to obtain quantitative information from the PSD's that can be used to make the comparison between samples easier.This quantitative information can be extracted with the help of mathematical models.

A. Overview of Models Applied in Surface Characterization
PSD functions contain a complete description of the surfaces' statistics in the sense that they do not only take into account the height statistics but also contain information on the surface lateral details.A further step in the PSD interpretation is to model them as analytical functions that depend on a reduced set of characteristic parameters.Provided that these parameters possess a physical meaning, their value is of use to characterize the surface.In this section the mathematical models available for the representation of PSD functions and their relation to the types of surfaces commonly found in optical coating technology are presented.
Several statistical models have been used in the past for roughness description of optical surfaces and thin films.These models consist of a function or combination of functions approximating the experimental PSD behavior.The most extended model for the PSD of a coating is the sum of Hankel transforms of a Gaussian and an exponential autocorrelation function. 6 -11,20The corresponding PSD's have the form In this model the characteristic parameters are the surface rms roughnesses and the correlation lengths for both the Gaussian and the exponential functions.The and the are the result of applying integral expressions ͑4͒ and ͑5͒ of Ref. 12 for all the spatial frequencies from 0 to infinity.This means that these rms roughnesses and correlation lengths become independent of the scale, and thus they are suitable for comparing different surfaces.The cor-relation length is a measure of the characteristic lateral size of the structures on the surface.As a result of the fast decrease of the Gaussian function, it is adequate for surface structures with short-range correlation length ͑s subscript͒.However, the exponential model has a much slower decrease, thus making it suitable for long-correlated surface structures ͑l subscript͒.This model has been extensively used and has led to satisfactory results.However, it is not completely correct when a wide range of spatial frequencies are considered.It has been demonstrated by AFM and scattering measurements 13 that the PSD of a coating can be actually approximated by the sum of the PSD of the substrate and the PSD of the pure thin film.The PSD's of the substrates follow an inverse power law that cannot be approximated by either the Gaussian or the exponential PSD's.To this end the fractal 12,17 model is used to describe these roughnesses, where K has units of length to the power of ͑3 Ϫ n͒.
Here rms roughness and correlation length cannot be defined for the fractal model, and consequently the parameters K and n are the values taken to characterize the surface.This PSD form is obtained when it is assumed that the surface is self-affine. 21This is a common characteristic of highly finished optical surfaces.
An alternative function for describing the PSD of the pure thin film is the ABC model 21,22 ͑also called the k-correlation model in some references͒: where C is a constant greater than 2. Unlike the previous models, the ABC model does not correspond to any defined autocovariance function.Nevertheless, specific information about the surface microstructure can be extracted from the parameters A, B, and C. Following Eqs.͑4͒ and ͑5͒ of Ref. 12, equivalent rms roughnesses and correlation lengths that depend on the three parameters can be derived: From analyzing expressions ͑5͒ and ͑6͒ it can be deduced that the parameter C provides the possibility of having a continuous transition model between the exponential ͑C ϭ 2͒ and the gaussian ͑C Ͼ Ͼ 2͒ models.All models presented up to this point are monotonically decreasing functions of the spatial frequency.However, it has been observed that several coatings show the formation of superstructures uniformly distributed along the surface. 23,24This induces a local maximum in the lower frequencies of the PSD that cannot be described satisfactorily by any of the previous models.An example is represented by the PSD of Fig. 1, where the local maximum can be recognized at a frequency near 3 m Ϫ1 .A convenient PSD for the modeling of this local maximum is a Gaussian function with its maximum shifted to a nonzero spatial frequency 25 : This PSD corresponds to an autocovariance function with the form of a Gaussian multiplied by a cosine.The period of the cosine is related to the periodicity of the superstructures in the surface.This period is translated into the spatial-frequency domain as the shift of the PSD maximum to the frequency f sh .The meaning of the other model parameters can also be related to other characteristics of these superstructures: The sh corresponds to their size; sh , to their height.So far, no combinations of these models have been used for a quantitative surface-roughness description, even when the experimentally obtained PSD's suggested that such a combination would be useful.The reason might have been the lack of a method indicating the steps to find the correct combination and providing prescriptions for extracting meaningful values.In this study we suggest such a method, which is based on the use of the models outlined above.

B. Analysis Method
The surface-roughness analysis of a given optical coating or substrate is accomplished by means of fitting the experimentally obtained PSD to a combination of models.This combination ͑and thus the number of parameters that characterize the surface͒ will depend on each particular sample.The combination has to be chosen through examination of the experimental PSD and recognition of the different roughness contributions.
In this study, for instance, we use the following particular combinations.For the substrates the fractal model is applied, where only two parameters are needed for the description.A convenient model for a large variety of coatings is the combination of the fractal and the ABC models: The fractal function takes into account the substrate influence, whereas the ABC model is directly related to the intrinsic roughness of the coating.The ABC model has been found to be more adequate than the Gaussian, the exponential, or their sum, because the exponent C enables better matching of the film PSD.This model needs five parameters to describe the surface ͑K, n, ABC , ABC , and C͒.Finally, as is demonstrated in detail in Subsection 3.B, a more elaborated model is needed to fit the PSD of coatings showing the superstructures mentioned above.In this case the sum of three PSD functions is considered: where as many as seven parameters are necessary.The fitting procedure is carried out by minimization of a merit function that is a measure of the difference between the experimental PSD and the theoretical model.Thus if P is the set of parameters that characterize the model under consideration, the merit function is defined as In this expression the f i are the spatial frequencies at which the measured PSD is evaluated and N is the number of points at which the PSD is sampled.This merit function is a customized form of the chi-square function, widely used in the fitting of experimental data. 26The customization consists of substracting the logarithms of the PSD's instead of the PSD's themselves.This is necessary because the values of the PSD function can extend over several orders of magnitude and because a direct comparison would give more weight to the spatial frequencies where the PSD value is higher.This form of merit function has been successfully applied to the fitting of angleresolved scattering measurements. 27 downhill simplex algorithm 26,28 was applied to accomplish the merit function minimization.The result of this minimization is a parameter set P providing the best fit of the model to the experimental data.The values of these parameters are then taken as the surface characteristic values.
The main advantage of this method is that it can be adapted easily to any kind of surface, regardless of its particular characteristics, by means of choosing the combination of PSD models that matches the exper- imental PSD behavior without making any a priori assumption.
As an example of this analysis method we present in Fig. 2 the roughness characterization of the sample corresponding to the PSD presented in Fig. 1.The pluses represent the PSD function resulting from the combination of the different measurements.The model components are represented separately as discontinuous lines.The straight dashed line corresponds to the fractal PSD modeling the substrate.The dotted curve is the ABC PSD used to represent the intrinsic film roughness.Finally, the dasheddotted curve is a shifted Gaussian taking into account the influence of the superstructures.The thick continuous curve is the sum of the three functions: the best fit obtained for these specific measured data.The corresponding parameters characterizing the surface roughness are presented in Table 1.
In summary, the method consists of the following steps: ͑i͒ Calculate the PSD from each measurement and combine all the PSD functions into a single one, with the combination procedure.
͑ii͒ Choose the appropriate combination of theoretical PSD functions that is likely to best match the PSD behavior.
͑iii͒ Fit this combination to the experimental PSD to obtain the values that characterize the sample.

A. Atomic Force Microscope Measurements and Calculation of Experimental Power Spectral Densities
The method described in Subsection 2.B was applied to a set of samples consisting of single-layer and mul-tilayer fluoride coatings deposited by IBS.The experiment was intended to study the elastic lightscattering properties in the VUV spectral region and the relation of these properties to the film microstructure.The results of this study have been published elsewhere. 23,24Table 2 summarizes the produced designs, where the thickness units are optical quarter-waves and the design wavelength is 193 nm.MgF 2 and LaF 3 were used as the low-and high-index materials, respectively.In the table, L ͑H͒ stands for a quarter-wave optical thickness layer of lowindex ͑high-index͒ material.The substrates used were superpolished UV-grade fused silica ͑FS͒ and CaF 2 .The designs were chosen to include the coating thickness ͑or equivalently, the number of layers͒ as variables in the study.
The microroughness was measured with an AFM ͑Digital Instruments, Model dimension 3000͒ operated in the proprietary Tapping Mode.Silicon tips with 10-nm radius were employed.The measurements consisted of matrices of 512 ϫ 512 points with scan sizes of 0.2 m ϫ 0.2 m ͑on some selected samples͒, 1 m ϫ 1 m, and 10 m ϫ 10 m.The columnar structures revealed a tendency to aggregate and form superstructures uniformly distributed, resulting in an increase in rms roughness.This tendency to aggregate is a particular character-  istic of the fluoride coatings studied in this paper and is not commonly observed in other types of coating.
For this reason we will distinguish throughout this paper between the roughness induced by these aggregates and the roughness induced by the granular structure of the coating ͑which will be called the intrinsic roughness of the coating͒.Figure 5 displays a selection of the PSD functions for the designs with quarter-wave layers.Graph ͑a͒ represents the data for the samples deposited on FS, and graph ͑b͒ corresponds to the CaF 2 substrate.The PSD's of the bare substrates have been included as a reference.First we observe that the PSD's of both substrates follow the expected inverse power law of the fractal model with the exception of the lower frequencies where the scanner bow effect 13 can give rise to an upward deviation.The PSD functions of quarterwave single-layer designs reveal a roughness component related to the fluoride columnar structure observed at high spatial frequencies.In the lowspatial-frequency range the roughness statistics of the substrate is reproduced by the films.
The PSD of the S͞L͞H coating is similar to that of the S͞L sample for both substrates.This indicates that the high-index quarter-wave layer adds a small roughness component.The addition of a second lowindex quarter-wave layer induces the formation of aggregates as seen in Fig. 4.These aggregates lead to the appearance of a local maximum in the PSD near 3 m Ϫ1 .By observing the PSD of the S͑͞L͞H͒ 10 samples, where the local maximum is overridden by the intrinsic roughness component of the coating, it can be concluded that the aggregates' height does not vary when new layers are added.To illustrate this fact, Fig. 6 presents the measurement of sample S͑͞L͞H͒ 10 onto CaF 2 : the aggregates are still present but with a much lower contrast as compared with Fig. 4, a fact that makes it difficult to distinguish them from the coating roughness.
To complete this presentation of experimental results, a selection of the PSD functions for the halfwave designs is presented in Fig. 7.As in the previous case, the PSD of the substrate has been included for reference.The presence of aggregates can be observed in samples S͞2L͞2H͞2L and S͞2L͞2H͞2L͞2H͞2L on FS and S͞2L͞2H on CaF 2 .The analysis method presented above has been used for a complete roughness characterization of the samples described in Subsection 3.A.Each PSD has been fitted to the suitable model; this includes a fractal for the substrates, a fractal and an ABC-PSD for the samples without aggregates, and finally a fractal plus an ABC plus a shifted Gaussian for the rest of samples.
The results are summarized in Tables 3 and 4. The tables contain the resulting rms roughnesses and   correlation lengths corresponding to the ABC model as well as the value of the exponent C. For the samples with aggregates the shifted-Gaussian characteristic parameters are included as well.The K and the n values obtained from the characterization of the samples were similar to those obtained for the bare substrates.This is a proof of the consistence of the analysis method, since it is able to separate correctly the substrate roughness component from the total roughness of a coated sample.Consequently, only the K and the n values for the substrates are presented in Table 5.
The results of the method allow for access to observations that would not be possible from simple PSD examination.First, a comparison of the growth of film roughness with thickness depending on the substrate type can be performed.This comparison is presented in Figs. 8 and 9.In Fig. 8 the intrinsic rms roughness of the coating ͑resulting from the ABC component͒ is represented for the quarter-wave samples.It can be seen that the roughness grows with coating thickness and that this growth is not influenced by the substrate, since the values are similar for both substrates.It is important to note that the high-index quarter-wave layer adds a smaller com-ponent of roughness than the low-index one.This can be seen when we compare the values for the samples S͞L and S͞L͞H.This small increase is related to the combination of two effects: On one side, the physical thickness of a high-index layer is smaller.On the other side, the LaF 3 films show a globular structure in opposition to the columnar structure of the MgF 2 films.This globular structure seems to promote a lower roughness.
From Fig. 8 it can also be observed that the rms roughness increases with physical thickness following a power law.More precisely, if T is the physical thickness of the coating, the intrinsic rms roughness ͑͒ follows ͑T͒ ϰ T ␤ , with ␤ ϭ 0.52 for the samples with FS substrates and ␤ ϭ 0.51 for those with CaF 2 substrates.This result has to be taken as an average for both MgF 2 and LaF 3 materials.Furthermore, these values are in good agreement with the results obtained from total scattering measurements for PbF 2 coatings deposited by plasma chemical vapor deposition. 29he correlation length of the quarter-wave samples is presented in Fig. 9. Again, the behavior of this parameter is similar for the samples on FS and for the CaF 2 substrates: a gradual increase with total  coating thickness.However, the establishment of a model for this behavior is not as clear as in the case of the rms roughness.The analysis procedure presented here enables us to extract information about the aggregates as well: In the shifted-Gaussian model the aggregate size is characterized by the parameter sh ; their periodicity is related to the central frequency f sh ; and, finally, their height is given by sh .As stated above in the qualitative description of the experimental results of Section 3, the height of the aggregates remains at the same level when the coating thickness increases.This qualitative information can now be quantitatively confirmed: The values of sh for the quarterwave samples are all in the range of 0.19 -0.25 nm and do not show any clear tendency of increase with thickness.Two further conclusions can be extracted from the characteristic parameters of aggregates: First, their size tends to increase slowly with the coating thickness; and second, the center frequency decreases slowly with thickness, indicating that periodicity of the aggregates on the surface increases.

Conclusions
In this study a method for comprehensive analysis of the microroughness of optical surfaces and coatings has been presented.The method is capable of extracting a reduced set of parameters that characterize a surface on a microscopic scale.The characterization procedure is based on power spectral densities ͑PSD's͒ determined from microtopography measurements and subsequent combination of all PSD functions obtained into a single PSD.This provides a surface description over a large bandwidth in the spatial-frequency space, since it allows for the combination of measurements performed at different scales or even with different experimental systems.The surface analysis is accomplished by means of fitting the combined PSD into a model dependent on a reduced set of physically meaningful parameters that quantitatively describe the surface roughness.This enables modeling of PSD's with an arbitrary combination of theoretical models, which can be chosen to match the different roughness contributions that measured PSD's usually contain.
With the results of this analysis procedure it is possible to raise conclusions that would not be available from the plain qualitative observation of the PSD.To illustrate this, the method was applied to coatings with successive layers of MgF 2 and LaF 3 deposited by ion beam sputtering ͑IBS͒ onto fusedsilica ͑FS͒ and CaF 2 substrates.Employing the proposed analysis procedure provided information on the development of the intrinsic film roughness independently of the substrate characteristics or other roughness contributions such as aggregate formation.
It was also shown that an ABC model is a good approximation for describing the intrinsic roughness of a coating.Also, the aggregates can be adequately modeled with a shifted-Gaussian PSD.The results reveal that the rms roughness and correlation length of the coating increase with increasing number of layers and that the substrate type does not influence this behavior.More specifically, the intrinsic rms roughness increases with the square root of the physical thickness, which is in agreement with other experimental results obtained for fluoride coatings.Concerning the aggregates, the use of the procedure has confirmed that their heights remain approximately constant with increasing number of layers, whereas their sizes increase and their periods increase slightly.
The authors thank Jo ¨rg Steinert for his support of the measurements and for fruitful discussions and Aure ´lien Petit dit Dariel for his contribution to the deposition of fluoride coatings.The authors also gratefully acknowledge the support of the European Commission ͑TMR network "High-quality UVcoatings," contract ERBFMRX-CT97-0101͒.

Fig. 1 .
Fig. 1.Example of the combination of PSD functions corresponding to different scan sizes.

Fig. 2 .
Fig. 2. Example of fitting of a measured PSD to a theoretical model.See text for the details.
Figures 3 and 4 are examples of two of these measurements.Figure 3 corresponds to the S͞L design onto CaF 2 .A granular structure related to the tops of the columnar structures, which are typical for fluoride coatings, can be observed in the left-hand image ͑L ϭ 1 m͒. Figure 4 corresponds to the S͞L͞H͞L design onto CaF 2 and shows a particular phenomenon of IBS fluoride coatings that was observed on some samples:

Table 2 . Designs of the Samples Studied in This Paper
a Quarter-wave optical thickness layer.b Quarter-wave.c Half-wave.