Quality control of oblique incidence optical coatings based on normal incidence measurement data

Tatiana V. Amotchkina; Michael K. Trubetskov; Alexander V. Tikhonravov; Sebastian Schlichting; Henrik Ehlers; Detlev Ristau; David Death; Robert J. Francis; Vladimir Pervak

doi:10.1364/OE.21.021508

1. Introduction

The determination of layer parameters of produced multilayer coatings (reverse engineering (RE) of optical coatings) provides a feedback for the design-production chain. The main intention of the RE is to detect systematic and random errors in layer parameters, to improve deposition control and, therefore, to raise the quality of optical coating production [1–4]. In order to properly characterize produced coatings, a reliable RE approach should be chosen.

We refer to a combination of the experimental data set, coating model, corresponding numerical algorithm, and verification method as a RE approach. It is known that optical coatings parameters (levels of thickness errors, stability of optical constants of layers, degree of inhomogeneity, etc.) are dependent on the deposition process and on the monitoring technique used in the course of the deposition. Hence, coatings deposited under different deposition conditions correspond to different mathematical models. RE procedures, therefore, are different. Evidently, the RE approach should be the same for all coatings deposited under the same deposition conditions. Once carefully chosen, the approach can be applied further to post-production characterization of all similar optical coatings. Such selection is especially important for mass production of optical coatings, allowing one to improve the quality of manufactured coatings and the production yield.

The most typical RE approaches are based on the numerical analysis of normal incidence or quasi-normal incidence transmittance ( $T$ ) and/or reflectance ( $R$ ) data related to a produced coating sample. This approach is quite simple from an experimental point of view because normal incidence transmittance and quasi-normal incidence reflectance can be measured using a wide set of spectrophotometers. The approach, however, may easily lead to unreliable results. This unreliability can be attributed to an insufficient information content of some experimental data sets and to the influence of measurement errors on the obtained results. Data related to broadband reflectors or to broadband antireflection coatings (AR) can be considered as examples of such low-informative data sets. Some examples of insufficient information content of measurement data are presented in our recent publication [2].

Modern state-of-the-art thin film coatings are becoming more complicated and as demands on optical coating performance continue to increase, the ambiguity of RE continues to grow with the typically growing number of layers. Generally, in order to overcome ambiguity in RE, approaches utilizing larger sets of measurement data should be developed. Recently, RE approaches based on in situ data analysis have been actively developed [2,3,5,6].

In order to obtain reliable RE results one should take care that not only a sufficient amount of input information is considered but also that reasonable optical coating model and numerical algorithm is used for data analysis. All a priori information on a produced coating, such as the course of the deposition process, deposition conditions and accuracy of measurement data should be taken into account by the model and algorithm. In the first turn, the algorithm should provide RE results which can be used for adjusting a deposition process (for example, calibration of quartz crystal monitors) or for correcting the optical coating model (for example, changing optical constants of layer materials). Achievement of perfect fitting of experimental data by model spectral characteristics is not necessarily an indication of good quality RE results. For example, in many cases a perfect fit can be achieved with physically non-sensible models.

The important issue concerning RE process is the validation of the results. For this purpose, various approaches can be used. Recently, we reported on verification of RE results using specially produced coatings with imposed errors to layer thicknesses [2,5]. In [6] the verification of the RE results with the help of group delay measurements was demonstrated.

In the present paper we consider the selection of reliable RE approaches in the case of multilayer coatings working at oblique incidence. Typically, RE processes for such coatings are based on the analysis of normal incidence data (ex situ and/or in situ). It is well known, however, that a good fit of normal incidence data does not guarantee the correct spectral performance at oblique incidence. Ideally it would be useful to increase the amount of input information by adding oblique incidence measurement data. Unfortunately, as discussed below, respective measurements are more complicated than normal incidence ones and it is therefore desirable to minimize them. Therefore, development of RE approaches which allow to obtain reliable oblique-incidence spectral characteristics on the basis of normal or quasi-normal incidence measurements only is an actual problem.

Accurate oblique-incidence $R$ and $T$ measurements are typically more complicated than normal and quasi-normal incidence measurements [4,7]. Multiangle photometric measurements of real samples typically need to measure not only the directly transmitted or reflected light beam but also sufficient of the internally reflected beams in order to provide an accurate measurement of the total transmittance or reflectance of the sample being studied. At normal and near-normal incidence the aperture required to collect the internal reflections is typically not much larger than that needed to acquire the directly transmitted or reflected beam. However, as the angle of incidence at the sample surface increases, the lateral separation of the internally reflected beams increases through a maximum, near 50° angle of incidence (AOI), before decreasing again as the beam approaches grazing incidence. Further, the lateral separation of the beams depends not only on the AOI but also on the thickness of the sample to be measured. The lateral separation of the internally reflected beams will be greater for a thick sample than for a thin sample. Lastly, as the AOI increases, the area on the surface of the sample on which the incident beam impinges grows as the lateral dimension increases as $1 / \cos (AOI) .$ Hence optical systems for accurate measurement in the range from normal to oblique incidence need to incorporate sufficient detector aperture size to collect the wide range of lateral beam spread that will be encountered. As a result careful consideration needs to be given to the physical dimensions of the sample so that all the expected internal reflections can be generated and escape the sample surface (front and rear) and be measured. For example, the direct reflection from a fused silica flat plate is approximately 3.5% of the incident flux; the first internal reflection is nearly as great at 3.2%. The second internal reflection falls dramatically to well below 0.01%. Similarly the direct transmission through a fused silica plate is approximately 93.2% and transmitted portion of the first internal reflection is approximately 0.1% and the second is essentially zero. Clearly, measurement systems that collect only the directly reflected or transmitted beam but neglect the internally reflected and transmitted portions provide a skewed estimate of the total reflectance and transmittance of the sample. Ideally an instrument or measurement technique specially configured for oblique incidence measurements or special accessories to devices typically used for normal incidence measurements, are required.

In our present work we propose to perform careful oblique-incidence measurements only for several test coatings and to use the obtained experimental data for adjusting RE process. The RE process is to be performed on the basis of normal incidence data, oblique incidence measurements are used only for verification of the results. For further oblique incidence coatings produced under the same deposition conditions, the adjusted RE procedure can be used without additional verification.

In Section 1 we show an example of unreliable RE approach. In the rest of the current work we demonstrate selection of reliable RE approach using three quite complicated problems. We perform post-production characterization of coatings deposited under different deposition conditions and, therefore, obtain different RE approaches. In Sections 2−4 we characterize oblique incidence edge filter, oblique incidence beam splitter and oblique incidence 43-layer quarter-wave mirror, respectively. Our final conclusions and recommendations are presented in Section 5.

2. Example of unreliable reverse engineering of 21-layer quarter-wave mirror

In order to demonstrate an unreliable RE approach we consider an example of post-production characterization of 21-layer quarter-wave mirror for the central wavelength of 400 nm. We used HfO₂ and SiO₂ as high (H) and low (L) index materials, respectively. Substrate is BK7 of 1 mm thickness. Nominal refractive indices of the materials and the substrate are presented in Table 1.

Table 1. Refractive indices of layer materials and substrates ( $λ$ should be expressed in µm).

View Table | View all tables in this article

The first layer of the mirror is a HfO₂ layer. The experimental sample was prepared at Rudjer Boskovic Institute (Zagreb, Croatia). We denote this sample as HR400-BK7. Layer thicknesses were controlled by a quartz crystal monitor. Reflectance and transmittance photometric data were taken in the spectral range from 300 nm to 2500 nm at AOI of 7°, 10°, 20°, 30°, 40°, 50°, and 60° for s- and p-polarized light. The accuracy of measurement data, estimated in [4], is 0.2%.

In Fig. 1(a) we compare theoretical and experimental reflectance data (s-polarization case, AOI = 7°). In Fig. 1(a) one can observe deviations between theoretical and measured data. These deviations can be attributed to various reasons, such as imprecise calibration of a quartz crystal monitor, inaccurate knowledge of high and low refractive indices, instability of the layer materials, etc.

Fig. 1 Comparison of near-normal incidence experimental and theoretical reflectance data related to HR400-BK7 sample (a). Fitting of experimental reflectance data by model reflectance data achieved by RE algorithm (b).

Download Full Size | PDF

In order to estimate closeness between model and experimental data we introduce a discrepancy function:

D F^{2} = \frac{1}{L} \sum_{j = 1}^{L} {[R (X; λ_{j}) - \hat{R} (λ_{j})]}^{2},

where

X

is the vector of model parameters describing the considered coating. Here and in what follows, we shall refer to “theoretical spectral curves” as to spectral curves of the initial design and to “model spectral curves” as to curves of the model design. In other words, theoretical and model can be called “unfitted” and “fitted” spectral characteristics. We perform post-production characterization of the HR400-BK7 sample on the basis of an analysis of experimental reflectance data taken in the spectral range

[300; 1100]

nm on the wavelength grid

{λ_{j}}, j = 1, \dots, L,

L = 537.

If we know refractive indices,

n_{H} (λ)

and

n_{L} (λ),

with high accuracy and assuming also that the layers are homogeneous, in our RE algorithm we minimize function Eq. (1) and search for relative errors in layer thicknesses only. In this case:

R (X, λ) = R ((1 + δ_{1}) d_{1}, \dots, (1 + δ_{N}) d_{N}; n_{H} (λ), n_{L} (λ); λ),

where

δ_{1}, \dots, δ_{N}

are relative errors in layer thicknesses,

N

is the number of layers.

In Fig. 1(b) we demonstrate the achieved excellent fit of the experimental data by model data. The estimated errors are shown in Fig. 2.

Fig. 2 Estimated relative errors in layer thicknesses of 21-layer quarter-wave mirror.

Download Full Size | PDF

At the same time the correspondence between oblique-incidence experimental and model reflectance data, presented in Fig. 3, is not good. In Fig. 3 one can observe noticeable deviations between model and measured data indicating unreliability of RE results. There are at least two reasons for this unreliability. The first reason is that input information is not sufficient for post-production characterization of the considered coating because the most informative ripples at the left side of high-reflection zone are not included to the measurements. The second reason is insufficiency of the applied coating model. More accurate model should include variations of the refractive indices, inhomogeneities of layers, etc. Further attempts of the RE of the sample are not possible because we cannot provide sufficient input experimental information (for example, multiscan in situ data or ex situ data in the UV spectral range). We show this example only as an illustration of unreliable RE approach.

Fig. 3 Comparison of oblique incidence experimental and model reflectance data related to HR400-BK7 sample.

Download Full Size | PDF

3. Reverse engineering of an edge filter

The second problem is the RE of an edge filter working at the AOI of 45°. The filter should provide the transmittance more than 93.5% in the spectral range from 670 nm to 710 nm, and reflectance more than 99% in the range from 585 nm to 643 nm. Substrate back side reflections are taken into account. Incidence light is unpolarized. In the design process TiO₂ as high-index material and SiO₂ as low index material were used. Substrate was Suprasil of 1 mm thickness. Refractive indices $n_{H} (λ),$ $n_{L} (λ)$ of the materials, and substrate refractive index $n_{s} (λ)$ are presented in Table 1. Extinction coefficient $k (λ)$ of TiO₂ is estimated as $k (400) = 0.0036,$ $k (450) = 0.0015,$ $k (500) = 0.0006,$ $k (550) = 0.0003,$ $k (600) = 0.0001,$ $k = 0$ for $λ > 600$ nm with piece-wise linear interpolation to intermediate $λ .$ In [8] a good solution to this design problem was found using OptiLayer thin film software [9]. This solution is a 28-layer design with total physical thickness of 4032.5 nm. The design structure is presented in Table 2.

Table 2. Design structures considered in Sections 3−5, layer numbers LN starts from the substrate, physical thicknesses and layer materials are listed.

View Table | View all tables in this article

The coating was produced using e-beam evaporation with ion-assistance at Leybold SYRUSPro 1100 deposition plant equipped with APS Pro. For controlling layer thicknesses broad band monitoring system (BBM) developed at the Laser Zentrum Hannover was used [10–12]. The produced sample is named as EF-Suprasil. In Fig. 4(a) theoretical and experimental normal incidence transmittance data of EF-Suprasil sample are compared.

Fig. 4 Comparison of experimental and theoretical near-normal incidence data (EF-Suprasil sample) (a). Fitting of experimental normal incidence transmittance data by model data (b).

Download Full Size | PDF

We perform RE on the basis of analysis of transmittance in situ spectra ${\hat{T}}^{(i)} (λ_{j})$ recorded after deposition of each layer $i = 1, ..., N .$ These data were acquired using BBM device at the spectral range $[400; 1000]$ nm on the wavelength grid ${λ_{j}},$ $j = 1, \dots, L,$ $L = 1354.$

Denote as $T (X_{i}; λ_{j})$ the model transmittance data for the first $i$ deposited coating layers, $X_{i}$ is the vector of parameters describing considered coating. In our RE approach we utilize all transmittance scans simultaneously and use the triangular algorithm for layer thicknesses determination [5]. In order to estimate closeness between model and experimental data, we introduce a multiscan discrepancy function:

M D F^{2} = \frac{1}{N L} \sum_{i = 1}^{N} \sum_{j = 1}^{L} {[T^{(i)} (X_{i}; λ_{j}) - {\hat{T}}^{(i)} (λ_{j})]}^{2} .

It is known from previous multiple experiments, that slight variations of optical constants of TiO₂ and SiO₂ materials are possible in the course of the deposition process [13]. Hence, in our RE algorithm we assume that refractive indices can be shifted with respect to the nominal refractive indices used in the theoretical design. We introduce two unknown parameters, namely, refractive index offsets $h_{H}$ and $h_{L} :$

T (X_{i}; λ) = T (d_{1}, \dots, d_{i}; n_{H} (λ) (1 + h_{H}), n_{L} (λ) (1 + h_{L}); λ) .

After minimization of the

M D F

function with respect to

h_{H}

and

h_{L}

parameters, we obtain corrected refractive indices

{\tilde{n}}_{H} = n_{H} (1 + h_{H})

and

{\tilde{n}}_{L} = n_{L} (1 + h_{L}),

where

h_{H} = - 0.0112

%,

h_{L} = 0.0207

%. As BBM provides self-compensation mechanism [14–16], then relatively large errors in layer thicknesses are possible. Hence, at the second step of the RE algorithm we assume non-correlated random errors in layer thicknesses and present them in the form:

T (X_{i}; λ) = T (d_{1} (1 + δ_{1}), \dots, d_{i} (1 + δ_{i}); {\tilde{n}}_{H} (λ), {\tilde{n}}_{L} (λ); λ) .

Determined relative errors in layer thicknesses are shown in Fig. 5. It is seen from Fig. 5(a) that the largest error of about 5.5% is found in the thinnest layer with number 25. Compensation of this error is observed in the subsequent three last layers 26-28. Obtained fitting of measured normal incidence transmittance values by model data is shown in Fig. 4(b).

Fig. 5 Estimated relative errors in layer thicknesses of the edge filter coating (a). Comparison of oblique incidence experimental transmittance data related to EF-Suprasil sample with model transmittance (b).

Download Full Size | PDF

As design requirements are specified at AOI = 45°, the correspondence between experimental and model oblique incidence characteristics must be checked. Perkin Elmer Lambda 950 allows one to measure transmittances ${\hat{T}}^{(s)} (λ_{j})$ and ${\hat{T}}^{(p)} (λ_{j})$ for s- and p-polarized light. In Fig. 5(b) we compare transmittance, calculated for model coating found as a result of applied RE algorithm, and experimental oblique incidence transmittance data $\hat{T} (λ_{j})$ calculated as $({\hat{T}}^{(s)} (λ_{j}) + {\hat{T}}^{(p)} (λ_{j})) / 2.$ In Fig. 5(b) one can observe a good correspondence between measured and model transmittance data. This correspondence confirms the reliability of our RE results and, more general, validates our RE approach. It can be checked that transmittance values in the spectral range from 670 to 710 nm are greater than 93.5%, reflectance values in the range $[585; 643]$ nm, calculated as $\hat{R} (λ_{j}) = 100 % - \hat{T} (λ_{j}),$ are more than 99%. For the next oblique-incidence coatings, it is not necessary, therefore, to perform oblique-incidence measurements.

4. Reverse engineering of a beam splitter

The third task is post-production characterization of a specialized beam splitter working at the AOI of 45°. The target specifications are the following. Target transmittance of the sample is more than 98% in the range from 935 nm to 945 nm, target reflectance is more than 98% in the range from 967 nm to 971 nm. Substrate is Suprasil of 1 mm thickness. In the design process we used Nb₂O₅ as high index material and SiO₂ as low index material. Nominal refractive indices of these materials and substrate refractive index are presented in Table 1. The solution to this problem is a 52-layer design for the substrate front side and 10-layer AR coating for the substrate back side [9]. Structures of both designs are presented in Table 2.

The experimental samples were produced with magnetron-sputtering plant (Helios, Leybold Optics GmbH). The plant was equipped with a BBM system [10–12]. As the sample is to be coated from its both sides, two deposition runs were performed.

In the first deposition run, we deposited the 52-layer coating at two samples. We placed two substrates, namely, Suprasil of 1 mm thickness and Glass B260 substrate of 1 mm thickness exactly at the same distance from the rotation center of turntable. At these positions two corresponding samples denoted as BS-Suprasil and BS-Glass were produced. The sample BS-Glass was placed on the BBM measurement position.

In the second run we deposited the AR coating at two samples. We turned up the BS-Suprasil sample and deposited 10-layer AR coating on its back side. We refer to this sample as to sample BS-AR-Suprasil. Also, a new uncoated B260 Glass substrate was placed on the position where the BBM measured transmittance and the sample, denoted as AR-Glass of AR coating on B260 Glass substrate, was produced. As a result of two deposition runs we had three samples: BS-AR-Suprasil (Suprasil substrate coated from its front and back sides), BS-Glass and AR-Glass samples (B260 Glass substrates coated with 52-layer beam splitter and 10-layer AR, respectively). For the latter two samples in situ transmittance data were recorded. In the course of both depositions, the BBM device worked in a passive mode for data acquisition only. Layer thicknesses were controlled using well-calibrated time monitoring [17,18].

We perform our RE process on the basis of analysis of BBM measurements. The nominal refractive indices (see Table 1) are known quite reliably because they had typically been used in the course of the deposition of complicated dispersive mirrors which are very sensitive to refractive index variations [17–20]. For this reason, we apply RE algorithm assuming random errors in layer thicknesses only. This algorithm is similar to one described in Section 3. As in this case we used accurate time monitoring, errors in layer thicknesses $δ_{i}$ are expected to be small [21]. We consider generalized discrepancy function which is obtained by adding Tikhonov's stabilizing functional [22] to the multiscan discrepancy function defined by Eq. (3):

G M D F^{2} = \frac{1}{N L} \sum_{i = 1}^{N} \sum_{j = 1}^{L} {[T^{(i)} (X_{i}; λ_{j}) - {\hat{T}}^{(i)} (λ_{j})]}^{2} + α \frac{1}{N} \sum_{i = 1}^{N} δ_{i}^{2},

where

α

is a regularization parameter that is selected to provide a compromise between data fitting and stability of the solution. The parameter

α

was taken equal to 1. In Eq. (6)

L = 1356,

λ_{1} = 400,

λ_{L} = 1000,

the model transmittance

T^{(i)} (X_{i}; λ_{j})

is:

T^{(i)} (X_{i}; λ) = T^{(i)} (d_{1} (1 + δ_{1}), \dots, d_{i} (1 + δ_{i}); n_{H} (λ), n_{L} (λ); λ) .

We perform separate RE processes of produced BS-Glass and AR-Glass samples because BBM data correspond to them. Achieved fitting of experimental in situ transmittance data related to BS-Glass sample by model data is shown in Fig. 6. Final fitting of normal incidence ex situ transmittance data related to AR-Glass sample by model transmittance is presented in Fig. 7(a). Estimated errors in layer thicknesses of BS-Glass and AR-Glass coatings are shown in Fig. 7(b). It is seen that the errors do not exceed 2−3% that is in correspondence with estimated accuracy of time monitoring.

Fig. 6 Fittings of in situ experimental transmittance data (red crosses) by model transmittances (black solid curves) related to sample BS-Glass after the deposition of layer 13, 26, 39, and 52.

Download Full Size | PDF

Fig. 7 Fittings of normal incidence experimental transmittance data by model transmittance related to the sample AR-Glass (a). Estimated relative errors in layer thicknesses of the samples BS-Glass and AR-Glass (b).

Download Full Size | PDF

For the sample BS-AR-Suprasil, multiangle spectral photometric measurements were performed using an advanced automated double beam UV-VIS-NIR multi-angle spectrophotometer, the Cary 7000 Universal Measurement Spectrophotometer (UMS) from Agilent Technologies [23]. In Fig. 8(a) we compare 45 degree transmittance data related to BS-AR-Suprasil sample and transmittance calculated for Suprasil substrate coated with obtained model beam splitter and model AR designs. Experimental transmittance data was calculated as $({\hat{T}}_{s} (λ_{j}) + {\hat{T}}_{p} (λ_{j})) / 2.$ For additional validation of the obtained results we also compare oblique incidence experimental and model transmittance data for both s- and p-polarizations, AOI = 30° [Fig. 8(b)]. Good correspondence between experimental and model data observed in both Figs. 8(a) and 8(b) validates our RE results. We conclude, therefore, that for further oblique incidence coatings of this type, the RE algorithm described in this section can be applied without additional verification of the results.

Fig. 8 Comparison of oblique incidence experimental transmittance data of the sample BS-AR-Glass with model transmittance: (a) non-polarized light at 45°, (b) s- and p-polarizations at 30°.

Download Full Size | PDF

5. Reverse engineering of a 43-layer quarter-wave mirror

In the fourth RE problem we characterize a 43-layer quarter-wave mirror for the central wavelength of 800 nm. AOI is 45°, light is unpolarized. Odd layers are HfO₂ layers and even layers are SiO₂ layers. Corresponding nominal refractive indices are presented in Table 1 (rows 8 and 9). Physical thicknesses of high and low index layers are 114.56 nm and 156.58 nm, respectively. The experimental samples were produced with e-beam evaporation using SYRUSPro 710 (Leybold Optics GmbH). In the course of the deposition process the substrate temperature was 120°C. Layer thicknesses were controlled with the help of a quartz crystal monitor. SYRUSPro 710 is also equipped with BBM device [10–12]. This device was used for data acquisition and for on-line recalibration of the quartz crystal monitor as well.

We produced two samples of the 43-layer quarter-wave mirror. Fused Silica substrate of 6.35 mm thickness was placed on one of the calotte positions, and B260 Glass substrate of 1 mm thickness was placed on the position where BBM performed transmittance measurements. Two deposited samples are denoted as HR800-FusedSilica and HR800-Glass. In Fig. 9(a) we compare experimental in situ transmittance data measured by BBM device, experimental ex situ transmittance data taken by Cary 7000 UMS and theoretical transmittance data related to HR800-Glass sample. In Fig. 9(a) one can observe quite large deviations between all three transmittance curves. The main deviations are shifts of the experimental curves to the shorter wavelengths with respect to theoretical curve, and differences in widths of the high-reflection zones in measured and theoretical data. The shifts to the shorter wavelengths are explained by underestimation of layer optical thicknesses. This underestimation can be connected with underestimating layer geometrical thicknesses and/or underestimation of nominal refractive indices of layer materials. Underestimation of layer geometrical thicknesses can be attributed to inaccurate calibration of the quartz crystal monitor.

Fig. 9 Comparison of in situ experimental transmittance data, ex situ experimental data and theoretical transmittance related to HR800-Glass sample (a). Fitting of in situ experimental transmittance data by model transmittances related to HR800-Glass sample (b).

Download Full Size | PDF

The nominal refractive indices of HfO₂ were used only as starting assumption, before we did not have reliable refractive index for the case of deposition at low substrate temperature of 120°C. It is known [24] that the width of high-reflection zone of quarter-wave mirror is dependent on the ratio of high and low refractive indices. Theoretical width of high-reflection zone is about 126 nm, width of high-reflection zone according to in situ data is 133 nm and the width of high-reflection zone according to ex situ measurements is 143 nm. If we neglect small effect of porosity of SiO₂ and assume that SiO₂ index is quite stable, then we should expect that refractive index of HfO₂ layers in vacuum is larger than the assumed nominal refractive index. Also, the refractive index of HfO₂ layers in the atmosphere is higher than refractive index of HfO₂ in vacuum. The latter statement can be explained by porous structure of HfO₂ layers [25]. After exposition to atmosphere, pores in HfO₂ layers are filled with water, and the refractive index of the HfO₂ layers increases. This phenomenon is known as a vacuum shift [25,26].

Taking into account the qualitative issues presented above, we propose the following RE algorithm. At the first step of the algorithm we analyze BBM data related to HR800-Glass sample. We choose coating model assuming random errors in layer thicknesses and random offsets of HfO₂ refractive indices. The discrepancy function is defined by Eq. (3) with

T^{(i)} (X_{i}; λ) = T^{(i)} (d_{1} (1 + δ_{1}), \dots, d_{i} (1 + δ_{i}); n_{H} (1 + h_{H, 1}), n_{L}, ..., n_{H} (1 + h_{H, i}); λ) .

Here

h_{H, 1}, \dots, h_{H, i}

are offsets of refractive indices of HfO₂ layers. In Eq. (3)

L = 1370,

λ_{1} = 400

nm,

λ_{L} = 1000

nm.

In Fig. 9(b) we present achieved fitting of in situ transmittance data by model transmittance data. In Fig. 10(a) we compare average refractive index of HfO₂ layers calculated as ${\tilde{n}}_{H} = n_{H} + \sum_{i = 1}^{22} h_{H, i} / 22$ and their nominal refractive index. As it was expected on the basis of the qualitative analysis, estimated refractive index is higher than nominal refractive index. In Fig. 10(b) estimated relative errors in layer thicknesses are shown. It is seen that all layers, except the first layer, are underdeposited. We call the design defined by the vector $X = ({\tilde{d}}_{1}, \dots, {\tilde{d}}_{N}; {\tilde{n}}_{H}, n_{L})$ with actual thicknesses ${\tilde{d}}_{1} = (1 + δ_{1}) d_{1}, \dots, {\tilde{d}}_{N} = (1 + δ_{1}) d_{N}$ as an intermediate design.

Fig. 10 Refractive indices of HfO₂ layers: nominal, average in situ and ex situ (a). Estimated relative errors in layer thicknesses of 43-layer quarter-wave mirror (b).

Download Full Size | PDF

In Fig. 11 we compare experimental normal and oblique incidence transmittance data related to HR800-FusedSilica sample, and transmittance values, calculated for intermediate design and Fused Silica substrate. In Fig. 11 one can observe that experimental data are shifted to the longer wavelengths with respect to model data. The widths of high-reflection zones of both curves are different which can be explained mainly by differences in refractive indices.

Fig. 11 Comparison of normal (a) and AOI = 45° (b) experimental transmittance data related to HR800-FusedSilica sample with model transmittances calculated for intermediate design.

Download Full Size | PDF

At the second step of the RE process we assume that actual layer thicknesses have been reliably determined at the first step of the RE process. We expect offset of HfO₂ refractive index. We assume also that HfO₂ layers are inhomogeneous, i.e., refractive indices of layers are varying with respect to physical thickness. We search for offset in refractive index of HfO₂ layers $h_{H}$ and degree of inhomogeneity $δ_{H}$ of these layers. Degree of inhomogeneity $δ$ is defined as

δ = (n^{+} - n^{-}) / n_{a v},

where

n^{+}

and

n^{-}

are refractive indices of a layer at its outer and inner boundaries, and

n_{a v}

is the average layer refractive index

n_{a v} = (n^{+} + n^{-}) / 2.

We perform RE on the basis of normal incidence transmittance and reflectance data. The algorithm is based on the minimization of discrepancy function taken in the form:

D F^{2} = \frac{1}{2 L} \sum_{j = 1}^{L} {[T (X; λ_{j}) - \hat{T} (λ_{j})]}^{2} + \frac{1}{2 L} \sum_{j = 1}^{L} {[R (X; λ_{j}) - \hat{R} (λ_{j})]}^{2},

where

L = 701,

λ_{1} = 400

nm,

λ_{L} = 1500

nm. In Eq. (10)

T (X; λ)

is

T (X; λ) = T ({\tilde{d}}_{1}, \dots, {\tilde{d}}_{N}; n_{H} + h_{H}, n_{L}; δ_{H}; λ) .

Corrected refractive index of HfO₂ layers is shown in Fig. 10(a) (blue curve). As it was expected, corrected refractive index is higher than nominal refractive index and also higher than refractive index of HfO₂ layers in vacuum. The estimated degree of bulk inhomogeneity is about −8%. Such high degree of inhomogeneity may be explained by low deposition temperature.

The final fitting of normal incidence experimental transmittance data by model data is shown in Fig. 12(a). The fitting is not ideal because the applied model is not accurate enough. At the same time, this model takes into account all main features of the deposited coating. More accurate model would assume different degrees of inhomogeneities of each layer as well as possible interlayers caused by interdiffusion of HfO₂ and SiO₂ materials. Multiple model parameters, however, cannot be found reliably from single ex situ transmittance/reflectance spectra.

Fig. 12 Final fitting of normal (a) and oblique incidence AOI = 45° (b) experimental transmittance data related to HR800-FusedSilica sample by model transmittances.

Download Full Size | PDF

For verification of the RE results we compare transmittance data measured at AOI = 45° using Cary 7000 UMS and model transmittance data [Fig. 12(b)]. In Fig. 12(b) one can observe good agreement between experimental and model data. Also, we observed good correspondences for experimental transmittance/reflectance data measured at all incidence angles from 0° to 45° with 5° angular step. Good correspondence between experimental and model oblique incidence data validates our RE approach.

We used obtained RE results in order to adjust the deposition process for the next deposition run. Namely, we changed tooling factors of the quartz crystal monitor, specified corrected refractive index of HfO₂ layers in the theoretical design, recalculated corresponding layer thicknesses, and took into account the vacuum shift. In Fig. 13(a) we compare theoretical transmittance values of recalculated 43-layer quarter-wave mirror and transmittance data measured at PerkinElmer Lambda 950. It is seen from Fig. 13(a) that the correspondence between theoretical and experimental data is significantly improved. In order to further improve the deposition process, we performed RE again and calculated model reflectance at 45 degrees [Fig. 13(b)]. We can judge now the performance of our sample on the basis of this data: reflectance in the vicinity of the wavelength 800 nm is larger than 99.95%.

Fig. 13 Comparison of normal incidence experimental data related to the second deposition run and theoretical transmittance data (a). Model reflectance at AOI = 45° after additional RE procedure (b).

Download Full Size | PDF

6. Conclusions

We have discussed choices of reliable approaches for the post-production characterization of complicated oblique incidence coatings. Each approach includes the choice of experimental data set, selection of the numerical algorithm, coating model, and verification of the obtained RE results. Since accurate oblique-incidence $R$ and $T$ measurements are more complicated than normal and quasi-normal incidence measurements we propose to lessen the first ones and in the mass-production to rely on the second ones. To achieve this goal we propose to perform careful oblique-incidence measurements for several test coatings only and to use the obtained experimental data for adjusting RE process. The RE process is to be performed on the basis of normal incidence data, oblique incidence measurements are used for verification of the results only. For further oblique incidence coatings produced under the same deposition conditions, the adjusted RE procedure can be used without additional verification. The discussed concept was successfully applied for the post-production characterization of complicated multilayer coatings.

Acknowledgments

This work was supported by the DFG Clusters of Excellence “Munich Centre for Advanced Photonics,” (http://www.munich-photonics.de) and “QUEST - Centre for Quantum Engineering and Space-Time Research.” The authors thank Dr. Vesna Janicki from Rudjer Boskovic Institute (Zagreb, Croatia) for HR400-BK7 sample.

References and links

1. A. V. Tikhonravov, M. K. Trubetskov, I. V. Kochikov, J. B. Oliver, and D. J. Smith, “Real-time characterization and optimization of E-beam evaporated optical coatings,” in Optical Interference Coatings, OSA Technical Digest Series (2001), ME8.

2. T. V. Amotchkina, M. K. Trubetskov, V. Pervak, B. Romanov, and A. V. Tikhonravov, “On the reliability of reverse engineering results,” Appl. Opt. 51(22), 5543–5551 (2012). [CrossRef] [PubMed]

3. 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] [PubMed]

4. A. V. Tikhonravov, T. V. Amotchkina, M. K. Trubetskov, R. J. Francis, V. Janicki, J. Sancho-Parramon, H. Zorc, and V. Pervak, “Optical characterization and reverse engineering based on multiangle spectroscopy,” Appl. Opt. 51(2), 245–254 (2012). [CrossRef] [PubMed]

5. 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] [PubMed]

6. T. V. Amotchkina, M. K. Trubetskov, A. V. Tikhonravov, and V. Pervak, “Reverse engineering of an output coupler using broadband monitoring data and group delay measurements,” in Optical Interference Coatings, OSA Technical Digest Series (2013), WB.2.

7. P. A. van Nijnatten, “An automated directional reflectance/transmittance analyser for coating analysis,” Thin Solid Films 442(1-2), 74–79 (2003). [CrossRef]

8. T. V. Amotchkina, S. Schlichting, H. Ehlers, M. K. Trubetskov, A. V. Tikhonravov, and D. Ristau, “Computational manufacturing as a tool for the selection of the most manufacturable design,” Appl. Opt. 51(36), 8677–8686 (2012). [CrossRef] [PubMed]

9. A. V. Tikhonravov and M. K. Trubetskov, “OptiLayer software,” http://www.optilayer.com. [CrossRef]

10. D. Ristau, H. Ehlers, S. Schlichting, and M. Lappschies, “State of the art in deterministic production of optical thin films,” Proc. SPIE 7101, 71010C, 71010C-14 (2008). [CrossRef]

11. D. Ristau, H. Ehlers, T. Gross, and M. Lappschies, “Optical broadband monitoring of conventional and ion processes,” Appl. Opt. 45(7), 1495–1501 (2006). [CrossRef] [PubMed]

12. H. E. Ehlers, S. S. Schlichting, C. S. Schmitz, and D. R. Ristau, “Adaptive manufacturing of high-precision optics based on virtual deposition and hybrid process control techniques,” Chin. Opt. Lett. 8, 62–66 (2010). [CrossRef]

13. T. V. Amotchkina, S. Schlichting, H. Ehlers, M. K. Trubetskov, A. V. Tikhonravov, and D. Ristau, “Computational manufacturing as a key element in the design-production chain for modern multilayer coatings,” Appl. Opt. 51(31), 7604–7615 (2012). [CrossRef] [PubMed]

14. B. Vidal, A. Fornier, and E. Pelletier, “Wideband optical monitoring of nonquarterwave multilayer filters,” Appl. Opt. 18(22), 3851–3856 (1979). [PubMed]

15. H. A. Macleod, Thin-film Optical Filters, 4th ed, Series in Optics and Optoelectronics (CRC Press/Taylor & Francis, 2010).

16. 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] [PubMed]

17. V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, S. Naumov, F. Krausz, and A. Apolonski, “1.5-octave chirped mirror for pulse compression down to sub-3 fs,” Appl. Phys. B 87(1), 5–12 (2007). [CrossRef]

18. V. Pervak, I. Ahmad, J. Fulop, M. K. Trubetskov, and A. V. Tikhonravov, “Comparison of dispersive mirrors based on the time-domain and conventional approaches, for sub-5-fs pulses,” Opt. Express 17(4), 2207–2217 (2009). [CrossRef] [PubMed]

19. V. Pervak, M. K. Trubetskov, and A. V. Tikhonravov, “Robust synthesis of dispersive mirrors,” Opt. Express 19(3), 2371–2380 (2011). [CrossRef] [PubMed]

20. V. Pervak, “Recent development and new ideas in the field of dispersive multilayer optics,” Appl. Opt. 50(9), C55–C61 (2011). [CrossRef] [PubMed]

21. T. V. Amotchkina, M. K. Trubetskov, V. Pervak, and A. V. Tikhonravov, “Design, production, and reverse engineering of two-octave antireflection coatings,” Appl. Opt. 50(35), 6468–6475 (2011). [CrossRef] [PubMed]

22. A. N. Tikhonov and V. I. Arsenin, Solutions of Ill-posed Problems (Winston, 1977).

23. D. Death, R. J. Francis, C. Bricker, T. Burt, and C. Colley, “The UMA: A new tool for multi-angle photometric spectroscopy,” in Optical Interference Coatings, OSA Technical Digest Series (2013), ThC.3.

24. S. A. Furman and A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontières, 1992).

25. O. Stenzel, S. Wilbrandt, S. Yulin, N. Kaiser, M. Held, A. Tünnermann, J. Biskupek, and U. Kaiser, “Plasma ion assisted deposition of hafnium dioxide using argon and xenon as process gases,” Opt. Mater. Express 1(2), 278–292 (2011). [CrossRef]

26. P. Baumeister, Optical Coating Technology (SPIE Optical Engineering Press, 2004).

Material	Deposition process/plant	Dispersion formula
HfO₂	e-beam evaporation/ Varian 3117	$n (λ) = 1.876067 + 1.0444878 \cdot 10^{- 2} / λ^{2} + 1.7964229 \cdot 10^{- 4} / λ^{4}$
SiO₂	e-beam evaporation/ Varian 3117	$n (λ) = 1.44765 + 3.8009553 \cdot 10^{- 3} / λ^{2} - 5.379729 \cdot 10^{- 5} / λ^{4}$
TiO₂	e-beam evaporation/ SYRUSPro 1100	$n^{2} (λ) = 1 + 4.062408 λ^{2} / (λ^{2} - 0.048857) + 1.40166 λ^{2} / (λ^{2} - 80)$
SiO₂	e-beam evaporation/ SYRUSPro 1100	$n^{2} (λ) = 1 + 1.158586 λ^{2} / (λ^{2} - 0.010727) + 0.5 λ^{2} / (λ^{2} - 61)$
Nb₂O₅	Magnetron sputtering/ Helios	$n (λ) = 2.218485 + 0.021827 / λ^{2} + 0.003999568 / λ^{4}$
SiO₂	Magnetron sputtering/ Helios	$n (λ) = 1.465 + 4.7108 \cdot 10^{- 4} / λ^{4}$
HfO₂	e-beam evaporation/ SYRUSPro 710	$n (λ) = 1.871707 + 6.8452516 \cdot 10^{- 3} / λ^{2} + 4.8332406 \cdot 10^{- 4} / λ^{4}$
SiO₂	e-beam evaporation/ SYRUSPro 710	$n (λ) = 1.456819 + 1.993508 \cdot 10^{- 3} / λ^{2} + 7.181666 \cdot 10^{- 10} / λ^{4}$
BK7		$n (λ) = 1.504167 + 4.7986293 \cdot 10^{- 3} / λ^{2} - 4.178326 \cdot 10^{- 5} / λ^{4}$
Fused Silica		$n (λ) = 1.448187 + 3.5607597 \cdot 10^{- 3} / λ^{2} - 7.487681 \cdot 10^{- 6} / λ^{4}$
Glass B260		$n (λ) = 1.502671 + 5.3319 \cdot 10^{- 3} / λ^{2} + 4.9185 \cdot 10^{- 4} / λ^{4}$
Suprasil		$n (λ) = 1.443268 + 4.06 \cdot 10^{- 3} / λ^{2} + 6.9481764 \cdot 10^{- 6} / λ^{4}$

Material	Deposition process/plant	Dispersion formula
HfO₂	e-beam evaporation/ Varian 3117	$n (λ) = 1.876067 + 1.0444878 \cdot 10^{- 2} / λ^{2} + 1.7964229 \cdot 10^{- 4} / λ^{4}$
SiO₂	e-beam evaporation/ Varian 3117	$n (λ) = 1.44765 + 3.8009553 \cdot 10^{- 3} / λ^{2} - 5.379729 \cdot 10^{- 5} / λ^{4}$
TiO₂	e-beam evaporation/ SYRUSPro 1100	$n^{2} (λ) = 1 + 4.062408 λ^{2} / (λ^{2} - 0.048857) + 1.40166 λ^{2} / (λ^{2} - 80)$
SiO₂	e-beam evaporation/ SYRUSPro 1100	$n^{2} (λ) = 1 + 1.158586 λ^{2} / (λ^{2} - 0.010727) + 0.5 λ^{2} / (λ^{2} - 61)$
Nb₂O₅	Magnetron sputtering/ Helios	$n (λ) = 2.218485 + 0.021827 / λ^{2} + 0.003999568 / λ^{4}$
SiO₂	Magnetron sputtering/ Helios	$n (λ) = 1.465 + 4.7108 \cdot 10^{- 4} / λ^{4}$
HfO₂	e-beam evaporation/ SYRUSPro 710	$n (λ) = 1.871707 + 6.8452516 \cdot 10^{- 3} / λ^{2} + 4.8332406 \cdot 10^{- 4} / λ^{4}$
SiO₂	e-beam evaporation/ SYRUSPro 710	$n (λ) = 1.456819 + 1.993508 \cdot 10^{- 3} / λ^{2} + 7.181666 \cdot 10^{- 10} / λ^{4}$
BK7		$n (λ) = 1.504167 + 4.7986293 \cdot 10^{- 3} / λ^{2} - 4.178326 \cdot 10^{- 5} / λ^{4}$
Fused Silica		$n (λ) = 1.448187 + 3.5607597 \cdot 10^{- 3} / λ^{2} - 7.487681 \cdot 10^{- 6} / λ^{4}$
Glass B260		$n (λ) = 1.502671 + 5.3319 \cdot 10^{- 3} / λ^{2} + 4.9185 \cdot 10^{- 4} / λ^{4}$
Suprasil		$n (λ) = 1.443268 + 4.06 \cdot 10^{- 3} / λ^{2} + 6.9481764 \cdot 10^{- 6} / λ^{4}$

Quality control of oblique incidence optical coatings based on normal incidence measurement data

Abstract

1. Introduction

2. Example of unreliable reverse engineering of 21-layer quarter-wave mirror

3. Reverse engineering of an edge filter

4. Reverse engineering of a beam splitter

5. Reverse engineering of a 43-layer quarter-wave mirror

6. Conclusions

Acknowledgments

References and links

Cited By

Figures (13)

Tables (2)

Equations (11)

Optics Express

Edge filter			Beam splitter						AR coating
LN	Th.	Mat.	LN	Th.	Mat.	LN	Th.	Mat.	LN	Th.	Mat.
1	189.02	TiO₂	1	114.27	Nb₂O₅	29	99.59	Nb₂O₅	1	241.76	SiO₂
2	314.64	SiO₂	2	185.02	SiO₂	30	179.48	SiO₂	2	170.17	Nb₂O₅
3	74.08	TiO₂	3	105.92	Nb₂O₅	31	112.33	Nb₂O₅	3	339.81	SiO₂
4	139.71	SiO₂	4	126.83	SiO₂	32	187.63	SiO₂	4	117.87	Nb₂O₅
5	188.73	TiO₂	5	44.84	Nb₂O₅	33	113.97	Nb₂O₅	5	140.52	SiO₂
6	324.15	SiO₂	6	118.57	SiO₂	34	187.54	SiO₂	6	68.74	Nb₂O₅
7	78.93	TiO₂	7	103.91	Nb₂O₅	35	112.26	Nb₂O₅	7	117.58	SiO₂
8	134.9	SiO₂	8	184.17	SiO₂	36	177.45	SiO₂	8	84.53	Nb₂O₅
9	87.48	TiO₂	9	114.71	Nb₂O₅	37	92.98	Nb₂O₅	9	197.91	SiO₂
10	270.27	SiO₂	10	189.85	SiO₂	38	87.27	SiO₂	10	22.22	Nb₂O₅
11	92.3	TiO₂	11	114.57	Nb₂O₅	39	56.45	Nb₂O₅
12	144.39	SiO₂	12	185.51	SiO₂	40	153.63	SiO₂
13	172.38	TiO₂	13	107.92	Nb₂O₅	41	108.68	Nb₂O₅
14	139.39	SiO₂	14	152.95	SiO₂	42	185.52	SiO₂
15	76.85	TiO₂	15	59.12	Nb₂O₅	43	113.88	Nb₂O₅
16	154.46	SiO₂	16	87.89	SiO₂	44	186.96	SiO₂
17	152.11	TiO₂	17	89.86	Nb₂O₅	45	111.19	Nb₂O₅
18	156.86	SiO₂	18	174.97	SiO₂	46	170.4	SiO₂
19	76.44	TiO₂	19	111.43	Nb₂O₅	47	78.36	Nb₂O₅
20	149.57	SiO₂	20	186.86	SiO₂	48	399.82	SiO₂
21	163.40	TiO₂	21	113.71	Nb₂O₅	49	95.56	Nb₂O₅
22	155.08	SiO₂	22	187.52	SiO₂	50	176.15	SiO₂
23	70.33	TiO₂	23	112.4	Nb₂O₅	51	110.07	Nb₂O₅
24	122.78	SiO₂	24	180.15	SiO₂	52	118.71	SiO₂
25	33.82	TiO₂	25	100.79	Nb₂O₅
26	102.97	SiO₂	26	123.45	SiO₂
27	58.05	TiO₂	27	51.56	Nb₂O₅
28	209.38	SiO₂	28	118.92	SiO₂