Modified sequential algorithm for the on-line characterization of optical coatings

A. V. Tikhonravov; A. Gorokh

doi:10.1364/OE.23.023561

1. Introduction

Starting from the famous works of the late seventies of the last century [1–3] broadband optical monitoring (BBOM) has become one of the most important tools for production of high-quality optical coatings [4–8]. This type of monitoring is often used in the form of direct optical monitoring when reflectance or transmittance spectra are measured on at least one of the coatings to be manufactured [9]. The content of this paper is primarily related to the direct BBOM.

Essential advantages of all types of BBOM techniques are their low sensitivities to random measurement errors as compared to monochromatic monitoring techniques [10]. In some early works it was also indicated that direct BBOM may possess an error self-compensation mechanism [1, 11]. In the case of BBOM this effect is not so well pronounced as in the case of turning point optical monitoring of narrow band pass filters [12–15]. Nevertheless, for the direct BBOM its existence was confirmed for various types of optical coatings [16]. Along with these advantages of direct BBOM its main challenge has become obvious starting from the very first works on its applications. This is an existence of the effect of accumulation of thickness errors [1–3, 10, 17]. Reducing of the development of this effect may help to raise quality of optical coating production.

The development of the cumulative effect of thickness errors is tightly connected with algorithms used for the analysis of on-line BBOM measurement data. Variety of these algorithms should be subdivided into two classes: algorithms used for the determination of thicknesses of already deposited layers and algorithms used for predicting termination instants for layer depositions. Algorithms of the first class are usually referred to as on-line characterization algorithms [18] while algorithms of the second class could be called on-line monitoring algorithms. Typically algorithms of the second class use information provided by the algorithms of the first class. There are various approaches to raising accuracy of the on-line monitoring algorithms [5, 19, 20] but the key issue is an accurate determination of thicknesses of already deposited layers (on-line characterization of deposited layers). In this paper we concentrate our attention on this issue, i.e. on the accuracy of on-line characterization.

There are two most widely used on-line characterization algorithms. These are sequential and triangular algorithms (S- and T- algorithms) [18, 19, 21]. In the case of the S-algorithm measurement data recorded at the end the of deposition of each new layer are used to determine actual thickness of this layer, thicknesses of all previously deposited layers are fixed at the values found at the previous steps of algorithm operation. In the case of the T-algorithm all measurement data collected at the ends of depositions of all already deposited layers are used to determine thicknesses of all these layers. Thus in this case thicknesses of previously deposited layers are not fixed and are recalculated after the deposition of each new layer. It has been demonstrated that the T-algorithm provides a better accuracy of on-line characterization than the S-algorithm [18, 19]. At the same time the T-algorithm is more time-consuming as compared to the S-algorithm.

In this paper we propose a modification of the S-algorithm providing an accuracy of on-line characterization close to that of the T-algorithm. At the same time the new algorithm requires less computations as compared to the T-algorithm. The main idea and the description of the algorithm are presented in Section 2. In Section 3 we use simulation and computational manufacturing experiments [22] to demonstrate a high accuracy of the proposed algorithm. Final conclusions are presented in Section 4.

2. Idea of the algorithm

Denote $d_{1}^{t}, ..., d_{m}^{t}$ as the theoretical layer thicknesses of a coating design. Here $m$ is the number of coating layers. For definiteness we assume that measurement data are transmittance data. Denote $T_{m e a s}^{j} (λ)$ array of transmittance data measured after the deposition of layer number $j$ on the wavelength grid { $λ$ }. Let $d_{1}^{a}, ..., d_{j}^{a}$ be actual thicknesses of the first $j$ coating layers. The difference

δ T_{m e a s}^{j} (λ) = T_{m e a s}^{j} (λ) - T^{j} (d_{1}^{a}, \dots, d_{j}^{a}, λ)

is the measurement error at the end of deposition of layer number

j

.

Consider the discrepancy function

F^{j} = \frac{1}{L} \sum_{{λ}} {[T^{j} (d_{1}, \dots, d_{j}, λ) - T_{m e a s}^{j} (λ)]}^{2}

Here

L

is the total number of wavelength points where transmittance spectrum is measured,

d_{1}, ..., d_{j}

are variable thicknesses of coating layers.

The standard S-algorithm works as follows. In Eq. (2) thicknesses of the previously deposited layers $d_{1}, ..., d_{j - 1}$ are taken equal to the values found at the previous algorithm steps. Denote them as $d_{1}^{e}, ..., d_{j - 1}^{e}$ . The discrepancy function is then minimized with respect to the variable $d_{j}$ and the $d_{j}^{e}$ value providing the minimum of this function is taken as an estimation for the thickness of layer number $j$ . As shown in [10] errors in thicknesses of the previously deposited layers influence accuracy of $j$ -th layer thickness determination and cause a cumulative effect in thickness errors with growing number of deposited layers.

The main idea of the Modified S-algorithm is to re-calculate thicknesses of some of the previously deposited layers along with the determination of the $j$ -th layer thickness. It may be worth noting here that on the contrary with the T-algorithm not all thicknesses are recalculated but only those which accuracy could be improved according to the considerations presented below.

Assume that $d_{i} = d_{i}^{a} + h_{i},$ where $h_{i}$ are supposed to be small deviations of thicknesses $d_{i}$ from the actual layer thicknesses $d_{i}^{a}$ .

We can then represent $T^{j}$ in Eq. (2) as

T^{j} (d_{1}, \dots, d_{j}, λ) = T^{j} (d_{1}^{a}, \dots, d_{j}^{a}, λ) + \sum_{i = 1}^{j} \frac{\partial T^{j}}{\partial d_{i}} h_{i}

Transmittance derivatives in Eq. (3) should be taken at $d_{i} = d_{i}^{a}$ . But $d_{i}^{a}$ values are not known. One of the main assumptions of the performed derivations is that $d_{i}^{a}$ values are not too far from the theoretical layer thicknesses $d_{i}^{t}$ and we can calculate all derivatives in Eq. (3) at $d_{i} = d_{i}^{t}$ .

Using Eq. (1) and Eq. (3), we can re-write Eq. (2) in the following form

F^{j} = \frac{1}{L} \sum_{{λ}} {[\sum_{i = 1}^{j} \frac{\partial T^{j}}{\partial d_{i}} h_{i} - δ T_{m e a s}^{j} (λ)]}^{2}

Let us isolate in this equation the term related to the error in layer number

i

:

F^{j} = \frac{1}{L} \sum_{{λ}} {[\frac{\partial T^{j}}{\partial d_{i}} h_{i} + \sum_{\begin{matrix} l = 1 \\ l \neq i \end{matrix}}^{j} \frac{\partial T^{j}}{\partial d_{l}} h_{l} - δ T_{m e a s}^{j} (λ)]}^{2}

Consider

F^{j}

as function of

h_{i}

. This is a quadratic function with respect to

h_{i}

and using standard derivations it is easy to find that the minimum of

F_{j}

with respect to

h_{i}

is achieved at

h_{i} = [\frac{1}{L} \sum_{{λ}} \frac{\partial T^{j}}{\partial d_{i}} δ T_{m e a s}^{j} (λ) - \sum_{\begin{matrix} l = 1 \\ l \neq i \end{matrix}}^{j} (\frac{1}{L} \sum_{{λ}} \frac{\partial T^{j}}{\partial d_{i}} \frac{\partial T^{j}}{\partial d_{l}}) h_{l}] / [\frac{1}{L} \sum_{{λ}} {(\frac{\partial T^{j}}{\partial d_{i}})}^{2}]

Equation (6) forms a basis for the following considerations. Let us denote

b_{i}^{j} (λ) = \frac{\partial T^{j}}{\partial d_{i}} / \sum_{{λ}} {(\frac{\partial T^{j}}{\partial d_{i}})}^{2}, c_{i l}^{j} = - \sum_{{λ}} \frac{\partial T^{j}}{\partial d_{i}} \frac{\partial T^{j}}{\partial d_{l}} / \sum_{{λ}} {(\frac{\partial T^{j}}{\partial d_{i}})}^{2}

and re-write Eq. (6) as

h_{i} = \sum_{{λ}} b_{i}^{j} (λ) δ T_{m e a s}^{j} (λ) + \sum_{\begin{matrix} l = 1, \\ l \neq i \end{matrix}}^{j} c_{i l}^{j} h_{l}

We will use this equation for estimating an accuracy of the determination of $i$ -th layer thickness at various steps of the constructed Modified S-algorithm. Our considerations will be based on the statistical analysis with some additional assumptions concerning measurement and thickness errors. These assumptions are required for simplifying the following derivations and obtaining practical formulas that can be used by the constructed algorithm.

First of all we assume that measurement errors are random errors distributed by the normal law with zero mathematical expectation and standard deviation $δ T_{m e a s}^{j} (λ) = σ_{m e a s}$ that doesn't depend on the wavelength $λ$ and on the number of series of measurement data $j$ . Next, we suppose that in a big series of experiments thickness errors are also distributed by the normal laws with zero mathematical expectations. Finally we assume that standard deviations of all layer thicknesses in the right part of Eq. (7) can be estimated by the same value $σ$ . Obviously, this is the most restrictive assumption especially taking into account the above discussed effect of accumulation of thickness errors. Currently we can provide the following reasoning for this assumption: we hope to construct an algorithm that will not give a rise to significant cumulative effect of thickness errors. But the final practical approval of this assumption and all following results will be done in the Section 3 based on the comparison of the new algorithm with the standard S-algorithm.

Equation (7) can be used for the statistical estimation of the accuracy of determination of $i$ -th layer thickness when layer thicknesses are determined by the minimization of the discrepancy function Eq. (2). According to the above assumptions, the mathematical expectation of $h_{i}$ is equal to zero and its standard deviation is

σ^{j} (h_{i}) = {[{(B_{i}^{j})}^{2} σ_{m e a s}^{2} + {(C_{i}^{j})}^{2} σ^{2}]}^{\frac{1}{2}}

where

B_{i}^{j} = 1 / {[\sum_{{λ}} {(\frac{\partial T^{j}}{\partial d_{i}})}^{2}]}^{\frac{1}{2}}, C_{i}^{j} = {[\sum_{\begin{matrix} l = 1, \\ l \neq i \end{matrix}}^{j} {(\sum_{{λ}} \frac{\partial T^{j}}{\partial d_{i}} \frac{\partial T^{j}}{\partial d_{l}})}^{2}]}^{\frac{1}{2}} / {[\sum_{{λ}} {(\frac{\partial T^{j}}{\partial d_{i}})}^{2}]}^{\frac{1}{2}}

In the following we shall refer $σ_{m e a s}$ and $σ$ as levels of transmittance and thickness errors. Let us set a relation between these levels $σ_{m e a s} = α σ$ . For example, if we estimate $σ_{m e a s}$ as 0.01 in absolute transmittance data and sigma as 1 nm then alpha will be 0.01 in inverse nanometers. Using this relation factor, we can re-write Eq. (8) as

σ^{j} (h_{i}) = A_{i}^{j} σ,

where

A_{i}^{j} = {[α^{2} {(B_{i}^{j})}^{2} + {(C_{i}^{j})}^{2}]}^{\frac{1}{2}}

According to Eq. (9), the coefficients $A_{i}^{j}$ can be used for the estimations of expected accuracies of layer thicknesses if these thicknesses are determined by the minimization of the discrepancy function (2). It was already mentioned at the beginning of this section that in the modified S-algorithm the discrepancy function (2) is used not only for the determination of the thickness of the last deposited layer $d_{j}$ but also for the recalculation of thicknesses of some previously deposited layers. It is important to note that only the last measured spectra is used for this determination (see Eq. (2)). For the identification of layers which thicknesses are to be determined we compare the coefficients $A_{i}^{j}$ of Eq. (9) with analogous coefficients found at the previous algorithm steps. Suppose that for the layer

A_{i}^{j} < c \min_{1 \leq k \leq j - 1} A_{i}^{k}

Here c is an additional algorithm parameter that will be discussed in Section 3. Now we indicate only that c < = 1. In this case one should expect that at the $j$ -th step of algorithm operation the layer thickness $d_{i}$ may be determined more accurately than it has been determined before. So the main idea of the Modified S-algorithm is to use the discrepancy function (2) to determine all layer thicknesses $d_{i}$ for which inequality (10) is valid along with $d_{j}$ . All these thicknesses are determined by the minimization of the discrepancy function. Of course for this purpose analytically accurate computations of this function and its derivatives are performed.

3. Simulation experiments

We start this section with the most straightforward simulation experiments allowing to illustrate the operation of the Modified S-algorithm and to demonstrate its superiority over the standard S-algorithm. As an example we consider one of the 40-layer hot mirror designs discussed in [23]. This design provides high transmission in the 400-690 nm wavelength region and high reflection in the 710-1200 nm region. Physical thicknesses of the considered hot mirror design and its transmittance are presented in Fig. 1, refractive indices of high and low index layers are 2.35 and 1.45, substrate refractive index is 1.52. The first design layer next to the substrate is a high index layer.

Fig. 1 Physical thicknesses (a) and spectral characteristic (b) of the 40-layer hot mirror design

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In the first series of simulation experiments we consider only random errors in transmittance data. Errors in layer thicknesses were not introduced and we compare on-line characterization algorithms by calculating differences between determined layer thicknesses and actual layer thicknesses. We assume that BBOM is performed in the 400-900 nm spectral region with 1 nm wavelength step. The level of simulated errors, i.e. the standard deviation of these errors in percentage is 1%.

The columns 1 and 4 of Table 1 list the numbers of layers after which the Modified S-algorithm operates. Numbers of layers which thicknesses are determined at the respective algorithms steps are indicated in the columns 2 and 5 for c = 0.5 and in columns 3 and 6 for c = 1.0. One can see that in the case c = 0.5 thicknesses of less layers should be calculated and as a sequence the Modified S-algorithm works faster than in the case of c = 1.0. This is the main reason for including additional algorithm parameter c in Eq. (10).

Table 1. Numbers of layers which thicknesses are determined at various algorithm steps for two different values of the coefficient c in Eq. (10): c = 0.5 and c = 1.0.

View Table

Let's analyze the levels of errors in layer thicknesses connected with the Modified S-algorithm compared to standart S- and T-algorithms. As discussed in Section 2 thickness errors are random errors specified by other random factors, in our first experiments by the random errors in transmittance data. To get reliable statistical information we perform three series of 50 simulation experiments with Modified S-algorithm, S-algorithm, and T-algorithm. For each experiment errors in layer thicknesses are found by comparison of determined layer thicknesses with actual layer thicknesses and then root mean square (RMS) values of these errors are calculated. We refer these values as levels of characterization errors.

Figure 2(a) compares levels of characterization errors in the case of Modified S-algorithm and S-algorithm, while Fig. 2(b) compares those in the case of Modified S-algorithm and T-algorithm. For the Modified S-algorithm the parameter c in Eq. (10) is taken equal to 1. In the case of T-algorithm levels of characterization errors are calculated based on the thickness values determined by the algorithm at the last step of the characterization procedure, i.e. after the deposition of all 40 layers.

Fig. 2 Comparison of levels of characterization errors: (a): Modified S-algorithm (red bars) and S-algorithm (blue bars), (b): Modified S-algorithm (red bars) and T-algorithm (green bars).

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In Fig. 2(a) the cumulative effect of thickness errors is clearly observed in the case of standard S-algorithm. This correlates with the results of Ref. 10 where this effect was analytically investigated just for this type of characterization algorithm. It is seen from Fig. 2(b) that levels of characterization errors are comparable in the cases of T-algorithm and Modified S-algorithm and that both algorithms do not give rise to any noticeable cumulative effect of thickness errors for this type of coating design. We have performed analogous experiments with other types of designs including antireflection coatings, various filters, polarisers. Our designs had numbers of layers ranging between 10 and 60 and in all cases analogous conclusions regarding the considered algorithms have been obtained. The absence of cumulative effect of thickness errors in the case of Modified S-algorithm serves as an approval for the most restrictive assumption made in Section 2 for the construction of this algorithm.

In our simulation experiments we also experimented with various values of the parameter c in Eq. (10). It has been found that typically it is advisable to use c values of about 0.5 - 0.6 which additionally accelerates operations of the Modified S-algorithm without noticeable influence on its accuracy.

At the end of this section we present results of some of our computational manufacturing experiments that further confirm the robustness of the proposed Modified S-algorithm. The results are presented for the same hot mirror design as before. In the experiments instabilities of deposition rates of high and low index materials were simulated. The mean rates of both materials were 3A/sec and RMS fluctuations of both rates were 1A/sec. BBOM was performed based on transmittance measurements in the spectral range 400-900 nm with the wavelength step of 1 nm. Time intervals between measurements were taken equal to 1 sec and random errors with 0.5% level were simulated in transmittance data.

In the course of computational manufacturing experiments a monitoring algorithm for predicting termination instants should be used along with the characterization algorithm. We apply the monitoring algorithm based on estimating the discrepancy between simulated BBOM data and theoretical transmittance at the end of a layer deposition. The discrepancy dependence on time for the last 20 measurement instants is approximated by a parabola and the expected termination instant is estimated according to the position of the minimum of this parabola.

Figure 3 presents statistically averaged results of the 30 computational manufacturing experiments. Levels of errors are again calculated as RMS values of respective errors obtained in these experiments. In Fig. 3(a) levels of deposition errors calculated as differences between thicknesses of deposited layer and theoretical layer thicknesses are presented. When analyzing this figure one should keep in mind that in the case of computational manufacturing experiments deposition errors are connected not only with the inaccuracies of characterization algorithm but also with those of the monitoring algorithm. For this reason we present in Fig. 3(b) also “pure” characterization errors. They are calculated as differences between thickness values determined by the respective characterization algorithm and “actual” layer thicknesses produced in the course of computational manufacturing experiments.

Fig. 3 Levels of errors in the case of Modified S-algorithm (red bars) and S-algorithm (blue bars): (a) errors in thicknesses of deposited coatings, (b) characterization errors.

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As one can see Figs. 3(a) and 3(b) clearly demonstrate superiority of the Modified S-algorithm over the standard S-algorithm. Computational manufacturing experiments with the designs of other types also confirm this conclusion.

4. Conclusions and discussion

Effectiveness of algorithms used for the on-line determination of thicknesses of already deposited layers (on-line characterization algorithms) is essential for the successful implementation of BBOM in optical coating production. Currently the most widely used on-line characterization algorithms are the S- and T-algorithms. The second one provides a better accuracy than the first one but is more time consuming. In this paper we presented a modification of the S-algorithm that provides an accuracy of on-line characterization close to that of the T-algorithm. At the same time the new algorithm requires less computations as compared to the T-algorithm.

The main idea of the Modified S-algorithm is to re-calculate thicknesses of some of the previously deposited layers along with the determination of the thickness of the last deposited layer. On the contrary with the T-algorithm not all thicknesses of previously deposited layers are recalculated but only those which accuracy could be improved according to the estimations presented in this paper. Simulation and computational manufacturing experiments with the new algorithm confirm a high accuracy of the proposed algorithm.

Acknowledgments

Authors are grateful to Dr. Michael Trubetskov for fruitful discussions and multiple valuable suggestions related to the content of this work.

This work was supported by RFBR, research project No. 13-07-00846 А (“Mathematical modeling and simulation of optical nano-coatings deposition”) and by the Cluster of Excellence “Munich-Centre for Advanced Photonics” (http://www.munich-photonics.de/) of the German Research Foundation (DFG).

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Modified sequential algorithm for the on-line characterization of optical coatings

Abstract

1. Introduction

2. Idea of the algorithm

3. Simulation experiments

4. Conclusions and discussion

Acknowledgments

References and links

Cited By

Figures (3)

Tables (1)

Equations (13)

Optics Express

Step	c = 0.5	c = 1.0	Step	c = 0.5	c = 1.0
1	1	1	21	20, 21	10-21
2	2	2	22	22	22
3	3	2, 3	23	22, 23	12-18, 22, 23
4	4	4	24	24	11-,22, 24
5	5	3-5	25	24, 25	8 −10, 23-25
6	6	6	26	26	7-10, 22, 23, 26
7	6, 7	4-7	27	26, 27	13, 26, 27
8	8	8	28	28	28
9	8, 9	6-9	29	28, 29	6-8, 10, 28, 29
10	4, 10	4, 10	30	30	27, 30
11	10, 11	8-11	31	30, 31	14, 26, 30, 31
12	12	5, 7, 12	32	32	32
13	12, 13	9-13	33	32, 33	10, 11, 32, 33
14	14	6, 14	34	34	6-11, 28, 34
15	14, 15	7, 10-15	35	34, 35	16, 17, 34, 35
16	16	7, 16	36	36	16, 33, 36
17	16, 17	8, 11-17	37	36, 37	36, 37
18	18	8, 18	38	38	11, 36-38
19	18, 19	9-19	39	38, 39	15-21, 27, 38, 39
20	20	20	40	40	40