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Abstract
The presence of the error self-compensation effect in optical coating production with monochromatic monitoring other than turning point optical monitoring, is demonstrated. For the demonstration, a series of computational manufacturing experiments with a UV-IR edge filter are performed. The strength of the error self-compensation effect is estimated using the approach previously applied to the case of broadband optical monitoring.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The effect of errors self-compensation was discovered almost 50 years ago [1–3] and since then it has been successfully used in the production of narrow-band optical filters. Application of this effect requires control of thicknesses of filter layers by turning point optical monitoring at the filter central wavelength [2,4]. Unfortunately, this type of monochromatic optical monitoring provides an error self-compensation effect only in the case of narrow-band filters composed of layers with a quarter-wave or multiple of a quarter-wave optical thickness. An explanation to this fact is given in [5]. The existence of the error self-compensation effect in the case of other optical coatings produced using other monochromatic monitoring methods was assumed in [6,7]. However, until now, a detailed study of this effect in the case of using monochromatic methods other than turning point optical monitoring has not been carried out.
In recent years, an extensive study of the error self-compensation effect associated with broadband optical monitoring was carried out [8–11]. It was shown that this effect is a positive consequence of the correlation of thickness errors caused by any optical monitoring method. In [12], a general approach to the study of the error correlation was proposed, based on the investigation of thickness errors obtained in the course of computational manufacturing experiments. The results of these experiments can also be used to check the presence of the error self-compensation effect [11].
In one respect, studying of the effects associated with monochromatic optical monitoring is more difficult than studying the corresponding effects in the case of broadband monitoring. The problem is that special algorithms are often used to correct raw monochromatic data during the coating production [6,13,14]. These special algorithms are designed to adjust the cut-off level for the deposition of the layer to prevent the rapid development of the cumulative effect of increasing thickness errors [15]. The operation of such algorithms affects the correlation of thickness errors, and, as a consequence, affects the strength of the error self-compensation effect. Nevertheless, the general approach to the study of error correlation, developed in [12], can also be applied in the case of monochromatic monitoring with various cut-off correction algorithms.
In this article, we demonstrate the application of the general approach using the example of a UV-IR edge filter, for which the existence of an error compensation effect was assumed in [6]. The filter design, the cut-off correction algorithm used, and manufacturing and monitoring errors modeled in computational manufacturing experiments are described in Section 2. A discussion of the results of computational manufacturing experiments and a proof of the existence of error self-compensation effect are presented in Section 3. Final conclusions are given in Section 4.
2. Edge filter design and computational manufacturing experiments
We did not have accurate data on the parameters of the 42-layer edge filter presented in [6]. For this reason, we designed our own filter using the same pair of high and low index materials, namely $\mathrm {TiO_2}$ and $\mathrm {SiO_2}$, as in [6]. BK7 glass was used as a substrate material. For the design, the OptiLayer thin film software was used [16]. The theoretical transmittance of the filter and its layer optical thicknesses are shown in Fig. 1 and 2.
Fig. 2. Layer optical thicknesses of the designed edge filter, odd layers are high index layers (blue color) and even layers are low index layers (pink color).
Our filter has 36 layers instead of 42 layers in the case of the edge filter from [6]. This may be due to a small difference in the $\mathrm {TiO_2}$ and $\mathrm {SiO_2}$ indices, some difference in the specification of high and low transmittance regions, as well as a difference in the design methods of constructing filters in our work and in [6]. At the same time, both designs have such an important identity as the two rather thin first layers of the design (to check this see Fig. 3 in [6]). As indicated in [6], the presence of such layers complicates monitoring of coating deposition.
Fig. 3. The theoretical monitoring signal depending on the physical thickness of the filter: the first two layers are monitored at a wavelength of 400 nm, and all other layers are monitored at a wavelength of 650 nm.
Figure 1 shows the transmittance of the filter without taking into account reflection from the back side of the substrate. Of course, this reflection is taken into account in the simulation experiments described below.
The most common form of monochromatic optical monitoring is the so-called level monitoring. In its original form [17,18], the deposition of a layer terminates when the monitoring signal at a certain monitoring wavelength reaches a theoretically predicted level, which is called the termination level or cut-off level. Unfortunately, this form of level monitoring leads to a strong cumulative effect of increasing production errors, which reveals itself in a rapid increase in thickness errors with an increase in the number of a deposited layer [19]. To prevent the rapid development of the cumulative effect of increasing thickness errors, it was proposed to use the registered extrema of the monitoring signal for the on-line correction of cut-off levels [20]. Advanced optical monitors now include cut-off correction algorithms in their software [13].
Below we provide a brief description of the cut-off correction algorithm used in our simulation experiments. A detailed description of this algorithms can be found in [21]. The algorithm is called the quasi-swing algorithm. The equation for adjusting the cut-off level is as follows
Here $T^{term, th}$ and $T^{term,act}$ are the theoretically predicted and corrected termination levels, $T^{last\,extr, th}$ and $T^{last\,extr, act}$ are the theoretically predicted and actual extremum values of the transmission coefficient just before the expected end of layer deposition, $A^{th}$ and $A^{act}$ are the theoretically calculated and actual amplitudes of the transmission coefficient variation during layer deposition. All theoretical values are calculated using the layer thicknesses of the theoretical coating design.
In the case of thick layers, where at least two signal extremes are recorded during layer deposition, Eq. (1) is the same as the equation for monitoring using signal swing values (see [15], Section 3.5). In the case of thinner layers, when only one or no extreme value is recorded, the measured extreme values are replaced by virtual extreme values, which are calculated using the non-local measurement data analysis algorithm from [21,22]. This explains the name «quasi-swing algorithm».
In our simulation experiments, we specified the same deposition rates for high and low index materials as in [6], i.e., 0.5 nm/s for the $\mathrm {TiO_2}$ layers and 0.8 nm/s for the $\mathrm {SiO_2}$ layers, with the same as in [6] standard deviations of these rates: 5 % and 10 % of the mean rate in the cases of $\mathrm {TiO_2}$ and $\mathrm {SiO_2}$, respectively. The errors in the refractive indices of the layers and in the shutter delays were also the same as in [6], that is, the errors in the indices of the $\mathrm {TiO_2}$ layers with a systematic error component of 1% plus random errors with a standard deviation of 0.5% and shutter delays in all layers with systematic and random errors of 0.5 s and 0.1 s, respectively. As in [6], BK7 glass was used as the substrate material. The random noise in the optical monitoring signal was 0.1% instead of 0.015% in [6].
In [6], two different monitor wavelengths were used. We also used a monitoring strategy with two monitoring wavelength that were selected using the Monitor option of the OptiLayer software [16]. Figure 3 shows the theoretical monitoring signal versus the physical filter thickness.
Computational manufacturing experiments were performed using the cut-off correction algorithm described above and the algorithm using level monitoring [17,18]. The second algorithm was used mainly to demonstrate the problems that arise when the on-line correction of cut-off levels is not performed.
3. Results and discussions
Two series of ten thousand computational experiments were performed using two algorithms as indicated at the end of the previous section. First, the correlation of errors in these series of experiments was investigated. According to [12,23], the study of the correlation of errors in the thicknesses of coating layers is based on the statistical analysis of a large number of error vectors obtained in the course of computational manufacturing experiments. The components of the error vectors are the thickness errors of individual layers, and the dimension of the error vectors is equal to the number of coating layers. Due to error correlation, error vectors have a specific distribution in $m$-dimensional space with some preferred directions. Here $m$ is the number of coating layers, in our case $m=36$.
In [12,23], it was shown that the distribution of error vectors in the $m$-dimensional space is described by a Gaussian distribution function, whose level surfaces in this space are multidimensional ellipsoids. It was also shown that the higher the correlation of the error vectors, the more elongated these ellipsoids are. The elongation of ellipsoids is characterized by the correlation coefficient $\beta$, which is calculated by the formula given in [12,23]. According to [23], $\beta$ values exceeding 3-4 indicate the presence of a very strong error correlation effect.
The correlation coefficient $\beta$ calculated using 10 K error vectors obtained in computational manufacturing experiments with the cut-off correction algorithm and 10 K error vectors obtained in experiments with the level monitoring algorithm is equal to 1.62 and 10.70, respectively. As pointed out by Angus Macleod [4], error correlation is a direct consequence of the effect of errors in previously deposited layer on the monitoring signal of the currently deposited layer. In the case of level monitoring, this signal is not additionally processed and error correlation appears in the strongest form, which explains the much higher $\beta$ value in the second case. It is shown below that the much stronger error correlation in the case of the level monitoring algorithm results in a much stronger cumulative effect of thickness error growth.
Figure 4 shows an example of an error vector obtained in computational manufacturing experiments using the cut-off correction algorithm, Fig. 5 shows the rms values of thickness errors in individual layers, and Fig. 6 shows a histogram of error vector norms obtained in all 10K experiments. The histogram is presented with the step of the error vector norm of 0.1 nm.
Fig. 4. Results of computational manufacturing experiments with the cut-off correction algorithm: an example of an error vector obtained in one of these experiments.
Fig. 5. RMS values of thickness errors in individual layers, calculated from 10 K computational manufacturing experiments.
Fig. 6. Results of computational manufacturing experiments with the cut-off correction algorithm: a histogram of the error vector norms calculated using all 10K experiments.
A slight tendency for the thickness errors growth with an increase in the number of coating layer is noticeable in Fig. 4 and in Fig. 5 as well as in the error vectors obtained in other experiments with the cut-off correction algorithm. But unambiguously, the cumulative effect of thickness errors growth in this case is not very strong. The mean value of the norms of the error vectors in experiments with the cut-off algorithm, calculated from the histogram in Fig. 6 is 6.08 nm.
In the experiments with the level monitoring algorithm, a tendency to increase the thickness errors is sharply higher. The cumulative effect of thickness errors growth is clearly observed, and the thickness error are several times higher than in the case of the quasi-swing algorithm. An increase in the level of thickness errors is characterized by the mean value of the norms of the error vectors, which increases to 34.94 nm. The much stronger cumulative effect of thickness errors is a direct consequence of the much stronger error correlation in the case of the level monitoring algorithm. Data related to Figures 4-6 are provided in Dataset 1 [24] and Dataset 2 [25].
The error self-compensation effect is also a consequence of the correlation of thickness errors [23]. The strength of this effect is estimated by comparing the influence of correlated and uncorrelated thickness errors on the merit function, which was used to design the edge filter shown in Fig. 1. The merit function has the form
Here $T(\lambda )$ is the normal incidence transmittance of the filter, $\tilde {T}(\lambda )$ is the target transmittance, equal to 100% and 0% in the spectral bands of 400–640 nm and 680–800 nm, respectively (see Fig. 1). The spectral grid has a step of 2 nm in both spectral bands, and the total number of L points of the spectral grid is 182. The minimum $MF$ value achieved for the design shown in Fig. 2 is 0.141.
Let $\Delta$ denote one of the error vectors obtained using computational manufacturing experiments and let $\delta MF(\Delta )$ be the variation of the merit function corresponding to this error vector. To compare the effects of correlated and uncorrelated thickness errors, we generate latter in such a way so as to have the same mean value of their norms as the mean value of the norms of correlated error vectors. We consider uncorrelated additive thickness errors, and the thickness errors of all filter layers have a normal distribution with zero mean values and the same standard deviations. It is shown in [23] that such errors are uniformly distributed in the the m-dimensional space and their correlation coefficient $\beta$ is equal to 1.
Let us denote by $\mathbb {E}\, \delta MF$ the mean value of variations of the merit function corresponding to the generated set of uncorrelated thickness errors. To check the presence of the error self-compensation effect, the error self-compensation coefficient $c$ was introduced in [23]. It is calculated by the formula
According to [23], the error self-compensation effect is present if $c<1$. Figure 7 shows a histogram of the error self-compensation coefficient $c$ calculated from 10K experiments with the cut-off correction algorithm. The histogram is presented in increments of 0.02. It is seen that $c<1$ in all experiments, i.e. there is always the error self-compensation effect.
Fig. 7. Histogram of the error self-compensation coefficient calculated using 10K computational manufacturing experiments with the cut-off correction algorithm.
In [23], to estimate the strength of the error self-compensation effect, it was proposed to use the mean value of c. This value was called the error self-compensation factor. For the cut-off correction algorithm, this factor turns out to be equal to 0.353. Experiments with the level monitoring algorithm also show the presence of the error self-compensation effect. But this effect is somewhat weaker than in the case of the cut-off correction algorithm – the error self-compensation factor is now equal to 0.425.
It is also interesting to compare the strength of the error self-compensation effect in cases of broadband and monochromatic optical monitoring. In [23], a 50-layer non-polarizing edge filter for 45-degree light incidence was considered as an example to illustrate the effect of error-self compensation in the case of broadband optical monitoring. The error self-compensation factor turned out be 0.053, which is almost 7 times less than in the case of monochromatic monitoring with the cut-off correction algorithm. Thus, in the case of broadband monitoring, the effect of error self-compensation is much stronger. This probably explains why this effect is predominantly mentioned in connection with broadband monitoring.
4. Conclusions
In this paper, we demonstrate the presence of the error self-compensation effect in optical coating production with monochromatic monitoring other than turning point optical monitoring. For the demonstration, we use the edge filter similar to the one previously discussed by A. Zoeller et.al. in connection with this effect. To prove the existence of the effect, a series of computational manufacturing experiments are carried out. The strength of the error self-compensation effect in estimated using the approach previously applied to the case of broadband optical monitoring. It is shown that the error self-compensation effect definitely exists in the case of monochromatic monitoring of the considered edge filter.
In our experiments, we used two algorithms for terminating layer deposition, they are the cut-off correction algorithm and the level monitoring algorithm without correcting the termination level. It is shown that the correlation of errors is much stronger in the case of the level monitoring algorithm, which leads to a much stronger cumulative effect of thickness error growth. At the same time, a much weaker error correlation in the case of the cut-off correction algorithm does not prevent the existence of the error self-compensation effect, but this effect manifests itself even in a stronger form than in the case of level monitoring.
In this paper, we discussed the error self-compensation effect of the monitoring procedure itself, and did not discuss the error compensation associated with on-line re-optimization of optical coatings. The latter requires separate consideration.
Funding
Russian Science Foundation (21-11-00011).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Layer error thicknesses obtained in simulations with quasi-swing and level monitoring algorithms, respectively, are provided in Dataset 1 [24], and Dataset 2 [25].
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