**3. Comparison of Various Monitoring Strategies**

In this section, we used the estimates of the previous section to compare four different monitoring strategies. The first one was the direct monitoring strategy with all filter layers monitored using one of the samples to be produced. The next strategy used two subsequent monitoring chips, so that filter layers with the numbers from 1–24 were monitored using the first chip, and layers 25–50 were monitored using the second chip. The third strategy applied four monitoring chips that were used to monitor layers 1–12, 13–24, 25–36, and 37–50. The fourth strategy used two chips that were moved out of the measurement position and returned back to this position so that the first chip monitored layers 1, 2, 4, ... , 50, while the second chip monitored layers 3, 5, ... , 49. Monitoring the first two layers with a single chip allowed us to increase the optical contrast for monitoring low-index layers (even layers) by applying the first high-index layer to this chip. All four strategies caused correlation of thickness errors, but we expected that, in the first case, parameter α would have lower values than in the other cases. Recall that parameter α is smaller when the correlation of errors is higher.

In all four cases, 1,000,000 error vectors were generated to calculate α and *S* values. Figure 4 shows the probability density functions for the degree of thickness error correlation α and the strength of the error self-compensation effect *S.*

As one may expect, parameter α had lower values in the case of direct monitoring. This reflects a stronger correlation of thickness errors when all layer thicknesses are monitored using a single sample. In the case of direct monitoring, the average *S* value was the largest, and it was equal to 16.6.

For predictive comparisons of various monitoring strategies, we also needed to compare thickness error levels for all four cases. Recall that for the comparison with uncorrelated errors, all error vectors in Equation (7) were normalized to the same value. Following this, strategies with several monitoring chips were introduced to reduce thickness error levels. Figure 5 shows the probability density functions of the distributions of norms of error vectors for the considered strategies. As before, calculations were performed based on 1,000,000 simulation tests in each case.

**Figure 4.** Probability density functions for α (**a**) and *S* (**b**) in the cases of different monitoring strategies. Full—direct strategy, 2seq—strategy with two subsequent chips, 4seq—strategy with four subsequent chips, and 2ret—strategy with two returning chips.

**Figure 5.** Probability density functions for the norms of error vectors: (**a**) direct strategy, (**b**) strategy with two subsequent chips, (**c**) strategy with four subsequent chips, and (**d**) strategy with two returning chips.

Indeed, levels of thickness errors were noticeably reduced when the strategies with several monitoring chips were applied. The average values of error vector norms in Figure 5 were equal to 17.25 nm in the case of direct monitoring and 9.32, 6.92, and 10.55 nm in the cases of strategies with several monitoring chips.

Despite somewhat weaker error self-compensation effects, the strategies with several monitoring chips may be preferable because of the lower levels of thickness errors. To evaluate a positive effect of error self-compensation, taking into account the expected levels of thickness errors, we made the following considerations. From a theoretical point of view, in the first approximation, the deviation δ*MF*(Δ) grew linearly with an increase in the norm of the error vector Δ. The strength of the error self-compensation effect *S* was estimated by Equation (7) for the error vectors Δ with the norms equal to 0.01*D*, where *D* is the norm of the design vector. Let us denote as Δ the average values of the error vector norms in the distributions shown in Figure 5. Using these average values, we introduced the effective strength of the error self-compensation effect for a given monitoring strategy by the equation

$$S\_{eff} = \langle S \rangle \frac{0.01D}{\langle \|\Delta\| \rangle} \tag{8}$$

Table 1 compares average *S* values in Figure 4b, average Δ in Figure 5, and *Se*ff values for the considered four monitoring strategies.

**Table 1.** Average *S*, average norm Δ, and *Se*ff values for the four discussed monitoring strategies.


The discussion of the obtained results is provided in the next section.

## **4. Discussion**

Recent achievements in the development of broadband monitoring hardware allow one to combine advantages of direct and indirect monitoring strategies through the use of several monitoring chips that are located on the main wheel of the deposition chamber. Even more, it is also possible to remove monitoring chips and bring them back to the measurement position many times during the coating deposition. This opens a way for using various broadband monitoring strategies. Thus, the question of comparing various strategies and choosing the most appropriate one becomes important. The presented research outlines a way for answering this question.

When considering optical monitoring strategies, we should take into account the correlation of thickness errors by optical monitoring procedures. This correlation causes both negative and positive effects. On one hand, it can lead to the development of a strong cumulative effect of thickness error growth, but on the other hand, it can result in the self-compensation of thickness errors. In this paper, we proposed a computational approach to assess the degree of thickness error correlation and the strength of the error self-compensation effect. The proposed approach was used to compare four strategies of broadband monitoring. It was shown that in the case of a 50-layer, nonpolarizing edge filter, the direct monitoring strategy provided the strongest correlation of thickness errors and the strongest error self-compensation effect. At the same time, in this case, one should expect the highest level of thickness errors caused by the negative cumulative effect of error growth. This reduces the effective strength of the error self-compensation effect. In the case of monitoring strategies with two and four subsequent monitoring chips, the strength of the error self-compensation effect was lower, but the expected levels of thickness errors were also lower. To evaluate the combined effect caused by the correlation of thickness errors, the effective strength of the error self-compensation effect *Se*ff was introduced by Equation (8). Table 1 shows that, in the case of strategies with several subsequent monitoring chips, *Se*ff was a bit higher than in the case of direct monitoring. However, taking into account the approximate nature of statistical estimates, on this basis, one should not conclude that the strategies with several subsequent chips have an absolute advantage in the case of a nonpolarizing edge filter. In the case of this design, all of the first three strategies deserve attention.

As for the fourth considered strategy, in the case of the nonpolarizing edge filter, it was clearly less suitable than the first three. However, this does not mean that the fourth strategy cannot be the best option for other designs. It is worth noting that the advantage of this strategy was discovered earlier [22] for a design with layers that were essentially thinner than the layers of the discussed filter.

The presented computational approach to comparing various broadband monitoring strategies is general and can be applied to check the prospects of the production of various types of optical coatings. **Author Contributions:** Conceptualization, A.T.; Methodology, A.T.; Software, T.I. and I.M.; Validation, I.K. and I.M.; Formal Analysis: A.T., I.K., and A.Y.; Investigation: A.T. and T.I.; Writing—Original Draft Preparation, A.T.; Writing—Review and Editing, A.T. and I.K.; Funding Acquisition, A.T. and A.Y.; Supervision, A.T. and A.Y.

**Funding:** This research was funded by a grant from the Russian Scientific Foundation (No. 16-11-10219).

**Conflicts of Interest:** The authors declare no conflicts of interest.
