**2. The Computational Approach to Assessing the Degree of Thickness Error Correlation and the Strength of the Error Self-Compensation E**ff**ect**

To illustrate the proposed computational approach, we analyzed a design of 50-layer, nonpolarizing edge filter with a 45◦ light incidence. Its theoretical spectral characteristics and layer physical thicknesses are presented in Figure 1. The filter used model high- and low- index materials with refractive indices of 2.35 and 1.45 (for example, model TiO2 and SiO2 indices). The first layer counting from the substrate was the high-index material layer, and the substrate refractive index was 1.52. It was designed using OptiLayer thin film software (v12.12) [23]. Computational manufacturing experiments with this filter [24] demonstrated the presence of a strong error self-compensation effect in the case of broadband monitoring in the normal incidence transmittance mode.

**Figure 1.** Theoretical *s*- and *p*-reflectances (**a**) and thicknesses (**b**) of the 50-layer, nonpolarizing edge filter.

Let *dt* <sup>1</sup>, ..., *<sup>d</sup><sup>t</sup> <sup>m</sup>* be physical thicknesses of a coating design. Here, *m* is the total number of coating layers, and *m* = 50 in the case of the considered nonpolarizing filter. In the course of production, actual layer thicknesses *d<sup>a</sup>* <sup>1</sup>, ..., *<sup>d</sup><sup>a</sup> <sup>m</sup>* differed from the planned values. Consider broadband monitoring using measured transmittance spectra. When using modern broadband monitoring devices, such spectra usually have hundreds or even thousands of spectral points. Let *d* be the growing thickness of the *j*-th coating layer. The measured transmittance is

$$T\_j(d) = T\_j(d\_1^a, \dots, d\_{j-1}^a, d) + \delta T\_{\text{meas}} \tag{1}$$

Here, δ*T*meas is the error in measured transmittance data. In Equation (1) and the following equations, we omitted the indication of an obvious dependence on wavelength λ.

With broadband monitoring, the deposition of the *j*-th layer is terminated in accordance with the condition that the minimum is reached by the discrepancy function

$$\Phi\_{\dot{\jmath}}(d) = \sum\_{\lambda} \left[ T\_{\dot{\jmath}}(d\_{1'}^{a}, \dots, d\_{\dot{\jmath}-1'}^{a}, d) + \delta T\_{\text{meas}} - T\_{\dot{\jmath}}(d\_{1'}^{t}, \dots, d\_{\dot{\jmath}}^{t}) \right]^2 \to \min \tag{2}$$

Here, the summation is carried out over the wavelength grid at which the transmittance is measured.

It follows from Equation (2) that the actual thickness of the deposited *j*-th layer is associated not only with the errors in transmittance data but is also determined by the actual thicknesses of all previously deposited layers. This is the reason for the correlation of errors in layer thicknesses.

As outlined above, a rigorous mathematical investigation of the correlation of thickness errors was provided in [19]. To present the main result of this investigation, we introduced the vector of thickness errors Δ = {δ*d*1, ..., δ*dm*}. When considering Equation (2) for all coating layers, starting from layer *j* = 2, the following matrix appears:

$$S\_{\bar{j}} = \|\sum\_{\lambda} \frac{\partial T\_{\bar{j}}}{\partial d\_{\bar{i}}} \frac{\partial T\_{\bar{j}}}{\partial d\_{k}}\|. \tag{3}$$

Here, ∂*Tj*/∂*di* are partial derivatives of the intensity transmission coefficient for the subsystem of layers with the numbers from 1 to *j*.

Let λ*<sup>j</sup> <sup>i</sup>* and *Pi <sup>j</sup>* be eigenvalues and eigenvectors of the matrix Cj, and *p ij* <sup>1</sup> , ..., *p ij <sup>j</sup>* be the elements of the eigenvectors *Pi j* . With their help, the following raw vectors are introduced for all i from 1 to j and all j from 2 to m:

$$\mathcal{W}\_{ij} = \sqrt{\lambda\_i^j} \{p\_1^{ij}, \dots, p\_j^{ij}, 0, \dots, 0\}. \tag{4}$$

These raw vectors were then used to form the rectangular matrix *W* with the dimensions *k* × *m*, where *m* is the number of coating layers, and *k* = (*m* − 1)/(*m* + 2)/2. In accordance with the results of Ref. [19], the correlation of thickness errors led to a small norm of the vector *W*Δ. To formalize the concept of the smallness of this norm, the parameter α was introduced in [20]. It is calculated as

$$\alpha = \left\| \mathcal{W} \Delta^0 \right\|^2 / \left\langle \left\| \mathcal{W} \Delta^r \right\|^2 \right\rangle \tag{5}$$

Here, <sup>Δ</sup><sup>0</sup> is the normalized error vector <sup>Δ</sup> (i.e., <sup>Δ</sup><sup>0</sup> <sup>=</sup> <sup>Δ</sup>/Δ), and *W*Δ*<sup>r</sup>* 2 is the averaged square norm of the vectors *W*Δ*<sup>r</sup>* over all vectors Δ*<sup>r</sup>* with the norm Δ*<sup>r</sup>* = 1.

The introduced parameter α compares the value of the norm *W*Δ*<sup>0</sup>* with the average value of the norm *W*Δ*<sup>r</sup>* for all random vectors of the unit length. The introduced parameter is called the degree of thickness error correlation. The smaller this parameter the stronger the correlation of errors.

In [20,21], the degree of thickness error correlation was estimated for two cases in which a strong error self-compensation effect was observed either practically [18] or in the course of simulating the coating deposition [24]. The Brewster's angle polarizer and the nonpolarizing edge filter were considered here. In both cases, the smallness of parameter α was confirmed. We consider the related results for the edge filter later in this document.

Subsequently, we proceeded to assess the strength of the error self-compensation effect. Despite the correlation of thickness errors, error vectors Δ are also random in nature since they are determined by various random factors. Therefore, the estimation of the strength of the error self-compensation effect should have a statistical form. Since this effect is caused by the correlation of thickness errors, it is also natural to estimate it by comparing it with the influence of uncorrelated thickness errors.

To assess the impact of errors on coating spectral characteristics, we used the merit function to solve the respective design problem. In the case of the nonpolarizing edge filter, it has the form

$$MF = \left\{\frac{1}{2L} \sum\_{\lambda} \left[ \left( T\_s - \hat{\mathcal{T}} \right)^2 + \left( T\_p - \hat{\mathcal{T}} \right)^2 \right] \right\}^{1/2} \tag{6}$$

Here, *Ts* and *Tp* are transmittances for the *s*- and *p*-polarized light at 45◦ light incidence; *T*ˆ is the target transmittance, equal to 0% in the 900–990 nm spectral band and equal to 100% in the 1010–1100 nm band; and *L* is the total number of spectral grid points that has the step of 1 nm in the two target spectral bands.

Let us designate δ*MF*(Δ) as the deviation of merit function corresponding to the error vector Δ. We wanted to compare the deviations that are caused by the correlated and uncorrelated thickness errors.

The uncorrelated thickness errors are most consistent with stable production processes using time and quartz crystal monitoring. It is generally accepted that when using these monitoring techniques, the best accuracy in controlling layer thicknesses is about 1% of the planned thickness values. Let us denote δ*MF* as the root-mean-square value of merit function deviations calculated for the large number of random error vectors Δ that are set so their coordinates Δ*dj* are distributed according to Gaussian law with zero mathematical expectations and standard deviations equal to 1% of the thicknesses of the corresponding layers. Thus, δ*MF* is an estimate of the effect of uncorrelated errors. Further, to obtain this estimate, we generated 1 million uncorrelated error vectors.

To obtain the vector of correlated thickness errors, one can use computational manufacturing experiments to simulate the deposition process with broadband optical monitoring. In [24], OptiLayer software [23] was used for this purpose. In [21], a simplified simulator of this process was proposed that, on the one hand, fully reflected the process of thickness error correlation and, on the other hand, allowed error vectors to be generated much faster than full simulators of the deposition processes. In this paper, we applied the simplified simulator from [21].

In order to evaluate the strength of the error self-compensation effect *S*, two possible assessments were considered in our previous works [20,25]. They were also based on a comparison of the effect of correlated and uncorrelated thickness errors. In Ref. [20], a special *D*α region was considered in the unit sphere in the space of error vectors, and an estimate for parameter *S* was introduced using all correlated error vectors with a degree of correlation of thickness errors less than α. In Ref. [25], an estimate for *S* was introduced with the normalization of all error vectors to unit norm vectors. Here, we introduce a new evaluation form for *S,* which we hope is more consistent with practice.

Depending on the levels of simulated error factors, generated error vectors Δ will have various norms. In general, with a lower norm of the error vector, lower values of the merit function variations should be expected. For a more objective comparison of correlated and uncorrelated thickness errors, it is advisable to consider, in both cases, error vectors of the same norm. For this reason, we normalized all correlated errors vectors so that their norms were 1% of the design vector norm. With this normalization, the strength of the error self-compensation effect for a specific vector of correlated errors will be estimated as

$$S = \langle \delta MF \rangle / \delta MF(\Delta) \tag{7}$$

Figure 2 shows the probability density functions of the distributions of the degree of correlation of thickness errors and the strength of the error self-compensation effect in the case of direct transmittance monitoring in the 450–950 nm spectral band. These distributions are calculated using 1,000,000 vectors of correlated errors.

**Figure 2.** Probability density functions for the degree of thickness error correlation α (**a**) and for the strength of the error self-compensation effect *S* (**b**).

In full accordance with previously obtained results [18,24], Figure 2 demonstrates the smallness of parameter α. At the same time, almost all calculated *S* values were large enough, which indicates the presence of a strong error self-compensation effect. The average *S* value was equal to 16.6.

An even more visual representation of a strong correlation of thickness errors and the associated error self-compensation effect is given by Figure 3, where pairs of α and *S* values are presented for correlated and uncorrelated error vectors. The pairs of values corresponding to these two types of errors are located in significantly different parts of the (α, *S*) plane.

**Figure 3.** Comparison of pairs α and *S* for correlated (blue) and uncorrelated (red) error vectors. Dots represent 10,000 tests randomly selected from calculated sequences of 1,000,000 vectors.
