## Abstract

The main theoretical results related to the investigation of the error self-compensation mechanism associated with direct broad band monitoring of optical coating production are presented. The presented results are illustrated using the production of Brewster angle polarizer where this effect is especially strong. Specific properties of the design merit function required for the presence of the error self-compensation effect are discussed and the mechanism of thickness errors correlation by the direct broad band monitoring is described. It is also discussed how one can check whether a strong error self-compensation effect may be expected for a given coating design and specific parameters of the monitoring procedure that will be used for coating production.

© 2017 Optical Society of America

## 1. Introduction

The first works on optical coatings production with direct broad band monitoring (BBM) were published four decades ago [1–3] but the two important features of this type of monitoring were already mentioned in these early publications. The first, negative feature is the accumulation of thickness errors with the growing number of deposited layers [1–5]. At the same time it was indicated that direct BBM may produce the positive effect of error self-compensation [1, 6]. In the case of direct BBM monitoring spectra are measured on one of the coatings to be manufactured [7] and thickness errors in the previously deposited layers influence measurement data for the currently deposited layer. This causes the correlation of thickness errors by the monitoring procedure.

The correlation of thickness errors is associated with all types of direct optical monitoring techniques. The remarkable positive result of this correlation is a very strong error self-compensation effect observed in the production of narrow band pass filters with turning point monochromatic optical monitoring [8–10]. The mechanism of this effect was studied in [11] and it was shown that it is related to the maintenance of phase properties of filter cavities provided by this type of monitoring. The existence of the error self-compensation effect in the case of optical coating production with direct BBM was numerically investigated in [12] for several types of optical coatings and this effect was always observed for the designs that were considered in this reference. It was found that the strength of the error self-compensation effect varied noticeably depending on the type and specific structure of the investigated optical coating design.

Recently the existence of a very strong error self-compensation effect associated with direct BBM was discovered in the case of manufacturing Brewster angle polarizers [13]. This discovery gives a new stimulus for the investigation of this effect. It is important to understand how the mechanism of error self-compensation operates in the case of direct BBM because such understanding may answer the question when this type of monitoring is advantageous compared to other monitoring techniques that can be used for optical coating production.

This paper presents the main results obtained in the course of mathematical investigation of the error self-compensation mechanism associated with direct BBM. The mathematical investigation itself is performed using the vector and matrix algebra in *m*-dimensional space and its detailed and rigorous description is the subject of separate mathematical paper [14]. Here all presented mathematical results are illustrated using the Brewster angle polarizer design and the direct BBM procedure described in [13].

The main goal of this paper is to show how one can investigate the existence of error self-compensation effect if he knows a design and parameters of direct BBM procedure that will be used for its production. In Section 2 we discuss specific properties of the design merit function required for the existence of the error self-compensation effect. In Section 3 the mechanism of thickness errors correlation by the direct BBM procedure is considered. Section 4 describes the algorithm that one can use to check whether a strong error self-compensation effect should be expected for a given coating design and specific parameters of BBM procedure that will be used for coating production. Final conclusions are presented in Section 5.

## 2. Merit function properties and error self-compensation

In the following the polarizer design from [13] will be used as an example. Its theoretical layer thicknesses are shown in Fig. 1(a). Theoretical *s*- and *p*-transmittances for the incidence angle of 55.6 deg. are presented in Fig. 1(b). The merit function used in the course of polarizer designing has the form:

Here *T _{s}* and

*T*designate

_{p}*s*- and

*p*-transmittances calculated at 55.6 deg. light incidence. Both transmittances are expressed in Eq. (1) in absolute values. The first term in the curled brackets of Eq. (1) expresses the requirement of

*T*being close to zero while the second term represents the requirement of

_{s}*T*being close to 1.

_{p}In Eq. (1) the summation is performed over the wavelength grid with the 1 nm step around the laser wavelength of 1064 nm. The total number of grid points is equal to 31. Both *s*- and *p*-transmittances in Eq. (1) are calculated in absolute values.

Let ${d}_{1}^{t},\mathrm{...},{d}_{m}^{t}$ be the theoretical layer thicknesses of coating design. Production errors in layer thicknesses cause variations of layer thicknesses from these values and as a consequence variation of the merit function from its minimum value achieved in the course of coating designing. Typically all modern optimization techniques [15] perform a search for merit function minima in such a way that gradients of the merit function become equal to zero. Due to this fact the deviation of the merit function from its minimum value caused by the production errors can be written down in the form

Let us introduce the matrix

This matrix is known as Hesse matrix of the merit function. It describes the merit function variations from its minimum value and has remarkable properties that allow one to apply powerful algebraic results for studying these variations. First of all *A* is a symmetric and positive definite matrix. Such matrices have *m* positive eigenvalues where *m* is the matrix dimension [16]. If we introduce the column vector Δ whose coordinates are variations $\delta {d}_{1},\mathrm{...},\delta {d}_{m}$ of actual layer thicknesses from theoretical thicknesses then the variation of the merit function presented by Eq. (2) can be written down as follows

Here *μ _{i}* are the eigenvalues and

*Q*are the eigenvectors of the matrix

_{i}*A*. The brackets in Eq. (4) denote scalar products of the vectors

*Q*and Δ. These scalar products are linear combinations of thickness errors with the coefficients being coordinates of the respective eigenvectors in Eq. (4).

_{i}It turns out that in the case of a strong error self-compensation effect only a few matrix *A* eigenvalues *μ _{i}* differ noticeably from zero. The matrix

*A*eigenvalues in the case of the polarizer design from [13] are presented in Table 1. We see that only first three eigenvalues are noticeably different from zero. This means that only the first three terms in the sum of Eq. (4) give essential contributions to the merit function variation. Suppose now that errors in layer thicknesses are correlated by the monitoring procedure so that the first three scalar products in Eq. (4) are zero or close to zero. In this case the merit function variation is also close to zero and thus the error self-compensation effect is observed. Thus a small number of essentially non-zero eigenvalues of the matrix

*A*is a necessary condition for the existence of this effect.

## 3. Direct BBM and correlation of thickness errors

This section provides the mathematical description of the mechanism of thickness errors correlation by the direct BBM procedure. Denote ${T}_{j}^{\text{meas}}(\lambda )$array of transmittance data measured after the deposition of the *j*-th layer on the wavelength grid {*λ*}. Note that this grid may be absolutely different from that used in the definition of the merit function given by Eq. (1). For example in [13] BBM measurements were performed at 2036 wavelength points in the spectral region from 658 nm to 1172 nm.

Let ${d}_{1}^{a},\mathrm{...},{d}_{j-1}^{a}$ be actual thicknesses of the first *j*–1coating layers. The transmittance spectrum measured during the deposition of the *j*-th coating layer can be presented in the form

*d*is the thickness of the growing layer and $\delta {T}_{\text{meas}}$ are measurement errors.

We assume that the deposition of the *j*-th layer is terminated at an instant when the minimum of the discrepancy function

*j*-th layer thickness from its planned theoretical value.

The discrepancy function Ф* _{j}*(

*d*) is the quadratic function with respect to the thickness errors and we can rewrite it in the form

Nowadays BBM spectra are acquired on wavelength grids with many hundreds or thousands of points. The derivatives $\partial {T}_{j}/\partial {d}_{i}$ are smooth functions of the wavelength. In the following we assume that measurement errors *δT*_{meas} are normally distributed random errors and neglect possible systematic errors in transmittance measurement data. Under these assumptions the summation of the terms $\frac{\partial {T}_{j}}{\partial {d}_{i}}\delta {T}_{\text{meas}}$ over the wavelength grid in Eq. (8) is expected to give values close to zero and we shall neglect these terms in the following considerations. This means that we assume that the minimum of Ф* _{j}*(

*d*) is achieved when the first term in the right hand part of (8) reaches its minimum value. This term is the quadratic form with respect to the thickness errors

*δd*:

_{j}Let us introduce the matrix of this quadratic form

As in the case of the matrix *A* considered in the previous section we can rewrite Eq. (9) using the eigenvalues and eigenvectors of the matrix *C _{j}*. Let ${\lambda}_{1}^{j},\mathrm{..},{\lambda}_{j}^{j}$ be these eigenvalues and ${P}_{1}^{j},\mathrm{..},{P}_{j}^{j}$ be these eigenvectors. Introducing the vector

*D*whose coordinates are thickness errors in the layers from 1 to

_{j}*j*, we can rewrite Eq. (9) as

The brackets in Eq. (11) denote scalar products of the vectors ${P}_{i}^{j}$ and *D _{j}*. These scalar products are linear combinations of thickness errors with the coefficients being coordinates of the respective eigenvectors in Eq. (11).

According to the above considerations the deposition of the *j*-th layer is terminated when the minimum value of the quadratic form in Eq. (9) is achieved. Thus the condition

*j*-th layer thickness error

*δd*with the thickness errors $\delta {d}_{1}^{a},\mathrm{...},\delta {d}_{j-1}^{a}$ that have been already made during the deposition of previous layers.

_{j}In the next section we will discuss how Eq. (12) can be used for predicting the existence of a strong error self-compensation effect in the case of optical coating production with BBM. In this section we provide examples showing that the correlation of thickness errors by the direct BBM gives specific thickness error vectors Δ that have almost zero scalar products with the most essential vectors *Q _{i}* in Eq. (4) and thus according to Section 2 have only a small influence on the resulting performance of optical coating.

It should be noted here that thickness errors correlated by the direct BBM also have a random nature. Errors in deposited layer thicknesses are caused by multiple factors [17] and always contain random components. For this reason the error vector Δ may vary essentially from one deposition run to another. Figure 2(a) presents the set of thickness errors obtained in the first test polarizer deposition run in [13]. This set of errors was found with the help of reverse engineering of 28 on-line transmittance spectra recorded after the deposition of each polarizer layer. Recall that in [13] direct BBM was performed by measuring normal incidence transmittance data at 2036 spectral points in the wavelength region from 658 nm to 1172 nm. For the determination of 28 parameters – errors in layer thickness – the triangular algorithm [18,19] was applied. This algorithm has been proved to provide reliable determination of thickness errors when multiple sets of measurement data are used as input data for the reverse engineering procedure.

Despite a high level of thickness errors shown in Fig. 2(a), spectral properties of the polarizer are excellent. Figure 2(b) compares *s-* and *p-*transmittances calculated for the design with errors shown in Fig. 2(a) (solid curves) with the respective transmittances of the unperturbed design (dashed curves). Superior spectral properties of the produced polarizer were confirmed by measurements of its spectral transmittances with Cary 7000 spectrophotometer [13]. The explanation of this remarkable result is that the set of errors shown in Fig. 2(a) is almost orthogonal to the first three eigenvectors of the matrix *A*, i.e. the first three scalar products in Eq. (4) are close to zero. Recall that according to the results of Section 2 this means that the merit function variation is also close to zero and the error self-compensation effect is present. Angles between the error vectors from Fig. 2(a) and the first three matrix *A* eigenvectors are presented in the second line of Table 2.

Figure 2(c) presents one more set of thickness errors obtained in the course of one of the subsequent polarizer deposition runs. One can see that the general pattern of these errors differs essentially from that of errors shown in Fig. 2(a). However the second set of errors is also almost orthogonal to the first three eigenvectors of the matrix *A*. Angles between this second error vector and the first three matrix *A* eigenvectors are presented in the third line of Table 2. As one would expect spectral properties of the polarizer with such thickness errors are also good. This is confirmed by Fig. 2(d) presenting *s-* and *p-*transmittances calculated for the design with errors shown in Fig. 2(c) (solid curves) and respective transmittances of the unperturbed design (dashed curves).

An outstanding strength of the observed error self-compensation effect is illustrated by the comparison of Fig. 2(b) with Fig. 3 presenting calculated *s-* and *p*-transmittances in the case of uncorrelated thickness errors. These errors were generated as normally distributed random errors with zero mathematical expectations and standard deviations equal to errors shown in Fig. 2(a) (different levels of errors for different layers). Figure 3 presents examples of *s*-transmittances and *p*-transmittances for five designs with such errors. In all cases there is nothing close to the remarkable results presented by solid curves in Figs. 2(b) and 2(d).

## 4. The algorithm for predicting the error-self compensation effect

According to the results of the two previous sections one should expect the presence of a strong error self-compensation effect if thickness errors are correlated by the direct BBM procedure in such a way that the resulting error vector is almost orthogonal to the main eigenvectors of the matrix *A* corresponding to the essentially non-zero eigenvalues of this matrix. Naturally the question arises how one can check whether this effect will be present knowing the coating design and specific parameters of BBM procedure that will be used for coating production.

The detailed description of the algorithm for predicting the existence of a strong error self-compensation effect requires intensive mathematical derivations and such description is the subject of a special mathematical paper [14]. Here we only present final results that can be useful for those who are interested in their own investigations of this effect.

Recall that according to Section 3 the correlation of thickness errors by the direct BBM is described by Eq. (12). This equation presents conditions that are fulfilled at all steps of the deposition procedure starting from *j* = 2 because the correlation of thickness errors starts with the deposition of layer number 2. Let ${p}_{1}^{ij}.,,,.{p}_{j}^{ij}$ be elements of one of the eigenvectors ${P}_{j}^{i}$. Consider the *m*-coordinate row vector *W _{ij}* with the coordinates

We can now write down Eq. (12) as

where Δ is the full*m*-coordinate vector of thickness errors.

Let us compose the matrix *W* whose rows are row vectors *W _{ij}*. This is the matrix of the dimension

*k*×

*m*where

*k*is the total number of all row vectors

*W*. Obviously

_{ij}*k*exceeds

*m*significantly. Let us also compose the matrix $\widehat{W}$ by adding

*m*row vectors${\mu}_{i}{Q}_{i}^{T}$ to the matrix

*W*. This is (

*k*+

*m*) by

*m*matrix. The algorithm for predicting the existence of the error self-compensation effect is based on the consideration of the singular value decompositions [16] of the matrices

*W*and $\widehat{W}$. It is shown in [14] that the error self-compensation effect is observed when the matrix $\widehat{W}$ has the same number of non-negligible singular values as the matrix

*W*and these values are nearly the same as the main singular values of the matrix

*W*. On a qualitative level this means that all vectors

*Q*added to form the matrix $\widehat{W}$are close to their projections on the subspace formed by the matrix

_{i}*W*rows and due to this fact the correlation of thickness errors gives the error vector Δ that is almost orthogonal to the vectors

*Q*.

_{i}The algorithm for predicting the existence of error self-compensation effect is formulated in terms of singular values of rectangular matrices. Calculation of these values is a rather routine procedure now because it can be accomplished with the help of all widely used mathematical packages such as MATLAB [20], Maple [21] and Mathematica [22].

## 5. Conclusion

Direct broad band monitoring of optical coating production results in the correlation of thickness errors. This correlation may cause a positive effect of error-self compensation that enables one to obtain coatings with high quality spectral properties even when thickness errors in individual layers are quite high. The existence of this effect was first mentioned nearly 40 years ago [1–3] but its detailed mathematical investigation was performed only recently [14]. This paper presents the main practically important conclusions of the mathematical investigation of the error self-compensation effect and illustrates them using the results of production of Brewster angle polarizer from [13].

It is shown that for the existence of a strong error self-compensation effect two requirements must be fulfilled. First of all the design merit function must have special properties that are formulated in terms of specific properties of the merit function’s Hesse matrix (matrix of the merit function second order derivatives). Then the direct BBM procedure must correlate thickness errors in such a way that the resulting vector of thickness errors is almost orthogonal to the main eigenvectors of the merit function Hesse matrix.

The mechanism of thickness errors correlation is mathematically described by the condition Eq. (12) presented in Section 3 and this mathematical presentation enables formulating the algorithm for predicting the existence of a strong error self-compensation effect. This algorithm is briefly outlined in the Section 4.

Naturally the question arises whether there are designs of other types that also exhibit the error self-compensation effect. The authors have some reasons to believe that such designs exist. However, this question requires further serious investigations.

It is worth noting that the results of this paper are obtained under the assumption that terminations of layer depositions are based on the criterion presented by Eq. (6). Not all available broad band monitors exploit this criterion and the existence of the error self-compensation effect in the case of other deposition termination scenarios requires additional investigations.

## Funding

Russian Science Foundation (16-11-10219).

## Acknowledgments

Authors are grateful to Dr. Michael Trubetskov for his valuable comments.

## References and links

**1. **B. Vidal, A. Fornier, and E. Pelletier, “Optical monitoring of nonquarterwave multilayer filters,” Appl. Opt. **17**(7), 1038–1047 (1978). [PubMed]

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

**3. **B. Vidal and E. Pelletier, “Nonquarterwave multilayer filters: optical monitoring with a minicomputer allowing correction of thickness errors,” Appl. Opt. **18**(22), 3857–3862 (1979). [PubMed]

**4. **B. T. Sullivan and J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings. I. Theoretical description,” Appl. Opt. **31**(19), 3821–3835 (1992). [PubMed]

**5. **A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Investigation of the effect of accumulation of thickness errors in optical coating production by broadband optical monitoring,” Appl. Opt. **45**(27), 7026–7034 (2006). [PubMed]

**6. **H. A. Macleod, “Monitoring of optical coatings,” Appl. Opt. **20**(1), 82–89 (1981). [PubMed]

**7. **H. A. Macleod, *Thin-Film Optical Filters*, 4th ed. (Taylor & Francis, 2010).

**8. **P. Bousquet, A. Fornier, R. Kowalczyk, E. Pelletier, and P. Roche, “Optical filters: monitoring process allowing the autocorrection of thickness errors,” Thin Solid Films **13**, 285–290 (1972).

**9. **H. A. Macleod, “Turning value monitoring of narrow-band alldielectric thin film optical filters,” Opt. Acta (Lond.) **19**, 1–28 (1972).

**10. **H. Macleod and E. Pelletier, “Error compensation mechanisms in some thin-film monitoring systems,” Opt. Acta (Lond.) **24**, 907–930 (1977).

**11. **A. V. Tikhonravov and M. K. Trubetskov, “Automated design and sensitivity analysis of wavelengh-division multiplexing filters,” Appl. Opt. **41**(16), 3176–3182 (2002). [PubMed]

**12. **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). [PubMed]

**13. **V. Zhupanov, I. Kozlov, V. Fedoseev, P. Konotopov, M. Trubetskov, and A. Tikhonravov, “Production of Brewster angle thin film polarizers using a ZrO_{2}/SiO_{2} pair of materials,” Appl. Opt. **56**(4), C30–C34 (2017). [PubMed]

**14. **A. V. Tikhonravov, I. V. Kochikov, and A. G. Yagola, “Mathematical investigation of the error self-compensation mechanism in optical coating technology,” IPSE (to be published).

**15. **A. V. Tikhonravov and M. K. Trubetskov, “Modern design tools and a new paradigm in optical coating design,” Appl. Opt. **51**(30), 7319–7332 (2012). [PubMed]

**16. **G. H. Golub and C. F. Van Loan, *Matrix Computations*, 3rd ed. (John Hopkins University, 1996).

**17. **A. V. Tikhonravov and M. K. Trubetskov, “Computational manufacturing as a bridge between design and production,” Appl. Opt. **44**(32), 6877–6884 (2005). [PubMed]

**18. **A. V. Tikhonravov and M. K. Trubetskov, “Online characterization and re-optimization of optical coatings”, in *Advances in Optical Thin Films* (SPIE, 2003), pp. 406–413.

**19. **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). [PubMed]

**20. **MATLAB version 7.10.0. Natick, MA: The MathWorks Inc. (2010).

**21. **Maple 2016. Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.

**22. **Wolfram Research, Inc., Mathematica, Version 11.2, Champaign, IL (2017).