Open Access
Add to BibSonomy
Abstract
Computational manufacturing experiments are used to detect the types of optical coatings that are showing the presence of a strong error self-compensation effect in the coating production with direct broad band monitoring. It is shown that predictions made on the basis of these experiments coincide with the predictions of the previously developed rigorous mathematical approach to the investigation of the error self-compensation effect.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
It is well known that all types of direct optical monitoring cause correlation of errors in thicknesses of layers of manufactured coatings [1]. This correlation is the main reason for the cumulative effect of thickness errors exhibiting itself as the growth of thickness errors with the growing number of deposited layers [1,2]. Along with this negative effect, the correlation of errors may produce also a positive effect of error self-compensation. This effect was discovered nearly five decades ago for the case of production of narrow band pass filters with a special type of direct monochromatic monitoring known as turning point optical monitoring [3–5]. The mechanism of error self-compensation effect was explained in [6] and it was shown that it is connected with maintaining of full phase retardance in each cavity layer (spacer layer) of a narrow band filter. This mechanism is automatically provided by turning point monitoring and that's why this type of monitoring is the most suitable one for the production of WDM and other types of multilayer narrow band pass filters.
According to the basic classification of monitoring techniques, the term direct monitoring is applied in the situations when “all layers are monitored on the same substrate, usually the actual filter being produced” [1]. The very first comments about the existence of error self-compensation effect in the case of coating production with direct broad band monitoring (BBM) were made four decades ago in the series of publications [7–9]. An attempt of rigorous investigation of the presence of this effect in the case of BBM was made in [10]. It was based on the comparison of influence of correlated and uncorrelated thickness errors of the same level on a spectral performance of investigated optical coating. Five different types of optical coatings (hot and cold mirrors, band pass filters) and various designs of coatings of the same type were considered in this work and it was found that the error self-compensation effect was always present. However, the strength of this effect varied considerably, depending not only on the type of the investigated coating, but also on its specific design structure. In many situations this effect was not significant.
In [11] the existence of a very strong error self-compensation effect was reported for the production of Brewster angle polarizer with ZrO2-SiO2 pair of thin film materials. Due to the correlation of thickness errors by direct BBM, spectral characteristics of the produced polarizer were excellent even in the case of high thickness errors in individual layer thicknesses. The publication [11] stimulated the mathematical investigation of the error self-compensation mechanism that was performed in [12] using the algebra in m-dimensional space. The results of this investigation were also presented in [13] where the algorithm for predicting the existence of the error self-compensation effect was presented and illustrated using the Brewster angle polarizer from [11].
An existence of a strong error self-compensation effect may be an important argument in favour of choosing BBM as a monitoring technique for coating production. That's why it is relevant to be able to investigate a potential strength of this effect for a given coating design. The mathematical algorithm presented in [12,13] requires calculating singular values of special matrices and can be difficult for practical use. At the same time, there is such a tool as computational manufacturing experiments [14] that becomes more and more popular nowadays and that is available not only as a part of commercial thin film software but also as an option of some deposition equipment.
In this paper, we consider three different types of optical coatings for which computational manufacturing experiments predict the existence of a strong error self-compensation effect. The designs chosen for this publication present the cases of coatings with narrow band and wide band spectral characteristic demands. We also consider various types of direct BBM procedure when the monitoring spectral region covers the spectral region of target coating characteristic and when it is essentially different from the target one. The results of computational manufacturing experiments are presented in Section 2. The discussion of the obtained results and their comparison with the results of rigorous mathematical investigations are presented in Section 3. Final conclusions are given in Section 4.
2. Computational manufacturing experiments for checking the presence of error self-compensation effect
All designs considered in this section serve for demonstration purposes. For this reason, we use only model non-dispersive refractive indices of thin films and substrates. Similar to [10], a presence of some error-self compensation effect has been observed for various types of optical coatings but for this publication we have selected only three designs that exhibit distinctly strong effect. All computational manufacturing experiments have been performed using the OptiLayer software option for simulating optical coating production with direct BBM [15]. For all designs presented in this Section, the layer number starts from the substrate side.
2.1 Non-polarizing edge filter
In this subsection we consider non-polarizing short-wave-pass edge filter with the working range from 900 nm to 1100 nm and angle of light incidence equal to 45 deg. The refractive indices of high and low index materials are 2.35 and 1.45 respectively, the substrate refractive index is 1.52. Layer optical thicknesses and s- and p- reflectances of the theoretical filter design are shown in Fig. 1. The filter was designed using the OptiLayer thin film software [15]. The obtained design is later referred to as NPEF.
Fig. 1 Optical thicknesses (a) and theoretical s- and p-reflectances (b) of the 50-layer non-polarizing edge filter. See Data File 1 for underlying values.
In the course of computational manufacturing experiments, the instabilities of deposition rates of high and low index materials were simulated. The mean rates for H and L materials were 0.4 and 0.5 nm/sec, respectively, and the instabilities of these rates had 3 sec. correlation times and 0.05 and 0.1 nm/sec standard deviations. In OptiLayer software layer thickness monitoring is based on the estimation of the discrepancy between measured transmittance data and theoretical transmittance data expected at the end of layer deposition. Layer deposition is terminated when discrepancy minimum is achieved. Production errors associated with this widely used approach are automatically taken into account by the software. But there could be also additional errors associated with closing shutters above evaporation sources. Inaccuracies of closing shutters for terminations of layer depositions were specified by the random shutter delays with 1 sec rms. The monitoring range for BBM measurements was 400-900 nm with 1 nm step, i.e. with 501 evenly distributed wavelength points. Monitoring was performed in the transmittance mode and scans of arrays of BBM data were acquired with the intervals of 3 sec. Random errors in transmittance data were set to be 1%. Such error factor as spectral resolution of the measurement device was not taken into account in simulation experiments.
Figure 2 presents results of a typical computational manufacturing experiment. Relative errors in the thicknesses of computationally manufactured coating are shown in Fig. 2(a) and s- and p-reflectances of this coating are presented in Fig. 2(b). Relative errors in Fig. 2(a) are calculated as relative deviations of manufactured coating thicknesses from the theoretical design thicknesses with respect to these ideal design thicknesses. One can see that despite of a high level of errors in many coating layers (up to 10 percent and more) spectral properties of the edge filter are still quite good. The effect on non-polarizing properties of the filter at the transition zone around 1000 nm is very slight and there are only small ripples of the Rs in the transition zone. We have performed 100 simulation experiments and in all cases variations of s- and p-reflectances were similar to or smaller than those shown in Fig. 2(b).
Fig. 2 Relative errors in the thicknesses of computationally manufactured edge filter (a) and s- and p-reflectances of this filter (b). See Data File 1 for underlying values.
To demonstrate the presence of a very strong error self-compensation effect statistical error analysis with uncorrelated thickness errors of the same level as in Fig. 2(a) was performed. Errors in layer thickness were generated as normally distributed errors with zero mathematical expectation and standard deviations equal to modules of correlated errors shown in Fig. 2(a). Figures 3(a) and 3(b) present results of five subsequent tests of this analysis. It is seen that in the case of uncorrelated errors spectral properties of the edge filter are totally destroyed.
Fig. 3 Results of 5 tests with uncorrelated thickness errors of the same level as correlated errors shown in Fig. 2(a) (a - Rs, b – Rp) and results of 5 tests with uncorrelated thickness errors of the same 2% levels for all coating layers (c – Rs, d – Rp).
One can see in Fig. 2(a) that relative levels of thickness errors are quite high for some layers. This is also a consequence of correlation of thickness errors by direct BBM. It is known as cumulative effect of thickness errors [1,2]. Uncorrelated thickness errors may be a good approximation for indirect monitoring procedures [1] but for such procedures one should expect more uniform levels of thickness errors than those shown in Fig. 2(a). For this reason we performed also experiments with uncorrelated thickness errors where levels of errors were the same for all coating layers. In these experiments errors in layer thicknesses were also normally distributed errors with zero mathematical expectations but standard deviations were set equal to 2% for all layers. This level is almost two times less than the rms level of thickness errors in Fig. 2(a). Results of five subsequent tests of the analysis with such errors are presented in Figs. 3(c) and 3(d). It is seen that despite the lower level of errors than the rms level of errors in Fig. 2(a), spectral properties of the edge filter are still totally destroyed.
2.2 Three-line filter
In this subsection we consider the three-line filter with narrow transition zones located near 450 nm, 510 nm, and 640 nm. The refractive indices of high and low index materials and substrate refractive index are the same as in the previous section. Layer optical thicknesses and transmittance of the theoretical filter design are shown in Fig. 4. As before the filter was designed using the OptiLayer thin film software [15]. Computational manufacturing experiments were performed with the same error factors and with the same BBM settings as in the previous example.
Fig. 4 Optical thicknesses (a) and theoretical transmittance (b) of the 51-layer 3-line filter. See Data File 2 for underlying values.
Figure 5(a) presents relative errors in the thicknesses of 3-line filter obtained in one of the computational manufacturing experiments. Respective transmittance of the computationally manufactured filter is shown in Fig. 5(b). As in the previous example spectral properties of the filter are quite good despite of the high level of thickness errors. In the intended transmission zones only, small degradations of filter spectral properties are observed. But the situation is absolutely different when the tests with uncorrelated thickness errors are performed. This is clearly demonstrated by Fig. 5(c) presenting results of the error analysis with uncorrelated thickness errors. As before these errors are normally distributed values with zero mathematical expectations and standard deviations equal to modules of correlated errors shown in Fig. 5(a). Figure 5(c) presents results of the 1000 tests with such errors. The red curve in this figure shows the mathematical expectation of the transmittances of the perturbed designs and the black curves denote the boundaries of the corridor of transmittance standard deviation. It is obvious that in the case of uncorrelated errors spectral properties of the filter are totally destroyed. Analogous tests with more uniform uncorrelated thickness errors (the same standard deviations for errors in thicknesses of all layers) also demonstrate the failure of filter spectral properties even when the level of errors is lower than the average level of errors in Fig. 5(a).
Fig. 5 (a) – relative errors in the thicknesses of computationally manufactured 3-line filter; (b) – transmittance of the computationally manufactured filter; (c) – results of the statistical error analysis with uncorrelated thickness errors of the same level as in Fig. 5(a): red curve –mathematical expectation of the filter transmittance, black curves show the corridor of standard deviations. See Data File 2 for underlying values.
2.3 Gain flattening filter
This subsection presents results obtained for the gain flattening filter (GFF). Layer optical thicknesses and transmittance of the theoretical filter design are shown in Fig. 6. The refractive indices of high and low index materials and substrate refractive index are 2.1, 1.45, and 1.52, respectively. The filter was designed using the OptiLayer thin film software [15]. Computational manufacturing experiments were performed with the same error factors of the deposition process and the same level of errors in monitoring data as in the two previous examples. But in this set of experiments three different spectral regions of BBM measurements were considered: 450-950 nm, 650-1150 nm, and 1100-1600 nm. Note that only the third spectral region included the spectral band where the GFF target was specified. The wavelength step of transmittance measurements was 1 nm for the all three spectral regions and the time interval between subsequent BBM measurements was 3 seconds as in the previous examples.
Fig. 6 Optical thicknesses (a) and theoretical transmittance (b) of the73-layer gain flattening filter (crosses in Fig. 6(b) present target transmittance values). See Data File 3 for underlying values.
Results of the computational manufacturing experiments with BBM in the first two considered spectral regions were noticeably better than those in the case of monitoring in the 1100-1600 nm region. Figure 7 presents results obtained in the course of experiments with 650-1150 nm monitoring region. Relative errors in the thicknesses of one of the computationally manufactured GFF filter and its transmittance in comparison with the transmittance of the theoretical design are shown in Fig. 7(a) and Fig. 7(b) respectively. The presence of the error self-compensation effect is clearly demonstrated by the comparison of Fig. 7(b) with Fig. 7(c) where the results of statistical error analysis with uncorrelated thickness errors are presented. Errors simulated in the course of this analysis were even lower than those shown in Fig. 7(a): normally distributed uncorrelated thickness errors had zero mathematical expectations and standard deviations equal 1%. The number of tests for calculating the transmittance mathematical expectation and the corridor of transmittance standard deviation was 1000. Figure 7(c) shows that even with the 1% level of uncorrelated thickness errors spectral properties of the GFF filter are totally destroyed.
Fig. 7 (a) – relative errors in the thicknesses of computationally manufactured gain flattening filter; (b) – transmittance of the computationally manufactured filter (red curve) and transmittance of the theoretical filter design (black curve); (c) – results of the statistical error analysis with 1% uncorrelated thickness errors: red curve – mathematical expectation of the filter transmittance, black curves designate the corridor of standard deviations. See Data File 3 for underlying values.
3. Discussion of the obtained results
The designs considered in the previous section have essentially different structural properties. One of them, namely the GFF design, is close to a quarter-wave mirror while the two other designs are distinctly different from the quarter-wave ones. Two of the considered designs (GFF and NPEF) exhibit almost periodic modulations of optical thicknesses of design layers. It is known [16,17] that designs with periodic modulations of layer thicknesses are described by a small number of specific design parameters. At the same time the 3-line design is obviously different from such low-parametric designs. So it is necessary to conclude that the presence of a strong error self-compensation effect cannot be attributed to some specific structural properties of an optical coating design.
In [13] the presence of a strong error self-compensation effect was discovered for the Brewster angle polarizer with a narrow spectral zone of target characteristics. Only one of the designs from the previous section (GFF design) has also a narrow band spectral target. Two other designs have spectral targets specified in much wider spectral regions. Thus we see that the discussed effect can be utilized also for some coatings with wide band spectral characteristics.
It is important to note that the presence of a strong error self-compensation effect doesn't require that the BBM spectral region covers the spectral region of target spectral characteristic. Even more experiments with the GFF design demonstrate that a stronger effect can be observed when these spectral regions are different. The results of [13] and the experiments with NPEF demonstrate that the normal incidence transmittance monitoring can provide a strong effect in the cases when coating is intended for operation at an oblique light incidence.
The mechanism of error self-compensation effect is connected with the correlation of thickness errors by the direct BBM procedure. One of the consequences of this correlation is clearly seen in Figs. 2(a), 5(a), and 7(a). Errors in optical thicknesses are arranged so that they compensate each other and the variation of total optical thickness is close to zero. Of course, this is not the only consequence of the correlation of errors by the direct BBM. A more complete picture of this process can be given by considering vectors in a multi-dimensional space [12]. It is shown there that error vectors tend to be orthogonal to some subspace Q in the m-dimensional space where m is the number of layers of the considered design.
Let F be the merit function estimating the closeness of the design spectral characteristic to the target characteristic. According to [12] a strong error self-compensation effect takes place if variations of the merit function F along all error vectors belonging to the above-mentioned subspace Q are close to zero. For the mathematical description of this effect two rectangular matrices W and were introduced in [12]. We do not provide their exact presentations here because, if necessary, these presentations can be found also in [13]. In [12] the algorithm for predicting the existence of a strong error self-compensation effect is formulated in terms of singular values of the matrices W and . It is shown that such effect is present when these matrices have the same number of non-negligible singular values and these values are nearly the same for both matrices.
Figure 8 presents singular values of the matrices W and for the case of NPEF design. It is seen that the conclusion of the previous section about the presence of a strong error self-compensation effect is in full correspondence with the results of the rigorous mathematical investigations. Indeed, the singular values are almost identical for both matrices. The situation is exactly the same in the cases of 3-line filter and GFF designs.
4. Conclusion
Computational manufacturing experiments may serve as a tool for the investigation of a possible presence of a strong error self-compensation effect in optical coating production with direct BBM. They show that this effect can be present both in the cases of coatings with narrow band and wide band spectral characteristics. For the existence of a strong effect it is not necessary that the monitoring spectral region covers the spectral region of target coating characteristic. Even more the monitoring spectral region and angle of monitoring beam incidence can be essentially different from the target spectral region and working incidence angle.
Predictions made on the basis of computational manufacturing experiments coincide with the predictions of the rigorous mathematical analysis. But from a practical point of view it is essential that these experiments are more obvious than rigorous considerations. Conclusions made on the basis of computational manufacturing experiments can be strong arguments for choosing BBM as a monitoring technique for coating production.
Funding
Russian Science Foundation (16-11-10219).
Acknowledgments
Authors are grateful to Dr. Michael Trubetskov for his valuable support.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
1. H. A. Macleod, Thin-Film Optical Filters, 4th ed. (Taylor & Francis, 2010)
2. 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). [CrossRef] [PubMed]
3. 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(2), 285–290 (1972). [CrossRef]
4. H. A. Macleod, “Turning value monitoring of narrow-band all-dielectric thin film optical filters,” Opt. Acta (Lond.) 19(1), 1–28 (1972). [CrossRef]
5. H. Macleod and E. Pelletier, “Error compensation mechanisms in some thin-film monitoring systems,” Opt. Acta (Lond.) 24(9), 907–930 (1977). [CrossRef]
6. A. V. Tikhonravov and M. K. Trubetskov, “Automated design and sensitivity analysis of wavelengh-division multiplexing filters,” Appl. Opt. 41(16), 3176–3182 (2002). [CrossRef] [PubMed]
7. B. Vidal, A. Fornier, and E. Pelletier, “Optical monitoring of nonquarterwave multilayer filters,” Appl. Opt. 17(7), 1038–1047 (1978). [CrossRef] [PubMed]
8. B. Vidal, A. Fornier, and E. Pelletier, “Wideband optical monitoring of nonquarterwave multilayer filters,” Appl. Opt. 18(22), 3851–3856 (1979). [CrossRef] [PubMed]
9. 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). [CrossRef] [PubMed]
10. 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). [CrossRef] [PubMed]
11. V. Zhupanov, I. Kozlov, V. Fedoseev, P. Konotopov, M. Trubetskov, and A. Tikhonravov, “Production of Brewster angle thin film polarizers using a ZrO2/SiO2 pair of materials,” Appl. Opt. 56(4), C30–C34 (2017). [CrossRef] [PubMed]
12. A. V. Tikhonravov, I. V. Kochikov, and A. G. Yagola, “Mathematical investigation of the error self-compensation mechanism in optical coating technology,” Inverse Probl. Sci. Eng. 26(8), 1214–1229 (2018). [CrossRef]
13. A. V. Tikhonravov, I. V. Kochikov, and A. G. Yagola, “Error self-compensation mechanism in the optical coating production with direct broad band monitoring,” Opt. Express 25(22), 27225–27233 (2017). [CrossRef] [PubMed]
14. A. V. Tikhonravov and M. K. Trubetskov, “Computational manufacturing as a bridge between design and production,” Appl. Opt. 44(32), 6877–6884 (2005). [CrossRef] [PubMed]
15. www.optilayer.com.
16. B. E. Perilloux, “Discrete thin-film thickness-modulated designs: spacing of all possible stopbands,” Appl. Opt. 38(13), 2911–2915 (1999). [CrossRef] [PubMed]
17. O. Lyngnes and J. Kraus, “Design of optical notch filters using apodized thickness modulation,” Appl. Opt. 53(4), A21–A26 (2014). [CrossRef] [PubMed]
