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Abstract
The feasibility of using direct broad band optical monitoring control in the fabrication of the ultra-steep dichroic filters based on resonant structures is investigated. Using computational manufacturing and deposition experiments, the role of the errors self-compensation effect is clarified by comparing the results of direct broad band optical monitoring and time monitoring. The errors correlation strength of ultra-steep dichroic filter is analyzed and it shows that the correlation calculated by the current model is not strong. The relationship between errors correlation and errors self-compensation effect for the ultra-steep dichroic filter is discussed.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Ultra-steep dichroic filters (USDF) are among the key thin film elements required for Raman spectroscopy, super-resolution and fusion laser optics [1–3]. The high spectral efficiency and narrow transition width is an ideal condition for application in optics system. Many investigations have been conducted to design a dichroic filter. One of the universal design approaches is the Fabry-Perot interference filter [4]. However, this structure requires all layers are multiple quarter-wave, which leads to limitations when it comes to wide passband bandwidth and non-polarizing for the wavelength in the case of oblique incidence [3]. Another kind of dichroic filter designed based on resonant cavity structure can provide highly desirable optical performance. The structure is based on the concept of the detuned Fabry–Perot interference filter [5,6], through the combination of multiple resonators cavities and the optimization of the irregular structure to satisfy the broadband and high-efficiency passband requirements. It is possible to achieve zero separation between the two polarization components through a suitable refinement. However, there are serious difficulties in coating USDF based on cavity resonant structures. Firstly, such a dichroic filter is constructed using many layers to achieve higher steepness, and the irregular structure is optimized to obtain high transmittance. Secondly, due to the high sensitivity of the spectra to thickness errors would result in a low passband efficiency with the accumulation of the deposition system errors and random errors. Obviously, it is practically important to study how to fabricate dichroic filters precisely.
As the spectral characteristics of USDF are extremely sensitive to coating errors, reliable thin film thickness monitoring is critical for successful fabrication. Direct optical monitoring is an important means to precisely control thin film thickness [7,8]. A negative feature of this type of monitoring is the accumulation of thickness errors with the growing number of deposited layers [9,10]. Another important feature is the thickness errors correlation, and the thickness errors of the previously deposited layer affect the measurement data, thus resulting in the error of the current layer depending on the previous layers thickness errors. The positive effect of the thickness error correlation is to cause a self-compensation effect of the thickness error [11–13]. The strong error self-compensation effect in the preparation of narrow bandpass filters using turning-point monochromatic optical monitoring has been explained in [12]. Unfortunately, the application of this monitoring technique requires that all layers are multiple quarter-wave of the filter center wavelength. This means that the turning-point monochromatic optical monitoring strategy is not applicable to non-quarter-wave resonant structures. Direct monochromatic level monitoring is one of the strategies used to produce optical coating with layers of non-quarter-wave optical thickness [14]. The latest research results show the presence of the error self-compensation effect in optical coating production with the direct monochromatic level monitoring. However, when comparing the strength of the error self-compensation effect in cases of broad band and monochromatic optical monitoring, it is found that the broad band optical monitoring (BBOM) is much stronger [15]. In recent years, the advantages of the error self-compensation characteristic of direct BBOM have become more and more prominent. The results of recent manufacturing experiments showed that Brewster angle polarizers with direct BBOM have strong thickness errors correlation and strong error self-compensation effect [16]. It was investigated that several types of optical coatings (hot and cold filters, bandpass filters) have an error self-compensation effect under the condition of BBOM. However, the strength of this effect depends not only on the type of coating but also on the design structure [17,18]. This effect was associated with the thickness errors correlation of different types of coatings. The study of thickness error correlation is practically important for coating the most complex optical thin films. In [19], the intensity of thickness error correlation was estimated by a geometric approach, and it was demonstrated that the 50-layer non-polarizing edge filter exhibits a strong error self-compensation effect in production with direct BBOM. Therefore, it may be a feasible way to fabricate USDF by direct BBOM strategy.
In this paper, the design and production of USDF for wavelengths of 1080 nm are considered. In Section 2, we demonstrate a direct BBOM computational production experiment to predict whether there is a strong error self-compensation effect in coating design. The coating process and manufacturing results of USDF are confirmed in Section 3. In Section 4, we estimate the strength of error correlation of USDF by using the geometric approach proposed in [20]. Final conclusions are presented in Section 5.
2. Computational experiments with ultra-steep dichroic filter
In this section, the investigation results of USDF computational manufacturing experiments under direct BBOM conditions are presented. Before starting the computational manufacturing experiments, it is necessary to provide a feasible design. In this work, we consider USDF with the working range from 1040 nm to 1110 nm and angle of light incidence equal to 15 deg, and the transition width is 8 nm. The filter is designed using the OptiLayer thin film software [21]. We use Ta2O5 as a high refractive index material and SiO2 as a low refractive index material, and the substrate is fused silica. The refractive index wavelength dependencies of thin-film materials and substrates are described by a well-known Cauchy formula:
where A0, A1, and A2 are dimensionless parameters, λ0 = 1000nm, and λ is specified in nanometers. The values of Cauchy parameters of thin-film materials and substrates are presented in Table 1. It is noted that the refractive index of the material in Table 1 is derived from our laboratory's ion beam sputter coating equipment. The reliability of the refractive index parameters of materials and substrate has been verified experimentally, so in the computational manufacturing experiments variables in this paper, the error of refractive index parameters was not considered.
Table 1. Cauchy parameters of thin film materials and substrates
Layer optical thicknesses and theoretical spectral characteristics of the USDF design are shown in Fig. 1. Obviously, there is no thin layers in design, and optical thickness of the thinnest layer is greater than 100 nm. To examine the resonance structures around cavity layers, we calculated and plotted the electric field distribution of this structure at 1100 nm as shown in Fig. 1(c). As we described in the introduction, the spectra based on resonant structures are high sensitivity to thickness errors. It is clearly seen in Fig. 1(d) that the influence of errors is high at cavity layers.
Fig. 1. Optical layer thicknesses of 81-layers USDF design (a). Theoretical spectral characteristics of USDF design (b). Electric field distribution at 1100 nm (c). Layer sensitivity of USDF design (d).
The computational manufacturing experiments have been performed using the broad band monitoring simulation module of OptiLayer software [21]. In the process of computational manufacturing experiment, the simulation parameters are set according to the actual deposition situation. The mean deposition rates for Ta2O5 and SiO2 materials are 0.24nm/s and 0.18nm/s, respectively. The instability correlation times for these rates are 3 seconds with standard deviations of 0.012nm/s and 0.018 nm/sec, respectively. The monitoring range for BBOM measurements is 500-1050nm with 1320 evenly distributed wavelength points. Monitoring is performed in the transmittance mode with the normal incident and the BBOM data arrays are acquired every 3 seconds. The random errors of monitoring are set to be 3%. Figure 2 presents the results of a computational fabrication experiment. Relative errors in the thickness of computationally manufactured coating are shown in Fig. 2(a), and the transmission spectra of coating design and simulation are shown in Fig. 2(b). It can be seen that the simulated spectra of computationally manufactured coating are in high agreement with the design spectra, especially in the 1000-1100 nm band we care about, which is definitely an exciting result. From the investigation results of previous publications [17], it is reasonable to assume that USDF has a strong error self-compensation effect under BBOM conditions.
Fig. 2. Correlated errors in the thicknesses of computationally manufactured experiment (a). Theoretical and simulate transmittance spectra (b).
The existence of an error self-compensation effect is investigated by comparing the effects caused by correlated and uncorrelated thickness errors. It is well known [10] that all optical monitoring methods cause correlation of layer thickness errors. Uncorrelated thickness errors are typical for time monitoring and quartz crystal monitoring, where thickness errors in subsequent layers can be considered independent of errors in previous layers. In order to demonstrate the existence of strong error self-compensation effect, we used the uncorrelated thickness errors with the same error level, as shown in Fig. 2(a) for the computational production experiments. Figure 3 shows the characteristic spectrum when the thickness errors is uncorrelated. These errors are generated as normally distributed, and the mathematical expectation and standard deviations are equal to modulus of correlation errors in Fig. 2(a). It shows that the spectral properties of the USDF are totally destroyed in the case of uncorrelated errors.
Fig. 3. Results of the statistical error analysis with uncorrelated thickness errors of the same level, as shown in Fig. 2(a). Red curve is the mathematical expectation of the USDF transmittance, and black curves show the corridor of standard deviations.
To further evaluate the strength of error self-compensation, we need to statistically analyze a large number of correlated and uncorrelated error vectors. By using the broad band monitoring simulation in OptiLayer software, a set of correlated thickness error vectors can be obtained from one calculation and manufacturing experiment, which would take several minutes, whereas it would obviously take a lot of time to obtain thousands of sets of correlated thickness error vectors. Therefore, we use a special simulator [20] that can generate a large number of correlated error vectors in a reasonable amount of time. The uncorrelated error vectors can be obtained by a random value generator.
Following Ref. [19], the strength of the error self-compensation effect is estimated by comparing the influence of correlated and uncorrelated thickness errors on the merit function. In the case of the USDF with BBOM in the transmittance mode, this merit function has the form
To numerically characterize the strength of the error self-compensation effect, the error self-compensation coefficient was introduced in [19]. A detailed description of its definition can be found in this link. Here we present only the final formula for this coefficient. Let $\delta MF(\Delta )$ denote the effect of the correlated error vector on the merit function. Let us denote by $E({\delta MF} )$ the mathematical expectation of the change in the merit function corresponding to the generated set of uncorrelated thickness errors. The formula of the error self-compensation coefficient c can be expressed as
It can be understood that the error self-compensation effect is present if c < 1 and absent if c > 1. In other words, at smaller values of c, the error self-compensation is stronger.
Figure 4 represents the histogram of the coefficient c distribution of USDF, which is calculated using 1,000 vectors of correlated and uncorrelated errors, and gets the mean value of the coefficient c to be 0.1141. The results show that the correlated thickness errors of USDF design have strong error self-compensation effects.
3. Coating process and manufacturing results
We produced designed coatings using Veeco ion beam sputtering (IBS) Spector-HT plant. The deposition system is equipped with a 16 cm ICP RF ion source and a 12 cm ICP RF assist ion source. During the IBS deposition process, the parameters of the 16 cm ion source are as follows: beam voltage 1350 V, beam current 900 mA and accelerator voltage 600 V. The parameters of the 12 cm ion source are as follows: beam voltage 400 V, beam current 200 mA and accelerator voltage 250 V. Silicon dioxide and Tantalum targets of 99.99% purity were used for the SiO2 and Ta2O5 films respectively. The deposition mean rates for Ta2O5 and SiO2 materials are 0.24 nm/s and 0.18 nm/s, respectively. The systems were equipped with a 2 planet 425 mm diameter planetary fixture. The systems were equipped with a broadband optical monitoring system (Veeco Instruments Quest OMS) using intermittent transmission monitoring through the center of one of the planets. The sun gear rotation speed was 25 rpm. In IBS plant shadow masks are used to achieve uniformity. The direct BBOM is set to measure the transmittance data of 1320 evenly distributed wavelength points in the wavelength spectrum of 500-1050 nm. We performed two deposited runs of USDF design, and the samples for both coating are deposited on Corning 7890 substrates (φ50 mm). The details of the coating design have been described in Section 2.
In the first run, we use a direct BBOM strategy to control the thickness of the 81-layers USDF. The designed and experimental transmittance spectra of USDF is plotted in Fig. 5. Transmittances are measured using an Agilent Cary7000 spectrophotometer. Figure 5(a) is the transmittance spectra in the wavelength region from 500 to 1200 nm with 1 nm step. Figure 5(b) presents measured average light transmittance of the deposited sample at 15 deg of light incidence. We have performed a high accuracy test on the target spectrum. The test range is 1060-1120 nm, and the test step is 0.1 nm. The prepared USDF exhibits prefect spectral efficiency with a high transmittance of greater than 95.01% at 1080nm and a high reflectance of greater than 99.52%. Respective values for the theoretical design are 96.3% and 99.91%. The fabrication result is in good agreement with the theoretical design. The results indicate that a strong error self-compensation effect is observed when USDF is produced using direct BBOM.
Fig. 5. Experimental and designed spectra of USDF for run1 (a). High accuracy measurement of target spectra (b).
In the second run, we used a hybrid strategy of direct BBOM and time monitoring to fabricate USDF, which enabled deposition experiments with uncorrelated partial layer thickness errors. The monitoring process is as follow: the thicknesses of the first 40 layers are controlled by direct BBOM, and the thicknesses of the layers from the 41st to 81st are controlled by time monitoring. At the beginning of deposition, the deposition rate of the material is very unstable, which causes significant thickness errors. In order to avoid excessive thickness errors caused by deposition rate fluctuation, the direct BBOM is used for the first 40 layers, then the time monitoring parameters are calculated from the 26-31 layers. By using time monitoring after 41st layer, the IBS process can provide a very stable deposition rates, so we can suppose that the uncorrelated thickness errors deposition is performed from the 41st layer to the 81st layer.
The designed and experimental transmittance spectra of USDF are shown in Fig. 6. Figure 6(a) is the normal incidence transmittance spectra in the wavelength region from 500 to 1200 nm with 1 nm step. Figure 6(b) presents measured average light transmittance of the deposited sample at 15 deg of light incidence. As shown in Fig. 6(b), the experimental results have an obvious deviation with the designed spectra, and the transmission at the wavelength of around 1080nm is decreased significantly.
Fig. 6. Experimental and simulated spectra of USDF for run2 (a). High accuracy measurement of target spectra (b).
In order to compare the real thickness error vectors in layers of the two deposition runs, we performed reverse engineering of the produced coatings, which is done in the OptiRE module of the OptiLayer commercial software [21]. We have transmittance data recorded after the deposition of each layer by the BBOM device. These data help us to minimize the uncertainty of thickness errors obtained by reverse engineering. In Fig. 7, we present the model spectra of inversion and the measure data in the working band, from which we can observe the good fitting of the model data to the measure data. Small differences in ripples are acceptable. Because for sensitive layers, even if the thickness relative error of the inversion is less than 0.5%, it may lead to ripples differences.
The investigation results of the relative thickness errors for the two deposition runs are shown in Fig. 8. Review monitoring strategies of the two deposition runs. In run 1, direct BBOM is used for all layers to control the deposition thickness, where the thickness errors are correlated. In Fig. 8, the relative thickness errors of run1 are complementarily distributed, especially after coating several tens of layers. It shows that the deposition error increases significantly after 75 layers, which is due to the cumulative effect of the error. With the thickness errors at such a level, the spectral properties of USDF should be destroyed. However, spectral properties of this sample are in fact very good. This is because of the positive effect of the error self-compensation effect. In run 2, the first 40 layers are controlled by direct BBOM, and the 41st to 81st layers are monitored by time. We observe that the thickness error vectors for the first 40 layers of the two depositions are very close, it indicates high reproducibility and a stable process. The uncorrelated thickness errors are shown for layers 41 to 81 of run 2 in Fig. 8. Based on the high stability of ion beam sputtering deposition, the thickness error is controlled within 2%. Overall, the relative thickness errors of run 2 are smaller than run 1, but eventually the spectral properties of the USDF are destroyed.
The presence of the error self-compensation effect is clearly demonstrated by the comparison of thickness errors correlation with uncorrelation deposition experimental results. When using direct BBOM to fabrication USDF in run 1, the positive effect of the thickness errors correlation causes the strong error self-compensation effect, and it only observes the small degradations of the USDF spectral characteristics. When using the hybrid monitoring strategy to coating USDF in run 2, the errors caused by time monitoring can be regarded as uncorrelated thickness errors. The uncorrelated thickness errors cannot provide an effective error self-compensation effect. It is obvious that spectral properties of the USDF are destroyed in the case of uncorrelated errors.
4. Estimate for the strength of error correlation under direct broad band optical monitoring
The work in Sections 2 and 3 has shown that the USDF has a strong error self-compensation effect under the condition of direct BBOM, which is related to the thickness error correlation caused by direct monitoring. Therefore, characterizing the degree of errors correlation is helpful to more adequately study the error self-compensation mechanism. Mathematical investigation of the thickness errors correlation was performed in [17–19]. These publications illustrated that the estimation of thickness errors correlation strength is related to direct optical monitoring strategies and coating designs. In [18], a geometric approach for investigating the correlation of thickness errors was proposed. By exploring the structure and shape of the cloud, the correlation strength of the errors is evaluated. The simulator of coating computational manufacturing experiment is described in [20]. In this section, we use the same simulator to perform the computational manufacturing experiment for USDF.
The strength of correlation of thickness errors is studied by statistical analysis of a large number of thickness error vectors obtained in the coating computational manufacturing experiments. For clarification, all coating layers are monitored by a single monitoring chip. Denote $\Delta = {\{{\delta {d_1}, \ldots ,\delta {d_m}} \}^T}$ be the vector of thickness errors in the coating layers. Let the total number of layers be equal to m. The large number of error vectors form a cloud in m-dimensional space. This cloud is described by a multidimensional Gaussian distribution function [19] whose level surfaces are multidimensional ellipsoids in this space. The orientations of the multidimensional ellipsoids are determined by the matrix $\mu $. $\mu $ is $m\; by\; m$ matrix. We can write down the elements of the matrix $\mu $ as
In this equation ${\sigma _{av}}$ is
which is equal to the average error norm divided by $m.$The histograms are given in Fig. 9, which shows the distribution of the norm of the error vectors obtained in these manufacturing experiments. The mean square error vector norm value is 3.3167nm.
Fig. 9. Statistical analysis of the error vectors obtained in the computational manufacturing experiments with direct broad band optical monitoring: Error vectors norms distribution of USDF.
From the proposed geometric approach to assess the errors correlation of the coating, it can be seen that, the lager the correlation coefficient $\beta $ is, the more elongated the ellipsoid formed by the error vector in m-dimensional space, which also means the stronger the errors correlation. Conversely, when the correlation coefficient $\beta $ is close to 1, the error vectors form a near-sphere in m-dimensional space and the thickness errors are uncorrelated. The calculation of the correlation coefficient $\beta $ of USDF is based on the results of 1,000 computational manufacturing experiments, and the correlation coefficient $\beta $ in the case of direct BBOM is 1.8796. This is not a big value compared with the correlation coefficient $\beta $ of non-polarizing edge filter in [18]. The results of these experiments are summarized in Table 2.
Table 2. Minimum and maximum standard deviations along the $\sigma_{min}$, $\sigma_{max}$, $\sigma_{av}$ and coefficients $\beta$ for the correlated and uncorrelated errors
The third column of Table 2 are related to uncorrelated errors. These values are calculated for additive random thickness errors and generated as normally distributed errors with the same normal distribution parameters for all layers of the design. The values of ${\sigma _{min}}$, ${\sigma _{max}}$, and ${\sigma _{av}}$, in the uncorrelated error columns are closer to each other, and as expected, the coefficient $\beta $ values are closer to 1.
In the previous research on the error self-compensation effect, it is generally believed that the strong thickness errors correlation could cause a strong error self-compensation effect. However, from the investigation results of this paper, it is found that the error correlation is not very big for USDF design, while the error self-compensation effect is very strong in case. The reason for this phenomenon can be explained that the dichroic filter designed in this paper is based on the resonant cavity structure, just like the Fabry-Perot cavity. Exploring sensitivity of spectral variations with respect to thickness errors, it is found that the distribution of the sensitive layers is very characteristic, the most sensitive layers are always distributed near the cavity layer, and extremely low error sensitivity is far from the cavity layer. The relatively low sensitivity of these layers results in a low error correlation since the error correlation is related to the spectral variation for thickness errors. For the highly sensitive cavity layer, the error correlation effect is still strong, so such a design has a strong error self-compensation effect in the production of direct BBOM.
5. Conclusion
In this work, the feasibility of using direct BBOM control in the process of the optical coatings based on resonator structure is studied in detail. Based on the previously developed simulator, we demonstrate that the USDF has a strong error self-compensation effect under the condition of direct BBOM. Moreover, the positive effect of thickness errors correlation in the error self-compensation effect is proved by the comparison of thickness errors correlation and uncorrelation deposition experiments. Based on the previously proposed geometric approach, the errors correlation effect strength of USDF in direct BBOM is analyzed. The errors correlation effect strength evaluated with this approach is not very big, and the explanation of this phenomenon is due to the unique sensitivity layers distribution of the resonator structure. It illustrates the limitations of this evaluation approach, and requires further serious investigations to improve. The prepared USDF under the case of direct BBOM exhibits perfect spectral efficiency, but when you carefully examine Fig. 5(b) you will find discrepancies between the theoretical and actual passband ripples. It is reasonable to assume that a design with a stronger error self-compensation effect could reduce the influence of deposition errors on spectral properties. The better designs are possible through computational manufacturing and multilayer design collaborative optimization. The simulation and experimental results demonstrate the excellent performance of direct BBOM in the fabrication of USDF. The successful production of USDF was demonstrated in the IBS system, but it is reasonable to conjecture that this fabrication approach is equally applicable to other deposition techniques under the same direct BBOM conditions, such as magnetron sputtering technique and ion assisted deposition technique. The research approach presented in this paper has practical application value and it is aimed at providing strong arguments for depositing the most complicated and technically challenging multilayer coatings.
Funding
National Natural Science Foundation of China (61621001, 61925504, 62061136008); Innovation Program of Shanghai Municipal Education Commission (2017-01-07-00-07-E00063); Russian Science Foundation (Grant 21-11-00011).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data underlying the results presented in this paper can be obtained from the authors upon reasonable request.
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