Vol 17, No 12 (2017) / Wang

Preliminary study on broadband antireflection coatings for large aperture telescopes

Preliminary study on broadband antireflection coatings for large aperture telescopes

Wang Ye-Ru1, 2, 3, , Li Xin-Nan1, 2, Wang Jin-Feng1, 2, Tian Jie1, 2

National Astronomical Observatories/Nanjing Institute of Astronomical Optics & Technology, Chinese Academy of Sciences, Nanjing 210042, China
Key Laboratory of Astronomical Optics & Technology, Nanjing Institute of Astronomical Optics & Technology, Chinese Academy of Sciences, Nanjing 210042, China
University of Chinese Academy of Sciences, Beijing 100049, China

† Corresponding author. E-mail: yrwang@niaot.ac.cn

Abstract: Abstract

A broadband anti-reflective (AR) coating design for astronomical large-aperture telescopes is proposed. We give simulations of two-, three- and four-layer silica sol-gel on fused silica and finally get the optimal optical constants. As a comparison, we discuss the traditional dielectric material that has been applied to broadband AR coatings. To better guide the following experiment, we also conduct error analysis and feasibility analysis, combining with the technological characteristics of sol-gel. The analytical method is suitable for other wavebands and substrates. It is also instructive for large area AR coatings in the field of solar cells.

Keywords: instrumentation: miscellaneous;methods: numerical;techniques: spectroscopic

1 Introduction

It is more and more difficult to apply appropriate anti-reflective (AR) coatings onto transmission optical elements, the apertures of which are gradually increasing as large ground-based optical/infrared telescopes develop (Tokunaga 2014 ). Despite the fact that most large-scale telescopes adopt segmented mirror active optics and reflective optical structures, many large-aperture prisms, which could improve the light gathering power and collect more information from observed objects, are used in terminal equipment such as various spectrographs (Angeli et al. 2004 ). In addition, the hermetic windows of many telescopes need AR coatings to be applied. Unlike AR coatings that are used in solar cells and other devices which only focus on their transmission properties (Kanamori et al. 2002 ; Liu et al. 2012 ), rigorous surface precision is desired for the optical lens in a telescope, especially the correctors. Thus, we hope that the thickness of a coating would be as small as possible and the whole coating growth process could be carried out under lower temperature, which will have less influence on the associated surface precision. In addition, a good spectral reproduction ability is very important, which makes the subsequent spectral analysis more precise and easier. In view of the fact that most dielectric materials absorb in the ultraviolet wavelength region, the limited number of alternative materials in coating design yields an unsatisfactory spectral transmittance when they are applied to telescopes whose working waveband ranges from ultraviolet to near infrared.

At the beginning of this century, sol-gel coatings were firstly applied in large-aperture telescopes and an encouraging ultra-broadband AR result was achieved (Phillips et al. 2008 ). The Atmospheric Dispersion Corrector (ADC) for the Keck-I Cassegrain focus with fused silica prisms is about 1022 mm in diameter. This is a simple structure of silica sol-gel over an MgF2 layer, which was conventionally obtained by vacuum deposition, producing a well-formed coating from 310 nm to 1100 nm (Phillips et al. 2006 ). The relatively-low refractive index (RI) of silica sol-gel is perfectly matched with the fused prisms, and there is a smaller RI difference between the silica sol-gel and incident medium (i.e. air). The features above make it possible to achieve a good broadband transmittance with only two layers. On the contrary, many more layers should be applied if traditional dielectric materials are used. Therefore, we reduce the thickness and number of layers by employing sol-gel coatings, and moreover, the transmittance results become even better. In addition, the sol-gel process has many advantages compared to traditional vacuum dielectric films, such as a minor restriction in the substrate shape and size, better homogeneity, simple techniques and processing at a lower temperature. There are many larger optics in the Thirty Meter Telescope (TMT), which is under construction, and in the Large Optical/Infrared Telescope (LOT), which is planned in China (Zhang et al. 2016 ). Problems that exist in the process of dielectric film deposition, such as material limitations in ultraviolet region, film uniformity and process safety, would be perfectly eliminated on condition that the dielectric layer is replaced by silica sol-gel. An exciting prospect for applying sol-gel coatings in astronomical telescopes could be possible.

In this paper, this problem will be discussed in detail. We firstly carry out numerical simulations for sol-gel broadband AR coatings, focusing on the coating design, and we also hope to propose an ideal coating structure in theory and finally obtain some good results. These simulations are based on existing experimental facts rather than a theoretical possibility. Then, we conduct feasibility and error analyses combined with specific processes, and finally discard some solutions that are extremely difficult to implement. The results can give some basic guidelines for subsequent experiments.

2 Theory and Hypothesis

Reflection and transmission occur when light arrives at the interface of two different media, the RIs of which are n 0 and n 1, respectively (Born & Wolf 1959 ). The reflectivity of the normal angle of incidence is , supposing that the media are nonabsorbent. For common BK-7 glass, the RI of which is about 1.52, the reflectivity is as high as 4% from one side; in other words, the transmittance is only about 92%. An ideal homogeneous single-layer AR coating could achieve zero reflection at the reference wavelength while satisfying the following conditions: the optical thickness of the coating should be , where λ is the reference wavelength; and , where , and are the RIs of the air, coating and substrate, respectively. For a dielectric material, in order to achieve near zero reflection of the monolayer, the RI of the substrate, which is calculated under the above conditions, should be at least 1.9, and it is clear that most substrates with a low RI do not satisfy this condition. Due to its loose and porous characteristics, the RI of silica sol-gel is very low, which perfectly matches the low RI substrate, and thus zero reflectance at a special wavelength can be achieved. However, an effective near-zero reflectance is achieved only for a narrow spectral width around the reference wavelength because of the uniform RI throughout the whole coating. In the rest of the waveband, the reflectance will increase rapidly up to the bare substrate at half-wavelength.

Let us go back to the reflectance formula concerning light traveling between two media with different RI, . We can draw a conclusion that the smaller the difference in RI between adjacent layers is, the smaller the reflection loss is. It is conceivable that there will be nearly no reflection loss when the RI of the coating changes slowly from the substrate to air, that is to say, a sharp variation of RI between adjacent layers should be avoided. The ideal condition would be that the RI continuously decreases from the bottom to the top of the coating, which is the so-called gradient-index AR coating.

Since the 1960s, a great many of papers have focused on design and application of optical gradient-index coatings (Anders & Eichinger 1965 ; Southwell 1983 , 1985 ; Dobrowolski et al. 2002 ). There are many advantages besides perfect broadband and omnidirectional spectral performance, e.g. excellent mechanical properties (Rats et al. 1999 ), fewer scattering losses (Arnon 1977 ), less residual stress (Tang et al. 2011 ), enhanced resistance to laser damage (Ristau et al. 2008 ) and enhanced temperature stability (Tang et al. 2011 ). However, it is very difficult to manufacture gradient-index coatings that can be used in optical and infrared bands. Although some special process may be used to produce materials with continuous changes in RI (Li et al. 2010 ; Huang et al. 2007 ), the production process is complicated, which is not suitable for large scale fabrication, and the scope of application is limited. A multilayer coating can be viewed as a discrete approximation of the gradient-index AR coating, which is described as a quasi-gradient coating, so a more feasible method is to use several uniform sub-layers to approximate the inhomogeneous gradient-index coating. Until now, the oblique-angle deposition and chemical etch-leach process have been applied to successfully manufacture quasi-gradient coating (Xi et al. 2007 ; Liu et al. 2012 ), but they are not suitable for an astronomical lens. Through our analysis, three layers, or even two layers of sol-gel film, achieve excellent broadband AR effect. This method has excellent prospects for application in the field of broadband large-aperture telescopes due to its advantages in film uniformity, low-stress and suitability for large area plating. At the same time, sol-gel coatings can also be applied to the field of solar cells, so that the photoelectric conversion efficiency could be widely improved.

The sol-gel process is a kind of chemical coating method. The usual practice is to have organic metal alkoxide dissolved in ethanol or methanol and then to add water as a hydrolytic reagent, which is accelerated by catalysts. Here we use tetraethyl orthosilicate as a reactant and the overall reaction is shown as follows (Thomas 1986 , 1992 )

where is the value of the RI at the center of the band (i.e. 700 nm). Of course, you can also ignore the dispersion, which has little effect on spectral performance.

Fig. 1 The solid line represents measured data by an ellipsometer, and the dashed one is the fitted result.

With the assistance of Essential Macleod, we will start our simulations from the simplest two-layer film, to find the best combination of RI and film thickness and finally give the optimum solution. The analysis method can be applied to more layers and the target band can also be changed as needed, but the analysis method remains unchanged.

3 Numerical Simulation and Analysis

By means of the acid-base two-step method, the RI can be continuously adjusted from the alkali-catalyzed film (with n about 1.18) to the acid-catalyzed film (with n of 1.42 or so). Our basic idea is to discretize the RI with the sample interval and the results are as follows

For multilayer coatings, different RI values are assigned to each layer, and then we optimize thickness for each combination, the aim of which is to make the transmittance in each point as high as possible in the target band. The best coating structure would be sought out after comparing the optimal solutions of all RI combinations. Since the transmittance varies continuously with the RI and thickness, i.e., there is no abrupt change, our results could be accordingly considered as the optimal solution with a certain number of layers.

3.1 Two-layer System

Assuming that the RI of the outer layer is n 1, and the film thickness is d 1, similarly, n 2 and d 2 are the optical constants of the inner layer. By means of the control variable method, n 1 and n 2 are fixed respectively to investigate the impact of RI variation of the other layer on transmittance. The following results, unless particularly stated, are for single-sided transmittance.

Firstly, the RI of the inner layer is fixed, the structure of which is Sub/n 2/⋯/Air. The transmittance curves are shown in Figure 2. In general, the curves decline with an increase of RI for the outer layer in the visible region; whereas, in the ultraviolet and near infrared regions, the condition is complicated in that the curve is crossed, which is not serious. Therefore, it can be approximated that the smaller the outer layer RI is, the higher the transmittance is. Except in Figure 2a, each figure has a unimodal curve (red lines) that appears to be different from others, which results when the outer layer RI is greater than that of the inner layer. The simulation results show that the outer layer thickness is zero, that is, the two-layer system degenerates into a single layer, so the inequality must be implemented when applying two-layer AR coatings.

Fig. 2 Transmittance that changes with n 1 when n 2 is fixed. The curves go down with an increase of n 1 in the visible region, and the red lines represent conditions in which .

Then we fix n 1, that is, Sub/⋯/n 1/Air. In view of the above conclusions drawn when n 2 was fixed, several RI combinations that have a small value of n 1 were chosen to be discussed, with the restricted condition , which is illustrated in Figure 3. It is obvious that the transmittance curves rise with an increase of RI for the inner layer in the visible region, whereas, there is a converse variation in the ultraviolet and near infrared regions.

Fig. 3 Transmittance that changes with n 2 when n 1 is fixed. The curves go up with an increase of n 2 in the visible region, and the red lines represent the best results.

Among these different curves, we should choose the optimal one, which is dependent on the specific spectral requirements for transmittance, e.g., the requirement of high transmittance in a relatively narrow band, or that an average transmittance is as high as possible in the whole band, and there would be different optimization results with regard to various spectral performance requirements. Here, our optimization target is maximizing the lowest transmittance in the region from 300 nm to 1100 nm, that is,

According to the optimization target, we choose three best results for from Figure 3b, c and d, respectively, namely, 1.34/1.22, 1.31/1.19 and 1.28/1.16 (Fig. 4). Apparently, the combination of 1.28/1.16 is the best. In order to make the results more accurate, we have RIs in the range 1.25−1.31 discretized by ; in view of practical feasibility, the lowest RI can be set as 1.16. With the same optimization method, we get the best result, 1.29/1.16, as shown in Figure 5. The minimum transmittance is , which is the highest value we believe that could be theoretically achieved.

Fig. 4 Three best results from Fig. 3.
Fig. 5 The best result of the two-layer system, i.e. Sub/1.29/1.16/Air. A, B and C are three minimum points.

Figures 4 and 5 show curves in the shape of an “M” in the target waveband, which has a minimal value in the visible region, and two maximal values in the ultraviolet and near infrared regions. In the visible region, the curves meet the conditions and , but there is an opposite variation trend in the other wavebands, bounded by wavelengths at two maximal transmittances. In other words, the trend of the middle part of the “M” curve is opposite to that of the two sides. So, the process of seeking the optimal solution could be intuitively expressed as finding a curve that makes the minimum point equal to the edge transmittance on both sides of the band, that is, having the three points A, B and C on the same line but as far apart as possible (Fig. 5).

Since the sol-gel method is affected by many technological conditions it is impossible to precisely adjust the RI and film thickness during the actual fabrication. In the following part of this section, we will discuss the effect of small changes in the optical constant on the transmittance curve.

We take the best coating structure 1.29/1.16 for example. Several cases with are shown in Figure 6. The small error of the RI makes a transmittance variation of about 0.1% in the short wave region, and 0.05% in the infrared region, which are acceptable.

Fig. 6 Influence of minor error in RI on transmittance. The black line in the middle represents 1.19/1.16, the two red lines above share the same difference in RI of 0.14 and the two blue lines are 0.12.

It is well known that changes in the thickness of optical coating could lead to horizontal movement of the transmittance curve. We still choose the combination 1.29/1.16 to do our analysis, because the designed value of its physical thickness is and .

Figure 7 shows variation of transmittance with the film thickness error . The average transmittance of the whole band remains unchanged, but the curve shifts to the infrared region as the thickness increases. As a result, the main variation of occurs at the edges of the band, i.e., 300 nm and 1100 nm. There is a decline in transmittance of 0.3% at 300 nm when , yet only a decline of 0.1% at 1100 nm when . This is because the slope of the curve near 300 nm is larger than that near 1100 nm. As a consequence, there is a greater influence in the ultraviolet region, so we prefer the positive physical thickness errors supposing that the error exists. The inner layer shares a similar error rule with the outer layer. Another significant practical guiding conclusion is that the thickness error of a single layer (which is as small as possible) does not affect the transmission curve if we keep the total thickness of the two layers constant. In the actual process of applying the coating, therefore, we can test the thickness after one layer is applied, and then eliminate the thickness error by adjusting subsequent layers.

Fig. 7 Influence of film thickness error on transmittance. (red line) and (blue line).

Let us go back to Figure 6. The two clusters of curves on both sides of the optimal curve are almost coincident with no crossings, and similar features are also found in Figure 4 for a total of three clusters of curves. These three clusters of curves are characterized by a fixed RI difference between two layers. Then we draw a series of curves with an RI difference of 0.13, as shown in Figure 8. When and n 2 change by 0.03 simultaneously, . Compared with the results in Figure 6, we believe that the optimal RI difference between two layers is 0.13, and the more deviation there is from this value, the farther the transmittance deviates from the optimal solution. However, the synchronous RI errors from two layers will introduce less variation in transmittance than the individual error from one layer. Under the premise of maintaining , the lower the RI of the outer layer is, the higher the transmittance becomes. This conclusion is of great significance in guiding the actual process.

Fig. 8 A cluster of curves with an RI difference of 0.13. n 1 and n 2 change by 0.03 simultaneously.

3.2 Comparison with Vacuum Dielectric Film

Due to good performance of the two-layer sol-gel coatings, we believe a better result can be produced by increasing the number of layers with different RIs. However, before introducing a three-layer structure, we will make a comparison between the vacuum dielectric coating and the sol-gel coating.

Parameters used to construct a typical 8-layer structure with a substrate of fused silica are listed in Table 1, and the transmittance curve is shown in Figure 9. As we can see, the curve greatly decreases near 400 nm, which is caused by absorption from the TiO2 layer. If we extend the band to 300 nm, the absorption will be more severe (Fig. 9 inset). A number of conventional dielectric materials have absorption in the ultraviolet region, which makes the vacuum dielectric film greatly disadvantageous when applied to wideband AR coatings.

Fig. 9 The transmittance of single-side coated films in Table 1. The absorbance of quarter-wave TiO2 coatings from 300 nm to 500 nm is shown in the inset.
Layer Material RI Extinction coefficient Optical thickness (FWOT) Physical thickness (nm)
Medium Air 1 0
1 MgF2 1.38542 0 0.28857433 106.23
2 TiO2 2.34867 0.00037 0.09577107 20.8
3 SiO2 1.4618 0 0.0612797 21.38
4 TiO2 2.34867 0.00037 0.51796575 112.47
5 SiO2 1.4618 0 0.04651835 16.23
6 TiO2 2.34867 0.00037 0.1164711 25.29
7 SiO2 1.4618 0 0.12799445 44.66
8 TiO2 2.34867 0.00037 0.04589671 9.97
Substrate SiO2 1.4618 0
Total 1.30047145 357.02

Table 1 Parameters used to construct a typical 8-layer dielectric film stack with a substrate of fused silica. The reference wavelength is 510 nm.

Here, the TiO2 layers are replaced with HfO2 that has no absorption in the ultraviolet region, and we get a better result (Fig. 10). However, there are big challenges in coating homogeneity, coating stress and controlling the process as the number of layers increases. Therefore, it is almost the best result for the vacuum dielectric coating. However for the KECK telescope, a combination of silica sol-gel over dielectric MgF2 was used, which provides a better result than traditional vacuum dielectric multilayer coatings.

Fig. 10 The double-sided transmittance of 8-layer dielectric, KECK and sol-gel coatings as a function of wavelength from 300 nm to 1100 nm. The table in the inset lists statistical data on average and minimum transmittance.

Figure 10 shows the double-sided transmittance of a vacuum 8-layer coating, the KECK coating and sol-gel coating from 300 nm to 1100 nm, and the associated statistical data are listed in the inset of Figure 10. Sol-gel coating has the highest average and minimum transmittance in the whole band, followed by KECK, and the worst is the dielectric coating, but there is a certain advantage in the visible waveband of the KECK coating. The reason why the KECK AR coatings have a somewhat poor result is that the MgF2 layer has an RI of 1.37, far from the best n 2 (1.29). Therefore, it is possible to get a better result using the sol-gel two-layer structure than using the vacuum 8-layer or KECK coatings. The difficulty in achieving the theoretical value lies in the precise regulation of RI and thickness, which is the focus of the following experiment.

3.3 Three-layer System

The optimization methods are also applicable to three-layer structures. Firstly, we draw several curves to understand how transmittance changes with the RI of the three layers (Fig. 11), and then do a simple partition of it and give a summary of the variations in Table 2. From this rule, we can gradually adjust the curve for any combination of RIs until the target requirements are satisfied, that is, having the minimum points on the same horizontal line, as much as possible.

Fig. 11 Transmittance variation for different fixed RI of three-layer AR coatings.
Table 2 Variation rules for partition in Fig. 12

Figure 12 shows the optimized curve, which is 1.40/1.29/1.16, and . We also make a comparison between the two-layer and three-layer systems in Figure 13b. The two have approximately the same average transmittance (99.68% and 99.76% for the two-layer and three-layer cases, respectively), but the three-layer system possesses a better spectral reproduction ability and smaller fluctuations, with single-sided transmittance being more than 99.6% in the whole band.

Fig. 12 The optimized curve for a three-layer coating, which uses 1.40/1.29/1.16.
Fig. 13 (a) Comparison of three-layer coating with different n 1 which is available. (b) Optimum transmittance of 2-layer, 3-layer and 4-layer coatings.

In consideration of the experiences above, we may expect a better spectral performance with ever-increasing layers, however, the results are not notably better than the three-layer system after a large number of simulation tests. Moreover, difficulties in manufacturing increase rapidly with an increasing number of layers, so it is not practical. Let us go back to the simulations related to the two-layer and three-layer systems. The lowest RI is 1.16, and there is a big gap from the air, which limits the ultimate performance. In fact, the degree that transmittance performance might achieve is mainly determined by the outermost film, so we would expect a better result as is decreasing, as shown in Figure 13a.

Comparing the results of two-layer and three-layer structures, we found that the difference in RI from adjacent layers is about 0.11–0.13. Without considering the actual feasibility, the silica sol-gel on a fused silica substrate is no more than four layers. Figure 13b shows one of the results, with a minimum transmittance of 99.93% and an average transmittance of 99.98%, indicating an ideal approximately zero reflectance of no more than 0.07% in the whole band. This is a big gain in transmittance compared to the three-layer system. However, the most encouraging results stay in theory for now since the sol-gel film, with an RI less than 1.1, is difficult to produce, and by reducing the outermost RI, a three-layer film could also yield a perfectly acceptable result. A comparison of two, three and four-layer coating single-sided transmittance is shown in Figure 13b.

Besides perfect broadband AR performance, the graded-index multilayer coatings also have great potential in suppressing reflection for a wide range of angles of incidence. The relationship between transmittance and angle of incidence for the two-layer, three-layer and four-layer sol-gel is shown in Figure 14. The curves corresponding to two and three layers are close to each other, and , . For the four-layer structure, , .

Fig. 14 Transmittance of bare substrate, 2-layer, 3-layer and 4-layer as a function of angle of incidence.

4 Conclusions

In this paper, we describe the structural designs for ultra-wideband AR coatings used in large aperture astronomical telescopes, and obtain the optimal solution by numerical simulations. The error analysis and feasibility analysis are carried out according to the actual process of producing sol-gel which will provide abundant data that can support future experiments. In the end, we list several notes related to broadband multilayer coatings based on fused silica. The target band is 300–1100 nm.

The RI of the coating gradually decreases from the substrate to the air, that is, , where k is the number of sublayers.

The difference in RI between adjacent layers is about 0.11–1.13.

On the premise of note 2, the lower the RI is of the outermost layer, the better the performance.

Satisfying the above conditions, silica sol-gel coatings are applied with a maximum of four layers, and the optical thickness of each layer is about one quarter of the reference wavelength. More layers cannot achieve remarkably better results. Taking into account practical feasibility, two or three layers can achieve a perfect result with an average transmittance of 99.7% in the region from 300 nm to 1100 nm.

The method used for our analysis is suitable for other bands and substrates. It is also instructive for large area AR coatings in the field of solar cells. Sol-gel films, due to their good broadband anti-reflection capability (especially in the ultraviolet band), simple manufacturing process, suitability for large area coating, high cost-performance ratio, etc, are believed to have prospects for wide application in the field of large aperture astronomical telescopes.


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Cite this article: Wang Ye-Ru, Li Xin-Nan, Wang Jin-Feng, Tian Jie. Preliminary study on broadband antireflection coatings for large aperture telescopes. Res. Astron. Astrophys. 2017; 12:124.


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