2.1.

Design Approach

The designs consist of a set of absorptive-interferometric AR coatings that have low reflectivity over SRM or HabEx bands. Each design emphasizes a different combination of optimal wavelength, bandwidth, angle of incidence, and polarization properties. Several of the designs were not intended for a particular mission band, but rather were to study the effects of different reflectances and coating thicknesses on the resulting edge scatter. In a space mission, out-of-band reflectivity is limited by dichroic filters in the cameras; we are only concerned about in-band performance.⁴ The goal of this shotgun approach was to find a coating that best compensated for the combined diffraction and reflection from the terminal edge.

We arrived at these designs using standard thin-film design software, assuming a flat surface, a given angle of incidence, and unpolarized light. This software is blind to diffraction and curved-edge reflection, yet it still resulted in coatings that greatly reduced the overall glint. Improvements to the designs accounting for edge physics with the FDTD analysis are presented in Sec. 5.

Table 2 lists the 21 coatings that we designed and analyzed. The naming scheme X:Y-Z represents the low:high ends of the targeted waveband and indexing for variations in bandpass weighting and polarization uniformity. Additionally, where it was a design parameter, a specific angle of incidence is listed. Coatings which were manufactured and tested are indicated, with results presented in Sec. 4.1.

Table 2

Coating characteristics and results for the assortment of coatings designed and evaluated. Reflectance and scatter results are averaged over their respective mission bands and simulated using both the LoS model and the uniform coating model, with the best performing member of each column emphasized in bold. The reflectance and scatter results are also presented for an uncoated AM edge, providing a starting point for each coating.

2.2.

Reflectance and Scatter Calculation

The reflectances listed in columns 2 and 6 of Table 2 are the average values for a uniform coating on an infinite flat substrate integrated over a range of angles and wavelengths as follows: the reflectance of each coating is averaged over the mission’s waveband and Sun angle range, specified in Table 1. The Sun angle $\varphi$, measured from the starshade normal, is shown in Figs. 1(a) and 4(b). For HabEx, with a single continuous band, the coating reflectance is averaged from 300 to 1000 nm and $40\mathrm{deg}<\varphi <85\mathrm{deg}$. For SRM, with two bands operated at different distances, the average is first performed individually over the blue and green bands, and the two bands are then weighted by the inverse square of their operating distance. Thus, the green band is weighted by a factor of 2.1 relative to the blue band.

The reflectances calculated on a flat surface serve as a heuristic for coating performance on the curved terminal edge. Generally, the darker the coating is, the better the overall suppression of edge-scattered light is. But, as will be shown, a significant performance improvement is seen when the layer thicknesses are adjusted to compensate for the combined diffraction and reflection at the curved terminal edge.

The scatter values in Table 2 represent the average of the incident intensity of the combined s- and p- polarized states, per meter of edge, scaled to an observation distance of 1 m, and weighted across the bands, for a specular edge orientation. To calculate the apparent magnitude of a 1 m long specular edge as it would appear to the Roman Space Telescope, we use a representative edge fractional intensity of $2\times {10}^{-9}$, a distance of 26 Mm, and account for the angular diameter of the Sun which spreads the light by 0.01 radians along the length of the edge, resulting in an observed fractional intensity of $2\times {10}^{-9}/0.01/{(26\times {10}^6)}^2=3\times {10}^{-22}$ of the incident sunlight on the edge. This is an attenuation of 53.8 magnitudes which given the Sun's visual magnitude of $-26.7$ would lead to an integrated magnitude for the edge of $V=27.1$.

2.3.

Manufacturing Approach

ZeCoat’s standard motion-controlled coating system makes flat coatings by moving an evaporation source down the radius of the coating area, while the substrate is rotated above the source. To coat the edge of a blade, which can be considered a two-dimensional surface, the approach was to move the source linearly past the substrate, rather than rotating the substrate, as shown in Fig. 3.

Fig. 3

Motion-controlled coating system employed at ZeCoat, Inc.

A quartz-crystal microbalance tracks the amount of mass applied to the substrate as a function of position on the coated substrate. The translation speed is adjusted with a closed-loop feedback system based on changes in coating flux leaving the evaporation source. Noise from the crystal sensor is filtered with software. The entire coating plume is moved completely past the blade (to account for plume nonuniformities) so that all points on the blade’s edge receive the same volume of coating. Edges were held at angles, $\alpha$, between 30 deg and 60 deg, for this study. Mass tracking (as opposed to the industry standard, “optical monitoring,” which uses interference to determine coating thickness) is particularly useful for precisely coating very large substrates greater than 1 m in diameter or length. This process achieves a coating uniformity of $\pm 2\%$ over 2-m diameters.

2.4.

Environmental Tests

We performed tape adhesion, abrasion, 24-h humidity, and thermal cycling tests to evaluate robustness on coating 450:1000-57.5°-2. This coating is representative of the others with the same manufacturing process and materials, with some differences in the number of layers and thicknesses. The coating was tested for adhesion to the substrate by applying 3M Magic 810™ tape to the surface and slowly removing the tape by pulling at 90 deg. The coating tightly adhered to the AM foil surface and showed no loss of adhesion.

The smooth coatings are also hard and durable. The coating was tested per the Mil-PRF-13830B paragraph B.4.4.5 moderate abrasion test and showed no damage or particulate contamination.

The coating passed a 24-h humidity test per Mil-PRF-13830B paragraph B.4.4.7, which calls for $>95\%$ humidity and a temperature of $120\pm 4°\mathrm{F}$. The certified testing was performed at Environmental Associates, Inc., California.

Thermal cycling tests were performed at ZeCoat and Environmental Associates, Inc. In ZeCoat’s test, we rapidly heated a coated sample by placing it in a preheated oven at 170°C for 1 h. The sample was then removed and cooled on the lab bench back to room temperature. The coating was tape-tested after cooling, with standard cellophane tape and a very aggressive tape that far exceeds forces generated with cellophane tape, and showed no loss of adhesion and no physical changes. The same coated foil sample was then wrapped in bubble wrap with a thermocouple, and both were heat-sealed into a plastic bag. The bag was placed in an insulated box (a cooler) filled with frozen carbon dioxide and rapidly cooled to $-75°\mathrm{C}$. Afterward, the sample was removed, warmed to room temperature, and tape-tested with aggressive tape. The coating showed no loss of adhesion and no physical changes.

A second, relatively benign certified test, comprising three cycles from $-20°\mathrm{C}$ to $+50°\mathrm{C}$, was administered at Environmental Associates, Inc. and resulted in no apparent damage.

3.1.

LoS Model

The AM substrate is constructed using a 150-nm radius circle as the terminal edge and a 300-nm wide rectangle making up the body of the substrate. Each coating layer is then added to the substrate by copying the substrate geometry and shifting it by the layer thickness in the direction of the coating LoS. Where the coating is curved along the top edge, it is necessary to add a rectangle, rotated by $\alpha$, to smoothly bridge from the current layer to the underlying layer, imitating the process of building up the coating along the LoS. The outermost layer with the highest mesh order is created first, then the next layer inward is added, overlapping and overwriting the material within, followed by each subsequent layer, and ending with the substrate core.

3.2.

Uniform Model

This model consists of a nested set of circular edges with rectangular bodies with widths that match the diameters of the edges. The circular edges are concentric at the origin with bodies that extend down to the bottom of the simulation region.

3.3.

Boundary Conditions

The simulation region consists of a $32\times 32\mu {\mathrm{m}}^2$ area using perfectly matched layers (PML) at each boundary. This size is determined by the detector range (Sec. 3.6), allowing the scatter to be analyzed at any angle. The PML begins at the outside edge of the simulation region and uses the steep angle profile with 12 layers, ensuring that obliquely scattered light is sufficiently absorbed at the boundary and does not cause extraneous reflections. Other settings, such as the parameters that fine-tune the PML properties, were left at their default values.

3.4.

Mesh Settings

Unless otherwise noted, all results were simulated using an auto-nonuniform mesh with a mesh accuracy setting of one, corresponding to a minimum sampling of six points/wavelength. To capture the finer detail present in the layered edge, a mesh override region that provided a finer uniform mesh spacing of 2 nm was placed around the tip. To reduce simulation run time, this tip mesh extended only past the outermost layer, rounded to the nearest tenth micron. The auto-nonuniform mesh increases the sampling near the border of the mesh override region. This serves to smooth out the transition from the relatively low-resolution background mesh to the high-resolution region around the edge tip.

A convergence test with varying tip mesh spacings was performed on coating 450:800-2 to verify the accuracy of the simulation results. We compared the scatter averaged over SRM bands and solar angles and summarize the results in Table 3. The scatter results are normalized to the highest accuracy setting of 0.5 nm. These results show that scatter results obtained using a 2-nm mesh differ by 1.3% from the highest accuracy setting. Results reported below use a 2-nm mesh unless otherwise stated.

Table 3

Convergence test results.

3.5.

Source Settings

The source consists of a Gaussian beam with a $5\text{-}\mu \mathrm{m}$ waist radius. The bandpass spans the wavelengths 300 to 1200 nm. The beam is generated using the scalar approximation setting and is placed $5\mu \mathrm{m}$ away from the edge, with the beam waist focused on the origin, coinciding with the edge’s center of curvature. The $5\text{-}\mu \mathrm{m}$ waist radius is wide enough to ensure that the tip of the edge receives spatially uniform incident light and that the beam does not significantly diverge between the edge tip and the detectors, while also being narrow enough to allow the source to taper off sufficiently before encountering the top and bottom edges of the simulation region, preventing stray reflections from the PML region.

The polarization angle of the source is set 45 deg to the edge in the $yz$-plane to generate equal light in the s- and p-polarizations. In this configuration, each polarization can be extracted and analyzed separately by examining the separate $x\text{-}/y\text{-}$, or $z$-components of the scattered field. The $z$-component of the electric field makes up the s-polarized light, and the $x$- and $y$-components make up the p-polarized light. Examining the separate polarizations allows us to analyze the polarization-dependent effects of different designs.

3.6.

Detector Settings

The detectors consist of field and power point monitors placed along a $15\text{-}\mu \mathrm{m}$ radius ring centered on the origin and spaced evenly every quarter degree. The detectors span the full 360 deg around the edge. The detectors collect the field profile in the frequency domain and report it at 19 points spaced every 50 nm, spanning the 300- to 1200-nm wavelength range. Collecting the field, rather than the intensity, allows for the incident pulse to be coherently subtracted from the total scattered field.

The detectors must be placed far enough away to measure the far-field behavior, but close enough to limit simulation run times to a reasonable level. Table 3 summarizes the results of a convergence test comparing the average scattered signal at a range of distances from a 150 nm radius uncoated AM edge. Each result was scaled linearly to $20\mu \mathrm{m}$ and normalized to the scatter at that distance. Table 3 indicates that the signal stabilizes at a distance of $\sim 10\mu \mathrm{m}$ and changes by just 0.5% beyond $15\mu \mathrm{m}$. All results presented in this paper were computed at a range of $15\mu \mathrm{m}$, where run times were reasonable, typically $\sim 10$ minutes per coating on a modern laptop computer.

4.1.

Comparison with Experimental Data

Preliminary scatter data were gathered experimentally using the single-angle scatterometer (SAS) at JPL. Details of the design and calibration of the SAS are given in Shaklan et al.,⁹ with a brief description given here. Figure 6 shows the setup of the instrument. A linearly polarized laser illuminates the edge of a test coupon from below and scatters light into a high-resolution camera. The camera gathers s-polarized, 635-nm light with a long working distance $10\times$ microscope objective with a 30 deg acceptance angle that is held at 60 deg relative to the plane of the test coupon. The instrument forms an image of a 1-mm length of the edge on the camera. The instrument software centers and refocuses the camera on the edge at each point before taking an image. The pixel values in the resulting image are summed and normalized by the exposure time and the incident laser power. To determine the absolute level of scatter, the data are compared with a reference coupon with an absolute scatter ratio that was measured with another instrument, the MAS.⁹ The SAS then moves along the length of the test coupon to the next point and repeats the alignment and measurement process, eventually measuring the scatter performance of the test coupon along its entire length.

Fig. 6

(a) Schematic of the single-angle scatterometer and (b) picture of the assembled instrument.

Figure 7 shows the results for four manufactured coatings, each applied to three coupons at different coating angles, as well as a representative uncoated coupon, labeled B27 for its manufacturing lot and sequence. Table 4 summarizes the median performance of each edge. Both Fig. 7 and Table 4 include the average value of the median performance of 13 representative uncoated coupons comanufactured with the ones presented here. We also show the Sommerfeld diffraction limit¹² for a semi-infinite, perfectly conducting, infinitesimally thick half-plane. The uncoated edges scatter light at close to this value, while the coated edges scatter much less light. The measurement errors in Table 4 include the quadrature addition of a 1σ value of $\pm 16\%$ determined from instrument calibration and coupon statistics reported in our previous work⁹ and a $\pm 30\%$ estimate of the standard error of the median. The standard error of the median was determined by measuring a 500-mm long -coated edge (Fig. 8) and computing the standard deviation of the median of 10 adjacent 50-mm long subsamples, each representing a potential coupon. The final estimated measurement error for the individual coupons was then 34%.

Fig. 7

SAS measurements of coated coupons and a representative uncoated coupon serving as a reference point. The average performance of 13 coupons, as well as the value for Sommerfeld half-plane diffraction, is marked.

Table 4

Summary of SAS measurements (s-polarization) of preliminary coated edges as well as a representative uncoated edge, which serves as a baseline to gauge performance improvement.

Fig. 8

Before and after SAS measurements of a 50 cm AM edge segment that was coated with 450:800-2 at an angle of 45 deg. Also included in this plot is the representative uncoated coupon B27, as well as the coupon B33, which was coated in the same manner as the segment.

The variability of the scatter along the 50-mm length of coupon B27 is typical of all uncoated coupons that we tested. The relative variability in the coated edges is higher. From the SAS images, each of which covers 1 mm of length along the edge, we find that the bright spots, visible in Fig. 7, on coupons B43 (0 to 10 mm) and B40 (33 to 43 mm) are both from physical damage to the substrate. The root cause of this damage is unknown, although it may have happened during installation in the mount. Other defects such as those on coupon B41 (13 to 18 mm) and coupon B42 (30 to 40 mm) do not exhibit obvious signs of damage. Instead, these are likely to be the result of dust or fibers landing on the edge prior to coating when the edges were handled in standard laboratory environments. Since performing these tests, the coating chamber has been relocated to a new cleanroom and we have developed new safe handling procedures, which should reduce the number of defects and improve scatter performance along the coupon edges and, ultimately, the entire starshade optical edge.

Along with the 12 coated 50-mm coupons, we also measured a 50-cm AM edge segment before and after coating with 450:800-2 at 45 deg (Fig. 8). The results for coupons B27 (uncoated) and B33 (450:800-2 @ 45 deg) are also plotted for reference. The coating reduced measured scatter by nearly a factor of $8\times$ on average with a range of $1.5\times$ to $20\times$. There are several scatter features present in the uncoated scatter measurement that are still identifiable in the coated segment. However, the two largest spikes at 56 and 252 mm cannot be seen postcoating. These spikes are most likely due to debris on the segment, which was cleaned prior to coating.

We compare our experimental results with the model results in Fig. 9. These results validate the overall accuracy of the models and suggest that coating formulations and application angles could be optimized to further reduce scatter. Model error bars include the error associated with substrate radius uncertainty as well as the error associated with manufacturing tolerances and refractive index uncertainties. As evidenced in Fig. 2, the terminal edge radius of the AM substrate varies along its length; we estimate that the radius ranges between 100 and 200 nm, with a mean value of 150 nm. We used Lumerical to simulate a coating over this range of edge radii and found that the scatter changed by $\pm 25\%$.

Fig. 9

Comparison between the s-polarized SAS experimental data and Lumerical results for a wavelength of 635 nm for coatings (a) 450:800-2, (b) 450:800-3, (c) 450:1000-1, and (d) 450:1000-57.5°-2. Experimental error bars are $\pm 34\%$, and model error bars are $\pm 29\%$ as per Sec. 4.1.

The individual coating layers are manufactured to a tolerance that is specific to the material being evaporated. All materials used here can currently be manufactured to within $\pm 1.5\mathrm{nm}$ $1\sigma$. Errors in the refractive index data of 3% (also material-dependent, measured by ellipsometry on several sample coatings) are included as a first-order effect on the layer thicknesses. To evaluate the effect of refractive index and layer thickness errors on the scatter, we evaluated a set of 25 simulated coatings in a Monte-Carlo analysis, resulting in a $\pm 15\%$ scatter standard deviation. Combining in quadrature the scatter deviation due to index and thickness errors with the deviation due to radius uncertainty, we find total model uncertainty of $\pm 29\%$. These error bars are plotted in Fig. 9.

4.2.

LoS Coating Study

We studied the effect of coating at different angles using coating 450:800-1, which had the best performance in our initial study. Figure 10 shows the scatter results for both SRM and HabEx. A shallow minimum is achieved near $\alpha =60\mathrm{deg}$. This minimum roughly coincides with the half-angle between the incident light direction and the scattered light direction, $40\mathrm{deg}<\varphi <85\mathrm{deg}$. The scatter is maximized at $\alpha \sim 150\mathrm{deg}$, with the coating on the sunlit side of the edge. At this angle, the sunlight reflects directly from the coating into the telescope, and the edge glint is brighter than the uncoated substrate. From these results, a coating angle of 60 deg was chosen to analyze the set of coatings presented in Table 2.

Fig. 10

Plot of the average in-band scatter as a function of coating angle. Note the minimum around $\alpha =60\mathrm{deg}$ and the maximum near $\alpha =150\mathrm{deg}$. The average in-band scatter from an uncoated substrate is also marked for each mission.

Compared with the scatter from an uncoated 150-nm radius AM substrate, all coating formulations provide a significant reduction, with the best result for SRM coming from 450:800-1, which reduced edge scatter by a factor of 25. The best result for the HabEx mission was 450:800-60°-1, reducing the edge scatter by a factor of 8.

The relationship between reflectance and scatter, as a function of wavelength and Sun angle, is shown in Fig. 11 for one particular coating, while the angle and wavelength-averaged reflectance is plotted for all of the coatings in Fig. 12. For the LoS coating, Fig. 11(b) shows a weak correlation between reflectance and scatter through the band and over the range of Sun angles. This is not surprising because the coating geometry differs strongly from the intended planar geometry.

Fig. 11

Surface plots showing (a) the unpolarized reflectance, (b) the LoS coating scatter, and (c) the uniform coating scatter for coating 450:800-2.

Fig. 12

Plot of the average reflectance versus scatter for the LoS model for both SRM and HabEx, showing a weak positive correlation between reflectance and scatter. Coatings that were manufactured and tested, as well as the starting and ending point for optimized SRM and HabEx designs, are labeled.

Likewise, Fig. 12 shows a weak positive correlation between reflectance and scatter for all coatings. For the purpose of identifying design principles that would afford an efficient coating optimization and tolerancing approach, we considered different reflectance calculations that might better correlate with the scatter. We separately assessed shifting the angles of incidence to account for the coating angle, treating the coating as being thinned by $\cos \alpha$, and calculating the reflectance at normal incidence because the telescope is normal to the starshade surface. None of these variations, however, improved the correlation for the LoS coating.

4.3.

Uniform Coating Results

We found that the correlation between reflectance and scatter over wavelength and Sun angle is strong for the uniform coating [Figs. 11(a) and 11(c)]. The correlation is worse at low angles, where the diffraction component is strongest, but for $\varphi >50\mathrm{deg}$, the correlation is clear. However, the correlation between average scatter and average reflectance for all coatings is weak. This is due to a key difference between the uniform and LoS coatings; the uniform coating surface area grows with coating thickness. When we plot the average scatter as a function of reflectivity and edge radius, where radius is the substrate radius plus the coating thickness, we find a clear linear relationship, with all of the scatter points falling on a plane (Fig. 13). The plane is a good fit, with correlation coefficient $r^2=0.957$. A similar planar fit is also found for the HabEx band.

Fig. 13

Scatter plots of the uniform SRM scatter versus reflectance versus coating thickness for the coatings listed in Table 2. A planar fit is achieved describing the average scatter as the linear combination of average reflectance and coating thickness.

The linear dependence on thickness and reflectivity was surprising as we had expected the scatter to be a function of the product of the reflectance and radius. While this must be the case in the geometric regime, the results here show that the combination of subwavelength structures and significant levels of diffraction leads to a linear behavior for the uniform coated case. The plane indicates that low scatter can be obtained from a coating that is both very thin and has very low average reflectivity. Typically, in AR coating design, these two properties are at odds with one another, and it has proven difficult to design a coating that lies in this low-scatter regime. While we find a good planar fit to the simulated coatings, we do not expect to be able to populate the thin/low-reflectivity region.

5.1.

Uniform Coating Optimization

Optimization of the uniform coating designs took advantage of the linear dependence of scatter on coating thickness and reflectance. Using reflectance as a proxy for scatter, we optimized scatter performance by adjusting the coating layer thicknesses using MatLab’s standard “fminsearch” function while calling a vectorized version of the MatLab “jreftran” routine¹⁵ to efficiently evaluate reflectance. Once an optimized design was determined, it was evaluated in Lumerical to accurately measure the level of scatter.

Starting with 450:800-2 for both missions, the average scatter for SRM was reduced by 14%. The improvement for HabEx was just 3%.

5.2.

Manufacturing Tolerances

The coating performance will be limited by manufacturing limitations. As mentioned in Sec. 2.3, the manufacturing process is capable of achieving a coating uniformity of $\pm 2\%$ over a 2-m length. The tolerance on individual layer thicknesses, as stated in Sec. 4.1, is currently $\pm 1.5\mathrm{nm}$, and in the future, with experience and refinements to the coating process, we expect to improve layer thickness tolerances to $\pm 1\mathrm{nm}$.

We performed Monte-Carlo simulations of the optimized coating designs to estimate the as-manufactured expected performance and its uncertainty, using the same process described in Sec. 4.1. Here, however, we do not account for refractive index uncertainty because the coating is assumed to be optimized for measured index data at the time of manufacture. For the LoS coating, which requires repeated Lumerical simulations, we ran 25 trials for each mission. For the uniform coating, the reflectivity-based scatter estimation function allowed a sample size of 10,000 trials. The performance of the toleranced coatings is presented in Table 5. All of the designs are expected to perform to within a few percent of the design point.

We note that the toleranced, optimized coatings are only modestly better than the best measured coatings. For SRM, we measured coating 450:1000-1, which has a predicted scatter level of $1.96\times {10}^{-9}$ (Table 2), compared with the optimized coating 450:800-1 with a predicted scatter value of $1.17\times {10}^{-9}$, a $1.7\times$ improvement. For HabEx, coating 450:1000-1 has a predicted scatter level of $5.06\times {10}^{-9}$ compared with the optimized coating 450:800-60°-1 with a predicted performance of $3.79\times {10}^{-9}$, a $1.3\times$ improvement. The laboratory measurements are consistent with performance that is within a factor of 1.7 of the best coating designs presented here.