Apodizing

Apodizing Screen Design Charts Optimized for Planetary Contrast

John Gibb

Revised January 2015

Foreword

These articles by Dick Suiter, author of Star Testing Astronomical Telescopes, are an excellent orientation to apodizing telescopes, covering both design and construction. They are the starting point of this article, to which this makes frequent reference:

http://home.digitalexp.com/~suiterhr/TM/apodize.htm.

This version is revised and improved in light of comments Dr. Suiter graciously provided on review of a previous version. Further improvement remains an ongoing project. Comments welcome at jagua703 (at) gmail (dot) com.

Further reading: http://telescope-optics.net/apodizing_mask.htm

Abstract (technical summary)

This article identifies and examines a type of apodization that maximizes modulation transfer at low special frequencies, encircled energy within the Airy pattern’s second minimum (instead of the first), and minimizes edge spread beyond the second minimum’s radius. (It is reportedly similar to a criterion by Hopkins; citation and findings to be reported when available.) It is considered to improve the contrast of planetary features, especially through larger-aperture telescopes. It is applied and optimized with rings of discrete transmissivity such as from mesh screens, and a given secondary obstruction size. Optimal solutions for any of these measures tend to be near-optimal of the others. This type of apodization makes trade-offs accepting central-disk and first-ring bloat for outer ring diminution, and against modulation transfer near maximum special frequency for increased transfer at low frequencies. An encouraging finding is that the edge spread and encircled energy distributions, compared to no apodization, improve significantly with no appreciable sacrifice within those distributions. Apodizations optimizing encircled energy within the first minimum achieve a lesser extent of these goals. (tentative finding; further study and reporting required.) As a practical aid for amateur astronomers to apply the studied apodization, design charts give mesh ring sizes optimized for a range of mesh transmissivities and secondary obstruction sizes.

Introduction

When a telescope or camera focuses a circular aperture to a point, light diffracts in the well-known Airy pattern consisting of a central disk and a series of concentric rings of diminishing intensity outward. The well-known Dawes’ and Rayleigh limits of double-star resolution arise from the size of the central disk of this diffraction pattern.

Diffraction has a more complex and subtle effect on planets through a telescope. Many planetary features and details, such as swirls on Jupiter and especially Saturn, and surface features of Mars, have inherently low contrast, unlike double-stars against the black background of space. The diffracted light from the rings smears across planetary details, reducing their contrast and sharpness, even if the telescope otherwise resolves them.

For the preservation of contrast where already subtle, the Airy disk size matters less than the distribution of light to the diffraction rings. It’s fairly well known that a large central obstruction worsens planetary contrast. Even though it shrinks the central disk, it spreads more light around to the rings.

Apodizing refers to a variety of schemes to shade, screen, shape, or vary phase within an aperture to manipulate the diffraction pattern. Most aim to suppress rings some way or another. An extensive scientific literature on apodizing focuses on continuously-varying shading, usually of unobstructed apertures, according to analytic mathematical functions (most notably the Gaussian distribution). Construction of these is impractical for most amateur astronomers. However, quite practical apodization uses a stack of mesh window screens yielding discrete transmission intensities within ring-shaped parts of the aperture.

Custom-built apodizing screen lays on top of the upper cage.

Window-screen apodizers are mainly suited to observation of small-field objects such as planets, because the array of rainbow-smeared diffraction artifacts distracts from most other astronomical views but leaves room for any of the planets. Apodizing an obstructed aperture can suppress the light distributed to the outer diffraction rings below that of an unobstructed aperture of the same size.

A common argument against apodizing is that putting anything in front of a perfect aperture cannot improve it. After all, secondary mirrors, their support vanes, mirror clips, etc. all degrade views to some extent. Such obstructions spread and smear light away from the central diffraction disk, reducing contrast. Furthermore, as seen below, apodizing almost always enlarges the central disk from the Airy size. The size of the central diffraction disk is not the sole determinant of a telescope's performance, however. Effective apodizing makes trade-offs among different aspects of optical performance. “Improvement” occurs when these trade-offs are favorable for the target and the observing purpose and conditions.

A brief explanation of how they work is that they create around 4 or 5 superimposed, concentric apertures. The amplitude and phase of their patterns are superimposed. Odd-numbered rings are in opposite phase to the disk and the even-numbered rings. The secondary shadow also superimposes a pattern of reversed phase, since it subtracts from the other apertures. The diffraction patterns from these apertures each have different sizes. Consequently, if one's first ring overlays another's second, they will partially cancel each other out, rather than reinforce. But with several superpositions, it is difficult to actually design specific cancel-out relationships across a whole diffraction pattern.

As Suiter points out, apodizing screen designs depend on the particular telescope’s aperture diameter, secondary obstruction diameter, and the transmissivity of the screening material. Screens built arbitrarily or according to rumored recipes might work, but are likely to be as good as none at all, or even worse. While construction is in reach of anyone with some common hand tools, an informed choice of diameters for the screen layers depends on some rather involved mathematics. The difficulties of design have likely hampered more widespread use among amateur astronomers. Screens built without a proper design, or even an effective design objective, likely account for the mixed levels of success and their cycles of trending and falling out of favor.

To help overcome the design difficulties, Suiter provided (in that same website) an analysis spreadsheet that calculates the diffraction characteristics of any screen apodizer with up to three layers. After specifying the screen material transmissivity, and the secondary obstruction ratio, the user iteratively varies the relative sizes of the screen rings toward maximum improvement.

However, some rather technical matters remain. Suiter and others discuss them in various books and articles, but some readers might not follow the chain of development. These are:

measuring the transmissivity of the screen material,
choosing a measure of effectiveness - the measure of improvement or quality expected by a screen design, and
maximizing effectiveness efficiently - finding the screen design of maximum improvement.

First below is presented an alternative means to measure a screen’s transmissivity. Then this article compares various performance measures of apodizing schemes aiming to improve planetary contrast, giving the background and reasoning behind the designs that follow. Finally come the design charts, from which one can look up the ring sizes (in proportion to the aperture), knowing the obstruction ratio and the screen transmissivity. It is hoped that amateur astronomers can enjoy noticeably clearer planetary views using these charts, or through better-informed designs of their own.

To measure the energy-transmissivity of a screening material

When Suiter wrote his articles on apodization (2001), a manual SLR camera's light metering system was readily available to a serious amateur photographer. But when I started working on my screens (~2008), I had only a small digital camera, and some telescope eyepieces. I suspended and backlit a piece of screen, and took several pictures through a 26mm eyepiece held backwards with the barrel around the camera's lens nose (such that no lens surfaces were in danger of being scratched). Most were blurry, and some backlighting worked better than others, but some were clear enough to show the gritty jagged details of the screen wire. I viewed several such pictures on my computer screen, zoomed in enough to measure the wire width and spacing in both directions (in relative units, holding an engineer's scale or ruler up to the screen). At several sample mesh-cells, I calculated the transmissivity ratio, and then computed the average of all the “reasonably-in-agreement” sample ratios.

Two pictures of window-screen mesh taken through a telescope eyepiece. (Left and below are a rubber material, right is metal.)

Sample cell's transmissivity = (area of red rectangle) / (area of green rectangle)

Many people now own or have access to a flatbed scanner. It's easy to lay a piece of screen material on the bed, and scan it. Accurate determination of transmissivity requires a very fine scanning resolution. My scanner's maximum of 4800 dpi is comparably detailed as the above zoomed-in picture. A problem, however, is the gigantic sizes of the scan files. My scanner has a feature to pre-select the area to scan at that resolution. I picked an area of only 1 square inch, and still got a 30 MB file! The material didn't all sit down flat on the glass; portions not touching were out of focus.

Flatbed scanner image at 4800 dpi (small zoomed-in portion of a square inch sample) of wire window screen

Measures of effectiveness

Now for the technical discussion. If you just want to get an effective screen design, then at least read the Suiter articles, measure your screen material's transmissivity, then skip below to the design charts.

To judge the improvement or quality expected of a screen design, the article and the spreadsheet identify a “quality factor” being the encircled energy fraction within the unobstructed aperture's Airy disk radius. The article also mentions the central Strehl ratio, or relative energy concentration at the exact middle. The article then points out correctly that no one-number criterion really tells the whole story of optical quality.

Suiter’s analysis spreadsheet also gives graphs for its encircled energy distribution and modulation-transfer function (MTF), in comparison to an unobstructed, unshaded aperture. These are familiar measures of optical transmission, especially to readers of Suiter's Star Testing. It is a favorite way to express contrast as a function of resolution. (There's a common usage of the words "contrast" and "resolution" in which "contrast" applies to lower resolution levels, and "resolution" means contrast at high resolution levels. But actually they are a continuum.)

The ability to distinguish subtle changes in color among regions of Mars, or bands and swirls of Jupiter and Saturn, depends on contrast at low spacial resolutions, the upper-left of the MTF curve. It's not that these features have low spacial resolutions, its that only low resolutions preserve enough contrast to perceive. Improving the low contrast levels at higher resolutions still results in imperceptibly low contrast levels, and might not help show low-contrast targets.

Another reason to emphasize the low resolution MTF is that at most places, seeing is seldom perfect enough for a large-aperture telescope to realize its fine-resolution potential anyway. It makes sense to pick your battles with what you actually can improve significantly through apodizing - contrast at lower resolutions, rather than fine resolution requiring better seeing anyway. (Apodizing screens are sometimes called "seeing filters" for this reason, although they improve planetary contrast in perfect seeing, too.)

Besides the MTF and encircled energy distribution functions, another evaluation function is the edge-spread function. The edge-spread function is the intensity at a distance away from a boundary between a lit area (source intensity 1) into a dark area (source intensity 0). In almost all telescopes it is symmetric about the point (0, 1/2), so one side sufficiently describes the function. It profiles the blurring of a boundary.

Edge-spread illustration: (a) a sharp boundary, (b) as spread according to function graphed in (c); (d) poorly-contrasting colors, similarly spread in (e)

Human vision is great at perceiving contrasting boundaries, but not so good when they are blurred into gradients. When planet features have low contrast to begin with, any blurring of their boundaries into a color gradient makes it difficult to perceive the features themselves. (I think this is why excessive magnification makes planets look dull. Dimness might have a role too, but probably not, because visual astronomers often prefer to magnify faint deep-space objects, making them dimmer, along with the surrounding sky brightness.)

Maximizing effectiveness

To adjust a screen design for maximum improvement, the user is instructed to start with a single screen at the outside, then pull it in, and successively the second and third screens in, and repeat, iterating until no improvement results in the chosen quality measure. This is a reliable procedure for one design and a pre-chosen quality criterion, even if it takes a bit of time (which anybody pursuing this should have or make).

I used Excel’s Solver add-in, a black-box optimization tool included with Excel (but not active in the default installation). Let me warn that Solver is a bear trap if you are not familiar with the methods and alert to the dangers of numeric optimization, especially with black-box tools. (In my work, I run some rather large and complex optimization problems.) One can get different answers depending on the starting values, and even degenerate answers, such as the inner radius equal to the obstruction. (Using Solver on linked spreadsheets often leads to Excel crashing, too.) Aside from having to be alert and careful to interpret what I get, Solver lets me quickly solve and evaluate lots of designs, for several different optimization objectives, and can verify them at different starting points. Sorry, this article will not teach anyone how to use Solver for apodizing or any other problem – there are whole books and advanced degree programs on numerical optimization.

Exploratory analysis

I started exploring what happens to MTF, edge-spread, encircled energy, and the intensity at specific radial distances, given my particular obstruction ratio and screen material, first by manually adjusting screen radii. (My obstruction ratio is close to 15%, and the material measures very close to 70% transmissivity.) Then I started picking different points on these functions to optimize using Solver, and to see how different objectives either go together or conflict. Later, I broadened the exploration to different obstruction and screen-transmissivity ratios.

The suggested “quality factor” on Suiter’s spreadsheet is based on the fraction of encircled energy within the unobstructed Airy pattern's first minimum, i.e., within the disk. Optimizing this with typical obstructions and materials increases this by around 10% compared to no apodizing. Edge-spread distributions and increase low-resolution MTF improve moderately too.

Suiter clearly did not dictate this factor as the single bottom-line measure of quality. MTF at a selected spacial frequency is suitable. Encircled energy at other radii, and points selected on the edge-spread distribution are too.

Optimizing low-resolution MTF makes it gain significantly more than the original quality factor. Or, looking at the inverse of the function, it increases the resolution at which high contrast is preserved - needed for distinguishing planet features having low contrast of their own. It strongly suppresses energy to rings beyond the first, though it broadens the disk and first ring area. Optimizing encircled energy or edge-spread at distances beyond the first ring achieves significant gains.

Trying higher-resolution goals didn't yield much improvement to them or other measures. Optimizations for minimum near-ring intensities tend to result in lower MTF at all resolutions, and wider encircled-energy and edge-spread distributions. Optimizing higher resolution MTF at, say, 0.30 relative spacial frequency, resulted in worse edge spread profiles, worse MTF at the lower resolutions, and probably imperceptible improvement at that target resolution level. Optimizations of the edge spread's central gradient, or closer-in edge-spread values, failed to push any of the edge-spread curve down nearly as appreciably.

Following various experiments, one promising measure became the sum of the MTFs at 0.05 and at 0.10. There is nothing theoretically special about this peculiar compound objective; it achieves nearly the same design and performance results as effective optimizations of edge-spread and encircled energy functions. (Suiter tells me this MTF goal resembles a criterion by Hopkins; I will add a citation and more info when available.) Compared to Suiter's suggested quality factor, it takes a different trade-off, accepting growth of the central disk and first ring in exchange for a stronger suppression of the outer rings, in a way that I (tentatively) think makes the most difference for a large-aperture telescope whose fine-resolution is usually seeing-limited anyway. I suspect that the disk-energy criterion might best serve mid-size scopes and/or the best seeing conditions. Optimizing encircled energy or edge spreads within the second minimum (the disk plus the first ring) yielded very similar designs and performance statistic improvements. (More details coming soon.)

Below is the MTF from an optimization with 15% obstruction and 70% transmissivity. The screen diameters from this optimization are 0.726, 0.843, and 0.912 of the aperture. The upper-left portion of the MTF increases significantly towards 1 even exceeding the unobstructed aperture, much more than making up for the damage due to the obstruction. These gains come at the expense of fine resolution at the lower right. It might not help resolve the finer gaps in the rings of Saturn, but it's easy enough to remove this apodizer when looking for them. I could roughly and arguably compare my apodized 12.5-inch scope's fine resolution to an 11-incher, but the contrast to a 16-inch scope.

(Horizontal axis to be better labeled "Spacial Frequency")

Next is its intensity distribution, compared to the full unobstructed aperture, and the same obstructed but unapodized aperture. (Intensities are normalized, so the total energy left after screening and/or obstruction is factored up to the unobstructed case's total energy.) Notice the disk and first ring are wider; the first ring is only slightly less bright than the base obstructed case. The higher-order rings, however, are significantly suppressed.

The encircled energy function is presented below as the remaining fraction of energy not encircled - the “excircled” energy, on a logarithmic scale. By this measure, for this case, apodization more than undoes the damage of the obstruction, improves the distribution even at the first ring, and cuts by more than half the energy distributed to the higher-order rings.

The measure discussed in Suiter’s article, encircled energy at the first unobstructed minimum, also improves, even over the unobstructed case, though not by as much as if we optimize that measure directly.

The edge-spread function has a similar result. While the boundary retains the same central gradient as the unobstructed scope, the amount of wider-spread smearing is significantly suppressed. The central gradient is difficult to steepen by any apodization. Directly optimizing for minimum edge-spread at reduced radii from around 2 to 4 yields almost the same design and performance. Optimizing at any smaller reduced radius fails to change it much. While the MTF charts above appear to imply improvment mainly beyond a radius of 5 or so (since spacial frequency = 1/reduced radius), the edge spread clearly improves much closer in than that.

A case of large obstruction, 30%, is next considered. As well known, the larger obstruction does more damage to low-resolution contrast, though not to fine resolutions. Apodization still more than makes up for obstruction at the lowest resolutions, and improves upon the obstructed base case without damaging intermediate resolutions. Intensity is again suppressed from the higher-order rings, and shifted onto the disk and first ring. The encircled energy and edge-spread functions show improvement even over the unobstructed case beyond the second ring. Contrast improvements are not as great as with the small obstruction, but are still significant.

(Horizontal axis to be better labeled "Spacial Frequency")

Next, we consider the case of no obstruction at all. Large-enough apertures to use apodizing screens are almost always obstructed; smaller telescopes may not be able to use enough periods of mesh to diffract in discrete arrays rather than a smear. Large refractors are extremely expensive, and most tilted-mirror systems are limited to small apertures. Some such designs are not "diffraction-limited" enough for designed diffraction to work. But unobstructed well-corrected larger scopes do exist, that stand the most to gain from apodizing screens, such as the Stevick-Paul, the Jones “CHief”, and larger off-axis Newtonians. Contrast-optimizing an unobstructed apodization yields the following MTF, intensity, encircled energy, and edge-spreads. (This is still using window screens, not continuously-varying transmission functions as in the optics literature.)

Apodization offers the most contrast gain to unobstructed scopes, significantly more than possible with typical obstructions.

The above examples illustrate the gains in contrast offered by apodization screens. If I had the software handy to generate spot diagrams, I would. However, spot diagrams tend to deemphasize the outer rings, so one might dismiss them as too dim to matter. But with television screens now being sold by contrast ratios in the thousands and millions, maybe such contrast is perceptible, even if it takes logarithmic charts to show. Better comparisons might come from true-color space-probe photos blurred according to the point-spread functions without and with apodizing (for chosen apertures).

Suiter remarked to me what might be a limiting factor of these outer-ring-suppressing apodization designs: light scatter in the optics and the eye could outshine the suppressed outer diffraction rings. I suspect if we know how far to stop ring-suppression until it is "scatter-limited", we could trade off any more than that for some kind of improvement on the inner part of the diffraction pattern.

Where I observe the seeing is usually not good enough to get meaningful photographic comparisons. But visually, features on Mars, Jupiter, and Saturn stand out clearer with my apodizing mask than without on my 12.5-inch f/6.3 Newtonian, whether the seeing is good or mediocre, no matter how otherwise filtered. In mediocre seeing (or with uncooled mirror), without my screen, a bright star in high magnification might look like a bunch of dancing arcs and fireflies; with it in place, the dancing light area appears much more compact and confined. A continuous "null" screen instead of the apodizing screen does not achieve these effects, but only dims and dulls the view. Of course, "your mileage may vary."

Suiter left open the question of whether apodizing screens really work. Partly this is up to whether a telescope delivers an accurate enough wavefront for this theory to apply, without having to know and account for phase differences. And even if a telescope-apodizer performs as theoretically predicted, the remaining question is whether the difference is perceptable. That leads deeper into vision sciences. But for practical tests, we could ask it this way: Which apodizations better show which features on which targets? Those boosting resolution, low-resolution contrast, or some other objective? This study focuses on low-resolution contrast for reasons stated, but this is nowhere near the last word on the matter. (Next to the technical literature on apodizing, this is still quite primitive, with a brute-force optimizer and stuff from the hardware store.) Someday I hope to build and compare apodizing screens optimized for different goals. (You can increase fine resolution just by using a bigger central obstruction. That may help split double-stars, but I've never heard of anybody recommending that for planets.)

Design charts

To make contrast-optimized apodization designs generally available, I optimized a series of cases of different obstruction ratios and screening material transmissivity. The resulting designs appear in the following three charts, as the fraction of the aperture to cut out of each of two or three rings. [Coming soon: optimizations for EER(1.22) - the "quality factor" in Suiter's design spreadsheet, and performance charts for that often-repeated 90% - 78% - 55% recipe.]

It can be seen that no one-size-fits-all or rule-of-thumb applies to contrast-optimized design. Contrast-optimized designs vary widely depending on both the obstruction ratio and the screen transmissivity.

Some notes on the use of the design charts:

As in Suiter’s spreadsheet, the obstruction and the two or three screen ring radii are relative to the aperture radius. (Or, we could say the obstruction and ring diameters are relative to the aperture diameter.)

For the primary mirror size, measure across only the active optical surface, exclusive of any bevel or edge-mask. But for the secondary obstruction, measure the whole circular obstruction diameter including any support shrouds or hubs, rather than just its optical surface.

Do not mix and match screening materials with these charts; they only apply with the same material in every layer, or at least, materials of the same transmissivity. (The Suiter spreadsheet lets you provide different transmissivities in each layer, but these would need to be optimized case-by-case.)

Two-layer designs are given where it's either the optimized design, or as an option if the third ring's relative radius is very close but less than 1. (For example, with a 60%-transmissive material and a 25% obstruction, screen 3's optimized size is over 98% of the aperture. That screen obstructs little, with fewer effective periods of mesh. It's practical and effective to just make two screens, according to the dashed green lines, for this case.)

These charts may be interpolated. For example, if you have a 18% obstruction, and 69% transmissive screen, then on the vertical line at 69% transmissivity, estimate about 3/5 of the way from the 15% curve and the 20% curve (because 18 is 3/5 of the way from 15 to 20) to get approximate sizes of 0.742, 0.854, and 0.922. Interpolate between curves of the same “family” of whether 2- or 3-layered. Do not interpolate, say, r3 between a curve and a 1.00 truncation. Also, I think extrapolating a little below 15% or above 30% obstruction should work fine. (You might want to make your own plot of radius as a function of obstruction, for your screen transmissivity.)

Higher-transmissive materials tend to result in better optimizations.

While designs from the charts are optimized for the stated objective measure, this does not necessarily make them optimal for every telescope, target, or observing condition or objective. If you have additional or alternative goals, you may put a design from these charts into your copy of Suiter’s spreadsheet (or other analysis tool) as a starting point, and adjust it in favor of your other objectives.

A similarly-optimized design chart for an unobstructed scope is below.

Outstanding questions

Some things that I don't know but are probably known "out there" might answer questions such as:

How narrow can a ring be to be useful? By what criteria should a third ring design of, say, 98% of aperture, be rejected in favor of a two-ring design? (I think it's largely a matter of off-axis width variation being a large percentage of the width used.)
How small a scope can use an apodizing screen?
Which screen materials hold up better? (My metal one is getting frayed, so I might start using again my earlier but less-optimized rubber one.)

Dick Suiter suggested a 1964 review article by Jacquinot and Roizen-Dossier on types of apodization that have been identified. This may take some digging at the university library, but it's interesting just to get a sampling of the technical literature that comes up on an internet search for it.