In the 1960’s “large” telescopes that could be purchased ...



Optics - The Series

Angular resolution and quality requirements.

Shane Santi – President

Dream Telescopes & Accessories, Inc.

Copyright 2016 – v2

In the 1960’s a “large” telescope that could be purchased on the open market was 8” in diameter. Now a large telescope might be considered 24”, 32” or even 40”. In 1917 the newest and largest professional telescope was 100”; the Hooker telescope. Around the beginning of the next decade the world should see its first telescope that crosses the 1000” diameter mark; the Thirty Meter Telescope (TMT).

Although telescope sizes have drastically changed over the last 50 to 100 years the formula for angular resolution has not. Diffraction, interference caused by the wave-nature of light, prevents optics (lenses and mirrors) from showing details below a specific level. This is why we cannot see the Lunar Lander on the Moon using a 4” telescope from the Earth. The reason we can’t see it with really large ground-based telescopes is not due to diffraction but (mostly) refraction; our atmosphere. The largest modern telescopes use adaptive optics (AO) systems to get around the atmospheric problem, but are still bound by diffraction.

The f-ratio of the mirror has no affect on angular resolution. It is driven solely by the diameter of the mirror and the specific wavelength of light we want to evaluate. Chart 1 shows numerous mirror diameters, their diffraction-limited arc-second angular resolution and how the wavelength of light affects that angular resolution.

Diameter 400nm wavelength 500nm wavelength 632.8nm wavelength

4” 0.99 arc-seconds 1.24 arc-seconds 1.57 arc-seconds

5” 0.79 0.99 1.25

6” 0.66 0.83 1.04

8” 0.50 0.62 0.78

10” 0.40 0.50 0.63

12” 0.33 0.41 0.52

16” 0.25 0.31 0.39

20” 0.20 0.25 0.31

24” 0.17 0.21 0.26

28” 0.14 0.18 0.22

32” 0.12 0.15 0.20

36” 0.11 0.14 0.17

40” 0.10 0.12 0.16

HST 0.042 0.052 0.066

Chart 1: Angular resolution of different diameter mirrors at different wavelengths of light.

The resolution stated is what can be achieved when the mirror is diffraction-limited. A diffraction-limited on-axis primary mirror is defined as:

• L/4 PV wavefront and L/14 RMS wavefront or

• L/8 PV surface and L/28 RMS surface

To turn the L/4 PV wavefront fraction into nanometers (nm), simply take the wavelength and divide it by 4; 400nm/4 = 100nm. The largest error in a L/4 PV wavefront mirror, when evaluating it at 400nm, is 100nm. However, at the physical surface of the mirror the height of that error is 50nm; 400nm/8 = 50nm.

For a primary mirror wavefront means the light at the eyepiece or focal plane, whereas surface means the actual surface of the mirror, like a topographical map. Because a primary mirror reflects the light directly back toward the source, it’s wavefront error is twice as large as the surface error; L/4 is twice as large as L/8. It is imperative to call out surface or wavefront in conversations and in specifications so that both parties understand what is being discussed.

A plano (flat) elliptical secondary mirror is different than the on-axis primary mirror. The light is not reflecting back to the source so the degradation is less for this type of mirror. If the PV surface of that mirror is L/8, then the PV wavefront is L/5.6, not L/4. Instead of dividing by 2, like the primary mirror, an ellipse needs divided by 1.42.

It is vital to understand that even with a cutting edge adaptive optics system, even those costing more than 1,000,000 dollars, we cannot beat the diffraction-limit of a given diameter mirror at a given wavelength of light. This means that even in space the mirror cannot achieve better than diffraction-limited performance. This may seem counterintuitive but in space there are still many of the same issues that we face on the ground. Some of those are;

• Optical alignment errors

• Tracking errors

• Vibration issues

• Thermal issues

• Distortion of the mirror(s) from the supports

Chart 1 shows that wavelength has a linear relationship with resolution. As wavelength increases by 25% (400nm to 500nm), for example, the resolution changes by the same amount. We can also see that each time we double the diameter of the mirror, 4” to 8” to 16” and finally to 32”, the angular resolution (in arc-seconds) is cut in half. In this series of diameters, at 400nm wavelength, it goes from roughly 1.0 arc-second to 0.5 to 0.25 to 0.125 respectively.

The wavelengths chosen represent the bottom of the visual spectrum (400nm), roughly the wavelength (500nm) human eyes are most sensitive to when dark-adapted and the National Institute of Standards and Technology (NIST) standard wavelength (632.8nm) for testing optics. The rest of the article will discuss resolution at 400nm.

Back when a large telescope was around 8” in diameter requesting that the primary mirror be diffraction-limited made sense. 50 to 100 years ago there was far less light pollution and the population was still fairly rural. It is plausible that the average telescope might be used in 1.0 arc-second atmospheric seeing, while having fairly faint limiting magnitude skies. At 1.0 arc-second seeing, the 8” mirror finished to the diffraction-limit is twice as good as the atmospheric seeing allows. Ignoring a host of real-world degrading factors we would say the mirror is (atmospheric) seeing-limited, not mirror-limited.

All things being equal (f-ratio, conic, type of mirror, polishing tool, polishing compound, etc.) the time and effort to finish a paraboloidal 8” mirror is substantially less than that required to finish a 32” mirror to the same quality level. This has to do with three main factors, although there are plenty of other factors;

• Departure from sphere

• Mechanical stiffness of the mirror substrate itself

• Ease of testing & iterations

Departure From Sphere -

Departure from sphere is the maximum deviation between a best fit sphere and the finished, aspheric mirror figure. Even when f-ratio and conic, say –1 for a paraboloidal primary mirror, are held constant, smaller mirrors have less departure from sphere than larger diameter mirrors. Chart 2 compares the departure from sphere of f5 paraboidal mirrors of different diameters. As departure goes up the correction in that aspheric mirror becomes more difficult because the slopes are steeper. Steeper correction is also concentrated near the outer edge, which is a notorious region for (human) errors. All of this adds up to a mirror that not only takes far more time but also far more skill to finish to the same quality level as the small mirror.

Diameter f5 Difference

8” 0.40 microns 1x (baseline)

12” 0.59 1.5x

24” 1.19 3x

32” 1.58 4x

40” 1.97 5x

Chart 2: Departure from sphere amounts, in microns, of different diameter mirrors of the same f-ratio and conic.

The diameters have not been the only thing to change over the years. The f-ratios of Cassegrain primary mirrors, Cassegrain system f-ratios and Newtonian primary mirrors have changed too. Everything has become “faster,” which means a lower f-number. The affect on departure from sphere is shown in Chart 3.

Diameter f5 (50-100 years ago) f3 (modern mirror) Difference

8” 0.40 microns 1.83 microns 4.6x

12” 0.59 2.75 4.6x

24” 1.19 5.50 4.6x

32” 1.58 7.33 4.6x

40” 1.97 9.16 4.6x

Chart 3: Comparison of departure from sphere amounts of f3 and f5 mirrors of different diameter mirrors.

Chart 4 shows the difference when we compare a smaller diameter and slower f-ratio mirror to larger, faster mirrors. The 8” f5 mirror is not technically challenging because the aspheric correction is shallow and broad, which makes it fairly easy to produce a smooth (low Mid-Spatial Frequency (MSF) errors) mirror that has a good overall figure (low Low Spatial Frequency (LSF) errors). Finishing the 8” f5 mirror to the diffraction-limit makes sense for both sides, opticians and buyers, because it does not take the opticians much more time or skill to get it there and it is only twice as good as 1.0” seeing allows.

Diameter f5 Diameter f3 Difference

8” 0.40 microns 24” 5.50 microns 13.8x

8” 0.40 microns 32” 7.33 microns 18.3x

8” 0.40 microns 40” 9.16 microns 22.9x

Chart 4: Departure from sphere amounts of different diameter mirrors and of different f-ratios.

Larger, faster mirrors are entirely different. To get a 32” f3 paraboloidal mirror to the diffraction-limit, in the real world, takes a very large effort and has to account for factors that the small mirror, for the most part, did not deal with. The difference column does not account for a series of other factors that enlarge that difference even further.

Mechanical Stiffness -

The notion that a 6:1 aspect ratio 40” solid mirror is the same stiffness as a 6:1 aspect ratio 4” solid mirror is unfortunately not true. Far more care is needed when processing large mirrors that have lower stiffness than their smaller, stiffer counterparts. This statement is true for 6:1 aspect ratio mirrors and it is even more of an issue for higher aspect ratio mirrors; 8:1, 10:1, 12:1, etc.

Scrutiny of how the mirror is tested and supported during tests should not be ignored in smaller mirrors. Especially if they are thinner and/or are finished to a stated specification of better than L/4 PV wavefront (L/14 RMS wavefront). The magnitude of the errors caused by gravity, poorly designed mirror mounts and numerous other issues would shock most outside of truly quantified test methods. I saw this firsthand more than 15 years ago using the simple and inexpensive Ronchi test (qualitative) of a 12.5” conical solid mirror while on two pegs. It is such an inexpensive and simple test to conduct that anyone can and should experiment with it. Just realize that when you can see the error in this type of inexpensive test, the error is already well, well past L/4 PV wavefront.

Often very little discussion is given to testing the mirror in the final (actual) mirror mount and at more appropriate final use angles. For example, the difference in gravity performance of a horizontal test while on two pegs or a sling can be wildly different than a properly designed and produced kinematic (floatation) support used with the mirror at something closer to the zenith (vertical) angle. The vast majority of astronomical telescopes are used at some upward angle, not horizontally.

As mirror diameter goes up, this aspect of testing becomes more and more important because it is influencing the mirror’s figure more and more. Even for L/4 PV wavefront (L/14 RMS wavefront) error levels, modern large mirrors need to be tested in a far more realistic manner than small, stiffer mirrors, in order to quantify, not qualify, their real errors. Otherwise quality guarantees are based on feelings and a lack of proper testing, instead of facts. Given large thin mirrors can bend with full waves of error, there should be far more realistic testing done on these mirrors than is currently the norm. What can easily be seen in Finite Element Analysis (FEA), and with legitimate interferometers, appears to be ignored by many. It is up to the consumer to protect him or herself and to request the real-world quality, not advertising claims.

Ease Of testing & Iterations -

As just mentioned, a solid smaller mirror of the same aspect ratio normally will not require elaborate means for testing. We’ve seen issues with a 9” solid mirror so “smaller” does not mean 12”-16” in diameter. This is for a real-world requirement of “only” L/4 PV wavefront (L/14 RMS wavefront). As the real-world PV wavefront error goes past L/4, more and more scrutiny and time will be required to both work the surface and to test it, in a quantified, traceable way. There are no free lunches in optics. If proper testing is not done, then the mirror is an unknown.

Physically an 8” mirror is far easier to move, clean & rinse, set up for testing, then replace on the polishing machine than say a 32” mirror. The 8” mirror will also equalize far faster than the 32” mirror. For glass-based solid mirrors this also means the 8” will have less figure distortion due to internal temperature variance. Testing is therefore substantially faster and has less risk to the optic (dropping or banging it into a hard object). This creates less stress and fatigue for the optician. It allows more iterations in a given amount of time than the 32” mirror. Fatigue is rarely discussed in optics but it is the human factor and it has just as large a roll as any other in determining the final quality of the mirror.

The Move To Cities -

Our modern world is now full of light pollution and the population continues to flock to and around large cities. Cities create heat from machines/industry, vehicles, people, etc., as well as from the solar heating of buildings, streets, sidewalks, etc. With so many now living in and around cities the average person’s seeing conditions have deteriorated over the past 50 to 100 years.

When average seeing and aperture growth are considered, it no longer makes sense to request truly diffraction-limited performance, not in the larger apertures and not when you believe in diffraction. The right quality level is that which is always better than seeing allows, although this is only one important variable. There are many.

Simplified AO systems do not offer the same quality improvement that expensive, small-field AO systems can offer. But in either AO case the buyer can determine what quality level they need by using the same methods or information outlined within this article.

If the mirror is finished to a quality that is truly well past the level required, then the buyer has wasted their money. If that “added quality” didn’t cost extra money, or did not take extra time, then the buyer should be wondering why it didn’t cost extra. A company will not survive financially if they are loosing money on each optic. Loosing money on one or two is one thing. But it’s irrational to believe a company is giving away their money to strangers on every optic, on a daily basis. Advertising wants you to believe this but a wise consumer knows the difference between marketing and facts.

Chart 5 shows the same information as Chart 1 but three different seeing levels have been added. You can see that a diffraction-limited (L/4 PV wavefront, L/14 RMS wavefront) 40” mirror is a full magnitude beyond what 1.0” seeing conditions will allow.

Diameter Resolution at 400nm 1.0” seeing 1.5” seeing 2.0” seeing

8” 0.50 arc-seconds 2x 3x 4x

10” 0.40 2.5x 3.75x 5x

12” 0.33 3x 4.5x 6x

16” 0.25 4x 6x 8x

20” 0.20 5x 7.5x 10x

24” 0.17 6x 9x 12x

28” 0.14 7x 11x 14x

32” 0.12 8x 12.5x 16x

36” 0.11 9x 14x 18x

40” 0.10 10x 15x 20x

Chart 5: Diffraction-limited mirror and factor it is above three different seeing levels.

Put another way the 40” mirror finished to L/4 PV wavefront (L/14 RMS wavefront) could provide 0.1” angular resolution if 0.1” seeing conditions could be achieved. Think about that. This isn’t a “L/20” PV wavefront mirror but a mirror that meets the diffraction-limit requirement. It’s ten times better than 1.0” seeing will allow. If your site’s seeing is 1.5”, then it is 15 times better than your seeing will allow.

Diameter Quality 0.1” seeing 1.0” seeing 1.5” seeing 2.0” seeing

40” L/4 PV wavefront 1x 10x 15x 20x

40” L/20 5x 50x 75x 100x

40” L/40 10x 100x 150x 200x

Chart 6: 40” mirror of different quality levels and the proportion to the seeing level.

If the 40” mirror were finished to a quantified and traceable L/20 PV wavefront quality, it would be 5x better than the diffraction-limit and therefore 50x “better” than 1.0” seeing. Does that mean it will deliver 0.02” details (0.10/5)? No. Diffraction prevents this. Will it provide better quality than a 40” mirror finished to L/4 PV wavefront? If the MSF errors and RMS Surface Roughness (High Spatial Frequency (HSF) errors) are the same for both mirrors, there will be no improvement in performance. Remember, even at L/4 PV wavefront (L/14 RMS wavefront) it is still 10x better than 1.0” seeing conditions will allow.

If a person did either a visual or imaging star test and claimed it showed truly perfect intra and extra focus views with a 40” L/4 PV wavefront mirror, then it implies seeing was 0.1” (without expensive AO, this is similar to seeing Bigfoot) and there were absolutely no errors anywhere else in the system (similar to reaching infinity). As the old saying goes, if it sounds too good to be true, it probably is.

Diameter Quality Angular Resoluton at 400nm

40” L/4 PV wavefront 0.1 arc-seconds (diffraction-limited)

40” L/20 0.02 (fantasy)

40” L/40 0.01 (fantasy)

Chart 7: 40” mirrors of different quality levels and the corresponding “resolution.”

A decade ago an optician admitted they had to start saying “L/20” PV wavefront because they were loosing too many sales to others who were willing to say it. They knew and admitted their mirrors were not L/20. The past five years has shown this battle of the Lambda has gone ever further, pushing even more outlandish numbers, like L/40 PV wavefront. If those in high-level metrology, testing optics in a truly quantified and traceable manner, could pass on their 30-50 years of knowledge, the reader would realize how ridiculous it is to believe these types of claims. If the optician truly believes they have achieved that quality level, using qualitative tests and as an island, then they should have no problems with outside, quantified and traceable testing.

It will probably surprise most that the Hubble Space Telescope’s (HST) flight secondary mirror is L/10 PV surface (L/5 PV wavefront) or 25% better than the diffraction-limit. The corrective optics, to fix the spherical aberration in Hubble’s primary mirror, are still governed by diffraction. They cannot perform magic to “increase” the quality of the secondary mirror. Although there are still plenty to want to point to HST’s initial problems, it ended up setting the resolution standard for more than a decade.

Modern, ground-based multi-meter telescopes using AO systems are operating around the diffraction-limit; L/4 PV wavefront (L/14 RMS wavefront). For the past 10-15 years these (ground-based) telescopes, due to their much larger diameter and use of AO, have been taking images with higher resolution than HST. See the VLT’s AO work here: and general information here: . A Keck AO sample image can be seen here from 1999:

The market places huge pressures on companies to make outrageous quality statements. This practice is detrimental to consumers who know nothing about finishing or testing optics but actually want the utmost in real-world quality. It leads consumers to believe that “more” is actually better and a host of other factors in and around the telescope have no affect on the quality of the optical system. Although everything does start with the primary mirror, a telescope is only as good as its weakest link.

The utmost in quality has to account for more than just one data set taken of the naked mirror, on a horizontal-pointing test stand while the mirror is resting on two pegs or a sling, especially for larger diameter mirrors. Any first year engineering student doing FEA can see what the large mirror does when it is supported in this way. Anyone using real intereferometry can see it in person. Yet this is rarely discussed. It is noticeable at L/4 PV wavefront with a fairly small mirror and it is Mt Everest at a true, quantified and traceable, L/20 PV wavefront (L/40 PV surface), although real-world testing to this level of quality is the realm of very few of the world’s top optical shops.

If the optician who routinely finishes the large mirrors using qualitative testing normally produces “L/20” PV wavefront mirrors, then they should be able to offer you a large discount on the price, given how much additional time and effort it takes to push a large mirror, in reality, from L/4 to L/20 PV wavefront error. The level of metrology expertise, amount of variables that seemingly come out of the woodwork that greatly influence such low level testing and the time it takes to validate mirrors of any diameter to L/20 PV wavefront (L/40 PV surface) is on a scale that very few understand. Some throw these numbers are around like a rubber ball and that should be a red flag. The scale of difference between L/4 and L/20 cannot be properly impressed on the reader.

Those of us who have been around quantitative, traceable testing have seen just how easy it is to bend all mirrors. We’ve also seen just how difficult it is to validate the test data. This is for quantitative and traceable test equipment. Qualitative testing is another large set of additional challenges but can never be truly quantitative. Ignoring fundamentals and physical realities of optics means a person is no longer discussing reality.

A mirror truly finished well beyond the diffraction-limit will not make up for other errors. It can’t make up for optical alignment errors, or compensate for mirror seeing, or distortion from the mirror mount, or tracking errors, or dozens of other real-world, highly influential factors. There is no voodoo here. If the optical alignment is producing a full wave of error, a “perfect” mirror cannot correct that out. The system will have at least a full wave of error. Period. Regardless of how “perfect” a given mirror is. This is why educating yourself with the fundamentals of optics, like diffraction and basic mechanics, is so important. The irony is that when a person stays grounded and educates himself or herself, they can learn how to make their “L/20” telescope show real-world improvements.

Diameter Resolution at 400nm 0.25” seeing 0.5” seeing 1.0” seeing

8” 0.50 arc-seconds L/4 L/4 L/2

10” 0.40 L/4 L/3.2 L/1.6

12” 0.33 L/4 L/2.64 L/1.32

16” 0.25 L/4 L/2 1L

20” 0.20 L/3.2 L/1.6 1.25L

24” 0.17 L/2.72 L/1.36 1.47L

28” 0.14 L/2.24 L/1.12 1.79L

32” 0.12 L/2 1L 2.1L

36” 0.11 L/1.76 1.14L 2.27L

40” 0.10 L/1.6 1.25L 2.5L

Chart 8: PV wavefront requirements for different mirror diameters, at three different seeing conditions.

Chart 8 provides guidelines for the real-world, as-installed primary mirror PV wavefront requirement. If a 1m telescope will see 1.0” seeing for less than 1-5% of its life, then 0.5” seeing requirement could be used. This provides a 1m mirror that is twice as good as seeing will allow. 1.25L at 400nm light is a PV wavefront error of 500nm, which is 250nm PV at the surface of the mirror. Low-level, inexpensive AO systems improvement is normally in the 20-40% range. Even with the use of such a system the 40” primary specified for 0.5” seeing will still be more quality than AO will allow in 1.0” seeing conditions.

Human emotion wants to believe that adding “quality” is always the answer and always a plus. The problem is that this pushes the conversation into a fictional world where ridiculous claims abound. Understanding the basics outlined in this article will go a long way in allowing you to weed through what is a claim and what is closer to reality. This article is also touching on a much larger issue; isolated qualitative testing versus quantitative, traceable testing of optics.

Consumers don’t need to be experts in judging optical quality. A simple, readily available web cam can be used to document the intra and extra focus star images. A diffraction-limited telescope should show a perfectly round out of focus donut and should show those intra and extra focus donuts quite close to focus. Assuming the optician had spherical aberration (correction) under control, astigmatism is the most common error. This will show as anything from a slight ellipse to a flat line in more extreme cases.

As diameter grows too many things change for us to continue using our over-simplified knowledge that was based on 6-8” optics. In order to optimize a given telescope we need to always push for current knowledge so we can understand what is reality and what is emotion. This will help to reduce the degrading variables as much as possible. To focus on only one thing is to ignore the forest of additional errors that exist in a telescope.

Copyright 2016 Dream Telescopes & Acc., Inc. -

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