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Wild Card 003: Making Some Sense of the Strehl Ratio

Posted by Rick Shaffer   06/21/2004 12:00AM

Wild Card 003:  Making Some Sense of the Strehl Ratio
[ARTICLEIMGL="1"]Many years ago, while I was in the Army, I had a part-time job repairing and selling audio equipment at the local Lafayette Radio store in Huntsville, AL. It was there that I learned about the various ways of rating the power output of a stereo amplifier. Most of the ratings were designed to make the amplifier look as good as it could. We were, after all, trying to sell the things.

And so it is with telescope optics. There are a number of different ways to characterize the accuracy of a telescope mirror or a refractor objective. We read about “diffraction limited”, “1/4-wave accurate”, and the ever-popular “tack-sharp” optics. What does it all mean?

This edition of “Wild Card” will focus on the various ways we think about the quality of a reflector optic. To keep the discussion simple, I’ll be just addressing a telescope with a single curved mirror and a flat mirror: the Newtonian reflector. But the basic concept will work fine for Cassegrains, Schmidt-Cassegrains and Maksutov-Cassegrains, as well. In it’s “sister” column next week, I’ll go over some concepts I’ve borrowed from my days doing Radio Astronomy that I hope will allow amateur astronomers to better evaluate the relative merits of different optical telescopes. (But it won’t answer the age-old question: “Who was the best ‘Star Trek’ captain?” That’s later….)

Refractors present special problems, because the focal length of a refractor is dependent on the wavelength of light passing through it. I’ll devote a future column just to evaluating refractors.

Right now, we’ll just concern ourselves with how to evaluate the image formed by a concave mirror. The source of light is a star so far away that the “rays” of light coming from it are all parallel to one another. That means that, if we could tag each individual wave of light at the same point on the wave, we’d find that they’re all marching along together, creating a perfectly flat plane. We call this the wavefront. In this discussion the wavefront is perpendicular to the optical axis of the mirror.

It’s the job of the optic to take the plane wavefront and make it into a spherical wavefront. The center of that spherical wavefront is, of course, the focal point of the telescope. If the mirror does its job perfectly, the image of a star will be a tiny, perfect disk of light surrounded by several rings of light, each of which is dimmer as you look farther and farther from the center. This is called the diffraction pattern, because it’s caused by a physical phenomenon by that name. And, of course, diffraction is a consequence of the wave nature of light.

That central disk of light is known as the ‘Airy Disk’, after Sir George Biddle Airy, the British Astronomer Royal who investigated this phenomenon. Because of diffraction, there is a limit as to what angular size object a telescope can resolve. The bigger the aperture of a telescope, the better it will be its resolution, at least theoretically. But, of course, the atmosphere limits what we can see in our telescopes more than any other factor.

The only surface shape that can take a plane wavefront and reflect it into a perfectly spherical wavefront is a paraboloid. The perfect paraboloid doesn’t exist. Instead, we have optical surfaces which are very close to a perfect paraboloid. How close is close enough is a matter of considerable debate.

If a mirror isn’t a perfect paraboloid, what happens? I’m not going to try to explain it from a Physics standpoint, but the result is that some of the light that would be found in the Airy Disk makes its way into the rings. You might ask, “So, the Airy Disk of a particular star I see in my telescope isn’t as bright as it would be in a perfect telescope. What’s the big deal?”

If you’re just looking at individual stars, the answer might be that it’s not a big deal. However, think about the fact that there’s a diffraction pattern that corresponds to every point in the field of view of your telescope. An imperfect mirror will put more light into all the various sets of rings than a perfect mirror would. That will lower the contrast of your view. For mirrors that are just slightly imperfect, the view of an object that has many low-contrast features won’t look quite so sharp or “snappy” as it would with a perfect mirror. In the extreme, so much light would get transferred into the rings that the whole view would look out of focus, which is what happened to the Hubble Space Telescope.

The way that most opticians measure the quality of a mirror is to look at the differences in the length of the optical path the light must take through the telescope to the image. Rayleigh postulated that, if the difference in the optical path is ¼-wave or less, the optic is “diffraction-limited”. The implication is that the view wouldn’t be improved by making the optic any better. Several manufacturers of telescopes state explicitly that their optics are “diffraction limited”. What does that mean? If their telescopes were better than that, could we tell?

Instead of answering that question directly, lets take a look at some raw data from some “cybermirrors” I’ve created.

I created all these mirrors in an ancient DOS-based optical design program called SODA. I set it up to trace 81 rays through a reflector with a 20” f/4 mirror. Each ray represents an equal area of the mirror’s surface. The plot you see is of the differences in the optical path through the system to the focal plane. (And, I’m sorry if the plots aren’t fancier. But, remember that this is a “reality show”, and we don’t have a big budget!) For convenience this program assigns the central ray a value of zero-OPD. By convention, all the other rays have either the same path length through the system or a little shorter path. However, if we set the reference point a little differently, the values of the OPDs of the rays could be both + and -.

OK. On with the cybershow!

In case #1, you’ll also notice that the value for “RMS” is 0.0821-wave. RMS stands for “root-mean-square”. It means that the computer takes the square root of the mean of the squares of the OPDs. The goal is to determine the mean, or average of the OPDs so we don’t get fooled into thinking that the mirror is worse than it is.

So, here’s what the computer does: (“Compute along” with your calculator if you’d like!)

1. It takes each of the values you see displayed above and squares it. (It’s important to do this because the OPDs can be both positive and negative, depending on where the “zero-point is set.)
2. It then adds them up and divides by 81, to get the mean (or average) of the squares.
3. It then takes the square root of that value of the mean. Voila! We have the RMS of the OPDs, which is what most folks call “the RMS”.

So, this looks pretty good. But, anyone who’s looked through 20” f/4 with a mirror like this on a steady night at high power will likely tell you that the planetary images were a bit “soft”, and that the star images didn’t “snap into focus”. I had a 19” mirror for a good while that wasn’t quite as good as this one. I could definitely tell that it wasn’t quite right….

As you can see by looking at the plot in case #2, the Peak-to-Valley with the above mirror is 0.17-waves, or about 1/6-wave. This is a good, but not great mirror.

The Peak-to-Valley here in case #3 is 0.12-wave, so this is just a bit better than a 1/8-wave mirror. There are a lot of amateurs out there who are very happy with mirrors of this quality.

Case #4 is a 1/10-wave mirror, because its max. Peak-to-Valley error is 0.10-waves, as you can read for yourself on the plot. This is a bit better than the previous one, but you’d be hard-pressed to prove it at the eyepiece.

Case #5: Did you expect anything but a lot of zeroes? Any one of us would kill for this mirror! But, the truth is that it would be difficult to tell the difference between this perfect mirror and the ones shown in the precious two cases. There are observers who seem to be able to do so, but it would take the “night of your life” to get the seeing out of the way to do it.

I could keep going “through the paraboloid”, and show you more OPD plots. But, I think you get the idea. But there’s one more you ought to look at.

CASE #6: I can’t create many types of “problem” mirrors using SODA. It’s designed for analyzing designs, not existing mirrors. However, I’ve “cooked the books” of the OPD plot for a perfect paraboloid by just editing the plot and redoing the math on my calculator. Take a look at this poor thing (Case #6):

This, of course, is the notorious “turned-down-edge”. I’ve made it wider than you’ll usually find it, but note that, since the OPD is only a quarter-wave different, that means that the mirror is only turned down by half that, or an eighth-wave. Note that, even though this mirror has the same P2V of the mirror in CASE1, its RMS is much worse. That’s the power of the RMS concept. It can tell you useful things that the P2V cannot. But, of course, it also makes any mirror, even the one with the turned-down edge, look pretty good.

Let’s sum up the analysis of the group of cybermirrors I’ve assembled at the bottom of this page for your viewing pleasure:

See Cyber Mirror Analysis Table at bottom of page

A lot of amateur astronomers are content with the concept of rating mirrors and other optical systems by quoting the RMS of the OPDs. The objection to using this rating is that it addresses the “average lumpiness” of the mirror relative to the perfect paraboloid. And, it uses small numbers to represent high quality, and it can make a mirror look a bit better than it is. But it does tell us more about a mirror with a turned edge than the P2V does.

Instead of characterizing the mirror’s shape compared to a perfectly shaped mirror, why not use a rating system that measures how well a mirror performs in imaging that most difficult of all targets, a star? About a hundred years ago, a German named Strehl came up with a relation that does just that. He reasoned that, what we’d really like to do is compare the intensity of a mirror’s Airy Disk with the intensity of the Airy Disk of a perfect mirror. Here’s the equation for the Strehl Ratio:

Strehl Ratio = (Intensity of ACTUAL Airy Disk) / (Intensity of PERFECT Airy Disk)

The Strehl Ratio is always less than or equal to 1. (If you, or anyone you know can figure out a way to get a Strehl Ratio greater than 1, please let me know. You’d be getting more energy out of your telescope than is entering it! I’ll sell everything I own and invest in your company!)

You might say, “OK, that’s real nice, there, Rick. Do I have to measure the intensity of my diffraction pattern in order to determine how good my optics are?” No, but you actually could do that, and some folks working on adaptive optics do just that. (I wonder if there’s anyone out there who’d like to try doing CCD-photometry on the Airy Disk of his telescope mirror?….)

Fortunately, several other theorists worked out a way to take the RMS of the OPDs and determine the Strehl Ratio of a mirror. There are several formulas for approximations of the Strehl Ratio if we know the RMS. The one that seems to cover the range of amateur mirrors best was put forward by Mahajan:

Strehl Ratio = EXP [–(2Pi*RMS)^2]


Strehl Ratio = (2.7183)^[-(6.2832*RMS)^2]

Don’t worry. You won’t have to actually grind out the numbers yourself. Nor will there be “a quiz at the end of the hour”. Instead, I’ve produced a “handy-dandy” table for your Strehling Pleasure.

See Handy Dandy Table at the bottom of this page

I’ve also included some comments on the quality of an optic of the quality implied by the Strehl Ratio.

If you take a really close look at the two tables, you’ll notice that they don’t match perfectly. That’s because my optical design program uses a fairly simple algorithm for computing RMS of the OPDs. Other, more sophisticated methods yield slightly different values for the relationship between the P2V and the RMS of a mirror.

So, we now know what the Strehl Ratio is. Do we need it to accurately know a mirror’s quality? I believe that it helps to know it, but you should know that not everyone who makes good mirrors quotes the Strehl Ratio in describing a mirror’s quality. In fact, some makers don’t quote ANY quality figure. For example, the man who made my 20” f/4 mirror told me that he supposed that it was a 1/10-wave mirror (P2V), but that he really couldn’t say. What he did say was that he was sure I’d be happy with it. He wasn’t wrong. (I won’t mention his name because he doesn’t normally do mirrors for amateurs.) There are other makers of high quality mirrors who refuse to quote the Strehl Ratio or any other figure. I’ve looked through telescopes equipped with their mirrors, and can’t find a thing wrong with them. I conclude that they’re making very good mirrors.

Folks who design very large optical systems like it because they know that the Strehl Ratio gives one number that describes an optic’s quality that’s not sensitive to what kind of aberration is present in the system. So, in that sense, the Strehl Ratio has become the new “gold standard” for large telescope mirrors.

But, the real power of the Strehl Ratio is how it fits into an easy way to quantify the quality of an optical system, which includes the mirror mount, diagonal, central obstrction, coatings, the works! I’ll bash you over the head with that next week….RICK SHAFFER is an astronomer, teacher, writer, and designer/builder of telescopes and museum exhibits. He lives and works in Sedona, AZ, where the Strehl Ratio of rubber tomahawks to Red Rocks is 0.999….