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Wild Card 003.1 About the System(?) Strehl Ratio

Posted by Rick Shaffer   06/27/2004 00:00:AM

Wild Card 003.1 About the System(?) Strehl Ratio
[ARTICLEIMGL="1"]The first “half” of this column related the three main ways opticians rate an optic. I promised in this column to show how knowing the Strehl Ratio of an optic would allow us to characterize the quality of an entire optical system.

Just to get us all on the same page, remember that to arrive at the Strehl Rato, we compare the intensity of the Airy Disk as produced by the real optic with the intensity of the disk produced by a perfect optic. So, a perfect optic would have a Strehl Ratio of 1.000. That leads to any “real” optic having an SR just slightly less than 1.000.

I’ve read forum posts that suggest that we can’t quote the Strehl Ratio of a flat or of another aspect of a system because that individual part can’t form an image with an Airy Disk. No Airy Disk, no Strehl Ratio. That’s technically true, but irrelevant.

Actually, any aspect of an optical system that diminishes the intensity of the Airy Disk the system produces in the image plane can have a Strehl Ratio assigned to it. (If you require that each of these individual factors be called something other than “the Strehl Ratio”, be my guest. May I suggest “Larry”, “Curly”, or “Moe”?)

OK, what are these factors? Well, before we think about them, lets get all the wailing and gnashing of teeth out of the way. Here’s the FORMULA:

SystemSR =

ObjectiveSR * DiagonalSR * ShadingSR * ObstructionSR * CorrectorLossSR *Coating#1SR* Coating#2SR * Coating#3SR * Coating#4SR *Support#1SR* Support#2SR* Support#3SR * FilterSR

NOTE0: Have I included every possible factor? No. For instance, we might have factors for tube currents, mirror figure deformation due to temperature affects, number of home-runs hit on successive Sundays in May, factor due to slight miscollimation of the optics, etc., etc., etc. If you think of a factor I haven’t mentioned, please write me. I (might) want to hear about it.

NOTE1: If you don’t have all these various components, either take the factor of the missing component out, or just set it equal to 1.000.

NOTE2: All the asterisks are “times signs” written the way computers like to write them.

OK. Let’s go over this, term by term.

The ObjectiveSR is just the Strehl Ratio the optician who tested the mirror or set of optics determined....For a Newtonian, it’s the SR of the primary mirror. For a Cassegrain, it’s the SR of all the elements that form an image, including the corrector(s) of an SCT or a Mak-Cass.

The DiagonalSR is the SR assigned to the diagonal. (Note that most diagonals are a bit better than the SR they’re assigned. That’s because almost all of them have a slight turned-down edge.. If you’re using only the central 80% to reflect the on-axis image of your telescope, and if that portion is a lot flatter than its test results would indicate, then it will perform better than its SR would predict.)

The ShadingSR accounts for the loss of light because the diagonal or other secondary mirror is in the light path. It is = 1 – [(a/A)^2}], where a is the aperture of the diagonal’s obstruction, and A is the aperture of the system.

The ObstructionSR is the ratio of the intensity of the obstructed system’s Airy Disk to the same system without an obstruction. This “non-table” will give you the approximate SR for various percent obstructions: 0% = 1.000, 10% = 0.976, 20% = 0.905, 30% + 0.810, 40% = 0.690, 50% = 0.571.

The CorrectorLossSR is the SR that corresponds to the percentage of light that gets through the corrector. You could even use this term to describe the very modest loss due to absorption of your Paracor or Barlow Lens!

The CoatingSRs are the efficiencies of the various coatings of the mirrors. Since they do affect the intensity of the Airy Disk, they figure into the SR of the system.

The SupportSRs are those that are due to the support system of the various optical elements not supporting those elements perfectly. Some of the online software that allows you to design multipoint systems will also calculate the SR for the design for you.

The Filter SR accounts for that portion of the drop in intensity of the Airy Disk that would occur if a solar filter were placed in the system without its reflective coating. This is an attempt to quantify how badly a solar filter affects the image due to the quality and figure of the filter substrate (or lack thereof!).

A few thoughts about all this:

1. The easy way to do this is to use a spreadsheet program like Excel or its equivalent. That will allow you to evaluate the choices you might be making about what quality mirror to buy, what coatings to put on your mirrors, what size diagonal you might want to install, etc.

2. You can use this equation to think about a refractor, but you’d want to think about the ObjectiveSR over a limited range of wavelengths. (With an apo, you’d widen that range a lot….) It’s harder to do, but it can be done. I’ll be writing about this in a later column.

3. Just inspecting this equation should lead anyone to the conclusion that the simpler the optical system, the higher its SystemSR will be, everything else being equal. And that’s true. The maker of a catadioptric telescope of any kind must work harder on the numbers to achieve the same SystemSR as a really good refractor or Newtonian. As an example, let’s think about two telescopes with the same EFL, one an 8” f/10 Newtonian, and the other an 8” f/10 Schmidt-Cassegrain-Telescope.

The Newtonian has an SR = 0.95 primary mirror, an identical quality diagonal, 96% coatings on its two surfaces, and a 20% obstruction. For the sake of this discussion, we’ll assume that the supports for all the optics are perfect. Our Newtonian looks like this:

8f10NewtSystemSR = 0.95 * 0.95 * 0.960 * 0.905 * 0.96 * 0.96 = 0.720

Our SCT has an SR = 0.95 image-forming set of optics, an identical-quality star diagonal, 96% coatings on each of 3 surfaces, and a 38% central obstruction. It also allows 98% of the light that hits the corrector through the corrector.

8f10SCTSystemSR = 0.95 * 0.95 * 0.856 * 0.714 * 0.96 * 0.96 * 0.96 * 0.980 = 0.478

Will the SCT really look as bad as the SystemSR would imply? Well, yes and no. Yes, in that the Airy Disk of every point in the image will only be about 2/3 as intense with the SCT as it will be with the Newtonian. And, yes, in that the diffraction rings in the SCT’s image will be stronger than those in the Newtonian’s. But some of the factors in this calculation don’t affect the brightness of the Airy Disk relative to the brightness of the rings at all. They just diminish the diffraction rings as a whole, so they don’t have quite so bad an affect on the contrast of the image as the difference in SRs might imply. But it would surprise me if anyone would look through the Newtonian and want the SCT if the quality of the image were the only factor in his/her decision. (But, of course, the SystemSR doesn’t have a term that rates the portability, versatility, and ease of use of an 8” SCT vs. the equivalent Newtonian.)

I’ve just shown you how you can compare two telescopes of the same aperture. Can you use this concept to compare two telescopes of different apertures? Yes, but to make it convenient to do this, let's compare two telescopes of exactly the same focal length.

Consider the case of “Sally Planetary”, a person who has a 6” f/13.333 paraboloidal mirror of exquisite quality. It’s Strehl Ratio is 0.990. (At this focal ratio, such mirrors can and do exist!) She wishes to make this into the finest planetary telescope she can. So, she obtains the finest 1” diagonal she can, which also has an SR of 0.99, and she has both primaries coated with expensive 99% reflective coatings. Just as with the other systems we’ve discussed, we’ll assume for this one that she has her optics mounted in perfect cells. Let’s grind out the numbers for Sally Planetary’s planetary scope:

6f13NewtonianSystemSR = 0.990 * 0.990 * 0.990 * 0.955 * 0.990 * 0.990 = 0.894

So, by extremely judicious choice of optical quality, coatings, and central obstruction, Sally has a 6” planetary Newtonian that has a SystemSR of 0.894. So, she’s diminished her Airy Disk by only 10% and only some of that loss has been transferred to the diffraction rings. This should be a very high-contrast instrument, indeed, just what she needs for Planetary planetary viewing.

How does Sally’s telescope compare to the 8” SCT? Well for starts, we know that the ratio of the amount of light intercepted by two telescopes is proportional to the square of their two diameters, so we’d just square the two apertures to obtain their respective “light – gathering powers”. We would then multiply the square of each aperture by its SystemSR to obtain a comparison “figure of merit”:

For the 6”, we have: 6^2 = 36, and 36 * 0.984 = 32.2

For the 8”, we have: 8^2 = 64, and 64 * 0.478 = 30.6

The above analysis would suggest that Sally’s optimized 6” planetary telescope will perform slightly better than the 8” SCT. Is this possible?

Here’s where this concept doesn’t provide a definite result. It’s certain that there will be a bit more light in the Airy Disk of Sally’s 6” than there will be in the 8” SCT, but the Airy Disk in the 6” will be bigger than in the 8”, so there will be some fine details the 8” will resolve that the 6” won’t.

Complicating this analysis is that one of the reasons the SystemSR of the 8” is so low is that the central obstruction has transferred a lot of light out of the Airy Disk into the diffraction rings. This will lower contrast, which isn’t what Sally wants in her Planetary planetary scope. I suspect that Sally would be very happy with her 6”, and would prefer it to an 8” SCT for critical planetary viewing.

I’d likely prefer Sally’s scope over an 8” SCT for critical planetary observing. And, others might mention that a really well-made 6” refractor (achro or apo), would also do well in this analysis. They’d be right. But, if I had to pick between these two telescopes as my only telescope, I’d pick the 8” SCT every time. Reason is that it does a lot of things well that the long Newtonian doesn’t do well. It’s faster photographically, is much more compact, the eyepiece position is generally more convenient, the “back-end plumbing” is extremely versatile, and it shows a wider field for DSO observation. As a second telescope for lunar and planetary viewing, however, I’d take the 6” “long-Newtonian” every time. And, I’d also be likely to make the same choice if I were offered a quality 6” refractor.

But it’s important to realize that what Sally or I might pick, you might not. The reason is that there are a lot of subtleties involved. The analysis tool I’ve presented here isn’t perfect in evaluating an optical system, but it’s very useful, especially in making decisions about either designing a telescope system from the ground up or in deciding where you can get the most improvement of a system for the least amount of money or effort.

If this tool isn’t perfect, what is? Well, I suspect that there isn’t a perfect system. This one has the advantage that it doesn’t require a lot of heavy-duty math for the average ATM to understand.

There’s another way of looking at the quality of an optical system. It’s called the Modulation Transfer Function. It relates the contrast a telescope provides to the size of the detail it’s expected to resolve. When my head stops hurting from writing this column, I’ll write one on the MTF. It’ll be a while, though….

RICK SHAFFER is an astronomer, teacher, writer, consultant and designer/builder of telescopes, museum exhibits and prototype flying disks. He lives and works in Sedona, AZ, with his 20” Newtonian, Dora.