Notes from Under Sky

The Error in the Wave

What does it really mean to be 1/4-wave?

Someone asked:

I often seen optics quality measured as "1/8th wave or better". What does this mean? Is for example 1/8th wave better than 1/4th wave?

It depends. All other things being equal, yes, 1/8-wave is better than 1/4-wave. But all other things are not always equal.

Here's what it means. Light from a distant star arrives at the objective (lens, mirror, or both) in parallel rays. A perfect objective would focus all of these light rays down to one point, but perfection is impossible: invariably, some of the light rays overshoot the mark, some undershoot it. One measure of quality of an objective is how far the light rays miss the mark.

The precision of even a so-so set of telescope optics is astonishing: the error is on the order of 0.00001 inches. Now obviously, measuring something that small in inches or even millimeters would be a pain in the tush, so instead, the wavelength of light is used. Then that error of 0.00001 inches can be said instead to be about 1/2-wave, since the wavelength of visible light is on the order of 0.00002 inches.

Now I'm going to make that same error be about 1/6-wave.

The trick is that we forgot to specify the color, or wavelength, of light that we used as our yardstick. Light of a blue-green color is about 500 nanometers, or about 0.00002 inches, and using that wavelength as our yardstick, the error of our objective is indeed 1/2-wave. But let's suppose we use red light of about 700 nanometers. Then the error drops to 1/3-wave, simply because the wave we're using as our yardstick is longer. With me so far?

(Actually, it's worse than that in some cases. The above analysis works for reflectors, but in the case of refractors, they're often very well-shaped for some colors, but not for others. For visual use, refractors should be corrected in the middle of the visible light range, about 550 nanometers. Testing them in red light may set you off by much more than the 50 percent or so in the case of mirrors.)

Then there's concern over where we're measuring the error. I've measured it at the wavefront—what happens after the light has already passed through the objective. But let's suppose we measure the error at the surface of, say, an objective mirror. If we have a pit of depth x in the mirror, then light hitting that pit is delayed by 2xx on the way in, and x on the way out. In other words, the surface error is half of the wavefront error. Now our 1/3-wave error drops to just 1/6-wave.

(This "half" rule only works for mirrors—it doesn't apply to objective lenses, so refractors aren't subject to quite such a drop.)

Wait, it gets better. I've talked about measuring the extent to which light rays overshoot or undershoot the mark, called P-V (peak-to-valley) error. What if what we want to know is how smooth the wavefront is, not in the worst cases, but just on average? Then we measure the error in terms of RMS (root mean square). Because of the way the RMS error is derived, it's impossible to make a hard rule about the relation between it and P-V error, but we might reasonably see that a P-V of 1/6-wave might become a 1/20-wave RMS.

So you see, if you want to be hard on an objective, you measure it P-V, at the wavefront, in a short wavelength like blue-green. If you want to be kind to the objective, you measure it RMS, at the surface, in a long wavelength like red. The difference can be something like an entire order of magnitude, and you need to be sure how your particular error is being measured. Measuring errors RMS without mentioning it is pretty slimy and I don't think anyone big does that, but measuring at the surface in a long wavelength is probably at least somewhat extant.

1/2-wave and higher errors are considered poor. A 1/3-wave error, I suppose, would be considered OK, though you'd hardly get anyone on this newsgroup to cop to that 1/4-wave error is considered minimally good, and 1/8-wave error or better, I would consider brilliant. These are all wavefront errors, measured P-V with a reasonable wavelength (typically something like 550 nm or so, near the middle of the visual range).

The quality of the image is also affected by other factors. Some on sci.astro.amateur would have you believe that the central obstruction on reflectors (non-tilted) and catadioptrics like SCTs and Maks is the all-important, overriding factor, but it is not; it is just one factor, like many others. It just happens to be very easily measurable, and there is plenty of theory to back up the impact. All in all, optical quality—smoothness and precision of the figure—is the most important.

Copyright (c) 1999 Brian Tung