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Wave Errors: Getting to the Root

EXPERIENCED AMATEURS have learned to pay attention not just to the wave
rating of a telescope but to how that number was measured. Late in the 19th century,
when Lord Rayleigh established his 1/4-wave tolerance as a criterion of telescope performance, he meant there should be no more than a 1/4-wave difference between the extreme "peak" and "valley" in the wavefront converging toward focus.
An alternative measure, the root-mean-square (r.m.s.) error, has gotten lots of bad press among amateurs because of a concern that manufacturers might use it to make their wares seems better than they really are. In reality, an r.m.s. error, when identified as such, offers a better "figure of merit" for comparing instruments because it takes into account both the amount and area covered by any defects. It is simply the square root of the sum of the squares of the error values sampled at hundreds of random points in the wavefront. Just keep in mind that the r.m.s. value is numerically 30 percent as large as the peak-to-valley number when a telescope suffers from pure spherical aberration. Such an instrument meets the Rayleigh criterion only if the r.m.s. value is 0.075 wave or less. It is easy to convert either measure of wavefront error
(peak-to-valley or r.m.s.) to its equivalent on the glass surface. For a mirror, divide by 2. For a lens, divide by (n - 1), where n is the refractive index of the glass. Since n is about 1.57 (on average) for the lenses used in astronomical telescopes, it follows that a lens may have 3.5 times as much surface error as a mirror before the same harm is done to the wavefront. The wavelength is a convenient unit of measure because it is so small. For the yellow-green light to which the human eye is most sensitive,
1 wavelength is 0.000022 inch or 0.00056 millimeter.

Reprinted with permission from Sky Publishing Corporation as excerpted from the September 1993 issue of Sky & Telescope, pages 83-87. Featured in Telescope Making edited by Roger W. Sinnott under the article titled "Rules of Thumb for Planetary Scopes --- II" authored by William P. Zmek.