Recently, a reader of these pages asked me what I meant by “distortion” — a word I use frequently here. My initial impuse was to ask, “Don't you have a dictionary?” But when I looked at several on-line dictionaries, I was dismayed to find that most of them never explain the uses of this word in optics.

Two kinds of “distortion”

Atmospheric optics

To begin with, it's used in two different ways. In connection with mirages and other phenomena of atmospheric optics, it means the changes in shape of the images of distant objects, which are usually either stretched or compressed by changes in lapse rate with height. It's important to notice that atmospheric refraction displaces images only in the vertical direction, because the surfaces of constant density in the air are very nearly horizontal. It's the gradient in refractivity that shifts images; and that gradient is nearly always close to vertical. (The main exception is very close to inclined or vertical surfaces, where the direction of the gradient is perpendicular to the surface — as in mural mirages, for example.)

In this context, distortion  refers to the vertical compression of images in stooping, and the vertical expansion of images in towering. But the atmospheric distortion of images is usually a lot more complicated than that; see the simulations of mirages — especially, those of superior mirages — for examples.

Lens design

The word is also used in connection with lens design, where “distortion” is one of the five classical aberrations of lenses. Here again, the term refers to the deformation of images; but in the optics of lens systems with rotational symmetry, the image displacements are in the radial direction, toward or away from the axis of the system. Distortion in lens systems changes the magnification with distance from the axis, so that straight lines in object space become curved in image space.

If we regard the refracting atmosphere as a lens, it's a very astigmatic one: almost all of its power in in the vertical, and the horizontal power is negligible. From that point of view, we can say that the vertical deformations produced by the atmosphere are something like the lens-makers' astigmatism; but in the context of lenses, the word distortion  is restricted to a particular effect of the third-order term in the expansion of the sine function, while the atmospheric kind of distortion encompasses all orders, and usually involves such high-order terms that the idea of a series expansion is useless.

Other ambiguous words

While we are thinking about ambiguous terminology, maybe it's worth mentioning the word aberration , which appears in two different contexts here.

Optical aberrations

In the previous section, I mentioned optical aberrations, one of which is distortion. There are five different aberrations — to be precise, five different third-order  aberrations — that texts on lens design deal with: spherical; coma; astigmatism; distortion; and field curvature. Each of these arises from the third-order term in the series expansion of the sine function. (There are two angles to be considered, so we have several effects that result from their interactions.)

Notice that distortion is just one of the five classical third-order aberrations. I was disgusted to find an on-line dictionary that said the words aberration  and distortion  were synonyms; distortion is just one of five third-order aberrations.

In addition, lens designers have to deal with chromatic aberration, which is due to the dispersion of all transparent physical media, whether solid, liquid, or gaseous. (Green flashes are caused by a combination of the dispersion and the distortion of atmospheric refraction.)

Astronomical aberration

But astronomers also deal with a different image displacement, which is due to the relative motion between a light source (such as a star or planet) and the observer. Historically, this effect was named the aberration of light . It was discovered by James Bradley in 1727, though its effects had been noticed earlier; Bradley managed to find a sensible explanation for it. (You will find that Wikipedia has a long page that will tell you more than you want to know about this.)

Today, Bradley's explanation is regarded as obsolete and incorrect. Properly, the aberration of light is a relativistic effect that requires the Special Theory of Relativity to be understood correctly. Fortunately, we need not go into its details here.

Just remember that most of the mentions of aberration  in the bibliography refer to this kind of “aberration”, not the kinds lens designers have to deal with.

© 2023 Andrew T. Young

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