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First of all, what is distortion? Technically, there are quite a few types of distortion in optical instruments, but radial distortion is of most relevance to rifle scopes. Essentially, it describes an instrument’s ability to show you straight lines: if lines that are supposed to be straight, look curved near the edges of the image (and the further from the center, the more curved they appear) you have distortion. There are two types of it: pincushion (lines curve inward) and barrel (lines curve outward). Most types of sporting optics actually build slight amount of pincushion distortion into the image in order to counteract the rolling ball effect when panning. Generally, a lot of variable magnification riflescopes have visible edge distortion at lower magnifications and, by and large, it is not very interesting. We aim with the center portion of the image, so whatever is at the edge is not all that important for as long as it is not strong enough to distract your attention from the point of aim.
As far as optical aberrations go, they can be sub-divided into two types: monochromatic aberrations and color aberrations.
I will not spend a whole lot of time on monochromatic aberrations. They include things like spherical aberration, coma, astigmatism, etc. For the most part these aberrations are a by-product of paraxial approximation in optical design and/or of the fact that most lenses have spherical surfaces. Paraxial approximation simply implies that when the original optical design was done, the designer assumed that the light rays are not going to deviate too much from the optical axis of the scope, and in most cases that holds true. However, some riflescope configurations really push the limits of this approximation, for example compact scopes with large objective lenses.
Spherical lens surfaces have both strengths and weaknesses. Truthfully, they only have one advantage, but it is an important one: they are easy to manufacture very precisely. The main weakness is that the light that goes through the part of the lens that is away from the optical axis is focused at a slightly different spot than the light that is close to the optical axis. Hence, the larger the lens the more apparent this, appropriately named, spherical aberration becomes. Most modern scopes do a very good job of correcting for spherical aberrations, but if you stumble onto a scope with the image near the edges looking a bit out of focus, you are likely looking at some residual uncorrected spherical aberration. A lot of other optical aberrations are also a result of off-axis performance.
By making lens systems a bit more complicated (i.e. by adding appropriately shaped lenses), a lot of the aberrations (including ones I did not mention here) can be corrected, but every extra optical element adds size, weight and cost. It also adds additional surfaces that can cause reflections. Just like everywhere else, there is no free lunch here.
Color aberrations usually stem from the fact that most materials have different indices of refraction at different wavelengths. Refraction is the material property that makes lenses work: when light goes from air into glass, it changes direction a little bit (“bends”, for lack of a better word). How much light changes direction as it enters the glass is determined by the glass’ index of refraction. If we take into consideration primary colors only (blue, green and red), it turns out that for most typical glass, the refractive index for blue light is higher than for green light, which is higher that that for red light. That makes the focal length of the lens slightly shorter for blue light than for green (and red). For similar reasons, the image details of different color can end up reconstructed at different distance from the optical axis. Both of these effects are referred to as Chromatic Aberration (often shortened as CA). The effects of different focal length for different colors are called longitudinal CA, and the color-dependent variations in detail reconstruction around the optical axis are referred to as lateral CA.
In riflescopes, CA is usually an artefact of objective lens system and can be largely corrected with sufficiently sophisticated design. It is also more prominent in higher magnification scopes where the tolerances are generally tighter and various aberrations are more visible. This is where Extra Low Dispersion (ED) glass comes in. When properly used, an addition of an ED glass element can help eliminate any noticeable CA from an optical system. The use of ED glass in riflescopes is not very prominent at the moment, but it is becoming more common. If you happen to need a riflescope with the highest possible magnification, it is worth your while to look at offerings with ED glass in the design.
Both types of chromatic aberration (if visible in your scope) show up as colored fringed at the interface between bright and dark objects within the field of view. The actual color varies depending on the scope design. Red or violet fringes are most common, but I have seen CA of almost every color of the rainbow over the years. Lateral CA, as the name suggests, becomes most pronounced the further you get from the center of the image, while longitudinal CA is visible along the optical axis as well.
My take on Chromatic Aberration in riflescopes, by and large, is the same as that on all aberrations and distortions: I want the center of the image (i.e. the aiming point) to be clear, sharp and as devoid of undesirable artefacts as possible. I am, however, far more forgiving toward edge effects as long as they are not strong enough to be distracting.