I am a little bit confused about the precise statement of the
paraxial approximation. For simplicity, let's restrict the discussion
to image formation by a single curved mirror. Consider the following
two statements:
(A) optical rays make small angles relative to the principal axis
(B) optical rays strike the mirror near the vertex V (defined as the
point of intersection of the principal axis with the mirror)
QUESTION I. The paraxial approximation is defined to hold whenever:
(1) rays satisfy (A) regardless of whether or not they satisfy (B)
(2) rays satisfy (B) regardless of whether or not they satisfy (A)
(3) rays satisfy both (A) and (B)
WHICH OF THE THREE CHOICES (1) TO (3) IS CORRECT?
Before answering, consider a second, application question. (After
all, one can make any definition one likes. Without an application in
mind, it just becomes semantics instead of physics.)
QUESTION II. If the paraxial approximation is valid for a spherical
mirror, then one can state:
(4) there will be no aberrations of any kind
(5) some kinds of aberration will be eliminated, but other kinds will remain
(6) the paraxial approximation isn't very useful/relevant/interesting
when applied to spherical mirrors; it's primary application is to
other shapes (perhaps parabolic, elliptical, etc)
WHICH OF THE THREE CHOICES (4) TO (6) IS CORRECT?
I hope you see where I'm going with this. My reading of typical intro
textbooks is that their answers are (1) and (4). (Check out the intro
text *you* use to see.) But these two answers are inconsistent with
each other! For example, a collimated beam can be incident [so that
every ray is parallel to the principal axis and (A) certainly holds]
and yet one will get spherical aberrations if some incident rays are
far off-axis. I believe some texts try to avoid this by instead
choosing answer (3). But now I'm wondering whether that's overkill.
Wouldn't choice (2) avoid spherical aberrations? One might think
you'd get coma, but I'm not sure that's correct. It's true the focal
surface is curved, but provided I use a mirror of small diameter (or
equivalently aperture it right near it), won't an off-axis object
give a reasonably sharp image nonetheless (albeit located off-axis)?