[ATM] what causes coma?

[djv at bedford.net (Woodchuck)]

On Mon, 9 Feb 2004, Mark Holm wrote:

> The explanation at the OMI web site is probably technically correct, but
> I find it somewhat incomplete because, the way it describes coma makes
> it seem that coma could arise from a parabolic or spherical mirror.  I
> think spheres are coma free (if I am remembering what I have read
> correctly) but of course spheres have lots of spherical abberation.

Coma can be present in all-spherical systems, don't worry.

Spheres do not have "lots" of spherical aberration, they just have
what they have.  At some distances, this amount is zero.

> Maybe I am just not understanding what has been written.  Anybody else
> want to try an explanation that doesn't involve too much math?

Coma and astigmatism (and the other primary aberrations) are
mathematical abstractions, rather than physical realities, so
discussing them without mathematics is a bit hard.  Can I talk about
square roots or derivatives without mathematics?  To a degree, I
suppose.  It is the lack of drawings that *really* makes it hard
to talk about.

The two main (from our point of view -- narrow-angle visual
instruments) off-axis aberrations are coma and astigmatism.  For
coma, the different zones of the system focus at different distances
(focal lengths) and also at different angles.  The central rays
focus to the "point" of the coma, the larger zones at distances
farther away.

Astigmatism is different; it can be thought of as looking at the
system (off-axis) with a cross-shaped mask opening, the vertical
and horizontal bands of the cross have different focal lengths.
But those foci lie on a line with the center of the mirror and the
central focus.  Thus in a system with only astigmatism, there is a
"compromise" focus that has a symmetrical (circular) disk of
confusion; with coma there is no focus with a disk-shaped region
of confusion.  This makes coma a very serious fault for astrographic
instruments.

Astigmatism-the-aberration should be distinguished from astigmatism-
the-manufacturing-fault.  Perfectly made and mounted optics can
show the aberration.  It is unfortunate that the same word is used
for two distinct conditions.

That's a completely inadequate discussion.  A book on optical design
using geometric methods (A. E. Conrady, R. Kingslake, others) should
be consulted.

Thought experiment:  Picture the standard drawing illustrating primary
spherical aberration:  the axial rays come to a focus farther from the
mirror than the zonal (edge) rays.  Notice that this is a two-dimensional
picture.  Still in two dimensions, imagine the picture with an
off-axis bundle of rays.  The central rays reflect, come to a focus.
The edge rays come to a focus, too, but not on the line connecting
the central focus with the center of the mirror.  That is coma --
each radial zone of the mirror reflects light at a different angle
for an off-axis bundle.

Now imagine that this coma has been somehow corrected, so that the
central rays and the edge rays come to a focus along the same line,
and that this line passes through the center of the mirror.  But
the two foci don't coincide -- imagine they are like spherical
aberration, one closer to the mirror than the other.  This is, in
fact, spherical aberration off-axis, if this were the only aberration
present.  Now imagine the drawing becoming three dimensional.  The
central rays still come to the same focus.  But the edge rays are
now of two kinds.  Imagine the edge zone of the mirror to have a
clock face.  Consider a pair of zonal rays that come from the 12
o'clock and 6 o'clock, and that they come to a focus, say, closer
to the mirror than the axial rays.  Consider a pair now from 9
o'clock and 3 o-clock, and imagine that they come to a focus, still
along the "axis", but farther from the mirror than the axial rays.
This is astigmatism.

... or so I delude myself.

Dave