[ATM] what causes coma?

[dgood at aha.org (Good, Donald)]

Eureka, I've got it.  After studying the article About Coma at:
http://www.opticalmechanics.com/about_coma.htm
I suddenly realize what coma is!!

Mark, you are right, what I described previously was not coma, although I am
not sure that it is exactly  curvature of field.  Maybe it was a mix of
curvature, coma, and caustic.  A definite error was the  implication that
the "focus someplace else" was a focus to a point.  But lets let that go and
start over.

In About Coma, James Mulherin describes several relationships of coma
prominence to the angular distance from  the center of the field of view
(FOV), to the F number of the mirror, and to other parameters.  And I
believe  that, while he refers to parabolic and Newtonian and in one case,
to "all telescopes of a given focal ratio",  he is actually referring to the
classical (parabolic) Newtonian in all cases.  As to Mark's question about
coma in a spherical mirror, I will answer Yes, because I am going to relate
coma directly to spherical  aberration.  I am also going to describe (I
hope, in a simple, qualitative way - no math) the shape of coma  and the
peculiar double ring in Figure 2.  In this explanation, the airy disk and
diffraction effects will be  ignored.  Here goes.

Point 1 - A column of parallel incoming light, the diameter D of the mirror
and on axis with the mirror  (light from an infinitely far point source on
axis) will focus to a point on the axis in a parabolic mirror.   This
represents the object at the center of FOV.

Point 2 - That same column of light in a spherical mirror will not focus to
a point, but to a range of points  along the axis at approximately one half
of the radius of curvature of the sphere.  This range of points is  the
cause of spherical aberration.  Light reflected from the edge of the mirror
will focus closer to the  mirror than light reflected from the central
region of the mirror.

Lets look more closely at spherical aberration.  Light reflected from the
edge zone forms a cone (shell, not  solid) of light that focuses at a
distance E from the mirror on the axis.  Light reflected from the central
zone forms another cone of light that focuses at a distance C from the
mirror on the axis.  C is a little  greater than E.  The "best" focus is
some point B on the axis between E and C which represents the focus of a
cone from some intermediate radius of the mirror.  Lets put a small circular
piece of paper between E and C  perpendicular to the axis and on center,
representing the focal plane.  Close to E, the "E" cone focuses to a  point,
but the "C" cone has not reached focus and is cut off, forming a circle.
The "B" cone also forms a  smaller concentric circle.  As the paper is moved
toward C, the "B" and "C" circles get smaller until at B,  the "B" cone
reaches a point and then at C, the "C" cone reaches a point.  At the same
time, the "E" cone starts to expand past its focus forming a circle of
increasing diameter.  Between B and C, the "B" circle also starts to expand,
and beyond C all circles are expanding.  At B, we have a point focus of the
"B" cone, an expanding circle of the "E" cone and a shrinking circle of the
"C" cone.  Of course, all the other  intermediate cones are either expanding
in proportion between the edge zone and the "B" zone and shrinking in
proportion between the "B" zone and the central zone.  This lack of common
focus IS spherical aberration.

Now lets tilt the column of light (still parallel rays) a little over the
spherical mirror, representing a  far point object near the edge of FOV.  We
can easily make the tilt small enough that there is still a point  on the
mirror away from the mirror center that is perpendicular to the axis of the
light column although not  on that axis.  In other words, there is a
parallel ray somewhere in the light column that strikes the mirror
perpendicularly.  What is the effect of points E, B, and C along this new
ray?  NO DIFFERENCE!!  Due to the  symmetry of a sphere, any ray
perpendicular to the surface is equivalent.  The spherical aberration is
exactly  the same.

HERE IT IS!!
Now lets change the mirror surface under the tilted light column very
slightly, from a sphere to a parabola  along the original axis of the
mirror.  What happens to these circles from the various cones centered on
that  perpendicular (off-center) ray?  They are displaced perpendicular to
that ray slightly, forming a series of  off-center circles (almost circles,
there is a little flattening) that you see in the spot diagrams.  Oddly
enough, both the inside focus and outside focus circles are displaced in the
same direction in proportion to their sizes, giving the teardrop shaped spot
diagram.  The small end of the teardrop represents the point focus of the
"B" cone.  At best focus, the "E" cone and the "C" cone are represented by
the large end of the teardrop, being nearly the same size and nearly
concentric.  As the focus is change, one will get bigger and the other will
get smaller.  So coma is the slight distortion of spherical aberration due
to a parabaloid mirror instead of a spherical one.

At least that is how I see it.  It seems so simple that I feel it must be
the truth.
Don Good




-----Original Message-----
From: Mark Holm [mailto:mdholm@telerama.com]
Sent: Monday, February 09, 2004 8:56 PM
To: ATM Mailing List
Subject: RE: [ATM] what causes coma?


I think Donald Good's explanation is really a description of field 
curvature, an abberation also present in many reflecting telescope 
designs.  With field curvature, if you could warp your detector to fit 
the curvature (possible with photographic film and plates if the 
curvature isn't too strong), the abberation would vanish.  With coma, 
the abberation would still be there, even with field curvature 
corrected, no?  Schmidt cameras have no coma, and, I think, no 
astigmatism, and most or all of the spherical abberation canceled by the 
corrector plate, but, at least in the "pure" design, there is major 
field curvature.

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.

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

Mark Holm
mdholm@telerama.com

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