In this little exposition, I'm going to explain coma and other geometric aberrations from first principles and elementary geometry.

Suppose we have a spherical mirror, whose cross section we represent
by the unit circle centered at the origin *O*. Consider three
incoming rays of light; we will represent these rays by the following
three chords:

AB, which passes horizontally through the origin;

PQ, which passes horizontally, but aboveAB; and

P'Q', which passes horizontally and an equal amount belowAB.

These three rays, after reflecting off the right semicircle,
intersect the line *AB* at a point *D*, which can be
constructed straightforwardly. Simply draw the perpendicular bisectors
of *OQ* and *OQ'*. These will both intersect *AB*
at *D*. The following diagram should make this clear:

Figure 1.

From this diagram, we can see that by constructing point *D*
where the perpendicular bisector of (for example) *OQ* intersects
*AB*, we have constructed an isosceles triangle, *OQD*.
From this, it follows that angle *OQD* is congruent to angle
*QOD*. It is also evident that angle *QOD* is congruent
to angle *OQP*. Thus, angle *OQD* is congruent to angle
*OQP*, and the law of reflection is satisfied. A similar argument
applies for the reflection of the ray of light at *P'Q'*.

Next, let's consider what happens to these three rays if they are
still parallel, still strike the right semicircle at *Q*,
*B*, and *Q'*, but are no longer horizontal. Let's call
these three new rays, *RQ*, *CB*, and *R'Q'*. We
shall see that the reflections of these three new rays no longer
intersect at one point. If we follow the reflection of *R'Q'*,
we see in Figure 2 that it intersects the reflection of *RQ* at
*E*, then intersects the reflection of *CB* at
*V*.

Figure 2.

Suppose we try to determine the size of angle *QEQ'*. We see
that it is larger than angle *QDQ'* by an amount equal to angle
*DQE* minus angle *DQ'E*. But angle *DQE* is
congruent to angle *RQP*, and angle *DQ'E* is congruent
to angle *R'Q'P'*. Since the incoming rays are still parallel,
these angles must all be equal, so that angle *QDQ'* is congruent
to angle *QEQ'*.

But the locus of all points *E* such that angle *QEQ'*
is congruent to angle *QDQ'* is simply the unique circle containing
the points *Q*, *D*, and *Q'*! In other words, no
matter the angle *CBA*—that is, no matter how far off axis the
object and image are—the intersection of the rays at *RQ* and
*R'Q'* must land somewhere on this circle. Let the center of this
circle be called *S*. (Note that if angle *QOQ'* is equal
to 90 degrees, then *Q*, *D*, and *Q'* all lie on
a straight line, so that *S* is at infinity, and if angle
*QOQ'* is greater than 90 degrees, then *S* is actually
to the *left* of *D*.) Similar reasoning shows us that angle
*BVQ'* is congruent to angle *BDQ'*, so the locus of all
points *V* is simply the unique circle *U* containing the
three points *B*, *D*, and *Q'*. This is all shown
in Figure 3.

Figure 3.

Now, let's consider what happens if we return to the horizontal
rays, but move *PQ* and *P'Q'* steadily closer toward the
diameter *AB*. In that case, the point *D* steadily moves
toward the point *F*, halfway between *O* and *B*,
which is the nominal focal point of the mirror. In fact, from the above
discussion, *D* is precisely as far from *O* as is the
point *T*, which is the the point on *OQ* directly above
*F*. (See Figure 4.)

Figure 4.

In the limit, as *PQ* and *P'Q'* both move toward
*AB*, the circles *S* and *U* converge to the
circle *G*, which has radius 1/4, and intersects *AB*
at *F* and *B*. (See Figure 5.)

Figure 5.

We can now characterize various aberrations in terms of these
geometrical figures. The movement of *D* toward *F*
as *Q* and *Q'* approach each other, and toward
*B* as they separate, is spherical aberration. The curvature
of circle *G* is the field curvature. And the amount
by which *E* is under the reflection line *BV* is what
we call coma (tangential coma, to be more specific). Note that coma
varies with both off-axis angle (the angle *CBA*) and the
aperture (the separation of *Q* and *Q'*). Finally,
note that in Figure 3, *D* is to the left of *E*; the
horizontal distance between them (ignoring the vertical separation)
is the astigmatism.

An interesting observation from this derivation is that a
spherical mirror has both coma and, of course, spherical aberration.
A paraboloidal mirror would have *D* coincident with *F*
no matter how far apart *Q* and *Q'* were, so it does
not possess spherical aberration. However, since the circles
*S* and *U* still exist, it does have coma, and since
they both converge to *G* as *Q* and *Q'*
approach each other, it has field curvature as well.

Copyright (c) 2002 Brian Tung