From: Jerry Hudson 
To: atm{at}shore.net
Reply-To: Jerry Hudson 


The formula for the longitudinal aberration for any zone for a mirror under
test at the center of curvature is

    L.A. = e^2 z,

where
    L.A. = Longitudinal aberration: the distance
behind the central zone where a normal to the surface would intersect the
axis, in this case,
    e = Eccentricity of the conic section:
            Sphere => e = 0
            Prolate ellipse => 0 < e < 1
            Parabola => e = 1
            Hyperbola => e > 1.
    z = Sagitta for the mirror.  For any conic, this
is
        z = c r^2 / 2  - c^3 K r^2 / 8 + ...
where
    c = Central zone's curvature = 1/R
    r = Radius of zone from center of mirror
    K = Conic constant = e^2 - 1, e the eccentricity


Note the reading for fixed source and moving knife edge will measure TWICE
the L.A. given above.

Note that this result is in contradiction to Texereau's, which I believe to
be in error.  Accordingly, I enclose a derivation of my own formula.

- Jerry Hudson


Appendix:  Derivation of L.A. for conic section under test

From Herzberger's formula for a conic section:

    c K z^2 + 2 z - c r^2 = 0

where
    c = Curvature = 1/R, R the vertex radius of curvature
    K = Conic constant = e^2 - 1
    z = Sagitta
    r = Zone radius

Note that this can represent any conic section that is a surface of
revolution about the axis, with the vertex at r=0, z=0.

Differentiate the above with respect to r:

    dz/dr = c r / (R + cKz)

Note that this is also the slope of the surface normal at r, which is equal to

    slope = r / ( R - z + L.A. ).

It helps to draw a diagram, which I can't submit to the List,
unfortunately.  Here's an attempt:

         (Point r,z)
       *-      Surface normal:
      /  \-    Slope = r / (R - z + L.A.)
     | |   \-
     /       \-
    |  |       \-
   -A--B------C--D

where the points
    A = Vertex of curve
    B = Point directly under the zone
    C = Vertex center of curvature
    D = Normal (ray) intercept
Note that AB = sag = z
          AC = radius of curvature, R
          CD = L.A.
          B* = r
          slope = B* / BD

    Equating the two slopes, we obtain

    r / (R - z + L.A.) = c r / (R + cKz),

    L.A. = -R + z + (1 + cKz)/c

        = (1 + K) z = e^2 z (Q.E.D.)

--- BBBS/NT v4.00 MP