In this article we present a simple method of determining how far a
"parabolic" mirror actually departs from parabolic figure, followed by
a discussion on recommended tolerances.

**The chart in Figure 1 shows the surface error on the mirror corresponds to a
given measured departure from the desired knife-edge readings. It works
most conveniently with readings from a Foucault tester which measures knife-edge
position in thousandths of an inch ( such as that described in
Texereau's book*
).**

Apparently a chart of this sort was first made in 1945 by Paul E. Luce, who
had been a member of the optical division of the Amateur Astronomers Association
of New York City. I have modified his chart to make it more versatile.

HOW TO USE THE CHART:

**We make a set of knife-edge measurements for various zones of the
mirror. Beside these we tabulate the desired knife-edge positions ( r**^{2}
/ R ) for the measured zones. To the measurements we add or subtract a
constant so that the resulting numbers agree with the r^{2} /
R values most closely for the outer
zones of the mirror (
where a mirrors figure is most important ). We then subtract these
modified measurements from the r^{2} / R values: The largest difference
thus found ( which usually will be for the central zone of the mirror ) is what
we call the maximum knife-edge error. We draw a line across the chart from
point representing the knife-edge error and note where this line intersects the vertical
line representing the mirrors focal ratio. Comparing this intersection
point with the lines running diagonally from the lower left to the upper right
tells us the maximum size of the mirror's surface error, in wavelengths of
light.

**This procedure is simpler than it may sound, as a couple of examples will
make clear. Suppose, for instance, that we test a 6 inch f/8 mirror,
making careful knife-edge settings for the following four zones: r = .3 inch,
1.2 inch, 2.1 inch, 2.7 inch.. ( Note that the last zone is about 1/4 inch
in from the mirror's edge, as the extreme edge is difficult to measure
accurately. ) And suppose that we get the results listed in column C of
Table 1. The values of r**^{2} / R for the four zones are
listed in column D.

**We find a constant ( in this case 0.124 inch ) which when subtracted from the
measurements in column C gives numbers in column E which are nearly identical in
zones 3 and 4 to the r**^{2} / R values. Finally, column F lists the
knife-edge errors ( column D minus column E ). The largest residual,
-0.011 inch is that for zone 1 as usual. In Figure 2 I have drawn a heavy
horizontal line at f8, the mirror's focal ratio. These lines intersect
very near the 1/32 wave chart line, indicating that the largest surface error on
the mirror is approximately 1/32 wave.

As a second example, we test an 8 inch mirror of 40 inch focal length, using
a Foucault tester adjusted to give a zero reading for the central zone.
Table II presents the results. The constant in this case is 0.023.
The maximum knife-edge error turns out to be+0.025 inch. Plotting this
value on the chart with the mirror's focal ratio, f5, we find the surface error
to be a bit less than 1/4 wave-- say 1/5 wave.

**An easy way to find a constant which is to be added to or subtracted from the
knife-edge reading is as follows: for zone 3 subtract r**^{2}
/ R from the knife-edge reading; do the same for zone 4; the average of the
numbers so obtained is the desired constant. Thus for the 6 inch mirror of
Table I, we have 0.172 - 0.046 = 0.126 and 0.198 - 0.076 = 0.122, so the constant
is 0.124. As a check on the arithmetic, when the procedure is followed correctly, the final knife-edge errors for zone 3 and 4 will be opposite in sign
and approximately equal in size.

When the knife-edge error for zone 1 is a negative number, the mirror is
under corrected ( a prolate spheroid, or a "partially parabolized"
mirror ) ; when the zone 1 error is positive, the mirror is overcorrected ( a
hyperboloid ).

**A question often asked, especially by neophyte mirror makers who wisely
would rather not rely upon uncertain judgments of Foucault shadows, is,
"At what focal ratio may a mirror be left spherical and still perform
satisfactorily?" This question may be answered by referring to the
chart lines drawn from upper left to middle right, which are plots of r**^{2} / R
for common mirror diameters. The focal ratio at which a spherical mirror
is within a given tolerance of a parabola is the focal ratio at which the
mirror's r^{2} / R line and the desired surface error line
intersect. Thus, for example, a 6 inch sphere is parabolic within 1/4
wave at f8.5 and a 4.25 inch sphere is within 1/16 wave surface error of a
parabolic figure at f12, as the chart shows.

When may we consider a mirror finished? Answering this involves
answering three questions, as follows:

( 1 ) In the Foucault test, does the mirror present a good "doughnut"
figure, with smoothly blending shadows? Any irregularities in the figure
should be polished out.

( 2 ) What about the mirror edge? Usually this is the least
satisfactory part of an amateur-made mirror, thanks to the cult of the perfectly
figured edge which has been promulgated in all of the books on telescope
making. Turned edges ruin more Newtonian telescope images than any other
cause except bad diagonals and metal telescope tubes. Rare is the ATM
who can produce a parabolic mirror with a perfectly figured edge.

But many ATM's don't know a defective edge when they see one, because
the Foucault test is not adequate for detecting edge defects. The
mirror's edge must be tested either by a Ronchi test or by the knife-edge
diffraction test described in section II-25 of Texereau's book.

In spite of what all the telescope making books say, the mirror need not
have a good edge at the end of figuring. There is no point in ruining
the figure of the rest of the mirror in an attempt to correct a narrow turned
edge. But if the edge is not absolutely perfect, it must be either
ground off of masked before the mirror is installed in the telescope.
Most professionals bevel the edge after figuring, thus grinding off edge
defects.

( 3 ) Is the mirror's surface sufficiently close to the desired
parabola. as determined, for example, by the chart introduced in this article
or by the method described in Texereau's book? Here, of course, we must
first decide how close to parabolic figure is "sufficiently
close". The figuring tolerance depends critically upon what the
telescope will be used for. For the all-round observing in which the
average amateur engages, the classic Rayleigh limit is an adequate tolerance.

Despite popular misconception, the Rayleigh limit does not refer to a
1/4 wave maximum surface error on the primary mirror. Rather, it
specifies a maximum "optical path difference" or "wavefront
error" of one quarter wavelength of light. For practical purposes
we may say that the wavefront error is equal to 1 1/2 times the primary
mirror's surface error. Thus it turns out that a telescope consisting of
a primary mirror which is parabolic within 1/10 wave and a diagonal mirror
which is flat within 1/10 wave barely meets the Rayleigh criterion, since

1.5 ( 1/10 ) + ( 1/10 ) = 0.15 + 0.10 = 0.25 = 1/4 wave.

**A telescope which is to be used extensively for observation of fine planetary
detail should have a maximum wavefront error of 1/10 wave, which implies a
1/30 wave primary and a 1/20 wave diagonal. Such instruments are rare,
of course. At the other end of the scale, a telescope with a maximum
wavefront error of 1/2 wave ( e.g. a 1/4 wave primary and a 1/8 wave diagonal
) will fulfill the needs of many casual observers, as it will produce a sharp
star image at low and medium magnifications and will provide pleasing views of
the moon, wide double stars, Jupiter's moons and Saturn's rings, prominent
star clusters, bright nebula and galaxies and so forth.**

*Jean Texereau, How to Make a Telescope, Dover ( clothbound ); also
available in cloth bound reprint.