Limits on use of spherical mirrors on Newtonian telescopes

From cosmoquest

In a Newtonian telescope, a parabolic mirror is usually used, since that
surface focuses light from infinity to a nearly perfect on-axis image. Some
telescope manufacturers (and a few amateurs) substitute longer f/ratio
spherical mirrors, since they may sometimes be easier for some people to
figure and test rapidly. However, if the f/ratio isn't long enough, the
performance (especially at high power and for planetary viewing) may suffer.
One way to rate telescope mirrors is by seeing how much their surfaces
deviate from a perfect parabolic shape. One common rule of thumb states
that the telescope's optics must not produce a wavefront error of more than
1/4 wave in order to prevent optical degradation. This requirement is
sometimes extended somewhat to require that the mirror's surface must not
deviate from a "perfect" paraboloid surface by more than an eighth wave
(approximately 2.71 millionths of an inch) in order for the mirror to be
considered for astronomical use. By comparing the sagittal depths of a sphere
and a parabola of equal focal length, it can be seen that the difference
between the two often exceeds the rule of thumb by quite a margin for short
and moderate f/ratios. A spherical surface can be "fudged" into deviating
less strongly from a parabolic shape by extending the focal length very
slightly, such that its surface would "touch" a similar parabolic mirror's
surface at its center and at its outside edges. This minimizes the surface
difference between the two. Such spherical mirrors must have a minimum
f/ratio in order to achieve this. According to Texereau (HOW TO MAKE A
TELESCOPE, p.19)

the formula is 88.6D**4 = f**3 (** means to the power of:
ie: 2**3 = "two cubed" = 8),

where f is the focal length and D is the aperture (in inches).
Substituting F=f/D to get the f/ratio, we get:
F = cube-root (88.6*D).

The following minimums can just achieve the 1/8th wave surface rule of thumb:

APERTURE TEXEREAU MINIMUM F/RATIO
3 inch f/6.4
4 inch f/7.1
6 inch f/8.1
8 inch f/8.9
10 inch f/9.6
12 inch f/10.2

The above f/ratios might be fairly usable for an astronomical telescope's
spherical primary mirror, as they do just barely satisfy the 1/4 wave
"Rayleigh Limit" for wavefront error. However, amateurs looking for the best
in high-power contrast and detail in telescopic images (especially those
doing planetary observations) might be a little disappointed in the
performance of spherical mirrors with the above f/ratios. Practical
experience has shown that at high power, the images produced by spherical
mirrors of the above f/ratios or less tend to lack a little of the image
quality present in telescopes equipped with parabolic mirrors of the same
f/ratios.
In reality, it is more important to consider what happens at the focus of
telescope, rather than just how close the surface is to a parabolic shape.
In general, spherical mirrors do not focus light from a star to a point.
Their curves and slopes are not similar enough to a paraboloid to focus the
light properly at short and moderate f/ratios. This effect is known as
"Spherical Aberration" and causes the light to only roughly converge into
what is known as "the Circle of Least Confusion", (see: ASTRONOMICAL OPTICS,
by Daniel J. Schroeder, c. 1987, Academic Press, p.48-49).

This "circle" is a blur the size of about (D**3)/(32R**3),

where D is the diameter of the mirror and R is its radius of curvature.
The larger the radius of curvature is, the smaller the circle of least confusion is.
 If the circle of least
confusion is a good deal larger than the diffraction disk of a perfect
imaging system of that aperture, the image may tend to look a little woolly,
with slightly reduced high power contrast and detail. For example, for the
Texereau use of a 6 inch f/8.1 spherical mirror, the circle of least
confusion is nearly *1.7 times* the size of the diffraction disk produced by
a perfect 6 inch aperture optical system.
For most spherical mirrors focusing light from infinity, the focal length
is about half the mirror's radius of curvature. Thus, to improve the image,
we can use f/ratios longer than Texereau's limits to reduce the size of the
circle of least confusion to a point where it is equal to the size of a
parabolic mirror's diffraction disk (ie: "Diffraction-limited" optics).
NOTE: the term "Diffraction-limited" has a variety of interpretations, such
as the Marechal 1/14 wave RMS wavefront deviation, as well as the more
commonly referred to 1/4 wave P-V "Rayleigh Limit". If we set the angle the
confusion circle subtends at a point at the center of the mirror's surface
equal to the resolution limit of the aperture of a "perfect" paraboloidal
mirror (which is 1.22(Lambda)/D, where Lambda is the wavelength of light), we
can come to a formula for the minimum f/ratio needed for a sphere to produce
a truly "diffraction-limited" image. That relation is: D = .00854(F**3)
(for D in centimeters and F is the f/ratio), and for English units:
D = .00336(F**3). Thus, the minimum f/ratio goes as the cube root of the
mirror diameter, or the DIFFRACTION-LIMITED F/RATIO: F = 6.675(D**(1/3)).
For example, the typical "department store" 3 inch Newtonian frequently uses
a spherical f/10 mirror, and should give reasonably good images as long as
the figure is smooth and the secondary mirror isn't terribly big. For common
apertures, the following approximate minimum f/ratios for Diffraction-Limited
Newtonians using spherical primary mirrors can be found below:

APERTURE F/RATIO FOR DIFFRACTION-LIMITED SPHERICAL MIRRORS
-----------------------------------------------------------------------------
3 inches f/9.6 (28.8 inch focal length)
4 inches f/10.6 (42.4 inch focal length)
6 inches f/12.1 (72.6 inch focal length)
8 inches f/13.4 (107.2 inch focal length)
10 inches f/14.4 (144 inch focal length)
12 inches f/15.3 (183.6 inch focal length)

Using f/ratios fairly close to those above for spherical mirrors in Newtonian
telescopes should yield very good low and high power images. However,
spherical mirrors with f/ratios significantly smaller than those listed above
or given by our second formula can yield high power views which may be a bit
lacking in sharpness, contrast, and detail. Indeed, a few commercial
telescope manufacturers routinely use spherical mirrors at f/ratios even
shorter than those given by Texereau, and these products should be avoided.
An eight inch Newtonian using an f/13.4 spherical mirror could produce good
images, but would also have a tube length of nearly 9 feet, making it harder
to mount, use, store, and keep collimated(due to tube flexure under its own weight). Thus, using spherical mirrors for
diffraction-limited Newtonians with the above f/ratios for apertures above 6
inches is probably somewhat impractical. The old argument about eyepieces
performing better with long-focal length telescopes has been all but negated
by the recent improvements in eyepiece design. Those who are grinding their
own mirrors might wish to make spherical mirrors with f/ratios between the
Texereau values and the fully-diffraction-limited numbers, as these could
still yield fairly good performance without the need for parabolizing.

In the long run, it is probably better to use a well-figured (1/8th wave
wavefront error or less) parabolic primary mirror for moderate focal ratios
and a small secondary mirror (obstructing 20 percent or less of the primary
mirror diameter) rather than using a spherical mirror in moderate to large-
sized Newtonians designed for planetary viewing.

David Knisely, Prairie Astronomy Club, Inc.

from cloudynights

OTOH, Roger Sinnott on skyandtelescope gives a less strict formula:

Here is the formula for the wavefront error, e, when a spherical mirror is used as a Newtonian primary:

e = 22 D / F^3,

where D is the primary mirror’s diameter in inches and F is the focal ratio. The error of your 4.5-inch f/7.7 sphere works out to be 0.2 or 1/5 wave, so your spherical mirror already meets the Rayleigh 1/4-wave tolerance for optical performance. You may notice a slight improvement with the paraboloid, but not much.