(Reprinted from David Knisely, Prairie Astronomy Club, Inc. to ensure it doesn’t disappear into the ether)

In a Newtonian telescope, a parabolic mirror is usually used, since thatsurface 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:

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:

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.

By Gord

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