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Parabolic v. Spherical Mirrors

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#1 mnpd

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Posted 09 October 2005 - 04:07 PM

I think I understand the effect mechanics of parabolic v. spherical primary mirrors, but I really don't understand just how the surface figures differ between the two.

It's too tall of an order to ask anyone to explain theory on a forum, but I do have one question. Absent test equipment, is there a simple way to determine whether any given in-hand primary is parabolic or spherical?

#2 sixela

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Posted 09 October 2005 - 05:20 PM

By star testing it. Unsurprisingly, a spherical mirror will show spherical aberration (obviously, by definition, the mirror is not corrected, i.e. very uncorrected).

Figure four on e.g.:

http://www.astrosurf.../startest_e.htm

Of course, the smaller the mirror and the slower the f/ratio, the less you're going to see the difference - which is precisely why small large focal length mirrors *can* be left spherical and still yield a good image.

But an 8" f/4, on the other hand,... :foreheadslap:

#3 David Knisely

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Posted 10 October 2005 - 12:19 AM

LIMITS ON THE USE OF SPHERICAL MIRRORS FOR NEWTONIAN TELESCOPES

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) subtitute 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 due to the spherical aberration spherical mirrors have, which gets larger as the f/ratio gets shorter. 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 degredation. This requirement is somtimes extended somewhat to require that the mirror's surface must not deviate from a "perfect" paraboloidal 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 sagital 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 parbolic 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: i.e.: 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 somewhat 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. 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. Clear skies to you.

#4 mnpd

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Posted 10 October 2005 - 01:00 AM

Thank you, Sir. It'll take me a while to digest all that, but I'll much smarter after I'm fnished.

#5 werewolf6977

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Posted 10 October 2005 - 06:20 AM

So does this mean that my 130mm F/6.9 Orion spherical primary is no good? To repeat myself, this is the scope that I saw the Cassini Div with first, not my N8.

#6 sixela

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Posted 10 October 2005 - 08:06 AM

So does this mean that my 130mm F/6.9 Orion spherical primary is no good?


No, but it does mean that there's slightly more than 1/4th wave spherical aberration at the wavefront just because of this property.

Whether a scope that has this amount of spherical aberration is "no good" depends on how good good has to be.

#7 werewolf6977

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Posted 10 October 2005 - 10:06 AM

Refer to my comment about the Cassini Div. I have notice that above 45X/inch, the image gets iffy, but since most of the time you can't go over about 200X around here, it doesn't matter. That is why I do recommend one for a beginner.

#8 BluewaterObserva

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Posted 10 October 2005 - 12:25 PM

In any properly designed scope. The better mirror usually wins, Sphere -vs- parabola isn't the factor that makes or breaks them in all honesty. :)

#9 werewolf6977

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Posted 10 October 2005 - 12:45 PM

I musta got a good one then...

#10 David Knisely

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Posted 10 October 2005 - 01:48 PM

Well, to directly answer your question, a spherical mirror is just like if you placed a perfectly round ball into molten glass and let things cool off, giving you an "imprint" of that ball. A parabola is the mathematical curve that is found in things like the trajectory of a projectile (if not thrown straight up that is). For paraboloidal mirrors, it's like taking that spherical mirror and deepening it towards the center so that its curve in two dimensions resembles that of a parabola. Paraboloidal mirrors focus light from infinity to a point regardless of the f/ratio of the mirror, while spherical mirrors don't quite do that. If you make a spherical mirror of a long enough focal ratio however, the differences in where the light all comes to a focus from different parts of the mirror may become small enough to make the mirror perform almost as well as a true paraboloidal one. However, this is only generally practical if the mirror is a relatively small aperture one. Clear skies to you.

#11 sixela

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Posted 11 October 2005 - 07:01 AM

Refer to my comment about the Cassini Div.


Now's the time to look at modulation transfer function (MTF) or contrast transfer function (CTF) graphs...

Spherical aberration does not degrade contrast equally at all frequencies. The resolving power of the scope is degraded less than its contrast is. In other words, small high contrast features (certainly linear ones) are affected less than larger low contrast features. In other words, you have more trouble seeing features on Mars and in the Jovian cloud bands than you have seeing the Cassini division.

See e.g.:

http://aberrator.ast...l/body_mtf.html

Notice that the "ideal" graph is very close to the one with SA at high frequencies (but the amount of contrast necessary to perceive a feature is higher at these frequencies).

http://aberrator.ast..._spherical.html shows a simulation on Mars.

#12 werewolf6977

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Posted 11 October 2005 - 11:11 AM

Will try the Spaceprobe 130 on Mars, and see what happens.


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