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OPTICAL ABERRATIONS

However carefully and accurately a telescopes optics are made, there will always be image errors which arise out of the laws of optics, rather than the quality of the optical surfaces. We have already mentioned one of these errors which is caused by the central obstruction in a Newtonian telescope. Scmidt-Cassegrains and Maksutov will have similar problem.

But the main class of optical errors are called "aberrations". There are five main types: Chromatic Aberration, Spherical Aberration, Coma, Astigmatism, Distortion. The first three are of most importance in Astronomical telescopes.


CHROMATIC ABERRATION


White light consists of all the colors of the rainbow. In a refractor telescope, light passes through a glass lens, and in doing so the light is bent so that it all comes to a focus, and forms an image there. Unfortunately, if the lens is made of a single piece of glass, each of the colors of the rainbow is bent by a different amount in passing through the glass. This means that red light and blue light (for instance) will each form an image, but at a different places. This results in an unsharp image, which appear surrounded by color fringes.
A single element lens (as shown right) would be useless for astronomy.
To overcome this problem, refractor lenses are usually made of two or more pieces of different types of glass. If two pieces of glass are used the lens is called achromatic. The simplest achromatic lenses are made of flint and crown glass. Such lenses are a big improvement on a single lens. But the correction afforded is for two colors only, say red and blue, other colors may still be slightly out.

The problem is most noticeable if the focal ratio (F number) is low. That is, all things being equal, a 100mm f5 lens will have more color fringing than a 100mm f8 lens.

The problem is also worse with increasing aperture. That is, all things being equal, a 150mm f8 lens will have more color fringing than a 100mm f8 lens.

Of course, achromatic object lenses suffer from other aberrations as well, but chromatic aberration is the dominant one. The Chromatic Aberration Index (CAI) (which equals the Focal ratio of a lens divided by its aperture in inches, ie F/D(in) ) is a useful guide to the aberration errors in an achromatic lens.

Most experienced observers agree that for false color to be reduced to an insignificant level, the Chromatic Aberration Index (CAI) needs to be 3.00 or greater. This is also called the Sidgwick Standard.
ie, F/D(inches) > 3.00. Another way of saying this is that focal ratio (F) must be greater than 3 times diameter (inches).

While the Sidgwick standard is the ideal, it is not essential for diffraction=limited performance. Most recently, Vladimir Sacek has demonstrated that Achromatic Lenses can theoretically still achieve diffraction-limited performance provided that the Chromatic Aberration Index (CA) is more than 2.25.

So an 4 inch diameter achromat will be diffraction-limited provided its focal ratio is 9 or greater.

There is a lower standard, where the CA is less than 2.25, but greater than 1.5 - the lenses at this standard will not be diffraction limited, but will still produce good images, provided it is used with a minus-violet filter, or if the magnification used is moderate.

The attached table sets out the Chromatic Aberration Index for various refractors, measured against the three standards mentioned above.

Standards for Achromatic Telescope Objective Lenses - Minimum Focal Length/Focal Ratio

Diameter of Achromatic Telescope Objective Highest Standard

Minimum Focal Length/Ratio
- for Insignificant Color Fringeing
(Sidgwick Standard)
Diffraction Limited Standard

Minimum Focal Length/Ratio
- for Diffraction-Limited
Performance
Lower Standard

Minimum Focal Length/Ratio
- Provided Minus-Violet Filter used, or moderate magnification only
70 mm FL 600mm, F 8.5 FL 420mm, F 6 FL 280mm, F 4
80 mm FL 750mm, F 9.5 FL 560mm, F 7 FL 400mm, F 5
90 mm FL 1000mm, F 10.5 FL 720mm, F 8 FL 500mm, F 5.5
100 mm FL 1200mm, F 12 FL 900mm, F 9 FL 600mm, F 6
125 mm FL 1800mm, F 15 FL 1400mm, F 11 FL 900mm, F 7
150 mm FL 2700mm, F 18 FL 2000mm, F 13 FL 1300mm, F 9

A further improvement comes from the use of triplet lenses, or from the use of special glass (eg Fluorite, ED). But such telescopes are more expensive. Chromatic Aberration affects refractor telescopes only. Reflector telescopes of all kinds (whose main element is a mirror, not a lens) do not suffer from chromatic aberration.

SPHERICAL ABERRATION

A spherical mirror will focus light to form an image. But, even if the mirror is polished to a perfect spherical shape, rays of light reflecting off the edge of the mirror will form a different focus from rays of light reflecting off the center of the mirror.
As the diagram shows, light from the edge of the field is focused closer to the mirror along the optical axis than is light from the center of the field. This error is called spherical aberration. This means it can be impossible to find a single point of best focus, only a point where the image is smallest but still not sharp.

Every telescope design sets out to eliminate spherical aberration.

Commercial Schmidt-Cassegrain telescopes use spherical mirrors, which would, on their own, create spherical aberration. The Schmidt corrector lens on the front of an SCT eliminates the spherical aberration inherent in the mirror design.

Most Maksutov-Cassegrains work the same way.

Refracting telescopes normally use spherical lenses, due to the extreme difficulty and cost associated with constructing aspherical lenses. A single spherical lens of course suffers from spherical aberration. However, a refractor eliminates spherical aberration by combining two lenses with equal but opposite amounts of spherical aberration. More complex refractor designs may use three or four lenses, but the basic idea is the same. These lenses must also work to eliminate a number of other aberrations, so the design process is tricky, but in the end spherical aberration must be the smallest residual aberration if the telescope is to provide a good image.

In the case of the Newtonian telescope, spherical aberration is eliminated simply done by making the primary mirror parabolic. However, this is not necessary in all cases. Provided the telescope aperture is small, or the focal ratio long, spherical aberration can be acceptably small, even within the diffraction limit.

The Table below shows the minimum focal ratios (focal lengths) required to ensure that a spherical mirror is diffraction-limited.

Diffraction-Limited Spherical Mirrors

Diameter Minimum Focal Ratio for Diffraction-Limited*
F=(3.55D)1/3
Minimum Focal Length for Diffraction-Limited
100 mm F 7 700 mm
150 mm F 8 1200 mm
200 mm F 9 1800 mm
*formula from http://www.telescope-optics.net/reflecting.htm



Unfortunately, focal ratios of F8 and above are not always acceptable for larger Newtonian scopes. We want more compact, lighter, scopes of f6, f5 or even f4 focal ratios.

The solution to this problem is to make reflector mirrors in the shape of a parabola, and not a sphere. The more accurate the parabola shape of the mirror, the better quality of the mirror will be.

So, provided our mirror is figured to a parabola shape, the telescope can produce a sharp focus.

COMA

Coma is an off-axis aberration. Stars in the center of the field are not affected by coma, but the effect grows stronger toward the edge of the field. Stars affected by pure coma are shaped like little comets (hence the name) pointed toward the center of the field.

what occurs to cause coma is that light passing through the center of the field (but at an angle so it focuses off-axis) does not focus at the same distance along the optical path as light from farther off axis.

Since coma affects the edges of a field, and grows larger with increasing distance from the optical axis, it is a significant aberration with regards to wide-field viewing and imaging. For professional astronomers and advanced amateurs who are interested in scientific study, coma can be very problematic because it is an asymmetrical aberration.

This is a problem because it makes it impossible to accurately measure the position of stars (astrometry). For this reason, most professional instruments are designed specifically to eliminate coma (although sometimes by introducing another less problematic aberration). For most amateur astronomers, a small amount of coma is tolerable. For wide-field applications, some telescope designs eliminate or minimize coma, while coma-correcting lenses are available for other designs to minimize the effect if desired.

Telescope Designs with Coma: 1. Newtonian

The classic example of coma occurs in the Newtonian telescope. Coma is the primary aberration inherent in the Newtonian design and is the limiting factor in this design. Most Newtonian designs show coma at the edge of the field, but it is not a serious problem, unless the focal ration is short. Coma is a function of both off-axis distance and focal ratio, meaning faster-focal-ratio (smaller f-number) telescopes will have more coma than a similar size but slower telescope.

Therefore, an 8" f/4 Newtonian has more coma than an 8" f/6 Newtonian. f/4 is usually considered the fastest a Newtonian can be made without having excessive coma. Coma correctors are available that can minimize the amount of coma in a Newtonian design. These lenses fit into the focuser ahead of the eyepiece of camera. They are typically used on f/5 and faster telescopes.

Diffraction-Limited Field Radius
for Newtonian Telescopes

Focal Ratio, F Diffraction-Limited Field Radius
=F3/90
Diffraction-Limited Field Diameter
F 4 0.71 mm 1.42 mm
F 5 1.39 mm 2.78 mm
F 6 2.4 mm 4.8 mm
F 7 3.81 mm 7.62 mm
F 8 5.69 mm 11.38 mm


Telescope Designs with Coma: 2. Schmidt-Cassegrain and Maksutov-Cassegrain

Most commercial Schmidt-Cassegrain and Maksutov-Cassegrain designs also suffer from coma. Since they typically have long focal ratios, in the range of f/10 to f/15, the coma is less than in a similar-sized Newtonian. The amount of coma is not normally problematic when observing, but can appear at the edges of a large photographic field. Note that coma is not necessarily inherent in the Schmidt- and Maksutov-Cassegrain design, but exists because of the choice of optical parameters chosen to minimize the cost of manufacturing these commercial scopes.

Note: DL radius for a f10 schmidt-Cassegrain is about same as for f6 Newtonian. As is the case for Newtonians, the Diffraction-Limited radius (in mm) for Scmidt-Cassegrains (SCT) is dependent on the focal ratio only. The Diffraction-Limited radius for the standard F10 SCT is 2.33mm. The Diffraction-Limited radius (in arc minutes) varies with the focal length and is set out in the next table -

Coma Diffraction-Limited radius for Schmidt-Cassegrains

Diameter mm Focal Ratio Diffraction-Limited radius (arc-mins) Diffraction-Limited radius mm
200 mm F 10 4 2.33 mm
250 mm F 10 3.2 2.33 mm
300 mm F 10 2.7 2.33 mm
350 mm F 10 2.3 2.33 mm


Telescope Designs without Coma

For visual use, coma is not really a serious issue with Schmidt-Cassegrains, and Newtonians of moderate focal ratio.

Coma in Newtonians and Schmidt-Cassegrains can be reduced by placing a special lens into the eyepiece focuser. This lens is called a coma corrector. This will sharpen the image off-center, but at the expense of introducing some on-axis aberrations.

Also, in Schmidt-Cassegrains, by slightly changing the configuration of the mirrors at the design stage, and the parameters of the optical design, it is possible to create a design which is largely coma-free. But such designs are more expensive to produce, especially if on-axis sharpness is to be retained. But to many professionals, some symmetrical degredation of the on-axis image is acceptable provided off-axis coma is reduced. This compromise is made because symmetrical aberrations still allow astronomers to make accurate positional measurements. For amateur astronomers interested in viewing or taking nice looking images, there is little advantage to one aberration over the other.

Maksutov-Cassegrain telescopes have little coma, due to their long focal ratios (anything up to F14).

Most refractors have little or no coma, contributing to their being well-suited to wide-field viewing and imaging.

Other imaging systems such as hyperbolic astrographs, Schmidt cameras, and other uncommon designs are usually designed to be free from coma.






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