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Optical aberrations

Astigmatism (III)

Astigmatism is similar to coma but arises for small objects at the edges of the field striking an uncorrected lens asymmetrically. It is not as sensitive to aperture as coma.  The incoming rays passing through the lens in oblique angle with respect to the optical axis focus differently from those paraxials. Depending on the incidence angle of the off-axis rays entering the lens, the refracted plan is oriented either tangentially or sagittally. So the resulting image depends upon the location of the focal plan and thus produces blurry images more or less elongated which intensity and contrast decrease as the distance from the center increases.

When an optical system is well corrected for astigmatism, tangential and sagittal images coincide on the lens curvature surface also called Petzval surface. A way to represent these two components is to refer to the field angle, the angle between the chief ray and the main axis. If the lens is uncorrected, the curves of tangential and/or sagittal images will wander from from the main axis, representing the Petzval surface. So it is easy to calculate at what distance images becomes blurry and degrades.

Side view and Airy disc of a bright star coming 25° off axis showing astigmatism; the tangential and sagittal line image do not coincide in the area of less confusion and in this case display a cross shape.

Like field curvature, this aberration is proportional to the diameter of the optics and, in first approximation, to the square to the distance to the central axis. Therefore astigmatism is much less apparent than coma. Its effect is however more pronounced on Cassegrain and related systems (including Ritchey-Chrétien's) than on newtonians of the same aperture.

How to correct astigmatism ? Depending the thickness of the lens system and the position of the curves with respect to the Petzval surface (negative, positive, undercorrected, overcorrected, inward or backward-curving) the effect can be neutralized by adding a lens displaying the opposite effect (negative lens for example to balance a positive curving of the Petzval surface) in order to reduce the astigmatic difference or the distance from the tangential and sagittal images from the Petzval surface.

According optical engineers, astigmatism mainly occurs when there is poor alignment between the objective and the eyepiece system or when the objective lens is poorly centred. A similar effect occurs with coma. Most of the time when such a system is mis-aligned or badly reassembled, images show severe asymmetry over the entire field of view, even using the lower eyepieces as this aberration may occur as low as 5' off-axis.

On low cost systems, astigmatism often occurs in wide field eyepieces of long focal (say 12 mm and longer) using few lens-elements. Tele Vue Nagler's Type I for example display the famous "kidney bean" aberration on all long focals eyepieces. The problem is enhanced during daytime observations. Therefore Al Nagler produced the Type II series that corrects this aberration thanks to 8 lens-elements.

At last, it is a serious error to remove the spacer which sometimes separate individual lens-elements. These insignificant objects are as important as the lens curvature or the index of refraction of the lens as they participate in the image correction.

Field curvature

For all rays, paraxials or coming from the periphery of the lens, and converging to the focal point through a curved lens the focal length of the lens is constant, the image plane is placed on a Petzval surface which is curved not a flat one. This effect leads to the problem of field curvature. 

In the reality there are not only one focal plan but an infinity, one for each value of lens bending (front and back curvatures of a lens-element). But even with a single plan to consider if the image is well focused in the center of eyepiece field, it is not so sharp at the edges.

This effect occurs because a ray coming from the periphery of the lens (off-axis) has to cross a longer distance to reach the Petzval surface than a paraxial ray. This effect is still more visible using some image intensifiers.

This aberration varies with the lens curvature and affects lens quality. For this reason optical designers use this "bending" factor with caution, most of the time in combination with factors easier to handle as the index of refraction or the density of the glass.

The field curvature is mainly visible at the edges of the field.

Another way to suppress this aberration is to artificially reduce the field of view like we do with vignetting. But the best eyepieces are corrected for this aberration and display a true "flat field" all over the field as wide it could be. The lenses-elements of these wide field eyepieces must however be fully multicoated as the gain in flatness is paid with a loss of brightness and the rising of some monochromatic (Seidel) or polychromatic aberrations.

Usually, due to their aspherical mirrors SCT scopes are more sensible to this aberration than for example a newtonian of the same diameter that uses a paraboloïdal mirror or a Ritchey-Chrétien that uses aplanatic paraboloïdal mirror. If you consider this aberration is too visible at the edges of the field in your SCT  I suggest you to use a focal reducter/corrector 0.63x. This positive Shapley lens is able of cleaning up the field edge for this aberration as well as for coma.

Distortion

Distortion represents the inability of a lens to create a rectilinear image of the subject. It does not modify the colors or the sharpness of the image but rather its shape. This distorsion happens because the focal length of the lens varies over the Petzval surface (transverse magnification) and as parts of the image are more magnified than others. Distortion occurs in two main effects : barrel and pincushion, also called positive and negative distortion.

An image free of distortion, with barrel, and pincushion.

You probably know these effects as they are also visible on TV screens and computing monitors where they can be rectified manually. The barrel shows a central image bigger than the edges, hence its name. The magnification decreases with the off-axis object distance. The pincushion is the inverted phenomenon and is less common; the edges of the field are bigger than the central area, the magnification increasing with the off-axis distance of the object. 

These two types of distortions can be present in eyepieces that yield so-called "very sharp" images that are otherwise corrected for all other aberrations (super Plössl's for example but quality wide fields are not excluded). They occur mainly with eyepieces using thick lens-elements whereas thin lenses show little or no distortion. At last they occur often on compound lenses like zoom "eyepieces", optical systems containing meniscus or telephoto and retrofocus.

For more information

How to correct coma ? (on this site)

ATMOS software

Telescope Optics, evaluation and design, H.Rutten/M.van Venrooij, Willman-Bell, 1992

Star Testing Astronomical Telescope, H.R.Suiter, Willmann-Bell, 1994

Edmund Industrial Optics

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