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telescopeѲptics.net
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As a wavefront aberration, distortion results from the actual wavefront being formed tilted with respect to the perfect reference sphere. As a result, actual image point is shifted in the image space, distorting the image's geometric form, but otherwise not affecting image quality. In effect, distortion induces a variable field-point magnification. The aberration function of distortion is given by: Wt= Grcosq (27) with G=gd being the peak distortion aberration coefficient, g being the aberration coefficient, containing α3 factor (α is the field angle and d the aperture radius), and q is the pupil angle. Since ray aberration caused by distortion is independent of the pupil coordinates (r,q) all rays meet at the image point, which is displaced radially in proportion to the cube of the point field angle α (FIG. 28).
FIGURE 28: Illustration of the effect of image distortion: Gaussian image of a square centered in the field has its corners (C) farther away from the field center O than its mid-side points (M) by a factor of 21/2. Since distortion increases with the third power of off-center distance in the image plane (strictly talking, with the field angle, but for small angles the difference between the rate of change of the two is negligible) the corner point C is shifted away from its perfect coordinate by a factor (21/2) more than the mid-side point M (this proportion remain constant, only the magnitude of deformation changes). In other words, the length of the aberrated extension CC' or CC" of the aberrated image of the square is larger than MM' or MM", respectively, by a factor of (OC/OM)3. As a result, the image is deformed, either inward (negative, or barrel distortion), or outward (positive, or pincushion distortion). Since the amount of shift from the perfect coordinate is in proportion to the cube of off-axis angle, α3, linear distortion of any non-circular form centered in the field increases with the cube of its linear diameter; in effect, shape distortion increases with α2.
Aberration coefficient is zero for both, concave mirror and lens objective with the aperture stop at the surface. Distortion is introduced if the stop is displaced, which makes it present in multi-surface systems with the elements at more than insignificant separation (due to the stop being effectively displaced for surfaces farther away from the primary surface, even if it acts as an aperture stop). An exception is a sphere with the stop at the mirror center of curvature, when it also has zero distortion, due to its unique symmetry. In general, distortion is negligible
for small angular images, such are those of telescope objectives.
However, it
becomes significant at large angles, characteristic of the images viewed
through telescope eyepieces. |
