|
telescopeѲptics.net
▪
▪
▪
▪
▪▪▪▪
▪
▪
▪
▪
▪
▪
▪
▪
▪ CONTENTS
◄
4.4. Defocus
▐
4.6. Field curvature
► 4.5. IMAGE DISTORTIONAs a wavefront aberration, image distortion is caused by the change of magnification with the incident angle, with the actual wavefront being formed tilted with respect to a perfect (Gaussian) reference sphere. It is a consequence of the light cones for oblique angles coming in at a different angles, and using different portions of optical surface(s) than the near-axis cones. It changes the effective focal length for those converging cones, and with it their magnification, i.e. tilt angle. Some surfaces make focal length longer toward the edge, some shorter, and the final sum is the system distortion. In effect, the actual wavefront is tilted with respect to the Gaussian reference sphere, and the actual image point is shifted in the image space. The magnitude of shift increases with the third power of the incident angle, effectively inducing a varying point height magnification in the image space. The result is distortion of the image's geometric form but, since the wavefront remains spherical - or aberrated as determined by other factors - point-image quality itself is not affected. The aberration function of distortion is given by:
Wt
=
Gρcosθ (27) with
G=gdα3
being the peak distortion aberration coefficient, g
being the aberration coefficient (α
is the field angle and d the aperture radius), and
θ is the
pupil angle.
Since ray aberration caused by distortion is
independent of pupil coordinates (ρ,θ)
all rays meet at the image point, which is displaced radially in
proportion to the cube of the point field angle
α (FIG.
59).
FIGURE 59:
Illustration of the effect of image distortion: Gaussian image
of a square centered in the field has its corner point C farther
away from
field center O than its mid-side point M by a factor of 21/2.
Since distortion increases with the third power of off-center
distance (strictly talking, field angle, but for small angles
the difference between the rate of change of the two is
negligible) in the image plane the corner point C is shifted away from
its perfect coordinates by
a factor (21/2)3
more radially 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, blue), or outward
(positive, or pincushion distortion, red). 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 of distortion for a
single surface, refractive or reflective, with the stop at the
surface, is given by:
with the peak
aberration coefficient,
representing the
peak
wavefront error of tilt with respect to the reference sphere centered at
Gaussian focus along the axis of aberration (n and n' are
the refractive index of incident and refractive/reflective medium,
respectively, and D is the aperture diameter).
The aberration coefficient is zero for both, concave mirror
(n=1, n'=-1) and a thin lens with the aperture stop at the surface. Distortion is
introduced if the stop is displaced, which means that it is present in
multi-element systems with the elements at more than insignificant
separation. An exception is a sphere with the stop at
its 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. Picture below shows raytrace cross section of a 110-degree AFOV eyepiece,
with about 50% positive (pincushion) distortion (entering cones are not near orthogonal to the Smyth lens because this particular eyepiece was designed for a binocular objective).
The cones entering (actually exiting, since it is reverse raytracing)
eyepiece on the right are different in their width, with the field edge
cones being noticeably narrower. That is a consequence of their increased
magnification (narrower cone forms larger Airy disc, i.e. image scale);
in actual use, the entering cones would be of identical width, and the
edge cone's exit pupil smaller, according to their increased magnification.
Note that distortion in reverse raytracing has opposite sign to that in actual use.
◄
4.4. Defocus
▐
4.6. Field curvature
►
|
