Astigmatism
What is astigmatism?
Astigmatism
is the aberration that occurs when rays from an off-axis object that strike in
different planes are imaged at different distances. The parts of an object that are radially outward from the optical
axis are focused in a different location than the parts of the objectthat are
tangental to the axis (figure 1).
figure 1. Astigmatism - the
green parts of the object are focused on the green plane while the red
part of the object is focused on the red plane.
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Where does
astigmatism come from?
As an
object point is moved off axis, it's incident rays start to strike the lenses
asymmetrically (figure 2).
figure 2. When the circle is
moved off axis, the rays from it strike the lens in an oval shape
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The incident rays strike in two
planes on the axis: the meridional
plane and the sagittal plane (figures 3 and
4). The meridional plane is the plane
that contains both the optical axis and the chief
ray. The sagittal plane is perpendicular to the
meridional plane, and contains the chief ray.
figure 4. Side profile of figure
3
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figure 3. Rays strike in either
the meridional plane or the sagittal plane
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As the object is moved off the axis, the sagittal plane tilts because
the chief ray's slope is altered (figure 5).
The slope of the ray can cause the incident rays to strike at different
angles on the sagittal and meridional planes.
This angle at which the rays strike affects the focal
length, creating
two distinct focal lengths, one for the sagittal plane and one for the
meridional plane.
figure 5. The sagittal plane
tilts as the chief ray's slope is altered
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The difference between these focal lengths becomes greater with a
greater difference in angle, which happens as the object moves further off
axis. This difference is also affected
by the power of the lens.
Th
figure 6. The Circle of Least
Confusion occurs between the primary and secondary images
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Circle of
Least Confusion
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Pedrotti
& Pedrotti, Introduction to
Optics, 2nd ed.,
1987.
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e rays in the meridional plane focus first. However, because the rays in the sagittal plane are not yet
focused, the image is a line in the sagittal plane called the primary
image
(figure 6). Moving beyond this line,
the meridional rays spread out again and the sagittal rays converge creating
the image of a line in the meridional plane called the secondary
image (figure
6).
Between the primary image and the
secondary image, there is a location where the spread of meridional and the
spread of sagittal rays is exactly the same.
The image at this location is an even spread out circle of the original
point object. This location is called
the circle of least confusion (figure 6).
The circle of least confusion
increases in diameter as the astigmatism increases (which occurs as the object
moves further off axis). This means
that the image will continue to lose definition as the object moves off axis.
How can we see
astigmatism?
If a round
object with a radial grating is imaged through a spherical lens, the circle
around the grating will be in focus at a different location than the grating.
This occurs
because when you image a point off axis, the point forms an ellipse on the
lens, with the major axis radial to the lens and the minor
axis tangential. The radial projection is
therefore in the meridional plane and the tangential projection is in the
sagittal plane. The difference in focal
lengths between the meridional (tangential) and sagittal planes causes the
difference in locations where the circle focuses and where the grating focuses.
What does astigmatism
look like?
Primary Image
(meridional focus)
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Secondary Image
(sagittal focus)
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Meyer-Arendt,
Introduction to Classical and
Modern Optics, 4th ed., 1995
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Meyer-Arendt, Jurgen,
Introduction to Classical & Modern Optics, 4th edition, New Jersey:
Prentice Hall, 1995.
Pedrotti, Frank & Leno Pedrotti, Introduction
to Optics, 2nd edition, New Jersey: Prentice Hall, 1993.