Page 97 - Modern Optical Engineering The Design of Optical Systems

80 Chapter Five

Tangential
focal line Sagittal
focal line
Tangential ray
fan
y
z
Optical axis

Optical
axis
Sagittal ray
fan
z x

Figure 5.7 Astigmatism (negative or inward-curving).

in the sagittal plane. Conversely, the image formed by the rays of the
sagittal fans is a line which lies in the tangential plane.
Astigmatism occurs when the tangential and sagittal (sometimes
called radial) images do not coincide. In the presence of astigmatism,
the image of a point source is not a point, but takes the form of two
separate lines as shown in Fig. 5.7. Between the astigmatic foci the
image is an elliptical or circular blur. (Note that if diffraction effects are
significant, this blur may take on a square or diamond characteristic
because the line images are effectively acting as slit apertures.)
Unless a lens is poorly made, there is no astigmatism (or coma) when
an axial point is imaged. As the imaged point moves further from the
axis, the amount of astigmatism gradually increases. Off-axis images
seldom lie exactly in a true plane; when there is primary astigmatism
in a lens system, the images lie on curved surfaces which are paraboloid
in shape. The shape of these image surfaces is indicated for a simple
lens in Fig. 5.8. This drawing is to scale; it is not exagerated or distorted.
The amount of astigmatism in a lens is a function of the power and
shape of the lens and its distance from the aperture or diaphragm
which limits the size of the bundle of rays passing through the lens. In
the case of a simple lens or mirror, where its own diameter limits the
size of the ray bundle, the astigmatism is equal to the square of the dis-
tance from the axis to the image (i.e., the image height) divided by the
focal length of the element, i.e., h /f.
2
Every optical system has associated with it a sort of basic field
curvature, called the Petzval curvature, which is a function of the