Page 98 - Modern Optical Engineering The Design of Optical Systems

The Primary Aberrations 81

Figure 5.8 The primary astigma-
tism of a simple lens. The tan-
gential image is three times as
far from the Petzval surface as
the sagittal image. Note that the
figure is to scale.

index of refraction of the lens elements and their surface curvatures.
When there is no astigmatism, the sagittal and tangential image sur-
faces coincide with each other and lie on the Petzval surface. When
there is primary astigmatism present, the tangential image surface
lies three times as far from the Petzval surface as the sagittal image;
note that both image surfaces are on the same side of the Petzval surface,
as indicated in Fig. 5.8.
When the tangential image is to the left of the sagittal image (and
both are to the left of the Petzval surface) the astigmatism is called
negative, undercorrected, or inward-(toward the lens)curving. When the
order is reversed, the astigmatism is overcorrected, or backward-curving.
In Fig. 5.8, the astigmatism is undercorrected and all three surfaces are
inward-curving. It is possible to have overcorrected (backward curving)
Petzval and undercorrected (inward) astigmatism, or vice versa.
Positive lenses introduce inward curvature of the Petzval surface to
a system, and negative lenses introduce backward curvature. The
Petzval curvature (i.e., the longitudinal departure of the Petzval sur-
face from the ideal flat image surface) of a thin simple element is equal
to one-half the square of the image height divided by the focal length
and index of the element, h /2nf. Note that “field curvature” means
2
the longitudinal departure of the focal surfaces from the ideal image
surface (which is usually flat) and not the reciprocal of the radius of
the image surface. The radius of the Petzval surface (at the axis) for a
simple element is given by nf.

Distortion
When the image of an off-axis point is formed farther from the axis or
closer to the axis than the image height given by the paraxial expres-
sions, the image of an extended object is said to be distorted. The
amount of distortion is the displacement of the image from the paraxial