Page 505 - The Mechatronics Handbook

0066_frame_C19 Page 127 Wednesday, January 9, 2002 5:32 PM

FIGURE 19.106 The cardinal points of a multi-element lens operating in a single medium (usually air) are indicated.
The principal points and nodal points then coincide at N 1 and N 2 and the front and rear focal lengths are equal (f ).
Three rays from an object point are traced through the lens to the corresponding image point using the properties
of the cardinal points. In the case shown the image magnification is 0.5, so the image is some distance behind the
rear focal point F 2 . For distant objects the image plane would coincide with the plane transverse to the optical axis
passing through F 2 . Lenses are normally corrected for aberrations assuming that the object distance will be greater
than the image distance. In this case, for close-up work when the image distance is greater than the object distance,
the image quality is improved by reversing the lens.

Image Formation
Although perfect images are formed by small pinholes, lenses are needed to form bright images and
range from simple single-component lenses used to increase the amount of light falling on a single
detector or in low-cost cameras to complex zoom lenses, with between 14 and 20 components, capable
of producing high quality images of varying size. The two most important properties of a lens are its
focal length f, which determines the imaging behavior, and its light-gathering power or speed, specified
by an f-number f # . A lens has an optical axis passing through the central axis of each of its components
along which a ray of light passes without deviation. A lens is characterized, regardless of its complexity,
by six cardinal points [2] spaced along the optical axis as illustrated in Fig. 19.106, for a positive converging
lens. The position and magnification of the image of an object can be determined using these cardinal
points, which include two focal points, two nodal points, and two principal points. The nodal points
have the property that a ray outside the lens travelling towards one nodal point emerges from the lens
in a parallel direction, appearing to come from the other nodal point. The focal point is the point which
a ray of light incident on the lens parallel and close to the optical axis converges to (positive lens) or
diverges from (negative lens) after passing through the lens. The point where the lines colinear with the
ray on the two sides of the lens intersect defines a point on the principal plane. The point where the
optical axis intersects the principal plane is called the principal point. There are two nodal points, focal
points and principal planes, because light can be incident on the lens from either side. The front and
back focal lengths of the lens are the distances between the front and back focal points and their
corresponding principal planes. In the normal situation, when the lens is operating in a single medium,
such as air, the positions of the nodal points and principal points coincide and the front and back focal
lengths are equal. In general, when the lens construction is asymmetric, the front and back focal points
are at different distances from the corresponding external lens surface. In the case of an ideal thin lens,
the principal planes coincide with the lens center but in multi-element lenses they may be separated by
+20 to −10 mm, depending on the lens design. A lens of focal length f produces an image in best focus
at a distance v when the object is at distance u where

1 1 1
--- + -- = -- (19.81)
u v f
and the distances are measured to the corresponding principal planes. The image magnification m, defined
as the ratio of image to object sizes, is equal to the ratio of the image to object distances and is related to

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