The German mathematician and physicist Carl Friedrich Gauss (1777-1855) made a contribution to optics that made possible the design of modern optical objectives; his designs are still the basis of a whole family of camera lenses. He extended Newton's laws of optics, which originally considered only ‘thin’ lenses (i.e. lenses that could be considered as existing in a single plane), to real lenses that could be thick or contain several components. He showed that any lens could be treated by Newtonian methods when described in terms of six cardinal points, namely the front and rear principal foci, principal points and nodal points, and the planes through these points. If the lens is wholly in air, the principal and nodal points coincide with one another.
Gauss began with the premiss that the actual path of the rays through the lens could be ignored: it was the path of the rays into, and out of, the lens that mattered. He then showed that the entry and exit paths of a ray could be established in terms of the principal (or nodal) planes. A ray entering the lens, and (when produced) crossing the front principal plane at a particular point, would emerge as if it had originated at the corresponding point on the rear principal plane (Fig. 1). The normal rules for geometrical optics are now followed, the space between the principal planes being ignored.
The positions of the nodal points are established using a different procedure. A ray entering the lens that emerges without change of direction, when produced forwards from the point of entry and backwards from the point of exit, intersects the optic axis at the front and rear nodal points respectively (Fig. 2).
The position of the rear nodal point is a critical factor, as the focal length is measured from the rear nodal point. In a conventional (prime) lens the nodal points are about one-third of the way inside the lens, though this may vary (in the Rapid Rectilinear lens of the 1860s the nodal points were actually crossed). In a telephoto lens the rear nodal point is well in front of the lens itself, so that the physical dimensions of the lens are less than its focal length. In most wide-angle lenses the rear nodal point is well behind the lens (a retrofocus configuration) to give space inside the camera for the reflex mirror (Figs. 3a and b). A zoom lens operates by shifting the components along the optic axis to move the rear nodal point from behind the lens (short focus) to in front of it (long focus).
Fig. 1
Fig. 2
Fig. 3a
Fig. 3b
— Graham Saxby
Bibliography
- Buchler, W. K., Gauss: A Biographical Study (1987).
- Saxby, G., The Science of Imaging: An Introduction (2002)
