By A. Schuster, J. Nicholson
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04ϫ. tudinal magnification is approximately (by Eq. 04 in. Gaussian Optics: The Cardinal Points 33 Note that a more exact value for the image thickness and the longitudinal magnification can be obtained by calculating s′ for two object distances, 1 in apart. 038462 in. 041667 in. 040016 in. 4 The first principal point of a 2 in focal length lens is 1 in from the object. Where is the image, and what is the magnification? ANSWER: Using Eq. 5, we find the image at s′ ϭ s и f/(s ϩ f ) s′ ϭ Ϫ1 и 2/(Ϫ1 ϩ 2) ϭ Ϫ2/1 ϭ Ϫ2 in The image is 2 in to the left of the lens, and the magnification m ϭ s′/s ϭ Ϫ2/(Ϫ1) ϭ ϩ2ϫ.
By ruling the paper in squares, a simple arrangement of the constructional parameters at the top of the sheet and the ray data below helps to speed the calculation and eliminate errors. The following table (Fig. 5) sets forth the curvatures, thicknesses, and indices of the lens in the first three rows; the next two rows contain the ray heights and index-slope angle products of the calculation worked out above. The image height can now be found by tracing a ray from the top of the object and determining the intersection of this ray with the image plane we have just computed.
These are illustrated in Fig. 1. The nodal points are two axial points such that a ray directed toward the first nodal point appears (after passing through the system) to emerge from the second nodal point parallel to its original direction. The nodal points of an optical system are illustrated in Fig. 2 for an ordinary thick lens element. When an optical system is bounded on both sides by air (as is true in the great majority of applications), the nodal points coincide with the principal points.