In Section 1.6, we showed how to trace rays exactly from one surface to the next, and when they are refracted or reflected by a refracting or a reflecting surface. We then considered paraxial ray tracing, i.e., when the rays make small angles with the surface normals and the optical axis. We showed that, in the paraxial approximation, the curved surface could be replaced by a planar surface that is a tangent to the surface at its vertex, called the tangent plane or the paraxial surface. The approximation led to Gaussian optics, which represents imaging equations for obtaining the image of an object, i.e., the size and the location of the image in terms of the size and location of the object and the parameters of the imaging system.
We begin this chapter by rederiving the imaging equations for a refracting surface by assuming small angles of incidence and refraction and small slope angles of the rays (as in Section 1.8.2). How to determine the image graphically is also considered. We use standard notation suitable for a multisurface imaging system. Both the Gaussian and Newtonian forms of the imaging equations are given. These equations are used to obtain the corresponding equations for a thin lens. The imaging equations for a multisurface refracting system are derived next. The principal, focal, and nodal points, collectively called the cardinal points of such systems, are discussed. It is shown that simple imaging equations, similar to those for a single refracting surface, are obtained, provided the object and image distances are measured from the respective principal points of the system in the Gaussian form, and from the focal points in the Newtonian form of the imaging equations. The concept of the Lagrange invariant is discussed in each case.
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