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Chapter 4:
Paraxial Ray Tracing
Abstract
Paraxial ray tracing was introduced in Chapter 1 (see Section 1.7) and utilized in Chapters 2 and 3 to show that an imaging system could be characterized by its principal points and focal lengths. Although a system has six cardinal points, only three are independent. If the refractive indices of the object and image spaces are equal, which is often the case in practice, then the nodal points coincide with the corresponding principal points, and the object- and image-space focal lengths are equal in magnitude. We showed that the image of a point object formed by a system can be determined graphically by tracing any two of the three specific object rays: a ray incident parallel to the optical axis of the system and emerging from it passing through the image-space focal point; a ray incident passing through its object-space focal point and emerging parallel to the optical axis; and a ray incident passing through its object-space nodal point and emerging from the system passing though its image-space nodal point. The point of intersection of these rays in the image space determines the image point. However, before any of these three rays can be traced, we must know the location of the cardinal points. Of course, we need their locations in order to apply the Gaussian imaging equations as well. In the case of a single refracting surface, the principal points coincide with its vertex, and its nodal points coincide with its center of curvature. The principal and the nodal points of a thin lens (in air) coincide with its center. In this chapter, we develop the paraxial ray-tracing equations and demonstrate their utility by determining the cardinal points of simple imaging systems. Starting at an object point, a ray undergoes rectilinear propagation to the first surface of the system; it is refracted or reflected at the surface, depending on whether it is a refracting or a reflecting surface; it undergoes rectilinear propagation again until it reaches the next surface; and the process repeats itself until the ray reaches the image plane. We first develop the paraxial ray-tracing equations for a refracting surface and demonstrate their utility by determining its focal length. Ray tracing of a general system to determine its cardinal points is considered next. How to determine the cardinal points of a combination of two systems is also discussed. As examples of simple systems, a thin lens, a thick lens, and a two-lens system are considered. The ray-tracing equations for a mirror are derived next and applied to a two-mirror system, and a catadioptric system consisting of a thin lens and a mirror. Some of the results obtained in Chapters 2 and 3 on these simple systems are rederived to gain familiarity with the use of the ray-tracing equations. In practice, the ray-tracing equations are used to determine not only the Gaussian properties of a system but also the size of the imaging elements and apertures, vignetting of the rays, and obscurations in mirror systems. This is illustrated by determining the obscuration ratio of a two-mirror system, and the relative size of its secondary mirror and the hole in its primary mirror.