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Chapter 4:
Aberrations of a Spherical Mirror
Abstract
4.1 Introduction So far, we have considered refracting imaging systems: a spherical refracting surface in Chapter 1, a thin lens in Chapter 2, and a plane-parallel plate in Chapter 3. Now we consider the imaging properties of a spherical reflecting surface, i.e., a spherical mirror. These properties can be obtained in a manner similar to that for a spherical refracting surface. However, the geometry of the problem is different since now a ray incident on the surface is reflected back into the same medium containing the incident ray, instead of being refracted into another medium. Accordingly, it is instructive to draw object and image rays and not blindly use the imaging and aberration relations appropriate for a reflecting surface. In this chapter, we give the relations describing the primary aberrations of a spherical mirror for an arbitrary position of the aperture stop. These relations are applied to specific cases, one when the aperture stop is located at the mirror and the other when it is located at its center of curvature. It is shown that, in the first case, field curvature and distortion are zero. In the second case, coma, astigmatism, and distortion are zero. A numerical problem illustrates these results. 4.2 Primary Aberration Function Consider an imaging system consisting of a spherical mirror of radius of curvature R and focal length f. Let the aperture stop and the corresponding exit pupil of the system be located as indicated in Figure 4-1. The line joining the center of curvature C of the mirror and the center of the aperture stop (and, therefore, the center O of the exit pupil) defines the optical axis of the system. Consider an object lying at a distance S from the vertex V 0 of the mirror. Let the height of an object point P from the optical axis be h. The distance S ′ and height h ′ of its Gaussian image P ′ are given by 1 S +1 S ′ =−2 R =1 f and M=h ′ h =S ′ +R S+R =−S ′ /S, respectively, where M is the magnification of the image.