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Chapter 6:
Calculation of Primary Aberrations: Reflecting and Catadioptric Systems
6.1 INTRODUCTION In Chapter 5, we discussed with examples how to determine the aberrations of an imaging system consisting of refracting surfaces imaging a point object. In this chapter, we consider imaging systems with reflecting surfaces, i.e., catoptric or mirror systems. Catadioptric systems, i.e., those consisting of reflecting and refracting elements are also discussed. We start with a system consisting of a single reflecting surface. Although its aberrations may be derived by using the technique used in Sections 5.2 through 5.5 for a refracting surface (and it is quite instructive to do so), detailed derivations are not given here. Instead, we obtain the results for a conic reflecting surface from those for a corresponding refracting surface by substituting the refractive index associated with the reflected rays equal to the negative of the refractive index associated with the incident rays. The aberrations of a spherical mirror with the aperture stop located at its center of curvature are discussed and the results are utilized to describe catadioptric systems such as the Schmidt and Bouwers-Maksutov cameras. Next a beam expander with two confocal paraboloidal mirrors is discussed. It is shown that such a system is anastigmatic, i.e., it is free of spherical aberration, coma, and astigmatism. Finally, the aberrations of a two-mirror system imaging an astronomical object are discussed and the aberration properties of telescopes such as classical and aplanatic Cassegrain and Gregorian, Couder, and Schwarzschild are described. Finally, the aberrations of aspheric plates used in astronomical telescopes are discussed. 6.2 CONIC REFLECTING SURFACE In this section, we discuss conic reflecting surfaces briefly and give expressions for their primary aberrations as obtained from the corresponding expressions for a refracting surface by substituting for the refractive index associated with the reflected rays equal to the negative of the refractive index associated with the incident rays. Of course, these expressions can also be derived in the same manner as we did for a refracting surface in Sections 5.2 through 5.5. 6.2.1 Conic Surface A Cartesian reflecting surface is one for which light rays from a given point object pass through the same image point after reflection from it. The image is thus aberration free. The Cartesian surfaces for reflection are the conics of revolution. As discussed in Section 5.4.1, a conic represents the locus of a point P such that its distance from a fixed point F, called geometrical focus, bears a constant ratio e, called eccentricity, to its distance from a fixed straight line called the directrix.
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