Calculation of Primary Aberrations: Refracting Systems

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5.1 INTRODUCTION In Chapter 1, we discussed how to determine the Gaussian image of an object formed by an imaging system. In Chapter 2, we defined the entrance and exit pupils of a system, which determine the light cone diverging from a point object that enters the system and the light cone that exits from it converging to the Gaussian image point, respectively. Although we defined wave and ray aberrations of a system in Chapter 3 and determined the image spot shapes and sizes in Chapter 4, we did not discuss the quality of the images formed by it. The quality of an image formed by a system depends on its wave aberrations for the point object under consideration. Thus, before we can discuss the quality of an image, we must first determine the aberrations of the system corresponding to this image. This is done by considering the wave aberration of a ray as the difference between its optical path length from the object point to the Gaussian image point and that of the chief ray. It is possible to determine the shape of a refracting surface for which the optical path length of all the rays from a given point object to its Gaussian image point is the same. (See Section 5.4 and Problems 5.1 and 5.2 for an example.) For such a surface, called a Cartesian surface, the rays from the given point object all pass through the image point after refraction by it, and the image is said to be perfect or aberration free. However, this is not true for any other point object; the images of other point objects are aberrated. Accordingly, such surfaces are not very practical for imaging of extended objects. In this chapter, we give a step-by-step derivation of the monochromatic primary (or Seidel) aberrations of systems with an axis of rotational symmetry and express them in the plane of the exit pupil in terms of the pupil coordinates and the image height. The term “monochromatic” implies that the refractive indices used in the derivation are for a certain optical wavelength of the object radiation. We start with a derivation of the primary aberrations of a spherical refracting surface with the aperture stop located at the surface so that the exit pupil is also located there. An axial point object is considered first and because of the rotational symmetry of the problem, only spherical aberration is obtained. Next, the aberrations are determined for an off-axis point object with respect to its Petzval image point. Corrections are then determined and applied to obtain the aberrations with respect to the Gaussian image point. Next, the aberrations are determined at the exit pupil when the aperture stop is not located at the surface. These expressions are used to determine the aplanatic points of a spherical refracting surface. The aberrations introduced by a (aspheric) conic surface are determined by considering the additional optical path lengths of the rays due to the difference in its shape from a spherical surface whose radius of curvature is equal to the vertex radius of curvature of the conic surface.