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Chapter 7:
Ray Spot Sizes and Diagrams

In Chapters 2-6, we have determined the primary wave aberrations of simple optical imaging systems. In this chapter, we use the relationship between the wave and ray aberrations given in Section 1.2 to determine the ray distribution for a point object, called the ray spot diagrams, in the Gaussian image plane. For each primary aberration, we determine the extent or the size of the image spot in terms of its peak value and the focal ratio of the image-forming light cone. In the case of spherical aberration and astigmatism, we consider the ray distributions in defocused image planes as well, and we determine the plane in which the spot size is minimum. These minimum-size spots are referred to as the circles of least confusion and represent the best aberrated images based on geometrical optics.

We define the centroid and the standard deviation of the ray distribution and calculate them for the primary aberrations. In lens design, one often tries to minimize the spot sigma rather than the spot radius. However, we will see in Chapter 8 that, in reality, which is based on diffraction of light at the exit pupil of the system, an image distribution is not given by the corresponding ray spot diagrams. For example, the aberration-free image of a point object is a point according to geometrical optics, but its diffraction image for a circular pupil consists of a bright spot surrounded by concentric dark and bright rings. Even so, it is quite common practice in lens design to look at the spot diagrams in the early stages of a design, as discussed in Section 7.10.


Consider an optical system consisting of a series of rotationally symmetric coaxial refracting and/or reflecting surfaces imaging a point object. We have discussed in Chapter 1 that the primary aberration function representing the wave aberration at its exit pupil can be written


where (r, θ) are the polar coordinates of a point in the plane xy of the exit pupil of the system, h′ is the height of the Gaussian image point P′, and as, ac, aa, ad, and at represent the coefficients of spherical aberration, coma, astigmatism, field curvature, and distortion, respectively. The angle q is equal to zero or p for points lying in the tangential or meridional plane (i.e., the zx plane containing the optical axis and the point object and, therefore, its Gaussian image). The chief ray, which, by definition, passes through the center of the exit pupil, always lies in this plane. The plane normal to the tangential plane but containing the chief ray is called the sagittal plane. As the chief ray bends when it is refracted or reflected by a surface, so does the sagittal plane.

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