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Chapter 4:
Geometrical Point-Spread Function
4.1 INTRODUCTION In this chapter, we discuss the distribution of rays in the image of a point object formed by an aberrated optical system. Such a distribution is referred to as the spot diagram and its extent is called the spot size. The distribution of the density of rays is called the geometrical point-spread function. We define its centroid and standard deviation or spot sigma and calculate them for primary aberrations. In the case of spherical aberration and astigmatism, ray distribution and spot size are considered in image planes other than the Gaussian as well, thereby introducing the concept of aberration balancing. In the early stages of the design of an optical imaging system, one often considers its transverse ray aberrations in an image plane for a set of rays lying along a certain line in the plane of the exit pupil and passing through its center. Such a set of rays is called a ray fan and, often, rays along the x and y axes are used for investigating the ray aberrations and thereby the quality of the system, where the point object is assumed to be along the x axis of the object plane. The set of rays along the x axis of the exit pupil plane is called the tangential ray fan, and the one along its y axis is called the sagittal ray fan. The wave and ray aberrations for the two types of ray fans are discussed for each primary aberration. Also discussed are the balanced aberrations for minimum spot sigma in terms of Zernike circle polynomials. The characteristics of the ray spots and tolerance for primary aberrations are summarized in the last section of this chapter. 4.2 THEORY Consider an optical system consisting of a series of rotationally symmetric coaxial refracting and/or reflecting surfaces imaging a point object P lying at a height h from the optical axis. As in earlier chapters (see Figure 3-3), we assume, without loss of generality, that the point object lies along the x axis, and the z axis is along the optical axis of the system. In Chapter 3, we showed [see Eq. (3-38)] that the primary aberration function at its exit pupil may be written W(r,θ;h ′ )=a s r 4 +a c h ′ r 3 cosθ+a a h ′ 2 r 2 cos 2 θ+a d h ′ 2 r 2 +a t h ′ 3 rcosθ, 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 a s , a c , a a , a d , and a t represent the coefficients of spherical aberration, coma, astigmatism, field curvature, and distortion, respectively. The angle θ is equal to zero or π 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. The angle θ is equal to π/2 or 3π/2 for points lying in 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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