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In Part I of Optical Imaging and Aberrations, we showed how to determine the location and size of the image of an object formed by an imaging system in terms of the location and size of the object and certain parameters of the system. We discussed the relationship between the irradiance distribution of the image and the radiance distribution of the object, including the cosine-to-the-fourth-power dependence on the field angle and vignetting of the rays by the system. We discussed the ray distribution of the aberrated image of a point object, called the geometrical point-spread function (PSF) or the spot diagram. We showed how to design and analyze imaging systems in terms of their primary aberrations. We also showed how to calculate the primary aberrations of a multisurface optical system in terms of their radii of curvature, spacings between them, and the refractive indices associated with those spacings. We pointed out that the image obtained in practice differs from that predicted by geometrical optics. For example, when the system is aberration free, all of the rays from a point object transmitted by the system converge to its Gaussian image point according to geometrical optics. In reality, however, the image obtained is not a point. Because of diffraction of object radiation at the aperture stop of the system or, equivalently, at its exit pupil, the actual aberration-free image for a circular exit pupil is a light patch surrounded by dark and bright diffraction rings. The determination of the characteristics of the diffraction image of an object formed by an aberrated system is the subject of Part II.
In this chapter, we first describe the diffraction theory of image formation of incoherent objects, i.e., objects for which the radiation from one of its parts is incoherent with that from another. We start with a brief account of Rayleigh-Sommerfeld theory of diffraction from a Fourier transform standpoint and derive the Huygens-Fresnel principle from it. First, we consider the Gaussian image of an object, i.e., the aberration-free image according to geometrical optics, and show that it is an exact replica of the object scaled by its (transverse) magnification. Next, we consider the diffraction image based on the Huygens-Fresnel principle. We introduce the concept of a diffraction PSF, i.e., the diffraction image of a point object, and show that it is proportional to the modulus square of the Fourier transform of the complex amplitude across the exit pupil, called the pupil function, of the system.
Geometrical optics is assumed to hold from the point object to the exit pupil in that rays are traced through the system to determine the shape of the pupil and the aberration across it for the point object under consideration, as discussed in Section 3.2 of Part I. The amplitude associated with a ray is obtained by consideration of the transmission and reflection characteristics of the imaging elements of the system. In many applications, the amplitude variation across the pupil is negligible.
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