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Chapter 7:
Calculation of Primary Aberrations: Perturbed Optical Systems
7.1 INTRODUCTION The image quality of an optical system is limited not only by its inherent design aberrations, discussed in Chapters 5 and 6 for rotationally symmetric systems, but also by the fabrication and assembly errors of its elements. New aberrations arise when its elements are misaligned with respect to each other owing to lack of the rotational symmetry of the resulting perturbed system. The misalignment of an element may be the decentering of its vertex and/or tilting of its optical axis. The decenter of an element usually refers to a misposition of its vertex in a plane normal to its intended optical axis. The decenter along its optical axis is usually called despace in that the spacing between it and its adjacent element is incorrect. It should be evident that when one or more elements of a system are decentered or tilted, it loses its rotational symmetry since it no longer has a common optical axis. However, when the elements are only despaced, the system retains its common optical axis and, therefore, its rotational symmetry. In this chapter, we discuss how to determine the primary aberrations of a perturbed optical system assuming that they are known for the unperturbed system. The first-order effect of a decenter or a tilt of a surface of a system is to produce a transverse displacement of the image formed by the unperturbed system. Its second-order effect is to introduce some new aberrations. It is shown that a small decenter or a tilt does not change the primary spherical aberration of a system. However, if the spherical aberration of the unperturbed system is not zero, it introduces coma that is independent of the image height but depends on the pupil coordinates in the same manner as the primary coma. Since it exists for an on-axis point object, it is called axial coma. The other primary aberrations generate aberrations in addition to their own kind in pupil coordinates. For example, coma of the unperturbed system produces coma, astigmatism, and field curvature when it is perturbed. The degree of a new aberration in the image height is one less than the degree of the corresponding aberration of the unperturbed system. Thus, the additional coma is independent of the image height, astigmatism varies linearly with it, and distortion varies quadratically. A despace error displaces the image and the exit pupil (unless it is also the aperture stop) longitudinally and changes the values of the image distance and the distance of the image from the exit pupil. Accordingly, it changes the value of the aberrations introduced by the despaced element. In a multisurface system, the positions of the image and exit pupil change for each surface that follows the despaced surface. The change in the aberrations introduced by each surface can be calculated in a similar manner. The general equations for the aberrations introduced by a misaligned surface are derived and the results are applied to two-mirror telescopes discussed in Section 6.8.
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