In this chapter we consider the line of sight (LOS) of an aberrated optical system. The LOS is assumed here to coincide with the centroid of its diffraction point-spread function (PSF). For an aberration-free system, it coincides with the center of the PSF. For an aberrated system, it depends on the various orders of its coma aberrations. Thus, a coma aberration not only reduces the central value of the PSF like any other aberration, but it also shifts its centroid. We consider here PSFs aberrated by primary coma and give numerical results on the location of their peaks and centroids.
The LOS of an aberration-free optical system coincides with the center of its diffraction PSF. For an aberrated system, let us define its LOS as the centroid of its aberrated PSF. Thus, if I (x, y) represents the irradiance distribution of the aberrated image of a point object, its centroid x, y representing the LOS error of the system is given by
where P is the total power in the image. It can be shown that the centroid thus obtained is identical to that obtained from the geometrical PSF. Let the aberration function in terms of the Zernike circle polynomials (see Section 8.3.7) for a system with a circular exit pupil be given by
where cnm and snm are the Zernike aberration coefficients representing the standard deviations of the corresponding aberration terms across the pupil (with the exception of the piston term n = 0 = m, which has a standard deviation of zero). It can be shown that the centroid of its aberrated PSF for a uniformly illuminated pupil is given by
where F is the focal ratio or the f-number of the image-forming light cone and a prime indicates a summation over odd integral values of n. We note that only those aberrations contribute to the LOS errors that vary with θ as cosθ and sinθ. Aberrations varying as cosq contribute to x and those varying as sinq contribute to y.