Image distortion, pupil coma, and relative illumination

Relative illumination is affected by image distortion, pupil coma, and pupil magnification. Image and pupil aberrations have a known relationship that also can be written in terms of pupil magnification. Pupil coma is the stretching or compression of the pupil along the field axis, which directly affects relative illumination. Front-stopped designs have pupil aberrations on the exit pupil while rear-stopped systems have pupil aberrations on the entrance pupil. Of the primary pupil aberrations, pupil coma can have the largest effect on the relative illumination in optical designs with significant image distortion (ex: fisheye lens). The Lagrange invariant holds for systems with image distortion using a paraxial approximation of pupil coma. Relative illumination can be written in terms of % distortion, pupil magnification, and object–space field angle.