Pupil function

The pupil function or aperture function describes how a light wave is affected upon transmission through an optical imaging system such as a camera, microscope, or the human eye. More specifically, it is a complex function of the position in the pupil or aperture (often an iris) that indicates the relative change in amplitude and phase of the light wave. Sometimes this function is referred to as the generalized pupil function, in which case pupil function only indicates whether light is transmitted or not. Imperfections in the optics typically have a direct effect on the pupil function, it is therefore an important tool to study optical imaging systems and their performance.

The complex pupil function ${\displaystyle \mathrm {P} (u,v)}$ can be written in polar coordinates using two real functions:

where ${\displaystyle \mathrm {\Theta } (u,v)}$ is the phase change (in radians) introduced by the optics, or the surrounding medium. It captures all optical aberrations that occur between the image plane and the focal plane in the scene or sample. The light may also be attenuated differently at different positions ${\displaystyle (u,v)}$ in the pupil, sometimes deliberately for the purpose of apodization. Such change in amplitude of the light wave is described by the factor ${\displaystyle \mathrm {A} (u,v)}$.

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