The pupil function is a mathematical representation used in optics to describe how light passes through an optical system, such as a lens or an aperture. It characterizes the system's ability to focus light and is essential for understanding image formation. The pupil function takes into account factors like the shape and size of the aperture, as well as any aberrations that may affect the light's path. In imaging systems, the pupil function helps determine the quality and resolution of the resulting image. By analyzing the pupil function, optical engineers can optimize designs for devices like cameras and microscopes, ensuring that they capture clear and accurate representations of the subject being observed.