In the last article we learned that a complex lens can be modeled as just an entrance pupil, an exit pupil and a geometrical optics black-box in between. Goodman[1] suggests that all optical path errors for a given Gaussian point on the image plane can be thought of as being introduced by a custom phase plate at the pupil plane, delaying or advancing the light wavefront locally according to wavefront aberration function 
as described there.
The phase plate distorts the forming wavefront, introducing diffraction and aberrations, while otherwise allowing us to treat the rest of the lens as if it followed geometrical optics rules. It can be associated with either the entrance or the exit pupil. Photographers are usually concerned with the effects of the lens on the image plane so we will associate it with the adjacent Exit Pupil.
Continue reading Fourier Optics and the Complex Pupil Function
twice in sequence.
space to a Fast Fourier Transform routine and, presto, it produces MxN numbers representing the amplitude of the PSF on the 
sensing plane. Figure 1a shows a simple case where pupil function 
arrives at the front of the lens. It goes through the lens being partly blocked and distorted by it as it arrives at its virtual back end, the Exit Pupil, we’ll call this blocking/distorting function 
. Other than in very simple cases, the Exit Pupil does not necessarily coincide with a specific physical element or Principal surface.
as shown below (not to scale, the product of the two arrays is element-by-element):
![\[ MTF_{2D} = \left|\widehat{ PSF_{lens} }\cdot \widehat{PIX_{ap} }\right|_{pu}\ast\ast\: \delta\widehat{\delta_{pitch}} \]](https://i0.wp.com/www.strollswithmydog.com/wordpress/wp-content/ql-cache/quicklatex.com-3409f065a49bb47fd4618ed595be72fc_l3.png?resize=320%2C37&ssl=1 "Rendered by QuickLaTeX.com")