We have previously discussed the concept of effective aperture and have found that it is related to the size of the aperture and the distance the aperture is from the detector. A small aperture that is close to the detector is equivalent to a larger aperture farther away, i.e. as long as they make the same angle with the detector they will have the same effective aperture.
We have been talking about the "aperture" previously for this discussion, but the real work of the effective aperture is done at the exit pupil. As previously defined, the exit pupil the apparent size and position of the aperture as seen through the lens. This all means that the pupillary magnification have an effect on the effective aperture, although predominantly with close-up and high magnification.
To set the stage for this discussion, you need to know how the pupil moves in relation to the rear principal plane as the pupillary magnification is changed. With a symmetric lens (P=1), the entrance and the exit pupils are equal in size. So, a 100mm f/4 lens will have a 25mm entrance pupil and a 25mm exit pupil. In this case the entrance and the exit pupils will be positioned at the principal planes. The effective aperture at infinite focus will be focal length divided by the exit pupil, 100/25 or f/4.
For non-symmetric lenses, where P does not equal 1, with the same f/number, the exit pupil will travel along this same cone forward or backward depending upon the pupillary magnification, its size and position proportional to the pupillary magnification. This concept is easier to see in a diagram. A point that took me a while to figure out is that the diameter of the cone as it crosses the rear principal plane will always be the same same size as the entrance pupil, no matter what the pupillary magnification is.
At infinity focus (focus at the rear focal point of the lens), the effective aperture is always the same as the aperture setting. The exit pupil may be only half the distance from the focal point with a P=1/2 lens, but the size is also halved and the angle it makes with the detector is unchanged (see above diagram). At any focus closer than infinity, the front aperture setting and the effective aperture will start to diverge as you would expect with the effective aperture. The difference is that the resulting effective aperture will vary depending upon the pupillary magnification. Again, this is best seen with a diagram.
The formula that defines this relationship of the effective aperture to the pupillary magnification is:
N' = N*((m/P) + 1) (where N' = effective aperture, N = lens aperture setting, m = magnification, and P = pupillary magnification)
The long and the short of it is that with a P>1 you will lose less aperture as the magnification is increased and the opposite for a P<1. A larger effective aperture (smaller number) means more potential image resolution. Sounds like all high magnification lenses should have a P>1 - well maybe not as we will discuss the plusses and minuses of this situation in the next installment.