In the previous entry we discussed the numerical aperture (NA) in similar terms to the f-number, referring the calculation to the front focal point. Measured at the front focal point, the resulting NA is the maximum possible for the lens. Imaging at the front focal point implies an infinite magnification and infinite image distance - not possible with any camera that I know.
That leaves us with calculating an effective or working NA - the numerical aperture that the entrance pupil forms with the object being photographed. Like the effective aperture on the image side, the working NA is determined by f/#, magnification, and pupillary magnification. The equation is very similar to the equation from the previous entry but with an added multiplier that takes into account the magnification and the pupillary magnification.
NA = 1/(2*N*((1/m)+(1/P))) (where N=aperture setting, m=magnification, P=pupillary magnification)
You will notice that the f/4 P=2 lens has an Effective NA advantage on the front end just as it does on the back end of the lens with the Effective aperture. The same applies with the disadvantage with the f/4 P=1/2 lens. It turns out that there is an association between the NA on the front of the lens and the NA on the rear of the lens. That will addressed in the next installment.