As with the front of the lens, the effective aperture can be easily converted into a numerical
aperture (NA).
NA_rear = 1/(2*N*((m/P) + 1)) (where N = f-number of lens, m = magnification, P =
pupillary magnification)
Computing a working NA for the rear of the lens isn't particularly helpful for me as I prefer to use
effective aperture. The interesting part is that the working NA for the front and the rear of the
lens are tied together mathamatically.
NA_front = (NA_rear)*m
It's a very simple association and it is completely independent of pupillary magnification. That
means that with any lens of whatever pupillary magnification and f/number, any gains or losses in
NA on the back of the lens associated with the pupillary magnification will have
an proportional gain or loss in the NA on the front of the lens.
This association can also be tranferred into terms of f/numbers where it is probably more useful
to most photographers (microscopists work predominantly in NA terms).
Working f_num = Effective_aperture/m
As an example, if I have a lens set to f/4 with a P=1 and an m=5, the effective aperture is 4*(5+1)
or f/24. Applying the above equation gives you a working f/number of 24/5 or f/4.8. That
translates into a NA of 1/(2*4.8) or about 0.10.
Another example is with microscope objectives. The NA listed on the objective is a working NA at
the listed magnification. A 4x/0.10 objective has a working f/number at m=4 of 1/(2*0.1) or f/5.
The effective aperture is (5*4) or f/20.
Effective NA - front to rear
realtionship
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