# Effective NA - front to rear realationship

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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 a 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.