The-number describes the light-gathering ability of the lens in the case where the marginal rays on the object side are parallel to the axis of the lens.
2.
Where is the working-number, is the lens's magnification for an object a particular distance away, is the pupil magnification, and the NA is defined in terms of the angle of the marginal ray as before.
3.
Also, for endoscope applications, where small tube diameter is desirable, most of the tube is filled with glass, with thin air gaps to allow for powered surfaces; because marginal ray angle is smaller at a given numerical aperture the higher the index of refraction, this allows the relay to have higher NA for a given diameter.
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