For optical fibre as used in telecommunication, numerical aperture (NA) has the following meanings:
1. The sine of the vertex angle of the largest cone of meridional rays that can enter or leave an optical system or element, multiplied by the refractive index of the medium in which the vertex of the cone is located.
Note: The NA is generally measured with respect to an object or image point and will vary as that point is moved.
2. For an optical fiber in which the refractive index decreases monotonically from n 1 on the axis to n 2 in the cladding, an expression of the extent of the fiber's ability to accept, in its bound modes, non-normal incident rays, given by NA = (n 12-n 22)½.
Note: In multimode fibers, the term equilibrium numerical aperture is sometimes used. This refers to the numerical aperture with respect to the extreme exit angle of a ray emerging from a fiber in which equilibrium mode distribution has been established.
3. Colloquially, the sine of the radiation or acceptance angle of an optical fiber, multiplied by the refractive index of the material in contact with the exit or entrance face.
Note: This usage is approximate and imprecise, but is often encountered.
Source: from Federal Standard 1037C and from MIL-STD-188
where is the index of refraction of the medium in which the lens is working (1.0 for air, up to 1.56 for oils), and is the angular aperture of the lens. The numerical aperture basically is a measure of the diameter of the aperture compares to the focal length. In photography, the f-number expresses the same relationship.