Numerical aperture describes the cone of light an optic can accept or emit, as the sine of the maximum half-angle scaled by the refractive index of the medium the cone travels in.

n is the refractive index of the medium, close to 1.00 for air. θ is the half-angle of the cone measured from the optical axis, so the full cone angle is 2θ. Confusing the two is the most common error with this quantity, which is why both are shown. NA cannot exceed n, and at NA = n the half-angle reaches 90 degrees.

Two f-numbers, and they are not the same

The first form holds for a well corrected system obeying the Abbe sine condition and is the one used for objectives and fibre optics. The second follows from the ray geometry of a thin lens and is the working f-number of classical lens design. They agree to within a percent below about 10 degrees and separate quickly after that: at a half-angle of 30 degrees they differ by a quarter. In air the sine form cannot go below f/0.5, which is the physical limit at NA = 1.

Fibre numerical aperture

A fibre's NA comes from the index step between core and cladding, and it is defined for the acceptance cone in air. Enter that NA with the medium left at air to get the acceptance half-angle. Selecting fused silica as the medium gives something different and usually not what is wanted: the ray angle inside the core rather than the acceptance angle outside the fibre.

Assumptions

Refractive index varies with wavelength, and the values in the list are nominal figures for visible light. In the near infrared where TDLAS operates the difference is small but real, so enter the index at your own wavelength if precision matters. The calculation is geometric throughout and says nothing about aberration, vignetting, transmission or the actual intensity distribution within the cone, which for a laser beam is set by its divergence rather than by an aperture stop.