Numerical aperture
NA = n sin θ_max. For a fibre, NA = √(n_core² − n_cladding²) gives the sine of the maximum acceptance half-angle. For a microscope objective, NA determines the diffraction-limited resolution λ/(2·NA).
Definition
The numerical aperture NA of an optical system is a dimensionless measure of the cone of rays that the system can accept or emit. For a microscope objective, NA = n sin θ_max, where n is the index of the medium between the objective and the specimen (1 for air, 1.5 for immersion oil) and θ_max is the half-angle of the cone of rays the objective can collect from the focal plane. For an optical fibre, NA = √(n_core² − n_cladding²) gives sin θ_max, the sine of the maximum acceptance half-angle in the external medium.
NA controls two of the most important properties of any imaging or fibre-coupling system. In microscopy, the diffraction-limited transverse resolution is r = 0.61 λ/NA (Rayleigh criterion), and the depth of field is DoF ≈ λ/NA² — both improve with increasing NA, which is why immersion-oil objectives (NA up to 1.4) resolve more than dry objectives (NA limited to ~1). In fibre optics, NA determines how much light a source (LED, laser, SMA connector) can launch into the fibre: the accepted power scales as NA². Higher-NA fibres couple more cheaply to cheap light sources but have more modal dispersion in multimode operation and tighter bend-loss constraints; telecom single-mode fibres use NA ≈ 0.12, while high-NA fibres for short-distance illumination reach NA ≈ 0.5. NA is quoted on every objective's and fibre's datasheet because it sets the performance envelope of the subsequent system.