<Dl> <Dd> NA i = n sin ⁡ θ = n sin ⁡ (arctan ⁡ (D 2 f)) ≈ n D 2 f, (\ displaystyle (\ text (NA)) _ (\ text (i)) = n \ sin \ theta = n \ sin \ left (\ arctan \ left ((\ frac (D) (2f)) \ right) \ right) \ approx n (\ frac (D) (2f)),) </Dd> <Dd> thus N ≈ 1 / 2NA, assuming normal use in air (n = 1). </Dd> </Dl> <Dd> NA i = n sin ⁡ θ = n sin ⁡ (arctan ⁡ (D 2 f)) ≈ n D 2 f, (\ displaystyle (\ text (NA)) _ (\ text (i)) = n \ sin \ theta = n \ sin \ left (\ arctan \ left ((\ frac (D) (2f)) \ right) \ right) \ approx n (\ frac (D) (2f)),) </Dd> <Dd> thus N ≈ 1 / 2NA, assuming normal use in air (n = 1). </Dd> <P> The approximation holds when the numerical aperture is small, but it turns out that for well - corrected optical systems such as camera lenses, a more detailed analysis shows that N is almost exactly equal to 1 / 2NA even at large numerical apertures . As Rudolf Kingslake explains, "It is a common error to suppose that the ratio (D / 2f) is actually equal to tan θ, and not sin θ...The tangent would, of course, be correct if the principal planes were really plane . However, the complete theory of the Abbe sine condition shows that if a lens is corrected for coma and spherical aberration, as all good photographic objectives must be, the second principal plane becomes a portion of a sphere of radius f centered about the focal point". In this sense, the traditional thin - lens definition and illustration of f - number is misleading, and defining it in terms of numerical aperture may be more meaningful . </P>

Determination of numerical aperture and acceptance angle of an optical fibre