Gaussian beam

Numerical aperture

From the definitions for the Rayleigh range \(z_R\) and numerical aperture \(NA\): \[z_R = \frac{\pi w_0^2 n}{\lambda_0} \tag{1}\] \[ NA = n \sin\theta \tag{2}\] and from the derivation of the beam divergence angle \(\theta\) in the paraxial limit: \[ \theta \approx \frac{w_0}{z_R} \tag{3}\] we can combine (1) and (3) to get: \[ \theta \approx \frac{\lambda_0}{\pi w_0 n} \tag{4}\] and therefore the numerical aperture, also in the paraxial limit \(\sin\theta \approx \theta\), is: \[ NA \approx \frac{\lambda_0}{\pi w_0} \tag{3}\]