Gaussian Beam

F-number and numerical aperture for Gaussian beams

The useful f-number is set by the part of the lens your beam actually fills. This short guide connects that illuminated diameter to effective NA and to the paraxial design limit.

Published by Gaussian Beam · Updated August 4, 2026

The effective f-number

For a Gaussian beam incident on a focusing lens, define \(D_{\mathrm{beam}}=2\omega_L\) using the 1/e² beam radius \(\omega_L\) at the lens. The effective f-number is

\[ N\equiv f/\#=\frac{f}{D_{\mathrm{beam}}}=\frac{f}{2\omega_L} \]

A smaller \(N\) means a wider converging cone and tighter focusing; a larger \(N\) means a gentler, more paraxial cone.

Relation to numerical aperture

With refractive index \(n\) around the focus and cone half-angle \(\alpha\),

\[ \mathrm{NA}_{\mathrm{eff}} =n\sin\alpha =n\sin\!\left[\arctan\!\left(\frac{1}{2N}\right)\right] \approx\frac{n}{2N} \]

The last form is the small-angle approximation. In air, \(N=2\) gives \(\mathrm{NA}_{\mathrm{eff}}\approx0.24\).

Use the illuminated diameter

A lens datasheet usually computes f-number from its clear aperture. That is the correct effective value only when the beam fills the aperture. If the beam underfills the lens, use \(2\omega_L\): the effective NA is lower and the effective f-number is higher than the datasheet value.

F-number describes focusing geometry, not beam quality. A non-ideal spatial mode can focus worse than its f-number suggests; that degradation is described by M².

A practical design check

For \(N\gtrsim2\), equivalently \(\mathrm{NA}_{\mathrm{eff}}\lesssim0.25\) in air, paraxial Gaussian formulas are generally reliable for design. Below that, use them as a first-order estimate and consider a vector diffraction model when accurate high-NA focal fields matter.