How To Measure A Beam Waist

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It is often necessary to measure the waist of a Gaussian beam in lab. The simplest technique is the razor blade method: a straight razor blade is drawn across the beam and the intensity recorded at several points. For an input power P and a beam with waist w,

$x=-w/2\implies p=.84P$
$x=0\implies p=.5P$
$x=w/2\implies p=.16P$

This is used to find the beam measurements that are then used to calculate the beam parameters.

Derivation

We begin with the intensity equation for a Gaussian beam,

$I(x,y)=\frac{2P}{\pi w^2} e^{-\frac{2(x^2+y^2)}{w^2}}.$

The power blocked by the razor simply the double integral

$P(x)=\frac{2P}{\pi w^2}\int_{-\infty}^{\infty}dy\int_{-\infty}^{x}dx' e^{-\frac{2(x'^2+y^2)}{w^2}}.$

The two integrals can be performed separately. If the beam is elliptical and the razor is drawn along either the major or minor axis, perfectly sensible results for the beam parameters along that axis will be given. The y integral is a simple gaussian ( $\int_{-\infty}^{\infty}dy \exp[-y^2]=\sqrt{\pi}$ ) and the x integral is an error function, resulting in