= Radiant exposure of a Gaussian beam
{title2=$H(r)=2Ue^{-2r^2/w^2}/(\pi w^2)$}
The \b[radiant exposure of a Gaussian beam] is its time-integrated transverse <irradiance>. At a fixed plane normal to a circular <Gaussian beam>, suppose the <Gaussian beam radius> $w$ is unchanged during a pulse and denotes the radius where irradiance is $e^{-2}$ of the on-axis value. With total <laser pulse energy> $U$,
$$
H(r)=\frac{2U}{\pi w^2}e^{-2r^2/w^2}.
$$
The primary beam account specifies this radius convention and the corresponding irradiance normalization; the <radiant exposure> parent derives its time integral.<Propagation of a Gaussian Beam Through a Thin Lens by Francesco D'Angelo>[Gaussian Beam: Definition and Characteristics, equation1 and radius definition]{ref}
An original aperture example asks how much of that energy a finite receiver collects. Integrating over a centred disk of radius $R$ gives the <encircled radiant energy of a Gaussian beam>:
$$
Q(R)=\int_0^R H(r)\,2\pi r\,dr
=U\left(1-e^{-2R^2/w^2}\right).
$$
For stipulated $U=2\,\mathrm{\mu J}$, $w=0.80\,\mathrm{mm}$ and $R=0.40\,\mathrm{mm}$, the receiver collects only $39.35$percent, or $0.7869\,\mathrm{\mu J}$. Its <spatially averaged radiant exposure> is about $1.566\,\mathrm{J\,m^{-2}}$, compared with on-axis $H(0)\approx1.989\,\mathrm{J\,m^{-2}}$. Dividing the total $U$ by this receiver's area would count energy that missed it.
\Image[/gaussian-aperture-capture.png]
{title=Local Gaussian radiant exposure and cumulative energy capture describe different functions of radius.}
To collect a chosen fraction $0<f<1$ under this ideal model, the centred aperture needs $R=w\sqrt{-\log(1-f)/2}$. For95percent, $R\approx1.224w$. A possible experimental check measures integrated received energy through several known apertures and compares their ratios. Aperture centring, detector linearity and any profile change during the pulse affect that comparison. A time-varying radius must remain inside the time integral rather than being replaced by an unspecified average radius.
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