 Encircled radiant energy of a Gaussian beam

The encircled radiant energy of a Gaussian beam is the radiant energy inside a centred circular receiver at a specified transverse plane. For fixed Gaussian beam radius $w$ and total laser pulse energy $U$,The radiant exposure of a Gaussian beam contains the radial integration and aperture-capture plot. Its $w$ is the irradiance $e^{-2}$ radius, as specified in the primary beam account.^[ref]

An original annular example distinguishes energy near an axis from energy between two radii. Subtracting two encircled values givesChoose $U=5\mu \mathrm{J}$, $w=0.50\mathrm{mm}$, $R_1=0.25\mathrm{mm}$ and $R_2=0.50\mathrm{mm}$. The annulus receives $5(e^{-1/2}-e^{-2})\approx 2.356\mu \mathrm{J}$, or 47.12 percent of the full beam energy. A bright central point does not imply that most energy lies in an arbitrarily small disk around it.

Indeed, the radial radiant-energy density of a Gaussian beam is $dQ/dR=2\pi RH(R)$, where $H$ is local radiant exposure. For this profile it is largest at $R=w/2$, even though $H$ is largest at zero radius. Increasing a centred aperture and comparing integrated detector readings can test the cumulative prediction. Detector linearity, aperture alignment and the finite collection boundary must be controlled; a displaced aperture is not described by the centred formula.