= Radial radiant-energy density of a Gaussian beam
{title2=$dQ/dR=2\pi R H(R)$}
The \b[radial radiant-energy density of a Gaussian beam] gives energy per increment of radius, rather than per unit surface area. For local <radiant exposure> $H(R)$ and <encircled radiant energy of a Gaussian beam> $Q(R)$,
$$
g(R)=\frac{dQ}{dR}=2\pi R H(R)=\frac{4UR}{w^2}e^{-2R^2/w^2}.
$$
Its units are $\mathrm{J\,m^{-1}}$, while $H$ has units $\mathrm{J\,m^{-2}}$. The primary Gaussian profile uses $w$ as the irradiance $e^{-2}$ radius.<Propagation of a Gaussian Beam Through a Thin Lens by Francesco D'Angelo>[Equation1 and radius definition]{ref}
For an original narrow-annulus comparison, take $U=1\,\mathrm{\mu J}$ and $w=0.80\,\mathrm{mm}$. The central disk from zero to $0.020\,\mathrm{mm}$ receives about $1.249\,\mathrm{nJ}$. The annulus from $0.390$ to $0.410\,\mathrm{mm}$ receives about $30.32\,\mathrm{nJ}$. Both span the same radial width, but the outer annulus has much more area. Equal radial width is not equal receiving area.
The shell density is maximal at $R=w/2$ and vanishes at $R=0$, despite the local exposure being largest on axis. Dividing $g$ by $U$ gives a <probability density function> over nonnegative radius: it is a <Rayleigh distribution> with scale $w/2$. This is an energy-weighted mathematical representation, not an additional photon-counting assumption. Differences between successive centred-aperture energy measurements estimate shell energies; dividing by their radial separation approximates $g$ when that separation is sufficiently small.
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