Radial radiant-energy density of a Gaussian beam ($dQ/dR=2\pi RH(R)$)

The 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 $Q(R)$,Its units are $\mathrm{J}{\mathrm{m}}^{-1}$, while $H$ has units $\mathrm{J}{\mathrm{m}}^{-2}$. The primary Gaussian profile uses $w$ as the irradiance $e^{-2}$ radius.^[ref]

For an original narrow-annulus comparison, take $U=1\mu \mathrm{J}$ 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.