= Energy-normalized pulse profile
{title2=$p(t)=P(t)/U,\ \int p(t)\,dt=1$}
An \b[energy-normalized pulse profile] separates an optical pulse's temporal shape from its total <laser pulse energy>. For nonnegative <radiant power> $P(t)$ and finite $U=\int P(t)\,dt>0$, define $p(t)=P(t)/U$. Then $p$ has units inverse time and integrates to one. The <radiant energy> in a time interval is $U$ times the integral of $p$ over that interval.<International Lighting Vocabulary by CIE>[Entry17-21-041, note1, radiant energy as integrated radiant flux]{ref}
For an original two-lobe example, let $p=1/(8\,\mathrm{ns})$ during $0<t<2\,\mathrm{ns}$ and $p=3/(8\,\mathrm{ns})$ during $5<t<7\,\mathrm{ns}$, with zero values elsewhere. The two integrals are $1/4$ and $3/4$. Choosing $U=4\,\mathrm{nJ}$ gives powers $0.5$ and $1.5\,\mathrm W$, and energies $1$ and $3\,\mathrm{nJ}$. Multiplying $U$ by two doubles both powers without changing $p$.
The <temporal energy centroid of an optical pulse> is $\int t p(t)\,dt$ when this moment exists. Here it is $(1/4)(1)+(3/4)(6)=4.75\,\mathrm{ns}$, in the gap between lobes. An energy centroid need not be a location of maximum power.
Mathematically, $p$ is a <probability density function>, but its weighting here is energy over time. It is not automatically a measured photon-arrival distribution, particularly when photon energies vary. A zero-energy trace cannot be normalized this way, and a noisy baseline-corrected trace with negative excursions is not automatically a nonnegative density. The <Gaussian pulse energy and peak power> article gives one normalized shape family.
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