 Energy-normalized pulse profile

An 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.^[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 $\int tp(t)dt$ 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.