= Laser pulse energy
{title2=$U=\int P(t)\,dt$}
<Laser pulse energy> is the <radiant energy> carried by one specified <laser> <pulse> across a measurement plane. For an isolated pulse whose total <radiant power> is $P(t)\geq0$ and whose background is zero,
$$
\boxed{U=\int_{-\infty}^{\infty}P(t)\,dt.}
$$
Here $P$ is already integrated over the transverse beam <area>; it is not local <irradiance>. The unit of $U$ is the <joule>. <Radiant exposure> instead divides received <energy> by a receiving <area>, so a beam's $U$ alone does not specify the exposure at its centre. The <radiant exposure of a Gaussian beam> makes that spatial distinction quantitative.
The <Gaussian pulse energy and peak power> relation gives $P_{\rm pk}\simeq0.939437\,U/\tau$ when $\tau$ is the power <full width at half maximum>. The <pulse energy shape factor> records pulse-shape dependence, while the <Gaussian pulse field-envelope width> is $\sqrt2$ times the power width.
For an original example, stipulate $U=0.50\,\mathrm{\mu J}$ and a <Gaussian pulse> power <full width at half maximum> $\tau=10\,\mathrm{ns}$. Then $P_{\rm pk}=46.97\,\mathrm W$. Stretching the same model pulse to $40\,\mathrm{ns}$ while keeping $U$ unchanged reduces its <peak power> to $11.74\,\mathrm W$. Both curves below integrate to the same <energy>; their peak heights and widths differ.
\Image[/gaussian-pulse-energy-width.png]
{title=Original Gaussian power profiles with equal pulse energy: a longer pulse has lower peak power.}
A finite <detection time gate> collects the <Gaussian pulse time-gate energy fraction>, which depends on its width and alignment. A centred gate as wide as the power <full width at half maximum> receives only $76.10\%$ of $U$, because Gaussian tails extend beyond it. For a $\tau$-wide gate, a <Gaussian gate centring error> of $\tau/2$ reduces capture to $49.07\%$.
The <average power of a laser pulse train> follows by counting complete pulses. If a long observation of duration $T$ contains $N$ pulses of <energies> $U_1,\ldots,U_N$, with negligible edge contributions and no continuous background,
$$
\overline P=\frac1T\sum_{j=1}^N U_j
=\frac NT\,\overline U
=f_{\rm rep}\overline U.
$$
Here $f_{\rm rep}$ is the <pulse repetition frequency>, not the inverse pulse width. An <average power> readout constrains $\overline U$ when $f_{\rm rep}$ is known; it does not resolve <pulse-to-pulse energy fluctuation>. Identical, well-separated $0.50\,\mathrm{\mu J}$ pulses at $1.0\,\mathrm{MHz}$ give \b[$0.50\,\mathrm W$ average power], even though the stipulated $10\,\mathrm{ns}$ pulses peak near $47\,\mathrm W$.
If a steady optical background is present, state whether the reported <energy> includes it. For pulse energy above a measured background $P_{\rm bg}$, the <pulse energy baseline correction> uses $\int_{\rm gate}[P(t)-P_{\rm bg}]\,dt$. Failing to subtract a constant background adds $P_{\rm bg}T_g$ to the inferred <energy>. This optical-background model is distinct from an offset introduced later by detector electronics.
Historical NIST equipment illustrates what the readouts measure. Its TC-24 reference absorbed the beam in a <glass> plate, sensed a plate <temperature> difference with <thermocouples>, and integrated the <thermopile>'s <electric potential difference> during beam injection. Before/after readings enabled <background subtraction> and compensated thermal drift. Counting pulses converted total injected <energy> to mean pulse <energy>. The <semiconductor> pulse <photodetector> used an <integrating amplifier>; calibration related the peak output <electric potential difference> to pulse <energy>. That electrical peak did not by itself establish optical <peak power>.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[§3.5, printed p. 8; §4, printed pp. 9–10: thermal reference, baselines, pulse counting and integrated readout]{ref} <Pulse energy measurement with a photodiode> connects this kind of measurement to collected <electric charge>; validating the detector response under the actual pulse conditions remains part of that interpretation.
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