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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.

For an isolated Gaussian pulse, the relation between laser pulse energy $U$ and peak power $P_{\mathrm{pk}}$ depends on the pulse-width convention. Here $P(t)$ is the total incident radiant power, the background is zero, and $\tau >0$ denotes the power full width at half maximum. Integrating $P$ over the entire pulse defines its positive energy $U>0$.

Pulse energy, peak power and average power resolve different information. Consider a Gaussian pulse withThe chosen pulse duration $\tau$ is the full width at half maximum of radiant power: $P(\pm \tau /2)=P_{\mathrm{pk}}/2$. It refers to radiant power, rather than the wave amplitude of the electric field. The Gaussian integral gives the relation between Gaussian pulse energy and peak power:The pulse energy shape factor is $\kappa =U/(P_{\mathrm{pk}}\tau )$. It equals $1.064467$ for this power profile; a rectangular pulse of duration $\tau$ has $\kappa =1$. Knowing only a width and an integrated energy therefore requires a pulse-shape assumption before inferring peak power.

An equivalent form makes the normalization transparent. Write the pulse centre as $t_0$, and let $\sigma$ denote the standard deviation of its energy-normalized pulse profile:The ratio $p(t)=P(t)/U$ is a probability density function in the mathematical sense: it is nonnegative, has units ${\mathrm{s}}^{-1}$, and integrates to one. Here it weights the pulse's energy over time. Its normal distribution profile has mean $t_0$ and variance ${\sigma }^2$. This gives $P_{\mathrm{pk}}=U/(\sqrt{2\pi }\sigma )$ without a separate area estimate.

For a stipulated $\sigma =2\mathrm{ns}$ and $U=1\mathrm{nJ}$, the power full width at half maximum is $4.70964\mathrm{ns}$ and the peak power is $0.19947\mathrm{W}$. Confusing the standard deviation with the full width at half maximum would insert the wrong duration into the conversion.

The Gaussian pulse field-envelope width is a second distinct convention. If a fixed spatial field profile gives $P(t)=K|A(t){|}^2$ for a constant $K>0$, the wave amplitude envelope obeysConsequently its amplitude full width at half maximum is ${\tau }_E=\sqrt{2}\tau$. Inserting ${\tau }_E$ into the formula for a power-width input underestimates the true peak power by a factor $1/\sqrt{2}$. State which measured waveform defines the width.

An electrical photodetector trace can also be broadened by its impulse response. The optical pulse-width inference therefore needs a characterized detector response.^[ref]

The Gaussian pulse field-envelope width is the full width at half maximum of the magnitude of a Gaussian electric-field envelope. It differs from the FWHM of radiant power. For a fixed spatial mode with $P(t)=K|A(t){|}^2$ and constant $K>0$,The Gaussian pulse energy and peak power article derives this relation. Power-FWHM is also the convention explicitly used for PULSELENGTH in the primary source documentation.^[ref]

For an original example, a power width of $12\mathrm{ns}$ corresponds to field-envelope width $16.97\mathrm{ns}$. At the field-envelope half-maximum points, the power is one quarter of its peak, since $(1/2{)}^2=1/4$. At the power half-maximum points, the field magnitude is instead $1/\sqrt{2}$ of its peak. A threshold applied to one measured quantity therefore does not define the other quantity's FWHM.

The envelope magnitude is distinct from the rapidly oscillating signed electric field. A direct photodetector generally supplies a power-related electrical trace, further modified by its impulse response; it is not a direct recording of the optical carrier's field envelope. Inserting an amplitude width into a power-width peak power formula consequently uses the wrong convention. State the observable and the response model with the reported pulse duration.

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 $P(t)\ge 0$ and whose background is zero,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 $P_{\mathrm{pk}}\simeq 0.939437U/\tau$ 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 $\sqrt{2}$ times the power width.

For an original example, stipulate $U=0.50\mu \mathrm{J}$ and a Gaussian pulse power full width at half maximum $\tau =10\mathrm{ns}$. Then $P_{\mathrm{pk}}=46.97\mathrm{W}$. 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.

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 $U_1,\dots ,U_N$, with negligible edge contributions and no continuous background,Here $f_{\mathrm{rep}}$ is the pulse repetition frequency, not the inverse pulse width. An average power readout constrains $\overline{U}$ when $f_{\mathrm{rep}}$ is known; it does not resolve pulse-to-pulse energy fluctuation. Identical, well-separated $0.50\mu \mathrm{J}$ pulses at $1.0\mathrm{MHz}$ give $0.50\mathrm{W}$ 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 $P_{\mathrm{bg}}$, the pulse energy baseline correction uses ${\int }_{\mathrm{gate}}[P(t)-P_{\mathrm{bg}}]dt$. Failing to subtract a constant background adds $P_{\mathrm{bg}}{T}_g$ 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.^[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.

Pulse duration quantifies a pulse's temporal extent using a stated observable and width convention. A common optical convention is the full width at half maximum of the background-subtracted radiant power profile. The Ansys source documentation explicitly defines its PULSELENGTH parameter as the power-profile FWHM, rather than the width of the electric-field amplitude.^[ref]

For an original example, choose a triangular power pulseIts nonzero support extends from $-T$ to $T$, giving total support width $16\mathrm{ns}$. Its half-maximum crossings occur at $t=\pm T/2$, giving a power FWHM of only $8\mathrm{ns}$. Neither number is wrong when its convention is specified. The pulse energy shape factor additionally relates a chosen width to integrated energy and peak power.

A fast photodetector and oscilloscope record an electrical response to an optical pulse. For a linear response model, the recorded trace is a convolution with the instrument's impulse response; a slow response can broaden the measured feature. The readout width need not equal the optical width. State the observable, background treatment and instrument response when reporting a duration. Pulse energy baseline correction affects where the half-maximum threshold lies, while Gaussian pulse energy and peak power treats the conversion for a specific power-profile family.

The pulse energy shape factor is the dimensionless ratiowhere $U$ is laser pulse energy, $P_{\mathrm{pk}}$ is background-subtracted peak power and $\tau$ is a stated pulse duration. It belongs to the chosen profile and width convention. Here $\tau$ denotes the power full width at half maximum, the convention explicitly used by the primary source documentation.^[ref]

An original counterexample compares a rectangular pulse with a bilateral exponential pulse. Give both the same $P_{\mathrm{pk}}$ and power FWHM $\tau$. The rectangle is nonzero over a width $\tau$, so $U_{\mathrm{rect}}=P_{\mathrm{pk}}\tau$ and ${\kappa }_{\mathrm{rect}}=1$. Forthe half-maximum points are again $\pm \tau /2$, but integration of the two tails gives $U_{\exp }=P_{\mathrm{pk}}\tau /\ln 2$ and ${\kappa }_{\exp }=1/\ln 2\approx 1.4427$.

With $P_{\mathrm{pk}}=2\mathrm{W}$ and $\tau =5\mathrm{ns}$, the energies are $10$ and $14.43\mathrm{nJ}$. The same measured peak and FWHM therefore leave a 44.3 percent difference in this example. This nonuniqueness of pulse energy at fixed peak power and FWHM is a profile ambiguity, not an instrument-unit conversion.

The Gaussian pulse energy and peak power article gives the Gaussian-family relation. Reconstructing an arbitrary pulse's energy requires the full calibrated profile or an additional justified shape model; a width and peak alone cannot supply it.