= Pedestal energy fraction
{title2=$U_{\rm pedestal}/U$}
The \b[pedestal energy fraction] is the fraction of an <optical pulse>'s <laser pulse energy> assigned to its <optical pulse pedestal>. Specify the decomposition and integration interval before quoting it. For a nonnegative additive <radiant power> model $P(t)=P_m(t)+P_p(t)$,
$$
U=\int P(t)\,dt,\qquad U_p=\int P_p(t)\,dt,\qquad
f_p=\frac{U_p}{U},\qquad U>0.
$$
The assumptions $0\leq P_p\leq P$ imply $0\leq f_p\leq1$ by <integration>. A pedestal fit is a model of the measured <radiant power>; it is not automatically a unique physical separation into optical fields. Overlapping fields with <optical coherence> can have <wave interference> terms, so their separate field energies cannot generally be added as though those terms were absent.
The distinction from <optical pulse temporal contrast> is quantitative. Use the stipulated rectangular model of <optical pulse pedestal>: a main contribution has height $P_0$ and duration $\tau$, while a centered pedestal has height $\alpha P_0$ and duration $T\geq\tau$. For $0<\alpha<1$, the total peak is $(1+\alpha)P_0$ and the power <full width at half maximum> remains $\tau$. Define peak-to-pedestal contrast $C=(1+\alpha)/\alpha$. Then
$$
\alpha=\frac1{C-1},\qquad
\boxed{f_p=\frac{\alpha T}{\tau+\alpha T}
=\frac{T}{(C-1)\tau+T}}.
$$
Half the <laser pulse energy> is in the modeled pedestal when $T=(C-1)\tau$. A large peak-to-pedestal ratio therefore does not by itself imply a small energy contribution.
For an original comparison, fix $C=1001$ and the same $P_0,\tau$ in both traces. With $T=10\tau$, the pedestal fraction is $10/1010\approx0.00990$; with $T=1000\tau$, it is $1/2$. Thus \b[the same contrast and power <FWHM> can coexist with about1% or50% pedestal energy]. At fixed finite $C$, increasing the pedestal duration can make $f_p$ approach one without changing that <FWHM>.
\Image[/optical-pedestal-energy-and-contrast.png]
{title=Pedestal energy depends on duration as well as peak-to-pedestal contrast}
{description=Original additive rectangular-power model with $C>2$ and pedestal duration $T\geq\tau$. White contours mark10%,50% and90% pedestal energy. The two markers have identical contrast1001 but different pedestal durations.}
<Optical energy outside a time gate> is a related observable that avoids assigning a pedestal underneath the main peak. For a specified gate $G$ containing that peak,
$$
f_{\rm out}=1-\frac{\int_G P(t)\,dt}{U}.
$$
In the rectangular example with $G$ equal to the main interval, $f_{\rm out}=\alpha(T-\tau)/(\tau+\alpha T)$. At $\alpha=10^{-3}$ and $T=1000\tau$, this is $0.4995$, while $f_p=0.5000$. The difference is pedestal power inside the gate, which the outside-gate measurement does not label separately.
The analysis must also specify its temporal coverage and <detector baseline subtraction>. Subtracting actual weak <light> as though it were detector background changes the inferred <laser pulse energy>. A short <detection time gate> omits outlying light instead. Neither operation is justified solely by knowing the peak-to-pedestal contrast. The Sequoia technical sheet illustrates why temporal coverage and resolving power are separate instrument properties: it lists scan range, scan resolution and optical temporal resolution independently.<Sequoia temporal contrast measurement by Amplitude>[January2024 sheet, PDFp2, Specifications]{ref}
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