= Optical pulse temporal contrast
{title2=$C(t)=P_{\max}/P(t)$}
\b[Optical pulse temporal contrast] compares a pulse's peak with weaker light at another time. For a <radiant power> trace with peak $P_{\max}>0$, explicitly choose either the peak-to-local ratio $C(t)=P_{\max}/P(t)$ or its inverse $r(t)=P(t)/P_{\max}$. The first is at least one where the local power is positive; the second lies between zero and one. With a fixed transverse beam shape, the same ratio can be formed from <light intensity>. If the beam shape changes with time, specify the spatial sampling instead of identifying local intensity and total power.
A plateau at $10^{-3}$ of the peak has $r=10^{-3}$ and $C=1000$. Expressing this power ratio as <optical contrast in decibels> gives
$$
L=10\log_{10}C=30\,\mathrm{dB}.
$$
For an <amplitude> ratio $A$ whose power ratio is $A^2$, the same definition gives $10\log_{10}(A^2)=20\log_{10}A$. Contrast also needs a delay or time range: an <optical prepulse>, an <optical postpulse> and a long <optical pulse pedestal> can have the same relative height but different positions and <laser pulse energies>. The <pedestal energy fraction> additionally requires integration over the contribution's duration.
A useful nonlinear measurement is <third-order intensity autocorrelation>. In an ideal instantaneous model, an input <light intensity> $I(t)$ generates a second-harmonic gate proportional to $I^2(t)$. <Sum-frequency generation> with a delayed fundamental replica gives a recorded signal proportional to
$$
A_3(\tau)=\int I^2(t)I(t+\tau)\,dt.
$$
The instrument measures this <cross-correlation>, rather than simply reading $I(\tau)$. Finite gate width, <phase matching>, spatial overlap and detector response enter the interpretation.
The1998/99 instrument of <J. Collier (optical physicist)> and colleagues used <second-harmonic generation>, a <Wollaston prism> and an <optical biprism> to form crossing fundamental and harmonic beams. A second crystal generated the sum-frequency signal, imaged through <optical filters> onto a <charge-coupled device>. Position encoded delay. An inserted glass <etalon> produced a delayed replica near $1.5\times10^{-3}$ of the main signal, recorded on one side of the correlation peak. Figure4 combines window settings shifted with <etalons>.<A Single Shot 3rd Order Cross-Correlator for Pulse Contrast and Pulse Shape Measurements by J. Collier and others>[Printedpp194–195, Description/Figure2 and Results/Figure4]{ref}
An independent ideal calculation explains both the asymmetry and a possible <third-order pulse correlation ghost>. Let $I(t)=g(t)+\varepsilon g(t-d)$ represent two identical-shape pulses with well-separated supports and correlation lobes, with the second weaker and delayed. Squaring introduces no overlap term. If $A_g(\tau)=\int g^2(t)g(t+\tau)\,dt$, expansion gives
$$
A_3(\tau)=(1+\varepsilon^3)A_g(\tau)
+\varepsilon A_g(\tau-d)+\varepsilon^2A_g(\tau+d).
$$
The delayed replica produces a lobe of relative order $\varepsilon$, while the opposite-delay ghost has order $\varepsilon^2$. For $\varepsilon=0.01$, the scales are about $10^{-2}$ and $10^{-4}$ respectively. These values are predictions of the stated ideal model. Assigning every small correlation feature directly to a separate input pulse would miss this distinction.
A nondetection below the measurement floor establishes a bound under a detection model, rather than demonstrating that no light exists there. Finite <dynamic range>, <detector baseline subtraction>, delay coverage and resolution should accompany a reported contrast trace. The Sequoia manufacturer likewise distinguishes optical temporal resolution from the scan step; making that step smaller does not alone narrow the optical response.<Sequoia temporal contrast measurement by Amplitude>[January2024 sheet, PDFp2, Specifications]{ref}
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