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Attenuation scaling of a third-order correlation ghost tests the nonlinear origin of a side lobe by changing a real delayed replica's strength while preserving the main pulse. The diagnostic prediction differs from attenuating the entire input. The quantity attenuated must be stated as an intensity or energy ratio, rather than confused with a field-amplitude ratio.

Use the nonoverlapping equal-shape model $I(t)=g(t)+\epsilon g(t-d)$, with $d>0$ and $0<\epsilon <1$. Let an independently controlled intensity transmission $a$ act on the delayed replica only. With the main pulse fixed, the third-order intensity autocorrelation has central, real-replica and third-order pulse correlation ghost lobe weights proportional toThe common factor is the same single-pulse correlation amplitude. Thus the real lobe has logarithmic slope 1 and its ghost slope 2 versus replica transmission, before normalization. Their exact ratio is $G/S=a\epsilon$.

An original example starts with $\epsilon =0.08$. Initially the weights are $(1.000512,0.08,0.0064)$. Halving the delayed replica's intensity gives $(1.000064,0.04,0.0016)$. The real side lobe halves while its ghost quarters. If each trace is divided by its own central peak, the predictions becomeThese normalized curves have the small additional central-peak dependence; treating them as exact pure powers would omit it.

In contrast, common attenuation in third-order pulse correlation sends $I\to kI$ for the complete two-pulse input. All three factors in $\int I^2(t)I(t+\tau )dt$ acquire the same scaling, so every unnormalized lobe scales as $k^3$. Halving the entire input therefore reduces every lobe eightfold while leaving the normalized shape unchanged. That experiment cannot demonstrate selective quadratic ghost scaling.

A suitable controlled test forms a delayed replica in a separate optical path, varies that path's transmission without changing its shape or delay, then recombines it with a monitored main pulse. Retain background-corrected, unsaturated readings and verify comparable detector response at both delays. The historical experiment's inserted-etalon postpulse supplied a physical delayed reference, not this particular attenuation sweep.^[ref]

Agreement with the scaling model supports the ghost interpretation under its assumptions. It does not exclude an additional real optical prepulse, unequal replica shapes, overlapping fields or an attenuation-dependent background. Those possibilities require further independent observations.

A ghost-level consistency test for a delayed pulse replica checks whether measured side lobes agree with the simple model of one real delayed pulse and its nonlinear correlation ghost. It can reject that model; agreement alone does not prove that no real prepulse exists.

Use the separated, identical-shape replica model in third-order pulse correlation ghost. Let $a$ be the positive-delay lobe and $b$ the opposite-delay lobe, each divided by the central correlation peak. For delayed-to-main intensity ratio $0<\epsilon <1$,Thus the measured ratio gives $\epsilon =b/a$, and the first equation supplies an additional consistency condition. The frequently useful approximation $b\simeq a^2$ omits the small central-peak correction; exactly, $b/a^2=1+{\epsilon }^3$.

For an original pair of measured relative heights $a=0.080$ and $b=0.020$, their ratio would require $\epsilon =0.25$. That value predicts $a=0.25/1.015625\approx 0.246$, not $0.080$. These two readings therefore cannot both come from the stated single-replica model with equal instrument sensitivity at the two delays. A real earlier pulse is one possible explanation, but background, unequal pulse shapes or additional replicas can also invalidate the model.

An independent check varies the delayed replica while holding the main pulse fixed. Before central-peak normalization, halving $\epsilon$ halves the real side lobe and quarters its ghost. This attenuation scaling of a third-order correlation ghost probes the nonlinear origin rather than only one accidental height match. The existing parent diagram shows a direct numerical example without an actual pulse at the ghost's delay.

The historical correlator's inserted-etalon experiment provided a controlled physical postpulse and a known temporal direction.^[ref] The algebraic height test here is an independent ideal calculation. Use unsaturated, background-corrected readings with comparable response and lobe resolution; otherwise the measured ratios need not represent the model's intensities.

An optical postpulse is a secondary optical pulse arriving after the chosen main pulse. Specify the delay, relative peak radiant power, duration and collected laser pulse energy: a relative peak alone does not determine the energy of a differently shaped replica. It is distinct from an extended optical pulse pedestal and from electronic ringing from a photodetector.

A transparent plate provides a concrete mechanism. At normal incidence, a transmitted replica that makes one additional internal round trip has approximate delaywhere $\ell$ is plate thickness and $n_g$ is its group refractive index. Neglect absorption, dispersion and overlap, and let each internal reflection retain power fraction $R$. The first delayed replica then has approximately $R^2$ times the directly transmitted energy. This etalon round-trip delay can create a physical optical echo.

For an original chosen $\ell =1.2\mathrm{mm}$, $n_g=1.5$ and $R=0.04$, the delay is about $12\mathrm{ps}$ and the energy ratio is $0.0016$. A $2.0\mathrm{mJ}$ main pulse would be followed by a $3.2\mu \mathrm{J}$ replica under that model. The historical single-shot correlator experiment deliberately inserted a BK7 etalon to generate and detect such a delayed pulse.^[ref]

Changing the plate thickness tests the delay prediction; changing optical alignment tests whether a stray reflection is involved. A third-order intensity autocorrelation can distinguish temporal sides under its model, while a symmetric second-order intensity autocorrelation cannot alone assign a feature to a prepulse or postpulse. An opposite-delay correlation feature may instead be a third-order pulse correlation ghost; it need not identify another physical pulse.

An optical prepulse is a secondary optical pulse arriving before the reference main pulse. The sign of its delay is part of the definition: choose the main pulse at $t=0$, so a prepulse has negative arrival time. A weak earlier pulse and an equally weak optical postpulse can affect a target differently when the material retains a response to the earlier illumination. Temporal order therefore carries information that one total laser pulse energy does not.

An original separated rectangular-profile example uses a main pulse of duration $100\mathrm{fs}$ and a prepulse of duration $10\mathrm{ps}$ centred $50\mathrm{ps}$ earlier. Let the prepulse's peak radiant power be $10^{-4}$ of the main peak. The optical pulse temporal contrast is $10^4$, but the energy ratio isThus the prepulse carries $1/101\approx 0.990\%$ of the two-pulse energy. If the main pulse carries $1\mathrm{mJ}$, the prepulse carries $10\mu \mathrm{J}$. These stipulated disjoint power profiles have no field-overlap term, and the main pulse's full width at half maximum does not reveal the earlier contribution.

A temporal diagnostic needs both sufficient delay coverage and calibrated dynamic range. The single-shot third-order correlator paper motivates its asymmetric response by contrasting it with the symmetric second-order trace, which loses the distinction between before and after.^[ref] Its delay axis still needs a known physical reference. A third-order pulse correlation ghost from a real postpulse can appear on the prepulse side; interpreting every small lobe as earlier light would be incorrect.

Repeating the measurement while varying an independently characterized optical delay helps assign temporal order. Record the detector baseline subtraction, coverage and response width along with the trace. A nondetection means a limit within those measurement conditions, rather than proof of zero early optical energy.

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 givesFor 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 toThe 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 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 $1.5\times 10^{-3}$ of the main signal, recorded on one side of the correlation peak. Figure4 combines window settings shifted with etalons.^[ref]

An independent ideal calculation explains both the asymmetry and a possible third-order pulse correlation ghost. Let $I(t)=g(t)+\epsilon g(t-d)$ 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 givesThe delayed replica produces a lobe of relative order $\epsilon$, while the opposite-delay ghost has order ${\epsilon }^2$. For $\epsilon =0.01$, 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.^[ref]

A third-order intensity autocorrelation measures the delay-dependent intensity productThe name counts three intensity factors. A physical implementation can use two second-order nonlinear processes: second-harmonic generation creates a gate proportional to $I^2$, and sum-frequency generation mixes it with a delayed fundamental replica. It therefore need not use a third-order material susceptibility. The historical device spatially encoded delay in crossing beams and imaged the sum-frequency signal onto a charge-coupled device.^[ref]

An original Gaussian calculation illustrates response width. Set $I(t)=I_0{e}^{-t^2/(2{\sigma }^2)}$. Completing the square in the integral givesHence the third-order correlation width of a Gaussian pulse obeys ${\tau }_3=\sqrt{3/2}{\tau }_I$ for intensity full widths at half maximum. A $90\mathrm{fs}$ correlation width would imply ${\tau }_I\approx 73.5\mathrm{fs}$ within this ideal Gaussian model. The square-root-of-two conversion appropriate to second-order Gaussian intensity autocorrelation would be different.

For a general pulse, $A_3(\tau )$ need not equal $A_3(-\tau )$; this asymmetry can distinguish a delayed feature from an earlier one. It is still a cross-correlation with a finite-width nonlinear gate, rather than a direct reading of $I(\tau )$. The existing optical pulse temporal contrast derivation shows how a delayed weak replica also produces an opposite-delay third-order pulse correlation ghost.

Characterize phase matching, beam overlap, spatial uniformity and the temporal response before interpreting a narrow feature. Scaling the input intensity by $k$ scales this ideal signal by $k^3$, so input fluctuations also matter. The measured trace does not uniquely reconstruct an arbitrary complex optical electric field.