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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.

Optical contrast in decibels expresses a specified ratio logarithmically. For peak-to-local radiant power contrast $C=P_{\max }/P(t)>0$,The inverse convention $r=P(t)/P_{\max }$ gives $10{\log }_{10}r=-L$. State the convention and the time of the weak feature. A $60\mathrm{dB}$ peak-to-feature contrast means one million to one in power, not in field amplitude.

An original readout example uses a linear photodetector whose baseline-corrected output is proportional to optical power. A main reading of $0.80\mathrm{V}$ and weak-feature reading of $80\mu \mathrm{V}$ imply $C=10^4$ and $L=40\mathrm{dB}$. Applying $20{\log }_{10}$ to those readings would incorrectly report $80\mathrm{dB}$: the electrical signal represents power, not optical wave amplitude.

If instead a measurement is proportional to an optical field amplitude $A$, with the same impedance and spatial normalization for both fields, power is proportional to $|A{|}^2$. ThenA field-amplitude ratio of $100$ and a power ratio of $10^4$ therefore express the same $40\mathrm{dB}$ contrast. The correct multiplier follows the measurement model, not whether the readout unit happens to be volts.

Contrast does not specify pedestal energy fraction, optical prepulse duration or delay coverage. The Sequoia primary sheet specifies dynamic range, scan range and optical temporal resolution separately.^[ref] A signal beneath a known measurement floor supports a contrast bound within that window; subtracting an uncertain background does not establish unlimited contrast.

Optical energy outside a time gate is the radiant energy excluded by a specified ideal temporal acceptance set $G$. For a nonnegative radiant power profile of finite positive total energy,The time-integral definition is the radiometric one.^[ref] This observable needs a gate boundary but no fitted separation into a main field and an optical pulse pedestal.

For an original three-pulse example, let an optical prepulse, main pulse and optical postpulse carry $5$, $12$ and $3\mathrm{nJ}$ respectively, with disjoint supports. A gate containing only the main pulse admits $12\mathrm{nJ}$ of $20\mathrm{nJ}$, so $f_{\mathrm{out}}=0.40$. Widening that gate to include the postpulse reduces the fraction to $0.25$. The optical waveform has not changed; the admitted observable has.

For a translated interval $G_{\delta }=[a+\delta ,b+\delta ]$ with continuous boundary values,Unequal power at the boundaries produces first-order sensitivity to a timing shift. Equal boundary values eliminate this derivative but need not eliminate higher-order loss; quadratic energy loss from centred Gaussian gate timing error gives an existing example.

The total-energy denominator must cover the excluded light too. If the acquisition window misses an unknown tail, pulse-energy interval from incomplete tail measurements supplies conditional bounds rather than an exactly known fraction. A detection time gate on an electrical detector trace also includes the detector impulse response and is not automatically an ideal gate on incoming optical power. A characterized optical response and a stated detector baseline subtraction are needed to compare the two.

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 pulse correlation ghost is a correlation lobe generated by a real feature in the nonlinear gate, appearing at a delay where no separate input pulse need exist. It is a property of the measurement's cross-correlation, not automatically an optical prepulse or optical postpulse.

The optical pulse temporal contrast article derives the separated-replica model $I(t)=g(t)+\epsilon g(t-d)$ for $d>0$, $0<\epsilon <1$ and nonoverlapping identical-shape pulses. In third-order intensity autocorrelation, the main, positive-delay and opposite-delay lobe weights are respectivelyThe physical delayed replica enters linearly when it meets the strong squared gate; its squared contribution to the gate creates the opposite-delay ghost. Lobe separation and negligible pulse overlap are assumptions of this simple decomposition.

For an original $\epsilon =0.04$, the real postpulse has four percent of the main intensity and equal duration. Normalized to the correlation's main peak, its lobe is $0.04/1.000064\approx 0.0399974$, while the ghost is $0.0016/1.000064\approx 0.00159990$. A feature at negative delay near that latter level does not establish a physical prepulse. The ghost-to-real-lobe ratio is exactly $0.04$ in this model. This ghost-level consistency test for a delayed pulse replica can check an interpretation, although an actual prepulse could also be present.

The historical correlator's inserted-etalon test supplies a controlled physical postpulse for checking the delay orientation.^[ref] Independent changes to the replica strength, or a reference with known temporal structure, test the ghost interpretation. Overlapping pulses, unequal shapes and instrument backgrounds require a richer model.