Optical pulse ($\tau$)

An optical pulse is a finite-duration envelope of light. A nearly monochromatic pulse has many carrier oscillations within its duration $\tau$: in units with light speed one, its wavelength obeys $\lambda \ll \tau$. A slowly varying propagation-time map stretches the pulse duration by its derivative and divides the observed frequency by the same factor.

// Target: general-relativity.bigb

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 etalon round-trip delay is the extra envelope travel time associated with one additional internal round trip in an etalon. At normal incidence, for a transparent plate of thickness $L$ and a sufficiently narrowband optical pulse,The relevant quantity is group refractive index, since the delay concerns the pulse envelope. The extra propagation phase is $\varphi (\omega )=2\omega n(\omega )L/c$; its derivative gives $d\varphi /d\omega ={\tau }_{\mathrm{rt}}$. This assumes negligible frequency-dependent reflection phase and treats dispersion locally.

For an original chosen plate with $L=3.0\mathrm{mm}$ and $n_g=1.60$, the delay is about $32.0\mathrm{ps}$. A pulse transmitted after one extra round trip appears that much later than the directly transmitted pulse. Another extra round trip adds the same delay in this local model, producing an echo sequence. The optical postpulse article separately describes the reflection losses that determine echo energies.

The frequency-domain connection gives an independent consistency check. Neighboring transmission resonances differ in round-trip phase by $2\pi$, so the local free spectral range satisfiesThe example predicts about $31.2\mathrm{GHz}$. This etalon delay inferred from free spectral range uses the same local dispersion assumptions; the spacing can vary across a broad spectrum.

The historical third-order correlator calibrated its delay coordinate by inserting a BK7 etalon of known thickness into an arm and measuring the trace shift.^[ref] A plate inserted into an arm also changes the direct propagation delay relative to the displaced air path; that single-pass insertion delay is distinct from the echo's extra internal round trip. Oblique incidence, dispersive coatings and broad-band pulse reshaping require a more complete propagation model.

An etalon round-trip phase is the phase accumulated by a circulating optical field after one complete internal round trip, relative to its earlier passage. At normal incidence through a uniform plate of thickness $L$,where the last terms are the phases of the two internal amplitude-reflection coefficients. Propagation and reflection both contribute; a coating phase is not automatically zero.

The resonance condition follows from a field sum. If $r_1{r}_2{e}^{2ikL}=a{e}^{i{\varphi }_{\mathrm{rt}}}$, with $0\le a<1$, successive transmitted paths contain factors $1,a{e}^{i{\varphi }_{\mathrm{rt}}},a^2{e}^{2i{\varphi }_{\mathrm{rt}}},\dots$. Their geometric series has denominator $1-a{e}^{i{\varphi }_{\mathrm{rt}}}$. Its squared magnitude containsFor slowly varying amplitudes, transmission peaks occur near ${\varphi }_{\mathrm{rt}}=2\pi m$. The primary manufacturer tutorial describes the resulting etalon transmission peaks and distinguishes their spacing from their width.^[ref]

A constant phase offset shifts resonance positions without changing the local slope. In an original model with constant round-trip delay $20\mathrm{ps}$, adding reflection phase $0.4\mathrm{rad}$ shifts a given resonance by approximately $-0.4/(2\pi \times 20\mathrm{ps})=-3.18\mathrm{GHz}$. The free spectral range remains $50\mathrm{GHz}$ in this constant-slope model.

Frequency-dependent phase has a different effect. Differentiating with respect to angular frequency givesThe last two terms form an etalon coating group delay. A stipulated $20\mathrm{ps}$ propagation contribution plus $2\mathrm{ps}$ coating contribution gives local resonance spacing about $1/(22\mathrm{ps})=45.45\mathrm{GHz}$, rather than 50 GHz. These are illustrative phase functions, not measured coating specifications.

Use a continuous branch of wave phase when differentiating: jumps of $2\pi$ in a wrapped display are not physical delay spikes. The etalon delay inferred from free spectral range article derives the interval relation between phase slope and adjacent resonances. Finite pulse bandwidth, angular spread and frequency-dependent reflection amplitudes determine how closely the simple resonance model describes an actual optical pulse.

An etalon coating group delay is the frequency derivative of the reflection phase contributed by the two coatings during one internal round trip. At fixed incidence and polarization, write the internal reflection coefficients as $r_j(\omega )=|r_j(\omega )|e^{i{\theta }_j(\omega )}$. With the phase convention of etalon round-trip phase,The phases must refer to consistently chosen surfaces: moving a reference plane transfers a propagation contribution between the terms. Only the consistently defined total is directly comparable with the resonance spacing.

A constant coating phase changes the positions of resonances but contributes zero group delay. A linear phase produces constant group delay. Curvature produces group delay dispersion:A highly reflecting coating therefore cannot be characterized by reflectance alone. The manufacturer's ultrafast-mirror documentation separately specifies reflection dispersion and shows multiple-bounce arrangements for compensation.^[ref]

For an original constant-slope example, suppose propagation contributes $10\mathrm{ps}$ and the coatings together contribute $0.20\mathrm{ps}$. The local free spectral range is approximately $1/(10.20\mathrm{ps})=98.04\mathrm{GHz}$, rather than $100\mathrm{GHz}$. Ignoring coating delay would overassign that $0.20\mathrm{ps}$ to propagation when inferring a spacer thickness or group index. The values are stipulated, not a coating specification.

For an independent curvature example, let the summed coating phase contain $D(\omega -{\omega }_0{)}^2/2$, with $D=100\mathrm{f}{\mathrm{s}}^2$. Across a frequency interval $10\mathrm{THz}$, its group delay changes by $D2\pi \Delta f\approx 6.28\mathrm{fs}$. The derivative is with respect to angular frequency; omitting $2\pi$ changes the result.

Experimentally, one may measure adjacent resonance frequencies and compare their spacing with an independently characterized spacer and group index. A discrepancy alone cannot uniquely identify coating delay: spacer dispersion, incidence angle and geometry also affect the result. Etalon delay inferred from free spectral range gives an interval-averaged phase slope; identifying it with a single-frequency derivative requires sufficiently slow variation. The simple manufacturer's spacer relation explicitly neglects spacer dispersion.^[ref]

Etalon reference-plane dependence is the change in the assigned propagation and reflection phases when the mathematical planes defining an etalon's field amplitudes are moved. A consistently computed etalon round-trip phase and its resonance frequencies remain unchanged. The decomposition into spacer delay and etalon coating group delay depends on those planes.

Use time dependence $e^{-i\omega t}$ and a lossless uniform spacer with wavenumber $k(\omega )=n(\omega )\omega /c$. Let the physical interfaces be separated by $L$, with internal reflection coefficients $r_1,r_2$ referred to their surfaces. The round-trip multiplier isThe multilayer formalism explicitly assigns forward and backward amplitudes at specified interfaces and propagates their phases through each layer. Its sign conventions must remain consistent when comparing reflection phases.^[ref]

Move the left mathematical reference plane a distance $\delta >0$ into the spacer, leaving the actual coating in place. A wave launched from that plane travels to the coating and back, so the effective left reflection coefficient becomes $r_1^{\prime }=r_1{e}^{2ik\delta }$. The explicit remaining distance is $L^{\prime }=L-\delta$. HenceThis original derivation changes the representation, not the physical cavity. Moving a real mirror would instead change the system.

Let ${\tau }_c=d\arg (r_1{r}_2)/d\omega$, using continuous wave phase. Since $dk/d\omega =n_g/c$, the new assigned terms obeyTheir sum equals ${\tau }_c+2n_{g}L/c$. If the spacer is dispersive, the corresponding second-derivative contributions likewise transfer between the two terms; the total group delay dispersion is unchanged.

For an original example, take $n_g=1.5$ and $\delta =15\mu \mathrm{m}$. The assigned reflection delay increases by about $150.1\mathrm{fs}$ while the assigned propagation delay decreases by the same amount. The free spectral range cannot reveal a change of bookkeeping. An apparent discrepancy of that size between two coating calculations can arise solely from different reference planes.

Report the incident medium, polarization, reference surfaces and phase convention with a coating-delay result. A measured resonance spectrum constrains the total phase; independently identifying a coating contribution requires a compatible spacer model. Etalon delay inferred from free spectral range therefore should not be compared with an isolated coating derivative until both calculations use the same reference planes.

An etalon delay inferred from free spectral range estimates an etalon round-trip delay from the frequency spacing of adjacent transmission resonances. When round-trip group delay varies little across that spacing,The measured quantity is resonance spacing, not the width of one resonance. The manufacturer tutorial explicitly distinguishes free spectral range from optical finesse, which compares that spacing with an individual peak's width.^[ref]

The relation follows from wave phase. Let $\varphi (\nu )$ be the total etalon round-trip phase, including relevant reflection phases. Adjacent resonances satisfy $\varphi ({\nu }_{m+1})-\varphi ({\nu }_m)=2\pi$. Defining ${\tau }_{\mathrm{rt}}(\nu )=(2\pi {)}^{-1}d\varphi /d\nu$ gives the exact interval relationThus the reciprocal spacing is the delay averaged over that frequency interval. Interpreting it as a local pulse-envelope delay requires sufficiently weak variation across the interval and relevant pulse spectrum.

For a uniform plate at normal incidence with negligible frequency-dependent reflection phase, $\varphi =4\pi \nu n(\nu )L/c$. Differentiation gives ${\tau }_{\mathrm{rt}}=2n_{g}L/c$, where $n_g=n+\nu dn/d\nu$ is the group refractive index. Treating $n$ as frequency independent recovers the manufacturer tutorial's elementary plate formula; using a phase index in a strongly dispersive material can instead give the wrong delay.

An original measured-spacing example takes $\Delta {\nu }_{\mathrm{FSR}}=25.0\mathrm{GHz}$. The inferred delay is $40.0\mathrm{ps}$. A spacing standard uncertainty of $0.2\mathrm{GHz}$ contributes approximately $0.32\mathrm{ps}$ to the delay uncertainty through $u(\tau )=u(\Delta \nu )/(\Delta \nu {)}^2$. With stipulated $n_g=1.50$ and negligible coating delay, the corresponding plate thickness is about $4.00\mathrm{mm}$.

In a spectral experiment, identify adjacent peaks of the same resonance family and calibrate the frequency coordinate. If the coordinate is wavelength, use $\Delta \nu =c|1/{\lambda }_1-1/{\lambda }_2|$ rather than treating a wavelength interval as a frequency interval. Beam angle, dispersion and reflection phase belong in the propagation model.

A time-domain check looks for replicas separated by the predicted extra round trip. The historical single-shot correlator experiment inserted a known-thickness BK7 etalon to calibrate delay and separately placed an uncoated etalon in the input to create a double-reflection postpulse.^[ref] The single-pass delay caused by inserting glass into an arm is distinct from the additional round-trip separation of successive transmitted replicas.

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.

An optical pulse pedestal is lower-level optical power extending around a pulse's dominant short peak. It can precede the peak, follow it or extend on both sides. A broad pedestal can carry substantial laser pulse energy while barely changing the peak's full width at half maximum. Optical pulse temporal contrast describes the relative power levels; integrated energy also depends on how long each contribution persists.

For an original power-profile model, choose a central rectangular contribution of height $P_0$ and duration $\tau$, together with a pedestal of height $\alpha P_0$ and duration $T>\tau$, centred at the same time. Let $0<\alpha <1$. The peak power is $(1+\alpha )P_0$, while the pedestal lies below half of that value. The power FWHM is therefore still $\tau$, butThese are stipulated additive power contributions, not a rule for adding coherent electric fields.

Set $\alpha =10^{-3}$ and $T=1000\tau$. The pedestal then contains half the total energy even though its power is about a thousand times below the main peak. With $P_0=1\mathrm{MW}$ and $\tau =1\mathrm{ps}$, each contribution carries $1\mu \mathrm{J}$. The peak-to-pedestal power ratio is exactly $(1+\alpha )/\alpha =1001$ in this example. An inverse convention would quote a small pedestal-to-peak ratio instead; state which contrast definition is being used.

A measurement needs temporal range as well as dynamic range. A narrow scan can miss broad wings, while a readout that resolves only the peak can bury a pedestal in its own background. A commercial example is the Sequoia third-order cross-correlation instrument, whose manufacturer identifies temporal-contrast measurement as its purpose and separately specifies scan range and temporal resolution.^[ref] The cited instrument description establishes the measurement approach; the numerical profile above is an independent model.

A real optical pedestal must also be distinguished from detector baseline subtraction errors. Blocked-beam and reference measurements characterize the detector background, while a suitable optical measurement tests whether low-level light accompanies the pulse. A detection time gate that excludes a pedestal changes the admitted energy relative to a detector integrating the full waveform.

The 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 $P(t)=P_m(t)+P_p(t)$,The assumptions $0\le P_p\le P$ imply $0\le f_p\le 1$ 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 $T\ge \tau$. 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 $C=(1+\alpha )/\alpha$. ThenHalf 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 $10/1010\approx 0.00990$; with $T=1000\tau$, it is $1/2$. Thus 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.

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,In the rectangular example with $G$ equal to the main interval, $f_{\mathrm{out}}=\alpha (T-\tau )/(\tau +\alpha T)$. 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.^[ref]

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]

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.

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.

The temporal energy centroid of an optical pulse is its energy-weighted mean time. For nonnegative radiant power $P(t)$ with finite positive energy $U$ and finite absolute first moment,The normalization uses radiant energy as integrated radiant flux.^[ref] The centroid is a moment of the energy-normalized pulse profile; it need not coincide with the largest power value, a threshold crossing or the middle of the FWHM interval.

For an original two-component example, let ninety percent of the energy belong to a symmetric narrow component centred at zero and ten percent to a symmetric component centred at $20\mathrm{ns}$. Linearity of the integral gives $t_E=0.9(0)+0.1(20)=2\mathrm{ns}$. Weak delayed energy can move the centroid even when the dominant peak remains near zero. Multiplying the whole power trace by a positive constant leaves $t_E$ unchanged; translating it by $t_0$ shifts its centroid by $t_0$.

A detector response can shift the measured centroid too. Suppose a normalized impulse response $h$ and normalized optical profile $p$ have finite absolute first moments. For their convolution $q=p\ast h$, direct integration with $t=s+u$ givesThus a causal exponential detector response of time constant $2\mathrm{ns}$ shifts the recorded centroid by $2\mathrm{ns}$ in this model. NIST's pulsed-radiometer characterization explicitly measures detector impulse responses and uses convolution to assess reproduced waveforms.^[ref] The centroid identity is the independent moment calculation above.

Baseline drift can cause a separate shift. Suppose a true pulse has $U=1\mathrm{nJ}$ and $t_E=0$, but a positive residual baseline of $1\mathrm{mW}$ remains across the analysis window from $-10$ to $90\mathrm{ns}$. Its added energy is $0.10\mathrm{nJ}$, centred at $40\mathrm{ns}$. The resulting centroid isA waveform measurement should therefore record the time window, baseline estimate and detector timing response with its reported centroid. A negative background-subtracted trace also needs care: it is not automatically a nonnegative energy distribution.

A flying focus is an optical pulse whose position of maximum focusing varies with time because different frequencies focus at different axial locations and arrive at different times. For a broad incident Gaussian beam, $f(\omega )=f_0+\beta (\omega -{\omega }_0)$ and spectral phase $\psi =\eta (\omega -{\omega }_0{)}^2$ give $z=f(\omega )$ and $t={\psi }^{\prime }(\omega )+z/c$, hence $v_f/c=[1+2\eta c/\beta {]}^{-1}$. This is the velocity of a focusing pattern, which can exceed $c$ or be negative without individual portions of the field propagating faster than light.

The carrier-envelope phase locates the carrier oscillation relative to the envelope of an optical pulse. In $E_{\mathrm{real}}(t)=F(t)\cos ({\omega }_{0}t+{\varphi }_{\mathrm{CE}})$ with envelope maximum at zero, it is the displayed phase modulo $2\pi$. Few-cycle dynamics can depend on it even when cycle-averaged light intensity is unchanged. A pulse train's phase increment is related to its carrier-envelope offset frequency; signs depend on the complex-field convention.

Measure the spectrum of a nonlinear optical gate signal as its relative delay is varied. A retrieval algorithm seeks a complex pulse consistent with the resulting two-dimensional intensity record and the particular gate law. Different gate laws have different ambiguities and response limitations; a small fitting residual alone does not guarantee stable pulses or correct calibration.

An optical Kerr gate rotates a probe's polarization in proportion to a delayed replica's intensity in the weak-gate regime. Resolving the gated spectrum gives a pulse-characterization record with a third-order gate law, distinct from a second-harmonic gate.

The self-gated second-harmonic spectrogram has the displayed ideal form. Its delay marginal is proportional to the intensity autocorrelation by the Parseval identity. Absolute time and wave phase are undetermined; $A(t)$ and $A^{\ast }(-t)$ also give the same record. Finite crystal acceptance and spatial walk-off modify the ideal gate.

A Gaussian pulse has a field or intensity envelope specified by a Gaussian function; the distinction must be stated. For the displayed intensity, the temporal full width at half maximum is $2\sqrt{2\ln 2}{\sigma }_t$. A transform-limited pulse with this shape has intensity spectral and temporal widths satisfying $\Delta \nu \Delta t=2\ln 2/\pi$. A quadratic spectral phase can lengthen the pulse without changing its spectral intensity.

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.

The Gaussian pulse time-gate energy fraction is the share of an isolated pulse's laser pulse energy admitted by a finite time window. The result below acts on incident optical radiant power. Let $\tau$ be the power full width at half maximum, and first place the pulse centre and the detection time gate centre at zero.

A detection time gate need not collect the whole pulse. Integrating the same profile only over $-T_g/2\le t\le T_g/2$ gives the Gaussian pulse time-gate energy fractionwhere $\mathrm{erf}$ is the error function. A gate as wide as the power full width at half maximum collects only $\mathrm{erf}(\sqrt{\ln 2})\simeq 0.760968$, or 76.10% of the pulse energy. The pulse has nonzero Gaussian tails outside that interval. A real readout's impulse response can extend the required integration window further.

Alignment adds a parameter. Let the pulse centre be $t_0$, the gate centre be $t_c$, and write the Gaussian gate centring error as $d=t_c-t_0$. Put $\sigma =\tau /(2\sqrt{2\ln 2})$. Integrating the normalized Gaussian pulse profile over the gate givesAt $d=0$ the two error function terms reduce to the centred formula above. With $T_g=\tau$ and $d=\tau /2$, the captured fraction drops to $0.490734$. The pulse centre then lies on a gate edge. Merely keeping the gate width fixed does not keep its collected energy fixed.

The gate works best when centred. With the centred energy-normalized pulse profile $p(s)=P(t_0+s)/U$ and $a=T_g/2$, differentiation of $F(d)={\int }_{d-a}^{d+a}p(s)ds$ givesA Gaussian function is an even function about its centre and decreases with distance from that centre. Thus $F^{\prime }(d)\le 0$ for $d>0$, and symmetry gives the opposite sign for $d<0$. The same argument proves that a centred interval maximizes mass for an even unimodal density; unimodality supplies the required monotonicity.

A centred gate of width $2\tau$ captures $0.981468$ of the energy. Solving $\mathrm{erf}(\sqrt{\ln 2}{T}_g/\tau )=0.99$ gives $T_g\simeq 2.18771\tau$ for 99% capture in this ideal model. The requirement changes if the centring error is nonzero.

Pulse timing jitter can therefore create variation in gated energy even when every full pulse has the same $U$. A wider gate reduces this loss but also changes admitted background and the chance of including neighbouring pulses. Measure the timing and detector response associated with the chosen gate.

For a detector modeled as a linear time-invariant system, its recorded waveform is a convolution of the optical input with its impulse response. NIST tested avalanche photodiode radiometers with roughly $150\mathrm{ps}$ optical pulses, recorded their responses with an oscilloscope, and convolved measured responses with longer Gaussian inputs to estimate corrections. The detector and amplifier were characterized together.^[ref] A gate applied to that electrical waveform must use the readout model rather than assuming the optical fraction is unchanged.

The Gaussian gate centring error is $d=t_c-t_0$, the time displacement between a gate centre and a Gaussian pulse centre. The Gaussian pulse time-gate energy fraction gives the full displaced-gate formula. Equal positive and negative errors collect equal energy in that symmetric model, so an energy deficit alone does not reveal the direction of the misalignment.

A focused expansion explains sensitivity close to the optimum. Let $p$ be the centred energy-normalized pulse profile, let $a=T_g/2$, and define $F(d)={\int }_{d-a}^{d+a}p(t)dt$. For the Gaussian standard deviation $\sigma$,Thus the leading quadratic energy loss from centred Gaussian gate timing error is second order, rather than linear in $d$.

For an original example, take a gate as wide as the power FWHM $\tau$. Its centred fraction is $0.760968$. An error $d=0.10\tau$ gives $0.748042$ from the full formula; the quadratic approximation gives $0.747945$. For $\tau =10\mathrm{ns}$ this is a $1\mathrm{ns}$ displacement, with gate width and total pulse energy unchanged.

A delay scan of a gated readout can locate the maximum, provided the source and gate widths remain stable. Pulse timing jitter produces a distribution of offsets rather than a single deterministic $d$. Also distinguish a gate on optical power from one on a detector's electrical waveform: NIST measured detector impulse responses and used convolution to assess pulse reproduction.^[ref]

Quadratic energy loss from centred Gaussian gate timing error is the leading decrease in collected energy when a symmetric gate is displaced slightly from the centre of a Gaussian pulse. At the optimum the first-order change vanishes, but the second-order change is negative. The Gaussian pulse time-gate energy fraction gives the exact displaced-gate integral; here the local loss and its effect on jitter are derived explicitly.

Let $p(t)=e^{-t^2/(2{\sigma }^2)}/(\sqrt{2\pi }\sigma )$ be the centred energy-normalized pulse profile, let $a=T_g/2>0$, and setThe Leibniz integral rule gives $F^{\prime }(d)=p(d+a)-p(d-a)$. Since $p$ is even and $p^{\prime }(t)=-tp(t)/{\sigma }^2$,The Taylor expansion therefore yieldsThis is an absolute fraction of the full pulse energy. The relative loss compared with the centred gate is instead $[F(0)-F(d)]/F(0)$.

For a gate as wide as the power FWHM $\tau =2\sqrt{2\ln 2}\sigma$, $F(0)=0.760968$ and $C{\tau }^2=1.302337$. An offset $d=0.05\tau$ gives leading loss $0.00325584$, or about $0.428$ percent of the centred gate's collected energy. Neither the full pulse energy nor the gate width has changed.

A zero mean offset does not eliminate this loss. For a sufficiently narrow random offset $D$ with mean ${\mu }_D$ and variance ${\sigma }_D^2$, the leading expectation isIf $D$ has a centred normal distribution, the quadratic model also gives $\mathrm{SD}(F(D))\simeq \sqrt{2}C{\sigma }_D^2$. At ${\sigma }_D=0.05\tau$, the leading mean loss is $0.00325584$ and this standard deviation is $0.00460446$. These formulas neglect fourth-order effects; they are local approximations, not bounds for large timing excursions.

An apparatus check scans the gate delay about its optimum and compares the measured decrease with the model, while monitoring gate width and the pulse. A gate on the detector's electrical output need not act directly on optical power: the detector impulse response and readout convolution must be included. NIST's impulse-response measurements illustrate that distinction.^[ref]

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 product of a specified temporal width and spectral frequency width. State whether each width refers to power or field amplitude and whether it is a full width at half maximum or a standard deviation.

A transform-limited pulse needs coherent phases across its contributing frequency range. A laser's usable gain bandwidth therefore sets a duration scale through the time-bandwidth product, as displayed. The wavelength interval must first be converted to frequency; the simple conversion $c\Delta \lambda /{\lambda }^2$ is only a narrow-band approximation. Numerical constants depend on pulse shape and width convention: a Gaussian intensity envelope has $\Delta t_{\mathrm{FWHM}}\Delta {\nu }_{\mathrm{FWHM}}=2\ln 2/\pi \simeq 0.441$. A hard gain interval is not automatically a spectral FWHM. Mode locking can arrange the longitudinal modes' phases, while group velocity dispersion must be controlled to approach the transform limit. The cavity round-trip time sets pulse spacing, rather than this bandwidth-limited width.

A diagnostic proportional to $\int I(t)I(t-\tau )dt$ for delay $\tau$. It does not uniquely reconstruct a complex optical electric field; extracting pulse duration requires a shape model.

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.

The third-order correlation width of a Gaussian pulse converts a measured ideal third-order intensity autocorrelation width to the width of its assumed Gaussian intensity profile. If the pulse intensity has temporal standard deviation ${\sigma }_I$, its squared intensity gate has deviation ${\sigma }_I/\sqrt{2}$. Their cross-correlation haswhere both $\tau$ values are intensity full widths at half maximum. The parent third-order intensity autocorrelation article derives the complete Gaussian trace by integration. The factor concerns that shape family and that correlation definition; it is not a universal pulse-width conversion.

A focused extension separates a known instrument response. Suppose an additional normalized Gaussian response of width ${\tau }_R$ convolves the ideal trace. Addition of Gaussian variances givesFor an original chosen ${\tau }_{\mathrm{measured}}=120\mathrm{fs}$ and ${\tau }_R=60\mathrm{fs}$, the inferred pulse width is $84.9\mathrm{fs}$. Applying only the ideal conversion would give $98.0\mathrm{fs}$. This Gaussian instrument broadening in third-order pulse correlation assumes that the additional response really acts as a Gaussian convolution; it is not an empirical correction for every correlator.

The historical physical implementation used second-harmonic generation followed by sum-frequency generation and spatial delay encoding.^[ref] Optical gate width, phase matching and spatial overlap belong to its response model. A finer scan step alone does not demonstrate a smaller optical response width.

If the assumed response width equals or exceeds the measured width, the displayed subtraction does not provide a resolved positive pulse width. Non-Gaussian traces, satellites and chirped fields also require explicit modeling; an intensity-width conversion does not reconstruct the optical wave phase.

Gaussian instrument broadening in third-order pulse correlation is the widening produced when an ideal third-order intensity autocorrelation is convolved with an additional Gaussian instrument response. It is a model of the measurement, not a universal correction for every correlator. The finite width of the optical nonlinear gate is already part of the ideal third-order trace and must not be counted again as an independent response.

Let a Gaussian pulse have intensity standard deviation ${\sigma }_I$. Its ideal trace is a Gaussian $C(\tau )=C_0\exp [-{\tau }^2/(2{\sigma }_3^2)]$ with ${\sigma }_3^2=3{\sigma }_I^2/2$. Take the additional normalized impulse response to beThe convolution $M=C\ast R$ isThe response preserves integrated trace area while reducing peak height and increasing variance. Since every Gaussian full width at half maximum has the same conversion factor from standard deviation,

For an original example, ${\tau }_I=80\mathrm{fs}$ and ${\tau }_R=60\mathrm{fs}$ give an ideal correlation width about $98.0\mathrm{fs}$ and a measured width about $114.9\mathrm{fs}$. The peak falls to approximately 0.853 of the ideal peak under this unit-area response. Dividing the measured width only by $\sqrt{3/2}$ would overestimate the pulse duration.

The correct Gaussian inversion is ${\tau }_I=\sqrt{(2/3)({\tau }_M^2-{\tau }_R^2)}$ when the radicand is positive. Its conditioning worsens near the response floor: the derivatives with respect to ${\tau }_M$ and ${\tau }_R$ are $(2/3){\tau }_M/{\tau }_I$ and $-(2/3){\tau }_R/{\tau }_I$. Small width errors can therefore cause large errors in a narrowly resolved pulse. Gaussian pulse-width deconvolution uncertainty must include uncertainty of the characterized response, with covariance if relevant.

The primary correlator used second-harmonic generation, sum-frequency generation, crossed beams and spatial imaging to encode delay. It did not assert that every additional broadening mechanism was a Gaussian convolution.^[ref] To apply this model, characterize the response under the same optical alignment, spectral acceptance and detector conditions. Non-Gaussian wings, clipping or unresolved side lobes require a different model; a smaller sampling step cannot by itself remove optical broadening.

Gaussian pulse-width deconvolution uncertainty is the uncertainty in an inferred pulse duration after subtracting a characterized Gaussian instrument contribution. For the Gaussian instrument broadening in third-order pulse correlation model, let $M$ be measured correlation full width at half maximum, $R$ the additional response width and $P$ the inferred pulse width:These widths must use the same convention. The relation already includes the ideal nonlinear gate; $R$ represents an additional response.

First-order propagation uses the sensitivity coefficientsThus the standard uncertainty obeysThe negative covariance term follows from the subtraction, not a rule that uncertainties should simply be subtracted. The GUM's general law retains covariance and requires considering higher-order terms when nonlinearity is significant.^[ref]

For an original example, $M=120\mathrm{fs}$ and $R=60\mathrm{fs}$ give $P\approx 84.85\mathrm{fs}$. With independent $u(M)=u(R)=2\mathrm{fs}$, the formula gives $u(P)\approx 2.11\mathrm{fs}$. If instead the two width estimates have correlation coefficient 0.8 with those same uncertainties, the covariance is $3.2\mathrm{f}{\mathrm{s}}^2$ and $u(P)\approx 1.26\mathrm{fs}$. The assumed relationship between the estimates changes the result.