Wiki / Detectors / Autocorrelator

Autocorrelator

Measures the intensity autocorrelation of a pulse and infers its duration by dividing out a deconvolution factor — which means it only reads correctly if the assumed pulse shape matches the real one.

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Click the autocorrelator to see its live specs and try its parameters — this mini canvas can't be moved, deleted, or added to.

The same instrument in cross-correlation mode

Two synchronized sources — 790 nm and 1030 nm — combined on a dichroic and read by one cross-correlator. The delay line sweeps through time zero at 100.000 mm and back, so the two pulses walk across each other every ten seconds while the sum-frequency peak flares up between them at the crossing. Click the delay line to stop the sweep and hunt time zero by hand.

In the real world

A femtosecond pulse cannot be timed by anything electronic. The fastest photodiodes and sampling oscilloscopes reach a few picoseconds; a 100 fs pulse is two orders of magnitude shorter than that, and no detector exists whose response is short enough to resolve it^[1]. The way out is to stop looking for a faster clock and instead use the pulse to measure itself.

That is what an autocorrelator does. A beam splitter makes two copies of the incoming pulse; one travels through a variable delay line; the two are then brought together in a medium with a χ⁽²⁾ nonlinearity — typically a thin second-harmonic crystal — where they mix only while they physically overlap in time^[1]. Sum-frequency light appears at a new, shorter wavelength, and its power depends on how much of the two envelopes coincide. Sweep the delay, record that power, and the resulting curve — the autocorrelation trace — is about as wide as the pulse is long. Nothing in the detection chain needs to be fast: the photodiode only has to read an average power for each delay setting, because a mode-locked laser supplies a regular train of nominally identical pulses^[1].

Intensity autocorrelation

In the standard arrangement the two copies cross at a small angle in the crystal, so the sum-frequency beam leaves along its own direction, between the two inputs. Because that beam only exists where the pulses overlap, the signal falls to zero at large delay: the measurement is background-free^[1], and this non-collinear geometry is what gives an intensity autocorrelator its high dynamic range^[2]. The trace it records is

I_{\mathrm{ac}}(\tau)=\int_{-\infty}^{\infty} P(t)\,P(t+\tau)\,\mathrm{d}t

The intensity autocorrelation: the optical power of the pulse multiplied by a delayed copy of itself, integrated over time, as a function of the delay τ set by the moving arm.

Why the trace is always wider than the pulse

Look at that integral at zero delay: the two copies sit exactly on top of one another and the product is maximal. Now shift by a delay smaller than the pulse duration. The overlap has shrunk, but it has not vanished — the trailing part of one copy is still sitting on the leading part of the other, so a real signal is still produced. Only when the delay exceeds roughly the pulse length does the product finally go to zero. The curve cannot collapse to something narrower than the pulse, and this is not an instrumental defect that better optics would remove; it is a property of the operation.

Made exact, the statement is about second moments: correlating a function with itself doubles the variance, so the root-mean-square width of the trace is larger than the pulse's by exactly √2 — for every envelope, with no assumption at all. Full width at half maximum, which is what an instrument actually reads off, is the shape-dependent one. It is worth knowing that the broadening can vanish entirely under that measure: a rectangular pulse of width T has a triangular autocorrelation whose FWHM is also T, even though its rms width has still grown by √2. For the smooth envelopes real mode-locked lasers produce, the trace is genuinely wider.

How much wider depends on the shape of the envelope. For a Gaussian pulse the autocorrelation is itself Gaussian and about 1.41 times wider — exactly √2, because Gaussian widths add in quadrature and √(τ² + τ²) = √2 τ^[1]. For a sech² pulse, the shape most mode-locked oscillators actually produce, the pulse duration is about 0.65 times the width of the trace^[1] — a factor of roughly 1.543 the other way. So the instrument never reports a duration directly. It reports a trace width, and someone must divide out a deconvolution factor:

\tau_{\mathrm{p}}=\frac{\Delta\tau_{\mathrm{ac}}}{k},\qquad k_{\mathrm{Gauss}}=\sqrt{2}\approx 1.414,\qquad k_{\mathrm{sech}^2}\approx 1.543

The pulse duration is the measured autocorrelation FWHM divided by a factor that depends entirely on the pulse shape you assume it has.

And there is the catch that defines the technique. The factor depends on a shape the measurement itself cannot establish. Gaussian and sech² traces do not look dramatically different, so fitting one to the data is a sanity check rather than a proof^[1]. Yet the two factors differ by 9%, so assuming the wrong one puts the answer out by 9% before any other error is counted — and for genuinely odd pulse shapes, by far more. A quoted "150 fs, assuming sech²" is an honest reading; a quoted "150 fs" is an incomplete one.

What an autocorrelation cannot tell you

The deeper limitation is structural: the autocorrelation trace is always symmetric about zero delay, even when the pulse is not^[1]. Swapping t → −t in the integral leaves it unchanged, so a pulse with a steep rise and a slow decay produces exactly the same trace as its mirror image. The direction of time is simply not in the data^[2]. Neither is the phase: an intensity autocorrelation responds only to optical power, so it carries no information about chirp, and different pulses can yield indistinguishable traces^[1,3]. Usefully, the symmetry works as a diagnostic in reverse — an asymmetric trace means a misaligned autocorrelator, not an asymmetric pulse^[1].

Noise makes this worse in a specific and notorious way. When a laser is not mode-locking cleanly, each pulse in the train differs from the last, and the averaged trace can show a narrow spike sitting on a much broader pedestal. Taking that spike as the pulse duration is wrong: it is a coherent artifact, and in such a situation the trace conveys very little about the real pulse^[1,5]. A distorted train can look like a beautifully short pulse.

Interferometric autocorrelation

Send the two copies collinearly instead — same path, same polarization — and they interfere before the crystal sees them. The recorded signal then resolves the optical fringes: successive constructive peaks are one optical period apart on the delay axis, which in a double-pass arm is reached by moving the mirror only half a wavelength, since the mirror changes the path twice over^[1]. Plots are labelled in both coordinates, so it is worth checking which one an axis means:

I_{\mathrm{iac}}(\tau)=\int \bigl(E(t)+E(t+\tau)\bigr)^{4}\,\mathrm{d}t

The interferometric (fringe-resolved) autocorrelation. Because the fields add before being squared twice, perfect constructive interference gives four times the intensity and sixteen times the second-harmonic signal — against a background of twice that from one arm alone.

That arithmetic gives the technique its built-in alignment check: a properly aligned interferometric autocorrelator always produces a trace whose peak is exactly eight times its wings^[1]. If the fringes are averaged out, as they are for longer pulses, the ratio becomes 3:1 rather than 4:1, because the oscillation is not sinusoidal^[1]. Unlike the intensity version, this trace is sensitive to chirp — although a chirped pulse's duration is underestimated if one simply reads off the width, and post-processing methods such as MOSAIC exist to make the chirp legible^[1]. The collinear geometry avoids the geometric smearing that a crossing angle causes, which is why interferometric designs dominate at the few-femtosecond end^[1,2].

Practical variants

Scanning versus single-shot. Most traces are built from many pulses, one or more per delay setting, which quietly assumes the train is regular — fine for a mode-locked oscillator, unreliable for a low-repetition-rate amplifier. A single-shot autocorrelator instead focuses with a cylindrical lens so that position across the crystal maps to delay, and reads the whole trace off a camera from one pulse^[1]. Scanning units suit stable high-rate trains; single-shot units are what a 10 Hz or 1 kHz amplifier needs, and the only way to see shot-to-shot fluctuation^[1].

Two-photon detectors. A photodiode with a band gap too large to absorb the light linearly still responds through two-photon absorption, which is itself the required nonlinearity — so the crystal disappears entirely, and with it the phase-matching alignment^[1]. LEDs run backwards as detectors work too^[1]. These are the compact, nearly alignment-free instruments, at the cost of sensitivity: quoted as the product of average and peak power, a TPA head reaches around 10⁻² W² where a photomultiplier-based unit reaches 10⁻⁶ W²^[2].

Dynamic range. Weak pedestals and satellite pulses — a speciality of mode-locked fiber lasers — need far more range than a standard trace offers. Type-II phase matching, two-frequency chopping with lock-in detection, and photomultiplier detection push background-free measurements to 80 or even 100 dB^[1]. A third-order autocorrelator, mixing the light with its own second harmonic, breaks the symmetry altogether and can distinguish a pre-pulse from a post-pulse — at much lower sensitivity^[1].

When to stop autocorrelating. Below about 10 fs the phase-matching bandwidth of even a very thin crystal becomes the limit, and frequency-resolved optical gating (FROG) and spectral phase interferometry (SPIDER) are both more accurate and able to return the phase the autocorrelation discards ^[1,3,6]. FROG is in one sense just an autocorrelator that spectrally resolves its output — spectrum versus delay instead of energy versus delay — and that one extra axis is enough to lift the ambiguity^[2].

Cross-correlation: two different pulses

Nothing in the layout requires the two arms to carry copies of the same pulse. Feed the nonlinear crystal from two different beams and the same delay scan measures their cross-correlation:

I_{\mathrm{cc}}(\tau)=\int_{-\infty}^{\infty} I_1(t)\,I_2(t+\tau)\,\mathrm{d}t,\qquad \Delta\tau_{\mathrm{cc}}=\sqrt{\tau_1^{2}+\tau_2^{2}}\ \ (\text{Gaussians})

The cross-correlation of two pulses, and — for Gaussian envelopes — the width of the resulting trace, which adds the two durations in quadrature.

Two things change, and both are improvements. First, the trace is no longer forced to be symmetric, so an asymmetric pulse now shows its asymmetry and the direction of time survives the measurement. Second, if one of the two pulses is already known and much shorter than the other, it acts as a fast optical gate: I₁ approaches a delta function, the integral collapses to I₂(τ), and the trace is the unknown envelope, sampled directly rather than deconvolved^[1]. This is why a characterized reference pulse is worth so much, and why the quadrature relation above matters — with τ₁ ≪ τ₂ the measured width is just τ₂.

Finding time zero for multi-beam overlap

The most common use of a cross-correlation in a working laboratory is not measuring a duration at all. It is answering a blunter question: when do these two beams actually arrive at the same place at the same time?

Any experiment driven by two or more synchronized pulses has this problem. The beams travel different paths — different numbers of mirrors, different lengths of glass, an optical parametric oscillator in one arm and none in the other — and a single millimetre of path difference is 3.3 ps of timing error, which for 100 fs pulses means no overlap whatsoever. Spatial alignment can be judged by eye or on a camera; temporal alignment cannot be seen at all. Worse, the search space is large and the signal is exactly zero everywhere outside it, so scanning blind is hopeless without a signal that appears the moment the pulses coincide.

The cross-correlation provides exactly that. Combine the two beams on a dichroic mirror, focus them into a thin nonlinear crystal, and scan one arm's delay while watching for sum-frequency light. Because 1/λ_SF = 1/λ₁ + 1/λ₂, that light appears at a wavelength lying between the two second harmonics — a colour that only exists when both beams are present together, which makes it unmistakable. The delay-stage position that maximises it is time zero, and the width of the peak around it tells you how much timing slop the experiment can tolerate^[4].

Coherent Raman microscopy is the textbook case. In coherent anti-Stokes Raman scattering (CARS), a pump photon and a Stokes photon drive a molecular vibration whose frequency is their difference, and a third photon probes it — so the signal exists only where and when both beams overlap in the focal volume^[4]. The pump typically comes from a femtosecond oscillator and the Stokes from an optical parametric oscillator pumped by it: synchronized by construction, but arriving at the sample at quite different times until a delay line is set. The standard procedure is to focus the combined beams into a type-I BBO crystal and maximise the sum-frequency signal^[4].

Two subtleties make this more than an alignment step. The overlap that matters is at the focus of the objective, not at the entrance to the microscope, and a high-NA objective is a substantial piece of glass — so a measurement made on the bench with an external autocorrelator does not describe the pulses that actually reach the sample^[4]. And when the pulses are deliberately chirped for spectral focusing — stretched so that their instantaneous frequency difference stays constant across the overlap — the delay no longer merely switches the signal on. It tunes the Raman shift. Time zero then defines the origin of the spectroscopic axis, and getting it wrong shifts every measured vibrational frequency^[4].

A neat consequence, exploited by Piazza and co-workers, is that the delay line already present in every such microscope is enough to characterize both pulses without any autocorrelator at all. Scanning it while recording two different nonlinear signals from a sample — the sum-frequency signal, which mixes one pump photon with one Stokes photon, and the non-resonant four-wave-mixing signal, which takes two pump photons and one Stokes photon — gives two cross-correlation widths that depend differently on the two durations. Two equations, two unknowns: both durations fall out, and tracking how the centre wavelength of each signal drifts with delay yields each pulse's chirp as well^[4].

In OpticalSetup

The Autocorrelator reports the pulse duration of whatever pulse train reaches its face — and reports it the way a real instrument does, as a trace width with an assumption divided out, rather than as a number read off the source.

Time span sets the horizontal axis in both modes — ±0.5, ±1, ±5, ±10 or ±25 ps — and it is a setting rather than an automatic, so two traces of different duration on the same span look as different as they are. A trace too wide for the window is reported rather than clipped.

The one control that matters is Assumed pulse shape: Gaussian (÷1.414) or sech² (÷1.543). This is deliberately a user choice and not something the instrument works out for itself, because in a laboratory it is not something the instrument can work out for itself. Set it to the wrong shape and the reading changes — a Gaussian assumption on a sech² source reads about 9% long, and the inspector says so explicitly, naming the true duration beside the inferred one. That disagreement is the lesson the component exists to teach.

For a transform-limited Gaussian source, the reading is taken from the pulse that arrives rather than the one that was emitted. Put a glass rod in the path and the autocorrelator measures the stretched duration; add a pulse compressor with the opposite group delay dispersion and it measures the pulse recovering. The bundled Ultrashort pulse chirping example is built around exactly that comparison, with three autocorrelators reading the same pulse under three different dispersion conditions.

That qualification is not decoration. The broadening is computed from a closed-form Gaussian result, so it is only derived when the source is both transform-limited and Gaussian. Switch the source to sech², or clear its transform-limited box, and no stretched duration exists to report: the instrument falls back to the duration configured on the source, and the inspector says so in as many words rather than letting a dispersion measurement be read out of a number that never moved.

Cross-correlation mode

Measurement mode switches the same box between correlating one source against itself and correlating two sources against each other. In cross-correlation mode the assumed-shape control disappears, because the instrument is no longer inferring a duration — it is reporting a timing relationship.

The screen also changes what it is plotting, and the change is worth stating carefully. An autocorrelation is a scan-delay plot: the instrument sweeps one arm against the other and the peak sits at zero by construction. A cross-correlation screen here is a laboratory arrival-time plot instead — the view you get on a scope while walking a delay line, and the reason the two peaks move. The axis is labelled so the two cannot be confused.

On it are the two pulses, each at its own arrival time, each drawn at constant height: a beam's own second harmonic does not care where the other beam is. What grows between them is the sum-frequency signal, which exists only where the two overlap, so it appears at the midpoint and rises as the arms converge. Bring the pulses together and watch the middle peak light up — that is the whole procedure, and it is what a real cross-correlator detects. At the meeting point the readout says TIME ZERO and the overlap reads 100%.

Switching to cross-correlation also picks a sensible time span once, framing whatever separation the arms currently have — a 3 ps mismatch selects ±5 ps, a merged pair selects ±0.5 ps. After that the setting is yours, and it does not move again. That combination is deliberate: re-ranging on every frame would rescale the axis under the pulses exactly as they approached, so they would never appear to travel, and fixing the axis is what lets you watch them walk. It is the same reason a real oscilloscope makes the timebase a knob rather than an automatic, and the same reason it is worth setting once for you rather than leaving you to find the pulses in an arbitrary window.

Pick a wide span to find the pulses, then narrow it as they close. When they sit beyond the window the screen stops drawing and reports the gap instead — how far apart they are, and how many millimetres to take out of which arm, since a delay line is set in millimetres rather than femtoseconds. With only one beam arriving there is nothing to correlate, and it says only one beam present rather than quietly showing an empty axis.

The bench below the autocorrelation example does the whole thing on its own: time zero sits at 100.000 mm and the delay line sweeps ±0.2 mm either side of it, so the pulses walk through each other and back every ten seconds while the sum-frequency peak flares up at the crossing. Stop the sweep and hunt it by hand to see how sharp the merge is — a hundredth of a millimetre either way is 33 fs.

Beside the plot the inspector carries the numbers the screen has no room for, including the autocorrelation each arm would give on its own. Those widths deliberately stay off the plot: on an arrival-time axis what physically sits at each peak is the pulse, not its autocorrelation.

Two details are modelled because leaving them out would teach the wrong lesson. Trains with different repetition rates are reported as unsynchronised rather than given a trace, since without a fixed phase relationship there is nothing stable to average up. And the mismatch is measured against the nearest pulse of the other train, not the nominally corresponding one: pulses repeat, so arms can only ever be nulled modulo the repetition period, and an arm 12.5 ns long at 80 MHz is perfectly overlapped rather than hopelessly late.

Wired to a Detector screen, it draws the trace: delay on the horizontal axis rather than laboratory time, the curve symmetric about zero delay as a real autocorrelation always is, the half-maximum chord that is the measurement marked across it, and the inferred duration printed above. A continuous-wave source produces no trace and says so; and pulse trains whose timing disagrees — different repetition rate, duration, or phase — are reported as mixed rather than averaged into a meaningless number.

Simplified vs. reality

No scan is simulated. The trace is drawn from the arriving duration and the assumed shape rather than being accumulated by stepping a delay line through a nonlinear crystal, so there is no scan time, no delay-line travel limit setting a maximum measurable duration, and no acquisition noise. Everything downstream of that choice follows: no crystal, no phase matching, no group velocity mismatch, and therefore none of the difficulties that dominate real measurements below about 20 fs.

Only the intensity autocorrelation is modelled. There is no interferometric mode, so the fringes, the diagnostic 8:1 peak-to-background ratio, and the chirp sensitivity that comes with a collinear geometry have no counterpart here. There is no dynamic range and no noise floor, so pedestals, satellite pulses, and the coherent artifact cannot appear — the trace is always the clean curve of a well-behaved pulse. Pulse shapes other than Gaussian and sech² are not available, and since the modelled envelope is symmetric, the asymmetry that a real autocorrelation famously hides is not there to be hidden.

Mixing is detected by timing settings only. Two sources agreeing in repetition rate, pulse duration, and phase are treated as one train even when their shapes, path delays, or accumulated dispersion differ, in which case the trace is drawn from the first of them and the averaged group delay dispersion. That is a real gap: two genuinely different pulses can be measured as one.

Cross-correlation is between exactly two arriving trains. One is not enough and three cannot be reduced to a single pair, and both cases say so rather than picking two. The two arms are two sources whose light lands on one detector face, not two ports the instrument delays against each other, so the delay is whatever the scene builds rather than something the box scans internally — which is why nulling the mismatch is a job for a path or a delay line rather than a control on the instrument.

The trace width uses the exact result that variance adds under correlation, and is scaled so that both limiting cases come out right: two matched pulses reproduce their own autocorrelation factor, and a reference much shorter than the pulse returns the pulse's own width, since a short enough gate samples the envelope directly. Between those limits it is an interpolation, within a couple of percent of a numerically integrated sech² correlation. Mixed shapes — a Gaussian against a sech² — have no closed form at all, and are flagged as approximate. Chirp is not carried into the width: the arriving durations are used as they stand, so matched-chirp spectral focusing, where the delay tunes the Raman shift rather than merely switching the signal on, is described above but not modelled.