Finding time zero — sum frequency of two beams
Two colours in one crystal: the second harmonics are always there, and the line between them appears only when the pulses coincide.
Open in the canvas →Background
Two beams that have to work together — a pump and a probe, the two colours of a coherent Raman source, an amplifier and its seed — must reach the sample at the same instant. Position is easy to see; timing is not. A picosecond of error is 0.3 mm of path, and nothing on the table shows it.
A χ⁽²⁾ crystal can show it. The same second-order polarization that doubles a beam also sums two of them: both processes are allowed by the one susceptibility, though whether each is observable in a given setup depends on the wavelengths, the polarizations, the crystal's orientation and the detection. What does not depend on the setup is the timing: doubling needs one beam and happens whatever the delay, while mixing is instantaneous and needs both pulses inside the crystal at once. So the spectrum behind the crystal has a fixed reference and a variable peak — two second harmonics that sit where they are, and a third peak between them that appears only as the delay approaches zero[1].
That is the standard bench recipe. Focus both beams into one crystal, look at the spectrum, and scan the delay until the third line appears. Its peak is time zero, and its width is the cross-correlation of the two pulses. The beams must be synchronous on every shot, so they come from one locked source or from oscillators locked to a common clock. Sum-frequency mixing between two such beams is a working technique in its own right, used to reach wavelengths neither beam has[2].
What this setup demonstrates
A 1032 nm and a 790 nm laser, both 200 fs at 80 MHz, are combined on a shortpass dichroic and focused into one crystal by a 150 mm lens that sits one focal length in front of it. A shortpass dichroic behind the crystal dumps the two fundamentals, so the spectrometer sees only what the crystal made: 516 nm and 395 nm, the two second harmonics, and — at time zero — 447.5 nm, their sum frequency. The peak shapes are drawing conventions, not calculated nonlinear spectra: each harmonic carries its own beam's spectrum scaled with the wavelength, and the mixed peak is given the two inputs' widths added in quadrature, so that all three can be read on one intensity axis.
The delay. The 1032 nm beam is folded down onto the combiner and so carries 200 mm of path the other arm does not have. The delay line in the 790 nm arm is set to exactly that, so the two pulses reach the crystal together and the example opens with the third line present: the crystal's Two-beam mixing readout says 0 fs apart, 100 % temporal overlap. A real mechanical stage carries a fold, so moving it by Δx changes the path by 2Δx; this delay line adds its ΔL directly.
Scanning it. Retune the delay line and watch the spectrum. At 0.02 mm — 67 fs — the sum-frequency peak keeps 86 % of its height; at 0.06 mm, 200 fs, a quarter; at 0.2 mm it is gone and the readout says the pulses are 667 fs apart. Throughout, the two second harmonics do not move at all, which is what makes the third line a measurement rather than a brightness change. For two Gaussians of FWHM τ₁ and τ₂ arriving Δt apart, the line follows exp(−4 ln2 Δt² / (τ₁² + τ₂²)). That 2 % floor is where the drawing stops, not a physical edge: Gaussian pulses never stop overlapping abruptly.
Both lasers are set to 80 MHz, and the scene assumes they are locked to one clock — two sources that merely share a nominal rate would drift through each other, and equal numbers in the inspector do not by themselves establish synchronisation. This model mixes only trains at the same repetition rate. Difference-frequency generation is a checkbox on the crystal, left off here because its 3.4 µm line falls outside the range this bench would look at.
The crystal gates mixing on arrival time and nothing else. There is no phase matching, so the polarizations, crystal cut and angle each process would need are absent. This idealised χ⁽²⁾ proxy enables every beam's own second harmonic and every pair's sum frequency at once; their relative strengths, and whether all three would be visible together in any particular crystal and geometry, are not predicted here. A bench normally aligns for the line it wants. The reverse reading is not safe either — on a real bench a missing sum-frequency peak can mean a polarization, angle or overlap problem rather than a timing one. The focusing lens is drawn because a real setup focuses, but the overlap of the two foci is not calculated, and neither is the conversion's dependence on intensity: each harmonic is an authored 30 % of its beam and the mixed peak 30 % of what doubling leaves of both beams, scaled by the temporal overlap: 0.30 in each harmonic and 0.42 in the sum frequency, for two equal beams. That proportion is a drawing convention chosen so the three peaks sit in the same range, as they do on a bench, rather than a calculated efficiency: both beams contribute the same fraction of themselves, which is not the photon-energy-weighted depletion a real stage would show. The workbench caps authored conversion fractions at 60 %, which is a conservative application limit rather than a physical one. Holding the two second harmonics fixed while the mixed peak rises is the weak-conversion convention this model draws; a strongly depleted experiment would show them change too.
The drawn width of the correlation is therefore a timing proxy built from the two authored pulse durations, not a measurement: it cannot be used to retrieve a pulse width the way a real cross-correlation can, and it assumes ideal Gaussian envelopes with no dispersion between the arms. The two beams are drawn as single chief rays with no beam size, and the delay line is an ideal added path with no alignment or beam walk.