Chirped-Pulse Amplification

What is modelled

A femtosecond pulse is too intense to amplify directly, so it is first stretched in time by a dispersive element, amplified, and then compressed again. A pair of parallel diffraction gratings does the compression because red light takes a longer path than blue between them, giving a negative group-delay dispersion that cancels the positive chirp of the stretcher. The shortest possible pulse is set by the bandwidth.

Equations

• Grating equation sin i + sin d = λ / d with groove spacing d.
• Compressor dispersion (Treacy) GDD = −N λ³ L / (2π c² d²) · [1 − (λ/d − sin i)²]−3/2, with L = G / cos d the slant distance and N passes through the pair (two here).
• Transform-limited Gaussian pulse τ0 = 0.441 λ² / (c Δλ).
• Output duration τ = τ0 √(1 + (4 ln2 · GDD / τ0²)²), with GDD = GDDstretcher + GDDcompressor.

Checks run in the page

• The angular dispersion of the grating equation by numerical differentiation.
• A spectral phase applied to a Gaussian spectrum and transformed with an FFT, against the closed-form duration.
• The pulse energy before and after the phase.
• The matched separation that cancels the stretcher, and the time-bandwidth product 0.441 of the numerical spectrum.

Honest simplifications

The pulse is Gaussian, only second-order dispersion is included (third-order phase, which limits real compressors to about 10 fs, is ignored), the gratings are treated as ideal and lossless with a single central wavelength for the geometry, and the stretcher is represented by a single GDD number instead of a real lens or mirror design. The GDD constant follows the usual Treacy form for a folded two-pass layout; use the trends rather than the absolute figures.

• Press Matched compressor: the output pulse returns to the bare duration and the peak power gain equals one.
• Press Under-compressed and then Over-compressed, and watch the pulse lengthen again on either side of the match in the first graph.
• Raise the bandwidth in Broad bandwidth and see that a wider spectrum gives a shorter pulse but needs a much more accurate match.