Pulse compressor

A pulse compressor is an optical system whose group delay decreases with wavelength: it delays blue light more than red light, the reverse of ordinary glass. It shortens a pulse that carries a positive chirp, whether that chirp comes from lenses and windows, from self-phase modulation in a fiber, or from the deliberate stretcher of a chirped-pulse amplifier. The three common forms are a pair of diffraction gratings, a pair of prisms, and chirped dielectric mirrors. Typical amounts of negative group-delay dispersion (GDD) range from a few hundred fs² inside a femtosecond oscillator to several ps² after a CPA amplifier.

Why compression is needed

Every transparent material near 800 nm has positive group velocity dispersion: about 36.2 fs²/mm for fused silica and 58.0 fs²/mm for sapphire, from their Sellmeier equations. A transform-limited Gaussian pulse of 30 fs that passes through 10 mm of fused silica acquires 362 fs² and broadens to about 45 fs. The broadened pulse has the same spectrum; only the spectral phase has changed, so an element with −362 fs² restores the original duration.

Grating pair and the Treacy formula

Two parallel gratings used in the same diffraction order make the longer wavelengths, which diffract at larger angles, travel a longer path between them. E. B. Treacy analysed the arrangement in 1969. For a double pass, with the beam returned by a roof mirror so that the spatial chirp is removed, the GDD is

$$\mathrm{GDD} = -\frac{\lambda^3\,G}{\pi c^2 d^2 \cos^3\theta_d},$$ $$\sin\theta_d = \frac{\lambda}{d} - \sin\theta_i ,$$

where $G$ is the perpendicular separation of the gratings, $d$ the groove period, $\theta_i$ the angle of incidence and $\theta_d$ the first-order diffraction angle. For 1200 lines/mm gratings at 800 nm and 30° incidence, $\theta_d = 27.4°$, close to the Littrow angle of 28.7°. A separation of 10 cm then gives −3.73 × 10⁵ fs², or about −37,300 fs² per centimetre. The same pair has third-order dispersion of +7.41 × 10⁵ fs³, so the ratio TOD/GDD is about −2.0 fs. Glass has a TOD of the same sign as its GDD, so a grating compressor corrects the second order but adds to the third-order error of the material it compensates.

Grating pairs provide the large values needed after CPA; cancelling 3.25 ps² with these gratings needs a separation of about 87 cm. They are poorly suited to small corrections: cancelling the 362 fs² of the glass example above would need a separation of about 0.1 mm. Their losses are also significant, since four diffractions at 90–95 % efficiency transmit 66–81 %.

Prism pair

Two prisms at Brewster's angle, the second inverted relative to the first, produce negative GDD from angular dispersion in the same way. The magnitude grows linearly with the apex-to-apex separation $L$ and with the square of $dn/d\lambda$. In the leading approximation for a double pass,

$$\mathrm{GDD} \approx -\frac{4L\lambda^3}{\pi c^2}\left(\frac{dn}{d\lambda}\right)^2 ,$$

which for fused silica at 800 nm ($dn/d\lambda = -0.0173$ µm⁻¹) gives about −650 fs² at 30 cm. The glass the beam crosses inside the prisms adds positive GDD, so translating one prism into or out of the beam tunes the net dispersion continuously. This tuning and the low loss at Brewster incidence made prism pairs the standard intracavity compensator in Ti:sapphire oscillators. Their third-order dispersion limits them for pulses below about 10 fs.

Chirped mirrors

A chirped mirror is a multilayer dielectric stack whose layer period varies with depth, so that longer wavelengths penetrate further before reflecting and acquire more delay. Each reflection contributes a designed negative GDD, typically tens of fs², over a bandwidth that can reach several hundred nanometres. Several bounces, often on mirrors designed in complementary pairs to cancel ripple in the GDD, replace a prism pair in few-cycle oscillators and in the compressors that follow hollow-fiber broadening. Their dispersion is fixed, so a pair of thin glass wedges is usually added for fine adjustment.

Pitfalls

A misaligned grating or prism compressor leaves spatial chirp and pulse-front tilt, and the pulse is then longer than its transform limit across most of the beam. Compressors should be aligned with the pulse measured at the location where it is used, since every window between compressor and sample adds GDD; a FROG or SPIDER measurement shows residual higher-order phase that an autocorrelation hides. In fiber systems, dispersion compensation uses the same principle over much larger GDD values.

Common questions

Why does a grating pair give negative dispersion?

Longer wavelengths diffract at larger angles and travel a longer geometric path between the gratings, so they arrive later than shorter wavelengths, which is the opposite of the ordering in normal-dispersion glass.

Grating pair or prism pair?

Gratings for large GDD, of order 10⁴–10⁷ fs², and prisms for hundreds to a few thousand fs² with low loss and continuous tuning.

References: E. B. Treacy, IEEE J. Quantum Electron. 5, 454 (1969); R. L. Fork, O. E. Martinez, J. P. Gordon, Opt. Lett. 9, 150 (1984); R. Szipöcs, K. Ferencz, C. Spielmann, F. Krausz, Opt. Lett. 19, 201 (1994); J.-C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, 2006); I. H. Malitson, J. Opt. Soc. Am. 55, 1205 (1965).