Chirped mirror

A chirped mirror is a high-reflectance dielectric coating built from alternating high- and low-index layers whose thickness, and therefore local Bragg wavelength, changes gradually from the surface into the stack. Short wavelengths are reflected by the thinner layers near the surface; long wavelengths penetrate to the thicker layers deeper down and return later. The reflected group delay then decreases with frequency, which is negative group-delay dispersion. Designs for Ti:sapphire and other femtosecond lasers typically give −30 to −100 fs² per reflection, with reflectance typically above 99.5% over bandwidths from about 100 nm to several hundred nanometers. Chirped mirrors were introduced by Szipöcs, Ferencz, Spielmann and Krausz in 1994.

How the delay arises

A uniform quarter-wave stack, the distributed Bragg reflector, reflects strongly only inside a stop band around one design wavelength. Chirping the period spreads the stop band over a range of wavelengths, each reflected from the depth where the layers are a quarter wave thick for it. The extra round-trip path to that depth sets the delay.

An order-of-magnitude estimate shows how much dispersion is available. Suppose the Bragg wavelength runs from 700 nm near the surface to 900 nm at a depth of 5 µm, with an average index of 1.78 (as for alternating Ta₂O₅ and SiO₂). The round-trip delay difference is $2 \times 1.78 \times 5\ \mu\text{m}/c \approx 59$ fs, and the angular frequency changes by 0.60 rad/fs between the two wavelengths, so

$$\mathrm{GDD} \approx -\frac{59\ \text{fs}}{0.60\ \text{rad/fs}} \approx -99\ \text{fs}^2 .$$

This ignores the group index and the actual reflection profile, but it shows why tens of femtoseconds of delay spread over a few hundred nanometers give GDD of order −100 fs² per bounce. Real designs are found by numerical optimization of every layer thickness with the characteristic matrix method, starting from a chirped thin-film stack.

GDD ripple, double chirping and complementary pairs

A simple chirped stack produces GDD with strong oscillations across the band. The reflection from the front surface and the weak reflections from the outer layers form a low-finesse resonator with the deeper reflecting layers, a Gires–Tournois-type interference whose phase varies rapidly with wavelength. Two remedies are used. Double-chirped mirrors vary the thickness ratio of the high- and low-index layers near the surface as well as the period, matching the stack gradually to the ambient medium, and add an anti-reflection coating on top. Complementary pairs are two designs whose GDD ripples are of opposite phase, so that one bounce on each gives a smooth sum. Mirrors are therefore often specified and used in pairs, with an even number of bounces.

Worked example

A transform-limited Gaussian pulse of 20 fs at 800 nm passing through 10 mm of fused silica (GVD 36.2 fs²/mm) acquires 362 fs² and broadens to 54 fs. With mirrors of −50 fs² per bounce,

$$N = \frac{362\ \text{fs}^2}{50\ \text{fs}^2} = 7.2 ,$$

so seven bounces leave +12 fs², which lengthens the 20 fs pulse to 20.07 fs; eight bounces, an even number for a complementary pair, leave −38 fs² and 20.7 fs. If each reflection has 99.8% reflectance, seven bounces transmit 98.6%. Because the GDD comes in fixed steps, setups usually add a pair of thin glass wedges, whose insertion is adjustable, to trim the remainder.

Where they are used

In Kerr-lens mode-locked Ti:sapphire oscillators, chirped mirrors as cavity folding mirrors supply the net negative dispersion that balances the crystal's material dispersion and self-phase modulation. They replace the intracavity prism pair, which has larger third-order dispersion and requires tens of centimeters of path; with chirped mirrors, oscillators can produce pulses below 10 fs. They also compress spectrally broadened pulses after hollow-fiber or bulk broadening and precompensate the dispersion of microscope objectives. Mirrors can be designed to compensate the third-order dispersion of a given material as well as its GDD. For large amounts of dispersion, such as the ps² values after chirped-pulse amplification, a grating pulse compressor remains necessary.

Pitfalls

The GDD of a chirped mirror holds only at its design angle of incidence and polarization; at a larger angle the layers have a smaller effective optical thickness, and the band shifts to shorter wavelength. Residual ripple of a few to tens of fs² per bounce, multiplied over many bounces, can leave spectral phase structure that a FROG measurement shows as satellite pulses. Measured GDD, usually obtained by white-light interferometry, may differ from the design, since thickness errors of a fraction of a nanometer in individual layers shift the ripple. The coatings are thicker than ordinary laser mirrors, and contamination on the surface adds its own phase and loss.

Common questions

Why do chirped mirrors give negative dispersion?

Long wavelengths reflect from deeper layers and so travel farther in the coating than short wavelengths. Delaying the red relative to the blue is the reverse of what glass does, which is negative GDD.

How many bounces are needed?

The total material GDD divided by the GDD per bounce, rounded to a convenient, usually even, number: about 7–8 bounces at −50 fs² for 10 mm of fused silica at 800 nm.

Can a chirped mirror give positive dispersion?

Yes. Reversing the chirp, so that short wavelengths penetrate deeper, gives positive GDD, used for example to stretch pulses slightly or to offset negative dispersion elsewhere in a system.

References: R. Szipöcs, K. Ferencz, C. Spielmann and F. Krausz, "Chirped multilayer coatings for broadband dispersion control in femtosecond lasers," Opt. Lett. 19, 201 (1994); F. X. Kärtner, N. Matuschek, T. Schibli, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch and T. Tschudi, "Design and fabrication of double-chirped mirrors," Opt. Lett. 22, 831 (1997); J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, 2006).