@PaŭloEbermann No, optical tweezers are a completely separate side to this Nobel prize. The only connection is that they both involve lasers.

chirping is standard practice in radar and sonar. The laser implementation seems a little more difficult. (radar chirp compression can be done in software now-a-days).

@JEB Indeed it is, though with some differences - in radar, you send out a chirped pulse, get the reflection, and you only re-compress at the very end. (See the linked Electronics SE thread, as well as this question, for more details.) The radar precedent is explicitly recognised up-front in the initial Strickland & Mourou paper, though it is my understanding that the idea was developed independently and the analogy was only realised prior to publication. ([citation needed] on that, though.)

Does this not simply move the problem from the laser medium to the resered gratings medium? Dont you still have the same problem at the output gratings?

@Hakaishin Those are typically reflection gratings, so the damage threshold is much higher, and the propagation length is effectively zero so self-focusing isn't a concern.

How do you achieve this type of "time dispersion" with a dispersive element? Normally they produce angular dispersion, right? Do you just use two of them of them in series?

@jkej There's a wide variety of possible designs, and yes, most of them use two or more gratings. This example from Wikipedia captures the essence of a lot of such devices. (In the real world the stretcher and compressor, being the heart of the amplifier, are highly optimized, which often means that they evolve into highly complex equipment. But the basics are there.)

@EmilioPisanty Interesting. I was think you might need four gratings, but the example design just uses a mirror to make the beams go the same way back through the two gratings to produce the same effect I was thinking of. Seems like you would lose 75% of the light in the beamsplitter though. But as you said, the actual designs used are probably much more complex.

@jkej If that's what worries you, a mild tilt of the mirror will be enough to separate the in & out optical paths. Two gratings are enough (you can then re-focus later). And, of course, that schematic is mostly just proof-of-concept; real-world devices are much more complex, including things like multiple passes of the gain medium and phase shaping to optimize the pulse compression.

Maybe a stupid question but aren't lasers monochromatic? Does that just mean narrow range of frequencies but not too narrow to be dispersed?

@JollyJoker some lasers are monochromatic. Pulsed lasers are not - they simultaneously amplify a broad band of frequencies which then combine coherently to form the pulse.

So sharks with lasers on their heads will be possible soon...

So this one goes to 11?

@JollyJoker, a continuous wave has a specific frequency, but a short wave packet is a composition of a range of frequencies (verifiable via Fourier transform). The shorter the pulse, the wider the spectrum.