How to determine the laser duration in high harmonic generation?

Question

In high harmonic generation, how can we determine the laser width (duration)? Say, I have a Gaussian laser pulse with intensity $10^{14} \:\rm W/cm^2$ centered at 800 nm wavelength, interacting with an atom of ionization potential = 16 eV. What is the appropriate laser duration and why? I assume that one needs to use uncertainty relation but what is $\delta \omega$ in this case?

Answer 1

You can't ─ it's a free parameter, and it can take a bunch of different values: from the single- or few-cycle regime, say, if you want to do pulse-length gating, to the 100fs regime if you want to work with your HHG's spectral properties and you don't care so much about having an isolated attosecond pulse.

There are some limits, of course (you can't go shorter than one or two cycles of your IR, and if you go much longer than 100fs you will struggle to get the required intensity unless your pulse energy is very high) but within those limits, you have a broad spectrum of pulse-length choices that will affect the characteristics of the emitted radiation in many ways, and it is up to the experimental design to choose a driving-pulse length that is suitable for the science you want to do (and for your budget! few-cycle pulses don't come cheap in equipment, personnel, or time).

And, that said, there are some hard requirements on the bandwidth of the laser, in the sense that if you want to support a pulse length $\Delta \tau$, then your IR driver must have a bandwidth no smaller than $$ \Delta \omega \gtrsim \frac{2\pi}{\Delta\tau} $$ to support that pulse. As an example, if your laser oscillator and (CPA) amplifier are running on Ti:Sa gain media, then you will be restricted to a bandwidth of around 150-200 nm, i.e. some 600 THz, which then puts a hard limit of some 10 fs on your pulse length, i.e. some four cycles of full-width at half-maximum.

That's an immovable physical constraint, and that means that if you want to go lower than that and get to truly few- or single-cycle IR driving pulses, then you need more bandwidth. This is normally done using self-phase modulation in a gas-filled hollow-core optical fiber, where third-order nonlinear optical processes in the fiber are used to increase the bandwidth to the point where it can support the few-cycle pulses that you want, followed by extremely careful compensation of the added chirp to compress the pulse down to its Fourier-limited pulse length.