High-resolution Interferometric Diagnostics for Ultrashort Pulses

6.3 Theory of the macroscopic responsePhase (rad)4020Phase (rad)0−500(a)−100(c)Ampl.105(b)5000 10 20 30 40r (µm)IL (10 14 W/cm 2 )Ampl.2(d)100 1 2 3k T (rad/µm)Figure 6.8: Transverse profiles at the beam waist of harmonic 43 produced by one half-cycle of a5 × 10 14 W/cm 2 800 nm pulse with 50 µm beam waist in Argon. The short trajectory (blue), longtrajectory (green) and their uniform-approximation sum (red) are shown. (a) Near-field harmonicphase. (b) Near-field harmonic amplitude (left vertical axis), and laser intensity (black, right verticalaxis). (c) Far-field phase. (d) Far-field amplitude. The irregular shapes of the short and longtrajectory far-field profiles are largely caused by their cusps in the near field profile — these are anumerical artifact that disappears when the trajectories are summed using the uniform approximation.causes radial interference fringes in the sum of the long and short trajectories (Fig. 6.8(b)) untilthe intensity drops below the cutoff and the trajectories coalesce. After propagation to the farfield,(Fig. 6.8(c) and (d)), the short trajectory emission remains on-axis whilst the long trajectoryemission has an annular profile. At intermediate divergence angles, with significant contributionsfrom both trajectories, interference may also be observed [379].The most important longitudinal macroscopic effect is phase matching, which often playsa critical role in HHG. A crucial difference between HHG and perturbative processes such assecond-harmonic generation (SHG) is the intensity dependence of the phase in the former. Thisallows the intensity profile of the laser field to affect the phase matching. This in turn leads todifferent phase-matching conditions for the different trajectories. Therefore, macroscopic effectscan select certain quantum paths and suppress others, leading to a “cleaning” of the harmonicspectrum. Under certain conditions this selection can be controlled and exploited.145