High-resolution Interferometric Diagnostics for Ultrashort Pulses

6. HIGH-HARMONIC GENERATIONPhase matching is governed by the difference between the wave-vector of the spatially varyingsingle-atom response k S = ∇φ S (x,y ,z ,ω) and the wavenumber of freely propagating harmonicsβ = n(ω)ω/c. The generation is most efficient when k S = |k S | = β. The direction of the generatedradiation is k S . The coherence length L coh = π/|k S − β| is the distance over which the single-atomresponse becomes π out of phase with propagating harmonics. If, as is often the case, the coherencelength is shorter than other limiting distances such as the gas jet thickness or the absorptionlength, then the generation is phase-mismatched and the coherence length sets the effective generationlength.Much insight can be gained by considering the various contributions to the single-atom responsewave-vector [380, 381]. Generally, there are several significant contributions. Spatial variationsin the laser intensity leads to variation in the phase of the single-atom response, as discussedin section 6.2.5. This gives a contribution −ρ∇I L to k S . The phase φ L of the focused laser, whichincludes the converging/diverging wavefront, gives a contribution q∇φ L , where q is the harmonicorder. The dispersion of the gas at both the fundamental and harmonic wavelengths may alsoplay a role. Summing these effects enables one to determine the location and direction of efficientharmonic generation within the laser focus.Figure 6.9 shows the various contributions to the axial phase mismatch ∆k z for the same parametersas Fig. 6.8. The Gouy phase gives a negative contribution over the entire focus, whilstthe intensity-dependent phase changes sign at the focus as the beam goes from converging todiverging. For the chosen pressure of 0.01 atm, the dispersive contributions are small (they caneasily become significant at higher but experimentally realistic pressures). Under these conditions,on-axis phase matching may occur downstream of the focus, where the intensity-dependentand Gouy phase contributions can cancel. The higher intensity-dependence of the long trajectorymeans this cancellation occurs closer to the focus than for the short trajectory. Moving towardsthe cutoff (higher harmonics), the trajectories become more similar in character. However, theGouy phase contribution becomes larger. The interplay of these effects means that the conditionsfor optimal phase matching depend on both frequency and the desired trajectory.Additional phase-matching possibilities arise away from the axis, where the radial variation of146