High-resolution Interferometric Diagnostics for Ultrashort Pulses

6. HIGH-HARMONIC GENERATIONgiven in references [373, 375, 376]. The contribution from the j th saddle point is (j ) (ω)= (2π) 4r (ω) − t (j ) 3 (j ) (j )b(ω) detSst (j )r (ω),t (j )b (ω) ion(ω)reci t (j )= (j ) (ω)e iS(j )ω(ω)eiS(j)ω(6.19)where I have introduced the symbol (j ) (ω) for later purposes. Equation (6.19) permits a similarinterpretation to (6.13), with the additional prefactor accounting for spreading of the wavepacketdue to diffraction.One complication of the quantum orbit model is that not all of the stationary point solutionsare valid. In consequence, the net dipole response is not simply the sum of the individual contributions,each given by (6.19). Examining Figures 6.5 and 6.6, one observes that for frequenciesbetween the ionization potential and the classical cut-off, the long and short trajectories are distinct,both in terms of the exact TDSE solution as represented in the spectrogram and the quantumtrajectories. Approaching the cutoff, the two recombination events shown in the spectrogram coalesceinto one and this is also reflected in the real part of the recombination times of the quantumtrajectories. However, the trajectories do not actually merge into one — there are still two stationarypoints, even above the cutoff. One of these, the recessive trajectory, is unphysical — itscontribution to the amplitude diverges rapidly. Mathematically, this solution is no longer reachedby the steepest-descent contour integral used in the stationary-point approximation. The other,dominant solution, possesses the appropriate rapid decay and accurately approximates the integralabove the cutoff.The preceeding discussion shows that the stationary-point approximation to the net dipoleresponse is the sum over all trajectories well below the cutoff and the sum over all dominant trajectorieswell above the cutoff. Around the cutoff, the uniform approximation, which deals withcoalescing stationary-points, is necessary for an accurate result. The uniform approximation to142