High-resolution Interferometric Diagnostics for Ultrashort Pulses

8.5 Limitation of the perturbative quantum path analysisinterferometry, the size of the components in the Fourier domain is just as important as their locationin determining whether the interferometric component can be isolated. Since the perturbativeanalysis of section 8.3 culminating in (8.12) indicates that the trajectories are delta functionsi.e. infinitely small, their actual size in the Fourier domain must be related to the accuracy of thepertinent assumptions — namely the linearity of the change in action and the invariance of thetrajectory amplitudes with respect to the control field amplitudes. The resolution in the Fourierdomain is also limited by the range of control-field amplitudes over which the dipole response isknown. In this section I explore these limitations and relate them to the size of the trajectories inthe Fourier domain — and hence the success or failure of the present endeavor.8.5.1 Brute-force studyIn this section I verify the assumptions and predictions of the perturbative quantum orbit formalismusing exact quantum orbit calculations. I used the CW drive and control fields presented insection 8.4, and performed a quantum orbit analysis over a two-dimensional grid of control fieldamplitudes θ Q ,θ I ∈ [−0.1,0.1] in 0.01 increments. The zero control field case is compared withseveral extreme cases in Fig. 8.4. In most cases, the control fields cause only a minor change in thetime-frequency structure and the excursion times. However, in one of the cases (θ 1 = 0,θ 2 = −0.1),the cutoff is suppressed by approximately 15 harmonic orders by the control field.This results set enabled me to examine in turn the assumptions of a constant dipole amplitude,and the linear action variation.8.5.1.1 Variation of dipole amplitudeFrom the quantum orbits analysis, I calculated the control-field dependent dipole amplitude (j ) (ω; ¯θ )using (6.19). Figure 8.5 (a) and (b) shows the results for harmonic order 40, well below cutoff. Itis apparent that the variation of the dipole amplitude and phase with control field amplitude issmall, and predominatly linear, as will be quantified below. The conclusions are similar well abovethe cutoff.Figure 8.5 (c) and (d) shows the same results for harmonic order 70, around cutoff. Here, thedependence is not linear or smooth, with the in-phase control field causing a pronounced en-197