High-resolution Interferometric Diagnostics for Ultrashort Pulses

8.5 Limitation of the perturbative quantum path analysisRe[S (j )ω ]0−50−100(a) 403.53.453.4ω ]ω ]Im[S (j )Re[S (j )500−50(b) 70105Im[S (j )ω ]Re[S (j )ω ]40200(c) 9040200Im[S (j )ω ]−1503.35−0.1 0 0.1−1000−0.1 0 0.1−20−20−0.1 0 0.1Re[S (j )ω ]−70−75−80θ I(d) 4043.5Im[S (j )ω ]Re[S (j )ω ]−10−15−20θ I(e) 7043.5Im[S (j )ω ]Re[S (j )ω ]15105θ I(f) 9010.5109.5Im[S (j )ω ]−853−0.1 0 0.1θ Q−253−0.1 0 0.1θ Q09−0.1 0 0.1θ QFigure 8.8: Real (blue, left axis) and imaginary (red, right axis) parts of the action of the long (dominant)trajectory, subject to in-phase (subfigures (a)–(c) in the top row), and quadrature (subfigures(d)–(f) in the top row) control fields. The harmonic orders are 40 (subfigures (a) and (d), first column),70 (subfigures (b) and (e), second column), and 90 (subfigures (c) and (f), third column).in blue scaled according to the left vertical axes; the imaginary parts are in red and use the rightvertical axes. Moving to the interpretation of the figure, it is evident that in all cases, the exact resultsare tangent to the linear approximation at θ = 0, verifying the correctness of the formalism.Concerning the real parts, the linear approximation appears by eye to be reasonable for all thecases shown. The imaginary parts show similar agreement except for the in-phase control field atharmonic 70. Here, the exact results show a marked departure from the linear approximation atθ I = −0.02. This corresponds to the suppression of the cutoff by the control field and highlightsthe potential problems of this approach around cutoff.Figure 8.8 suggests that the linear approximation is accurate for most cases. However it in noway covers the entire dataset and doesn’t show the complete picture of the nonlinearity, which is atwo-dimensional surface dependent on θ I and θ Q . I required a quantitative measure of the curva-201