High-resolution Interferometric Diagnostics for Ultrashort Pulses

8.8 Extraction of orbit amplitudes using filteringInverse Fourier transforming ˜ (j ) (ω;ᾱ) gives (j ) (ω; ¯θ ). Evaluating this at ¯θ = 0 returns the trajectoryamplitude. In fact, using elementary properties of the Fourier transform one has (j ) (ω;0)=˜ (j ) (ω;ᾱ)dᾱ (8.27)where the integral is over all of the control-field sensitivity space.The filters, which depend on the trajectory and the frequency, must be chosen with some care.Ideally, one requires the filter to completely contain all the energy corresponding to its orbit, whilstexcluding that of the other orbits. The infinite support of the orbits clearly makes this an impossiblerequirement to fulfill exactly, and instead one must work down to an effective dynamic range.Here I retain the choice of 10 −4 (in intensity) used above. If the orbits are separate down to thislevel, then it is possible to choose suitable filters. However, there are two situations where thiswill not occur. The first is if orbits are coalescing long and short trajectories. In this case, thereis some arbitrariness in the assignment of energy between the two orbits. It may be argued thatthis is actually a problem of the definition, rather than the method, since as one approaches cutoffthe distinction between the long and short trajectories begins to fade. What is not arbitrary is thecombined amplitude of the coalescing orbits — this is a well-defined quantity that one may “expect”the analysis to accurately retrieve. This condition is satisfied if one chooses the filters suchthat (Sβm) (ω;ᾱ)+ (Lβm) (ω;ᾱ)=1, (8.28)i.e. regardless of the arbitrary assignment of energy into long and short trajectories, the total isconserved.The second situation is if the orbits are not coalescing, but are simply not sufficiently wellseparated in control-field sensitivity space, perhaps due to improper choice of control field. Inthis situation, the quantum path interferometry with the chosen dynamic range has failed — onecannot separate the trajectories objectively. This is a limitation of the method.For nonoverlapping orbits, the filter is a closed curve which encloses the orbit. The filter value209