High-resolution Interferometric Diagnostics for Ultrashort Pulses

7.2 Theory of the methodshears. In the current set of experiments, the worst violation of these assumptions arose from thespatial wings of the A arm, which undergo coherent interference with the B arm, altering the HHGprocess.Note that throughout this formalism, I have neglected terms constant with respect to x in theexpressions for the phase because the experimental setup is not stable enough to detect themreliably. In particular, we found that the lateral displacement of the roof-hat in arm B resultedin random time delay, typically a few femtoseconds, and probably caused by stage translationerrors. Because the resulting phase depends frequency but not position, this error does not affectquadratic and higher terms in the reconstructed spatial phase.7.2.1 Far-field diffraction regimeA useful interpretation is revealed when the detector is distant enough from the source to be inthe Fraunhofer diffraction regime. In that case, a stationary-phase approximation can be used torelate the field at the detector to the spatial Fourier transform of the (potentially virtual) field atz = 0:E (x,y , L,ω) ≈ 1 ik(x 2i λL exp + y 2 ) kxẼ2L L , ky L ,0,ω . (7.5)Here, Ẽ (k x ,k y ,0,ω) denotes the spatial Fourier transform of the field at z = 0. If the slit is sufficientlysmall and located in the Fraunhofer diffraction regime, then it selects the k y = 0 component.This is implied from here on. Under these conditions, if the beam is cylindrically symmetricthen the zeroth order Hankel transform relates the near and far-fields. However, the symmetryis broken by the shear and so for the derivation I shall work with the spatial Fourier transform.Substituting (7.5) into (7.3), one obtainsΓ(x,ω;X d )= ˜φ Bk x + X d,0,ω −L˜φ Bk x L ,0,ω(7.6)where ˜φ denotes the phase of Ẽ . Note that neither the quadratic phase factor of (7.5) nor the linearterm in (7.3) appear in (7.6) — they cancel each other exactly. Therefore in the Fraunhofer regimethe experiment represents LSI in the spatial Fourier domain, with the shear being K x = kX d /L.161