High-resolution Interferometric Diagnostics for Ultrashort Pulses

2.4 Extending ultrashort metrology to the space-time domainFigure 2.17: Spatial interferometry with (a) and without (b) a carrier, and the resulting fringe patterns.The wavefronts of the two beams are indicated; one is flat and the other is spherical.shop testing methods developed in the 1950s. For convenient evaluation of optical aberrations inthe workshop, the early devices were often designed to produce fringe patterns that were easilyinterpretable by eye. An example is shown in Fig. 2.17(a). The spherical wavefront is interferedco-linearly with a flat wavefront, and the resulting fringe pattern readily reveals the difference intheir curvature. However, the sign ambiguity discussed in the context of spectral interferometrysection 2.3.3.2, applies equally to the spatial case. In Fig. 2.17(a) the sign of the curvature cannot bedetermined, in the same way as one cannot distinguish a depression from a hill on a topographicmap using the contour lines only.Analogous to the spectral case, unambiguous phase recovery can be performed in several differentways. Multiple-trace methods require the phase of one of the arms to be modulated, enablingboth the in-phase (cosine) and quadrature (sine) components to be obtained. Here, I concentrateon the only known single-trace method, which employs a rapidly varying carrier, in theform of a tilt between the beams. The tilt is analogous to the time delay in Fourier-transformspectral interferometry. For beams inclined at an angle θ in the yz plane incident on a detectorperpendicular to the z -axis, the relative phase is θ ky. The local fringe phase is thereforeφ 2 (x,y ) − φ 1 (x ,y )+θ ky. Before the development of Fourier-transform interferometry [92], thefringe phase was extracted by a peak searching algorithm [231]. However, this has been almostcompletely superseded by Fourier-transform spatial interferometry, which uses the same algorithmas the spectral case described in section 2.3.3.2 except that a two-dimensional Fourier transformis used.One difference between spectral and spatial interferometry is the relative ease with which awell-characterised reference may be produced in the latter using spatial filtering. A pinhole may55