High-resolution Interferometric Diagnostics for Ultrashort Pulses

2. BACKGROUNDpulse duration. For a Gaussian pulse the factor is 2. The sideband terms are the field crosscorrelationfunction, with a duration which depends on the transform-limited duration and relativeamount of chirp of the two pulses. In general, the greater the difference in chirp, the longer thecrosscorrelation. The support of the crosscorrelation (down to some nominal noise level) does notexceed the sum of the supports of the two pulses.Regardless of the details, a well-resolved fringe pattern must be sampled at least three timesmore finely than required for the pulse spectra themselves. This challenges the resolution of manyspectrometers, and is the principal limitation of FTSI. For example, one immediate application isthe characterisation of the output of femtosecond pulse shapers. However, these typically havearound 512 independent channels, implying that an effective resolution of around 1500 independentchannels would be required on the spectrometer. This is one motivation for using a spatiallyresolved spectrometer, as I describe next.Fourier-transform spatio-spectral interferometry (FTSSI) adds a spatial dimension to the purelyspectral form [71, 93–96]. The beams are brought to interference on a spatially resolved spectrometer,such as an imaging spectrometer, with a tilt θ which produces a spatially varying linearphase θ ky between the beams. The interference fringes so produced are approximately parallelto the frequency axis. Figure 2.5 illustrates the concept. The Fourier-domain filtering algorithmis identical except that a two-dimensional transform is used. The advantages of achieving spatialresolution will be discussed in section 2.4. However, even for purely spectral measurements, thisapproach has the advantage of alleviating the spectral resolution requirement of spectral interferometrybecause it obviates the need for a temporal carrier. Although this in turn introduces acorresponding spatial sampling requirement, this is less commonly a problem because complexspatial structure is usually rare and undesired in an ultrashort pulse.Multiple-trace interferometry. Whilst the use of a carrier is common and used throughout thisthesis, it is also possible to perform interferometry without one. In general, carrier-free interferometryrelaxes the required resolution at the expense of requiring multiple shots and/or detectors.The procedure involves obtaining several interferograms, each with a different phase shift appliedto one of the fields. The interferometric contribution to the j -th interferogram is proportional to30