High-resolution Interferometric Diagnostics for Ultrashort Pulses

2. BACKGROUNDfrom which the field autocorrelation may be inferred using the fact that E (ω) contains only positivefrequencies. Elementary properties of the Fourier transform show that the spectrum is theFourier transform of the field autocorrelation.2.3.3.2 InterferometryWhere multiple fields E i (ω) are involved, one can generalise (2.19) to show that the signal acquiredon a slow integrating detector following a series of linear time-stationary operations can bereduced to a linear superposition of bilinear terms E i (ω)E ∗ j(ω) for all pairs of fields (i , j ). Undercertain circumstances it is possible to obtain each of these terms individually. Taking the complexargument of each term φ i (ω) − φ j (ω), one obtains the spectral phase difference between pairs offields. This is the basis of interferometry. For simplicity, I shall consider two fields from now on.Spectral phase differences acquired through interferometry can be put to several uses.1. If the two pulses originate from the same pulse, the difference in optical path lengths betweenthe beam splitter and the detector is obtained. This is a widely used means of measuringthe dispersion of transparent materials. Whilst not technically a pulse characterisationmethod, it is very closely related and uses a similar setup [83–86].2. If the spectral phase difference is related to the spectral phase of an unknown pulse bysome invertible mathematical operation, then the phase of the unknown pulse can be inferred.This represents self-referenced pulse characterisation, and according to the restrictionsabove, requires nonlinear or time-nonstationary elements to prepare. This will be discussedin the section on complete characterisation.3. If one of the fields is a well-characterised reference, then the spectral phase of the other canbe inferred.The third application represents a powerful externally referenced characterisation technique[87, 88], with one specific implementation known as temporal analysis by dispersing a pair of e -fields (TADPOLE). One of its main advantages is sensitivity; the unknown pulse heterodynes withthe reference, and detection is completely linear. Provided the reference is sufficiently intense, a26