High-resolution Interferometric Diagnostics for Ultrashort Pulses

2. BACKGROUNDamplitude Ē ( ¯ω)=E ( ¯ω + ω c ) is a shifted version of the spectral amplitude.Time-frequency domain. Together, the amplitude and phase, in either of the time- or frequencydomains,uniquely define the pulse. However they do not necessarily provide an intuitive representation,especially for complex pulses. In such cases, time-frequency functions, which representthe pulse on the two-dimensional (t ,ω) domain, are often helpful, despite the fact that the conjugatenature of time and frequency causes fundamental resolution and blurring issues related to theuncertainty principle. Often, not just a single pulse but the fluctuations of an ensemble of pulsesmay also be described in the time-frequency domain, in which case a time-frequency distributionis used.Many time-frequency distributions may be defined. They may be categorised according totheir theoretical significance, their intuitiveness and the practicality of measuring them.TheWigner function [73–75] has great theoretical significance because the equations of motion inoptics and quantum mechanics can be re-formulated to describe its evolution. It also has theproperty that interference fringes appear between well separated subpulses in the time-frequencydomain. For an ensemble, the fringes in the Wigner distribution represent the coherence betweenthe subpulses, whilst for a single pulse they show the relative phase of the subpulses. This is notnecessarily intuitive information, and the fringes become very complex when many subpulses arepresent. There are currently few means of directly observing the Wigner function.The spectrogram is a gated Fourier transform:B(τ,ω)= 1 2π ∞−∞g (t − τ)E (t )e i ωt dt . (2.11)It does not feature the interference fringes of the Wigner function, and can be measured directlyfor ultrashort pulses.16