High-resolution Interferometric Diagnostics for Ultrashort Pulses

2.2 Formal introduction to ultrashort pulses(x , ¯ω) (x,t ) (k x , ¯ω) (k x ,t )Ē (x, ¯ω)exp(i αx ¯ω) Ē (x,t − αx) Ē (k x − α ¯ω, ¯ω) —spatio-spectral phase position-dependentarrival timefrequency-dependenttiltĒ (x, ¯ω + βx) Ē (x,t )exp(i βxt) — Ē (k x − βt ,t )position-dependent spatio-temporaltime-dependent tiltcentre frequency phasearrivalĒ (x + γ ¯ω, ¯ω) — Ē (k x , ¯ω)exp(i γk x ¯ω) Ē (k x ,t − γk x )frequency-dependentbeam centroidwavenumberspectralphasewavenumberdependenttime— Ē (x + δt ,t ) Ē (k x , ¯ω + δk x ) Ē (k x ,t )exp(i δk x t )time-dependent wavefront-dependent wavenumbertemporalbeam centroid centre frequencyphaseTable 2.1: Lowest-order space-time phase couplings. Each row corresponds to a type of coupling,and each column corresponds to one of the four domains.Three types of space-time coupling are particularly common. Pulse-front tilt refers to any situationin which the arrival time is position dependent. Both a spatio-spectral phase (row 1 oftable 2.1) and a wavenumber-temporal phase (row 4) produce pulse-front tilt. Angular dispersionrefers to any situation in which the tilt of the beam depends on frequency. Again, both aspatio-spectral and a wavenumber-temporal phase coupling produce angular dispersion, and anyoperation which induces pulse-front tilt on a space-time factorable pulse will result in angular dispersion,and vice versa [58, 76, 77]. However, for general pulses, one does not necessarily implythe other [78]. The spatio-spectral phase coefficient is related to the angular dispersion ∂θ/∂ωbyα =(nω c /c)(∂θ/∂ω).Note the distinction between a tilt and a pulse-front tilt. Applying a tilt (e.g. a small rotation) toa pulse rotates the phase and group fronts by equal amounts, whereas the essence of pulse-fronttilt is that the phase front (the plane of constant phase at the centre frequency) and group fronts(the plane of constant group delay at the centre frequency) are different. Mathematically, a tiltis represented as a spatio-spectral phase of θωx/c whereas the simplest form of pulse-front tiltis represented by α ¯ωx i.e. in a tilt, the phase is proportional to the passband frequency whereas19