High-resolution Interferometric Diagnostics for Ultrashort Pulses

2. BACKGROUNDfactorability.2.2.3.3 Space-time coupled pulsesIf the pulse cannot be written in the form (2.15) for any rotation θ , it is said to possess space-timecoupling, and must be described in a combined space/wavenumber and time/frequency picture.For this discussion, coupling with only one spatial dimension, x, is considered. Fourier transformationmay be performed along either or both of the x and the t axes, resulting in four possibledomains in which the two-dimensional field of an ultrashort pulse may be described: spatiotemporal(x,t ), spatio-spectral (x,ω), wavenumber-temporal (k x ,t ), and wavenumber-spectral(k x ,ω).For cylindrically symmetric profiles E (r ), the appropriate conjugate representation is given bythe Hankel transform E (k T ), where k T is the transverse wavenumber.There are a vast range of possible space-time couplings. One useful starting point is to considerthe action of the lowest order couplings of the phase. There is one term for each of the four domains.For example, in the spatio-spectral domain, the lowest-order coupling term is exp(i αx ¯ω)where α is the coupling coefficient. Since the expansions are around the centre frequency, I usebaseband signals for this section. The physical significance of each coupling can be found by performinga Fourier transform to the other domains. For example, ∞12π−∞ ∞12π−∞Ē (x, ¯ω)e αx ¯ω e −i ¯ωt d ¯ω =Ē (x,t − αx ) (2.16)Ē (x, ¯ω)e αx ¯ω e −ixk xdx =Ē (k x − α ¯ω, ¯ω) (2.17)showing that a spatio-spectral coupling phase causes a shear along the time-axis in the spatiotemporaldomain, and a shear along the wavenumber axis in the wavenumber-frequency domain.A second Fourier transform produces a more complicated expression. Table 2.1 shows the mathematicalforms of each of the four couplings. Assuming that the original pulse Ē (x,t ) is space-timefactorable, each phase coupling in a given domain has a simple interpretation in the two otherdomains related by a single Fourier transform. These are given in table 2.1.18