High-resolution Interferometric Diagnostics for Ultrashort Pulses

2. BACKGROUNDfor a pulse-front tilt it is proportional to the baseband frequency. In the wavenumber-spectraldomain, a tilt is represented as E (k x − θω/c,ω), showing that the displacement in wavenumberis proportional to the passband frequency and hence represents a constant angle θ . On the otherhand the spatio-spectral phase is represented as Ē (k x − α ¯ω, ¯ω), showing that the frequencies arespread around the axis. This difference introduces a subtlety into the measurements of chapter 5.Spatial chirp refers to any situation in which the central frequency of the pulse varies withposition. A spatio-temporal phase (row 2) and a wavenumber-spectral phase (row 3) both producespatial chirp. Furthermore, a pulse with angular dispersion will acquire spatial chirp afterpropagation.For concreteness, I shall now discuss some physical operations which introduce space-timecoupling.Non-normal incidence at a dielectric boundary produces angular dispersion via Snell’s law. Ifthe central frequency of a pulse, is incident with angle θ 1 on a boundary between media withrefractive indices n 1 and n 2 , then the resulting angular dispersion is∂ n∂θ12∂ω = ∂ω sinθ 1 − ∂ n 2∂ω sinθ 2. (2.18)n 2 cosθ 2In most situations, this effect is quite small. For example, for fused silica at 800 nm, ∂ n/∂ω≈0.005fs, so a 5 fs pulse incident at 45deg experiences an angular spread of ≈ 0.1 mrad acrossits full-width at half-maximum bandwidth due to refraction. The equivalent pulse-front tilt is≈ 2.1fs/mm. Furthermore, for a parallel glass plate, the angular dispersion is completely reversedupon exiting, so that only a small spatial chirp acquired during propagation results. On the otherhand, dispersive prisms typical produce an angular dispersion of ∂θ/∂ω= 0.01fs, giving a pulsefronttilt of 100fs/mm, significant for femtosecond pulses.Frequency-dependent mode area. The focus of a space-time factorable pulse produced by a perfectlens has a frequency-dependent mode area because the mapping x = fk x /k between wavenumberand position x in the focus is frequency-dependent. This dependence becomes significant forbroadband pulses. Few-cycle oscillators can also produce such an effect [79].20