High-resolution Interferometric Diagnostics for Ultrashort Pulses

2. BACKGROUND2.3.7.1 Time magnifierA number of tomographic methods can be understood using familiar concepts in imaging. Asingle-lens imaging system has as its temporal analogy second-order dispersion φ (o)2 , a quadratictemporal phase ψ, and further second-order dispersion φ (i )2, related by Newton’s thin lens equation1φ (o)2+ 1φ (i ) = ψ. (2.43)2The temporal magnification is m = −φ (i )2 /φ(o) 2. The signal as detected on a photodiode is thereforea temporally magnified replica of the temporal intensity I sig (t ) ≈ I orig (t /m ). Ultrafast temporalprofiles can be “slowed down” to the extent that they can be detected on a photodiode. Magnificationsof 100 with 300 fs resolution have been achieved [206–209].2.3.7.2 Time-to-frequency converterAnother well-known spatial operation that has been reproduced in the time-frequency domain isthe spatial Fourier transform, yielding a time-to-frequency converter [210–214]. In space, this isachieved by propagation over a distance L, a lens of focal length f = L, and then another length Lof propagation. The analogous processes are quadratic dispersion φ 2 , quadratic temporal phaseψ = −1/φ 2 , and another application of dispersion φ 2 . The field is then detected on a spectrometer;the spectrum is related to the temporal profile of the original pulse by Ĩ sig (ω) ≈ I orig (−φ 2 t ).The best temporal resolutions achieved with time-to-frequency converters is about 220 fs [214].Both the time-magnifier and the time-to-frequency converter obtain the temporal intensity.An amplitude and phase which is consistent with both the temporal intensity and the spectrumcan be retrieved using the Gerchberg-Saxton algorithm [215, 216]. There are several exact or approximateambiguities in such a procedure.2.3.7.3 TomographyThe time-magnifier and time-to-frequency converter achieve their results through specific combinationsof dispersion and quadratic phase. Arbitrary arrangements of these operations can be understoodin terms of Wigner functions, also known as the chronocyclic representation. If W (t ,ω)50