High-resolution Interferometric Diagnostics for Ultrashort Pulses

2.3 Introduction to ultrashort pulse metrologyE 2 (t − τ)E 1 (t )E 1 (t )+E 2 (t − τ)Ĩ (ω)Figure 2.3: Apparatus for performing FTSI.B(ω)FTB(t )IFTφ(ω)ω−ττtωFigure 2.4: Fourier transform interferometry; in this example one of the pulses is transformlimitedand one has a linear chirp. The filter in the quasi-time domain is shown in green.ual intensities |E 1 (ω)| 2 and |E 2 (ω)| 2 upon Fourier transformation of (2.26). Rewriting (2.26), thiscan be made explicit:B(ω)=|E 1 (ω)| 2 + |E 2 (ω)| 2 + E 2 (ω)E ∗ 1 (ω)exp(i ωτ)+c.c. (2.28)where c.c. denotes complex conjugate. The first two terms, which are real and slowly varying,are located at t = 0 upon Fourier transformation to the time domain. These form the basebandcomponent. The first bracketed term is located at t = τ, and its complex conjugate is located att = −τ. These are the positive and negative sidebands. Multiplication by a rectangularly-shapedpassband filter and inverse Fourier transforming allows one of the sidebands to be isolated. Theprocess is illustrated in Fig. 2.4. The time-domain filtering step is only possible if the sidebands areseparate from the baseband component. As has been mentioned, there is a degree of arbitrarinesssince the components generally decay smoothly. However a useful practical criterion is for thesignals to drop below the background noise level in the quasi-time domain.The carrier amplitude required for separateness of the baseband and sidebands depends ontheir spread in the time domain, which can be derived using the correlation properties of theFourier transform of (2.28). In the quasi-time domain, the baseband terms are the field autocorrelationfunction, the duration of which is a factor of order unity times the transform-limited29