Ultrashort Laser Pulse Phenomena, 2nd Edition

Problems 485

MI



GDL

SF

S

Figure 9.17 Schematic diagram of a SPIDER apparatus. One replica of the pulse to be characterized

is stretched and chirped in an element with group delay dispersion (GDL), the other replica is split

into identical time-delayed pulses in a Michelson interferometer (MI). The sum frequency is produced

in a nonlinear crystal (SF) and recorded in a spectrometer (S). Adapted from Iaconis et al. [47].

The stretcher has to be designed so that the frequency of the stretched pulse

does not change (much) during the time of the original pulse duration. Pulses

consisting of few optical cycles have been successfully characterized using the

SPIDER technique [60]. By combining it with homodyne detection sensitivity

and versatility are improved [61]. True single shot implementation of SPIDER

has been demonstrated at a repetition rate of 1 KHz [62].

9.5. PROBLEMS

1. Show that the statistical average of the autocorrelation of Gaussian pulses

distributed in frequency [distribution given by Eq. (9.18)] is approximately

identical to the autocorrelation of a simple Gaussian pulse [perform the

integration of Eq. (9.19), and compare to Eq. (9.17)].

2. Consider a Gaussian pulse of 50-fs duration at 800 nm, with an upchirp cor-

responding to a = 1. 5. Determine analytically the result of a measurement

using the cross-correlator in which a block of BK7 glass has been inserted

in one arm. Calculate the amount of glass required for pulse broadening by

a factor 5, after double passage through the glass. Calculate the envelopes

A

1

(τ), A

2

(τ), and A

3

(τ) that will be obtained by measuring a second-order

cross-correlation, cf. Eq. (9.7). Write the expressions corresponding to the

various steps of the procedure leading to the reconstruction of the original

pulse following Section 9.2.2.2.

3. Derive Eq. (9.12), the expression for the third-order interferometric cor-

relation. Determine the peak to background ratio of the fringe resolved

autocorrelation and of the intensity autocorrelation. Assume equal pulses,

E

1

= E

2

.

4. Let us assume that the spectral phase of a pulse is given by () =

φ

2

( −ω



)

2

+φ

3

( −ω



)

3

and the complex field is to be retrieved from

the measurement of the pulse spectrum and (a) an amplitude unbalanced