Page 376 - Phase Space Optics Fundamentals and Applications

Phase Space in Ultrafast Optics 357

properties of the beam with the spatial and angular properties:
⎛ ⎞
A B
⎜ C D ⎟
⎜
⎟
T = ⎜ ⎟ (11.55)
⎝ a b⎠
c d
This matrix acts on the ray representation (x, k, t, ) in the (space,
wave vector, time, frequency) space. The 2 × 2 block diagonal ele-
ments are the spatial and temporal transfer matrices described above,
and the off-diagonal elements are those involved in space-time cou-
pling.Forexample,thematrixdescribingadiffractiongratingorprism
introducing angular dispersion (linear relation between wave vector
andopticalfrequency)hasanonzero .Arestrictedformofthismatrix,
where 7 of the 16 variables of the matrix of Eq. (11.55) are constrained,
has been applied to time-stationary filters by Kostenbauder. 44 Space-
time Wigner functions have been used, e.g., to describe the space-time
coupling induced by zero-dispersion line pulse shapers. 45,46

11.3 Metrology of Short Optical Pulses

11.3.1 Measurement Strategies
In principle, one can use an antenna to directly measure the oscillat-
ing electric field. For instance, low-temperature GaAs gated antennas
are routinely used to measure the oscillating field of electromagnetic
pulses whose carrier frequencies are on the order of 1 THz. But the
fastest antennas are far too slow to resolve the oscillations of optical
fields (the period of one cycle of an optical field in the visible spec-
tram is less than 3 fs). Detectors for the optical regime are square-law
(or energy) detectors which only respond to the intensity of the field.
State-of-the-art commercial photodiodes have response times on the
order of 10 ps, while streak cameras, by far the fastest electronic detec-
tion devices, offer a temporal resolution of about 1 ps. Herein lies the
problem of ultrashort pulse characterization: it is not possible to di-
rectly measure the temporal intensity of optical pulses with durations
less than 1 ps or so. The problem is especially acute for the few-cycle
optical regime and the XUV attosecond regime. Most conventional
photodetection schemes are also not sensitive to the phase of the elec-
tric field. These problems may be circumvented, however, by passing
the unknown (test) pulses through filters of known response func-
tions and then recording the average output energy as a function of
the parameters characterizing the filter response functions. As a gen-
eral proposition, pulse measurement techniques may be categorized