Page 385 - Phase Space Optics Fundamentals and Applications

366 Chapter Eleven

the pulse, although the properties of the spectrogram are identically
impacted by the chirp characteristics of E and g.
Type II devices are also common in ultrafast optics. Implementation
of a stationary amplitude filter can be done using filters such as a
slit in a zero-dispersion line. A high-resolution time-nonstationary
amplitude filter is more difficult to implement since the pulse under
test is usually the shortest event available in the laboratory, and that
filter should in theory be significantly shorter than that to ensure that
the experimental trace is the equivalent of Eq. (11.69) in the frequency
domain. Photodetection has been used for pulses in the picosecond
range, 56,57 and nonlinear cross-correlation with the pulse under test
has been used for shorter pulses. 58–60
Reconstruction of the electric field of the pulse under test from the
experimental trace of Eq. (11.69) can, in principle, be performed di-
rectly by deconvolution of the Wigner function of the pulse using the
Wigner function of the gate. However, this requires a good knowl-
A
edge of the function N (or equivalently, its Wigner function), which
is somewhat available for linear techniques but not available for non-
linear techniques with unknown gate pulses. Estimates of the group
delay or instantaneous frequency in the pulse can also be obtained
in some cases, using the properties of the spectrogram. 30 For a type I
A
device with a narrow function N , the weighted average time as a
function of frequency obtained using the spectrogram as the weight
function is the group delay of the unknown pulse as a function of
frequency. The practical use of this property, which is valid when the
gating function is significantly shorter than the pulse under test, is
hindered by the fact that the precision on the determination of the
group delay can be poor since the width of the spectrogram increases
dramatically in these conditions. The most practical approach to signal
reconstruction for type I and type II devices is based on iterative phase
retrieval. Electric field reconstruction from Eq. (11.69) is equivalent

to phase reconstruction of the two-dimensional quantity dt E in ×
A A
(t)N (t − ) exp(i C t) from its measured modulus | dt E in (t)N ×
(t − ) exp(i C t)|. Projections between ensembles of electric fields
matching different constraints allow, in most cases, convergence to
A
one possible solution of Eq. (11.69), whether the function N is known,
unknown, or a function of the unknown electric field itself. The princi-
pal component generalized projections algorithm can be used to invert
experimental trace obtained with type I and type II devices. 60,61

11.3.3.2 Tomographic Techniques
As with spectrographic methods, the so-called tomographic tech-
niques require in-series, time-stationary, and time-nonstationary fil-
ters so that the entire phase space can be explored. However, unlike
spectrographictechniques,thefirstfilterinatomographicapparatusis