Page 368 - Phase Space Optics Fundamentals and Applications

Phase Space in Ultrafast Optics 349

(s = 0, Fig. 11.2a) to the spectrogram (s = 2, Fig. 11.2c) and is positive
for values of s larger than 2. As s increases, the Wigner function is
smoothed out, but the resulting time-frequency distribution loses its
ability to display chirp.
In fact, it is possible to cast all measurement strategies in terms of
phase-space distributions, in a form that ensures the positivity of the
signal distribution. The requirement that the signal be positive arises
from the way in which optical detectors respond to the field. They
are square-law detectors in which the intensity or energy (depending
on the response time of the detector) gives rise to a photocurrent or
charge. This implies that the detected signal is positive. As we have
seen, not all phase-space distributions are positive, but a theorem due
to Jordan shows that the distribution

S(T, ) = d dt W(t, )W M (t − T, − ) (11.26)

with W M (t, ) the Wigner representation of the measurement appa-
ratus, is always positive. Apparatuses for which the measurement
function is a convolution, no matter how complex and negative the
Wigner function of the pulse and the measurement apparatus are,
yield a signal that is positive.
In general the measurement function may be written in the form

S(T, ) = d dt W(t, )W M (t, ; ,T) (11.27)
where the measurement Wigner function contains the apparatus pa-
rameters. This cannot always be written as a convolution, as in
Eq. (11.26), but nonetheless is always constrained to give a positive
signal. Further, it is clear from these formulas that any measurement
technique attempting to reconstruct either the Wigner or the ambigu-
ity function must be capable of exploring the entire two-dimensional
chronocyclic phase space.

11.2.4 Phase-Space Representation
of Paraxial Optical Systems
The representation of optical pulses in phase space can be understood
by analogy to the representation of optical ray trajectories in geometri-
cal optics. 34 This provides an important first-order design framework
for ultrafast optical systems as well as an intuitive appreciation of the
more formal representations of the chronocyclic distributions. Parax-
ial optical systems are specified by a 2 × 2 real transfer matrix T,

A B
T = (11.28)
C D