Page 366 - Phase Space Optics Fundamentals and Applications

Phase Space in Ultrafast Optics 347

indicative of the group delay of the pulse, i.e., a parabolic function of
the optical frequency.
There exist an infinite number of time-frequency distributions that
can potentially be used to represent a signal in the chronocyclic space.
One approach to obtain bilinear time-frequency distributions is to use
a signal-independent kernel to generate the set of functions 30

1 t t
R(t, ) = 3/2 d tdu d E u + E ∗ u −
(2 ) 2 2
× K( , t) exp(i t + i t − i u) (11.20)
For example, K uniformly equal to 1 leads to the Wigner function,
while K equaltoexp(−i | t|) leadstothePagedistributionwhichhas
also been used in the context of representing linear optical systems. 31
In the coherent case, if K is chosen as the ambiguity function A g of
an ancillary signal g as defined by Eq. (11.14), the time-frequency
distribution of Eq. (11.20) becomes

2
1
R g (t, ) = √ du E(u)g(u − t) exp(i u) (11.21)

2

Equation (11.21) indicates that R g (t, ) can be interpreted as the opti-
cal spectrum of the field after gating by the function g, represented as a
function of the optical frequency and the relative delay t between the
pulse and the gate. This representation is known as a (Gabor) spectro-
gram. This particular time-frequency distribution is evidently positive
and can be measured directly by applying a time gate g to the opti-
cal test pulse and measuring the resulting spectrum. The marginals
of the spectrogram are convolutions of the corresponding intensity
of the test pulse with the intensity of the ancillary function. There-
fore the marginals are not equal to the temporal and spectral inten-
sities of the pulse. It is interesting to note that Eq. (11.21) can also be
written as

1
R g (t, ) = du d W E (u, )W g (u − t, − ) (11.22)
2
which is valid regardless of the coherence of the ensemble of optical
pulses. This indicates that the spectrogram is obtained by convolution
of the Wigner function of the pulse with the Wigner function of the
gate in the chronocyclic space, and that its measurement requires a
full scan of the chronocyclic space with the Wigner function of the
gate. There exist other bilinear chronocyclic representations of pulses
that are positive definite, and therefore may be used as joint prob-
ability distributions. However, these do not have the property that
their marginal distributions are the temporal and spectral intensity.
Phase-space representations are not limited to the bilinear functions