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Phase Space in Ultrafast Optics 343

The two-frequency correlation function is linked to the two-time
correlation function of the ensemble by a double Fourier transform

7 8

˜ ˜ C( , ) = ˜ E + ˜ E ∗ −
2 2

1
= dt d tC(t, t) exp[i(t + t )] (11.12)
2
The center-frequency and difference-frequency coordinates in Eq.
(11.12) are given by = ( 1 + 2 )/2 and = 1 − 2 , respectively.
Similar arguments to those mentioned for the two-time-correlation
function apply to the two-frequency correlation function.
For a coherent train of pulses, Eqs. (11.10) and (11.11), and their
equivalent in the frequency domain, indicate that the time or fre-
quency representation of the analytic signal can be reconstructed from
a single line of the corresponding correlation function. Therefore, if the
ensemble is assumed a priori to be coherent, the amount of collected
data can be greatly reduced. This is a luxury afforded only to those
measurement techniques that directly measure one of the correlation
functions.

11.2.3 The Time-Frequency Phase Space
Time-frequency distributions are central to the characterization of
pulses in the optical domain, since they are straightforwardly related
to the measured data. In optics, direct measurement of the wave-
form is not possible. This is in contrast to the more usual application
of the distributions in signal processing, where they are commonly
used as mathematical tools for signal representation. It is frequently
useful to work with a representation of the correlation functions in the
chronocyclic phase space. The intuitive concept of chirp (that is, time-
dependent frequency in the pulse) can be most easily seen within
this space. The pulse ensemble may also be represented within the
chronocyclic phase spaces defined by the complementary variables
(t, ) and ( , t). The chronocyclic Wigner function W(t, ) and
ambiguity or Wigner characteristic function A( , t) provide two
particularly useful descriptions of the pulse train statistics in these
spaces. The relationship between the various representations of the
correlation function has been discussed in the context of spatially
localized fields, 13 and the Wigner function was originally applied
to problems in ultrafast optics. 14–16 Examples of applications of the
Wigner function, ambiguity function, and other time-frequency dis-
tributions in ultrafast optics can be found in Refs. 17 to 29. General
properties of the Wigner and ambiguity functions can, for example,
be found in Ref. 30.