Page 365 - Phase Space Optics Fundamentals and Applications

346 Chapter Eleven

(i.e., the spectral phase is at most a linear function of the optical fre-
quency), indicating no correlation between time and frequency local-
ization. Figure 11.1b is the Wigner function of a pulse with the same
spectrum and a parabolic spectral phase, i.e., a linear chirp. This leads
to stretching of the pulse in the time domain, as indicated by the
temporal marginal. Correlation between the temporal and spectral
components of the pulse can be inferred from the slope of the Wigner
function in the chronocyclic space. The amount of such chirp may
be quantified using the Wigner distribution in a way that respects
Fourier’s theorem, which may be thought to preclude the simulta-
neous specification of time and frequency. The instantaneous frequency
(t) may be defined as the instantaneous mean value of the frequency
of the distribution

d W(t, )
(t) = (11.17)
d W(t, )
This can be evaluated by using integration by parts, making use of
the compact support of the pulse field in the formula for the Wigner
function. This leads to the relation

d t

(t) =− (t) =− (t) (11.18)
t
dt
which embodies the intuitive result that the instantaneous frequency
is the temporal derivative of the temporal phase. The group delay can
be calculated similarly as

dt tW(t, ) d

T( ) = = ( ) = ( ) (11.19)
dt W(t, ) d
which corresponds to the common interpretation of the group delay.
The Wigner function also has some less intuitive features. For exam-
ple, it is tempting to consider W as a joint probability distribution of
the time at which different frequencies occur within the pulse ensem-
ble. But since W is not a positive definite function, it cannot play the
role of a probability distribution. Negativity of the Wigner function is
a common phenomenon. For example, Fig. 11.1c displays the Wigner
function of a pair of temporally delayed identical pulses. Interference
between the two pulses is indicated at the center of the chronocyclic
space by an alternation of positive and negative regions of the Wigner
function.Notethatthefrequencymarginal,i.e.,theopticalspectrum,is
positive, since negative regions of the cross-terms of the Wigner func-
tion at the center of the chronocyclic space are canceled by the positive
Wigner functions of each individual pulse. Finally, the Wigner func-
tion of a pulse with a third-order spectral phase is plotted in Fig. 11.1d.
This Wigner function also takes negative values, but its shape remains