Page 363 - Phase Space Optics Fundamentals and Applications

344 Chapter Eleven

The Wigner function is obtained by taking the one-dimensional
Fourier transform of C(t, t) over the time-difference coordinate

1
W(t, ) = √ d tC(t, t) exp(i t)
2
7 8
1 t t
= √ d t E t + E ∗ t − exp(i t)
2 2 2
7 8
1
= √ d ˜ E + ˜ E ∗ − exp(−i t)
2 2 2
(11.13)
The ambiguity function is obtained from C(t, t) by performing
the Fourier transform over the average-time coordinate

1
A( , t) = √ dt C(t, t) exp(i t)
2
7 8
1 t t
= √ dt E t + E ∗ t − exp(i t)
2 2 2
7 8
1
˜
= √ d E + ˜ E ∗ − exp(−i t)
2 2 2
(11.14)
These representations are uniquely and invertibly related to one
another by Fourier transformations.
The Wigner function has some features that make it useful in repre-
senting short optical pulses. For example, in contrast to the field and
correlation representations, it is real-valued. Moreover, its time and
frequency marginals (i.e., projections on the corresponding axis) are
the temporal and spectral intensity, respectively. The average time-
dependent intensity is obtained from the two-time correlation func-
tion by setting t = 0. This corresponds to a projection of the Wigner
function onto the time axis, or the Fourier transform of the t = 0
section of the ambiguity function

1
I (t) = C(t, 0) = √ d W(t, )
2

1
= √ d A( , 0) exp(−i t) (11.15)
2
Furthermore, the average pulse spectral intensity is obtained from the
two-frequency correlation function by setting = 0, by projecting