Page 378 - Phase Space Optics Fundamentals and Applications

Phase Space in Ultrafast Optics 359

It turns out to be necessary to consider only two classes 47 of linear
filter: time-stationary, in which the time of incidence of the input pulse
does not affect the output, and frequency stationary, the output of
which is unchanged by arbitrary frequency shifts of the input. A linear
filterofarbitraryresponsefunctionmaybesynthesizedfromthesetwo
classes. Moreover, they are the only classes of filter that have been used
to date in pulse shape measurement and are the easiest to implement
in practice. For a time-stationary filter, the output field is related to
the input field by

E out (t) = dt S(t − t )E in (t ) (11.57)

where the filter response function H(t, t ) is a function only of the dif-
ference in its arguments t − t . A frequency-stationary filter is defined

in an analogous manner in the spectral domain

˜ E out ( ) = d ˜ N( − ) ˜ E in ( ) (11.58)
where the filter transfer function ˜ N( − ) is a function only of the

difference in its arguments and the tilde represents a Fourier trans-
form. Frequency-stationary filters are time-nonstationary, since their
output depends on the time at which the pulse arrives at the input.
We use S and ˜ S, and N and ˜ N, to denote the response functions and
transfer functions of time-stationary and time-nonstationary filters,
respectively.
There are two further important filter specializations: amplitude-
only and phase-only. These filters behave as their names suggest; the
former provides amplitude modulation while the later modulates
only the phase. We distinguish amplitude-only and phase-only filters
with the superscripts Aand P, respectively. To be specific, we identify
six filters to be used in this analysis and their corresponding response
or transfer functions.
Time gate:
2 2
− (t − )
A
N (t; ) = exp (11.59)
2
Quadratic temporal phase modulator:
2
i t
t
P

N (t; ) = exp (11.60)
Q t
2
Linear temporal phase modulator or frequency shifter:

P

N (t; ) = exp i t (11.61)
t
L
t