Page 373 - Phase Space Optics Fundamentals and Applications

354 Chapter Eleven

Then the parameters describing the filter form a temporal transfer
matrix

a b
T = (11.44)
c d
which is again unimodular [det(T) = ad − bc = 1]. Define also a
column vector

t
= (11.45)

which describes time of occurrence and frequency of a temporal “ray.”
The properties of the pulse are described by a “bundle” of such rays,
which can be represented in the chronocyclic phase space to interpret
the effects of various linear filters. As in the case of the geometrical
optical rays, the output and input temporal rays are related by

out = T in (11.46)
Optical elements entirely analogous to free-space propagation, and
imaging may also be defined. The kernel for propagation in a dis-
persive medium, i.e., with a frequency-dependent index of refrac-
tion, or for double-passing a two-grating compressor as represented
in Fig. 11.5a,is

˜ ˜ H( , − ) = exp i 2 ( − ) (11.47)

2

in the limit where only second-order dispersion is considered in the

development of the introduced spectral phase . The corresponding
transfer matrix is

1
T dispersion = (11.48)
0 1
A pulse with such linear chirp is represented by a collection of ray
vectors in which t is a linear function of
$ %

t i = t + i
{ i }= (11.49)
i
This manifests itself in the phase-space representation of the ray
bundle by a slope related to (Fig. 11.5b).

Transfer through a quadratic temporal phase modulator (Fig. 11.5c)
is the temporal analog of the action of a paraxial lens on a ray. The
corresponding kernel is

t 2

H(t, t ) = exp i t (t − t ) (11.50)
2