Page 372 - Phase Space Optics Fundamentals and Applications

Phase Space in Ultrafast Optics 353

An important feature of the paraxial approximation is that it is
straightforward to propagate Gaussian beams using the transfer ma-
trices acting on the complex beam parameter

1 1 i
= − (11.38)
q R w 2
In Eq. (11.38), R is the radius of curvature of the beam at the reference
plane, w is the corresponding beam size, and is the index-dependent
wavelength in the medium. The complex beam parameters before and
after propagation, q and q , respectively, are linked by the formula

q Aq/n + B
= (11.39)
n Cq/n + D
where n and n are the optical index in the medium before and

after propagation, respectively. The elements of the transfer matrix
of Eq. (11.28) are then interpreted as modifying the beam waist and
radius of curvature, accordingly.

11.2.5 Temporal Paraxiality and the
Chronocyclic Phase Space
An optical pulse may be represented in a manner similar to optical
rays in geometrical optics. The analogy between space and time, the
space-time duality, has been very fruitful. 36−38 Consider the action of
a linear filter on a pulsed field. The relationship between input and
output fields for the filter is

E out (t) = dt H(t, t )E in (t ) (11.40)
which can also be written in the frequency domain as

˜

˜ E out ( ) = d ˜ H( , − ) ˜ E in ( ) (11.41)
Let us define a temporally paraxial approximation and postulate a
general linear filter function in the form of a temporal Fresnel kernel

1 i
2 2
H(t, t ) = √ exp − (at − 2tt + dt ) (11.42)
2 b 2b
where a, b, and d are real numbers. Here H is unitary and verifies

dt H(t, t )H (t, t”) = (t − t”) (11.43)
∗