130 Femtosecond Optics
interfering completely. As a result, the actual shortening factor is
√
1 + u
2
times
smaller than the theoretical one, as can be seen from the first exponential function.
The influence of such a filter can be decreased by using a large beam size.
A measure of this frequency filter, i.e., the magnitude of the quantity (1 + u
2
),
can be derived from the second exponent. Obviously the quantity (1 + u
2
)is
responsible for a certain ellipticity of the output beam which can be measured.
2.7. OPTICAL MATRICES FOR DISPERSIVE
SYSTEMS
In Chapter 1 we pointed out the similarities between Gaussian beam propa-
gation and pulse propagation. Even though this fact has been known for many
years [28, 42], it was only recently that optical matrices have been introduced
to describe pulse propagation through dispersive systems [43–47] in analogy to
optical ray matrices. The advantage of such an approach is that the propagation
through a sequence of optical elements can be described using matrix algebra.
Dijaili [46] defined a 2 × 2 matrix for dispersive elements which relates the
complex pulse parameters (cf. Table 1.2) of input and output pulse, ˜p and ˜p
,to
each other. Döpel [43] and Martinez [45] used 3 × 3 matrices to describe the
interplay between spatial (diffraction) and temporal (dispersion) mechanisms in
a variety of optical elements, such as prisms, gratings and lenses, and in com-
binations of them. The advantage of this method is the possibility to analyze
complicated optical systems such as femtosecond laser cavities with respect to
their dispersion—a task of increasing importance, as attempts are being made
to propagate ultrashort pulses near the bandwidth limit through complex optical
systems. The analysis is difficult since the matrix elements contain information
pertaining to both the optical system and of pulse.
One of the most comprehensive approaches to describe ray and pulse charac-
teristics in optical elements by means of matrices is that of Kostenbauder [47].
He defined 4 × 4 matrices which connect the input and output ray and pulse
coordinates to each other. As in ray optics, all information about the optical
system is carried in the matrix while the spatial and temporal characteristics of
the pulse are represented in a ray–pulse vector (x, , t, v). Its components
are defined by position x, slope , time t, and frequency v. These coordinates
have to be understood as difference quantities with respect to the coordinates of
a reference pulse. The spatial coordinates are similar to those known from the
ordinary
AB
CD
ray matrices. However, the origin of the coordinate system
is defined now by the path of a diffraction limited reference beam at the aver-
age pulse frequency. This reference pulse has a well-defined arrival time at any
Optical Matrices for Dispersive Systems 131
reference plane; the coordinate t, for example, is the difference in arrival time
of the pulse under investigation. In terms of such coordinates and using a 4 × 4
matrix, the action of an optical element can be written as
⎛
⎜
⎜
⎝
x
t
v
⎞
⎟
⎟
⎠
out
=
⎛
⎜
⎜
⎝
AB0 E
CD0 F
GH1 I
0001
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
x
t
v
⎞
⎟
⎟
⎠
in
(2.148)
where A, B, C, D are the components of the ray matrix and the additional
elements are
E =
∂x
out
∂v
in
, F =
∂
out
∂v
in
, G =
∂t
out
∂x
in
, H =
∂t
out
∂
in
, I =
∂t
out
∂v
in
.
(2.149)
The physical meaning of these matrix elements is illustrated by a few examples
of elementary elements in Figure 2.33. The occurrence of the zero elements
can easily be explained using simple physical arguments, namely (a) the center
frequency must not change in a linear (time invariant) element and (b) the ray
properties must not depend on t
in
. It can be shown that only six elements are
independent of each other and therefore three additional relations between the
nine nonzero matrix elements exists [47]. They can be written as
AD − BC = 1
BF − ED = λ
H (2.150)
AF − EC = λ
G.
Using the known ray matrices [48] and Eq. (2.149), the ray–pulse matrices for
a variety of optical systems can be calculated. Some example are shown in
Table 2.3.
A system matrix can be constructed as the ordered product of matrices corre-
sponding to the elementary operations (as in the example of the prism constructed
from the product of two interfaces and a propagation in glass). An important fea-
ture of a system of dispersive elements is the frequency dependent optical beam
path P, and the corresponding phase delay . This information is sufficient
for geometries that do not introduce a change in the beam parameters. Exam-
ples which have been discussed in this respect are four-prism and four-grating
sequences illuminated by a well-collimated beam.
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