102 Femtosecond Optics
where sin α = 0 and cos α = 1 if we take the derivatives at the center frequency of
the pulse, = ω
. The quantity (dα/d)|
ω
, responsible for angular dispersion,
is a characteristic of the actual optical device to be considered. It is interesting to
note that the dispersion parameter is always negative independently of the sign
of dα/d and that the dispersion increases with increasing distance L from the
diffraction point. Therefore angular dispersion always results in negative GVD.
Differentiation of Eq. (2.76) results in the next higher dispersion order
d
3
d
3
ω
=−
L
c
cos α
3
dα
d
2
+ 3
dα
d
d
2
α
d
2
+sin α
3
d
2
α
d
2
+
d
3
α
d
3
−
dα
d
3
ω
≈−
3L
c
dα
d
2
+
dα
d
d
2
α
d
2
ω
, (2.77)
where the last expression is a result of α(ω
) = 0.
The most widely used optical devices for angular dispersion are prisms and
gratings. To determine the dispersion introduced by them we need to specify not
only the quantity α() in the expressions derived previously, but also the optical
surfaces between which the path is being calculated. Indeed, we have assumed
in the previous calculation that the beam started as a plane wave (plane reference
surface normal to the initial beam) and terminates in a plane normal to the ray at
a reference optical frequency ω
. The choice of that terminal plane is as arbitrary
as that of the reference frequency ω
(cf. Section 1.1.1). After some propagation
distance, the various spectral component of the pulse will have separated, and a
finite size detector will only record a portion of the pulse spectrum.
Therefore, the “dispersion” of an element has only meaning in the context of
a particular application, which will associate reference surfaces to that element.
This is the case when an element is associated with a cavity, as will be considered
in the next section. In the following sections, we will consider combinations of
elements of which the angular dispersion is compensated. In that case, a natural
reference surface is the normal to the beam.
2.5.4. GVD of a Cavity Containing a Single Prism
Dispersion control is an important aspect in the development of fs sources.
The most elementary laser cavity as sketched in Figure 2.23 has an element
with angular dispersion. The dispersive element could be the Brewster angle
Elements with Angular Dispersion 103
L
3
L
g
A
h
B
(R)
C
Figure 2.23 Example of a cavity with a single right angle prism. The side of the right angle is
an end mirror of the cavity. The cavity is terminated by a curved mirror of radius of curvature R,
at a distance L from the Brewster angle exit face of the prism. Stability of the cavity requires that
L +
AB/n < R. Translation of the prism allows for an adjustment of the pathlength in glass L
g
. The
inset shows that this calculation applies to a symmetric cavity with a Brewster angle laser rod and
two spherical mirrors.
laser rod itself. The cavity will be typically terminated by a curved mirror.
The two reference surfaces to consider are the two end mirrors of the cavity.
We have seen that negative GVD is typically associated with angular dispersion,
and positive GVD with the propagation through a glass prism or laser rod.
9
One
might therefore expect to be able to tune the GVD in the arrangement of Fig. 2.23
from a negative to a positive value. An exact calculation of the frequency depen-
dence presented shows that this is not the case, and that the GVD of this cavity
is always positive.
A combination of elements with a tunable positive dispersion can also be
desirable in a fs laser cavity. We will consider the case of the linear cavity
sketched in Fig. 2.23, whose GVD can be determined analytically.
The cavity is terminated on one end by the plane face of the prism, on the other
end by a spherical mirror of curvature R. The prism–mirror distance measured
at the central frequency ω
is L. The beam originates from a distance h from the
9
It is generally the case—but not always—that optical elements in the visible have positive GVD.
Get Ultrashort Laser Pulse Phenomena, 2nd Edition now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
