Elements with Angular Dispersion 95
As discussed by Bor [24], the prism introduces a tilt of the pulse front with
respect to the phase front. As in lenses, the physical origin of this tilt is the
difference between group and phase velocity. According to Fermat’s principle
the prism transforms a phase front
AB into a phase front A
B
. The transit times
for the phase and pulse fronts along the marginal ray
BOB
are equal (ν
p
∼ ν
g
in air). In contrast the pulse is delayed with respect to the phase in any part of
the ray that travels through a certain amount of glass. This leads to an increasing
delay across the beam characterized by a certain tilt angle α. The maximum
arrival time difference in a plane perpendicular to the propagation direction is
(D
/c)tanα.
Before discussing pulse front tilt more thoroughly, let us briefly mention
another possible prism arrangement where the above condition for L
p
is not nec-
essary. Let us consider for example the symmetrical arrangement of four prisms
sketched in Figure 2.18. During their path through the prism sequence, different
spectral components travel through different optical distances. At the output of
the fourth prism all these components are again equally distributed in one beam.
The net effect of the four prisms is to introduce a certain amount of GVD lead-
ing to broadening of an unchirped input pulse. We will see later in this chapter
that this particular GVD can be interpreted as a result of angular dispersion and
can have a sign opposite to that of the GVD introduced by the glass material
constituting the prisms.
2.5.2. Tilting of Pulse Fronts
In an isotropic material the direction of energy flow—usually identified as ray
direction—is always orthogonal to the surfaces of constant phase (wave fronts)of
the corresponding propagating wave. In the case of a beam consisting of ultrashort
light pulses, one has to consider in addition planes of constant intensity (pulse
fronts). For most applications it is desirable that these pulse fronts be parallel to
Figure 2.18 Pulse broadening in a four prism sequence.
96 Femtosecond Optics
the phase fronts and thus orthogonal to the propagation direction. In the section
on focusing elements we have already seen how lenses cause a radially dependent
difference between pulse and phase fronts. This leads to a temporal broadening
of the intensity distribution in the focal plane. There are a number of other optical
components that introduce a tilt of the pulse front with respect to the phase front
and to the normal of the propagation direction, respectively. One example was
the prism discussed in the introduction of this section. As a general rule, the
pulse front tilting should be avoided whenever an optimum focalization of the
pulse energy is sought. There are situations where the pulse front tilt is desirable
to transfer a temporal delay to a transverse coordinate. Applications exploiting
this property of the pulse front tilt are pulse diagnostics (Chapter 9) and traveling
wave amplification (Chapter 7).
The general approach for tilting pulse fronts is to introduce an optical element
in the beam path, which retards the pulse fronts as a function of a coordinate
transverse to the beam direction. This is schematically shown in Figure 2.19 for
an element that changes only the propagation direction of a (plane) wave. Let us
assume that a wavefront
AB is transformed into a wavefront A
B
. From Fermat’s
principle it follows that the optical pathlength P
OL
between corresponding points
at the wavefronts
AB and A
B
must be equal:
P
OL
(BB
) = P
OL
(PP
) = P
OL
(AA
). (2.57)
Because the optical pathlength corresponds to a phase change = 2πP
OL
/λ,
the propagation time of the wavefronts can be written as
T
phase
=
ω
(2.58)
B
P
A
B
P
A
Figure 2.19 Delay of the pulse front with respect to the phase front.
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