494 Measurement Techniques of Femtosecond Spectroscopy
if the gating function I
g
(t) is known. Indeed, if I() is the Fourier transform of
the gate function I
g
(t), and S() is the Fourier transform of the measured signal
S(τ
d
), the Fourier transform f () of the physical quantity f (t) is just the ratio:
f () =
S()
I()
. (10.2)
The physical quantity f (t) can be calculated by taking the inverse Fourier
transform of Eq. (10.2). This deconvolution technique can be applied in numer-
ous cases where the gate function I
g
(t) does not depend on the phase of the
interaction.
1
10.3. BEAM GEOMETRY AND TEMPORAL
RESOLUTION
To obtain a better quantitative understanding of the influence of the beam
geometry on the temporal resolution, let us analyze a pump–probe experiment
as sketched in Figure 10.3. The pump pulse creates a small change of the trans-
mission coefficient, a(x, y, z, t), which is sampled by the time-delayed test
Test
Pump
Sample
A
B
D
PD
x
y
d
I
t
I
p
2
d
Figure 10.3 Schematic diagram of a pump–probe transmission experiment in noncollinear geom-
etry. The line AB shows the position of the pump pulse maximum at t = 0. Refraction at the sample
interfaces has been neglected.
1
The gate depends on the phase of the interaction in the case of coherent interaction treated in
Chapter 4. In that case the measured signal cannot be described by the simple expression (10.1).
Beam Geometry and Temporal Resolution 495
pulse of intensity I
t
(t). The signal measured by the detector PD as function of
the delay τ
d
can be written as the sum of the transmitted test pulse energy W
t0
in the absence of a pump and a pump-induced change W
t
(τ
d
):
W
t
(τ
d
) = W
t0
+ W
t
(τ
d
)
∝
∞
−∞
dt
dx dy dz
[
1 + a
o
+ a(x , y, z, t)
]
I
t
(x, y, z, t − τ
d
)
(10.3)
where a
o
is the transmission coefficient in the absence of the pump and |a
0
|1
has been assumed: e
a
= 1+a
0
+a. In the overlapping volume of the two beams,
a complex mixing of spatial and temporal effects occurs. We want to derive
conditions under which the excitation geometry does not affect substantially the
outcome of the experiment. For simplicity, the beam profiles of pump and test
pulse are assumed to be uniform and of rectangular shape, the temporal profiles
are Gaussian of equal FWHM τ
p
, and the overlapping region is symmetric with
respect to the sample center. The time axis is chosen so that the pump pulse
maximum reaches the origin of the coordinate system at t = 0, and the test pulse
reaches the origin at t = τ
d
. The sample response is assumed to follow the pump
pulse instantaneously, a ∝ I
p
(t), and we expect a signal W
t
(τ
d
) resembling
the pulse autocorrelation in the absence of geometrical effects. An increase of
the correlation FWHM is then a measure of the loss in temporal resolution of
any pump–probe experiment because of geometrical effects.
In the following considerations we will omit constants for the sake of brevity.
The delay-dependent part of the measured signal is
W
t
(τ
d
) ∝
∞
−∞
dt
dx
dyI
t
(x, y, t − τ
d
)I
p
(x, y, t) (10.4)
where we have already carried out the z-integration yielding a constant. The
pulses propagate through the sample with the group velocity ν
g
. Lines of
constant intensity (parallel to AB in Fig. 10.3) obey the equation
(x − ν
g
t sin α) =−(y − ν
g
t cos α)
cos α
sin α
(10.5)
for the pump pulse and
6
x + ν
g
(t − τ
d
)sinα
7
=
6
y − ν
g
(t − τ
d
) cos α
7
cos α
sin α
(10.6)
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