526 Measurement Techniques of Femtosecond Spectroscopy
Simple Fourier transformation of this equation leads to a solution for the
amplitude x(ω). Taking the inverse Fourier transform of that solution yields x(t):
x(t) =
F
0
2π
j
I
j
(10.44)
with
I
j
=
∞
−∞
e
iω(t−jT)
(K −mω
2
) + ibω
dω. (10.45)
The integrand I
j
in Eq. (10.45) has two poles at ω = i ± ω
21
, and ω
2
21
=
K/m −
2
and = b/2m. The stationary solution for the oscillator is found by
contour integration and summation over j of the geometric series:
x(t) =
1
2
A(T )e
−t +iω
21
t
+ c. c. (10.46)
with
A(T ) =
iF
0
ω
21
1
1 − e
T +iω
21
T
=
iF
0
ω
21
1 − e
T −iω
21
T
1 + e
2T
− 2e
T
cos ω
21
T
. (10.47)
A(T ) is essentially the amplitude of the first cycle of oscillation. Its value is
maximum and equal to iF
0
/[ω
21
(1 − e
T
)] when ω
21
T = 2nπ, and minimum,
equal to iF
0
/[ω
21
(1 + e
T
)] for ω
21
T = 2(n + 1)π. The modulation depth
(1 − e
T
)/(1 +e
T
) is thus determined solely by the damping rate and the period
of the driving force T. When driving a system at a subharmonic of the resonant
frequency, the term ω
21
T in Eq. (10.47) can be large (ω
21
T = 2nπ, with n a
large integer). The resonances (values of the periodicity T that satisfy the res-
onance condition) are closely spaced. The damping factor determines which
subharmonic N can still be used to drive effectively the resonance ω
21
. Each
δ function force sets off an oscillation, which should not be completely damped
before being reinforced by the next exciting pulse.
10.11. SELF-ACTION EXPERIMENTS
Pump–probe experiments are intended to provide information on linear and
nonlinear properties of matter. As noted previously, there is a fundamental
Self-Action Experiments 527
~
ε
r
()
~
ε
i
()
~
ε
t
()
z
Figure 10.23 For linear systems, and some simple nonlinear systems, the complex susceptibility
can be completely determined from single pulse transmission (reflection) measurements, provided the
amplitude and phase of the incident, transmitted, and reflected signals can be completely determined.
temporal limitation: For the measurement interpretation the pump or excitation
process should be completed before the medium is probed. One could try to
obtain information on the properties of matter by measuring the time resolved
fields of a single pulse incident, reflected and/or transmitted by a thin sample
(Figure 10.23), using some of the techniques outlined in Chapter 9.
In the case of a linear interaction with the medium, the problem is analogous to
the analysis of a linear circuit. For instance, referring to Chapter 1 [Eqs. (1.73)
through (1.79)], the complex dielectric constant ˜() =
0
[1 +˜χ()] can be
extracted by taking the Fourier transform
˜
E() of the incident (i) and transmitted
(t) fields:
1 +˜χ() =−
c
2
z
2
2
ln
2
˜
E
t
()
˜
E
i
()
(10.48)
where z is the sample thickness.
There is no simple algorithm that can solve the general problem of
retrieval of a nonlinear susceptibility χ
(n)
() from a series of measurements
of incident, transmitted, and reflected fields. Some assumptions have to be
made—for instance, that all nonlinear susceptibilities except the third order,
χ
(3)
, can be neglected. Within this approximation, measurement of the third
harmonic transmitted field
˜
E
3ω
leads to a determination of the third-order
susceptibility:
χ
(3)
() =
2cz
ω
˜
E
3ω
()
˜
E
2
()
. (10.49)
The transmission measurements provide information on the bulk properties
of the sample. Properties at the surface can be analyzed by measuring the
reflected field. For instance, at normal incidence, the reflection coefficient is
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