506 Measurement Techniques of Femtosecond Spectroscopy
efficiency experienced by the probe pulse and measured as function of τ
1
contain
information on T
2
.
The actual data evaluation in a transient grating experiment can be complex
and requires a detailed model of the processes involved. An example of determi-
nation of phase relaxation times using collinear counter-propagating pump pulses
is detailed in the next subsection.
10.6.2. Degenerate Four Wave Mixing (DFWM)
In this particular variation of transient grating experiment, the two pump
pulses are two strong counter-propagating waves
˜
E
p1
(t) exp[i(ω
t − k
p
z)] and
˜
E
p2
(t) exp[i(ω
t + k
p
z)]. The probe wave is sent along an intersecting direction
x and has as electric field
˜
E
3
(t −τ
d
) exp[i(ω
t −k
x
x)]. The nonlinear interaction
results in the generation of a signal
˜
E
4
(t) exp[i(ω
t +k
x
x)], which, for momentum
conservation, is counter-propagating to the probe direction (Figure 10.10). In the
case of continuous waves, and, for instance, a quadratic nonlinearity, it can
be shown that the wavefront of the generated signal wave
˜
E
4
is the reverse of
the wavefront of the probe
˜
E
3
[7]. This property of spatial phase conjugation
does not transpose directly in the time domain. Temporal phase conjugation is
chirp reversal, which can be shown to occur only when the following conditions
are met [8]:
• instantaneous nonlinearity,
• medium thickness than the pulse length, and
• weak interaction (|
˜
E
4
||
˜
E
3
|).
Probe
Pump
Pump
Signal
0Delay
DFWM
signal
T
2
T
2
t
t
t
~
ε
3
~
ε
p2
~
ε
p1
Figure 10.10 Coherent single-photon resonant DFWM. The probe pulse is trailed by a polarization
wave, that forms a population grating with one of the pump pulses that follows. The other pump
pulse scatters off that grating into the direction from which the probe originates. The rise of the signal
energy versus delay is thus a measure of the phase relaxation time of the single-photon resonance.
Transient Grating Techniques 507
It can easily be seen that if all but the second condition are met, each depth of
the medium will generate a DFWM signal, resulting in a square pulse
˜
E
4
with a
length equal to twice the sample thickness [8].
We have so far assumed that all three waves meet simultaneously in the non-
linear medium. Interesting information on the dynamics of the interaction can be
gathered from the study of the DFWM signal when all three waves are applied
in a particular time sequence.
We assume in the following discussion that the nonlinear medium is shorter
than the optical pulses and is either at single- or at two-photon resonance with
the radiation. Let us first consider the case of a single-photon resonant absorber
being excited first by a weak probe, followed by two simultaneous strong counter-
propagating pump pulses (Fig. 10.10). As we saw in Chapter 4, the short pulse
creates a pseudo-polarization
˜
Q
3
= w
0
sin θ
0
exp[−ik
x
x] that decays with a char-
acteristic time T
2
. If a strong pump pulse enters the interaction region within
that characteristic time, it will form a population grating [as seen from the Bloch
equation (4.7)] corresponding to the interferences between waves of vector k
x
and k
z
. If the second pump pulse impinges on this grating, it will be diffracted
along the opposite direction as the signal (wave vector −k
x
) according to the
Bloch equation (4.6). The longer T
2
is, the more the probe can be launched in
advance of the two pump pulses, and still produce a signal. As illustrated in
Fig. 10.10, the rise time of the signal versus delay is a measure of the phase
relaxation time of a single photon transition.
The same experiment performed on a two-photon resonant transition, as
sketched in Figure 10.11(a), leads to different results and interpretation. Because
the interaction is a two-photon process, the weak probe alone cannot have any
significant effect on the system, and there will be no signal if the probe is
ahead of the pump pulses. We saw in Chapter 4 that for a two-photon transition,
Bloch’s equations (4.6), (4.7) apply, except that the driving term is proportional
to the square of the field. The two counter-propagating pump pulses can produce
a two-photon excitation oscillating at 2ω
[see Eq. (4.95)], which will decay
with the phase relaxation time T
2(2ph)
of the two-photon transition. One compo-
nent of this two-photon excitation,
12
, with no spatial modulation (zero spatial
frequency), will interact with a probe to generate a counter-propagating signal
by two-photon stimulated emission. A probe pulse sent through the interaction
region with a subsequent delay τ
d
will induce a signal by two-photon stimulated
emission,
˜
E
4
∝
12
(τ
d
)
˜
E
∗
3
. Because the probe field corresponds to a phase factor
ω
+ k
x
x and the two-photon excitation to a phase factor 2ω
, the signal E
4
has
a phase factor 2ω
−ω
−k
x
x = ω
−k
x
x, which describes a wave propagating
in the direction opposite to the probe. Because the two-photon excitation
12
is the amplitude of an off-diagonal matrix element decaying with a two-photon
phase relaxation time T
2(2ph)
, the two-photon stimulated emission being propor-
tional to
12
will only exist within T
2(2ph)
of the pump excitation. In the case
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.
