Effect of propagation on pulsed four-wave mixing
P. Weisman, A. D. Wilson-Gordon, and H. Friedmann
Department of Chemistry, Bar-Ilan University, Ramat Gan 52900, Israel
Received 1 November 1999; published 17 April 2000
We examine the effect of propagation on the resonance Rabi sideband of the four-wave mixing FWM
spectrum, obtained when short temporally displaced pump and probe pulses interact with an optically thick
medium of two-level atoms. We find that the dependence of the time-integrated FWM signal on the pump-
probe delay is considerably altered by propagation. In particular, the logarithm of the FWM signal, for the case
where the probe precedes the pump, deviates from linearity and may even increase over a range of values. An
explanation is given in terms of the overlap of the pump envelope with the coherent response of the atomic
system to the probe, both of which are modified on propagation.
PACS numbers: 42.65.Hw, 42.50.Gy, 42.50.Md
I. INTRODUCTION
The steady-state excitation four-wave-mixing FWM
spectrum obtained when a two-level system interacts with a
coherent pump of arbitrary intensity and fixed frequency
1
and a weak probe of varying frequency
2
is symmetrical
and consists of three peaks 1,2. In addition to the central
peak at
2
=
1
, there are two Rabi sidebands at
2
=
1
1
where the pump generalized Rabi frequency
1
=(
1
2
+4 | V
1
2
| )
1/2
is determined by the pump detuning from
resonance
1
=
ba
-
1
and its Rabi frequency 2 V
1
= E
1
/ . In the limit of large detuning
1
2 | V
1
| one of
the peaks is situated near the resonance frequency
2
ba
and the other near the three-photon scattering TPS fre-
quency
2
2
1
-
ba
. However, a completely different
picture emerges when the cw pump and probe fields are re-
placed by short pulses 3,4. We have shown for temporally
overlapping pulses 3, that the spectrum becomes asym-
metrical with the relative widths and heights of the Rabi
sidebands determined mainly by the linewidths of the laser
pulses. We have also shown for temporally nonoverlapping
pulses 4 that the spectrum becomes extremely asymmetri-
cal and that only the sideband near the resonance frequency
survives, for the case where the probe precedes the pump.
For the opposite case where the pump precedes the probe, no
FWM spectrum is obtained at all. It should be noted that
these results hold provided the the pump and probe durations
D
1
and D
2
are much shorter than the transverse relaxation
time T
2
and also shorter than the time delay between the
pump and probe t
0
( t
0
0 implies that the probe precedes
the pump. We also showed 4 that T
2
can be determined
from the slope of the semilog plot of the FWM intensity,
near the resonance sideband, versus t
0
for t
0
0. In this
paper, we will study the effect of propagation in a thick
atomic medium on the FWM spectrum. In particular, we will
show that as propagation proceeds, the semilog plot which is
initially linear begins to deviate from linearity and may even
changes the sign of its slope for a range of values of t
0
.
This asymmetry of the resonance peak with respect to the
pump-probe time delay is well known for the case of degen-
erate FWM DFWM using weak, short, resonant pulses
(
1
=
2
=
ba
) in optically thin samples 5. DFWM has
been widely applied to femtosecond real-time probing of
gas-phase chemical reactions, including the the study of
transition-state dynamics 6, and also in studies of liquid
and solid systems 7,8 where the dephasing time T
2
has
been determined from the dependence of the time-integrated
FWM on the pump-probe time delay. By varying the pulse
sequences, it is also possible to measure the population de-
cay time T
1
6. Variations of FWM such as coherent time-
resolved anti-Stokes Raman spectroscopy CARS9 and
time-resolved transient-grating TG techniques 10 have
been used to explore the molecular dynamics both popula-
tions and coherences in gas-phase systems ranging from
single atoms to large polyatomic molecules.
The majority of theoretical treatments of pulsed FWM
and its variants, in gas-phase systems unlike those in solids
8, include neither saturation nor propagation effects and
are based on perturbation theory 11. Studies of saturation
effects include a nonperturbational approach to picosecond
experiments using spatial and frequency gratings 12 and an
extension to the transient regime 13 of the classic nonper-
turbative steady-state theory pf DFWM, developed by
Abrams and Lind 14. Propagation effects in DFWM for
time-delayed pulses have been studied both experimentally
and theoretically by Kinrot and Prior 15 for gaseous alkali
atoms. Their theoretical treatment is a combination of Ya-
jima and Taira’s 5 perturbative ‘‘standard model’’ of
DFWM in thin slabs and Crisp’s theory 16 of propagation
of 0 pulses, and is therefore only valid for weak pump
pulses. In the present paper, we are interested in the effect of
propagation on the asymmetry, with respect to pump-probe
time delay, of the Rabi sideband near the resonance fre-
quency in the FWM excitation spectrum 3. Our treatment is
based on the numerical solution of the Maxwell-Bloch equa-
tions for a short pump pulse of arbitrary intensity and detun-
ing and a short, weak probe pulse. Several of Kinrot and
Prior’s conclusions 15 will be of relevance here. For ex-
ample, they found that the DFWM signal becomes signifi-
cant, on propagation, even when the pump precedes the
probe. This has also been observed in solid-state DFWM
experiments 17 and has been explained by a wide variety
of mechanisms appropriate to the particular system being
studied. Kinrot and Prior also found that the slope of the In
of the FWM signal as a function of the pump-pulse delay
PHYSICAL REVIEW A, VOLUME 61, 053816
1050-2947/2000/615/0538166/$15.00 ©2000 The American Physical Society 61 053816-1