Three - University of Arkansas Physics Department

VOLUME 78, NUMBER 5 PHYSICAL REVIEW LETTERS 3 FEBRUARY 1997
Experimental Observation of the Continuous Pulse-Train Soliton Solution
to the Maxwell-Bloch Equations
John L. Shultz and Gregory J. Salamo
University of Arkansas, Physics Department, Fayetteville, Arkansas 72701
(Received 23 February 1996; revised manuscript received 2 July 1996)
We report the first experimental observation of the evolution of an arbitrarily shaped input optical
pulse train to the analytic, shape-preserving, Jacobi elliptic pulse-train solutions to the Maxwell-Bloch
equations including relaxation. [SO03 1-9007(96)02261-21
PACS numbers: 42.65.Tg
Coherent propagation of trains of optical pulses through
a two-level absorber has been investigated, from a theoretical
viewpoint, by many researchers. These investigations
have been based on the self-induced transparency
(SIT) equations [I], both with [2-51 and without [6] the
assumption of zero relaxation. Without relaxation, analytic
shape-preserving pulse-train solutions to the SIT
equations were found although they did require a rather
complicated initial atomic state which appeared difficult
to produce experimentally. Specifically, the necessary initial
state for each atom in a Doppler-broadened profile is
composed of a different superposition of ground and excited
states depending on the resonant frequency of the
atom. With relaxation, however, it was shown that the
atomic variables relax to the correct values needed to allow
shape-preserving propagation of the analytic pulsetrain
envelopes found as solutions to the zero-relaxation
problem. In this sense, the presence of relaxation effects
makes possible the experimental observation of the analytic
shape-preserving optical pulse-train solutions predicted
by the SIT equations with zero relaxation.
While these theoretical studies have been very thorough,
and experimental studies have demonstrated the features
of breakup and delay [7-91, the predicted Jacobi elliptic
soliton pulse-train solution has never been observed. In
this Letter we report the first experimental observation
of the Jacobi elliptic soliton shape-preserving solution to
the SIT equations including relaxation. In particular, it is
shown via experiment that an input train of optical pulses,
with rather arbitrary input envelope shape and input area
between rr and 37r, evolves through the interaction with
a two-level absorber to the analytical, shape-preserving,
Jacobi elliptic soliton-train solution to the SIT equations.
This is true, as predicted, even though the optical train
has been allowed to come to equilibrium for a time much
longer than the two-level atomic absorber relaxation time.
The basic apparatus consisted of a homemade modelocked
dye laser and sodium absorption cell. The number
of oscillating longitudinal modes in the cavity was controlled
using several intracavity etalons. For the experiment
these were limited to three longitudinal modes whose
separation was 167 MHz as determined by the cavity configuration
and monitored using a Fabry-Perot scan. The
phase locking of these three modes resulted in an optical
pulse train consisting of 2.9 ns (FWHM) pulses separated
by 5.8 ns, peak to peak. This approach to producing an optical
pulse train was chosen over using a cw laser coupled
with a nonlinear optical modulator to ensure the production
of an optical pulse train with both strong modulation and a
high repetition rate. An example of the three laser modes,
used to produce the input pulse train, as seen on an oscilloscope
display of a Fabry-Perot scan is shown in Fig. l (a)
along with a corresponding input optical pulse train in
Fig. l(b). A computer simulation of the pulse train using
the three mode-locked modes of relative amplitude shown
in Fig. l(a), and fixed relative phase, is shown in Fig. I (c).
The good agreement between Figs. l(b) and l(c) supports
the assumption that the train of optical pulses is produced
by mode locking three modes with fixed relative phase.
The output optical pulse train was then gated through a
Glan prism by a Pockels cell. This resulted in a train of
optical pulses that was about 1 ps long and was repeated
every millisecond. The long time between successive
pulse trains ensured that the experiment could be considered
as starting over every millisecond. Meanwhile,
the long length of the optical train, when compared to
the atomic relaxation times of TI and Ti, guaranteed that
equilibrium would be reached. The optical pulse train
was then focused by an f = 41 cm lens into a 5 mm
sodium cell. The sodium cell was housed in an oven and
placed in a magnetic field which could be varied from 0 to
10 kG. With a large magnetic field the sodium transitions
of different Mj are well resolved, but several MI transitions
are excited as a result of the large Doppler width.
However, since the dipole moments are independent of
MI, simultaneous excitation of several MI levels with the
same Mj quantum number can be treated as the excitation
of a single state. For our experiment we excited the
2~1/2 -+ 2~3/2 transition in sodium. In particular, using
circularly polarized light we selectively excited only the
3
+
2S~/2(~~ = 2 ~312(~J = Z) transition which could
I
then only decay back to the 2S1/2(~J = Z) ground state.
In this way we avoid optical pumping complications. For
this transition the energy decay time TI is equal to 16 ns
while the dipole dephasing time T; is equal to 32 ns. This
means that for our experiment the relaxation time is much
003 1 -9007/97/ 78(5)/855(4)$10.00 0 1997 The American Physical Society 855