Pulse Switching and Stability in FM ModeLocked Lasers

1
Pulse Switching and Stability in FM Mode-Locked
Lasers
Nicholas G. Usechak and Govind P. Agrawal
The Institute of Optics, University of Rochester,
Rochester, New York 14627
and Laboratory for Laser Energetics, Rochester,
New York 14623
CLEO, May 25, 2005
2
Pulse switching and stability in FM mode-locked
lasers are studied both numerically and
analytically
FM mode-locked lasers are prone to an instability
where the pulse randomly switches between the two
modulator extrema.1

• In practice, pulses favor one extrema over the
  other. A simple rule ?FM?2 lt 0 predicts the
  modulator cycle where the dominant pulses
  exist.2
• It has also been noted that the introduction of
  residual second-order dispersion (SOD) suppresses
  the pulse switching tendency.1,2

In this work, we use numerical simulations and
the moment method3,4 to

• gain a better understanding of pulse stability in
  FM mode-locked lasers
• investigate the effect of third-order dispersion
  (TOD) on pulse switching for the first time
• identify two different pulse switching mechanisms

3
All numerical simulations and analytic
investigations in this work used the master
equation of mode-locking
TOD
SPM
Propagation
SOD and gain filtering
Loss and saturated gain
FM mode-locker
Where
and
The master equation is written in terms of two
different time scales one with respect to the
local pulse time t and another with respect to
the round trip time T. The parameters are
averaged over the cavity.
4
A phase shift in the driving electronics causes
the mode-locked pulse(s) to temporally shift in
order to follow the ?FM?2 lt 0 cycle
In this case, the pulse switching is initiated
via the wings of the secondary pulse. However,
this type of switching can also be seeded through
noise.
5
By viewing the phase shifts effect on the
temporal and spectral widths, two distinct states
(and a selection mechanism) are found
This figure reveals the laser mechanism that
favors the dominant pulsespectral filtering.
Note the dominant pulse spectrum is 1/4 that of
the secondary pulse spectrum.
6
If the pulse energy is increased, the pulse can
survive the phase shift and will relocate under
the influence of TOD
Here, pulse switching is initiated via TOD. The
pulse remains intact while it resynchronizes with
the modulator extrema satisfying ?FM?2 lt 0.
7
Two different pulse switching mechanisms have
been identified through numerical simulations
noise and TOD
These cartoons identify the two different pulse
switching mechanisms. The green lines identify
the round trip where the phase was shifted. The
heavy black lines (top) identify the direction in
which pulse energy flows.
8
Using the moment method, we developed a
rate-equation approach that analytically predicts
the mode-locked pulse width and chirp
The pulse chirp is given by
The pulse width comes from
In these equations, constants Cn 1 and the
function ?1 depends on the pulse shape assumed
when applying the moment method.

• There are two steady-state chirps predicted one
  large and one small.
• These two possible chirps lead to multiple
  steady-state solutions for all FM mode-locked
  lasers!

9
Our rate-equation approach also allows us to
visualize the stability of the mode-locked pulses
Using the rate equations for pulse timing ? and
frequency shift ?, we can define the temporal
shift of the pulse center per round trip.
Shift of pulse center/round trip
By plotting TC as a function of pulse-modulator
detuning, using the steady-state pulse width ?
and chirp q, we find the modulator extrema where
the pulses exist.
Since TC includes the effect of TOD, we are also
able to analytically investigate its influence on
pulse stability.
10
Mapping the shift of pulse center for the
dominant mode-locked pulse(s) shows the effect
of TOD on stability

• These figures reveal that TOD creates an
  asymmetric velocity of pulse center about the
  stable/unstable modulator extrema.
• These results also agree with the rule ?FM?2 lt 0
  compare (b) and (d).

11
The shift of pulse center for the secondary
mode-locked pulses reveals they exist at the
alternate modulator cycles

• Figures (a) and (b) reveal that the secondary
  mode-locked pulses form at the alternate
  modulator extrema compare them with the previous
  slide.
• Figure (c) shows what happens when the TOD is
  increased the mode-locker eventually loses its
  ability to synchronize the pulses.

12
The results predicted by the moment method agree
with all prior work in the appropriate limits
In this figure both states are plotted for the
case ?2 ?3 ? 0. The pulses only differ in
the sign of the chirp and the modulator extrema
which they form under. As a result, both states
are stable and the laser can randomly switch
between the two.1 Note the velocity is much
weaker in this case!
13
Recent results obtained for pulse switching and
stability in FM mode-locked lasers have been
shown

• The role of third-order dispersion on pulse
  switching/stability was revealed.
• The effect of second-order dispersion on the
  location of the stable pulses was noted.
• Using both numerical simulations and the moment
  method, we obtained similar results for pulse
  stability in FM mode-locked lasers.
• In addition to agreeing with prior results, we
  identified two different pulse switching
  mechanisms.

References
1D. Kuizenga and A. Siegman, IEEE J. Quantum
Electron. QE-6, 694 (1970).
2K. Tamura and M. Nakazawa, Opt. Lett. 21, 1930
(1996).
3S. Vlasov, et al., Radiophys. Quantum Electron.
14, 1062 (1971).
4N. Usechak and G. Agrawal, Opt. Express. 13,
2075 (2005).
http//www.optics.rochester.edu/users/noodles