Simulations of a Semiconductor Laser with Filtered
Optical Feedback: Deterministic Dynamics and
Transitions to Chaos
Mirvais Yousefi, Daan Lenstra, Gautam Vemuri(1), Alexis P.A. Fischer(2)
Vrije Universiteit, FEW N&S, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
(mirvais@nat.vu.nl)
(1) Physics Department, Indiana University Purdue University, Indianapolis, IN 46202-3273, USA
(gvemuri@iupui.edu)
(2) Laboratoire de Physique des lasers-UMR CNRS 7538, Institute Galilee, Universite de Paris 13, Av. J.B.
Clement, 93430 Villetaneuse, France.
Abstract: Through simulations based on the rate equations for a diode laser with filtered external optical feedback,
we show that the laser's dynamical system attractors can be controlled through the filter parameters: the filter's
spectral width and its central frequency.
Simulations of a Semiconductor Laser with Filtered
Optical Feedback: Deterministic Dynamics and
Transitions to Chaos
Mirvais Yousefi, Daan Lenstra, Gautam Vemuri(1), Alexis P.A. Fischer(2)
Vrije Universiteit, FEW N&S, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
(mirvais@nat.vu.nl)
(1) Physics Department, Indiana University Purdue University, Indianapolis, IN 46202-3273, USA
(gvemuri@iupui.edu)
(2) Laboratoire de Physique des lasers-UMR CNRS 7538, Institute Galilee, Universite de Paris 13, Av. J.B.
Clement, 93430 Villetaneuse, France.
Abstract: Through simulations based on the rate equations for a diode laser with filtered external optical feedback,
we show that the laser's dynamical system attractors can be controlled through the filter parameters: the filter's
spectral width and its central frequency.
theory). A typical result is shown in Fig.1. In the lower
Introduction
part, the steady-state frequency is plotted versus the
The complexity of diode laser instabilities and dynamics,
solitary laser frequency  where the (fixed) filter centre
not just due to feedback but also to external injection or
frequency f is indicated by the vertical dashed line. In the
modulation of other parameters, such as the injection
upper part some examples of fixed points are given in the
current, has led many workers to develop a dynamical
power (P) versus external roundtrip phase difference ()
description of the laser response. The dynamical approach
phase sub-space ( = t-tis the phase of the
provides a perspective that complements other strategies,
electrical field and  is the external delay time).
such as studying time-series or spectral properties of the
emitted light. In the context of feedback effects in diode
lasers, the dynamical approach has been especially
successful in elucidating the mechanisms that are at play in
low-frequency fluctuation (LFF) phenomena [1], and in
understanding the wide range of behaviours that arise in
injection lasers [2]. In this paper we study filtered optical
feedback (FOF) from a dynamical perspective. This case
differs from conventional optical feedback (COF) in that
the delayed external light is frequency selectively filtered
before re-entering the diode laser. The filter restricts the
phase space that is available to the feedback laser system.
During COF, the dynamics of the system occupy a specific
region in phase space, which is determined, among others,
by the number of different external cavity modes (ECMs)
available. Introduction of the filter not only decreases the
number of ECMs, but also moves them around in phase
space, which leads to new and unexpected dynamics. Thus
the filter can be used to target a desired dynamical
behaviour in a specific region in phase space.
The present analysis focuses on how the systems dynamics
Figure 1: Fixed points of FOF system. (a)-(d)show the
depend on the "external" control "knobs", (i) the detuning
fixed points for some fixed values of the solitary laser
frequency between the solitary laser and the filter centre
frequencies as indicated. The lower plot shows the fixed
frequency and (ii) the spectral width. Since especially the
point frequencies versus solitary laser frequency. A
former parameter can most easily be controlled in an
indicates some saddle node points.
experimental situation by means of the laser pump current,
In
(a)
the
fixed
points have split into two groups, one group
we will present an overview of many (stable) dynamical
corresponding
to
laser operation close to the filter centre
attractors that show up.
frequency and the other group close to solitary laser
Model
frequency. Quite often, fixed points are not stable, but their
The model used for the simulation is for single longitudinal
position in phase space may give a first idea of where the
mode operation and consists of three rate equations. Two of
attractor orbits are likely to be found. Thus one may expect
them are complex and describe the temporal evolutions of
bi-stability in cases like (a) and (b).
the field in the laser and in the external cavity, respectively,
A subset of the fixed points corresponds to the compound
while the third is a real equation describing the temporal
cavity modes introduced due to feedback (ECMs). The
evolution of the electron-hole pair number [3]. In our
remaining points are (unstable) saddle points corresponding
model equations we also include the dependence of the
to anti-modes. The modes are asymmetric in power, in the
solitary-laser operation frequency on the bias pump current.
sense that for frequencies below the solitary laser frequency
In a first step the steady-state solutions are calculated
(red side) they have higher power, while for higher
(called fixed points in the language of non-linear dynamics
frequencies (blue side) the modes have lower power than
the solitary laser power. This is a consequence of the
phase-amplitude coupling introduced by the linewidth
enhancement factor.
Dynamics
Numerical integration of the rate equations is performed
using a modified Runge-Kutta method of fourth order. The
internal laser parameters were held fixed to the values
given in Table 1. The results are presented in terms of
bifurcation maps where the behaviour of the frequency of
the diode laser with FOF is depicted as a function of the
solitary laser frequency. Fig.2 gives an overview of the
various types of dynamics that simulations predict for a
case where the solitary laser operates about 40% above
threshold (in case of COF this laser would be in the
coherence collapse regime).
When inspecting Fig.2, it may be helpful to remark that in
regions of solitary laser frequency for which there are no
black dots, the system operates on a stable fixed point. This
corresponds to stable cw-emission and a black segment
covers the gray curve. In regions with black dots, the
system will be chaotic (irregular distribution of dots) or
operating on a limit cycle (dots on a line). If regions
overlap, there may be coexistence, meaning that the system
will operate in chaos, limit cycle or fixed point, depending
on initial conditions.
The line shape of a diode laser in case of relaxation
oscillation related limit cycle operation, i.e. when the
relaxation oscillations (RO) are undamped, consists of a
main peak and two side peaks located RO away in
frequency from the main peak. Here RO is the relaxation
oscillation angular frequency. When the detuning is set
equal to -RO, the low-frequency relaxation side peak is
resonant with the filter and couples back into the laser
diode. This results in the dynamical structure around -27
GHz in Fig.2. Again due to the amplitude-phase coupling
effect, there is no similar counterpart of this interesting
feedback mechanism via the high-frequency RO side peak.
This asymmetric RO-induced feedback was observed in our
laboratory in 1999 [4].
Figure 2: Bifurcation map of the FOF system in terms
of the actual instantaneous laser frequency vs. the
solitary laser frequency. The snake-like curve shows
the fixed points as in Fig 1, the light segments show the
mean values over trajectories when averaged over 50
roundtrips and the black dots show instantaneous
frequencies at intersections of phase space trajectories
with a certain plane (Poincare sections). The filter
central frequency is at -32.1 GHz and the filter FWHM
is 2 GHz.
In summary, different attractors can be accessed and routes
to chaos can be found by varying the detuning between the
filter centre and the solitary laser frequency. The
concentration of different attractors (i.e. the dynamical
complexity) in a specific region can be controlled through
the bandwidth of the filter.
Table 1. Parameter values used in the simulations
Quantity
Symbol
Linewidth enhancement factor
Feedbak rate
Ext. cavity roundtrip time
Differential gain coefficient
Photon decay rate
Carrier decay rate
Threshold pump rate
Pump rate





T1
Jthr
J
Value
2
11.18 x 109s-1
3 ns
2.142 x 104s-1
357 x 109s-1
0.167 ns
1.4 x 1017s-1
~1.4Jthr
References
/1/ van Tartwijk G.H.M. and Lenstra D., Asian J.
Physics 7 (1998)562-575.
/2/ Wieczorek S., Krauskopf B. and Lenstra D.,
Optics Commun. 172 (1999) 279-295.
/3/ Yousefi M. and Lenstra D., IEEE J. Quantum
Electron. QE-35 (1999) 970-976.
/4/ Fischer A.P.A, Andersen O.K., Yousefi M., Stolte S.
and Lenstra D., IEEE J. Quantum Electron. QE-36 (2000)
375-384.