2D-0D Direct Capture of Carriers in Quantum Dot Lasers under
Optical Feedback
George Andre Pereira The
Depto of Teleinformatic Engineering, Federal University of Ceara, Campus do Pici s/n, Bl 725, Fortaleza, Brazil
Keywords: Quantum Dot Lasers, Lyapunov Exponents, Direct Capture.
Abstract: In this paper the operation of quantum dot lasers under optical feedback is studied on the basis of calculated
Lyapunov exponents of the dynamic multi-population rate-equations. Influence of the direct capture path for
the wetting layer carriers on the sensitivity to initial conditions is discussed. Results show that positive
exponents are achieved for different current injection scenarios, and that negative exponents are mainly due
to carriers in the dot confined states. Furthermore, existence of hiperchaos is obtained with both cascade and
direct capture models.
1 INTRODUCTION
Semiconductor quantum dot (QD) lasers have been
intensively studied in the last years because of their
potential compared to quantum well and bulk lasers,
as well as due to the particular properties associated
with the 3D confinement in the quantum dots, such
as the high differential gain which should lead to
reduced linewidth enhancement factor (LEF) and
low chirp. Essentially, the LEF or alpha-factor is a
parameter which measures the coupling between real
(phase) and immaginary (gain) parts of the complex
refractive index of the active material; any
fluctuation of phase or amplitude of the laser field
(due to spontaneous emission or even reflected back
light, for instance) induces relaxation oscillations,
changing the immaginary component of the
refractive index and, consequently, the real part as
well. This means that gain modulation (fluctuation
of immaginary component) leads to phase
modulation (real part) and, therefore spectral
broadening of the laser linewidth (chirp) is observed.
Many papers have indeed addressed this issue
and shown there is significant dependence of the
alpha-factor on both internal (carrier scattering
dynamics, wetting layer carrier population, etc.) and
external factors (cavity length, temperature) (Carrol,
2005 and 2006, Melnik, 2006). From the above
mentioned, the device becomes sensitive to out-of-
phase optical field, and even the coupling to optical
fiber could be a problem due to the delayed optical
field reflected back to the cavity (Gioannini, 2008a).
Literature has previously identified the 2D carriers
present in the wetting layer carrier as responsible for
most of the frequency fluctuation observed in
quantum dot lasers when subject to optical feedback
(Gioannini, 2008a and 2008b). Those works
revealed different operating regimes of quantum dot
lasers, ranging from stable to chaotic-like solutions
as the injection current increased, but did not
considered a more rigorous analysis to confirm the
existence of chaos. On the other side, more recently,
carriers of the wetting layer reservoir have also been
shown to influence the steady state of solitary edge-
emitting lasers when the scattering to 0D states has a
non-negligible dynamics (The, 2012), thus justifying
the inclusion of such direct capture of carriers in the
rate-equations. This scenario naturally motivates one
to investigate whether the existence of the 2D-0D
direct capture channel may influence the laser
regimes previously reported and how it differs to the
case in which only the cascade scattering takes
place. To accomplish with that, in the present work
the quantum dot laser response to current injection
with both cascade and direct capture models is
analyzed from the calculated Lyapunov exponents of
the dynamic system (Monteiro, 2002). Such
exponents (each one associated to a given state
variable) represent a measure of how the
hypervolume of an n-dimensional sphere (thinking
of the state space system representation) changes
with time, being therefore essential to confirm
chaotic dynamic.
212
Andre Pereira The G..
2D-0D Direct Capture of Carriers in Quantum Dot Lasers under Optical Feedback.
DOI: 10.5220/0004794902120217
In Proceedings of 2nd International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS-2014), pages 212-217
ISBN: 978-989-758-008-6
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
This paper is organized as follows: in the next
section the rate-equations based model to take into
account the delayed optical field and the direct
capture path is presented, and a discussion of the
Lyapunov exponents is added. In section III results
are presented and discussed. Finally, we draw the
conclusions.
2 MODELING
2.1 Rate-equations
The model here used is based on that of (The, 2012),
which considers separate dynamics for electron and
hole populations of InAs quantum dots inserted in
InGaAs quantum well, with GaAs separate
confinement heterostructure (SCH). In order to
correctly include the inhomogeneous dot size
distribution which takes place in the Stranski-
Krastanow growth technique, a multi-population
approach has been adopted (Rossetti, 2007); this
means that quantum dots of an ensemble are
separated into n small groups of dots similar in size,
thus leading to n equations for carrier number of
each confined state (GS, ES1 and ES2).
Furthermore, it contains the scattering channel of
carrier from the wetting layer to the confined states.
To highlight this, for what concerns carriers of the
conduction band, in the following only the
population equations for wetting layer, first-excited
and ground-state are respectively reported.


2
2
2w
0
1
1
1
1
w
sw
w
sw ws
w2
w
w
w1 w0
n
ρ
n
nn
n= +
tt t t
n ρ
n ρ
tt




(1)
 

2101
1
21 01
10
11 2
12 10
11
1
11
w
w1
n ρ n ρ
n= +
tt
n ρ
+n ρ n ρ
ttt
 

 

(2)
 

10 0
0
10
01
01
11
1
w
w0
n ρ n ρ
n= +
tt
n ρ
t
 

(3)
In the above equations,
n
s
is the SCH carrier
number and
n
2
is the carrier number of electrons in
the second-excited state. Terms inside the brackets
typed with greek
ρ
are the average occupation of
confined states (2 for ES2, 1 for ES1 and 0 for GS).
Terms in the denominator indicate the average time
constants for the various scattering mechanisms
between carriers of the conduction band; for
example,
t
01
is the average escape time from level 0
(GS) to energy level 1 (ES1), whereas
t
w1
refers to
direct scattering from WL energy states to ES1
energy state. These scattering phenomena are
illustrated in Figure 1.
Figure 1: Schematic diagram of carrier dynamics in
conduction band. In dashed arrows the 2D-0D capture
channel.
In the valence band things are simpler, since hole
dynamics is governed by two equations only: one for
the dot states (index d) and one for holes in the SCH
(index s):
˙
n
s
=
I
e
n
s
t
s
d
+
n
d
t
ds
(4)
˙
n
d
=
n
s
t
s
d
n
d
t
ds
(5)
Notation of equations (4) and (5) is similar to the
previous ones, except that now they refer to holes,
and n
d
expresses the population of carriers in the
quantum dot and wetting layer altogether (2D and
0D holes constitute one ensemble). Term
I/e is the
mean number of holes injected per unit time in the
separate confinement heterostructure.
In the above equations non-radiative and
radiative recombination terms are omitted for
brevity.
The main difference between the model here
used and that of (The, 2012) is that the photon
equations for the cavity resonant modes have been
replaced by one equation for the internal electrical
field intensity,
E
0
and one for its phase, Φ. This
follows the classical Lang-Kobayashi model (Lang,
1980) used widely to study the effects of optical
feedback in semiconductor single-mode lasers (Otto,
2010; O'Brien, 2004). Here are the remaining two
equations:
2D-0DDirectCaptureofCarriersinQuantumDotLasersunderOpticalFeedback
213




0
0
0
00
2t 2n
cos
sp
ph r
nES2 nES1 nGS
dd
E
c
E= + +B
g
+g +g E
+k E t t w t + Δ t


(6)



00
0
0
sin
d
kE t t
Φ wt +Δ t
Et

(7)
In the above equations
t
ph
is the photon lifetime
in the laser cavity,
c is the free-space light velocity,
n
r
is the active material refractive index, k is the
intensity of feedback light,
w
0
is the angular
frequency of the solitary laser,
t
d
is the external
cavity roundtrip time and δf is the frequency chirp
calculated according to (Gioannini, 2007). Finally,
the terms inside the summation operator is the
material gain coupling cavity photons and carriers of
the ground-, first- and second-excited states.
2.2 Calculation of Lyapunov
Exponents
According to the theory of dynamical systems, only
nonlinear dissipative systems may experience
chaotic behavior, being chaos related to sensitivity
to initial conditions and characterized by a time
evolution towards a strange attractor in the phase
space (Monteiro, 2002). A widely used approach to
test sensitivity to initial conditions of nonlinear
systems and, therefore, conclude about the existence
of chaos requires the calculation of Lyapunov
exponents. Given a dynamic system with
p velocity
fields associated to state variables, there are two
requirements to be satisfied before concluding if the
process is chaotic:
a) at least one of the Lyapunov exponents
associated to the velocity equations is positive: this
is to guarantee divergence of adjacent trajectories
(those starting at slightly different initial
conditions);
b) the sum of all Lyapunov exponents associated
to the whole set of velocity field equations must be
negative: this is to ensure the system is dissipative
(and therefore phase space evolution towards a
strange attractor occurs).
In the present work the calculation of the Lyapunov
exponents is based on the following formula:
00
0
1
log
0
e
Fx+δ Fx
Λ =
N δ




(8)
In this expression,
N is the size of the discrete
time vector corresponding to the last time instants
of every simulation (transient is discarded), δ
0
is the
small deviation between two different initial
conditions and
F is the vector of state variables
representing the dynamical system.
3 RESULTS AND DISCUSSION
The model has been used to study the laser response
when it is subject to optical feedback, under
different current injection conditions. The device
considered is a 0.6 mm long edge-emitting device
emitting from the ground-state lasing line, at 1285
nm. This device has been chosen to allow for better
comprehension of already reported results,
especially the chaotic-like solutions which had not
been explained in terms of Lyapunov exponents
(Gioannini, 2008a and 2008b). Another goal of the
present analysis is to check whether the direct
capture carrier scattering phenomenon may
influence the conditions for chaos in the laser
device. Other parameters used in the simulations are
listed in Table 1.
Table 1: Quantum-dot material and laser parameters used.
Laser width: 4 microns QD density: 4.10
14
m
-2
Number of QD layers: 10 Energy separation (meV):
WL-ES2: 13 meV
ES2-ES1: 36 meV
ES1-GS: 37.3 meV
Internal loss: 1.5 cm
-1
Left/Right end-
reflectivity: 0.3/0.3
To accomplish with that, the following results
focus on the time-domain evolution of the Lyapunov
exponents. Figures 2 to 4 report for 150 mA, 300
mA and 600mA, respectively, in the top part the sum
of the Lyapunov exponents (for a total of 142
dynamic equations describing the laser under
feedback), and in the bottom the largest Lyapunov
exponent. First result is that at any of these injection
conditions there is always at least one positive
exponent and, besides, the whole sum of exponents
is always negative. This formally confirms,
therefore, that quantum dot lasers, as suggested by
previous theoretical and experimental works are
sensitive to feedback oscillations, entering chaotic
regime at different operating conditions.
Figures 2, 3 and 4 solid lines refer to results
obtained with a model including the direct capture
process, whereas dotted lines refer to a cascade
scattering based model. Simulation revealed two
interesting point about this: first, by comparison of
the largest Lyapunov exponent in Figures 2 and 4,
we see that at lower current injections higher
PHOTOPTICS2014-InternationalConferenceonPhotonics,OpticsandLaserTechnology
214
positive Lyapunov exponents are obtained when the
direct capture path is present, whereas at higher
electrical current levels the cascade model got higher
positive exponents. Since the positive Lyapunov
exponent is a measure of how fast two slightly
different initial conditions evolves to a divergence of
the velocity field, this result can be explained in
terms of the higher carrier density in the wetting
layer achieved with the cascade model (this is
because the direct capture represents an additional
drain for wetting layer carriers down to the dot
confined states).
Figure 2: Time evolution of Lyapunov exponents for 150
mA injection condition. Solid lines refer to direct capture
model and dotted lines to cascade only model.
Figure 3: Time evolution of Lyapunov exponents for 300
mA injection condition. Solid lines refer to direct capture
model and dotted lines to cascade only model.
Another important result in these figures is the
indication of hiperchaos in both direct capture and
cascade models, suggested by the existence, in
steady-state, of two positive Lyapunov exponents in
Figures 2 (solid lines, direct capture model) and 4
(dotted lines, cascade model).
To illustrate the sensitivity to initial conditions,
in Figure 5 it is shown the laser response in time-
domain, for a 150 mA driving current, when the
direct capture scattering process is included. Solid
line refers to the optical power evolution after a
certain initial condition, and the dotted line is the
same, except for a slight change in the initial
condition (from 10
-6
to 1.1 x 10
-6
) of some of the
state variables. Notice how the responses differ after
11000 integration steps.
Figure 4: Time evolution of Lyapunov exponents for 600
mA injection condition. Solid lines refer to direct capture
model and dotted lines to cascade only model.
Figure 5: Time-domain evolution of the photon number in
the laser cavity when the direct capture model is
considered. Dotted-line and solid-lines differ slightly in
the initial values for the state variables.
Finally, to complete this look at the Lyapunov
exponents, in Figure 6 the exponents associated to
the equations for carriers in the dot confined states
2D-0DDirectCaptureofCarriersinQuantumDotLasersunderOpticalFeedback
215
(ground-, first-excited and second-excited states) are
shown in color map. It refers again to the 150 mA
driving condition.
According to this figure we can see that the
exponents associated to the confined states are all
negative (thus being essential for the dissipative
feature of the system), and that there is a clear
dependence of the exponents on the quantum dot
grouping (carriers from the quantum dots more
likely in the ensemble lead to less negative
exponents). Additionally, if ES2 carriers are
compared to ES1 and GS ones, it can be pointed out
that carriers resonant with the spectral window of
lower material gain (that containing ES2 states)
contribute to less negative values of Lyapunov
exponents. This suggests that the way to get other
positive Lyapunov exponents is to operate with
shallow dots, favoring carrier escape up to wetting
layer, a scenario in which higher energy confined
states are less populated.
Figure 6: Time-domain evolution of the Lyapunov
exponents associated to the carrier number equations of
confined states. Color indicates magnitude of the
Lyapunov exponents.
4 CONCLUSIONS
In this work a numerical model for the direct capture
scattering process in quantum dot lasers under
optical feedback has been developed, and an
analysis of the possibility of chaotic operation has
been done after numerical calculation of the
Lyapunov exponents of the system.
Simulation results revealed that positive
Lyapunov exponents are achieved at different
driving conditions, for both direct capture and
cascade only models. Comparison between these
models showed different sensitivity to initial
conditions at different electrical driving levels: with
the direct capture included in the model, higher
positive exponents are obtained at lower currents,
and an inverse trend is obtained for the cascade
model.
Finally, results of the Lyapunov exponents
associated to the carriers in the dot confined states
showed that there is major tendency to negative
values and that the threshold of positive values may
be related to the spectrum window of lower material
gain.
ACKNOWLEDGEMENTS
Brazilian agency CNPq supported this work;
reference number 482393/2011-4.
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