Appl. Phys. Lett. 96, 011104 (2010); https://doi.org/10.1063/1.3279136 96, 011104
© 2010 American Institute of Physics.
Hybrid mode-locking in a 40 GHz
monolithic quantum dot laser
Cite as: Appl. Phys. Lett. 96, 011104 (2010); https://doi.org/10.1063/1.3279136
Submitted: 19 October 2009 • Accepted: 30 November 2009 • Published Online: 05 January 2010
G. Fiol, D. Arsenijević, D. Bimberg, et al.
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Hybrid mode-locking in a 40 GHz monolithic quantum dot laser
G. Fiol,1,a兲D. Arsenijević,1D. Bimberg,1A. G. Vladimirov,2,3 M. Wolfrum,2E. A. Viktorov,4
and Paul Mandel4
1Institut für Festkörperphysik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
2Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D—10117 Berlin, Germany
3Physics Faculty, St. Petersburg State University, Ulianovskaya 3, 198504 St. Petersburg, Russia
4Optique Nonlinéaire Théorique, Université Libre de Bruxelles, Campus Plaine CP 231,
B-1050 Bruxelles, Belgium
共Received 19 October 2009; accepted 30 November 2009; published online 5 January 2010兲
Hybrid mode-locking in monolithic quantum dot 共QD兲lasers is studied experimentally and
theoretically. A strong asymmetry of the locking range with respect to the passive mode locking
frequency is observed. The width of this range increases linearly with the modulation amplitude for
all operating parameters. Maximum locking range found is 30 MHz. The results of a numerical
analysis performed using a set of delay-differential equations taking into account carrier exchange
between QDs and wetting layer are in agreement with experiments and indicate that a spectral
filtering element could improve locking characteristics. © 2010 American Institute of Physics.
关doi:10.1063/1.3279136兴
Quantum dot 共QD兲mode-locked lasers are promising for
telecom applications since they demonstrate a number of im-
portant advantages as compared to standard quantum well
共QW兲devices.1–3However, tailoring the characteristics of
QD lasers for specific applications is a challenging issue. An
important characteristics of hybrid mode-locking 共ML兲,a
commonly used technique for improving quality of mode-
locked pulses, is the frequency locking range where the pulse
repetition frequency can be synchronized to that of the ex-
ternal signal. Since the first reports on 10 GHz hybrid ML
共Ref. 4兲a significant reduction of the pulse jitter has been
demonstrated using hybrid ML with promising results of 124
and 190 fs for 40 GHz mode-locked lasers.5,6However, the
so far achieved locking range for hybrid ML in QD lasers is
smaller than that reported for QW devices. This fact was
recently discussed in Ref. 7.
In this letter we study, experimentally and theoretically,
the characteristics of the hybrid mode-locked regime in QD
lasers with a pulse repetition rate of 40 GHz. We find that
stable hybrid ML in QD lasers shows a locking range up to a
few tens of MHz demonstrating a strong asymmetry with
respect to the frequency of the free running passively mode-
locked laser. We also discuss a possible improvement of hy-
brid ML locking range, and describe bifurcation mechanisms
responsible for locking and unlocking between the funda-
mental ML regime and the external RF modulation.
Experimental investigations have been performed using
a monolithic 40 GHz QD laser integrated in a module, com-
prising a standard single mode fiber pigtail and a microwave
port. It is based on a two-section QD laser diode with a total
length of 1 mm 共saturable absorber length is 1/10 of the total
length兲anda4
m wide ridge waveguide structure. The
active zone of the device contains 15 layers of self-organized
InAs QDs emitting at 1.3
m embedded in InGaAs quantum
wells.6The threshold current density of the diode is
360 A/cm2.
In the absence of the external modulation, the laser ex-
hibits passive ML, which remains stable in a wide range of
operating parameters. External modulation of the voltage V
is applied to the absorber section of the device and has the
form of V
¯
关1+acos共⍀t兲兴 where a,⍀, and V
¯
are the ampli-
tude, frequency, and dc component of the reverse bias, re-
spectively.
In order to investigate the locking range, the optical out-
put of the laser diode was measured using a high speed
55 GHz bandwidth photodetector and a 50 GHz electrical
spectrum analyzer. For various operating parameters 共injec-
tion current in the gain section, reverse bias applied to the
absorber section兲the external frequency was varied around
the cavity round trip frequency and from that the locking
range was determined. The criterion to consider operation as
locked to the external modulation was that the side band
suppression ratio in the electrical spectra was larger than
30 dB. The transition from locked to unlocked state is rather
sharp within our measurement resolution. Maximum locking
range of ⬃30 MHz was achieved for the smallest values of
current and reverse bias, requiring maximum modulation
amplitude. For a fixed set of injection current and absorber
bias 共80 mA, ⫺8V兲the variation of the modulation ampli-
tude results in a locking tongue shown in Fig. 1, starting with
about 7 MHz for the maximum applied modulation down to
a sub MHz range for 0.4 Vpp 共voltage peak-to-peak兲or less.
Linking the experimental modulation amplitude of 3.2 Vpp
to the modulation amplitude acannot be done with certainty,
due to the fact that the absorber section is an unmatched load
and external modulation power is partially reflected from it.
In Fig. 13.2 Vpp can correspond to a value of modulation
amplitude abetween 0.4 and 0.8.
The locking range is strongly red-shifted and asymmet-
ric with respect to the pulse repetition frequency of the pas-
sively mode-locked laser, as shown in the inset. The inset in
Fig. 1demonstrates the improvement of the jitter in hybrid
operation by reduction of the linewidth, while pulsewidth
and pulse shape stay the same, determined by autocorrelator
measurements. The asymmetry is maintained for all ampli-
tudes of the external signal. As discussed later, this asymme-
a兲Electronic mail: [email protected].
APPLIED PHYSICS LETTERS 96, 011104 共2010兲
0003-6951/2010/96共1兲/011104/3/$30.00 © 2010 American Institute of Physics96, 011104-1
try is related to the nonlinear frequency dependence of the
amplitude of the locked solution. An extension of the mea-
surement, where the injection current was kept fixed and the
reverse bias is varied, is plotted in Fig. 2. Here only the
overall locking range is evaluated. The values show a purely
linear dependence on the modulation amplitude for all oper-
ating parameters. First, from ⫺4to⫺8 V the locking range
decreases with bias. By increasing the bias further to ⫺10 V
the locking range gets larger again. This can be explained by
comparing the behavior of the locking range to the optical
pulse width, which has been obtained by measuring and fit-
ting the autocorrelations. As it is shown in the inset of Fig. 2,
shorter pulses with larger peak power exhibit smaller locking
ranges.
To explain these observations, we use the model of a QD
mode-locked laser described in Refs. 8and 9. This model is
described by a set of five delay differential equations
␥
−1
tA共t兲+A共t兲=冑
e共1−i
␣
g兲G共t−T兲/2+共1−i
␣
q兲Q共t−T兲/2A共t−T兲,
共1兲
t
g=Ng共1−
g兲/
g
cap −
g/
g
esc −
g/
g−sgIg,共2兲
tNg=共Ng0−Ng兲/
N−2Ng共1−
g兲/
g
cap +2
g/
g
esc,共3兲
t
gs =2
es共1−
gs兲/
q
cap −2
gs共1−
es兲/
q
esc −
gs/
q
−sqIq,共4兲
t
es =−
es/
w−
es共1−
q兲
q
cap +
gs共1−
es兲/
q
esc,共5兲
where A共t兲is the normalized complex amplitude of the
electric field at the entrance of the absorber section. Ig
=e−Q共eG−1兲兩A兩2and Iq=共1−e−Q兲兩A兩2.8The variables G共t兲
=2ggLg关2
g共t兲−1兴and Q共t兲=2gqLq关2
gs共t兲−1兴are the time-
dependent dimensionless cumulative saturable gain and ab-
sorption, and the parameters gg,qare the differential gains in
the corresponding sections. The parameters sq⬎sgare in-
versely proportional to the saturation intensities of the gain
and absorber sections.
In Eqs. 共1兲–共5兲, the delay Tis equal to the cold cavity
round trip time. The attenuation factor
⬍1 describes the
total nonresonant linear intensity losses per cavity round trip.
The dimensionless bandwidth of the spectral filtering is
␥
and the linewidth enhancement factor in the gain 共absorber兲
section is
␣
g共
␣
q兲.
The variables
g共t兲and Ng共t兲are the dot occupation
probability and the normalized carrier density in the wetting
layer of the gain section.
gand
Nare the recombination
rates in the wetting layer and in the dots, respectively. The
dimensionless parameter Ng0is a measure of carrier injection
in the amplifier section. For this section, we use a carrier
capture time of
g
cap=10 ps and an escape time
g
esc=80 ps.10
The carrier exchange dynamics for the absorber section
is described by the occupation probabilities
gs and
es of the
ground 共GS兲and the first excited 共ES兲state of a dot,
respectively.11,12 The parameters
q
cap,
q
esc, and
wdetermine
the time-dependent recovery of the QD absorber.
q
cap=2 ps
and
q
esc=10 ps,
w=
w0exp共兩V共t兲/V0兩兲, where
w0=18 ps
and V0⬇−2 V. The modulation of the applied voltage re-
sults in V共t兲=V
¯
关1+acos共⍀t兲兴, where V
¯
is the dc component
of the reverse bias applied to the absorber section. Other
parameters are:
␥
=20, s=sq/sg=20.
The numerical bifurcation diagram in Fig. 3shows the
locking tongue in two-parameter-plane, rf modulation fre-
quency ⍀, and amplitude a. The vertical dotted line indicates
the pulse repetition frequency of the free running passively
FIG. 1. 共Color online兲Experimentally measured change of the locking
range for a fixed injection current and absorber bias with varying modula-
tion amplitude 共locking tongue兲. The dots are marking the borders of the
locking range. The inset shows the asymmetry of the locking range 共black
and red line兲with respect to the passive mode locking frequency 共green line兲
for the maximum modulation amplitude of 3.2 Vpp.
FIG. 2. 共Color online兲Dependence of measured locking range on the modu-
lation amplitude. For a fixed gain current of 60 mA the absorber bias is
varied in four steps while the locking range is measured in dependence of
the modulation amplitude. The inset shows the behavior of the maximum
locking range in comparison to the width of the optical pulses. The values
for the locking range in the inset are for the highest modulation amplitude.
FIG. 3. Locking tongue calculated numerically. Thick lines indicate the
stability boundaries of the locking regime 共locking range兲.sn—saddle-node
bifurcation lines. H—Andronov–Hopf bifurcation line. Thin lines corre-
spond to saddle-node bifurcations of unstable solutions. Vertical dotted line
is the passive mode-locked laser frequency. The inset shows branches of
locked solutions. Stable 共unstable兲solutions are shown by solid 共dotted兲
lines. 共No.1isfora=0.5 and No. 2 for a=0.7兲. Empty dots indicate Hopf
共h兲and saddle-node 共sn兲bifurcations responsible for the destabilization of
the locked solutions.
011104-2 Fiol et al. Appl. Phys. Lett. 96, 011104 共2010兲
mode-locked laser 共a=0兲. Figure 3is very similar to the
experimental results shown in Fig. 1. For sufficiently large
modulation amplitudes the locking range is strongly asym-
metric with respect to this frequency. The asymmetry is re-
lated to the nonlinear dependence of the pulse repetition fre-
quency on the pulse amplitude, typical of nonlinear
resonance phenomenon13 共see inset in Fig. 3兲. The diagram
in Fig. 3demonstrates the fundamental difference between
bifurcation mechanisms, leading to locking at small and
large amplitude modulations. At small amplitudes of modu-
lation 共a⬍0.52兲the locking domain has the form of a nar-
row tongue bounded by two saddle node bifurcation lines.
This gives a sharp transition from locked to unlocked state in
agreement with our experimental observations. In this case
the branch of locked solutions is isolated in the parameter
space, see inset of Fig. 3. When crossing the saddle-node line
two locked solutions with periodic laser intensity, stable and
unstable, merge and disappear, giving rise to a quasiperiodic
motion on a torus. This is a standard mechanism of unlock-
ing, typical for small amplitude modulation.
Another bifurcation mechanism of unlocking appears at
sufficiently large modulation amplitudes a. The branch of
locked solutions becomes connected to the unstable branch
of modulated cw solutions, 共see curve 2 in the inset of Fig.
3兲, the left saddle-node bifurcation line disappears in the
cusp point, and the lower boundary of the locking range
becomes limited by an Andronov–Hopf bifurcation. It is seen
from the figure that this transition results in a significant
increase of the locking range asymmetry. Similar transforma-
tions take place at the right boundary of the locking range.
However, due to the strong asymmetry of the locking tongue,
the Andronov–Hopf bifurcation at the right boundary of the
locking tongue appears at larger modulation amplitudes than
that at the left boundary 共not shown in Fig. 3兲.
In agreement with the experimental data presented in the
inset of Fig. 2, our simulations predict that shorter pulses
with larger peak power have a smaller locking range. This is
illustrated by Fig. 4共b兲, which shows the dependence of this
range on the ratio s=sq/sgof the saturation intensities in the
gain and absorber sections. Increase of the parameter sleads
usually to an increase of the ML stability domain and im-
provement of the pulse quality.7,14,15 A similar decrease of
the locking range is observed with the increase of the spec-
tral filtering width
␥
which also leads to a decrease of the
pulse width, see Fig. 4共a兲. This figure also explains the dis-
crepancy between the values of the locking range widths
shown in Figs. 1and 3. For convenience in numerical simu-
lations we have used rather small values of the parameter
␥
.
Indeed, for
␥
−1=0 the derivative disappears from the field
Eq. 共1兲which becomes a discrete map. All the periodic so-
lutions of this map have a period equal to the delay time T.
Hence, such solutions will never synchronize to the external
modulation if the modulation period differs from T. Increas-
ing the locking range with the decreasing
␥
suggests that the
inclusion of an additional spectral filtering section into the
laser cavity can lead to an increase of the locking range 关See
Fig. 4共a兲兴. The experimentally determined optical spectra
were three to four times wider than those calculated theoreti-
cally. This suggests that the value of
␥
approximating most
closely the value of gain bandwidth is three to four times
larger than that used in simulations.
In conclusion, we studied hybrid ML in a two section
QD laser. The experiments demonstrate and the theory con-
firms, that the hybrid ML operation in QD lasers exhibits
a strongly asymmetric locking range which is smaller than
that for QW lasers. Our model explains the appearance of
a locked state as a result of saddle node and inverse
Andronov–Hopf bifurcations. Our numerical results indicate
that the locking range in hybrid mode-locked QD lasers
might be improved by incorporating an additional spectral
filtering section.
A.G.V is grateful to D. Rachinskii and D. Turaev for
stimulating discussions. The authors in Brussels acknowl-
edge support of the Fonds National de la Recherche Scienti-
fique 共Belgium兲. The authors at TU Berlin and WIAS would
like to acknowledge the funding of this work by the Grant
No. SFB787 of the DFG.
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FIG. 4. Plots illustrating the dependence of the locking frequency range on
the spectral filtering width
␥
共a兲and ratio of the saturation intensities in gain
and absorber sections s=sq/sg共b兲. Locking domain is shown by gray color.
011104-3 Fiol et al. Appl. Phys. Lett. 96, 011104 共2010兲
