CHAIRE PHOTONIQUE
Optical injection dynamics of frequency
combs
Yaya DOUMBIA, Tushar MALICA, Delphine WOLFERSBERGER, Krassimir PANAJOTOV and
Marc SCIAMANNA
January 10, 2020
We analyze the nonlinear dynamics of a semiconductor laser with optical injection from a frequency comb.
When varying both the injection parameters (detuning and injection ratio), and the comb properties (number
of comb lines and comb spacing), we identify and select several dynamics including (i) injection locking with
selective amplification of the comb line that shows the smallest detuning from the injected laser, (ii) unlocked
time-periodic dynamics that correspond to a new frequency comb with a broadened optical spectrum thereby
sharpening the pulse generated at the injected laser output, and (iii) unlocked chaotic dynamics.
Keywords: Semiconductor lasers, nonlinear dynamics, optical injection, frequency comb.
The injection locking (IL) properties of semiconductor
lasers have been analyzed since more than forty years: first,
as a way to control the coherence properties of the injected
laser, but more recently, as a technique to tailor specific non-
linear dynamics including optical chaos [1,2], time-periodic
self-pulsation [3], and also dissipative solitons in large aper-
ture laser diodes [4, 5]. These nonlinear dynamics found
applications in various fields such as physical security based
on optical chaos [6,7], optical sensing [8] and radio-over
fiber communications [9
–
11]. It is known that a multi-
frequency injected laser, unlike the case of a single mode
injected laser diode, brings additional nonlinear dynamics
and examples of which can be found in injection experi-
ments exploring polarization dynamics in VCSELs [12
–
14],
longitudinal mode dynamics in quantum dot lasers [3,15],
and the so-called two-colour laser diodes [16]. Much less
studied is the dynamics of a single mode laser diode injected
by a multi-frequency laser diode. This configuration has
raised recent interest within the context of optical frequency
combs, i.e. a coherent set of optical modes being mode-
locked and producing possibly very short pulses in the laser
output [17
–
19]. Recent experimental and theoretical work
have considered optical injection of such frequency combs
as a way to selectively amplify individual comb lines and
therefore, tailor the comb properties [20
–
23]. In addition,
the question of how optical injection induced nonlinear dy-
namics impacts the comb properties in the injected laser is
of great fundamental interest and has not been addressed
so far.
In this Letter, we make an in-depth analysis of single-
mode semiconductor laser dynamics induced by optical
injection of a frequency comb by varying both the injection
parameters and the comb properties. Besides the IL and the
selective amplification of the comb line with the smallest
detuning from the injected lines that has been reported in
several recent experiments [22,23], we reveal a large set
of nonlinear dynamics that bifurcate from the IL solution.
Of particular interest is the onset of a new time-periodic
dynamics in the unlocking regime that extends the injection
combs lines to a much broader range of optical frequencies,
hence, also significantly sharpening the pulse in the injected
laser output. The number of the resulting comb lines can
be tailored by varying the injection parameters and the
initial comb properties (number of lines and the comb
spacing). The laser system is modelled with the equations
from Ref. [24] adapted to injection from a comb. The
complex electrical field of the injected comb can be written
as:
E
M
(t) =
X
j
E
j
(t)e
i(2πν
j
t+ϕ)
, (1)
where
E
j
(t)
is the amplitude of the j-th comb-line,
ν
j
is the
frequencie of the j-th comb-line,
ϕ
is the initial phase of
each comb-line. We set the initial phase of each comb-line
to 0 and we suppose that the comb lines have the sames
amplitudes,
E
inj
. By doing the same calculation as in [24],
the injected laser’s equations can be written as:
˙
E(t) =
1
2
G
N
(N(t)−N
th
)E(t)+E
inj
X
j
cos(2π∆ν
j
t−φ(t)),
(2)
˙
Φ(t) =
1
2
αG
N
(N(t)−N
th
)+
E
inj
E(t)
X
j
sin(2π∆ν
j
t−φ(t)),
(3)
˙
N(t) = R
p
−
N(t)
τ
s
− G
N
(N(t) − N
th
)E(t)
2
−
E(t)
2
τ
p
. (4)
In these equations,
∆ν
j
is the detuning,
N
th
is the threshold
carrier density,
α
is the linewidth enhancement,
G
N
is the
differential gain,
R
p
is the pump rate and
τ
s
is the carrier
lifetime. In the following, we shall use
κ
for the injection
ratio with
κ =
E
inj
E
0
, where
E
0
is the field amplitude of the
injected laser without optical injection. The semicondutor
1
Yaya DOUMBIA et al. Optical injection dynamics of frequency combs
laser parameters are taken as
α
=5,
G
N
= 7.9 × 10
−13
m
3
s
−1
,
N
th
= 2.91924 × 10
24
m
−3
,
τ
s
= 2 × 10
−9
s,
τ
p
= 2 × 10
−12
s and
R
p
= 1.2R
th
with the pump rate
at threshold
R
th
= 1.8 × 10
33
s
−1
. In equations (2)-(4)
it is assumed that the bandwidth of the wavelength selec-
tive filter in the injected laser (gain bandwidth and cavity
bandwidth) is much larger than the comb width, hence not
preventing the new additional comb lines to be lasing as
will be discussed here after. Rate equations (2)-(4) are in-
tegrated using the fourth-order Runge-Kutta method with
a time step of 1.2 ps.
Figure 1:
Optical spectra shown in (a) shows the injected comb for
3 comb lines, ( b) free running injected laser and (c) injected laser
under IL condition.
Ω = 5
GHz is the comb spacing and
∆ν = 10
MHz is the detuning.
Figure 1 shows optical spectra of the injected frequency
comb [Fig. 1 (a)], the free running of the injected laser
[Fig. 1(b)] and the injected laser under IL [Fig. 1 (c)].
The optical spectra in Figs. 1 (b) and (c) are obtained by
discrete Fourier transform of the optical field from 80 ns
long time-trace. IL is achieved in Fig. 1 (c) for
∆ν = 10
MHz from the central comb line.
The frequency comb laser has 3 lines with equal am-
plitudes separated by
Ω = ν
j+1
− ν
j
=5 GHz and hence
the injecting diode laser has modulated output power with
pulse full-width at half maximum (FWHM) of 62.4 ps. The
pulse repetition rate of
Ω = 5
GHz correspond to the comb
spacing. In IL condition Fig. 1 (c), the injected laser selec-
tively amplifies the frequency comb line that is the closest to
the injected laser frequency and simultaneously suppresses
the side comb lines. The resulting injected laser output is
not steady but modulated at a frequency matching
Ω
. This
selective amplification of a comb line under IL matches
recent experimental observations [22,23].
We next analyze the emergence of complex dynamics
from the injection locked solution. For a fixed
∆ν =
0.2
GHz, and varying
κ
, the injected laser dynamics shows
additional bifurcations from the IL solution as shown in
Fig. 2 for 3 (top panels and 7 (bottom panels) of injected
comb lines. Figures 2 (
a
1
) and (
b
1
) show wave mixing,
i.e., the injected laser dynamics is intensity modulated as
a result of nonlinear mixing between two frequencies, the
detuning ∆ν and Ω.
Figures 2 (
a
2
) and (
b
2
) show IL for 3 and 7 comb lines
injection. In both cases, the sidemode suppression ratio is
the same (equal to 14 dB). However, the selective amplifi-
cation of the injected comb line is stronger for the 3-comb
lines injection. The appearance of the additional comb
lines is due to the nonlinear dynamics taking place in the
injected laser, namely, the injection frequency comb leads
to modulation of the carrier density at the frequency given
by the spacing of the comb lines. Figures 2 (
a
3
) and (
b
3
)
show two examples of chaotic dynamics. A close inspec-
tion of the corresponding power spectra shows that the
chaos bandwidth of the 7 comb lines case (10.57 GHz) is
slightly larger than for the 3 comb lines case (9.92 GHz),
computed as in [25]. Figures 2 (
a
4
) and (
b
4
) correspond
to the unlocked time-periodic dynamics with the injected
laser output displaying a new frequency comb. We con-
sider a new comb line when its amplitude lies above -30
dB from the maximum amplitude: we observe that the
7-comb lines injection leads to a wider optical spectrum
with 29 lines involved in the comb rather than 17 lines for
3-lines case. When further increasing
κ
, we find yet another
comb with even broaded optical spectrum see Fig. 2 (
a
5
)
3-comb lines and Fig. 2 (
b
5
) 7-comb line injection. The red
dashed line in each optical spectrum indicates the injection
locked frequency position, i.e., the frequency position of
the injected central comb line. It is interesting to note that
the unlocked comb is frequency shifted from the injection
locked solution, with a frequency offset that increases with
κ
. The increase of
κ
leads to a decrease of the average car-
rier density, yielding also more power in the injected laser
and therefore to a redshift of the instantaneous frequency
through equation (3). This explains the shift of the comb
main line to the red when increasing κ.
The 3-comb lines and 7-comb lines injection cases differ
by the range of injection parameters in which these new
comb are observed. In Fig. 3, we map in detail the region
of the time-periodic dynamics in the plane of the injection
parameters. To this aim, We follow the maxima of the time
series and compare the heights of the consecutive maxima
for all the maxima lying above the mean. We consider a
time series to be a comb when this difference is less than
0-2% of the maximum amplitude meaning that the dy-
namics is indeed time-periodic. The region shaded in blue
corresponds to the time-periodic dynamics, i.e., either an
injection locked, or an unlocked injected laser comb. The
region shaded in red corresponds to any other unlocked dy-
namics. We are not interested here by the peculiar chaotic
properties of the injected laser dynamics but, in the pa-
rameter range in which the injected laser shows an optical
comb dynamics and the corresponding comb properties.
Interestingly, the unlocked comb dynamics of the injected
laser extends to a much broader range of injection param-
eters than the IL solution. Figures 3 (a)-(c) correspond
to 3-comb lines injection with
Ω
equal to 3, 5 and 10 GHz,
respectively. Figures 3 (e)-(g) correspond to 7-comb lines
injection with
Ω
equal to 3, 5 and 10 GHz, respectively.
The region around zero
κ
is the IL region. We identify as
many IL regions as the number of the comb lines, labelled
’IL’ in the mapping. The IL region is limited when the ratio
between the power of the comb line that shows the smallest
detuning from the injected comb and the next strongest
comb line is more than 10 dB. The green points in the map-
2
Yaya DOUMBIA et al. Optical injection dynamics of frequency combs
Figure 2:
Optical spectra when varying the injection ratio
κ
.The top and bottom panels correspond to 3 and 7 comb line injection,
respectively. All the figures are obtained with
∆ν = 0.2
GHz except
b
5
which is obtained with
∆ν = 10
GHz. Evolution of dynamics is as
follows (
a
1
) and (
b
1
) wave mixing at
κ
=0.006, (
a
2
) and (
b
2
) injection locking at
κ
=0.012, (
a
3
) and (
b
3
) chaos at
κ
=0.1, (
a
4
) and
(
b
4
) first unlocked time-periodic regime for 3 and 7 comb line at
κ
=0.202 and
κ
=0.284, respectively and (
a
5
) and (
b
5
) second unlocked
time-periodic regime at κ=0.6. The red dashed line in each optical spectrum indicates the injection locked frequency position.
Figure 3:
Numerical mapping of the semiconductor laser dynamics when subject to optical injection with frequency combs is shown by
varying
κ
as a function of detuning frequency for for comb spacing of 3, 5 and 10 GHz respectively for (a)-(c) 3 comb lines injection and,
(d)-(f)7 comb lines. Different regions are observed: Injection locking (IL), unlocked time-periodic dynamics (Comb1, Comb2, Comb3,
Comb4 and Comb5). The green color corresponds to the limit of the injection locking region.
ping are the limit of the IL region. In the case of
Ω = 5
GHz,
we observe two regions of unlocked time-periodic dynamics
(see regions labelled, ’comb1’ and ’comb2’ in the mapping).
The areas corresponding to the new comb extend both with
κ
and with
∆ν
. The new combs are observed for
Ω
ranging
between below and up to several times relaxation oscil-
lation frequency (3.6 GHz). This confirms that the comb
appearance is not restricted to injected comb spacing
Ω
close to the oscillation relaxation frequency. Tongue-like
comb regions in the mapping (Fig. 3 (c) and (f)), are re-
dishifting when increasing
κ
and become more structured
and narrower when increasing
Ω
. We also notice that when
increasing the number of the injected comb lines, the areas
corresponding to the new comb extend more towards the
negative detuning as
κ
increases. As is made clear for ex-
ample in Fig. 3(b) for
Ω = 5
GHz, the new comb connects
to the IL solution, hence suggesting that this new comb
indeed originates from the IL solution. We have checked
that the transition from the IL to the comb is smooth with
an increased power of the comb lines and the newly created
equidistant frequency lines.
Tailoring the comb properties by varying the injection
parameters is better seen in Fig. 4: we plot the pulse of the
injected laser output when varying
κ
within the parameter
range corresponding to the unlocked injected laser comb
(comb1 and comb2) of the Fig. 3 (b). In Fig. 4 (a), the
blue and red curves correspond to the pulses in (comb1)
and (comb2) region for
κ = 0.35
and
κ = 0.6
respectively
with the same detuning
∆ν = 0.2
GHz. The pulse width
decreases when
κ
increases. In Fig. 4 (b) we show the
number of comb lines of the injected laser versus number of
lines of the injected comb. The new comb becomes wider
with the number of injected comb lines. It is interesting to
note that the new comb extends more when increasing the
3
Yaya DOUMBIA et al. Optical injection dynamics of frequency combs
Figure 4:
Control of comb properties for
Ω = 5
GHz for 3 comb
lines injection. (a) pulses in comb1 and comb2 for
∆ν = 0.2
GHz,
κ = 0.35
(red) and
κ = 0.6
(blue). (b) number of comb lines in the
injected laser versus number of lines of the injected comb for
∆ν = 0
GHz and
κ = 0.8
. (c) FWHM of the comb1 on the left vertical axis
and time-bandwidth product (TBP) on the right vertical axis when
increasing
κ
. (d) same as (c) for the comb2 also comparing the
cases of Fig. 3 (b) for
∆ν = −0.7
GHz (red),
∆ν = 0
GHz (blue)
and ∆ν = 0.7 GHz (brown).
drive current of the injected laser. For example, in the case
of Fig. 5 (a), when we increase the drive current of the
injected laser by 15%, the number of comb line goes from
21 to 27. In Fig. 4 (c) we show the pulse width on the left
vertical axis for comb1 and the time-bandwidth product
(TBP) on the right vertical axis when increasing
κ
. The TBP
is calculated from the product between the FWHM and
the spectral width at half maximum that is computed from
the simulated optical spectra. The pulse width decreases
with
κ
, but the TBP increases, suggesting that the pulse
width is not only determined by the increase of spectral
bandwidth, but also that there should be a dependency on
the phase of the comb line as will be discussed here after.
The TBP deteriorates as the pulses width decreases and
reaches about twice the Fourier limit
∼
0.44 for a Gaussian
shape pulse. Figure 4 (d) shows the pulse width for 3
detuning values of 0.7 GHz, 0 GHz and -0.7 GHz for the
case of comb2. The pulse width is smaller for the negative
∆ν than for the zero or positive ∆ν.
Our statement above on the phase of the comb lines is
further supported by the analysis of Fig. 5. Figures 5 (a)
and (b) show the optical spectra on the left vertical axis
and the phase of each comb line on the right vertical axis.
In both cases, we see that the comb lines have an almost
constant phase difference between them. The linear fit of
the phase difference versus frequency gets a little bit worst
when increasing the number of comb lines as shown in
Fig. 5 (c).
In summary, we show that it is possible to generate a
broadened frequency comb using injection locking. We
demonstrate this technique using a frequency comb as in-
jected laser and single mode laser as injected laser. When
Figure 5:
Optical spectra and phase for each comb line for
∆ν = 0
GHz,
κ = 0.8
and
Ω = 5
GHz. (a) and (b), optical spectra (left
vertical axis ), phase (right vertical axis) for 3 and 7 comb line,
respectively. (c) fitting of the linear part of the phase dependance
for 3 comb lines (red) and 7 comb lines (blue)
.
increasing the injection ratio, the injected laser first shows
wave mixing with its intensity modulated from the non-
linear mixing between the detuning frequency and a new
frequency that depends on the comb mode spacing. Increas-
ing the injection ratio, the injected laser gets locked with
selective amplification of the comb mode with smallest de-
tuning to the injected laser. Consecutively, more complex,
including chaotic, dynamics is observed. Most importantly,
new comb solutions take place for still increased injection
ratio. The number of lines involved in the new comb so-
lution, and therefore, the pulse width of the mode-locked
injected laser dynamics is controlled by injection parame-
ters, and by the number of the injected modes. In particular,
the injected laser pulse width decreases when increasing
the injection ratio and when moving the injected comb
towards more negative detuning.
Funding
. Chaire Photonique: Ministère de
l’Enseignement Supérieur, de la Recherche et de
l’Innovation; Région Grand-Est; Département Moselle;
European Regional Development Fund (ERDF); Airbus
GDI Simulation; CentraleSupélec; Fondation Centrale-
Supélec. Fondation Supélec; Metz Metropole, Fonds
Wetenschappelijk Onderzoek (FWO) Vlaanderen Project
No.G0E5819N.
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