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We study in detail the complex dynamics of an erbium-doped fiber laser that has been subjected to harmonic modulation of a diode pump laser. We introduce a novel laser model that describes perfectly all experimentally observed features. The model is

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Dynamics of an erbium-doped ﬁber laser withpump modulation: theory and experiment
Alexander N. Pisarchik, Alexander V. Kir’yanov, and Yuri O. Barmenkov
Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, Leon 37150, Guanajuato, Mexico
Rider Jaimes-Reátegui
Universidad de Guadalajara, Campus Universitario Los Lagos, Enrique Diaz, Paseo de Las Montanas, C.P. 47460,Lagos del Moreno, Jalisco, Mexico
Received September 28, 2004; revised manuscript received April 17, 2005; accepted May 4, 2005
We study in detail the complex dynamics of an erbium-doped ﬁber laser that has been subjected to harmonicmodulation of a diode pump laser. We introduce a novel laser model that describes perfectly all experimentallyobserved features. The model is generalized to a nonlinear oscillator. The coexistence of different periodic andchaotic regimes and their relation to subharmonics and higher harmonics of the relaxation oscillation fre-quency of the laser are demonstrated with codimensional-one and codimensional-two bifurcation diagrams inparameter space of the modulation frequency and amplitude. The phase difference between the laser responseand the pump modulation is also investigated. © 2005 Optical Society of America
OCIS codes:
140.3510, 140.1540
.
1. INTRODUCTION
In the past decades, revolutionary progress has beenachieved in research and commercialization of erbium-doped ﬁber lasers (EDFLs). The advantages of these la-sers are the long interaction length of pumping light withthe active ions, which leads to high gain and to single-transversal-mode operation produced by a suitable choiceof ﬁber parameters. These properties make EDFLs excel-lent light sources for optical communications, reﬂectom-etry, sensing, medicine, etc.
1,2
Meanwhile, these lasersare quite sensitive to any external perturbation that maydestabilize their normal operation. Therefore, knowledgeof the dynamic behavior of these lasers under externalmodulation is of great importance and can be importantfor many applications.From the viewpoint of nonlinear dynamics, rare-earth-doped ﬁber lasers with external modulation, along withsolid-state, semiconductor, and electric discharge CO
2
and CO lasers, are class-B lasers.
3
These are nonautono-mous systems in which polarization is adiabatically elimi-nated and the dynamics can be ruled by two rate equa-tions for ﬁeld and population inversion. In spite of animpressive array of research on complex dynamics in la-sers, the nonlinear dynamics of EDFLs has begun to bestudied only recently. The main features of the dynamicbehavior of these lasers are similar to those of otherclass-B lasers. Different conditions for the development of chaotic motion have been found in EDFLs. First, a period-doubling route to chaos was observed by Lacot
et al.
4
in abipolarized two-mode EDFL with harmonic pump modu-lation. Those authors have also developed a model basedon two coherently pumped coupled lasers. A quasi-periodic route to chaos was found by Sanchez
et al.
5
in adual-wavelength EDFL. Eventually, Luo
et al.
6
revealedthe coexistence of period-doubling and intermittencyroutes to chaos in a pump-modulated ring EDFL. Theyalso reported on bistability (the coexistence of two peri-odic attractors) in this laser.
6,7
More recently, optical bi-stability (coexistence of a limit cycle and a ﬁxed point)was detected by Mao and Lit
8
in the vicinity of the ﬁrstlaser threshold in a dual-wavelength EDFL with overlap-ping cavities.Previously, the authors and others have reported thecoexistence of multiple periodic attractors (generalizedmultistability) found both theoretically and experimen-tally in EDFLs subjected to loss
9
or pump modulation.
10,11
Many papers have been devoted to a study of self-pulsation behavior of EDFLs (see, for example, Refs.12–14). Such behavior has been suggested to be due to thepresence of a saturable absorber in the ﬁber in the form of ion pairs
12
or pump depletion.
15
The
Q
-switching behaviorcan also be attributed to excited-state absorption (ESA) atthe lasing wavelength
16
and to a thermo-lensing effectthat is due to ESA at the pump wavelength.
17
Only a few papers have been devoted to a study of thenonlinear response of the EDFL to parametric modula-tion. The dynamics of this laser were reported recently bySola
et al.
18
and by the present authors and others.
9–11
Sola
et al.
studied the dynamics of a ring 1533 nm EDFLwith a sinusoidally modulated 1470 nm pump diode laser.They developed a rather complicated model,
19
which de-scribes their experimental results well. Previously westudied a linear 1560 nm EDFL pumped by a 967 nm la-ser diode. Such a laser is commonly used in many labora-tories and serves for various applications. However, thespectroscopy of this laser is quite different from that of the laser studied by Sola
et al.
, and hence their modelcannot describe our laser.In this paper we study the dynamics of a 1560 nmEDFL with a Fabry–Perot cavity that has been subjected
Pisarchik
et al.
Vol. 22, No. 10/October 2005/J. Opt. Soc. Am. B 21070740-3224/05/102107-8/$15.00 © 2005 Optical Society of America
to harmonic pump modulation of a diode pumping laser.We develop a novel model that can be used to describesuch a laser and that, as we show below, perfectly ad-dresses all laser peculiarities observed experimentally.We investigate theoretically the laser dynamics over awide range of frequencies and amplitudes of pump modu-lation.The paper is organized as follows: In Section 2 we in-troduce the novel model of the EDFL, normalized equa-tions, and transform the laser equations into a singleequation of a nonlinear oscillator. Then in Section 3 wepresent the results of numerical simulations of experi-ments reported in our previous papers
10,11
for the casewhen the modulation frequency is higher than the relax-ation oscillation frequency of the laser and we simulatethe experiments presented in this paper for the case whenthe modulation frequency is smaller than the relaxationoscillation frequency. We demonstrate that the low-frequency range furnishes a rather interesting insightinto EDFL dynamics with external modulation because of the appearance of many ﬁne dynamic phenomena that be-come latent at higher modulation frequencies. In Section4 we describe our experimental setup and compare the re-sults of the numerical simulations with the experimentalresults. In the course of experiments, we determine di-rectly the structure of frequency-locked and phase-lockedstates (with respect to pump modulation) through bifur-cation diagrams in space of the modulation parameters.Finally, our main conclusions are given in Section 5.
2. THEORY
A. Laser Model
The model is based on the rate equations in which we usea power-balance approach applied to a longitudinallypumped EDFL, in which the ESA in erbium at the1.5-
m wavelength and the averaging of the populationalong the pumped active ﬁber are taken into account.Such a model would address the two most evident factors,i.e., the ESA at the laser wavelength and the depleting of the pump wave at propagation along the active ﬁber. Anenergy-level diagram of our model is shown in Fig. 1. Thismodel does not include the mechanisms that are respon-sible for establishing the self-pulsing regime in the laser,such as a thermo-lensing effect
17
and erbium ion pairs inthe ﬁber,
12
for the following reasons. First, in our experi-ments the pump power is too small to induce thermo lens-ing and second, the concentration of erbium is too low tomake the effect of the ion pairs signiﬁcant.The balance equations for intracavity laser power
P
(which is a sum of the powers of the contrapropagatingwaves inside the cavity, in inverse seconds) and the aver-aged (over the active ﬁber length) population
N
of the up-per (2) level (which is a dimensionless variable, 0
N
1)ared
P
d
t
=2
LT
r
P
r
w
0
N
1
−
2
−1
−
th
+
P
sp
,
1
d
N
d
t
=−
12
s
r
w
P
r
02
N
1
−1
−
N
+
P
pump
,
2
where
N
=
1/
n
0
L
0
L
N
2
z
d
z
(
N
2
is the population of up-per laser level 2,
n
0
is the refractive index of a colderbium-doped ﬁber core, and
L
is the active ﬁber length)and
12
is the cross section of the absorption transitionfrom ground state 1 to upper state 2. Here we assumethat the cross section of the return stimulated transitionis practically the same
12
=
21
as that which yields
1
=
12
+
21
/
12
=2.
2
=
23
/
12
=0.4 is the coefﬁcient thatstands for the ratio between the ESA
23
and ground-state absorption cross sections at the laser wavelength,
T
r
=
2
n
0
/
c
L
+
l
0
is the photon intracavity round-triptime [
l
0
is the total length of the ﬁber Bragg grating(FBG) coupler tails inside the cavity],
0
=
N
0
12
s
is thesmall-signal absorption of the erbium ﬁber at the laserwavelength (
N
0
=
N
1
+
N
2
is the total concentration of er-bium ions in the active ﬁber and
s
is the overlap factorfor EDFL radiation),
th
=
0
+
1/2
L
ln
1/
R
is the intrac-avity loss on threshold (
0
is the nonresonant ﬁber lossand
R
is the total reﬂection coefﬁcient of the FBG cou-plers),
is the lifetime of erbium ions in excited state 2,
r
0
is the ﬁber core radius,
w
0
is the radius of the fundamen-tal ﬁber mode, and
r
w
=1−exp
−2
r
0
/
w
0
2
is the factorthat addresses a match between the laser fundamentalmode and erbium-doped core volumes inside the active ﬁ-ber. In Eq. (1),
P
sp
=
N
g
w
0
2
r
02
0
L
4
2
12
s
T
r
10
−3
is the spontaneous emission into the fundamental lasermode. We assume here that the laser spectrum width is10
−3
of the erbium luminescence spectral bandwidth (
g
isthe laser wavelength). In Eq. (2),
P
pump
=
P
p
/
N
0
r
02
L
1−exp
−
p
L
1−
N
is the pump power, where
P
p
is thepump power at the ﬁber entrance and
p
=
N
0
14
p
is thesmall-signal absorption of the erbium ﬁber at the pumpwavelength (
14
is the cross section of the absorptiontransition from level 1 to level 4 and
p
is the overlap fac-tor for pump radiation). The system of Eqs. (1) and (2) de-scribes the laser dynamics without external modulation.The harmonic pump modulation is added as
Fig. 1. Erbium energy-level diagram.2108 J. Opt. Soc. Am. B/Vol. 22, No. 10/October 2005 Pisarchik
et al.
P
p
=
P
p
0
1+
m
sin
2
F
m
t
,
3
where
m
and
F
m
are the modulation depth and frequency,respectively, and
P
p
0
is the pump power without modula-tion (at
m
=0).The calculations are performed for the experimentalconditions described in Refs. 10 and 11 and in Section 5below. The parameter values are presented in Table 1.The value of
w
0
has been measured experimentally,and it is a bit higher than the value 2.5
10
−4
cm given bythe formula for a step-index single-mode ﬁber:
w
0
=
r
0
0.65+1.619/
V
1.5
+2.879/
V
6
, where
V
is related to nu-merical aperture NA and
r
0
, as
V
=
2
r
0
/
g
NA; the val-ues
r
0
and
w
0
result in
r
w
=0.308. The coefﬁcients
0
and
p
that characterize the resonant-absorption properties of the erbium ﬁber at the laser and pump wavelengths weremeasured directly in the heavily doped ﬁber with
s
=0.43 and
p
=1. The values
0
and
R
yield
th
=3.92
10
−2
. The lasing wavelength is taken to be
g
=1.56
10
−4
cm
h
g
=1.274
10
−19
J
, which corresponds to theexperimental values where the maximum reﬂection coef-ﬁcients of both FBGs are centered on this wavelength.The parameters that can be varied in experiments are (i)the excess over the ﬁrst laser threshold, deﬁned as
=
P
r
/
P
th
, where threshold pump power
P
th
=
N
th
N
0
L
w
p
2
/
1−exp
−
p
L
1−
N
th
−1
and thresholdpopulation of level 2
N
th
=
1/
1
1+
th
/
r
w
0
, with theradius of the pump beam taken, for simplicity, as thesame as the radius of the lasing beam, i.e.,
w
0
=
r
g
, and (ii)the parameters of pump modulation, i.e., modulation fre-quency
F
m
and modulation depth
m
.
B. Normalized Equations
To simplify and generalize the laser model, we transformthe complete system of Eqs. (1) and (2) into the simpleformd
x
d
=
xy
−
a
1
x
+
a
2
y
+
a
3
,
4
d
y
d
=−
xy
−
b
1
y
−
b
2
+
P
0
1−
b
3
e
y
,
5
where the following changes have been made in the vari-ables:
x
=
12
s
T
r
p
2
r
02
0
1
1
−
2
P
,
6
y
=
p
L
N
−1
1
,
7
=2
r
w
0
T
r
p
1
−
2
t
8
and in the parameters:
a
1
=
p
L
1
−
2
th
0
r
w
+
2
1
,
9
a
2
=
1
p
r
w
0
T
r
g
4
w
0
1
−
2
2
10
−3
,
10
a
3
=
L
r
w
0
T
r
g
p
4
w
0
1
−
2
2
10
−3
,
11
b
1
=
p
2
r
w
0
1
−
2
T
r
,
12
b
2
=
p
2
L
2
r
w
0
1
1
−
2
T
r
,
13
b
3
=exp
−
p
L
1−1
1
,
14
P
0
=
p
2
T
r
2
r
02
N
0
r
w
0
1
−
2
P
p
.
15
The variables
x
and
y
are the normalized laser power den-sity and inversion population, respectively, and
P
0
is pro-portional to the pump power. Pump modulation
P
p
isgiven by Eq. (3). The values of the new parameters arepresented in Table 2.Without external modulation
m
=0
, the variation of parameter
P
0
from 1.5
10
−3
to 4.8
10
−3
yields thechange in the relaxation oscillation frequency of the laser
f
0
from 30 to 50 kHz. For our parameters, the normalized
Table 1. Laser Parameters Used in theSimulations
Parameter Dimension Value
L
cm 70
n
0
1.45
l
0
cm 20
T
r
ns 8.7
r
0
cm1.5
10
−4
s10
−2
w
0
cm3.5
10
−4
12
cm
2
2.3
10
−21
23
cm
2
0.9
10
−21
0
0.038
R
0.8
N
0
cm
−3
5.4
10
19
0
cm
−1
0.053
p
cm
−1
0.025
Table 2. Normalized Parameters
a
1
a
2
a
3
b
1
b
2
b
3
P
0
2.46.9
10
−13
5.1
10
−13
3.5
10
−7
2.6
10
−7
0.52
10
−23
P
p
Pisarchik
et al.
Vol. 22, No. 10/October 2005/J. Opt. Soc. Am. B 2109
variables related to the real laser parameters as follows:
x
4
10
−23
P
,
y
=0.7
2
N
−1
, and
=2.8
10
8
t
.
C. Potential Function
The EDFL can be interpreted as a damped nonlinear os-cillator with a certain potential function. The potentialfunction can be evaluated by use of a perturbationmethod, and the laser can be described by a second-orderdifferential equation with a single variable
x
that is pro-portional to the laser power density:
x¨
−
Ax˙
−
V
,
16
where
A
is the damping parameter,
d/d
x
, and
V
is apotential function approximated by a ﬁnite Taylor seriesto be
V
=
c
1
x
2
+
c
2
x
−
c
3
exp
−
x
.
17
The parameters of Eq. (16) are related to the parametersof Eqs. (4) and (5) as follows:
A
=
b
1
−
a
1
−
x
0
+
y
0
,
18
c
1
=
1
2
y
0
a
2
+
x
0
+
1
2
b
1
−
x
0
y
0
−
a
1
,
19
c
2
=
b
1
−
x
0
a
3
−
a
1
x
0
+
y
0
a
2
+
x
0
−
a
2
+
x
0
P
0
−
b
2
−
y
0
b
1
+
x
0
,
20
c
3
=
b
3
a
2
+
x
0
2
y
0
−
a
1
exp
−
a
3
+
y
0
a
2
+
x
0
+1
−
a
1
x
0
a
2
+
x
0
P
0
,
21
=
y
0
−
a
1
a
2
+
x
0
,
22
where
x
0
,
y
0
=
P
0
1−
b
3
e
−
b
1
−
b
2
/
,
[where
is aroot of
b
1
−
a
2
Z
2
+
b
2
−
a
3
−
a
1
b
1
Z
−
a
1
b
2
+
a
1
−
Z
1−
b
3
e
Z
P
0
=0] is the ﬁxed point of Eqs. (4) and (5). The po-sition of the ﬁxed point depends on pump power
P
0
. Forour parameters (Table 2) this dependence is shown in Fig.2. The ﬁxed point for inversion population
y
0
becomespositive at
P
0
=5
10
−7
[Fig. 2(a)] and then increases rap-idly with increasing
P
0
and is saturated to
y
0
=0.74 for
P
0
10
−4
. In our experiments the relaxation oscillationfrequency is varied from 30 to 50 kHz, which correspondsto the variation of
P
0
from 1.5
10
−3
to 4.8
10
−3
(Subsec-tion 2.B). Therefore we can consider
y
0
to be constant. Asseen from Fig. 2(b),
x
0
increases linearly with
P
0
. The po-sition of the ﬁxed point
x
0
,
y
0
as a function of
P
0
is alsoshown in the three-dimensional plot in Fig. 2(c). The po-tential function [Eq. (17)] in
x
,
P
0
space is shown in Fig.3. One can see that below the threshold value
P
0
P
th
the potential
V
0 and the ﬁxed point is unstable. Withincreasing
P
0
P
th
the potential function becomes posi-tive and the well becomes narrower.Thus the EDFL can be simply represented as a nonlin-ear oscillator [Eq. (16)] with damped parameter
A
and po-tential function
V
described by Eq. (17). Because the lastterm in Eq. (17) is negligibly small for our parameters,the strength of nonlinearity,
c
1
, forms the shape of theparabolic potential well. The pump modulation [Eq. (3)]is, in fact, the parametric modulation applied to param-eters
c
2
and
c
3
that results in resonance phenomena inthe system.
Fig. 2. Positions of ﬁxed points of Eqs. (4) and (5) versus laserpump power for (a) the
y
0
coordinate, (b) the
x
0
coordinate at
y
0
=7.4, and (c) both coordinates.Fig. 3. Potential function [Eq. (17)] for pump power
P
0
=0−0.02. The ﬁxed point is stable for
P
0
P
th
.2110 J. Opt. Soc. Am. B/Vol. 22, No. 10/October 2005 Pisarchik
et al.
3. NUMERICAL RESULTS
The numerical calculations employing the system of Eqs.(1) and (2) or (4) and (5) allow us to obtain time series andbifurcation diagrams for characterization of the dynamicsof the pump-modulated EDFL. As was shownpreviously,
9–11
the dynamics of this laser, as well as of other class-B lasers (see, for example, Refs. 3 and 20 andreferences therein), is related to the main laser reso-nance, which appears close to the relaxation oscillationfrequency of the laser,
f
0
. A rich variety of attractorsarises in primary saddle-node bifurcations (SNBs). De-pending on the modulation frequency, the laser responsemay contain either subharmonics or higher harmonics of
F
m
. At the high-frequency range
F
m
f
0
, various SNBsgive rise to subharmonic laser oscillations, whereas at therelatively low modulation frequencies
F
m
f
0
the higherharmonics of
F
m
rule the laser dynamics. The dynamics of the pump-modulated EDFL in the high-frequency rangehas been investigated experimentally.
10,11
Therefore weaddress only numerical results obtained with our newmodel at this frequency range and compare them with ourprevious experiments, whereas the laser dynamics in thelow-frequency range are studied in detail both numeri-cally and experimentally.
A. High-Frequency Range
To simulate the laser dynamics in the high-frequencyrange
F
m
f
0
, we use the parameters that are close tothe experimental ones taken from Refs. 10 and 11. Wechose pump power
P
p
0
=7.4
10
19
s
−1
to get a relaxationoscillation frequency of the laser of
f
0
30 kHz. In Fig. 4we plot the bifurcation diagram of the peak-to-peak laserpower
P
max
with
F
m
as a control parameter for
m
=0.5. Weconstructed this diagram by taking different initial condi-tions for
P
and
N
that allow us to display all coexistingstable solutions in the same diagram.As can be seen fromthe ﬁgure, the subharmonic attractors, period 1 (P1), pe-riod 2 (P2), and period 3 (P3), may coexist within a certainfrequency range. Each attractor is born in the correspond-ing primary SNB. The comparison of this diagram withthe experimental ones displayed in Refs. 10 and 11 yieldsgood agreement, even in detail, between the experimentand our new theoretical model. Note that other models of the EDFL, which do not address the contributions in laserdynamics stemming from the two features mentioned (theESA at the laser wavelength and depleting of a pumpwave within the active ﬁber) are not able to arrive at sucha perfect match with the experiment.
B. Low-Frequency Range
To study the laser dynamics in the low-frequency range
F
m
f
0
, we chose the pump power to be
P
p
0
=2.4
10
20
s
−1
, which led to
f
0
50 kHz. We calculate bifurca-tion diagrams of the peak-to-peak laser intensity andphase difference
between the pulses of the ﬁber laserand the diode pump laser versus the modulation fre-quency for different modulation amplitudes. For example,in Fig. 5 we plot the bifurcation diagrams for
m
=0.5. Weﬁnd that, within certain frequency ranges, the laser dis-plays generalized bistability, i.e., the coexistence of twoattractors with the same ratio of pulses with respect tothe period of the pump modulation (winding number)(e.g., 1:1, 2:1, 3:1, and 4:1) but with different amplitudesand phases of the spikes. The insets of time series foreach attractor clear up the different pulsed regimes of thelaser in the chosen point of the parameter space (the co-existent regimes are marked by stars).In Fig. 5(a) different branches of the bifurcation dia-gram are labeled b1, b2, and b3. The common feature of the bifurcation diagrams in the low-frequency range [Fig.5(a)] and in the high-frequency range (Fig. 4) is that thebranches in both diagrams are born and die in the pri-mary SNBs that result from the resonant interaction of the modulation frequency with the relaxation oscillationfrequency of the laser. However, in the latter case this in-
Fig. 4. Numerical bifurcation diagram of laser peak power
P
max
with modulation frequency
F
m
as a control parameter at
m
=0.5.P1, P2, and P3 are the branches of the period-1, period-2, andperiod-3 attractors, respectively. The dashed line indicates thefrequency of relaxation oscillations of the laser,
f
0
.Fig. 5. Numerical bifurcation diagrams of (a) laser peak power
P
max
and (b) phase difference
between laser spikes and pumpmodulation versus modulation frequency
F
m
at
m
=0.5. Thedashed lines indicate the positions of SNB points
S
i
.Pisarchik
et al.
Vol. 22, No. 10/October 2005/J. Opt. Soc. Am. B 2111

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