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An erbium-doped fiber laser is shown to operate as a bistable or multistable nonlinear system under harmonic modulation of the diode pump laser. Phase- and frequency-dependent states are demonstrated both experimentally and in numerical simulations

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Dynamics of an erbium-doped fiber laser subjected to harmonicmodulation of a diode pump laser
A. N. Pisarchik*, A. V.Kir'yanov, Yu. 0. Barmenkov and R. Jaimes Reategui
Centro de Investigaciones en Optica, Loma del Bosque #1 15, Col. Lomas del Campestre,
Leon, Gto., Mexico
ABSTRACT
An erbium-doped fiber laser is shown to operate as a bistable or multistable nonlinear system under harmonic
modulation of the diode pump laser. Phase- and frequency-dependent states are demonstrated both experimentally and innumerical simulations through codimensional-one and codimensional-two bifurcation diagrams in the parameter space of
the modulation frequency and amplitude. In particular, generalized bistability results in doubling of saddle-node
bifurcation lines where different coexisting attractors born. The laser model describes well all experimental features.
Keywords: Erbium-doped fiber laser, nonlinear dynamics, pump modulation
1. INTRODUCTION
In the past decades, revolutionary progress has been achieved in research and commercialization of erbium-doped fiberlasers (EDFLs). The exclusive advantages ofthese lasers are the long interaction length ofpumping light with the active
ions that leads to very high gain and a single transversal mode operation given by a suitable choice of the fiber core
diameter and index step. These features make EDFLs excellent light sources for optical communications, reflectometry,
sensing, medicine, etc. Meanwhile, these lasers are quite sensitive to any external perturbation which may destabilize
their normal operation. Therefore, the knowledge ofthe dynamic behavior ofthese lasers under external modulation is of
great importance and can be prominent for many applications.
From t he v iewpoint o f nonlinear d ynamics, rare-doped fiber lasers w ith e xternal modulation among with s olid-state,
semiconductor, and gas lasers with electric discharge (like CO2 and CO lasers) belong to class-B lasers. These are
nonautonomous systems in which polarization is adiabatically eliminated and the dynamics can be ruled by two rate
equations for field and population inversion. Therefore, the main features of the dynamical behavior of these lasers arevery similar to those of other class-B lasers. Different scenario for development of a chaotic motion have been found in
EDFLs. In spite of an impressive array of research on complex dynamics in lasers, only few works are devoted to a study
of the nonlinear response of EDFL on parametric modulation. The dynamics of this laser were reported recently in the
work of Sola et al.1 and in our works.2 Sola et a!. studied the dynamics of a ring 1533-nm EDFL with sinusoidally
modulated 1470-nm pump diode laser. They developed a rather complicated model which describes well their
experimental results. In our previous works we studied a linear 1560-nm EDFL pumped by a 967-nm laser diode. Such alaser is commonly used in many laboratories and serves for various applications. However, the spectroscopy of this laseris quite different from that ofthe laser studied by Sola et a!., and hence their model cannot be used for our laser.
In this paper, we study the dynamics of a 1560-nm EDFL with Fabry-Perot cavity subjected to harmonic pump
modulation of the diode laser. We develop a novel simple model which can be used to describe such a laser and, as we
will s how below, p erfectly a ddresses all 1 aser p eculiarities o bserved experimentally. W e i nvestigate theoretically the
laser dynamics over a wide range of frequencies of pump modulation. Our model does not account for a saturable
absorber phenomenon in a fiber and hence the self-pulsing behavior does not appear. This is stipulated by the followingreasons. First, the pump power used in our experiments is too small to induce a thermo lens in the fiber, and second, the
pump modulation amplitude is much larger than noise.
2. THEORY
2.1. Laser model
Our model is based on the rate equations in which we use a power-balance approach applied to a longitudinally pumpedEDFL, where the excited state absorption (ESA) in erbium at the 1.5-tm wavelength and the averaging of the population
inversion along the pumped active fiber are accounted for. Such a model would address the two most evident factors, i.e.,
ESA at the laser wavelength and the depleting of the pump wave at propagation along the active fiber. This model does
5th Iberoamerican Meeting on Optics and 8th Latin American Meeting on Optics, Lasers,and Their Applications, edited by A. Marcano O., J. L. Paz, Proc. of SPIE Vol. 5622(SPIE, Bellingham, WA, 2004) · 0277-786X/04/$15 · doi: 10.1117/12.590735379
Downloaded from SPIE Digital Library on 06 Jul 2011 to 134.206.56.103. Terms of Use: http://spiedl.org/terms
not include mechanisms for a self-pulsing behavior of the laser because in our experiments the amplitude of self-
pulsations i 5 5 mall (
in
2 -3 o rders of m agnitude) as c ompared with t he a mplitude o f the 1 aser response d ue t op ump
modulation, and hence this effect has no influence of the laser dynamics. The possible mechanisms for the self-pulsing
behavior of EDFL (i.e., thermo-lensing effects and noisy pumping) are considered in other works5 where some
modifications are made in the model.
The balance equations for the intra-cavity laser power P (being a sum of the powers of the contra-propagating waves
inside the cavity, in s1 and the averaged (over the active fiber length) population y of the upper ("2") level (being a
dimensionless variable,
have been derived to be as follows:
4=P{rWaOy(-1/)-1]-aIh}+FP,
(1)
= —
12rP
_
i)
— + pump '
(2)
dt
,zr0
r
where y =
JN2 (z)dz is the cross-section of the absorption transition from the ground state "1" to the upper
n0L
state "2" (N2 is the population of the upper laser level "2", n0 is the refractive index of a "cold" erbium-doped fiber core,
and L is the active fiber length), a12 is the cross-section of the absorption transition from the ground state "1" to the
upper state "2". Here w e s uppose that t he c ross-section o f the return s timulated t ransition i s practically the s ame in
magnitude {des94} that yields = 12 21 2 .
i
= —
=
0.4 is the coefficient that stands for the ratio between
°12
a12
ESA (23) and ground-state absorption cross-sections at the laser wavelength, 7 =2n,Ic =(L+l) is the total
concentration of erbium ions in the active fiber (lo being the intra-cavity tails ofthe fiber Bragg grating (FBG) couplers),
a0
=
N0a12 is the small-signal absorption of the erbium fiber at the laser wavelength (N0 =
N1
+ N2 being the total
concentration of erbium ions in the active fiber), ah Yo
+
(1/ 2L)ln(1/ R) is the intra-cavity losses on the threshold (yo
being the non-resonant fiber loss and R is the total reflection coefficient ofthe FBG couplers), t is the lifetime of erbium
ions in the excited state "2", r0 is the fiber core radius, w0 is the radius of the fundamental fiber mode, and
t; =
1
—
exp[_
2(r I W0
)2
J
is
the factor addressing a match between the laser fundamental mode and erbium-doped core
.
.
.
1Oy('%g2
r2ct0L
.
.
.
volumes
inside the active fiber. In Eq. (1), p =
i
°
is the spontaneous emission into the
sp
lTr Lwo) 4,r2a12
fundamental laser m ode. We assume here that the laser s pectrum width is 1 O of the erbium luminescence spectral
bandwidth (Xg being the laser wavelength). In Eq. (2), p
p
exPE— fla0L(1— y)1
the pump power, where P is the
pump
p
n0,rrL
pump power at the fiber entrance (left-hand side in Fig. 1) and 3 =
a,/ao
is the dimensionless coefficient that accountsfor the ratio of absorption coefficients of the erbium fiber at pump wavelength X(a) to that at laser wavelength Xg(ao).
The system of Eqs. (1,2) describes the laser dynamics without an external modulation. In order to account for the
harmonic pump modulation, one needs to write the pump power at the active fiber facet as
P; =
+
m0 sin(2rFmt)}, where m0 and Fm are the modulation depth and frequency. The calculations were performed for
the experimental conditions described in Sec. 3. The following main parameters are used
L
no
1
Trr0
W0
cy12 =
a21
a23
t
Yo
R
cmcm
ns
cmcm
cm2cm2
s
70
1.45
20
8.71.5x1043.5x104
3x1021
O.9x1021
102
0.038
0.8
The value of w0 has been measured experimentally and it is a bit higher than the value 2.5x i0 cm given by the formula
for a step-index single-mode fiber w0 =
ro(0.65+1.619/V15+2.879/J"),
where the V-parameter is connected with
numerical aperture NA and r0 as V =
(2irro/Xg)NA;
the values r0 and wo result in r =
0.308.
The coefficients
380 Proc. of SPIE Vol. 5622
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characterizing the resonant-absorption properties of the erbium fiber at the laser and pump wavelengths are a00.4 cm'
and 3 =
0.5.
These values correspond to direct measurements for the heavily doped fiber with erbium concentration of
2300 ppm. The va'ues y and R yield ah =
3.92x102.
The lasing wavelength is taken to be Xg
1.56x104
cm (hvg
1 .274x iO'J), corresponding to the experiment, where the maximum reflection coefficients of both FBGs are centered
on this wavelength. The parameters varying in the simulations are (i) the excess over the first laser threshold is defined
as =
PpIPth,
where the threshold pump power Pth yh(nOL7rwP/T){1-exp[-aOL13(1-y(hY1} and the threshold population of
level "2"
=+
—_
with
the radius of the pump beam being taken, for simplicity, the same as the radius of the
ih
rcto)
lasing
beam, i.e., r =
rg,
and (ii) the control parameters, i.e., the modulation frequency Fm and depth m0.
2.2. Numerical results
In order to simulate the laser dynamics in the high-frequency range (Fm > fo), we use the parameters close to the
experimental ones.3'4 The pump power is estimated as P =
7.4x1019
s. For these parameters, we find the relaxation
oscillation frequency of the laserfo 30 kHz. The bifurcation diagrams of the peak-to-peak laser power with Fm as a
control parameter is shown in Fig. 1 .
The
diagram is constructed by taking different initial conditions for P and N that
allows us to plot all coexisting stable solutions in the same diagrams. It is seen from the figures, at high m0 different
subharmonic attractors are bom in the primary SNBs and within certain frequency range two or three attractors coexists.
The comparison o ft hese diagrams with experimental results3'4 gives a good a greement, e yen in details, between the
experiment and the developed theoretical model. Note, that other models of EDFL, which do not address the
contributions in laser dynamics stemming from the two mentioned features (ESA at the laser wavelength and the
depleting of a pump wave within the active fiber) are not able to arrive such a perfect match with the experiment.
Fig. 1 .
Numerical
bifurcation diagram ofpeak-to-peak laser power versus modulation frequency at m0 =
0.5.
To study the laser dynamics in the low-frequency range (Fm <fo) the pump power is chosen to be P, =
2.43.5x102°
srn'.
This yields j 50 kHz. Depending on initial conditions, the laser can oscillate in different periodic regimes with
different number ofpulses in the laser output with respect to the period ofthe pump modulation, e.g., 1:1, 2:1, 3:1, 4:1,
and so on. Within certain ranges ofthe modulation parameters, the laser displays generalized bistability, i.e., coexistence
oftwo attractors. In Fig. 2 we present the bifurcation diagram of the peak-to-peak laser power with the modulation
frequency as a control parameter. One can see that for some frequency ranges there coexist two different periodic
attractors (bistability), for example, there are two period-i regimes, 1:1 and 1:1. The insets of the time series for each
attractor clear up the different pulsed regimes of the laser (1:1, 2:1, and 3:1).
F (kHz)
Proc. of SPIE Vol. 5622 381
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3.02.52.0
1.5
1.0
0.50.0
-0.E
3:
3:1
1"
2:1*1:1*
—H
50
1:0
F (kHz)
Fig. 2. Numerical bifurcation diagram of peak-to-peak laserpower versus modulation frequency in low-frequency range
m0=0.5.
Fig. 3. Phase difference between the spikes of the pump laser
and fiber laser for different dynamical states.
The phase difference between the pulses of the pump laser and the pulses of the fiber laser also depends on Fm. This
dependence shown in Fig. 3 for different dynamical regimes represents the phase-locked regions where A 0 or t
and
almost independent on Fm.
The codimensional-two bifurcation diagram in the space of the modulation amplitude and modulation frequency are
shown in Fig. 4. The cross-hatched regions indicate the bistable ranges with the same winding numbers and the lines
Sl-S6 are the saddle-node bifurcation (SNB) lines where the attractors corresponding to different limit cycles are born
and dead. Inside these regions, the two different regimes with the same periodicity coexist.
E
Fig.
4. Numerical codimensional-two bifurcation diagram in parameter space of the modulation frequency and depth. The
crosshatched regions are the bistable ranges with the same winding number. P 1 and P2 are the boundaries of both the phase-locked
regions.
F (kHz)F (kHz)
382 Proc. of SPIE Vol. 5622
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3. EXPERIMENT
In our experiments, EDFL is pumped by a commercia' laser diode (wavelength 976 nm, maximum pump power 300
mW) through a wavelength-division multiplexing coupler and polarization controller. The laser cavity of a 1.5-m length
is formed by a piece of heavily doped erbium fiber of a 70-cm length and a core diameter of 2.7 tm, and two fiber
Bragg gratings with a 2-nm FWHM (full width on half-magnitude) bandwidth and reflectivity of 91% and 95% at a1560-nm wavelength. Output power of the pumping laser diode and fiber laser are recorded with photodetectors and
analyzed with an oscilloscope and a Fourier spectrum analyzer. The output power of the diode laser depends linearly on
the laser diode current.
The harmonic signal, A sin(2tfl) (where A and f are the driving amplitude and frequency), applied from a signal
generator to the laser driver causes harmonic modulation ofthe diode current withf In our experiments, the signal with
A =
800
mV results in 100% modulation depth ofthe pump power, while the average diode current is fixed at 40 mA.
Figure 5 shows the experimental bifurcation diagrams of the peak-to-peak laser intensity and phase difference versusthe modulation frequency at the 50% modulation depth. Within certain ranges of the modulation frequency, the laser
displays generalized bistability, i.e., coexistence of two attractors. Depending on the initial conditions, the laser canoscillate in different periodic regimes with the same ratio of the number of pulses in the laser output and modulation
signal (winding number), e.g., 1:1, 2:1 (compare with Figs. 1 and 2), and 3:1, 4:1, and so on.
—3
CD
-CD-
a,
C/)
CD
.
A
1.61.41.2
i
it
b3,III
/2II
-&E}O—---—o—---———_---E1---——-EJ
LT
it/2
/
.
..
b3I:
b2,I
:
:
-&&O-—--------O
0
f (kHz)
1•0
1'5
20 25 30 35 40 45 50
f(kHz)
Fig.
5.
Peak-to-peak
laser intensity and phase difference versus modulation frequency for A 230 mV. bi, b2, and b3 are the
branches correspondmg to different attractors. I, II, and III are the first, second, and third harmonics of the corresponding periodic
regime.
The experimentally measured frequency-locked regions are clearly seen in Fig. 6, that is the codimensional-two
bifurcation diagram in the space of the modulation amplitude and modulation frequency. Note that the horizontal and
vertical dashed lines indicate, respectively, the values of modulation amplitude and modulation depth for which the
diagrams in Fig. 5 are obtained. Within the tongues a period-doubling route to chaos occurs, whereas between the
tongues a quasi-periodic (QP) route to chaos is registered.
Proc. of SPIE Vol. 5622 383
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