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Pisarchik et al. Vol. 22, No. 10 /October 2005 /J. Opt. Soc. Am. B 2107

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.

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. INTRODUCTIONn the past decades, revolutionary progress has beenchieved in research and commercialization of erbium-oped fiber lasers (EDFLs). The advantages of these la-ers are the long interaction length of pumping light withhe active ions, which leads to high gain and to single-ransversal-mode operation produced by a suitable choicef fiber parameters. These properties make EDFLs excel-ent light sources for optical communications, reflectom-try, sensing, medicine, etc.1,2 Meanwhile, these lasersre quite sensitive to any external perturbation that mayestabilize their normal operation. Therefore, knowledgef the dynamic behavior of these lasers under externalodulation is of great importance and can be important

or many applications.From the viewpoint of nonlinear dynamics, rare-earth-

oped fiber lasers with external modulation, along witholid-state, semiconductor, and electric discharge CO2nd CO lasers, are class-B lasers.3 These are nonautono-ous systems in which polarization is adiabatically elimi-

ated and the dynamics can be ruled by two rate equa-ions for field and population inversion. In spite of anmpressive array of research on complex dynamics in la-ers, the nonlinear dynamics of EDFLs has begun to betudied only recently. The main features of the dynamicehavior of these lasers are similar to those of otherlass-B lasers. Different conditions for the development ofhaotic motion have been found in EDFLs. First, a period-oubling route to chaos was observed by Lacot et al.4 in aipolarized two-mode EDFL with harmonic pump modu-ation. Those authors have also developed a model basedn two coherently pumped coupled lasers. A quasi-eriodic route to chaos was found by Sanchez et al.5 in aual-wavelength EDFL. Eventually, Luo et al.6 revealedhe coexistence of period-doubling and intermittency

0740-3224/05/102107-8/$15.00 © 2

outes to chaos in a pump-modulated ring EDFL. Theylso reported on bistability (the coexistence of two peri-dic attractors) in this laser.6,7 More recently, optical bi-tability (coexistence of a limit cycle and a fixed point)as detected by Mao and Lit8 in the vicinity of the first

aser threshold in a dual-wavelength EDFL with overlap-ing cavities.Previously, the authors and others have reported the

oexistence of multiple periodic attractors (generalizedultistability) found both theoretically and experimen-

ally in EDFLs subjected to loss9 or pump modulation.10,11

any papers have been devoted to a study of self-ulsation behavior of EDFLs (see, for example, Refs.2–14). Such behavior has been suggested to be due to theresence of a saturable absorber in the fiber in the form ofon pairs12 or pump depletion.15 The Q-switching behavioran also be attributed to excited-state absorption (ESA) athe lasing wavelength16 and to a thermo-lensing effecthat is due to ESA at the pump wavelength.17

Only a few papers have been devoted to a study of theonlinear response of the EDFL to parametric modula-ion. The dynamics of this laser were reported recently byola et al.18 and by the present authors and others.9–11

ola et al. studied the dynamics of a ring 1533 nm EDFLith a sinusoidally modulated 1470 nm pump diode laser.hey developed a rather complicated model,19 which de-cribes their experimental results well. Previously wetudied a linear 1560 nm EDFL pumped by a 967 nm la-er diode. Such a laser is commonly used in many labora-ories and serves for various applications. However, thepectroscopy of this laser is quite different from that ofhe laser studied by Sola et al., and hence their modelannot describe our laser.

In this paper we study the dynamics of a 1560 nmDFL with a Fabry–Perot cavity that has been subjected

005 Optical Society of America

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2108 J. Opt. Soc. Am. B/Vol. 22, No. 10 /October 2005 Pisarchik et al.

o harmonic pump modulation of a diode pumping laser.e develop a novel model that can be used to describe

uch a laser and that, as we show below, perfectly ad-resses all laser peculiarities observed experimentally.e investigate theoretically the laser dynamics over aide range of frequencies and amplitudes of pump modu-

ation.The paper is organized as follows: In Section 2 we in-

roduce the novel model of the EDFL, normalized equa-ions, and transform the laser equations into a singlequation of a nonlinear oscillator. Then in Section 3 weresent the results of numerical simulations of experi-ents reported in our previous papers10,11 for the casehen the modulation frequency is higher than the relax-tion oscillation frequency of the laser and we simulatehe experiments presented in this paper for the case whenhe modulation frequency is smaller than the relaxationscillation frequency. We demonstrate that the low-requency range furnishes a rather interesting insightnto EDFL dynamics with external modulation because ofhe appearance of many fine dynamic phenomena that be-ome latent at higher modulation frequencies. In Sectionwe describe our experimental setup and compare the re-

ults of the numerical simulations with the experimentalesults. In the course of experiments, we determine di-ectly the structure of frequency-locked and phase-lockedtates (with respect to pump modulation) through bifur-ation diagrams in space of the modulation parameters.inally, our main conclusions are given in Section 5.

. THEORY. Laser Modelhe model is based on the rate equations in which we usepower-balance approach applied to a longitudinally

umped EDFL, in which the ESA in erbium at the.5-�m wavelength and the averaging of the populationlong the pumped active fiber are taken into account.uch a model would address the two most evident factors,

.e., the ESA at the laser wavelength and the depleting ofhe pump wave at propagation along the active fiber. Annergy-level diagram of our model is shown in Fig. 1. Thisodel does not include the mechanisms that are respon-

ible for establishing the self-pulsing regime in the laser,uch as a thermo-lensing effect17 and erbium ion pairs inhe fiber,12 for the following reasons. First, in our experi-ents the pump power is too small to induce thermo lens-

ng and second, the concentration of erbium is too low toake the effect of the ion pairs significant.The balance equations for intracavity laser power P

which is a sum of the powers of the contrapropagatingaves inside the cavity, in inverse seconds) and the aver-ged (over the active fiber length) population N of the up-er (2) level (which is a dimensionless variable, 0�N�1)re

dP

dt=

2L

TrP�rw�0�N��1 − �2� − 1� − �th� + Psp, �1�

dN

dt= −

�12�srwP

�r02 �N�1 − 1� −

N

�+ Ppump, �2�

here N= �1/n0L��0L N2�z�dz (N2 is the population of up-

er laser level 2, n0 is the refractive index of a coldrbium-doped fiber core, and L is the active fiber length)nd �12 is the cross section of the absorption transitionrom ground state 1 to upper state 2. Here we assumehat the cross section of the return stimulated transitions practically the same ��12=�21� as that which yields �1��12+�21� /�12=2. �2=�23/�12=0.4 is the coefficient thattands for the ratio between the ESA ��23� and ground-tate absorption cross sections at the laser wavelength,r= �2n0 /c��L+ l0� is the photon intracavity round-trip

ime [l0 is the total length of the fiber Bragg gratingFBG) coupler tails inside the cavity], �0=N0�12�s is themall-signal absorption of the erbium fiber at the laseravelength (N0=N1+N2 is the total concentration of er-ium ions in the active fiber and �s is the overlap factoror EDFL radiation), �th=0+ �1/2L�ln�1/R� is the intrac-vity loss on threshold (0 is the nonresonant fiber lossnd R is the total reflection coefficient of the FBG cou-lers), � is the lifetime of erbium ions in excited state 2, r0s the fiber core radius, w0 is the radius of the fundamen-al fiber mode, and rw=1−exp�−2�r0 /w0�2� is the factorhat addresses a match between the laser fundamentalode and erbium-doped core volumes inside the active fi-

er. In Eq. (1),

Psp = N� g

w02 r0

2�0L

4�2�12�s�Tr� 10−3

s the spontaneous emission into the fundamental laserode. We assume here that the laser spectrum width is

0−3 of the erbium luminescence spectral bandwidth (g ishe laser wavelength). In Eq. (2), Ppump= �Pp /N0�r0

2L��1exp�−�pL�1−N��� is the pump power, where Pp is theump power at the fiber entrance and �p=N0�14�p is themall-signal absorption of the erbium fiber at the pumpavelength (�14 is the cross section of the absorption

ransition from level 1 to level 4 and �p is the overlap fac-or for pump radiation). The system of Eqs. (1) and (2) de-cribes the laser dynamics without external modulation.he harmonic pump modulation is added as

Fig. 1. Erbium energy-level diagram.

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Pisarchik et al. Vol. 22, No. 10 /October 2005 /J. Opt. Soc. Am. B 2109

Pp = Pp0�1 + m sin�2�Fmt��, �3�

here m and Fm are the modulation depth and frequency,espectively, and Pp

0 is the pump power without modula-ion (at m=0).

The calculations are performed for the experimentalonditions described in Refs. 10 and 11 and in Section 5elow. The parameter values are presented in Table 1.The value of w0 has been measured experimentally,

nd it is a bit higher than the value 2.5�10−4 cm given byhe formula for a step-index single-mode fiber: w0r0�0.65+1.619/V1.5+2.879/V6�, where V is related to nu-erical aperture NA and r0, as V= �2�r0 /g�NA; the val-es r0 and w0 result in rw=0.308. The coefficients �0 andp that characterize the resonant-absorption properties ofhe erbium fiber at the laser and pump wavelengths wereeasured directly in the heavily doped fiber with �s0.43 and �p=1. The values 0 and R yield �th=3.9210−2. The lasing wavelength is taken to be g=1.5610−4 cm �h�g=1.274�10−19 J�, which corresponds to the

xperimental values where the maximum reflection coef-cients of both FBGs are centered on this wavelength.he parameters that can be varied in experiments are (i)he excess over the first laser threshold, defined as Pr /Pth, where threshold pump power PthNth�N0L�wp

2 /���1−exp�−�pL�1−Nth���−1 and thresholdopulation of level 2 Nth= �1/�1��1+ ��th/rw�0��, with theadius of the pump beam taken, for simplicity, as theame as the radius of the lasing beam, i.e., w0=rg, and (ii)he parameters of pump modulation, i.e., modulation fre-uency Fm and modulation depth m.

. Normalized Equationso simplify and generalize the laser model, we transformhe complete system of Eqs. (1) and (2) into the simpleorm

Table 1. Laser Parameters Used in theSimulations

Parameter Dimension Value

L cm 70n0 1.45l0 cm 20Tr ns 8.7r0 cm 1.5�10−4

� s 10−2

w0 cm 3.5�10−4

�12 cm2 2.3�10−21

�23 cm2 0.9�10−21

0 0.038R 0.8N0 cm−3 5.4�1019

�0 cm−1 0.053�p cm−1 0.025

Table 2. Norm

a1 a2 a3

.4 6.9�10−13 5.1�10−13 3.

dx

d�= xy − a1x + a2y + a3, �4�

dy

d�= − xy − b1y − b2 + P0�1 − b3ey�, �5�

here the following changes have been made in the vari-bles:

x =�12�sTr�p

2�r02�0

�1

�1 − �2P, �6�

y = �pL�N −1

�1 , �7�

� =2rw�0

Tr�p��1 − �2�t �8�

nd in the parameters:

a1 =�pL

�1 − �2� �th

�0rw+

�2

�1 , �9�

a2 =�1�p

�rw�0

Tr

� g

4�w0��1 − �2��2

� 10−3, �10�

a3 =L

�rw�0

Tr

� g�p

4�w0��1 − �2��2

� 10−3, �11�

b1 =�p

2rw�0��1 − �2�

Tr

�, �12�

b2 =�p

2L

2rw�0�1��1 − �2�

Tr

�, �13�

b3 = exp− �pL�1 −1

�1� , �14�

P0 =�p

2Tr

2�r02N0rw�0��1 − �2�

Pp. �15�

he variables x and y are the normalized laser power den-ity and inversion population, respectively, and P0 is pro-ortional to the pump power. Pump modulation Pp isiven by Eq. (3). The values of the new parameters areresented in Table 2.Without external modulation �m=0�, the variation of

arameter P0 from 1.5�10−3 to 4.8�10−3 yields thehange in the relaxation oscillation frequency of the laser0 from 30 to 50 kHz. For our parameters, the normalized

d Parameters

b2 b3 P0

7 2.6�10−7 0.5 2�10−23Pp

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2110 J. Opt. Soc. Am. B/Vol. 22, No. 10 /October 2005 Pisarchik et al.

ariables related to the real laser parameters as follows:�4�10−23P, y=0.7�2N−1�, and �=2.8�108t.

. Potential Functionhe EDFL can be interpreted as a damped nonlinear os-illator with a certain potential function. The potentialunction can be evaluated by use of a perturbationethod, and the laser can be described by a second-order

ifferential equation with a single variable x that is pro-ortional to the laser power density:

x − Ax � − �V, �16�

here A is the damping parameter, � d/dx, and V is aotential function approximated by a finite Taylor serieso be

V = c1x2 + c2x − c3 exp�− �x�. �17�

he parameters of Eq. (16) are related to the parametersf Eqs. (4) and (5) as follows:

ig. 2. Positions of fixed points of Eqs. (4) and (5) versus laserump power for (a) the y0 coordinate, (b) the x0 coordinate at y07.4, and (c) both coordinates.

A = b1 − a1 − x0 + y0, �18�

c1 = 12y0�a2 + x0� + 1

2 �b1 − x0��y0 − a1�, �19�

c2 = �b1 − x0��a3 − a1x0 + y0�a2 + x0��

− �a2 + x0��P0 − b2 − y0�b1 + x0��, �20�

c3 = b3

�a2 + x0�2

y0 − a1

�exp−a3 + y0�a2 + x0 + 1� − a1x0

a2 + x0�P0, �21�

� =y0 − a1

a2 + x0, �22�

here �x0 ,y0�= �P0�1−b3e��−b1�−b2 /�� ,� [where � is aoot of �b1−a2�Z2+ �b2−a3−a1b1�Z−a1b2+ �a1−Z��1b3eZ�P0=0] is the fixed point of Eqs. (4) and (5). The po-ition of the fixed point depends on pump power P0. Forur parameters (Table 2) this dependence is shown in Fig.. The fixed point for inversion population y0 becomesositive at P0=5�10−7 [Fig. 2(a)] and then increases rap-dly with increasing P0 and is saturated to y0=0.74 for0�10−4. In our experiments the relaxation oscillation

requency is varied from 30 to 50 kHz, which correspondso the variation of P0 from 1.5�10−3 to 4.8�10−3 (Subsec-ion 2.B). Therefore we can consider y0 to be constant. Aseen from Fig. 2(b), x0 increases linearly with P0. The po-ition of the fixed point �x0 ,y0� as a function of P0 is alsohown in the three-dimensional plot in Fig. 2(c). The po-ential function [Eq. (17)] in �x ,P0� space is shown in Fig.. One can see that below the threshold value �P0�Pth�he potential V�0 and the fixed point is unstable. Withncreasing P0�Pth the potential function becomes posi-ive and the well becomes narrower.

Thus the EDFL can be simply represented as a nonlin-ar oscillator [Eq. (16)] with damped parameter A and po-ential function V described by Eq. (17). Because the lasterm in Eq. (17) is negligibly small for our parameters,he strength of nonlinearity, c1, forms the shape of thearabolic potential well. The pump modulation [Eq. (3)]s, in fact, the parametric modulation applied to param-ters c2 and c3 that results in resonance phenomena inhe system.

ig. 3. Potential function [Eq. (17)] for pump power P0=00.02. The fixed point is stable for P �P .

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Pisarchik et al. Vol. 22, No. 10 /October 2005 /J. Opt. Soc. Am. B 2111

. NUMERICAL RESULTShe numerical calculations employing the system of Eqs.

1) and (2) or (4) and (5) allow us to obtain time series andifurcation diagrams for characterization of the dynamicsf the pump-modulated EDFL. As was shownreviously,9–11 the dynamics of this laser, as well as ofther class-B lasers (see, for example, Refs. 3 and 20 andeferences therein), is related to the main laser reso-ance, which appears close to the relaxation oscillation

requency of the laser, f0. A rich variety of attractorsrises in primary saddle-node bifurcations (SNBs). De-ending on the modulation frequency, the laser responseay contain either subharmonics or higher harmonics of

m. At the high-frequency range �Fm� f0�, various SNBsive rise to subharmonic laser oscillations, whereas at theelatively low modulation frequencies �Fm� f0� the higherarmonics of Fm rule the laser dynamics. The dynamics ofhe pump-modulated EDFL in the high-frequency rangeas been investigated experimentally.10,11 Therefore weddress only numerical results obtained with our newodel at this frequency range and compare them with our

revious experiments, whereas the laser dynamics in theow-frequency range are studied in detail both numeri-ally and experimentally.

. High-Frequency Rangeo simulate the laser dynamics in the high-frequencyange �Fm� f0�, we use the parameters that are close tohe experimental ones taken from Refs. 10 and 11. Wehose pump power Pp

0=7.4�1019 s−1 to get a relaxationscillation frequency of the laser of f0�30 kHz. In Fig. 4e plot the bifurcation diagram of the peak-to-peak laserower Pmax with Fm as a control parameter for m=0.5. Weonstructed this diagram by taking different initial condi-ions for P and N that allow us to display all coexistingtable solutions in the same diagram. As can be seen fromhe figure, the subharmonic attractors, period 1 (P1), pe-iod 2 (P2), and period 3 (P3), may coexist within a certainrequency range. Each attractor is born in the correspond-ng primary SNB. The comparison of this diagram withhe experimental ones displayed in Refs. 10 and 11 yieldsood agreement, even in detail, between the experimentnd our new theoretical model. Note that other models ofhe EDFL, which do not address the contributions in laserynamics stemming from the two features mentioned (theSA at the laser wavelength and depleting of a pumpave within the active fiber) are not able to arrive at suchperfect match with the experiment.

. Low-Frequency Rangeo study the laser dynamics in the low-frequency rangeFm� f0�, we chose the pump power to be Pp

0=2.41020 s−1, which led to f0�50 kHz. We calculate bifurca-

ion diagrams of the peak-to-peak laser intensity andhase difference �� between the pulses of the fiber lasernd the diode pump laser versus the modulation fre-uency for different modulation amplitudes. For example,n Fig. 5 we plot the bifurcation diagrams for m=0.5. Wend that, within certain frequency ranges, the laser dis-lays generalized bistability, i.e., the coexistence of twottractors with the same ratio of pulses with respect to

he period of the pump modulation (winding number)e.g., 1:1, 2:1, 3:1, and 4:1) but with different amplitudesnd phases of the spikes. The insets of time series forach attractor clear up the different pulsed regimes of theaser in the chosen point of the parameter space (the co-xistent regimes are marked by stars).

In Fig. 5(a) different branches of the bifurcation dia-ram are labeled b1, b2, and b3. The common feature ofhe bifurcation diagrams in the low-frequency range [Fig.(a)] and in the high-frequency range (Fig. 4) is that theranches in both diagrams are born and die in the pri-ary SNBs that result from the resonant interaction of

he modulation frequency with the relaxation oscillationrequency of the laser. However, in the latter case this in-

ig. 4. Numerical bifurcation diagram of laser peak power Pmax

ith modulation frequency Fm as a control parameter at m=0.5.1, P2, and P3 are the branches of the period-1, period-2, anderiod-3 attractors, respectively. The dashed line indicates therequency of relaxation oscillations of the laser, f0.

ig. 5. Numerical bifurcation diagrams of (a) laser peak powermax and (b) phase difference �� between laser spikes and pumpodulation versus modulation frequency Fm at m=0.5. The

ashed lines indicate the positions of SNB points Si.

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2112 J. Opt. Soc. Am. B/Vol. 22, No. 10 /October 2005 Pisarchik et al.

eraction gives rise to subharmonics of Fm in the laser re-ponse, whereas in the former case a new harmonic com-onent of Fm appears in each subsequent branch. Fornstance, in branch b2 the subattractor (labeled 2:1) isorn as a continuous degeneration of subattractor 1:1* byeans of the appearance of its second-harmonic compo-

ent; in branch b3 the third harmonic arises, and so on.he SNB points are marked by the dashed curves in Fig.and are labeled Si �i=1, . . . ,6�.In Fig. 5(b) we show the phase difference between theodulation signal from the pump laser and the spikes in

he output of the fiber laser. The Roman numbers I, II,nd III near the branch numbers indicate the spike num-ers in the laser response. One can see the existence ofhe phase-locked regions within certain range of Fm. Inhese locking regions the phase is locked to ��=0 or ��

� and is almost independent of Fm, whereas in thether regions of Fm the phase difference changes with Fm.

The codimensional-two bifurcation diagram in thepace of the modulation depth and modulation frequencys shown in Fig. 6. This diagram represents the SNB lineshat bound the dynamic regimes with different windingumbers. In the cross-hatched regions two different re-imes with the same winding number coexist. Lines P1nd P2 also indicate the boundaries where the windingumber changes, respectively from 1:1 to 2:1 and from 2:1o 3:1. The double SNB lines and the P lines form theongues of the regimes with the same winding number.he horizontal dotted line indicates the value of theodulation depth �m=0.5� for which the phase diagram

n Fig. 5(b) is obtained.As was shown in Subsection 2.C, the modulated laser

an be considered a periodically forced nonlinear oscilla-or. Consider, first, the simplest case of zero detuning,here the frequency of the force is the same as the fre-uency of natural oscillations, i.e., Fm= f0. Exploiting thenalogy with the motion of a light particle on a plane, wean say that the nonzero force m�0 creates a curved sur-ace potential with the maximum and minimum corre-ponding to unstable and stable equilibria, respectively.athematically, the stable equilibrium is asymptotically

ig. 6. Numerical codimensional-two bifurcation diagram in thearameter space of modulation frequency and depth. The cross-atched regions indicate the range of the parameters where gen-ralized bistability is observed.

table, whereas the phase in the unforced oscillator isarginally stable but asymptotically unstable. Whatever

he initial phase difference, the phase point moves towardhe stable equilibrium, so the phase of the oscillator isocked by the force even if the force is vanishingly small.his case is rather trivial, because the frequencies of thescillator and of the force are equal from the beginning, sohe synchronization manifests itself only in the appear-nce of a stable phase relation.When the frequency of the force differs from the natu-

al frequency �Fm� f0�, the effects of the force and the de-uning are opposite: The force tries to make the phasesqual (the phase point tends to the minimum in the po-ential), whereas the detuning drags the phases apart.epending on the relation between the detuning and the

orcing amplitude, one of the factors wins. Consequently,wo qualitatively different regimes are possible.

. EXPERIMENT. Experimental Setup

n our experiments the EDFL is pumped by a commercialaser diode (wavelength, 976 nm; maximum pump power,00 mW; Fig. 7). The laser cavity of a 1.5-m length isormed by a piece of erbium-doped fiber of a 70 cm lengthnd a core diameter of 2.7 �m, and two fiber Bragg grat-ngs (FBG1 and FBG2) with a 1-nm FWHM bandwidthnd reflectivities of 91% and 95% at a 1560 nm wave-ength. The output power of the pumping laser diode andhe fiber laser are recorded through a wavelength-divisionultiplexing coupler (WDM) with photodetectors D1 and2 and analyzed with an oscilloscope and a Fourier spec-

rum analyzer. The output power of the diode laser de-ends linearly on the laser diode current. The harmonicignal, Am sin�2�Fmt�, where Am and Fm are the ampli-ude and the frequency, respectively, of external modula-ion, applied from a signal generator to the laser driverauses harmonic modulation of the diode current with Fm.n our experiments the signal with Am=800 mV results in00% modulation depth of the pump power, while the av-rage diode current is fixed at 40 mA.

. Experimental Resultsn this paper we study only the case of low average pumpowers, when the amplitude of self-modulation of the la-

Fig. 7. Experimental setup.

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Pisarchik et al. Vol. 22, No. 10 /October 2005 /J. Opt. Soc. Am. B 2113

er without external modulation of the pump �Am=0� isegligibly small (2–3% of the magnitude of spikes in theresence of pump modulation). We also present experi-ental results only for relatively low modulation frequen-

ies �Fm� f0�, where f0=50 kHz occurs at the 40 mA cur-ent of the pump diode laser. The laser dynamics at highodulation frequencies �Fm� f0� was studied experimen-

ally in previous papers by some of the presentuthors.10,11

We measure experimentally the peak-to-peak laser in-ensity and phase difference versus the modulation fre-uency for different amplitudes of the pump modulation.he bifurcation diagrams for m=0.5 are shown in Fig. 8.ithin certain ranges of the modulation frequency, the la-

er displays generalized bistability, i.e., the coexistence ofwo attractors. Depending on initial conditions, the laseran oscillate in different periodic regimes with the sameumber of spikes in the laser output during the modula-ion period, e.g., 1:1, 2:1, 3:1, and 4:1. These results con-rm the theoretical prediction made in Section 3 (cf. Fig.). Again, as in Fig. 5, we plot, near each attractor branch,he corresponding time series that display the temporalynamics in the chosen point of the parameter space. Theabels (i runs from 1 to 6) indicate the associated SNBoints, where an attractor is born or dies.Phase difference �� shown in Fig. 8(b) displays the

hase-locking regions, i.e., the regions where �� is inde-endent of modulation frequency Fm. In these regions�=� or ��=0, depending on the number of the branchr spike. A small difference from the results of simula-ions [Fig. 5(b)] is that in the experiment the phase isompletely locked to � and does not depend on Fm,hereas in the simulations the phase is close to � but

ig. 8. Experimental bifurcation diagrams of (a) peak laser in-ensity Ip and (b) phase difference �� versus modulation fre-uency F .

m

hanges slowly with Fm. This difference may result fromhe finite lifetime of level 4 in Fig. 1 (7 �s; Ref. 1) and beue to the processes of the ESA at the pump wavelengthnd the spontaneous emission from the upper levels ofrbium17 (not shown in Fig. 1) that are not considered inhe theory. In the experiments the pump power is de-ected after the pump radiation has passed through thective laser; hence all the above processes have alreadyeen accomplished. Meanwhile, in the simulations thehase difference always depends on Fm. However, thislight discrepancy between the experimental and numeri-al results appears only in the phase diagram and has noffect on our understanding of general features of the la-er dynamics.

The global bifurcation structure of the EDFL withump modulation is shown by codimensional-two bifurca-ion diagrams in the �Fm ,m� plane in Fig. 9. We constructhis diagram by slowly increasing and decreasing Fm andeasuring the maximum amplitude of the laser pulses for

ifferent dynamic regimes. The horizontal dashed line in-icates the values of modulation depth for which the dia-rams in Fig. 8 are obtained. Inside the hatched regions,wo different regimes with the same winding number co-xist. One can observe good agreement between the ex-erimental diagram in Fig. 9 and the numerical diagramn Fig. 6.

. CONCLUSIONSe have investigated in detail, both experimentally and

heoretically, the dynamics of an erbium-doped fiber lasernder harmonic modulation of the diode pump laser. Alobal analysis of the bifurcation structure of phase spaceas been performed with the use of an enhanced theoret-

cal model for the EDFL. We have demonstrated a rich va-iety of bifurcations and coexistence of attractors that ap-ear in the primary saddle-node bifurcations. Thebserved generalized bistability in the low-frequencyange results in doubling of the SNB lines in the param-ter space of the modulation frequency and amplitude. Weave shown the existence of the phase-locked states inhich the phase difference between the laser spikes andump modulation signal does not depend on the modula-ion frequency. The novel laser model has been shown to

ig. 9. Experimental codimensional-two bifurcation diagram inFm ,m� parameter space.

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2114 J. Opt. Soc. Am. B/Vol. 22, No. 10 /October 2005 Pisarchik et al.

escribe our experimental results perfectly. Therefore weelieve that our model of the EDFL may have a definiteffect on future studies of more-complicated rare-earth-oped fiber laser systems with essentially nonlinear dy-amics. We believe that further improvement of theodel, with the thermo-induced lensing in the fiber17 and

on pairs12 accounted for, may help in explanation of otherxperimentally observed features of the diode pumpedDFL, including self-pulsations.

CKNOWLEDGMENTShis research has been supported through a grant fromhe Institute Mexico–USA of the University of CaliforniaUC-MEXUS) and Consejo Nacional de Ciencia y Tecnolo-ia of Mexico and through the CONACYT project 46973.

A. N. Pisarchik’s e-mail address is apisarch@cio.mx,nd A. V. Kir’yanov’s e-mail address is kiryanov@cio.mx.

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