theoretical analysis of a pulsed regime observed with a frequency-shifted-feedback fiber laser

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1302 J. Opt. Soc. Am. B/Vol. 23, No. 7 /July 2006 Maran et al.

Theoretical analysis of a pulsed regime observedwith a frequency-shifted-feedback fiber laser

Jean-Noël Maran

Centre d’Optique, Photonique et Laser, Département de Génie Electriquè et de Génie Informatique, Université Laval,Québec G1K 7P4, Canada

Pascal Besnard

École Nationale Supérieure des Sciences Appliquées et de Technologies, Unité Mixte de Recherches, FonctionsOptiques pour les télécommunication Laboratoire d’Optronique, 6, rue de Kerampont, 22300 Lannion,

France

Sophie LaRochelle

Centre d’Optique, Photonique et Laser, Département de Génie Électrique et de Génie Informatique, Université Laval,Québec G1K 7P4, Canada

Received July 29, 2005; revised January 31, 2006; accepted February 3, 2006; posted March 7, 2006 (Doc. ID 63740)

We study a spontaneous Q-switched laser regime obtained with a multiwavelength frequency-shifted-feedbackerbium-doped fiber laser. We have developed a traveling wave model to describe the dynamics of this ring cav-ity laser. The numerical results are in good agreement with experimental measurements. Furthermore, themodel gives insight into the origin of this pulsed emission. Unlike in most Q-switched lasers, Q-switched op-eration does not rely on amplitude modulation of the net cavity gain but is produced by modulation of theoptical frequency spectrum. To our knowledge, this is the first demonstration of a fiber laser with frequency-modulated Q-switched operation. © 2006 Optical Society of America

OCIS codes: 140.3430, 140.3500, 140.3540.

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. INTRODUCTIONn the past three decades, frequency-shifted-feedback la-ers have been extensively investigated. These lasers,haracterized by the insertion of an acousto-optic fre-uency shifter into the cavity, were first proposed tochieve electronic wavelength tuning of cw lasers.1,2 Thispproach, which was initially studied in dye laser sys-ems, was later applied to solid-state lasers, external cav-ty semiconductor lasers, rare-earth-doped fibers, and Ti-sapphire lasers.3–5 Later, mode locking of frequency-hifted-feedback lasers was demonstrated by matching ofhe acousto-optic frequency shift to the cavity modepacing.6

Recently the multiwavelength operation of frequency-hifted erbium-doped fiber lasers (EDFLs) has created aesurgence of interest in this type of laser.7,8 Sasamori etl.7 observed laser emission over several spectral bandsy incorporating a frequency shifter and a multibandpectral filter into a fiber ring cavity. It was later demon-trated, through numerical simulations, that the presencef the frequency shifter in the ring cavity inhibits cross-ain saturation of the gain medium and allows lasing op-ration to occur over several simultaneous wavelengths.9

n fact, Sabert and Brinkmeyer had already reportedhat, because of the presence of the frequency shifter, theaser operates in an unsaturated gain regime condition.8

ince the demonstration of multiwavelength EDFL opera-ion, several research groups9–14 have studied this laser

0740-3224/06/071302-10/$15.00 © 2

onfiguration to obtain the largest number of wavelengthsith uniform spectral power density. Although most of

hese studies focused on the optimization of these lasersor multiwavelength cw emission, there have been severaleports of frequency-shifted-feedback fiber lasers operat-ng in pulsed regimes.15–17 Spontaneous generation ofhort pulses has been observed in both single- andultiple-wavelength EDFLs with frequency-shifted

eedback.18 This spontaneous mode locking is explaineds the interaction between self-phase modulation and therequency shift. These lasers also present another type ofulsed activity that more closely resembles spontaneous

switching. In Ref. 18 Maran and LaRochelle reportedhe experimental observation of three kinds of emission:w, passive mode locking, and spontaneous Q switching.he last-named type is characterized by a pulse period

hat varies from 90 to 58 �s and a pulse’s full width atalf-maximum (FWHM) that varies from 20 to 9 �s.In this paper we investigate the origin of spontaneousswitching of frequency-shifted EDFLs. In Section 2 we

resent an experimental demonstration of this type of la-er in a ring cavity incorporating an acousto-optic fre-uency shifter and fiber Bragg gratings. In Section 3 weescribe a model based on a traveling wave formalismnd compare the numerical results with experimentaleasurements in terms of pulse width and repetition

ate. The numerical simulations also provide insight intohe physical origin of Q-switched pulse regimes. A thor-

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Maran et al. Vol. 23, No. 7 /July 2006 /J. Opt. Soc. Am. B 1303

ugh analysis of the relevant laser parameters is pre-ented in Section 4. We then discuss the origin of thepontaneous Q switching and explain it as being due to aodulation of the laser frequency spectrum.

. EXPERIMENTAL RESULTSelf-pulsing in a frequency-shifted fiber laser (FSFL) withwell-defined pulse width and pulse repetition rate18 was

ecently observed, in contrast to a previous observation ofnstable pulsing.8 Self-pulsing with EDFLs has also beenidely observed. This behavior was attributed to an ion-uenching process.19 When an erbium-doped fiber (EDF)s highly doped, erbium ions can form aggregates of two or

ore ions. These aggregates lead to dynamics similar tohose obtained with a saturable absorber and induce self-ulsing. In this section we briefly describe the operatingrinciple of a multiwavelength FSFL and the associatedxperimental setup. We then show that the pulsing in aSFL is not due to ion quenching and perform a detailedemporal characterization of the pulsed emission.

Figure 1 shows the experimental setup of the multi-avelength FSFL. It consists of a ring cavity incorporat-

ng an acousto-optic frequency shifter and a multibandpectral filter. The gain medium consists of a 14 m longDF (Lucent Technologies) pumped by a 980 nm laser di-de through a wavelength division multiplexing (WDM)oupler. An EDF is known to behave, at room tempera-ure, as a homogeneously broadened gain medium inhich multiwavelength emission is prohibited by strong

ross-gain saturation. In the FSFL the presence of thecousto-optic element, with a frequency shift of 80 MHz,nhibits gain saturation. In this case, the frequency of themitted radiation is shifted away from the maximumransmission peaks of the spectral filter. Rather, theteady-state frequency oscillates about a mean value cen-ered at the edge of the transmission peaks of the filterands, typically a few gigahertz away from the maximumet gain of the cavity.8,13 Therefore the acousto-optic fre-uency shifter (AOFS) compels the laser to operate withn unsaturated gain and permits simultaneous emissionver a large number of wavelength bands.9–14 By wave-ength bands, we mean channels that are defined by thepectral filter. When the frequency shifter is removed, theaser operates in a single-wavelength manner, at theavelength band with the maximum net gain.14 It should

Fig. 1. Frequency-shifted feedback EDFL setup.

e noted that the frequency shift �80 MHz� is not resonantith the free spectral range �3.12 MHz� of the cavity.In a previous experimental investigation of this multi-

avelength FSFL, Maran and LaRochelle noted that,hen the pulsed regime is obtained for one wavelengthand, it is also observed for all the other wavelengthands.18 Here, to simplify the numerical analysis, weave decided to operate the FSFL in a single-wavelengthonfiguration, i.e., by inserting a single narrowband25 GHz� filter into the cavity. We experimentally verifiedhat the self-pulsing regime obtained with the single-avelength configuration is identical to that observed forne wavelength filtered from the output of the multiwave-ength FSFL configuration. Consequently, we believe thathe scope of the present study regarding the dynamics ofhe self-pulsing configuration is not limited by single-avelength operation.In the experimental setup, the single-band spectral fil-

er is composed of a fiber Bragg grating (FBG) insertednto the cavity through an optical circulator. The peakavelength of the FBG is 1549.5 nm, its bandwidth5 GHz, and its out-of-band isolation 20 dB [Fig. 2(A)].hese characteristics are identical to the characteristicsf an isolated band of the multiwavelength filter used inef. 11. The frequency parameters involved in the opera-

ion and numerical simulation of the laser are as follows:requency shift, 80 MHz; cavity free spectral range,.12 MHz; gain bandwidth, 5 THz; pulse repetition rate,7–11 kHz. Other components of the FSFL shown in Fig.

include an optical isolator to ensure unidirectional

ig. 2. (A) Spectral transmission of the filter and (B) Emissionnd absorption cross sections of the doped fiber as a function ofavelength.

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1304 J. Opt. Soc. Am. B/Vol. 23, No. 7 /July 2006 Maran et al.

ropagation and prevent spectral hole burning, an outputoupler, a polarization controller, and a variable attenua-or.

To investigate the pulsed operation of this laser, weeasured the laser’s temporal output with a fast photodi-

de (bandwidth, 25 GHz� and an oscilloscope in single-hot mode (bandwidth, 1.5 GHz). The measurements wereerformed first for the laser ring cavity without the spec-ral filter and without the frequency shifter [Fig. 3(A)],hen with the frequency shifter alone [Fig. 3(B)], andhirdly with the spectral filter (FBG) but without the fre-uency shifter [Fig. 3(C)]. Finally, both the spectral filternd the frequency shifter were added [Fig. 3(D)]. Theulsed regime appeared only in the last case. For thehree other laser configurations the laser emitted in a cwegime for any pump current. The fact that the lasermission is cw in the free running configuration (withouthe spectral filter and the frequency shifter) demonstrateshat the origin of the pulsed condition cannot be attrib-ted to ion quenching.A temporal trace of a single pulse is shown in Fig. 4. We

ote that the shape of this pulse is symmetrical, which isarkedly different from pulses obtained in classic-switched lasers. This pulsed emission occurs with rep-

tition rates of typically tens of kilohertz and with pulseidths of tens of microseconds. The FWHM and the pe-

iod of the pulses are displayed in Fig. 5 as functions ofhe pump power.

. NUMERICAL MODELhe experimental investigations have revealed thatulsed operation is induced by the simultaneous presence

ig. 3. Experimental measurement of the FSFL output as a funrequency shifter, (B) without a spectral filter but including a freqD) with a spectral filter and a frequency shifter.

f a frequency shifter and a narrowband filter. To betternderstand the FSFL dynamics, we describe in this sec-ion a traveling wave model to describe the evolution ofhe population inversion, of the pump power, and of thepectrally resolved signal power. After a short discussionf the model assumptions, we present the main equationsnd define the relevant parameters. We also detail theumerical method used to solve these coupled equations,nd we validate the model against previously publishedata.

. Basic Assumptionshe rate equation formalism can be used to describe the

emporal behavior of a laser when the temporal dynamics

f time for several laser cavities: (A) without a spectral filter or ashifter, (C) with a spectral filter but without a frequency shifter,

ig. 4. Single pulse measurement obtained with a FSFL oper-ting in a Q-switched condition.

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Maran et al. Vol. 23, No. 7 /July 2006 /J. Opt. Soc. Am. B 1305

re slow in comparison with the coherence time (the in-erse of the laser transition linewidth). In the FSFL thisondition is fulfilled because the repetition rate of theulse train is tens of kilohertz whereas the FWHM is tensf microseconds, which is 3 orders of magnitude largerhan the laser transition linewidth. The proposed models an intensity model, which means that the phase of theeld is not considered. Moreover, nonlinear effects are notonsidered but the saturation of the gain medium is.

. Description of the Modeleveral models have been proposed to describe pulsednd Q-switched operation for various types of laser suchs gas, semiconductor, and fiber lasers.20–23 These modelsan be classified into two groups: point models and trav-ling wave models. For point models it is assumed thathe population inversion and photon densities are uni-ormly distributed along the gain medium. This hypoth-sis simplifies the numerical solution but leads to someritical limitations21 for a long gain medium. In the workescribed here we used a traveling wave model that con-ists of three basic equations: the first one describes thevolution of the population inversion, and the two othersescribe the evolution of the pump and of the signal in-ensities.

Erbium is often considered a three-level system.24 Inhis assumption, following the excitation of an ion in theround state (4I15/2; level 1) through absorption of a80 nm pump photon, a quick multiphonon interaction,hich can be assumed instantaneous, causes the relax-tion between the upper level (4I11/2; level 3) and the ex-ited metastable state (4I13/2; level 2). The transitions be-ween the excited state and the ground state are due topontaneous emission, which is characterized by a timeonstant �2 (typically 10 ms) and to stimulated emission.he stimulated transitions are controlled by the rates Wend Wa. They depend on signal intensity and on the emis-ion and absorption cross sections �e and �a. Finally, R13epresents the pump absorption transition rate. However,e have to keep in mind that in the case of a FSFL the

ignal is frequency shifted by the acousto-optic element atach cavity round trip. To describe correctly the dynamicsf these lasers, the model has to take into account the fre-

ig. 5. Pulse FWHM (triangles) and repetition period (circles)ersus pump power. Filled symbols represent experimental data,nd open symbols are numerical simulation results.

uency dependence of the different elements. For thisurpose the spectral range will be divided in several fre-uency slots; each of them will be treated as an indepen-ent signal coupled through the gain medium. The evolu-ion of the population of the excited state can be describeds follows24:

�N2�z,t�

�t= �R13�z,t� + Wa�z,t��N1�z,t�

− �We�z,t� +1

�2�N2�z,t�, �1�

N1�z,t� + N2�z,t� = N0, �2�

here Nj�z , t� represents the population density of level jt position z and at time t and N0 is the total erbium con-entration. Excited-state absorption of the pump is negli-ible with a pump wavelength of 980 nm. The emissionnd absorption cross sections �e��� and �a���, respectively,re frequency dependent, as shown in Fig. 2(B), so thetimulated transition rates are defined as

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� �e����Is+�z,t,�� + Is

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here Is±�z , t ,�� represent the spectral intensity along po-

ition z inside the gain medium at time t propagating inhe clockwise �+� and anticlockwise �−� directions, whilep

+�z , t� corresponds to the pump intensity. Also, �p repre-ents the absorption cross section at the pump wave-ength, h is Planck’s constant, and Aeff is the effective fi-er core area. The overlap integral between the LP01ode intensity distribution and the erbium density distri-

ution ��r� is ����, defined by

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o complete the model, two additional equations areeeded to describe the propagation of the signal and ofhe pump through the gain medium:

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1306 J. Opt. Soc. Am. B/Vol. 23, No. 7 /July 2006 Maran et al.

1

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here vg represents the group velocity in the gain me-ium, �s=�a /�e, and /4� represents the fraction ofpontaneous emission guided by the optical fiber core.he first term on the right-hand side of Eq. (7) defines the

evel of stimulated emission generated by the laser field,hile the second term represents the amplified spontane-us emission in each frequency slot.

To solve the partial differential equations described byqs. (1), (6), and (7), we have to specify some boundaryonditions at the extremities of the EDF that include thearious elements of the laser cavity:

Ip+�0,t� = I0

+, �8a�

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+�L,t,� + ��Tf���, �8b�

Is−�L,t,�� = ��coupler�isoIs

−�0,t,� + �Tf���, �8c�

here z=0 corresponds to the input of the gain mediumnd z=L to the output. The losses �, �coupler, and �iso rep-esent, respectively, the background fiber loss, couplerosses, and isolator losses. Also, Tf��� is the transmissionunction of the spectral filter, while Is

+�z , t ,�+�� ands−�z , t ,�+�� represent the two counterpropagating in-

ensities after they have passed through the frequencyhifter (� is the frequency shift). Equations (8) are givenor a ring cavity.

. Numerical Solutionhe solution of partial differential equations such as Eqs.

1), (6), and (7) is numerically complex. However, in thisase we can simplify the problem considerably by intro-ucing two independent space–time variables, u and ,efined as follows22:

u = �vgt + z�/2,

= �vgt − z�/2. �9�

pplying Eq. (9) to Eqs. (6) and (7), we obtain an equiva-ent equation system expressed in coordinates �u , �:

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quations (10) form a system of coupled differential equa-ions for which the numerical solution are well known.25

ote that Eq. (1) is not affected by the change of variablesEq. (9)].

Another important consideration in choosing the nu-erical technique to solve the coupled differential equa-

ions is the required frequency and time sampling. Toodel the frequency shift adequately, one has to sample

he spectral band with a step corresponding to at least therequency-shift value �80 MHz�. This sampling generatesarge arrays, so the choice of the numerical technique ismportant for minimizing computation time. Finally, weeed a fine time sampling, of the order of tens of nanosec-nds, to describe each pulse adequately and to integratever several milliseconds to go over the transient condi-ion and correctly analyze the Q-switched condition. Forll these reasons, to reduce the computation time we usedsecond-order Runge–Kutta method to solve Eqs. (10).

his method needs only one intermediate point for one tonow the function value at iteration step i+1 instead of 3n the case of a fourth-order method.25

. Validation of the Numerical Modele validated the model and the numerical technique by

erforming simulations of the experimental results re-orted in Ref. 26. In this case, the laser system is a linearavity, EDFL for which Q switching is achieved by modu-ation of the intracavity loss. For the simulations, theoundary conditions were modified in the following wayo describe a linear cavity with active modulation:

Is+�0,t,�� = R1M�t�Is

−�0,t,��, �11a�

Is−�L,t,�� = R2M�t�Is

+�L,t,��, �11b�

here R1 and R2 represent the reflectivity of each mirrornd M�t� defines the transmission function of the intrac-vity amplitude modulator.The results of the numerical simulations, shown in Fig.

, were found to be in good agreement with the experi-ental ones given in Ref. 26. We note that pulse asymme-

ry, which is characteristic of a classic Q-switched regimeith cavity loss modulation, is well predicted by theodel. The measured FWHM of the pulse was 13.5 ns,

ompared to 11.5 ns in the simulation. We performed theimulations using both fourth- and second-order Runge–utta algorithms, noting no differences between resultsxcept that the computation time is longer when theourth-order method is used.

. NUMERICAL SIMULATIONS OF THE-SWITCHED REGIME OF A FREQUENCY-HIFTED FEEDBACKRBIUM-DOPED FIBER LASER

n this section we present numerical simulations of theulsed emission from a FSFL. We demonstrate that the

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Maran et al. Vol. 23, No. 7 /July 2006 /J. Opt. Soc. Am. B 1307

ulsed condition is linked to the presence of both the fre-uency shifter and the spectral filter, and we examine theemporal characteristics of the pulsed emission. To gainore insight into the origin of the pulsed condition we

hen analyze in detail the evolution of the population in-ersion and of the laser spectrum.

. Simulations of the Pulsed Emissions mentioned above, we decided to do experimental char-cterization and numerical simulation of a FSFL in aingle-wavelength configuration to save computationime. Furthermore, to further reduce the computationime we varied the frequency sampling step across the op-ical spectrum. A fine sampling step, equal to the fre-uency shift �80 MHz), was used over 0.6 nm about theasing wavelength, while the rest of the spectral band1440–1640 nm� was sampled with a frequency step of25 GHz. In the simulations, the transfer function of theptical filter was defined from the measured spectralransmission data of the FBG that was interpolated withfrequency step of 80 MHz. The parameters used in the

umerical simulations are listed in Table 1.

ig. 6. Simulation results obtained with the parameters pub-ished by Chandonnet and Larose.26

Table 1. Laser and Numerical Parameters Usedin the Simulations

hysical Parameter Value

ain medium length (m) 15patial step (m) 1

0 �m−3� 1.54�1024

(ms) 10.3avity losses including AOFS losses (dB) 9ariable optical attenuator losses (dB) 1ffective area ��m2� 50

pN0 (dB/m) 2.04ump power (mW) 120ump wavelength (nm) 980pectral bandwidth (nm) 1450–1628umber of spectral points 1998emporal window (ms) 0; 8ime step (ns) 5

Using the model described by Eqs. (10) and (8), weimulated the experimental conditions presented in Fig.. The numerical results, shown in Fig. 7, confirm thatulsed emission appears only when the spectral filter andhe frequency shifter are inserted into the cavity, as is as-umed in all the following results.

In the simulations, we first consider an incident pumpower of 120 mW at z=0. At the doped fiber output, theesidual pump power is 22 mW. In this condition, theength averaged population of the metastable level, N2ave,s 77%. The threshold of the Q-switched regime was5 mW of pump power in the experiments, while it oc-urred for 40 mW in the numerical simulations. Figure 8hows the laser spectrum, the temporal trace of the laserutput, and N2ave as functions of time. The repetition ratef the pulse train is 22.2 kHz, and the pulses have aWHM of 3.98 �s. The peak power is 339 �W, and theulse energy is 3.7 nJ. Figure 8(B) shows that the emittedulses are symmetrical, as were observed in the experi-ents. In Q switching, the pulse’s FWHM and repetition

eriod typically decrease with increased pump power. TheWHMs of the pulses, calculated as a function of pumpower, are found to be in good agreement with the experi-

ig. 7. Numerical results of the FSFL output as a function ofime for various laser cavities: (A) without a spectral filter or arequency shifter (similar results would be obtained without apectral filter but including a frequency shifter, or with a spectrallter but without a frequency shifter), (B) with a spectral filternd a frequency shifter.

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1308 J. Opt. Soc. Am. B/Vol. 23, No. 7 /July 2006 Maran et al.

entally measured values, as can be seen from Fig. 5.he calculated pulse repetition rate was found to be un-erestimated by the numerical simulations, a discrepancyhat grows when the pump power is reduced. However, byrtificially increasing the level of spontaneous emission inq. (7), we were able to obtain better agreement for theepetition rate, which indicates that the spontaneousmission rate is not adequately estimated by the emissionnd absorption cross section data provided by the fiberanufacturer.

ig. 8. Numerical simulation results of the FSFL: (A) output la-er spectrum, (B) output laser pulse train, (C) length averagedopulation of metastable level N2 and output intensity versusime.

As we discuss in detail in Section 5 below, the classic-switching mechanism usually relies on modulation of

he net cavity loss, which results in large variations of theopulation inversion. Such is clearly not the case for thisaser, as shown by Fig. 8(C). In the Q-switched regime of aSFL, the population inversion remains almost constants pulsed emission occurs. In Subsection 4.B we shall ex-mine this behavior in more detail.

. Investigations into the Origin of Q Switchingo confirm the origin of the Q switching we calculated, atifferent times during the emission of a pulse (time pointspecified in Fig. 9), the net gain-spectrum:

G��� = Tf���exp��e����N2ave − �s�1 − N2ave��L − 2�L�.

�12�

igure 10 displays the evolution of the optical spectrumnd of the net gain as the pulse builds up inside the cav-ty. We previously observed from Fig. 8(C) that N2ave waslmost constant in time and that the inversion displayedeak variations. This is confirmed by the unchanged netain versus time, as shown in Fig. 10. Only small peak-o-peak changes, of the order of 1%, can be observed. Ob-iously, this small net gain variation is not able to explainsustained pulsed emission. This constant gain is easily

nderstandable if one considers the lifetime of the activeedium (of the order of 10 ms), which is not short enough

o respond to a train with a repetition rate of 22 kHz. Themplifying medium sees an average optical power, ashen an optical erbium-doped amplifier is used in high-it-rate optical telecommunications.To investigate further the mechanism of the origin of Q

witching in a FSFL, we examine in detail optical spectraalculated at different times. As observed by Sabert andrinkmeyer8 for a cw laser, the laser spectrum is localizedn the blue edge of the net gain, as schematized by theashed line in Fig. 10. When the laser is switched on, theptical spectrum is located at the center of the net gain toavor maximum optical power, as expected from a homo-eneously broadened gain medium. Afterward, at eachavity round trip, the emission line is shifted towardhorter wavelengths as the signal passes through thecousto-optic element. In a cw laser this shift stops when

ig. 9. Distribution of sampled time points along one outputulse.

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Maran et al. Vol. 23, No. 7 /July 2006 /J. Opt. Soc. Am. B 1309

he laser spectrum is centered at a frequency for whichhe net gain is equal to 1. A steady-state condition is thenstablished, which corresponds to equilibrium, reachedfter each round trip, between the gain of a pulse crossinghe active medium and the losses that are due to the blue-hift of the laser spectrum. In the pulsed condition, how-ver, no equilibrium is found, and the optical spectrum isonstantly changing close to the operating point (dashedine), revealing competition between two effects: the blue-hift that is due to the accousto-optic element and the red-hift that is due to the natural tendency of a homoge-eously broadened gain medium to operate at the centerf the gain line.

The position and the shape of the laser spectrum resultrom an interplay between the shape of the net gain spec-rum and the blueshift of the laser spectrum, operated byhe acousto-optic frequency shifter. Initially, the opticalaser spectrum (shown as the black curve of Fig. 10) isroad, and its maximum is located on the red side of thequilibrium point (schematized by the vertical dashedine). Note also that this spectrum has an asymmetrichape with a long tail on the red side because the net gains bigger at these longer wavelengths. After each roundrip there is a frequency shift. The energy, localized atigher wavelengths, is transferred to shorter wavelengthsnd will then participate in the peak spectrum growthhile some energy located at lower wavelengths is shiftedut of the amplifying region, where the net gain is lowerhan 1. In short, energy is lost at lower wavelengths andnergy is gained at longer wavelengths. The energy bud-et is unbalanced because of the asymmetry of the opticalpectrum with respect to the slope of net gain. It is thissymmetry that is at the origin of the buildup of theulses. When the energy budget is positive, the pulserows and corresponds to the red spectrum. At the maxi-um of the pulse, the associated spectrum (green curve)

ecomes narrower and more symmetric. It is also shiftedoward shorter wavelengths. The spectrum reaches a po-ition centered at a wavelength for which the energy bud-et becomes negative. After a round trip, a frequency shiftf this spectrum will send more energy outside the ampli-ying band than energy gained by the amplification. Allhese factors contribute to an increased loss. The pulsehen starts to decay (and corresponds to the blue and or-nge spectra).It is quite clear that the dynamics of the optical spec-

rum explain the pulsed regime. However, in this case

ig. 10. Calculation of the optical and net gain spectra at eachampled time.

hese different spectra are associated with the same levelf losses and of gain. Thus the Q switching is not accom-anied by a change in the net gain but is due to differentverlaps between the optical laser spectrum and the netpectral gain. The last quantity is given by the constantopulation inversion and is fixed by the average opticalntensity seen by the gain medium, which is unable to re-pond instantly to such a few-microsecond pulse. Qwitching is produced by energy redistribution among theifferent optical frequencies.To better understand this dynamics, we observed si-ultaneously the evolution of the pulse intensity and of

he FWHM of the laser spectrum. The results are given inig. 11(A). Also Fig. 11(B) presents the evolution of theavelength of the optical spectrum maximum and of theet gain at that wavelength. Note that the jumps in thevolution of the maximum wavelength are numerical ar-ifacts, that are due to the numerical frequency integra-ion step �80 MHz�. Analyzing these two figures, we dis-inguish a three-phase process in the Q-switchedynamics. In a first phase [step (a), Fig. 11], the laser out-ut power is low, the optical spectrum is broad, and itsaximum wavelength is located at longer wavelengths.he gain from the stimulated emission and from the en-rgy injected by the AOFS from the central part of thepectrum is more important than the energy lost athorter wavelengths. The output power increases, and theulse begins to build up as the spectrum narrows androws at the expense of the pedestal, i.e., of the accumu-ated spontaneous energy. The wavelength of the spec-rum maximum is shifted to shorter wavelengths by theOFS. This phase [step (a)] corresponds to the pulseuildup.

ig. 11. Study of the Q-switched dynamic: (A) output pulses andpectrum FWHM versus time and (B) spectrum maximum wave-ength and net gain versus time.

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1310 J. Opt. Soc. Am. B/Vol. 23, No. 7 /July 2006 Maran et al.

As the wavelength of the spectrum maximum is shiftedoward shorter wavelengths and the laser spectrum nar-ows, the energy that is lost becomes more importanthan the energy gained by the stimulated amplificationnd by the redistribution from longer to shorter wave-engths. This phase [step (b)] corresponds to the decay ofhe pulse. It starts when the spectrum’s FWHM is.125 GHz.The last phase [step (c)] is essential to the understand-

ng of the essence of the spectrum dynamics and of theeason that a redshift can exist despite the blueshift im-osed by the AOFS. The output power is low and comesrom the ASE generated in the EDF. As a matter of fact,he FSFL is not lasing and tends to amplify the longeravelengths for which the gain is higher, independentlyf the AOFS. This results in a redshift to maximize theutput optical power. The spectrum broadens by 2.75 GHznd shifts toward longer wavelengths. This process can beeen as an initialization during which there is an accumu-ation of energy over a wide spectrum. At the end of stepc), the FSFL starts to lase, the FWHM starts to decrease,nd the pulse to build up.We can note in conclusion that the pulse’s FWHM is

ontrolled by steps (a) and (b) while the pulse train’s rep-tition rate is governed by step (c).

. DISCUSSIONhe numerical simulations show that the Q-switched re-ime in a FSFL takes its origin in the modulation of itsptical spectrum, in contrast to most active Q-switchingechanisms based on amplitude modulation of cavity

osses.27 The schematic behavior of the Q-switched lasersith amplitude modulation of cavity losses is presented inig. 12(A). At the beginning of the process, the cavity

osses are high, thereby preventing laser action. Theopulation inversion and gain build up in the laser cavityntil the losses are suddenly decreased. The gain becomesuch larger than the cavity losses, and a laser pulse be-

ins to build up. This pulse saturates the gain and drivesown the population inversion below the new cavity lossevel. The laser oscillation dies. The losses are increasedgain, and the process is repeated. The loss modulationan be produced by different types of intracavity ampli-ude modulator. Note again that the pulses obtained withclassic Q-switching process are always asymmetric be-

ause pulse buildup is faster than pulse decay.In a FSFL we observe a modulation of the optical fre-

uency spectrum. A schematic representation of Q switch-ng is presented in Fig. 12(B). Initially, the optical spec-rum is broad and located toward the center of the opticallter. As the laser action begins, the spectrum becomesarrower and its central wavelength shifts toward thedge of the filter. Simultaneously, the output power in-reases and a pulse builds up. Because of the presence ofhe frequency shifter, the narrowing of the spectrum andts shift toward the edge of the filter result in increasedoss, and the pulse starts to decay. The process then startsgain from the ASE spectrum in the central portion of thelter. This mechanism is quite similar to frequency modu-

ation (FM) mode locking.27 We thus propose to call it FMswitching.

We stress the importance of ASE. In a nonshifted laserhe numerical model [Eqs. (1), (6), and (7)] permits theasing effect to occur even without ASE. For the FSFL,owever, when the ASE is removed the emission of the la-er dies out because after several round trips the coherentignal is totally shifted by the AOFS out of the gain band-idth. The reinjected signal that is due to the ASE en-

ures the buildup of the laser emission.Pulses generated by FM have a symmetric shape, un-

ike in amplitude modulated Q switching for which theulse shape is asymmetrical. In the latter case, when theulse grows, the medium is already inverted and thepontaneous noise quickly saturates the gain at a rateroportional to the photon’s lifetime. When the pulse in-ensity is high, however, the population inversion decayst a speed driven by the time constant of the cavity. In FM

switching, the net gain is always constant and it isever larger than its typical steady-state value. The edgesf the gain are shaped by the filter, and, in the presentase, the slope of Tf�v� is linear in the spectral regionhere the laser operates. Moreover, the frequency shift islso linear because it is induced at each cavity round trip.hese characteristics, combined with the fact that the la-er spectrum is located on the edge of the gain, explain,hy the pulses are symmetrical in the case of this FM-switched laser.

ig. 12. Schematic description of the operation of a Q-switchedaser with modulation of (A) the cavity losses and (b) (B) the op-ical frequency spectrum in a frequency-shifted-feedback laser.

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Maran et al. Vol. 23, No. 7 /July 2006 /J. Opt. Soc. Am. B 1311

. CONCLUSIONSe have proposed an intensity model to describe a pulsed

ondition observed with a multiwavelength frequency-eedback laser. Our simulations have allowed us to iden-ify the origin of Q-switching brought on by the associa-ion of an acousto-optic frequency shifter and aarrowband filter. Unlike for conventional Q-switched la-ers, there is no need for amplitude modulation of the cav-ty losses to produce a pulse train. The dynamics are ex-lained by a redistribution of the energy by the AOFSmong the different modes, about an operating point thats close to the lasing threshold. The optical laser spectrums not constant. This variation is due to competition be-ween the AOFS and the gain. The AOFS tends to pull thenergy toward the shorter wavelengths, while the gainends to favor modes at longer wavelengths because theaser is operating on a filter edge. The amplitude of thispectrum is modulated and generates a train of pulseshile the net gain and the inversion remain constant. We

dentify this new condition FM Q switching.

CKNOWLEDGEMENTShis work was supported by the Canadian Institute forhotonic Innovations. We thank Yves Jaouën for fruitfuliscussions about the numerical procedure. We thank onef the referees for drawing our attention to the contribu-ion of spontaneous emission to the repetition rate.

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