346 J. Opt. Soc. Am. B/Vol. 26, No. 2 /February 2009 Shtyrina et al.
Evolution and stability of pulse regimesin SESAM-mode-locked femtosecond fiber lasers
Olga Shtyrina,1 Mikhail Fedoruk,1 Sergey Turitsyn,2,* Robert Herda,3 and Oleg Okhotnikov4
1Institute of Computational Technologies, Siberian Branch of Russian Academy of Sciencies, 6 Lavrentyev Avenue,Novosibirsk 630090, Russia
2Photonics Research Group, School of Engineering and Applied Science, Aston University,Birmingham B4 7ET, UK
3TOPTICA Photonics AG, Lochhamer Schlag 19, D-82166 Graefelfing, Germany4Optoelectronics Research Center, Tampere University of Technology, P.O. Box 693, 33101 Tampere, Finland
. INTRODUCTIONemtosecond mode-locked fiber lasers have attracted areat deal of interest recently both as advanced photonicevices with a range of research and industrial applica-ions and also as an interesting nonlinear physical systemo study. Pulsed laser systems represent a distinctive ex-mple of practical applications of the soliton theories pro-iding a constructive impact on laser science and nonlin-ar physics. Typical femtosecond mode-locking fiberasers comprise the following key elements: active fibercting as an amplifying medium, a dispersion compensat-ng element (e.g., diffractive grating pairs, though in someecent designs it is shown that intracavity compensations not a prerequisite), and a saturable absorber elementroviding pulse formation from noise and a stable mode-ocking operation. The operation of a mode-locked laser inhe femtosecond regime is determined by the interactionetween dispersion and nonlinearity; the interplay ofhese parameters is becoming more complex, and the sen-itivity to small changes is increasing when aiming at theeneration of shorter pulses with higher pulse energies.he nonlinear nature of dynamics dependent on many pa-ameters makes the mode-locked regime complex result-ng in a strong spectral and temporal evolution duringulse propagation through the laser cavity. Therefore, de-ign optimization and modeling of pulsed lasers presentsmultiparametric nonlinear problem and despite recent
mpressive progress in femtosecond lasers many researchroblems are still open. Analysis of all cavity arrange-ent aspects is particularly essential for achieving a
ingle-pulse regime with high pulse energy and high sta-ility against cavity perturbations. In this paper we haveerformed laser optimizations for a specific class of
emtosecond systems using semiconductor saturablebsorbers.One of the key approaches in femtosecond lasers is the
echnology of semiconductor saturable absorber mirrorsSESAMs) [1–3] that is based on bandgap engineeringnd epitaxial growth. The SESAM technology has manydvantages compared to other methods, e.g., Kerr lensode-locking [4] and mode-locking by exploiting nonlin-
ar polarization evolution [5] since it offers a self-startingulse operation and good environmental stability. SESAMechnology has been widely used nowadays to initiate ando support the mode-locking regime in different solid-tate and fiber lasers. Fiber lasers have been mode-lockedsing SESAM technology in a broad wavelength rangerom 900 to 2000 nm [6–8].
The reliable start-up of the continuous-wave (cw) mode-ocked regime in fiber lasers is typically provided byESAMs with a high modulation depth and a low satura-ion fluence due to the high level of amplified spontaneousadiation inside the fiber cavity and the large value of dis-ersion. Quantum well absorbers could offer both a highodulation depth and a low saturation energy when
laced in a microcavity providing the resonant condition1,9,10]. For reliable self-starting further optimization ofhe recovery time is also essential. As-grown quantumells typically have a long recovery time of a few hun-reds of picoseconds and consequently cannot start mode-ocking because intracavity noiselike quasi-cw radiationould efficiently saturate the SESAM [11]. The recoveryime can be reduced by introducing fast recombinationenters using low temperature growth [12] or heavy ionrradiation [13]. Subsequent thermal annealing can re-uce the undesirable nonsaturable losses while preserv-
009 Optical Society of America
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Shtyrina et al. Vol. 26, No. 2 /February 2009 /J. Opt. Soc. Am. B 347
ng the high modulation depth of the fast absorbers fromegradation [14]. Quantum dot absorbers provide anotherromising option for fast structures with a low saturationnergy [15].
One of the goals of this paper is to study how SESAMarameters affect properties of steady-state pulses andhe overall laser system performance. We perform here aassive optimization of the laser system including total
ain, cavity dispersion, saturable absorber recovery time,nd saturation energy. In this paper we focus mainly onhe analysis of a single-pulse operation as this regime isost desired for high energy oscillators with pulse evolu-
ion in the laser cavity. We examine the stability of mode-ocking in terms of a parameter map.
. LASER SYSTEM SETUPigure 1 shows the setup of the laser system used in theumerical simulation similar to the experimental configu-ation presented in [7]. The evolution of the pulse haseen simulated by propagating the pulse through the dif-erent cavity elements consecutively [16]. The fiber cavityonsists of an output coupler, a 1.5 m long passive fiber,n active fiber with a length of 0.5 m, a grating pair thatrovides anomalous group-velocity dispersion (GVD) withegligible nonlinearity, and a SESAM. The output cou-ling is modeled as an 80% loss of energy at the beginningf each round trip. The cumulative dispersion of the pas-ive and active fibers is 0.08 ps2 on the one round trip.
The values of the nonlinear parameter and dispersionf the passive and active fibers used in the numericalodeling were �=5 W−1 km−1, �2=20 ps2/km, and �30.05 ps3/km. Both the passive fiber and the active fiberections have been modeled using the nonlinearchrödinger equation with gain
�A
�z= − i
�2
2
�2A
��2 +�3
6
�3A
��3 + i��A�2A + gA.
ere A�z ,�� is the electric field envelope and z and � arehe propagation distance and the pulse local time, respec-ively. The equation has been solved using the standardplit-step Fourier-transform method. The effect of gainpectral filtering is introduced in the frequency domainsing a Lorentzian line shape with a bandwidth of 40 nmnd a central wavelength of 1027 nm. Gain saturation isodeled according to
g�Pave� =g0
1 + Pave/Psat,G,
here Pave=Ep /TR is the average power, g0 is the smallignal gain, and Psat,G=10 mW is the saturation power.he pulse energy is then given by Ep=�−TR/2
TR/2 �A�z ,���2d�,
Input
Output
Coupling
1.5 m
Activefiber
Grating
SESAM
Passivefiber
0.5 m
Fig. 1. Setup of a mode-locked fiber laser.
here TR=20 ns is the cavity round trip time. The con-entional technique for intracavity dispersion compensa-ion in a fiber laser is a grating pair that provides suffi-ient anomalous GVD with negligible nonlinearity andinear losses below 20%. The typical values of the gratingair dispersion characteristics around which we havetarted optimization are second order dispersion of0.126 ps2 and third order dispersion of 0.000355 ps3. Inur simulations the accumulative cavity dispersion was aariable parameter, and we have varied the total intrac-vity dispersion by changing the value of the GVD of therating pair. Therefore, in what follows we will use theerms �2 and �3 for accumulated cavity dispersion param-ters rather than for local ones. The inclusion of third or-er dispersion is necessary because we compensate theormal dispersion of the fiber with the anomalous disper-ion of the grating pair. The total third order dispersionan therefore reach significant values because the fibernd the grating pair third order dispersion both have aositive sign. A neglect of the third order dispersion wouldead to unrealistically short pulse durations.
The saturable absorber loss q was modeled using thetandard rate equation [17]
dq���
d�= −
q��� − q0
�A−
q���P�z*,��
�APsat,
here P�z* ,��= �A�z* ,���2 is the incident pulse power, z* ishe fixed distance, q0 is the saturable loss (modulationepth), Psat is the saturation power, and �A is the recoveryime.
By solving this equation we determine the saturablebsorption q as a function of time and the input field. Theower transfer function of the saturable absorber couldhen be found as T�t�=1−q�t ,P�z* , t��. In terms of opticalower at the input and the output of the saturable ab-orber, it can be expressed in the form
he SESAM is assumed to have a nonsaturable loss of%.The initial field distribution in the laser cavity has
een assumed to be white noise. The simulations are runntil the field attains the steady-state after a certainumber of cavity round trips. The impact of initial back-round noise on a mode-locking process has been studiedtatistically for different start-up scenarios. Stability re-ions and, particularly, the gap between single-pulse andultiple-pulse regimes have been determined by control
f relative variations of the key signal characteristics dur-ng the last 200 round trips to the steady-state. In gen-ral, a given regime was regarded as a steady-state whenhe relative variation of the pulse energy outcoupled fromhe optical cavity does not exceed �Ep�10−3, and theariations of the pulse width and peak power areTFWHM�10−2 and �Pp�10−2, respectively. Thus, the ac-epted criteria for stability assume the variations of theulse parameters to be within 1%.
. MODELING RESULTSirst, we have performed extensive modeling to deter-ine regimes with single- and multiple-pulse genera-
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FrP
Frg
Fw=
348 J. Opt. Soc. Am. B/Vol. 26, No. 2 /February 2009 Shtyrina et al.
ions. The optimization in this paper concentrated on thexperimentally easy accessible parameters of small signalain, which can be adjusted by changing the pump powernd total cavity dispersion, which can be adjusted byhanging the grating separation. The optimization ofore fiber and SESAM parameters will be the scope of
urther study. Figures 2 and 3 show regions of stable mul-ipulse regimes in the parameters plane of saturationnergy—versus recovery time (Fig. 2) and gain—umulative GVD (Fig. 3) for the SESAM with a modula-ion depth of q0=0.1. The cumulative cavity dispersionould experimentally be varied by changing the disper-ion of the grating compensator. The curves in Figs. 2 andseparate the region with a different number of pulses in
he cavity. To demonstrate how the imposed requirementsn the relative variations of the pulse energy, pulse width,nd peak power outcoupled from the optical cavity impactones of steady-state solutions we plot also regions withore tough conditions. Namely, the pink and gray colored
reas in Figs. 2 and 3 correspond to the relative varia-ions of the energy satisfying �Ep�10−5, and the varia-ions of pulse width and peak power are �TFWHM�10−3
nd �Pp�10−2, respectively. Thus, the area confinedithin the red curve corresponds to a single-pulse genera-
ion while the area limited by the black curve correspondso two and three pulses (Fig. 2) and more than two pulsesFig. 3) circulating in the cavity. Note that multiple-pulseegimes are rather sensitive to initial noise and, conse-uently, in the same area (and with the same energy) two,hree, and more pulses could be generated depending onhe particular noise structure in a given trial of mode-ocking start-up. As can be seen in Figs. 2 and 3, theingle-pulse generation takes place in a wide region onhe parametrical planes.
It can also be seen that a high saturation energy and amall recovery time favor the single-pulse operation.owever one has to keep in mind that a SESAM with a
oo high saturation energy and too fast recovery could not
τA (ps )
Esa
t(pJ)
0 20 40 60 80 100
0
2
4
6
8
10
12
14
ig. 2. Pulse stability zones bounded by red curve—single-pulseegimes bounded by black curves—two and three pulse regimes.ink and gray zones are explained in the text.
e saturated by the initial low-amplitude and long-widthuctuations in the laser and, therefore, might not be ableo start the passive mode locking.
Figure 3 shows that a stable single-pulse regime devel-ps with anomalous cavity dispersion for the total gainetween 5 and 7.5 dB. Note that Figs. 2 and 3 have beenreated using extensive simulations with 1770 runs forig. 2 and 1620 runs for Fig. 3. Each numerical run cor-esponds to a single point in Figs. 2 and 3. Figure 4 pre-ents a typical evolution phase portrait in the (chirp,ulsewidth) plane. It is seen that initial intracavity noisevolves to the single-pulse regime with the followingsymptotic values: pulse duration TFWHM�0.56 ps, chirparameter C�−3.44 ps−2, and pulse peak power P0120 W. The pulse chirp parameter was measured as a
Gain g0 (dB )
Cum
ulativ
eβ 2
(ps2)
5 6 7 8 9 10
-2.1
-1.8
-1.5
-1.2
-0.9
-0.6
-0.3
0
ig. 3. Pulse stability zones bounded by red curve—single-pulseegimes bounded by black curve—multipulse (more than two) re-imes. Pink and gray zones are explained in the text.
TFWHM (ps)
Chi
rp(1
/ps2 )
0.52 0.54 0.56 0.58 0.6 0.62
-4
-3.5
-3
-2.5
-2
-1.5
ig. 4. Phase portrait of a pulse in the plane (chirp versus pulseidth) g0=5.5 dB, �2=−0.046 ps2, �3=5·10−4 ps3, q0=0.3, Esat0.5 pJ, and � =10 ps.
sat
hMaasilti1bstcpuaeFdrrrpceas
l�trs
psmcpptntnhiapsaoupbbsptsab
atuppsaucawp
FF
Fa
Shtyrina et al. Vol. 26, No. 2 /February 2009 /J. Opt. Soc. Am. B 349
alf second derivative of the phase near the pulse peak.athematically, the solution presents an attractor and
ny initial field distribution evolves to this stablesymptotic state. Temporal shape and optical spectrum ofuch a steady-state asymptotic pulse for a set of differentnitial noise realizations are depicted in Fig. 5. The wave-ength spectrum shows sidebands, which are caused byhe periodic perturbations of the solitons during each cav-ty round trip [18]. Note that here results are shown for7 different initial noise realizations. It can be seen thatoth temporal and spectral shapes of the steady-stateingle pulse are not dependent on the particular form ofhe initial noise indicating stability of such a regime. Itan also be seen that a temporal profile of the generatedulse can be well approximated by a Gaussian shape. Fig-re 6 illustrates the mode-locking evolution to the stablesymptotic pulse from the noise with simulation param-ters similar to those used in Fig. 4. As it can be seen inigs. 6 and 7, steady-state pulses represent a kind of aispersion-managed autosolitons (see, e.g., [19–22] andeferences therein) with temporal and spectral forms pe-iodically reproduced after each round trip and pulse pa-ameters fixed by the system characteristics. The keyulse characteristics, such as temporal width, energy,hirp, and bandwidth, are nearly identical at the end ofach round trip. Figure 7 shows details of the evolution oftypical quasi-steady-state single-pulse regime in the la-
er cavity over one round trip.The following parameters have been used in the simu-
ations presented in Fig. 7: g0=5.5 dB, �2=−0.046 ps2,3=5·10−4 ps3, q0=0.3, Esat=0.5 pJ, and �sat=10 ps. Notehat Fig. 7 shows a significant variation of key pulse pa-ameters during one cavity round trip. A strong evolutionhould be taken into account when considering the exact
ig. 5. Pulse waveform and spectral power; parameters as inig. 4.
osition in the cavity for coupling out a pulse. It can beeen in the Fig. 7 that the pulse duration reaches theinimum value at the chirp-free point located at 3 m, ac-
urately after propagation through the dispersion com-ensator. Outcoupling before or close to this chirp-freeoint would enable the shortest output pulses to be ex-racted from the laser system. Note that we typically didot find exact periodic solutions that are recovered en-irely after every round trip; instead the small oscillationsear the asymptotic values of the pulse characteristicsave been typically observed. This phenomenon is shown
n Fig. 8, which depicts the evolution of pulsewidth, chirp,nd peak power observed for a quasi-steady-state single-ulse regime during the last 150 round trips of numericalimulations with one round trip time corresponding to thectual distance of LR=4 m. It can be seen that the “long”scillation period of the parameters near asymptotic val-es for a single-pulse regime, shown in Fig. 8, takes ap-roximately 20 cavity round trip times. This is a typicalehavior observed for a single-pulse regime, as also cane seen in Figs. 2 and 3. Note that the magnitude of suchlow oscillations is very small and in practical terms theseulses can be considered as dispersion-managed autosoli-ons (which would be a true periodic solution in such aystem). This might have a certain impact on pulse jitternd methods of suppression of such slow oscillations coulde of significant interest.The results of modeling presented in Figs. 9–11 provide
n approach for generation of pulses with specific charac-eristics by adjusting parameters of the laser cavity. Fig-res 9–11 depict contour plots of pulsewidth, chirp, andeak power, respectively, for asymptotic pulses in thelane of gain-versus-cumulative dispersion for a quasi-teady-state single-pulse regime. Note that stable pulsesre shown to acquire a relatively low value of chirp. Fig-re 9 shows how much dispersion can be tolerated in theavity for a given pulse width. Note that some amount ofnomalous dispersion is required to achieve stable pulses,hereas no solutions were found with zero net cavity dis-ersion. It can also be seen in Fig. 11 that higher gain
Cavity round triptimes
550
600
650
700
750Tim
e(ps) 6
7
8
Po
we
r( W
)
0
100
ig. 6. Illustration of the mode-locking dynamics andsymptotic pulse generation; parameters as in Fig. 4.
Fs
Fa
350 J. Opt. Soc. Am. B/Vol. 26, No. 2 /February 2009 Shtyrina et al.
Grating pair + SESAM
Grating pair + SESAM
Distance (m)
Pea
kpo
wer
(W)
0 1 2 3 4
40 40
80 80
120 120
160 160
Distance (m)T
FW
HM
(ps)
0 1 2 3 40.3 0.3
0.4 0.4
0.5 0.5
0.6 0.6
0.7 0.7
Distance (m)
Chi
rp(1
/ps2
)
0 1 2 3 4-8 -8
-6 -6
-4 -4
-2 -2
0 0
2 2
4 4
6 6Distance (m)
Cum
ula
tive
β 2(p
s2)
Cum
ulat
ive
β 3(p
s3 )
0 1 2 3 4-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
ig. 7. (Color online) Evolution of the typical single-pulse regime during one round trip time. Here all parameters are as for the casehown in Fig. 4.
ig. 9. Contour plot of pulse width in the plane cumulativeispersion-gain.
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4WfslticmsasmsagssptowCrmol
R
1
1
Fd
Fl
Shtyrina et al. Vol. 26, No. 2 /February 2009 /J. Opt. Soc. Am. B 351
ould offer a higher soliton energy but only for accumu-ated cavity dispersion in the interval from approximately2=−0.1 ps2 to �2=−0.5 ps2, and it is required that thealue of gain is adjusted for a given value of dispersion. Ithould also be mentioned that with high gain and low dis-ersion, the laser becomes strongly dispersion-managed.his causes a larger chirp and a strong evolution of theulse shape during one cavity round trip as seen in Fig. 7.herefore, the pulses can acquire a fairly high chirp for
ow total dispersion values as it can be observed inig. 10.
Gain (dB )
Cumulativ
eβ 2
(ps2)
5 5.5 6 6.5 7-2
-1.5
-1
-0.5
0Chirp (1/ps2): -1.4 -0.8 -0.49 -0.15 -0.07 -0.02
ig. 10. Contour plot of pulse chirp in the plane cumulativeispersion-gain.
Gain (dB )
Cumulativ
eβ 2
(ps2)
5 5.5 6 6.5 7-2
-1.5
-1
-0.5
0Peak power (W): 15 25 40 60 75 100
ig. 11. Contour plot of output peak power in the plane cumu-ative dispersion-gain.
. CONCLUSIONe have performed extensive numerical modeling of a
emtosecond mode-locked fiber laser using SESAM as aaturable absorber element. The numerical simulation al-owed to identify stable regimes of a single-pulse genera-ion in the multidimensional space of parameters includ-ng saturation energy, recovery time, gain, andumulative intracavity dispersion. We have found nu-erically the conditions for generation of quasi-stable
ingle pulses and show that forming asymptotic pulsesre not perfectly periodic solutions but rather representlow oscillations of the pulse parameters with periods ofany round trips. Another important result of modeling
hows the effect of characteristics of quasi-stablesymptotic pulses on the key system characteristics, totalain and accumulated intracavity dispersion. Close in-pection of pulse parameters shows that in the case oftrong dispersion management, the location of the outcou-ling in the cavity requires careful adjustment to extracthe pulse, where it has a minimal width. For instance,utcoupling from the location with the chirp-free pulsesould enable the shortest output pulses to be obtained.areful modeling of total gain, cavity dispersion, satu-able absorber recovery time, and saturation energyight have a strong impact on pulse generation and the
verall performance of femtosecond mode-locked fiberasers.
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