Based on the extended nonlinear Schrödinger equation, the influences of the filter effect on pulse splitting in a passively modelocked erbiumdoped fiber laser with positive dispersion cavity are investigated theoretically. Numerical results show that, as the bandwidth of the spectral filter decreases, the nonlinear chirp appended to the pulse increases under the combined action of the filter effect of the superGaussian spectral filter and the selfphase modulation effect. On further decreasing the bandwidth, the wave breaking of the pulse takes place. In addition, by varying the pump power of the laser or the profile of the spectral filter, the influences of the filter effect on pulse splitting also change accordingly.
I. INTRODUCTION
Highrepetitionrate and large energy ultrashort laser pulses are very important for applications of nonlinear frequency conversion, ultrafast dynamics exploration, and optical precision procession
[1

3]
. Rareearth doped silicabased fibers are very good gain media for building modelocked lasers generating such ultrashort pulses
[4

8]
. However, the nonlinear Kerr effect in the longrange transmission fibers induced by the ultrashort pulses with high peak power is very serious. This may result in a very high nonlinear phaseshifting. When the dispersion cannot control or compensate the accumulated nonlinear phaseshifting, the nonlinear effect may induce nonlinear chirps to cause a pulse distortion [9], which makes it difficult to obtain large energy pulses. Previous research shows that a fiber laser with allnormal groupvelocity dispersion (GVD) or large normal GVD together with small anomalous GVD can achieve a dissipative soliton (DS) with large energy in the modelocked pulse shaping
[10
,
11]
. The formation mechanism of DSs is attributed to an integrated result of all kinds of effects in the laser cavity, such as the spectral filter effect of the gain medium, Kerr nonlinear effect, normal dispersion, nonlinear polarization rotation (NPR) or saturable absorber (SA), gain and loss of the laser cavity, etc. It has also been found that a spectral filter (SF) inserted in a passively modelocked fiber laser with a positive dispersion cavity can be used to balance the pulse chirp for achieving stable DSs. Moreover, the pulse shape (pulse duration, peak power, etc.) and the spectrum (spectral width, etc.) of the DS are closely related to the bandwidth of the SF
[11

14]
. In fact, the variation of the SF bandwidth would not only result in changes of the pulse shape and spectrum of the DS, but would also induce the pulse to cause the wave breaking, which affects the stabilization of the output pulse and the increment of the pulse energy. Therefore, the influences of the filter effect on pulse splitting in a passively modelocked fiber laser with positive dispersion cavity are worth indepth study.
In this paper, the influences of the filter effect on pulse splitting in a passively modelocked erbiumdoped fiber laser (EDFL) with positive dispersion cavity are investigated theoretically. Our results show that as the SF bandwidth decreases, the nonlinear chirp appended to the pulse increases under the combined action of the filter effect of the superGaussian SF and the selfphase modulation (SPM) effect. On further decreasing the bandwidth, the wave breaking of the pulse takes place. In addition, by varying the pump power of the laser or the profile of the SF, the influences of the filter effect on pulse splitting would change accordingly.
II. THEORETICAL MODEL
Figure 1
shows the propagation model of a passively modelocked EDFL with an SF. The pump light is launched into the EDF through a wavelength division multiplexing (WDM) coupler. The positive dispersion ring cavity is comprised of a singlemode fiber (SMF) and the EDF. The polarization additivepulse modelocked (PAPM) system is made of a polarizationsensitive isolator and two sets of polarization controllers. The PAPM is used to produce the NPR effect. An SF inserted in the cavity after the PAPM has a variable bandwidth. The 10% port of a 10:90 optical coupler (OC) is used for the laser pulse output.
Illustration of the fiber laser cavity elements used for the proposed model. EDF, erbiumdoped fiber; WDM, wavelength division multiplexing; PAPM, polarization additivepulse modelocked; OC, optical coupler; SF, spectral filter.
By solving the extended nonlinear Schrödinger equation (NLSE), the operation of a passively modelocked fiber laser with a positive dispersion cavity can be numerically simulated. The equation is given by
[15
,
16]
where
A
is the pulse envelope amplitude,
β
_{2}
represents the fiber dispersion,
γ
is the nonlinearity parameter, Ω
_{g}
is the bandwidth of the laser gain. The gain function for the EDF is described by
[17
,
18]
Here
g
_{0}
is the small signal gain coefficient.
E_{s}
is the gain saturation energy, which is related to the laser pump power. The pulse energy is given by
where
T_{R}
is the cavity roundtrip time.
The gain medium of the EDF has a gain bandwidth of 25 nm, with profile of superGaussian type. The PAPM element is a similar SA
[11]
, which can serve as the spectral and temporal filtering element
[11
,
12
,
19
,
20]
. The PAPM element has a superGaussian transmission function with the bandwidth of 70 nm.
A scalar model to simplify the propagation model with the PAPM function, so the intensitydepended transmission function of the PAPM element is given by
[7]
where
I
(
t
) is the pulse intensity distribution of the time
t
, and
I
_{max}
is the maximum value of
I
(
t
). An SF inserted in the cavity has a superGaussian profile, and the filtering function is given by
where
ω
is the angular frequency, Ω
_{SF}
is the bandwidth of SF. In fact, SF could also be seen as an effective SA, which cuts off the temporal wings of a pulse.
The total length of laser cavity is 14.5 m with the net cavity dispersion of ~0.44 ps
^{2}
. A pulse with Gaussian shape is used as an initial condition, and the initial chirp parameter C is zero. In accordance with Eq. (1), by using the standard split step Fourier method, the laser operation in a passively modelocked fiber laser can be simulated with the parameters displayed in
Table 1
. In the simulation, we use the shooting method to determine whether the laser reaches the steady state. Based on the selfreproduction features of the laser, only when the difference between the initial value and the calculated value is within given precision limitation do we believe that the laser reaches the steady state.
Fiber parameters used in the simulation of the laser cavity
Fiber parameters used in the simulation of the laser cavity
III. RESULTS AND DISCUSSIONS
Initially, the influences of the filter bandwidth on pulse output properties are investigated. In the simulation, the gain saturation energy
E_{s}
is fixed at 0.3 nJ.
Figure 2
shows the simulation results for different SF bandwidths. As shown, the output pulse spectrum has a flat top with steep edges and is similar to the trapezoidspectrum shape. The pulse duration and peak power can reach several picoseconds and above 100 watts, respectively. This means that the large energy DSs have been achieved in a passively modelocked fiber laser with a positive dispersion cavity. When the SF bandwidths are 50, 35 and 25 nm, the pulse durations are 11.64, 11.6 and 11.4 ps, the peak powers are 136.7, 137.2 and 139.3 W, and the spectral widths are 17.54, 17.6 and 17.76 nm, respectively. One can clearly see that the pulse shapes and the spectra vary slowly with decreasing SF bandwidth. The pulse duration is decreasing and the peak power becomes higher, and the pulse spectrum broadens accordingly. The reason for this may be explained as follows. When the SF bandwidth exceeds the pulse spectral width, the filter effect is very weak. Only the leading and trailing edges of the pulse spectral envelope can be filtered, which makes the central part of the pulse obtain higher gain than the edges. From the maps of the temporal evolution of the pulse width and spectral bandwidth in one round trip
[7
,
11]
, there is an abrupt change for the pulse temporal waveforms and the spectra. This corresponds to narrowing down the widths of both the pulse shape and the spectrum as the pulse passes through the filter. The narrower the bandwidth of the SF, the bigger the changes. When the pulse gets a balance, a stable pulse can be achieved with the broader spectrum, the narrower pulse shape and the higher peak power.
Temporal power profile (solid curve) and chirp (dashed curve) and spectral power profile at the output position for the different bandwidths of the SF. (a)(b): 50 nm, (c)(d): 35 nm, (e)(f): 25 nm.
It can also be seen that the pulse chirp doesn’t satisfy the linear evolution, but the accumulation of the nonlinear chirp is not enough to make the pulse split as the SF bandwidth decreases. The reason is that the filter has less impact on the pulse for the broader SF, and the pulse peak power increase is limited for the decrease of the SF bandwidth. Given that the nonlinear chirp is caused by the combination of the filter effect and the SPM effect, the impact on the nonlinear chirp accumulation of the pulse is limited.
Figure 3
shows the output pulse shape, chirp and spectrum with a decrease to 15 nm of the SF bandwidth. As shown, the output pulse spectral width increases to 21.04 nm rapidly. Meanwhile, the output pulse shape narrows down to 10.4 ps, and the peak power rapidly increases to 157.1 W. At this point, the SF bandwidth is less than the pulse spectral width. When the pulse passes the filter, not only does the pulse shape become narrower, but the spectral wings can be cut off, so both the pulse shape and the spectrum have obvious changes. It also indicates that the nonlinear chirp increases along with the decrease of the SF bandwidth. This is because the filter effect begins to be notable when the SF bandwidth can compare to the pulse spectral width. Meanwhile, the influences of the SPM effect are enhanced greatly for the rapid increase of the pulse peak power, which results in output pulse containing more nonlinear chirp. With the action of the combination of the filter effect and the SPM effect, the pulse chirp deviates seriously from the linear distribution. It is worthwhile to point out that, as the nonlinear chirp increases, the wave breaking is more likely to occur, so it can be seen that the pulse would split if the SF bandwidth further decreases.
(a) Temporal power profile (solid curve) and chirp (dashed curve) and (b) spectral power profile at the output position with the SF bandwidth of 15 nm.
Figure 4(a)

(b)
shows the output pulse shape, chirp and spectrum with the SF bandwidth of 9 nm. Here we can see that the output pulse spectrum further broadens and begins to deviate from the trapeziform distribution. Meanwhile, the pulse shape obviously changes and at the edges small fluctuations emerge. By comparing the results shown in
Fig. 4(c)

(d)
with the SF bandwidth of 7 nm to those in
Fig. 4(a)

(b)
, we clearly see that the small fluctuations have evolved into multipeak oscillation structures. This indicates that the pulse is beginning to split. The nonlinear chirp increases quickly with the further decrease of the SF bandwidth, so the different parts of the pulse spectral envelopes have the same instantaneous frequency, thus interfering with each other, which leads to the edges with multipeak oscillation structures.
Temporal power profile (solid curve) and chirp (dashed curve) and spectral power profile at the output position for the different bandwidth of the SF. (a)(b): 9 nm, (c)(d): 7 nm.
It is worth pointing out, as an SF is introduced in the laser cavity, it modifies the laser attracting state in some way
[11]
. When the SF band is wide, the roundtrip cavity length is long enough so that after filter application the pulse can return to its attracting state before encountering the SF again. With the decreasing of SF bandwidth, the laser attracting state would not change. Finally we can see that the pulse duration decreases, while the spectral width and the pulse peak power increase, but both the pulse temporal waveforms and the spectra are similar. When the SF band is very narrow, SF has a strong filtering effect. The pulse cannot return to its original attracting state when the pulse encountering SF again, so the laser attracting state would change. Both the pulse temporal waveforms and the spectra have big changes, which could eventually lead to the wave breaking.
The above results show that the pulse splitting in a passively modelocked fiber laser with positive dispersion cavity is influenced by the SF bandwidth when the laser pump power is constant. In fact, the pulse shape and spectrum are related to the laser pump power, the variation of the pump power can lead to the changes of the pulse splitting characteristics with a certain bandwidth SF.
Figure 5
shows the pulse shape, chirp and spectrum when
E_{s}
is 0.45 nJ and the SF bandwidth is 7 nm. Compared to the results in
Fig. 4(c)

(d)
, the pulse is broadened in both time and frequency domains, and the peak power is also increased. Meanwhile, the multipeak oscillation structures of the pulse edges become more obvious. As the laser pump power is increased, the phenomenon of wave breaking is clearer. The reason is that both the pulse duration and spectral width increase with the pump power, so the filter effect of the SF is strengthening for the same SF bandwidth and the SPM effect is enhanced for the increase of the pulse peak power.
(a) Temporal power profile (solid curve) and chirp (dashed curve) and (b) spectral power profile at the output position with the SF bandwidth of 7 nm when the gain saturation energy E_{s} is 0.45 nJ.
In addition, since the superGaussian SF brings in the nonlinear chirp, which can make the pulse split. In the case shown in
Fig. 6
, a Gaussian SF is used in the laser cavity instead while
E_{s}
of 0.45 nJ and the SF bandwidth of 7 nm remain unchanged. As shown, the nonlinear chirp decreases and the multipeak oscillation structures of the pulse edges are weakening to small fluctuations. Because the Gaussian SF does not induce the nonlinear chirp, the nonlinear chirp is only related to the SPM effect. The wave breaking effect may be weak with the decrease of the nonlinear chirp.
(a) Temporal power profile (solid curve) and chirp (dashed curve) and (b) spectral power profile at the output position with the Gaussian SF bandwidth of 7 nm when the gain saturation energy E_{s} is 0.45 nJ.
IV. CONCLUSION
We have investigated theoretically the influences of the filter effect on pulse splitting in a passively modelocked EDFL with a positive dispersion cavity based on the extended NLSE. Our results have shown that as the SF bandwidth decreases, the nonlinear chirp appended to the pulse increases under the combined action of the filter effect of the superGaussian SF and the SPM effect. On further decreasing the bandwidth, the wave breaking of the pulses takes place. In addition, by varying the pump power of the laser or the profile of SF, the influences of the filter effect on pulse splitting also change accordingly. As the laser pump power is increased, the phenomenon of wave breaking is clearer. By using a Gaussian SF in the laser cavity instead, the wave breaking effect may be weak with the decrease of the nonlinear chirp.
Schaffer C. B.
,
Brodeur A.
,
García J. F.
,
Mazur E.
2001
“Micromachining bulk glass by use of femtosecond laser pulses with nanojoule energy,”
Opt. Lett.
26
93 
95
Wang Y. M.
,
Shen H.
,
Hua L. Q.
,
Hu C. J.
,
Zhang B.
2009
“Predissociation dynamics of the B state of CH3I by femtosecond pumpprobe technique,”
Opt. Express
17
10506 
10513
Korte F.
,
Adams S.
,
Egbert A.
,
Fallnich C.
,
Ostendorf A.
,
Nolte S.
,
Will M.
,
Ruske J. P.
,
Chichkov B.
,
Tuennermann A.
2000
“Subdiffraction limited structuring of solid targets with femtosecond laser pulses,”
Opt. Express
7
41 
49
Ruehl A.
,
Wandt D.
,
Morgner U.
,
Kracht D.
2009
“Normal dispersive ultrafast fiber oscillators,”
IEEE J. Select. Topics Quantum Electron.
15
170 
181
Zhang H.
,
Tang D. Y.
,
Zhao L. M.
,
Wu X.
,
Tam H. Y.
2009
“Dissipative vector solitons in a dispersion managed cavity fiber laser with net positive cavity dispersion,”
Opt. Express
17
455 
460
Haboucha A.
,
Komarov A.
,
Leblond H.
,
Sanchez F.
,
Martel G.
2008
“Mechanism of multiple pulse formation in the normal dispersion regime of passively modelocked fiber ring lasers,”
Opt. Fiber Technol.
14
262 
267
Liu X. M.
2009
“Numerical and experimental investigation of dissipative solitons in passively modelocked fiber lasers with large netnormaldispersion and high nonlinearity,”
Opt. Express
17
22401 
22416
VazquezZuniga L. A.
,
Jeong Y. C.
2013
“Wavelengthtunable, passively modelocked erbiumdoped fiber masteroscillator incorporating a semiconductor saturable absorber mirror,”
J. Opt. Soc. Korea
17
117 
129
Agrawal G. P.
2007
Nonlinear Fiber Optics
Academic Press
Boston, USA
Akhmediev N.
,
Ankiewicz A.
2005
Dissipative Solitons
Springer
Berlin, Germany
Wise F. W.
,
Chong A.
,
Renninger W.
2008
“Highenergy femtosecond fiber lasers based on pulse propagation at normal dispersion,”
Laser Photonics Rev.
2
58 
73
Bale B. G.
,
Kutz J. N.
,
Chong A.
,
Renninger W. H.
,
Wise F. W.
2008
“Spectral filtering for highenergy modelocking in normal dispersion fiber lasers,”
J. Opt. Soc. Am. B
25
1763 
1770
Li X. H.
,
Wang Y. S.
,
Zhao W.
,
Zhang W.
,
Yang Z.
,
Hu X. H.
,
Wang H. S.
,
Wang X. L.
,
Zhang Y. N.
,
Gong Y. K.
,
Li C.
,
Shen D. Y.
2011
“Allnormal dispersion, figureeight, tunable passively modelocked fiber laser with an invisible and changeable intracavity bandpass filter,”
Laser Phys.
21
940 
944
VazquezZuniga L. A.
,
Jeong Y. C.
2013
“Powerscalable, subnanosecond modelocked erbiumdoped fiber laser based on a frequencyshiftedfeedback ring cavity incorporating a narrow bandpass filter,”
J. Opt. Soc. Korea
17
177 
181
Cabasse A.
,
Ortac B.
,
Martel G.
,
Hideur A.
,
Limpert J.
2008
“Dissipative solitons in a passively modelocked Erdoped fiber with strong normal dispersion,”
Opt. Express
16
19322 
19329
Li X. H.
,
Wang Y. S.
,
Zhao W.
,
Liu X. L.
,
Wang Y. G.
,
Tsang Y. H.
,
Zhang W.
,
Hu X. H.
,
Yang Z.
,
Gao C. X.
,
Li C.
,
Shen D. Y.
2012
“Allfiber dissipative solitons evolution in a compact passively Ybdoped modelocked fiber laser,”
J. Lightwave Technol.
30
2502 
2507
Zhao L. M.
,
Tang D. Y.
,
Tam H. Y.
,
Lu C.
2008
“Pulse breaking recovery in fiber lasers,”
Opt. Express
16
12102 
12107
Agrawal G. P.
1990
“Amplification of ultrashort solitons in erbiumdoped fiber amplifiers,”
IEEE Photon. Technol. Lett.
2
875 
877
Chong A.
,
Buckley J.
,
Renninger W.
,
Wise F.
2006
“Allnormaldispersion femtosecond fiber laser,”
Opt. Express
14
10095 
10100
Martel G.
,
Chedot C.
,
Hideur A.
,
Grelu P.
2008
“Numerical maps for fiber lasers mode locked with nonlinear polarization evolution: Comparison with semianalytical models,”
Fiber Integr. Opt.
27
320 
340