. INTRODUCTIONulse splitting is a well-known process occurring in theery initial moments of pulse propagation in nonlinearptical fibers [1,2]. According to the most accreditedheory of pulse splitting [1,2], in the femtosecond regime,igher-order solitons are affected by stimulated Ramancattering (SRS) and higher-order dispersion terms, be-oming unstable and eventually breaking up into severalundamental solitons. Explicit expressions for the peakower Pj and temporal width Tj of the jth fundamentaloliton created in the splitting process are given by1� j�N�
Pj = P0�2N − 2j + 1�2/N2, �1�
Tj = T0/�2N − 2j + 1�, �2�
here P0 and T0 are the peak power and temporal widthf the initial hyperbolic secant pulse and N is the so-alled soliton order (see [1,2]). After splitting, intrapulseRS causes fundamental solitons to shift continuously to
ower frequencies via the Raman soliton self-frequencyhift (RSFS) [3,4]. The rate of the RSFS is proportional to/Tj
4 [3,4]. Thus, pulse splitting together with the RSFSventually leads to the complete breakup of a higher-rder soliton, which ejects a stream of fundamental soli-ons one after another. According to Eqs. (1) and (2), soli-ons that are ejected earlier have higher amplitudes,horter duration, and—as a result—demonstrate strongerSFSs. This leads to a sequence of fundamental solitonsith different carrier frequencies, which constantly in-
rease their temporal separations during propagation dueo the RSFS-induced walk-off effect.
However, this well-known description ignores somerocesses that could affect the soliton dynamics, such asnteraction between solitons, and the emission of disper-ive wave radiation near a zero group-velocity dispersionGVD) point. In a series of recent experiments (see [5,6]),
hort and intense pulses have been launched in highlyonlinear photonic crystal fibers (PCFs; see also [7]). Thenexpected formation of long-lived two-peak solitontates for specific magic input pulse energies was ob-erved [5,6]. In these experiments, it was clear that thishenomenon is quite universal for a large variety ofighly nonlinear optical fibers, especially in the deepnomalous GVD regime far from zero-GVD points. Suchocalized states always show peaks of differing amplitudeith a well-defined peak power ratio (r�0.73 in the men-
ioned experiments). Moreover, the temporal separationetween the two peaks is also uniquely determined by rnd the Raman parameter �R. Thus, under certain cir-umstances a pair of solitons with unequal amplitudesan form a localized state, which can propagate over longistances even in the presence of Raman effect.Such soliton states were numerically discovered and
tudied in 1996 by Akhmediev et al. [8]. Single peak andwo-peak states were found by solving the simplified non-inear Schrödinger equation (NLSE); see Eqs. (3) and (4)elow. The analysis in [8] is limited to two aspects: (i)nding the localized two-peak solutions of Eq. (4) and (ii)he question of numerical stability of such solutions.
In this paper, inspired by the recent experimental ob-ervation of the Raman multipeak states in [5,6] and byhe early numerical findings in [8], we investigate the is-ue of Raman multipeak states in detail. We proposeualitative and quantitative explanations for all the ex-erimental observations, based on the powerful concept ofravity-like potential introduced in [8–11]. The generaloint-of-view introduced by our approach allows one toake several important predictions to be tested in future
xperiments, namely, the possibility of forming Ramanoliton states with more than two peaks in PCFs, thuseading to a complete violation of the soliton splitting lawf Eqs. (1) and (2) and to new ways to manipulate super-ontinuum generation (SCG) in microstructured fibers byontrolling exotic states of light in the fiber.
010 Optical Society of America
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1786 J. Opt. Soc. Am. B/Vol. 27, No. 9 /September 2010 Tran et al.
. MULTIPEAK SOLITONSsimplified form of the NLSE can be written, in dimen-
ionless units, as
i�z� +1
2�t
2� + ���2� − �R��t���2 = 0, �3�
here � is the electric field envelope rescaled with theundamental soliton power PS���2� / ��t0
2�, with �2 beinghe GVD coefficient at the reference frequency and � be-ng the fiber nonlinear coefficient. Variables z and t are di-
ensionless space and time, respectively, rescaled withhe second-order dispersion length LD2� t0
2 / ��2� and withhe input pulse duration t0. The last term of Eq. (3), re-ponsible for the Raman effect, produces a constant RSFSor solitons in silica fibers [3,4], with �R�TR / t0 being amall parameter, where TR is the Raman response time,pproximately equal to 2 fs in silica. Solitons subject tohe Raman effect shift continuously toward the red part ofhe spectrum due to the RSFS, leading to a constant ac-eleration of the soliton in the time domain [3,4]. In theeference frame of an intense accelerating soliton, one canperate the Gagnon–Bélanger transformation ��z , t�f���exp�iz�q−b2z2 /3+bt��, where �� t−bz2 /2 and b32�Rq2 /15. This leads to the following (in general com-lex) ordinary differential equation for f��� [8,9]:
1
2f�� − �q + b��f + �f�2f − �Rf��f�2�� = 0, �4�
here q is the wavenumber of the strongest soliton, pro-ortional to its peak power. If one neglects all the nonlin-ar terms, Eq. (4) would correspond to the stationarychrödinger equation for a unitary mass particle of en-rgy q subject to a “gravitational” potential U���=b�10,11]. Such a gravity-like behavior of the Raman effectn fibers has recently emerged as one of the most impor-ant effects in the development of the high-frequency partf the SCG (see [10,11]).
Numerical solitary-wave solutions of Eq. (4) with onend two peaks were found by Akhmediev et al. for real f8]. For any value of the soliton wavenumber q, such so-utions always show an Airy tail on the leading edge of theulse, with the peak power ratio r and temporal separa-ion �0 of the maxima in two-peak solutions being mainlyet by material parameters, in particular by �R. In [8],oreover, it was established that these Raman localized
tates are metastable, propagating for many tens of dis-ersion lengths before eventually disappearing due to in-ernal collapse.
In our investigation, we have found a variety of real so-utions of Eq. (4) with an arbitrary number of peaks. Inig. 1 we show the temporal profiles of one-, two-, three-,nd four-peak solitons found by solving the boundaryalue problem by means of a shooting method with appro-riate boundary conditions [12], for �R=0.1. Even thoughhese solutions have Airy tails on the leading edge ofulses (due to tunneling of the solutions of the linearizedchrödinger equation), such tails are quite small whensing the physically relevant parameters (see Fig. 1). Airyails will be more pronounced if the slope b (or parameterwhich is proportional to �b) of the gravity-like potential
ets larger. The construction of multipeak solutions of Eq.4) can be understood by looking at the potential term qb� exhibited by Eq. (4). In fact, once the peak power �2q�nd the position �=�1 of the strongest soliton are given,he second soliton, with peak power 2rq �r� �0,1�� and po-ition �=�2, can be arranged only at a temporal separa-ion [see Fig. 1(b)]:
�0 � �1 − �2 � q�1 − r�/b. �5�
his simple approximate geometrical reasoning allowsne to find a relation between the temporal separation �0nd the magic peak power ratio r. It is important to notehat one can in principle arrange any number of indi-idual solitons into a multipeak state, in an “organ-pipe”ashion, if the intensity of the strongest soliton is largenough. The strongest soliton creates the gravity-like po-ential in its leading edge into which all the subsequentolitons, with progressively decreasing peak powers, cane fitted according to the qualitative geometrical relationf Eq. (5) (see Fig. 1).
The temporal evolutions of one-, two-, three-, and four-eak solitons are shown in Figs. 2(a)–2(d), respectively,here the corresponding profiles shown in Fig. 1 are useds the initial input values for simulation of Eq. (3). Apartrom the single peak soliton, all other multipeak solutionsre not stable, and after some propagation length theyill collapse as clearly shown in Fig. 2. The more peaks
hese localized states have, the less robust they are, andhe smaller the propagation length required for them toollapse. In contrast to multipeak solitons, the one-peakoliton is quite stable, with its profile (including its am-litude and width) being stationary to a high degree of ap-roximation as shown in Fig. 2(a).
. CRITERIA OF TWO-PEAK STATEORMATION: MAGIC PEAK POWER RATIOND TEMPORAL SEPARATION
n order to understand the underlying physics that allowshe formation of the above Raman localized states we
(a) (b)
(c) (d)
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ig. 1. (Color online) Profiles of multipeak soliton solutions.lue dashed lines show the gravity-like potential U��� created by
he most intense soliton in its leading edge into which all otherolitons with decreasing powers can be progressively fitted in anrgan-pipe fashion.
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Tran et al. Vol. 27, No. 9 /September 2010 /J. Opt. Soc. Am. B 1787
nalyze the two-peak soliton using a Lagrangian method.wo mechanisms are involved in the evolution of two-eak solitons: (i) the intersoliton forces (ISFs) of twoingle initially in-phase solitons in the absence of the Ra-an effect [13,14] and (ii) the increasing separation of
hese two single unequal-amplitude solitons due to theSFS in time domain. Here we treat these two mecha-isms independently and require that they have to com-ensate each other in order to obtain a localized state. Fol-owing a theory formulated in [13,14], due to ISFs twonitially in-phase solitons of amplitudes �2q and �2qrith initial temporal separation �0 in the absence of theaman effect will pulsate with an oscillation period equal
o Lp:
Lp ��
2�2��2expy0
4 , �6�
here �=�2q�1+�r� /4 and y0=4��0. The two solitonsave been assumed to have a vanishing phase differenceapart from a constant global phase across the pulse thatan be assumed to be zero due to the gauge invariance ofhe complex NLSE) since exact numerical soliton solu-ions of Eq. (4) are always purely real. On the other hand,ue to the Raman effect, each single soliton after a propa-ation length Lp will acquire some additional temporaleparation t1,2 with respect to its own initial position: t1�1/2�bLp
2 ��16/15��Rq2Lp2; t2= �16/15��Rr2q2Lp
2. Thus, inrder to obtain the soliton state where the temporal sepa-ation of two single solitons is maintained during propa-ation, the following condition must hold true: t1− t2=�0,r
�216
15�R
1 − r2
�1 + �r�4exp1 + �r
�2�0�q = �0. �7�
his condition together with Eq. (5) leads to a transcen-ental equation for the peak power ratio r:
��16
15�R�22�1 + r�q
�1 + �r�4exp15�1 + �r��1 − r�
32�R�2q = 1. �8�
rom Eqs. (5) and (8) one can easily calculate the peakower ratio r and the temporal separation � for a given
propagation lengthpropagation length
propagation lengthpropagation length
dela
y
dela
ydela
y
dela
y(a) (b)
(d)(c)
ig. 2. (Color online) Temporal evolution of multipeak solitons.
0
avenumber q of the strongest soliton. Figure 3 showshe dependence of the peak power ratio r and the tempo-al separation �0 on the parameter q, where red solid linesndicate the analytical (approximate) results, obtainedrom Eqs. (5) and (8), whereas the blue dots show the nu-erical (exact) results, obtained by the shooting method
or solving Eq. (4). It is clear from Fig. 3 that the approxi-ate and exact results are in good qualitative agreement,
specially when the wavenumber q of the strongest soli-on gets smaller. This behavior is expected, because inhis case the slope of the potential U��� shown for the pa-ameter b q2 will also get smaller so that the Airy tailill be negligible. Under these circumstances, the en-
emble of two solitons that we use to analytically obtainqs. (6)–(8) will be closer to the two-peak numerical solu-
ions with smaller Airy tails.
. EXCITATION OF TWO-PEAK STATESn this subsection we study the excitation conditions foraman localized states by numerically integrating Eq.
3). As an input condition for Eq. (3) we use the combina-ion of two hyperbolic secant pulses with an initialrequency detuning as follows: input�t��2q sech��2qt�exp�−it�+�2qr sech��2qr�t−�0��. We
aunch the pulses for a dimensionless propagation length=160. For each value of the peak power ratio r we needo find the corresponding initial temporal separation �0uch that the temporal separation of the two pulses is astable as possible during propagation. The results of thisumerical analysis are depicted by the green curves withquare markers in Figs. 4(a)–4(d) for four values of q0.369, 0.3164, 0.1151, and 0.0616, respectively, for thease when the two solitons have initially the same fre-uency �=0�. When parameters �r ,�0� are deviated fromhe green curve, the two pulses are less stable duringropagation and they will collapse at shorter propagationengths. This situation is explained in detail in Fig. 2(b) of15], where an approach based on the analysis of thehase evolution of each soliton forming the localized stateas been adopted. The big black dots [we refer to thesepecial points �r ,�0� as the magic points] shown in Fig. 4epresent the exact numerical soliton solutions of Eq. (4)see also Fig. 1(b)].
The red dotted lines in Fig. 4 are calculated based onq. (5), and the blue solid curves are obtained from Eq.
7). The crossing points of these curves correspond to theolution of r as found by our qualitative model of Eq. (8).he discrepancy between the analytical curves (blue solid
soliton wavenumber q
(a) (b)
peak
pow
er
ratio
r
tem
po
ralse
pa
ratio
nξ 0
soliton wavenumber q
ig. 3. (Color online) (a) Plot of the peak power ratio r and (b)he initial temporal separation �0 as functions of q. Blue dots anded solid lines indicate the numerical and analytical results, re-pectively. Parameter is �R=0.1.
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1788 J. Opt. Soc. Am. B/Vol. 27, No. 9 /September 2010 Tran et al.
nes) and the numerical curves (green curves with squarearkers) gets smaller for larger r and �0.When the parameter q is large enough [see the green
urves with square markers in Figs. 4(a) and 4(b) with=0.369 and 0.3164, for propagation length z=160 and
nitial frequency detuning =0] we do not find any solu-ion �r ,�0� very close or below the magic points (single biglack dots in Fig. 4) so that the temporal separation of thewo pulses is preserved during propagation. Of course,he localized states can survive for shorter propagationengths (e.g., z�10), but they collapse during furtherropagation. This behavior takes place even exactly at theagic points. This means that the Raman localized states
tudied here (including magic points) are not mathemati-ally stable, but metastable as found in [8]. Nevertheless,hese metastable states can propagate for surprisinglyong distances, as experimentally observed in [5,6]. If weo upward along the green curves with square markers inig. 4, the localized states survive for longer propagation
engths. This is expected, because when the peak poweratio r and the temporal separation �0 become larger, themplitude difference between the two pulses falls. As a re-ult, the Raman effect is almost the same for them; theyre located further from each other, and thus their ISFset smaller. In the extreme case when two pulses have theame amplitude and are located infinitely far from eachther, this localized state will obviously survive for quite aong propagation length, because each fundamental soli-on is very robust [8].
The curve with red triangular markers in Fig. 4(b)hows the case when the two pulses have initially differ-nt frequencies (the initial frequency detuning =−0.2).n this case, for the same peak power ratio r, the initialemporal separation �0 between the two pulses is smallerompared to the case when =0. But during propagationhe temporal separation gets larger and asymptoticallytabilizes. In the case =0 the temporal separation is al-
(a) (b)
(c) (d)
pe
ak
po
we
rra
tio
rp
ea
kp
ow
er
ratio
r
temporal separation ξ0
pe
ak
po
we
rra
tio
rp
ea
kp
ow
er
ratio
r
temporal separation ξ0
temporal separation ξ0temporal separation ξ0
ig. 4. (Color online) Peak power ratio r versus initial temporaleparation �0. (a),(b),(c),(d) Results for q=0.369, 0.3164, 0.1151,nd 0.0616, respectively. Red dotted lines are calculated based onq. (5), blue solid curves are obtained from Eq. (7), and crossingoints of these curves correspond to the solution of Eq. (8). Singleig black dots are magic points. Green curves with square mark-rs are obtained by numerically modeling Eq. (3) for a propaga-ion length z=160 with two hyperbolic secant solitons initiallyaving the same frequency �=0� as initial input values. Theurve with triangular markers in (b) shows the case =−0.2.
ost the same during the whole propagation length.
. MAGIC INPUT POWERSnce the values of r and �0 have been established, one can
xplicitly calculate the “magic input power” of an intensenput pulse around which the formation of two-peak local-zed states will appear during the pulse splitting process.ccording to the accepted theory of pulse splitting under
he effects of Raman perturbation [1,2], an intenseth-order soliton �input=N sech�t� will split into N indi-
idual fundamental solitons of the form �Pj sech��Pjt�,here the powers Pj are given in Eq. (1). The condition forhich two successive solitons (for instance, j and j+1)ave a peak power ratio equal to r turns into the equationj+1/Pj=r, which has the physically relevant solutions
mag,j= �j−1/2�+ �1−�r�−1. The first two solitons withtronger peak powers (j=1 and j=2) will have the correctmplitude ratio for Nmag,1=1/2+ �1−�r�−1. For the case rR2�0.73, one obtains Nmag,1�7.4 which correspondsurprisingly well to the measured value �N�8� (see [5,6]).ccording to the above straightforward theory, only one
wo-peak soliton can be created starting with one intensenput pulse, since at each magic input power only one pairf solitons will possess the magic peak power ratio r. It islso clear from the above relation that, for a given valuef the peak power ratio r, a higher input soliton order N isequired to form a localized state between the jth and thej+1�th fundamental solitons with a higher value of j.
Figure 5 shows the dependence of the spectra at theutput for z=0.45 as a function of the input soliton order
when using a soliton of the form �input=N sech�t� as thenitial value for the numerical simulation of the general-zed nonlinear Schrödinger equation (GNLSE) [Eq. (9)]ith the full convolution for the nonlinear term:
i�z� +1
2�t
2� + ��z,t��−�
t
R�t�����z,t − t���2dt� = 0, �9�
here the response function R�t� includes both the elec-ronic (Kerr) and vibrational (Raman) contributions [16].
soliton order N
frequency
detu
nin
g
bound states
ig. 5. (Color online) Spectral evolution with increasing solitonumber N. The propagation length is z=0.45.
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Tran et al. Vol. 27, No. 9 /September 2010 /J. Opt. Soc. Am. B 1789
ne has R�t�= �1−�� �t�+�h�t�, where h�t����12+�2
2���1�2
2�−1exp�−t /�2�sin�t /�1� is the Raman response func-ion of silica, with �1=12.2 fs and �2=32 fs; �t� is theirac delta function. The coefficient � parameterizes the
elative importance between Raman (non-instantaneous)nd Kerr (instantaneous) effects, and for silica ��0.18.he parameter �R in Eqs. (3) and (4) is related to �1,2 as
ollows: �R=2��12�2 / �t0��1
2+�22��, with t0 being the input
ulse duration. With suitable approximations Eq. (9) cane shown to reduce to Eq. (3). In [5,6] the numerical pulseropagation analysis includes higher-order dispersionerms. In this paper, as seen from Eq. (9), we limit our-elves to second-order dispersion because we wish toliminate the phenomenon of resonant radiation emissionrom solitons [17], which was present in the experimentsf [5,6], but was not shown there. Operating far from anyero-GVD point increases the chance of observing Ramanocalized states, since their formation will not be dis-urbed by the small amplitude background waves gener-ted in the normal dispersion regime close to a zero-GVDoint. However, many solid-core PCFs have a zero-GVDn the near-visible and a broad region of anomalous GVDn the far infrared. In such fibers, even though solitonsmit radiation if launched near the zero-GVD point, theyhift continuously toward the deep anomalous GVD re-ion, where in many cases the GVD is nearly constant forbroad range of frequencies—hence the constant GVD
pproximation used in our model of Eq. (9).Within the three regions enclosed by white ellipses in
ig. 5 soliton pairs can be formed between adjacent fun-amental solitons ejected during the breakup of the inputoliton. Four values of soliton order N=9.63, 12.075,2.65, and 14.65 are used in these regions to show theulse temporal evolution in Figs. 6(a)–6(d), respectively.he localized pair formed by the first and the second fun-amental solitons is depicted in Fig. 6(a) [N=9.63 is lo-ated in the left-hand ellipse in Fig. 5], with the insethowing the details of this localized pair and the thirdeparate fundamental soliton. In this case, after a propa-ation length z=1, the peak power ratio of two pulsesorming the Raman state is r=0.7534, corresponding tohe calculated value of Nmag=8.07. Here we should men-ion that during pulse propagation the two pulses forminghe pair will exchange energy so that the peak power ratio
propagation length
de
lay
(b)
(c) (d)
(a)
propagation length
propagation length propagation length
de
lay
de
lay
de
lay
ig. 6. (Color online) Formation of two-peak localized states.a),(b),(c),(d) Temporal evolution of pulses with input soliton or-ers N=9.63, 12.075, 12.65, and 14.65, respectively.
will also slightly change during propagation. Evenhough the model used in this section to calculate theagic power is quite crude and qualitative, the correspon-
ence with the experiments is surprisingly good.Figures 6(b) and 6(c) with input soliton orders N
12.075 and 12.65 (middle ellipse in Fig. 5) show thetates formed by second with third, and third with fourthundamental solitons, respectively. The pair formed byourth and fifth fundamental solitons is depicted in Fig.(d) with the input soliton order N=14.65 (right-hand el-ipse in Fig. 5). All the localized states shown in Fig. 6 are
etastable, collapsing after sufficiently long propagationengths. The results shown in Fig. 6 are remarkable inhe sense that they demonstrate that one can generateairs for any selected adjacent solitons, simply by prop-rly adjusting the input pulse energy.
In the case of solitons represented by magic points (seeigs. 1 and 2), not only the temporal separation �0, butlso the intensity profiles of each pulse, and thus the peakower ratio r, will be preserved where they are stable. So,he localized states shown in Fig. 6 are not solitons pre-ented by magic points. Moreover, solitons correspondingo magic points are solutions of the simplified model ofq. (3), not of the more complete Eq. (9).So far we have only investigated the two-peak states of
he GNLSE [Eq. (9)]. Apart from two-peak Raman states,he simplified form of the NLSE [Eqs. (3) and (4)] alsoossesses localized solutions with an arbitrary number ofeaks [provided that there is enough “space” to arrangehem according to Eq. (5)] as shown in Fig. 1. It turns outhat the GNLSE [Eq. (9)] also shows evidence of the for-ation of three-peak states in its dynamics, contrary to
he oversimplified conclusions that can be made by usingqs. (1) and (2). This leads to the important prediction
hat in properly designed realistic PCFs there must be atrong violation of the soliton splitting law given by Eqs.1) and (2). Indeed, Fig. 7(a) shows the temporal evolutionf an intense pulse with the input soliton order N=13.81right-hand ellipse in Fig. 5). During the pulse splittingrocess, a three-peak state is formed by the second, third,nd fourth fundamental solitons. The intensity profile ofhis three-peak state at z=0.8 is shown in Fig. 7(b).gain, this intensity profile demonstrates that solitons or-anize themselves in an organ-pipe fashion, consistentith the gravity-like potential approach used here [seelso Fig. 1(c)].Two-peak Raman states were experimentally observed
nd reported in [5,6]. From the same experimental datae have also observed some indication of the existence of
hree-peak states. Figure 8 shows the power dependence
3-peakstate
propagation length
dela
y
delay
inte
nsity
(a)
(b)
ig. 7. (Color online) Formation of a three-peak Raman state.a) Temporal evolution of a pulse with input soliton order N13.81. (b) Intensity profile of this three-peak Raman state at=0.8.
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1790 J. Opt. Soc. Am. B/Vol. 27, No. 9 /September 2010 Tran et al.
f the spectra obtained experimentally at the end of aighly nonlinear PCF (18 cm long) [5,6]. At 100 mW, awo-peak Raman state is formed by the second and thehird fundamental solitons, where at 200 mW one canee some signature of a three-peak state formed by therst, second, and third fundamental solitons. These threeolitons are close to each other in frequency. At the sameime, the presence of strong spectral fringes at 200 mWnot shown here) shows that they are also close in time.s mentioned above, two-peak states are more robust
han three-peak states, but the existence of solitonictates with more than three peaks would unambiguouslyonfirm our theoretical model.
. CONCLUSIONhe formation of multipeak localized structures sup-orted by the Raman effect turns out to be a fundamentalnd surprisingly common process occurring in the regionf deep anomalous dispersion in photonic crystal fibersPCFs). The gravity-like potential approach turns out toe a fruitful tool, leading to a simple relationship betweenwo important parameters of two-peak states, namely, theeak power ratio r and the temporal separation �0 be-ween two pulses forming the pair. Our approach explainsn a natural way the organ-pipe arrangement of multi-eak states, where the strongest soliton creates a poten-ial in its leading edge and subsequent solitons with mo-otonously decreasing peak powers can be fitted withinhis potential (see Fig. 1). The formation of pairs can bexplained through the following simple mechanism: theSFS induces an increasing separation between twonequal-amplitude solitons that can only be counterbal-nced by intersoliton forces (ISFs). Analyzing these twoffects together with the results obtained from the poten-ial approach led us to a simple transcendental equationor the peak power ratio r, and thus also for the temporaleparation �0. This transcendental equation gives resultshat are in good agreement with the exact numerical so-utions. The physical interpretation of the two-peak stateormation in our model is thus quite simple and intuitive.
We have used a qualitative model to predict the magiculse power needed to generate a two-peak state duringhe pulse splitting process. The predicted values aregain in good agreement with experimentally measurednes. In addition to two-peak states, we have also inves-
Input power, mW
Wavele
ngth
,nm 2-peak state
3-peak state
ig. 8. (Color online) Power dependence of the spectra obtainedxperimentally at the end of a highly nonlinear PCF (18 cmong).
igated n-peak states with n�2. We have demonstratedumerically—using some hints from experiments—thathe complexity of the GNLSE allows the existence ofhree-peak states. Even though such states are much lessobust than two-peak states, they can survive for at leasteveral tens of dispersion lengths. It is hoped that suchhree-peak states can also be experimentally observed inn unambiguous way in the near future.We expect that the results reported here, apart from
heir obvious fundamental interest, will allow researcherso manipulate more effectively the dynamics of solitaryaves and supercontinuum generation (SCG) in solid-
ore microstructured fibers with a properly designedroup-velocity dispersion (GVD), and in hollow-core fiberslled with Raman-active gases.
CKNOWLEDGMENTShis work is supported by the German Max Planck Soci-ty for the Advancement of Science (MPG). The authorsould like to acknowledge several useful discussions withlexander Hause and Fedor Mitschke from the Univer-ity of Rostock, Germany.
EFERENCES AND NOTES1. K. Tai, A. Hasegawa, and N. Bekki, “Fission of optical soli-
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2. Equation (4) has to be interpreted as an eigen problem forf��� and q, with b=32�Rq2 /15 imposed by the physics ofRSFS. The boundary conditions that one has to impose tofind the soliton solutions of Eq. (4) by using the shootingmethod are as follows: (i) the function values at the leadingedge boundary �1 and the trailing edge boundary �2 are van-ishing: f��1�= f��2�=0; (ii) the first-order derivative at thetrailing edge boundary �2 is vanishing: f���2�=0. The nu-merical solution may shift the position of the maximumslightly with respect to �=0 in order to satisfy the aboveconditions, due to the presence of the oscillating and slowlydecaying Airy tail on the leading edge of the soliton.
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