robust pedestal-free pulse compression in cubic-quintic...

Robust pedestal-free pulse compression in cubic-quintic nonlinear media

K. Senthilnathan,1 Qian Li,1 K. Nakkeeran,2 and P. K. A. Wai11Photonics Research Center and Department of Electronic and Information Engineering, The Hong Kong Polytechnic University,

Hung Hom, Kowloon, Hong Kong, China2School of Engineering, Fraser Noble Building, King’s College, University of Aberdeen, Aberdeen AB24 3UE, Scotland, United Kingdom

�Received 18 June 2008; revised manuscript received 13 August 2008; published 24 September 2008�

We consider the evolution of nonlinear optical pulses in cubic-quintic nonlinear media wherein the pulsepropagation is governed by the generalized nonlinear Schrödinger equation with exponentially varying disper-sion, cubic, and quintic nonlinearities and gain and/or loss. Using a self-similar analysis, we find the chirpedbright soliton solutions in the anomalous and normal dispersion regimes. From a stability analysis, we showthat the soliton in the anomalous dispersion regime is stable, whereas the soliton in the normal dispersionregime is unstable. Numerical simulation results show that competing cubic-quintic nonlinearities stabilize thechirped soliton pulse propagation against perturbations in the initial soliton pulse parameters. We characterizethe quality of the compressed pulse by determining the pedestal energy generated and compression factor whenthe initial pulse is perturbed from the soliton solutions. Finally, we study the possibility of rapid compressionof Townes solitons by the collapse phenomenon and the exponentially decreasing dispersion. We find that thecollapse could be postponed if the dispersion increases exponentially.

DOI: 10.1103/PhysRevA.78.033835 PACS number�s�: 42.81.Dp, 42.65.Tg, 05.45.Yv

I. INTRODUCTION

The generation of ultrashort optical pulses has receivedmuch attention owing to their wide applications in manydifferent areas such as ultrahigh-bit-rate optical communica-tion systems, ultrafast physical processes, infrared time-resolved spectroscopy, optical sampling systems, etc. �1–3�.In general, it is difficult to produce very short optical pulseseven from the best available laser sources. Therefore opticalpulse compression techniques have been utilized to generateoptical pulses shorter than those produced by lasers or am-plifiers. Optical pulse compression can be classified into twodifferent types. The first is linear compression of chirpedpulses using a dispersive fiber delay line or grating pairs. Thesecond is nonlinear compression based on the interplay be-tween self-phase modulation and group velocity dispersionusing either a soliton or adiabatic effect �3�. In soliton effectcompression, the compressed pulse typically is a sharp, nar-row spike centered on a broad low-intensity pedestal whichcontains a large proportion of the pulse energy. In opticalcommunication systems, a broad pedestal is undesirable be-cause it will overlap with adjacent pulses and results in in-tersymbol interference. A number of pedestal suppressiontechniques have been proposed to block the low-intensitytails of the compressed pulses �4�. Much of the energy of theinitial pulse, however, is also discarded in the process. Adia-batic pulse compression, in principle, can achieve pedestal-free compression, but it is difficult to maintain the adiabaticcondition throughout the compression process.

Contrary to the above-discussed conventional pulse com-pression techniques, a technique has been proposed based onfilamentation and plasma generation in the high-intensity re-gion �5�. When the intensity of the incident field is highenough in the range of 1013–1014 W /cm2 or the intensityreaches the value where the Kerr nonlinearity saturates, im-portant physical effects like self-focusing, plasma defocus-ing, etc., come into play. When a pulse with power exceeding

the critical power for self-focusing, Pcr, propagates in gasesit supports a narrow coherent structure known as filaments.The filament is generated after the dynamical balance of twocounteracting physical effects: namely, focusing due to theKerr effect and defocusing due to plasma generated by mul-tiphoton ionization. The filament has been observed in gasesas air �6�, in solids as silica glasses �7�, and in liquids �8�.

Recently, much interest has been focused on the compres-sion of linearly chirped pulses. Moores suggested thatchirped solitary pulses can be compressed more efficiently ifthe dispersion decreases approximately exponentially �9�. Aself-similar analysis has been utilized to study linearlychirped pulses in optical fibers and fiber amplifiers. The self-similar analysis for the nonlinear Schrödinger �NLS� equa-tion with constant gain has revealed that the interplay ofnormal dispersion, nonlinearity, and gain produces a linearlychirped pulse with a parabolic intensity profile which resiststhe deleterious effects of optical wave breaking �10�. Thegeneration of bright and dark self-similar solitary pulses hasbeen investigated using a NLS-type equation in the presenceof gain �11�. Chirped solitary pulse compression has beendemonstrated in these optical amplifiers. Using the self-similar analysis, we have shown that chirped Bragg solitarypulses can be generated near the photonic band gap of non-uniform-fiber Bragg gratings. We have also investigatedpedestal-free Bragg soliton pulse compression �12�. Usingthe Hirota bilinear method and variational analysis, we haveshown that the intensity and the chirp of the chirped solitonfor an exponentially decreasing dispersive medium increaseexponentially, while its width decreases exponentially. Theseproperties are consistent with self-similar behavior �13�.

When an optical pulse is compressed, its peak intensityincreases, while the pulse width decreases. When the pulsepeak intensity is sufficiently large, the field-induced changeof the refractive index is no longer described by the usualKerr-type nonlinearity—i.e., n�� , I�=n0��+n2I, wheren�� , I� is the refractive index of the medium, n0�� is therefractive index of the medium for weak input optical power,

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� is the angular frequency, n2 is the Kerr constant, and I isthe intensity of the optical pulse. A higher-order nonlineareffect such as the quintic nonlinearity will have to be takeninto account and the refractive index will have to be modi-fied to be n�� , I�=n0��+n2I−n4I2, where n4 is the quinticnonlinearity coefficient. The coefficients n2 and n4 are re-lated to the third-order ��3� and fifth-order ��5� susceptibili-ties through n2=3��3� / �8n0� and n4=−5��5� / �32n0�. Pulsepropagation in the presence of competing cubic-quintic non-linearities has received much attention lately because com-petition between nonlinearities of different orders could leadto strong stabilization of pulse propagation �14�. Higher-order nonlinearities must be considered if the optical pulseintensity is high or the nonlinear coefficients of the materialsare large—for instance, in semiconductor doped glasses�14,15�. One example of the effect of competing nonlineari-ties is the stabilization of vortices and vortex tori in cubic-quintic nonlinear media �16–19�. In the constant-dispersioncase, the bright and dark solitons have been extensively in-vestigated in the cubic-quintic nonlinear media �20,21�. Thebright and dark quasi-solitons for the cubic-quintic nonlin-earity have also been studied �22�. In recent years, a numberof experiments reported the measurement of the cubic-quintic nonlinearity �23�. More recently, we have investi-gated the existence of chirped bright solitons in cubic-quinticnonlinear media with exponentially decreasing dispersion�24�. In this paper, we investigate pedestal-free pulse com-pression of the chirped bright soliton pulse under the influ-ence of cubic and quintic nonlinearities. Numerical resultsreveal that competing cubic and quintic nonlinearities canstabilize the pulse propagation and will lead to more robustpulse compression. As a special case, we also discuss chirpedTownes solitons in pure quintic media—i.e., power-lawnonlinearity—and study the possibility of rapid pulse com-pression by wave collapse and exponentially decreasing dis-persion. We note that the pure quintic nonlinearity aptlymodels the Tonks-Girardeau �TG� regime in Bose-Einsteincondensation �BEC�.

The rest of the paper is organized as follows. Section IIdiscusses the theoretical model and the origin of the fifth-order �quintic� nonlinearity. In Sec. III, we present thechirped bright soliton solutions. We then determine the rela-tions among the dispersion, cubic nonlinear, and quintic non-linear lengths. Using numerical simulations, we study theeffects of the perturbations in the soliton parameters such asthe peak power and chirp on the quality of compressedpulses. These numerical results show that the chirped brightsoliton in the anomalous dispersion regime is stable, whereasit is unstable in the normal dispersion regime. In Sec. IV, westudy the chirped Townes solitons in pure quintic nonlinear-ity and discuss possible applications to BEC. Finally, weconclude in Sec. V.

II. THEORETICAL MODEL

Cubic-quintic nonlinearity arises from a nonlinear correc-tion to the refractive index of a medium in the form �n=n2I−n4I2, and the coefficients n2, n4�0 determine the non-linear response of the medium. Formally, the expression for

�n may be obtained by an expansion of a saturable nonlin-earity of the form �n=n2I�1+ �n4 /n2�I�−1 under the assump-tion of self-focusing as d��n� /dI. However, the cubic-quinticmodel changes the sign of focusing at a critical intensity Ic=n2 /2n4. Experimental measurement of the nonlinear dielec-tric response in para-toluene salfonate �PTS� optical crystalconfirmed the observations �14�. Cubic-quintic nonlinearitiescan also be obtained by doping a fiber with two appropriatesemiconductor materials. One should be positive n2

�1��0 andhave a large saturation intensity Isat

�1�, and the other should benegative n2

�2��0 with nearly the same magnitude and have alow saturation intensity—i.e., Isat

�2�� Isat�1� �20�.

Pulse propagation in a medium with cubic-quintic nonlin-earities is governed by the generalized cubic-quintic NLS�CQNLS� equation

i�A

�z−

��z�2

�2A

��2 + �z��A�2A − ��z��A�4A − ig�z�

2A = 0,

�1�

where A�z ,�� is the slowly varying envelope of the axialelectrical field, � is the retarded time, z is the propagatingdistance, ��z� is the group velocity dispersion, and g�z� is thedistributed gain or loss function. The cubic and quintic non-linear parameters are given by =2n2 /�0Aeff and �=2n4 /�0Aeff

2 , where �0 is the central wavelength and Aeff isthe effective core area of the fiber. All physical parameters��z�, �z�, ��z�, and g�z� are functions of the propagationdistance z.

We now investigate the chirped soliton solutions of theCQNLS equation by a scaling analysis known as self-similaranalysis. We assume the complex function A�z ,�� in the form

A�z,�� = Q�z,��exp�i��z,�� , �2�

where Q and � are the amplitude and phase of the envelopefunction A, respectively. In order to study the generation ofchirped solitons of Eq. �1�, we assume a quadratic phasegiven by

��z,�� = 1�z� + 2�z�

2�� − �c�2, �3�

where 1�z� and 2�z� are functions of z and �c is the centerof the pulse. By self-similar scaling analysis, we assume thatthe amplitude depends on the scaling variable �, which is acombination of variables �−�c and some function ��z� of thevariable z. Since the self-similar solutions possess scalingstructure, we represent the amplitude Q�z ,�� as

Q�z,�� =1

��z�R��exp�G�z�

2� . �4�

The scaling variable � and the function G�z� are given by

� =� − �c

��z�, G�z� =

0

z

g�z��dz�. �5�

Here ��z� and R�� are some functions which have to bedetermined. We also assume that ��0�=1 without loss ofgenerality. Substituting Eqs. �2�–�4� into Eq. �1�, the qua-

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dratic phase coefficient 2�z� and the function ��z� are foundto be

2�z� = 20

1 − 20D�z�, ��z� = 1 − 20D�z� , �6�

where 20= 2�0��0 because the phase is assumed to be aquadratic function of the variable �−�c and the cumulativedispersion function D�z� is given by

D�z� = 0

z

��z��dz�. �7�

In addition to the above conditions, we also find

d2R

d�2 +2�2

�

d 1

dzR −

2�

�exp�G�z��R3 +

2�

�exp�2G�z��R5 = 0.

�8�

Note that the coefficients in Eq. �8� are functions of the vari-able z, but the function R�� depends only on the scalingvariable �. Therefore, Eq. �8� possesses nontrivial solutions�R��0� if and only if the coefficients in Eq. �8� are con-stants, i.e.,

−2�2�z��z�

d 1

dz= �1, �9�

��z��z��z�

exp�G�z�� = �2, �10�

��z��z�

exp�2G�z�� = �3, �11�

where �1, �2, and �3 are constants. Equations �9�–�11� yield

�1 = − 2

�0

d 1

dz

z=0, �2 =

0

�0, �3 =

�0

�0, �12�

because ��0�=1 and G�0�=0. The parameters �0=��0�, 0=�0�, and �0=��0�. Thus for the nontrivial case, Eq. �8� canbe written as

d2R

d�2 − �1R + 2�2R3 + 2�3R5 = 0. �13�

The solution of Eq. �9� is

1�z� = 10 −�1

2

0

z ��z��dz�

�1 − 20D�z��2 , �14�

where 10 is an integration constant. Next we proceed to findthe distributed gain function using Eqs. �10� and �11�,

g�z� =1

��z�d�

dz−

20��z��z�

, �15�

where we define the function ��z� as

��z� =�z��z�

, ��0� =0

�0. �16�

From Eqs. �10� and �11�, the condition for the variation ofthe quintic nonlinear parameter is

��z� =2�z��2�z�

��z��3

�22 . �17�

Equations �6�, �7�, and �12�–�17� are the required conditionsfor the existence of self-similar solutions in Eqs. �2�–�5� ofthe generalized CQNLS with distributed coefficients—i.e.,Eq. �1�. We observed that for the self-similar solutions of Eq.�1�, only two of the four parameters ��z�, �z�, ��z�, and g�z�in Eq. �1� are free parameters. For example, if ��z� and �z�are chosen to be the free parameters, then g�z� and ��z� willbe determined from Eqs. �15� and �17� respectively.

Different physical situations lead to different choices ofthe two free parameters. For example, consider that both thecubic and quintic nonlinear coefficients do not vary withdistance; i.e., both �z� and ��z� are constant. A self-similarsolution to Eq. �1� exists if the dispersion and gain or lossvary in the form ��z�=�0 / �1+�0 20z�2 and g�z�=�0 20 / �1+�0 20z�. If the cubic nonlinear parameter �z� is constantand there is no gain or loss �g�z�=0�, then the dispersion andthe quintic nonlinearity must vary exponentially—i.e., ��z�=�0 exp�− 20�0z� and ��z�=�0 exp�− 20�0z�. In this case,the function ��z�=exp�− 20�0z�. Thus, from Eq. �5�, thepulse width will vary exponentially as the dispersion ��z�.Since we are interested in pedestal-free compression of op-tical pulses, we will focus on the soliton solutions to CQNLSequation with exponentially varying dispersion in Sec. III.Another special case is pure quintic nonlinearity with no gainor loss—i.e., �z�=g�z�=0. From Eq. �11�, the quintic non-linearity ��z� is then directly proportional to the dispersion��z�, which can take any functional form. We will study thisspecial case in Sec. IV.

III. CHIRPED SELF-SIMILAR BRIGHT SOLITONS INTHE ANOMALOUS AND NORMAL DISPERSION

REGIMES

In this section, we assume that the cubic nonlinear param-eter �z� is constant and there is no gain or loss—i.e., g�z�=0. The dispersion and quintic nonlinearity therefore varyexponentially as ��z�=�0 exp�− 20�0z� and ��z�=�0 exp�− 20�0z�. From Sec. II, the phase and amplitude ofthe self-similar solutions of the generalized CQNLS equationwith distributed coefficients are given by Eqs. �3� and �4�.We determine the complex envelope of the bright solitarywave by integrating Eq. �13� and obtain

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A�z,�� =1

�0�1 − 20D�z�� 2�1

�1 +8�1

2

3�2�02�1 − 20D�z��2 cosh 2� � − �c

�0�1 − 20D�z�� 1�1/2

exp�i�� , �18�

where �1�z�=��z� /�z� and �2�z�=��z� /��z�. The integrationconstant �1 is chosen to be 1 /�0

2, where �0 is the initial pulsewidth parameter. The upper sign in Eq. �18� corresponds tothe soliton in the anomalous dispersion regime, while thelower sign corresponds to that in the normal dispersion re-gime. In the anomalous dispersion regime, the physical con-ditions ��0, �0, and ��0 �competing cubic-quintic non-linearities� or ��0, �0, and ��0 �cooperating cubic-quintic nonlinearities� should be maintained for the existenceof solitons. Similarly, ��0, �0, and ��0 �competingcubic-quintic nonlinearities� have to be valid for the solitonin the normal dispersion regime. Equation �18� is a linearlychirped self-similar bright solitary pulse since it propagatesin a self-similar manner in a fiber medium under the influ-ence of cubic-quintic nonlinearities. In semiconductordouble-doped optical fibers, self-similar solitons in both nor-mal and anomalous dispersion regimes are possible depend-ing on the doping materials, the operating frequency, and theoptical pulse intensities �21,24�.

A. Length scales

Unlike the conventional soliton in Kerr media, where thedispersion and nonlinear lengths are proportional to eachother, we found that the dispersion, cubic, and quintic non-linear lengths for the self-similar solitons in a cubic-quinticmedium follow a harmonic relationship

2 sgn��3Ln4

�z�−

sgn��Ln2

�z�−

sgn��LD�z�

= 0, �19�

where the dispersion �LD�, cubic �Ln2�, and quintic �Ln4

� non-linear lengths are given by

LD�z� =�0

2�2�z��z��

, Ln2�z� =

1

��z��P0�z�,

Ln4�z� =

1

��z��P02�z�

. �20�

Here P0= �Amax�2 is the peak power of the chirped soliton,where Amax is the maximum amplitude �which is given in Eq.�23��. Note that all the three length scales vary with distancebecause the coefficients of the CQNLS equation depend ondistance. We note that when the quintic nonlinearity isswitched off—i.e., Ln4

→�—Eq. �19� is reduced to the well-known condition LD=Ln2

for soliton formation in Kerr me-dia. In this case, the linearly chirped bright soliton in Eq.�18� will be reduced to the chirped soliton solution for Kerrnonlinearity �11–13�.

From Eq. �18�, the energy of the chirped bright soliton iscalculated as

W = −�

�

�A�2d� = −2�1�z�

�0��1 − b2ln

1 + �1 − b2

b�b � 1� ,

�21�

W = −�

�

�A�2d� = �2�1�z�

�0��b2 − 1�

2� sin−1 1

b�� b � 1� ,

�22�

where b=�1+8�3 /3�02�2

2 is a constant. In terms oflength scales, the parameter b can also be written as

b=�1+4Ln2�z�LD�z�2 �sgn��sgn��LD�z�+Ln2�z��. Equation �21�

represents the energy of the soliton in the anomalous disper-sion regime when b�1. The upper sign in Eq. �22� corre-sponds to the soliton energy in the anomalous dispersionregime when b�1. The lower sign corresponds to the solitonenergy in the normal dispersion regime for all values of b�0. The condition b�1 ��1� corresponds to �0 /�0�0��0�. For optical fibers, the cubic nonlinear coefficient �0. Thus, in the anomalous dispersion regime, the conditionb�1 means ��0 representing cooperating cubic-quinticnonlinearities, whereby the effect of dispersion is balancedby the cubic and quintic nonlinearities. The other conditionb�1 means ��0 representing the competing cubic-quinticnonlinearities, wherein the effect of cubic nonlinearity is bal-anced by that of dispersion and quintic nonlinearity. The con-dition b=1 is not of interest because it corresponds toLD�z�=Ln2�z�, which is possible only if Ln4

→�—i.e., �=0.In the normal dispersion regime, normally b�1 becausesgn��0. Thus, in the anomalous dispersion regime, theparameter b determines whether the nonlinearities are coop-erating or competing.

We find that the soliton solution in the anomalous disper-sion regime will approach the chirped soliton solution re-ported in �11–13� when the coefficient of the quintic nonlin-earity approaches zero ��→0�. However, the energy of Eq.�22� goes to infinity when �→0 ��3→0�. Thus, the quinticnonlinearity is crucial to the chirped soliton solutions in thenormal dispersion regime. Similarly, the peak intensity andthe full width at half maximum �FWHM� are given by

Amax2 �z� = P0 = �

1

�02�2

2�1�z�b � 1

, �23�

�� = �0��z�ln�2 � y + ��2 � y�2 − 1� , �24�

where y=1 /b. The upper sign in Eqs. �23� and �24� corre-sponds to the peak intensity and the pulse width of solitonsin the anomalous dispersion regime, while the lower signcorresponds to that in the normal dispersion regime. Note


033835-4

that Eq. �24� represents the FWHM of the chirped solitonpulse after the compression process. Further, from the samerelation, as we know the initial and final widths of the pulse,one can easily determine the pulse compression factor withthe relation

�0

��= �� ln�2 � y + ��2 � y�2 − 1��−1. �25�

B. Stability of the chirped solitons

Strictly speaking, the localized solution given in Eq. �18�is not a soliton, but rather a solitary wave. It is thereforecrucial to determine the stability of the chirped solitons. Ide-ally, analytical techniques such as the Vakhitov-Kolokolv�VK� criterion should be used to determine the stability ofthe solitary waves. The VK criterion has been well estab-lished for constant-coefficient NLS-type equations �for bothcubic and cubic and quintic nonlinearities�. However, math-ematically the VK criterion is applicable to the ground statesof NLS-type equations only. Thus, the VK criterion cannotbe applied to analyze the stability of the chirped solitongiven in Eq. �18�. As a result, we have to resort to usingnumerical simulations to determine the stability of thechirped solitons. From extensive numerical simulations, wefind that the chirped soliton in the anomalous dispersion re-gime is stable, whereas that in the normal dispersion regimeis unstable �for details see Sec. III C�. Therefore, hereafter,we focus on pulse compression in the anomalous dispersionregime only.

Before we leave this subsection, we would like to illus-trate the relationship between the three length scales dis-cussed in Sec. III A. We consider the case in which the cubicnonlinearity =const and gain or loss g�z�=0. Thus the dis-persion and quintic nonlinearity parameters are given by

��z� = �0 exp�− �z�, ��z� = �0 exp�− �z� , �26�

where �0�0, �0�0, and �= 20�0�0 for dispersion-decreasing fibers. As an example of pulse compression, weconsider a semiconductor-doped �chalcogenide AS2Se3� fiberof length L=400 m. The effective core area of the fiber isassumed to be 10 �m2. The cubic and quintic nonlinear co-efficients are assumed to be 0=0.2362 W−1 m−1 and �0=0.4724 W−2 m−1, respectively. The initial dispersion �0 ischosen to be −0.5 ps2 m−1. The other physical parameterschosen are �0=5 ps, �=0.005 m−1, 20=−0.01 THz2, and g=0. Figure 1 plots the variation of dispersion, cubic, andquintic nonlinear lengths for a soliton solution. From Fig. 1,in the beginning of the compression, the cubic nonlinearlength and the dispersion length dominate. As the pulsepropagates and compresses, its peak intensity increases andhence the quintic nonlinear length decreases and becomescomparable to the dispersion length and the cubic nonlinearlength.

Figure 2 shows the compression of the bright soliton un-der the influence of the competing cubic-quintic nonlineari-ties. The compression factor of the above compressor isfound to be 7.39. We also numerically integrated Eq. �1�using the exact soliton solution as an initial condition. The

numerical results agree very well with the analytical solutionas illustrated in Fig. 2.

C. Perturbations in the initial peak power and chirp

Since Eq. �1� is not integrable, for optical compressionapplications it is necessary to study the effects of perturba-tions on the evolution of the solitary-wave solution given inEq. �18�. In the following, we investigate the effect of per-turbations in the initial peak power and chirp on the pulseevolution. First, we consider variations in the initial peakpower and initial chirp and study the corresponding evolu-tion in peak power and chirp. We determine the quality of thecompressed pulse by monitoring the evolution of the pedes-tal energy generated and the compression factor of the per-turbed initial pulse.

Figure 3 shows the deviations in peak power and chirpcoefficient at L=400 m from the analytic solution when theinput peak power deviates from the ideal peak power from−20% to +20%, but the pulse width of the solitary pulseremain unchanged. The choice of fiber and pulse parametersare the same as those in Fig. 1. The dots represent results forcompeting cubic-quintic nonlinearities �b�1�, circles repre-sent the results for cooperating cubic-quintic nonlinearities�b�1�, and crosses represent the results for pure cubic non-

0 100 200 300 400

50

100

150

200

250

z (m)

Leng

th(m

)

Ln2

Ln4

LD

FIG. 1. �Color online� Variation of the dispersion length �dots�,cubic nonlinear length �solid line�, and quintic nonlinearlength �dashed line� for a chirped soliton in a cubic-quinticnonlinear medium. The physical parameters chosen are �0=5 ps,�0=−0.5 ps2 m−1, �=0.005 m−1, 0=0.2362 W−1 m−1, �0

=0.4724 W−2 m−1, 20=−0.01 THz2, g=0, and z=400 m.

−3 −2 −1 0 1 2 3

2

4

6

8

(τ−τc)/τ

0

Inte

nsity

(arb

.uni

ts)

z = 400 m

z = 0

FIG. 2. �Color online� Compression of a chirped bright solitonpulse. The dots and solid lines represent analytical and numericalresults, respectively. The parameters chosen are the same as those inFig. 1.


033835-5

linearity. Figure 3�a� shows that the deviations of the peakpower from the ideal solution is smallest for the case ofcompeting cubic-quintic nonlinearities and largest for coop-erating cubic-quintic nonlinearities. Here, it should be em-phasized that the larger deviation in the latter case is owingto the beam collapse which occurs in the case of cooperatingcubic and quintic nonlinearities. In Fig. 3�a�, the deviation inpeak power and the perturbation in initial peak power havebeen calculated with the expression �Pperturbed− Pideal� / Pideal�100%. From Fig. 3�b�, the same observation holds for thedeviations of the chirp parameter, but the variations in thechirp parameter are more sensitive to the perturbation. Here,the chirp of the perturbed solitary pulse is measured by car-rying out a polynomial fit of the phase. From the polynomialfitting, we observe that the chirp remains close to the qua-dratic chirp since the chirps contributed by all higher-orderterms are very small. Based on the numerical results, weobserve that the soliton solutions for competing cubic-quinticnonlinearities are more robust to perturbations. The amountof pedestal generated during pulse compression is an impor-tant parameter characterizing the quality of the compressedpulses. The pedestal energy is defined as the relative differ-ence between the total energy of the transmitted pulse andthe energy of a hyperbolic-secant pulse having the same peakpower and width as those of the transmitted pulse—i.e., ped-estal energy �%�= �Eperturbed−ECQ� /Eperturbed�100%. Here,Eperturbed is the energy of the perturbed pulse. Note that theenergy of a hyperbolic-secant type pulse �ECQ� in cubic-quintic media is calculated by using curve fitting to obtainthe best fit from the data. We use the function �A�2=�1 / ��2 cosh�2� /�3�+1� to fit the intensity of thehyperbolic-secant type pulse, where the values of the param-eters �1, �2, and �3 are determined by the curve fitting. Figure3�c� shows the amount of pedestal generated for the corre-sponding perturbation in peak power. The results show thatcompeting cubic-quintic nonlinearities generate only a smallamount of pedestal energy when compared to cooperating

cubic-quintic nonlinearities and pure cubic nonlinearity.However, the amount of pedestal generated in all three casesis very small. Even in the worst case in which the initial peakpower is only 80% of the ideal value for cooperating cubic-quintic nonlinearities, the pedestal energy is only 1.6%. Fig-ure 3�d� shows the compression factor for the correspondingperturbation in the peak power for all three cases. As ex-pected, the compression factor is the highest for cooperatingcubic-quintic nonlinearities and the lowest for the competingcase. Thus an optical pulse compressor using competingcubic-quintic nonlinearities are more robust to perturbationsof the input pulse parameters at the expense of the compres-sion factor.

Figures 4�a� and 4�b�, respectively, show the deviation inpeak power and the chirp when the initial chirp of the inputpulse deviates from the ideal value. Unlike the perturbationsin peak power, the deviation in peak power is small and isvery similar in all three cases. The differences in the devia-tions in chirp value for the three cases are more significant.Again, the deviation of the chirp value is the smallest forcompeting cubic-quintic nonlinearities and the largest for co-operating case. Figures 4�c� and 4�d� show the pedestal en-ergy and compression factor, respectively, for the corre-sponding perturbations in the initial chirp. The pedestalenergy generated in all three cases is only a fraction of apercent. The deviation in compression factor is small and issimilar for all three cases. Thus the soliton is very robust toperturbations in the initial chirp in all three cases. We havecarried out extensive numerical simulations on the perturba-tions of initial power and chirp by varying the decay rate ofdispersion and changing the strength of quintic nonlinearity.In all cases we have studied, we find that the compressedpulse in the case of competing cubic-quintic nonlinearitieshas the smallest deviations in peak power and chirp. Thepedestal energy is also the smallest, but the compression fac-tor is also the smallest when compared to the case of purecubic nonlinearity and cooperating cubic-quintic nonlineari-ties.

−20 −10 0 10 20

−20

0

20

40

perturbation in initial peak power (%)

Dev

iatio

nin

peak

pow

er(%

)

−20 −10 0 10 20

0

100

200


Dev

iatio

nin

α 2(z)

(%)

−20 −10 0 10 20

0

0.5

1

1.5

2


pede

stal

ener

gy(%

)

−20 −10 0 10 20

6

7

8

9

10


com

pres

sion

fact

or

(a)

(b)

(c)

(d)

FIG. 3. �Color online� �a� De-viations of the peak power fromthe exact solution, �b� the chirpcoefficient, �c� the pedestal en-ergy, and �d� the compression fac-tor at L=400 m when the inputpeak power deviates from theideal peak power from −20% to+20%. The physical parameterschosen are the same as those inFig. 1. The dots represent resultsfor competing cubic-quintic non-linearities ��0=0.4724 W−2 m−1�,circles represent the results for co-operating cubic-quintic nonlin-earities ��0=−0.4724 W−2 m−1�,and crosses represent the resultsfor pure cubic nonlinearity ��0

=0�.


033835-6

Figure 5 shows the evolution of the �a� peak power fromthe exact values, �b� chirp, �c� pedestal energy, and �d� com-pression factor when the peak power of the initial solution is+20% larger than the ideal solution given in Eq. �21� forcompeting cubic-quintic nonlinearties and Eq. �22� for coop-erating cubic-quintic nonlinearities. The distance traveled is400 m. The parameter LD0 is the initial dispersion length asz=0. The dashed lines represent the soliton solution of theCQNLS equation. The dots represent perturbation results forcompeting cubic-quintic nonlinearties, the circles representcooperating cubic-quintic nonlinearities, and the crosses rep-resent pure cubic nonlinearity. From Figs. 5�a� and 5�d�, thepeak power and pulse width �and thereby compression fac-tor� of the solitary pulse undergo periodic oscillations during

the evolution. We note that the oscillation periods are differ-ent for the three different cases. Without quintic nonlinearityand for constant dispersion, the oscillation has a period of8z0 for solitons where z0 �=LD0 /2� is the soliton period,because of the resonance of the perturbation with the solitonwave vector, 2 / �8z0� �25�. Figures 5�b� and 5�c�, respec-tively, represent the evolution of the chirp and pedestal en-ergy generated.

In what follows, for the completeness of the investigation,we predict the oscillations exhibited by the peak power andpulse width during the evolution. Figure 6�a� represents thebehavior of change in peak power �P= Pperturbed− Pideal alongthe propagation direction when the initial input peak powerdeviates from the ideal peak power from −20% to +20%. We

−20 −10 0 10 20

−1

0

1

2

perturbation in initial chirp (%)

Dev

iatio

nin

peak

pow

er(%

)

−20 −10 0 10 20

−20

0

20

40


Dev

iatio

nin

α 2(z)

(%)

−20 −10 0 10 20

0

0.1

0.2

0.3


pede

stal

ener

gy(%

)

−20 −10 0 10 20

7.3

7.4

7.5

7.6


com

pres

sion

fact

or

(a)

(b)

(c)

(d)

FIG. 4. �Color online� �a� De-viations of the peak power fromthe exact solution, �b� the chirpcoefficient, �c� the pedestal en-ergy, and �d� the compression fac-tor at L=400 m when the initialchirp deviates from the chirpvalue of the exact solutionfrom −20% to +20%. The physi-cal parameters chosen are thesame as those in Fig. 3. Thedots represent results for compet-ing cubic-quintic nonlinearities��0=0.4724 W−2 m−1�, circlesrepresent the results for cooperat-ing cubic-quintic nonlinearities��0=−0.4724 W−2 m−1�, andcrosses represent the results forpure cubic nonlinearity ��0=0�.

0 2 4 6 8

0.2

0.4

0.6

0.8

1

log 10

[pea

kpo

wer

(arb

.uni

ts)

]

z / LD0

1 3 6 9

−0.15

−0.1

−0.05

0

0.05

peak power (arb.units)

α 2(z)

(TH

z2 )

0 2 4 6 8

0

0.5

1

1.5

2

z / LD0

pede

stal

ener

gy(%

)

0 2 4 6 8

2

4

6

8

10

com

pres

sion

fact

or

z / LD0

(a)

(b)

(c)

(d)

FIG. 5. �Color online� Evolu-tion of the �a� peak power, �b�chirp, �c� pedestal energy, and �d�compression factor when the ini-tial peak power deviates +20%from the ideal peak power whenthe distance varies from z=0 to 400 m for the same physi-cal parameters used in Fig. 1. Theparameter LD0 is the initial disper-sion length. The dashed linesrepresent results for the exact so-lution, dots represent the compet-ing cubic-quintic nonlinearities��0=0.4724 W−2 m−1�, circlesrepresent the results for cooperat-ing cubic-quintic nonlinearities��0=−0.4724 W−2 m−1�, andcrosses represent the results forpure cubic nonlinearity ��0=0�.


033835-7

have halved the dispersion decay rate and doubled the pulsepropagation distance in Fig. 6�a� when compared to that usedin Fig. 4 in order to observe more oscillation periods. Thesolid lines represent the competing cubic-quintic nonlineari-ties, dashed lines represent the cooperating cubic-quinticnonlinearities, and dot-dashed lines represent the pure cubicnonlinearity. From Fig. 6�a�, the deviation �P is the smallestfor the competing cubic-quintic nonlinearities case. Figure6�b� shows that the period of oscillations decreases exponen-tially along the propagation direction. The solid lines repre-sent exponential fits for the three different cases. The rate ofchange of the period is found to be 0.002 474 m−1 for thecompeting cubic-quintic nonlinearities �dots�, 0.002 716 m−1

for the cooperating cubic-quintic nonlinearities �circles�, and0.002 637 m−1 for pure cubic nonlinearity �crosses�. All threedecay rates for the three different cases are close to the decayrate of the dispersion, which is 0.0025 m−1. Thus, the periodof oscillation decreases exponentially as the dispersion de-creases along the propagation direction. The oscillations in�P represent a gradual adjustment towards ideal pulse shape

under investigation. Figure 7 shows curve fitting of the evo-lution of �P for competing cubic-quintic nonlinearities withthe function f��=c1 exp�c2��+c3 exp�c4��sin�c5 exp�c6��,where �=z /LD0 and ci, i=1, . . . ,6, are constants. The solidlines represent the simulation results of the evolution of �Pfor competing cubic-quintic nonlinearities when the initialpeak power is +20% larger than that of the ideal values. Thedashed lines represent the curve-fitting results using themethod of steepest descent where c1=0.0917, c2=0.1238,c3=0.08, c4=−0.0177, c5=−7.6532, and c6=0.1043.

Figure 8 shows the evolution of the �a� peak power, �b�chirp, �c� pedestal energy, and �d� compression factor whenthe initial chirp is +20% larger than that of the ideal solution.The dashed lines represent the soliton solution of theCQNLS equation. The dots represent perturbation results forcompeting cubic-quintic nonlinearties, the circles representthose for cooperating cubic-quintic nonlinearities, and thecrosses represent those for pure cubic nonlinearity. We ob-serve that the peak power and the compression factor �pulsewidth� of the solitary pulse undergo only very small periodicoscillations. The deviations in peak power and compressionfactor are very small. The pedestal energy undergoes rela-tively larger periodic oscillations during the evolution, butthe magnitude of the variation is smaller than that for theperturbation in the peak power.

IV. CHIRPED SELF-SIMILAR TOWNES SOLITON

From the previous section, the soliton solution in the nor-mal dispersion regime of the CQNLS equation with expo-nentially varying dispersion is unstable. Further, in the ab-sence of quintic nonlinearity, the combination of cubicnonlinearity and normal dispersion, exponentially varying ornot, does not support soliton solutions. However, a purequintic nonlinear medium does support the soliton solutionknown as Townes solitons, which were discovered for thetwo-dimensional NLS equation in nonlinear optics and de-

0 4 8 12 16

−2

−1

0

1

2

3

∆P

z / LD0

120%

80%

0 4 8 12 16

0.5

1

1.5

z / LD0

log 10

(pe

riod

/LD

0)

(a)

(b)

FIG. 6. �Color online� �a� Evolution of �P= Pperturbed− Pideal

along the propagation direction. The solid lines represent the resultsfor competing cubic-quintic nonlinearities ��0=0.4724 W−2 m−1�,dashed lines represent the results for cooperating cubic-quintic non-linearities ��0=−0.4724 W−2 m−1�, and dot-dashed lines representthe results for pure cubic nonlinearity ��0=0�. The physical param-eters are chosen as �0=5 ps, �0=−0.5 ps2 m−1, �=0.0025 m−1, 0

=0.2362 W−1 m−1, 20=−0.005 THz2, g=0, and z=800 m. �b�Variation of the oscillation periods along the propagation direction.The dots represent the results for competing cubic-quintic nonlin-earities ��0=0.4724 W−2 m−1�, circles represent the results for co-operating cubic-quintic nonlinearities ��0=−0.4724 W−2 m−1�, andcrosses represent the results for pure cubic nonlinearity ��0=0�. Thesolid lines represent exponential fits for the three different cases.

0 4 8 12 16

0.2

0.4

0.6

0.8

z / LD0

∆P

FIG. 7. �Color online� Curve fitting the evolution of �P forcompeting cubic and quintic nonlinearities by the function f��=c1 exp�c2��+c3 exp�c4��sin�c5 exp�c6��, where �=z /LD0 and ci,i=1, . . . ,6, are constants. The solid line represents the simulationresults of the evolution of �P for competing cubic-quintic nonlin-earities when the initial peak power deviates +20% from the idealvalue. The dashed lines represent the fitting results using themethod of steepest descent where c1=0.0917, c2=0.1238, c3=0.08,c4=−0.0177, c5=−7.6532, and c6=0.1043, and the dashed linesrepresent the results for the fitting function.


033835-8

scribe the collapsing and dispersing of optical pulses �26�.Pulse propagation in a pure quintic NLS �QNLS� equationwith distributed dispersion and distributed linear gain isgiven by

i�A

�z−

��z�2

�2A

��2 − ��z��A�4A − ig�z�

2A = 0. �27�

Equation �27� describes nonlinear pulse propagation inmany fields of nonlinear science—for example, in nonlinearoptics under power-law nonlinearity. It is known that physi-cally, various materials, including semiconductors, exhibitpower-law nonlinearity. Spatial solitons have been investi-gated in media that have a power-law dependence on theintensity Iq for continuum values of q �with I being the in-tensity� �27�. However, in general, it may be difficult to findsuitable nonlinear optical media which exhibit pure quinticnonlinearity because manipulation of the magnitude and thesign of nonlinearity is not easy. However, the QNLS equa-tion aptly models BECs especially in the TG regime. Experi-mental generation of such a gas had also been reported �28�.Recently, it has been shown that the magnitude as well as thesign of nonlinearity, which are determined by the interactionsbetween atoms in the condensate, could be manipulated byvarying the external magnetic filed near the Feshbach reso-nance �29�. Therefore, it is physically relevant to discuss thechirped Townes soliton in BECs. The formation of solitonsin BECs is similar to nonlinear optics, where the bright anddark solitons are supported by focusing and defocusing non-linearities, respectively, whereas in BECs, the s-wave scat-tering interaction between atoms actually determines thesoliton formation. Thus, bright and dark solitons are found incondensates with attractive and repulsive interactions. Wenow discuss the chirped Townes soliton in pure quintic me-dia. The complex envelope of the chirped Townes soliton isobtained by applying the following physical condition ��z��0 and ��z��0 in Eq. �27�:

A�z,�� =1

�0�1 − 20D�z��

�� z��2��0��0�/3�0

2sech 2 � − �c

�0�1 − 20D�z��1/2

�exp�i�� . �28�

The relation between the dispersion and the quintic non-linear lengths is 2LD /3=Ln4. The energy and peak intensityof the chirped Townes soliton are given by

W =�3��z��

�1 − 20D�z��8��0��0�,

Amax2 =

�3��z��

�0�1 − 20D�z��2�2��0��0�. �29�

We then proceed to investigate the stability of the chirpedTownes soliton. For the constant-coefficient case, it has beendemonstrated that the Townes soliton is marginally stable inhomogeneous media �g=0� according to the VK criterion. Ashas been discussed in Sec. III B, the VK criterion does notapply to the chirped Townes soliton in Eq. �28�. However,unlike Sec. III B it is not possible to numerically establishthat a solution is marginally stable. We would try to infer thecharacter of the solution in Eq. �28� by numerically studyingthe evolution of inputs with amplitude above and below thatof the solution. First we study the evolution of the analyticchirped Townes soliton. Figure 9 plots the evolution of ana-lytic chirped Townes soliton solutions for constant �solidlines, �0=−0.5 ps2 m−1�, exponentially increasing �dot-dashed lines, �=−0.005 m−1�, and exponentially decreasing�dashed lines, �=0.005 m−1� dispersions. The dots representthe results obtained by dint numerical simulations. Note that

0 2 4 6 8

0.2

0.4

0.6

0.8

1

log 10

[pea

kpo

wer

(arb

.uni

ts)

]

z / LD0(a)

(b)

(c)

(d)1 2 4 6 8

−0.08

−0.06

−0.04

−0.02

0

peak power (arb.units)

α 2(z)

(TH

z2 )

0 2 4 6 8

0.1

0.2

0.3

0.4

z / LD0

pede

stal

ener

gy(%

)

0 2 4 6 8

2

4

6

8co

mpr

essi

onfa

ctor

z / LD0

FIG. 8. �Color online� Evolu-tion of the �a� peak power, �b�chirp, �c� pedestal energy, and �d�compression factor when the ini-tial chirp deviates +20% from theexact chirp value when the dis-tance varies from z=0 to 400 mfor the same physical parametersused in Fig. 1. The dashed linesrepresent results for the exact so-lution, the dots represent the com-peting cubic-quintic nonlinearities��0=0.4724 W−2 m−1�, circlesrepresent the results for cooperat-ing cubic-quintic nonlinearities��0=−0.4724 W−2 m−1�, andcrosses represent the results forpure cubic nonlinearity ��0=0�.


033835-9

the analytical results are in good agreement with the resultsobtained by dint numerical simulations for the distance simu-lated.

So far, we have discussed the conventional pulse com-pression technique whereby we utilized the cubic nonlinear-ity and dispersion to achieve pulse compression. A noncon-ventional pulse compression technique based on the wavecollapse phenomenon has also been reported �33�. It is wellestablished that the pure quintic model exhibits collapse �30�.However, the occurrence of collapse could either be arrestedor postponed by physical effects like damping, nonlinearsaturation, etc. �31,32�. Recently, it has been shown that thequintic damping effect, which arises from the three-body in-teraction of an imaginary component, and cubic nonlinearitymanagement help to arrest and/or suppress the occurrence ofcollapse �31�.

Figure 10 shows the deviations of peak power from theexact Townes soliton when the input peak power deviatesfrom the ideal peak power by �a� −5% and �b� +5% forconstant dispersion �solid lines�, exponentially decreasingdispersion �dashed lines�, and exponentially increasing dis-persion �dot-dashed lines�. The physical parameters used arethe same as those in Fig. 9. For constant dispersion �solidlines�, the pulse disperses because the input power is lessthan the critical power, as shown in Fig. 10�a�. When thedispersion increases exponentially �dot-dashed lines�, theself-similar effect and the wave collapse act together tobroaden the pulse. Thus the pulse undergoes fast broadening.When the dispersion decreases exponentially �dashed lines�,the self-similar effect and the collapse act oppositely. Thusthe pulse initially compresses slightly as the self-similar ef-fect dominates, but eventually the wave collapse effect takesover and the pulse begins to broaden. The self-similar com-pression in the exponentially decreasing dispersion can beused to postpone the wave dispersion in a pure quinticmedium.

Similarly, when the dispersion is constant, the pulse un-dergoes collapse when the input peak power is higher thanthe critical power, as shown by the solid lines of Fig. 10�b�.Rapid compression can be achieved by the combined actionof the collapse phenomenon and self-similar pulse compres-

sion in the exponentially decreasing dispersion �dashedlines�. The occurrence of collapse is postponed in the case ofexponentially increasing dispersion �dot-dashed lines� sincethe self-similar effect in the exponentially increasing disper-sion acts against the collapse. Our numerical results showthat the chirped Townes soliton exhibits a similar property asthe Townes soliton with constant dispersion when the inputpeak power deviates from the ideal peak power. Thus it islikely that the chirped Townes soliton is also marginallystable.

Finally, in general, it is difficult to realize a medium thatexhibits pure quintic nonlinearity. It is therefore important todetermine the tolerance of the chirped Townes soliton of aquintic medium in the presence of a small amount of cubicnonlinearity. Figure 11 shows the change in peak power ofthe Townes soliton in the presence of cubic nonlinearity. Thedots represent the results of pure quintic nonlinearity. Thesolid line, dot-dashed line, and dashed line represent the re-sults when the ratio of cubic nonlinear length to quintic non-linear length �Ln2

/Ln4� is 12 000, 1 200, and 120, respec-

tively. Figure 11 shows that the evolution of the chirpedTownes soliton is very sensitive to the presence of even asmall amount of cubic nonlinearity �Ln4

/Ln2�0.1% �. As the

cubic nonlinearity increases, the chirped Townes solitonpulse becomes unstable. The three different values of cubicnonlinearity are 0=10−5, 10−4, and 10−3 W−1 m−1. The otherphysical parameters are same as those in Fig. 1.

V. CONCLUSION

By using self-similar scaling analysis, we have found thechirped bright soliton solutions in the anomalous and normal

0 2 4 6 8

2

4

6

8

z / LD

peak

pow

er(a

rb.u

nits

)

FIG. 9. �Color online� Evolution of the peak power ofTownes solitons when the dispersion is constant �solid lines,�0=−0.5 ps2 m−1�, exponentially increasing �dot-dashed lines,�=−0.005 m−1�, and exponentially decreasing �dashed lines �=0.005 m−1�. The dots represent the results obtained by dint nu-merical simulation. The physical parameters are the same as thoseused in Fig. 1 except 0 �=0�.

0 2 4 6 8

−1

−0.5

0

0.5

z / LD0

log 10

[pea

kpo

wer

(arb

.uni

ts)

]

0 2 4 6 8

0

0.5

1

1.5

2

z / LD0

log 10

[pea

kpo

wer

(arb

.uni

ts)

]

(a)

(b)

FIG. 10. �Color online� The deviations of peak power from theexact Townes soliton solution when the input peak power deviatesfrom the ideal peak power by �a� −5% and �b� +5% when thedispersion is constant �solid lines�, exponentially decreasing�dashed lines�, and exponentially increasing �dot-dashed lines�. Thephysical parameters are the same as those used in Fig. 9.


033835-10

dispersion regimes of cubic-quintic nonlinear media. Bymeans of direct numerical simulations, we show that thechirped soliton in the anomalous dispersion regime is stable,whereas that in the normal dispersion regime is unstable. Forthese chirped solitons, the dispersion length, the cubic non-linear length, and the quintic nonlinear length are related. Ifthe quintic nonlinear length goes to infinity, the dispersionlength will equal the cubic nonlinear length. We observedthat by the use of exponentially decreasing dispersion, it ispossible to utilize these self-similar solitons to achievepedestal-free pulse compression in cubic-quintic nonlinearmedia.

We then studied the evolution of the chirped bright soli-tons when the initial peak power or the initial chirp is per-

turbed. We have found that the evolution of the soliton is notsignificantly affected even when the pertubation in the initialparameters is �20% of its ideal values. The solitons havebetter tolerance to perturbations in the initial chirp than ini-tial peak power. We have carried out extensive numericalsimulations by varying the decay rate of dispersion andchanging the strength of the quintic nonlinearity. From thenumerical simulations, we have observed that competitionbetween the cubic and quintic nonlinearities stabilizes thepulse against perturbations in initial pulse parameters.Hence, one can construct a stable pedestal-free optical pulsecompressor by using competing cubic-quintic nonlinearities.

Finally, we studied the chirped Townes soliton in purequintic nonlinear media. As the Townes solitons with con-stant dispersion are marginally stable, we studied the pertur-bation of the chirped Townes soliton in terms of peak power.We found that we can achieve fast pulse compression bycombining the wave collapse and exponentially decreasingdispersion when the perturbation is higher than the ideal one.The occurrence of collapse can also be postponed by usingexponentially increasing dispersive media. These two issues�fast and slow compression� could be implemented depend-ing on the physical situation and requirement. Applicationsin terms of BECs in the TG regime have also been discussed.

ACKNOWLEDGMENTS

The authors acknowledge the support of the ResearchGrant Council of the Hong Kong Special Administrative Re-gion, China �Project No. PolyU5289/07E�. K.N. appreciatesthe hospitality of the Department of Electronic and Informa-tion Engineering at The Hong Kong Polytechnic University.K.N. also wishes to thank the Royal Society for support inthe form of an International Joint Project Grant.

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0 2 4 6 8

2

4

6

8

z / LD0

peak

pow

er(a

rb.u

nits

)

FIG. 11. �Color online� Evolution of the peak powers of chirpedTownes solitons in the presence of cubic nonlinearity. The dotsrepresent the results of pure quintic nonlinearity. The solid line,dot-dashed lines, and dashed lines represent the results when theratio of cubic nonlinear length to quintic nonlinear length �Ln2

/Ln4�

is 12 000, 1200, and 120, respectively. The corresponding values ofcubic nonlinearity are 0=10−5, 10−4, and 10−3 W−1 m−1, respec-tively. The other physical parameters are same as those in Fig. 1.


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