Bright and dark solitary waves in the presence of third-harmonic generation

1488 J. Opt. Soc. Am. B/Vol. 15, No. 5 /May 1998 Sammut et al.

Bright and dark solitary waves in the presenceof third-harmonic generation

Rowland A. Sammut and Alexander V. Buryak

School of Mathematics and Statistics, University College, Australian Defence Force Academy,Canberra ACT 2600 Australia; Optical Sciences Centre, Research School of Physical Sciences and Engineering

The Australian National University, ACT 0200 Canberra, Australia

Yuri S. Kivshar

Australian Photonics Cooperative Research Centre, Research School of Physical Sciences and Engineering,The Australian National University, ACT 0200 Canberra, Australia

Received June 11, 1997; revised manuscript received December 5, 1997

We analyze the effect of phase-matched third-harmonic generation on the existence and stability of(1 1 1)-dimensional bright and dark spatial solitary waves in optical media with a cubic (or Kerr) nonlinearresponse. We demonstrate that parametric coupling of the fundamental beam with the third harmonic leadsto the existence of two-color solitary waves resembling those in a x (2) medium and that it can modify drasti-cally the properties of solitary waves due to effective non-Kerr nonlinearities. In particular, we find a powerthreshold for the existence of two-frequency parametric bright solitons and also reveal the soliton multistabil-ity in a Kerr medium that becomes possible owing to a higher-order nonlinear phase shift caused by cascadedthird-order processes. We also analyze dark solitary waves and their stabilities. We show that, in a certainparameter domain, parametric x (3) dark solitons may become unstable owing to the modulational instability ofthe supporting background or to other instability mechanisms caused by the parametric coupling between theharmonics. © 1998 Optical Society of America [S0740-3224(98)02004-9]

OCIS codes: 190.3270, 190.5530, 190.4160.

1. INTRODUCTIONIf a monochromatic beam of frequency v is launched intoa nonlinear medium with a Kerr-type response, the third-order contribution to the polarization is made up of twocomponents (see, e.g., Ref. 1 and references therein). Thefirst is a response at frequency v, which is usually inter-preted as an intensity-dependent refractive index. It isthe balance between this nonlinearity-induced change ofthe refractive index and diffraction (or dispersion) thatprovides the well-known physical mechanism of the for-mation of optical spatial self-guided beams (or temporalsolitons). The second polarization component, which isgenerally neglected in considering soliton propagation ina Kerr medium, is a response at frequency 3v. This con-tribution to the nonlinear polarization is known to lead tothird-harmonic generation provided that appropriatephase-matching conditions are satisfied. Thus launchinga monochromatic beam can result in copropagating beamsat the fundamental and third-harmonic frequencies.

It has been understood for some time that such a pro-cess can lead to a cascading effect whereby effectivehigher-order nonlinearities are generated (see, e.g., Ref. 2and references therein). Cascading in media with x (2)

nonlinearity has been the subject of renewed interest overthe past five years3 as a result of the realization that cas-caded second-harmonic generation provides a mechanismfor achieving an effective intensity-dependent phasechange, which may be larger than that caused by the in-herent x (3) nonlinear susceptibility of the medium. This

0740-3224/98/051488-09$15.00 ©

induced cubic nonlinearity can support both bright anddark parametric solitary waves resulting from the two- orthree-wave-mixing processes (see, e.g., Refs. 4–8), whichhave been recently observed experimentally in a KTPbulk crystal and a LiNbO3 slab waveguide.9

Subject to appropriate phase-matching conditions, anenhancement of nonlinearity that is due to cascading canalso be expected for third-order processes in optical mate-rials with the x (3) susceptibility. Indeed, Saltiel et al.10

have shown recently that the phase shift arising from thecascaded third-order processes can be understood as be-ing generated by a higher-order, quintic nonlinearity witha value that can exceed the inherent fifth-order suscepti-bility of the nonlinear medium. We believe this observa-tion indicates the possibility of a simple physical mecha-nism leading to effective, resonantly induced non-Kerrnonlinearities in an optical material with a purely cubicresponse, which can be used for enhancing the value ofthe medium’s nonlinear response. On the other hand,the mechanism of induced nonlinearities also raises thequestion of how these higher-order nonlinearities affectthe propagation of bright and dark solitary waves of thefundamental frequency under the condition of phase-matching with the third harmonic. Indeed, if a slightlymismatched process of third-harmonic generation leads toan effective quintic nonlinearity, from the results of clas-sification of bright solitary waves of the cubic–quinticnonlinear Schrodinger (NLS) equation11 we expect thatsolitary waves cease to exist when the effective quintic

1998 Optical Society of America

Sammut et al. Vol. 15, No. 5 /May 1998/J. Opt. Soc. Am. B 1489

nonlinearity becomes strongly defocusing. This simpleobservation suggests that solitary waves should be modi-fied near the point of phase matching with the third har-monic, and this explains the motivation of our study.

In this paper we analyze bright and dark spatial soli-tary waves under the phase-matched condition when thefundamental wave is coupled to its third harmonic. Thisis a particular degenerate case of the solitary waves sup-ported by four-wave-mixing processes,12 which is not com-pletely understood yet. We assume that the interactionbetween the fundamental and third-harmonic waves in-cludes parametric four-wave mixing, self-phase modula-tion effects, and cross-phase modulation. We analyzebright and dark solitary waves of this model and showthat the resonant coupling with the third harmonic leadsto several important physical effects, including a powerthreshold for the existence of bright solitons, multistabil-ity of bright solitary waves, and parametric modulationalinstability of dark solitary waves.

2. FUNDAMENTAL EQUATIONSWe are interested in the resonant interaction between alinearly polarized beam of frequency v and its third har-monic, which is assumed to be identically polarized. Wealso assume that the beams propagate in a slab wave-guide, so that we need be concerned only with one trans-verse dimension and analyze (1 1 1)-dimensional beampropagation. Hence we consider the scalar wave equa-tion

]2E

]x2 1]2E

]z2 2n2

c2

]2E

]t2 54p

c2

]2PNL

]t2 , (1)

where E is the electric field, n is the linear refractive in-dex, and PNL is the (third-order) nonlinear polarization.We write the electric field in the form

E 512 $E1 exp@i~k1z 2 vt !#

1 E3 exp@i~k3z 2 3vt !#% 1 c.c., (2)

where kj 5 jvnj /c and nj 5 n( jv) for j 5 1, 3. Eachfrequency component of the field then satisfies Eq. (1),where PNL(v) and PNL(3v) are the appropriate terms inthe expansion of the nonlinear polarization PNL5x (3)E3,namely,

PNL~v! 5x~3 !

8@3uE1u2E1 1 6uE3u2E1

1 3E1* 2E3 exp~2idkz !#exp@i~k1z 2 vt !#,

PNL~3v! 5x~3 !

8@3uE3u2E3 1 6uE1u2E3

1 E13 exp~idkz !#exp@i~k3z 2 3vt !#, (3)

where x (3) is a diagonal element of the third-order suscep-tibility tensor and dk 5 3k1 2 k3 .

Substituting Eqs. (2) and (3) into Eq. (1) and assumingthat the envelope functions E1 and E3 are slowly varying,we then find a pair of coupled nonlinear equations

2ik1

]E1

]z1

]2E1

]x2 6 x@~ uE1u2 1 2uE3u2!E1

1 E1* 2E3 exp~2idkz !] 5 0, (4)

2ik3

]E3

]z1

]2E3

]x2 6 9x@~ uE3u2 1 2uE1u2!E3

113

E13 exp~idkz !] 5 0, (5)

where the star denotes complex conjugation and the non-linearity parameter x 5 (3pv2/c2)ux (3)u is defined to bealways positive, whereas the sign of the Kerr nonlinearitydepends on whether the material is self-focusing (posi-tive) or self-defocusing (negative). These equations de-scribe a special case of a more general four-wave-mixingprocess (see, e.g., Ref. 12 for an example of the solitarywave dynamics).

To normalize Eqs. (4) and (5), we introduce a beamwidth x0 and diffraction length zd 5 2x0

2k1 and thereforedefine the dimensionless variables z 5 zdZ and x5 x0 X. This allows us to reduce the system of Eqs. (4)and (5) to the dimensionless equations

i]U

]Z1

]2U

]X2 6 F S 1

9uUu2 1 2uWu2DU 1

1

3U* 2WG

5 0, (6)

is]W

]Z1

]2W

]X2 2 DsW 6 F ~9uWu2 1 2uUu2!W 11

9U3G

5 0, (7)

where

U 5 3~k1x02x!1/2E1 ,

W 5 ~k1x02x!1/2 exp~2idkz !E3 .

After this rescaling, the resonant wave interaction is de-termined by two dimensionless parameters, D5 2k1dkx0

2 and s 5 k3 /k1 . 3. Since we consider thecase of spatial beam propagation (spatial solitary waves)with two harmonics of the identical polarization, no walk-off is taken into account.

A similar system of coupled equations for the plane-wave (PW) approximation (no spatial derivatives in X)has been considered by Saltiel et al.,10 in which the self-phase modulation of the third-harmonic has been ne-glected, but, instead, an inherent fifth-order susceptibilityhas been included. Our analysis indicates that self-phase modulation terms in both Eqs. (6) and (7) lead tosome important consequences for the existence and stabil-ity of solitary waves near the point of phase matching,and therefore both self-phase modulation terms should beincluded. The importance of the fifth-order nonlinearitycan be justified only far from resonance, where the cas-caded nonlinearity becomes negligible. Thus the inher-ent fifth-order nonlinearity can be omitted near the pointof phase matching as a higher-order effect.


3. BRIGHT SOLITARY WAVESA. Families of Stationary SolutionsWe are interested in finding stationary solutions of Eqs.(6) and (7), so we substitute

U 5 uAb exp~ibZ !, W 5 wAb exp~i3bZ !,

where b is the nonlinearity-induced propagation constantshift. It can be treated as an internal soliton parameterand also can be controlled by the beam total power.

Then the system of Eqs. (6) and (7) for solitary wavescan be rewritten in the following dimensionless form:

i]u

]z1

]2u

]x2 2 u 1 S 1

9uuu2 1 2uwu2Du 1

1

3u* 2w 5 0,

(8)

is]w

]z1

]2w

]x2 2 aw 1 ~9uwu2 1 2uuu2!w 11

9u3 5 0,

(9)

where z 5 bZ and x 5 b1/2X. Stationary beams are de-scribed by the real solutions u(x) and w(x), which are de-fined by the same system of Eqs. (8) and (9) but with thez derivatives omitted. These localized solutions dependon only a single dimensionless parameter, a 5 s(31 D/b).

We point out that the basic structure of Eqs. (8) and (9)is qualitatively similar to the familiar equations derivedin the case of parametric solitary waves supported by two-wave mixing in x (2) media.5,6 The definition of the effec-tive mismatch parameter a is almost identical to thatcase, in spite of the different structure and physicalmeaning of nonlinear coupling terms. Moreover we canemploy similar techniques for analyzing the two-frequency solitary waves that are due to cascading in cu-bic materials.

First of all, using such a direct analogy with the theoryof x (2) solitons, we investigate the solitary waves in theso-called cascading limit when uau @ 1. In this limit, theenergy conversion from the fundamental to the third har-monic is relatively small, i.e., uwu ! uuu. For uau @ 1,from Eq. (9) we find approximately w . u3/9a, and Eq.(8) becomes the cubic–quintic NLS equation allowing thesolutions for solitary waves in an explicit form (see, e.g.,Ref. 11). This suggests the structure of an asymptotic ex-pansion for the localized solutions of Eqs. (8) and (9) inpowers of e [ a21,

u~x ! 53A2

cosh x F1 2 2eS 2 21

cosh2 x D2 2e2S 70 2

29

cosh2 x2

12

cosh4 x D G 1 O~e3!,

(10)

w~x ! 56A2e

cosh3 x F1 2 3eS 1 210

cosh3 x D G 1 O~e3!.

(11)

This asymptotic solution of the cascaded limit is usedas a starting point in the search for families of localizedsolutions. In this section we consider positive values of a

only, because bright soliton solutions, which decay as-ymptotically to zero for large x, are not possible for a, 0. These spatially localized solutions have been found

with the help of a numerical relaxation technique (similarto that used elsewhere; see, e.g., Refs. 7 and 8), and theydescribe two spatially localized envelopes u(x) and w(x).To characterize these solutions, we use the normalized to-tal power

Ptot 5 E2`

`

~ uuu2 1 3suwu2!dx, (12)

which is one of the conserved quantities of the system ofEqs. (8) and (9). In all the results presented below weput s 5 3, which corresponds to the case of spatial soli-tons.

In Fig. 1 we show the variation of the normalized totalpower, Ptot , with the normalized mismatch parameter afor different types of two-wave localized solutions of thesystem of Eqs. (8) and (9). The inset figure shows an ex-panded portion of the dependence Ptot(a) for the range8.2 < a < 9.2. It can be seen from the form of Ptot(a)that near the point of phase matching between the funda-mental and third harmonics (i.e., a 5 9 at s 5 3) thereexist three distinct types of localized solutions for brightsolitary waves, which we discuss in detail below.

The most important class of two-wave bright solitons isdescribed by a family of localized solutions for coupledfundamental and third-harmonic fields. The distributionof power between the two frequencies varies from beingpredominantly in the third harmonic, for smaller a, topredominantly in the fundamental, at larger values of a.In this latter case, we can apply the cascading approxima-tion to find

Ptot~e! 5 36 2 192e 227072

5e2 1 O~e3!, e [ a21,

(13)

which is shown as the dashed curve in Fig. 1.

Fig. 1. Variation of the normalized total power, P tot , versus thedimensionless mismatch parameter a for the three distinct fami-lies of solitary-wave solutions of Eqs. (8) and (9). The dashedcurve corresponds to the asymptotic expansion found analyticallyin the cascading limit. Lower curves merge at the bifurcationpoint O (a 5 9). Points A–D indicate the particular examplespresented in Figs. 2(a)–2(d). Filled circle corresponds to the ex-act solution (14).


For larger values of the normalized mismatch param-eter a the amplitude of the beam at the fundamental fre-quency grows whereas that of the third harmonic van-ishes, in agreement with the prediction of the analyticalresults obtained in the cascading approximation. An ex-ample of the solution of this family is presented in Fig.2(a), corresponding to point A in Fig. 1.

A simple analysis shows that the family of two-frequency solitary waves bifurcates from the one-frequency solution for the third harmonic,

w~x ! 5A2a

3sech~Aax !, u~x ! 5 0,

Fig. 2. Examples of the fundamental (thin solid curve) andthird-harmonic (thick solid curve) profiles for several one-humpsolitary wave solutions, which belong to different families. Pro-files (a)–(d) correspond to points A–D in Fig. 1.

Fig. 3. Examples of the fundamental (thin solid curve) andthird-harmonic (thick solid curve) profiles for several multihump(higher-order) solitary wave solutions belonging to the familythat also includes the solution (14) given by filled circle in Fig. 1.

at the point of exact phase-matching (i.e., a 5 9 at s5 3 and D 5 0). This family of one-frequency solitarywaves is characterized by the normalized power, Ptot5 4(a)1/2, and it is described by the standard cubic NLSequation, which follows from Eq. (9) at u 5 0. It is clearthat this type of solitary wave is possible only because ofthe self-phase-modulation effect that is taken into ac-count for the third harmonic.

Finally, the third family of localized solutions shown inFig. 1 includes the simplest hyperbolic-secant-type ana-lytical solution.13 This solution (shown by the filledcircle in Fig. 1) exists only at a 5 1, and it has the follow-ing form:

us~x ! 5 a sech x, ws~x ! 5 bus~x !, (14)

where the parameter b is the real root of the cubic equa-tion

63b3 2 3b2 1 17b 1 1 5 0, (15)

and the parameter a is found from the following relation,

a2 518

~18b2 1 3b 1 1 !. (16)

In sharp contrast with the theory of x (2) parametricsolitons,5 the analytical solution of the models (8) and (9)and the asymptotic solution of the cascading limit a @ 1do not belong to the same family. Moreover, varying con-tinuously the effective mismatch parameter a along thisfamily of localized solutions shows that this class of soli-tary waves corresponds to multihump solitary waves, asis shown in Fig. 3. The point a 5 1 is special because itseparates two subfamilies of solitons with different num-bers of humps in the third harmonic. Thus, even thoughit is a one-hump solution itself, the hyperbolic-secant-typesoliton at a 5 1 belongs to a higher-order soliton family.It is not surprising that all solutions of this family are un-stable, a conclusion that we have verified by direct nu-merical simulations.

B. Multistability of Solitary WavesAs has been mentioned above, the most interesting fea-tures of two-wave parametrically coupled solitary wavesmanifest themselves near the point of phase matching.In the case of negative phase mismatch, corresponding tothe part of the curve Ptot(a) on the left of the bifurcationpoint O shown in Fig. 1, for any fixed value of the param-eter a we reveal the simultaneous existence of three local-ized solutions. Therefore in this case the propagationcharacteristics of two-frequency coupled self-guidedbeams become multivalued. Characteristic profiles ofthe solutions in this region are shown in Figs. 2(b), 2(c),and 2(d), corresponding to the points B, C, and D in Fig. 1,respectively.

We have investigated the stability of these solitarywaves, using both linear stability analysis and numericalsimulations of the beam propagation, and verified that, aswas expected, the pure third-harmonic branch and thelower two-frequency branch are stable, whereas the inter-mediate branch shown in the inset of Fig. 1 is unstable.Stability changes at the critical point d Ptot /da 5 `.


Therefore, in the problem of third-harmonic generationwe find that there exists more than one possible propaga-tion constant (and consequently more than one possibleshape) of the parametric spatial soliton for the samevalue of the total power Ptot defined by Eq. (12). Thisphenomenon is known as soliton bistability or multista-bility. Importantly, bistable solitons were firstpredicted14 for scalar wave propagation described by thegeneralized NLS equation, when the dependence of thenonlinear susceptibility on the light intensity changes itssign or its derivative has a sufficiently sharp peak (a step-like function) (see also Ref. 15). This means that multi-stable solitons were found not to be possible for a Kerrmedium. However, later it was noticed that parametricwave interaction in quadratic media can lead to multista-bility of solitary waves.8 Similarly, our results for third-harmonic generation indicate the importance of a novelphysical mechanism, which can lead to effective non-Kerrnonlinearities and multistable solitary waves. We be-lieve this observation may broaden the range of possibleoptical materials for experimental realization of all-optical soliton switching based on multistability of lightself-trapping.

As a particular example of the dynamics of unstablesolitons and solitary wave switching, in Figs. 4(a) and 4(b)we show how a two-frequency beam launched on the un-stable branch at a . 8.4 can be switched towards eitherthe left [Fig. 4(a)] or right [Fig. 4(b)] stable branch, de-pending on the sign of the applied perturbation. The fig-ure presents the evolution of the intensity at the funda-mental frequency, with Fig. 4(a) showing that theintensity in that beam initially falls as we switch towardthe left (purely third-harmonic) branch while Fig. 4(b)shows the reverse situation, in which we switch towardthe right branch, which contains a larger proportion ofthe fundamental frequency.

4. DARK SOLITARY WAVESA. Structure of Stationary SolutionsWe now turn our attention to searching for stationary so-lutions in self-defocusing media. We follow a very simi-

Fig. 4. Switching dynamics in the bistable region. The figureshows what happens when a beam is launched on the unstablebranch at a . 8.4 (point C of Fig. 1) when its amplitude is ini-tially (a) decreased or (b) increased.

lar procedure to that used for bright solitons but, sincethe effect of the nonlinearity is now to reduce the refrac-tive index, we substitute

U 5 uAb exp~2ibZ !, W 5 wAb exp~2i3bZ !.

This leads to the following system of normalized equa-tions for u and w:

i]u

]z1

]2u

]x2 1 u 2 S 1

9uuu2 1 2uwu2Du 2

1

3u* 2w 5 0,

(17)

is]w

]z1

]2w

]x2 2 aw 2 ~9uwu2 1 2uuu2!w 21

9u3 5 0.

(18)

Stationary beams are real solutions u(x) and w(x) of thesystem of Eqs. (17) and (18) with the z derivatives omit-ted, and again they depend on a single dimensionless pa-rameter, a 5 s(23 1 D/b), which differs only slightlyfrom the bright case.

It is important to notice that even though the resultingdimensionless Eqs. (17) and (18) look very similar to thecase of bright solitons, their meaning is different. In-deed, this time the parameter b is defined by the value ofthe background intensity at the fundamental frequency,which should be determined in a self-consistent way fromthe solutions of the corresponding equation for the PWpropagation. Such an analysis is presented in Section4.B below. Therefore the parameter b stands for thenonlinearly induced propagation constant of the PW back-ground field at the fundamental frequency, and it shouldbe determined from Eq. (24) together with the back-ground of the third-harmonic field. In particular, both band the amplitude of the third harmonic can be expressedfrom Eq. (24) as functions of the background amplitude ofthe fundamental beam.

After defining the system of dimensionless Eqs. (17)and (18), we can look for solitary waves in the cascadinglimit when uau @ 1 and uwu ! uuu. The asymptotic ex-pansion for the localized solutions of Eqs. (17) and (18) inpowers of e [ a21 is now

u~j! 5 3 tanh j 1 eS 9

2tanh j 1 3

tanh j

cosh2 j2

9j

2 cosh2 jD

1 e2S 2675

8tanh j 2

273 tanh j

2 cosh2 j1

18 tanh j

cosh4 j

11395j

8 cosh2 j2

27j

2 cosh4 j2

27j2 tanh j

4 cosh2 jD

1 O~e3!, (19)

w~j! 5 23e tanh3 j 1 e2S 81

2tanh j 2

189 tanh j

2 cosh2 j

145 tanh j

cosh4 j1

27j tanh2 j

2 cosh2 jD 1 O~e3!, (20)

where j 5 x/(2)1/2. The asymptotic solutions (19) and(20) of the cascaded limit have again been used as a start-ing point in the numerical search for localized solutions


over a broad range of both positive and negative values ofa. We characterize these solutions using the complemen-tary, normalized total power

Ptot 5 E2`

`

@~ uu0u2 1 3suw0u2! 2 ~ uuu2 1 3suwu2!#dx,

(21)

where u0 and w0 are the background amplitudes of thetwo parametrically coupled PW beams. The complemen-tary power (21) has more limited applicability than itsanalog Eq. (12) for bright solitons. In particular, it doesnot define the solution stability, which should be formu-lated in terms of the so-called renormalized soliton mo-mentum (see, e.g., Refs. 16 and 17).

In Fig. 5 we show the dependence of the normalizedcomplementary power (21) on the effective mismatch pa-rameter a for classification purposes only, presenting dif-ferent types of localized solutions with nonvanishingbackgrounds. Thick solid and dashed curves show thefamilies of one- and two-frequency dark solitary waves,respectively. The dashed curve extending to large nega-tive values of a is defined explicitly because it corre-sponds, similar to the case of bright solitons, to the one-frequency NLS solution for the third harmonic (a , 0),

w~x ! 5Auau

3tanhSAuau

2x D , u~x ! 5 0,

which is characterized by the complementary power, Ptot5 2(2uau)1/2. The two thin dashed–dotted curves in Fig.5 represent the asymptotic results for the soliton comple-mentary power calculated with the help of expansions(19) and (20).

Ptot 5 A2S 18 1 69e 223121

20e2D 1 O~e3!. (22)

The solid curve extending to large positive values of arepresents two-frequency solutions, which have the

Fig. 5. Variation of the normalized complementary power, P tot ,versus the dimensionless mismatch parameter a for solitary-wave solutions of Eqs. (17) and (18). Thick solid and dashedcurves show the families of one- and two-frequency dark solitarywaves, respectively. Two thin dashed–dotted curves correspondto the asymptotic expansions of the cascading limit. Note thatfor pointlike solutions (given by open circles) we scale P tot(a) bythe factor 1/2. Points A–D indicate the particular examples pre-sented in Figs. 7(a)–7(d). Filled circle at }521 corresponds tothe exact solution (23).

asymptotic form given by Eqs. (19) and (20). This branchcontains the simple analytical solution (shown as a filledcircle in Fig. 5), which exists only at a 5 21, namely,

us~x ! 5a

A2tanhS x

A2D , ws~x ! 5 bus~x !, (23)

where a and b are defined by the same expressions (15)and (16) as for the bright soliton given in Eq. (14).

Importantly, Fig. 5 does not display any continuousfamily of solitary-wave solutions merging with theasymptotic result of the cascaded limit for negative valuesof a. Instead, only localized solutions at some specialvalues of a are found. (A small number of these solutionsare shown as isolated open circles in Fig. 5. Note that forthese pointlike solutions we scale Ptot(a) by the factor1/2.) Such a phenomenon was first discovered for darksolitons of two-wave interaction in a x (2) medium,5 and ithas been described by employing the concept of boundstates of radiative solitary waves.18 In our case, thismeans that for negative values of a the dark solitons ap-pear to possess small-amplitude nonvanishing oscillatorytails, and the spatially localized solutions become possibleonly as a superposition of two (or more) solitons when, atcertain values of a, the outgoing radiation is cancelled bymeans of destructive wave interference. Some typicalfield profiles in the vicinity of the phase-matching point(a 5 29) are shown in Figs. 6 and 7. Note that thepointlike soliton solutions (including ones in Fig. 7) werefound by numerical shooting technique. For all other darksolitons the relaxation method was used.

B. Modulational Instability of the Plane-WaveBackgroundAs has been mentioned above, real PW solutions u0 andw0 , which define the background of dark solitons, can befound by solving the system of coupled algebraic equa-tions,

Fig. 6. Examples of the fundamental (thin solid curve) andthird-harmonic (thick solid curve) profiles for several darksolitary-wave solutions belonging to the family that also includesthe solution (23) given by filled circle in Fig. 5.


1 219

u02 2 2w0

2 213

u0w0 5 0,

aw0 1 ~9w02 1 2u0

2!w0 119

u03 5 0. (24)

The corresponding solutions are found analytically, andthey can be presented in two equivalent forms:

u0 5w0~a 1 15 2 21w0

2!

~7w02 2 1 !

,

where w02 is a root of the cubic equation

882w06 2 ~21a 1 945!w0

4 1 ~a2 1 27a 1 324!w02 2 9

5 0,

or

Fig. 7. Examples of some resonant dark-soliton structures fornegative values of a: two-soliton states (a) a 5 224.3172, (b)a 5 29.8142, (c) a 5 24.9254, and three-soliton state (d) a5 29.5423. Profiles (a)–(d) correspond to points A–D in Fig. 5.

Fig. 8. Variation of normalized intensity of the three branchesof PW solutions as functions of a. The normalized complemen-tary power of dark solitons in the cascading limit is also shownas the dotted curves.

w0 5u0

9 F98u04 1 ~49a 2 819!u0

2 2 ~513a 1 1944!

8a2 1 126a 1 405G ,

where u02 is a root of the cubic equation

98u06 1 ~105a 2 567!u0

4 1 ~36a2 2 729a 2 3645!u02

2 ~324a2 1 2916a 1 6561! 5 0.

As we would expect, there is a unique, real solution forlarge positive and negative values of a. However, in theinterval 214.509 , a , 24.334 containing the point ofthe exact phase matching, there are three distinct PW so-lutions. The normalized intensities of these solutions(u0

2 1 3sw02) are shown as functions of a in Fig. 8 to-

gether with the normalized complementary power of darksolitons in the cascading limit given by Eq. (22).

To analyze modulational instability of these PW solu-tions, we introduce perturbations of the form

u 5 u0 1 ~ur 1 iui!cos~kx !exp~Vz !, (25)

w 5 w0 1 ~wr 1 iwi!cos~kx !exp~Vz !. (26)

Substituting these expansions into Eq. (24) and lineariz-ing the resulting equations, we find that the vectorV 5 (ur , ui , wr , wi)

T satisfies the linear equation AV5 0 where the elements Aij of the matrix A are given by

A11 5 2A22 5 V, A33 5 2A44 5 sV,

A12 5 1 2 k2 2 u02/9 1 2u0w0/3 2 2w0

2,

A13 5 A24 5 A31 5 A42 5 0,

A14 5 A32 5 2u02/3,

A21 5 1 2 k2 2 u02/3 2 2u0w0/3 2 2w0

2,

A23 5 A41 5 2~4u0w0 1 u02/3!,

A34 5 2~a 1 k2 1 2u02 1 9w0

2!,

A43 5 2~a 1 k2 1 2u02 1 27w0

2!.

Therefore the parameters V and k are linked by the dis-persion relation det@A# 5 0, which is reduced to

V4 1 2BV2 1 D 5 0, (27)

where

B 512

A12 A21 11s

A23 A32 11

2s2A34 A43 , (28)

D 51

s2~A14

2 2 A12 A34!~A232 2 A21 A43!. (29)

Now modulational instability occurs if for any real k themodulation parameter V is either real or has a positivereal part.

We find that on the upper branch of PW solutions, thiscondition is never satisfied, so those solutions do not ex-hibit modulational instability. However, with the excep-tion of two isolated points, the other two branches aremodulationally unstable for all a. Figure 9 shows therange of wave numbers k for which modulational instabil-ity occurs. The diagonally striped region applies to thelower branch of PW solutions, and the horizontallystriped region applies to the intermediate branch. Fig-


Fig. 9. Bands of modulation frequencies, which induce instabil-ity of PW background waves. The diagonally striped region ap-plies to the lower (dashed curve) branch of PW solutions in Fig. 8while the horizontally striped region applies to the intermediate(solid curve) branch.

Fig. 10. Maximum growth rate of modulational instability foreach value of a. The dashed and solid portions of the curves forVmax relate to the corresponding branches of PW solutions in Fig.8.

Fig. 11. Numerical demonstration of the modulational instabil-ity of the approximate two-frequency dark soliton solution at a5 280.

ure 10 shows the maximum growth rate of the instabilityVmax for each value of a. Although this growth rate be-comes very small for large negative a (the cascadinglimit), modulational instability is still very easily ob-served, even in the range in which we might expect theinfluence of the third harmonic to be insignificant. As aresult, all possible two-frequency solutions for dark soli-tary waves found in the region of negative values of thephase-matching parameter a are unstable owing to modu-lational instability of the background.

This situation resembles the case of parametric spatialsolitons in x (2) media known to be unstable because ofmodulational instability of the background field.19 Anexample of the development of such an instability fromnoise is presented in Fig. 11 at a 5 280. Remarkably,the instability is important even in the region of large awhere, in principle, the fundamental and third harmonicscan be considered to be completely decoupled. This re-sult shows the crucial importance of the parametric inter-action between harmonics for the solitary-wave stability.

C. Instability of Dark Solitons Induced by ParametricInteractionThe results of the previous subsection show that weshould not expect to find stable two-frequency dark soli-tons for a , 214.509 (because of the modulationally un-stable background). To examine the inherent stability ofthe solitons for a . 214.509 [Fig. 5] when the back-ground is stable, we should perform a stability analysisbased on the soliton renormalized momentum.16 Thisanalysis requires us to find moving (gray) solitons inwhich the x dependence of our previous solutions is re-placed by a dependence of the form (x 2 Vz), where V isthe transverse velocity of the gray soliton,17 and the soli-ton envelopes u and w should be considered in a complexplane. Because we expect the black solitons (at V 5 0) tobe mostly unstable, similarly to the case of saturablenonlinearities,16 we need only the solutions for small V,but even these are difficult to find for arbitrary a. How-ever, for large values of a, we can develop a generalizationof the cascaded limit and find an expansion of the form

u~j; V ! 5 u~j; 0! 1 i3V

A2@1 1 e~2 2 tanh2 j!#

1 O~e2, V2!, (30)

w~j; V ! 5 w~j; 0! 2 i9V

A2e tanh2 j 1 O~e2, V2!,

(31)

where j 5 x/(2)1/2, and the functions u(j, 0) and w(j, 0)at V 5 0 are defined by the asymptotic expressions (19)and (20) for small values of e 5 a21.

Using these expressions we have calculated the renor-malized soliton momentum Mr(V) for small values of V(see Ref. 16 for definition and some examples). The de-rivative dMr /dV was found to be positive and, accordingto the stability criterion for dark solitons,16 this indicatesthat the dark solitons are stable for at least large positivevalues of a.

For moderate and small values of a, we must rely onnumerical simulations. We have performed a series of


direct numerical simulations of the stability of dark soli-tons for different values of a. These indicate that, forpositive and negative values down to a . 25 (see Fig. 5),the solutions for dark solitons propagate stably. How-ever, as we approach closer to the phase-matching point(at a 5 29; see Fig. 5), we observe the development ofthe dark soliton instability similar to that earlier reportedfor saturable nonlinearities.16 Indeed, as can be seenfrom the contour plot presented in Fig. 12, an unstabledark soliton undergoes a drift instability and transformsinto a gray soliton.

5. CONCLUSIONSWe have shown that phase-matched interaction betweenthe fundamental beam and its third harmonic leads to ef-fective higher-order (non-Kerr) nonlinearities, and there-fore it can have an important effect on the propagation ofspatial solitary waves. First, for bright solitary waves,such induced non-Kerr nonlinearities restrict the exis-tence region of allowed values of the soliton power, andalso they lead to multistable soliton propagation whenmore than one possible beam profile and propagation con-stant exist for a fixed value of power near the phasematching. For dark solitary waves, parametric couplingbetween the fundamental and third harmonics generatestwo types of instability parametric modulational instabil-ity of the PW background, which is found even fartherfrom the point of the phase matching, and inherent insta-bility of black solitons similar to that predicted earlier forscalar dark solitons of saturable nonlinearities. We be-lieve that all the effects described here can be found inother situations when resonant, phase-matched wave in-teraction generates a parametric coupling between thefundamental wave and one of the harmonic waves.

ACKNOWLEDGMENTSA. V. Buryak acknowledges financial support from theAustralian Research Council. A. V. Buryak and Yu. S.

Fig. 12. Numerical demonstration of the drift instability of thedark soliton at a 5 29 [presented in Fig. 6(b)]. In this case, thebackground is modulationally stable but the soliton itself oscil-lates and then moves off-axis and transforms into a gray soliton.

Kivshar thank Falk Lederer for his warm hospitality inJena and useful discussions. Yu. S. Kivshar is alsograteful to Solomon Saltiel for useful suggestions and ref-erences. The authors are indebted to Alan Boardman forproviding a copy of Ref. 10 prior to its publication.

REFERENCES AND NOTES1. D. L. Mills, Nonlinear Optics: Basic Concepts (Springer-

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2. I. V. Tomov and M. C. Richardson, IEEE J. Quantum Elec-tron. 12, 521 (1976).

3. See, for example, the recent review paper on cascading inx (2) materials, G. I. Stegeman, D. J. Hagan, and L. Torner,Opt. Quantum Electron. 28, 1691 (1996).

4. Yu. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP41, 414 (1976); A. A. Kanashov and A. M. Rubenchik,Physica D 4, 122 (1981); R. Schiek, J. Opt. Soc. Am. B 10,1848 (1993); A. V. Buryak and Yu. S. Kivshar, Opt. Lett. 19,1612 (1994); L. Torner, C. R. Menyuk, and G. I. Stegeman,Opt. Lett. 19, 1615 (1994).

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13. This exact solution has been presented previously in the pa-pers by K. Hayata and M. Koshiba, Opt. Lett. 19, 1717(1994), and Y. Chen, Phys. Rev. A 50, 5145 (1994). How-ever, unlike the theory of solitary waves in quadratic me-dia, in the case of third-harmonic generation the exacthyperbolic-secant-like solution does not give any useful in-formation about the families of proper (stable) two-wavesolitary waves.

14. A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985); IEEE J.Quantum Electron. 21, 1538 (1985).

15. A. W. Snyder, D. J. Mitchell, and A. V. Buryak, J. Opt. Soc.Am. B 13, 1146 (1996).

16. Yu. S. Kivshar and W. Krolikowski, Opt. Lett. 20, 1527(1995); Yu. S. Kivshar and V. V. Afanasjev, Opt. Lett. 21,1135 (1996); D. E. Pelinovsky, Yu. S. Kivshar, and V. V.Afanasjev, Phys. Rev. E 54, 2015 (1996).

17. See the review paper by Yu. S. Kivshar and B. Luther-Davies, ‘‘Dark optical solitons: physics and applications,’’Phys. Rep. (to be published).

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Bright and dark solitary waves in the presence of third-harmonic generation