1680 OPTICS LETTERS / Vol. 22, No. 22 / November 15, 1997

Bright spatial solitons in defocusing Kerr mediasupported by cascaded nonlinearities

Ole Bang and Yuri S. Kivshar

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

Alexander V. Buryak

School of Mathematics and Statistics, University College, Australian Defence Force Academy, Canberra, ACT 2600, Australia, and OpticalSciences Centre, Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT 0200, Australia

Received July 10, 1997

We show that resonant wave mixing that is due to quadratic nonlinearity can support stable bright spatialsolitons, even in the most counterintuitive case of a bulk medium with defocusing Kerr nonlinearity. Weanalyze the structure and stability of such self-guided beams and demonstrate that they can be generatedfrom a Gaussian input beam, provided that its power is above a certain threshold. 1997 Optical Society ofAmerica

It is well known that bright spatial solitons can existonly for focusing nonlinearity. In bulk Kerr [or x s3d]media their existence further requires a saturatingeffect to arrest catastrophic self-focusing. Defocusingnonlinearity leads usually to an enhanced beam broad-ening and does not support any localized structuresother than vortex and dark solitons, which require abackground beam.1 However, theoretical and experi-mental results indicate that multidimensional brightsolitons can exist in noncentrosymmetric materialswith quadratic [or x s2d] nonlinearity.2 In this Letterwe demonstrate an important feature of the para-metric self-trapping induced by resonant wave mixingin x s2d media. We show that even a weak quadraticnonlinearity can lead to self-focusing and stable brightsolitons in a bulk medium with defocusing Kerr non-linearity, provided that the fundamental and its sec-ond harmonic are nearly phase matched. This meansthat the parametric interaction in a x s2d medium can bestrong enough not only to suppress the beam broaden-ing that is due to diffraction but also to overcome thebroadening effect of the defocusing Kerr nonlinearity.

We consider beam propagation in noncentrosymmet-ric lossless bulk media with defocusing cubic nonlinear-ity, described by the dimensionless equations

i≠w≠z

1 ='2w 1 wpv 2 sjwj2 1 rjvj2dw ­ 0 ,

(1)

2i≠v≠z

1 ='2v 2 bv 1

12

w2 2 shjvj2 1 rjwj2dv ­ 0 ,

(2)

which are valid when spatial walk-off is negligibleand the fundamental frequency v1 and its second har-monic v2 ­ 2v1 are far from resonance. The slowlyvarying complex envelope function of the fundamen-tal w ­ wsr, zd and of the second harmonic v ­ vsr, zdare assumed to propagate with a constant polariza-tion, e1 and e2, along the z axis. The Laplacian ='

2

refers to the transverse coordinates r ­ sx, yd. Thephysical electric field is EsR, Z, T d ­ E0fw expsiu1de1 1

0146-9592/97/221680-03$10.00/0

2v expsi2u1de2g 1 c.c., where R ­ r0r, Z ­ z0z, andu1 ­ k1Z 2 v1T . The real normalization parame-ters are3 E0 ­ 4x

s2d1 yf3jx

s3d1s jg, z0 ­ 2k1r0

2, and r02 ­

3jxs3d1s jyh16m0v1

2f xs2d1 g2j, where m0 is the vacuum per-

meability and kp is the wave number at frequency vp.Furthermore, b ­ 2z0Dk, h ­ 16x

s3d2s yx

s3d1s , and r ­

8xs3d1c yx

s3d1s , where Dk ­ 2k1 2 k2 ,, k1 is the wave-

vector mismatch. The coeff icients xs j dp ­ x s j dsvpd de-

note the Fourier components at vp of the jth-ordersusceptibility tensor. Thus x

s2d1 ­ x

s2d2 represents the

quadratic nonlinearity, and xs3dps and x

s3d1c ­ x

s3d2c repre-

sent the parts of the cubic nonlinearity responsible forself- and cross-phase modulation, respectively. Thesystem of Eqs. (1) and (2) conserves the dimensionlesspower, P ­

Rsjwj2 1 4jvj2ddr, that corresponds to the

physical power P0P , where P0 ­ 0.5p

e0ym0 E02r0

2.Equations (1) and (2), but for focusing Kerr nonlin-

earity, were used in Ref. 4 for studies of collapse inarbitrary dimensions. After a simple transformationthese equations correspond to the s1 1 1d-dimensionalequations used in Ref. 5 and later derived rigorouslyin Ref. 3. Similar equations were recently shown toappear in the theory of self-focusing in quasi-phase-matched x s2d media.6

In this Letter we are interested in s2 1 1d-dimensional bright solitary waves, and thereforewe look for spatially localized solutions to Eqs. (1) and(2) of the formwsr, zd ­ w0srdexpsilzd, vsr, zd ­ v0srdexps2ilzd ,

(3)

where the real and radially symmetric functions w0srdand v0srd decay monotonically to zero as r ­

px2 1 y2

increases. The real propagation constant l must beabove cutoff, l . lc ­ maxs0, 2by4d, for w0 and v0to be exponentially localized. For a large class ofmaterials and experimental settings we can neglectthe dispersion of x s3d and set x

s3d1s ­ x

s3d2s , and it is

further reasonable to set xs3d1s ­ x

s3d1c . In this case we

1997 Optical Society of America

November 15, 1997 / Vol. 22, No. 22 / OPTICS LETTERS 1681

get h ­ 2r ­ 16, which we use below. These valueswere also used in earlier papers on competing x s2d andx s3d nonlinearities.3,5

The existence of localized solutions to Eqs. (1)and (2) is a nontrivial issue. When w0 ­ 0,Eqs. (1) and (2) reduce to the stationary nonlinearSchrodinger equation for the second harmonic,='

2v0 2 s b 1 4ldv0 2 hv03 ­ 0, which does not per-

mit spatially localized solutions for defocusing Kerrnonlinearity, h . 0. Similarly, when b .. 1 andb .. l, we obtain v0 ø w0

2y2b from Eq. (2). ThenEq. (1) gives the stationary nonlinear Schrodingerequation for the fundamental, ='

2w0 2 lw0 2 w03 ­ 0,

and localized solutions are not possible either.Therefore, if bright spatial solitons should existin defocusing Kerr media, both components shouldbe nonzero, w0 fi 0 and v0 fi 0 (combined or Csolutions5).

Using a standard relaxation scheme, we numericallyfound the families of localized C solutions [Eqs. (3)]to Eqs. (1) and (2) for the allowed values of l and b.In Fig. 1 we show examples of the profiles w0srd andv0srd for b ­ 0.1 and different values of l. When l issmall (i.e., low power), the profiles resemble the sech-shaped solitons that exist in self-focusing Kerr me-dia [Fig. 1(a)]. Increasing l also increases the beamamplitude [Fig. 1(b)]. However, for suff iciently largevalues of l (or of the power), the amplitudes satu-rate, and the beam broadens significantly [Fig. 1(c)]because of the defocusing effect of the cubic terms inEqs. (1) and (2). Above l ø 0.00656, no localized solu-tions exist. Similarly, near the cutoff lc ­ 0, the beamwidth increases rapidly and the amplitude decreases.Below the cutoff, no localized solution exist.

In Fig. 2 we show the amplitude of the C solutions,w0s0d and v0s0d, and their power P as function of l fortwo particular values of b. Clearly the solutions existin a limited region only, ranging from the cutoff lc toa certain upper limit at which the beam power tendsto infinity, even though the amplitude saturates, re-f lecting the defocusing effect of the Kerr nonlinearity.For b ­ 0.1 the power increases monotonically with l,whereas for b ­ 20.02 it decreases in a narrow inter-val above the cutoff. In both cases there is a powerthreshold below which no solutions exist.

The Vakhitov–Kolokolov stability criterion, ≠Ny≠l . 0, has been shown to apply to solitary waves sup-ported by pure x s2d nonlinearity.7 Its derivation forEqs. (1) and (2) is similar. According to this criterionthe solutions are stable in the whole region of existencefor b ­ 0.1, whereas for b ­ 20.02 they become un-stable in a narrow region above cutoff.

We made a series of calculations as shown inFig. 2(b), found ≠Ny≠l, and identified the regions ofexistence and stability of the C solutions in the sl, bdplane. The results are summarized in Fig. 3(a). Inregions I and II no localized solutions exist, in region Ibecause l is below cutoff and in region II because thedefocusing effect of the cubic nonlinearity becomesdominant. For b , 0 there is a narrow band [magni-fied f ive times in Fig. 3(a) to show it] where solutionsexist but are unstable. This result is to be expectedbecause the instability also exists for pure x s2d nonlin-

earity.7 Stable solitons exist in the hatched region.We have confirmed their stability to propagation bysimulation of Eqs. (1) and (2) for representative cases.In Fig. 3(b) we show the power of the stable solitons.As is also evident from Fig. 2(b), the stable solitonsexist for all powers above a certain threshold.

Two physical effects are apparent from Fig. 3 whenwe keep in mind that the effective mismatch b is pro-portional to both the wave-vector mismatch Dk and theratio of the x s3d and x s2d coeff icients, b ~ Dkx

s3d1s yf x

s2d1 g2.

When jbj increases, the existence region becomes nar-rower, ref lecting that the defocusing cubic nonlin-earity becomes progressively more dominant, eitherbecause jx

s3d1s j .. f x

s2d1 g2 or because the waves are not

phase matched (see also Ref. 5). Similarly, for f ixedb, solitons exist only when the maximum intensity, orequivalently, l is sufficiently small. When the inten-sity, or l, becomes too high, the defocusing Kerr nonlin-earity is again dominant, prohibiting the existence ofstable bright solitons. This phenomenon is due to the

Fig. 1. Profiles w0sr ­ xd (solid curves) and v0sr ­ xd(dotted curves) of solutions (3) for h ­ 2r ­ 16, b ­ 0.1,and (a) l ­ 0.003, (b) l ­ 0.006, (c) l ­ 0.00655.

Fig. 2. Characteristics of the C solutions [Eqs. (3)] forh ­ 2r ­ 16 and b ­ 20.02 (dotted curves) and b ­ 0.1(solid curves). (a) Amplitude versus l for the fundamentalw0s0d (upper curve) and the second harmonic v0s0d (lowercurve). (b) Dimensionless power versus l.

Fig. 3. (a) Region of existence of stable (hatched) andunstable (black) solitons in the sl, bd plane. (b) Powerregime (hatched) of stable solitons versus b. Parame-ters: h ­ 2r ­ 16.

1682 OPTICS LETTERS / Vol. 22, No. 22 / November 15, 1997

Fig. 4. Evolution of a Gaussian input beam at the funda-mental, wsr, 0d ­ 0.1 expf2sry30d2g, with vsr, 0d ­ 0 and (a)b ­ 0.29, (b) b ­ 0.1. Parameters: h ­ 2r ­ 16.

Fig. 5. Physical power threshold versus xs3d1s for h ­ 2r ­

16, l1 ­ 1.064 mm, and values of xs2d1 of 3 pmyV (solid

curves) and 5.6 pmyV (dashed curves). In each case theupper curve is for k ­ 21023 and the lower curve is fork ­ 1023.

competition between two kinds of nonlinearity and isqualitatively similar to, e.g., the effect of self-trappingin media with focusing cubic and defocusing quinticnonlinearity.8

We numerically analyzed the generation of brightsolitons from a Gaussian input beam with powerP ­ 14.1 at the fundamental, without seeding of thesecond harmonic, corresponding to the filled circlesin Fig. 3(b). The results, presented in Fig. 4, showtwo characteristic types of evolution. When the ef-fective mismatch is sufficiently large, b ­ 0.29, theinput power is below threshold, and the evolution isstrongly affected by the defocusing Kerr effect. There-fore the beam rapidly diffracts [Fig. 4(a)]. When themismatch is so small s b ­ 0.1d that the input poweris sufficiently above threshold (part of the power isalways lost to radiation), the parametric focusing isstronger, and we observe the formation of a localizedself-trapped beam [Fig. 4(b)].

To get a feeling for the powers, nonlinear coefficients,and degree of phase matching required for generationof these bright solitons, we consider a fundamentalwavelength of l1 ­ 1.064 mm, a phase mismatch ofk ­ 61023, and x

s2d1 ­ 3 or x

s2d1 ­ 5.6 pmyV. Here

k ; n12Dkyk1, where n1 is the refractive index at

the fundamental frequency. From the dimensionlessthreshold power shown in Fig. 3(b) and the parameterdefinitions given below Eqs. (1) and (2) we can thencalculate the physical threshold power as function ofx

s3d1s . The result is shown in Fig. 5. For example, for

KTP with xs2d1 ­ 5.6 pmyV and a phase mismatch of

k ­ 1023 the threshold is of the order of 10 kW fora rather wide range of x

s3d1s values and is thus of the

same order as the power in the solitons observed inKTP.9 When jx

s3d1s j increases or jx

s2d1 j decreases, the

required power increases. For a negative wave-vectormismatch, Dk , 0, the threshold power is alwayshigher and the x

s3d1s region in which stable solitons exist

is always narrower than for the corresponding positivemismatch.

Consider k ­ 1023 and the Gaussian initial conditionused in Fig. 4, which generated a soliton for b ­ 0.1.For a fixed effective mismatch of b ­ 0.1, the ratioof the nonlinearity coefficients is f ixed at x3yx2

2 ­24.2. Hence this input beam has a fixed FWHMwidth of 42 mm, a power of x2

22 3 42.5 kW, anda peak intensity of x2

22 3 4.2 GWycm2, where thenormalized nonlinearity coefficients are defined asx2 ­ x

s2d1 ys5.6 pmyVd and x3 ­ xs3d

1s ys103 pm2yV2d.In conclusion, we have analyzed the effect of

quadratic nonlinearity on the existence and stability ofbright spatial solitons in bulk media with defocusingKerr nonlinearity. We have shown that stable brightspatial solitons can exist because of parametricallyinduced self-focusing and that they can be generatedfrom Gaussian input beams with experimentallyreasonable powers. From a physical point of view thisphenomenon indicates an important feature of cas-caded nonlinearities, permitting self-focusing effectseven in a defocusing Kerr medium.

The authors acknowledge useful discussions with S.Trillo and W. Torruellas. Part of this research hasbeen supported by the Australian Department of In-dustry, Science and Tourism, through grant 74 un-der the International Science and Technology program.A. V. Burak acknowledges support of the AustralianResearch Council.

References

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