May 1, 2000 / Vol. 25, No. 9 / OPTICS LETTERS 643

Vector solitons in (2 1 1) dimensions

Johan N. Malmberg, Andreas H. Carlsson, Dan Anderson, and Mietek Lisak

Department of Electrodynamics, Chalmers University of Technology, S-412 96 Goteborg, Sweden

Elena A. Ostrovskaya and Yuri S. Kivshar

Optical Sciences Centre, The Australian National University, Canberra, ACT 0200, Australia

Received November 29, 1999

We address the problem of the existence and stability of vector spatial solitons formed by two incoherentlyinteracting optical beams in bulk Kerr and saturable media. We identify families of �2 1 1�-dimensional two-mode self-trapped beams, with and without a topological charge, and describe their properties analytically andnumerically. 2000 Optical Society of America

OCIS codes: 190.5530, 190.3270.

Recent experiments on multidimensional spatial opti-cal solitons1 have called for a detailed analysis of theself-trapping of light in higher dimensions. When two(or more) fields interact nonlinearly, they can formmulticomponent trapped states, known as vector soli-tons. Vector solitons, extensively studied theoreticallyin �1 1 1�-dimensional ��1 1 1�-D� models since the pi-oneering research,2 were experimentally observed inplanar waveguides and birefringent fibers.3 A fabri-cated waveguiding structure localizes such solitons inone of the two dimensions transverse to the directionof propagation; hence these solitons are effectively onedimensional. It is only recently that the theory andexperiments on truly two-dimensional self-trapping ofbeams in a bulk (saturable) medium merged, indicat-ing progress toward the study of different types of�2 1 1�-D vector soliton and their interactions.1

The possibility of performing such a study dependsgreatly on the soliton’s stability in the medium with re-alistic Kerr or saturable nonlinearity. It is known thatscalar (one-component), fundamental �2 1 1�-D solitonsare stable in saturable media, but they exhibit criti-cal collapse in Kerr-type media.4 However, both theexistence and the stability of multidimensional vectorsolitons are nontrivial issues, which to our knowledgewere not systematically addressed previously.

In this Letter we investigate �2 1 1�-D vector solitonsin Kerr and saturable media. We analyze two classesof such solitons. First, we considered bell-shaped soli-tons formed by the coupling of two fundamental modes.Second, we analyze the coupling between the funda-mental mode of one field and the first-order mode (i.e.,that which carries a topological charge) of the otherfield. Such vector solitons were recently introduced inRef. 5. They may possess a ring structure and are ex-pected to be analogous to the two-hump �1 1 1�-D vec-tor solitons that have proved to be stable in a saturablemedium.6

We consider two incoherently interacting beamspropagating along the z direction in a bulk, weaklynonlinear optical medium. For a Kerr medium the

0146-9592/00/090643-03$15.00/0

problem is described by the normalized coupledequations for the slowly varying beam envelopes, E1and E2:

i≠E1, 2

≠z1 D�E1, 2 1 �jE1, 2j

2 1 sjE2, 1j2�E1,2 � 0 , (1)

where D� is the transverse Laplacian and s mea-sures the relative strengths of cross- and self-phasemodulation effects. Depending on the polarization ofthe beams, and on the nonlinearity and anisotropyof the material, s varies over a wide range. For aKerr-type material with nonresonant electronic non-linearity, s $ 2�3; whereas for a nonlinearity owing tomolecular orientation, s # 7.

We look for solutions of Eqs. (1) in the form

E1 �p

b1 u�r�exp�ib1z�exp�inw�,

E2 �p

b1 v�r�exp�ib2z�exp�imw� , (2)

where b1 and b2 are propagation constants and n,m � 0, 61 are topological charges (or the phasewinding numbers). Measuring the radial coordinater in the units of

pb1 and introducing the ratio of

the propagation constants, l � b2�b1, from Eq. (1), wederive a system of stationary equations for the radiallysymmetric envelopes u and v:

Dru 2 u 1 �u2 1 sv2�u � 0 ,

Drv 2m2

r2v 2 lv 1 �v2 1 su2�v � 0 , (3)

where Dr � r21�d�dr� �rd�dr� and we assume thatn � 0. Following the notation introduced in Ref. 6,we describe all vector solitons [Eqs. (2)] by their statevectors jn, m�.

First we consider solutions j0, 0�. The families ofthese radially symmetric, two-components vector soli-tons are characterized by a single parameter l, and atany fixed value of s their existence domain is confined

2000 Optical Society of America

644 OPTICS LETTERS / Vol. 25, No. 9 / May 1, 2000

Fig. 1. Top, existence region (hatched) for the j0, 0� vectorsolitons. Solid curves, numerically obtained cutoff curvess1�l� and s2�l�; dashed curves, results of the variationalanalysis. Bottom, amplitudes of the u (solid curves) andv (dashed curves) components of j0, 0� vector solitons ats � 2, corresponding to points a and b, respectively, above.

between two cutoff values, l1 and l2. When l , l1or l . l2, self-trapping of coupled fields does not oc-cur, and there exist only scalar solitons for either theu or the v components. However, for l1 , l , l2, atwo-mode self-trapped state emerges. Near the cutoffpoints (l � l1 and l � l2) this state can be presentedas a waveguide created by one field component witha small-amplitude guided mode of the other field com-ponent. Examples of j0, 0� solitons are shown in Fig. 1(bottom) for s � 2. On the parameter plane �s, l�, theexistence domain is skirted by two curves, s1�l� ands2�l�, which are defined by the corresponding cutoffvalues, l1 and l2, for each of the soliton-induced wave-guides (Fig. 1).

For s � 1, the j0, 0� vector solitons exist only atl � 1, and their properties resemble those of the�1 1 1�-D Manakov2 vector solitons. They can beconstructed by the transformation u � U cos u andv � U sin u, where u is arbitrary and U satisfies thescalar equation d2U�dr2 1 r21�dU�dr� 2 U 1 U 3 � 0.

To describe the existence domain of the multidi-mensional vector solitons analytically, we employthe variational technique.7 We look for stationarytwo-component solutions of Eqs. (3) in the formu�r� � A exp�2r2�a2�, v�r� � B exp�2r2�b2�, wherethe parameters A, B, a, and b are defined byvariation of the effective Lagrangian of the model[Eq. (1)]. Here we mention that the coupled alge-braic equations of the variational analysis allowus to find the borders of the existence domain forthe j0, 0� vector solitons: s1�l� � �1 1

pl �2�4 and

s2�l� � �1 1p

l �2��4l�, shown in Fig. 1 by dashedcurves. One can see that the variational approachprovides an excellent alternative to numerics for

identifying the existence domains of the vectorsolitons.

An important physical characteristic of �2 1 1�-Dvector solitons is the total power, defined as P �Pu 1 Pv � 2p

R`

0 �u2 1 v2�rdr, where the partial powersPu and Pv are the integrals of motion for the model[Eq. (1)]. Figures 2(a) and 2(b) show the total powerof the j0, 0� solitons versus l for s � 2�3 and s � 2,respectively.

As can be seen from Fig. 2, the cases s . 1 and s , 1are qualitatively different. In the former case, the to-tal power of the vector soliton is lower than the power ofa scalar soliton, P0 � 11.7 at s � l � 1 [see Fig. 2(b)].This is an important and unexpected physical resultthat indicates, in contrast to the commonly held belief,that the excitation of vector solitary waves would re-quire low input power than would scalar solitons. Fors , 1, the situation is the opposite, and the vector soli-tons exist at higher powers than do the scalar solitons[see Fig. 2(a)]. A lower total power of the vector soli-ton, compared with the corresponding power of a scalarsoliton, allows us to explain an effective suppression ofthe blow-up instability that was numerically observedat s � 2 for the special case of l � 1 when an analyti-cal form of the j0, 0� vector soliton can be found by aHartree-type ansatz.8

Similarly to the well-studied case of the �1 1 1�-Dsolitons (see, e.g., Ref. 6), a �2 1 1�-D soliton-inducedwaveguide can guide higher-order modes. Such modescan carry a topological charge, i.e., m fi 0 in Eqs. (2).In the simplest case, we take m � 61 and solve thesystem of stationary equations (3) for the radially sym-metric wave envelopes. Examples of two-mode j0, 1�solitary waves are shown in Fig. 3.

Near the cutoff, the second component �v� appearsas a first-order guided mode (dashed–dotted curve) ofthe effective waveguide created by the scalar soliton in

Fig. 2. Total power of the j0, 0� vector soliton (solidcurves) and partial powers of its components (dashed anddotted–dashed curves).

Fig. 3. Intensity profiles, I �r� � u2 1 v2, of the j0, 1� vectorsolitons (solid curves) formed by scalar components with (v;dashed–dotted curves) and without (u; dashed curves) atopological charge �s � 3�.

May 1, 2000 / Vol. 25, No. 9 / OPTICS LETTERS 645

Fig. 4. (a) Existence domain (hatched) for j0, 1� solitons ina saturable medium. (b) Evolution of the total intensity atr � 0 for three values of s. l � 0.6, z � 50.

Fig. 5. Typical evolution of the j0, 1� solitons in a satu-rable medium �s � 0.65, l � 0.6�: (a) Intensity distribu-tion at z � 0, (b) evolution of the intensity profile at y � 0,(c) intensity distribution at z � 100.

the u component (dashed curve). The total intensity,shown in Fig. 3(a) by a solid curve, has a maximumat the beam center. Akin to the humps in the totalintensity profile of �1 1 1�-D solitons,6 the ring shape of�2 1 1�-D vector solitons develops far from the cutoff forthe first-order mode when the guided mode deforms thesoliton waveguide and creates a coupled state that hasits intensity maximum shifted from the beam center[see Fig. 3(b)].

Next, we conduct a numerical stability analysis andthe generalized Vakhitov–Kolokolov stability criterionfor j0, 0� and j0, 1� vector solitons.6 Our analysisreveals that neither the fundamental nor the first-order mode of the soliton-induced waveguide can arrestthe collapse of the scalar bright �2 1 1�-D solitons in aKerr medium (see also Ref. 9), and the correspondingj0, 0� and j0, 1� vector solitons are linearly unstable.

Given the established stability of �1 1 1�-D vectorsolitons in a saturable medium,6 a tempting task is tolook for the j0, 1� ringlike solitons in such a medium.Here we consider the model that corresponds, in theisotropic approximation, to the physically realized soli-tons in photorefractive materials. The normalized dy-namic equations for the envelopes of two incoherentlyinteracting beams can, in this case, be written approxi-mately in the form i�≠E1, 2�≠z� 1 D�E1, 2 1 E1, 2�1 1jE1j

2 1 jE2j2�21 � 0. Seeking stationary solutions in

the general form of Eqs. (2) and introducing the rela-tive propagation constant l � �1 2 b2���1 2 b1�, we ar-rive, after the corresponding renormalizations,6 at thefollowing system of equations for the j0, m� solitons:

Dru 2 u 1 uf �I � � 0 ,

Drv 2m2

r2v 2 lv 1 vf �I � � 0 , (4)

where f �I � � I �1 1 sI �21, I � u2 1 v2, and s �1 2 b1 acts as a saturation parameter. For s � 0,

this model describes the Kerr nonlinearity with s �1 [cf. Eqs. (3)]. In this case, the lowest-order bell-shaped j0, 0� solutions exist only at l � 1, and theyare stable. In the remaining region of the parame-ter plane �s, l�, the j0, 1� solutions, similar to thosedescribed above for the Kerr nonlinearity, are found[see Fig. 4(a)]. Again, the ring-shaped structure ofthese solutions develops only far from the cutoff for thevortex-type guided mode. Close to the cutoff, all j0, 1�vector solitons are bell shaped.

Although our numerical simulations confirmed thatthe saturation does have a strong stabilizing effect onthe j0, 1� vector solitons [cf. cases s � 0 and s � 0.6in Fig. 4(b)], vector solitons of this type appear tobe linearly unstable. The instability, although it islargely suppressed by saturation, triggers the decay ofthe solitons into a dipole structure (as shown in Fig. 5)for even small contribution of the charged mode. Asimilar type of vortex-ring decay has been observedfor a scalar beam propagation10 that is due to thedevelopment of modulational instability.

Quasi-stable dynamics exhibited by j0, 1� vectorsolitons over long propagation distances may permitthe experimental observation of these solitons. Be-cause for the current experiments on spatial solitons inphotorefractive media11 the propagation length z � 100corresponds to a crystal length of as much as40 mm, the slowly developing dynamic instability ofa j0, 1� soliton may be hard to detect definitively inexperiments.

In conclusions, we have analyzed multidimensionalvector solitons composed of two incoherently coupledbeams in both Kerr and saturable media and defined,analytically and numerically, the existence domains fornew classes of spatial solitary waves. A stabilizing ef-fect of nonlinearity saturation on �2 1 1�-D vector soli-tons with a topological charge has been demonstrated.

E. Ostrovskaya’s e-mail address is ost124@rsphysse.anu.edu.au.

References

1. See an overview by M. Segev and G. I. Stegeman, Phys.Today 51(8), 42 (1998).

2. S. V. Manakov, Sov. Phys. JETP 38, 248 (1974).3. J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. N.

Akhmediev, Phys. Rev. Lett. 76, 3699 (1996); Y. Baradand Y. Silberberg, Phys. Rev. Lett. 78, 3290 (1997).

4. L. Berge, Phys. Rep. 303, 259 (1998).5. Z. H. Musslimani, M. Segev, and D. N. Christodoulides,

Opt. Lett. 25, 61 (2000).6. E. A. Ostrovskaya, Yu. S. Kivshar, D. V. Skryabin, and

W. Firth, Phys. Rev. Lett. 83, 296 (1999).7. See, e.g., M. Desaix, D. Anderson, and M. Lisak, J. Opt.

Soc. Am. B 8, 2082 (1991).8. K. Hayata and M. Koshiba, Opt. Lett. 19, 1717 (1994).9. C. J. McKinstrie and D. A. Russell, Phys. Rev. Lett. 61,

2929 (1988).10. See, e.g., V. Tikhonenko, J. Christou, and B. Luther-

Davies, J. Opt. Soc. Am. B 12, 2046 (1995).11. M. Segev, G. C. Valley, S. R. Singh, M. I. Carvalho, and

D. N. Christodoulides, Opt. Lett. 17, 1764 (1995).