R. Khomeriki and L. Tkeshelashvili Vol. 21, No. 12 /December 2004 /J. Opt. Soc. Am. B 2175

Interaction of noncollinear spatial solitons in anonlinear optical medium

Ramaz Khomeriki

Department of Physics, Tbilisi State University, Chavchavadze Avenue 3, Tbilisi 0128, Republic of Georgia

Lasha Tkeshelashvili

Institute for Theory of Condensed Matter, University of Karlsruhe, P.O. Box 6980, 76128 Karlsruhe, Germany,and Institute of Physics, Tamarashvili strasse 6, Tbilisi 0177, Republic of Georgia

Received February 15, 2004; revised manuscript received May 16, 2004; accepted July 13, 2004

The effects caused by nonresonant nonlinear interaction between noncollinear self-focusing beams are consid-ered in two-dimensional optical samples by use of multiscale analysis. An analytical expression for beam tra-jectory shifts that are due to mutual interaction is derived, and the range of parameters is given, beyond whichthe mentioned consideration fails. We compare our results with the naive geometrical-optics model. It isshown that these two approaches give the same results. This justifies use of the geometrical-optics approachto describe elastic and almost-elastic collision processes both in Kerr and saturable nonlinear media. Theresults we obtained could be useful for the design of phase independent nonlinear photonic switches and all-optical logic elements. © 2004 Optical Society of America

OCIS codes: 190.4370, 250.5530, 080.2720.

1. INTRODUCTIONNonlinear localized waves or solitonlike excitations playan important role in many branches of physics, for ex-ample, nonlinear optics,1 plasma physics,2

hydrodynamics,3 and magnetic systems.4 In contrastwith linear excitations such nonlinear creations can beexceedingly stable. That is, they can propagate over longdistances without distortion. However the most excitingfeature of soliton phenomena is their interaction pro-cesses, particularly when they collide with other solitonsand exhibit particlelike behavior.5

Despite the fact that there is a large diversity of non-linear physical systems that exhibit solitonlike excita-tions, because of the universal properties of such cre-ations, nonlinear localization dynamics can be describedonly within a few theoretical models,2 which is of greatimportance. In particular, this allows one to study ex-perimentally nonlinear phenomena in most convenientphysical systems. The direct experimental investigationof a particular system might be more difficult or even im-possible. To date, such model experimental systems arenonlinear spin waves in ferromagnetic films4 and spatialoptical solitons.5 However, magnetic envelope solitonscan be observed only in quasi-one-dimensional (1-D)samples. Because of transverse instabilities they are un-stable at higher space dimensions.4 Thus it is doubtfulthat recently suggested interaction effects6–8 of noncol-linearly propagating 1-D envelope solitons in two-dimensional (2-D) magnetically ordered samples will beeasily realized experimentally. On the other hand, anonlinear optical medium is most appropriate for thementioned purpose, that is, the study of 1-D soliton non-collinear interaction in 2-D samples. In particular, spa-

0740-3224/2004/122175-05$15.00 ©

tial optical solitons exhibit a richness of characteristicsnot found for localized waves in other nonlinear media.5

Indeed, many theoretical and experimental investigationshave been performed for optical spatial solitons9: elasticinteraction of Kerr solitons,10–16 and almost elastic andinelastic collisions of solitons in saturable media includ-ing fusion, fission, annihilation, and spiralingoccurrences17–22 (see also Ref. 5).

Here we consider theoretically the problem of nonreso-nant interaction of Kerr spatial optical solitons. The in-teraction effects of noncollinear spartial solitons, as is em-phasized below, could be useful for the design of phase-independent all-optical logic elements and switches. Themethod used here has been suggested for an analyticaldescription of interaction of noncollinearly propagating1-D envelope solitons of magnetization in 2-D magneti-cally ordered samples.6–8 This method itself is a gener-alization of a well-known 1-D multiple-scale perturbationtheory23,24 for higher space dimensions. Since this ap-proach allows us to study the interaction of two spatialsolitons with different carrier wave frequencies, for thenonresonant interaction of solitons the results presentedhere are more general compared with those obtained inRef. 10, where the exact solutions were found but concernonly the case of spatial soliton interaction with the samecarrier wave number. Later these analytical resultswere used to suggest different applications of spatial soli-ton interactions, e.g., to produce nonlinear photonicswitching25 and all-optical logic elements.26 Optical soli-ton dragging and collision in the presence of absorptionhas also been studied27 by the application of numericalmethods. Here we obtain analytical results that describecollision processes of noncollinear spatial solitons with

2004 Optical Society of America

2176 J. Opt. Soc. Am. B/Vol. 21, No. 12 /December 2004 R. Khomeriki and L. Tkeshelashvili

different carrier wave numbers and indicate the possiblerelevance of this study to the above-mentioned applica-tions.

For simplicity we consider the interaction of spatial op-tical solitons in isotropic thin optical films and assumethat the electric field is normal to the film plane. If themedium is off resonance with respect to the optical signaland the optical film is thin enough, the dispersion can beneglected and modulations develop only along the (single)transverse direction.14–16 As a result, for an appropriatesign of the nonlinear coefficient, the so-called 1-D spatialsolitons (self-focusing beams) are formed in 2-D samples.Obviously one can consider the crossing of two beams andstudy analytically the influence of one self-focusing beamon another by using the above-mentioned method.6–8

One could try to understand the nonlinear effect of spa-tial soliton interaction with a naive geometrical-opticsmodel: propagation of the intense beam through thesample locally causes an increase in the refractive-indexthat is due to the nonlinear reaction of the medium (Kerreffect). This in turn generates a waveguide area and, asa consequence, the beam becomes self-focusing.5 At thesame time, as long as the refractive index within thebeam area is larger than the outside, it is natural to ex-pect that the second beam will be bent when it crosses thefirst beam area and eventually its trajectory will shift af-ter the interaction as is shown schematically in Fig. 1.From the same geometrical-optics consideration it followsthat this shift should be zero for perpendicular beams.However, the interaction process is much more compli-cated. Actually the second beam affects the inducedwaveguide of the first beam [the first (wide) beam in Fig.1 is slightly shifted as well]. This gives rise to self-actionof one beam through another during the interaction. Inaddition, interference effects can take place between thecarrier waves of the interacting beams. For Kerr spatialsolitons it is well established that, for large enough con-verging input angles, the solitons pass through each otherunaffected. The only effect of such nonresonant interac-tion is the trajectory shift of the interacting beams.5

Thus, qualitatively, the effect is the same as the result

Fig. 1. Schematic of the interaction process between self-focusing beams in an off-resonant optical medium. Solid curvesborders of the beams; a0 , the angle between the first (narrow)beam and the normal vector n2 of the second (wide) beam; k1 andk2 , carrier wave vectors, respectively; dl1 , a trajectory shift ofthe first beam, which was caused by nonlinear interaction effects.Note that the second beam trajectory is also slightly shifted.

from the naive geometrical-optics consideration. Sur-prisingly, we also found that the results are the sameeven quantitatively.

2. BASIC EQUATIONS AND WEAKLYNONLINEAR SOLUTIONSIn a nonlinear Kerr medium polarization P depends non-linearly on electric field E as follows:

P 5 x~1 !E 1 x~3 !E3, (1)

where x (1) and x (3) are, respectively, linear and nonlinearpolarizability constants. For simplicity we consider cen-trally symmetric materials. From the symmetry we de-termined that the second-order term in Eq. (1) is identi-cally zero. The wave equation for the electric field reads(see, e.g., Ref. 2 for more details):

¹2E 2 bEtt 5 g ~E3!tt , (2)

with coefficients b 5 @1 1 4px (1)#/c2 and g5 4px (3)/c2, where c denotes the speed of light and t rep-resents the time variable. The nabla operator acts in 2-Dspace as long as only film samples are considered here.

Let us consider the weakly nonlinear case, i.e., whenthe cubic term is much smaller than the linear term. Wedo not repeat all the calculations steps to obtain the 1-Dspatial soliton solution, let us just mention that a weaklynonlinear wave with a slowly varying envelope is soughtin the following way23:

E 5 (a51

`

ea (l52`

`

w l~a!~j, t!exp@il~kr 2 vt !#, (3)

where r is a 2-D radius vector and frequency v and wavevector k of the carrier wave are connected by the simpledispersion relation v 5 k/Ab; the envelopes w l

(a)

5 ( w2l(a))* are functions of slow variables j 5 e(r 2 vt)

and t 5 e2r/2k; v 5 dv/dk is a group velocity of linearwave vik and e is a formal parameter that defines thesmallness or slowness of the physical quantity beforewhich it appears. Then by building an ordinary pertur-bation model2,23 in the third approximation over e one ob-tains the 1-D nonlinear Schrodinger (NLS) equation:

i]w1

~1 !

]t1

]2w1~1 !

]j21 3gv2w1

~1 !u w1~1 !u2 5 0, (4)

with variables j [ j' 5 e(nr) and t [ ti 5 e2(kr)/2k2,where t plays the role of time and n is a unit vector per-pendicular to k. Equation (4) has the well-known spatialsoliton (self-focusing beam) solution if g . 0. Physically,the spatial soliton formation is the result of balance be-tween defocusing diffraction and focusing nonlinearity.It is worth noting that, since diffraction in general isstrong, the nonlinearities involved in spatial solitons aremuch stronger compared with nonlinearities involved intemporal solitons. We again emphasize that we obtaineda 1-D NLS because the longitudinal dispersion in an off-resonant optical medium could be neglected. In otherwords, wave group velocity v [ dv/dk does not depend onwave number k. When longitudinal dispersion is

R. Khomeriki and L. Tkeshelashvili Vol. 21, No. 12 /December 2004 /J. Opt. Soc. Am. B 2177

present, the physical process would be described2 by a 2-DNLS equation for which a 1-D soliton solution would beunstable.

Now we begin the main task of the paper, particularlythe analytical investigation of interaction between non-collinear self-focusing beams. For that purpose we wishto determine a solution following the general method de-veloped in Refs. 6–8:

E 5 (a51

`

ea (l1l252`

`

w l1l2

~a! exp@i~kl1l2r 2 v l1l2

t 1 eV l1l2!#,

(5)

where v l1l25 l1v1 1 l2v2 ; kl1l2

5 l1k1 1 l2k2 ; v1 , k1

and v2 , k2 are carrier frequencies and wave vectors, re-spectively; envelopes w l1l2

(a) and phase shifts V l1l2of the in-

teracting waves are functions of slow variables ( p5 1, 2):

jp 5 e@~npr! 2 ecp~j1 , j2 , t1 , t2!#, tp 5 e2~kpr!

2kp2

,

(6)

where n1 and n2 are unit vectors perpendicular to carrierwave vectors k1 and k2 , respectively; c1 and c2 , whichrepresent the wave-front shifts of the colliding beams, areunknown functions at the moment and should be deter-mined by use of our multiscale analysis. Proceeding withsimilar calculations as was done in Refs. 6–8, we arrive atthe leading approximation to the following solution:

E 5 w10~1 ! exp$i@k1r 2 v1t 1 V10

~1 !#% 1 w01~1 !

3 exp$i@k2r 2 v2t 1 V01~1 !#% 1 c.c., (7)

where c.c. denotes complex conjugate terms; w10(1) and w01

(1)

are the solutions of 1-D NLS equations [see Eq. (4)] withderivatives over a set of slow variables j1 , t1 and j2 , t2 ,respectively. For example, in the simplest case of one-soliton envelopes w10

(1) and w01(1) , we have

u w10~1 !u 5 uA1usech$uA1uA6gv1

2@~n1r! 2 c1~1 !#%,

u w01~1 !u 5 uA2usech$uA2uA6gv2

2@~n2r! 2 c2~1 !#%,

(8)

and A1 and A2 are amplitudes of the first and the secondself-focusing beams, respectively. Besides that, thephase and position shifts of the first self-focusing beaminduced by weakly nonlinear interaction with the secondbeam are defined by the following formula:

]c1~1 !

]j25

~n1n2!

~k1n2!

]V10~1 !

]j25 3gv1

2u w01~1 !u2

~n1n2!

~k1n2!2. (9)

According to the perturbative approach we have the fol-lowing scaling ]c1

(1)/]j2 ; e2. Taking into considerationthe dispersion relation (v1

2 5 k12/b), we derived the fol-

lowing restriction on the soliton parameters:

uA2uA3g

b

usin a0u

cos2 a0

! 1, (10)

where a0 5 (p/2) 2 u0 , and u0 is an angle between theself-focusing beams. From Eq. (9) it is easy to derive an

analytical expression for the trajectory shift of the firstbeam caused by the nonresonant interaction with anotherbeam:

dl1 5 c1~1 !~`! 2 c1

~1 !~2`!

53g

b

sin a0

cos2 a0E

2`

`

u w01~1 !~j2!u2dj2 . (11)

For one-soliton envelopes in Eqs. (8), from Eq. (11) onecan obtain an analytical expression for the trajectory shiftof the first spatial soliton:

dl1 5 uA2uA6g

bv2

sin a0

cos2 a0

. (12)

We now list the restrictions to the results given above.Since we have neglected the effects of dispersion, the fre-quencies of the carrier waves have to be of the same orderof magnitude. Another restriction is given by the factthat a multiscale analysis is valid only within the weaklynonlinear limit. If the amplitude of the nonlinear waveis not small enough, the effects of higher-order nonlineari-ties become important. In addition, the amplitude of thesoliton cannot be arbitrarily large since the width of thesoliton decreases when the amplitude increases [see Eq.(8)]. Note, that the width of the soliton must be muchlarger compared with the carrier wave wavelength basedon multiscale analysis.23 On the other hand, to form asoliton while it travels in a sample, the amplitude of thenonlinear wave cannot be arbitrarily small. Taking intoaccount the restrictions outlined above, we conclude thatfor real experiments the amplitudes as well as the widthsof the interacting spatial solitons must be of the same or-der of magnitude. These conditions guarantee that eachinteracting beam obeys a 1-D NLS equation. As a result,the validity of the analytical expression in Eq. (12) isguaranteed as well. Note that all these conditions can beeasily fulfilled in planar glass waveguides, where the in-teraction processes of collinear Kerr spatial solitons wereivestigated numerically (with a beam propagationmethod) and observed experimentally.15,16

In addition we emphasize that beyond the limit givenby inequality (10) the dynamics is governed by a set oftwo coupled NLS type equations, which, in general, is notan integrable model and yields qualitatively different be-havior for interacting solitons (see Ref. 28 and the discus-sion therein).

3. GEOMETRICAL-OPTICS APPROACHWe now compare the results obtained above with the pic-ture given by the naive geometrical-optics consideration.This provides a deeper insight into the problem. Firstsuppose that one has only one self-focusing beam [par-ticularly, the second (wide) beam] and let us calculate howit changes the refractive index (see Fig. 2). In view of thedependence of polarization on the electric field in Eq. (1),we can write the expression for the refractive index as s5 Ab 1 3gE2

2, where E2 denotes the electric field in theself-focusing beam area and it has the form of an envelopespatial soliton; see Eqs. (8). Averaging the refractive in-

2178 J. Opt. Soc. Am. B/Vol. 21, No. 12 /December 2004 R. Khomeriki and L. Tkeshelashvili

dex over fast variables r and t in a weakly nonlinear limit(the term proportional to E2

2 is small), we obtain the fol-lowing approximate formula for slowly varying (alongaxis x) averaged refractive index s(x) 5 Ab@11 (3g/2b)E2

2#.We now solve the following geometrical-optics problem,

particularly, how the optical rays refract when theypropagate through the area of the second beam with aslowly changing refractive index. For that purpose wenote that an angle a between the ray and the normal vec-tor (with respect to the beam) at any point could be calcu-lated with a simple refractive formula: s(x)sin a5 s0 sin a0 , where s0 [ Ab. Taking into account thefact that the trajectory shift of the ray could be calculatedas follows: dl1 5 cos a0 *2`

` dx(tan a0 2 tan a), we arriveexactly at Eq. (11), which we obtained from multiscaleanalysis.6–8 However, to avoid any misunderstanding wemust stress that the geometrical-optics approach to theproblem of spatial soliton interaction is not self-consistent. Indeed, as we have pointed out already, inthis model the self-action effects of the first solitonthrough another are neglected. Physically this meansthat the first beam is linear, but the diffraction in the non-linear problem considered here is not negligible. Thusthe first beam diffracts and the interaction picture givenby the geometrical-optics model is not meaningful underrealistic experimental situations of the spatial soliton in-teraction.

Although the geometrical-optics approach in somecases provides a complete understanding of the soliton in-teraction process,29 one generally expects that this ap-proach is valid only for a qualitative description of the in-coherent spatial soliton interaction.5 As is mentionedabove the nonlinear self-action of one beam through an-other is neglected without justification in the geometrical-optics model of soliton collisions. However, the analysispresented here shows that this additional nonlinear self-action during the interaction process does not affect thesoliton dynamics asymptotically. That is why such a na-ive geometrical-optics model gives correct results even foralmost-elastic collision processes between solitons insaturable media.5

4. CONCLUSIONSIn summary, we have considered the problem of nonreso-nant interaction of Kerr spatial solitons (self-focusing

Fig. 2. Optical ray refraction through the self-focusing beamarea with borders denoted by horizontal solid lines.

beams) with different carrier wave frequencies by use ofmultiscale analysis. We have shown that the beams tra-jectories are shifted due to mutual interaction. The ana-lytical expressions for these shifts were obtained as well.Surprisingly the naive geometrical-optics model of thesoliton collision is in complete agreement with the generaltheory results. This shows that the self-action of the soli-ton caused by a nonresonant interaction process does notchange its asymptotic behavior after collision, which inturn justifies use of the geometrical-optics model for a de-scription of elastic and almost-elastic collision processesboth in Kerr and saturable media.

The analysis presented here and the derived analyticalformula in Eq. (12) could be used to design phase-independent nonlinear photonic switches and all-opticallogic elements.25,26 It should be emphasized that, as out-lined in this paper, one can control the shifts of spatialsolitons by varying the collision angle, the amplitudes,and the carrier frequencies of the interacting beams.

ACKNOWLEDGMENTSR. Khomeriki (khomeriki@hotmail.com) is obliged to GregSalamo for the enlightening discussions with regard tothe physics of self-focusing beam interaction processes L.Tkeshelashvili (lasha@tkm.physik.uni-karlsruhe.de) ac-knowledges the financial support from the Deutsche For-schungsgemeinschaft (DFG) under Bu 1107/2-2 (theEmmy-Noether program) and the DFG Forschungszen-trum Center for Functional Nanostructures at the Uni-versity of Karlsruhe. R. Khomeriki is supported by theU.S. Civilian Research and Development Foundationaward GP2-2311-TB-02.

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