Volume148, number1,2 PHYSICSLETTERSA 6 August 1990

Maxwell optical solitonsin inhomogeneousmedia

JesusLifiaresLaboratorio deOptica, DepartamentodeFisica Aplicada,FacultadedeFisica, UniversidadedeSantiagode Compostela,E-15706Santiagode Compostela,Galicia, Spain

Received23April 1990; acceptedfor publication7 June1990Communicatedby J.P.Vigier

Maxwell opticalsolitonsareevaluated,in a non-linearmono-modeweaklyinhomogeneousmedium, undertheslowly-varyingenvelopeapproximationandby an“average”methodappliedto Maxwell eigenmodes.Differencesin thegroupvelocity,maxi-mum amplitudeandboundarybetweenbright anddark solitonsarefoundwith respectto thescalartheory. TheMukunda—Simon—Sudarshanformalism,basedon thefront formofrelativisticdynamics,constitutesthestartingpointfor derivingthiskindof non-linearpulses.

The dispersionis the deterministicfactor in de- hasbeenonly appliedto linear problems.ciding therateof pulsebroadening.To overcomethis Recently,thepath integral (P1) formalism (com-limitation the non-linear changeof the dielectric plementarytothat setupon a grouptheoreticalbasisconstant(the so-calledKerr effect) of theoptical fi- andwith canonicaloperators:the MSS formalism)berhasbeenusedto compensatefor the dispersion hasbeendevelopedto derivethe mentionedMax-effect. Whenthe frequencyshift due to the Kerref- well solutions[5]. Thiskind of solutionsgivesasat-fect is balancedwith that due to the dispersion,the isfactoryaccountof the polarizationin vectorwaveoptical pulsemay tend to form a stablenon-linear problems,becausethe Maxwell equationsare ful-pulse,calledoptical soliton. filled: evolutionandconstraintequations[3].

The opticalpulsepropagationin non-linearfibers In this paper,a newresultis shownconcerningthehasbeenexaminedin the so-calledslowly-varying envelopeoptical solitonspropagatingthrougha non-envelopeapproximation[1]. Moreover,an average linearweaklyinhomogeneousmedium,in particularmethod [2] hasbeenemployedto derivea one-di- a mono-modefiber. The averagingmethod is ap-mensionalnon-linear Schrödingertype differential plied to a Maxwell (or vector) lineareigenfunction.equation.A scalareigenfunctionis usedto obtainan Theelectric field is below thresholdfor the self-fo-averagetakenovereachcrosssection,however,the cusing,thereforethe useof the linear eigenmodeisanalysispresentedin this papershowsthateven in justified.ThisMaxwelleigenfunctionwill bederivedweakly inhomogeneousmedia (or weak guiding), from an exactBorn expansionappliedto Maxwellvectoreigenfunctionsmustbe usedto calculateav- wave optics (the resultscanbe alsoderivedby theerageeffectsoverthenon-linearpropagation.In this MSS formalism). The envelopesolution obtainedway, Maxwell optical solitonsare obtained, will becalleda Maxwellsoliton,which givesaccount

The startingpoint is the formalism developedby of the polarizationstatein non-linearpropagation.Mukunda,Simon andSudarshan[31(hereafterre- Thissolution is nota “vector” soliton,neverthelessferredto asthe MSS formalism) basedon the front it would be taken into accountto makesurethatform of relativistic dynamicsfor the treatmentof Maxwellconstraintequationsarenotviolatedunderparaxialwavepropagationin vectoroptics.TheMSS non-linearpropagation.formalismleadsto theso-calledMaxwellbeams[4]. It will be shownthatMaxwell solitonshavea dif-It mustbe stressedthat, up to now,this formalism ferent velocity group, amplitude and polarization

0375-9601/90/S03.50© 1990 — ElsevierSciencePublishersB.V. (North-Holland) 31

Volume 148, number1,2 PHYSICSLETTERSA 6 August 1990

state,with respectto scalarsolitons.The polariza- T~J=oJa,N+aJNaI, N=log[n(r, co)] . (8)tion propertiesare “transmitted” to the soliton bythe Maxwell eigenmode,which presentsparticular Applying the slowly-varying envelopeapproxima-vectorproperties[4,6]. Likewise,thetransitioncon- tion to eq. (7), that is,dition between bright and dark soliton will be

E,(r, 1)=A1(r, t) exp[i(/30z—w0t)] , (9)modified.

Startingfrom a three-dimensionalwave propaga- it follows thattion in a dispersivenon-linear medium,the scalarwave equationfor the electricalfield E~is given by (i3~~+ 2ifl0 ô z — fl~)A1+ T,A1[1,2] ~

aJJEl_ca~(D,=2n2nOca~L(EJEJEl), (l) =_2n2k~c2A7A~A

1, (10)

where n2 representsthe non-linear part of the re- wherefractive index

k~=2irn(a0),~~(co0)=2itn0A~ (11)n

2(r,w, E.)~rn(r,w)+n2E7E1 (2)

andthe prime indicatesderivativewith respecttoandthe inhomogeneityofthe mediumis takento be frequencyw. A canbe rewritten asrepresentedby

A,(r,t)=U1(r1)Ø(z,t) , (12)n

2(r,w)=n2(w)f(r) (3)where U, representsa Maxwell eigenmodeof the

In theslowly-varyingenvelopeapproximation,the waveguide,which fulfills the vectorwave equationelectric field E? canbe rewrittenas

~ . (13)E?(r, 1) =A?(r, t) exp[i(/3

0z—w0t)] , (4)To evaluateU1 werecalltheBorn expansion[7] (or

w0 being the carrier frequencyand fl0 the constant perturbativemethod).Therefore,if the scalareigen-propagation.By standardmethods,from eq. (1) a mode U? is knownthen the Maxwell eigenmodeisnon-linear wave equation for A~°can be derived, given byMoreover,by the “averaging”method,the final so-lution can berewritten as U1(r1 ) = U?(r±)

A?(r, 1) = U?(r1 )Ø(z, t) r1 =xi+vJ~ (5) —iX~’JK~(r,ro)T,m(ro) U~,(r01)dr0

whereU~?is the scalareigenmodeof the waveguide.Eq. (5) representsa separationof variables,pro- +O(2~

2), (14)videdthatnon-lineareffectsare small.Nevertheless, whereK~is the scalaropticalpropagator,which hasthe solutionof eq. (1) in the form (4) mustfulfill to be known. If we restrictour attentionto mono-the Maxwell constraintconditions mode optical fibers characterizedby the refractive

a~D3=0,8~B~=0. (6) index

Forthat,anewtermineq.(l)hastobeadded,in n2(r,w)=n(w)[l—(x2+y2)/L2], (15)

asucha waythat the solutionsgive a satisfactoryac- whereL is a gradientparameter,the optical propa-count of the polarizationunderpropagation(MSS gatorcanbe easily evaluated[5,81 for weakly in-approach).Therefore,eq. (1) hasto be replacedby homogeneousmediadescribedby eq. (15),

[5]K

11(r,r0)=~K°(r,r0), (16)ô33E1—c

28,~D1+ ~

with=2n2n0c

2ô,,(E~’E~E1) , (7)

K°(r,r0) =F(z, z0) exp{(ino/2)~0)[W(r,ro)]}where

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Volume 148, number1,2 PHYSICSLETTERSA 6 August 1990

2i,tn0 wherethe “radius” of crosssectionof the MaxwellF(z, z0)= )~0Lsin[(z—z0)/L] ‘ beamhasbeentakento be (L/k0)”

2 assuggestedby

eq. (18).W(r,r

0)={cos[(z—z0)/L](x2+y2+x~+y~) Now, multiplying eq. (21) by U7, given by eq.

—2xx0 _2yyo}{sin[(z—zo)/L]} —1• (19), andintegratingover T=XQXa, we obtain,after

a longbutstraightforwardcalculation,the followingOn the otherhand,the term Tim in eq. (14) can non-linearequation,

be approximated,for a weakly inhomogeneousme-diumcharacterizedby eq. (15), as {8~~+2ifl08~+(1 —a1/Ic0L)[2ik0k’~8~

Tim~oc[(Poxa)/L2]o~, (17) —(k~+k

0kg)o11]}Ø(z, I)

whereonlylineartermshavebeenconsidered(Greek = — 2n2k~c2a

20*Ø Ø(z,1) , (22)indicesindicatingvariablesx, y). Moreover,eq.(17) wherecanbe alsojustifiedby the slowly-varyingenvelopeapproximation. 1 1+ 1 /k0L

On the other hand,the scalareigenmodeU? is a1 = k0L 1+ l/2L2’ (23a)

givenbya

21+1/2L. (23b)U?(rjj=exp(—koxaxa/2L). (18)

Eq. (20) is now independentof the radialdirectionSubstitutingeqs.(16)—(18) into eq.(14), andper- r. The inhomogeneityin the r direction and con-forming the integral, we obtain the Maxwell straintconditions(6) have,however,left their effecteigenmode on thenon-linearpropagationthroughfactorsa1 andU,(r1)= U~(r1)[ô~i+ôz3(Xa/iL)ôaj] . (19) a2.Thesefactorshavea profoundeffecton theprop-

agationcharacteristicsof the pulse in this medium.It canbe shown that the constantpropagationfl~of Theeffect ofthe constraintconditionsis the centralthe Maxwell eigenmodesdoesnot undergochanges resultofthis work, andsomeof its consequencesarewith respectto thescalarcasein the approximation commentedupon below.definedby the perturbativeoperator(17). More- Twoparticularsolutionsof eq. (22) canbeeasilyover,termsof orderhigherthan 1 íA0 in theBornex- obtained,i.e.,thebright(b) anddark(d) one-solitonpansion(14) are zero.Thisresult couldbe alsode-

ø(b)(’)O(b) sech(~)rived by the MSS formalism (so-called MaxwellGaussianbeams)by the propertiesof theso-called Ø(d)(~)=Øo(d)tanh(~), (24)GSPIn matrix.

Substitutingeqs. (12), (13) and (15) into eq. with ~=(t—z/v8)/r, where r is the “width” of the(10) it follows that pulseandv~is the group velocity givenby

Uj(rj){özz+2ifloöz+(1—xaxa/L2)[2ikok’oôt v

5flo[(l_a1)kok’o]~1/k’o, (25)

— (k’0+k0kg)811]}Ø(z, t) wherefl0=k0(1—2/k0L)”2 hasbeenused.Approx-imation (25) is notnecessarybutallowsustoobtain

= _2n2k~c

2U~UjU1(rj)Ø*ØØ(z,t). (20) analyticalresults.Amplitudesare givenby

Sincewe are treatingthe xa dependencein a linear ø~b= — (1 —a1 )k0k’~+ a1 k’2

fashion,wecanreplaceU’ U~by its “average”value n2a2k~’r

2/n0

overboth the crosssectionS andthe numberof in-dependentpolarizationstates,i.e., two statescorre- 2 + (1—a,)k0k’~—a,k’

2 (26)00(d) =spondingto TE andTM modes: n2a2k~r

2/n0

~ J 2it UU it/k0L+2,tL/k0 If the constraintconditionsare not taken into ac-= r dr ~ = , (21) count,i.e.,ascalarapproximationis used,theresults2S 27tL/k0

becomethosederivedin previousworks [9]. There-

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Volume148,number1,2 PHYSICSLETTERSA 6 August 1990

fore,solution (12) givenby eqs.(19) and (24) can liton [111 as well asgain anddissipativeeffectscouldbe calleda Maxwell soliton. This solution mustbe beanalyzed.takeninto accountevenwhen“vector” solitonsarestudied.Work aboutthisproblemis in progress.The I would like to thankProfessorH.A. Ferwerda,Dr.polarizationeffectsof this soliton are providedby B.J. HoendersandDr. W. Kamminga(Groningenthe Maxwell eigenfunction(19). University), for a numberof helpfulandstimulating

Let us considersomeimportantconsequences.If discussionsaboutnon-linearoptical propagation.

n2> 0 andfor anomalousdispersionk~<0, the nu-

meratorfor b solitonsis alwayspositive. Therefore, Referencesthe consequenceis that a highervalue of intensitywould be neededthan when the constraintcondi- Li] N.TzoarandM. Jam,Phys.Rev.A 23 (1981)1266;

D. AndersonandM. Lisak,Phys.Rev.A 27 (1983) 1393.tionsarenot considered.Ontheotherhand, for nor-

[2] A. HasegawaandY. Kodama,Proc.IEEE 69(1981)1145.mal dispersion(k~>0), thetransitionconditionbe- [3] N.Mukunda,R. SimonandE.C.G.Sudarshan,Phys.Rev.tween bandd solitons is derivedfrom eq. (26), A28 (1983)2933;J.Opt.Soc.Am.A2(1985)416;

R. Jagannathan,R. Simon, E.C.G. Sudarshanand N.

(1—a1)k0k~=a,k’2, (27) Mukunda,Phys.Lett.Al34(1989) 457.

[4] R. Simon,E.C.G.SudarshanandN. Mukunda,J.Opt. Soc.Am.A3 (1986) 536.

therefore, the boundary line between d and b soli- [5]J. Liñares, Phys. Lett.A 141 (1989) 207;

tonshasbeenmodified when a consistentMaxwell J.LiñaresandP. Moretti, Nuovo Cimento 101B (1988)theory is used in the slowly-varying envelopeap-proximationandby the “average”method. [6] Y. FainmanandJ. Shamir,AppI. Opt. 23 (1989)3188.

[71R.P. FeynmanandA.R. Hibbs, QuantummechanicsandIn short,Maxwell solitonshavebeenobtained,and

path integrals (McGraw-Hill, New York, 1965).their influence in non-linear propagation has been [8] C. Gomez-Reino and J. Liñares, J. Opt. Soc. Am. A 4 (1987)

stressed.Thus, the group velocity, maximum am- 1337.plitudeandboundarybetweenb andd solitonshave [9] M. Jam and N. Tzoar, J. AppI. Phys. 49 (1978) 4649.

10] Y. Kodama,J. Phys.Soc. Japan45 ( 1978)311.changedwith respectto the scalarsoliton case.Fi- [11] J. Satsumaand N. Yajima, Suppl. Prog. Theor.Phys. 55

nally, higher-orderdispersion [10], Maxwell N-so- (1974)284.

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