Volume 154, number5,6 PHYSICSLETTERSA 8 April 1991

Waveguidepropagationof electromagneticwavesin gyrotropicmedia

V.1. KarpmanJZMIRAN,AcademicCity, Troitsk,MoscowRegion142092, USSR

Received2 January1991; acceptedfor publication21 January1991Communicatedby V.M. Agranovich

The theoryof waveguidingof electromagneticwavesin gyrotropicmediais considered.It is shownthat in thepresenceof anambientmagneticfield, a new nonguidedmodecanappearanda lineartransformationof theguidedmodeinto thenonguidedoneleadsto radiationof energyfrom thewaveguideand to attenuationof theguidedradiation.The trappingconditionsin theWKB approximationaswell asthetransformationcoefficientandattenuationrateareobtained.

1. Introduction extendedto anisotropiccrystals.Thesolutionsof thebasicequationsfor the homogeneousmediaarepre-

The problemin questionis connectedwith some sentedin section3. They contain,in particular,newinterestingeffectsof rathergeneralsignificance.We modesappearingin the magneticfield. The WKBshowthat in gyrotropicmedia,in the presenceof an solutionsfor thewaveguideswithsmoothandbroadambient magneticfield, there appearnew modes profilesof thedielectricpermeabilityareobtainedinwhich maydrasticallychangetheconditionsforwave sections4 and 5 anda theoryof transformationef-guiding. Mathematically,this is connectedwith an fects, leadingto irradiationfrom the waveguides,isincreaseof the orderof a systemof equationsde- givenin section5. Theresultsobtainedareusefulforscribingelectromagneticwave propagationin mag- the theoriesof the fiber propagationandnonlinearnetizedgyrotropicmedia.Thenewmodes,generally, self-focusingof electromagneticwavesin gyrotropicbelongto thecontinuousspectrumandwe show that media [4].theremaybea couplingbetweentheboundandnon-boundstatesin a waveguidewhich leadsto a trans-formation of guided waves into nonguidedones 2. Basic equationswhich resultsin radiationfrom the waveguide.Thenatureof this effect is similar to the tunneling in Our startingequationsarequantummechanicswhenanexternalfield convertsa perfectpotentialwell into a potentialbarrier.Sim- rot rot E= —~- D, (2.la)ilar effectsof “tunnelingtransformation”havebeen C

discussedin ref. [1] for whistler wavesin a low-fl D=(w; r)E+iExg, (2.lb)plasmaandsomemethodsdevelopedin refs. [1—3] 2 2 2

wherer= (x +y ) / andg=g(w) is the gyrationfor whistler waves (which havealways right-hand .vector proportional to the ambientmagneticfieldpolarization)areappliedto thetheorypresentedbe-

(see,e.g.,refs. [5,6]). We assumethat thez-axis islow with somewhatdifferent physics (dealing, in . . . . .

directedalong g. Using cylindrical variables, it isparticular,with both right- andleft-handedpolari- . .convenientto introduceinsteadofEr andE thenewzation).The basicequations(section2) aregiven, .

unknown functionsfor simplicity, for isotropic (withoutmagneticfield)media,but our approachcan straightforwardlybe F(r, ci, z) =E~—iE~,,G(r, ~, z) =Er +iE~,. (2.2)

230 ElsevierSciencePublishersB.V. (North-Holland)

Volume 154,number5,6 PHYSICSLETTERSA 8 April 1991

Then, due to the axial symmetryandhomogeneity [02 ln ( �—g) F+(~—g)alongthe2-axis,eqs.(2.1) aresatisfiedby theansatz Op2

F(r, ci, z)Fm(r;p) exp[imço+i(w/c)pz], Oln(~_~)(OFm— ~Fm)]+ Op Op p

G(r, ci, Z)=Gm(r,P) exp[imço+i(w/c)pz]

Ez(r,~,z)=Emz(r,p)exp[imço+i(w/c)pz] =2~(p2—~—g)Grn, (2.5b)(2.3) where

FromtheequationdivD=0,wehaveforEm~(r,p) 1 0 / 0) ~2

_—~ (2.6)Emz=~ and

0 ln(~—g)x[(E_g)(~rFm_mFm+ Or rFm) p=(w/c)r. (2.7)

+ (�+g) (~-rGm+ mGm+ ~ ln(~+g)rGm)]. 3. HomogeneousmediumOr

(2.4)An importantunderstandingof system(2.5) can

Thenfrom eqs.(2.1) and (2.4)wearriveat thefol- beobtainedfrom its solutionsina homogeneousme-lowing equationsfor FmandGm, dium. In this caseeqs. (2.5) havesolutionsof the

form (cf. ref. [3])(2�_g)L(m~ ~Fm

Fm(r;p)=A(p)Zm_~(qp),

+g(Lm÷’~+

2m0(pGm) 2m2p2 Or + ~Gm) Grnfr,p)B(p)Zm~i(qp), (3.1)

r0 2 ln (~ g) whereZ,.(qp) arecylindricalfunctionssatisfyingthe

+ (~_g)[ 02 Fm Bessel equation

L~”~Z~(qp)=—q2Z~(qp) (3.2)o ln(f—g)(~+ Fm)] andthe famousrelations

+ Op \0

r02lfl(~+g) 1 d[WZm+i(W)]+(~+~)[ 82 Gm W dW

o ln(f+~)(OGm+ ~ Gm)] = ±(Zm(W) ~Zm±i)~ (3.3a)+ Op 0 p

=2~(p2—~+g)Fm, (2.5a) 2mZm(W)= W[Zm+i(W)+Zm_i(W)] , (3.3b)

(2~±g)J~m±1)G~

2dZm(W

)

2m0(pFm) dW =Zm_i(W)Zm+i(W). (3.3c)

_g(~(m1)Fm_ p2 0~ + 2~2Fm) From (3.2) and (3.3) follows

ro2ln(~+g

)

2mO[pZm~i(qp)]+(�+~)[ 0,~2 Gm ~ 0

Oln(~+g)(oGm

m_________ 2m2+ 0 ~___Gm)] +Zm~j(qp)rrq2Zrn±i(qp). (3.4)

Substituting (3.1) into (2.5) andusing (3.3) and

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(3.4), we obtainlinearalgebraicequationsforA and Now, let us assumeB,

p=p±+~Sp, öp~<<p. (3.14)[2�(p2—e+g)+ (2�—g)q2]A—gq2B=0, (3.5a)

Thenfrom (3.7) we obtain two branchesof q2,gq2A+ [(2�+g)q2+2�(p2—�—g)]B=0. (3.5b)

2 4�p±öpTheconditionof solvabilityof thehomogeneoussys- q

1 = — 2�±g +0 ( (öii/~v)2) (3.15a)tern (3.5) gives a quarticequationfor q,

2_— g(2�±g)—

4p±~p2�±g

=0, (3.6) (3.lsb)

which definesa function q=q(p), wherethe signcoincideswith that in (3.14) andthenumberingof q is sodefinedthat I <I q2 I for small

q2= 2�(�—p2)—g2±g(4�p2+g2)”2 I~pI.thus, the signsin eqs.(3.13)—(3.l5) havea

(3.7)2� . meaningdifferentfrom thosein (3.7), (3.9), (3.11)

and (3.12). Note that smallnessof g/� is not as-Substituting (3.1) into (2.4) andusing (3.3), we sumedin eqs.(3.15).have

From (3.15) follows thatonly q1 vanishesiföp=0

Emz(Pp) whereasq2 remainsfinite. Also, if p is nearp~,thenonly onebranch,q1, canbe real (for ~p<O). If p is

= J9-. [(�+g)B_(�_g)A]Zm(qp) . (3.8) nearp_,thereare two realbranches:q1 (for~p<O)andq2 (for any sign of ~p). Theseconclusionscan

Up to now, we havenot assumedsmallnessof g. be generalized:from (3.7) follows that if p2 <� — g,

If g/ � << 1, which is the casefor naturalgyrotropic therearetwo real branchesof q, if �— g<p2<�

media(exceptforplasmas),oneobtainsfrom (3.7) thereis only one realbranch,andforp2>� +g bothbranchesare imaginary.

q2~E_p2±gp/~,/~+o(g2) (3.9) Returning to small ~p, substituting (3.15) into

Then, for the refractiveindex n, system(3.5) andtaking into account(3.1), oneob-tainsinformationaboutthepolarizationof themodes

n2=p2+q2, (3.10) (3.1). It is givenby the relations

we havethe well known equation [5] I G I B q2

(n~gp/2�)2~�+O(g2). (3.11) (p=p+~p,q=q1), (3.16a)

Also, it is easyto checkthat the solution of system(3.5), whereq

2 is givenby (3.9), canbewritten as —f--— (p=p + ~p,q=q2), (3.16b)

IFI B 2�—g

A q2

~ (3.12) ~— (p=p÷+~p,q=q1). (3.16c)

Thus at p=p+öp, brancheswith q=q1 andq=q2wherea is an arbitraryparameter.From eq. (3.6) have, essentially,right- and left-handed polariza-follows thatq=0 for tions. For p=p~+~p,the only real branch (with

q=q1) is approximatelyleft-handedpolarized.p

2=p~~�±g. (3.13) The resultsof this sectioncanbe appliedto theThisdefinestwo valuesof thewavenumberp for the waveguideswith sharpboundaries.In this casetheplanewavepropagatingalongtheambientmagnetic spectrumof ~pis definedby an equationfollowingfield, from the matchingconditionson theboundary.This

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Volume 154, number5,6 PHYSICSLETTERSA 8 April 1991

requiressomenumericalwork andwill not be dis-

2�” / g )cussedhere.Belowweconsidersufficientlybroadand 0 ~ ‘ii’ ~ 2 1 ~ . (4.6)smoothwaveguideswhichcanbetreatedanalytically (~) 2�(co)by the WKB method. The plots of q,(p) andq2(p), when (4.6) is satis-

fied, are shown in fig. 1.Now let us write the WKB solution for theupper

4. WKB approximation signin (4.5) whenthereis only onerealmode,q,(p).Startingfrom thewaveguidecenterregion,nearp= 0,

Considersufficientlysmoothandbroadprofilesof we assume that there, with sufficient accuracy,�(p)with g=const.Thenonecanneglectthe terms e(p)~�(O)(thiscanberealizeddueto (Oe/Op)~=O,containingderivativesof � in eqs.(2.5), andlook etc.).Then, at smallp onecanusethe solution forfor their solutionsin the WKB approximationwhich homogeneousmedia(3.1), andthe conditionof fi-are superpositionsof termswith phasefactors nitenessleadsto

Frn(p)AJm_1(qi(0)p),

exp(±iJqn(p’)dp’), (4.1) G~(p)=BJ~÷,(q1(0)p), (4.7a)

wherethe q~(p)are givenby the sameequationas A~2a(n—p), B~2a(n+p) (4.7b)forhomogeneousmedia,i.e. (3.7),butwith =�(p)andso the problemis reducedto finding the ampli- where.J,~( W) isthe BesselfunctionandA,Baretakentudesof (4.1). Onehasto remember,however,that from (3.12).In theregionq,(0 )p>> 1, but with suf-expressionscontaining (4.1) are not valid nearthe ficiently smallp, in orderto regard� (p) �(0), oneturningpointsPo, where q~(Po)= 0, as well as near canusethe asymptoticsof (4.7),the centerp=O.

To simplify the problem,let us assumethat �(p) Fm(P)~A( 2 \1/2

is a monotonicfunction atp~0and cos[q,(0)p— 1m~+~1,(4.8a)

I(0)—�(oo)I <--<<1. (4.2) Gm(p)~B( 2 \h/2\Itq,(0)p) cos[q1(0)p—~mic—~t]

�(co) 2�

Thetrappingconditionfor thebranchq~is q~(p) >0 (4.8b)at/.~<Poandq~(p) <0 atP>Po, inparticular,at~ Theseexpressionshaveto be matchedwith thoseDefining z~pandAe(p) by

p=p±(oo)+1~p,p~±(oo)=�(cc)±g, (4.3) 1 1(4.4)

~~.____.._______— 2

onecan easilyobtain from (3.7)

+0((g/�)2, (Ap/p)2) , (4.5a)

q~(p)=~2g—2�”2(oo)[l±g/�(oc)]L~p

+ [1 ±g/2�(oo)]&(p) (a) (b)

+O((g/�)2, (i~p/p)2). (4.5b)Fig. 1. Branchesofq(p) in awaveguide.(a) Positivesignin (4.3).

From (4.5) follows that only the branchq1 (p; L~p) (b) Negativesign in (4.3).Curves(I) and (2) depictq1 (p) and

canbe trapped,andthe trappingconditionis q2(p).

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containing factors (4.1). Thus in the region of va-Gm(p)=B[2~tp~.J_q~(p)]1/2

lidity of the WKB approximationatp <Poweobtainp po

F~(p)=A[2~pq1(p)]

112 (_J~dp—i Jqi dp+~i(m+2)~).X exp

p

xexp[i(JqidP_~m~+~)]+c.c., (4.9a) (4.l2b)

At thesametimewe obtain thecondition

Gm(p)B[27tpq1(p)]~”

2

p

xexP[i(JqidP_~m~_~)]+c.c. (4.9b) Jqi(p’;~p)dp=~(n+~m),0

0n=±l,±2 n+~m<0, (4.13)

Substitutinghere expressions(4.7b) with n=n(p) which is an equationfor the spectrumof i.~p. Eq.and �=�(p), one mustchoosea(p) in sucha way (4.13) coincideswith that found for whistler wavesthat rHr=constwhere TI,. is the r-componentof the in ref. [3].Poynting vector 11=(C/8~t)Re(ExR*). For run-ning wavesin the WKB approximation,with ac-count of (3.11) and (3.12),we have

5. Radiation from a waveguideC / gpn(p)) 2

pHr~2~(~lR ~ a~ Forthelowersignin (4.5),in additionto themode

q1(p), there is a nontrapped mode q2(p) (fig. lb).

+0(g2) (4.10) As far as q

1(p) and q2(p) do not itersectat real p,

whereq = ±q1,2. The conditionpH,.= const gives the mode q1 (p) is not connectedwith q2 (p). It isvalid, however, only in the adiabatic approximation.

a (p) 2 (1 + gp~~(P))x const In an inhomogeneous medium there is the possibil-— 2�

2(p) ity of the transformationq1—~q2which, from the

mathematicalpoint of view, may take placedue ton (p) Xconst the crossingof the branchesq1(p) and q2(p) in the

complexplaneofp. Thisprocesshasbeenstudiedin

andthereforeexpressions(3.12) can be written as refs. [1,2] for whistler wavesin plasmasandin thepresentcasecanbe treatedin a similar way. When

1/2

A= 2 ~ C+ 0(g2), the profile �(p) is smoothenoughto satisfy thecon-

e—g � ditions of the WKB approximation,the transfor-

B 2 n+pn’/2 mationq1~q2is weak andthusthe trappingof the— —~——~-~- C+O(g~), (4.11) modeq1(p) is almostperfect. Therefore,the trans-— formationq1 —~q2lookslike a smalltunneling(there-where c=c(p), n is a solution of eq. (3.11) and C fore it wascalledin refs. [1,2] “tunnelingtransfor-

is a constant.Neartheturningpoint (p—po),expres- mation”). It displays itself as radiation from thesions (4.9) are invalid. Usinga standardapproach waveguidewith the samefrequencyw andlongitu-(cf. ref. [7]) we havethe following asymptoticsat dinal wave numberp as the quasitrappedfield, butP>Po, whereq~(p) <0, with the perpendicularwavenumberq2. Also, the

Fm(P)=A [2itp,.J—q~(p)] — 1/2 tunnelingtransformationshouldresult in the energyattenuationof the quasitrappedmodeq1 (p) evenif

p thereis no dissipation.The WKB solution of eqs.xexp(_J~~ dp_i$qi dp+~imE). (2.5), describingtheseprocesses,hasthe form

P0 0Fm(p)F~(P)+F~(P),

(4.l2a)

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Volume 154,number5,6 PHYSICSLETTERSA 8 April 1991

Gm(P)G~(p)+G~(p), (5.1) p=p...(ct3)+~p,the lower sign in (3.9) leadstoq2=q~andthe upperone to q2=q~.)

where To find the relationsbetweenA ‘~, A ‘1’s A ‘1” andA2,

F5,,h)(p)=[2irpq,(p)]’I2 one canusean approachsimilar to ref. [2]. Con-

sider an analytic continuation of the functionsF (p) and ~ (p) in the complex plane. Evi-

x {A~exp[i(~ q1 dp_~mE+~1t)]

dently,the pointsp= 0 andp = cc aswell asthe rootso of the equation

p

+A’i’exp[_i(J q, dp_imE+~1L)]} q,(p)=q2(p) (5.6)o arebranchpointsof thefunctionsFm(p). Introduce

(P<Po), (5.2a) a cut alongthe negativeimaginaryaxis (i.e.betweenp=0andp=—icc) andletp1 betherootofeq. (5.6)

Ft,,’ (p)=A ‘1” [2~tp~J— qf(p) 1 —1/2 with thesmallestpositive imaginarypart.Then, after

P P0 a passalong thecontourL (fig. 2), the first terminxex~(_$~ dp—i $ q, dp+ iimit) ~ transformsinto ~ Thisgives

P0 0 P1

(P>Po), (5.2b) C~eXP(iJ(~i_~2)dP)C2. (5.7)0

F~(p)=A2[2npq1(p)}”2

Deriving (5.7), oneshould takeinto accountthatP

after a passround the branchpoint p,, the signbe-Xexp [~(J q

2 ~— ~m~+ ~)]~ (5.3) fore p in (5.5a)mustbe changedbecausethe tran-0

sition q1-~q2, accordingto (3.9), is attainedby theand similar expressionsfor G~ andG5,~),obtained changep—~—p.from (5.2) and (5.3) by the substitutions Introducingthe transformationcoefficientas

As’,A’1”—+B’,, B’1’, B’1”, A2—~B2, H,.(q2) 1C212

T=rlr(q) ~ (5.8)±~+R~t, (5.4)

we havewhere

1/2

c’1, Imp

�—g

1/2

�+g \..J~B~=2~_T-R(_~_)C’,, (5.5a)

etc.,and L

1/2

A2 = 2 ~ ~ C2,

1/2

B2=2~-±~P(_~_)C2. (5.5b)

In (5.5a),therefractiveindexn shouldbetakenfrom(3.11) with the lower sign andin (5.5b) with the Fig. 2. Contoursofanalyticalcontinuationin thecomplexplaneupper sign. (It is easy to check that for p.

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Volume 154, number5,6 PHYSICSLETTERSA 8 April 1991

From energy conservation followsp1

T__exp(_21m$(qi_q2)dp). (5.9) ICfl2=IC’,12+1C

212. (5.17)

0

From (5.8), (5.16) and (5.17) followsThe energy leakage from the waveguideis, propor-

P0

tional to T, andit resultsin intensityattenuationin (4 j q’1 dp)= 1 + T.

thez direction.Thismeansthatp musthavea p05- expitive imaginarypart (see(2.3)). Thereforeq~(p;p) 0

shouldalso havean imaginarypart. DenoteExpandingthe exponentand using (5.13) we ob-

p=p°+ip’ (p’>O) , tam,finally,PU —l

q~=q°~—iq’~,n=l, 2, (5.10) w __________

wherep and q’~are small quantities,becauseT is 0 ~ dP) T. (5.18)small. Then

Relation (5.18) expresses the attenuation rateq’~~— (Oq°~/Op°)p’. (5.11) throughthetransformationcoefficient(5.9) (cf. ref.The wave attenuationrateis [2]). An analysisof formula (5.9) showsthat for

waveguides with broad smooth profiles the atten-/2 (w/C)p’ . (5.12) uation rate is exponentiallysmall. However, for a

sufficientlynarrow waveguide,with theradiuscorn-After simplecalculationswe haveparableto the longitudinalwavelengthc/cop, Tmay

vgz(qn°,p°)C be of the orderof unity. This canbe important,in0

vgr(qn,p°) w~’ (5.13) particular,for nonlinearself-focusing[4].In conclusion,we haveshown that the ambient

where v5~and v5,. are componentsof the group ve-locity vector, magneticfield plays an importantrole in the wave-

guidepropagationof electro-magneticwavesin gy-Now,continuing(5.2b) into the regionp <Poalong rotropic media.Under certainconditions,a wave,contoursL1 and L2 (fig. 2) we have

which is guidedwithout a magneticfield, leaksoutP0 from the waveguidebecauseof the linear transfor-

C’, = C’1” exp(— 2i Jq~dp+ im7t), (5.14a) mationinto a newnonguidedmodeappearingin the0 presenceof a magneticfield. The wave leakage(or

radiation) is weakfor ratherbroadwaveguideswithC’1’=C1”, C~1’~=C~1’~*. (5.l4b)

smoothprofiles.Fora sufficientlynarrowwaveguideIn the zerothorder approximation,whenthe trans- theradiationmaybestrongenoughto resultin asig-formationq~—q2is neglected,oneshoulddrop q’1 in nificant attenuationof the intensity of the guided(5.14b) andwrite C’, C’,’”, as in (4.9). Thisgives wave.Thismayplayan importantrole, in particular,C’, C’,” and (cf. (4.13)) for nonlinearself-focusing [4].

PUJo Acknowledgement

n±l,±2 n+~tn>0. (5.15)Thiswork waspartially performedduring thestay

Taking into accountthe transformationq~—oq2,we of theauthorat the Departmentof Mechanicalandhave Aerospace Engineering of the Rutgers University of

P0 New Jersey.

IC~I=lC~IexP(2J~~dP). (5.16)

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Volume 154, number5,6 PHYSICSLETTERSA 8 April 1991

References [4] V.1. Karpman,Phys.Lett. A 154 (1991)238.[5] L.D. Landauand E.M. Lifshitz, The electrodynamicsof

[1) V.1. Karpmanand RN. Kaufman,Pis’ma Zh. Eksp. Teor. continuousmedia(Pergamon,Oxford, 1983).Fiz. 33 (1981)266 [JETP Lett. 33 (1981)252]. [6] V.M. AgranovichandV.L. Ginzburg, Crystal optics with

[2]V.1. KarpmanandR.N.Kaufman,J. PlasmaPhys.27 (1982) spatialdispersion,andexcitons(Springer,Berlin, 1984).225. [71L.D. LandauandE.M. Lifshitz, Quantummechanics;non-

[3]R.N. Kaufman, Izv. Vyssh Uchebn. Zaved. Radiofiz. 28 relativistictheory(Addison-Wesley,Reading,1965).(1985)566 [in Russian].

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