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Accurate

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Accurate

    Accurate

    TransactionsofTianjinUniversity

    Vo1.9No.3Sep.2003

    AccurateJonesMatofthePracticalFaradayRotator

    WANGLin-dou(王林斗),ZHUShong-~(tg弄翔),LIYu(李玉峰),

    )(INGWtm4ie(邢文;l!1),I/t~1Jing-zhi(魂秉芝

    (SchoolofElectronicInformationEngineering,TianjinUniversity,Tianjin300072,China) Abstract:TheJonesmatrixofpracticalFaradayrotatorsisoftenusedintheengineeringcalcul

    re ationofnon

    ciprocalopticalfield.Nevertheless,onlytheapproximateJonesmatrixofpracticalFaradayrotatorshasbeen

    presentedbynow.Basedonthetheoryofpolarizedlight,thispaperpresentstheaccurateJonesmatrixofprac

    ticalFaradayrotators.Inaddition,anexperimenthasbeencarriedouttoverifythevalidityoftheaccurate

    Jonesmatrix.ThismatrixaccuratelydescribestheopticalcharacteristicsofpracticalFaradayrotators,including

    rotation.1OSSanddepolarizationofthepolarizedlight.TheaccurateJonesmatrixcanbeusedtoobtaintheac

    curateresultsforthepracticalFaradayrotatortotransformthepolarizedlight,whichpavesthewayfortheaccu

    rateanalysisandcalculationofpracticalFaradayrotatorsinrelevantengineeringapplications.

    Keywords:polarizedlight;Jonesmatrix;Faradayrotator;Jonesvector

    ArticleID:10064982(2003)03019804

    TheFaradayrotatorisoneoftheessentialelements

    0f0pticalisolatorsandopticalcirculators_1.2],whichis

widelyusedinopticalfibercommunicationandopticalpre

    cisemeasurementsystems.Asisknown,itisconvenient toresearchthetransformationsofpolarizedlighttopass throughsomeopticalelementswiththehelpofJonesmatri

    cesoftl1eelements.Therefore.itisnecessarytoknowthe Jonesmatricesoftheopticalelements.FortheidealFara. dayrotator.itsJonesmatrixwhichisfamiliartouscanbe usedinsomesituationsL.ButthepracticalFaradayrotator isanon.idealrotatorwhoseperformanceisdifferentfrom thatofRnidealone.Uptonow.onlyaninaccurateJones matrixoftl1epracticalFaradayrotatorhasbeendeveloped andappliedL4J.Calculationsdemonstrateattl1ereareer.

    rorsintheresultsobtainedthroughtheuseofeinaccu.

    rateJonesmatrix.Therefore.tl1eresearchofanaccurate JonesmatrixofthepracticalFaradayrotatorisofconsider. ablesignificancenotonlyintheorybutinpractice.Inthis paper,anaccurateJonesmatrixofthepracticalFaraday rotatorisdeveloped.TheexpressionoftheaccurateJones matrixisderivedfromtl1eopticalpolarizationtheory.In addition,thedifferencebetweentheaccurateJonesmatrix andtheinaccurateoneisalsodiscussed.

    1Theoreticaldevelopment

    1.1DerivationoftheaccurateJonesmatrix

    ThepracticalFaradayrotatorsareallnon?-idealFara?- dayrotators.TheJonesmatrixofanidealFaradayrotator isonlyarotationmatrix[.

    However.theJonesmatrixofa

    practicalFaradayrotatorinvolvestheextinctionratioand insertionlossbesidestl1erotation.

    Firstofall,letusderivetheJonesmatrixofaFara.

    dayrotatorwhichhasarotationangleofFandextinction ratioofEF(dB).ThisFaradayrotatorsurelytransformsa linearlypolarizedlightintoallellipticallypolarizedlight withallazimuthalanglerotatedbyFandellipticityofeF,

    whereeF=10F/20.ItCanbeaSSumedthatthe0Derati0n oftheFaradayrotatorisdividedintotwooperations,rota

    tionandellipticalpolarization.Iftherotationoperationis expressedbymatrixFlandtheellipticalpolarizationoper

    ationbymatrixF2,theoperationoftheFaradayrotatorcan beexpressedbytheproductofFtandF2.TheFtisfami

    liartous.itis[]

    si

    CO

    n

    S

    

    4'F-

    

    sin

    4']

    AssumingthatFzisexpressedby

    J12

    (1)

    (2)

    whereJ(i,k=1,2)arestilltobedetermined.Ifaninci

    dentlightishorizontallypolarizedanditsopticalpoweris Accepteddate:20030606.

    WANGLindou,bornin1946,male,Prof

    Email:tedaz~x@163.tom

WANGLindouetal:Accurate.toesMatrixofthePracticalFaradayRotator

    umty,ltsJonesvectoris

    =

    ItistransformedbyF2into

    E=,Ei=[:]

    AccordingtotheVenconditions,

    polarizedandcanbeexpressedby E0h=

    whereZF=10-Lr/20.

    Eq.(13)isthederivedresult. ,.,

    1.2Adaptability

    ThecalculationshowsthattheidentityF=FTex

    ists,where

    c4)=](14)

    E.hstbeellipticallyTherefore,thematrixFcanbeappliedt.allincidentlight

    witharbitrarypolarizationdirection. (5)2Discussi0Ils

    wherejistheunitofimaginarynumber.Theaandbmust

    satisfy

    rb

    a

    =eF

    【?2+6=1

    Therefore

    J

    6eF

    SubstitutingEq.(7)intoEq.(5)yields 1j]

    (6)

(7)

    (8)

    1j(9)

    Similarly,replacingthehorizontallypolarizedincidentlight withaverticallypolarizedincidentlightwecanobtain 嘲志?

    AccordingtoEq.(2),Eq.(9)andEq.(10),F2be

    SotheJonesmatrixoftheFaradayrotatorisobminedas :=

    -

    .

    sin4'

    OS~)F2:cjJ

    (12)

    ThenforthepracticalFaradayrotator,itsinsertion lossshouldbeinvolved.Assumingthatitsinsertionlossis (dB),theaccurateJonesmatrixofthepracticalFara

    dayrotatorcanbeexpressedas

    ;-j

    2.1CalculationwiththeaccurateJonesmatrix Assumingthatanincidentlightbeamisalinearlypo- larizedlightwitharbitrarypolarizationdirectionanditsop- ticalpowerisunity,itsJonesvectoris

    E[

    cos

    

    O](15)

    whereshowstheazimuthalangleofthepolarizationdi- rection.

    AfterpassingthroughtthepracticalFaradayrotator,

    theJonesvectoroftheemerginglightbeamcanbecalcu- latedintermsofthetheoryofmatrixopticsas COS-sin4'1[.J-jLrsinc~s~l

    Thee

    (16)

    ofthe

    squaresumofeachabsolutecomponentinEq.(16),itis P.=()[(coscossinsin)+

    ?1+e;

    e

    2

    F(COSsin+sinCOS)+

    e;(COSCOSsinsin)+

    (sin(bCOSlf,+cossinlf,)]=(17)

    Accordingtotheemengopticalpower(P.f)and

    incidentopticalpower(unity),theinsertionlosscanbe calculated,itis

    L=10lg=?F(dB)(18)

    Eq.(18)showsthatthecalculatedresultisidentical withtheassumedloss.sotheresultcalculatedbytheac. curatematrixhasnotanyerror.

    2.2CalculationwiththeinaccurateJonesmatrix TheinaccurateJonesmatrix[4]mentionedaboveis zr

    Accordingtothe

    

    sin11

    c.sJJ](19)

    samecalculationprocedureofsection

.--——

    199?--——

    TransactionsofTianfinUniversityVo1.9No.32003 2.1,theJonesvectoroftheemerginglightbeamcanbe calculatedbytheinaccuratematrixas

    co

    ns-sin~b]雌一]eF]rc~s~1=

    r(COS(bCOSsin(bsin)]

    fFIjev(ncos~bf,LjeF(COSCOSsinsin)J

    Andtheemerngopticalpoweris

    Pi=[(COSCOSsinsin)+

    e2

    F(COSsin+sinCOS)+

    e2

    F(coscossinsin)+

    (sinCOS+COSsin)]=z;(1+e;)

    (21)

    Theinsertionlosscalculatedbytheinaccuratematrixis Li=10lg[f;(1+e;)](dB)(22)

    Eq.(22)showsthatthecalculatedresultisnotidenti

    calwi山山eassumedinsertionloss(LF),whichmeansthe resultcalculatedbytheinaccuratematrixhassomeerrors. Forconvenience,theerrorscanbeanalyzedintermsofthe valuesofthecalculatedemergingpower.Theabsolutee

    Foroftheresultcalculatedbytheinaccuratematrixis APsl=PtiPn=Z2Fe2F(23)

    Anditsre1ativeerroris

    Rl=(APsl/P.)×100%=e2F×100%(24)

    AccordingtoEq.(24),thecurareoftherelativeerroris

    showninFig.1.Fig.1showsthattheresultcalculatedby theinaccuratematrixhasobviouserrorsunlessthevalueof eFapproacheszero.Themaximumvalueoftherelativeer? lt'oris100%.

    .1

    EllipticityeF

    Relationshipbetweenrelativeerrorofemerging powerandellipticityeF

    ample.Ageneralisolatoriscomposedoftwopolarizersand aFaradayrotatorbetweenthem.Ifthepolarizingaxisof apolarizerisparalleltothehorizontaldirectionandthepo? larizerhasanextinctionratioofEP(dB)andinsertionloss ofLP(dB),theJonesmatrixofthepolarizercanbeex

    pressedas[]

    z,

    where1.:10~p/2oandep:10.

    Intheisolator.therelativeanglebetweenthetwopo? larizingaxesoftwopolarizersissetto45..Ifthepolari

    zingaxisofapolarizeris45.fromthehorizontaldirection, itsJonesmatrixcanbeobtainedfrom

    P=P0T

    

    whereToisgivenby

    (25).

    (26)

    Eq.(14)andP0isgivenbyEq.

    Substituting0=45.intoEq.(26)yields

    

    1

    2

p

    1_[1+ep-ep(27)

    Intheisolator.erotatingangleoftheFaradayrotator mustbe45..FromEq.(13),theJonesmatrixofF=45.

    shouldbe

    F45=

    2:-+ej-eeFeFJ+JJ(28)

    Assumingthattheincidentlightintheforwarddirec?

    tionoftlleisolatoriS

    Accordingtothetheoryofpolarizationlight,the

    lightthroughtheisolatorshouldbe P45F45Po

    (29)

    emery_ng

    (30)

    SubstitutingEq.(25),Eq.(27)andEq.(28)intoEq.

    (30)yields

    .=4~-/2

    +

    /F[

    .,

    j

    

    -

    epeF]

    2?l+e;L.,ePF

    Sotheemergingopticalpowercanbe tothesquaresumofeachabsolute (31),itis

    P0Iltl

Z4(1+e2Pe2F)

    1+e;

    (31)

    calculatedaccording

    componentofEq.

    (32)

    Bycomparison,theinaccurateJonesmatrixmen

    3Apracticalexampletionedab0veiswrittenas[]

    Apracticalisusedtoexaminetheaccurate COS~)

    

    F-

    

    sin~b

    j

    

    ,matrix?Iisc.nVenien.useageneralis.la.rasanexIn

    seningF:45.int0Eq.(33),thensubstitutingitf0r

    

    20o

    ,J0^I

WANGLindouetal:AccurateJonesMatrixofthePracticalFaradayRotator

    F45inEq.(30),theemergingopticalpoweroftheisolator

    is[4]

    Pout2=Z4P2F(1+eP2e2F)

    Iftheinsertionlossesofthe areallzero(dB),thatislP gesinto

    (34)

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