光纤的光束质量因子计算

Optik

Optik121(2010)452–456

Optics

www.elsevier.de/ijleo

Beamqualityfactorofmixedmodesemergingfromamultimodestep-index ber

YuqingFua,GuoyingFenga,Ã,DayongZhangb,JianguoChena,ShouhuanZhoua,b

ab

SchoolofElectronicandInformationEngineering,SichuanUniversity,Chengdu610064,ChinaNorthChinaResearchInstituteofElectro-Optics,Beijing100015,China

Received26March2008;accepted20August2008

Abstract

Thebeamqualityfactors(orM2parameter)ofcoherentandincoherentsuperpositionoftheseverallower-orderLPmodesemergingfromastep-index berhavebeencalculatedbyusingthesecond-ordermomentmethod.Theresultsindicatethat,foranindividualLPmode,theM2parametertakesitsmaximumvaluewhenthenormalizedfrequencyVofthestep-index berapproachesthecut-offfrequency,anditgraduallybecomesconstantasVincreases.Inthecaseofincoherentsuperposition,thelargerthefractionintensitycarriedbythehigher-ordermode,thelargerthebeamqualityfactorM2.Undercertaincircumstances,thevalueoftheM2parameterofthemixedmodethatcomprisesseveralLP-modescontentsmaybecomeevensmallerandclosertotheidealGaussianbeamthanthatofthefundamentalmodeinastep-index ber.However,inthecaseofthecoherentsuperposition,thevalueoftheM2parameterofthemixedmodemaybegreaterthanthatofthehigher-orderconstituentmode.Theresultsreportedherecouldbehelpfulfortheapplicationofthehigh-power berlasersystems.r2008ElsevierGmbH.Allrightsreserved.

Keywords:Electric elddistribution;Beamqualityfactor;Linearpolarizationmode(LP);Step-index ber

1.Introduction

Thedevelopmentrateofthe berlasersissoquickinrecentyearsthattheiroutputpowerisintherangeofkilowatt[1].Forsuchhighoutputpower,thenonlineareffectsinthe bersuchasstimulatedBrillouinscatteringandstimulatedRamanscatteringbecomethemajorfactors,limitingfurtherpowerupscaling[2,3].Inordertoreducethelimitationsthattheseeffectsimposeontheoutputpower,onecanincreasetheactive bercoreareawhilemaintainingahighbeamqualityofthelaseroutput.However,followingtheincreaseofthe bercore

Correspondingauthor.

area,itwillhappenthatthe bersupportsseveralspatialmodes,andthehigh-ordermodesmayeventuallydegradethebeamqualityoftheoutputradiation[4].So,itisnecessarytostudythe elddistributionsofthehigh-ordermodesofthemultimodestep-index bersandthepropagationcharacteristicsofthebeamsgeneratedfromthelasersbasedonthesetypesof bers.

2.Theoreticalcalculations

Thestep-index ber,undertheconsiderationinthiswork,ismadeofacore,acladdingandacoatedlayer;therefractiveindexofthecladdingisslightlysmallerthanthatofthecoreasshowninFig.1[5].Theradiusof

E-mailaddress:guoying_feng@http://(G.Feng).

0030-4026/$-seefrontmatterr2008ElsevierGmbH.Allrightsreserved.doi:10.1016/j.ijleo.2008.08.003

光纤的光束质量因子计算

Y.Fuetal./Optik121(2010)452–456

453

nn1

n2

covercladdingcore

n3

0abcr

Fig.1.Refractiveindexdistributionofthestep-index ber.

the bercoreisa,andthatofthecladdingandthecoatinglayerarebandc,respectively,andthecorrespondingrefractiveindexesarem1,m2andm3,respectively.Thebeamenergywillmostlybelimitedinthe bercoreifthemediumsofthecoreandcladdingmakeupawaveguidestructurewhenm14m2.

Accordingtothescalarmethod[5],theelectromag-netic eldsoftheLPmnmodesfortheweaklyguidedstep-index bersatisfythestandardBesselequation:d2R1dRdrþrdr

þðk2n2Àb2mnÀm2=r2

ÞR¼0(1)

wherethefunctionRdescribesthevariationoftheelectromagnetic eldalongtheradialcoordinater,kisthewavenumberinfreespace,bmnisthepropagationconstantthatdependsontheordermn.TherefractiveindexminEq.(1)ism1forthe bercoreareaandm2forthecladdingarea.

Usingthestandardprocedureandassumingay-polarizationwave,the elddistributioncanbeobtained,whichreads

E1

y¼

JJmðUmnr=aÞcosðmyÞexpðÀjb(2)

mðUmnÞmnzÞHx¼

om01

bJmðUmnr=aÞcosðmyÞexpðÀjb(3)

mðUmnÞ

mnzÞ

mnJforthe bercorearea,andE1

y¼

KWKmðWmnr=aÞcosðmyÞexpðÀjb(4)

mnÞ

mnzÞ

mðHomx¼

01

bKmðWmnr=aÞcosðmyÞexpðÀjbmðWmnÞ

mnzÞ

mnK(5)

forthecladdingarea.Intheaboveequation,JmandKmarethemth-orderBesselfunctionofthe rstkindandthemodi edBesselfunctionofthesecondkind,respectively,nidenti edthenthrootoftheBessel

function.Umn¼ðk21Àb2mnÞ

1=2

aisthenormalizedtrans-versepropagationconstantinthe bercorearea,whilek1¼m1kforthewavenumberinthe bercoreareaand

Wmn¼ðb2mnÀk2mnÞ

1=2

aisthenormalizedtransversepropagationconstantinthe bercladdingarea,while

k2¼m2kforthewavenumberinthe bercladdingarea.Manymethodscanbeusedtomeasurethebeamqualityfactorofthebeaminfreespace,suchastheaggregationtwo-pointmethod,waisttwo-pointmethod,focusingtwo-pointmethod,three-pointmethod,multi-point ttingmethod,etc.Inthiswork,weadoptthelastonetomeasurethebeamqualityfactorexitingthe berlaser,inaccordancewiththeISO9000standard.Thismethodinvolvesevaluatingthehalf-widthwofthebeamindifferentpositionsalongthepropagationaxesat rst,andthendeterminingthebeampro lebyusinghyperboliccurve tting,and nallycalculatingtheM2factor.Asiswellknown,thehyperboliccurve ttingequationofthebeamisgivenby[6]w2ðzÞ¼AþBzþCz2

(6)

Byusingtheknowledgeofmathematicalstatisticstoobtainthecoef cientofthehyperboliccurve,wehavethefollowingformulasforthebeamwaisto0,positionz0ofthebeamwaistanddivergenceanglefofthebeam:

wq ÀB=4C;zp0¼A2

0¼ÀB=2C;f¼ C(7)TheM2factorcanbespeci edbyusingtherelationM2¼pw0f/l.

Atthisstage,theissueofcalculatingthebeamqualityfactordeterminesthequantityw2(z).AccordingtoSiegman[7],thez-dependentbeamwidt …… 此处隐藏：10234字，全部文档内容请下载后查看。喜欢就下载吧 ……