university of southampton research repository eprints soton › 30238 › 1 ›...

University of Southampton Research Repository

ePrints Soton

Copyright © and Moral Rights for this thesis are retained by the author and/or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder/s. The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.

When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given e.g.

AUTHOR (year of submission) "Full thesis title", University of Southampton, name of the University School or Department, PhD Thesis, pagination

http://eprints.soton.ac.uk

http://eprints.soton.ac.uk/

UNIVERSITY OF SOUTHAMPTON

Faculty of Engineering and Applied Science

Department of Electronics and Computer Science

All-Fibre Devices for WDM Optical Communications

by

Carlos Feio Gama Alegria

A thesis submitted for the degree of

Doctor of Philosophy

DECEMBER 2001

UNIVERSITYOFSOUTHAMPTONABSTRACT

FacultyofEngineeringandAppliedScienceDepartmentofElectronicsandComputerScience

DoctorofPhilosophy

ALL-FIBREDEVICESFORWDMOPTICALCOMMUNICATIONSbyCarlosFeioGamaAlegria

This thesis is concerned with the study of two key technologies for enablingwavelength division multiplexed optical communication systems. The first is gainequalisation of the erbium-doped fibre amplifier and the second is the routing ofoptical channels through the network by means of all-fibre add-drop multiplexerconfigurations. Firstly, in order to flatten dynamically the EDFA gain spectrum, an AO filterbased on a multi-tapered fibre structure was demonstrated. The controlled taperprofilewasusedasanotherdegreeoffreedomfortailoringthefilterlossspectrum.Thecouplingbetweenthefundamentalandseveralcladdingmodeswasinvestigatedbystudyingtheevolutionoftheresonanceconditionsasthefibresareprogressivelytapered both theoretically and experimentally. The filter was demonstrated byequalising the EDFA gain spectrum for different saturation levels. The mainadvantage of this novel design when compared to alternative AO filters is itssimplicityduetothereducednumberoftuningparameters.Furthermore,amethodof determining the ideal filter loss spectrum and correct placement within theamplifierwasanalysed.ThisisbasedoncalculatingtheEDFwavelengthdependentbackgroundlossnecessarytoequalisetheamplifiergainspectrum,andintegratingitintoadiscretenumberof filtersplacedwithin theEDFA.Configurationsbasedonone and two equalising filters were compared. Additionally, this method allowednovelcomplexfilterdesigns,whichcouldcompensatefortheirowninsertionlossesaswellastheinsertionlossesofotherdevicesdistributedalongtheamplifier,whileachievingaflatgainspectrum. Secondly, all-fibre OADMs based on the inscription of Bragg gratings in thewaist of fused fibre-couplers were investigated. Design considerations of devicesbased on half- and full-cycle couplers were presented and their performancescompared. Inboththeseconfigurationstheexactpositioningof thegratingswithinthe fused coupler waist is critical to achieve optimum performance. An all-fibrecompact add-drop multiplexer based on a novel non-uniform half-cycle fusedcoupler is presented, providing an alternative OADM design with optimisedsymmetricoperation,whichisinsensitivetothepositionofthegratinginthecouplerwaist.Thespectralperformanceofthis3cmlongdeviceissimilartothatofadevicebasedonameter-longuniformhalf-cyclecoupler.Finally,atechniqueforthenon-destructivecharacterisationofcouplers isproposed, inorder todetermine the3dBpoints within the couplers waist. A CO2 laser beam is scanned along the couplerlength inducing a local perturbation to the coupler eigenmodes. Asymmetric andsymmetricperturbationscangiveaccuratemappingofpower-evolutionandcoupler-waistshape.

Contents

Acknowledgments………………………………………………………..…..viii

I INTRODUCTIONCONTENTS............................................................................................................... III

1THESISOVERVIEW............................................................................................ 2

1.1 WAVELENGTHDIVISIONMULTIPLEXING ..................................................... 3

1.2 MOTIVATION ................................................................................................ 4

1.3 MAINACHIEVEMENTS.................................................................................. 5

1.4 SUMMARYOFTHETHESIS............................................................................. 6

2INTRODUCTIONTOTHEEDFA...................................................................... 8

2.1 EDFAOVERVIEW ........................................................................................ 9

2.2 THEORY ..................................................................................................... 10

2.2.1 Energy levels ..................................................................................... 10

2.2.2 Numerical modelling of spectral properties...................................... 13

2.3 NOISEFIGURE............................................................................................. 15

2.4 LARGERBANDWIDTH ................................................................................. 17

2.5 GAINEQUALISATION .................................................................................. 18

2.6 SUMMARY .................................................................................................. 20

iv

3INTRODUCTIONTOADD-DROPMULTIPLEXERS.................................. 21

3.1 OPTICALADD-DROPTECHNOLOGY ........................................................... 22

3.2 ADD-DROPCONFIGURATIONS .................................................................... 23

3.2.1 ReconfigurableAdd-Drops ............................................................... 27

3.3 ADD-DROPPERFORMANCE ........................................................................ 28

3.3.1 IsolationandCrosstalk ..................................................................... 28

3.3.2 Insertionlosses.................................................................................. 29

3.3.3 Back-reflections................................................................................. 30

3.4 SUMMARY .................................................................................................. 31

4INTRODUCTIONTOFIBRE-COUPLERS..................................................... 32

4.1 COUPLERTECHNOLOGY ............................................................................. 33

4.2 THEORETICALCOUPLERDESCRIPTION....................................................... 33

4.3 FABRICATIONOFFUSEDFIBRECOUPLERS ................................................. 37

4.3.1 Flame-BrushTechnique .................................................................... 37

4.3.2 CO2Laser.......................................................................................... 40

4.3.3 HeatingOven..................................................................................... 41

4.3.4 ShapeoftheTaperedRegion ............................................................ 41

4.3.5 Effectofthetaperedtransitiononthecouplerpowerevolution....... 44

4.3.6 Couplercrosssection ........................................................................ 48

4.4 SUMMARY .................................................................................................. 48

5INTRODUCTIONTOFIBREBRAGGGRATINGS...................................... 50

5.1 PHASEMATCHINGCONDITIONS ................................................................. 51

5.2 MATHEMATICALDESCRIPTIONOFBRAGGGRATINGS................................ 53

5.2.1 Coupledmodeequations ................................................................... 53

5.3 APODISATION ............................................................................................. 57

5.4 TRANSFERMATRIX .................................................................................... 60

5.5 PHOTOSENSITIVITY .................................................................................... 61

5.6 SUMMARY .................................................................................................. 62

v

II EDFAGAINEQUALISATION

6ACOUSTO-OPTICTUNABLEFILTERDESIGN ......................................... 63

6.1 ACOUSTO-OPTICTECHNOLOGY .................................................................. 64

6.2 THEORY ..................................................................................................... 66

6.2.1 Propagationoftheacousticwave ..................................................... 66

6.2.2 Opticalmodesintaperedfibres ........................................................ 68

6.2.3 Acousto-opticinteraction .................................................................. 71

6.3 EXPERIMENTS............................................................................................. 77

6.3.1 Characterisationofthedispersionrelations..................................... 78

6.3.2 FlatteningtheEDFAASEspectrum.................................................. 81

6.4 SUMMARY .................................................................................................. 86

7IDEALFILTERDESIGNFOREDFAGAINEQUALISATION.................. 88

7.1 INTRODUCTION........................................................................................... 89

7.1.1 TheoreticalModel ............................................................................. 90

7.2 THEORETICALFILTERDESIGN: ................................................................... 90

7.2.1 Effectofthefibrebackgroundloss.................................................... 92

7.3 DESIGNOFPRACTICALFILTERS .................................................................. 97

7.3.1 Idealfilter–Noinsertionloss........................................................... 97

7.3.2 Inclusionofthefilterinsertionloss................................................. 108

7.3.3 Filterdesignscompensatingthedeviceowninsertionloss ............ 113

7.3.4 EDFAEqualisationbyusingtheinverseofthegainspectrum....... 120

7.3.5 Conclusions ..................................................................................... 122

7.4 GAINFLATTENINGFILTERSCOMPENSATINGFORTHEINSERTIONLOSSESOF

OTHERDEVICES .................................................................................................... 124

7.4.1 EqualisationoftheEDFAwithalumplosspositionedatZ=2m ... 125

7.4.2 Equalisationofa(EDFA+isolator)structure: ............................. 132

7.4.3 Conclusions ..................................................................................... 137

7.5 SUMMARY ................................................................................................ 137

vi

III Add-DropMultiplexers

8ALL-FIBREADD-DROPMULTIPLEXERS................................................. 139

8.1 OVERVIEW ............................................................................................... 140

8.2 NUMERICALMODEL................................................................................. 140

8.3 ADD-DROPCONFIGURATIONS .................................................................. 143

8.3.1 Grating-baseduniformhalf-cyclefibrecouplerOADM................. 144

8.3.2 Grating-baseduniformfull-cyclefibrecouplerOADM.................. 160

8.3.3 Grating-basednon-uniformfibrecouplerOADM. ......................... 165

8.4 SUMMARY ................................................................................................ 176

9CHARACTERISATIONOFFIBRE-COUPLERS ........................................ 178

9.1 INTRODUCTION......................................................................................... 179

9.2 LOCALPERTURBATIONCOUPLERCHARACTERISATIONTECHNIQUE ........ 180

9.2.1 GeneralDescriptionoftheProposedMethod ................................ 180

9.3 THEORETICALMODEL.............................................................................. 182

9.3.1 CouplerDescription........................................................................ 182

9.3.2 EffectofExternalPerturbation ....................................................... 183

9.3.3 Asymmetricperturbationsofnon-idealcouplers ............................ 191

9.3.4 OutputRelativePhaseMeasurements............................................. 193

9.4 NUMERICALSIMULATIONS....................................................................... 194

9.4.1 Overlapintegralsbetweenthecouplereigenmodesandthe

perturbationprofile. ........................................................................................ 194

9.4.2 CouplerPerturbationResults.......................................................... 200

9.4.3 Perturbationsofnon-idealcouplers ............................................... 206

9.4.4 OutputPhasePerturbation ............................................................. 212

9.5 EXPERIMENTALRESULTS ......................................................................... 213

9.5.1 Characterisationofahalf-cyclecoupler[φ(L)=π] ........................ 215

9.5.2 Characterisationofafull-cyclecoupler[φ(L)=2π] ....................... 217

9.5.3 Characterisationofacomplexnon-uniformπcoupler................... 220

9.6 SUMMARY ............................................................................................221~~

vii

IV SUMMARY

10SUMMARYOFTHESIS ................................................................................ 224

10.1 EDFAGAINEQUALISATION ..................................................................... 225

10.2 ADD-DROPMULTIPLEXERS ....................................................................... 226

10.3 FUTUREWORK......................................................................................... 226

Appendix A… … … … … … … … … … … … … … … … … … … … … … … … ..… … .228

Appendix B… … … … … … … … … … … … … … … … … … … … ..… … … … ..… ...230

Appendix C… … ..… … … … … … … … … … … … … … … … … … … … … … … .....232

Appendix D… … … ...… … … … … … … … … … … … … … … … … … … … .… … ...240

References… … … … ..… … … … … … … … … … … … … … … … … … … … … … .244

List of Publications...................................................................................254

viii

Acknowledgements

During my studies at the ORC of the University of Southampton I have had the

pleasuretoworkanddiscussdifferentaspectsofoptoelectronicswithextraordinary

people.IamgratefultoProf.D.Payneforgivingmetheopportunityofstudyingat

theORCand to thePortugueseFundaçãoparaaCiênciaeTecnologia for funding

myPhD.

Among other people that have passed by, or are still at the ORC, I’d like to

thank Prof. D. Richardson, Prof R. Eason and Dr. E. Tarbox for giving me

confidence in my work, M. Ibsen, Dr. Y. S. Kim, Dr. C. Renaud for useful

discussions,R.Haaksmanforallhislogisticalhelp,theORCsecretariesEveSmith

andHeatherSpencer forhelpingme innumerous situations. I’dalso like to thank

everyonewhichwhomIhaveworkeddirectlyinthelaboratoriesfromwhomIhave

acquiredmanytechnicalskills,namely,F.Ghiringhelli,G.Brambilla,M.Ibsen,Dr.

R.Feced,Dr.M.Gunning,Dr.M.Durkin,NielP.FaganandSimonButler.

In particular I couldn’t thank enough Dr. R. Feced for all his help during the

initialstagesofmyPhDandJ.MackenzieandDr.E.Tarboxforgoingoutoftheir

way, taking the task of proofreading my thesis. I am also grateful to Prof. M. N.

Zervas, forhisexcellent supervisionof theworkandcommentson the thesis, and

RichardLamingfororiginallyacceptingmeashisPhDstudent.

I am also very grateful to all my friends that in one way or another gave me

supportthroughoutmystayinSouthamptonandinparticular;Isabel,Ricardo,Jacob

and the Sparrows. Finally, I would like to thank my family for all their support

duringmyPhD,withoutwhomIwouldn’tbewritingtheselines.

1

ThesisOverview

1–ThesisOverview 3

1.1 WavelengthDivisionMultiplexing

The advent of the Internet and global spread of personal computers has

revolutionisedourwayoflifeinthelast10years.Theabilitytocommunicate,shop,

travel, find information, listen to radio, get medical support, and so many other

aspectsofthedaybydaylife,areaccessiblewithasimplemouse-click.Thedemand

forbettermultimediaservicesandtheincreasingnumberofInternetusershasgiven

rise to an increased demand on the optical network capacity and efficiency, in all

sectors - local area networks (LAN), metropolitan networks (METRO), and long-

haulsystems.Consequently,theneedtotransmitgreateramountsofinformationvia

a single optical fibre, coupled with the need for low cost and more efficient

distributionnodes inLAN[1],has led to the increasing importanceofwavelength

division multiplexed (WDM) systems. These networks transmit several channels

corresponding to different wavelengths in the same optical fibre as illustrated in

Figure 1.1. Different channels are launched in a single fibre by means of a

multiplexer and after transmission through an amplified link, separated using a

demultiplexer.Forthepracticalimplementationofthesemulti-wavelengthnetworks

several network key technologies have to be available; which include equalised

opticalamplifiers,opticalswitchesandcross-connects,andadd-dropmultiplexers.

Figure1.1-BasicrepresentationofaWDMtransmissionlink.

In LANs the technologyused is typically selectedon the cost-effectivenessof the

individual devices, due to the quantity of local nodes and components involved.

Therefore, utilisation of all-fibre devices is very attractive due to the wide


availability and relatively low cost of optical fibres; consequently significant

researchhasbeenaimedat thisarea.However, in long-haul transmissionsystems,

emphasis is given to long-term stability and performance of the technologies

employed. Recently the utilisation of fibre amplifiers operating at different

wavelength bands (S, L and C) led to a system trial that demonstrated a record

transmission capacity of 6.4Tbits/s using WDM technology [2]. In contrast, for

opticaltimedomainmultiplexing(OTDM)systemsthemaximumbitrateachieved

was1.28Tbit/s[3].

This thesis is aimed mainly at investigating two components used in WDM

systems namely; gain equalised erbium-doped fibre amplifiers (EDFAs) and all-

fibre add-drop multiplexer configurations. An acousto-optic tunable filter for the

dynamicequalisationoftheEDFAgainspectrumisdemonstratedandatheoretical

studyoftheidealfiltershapeandplacementintheamplifierisperformed.Different

add-drop configurations based on the inscription of gratings in the waist of fused

fibre-couplers are investigated and a novel device based on a non-uniform fibre

couplerisdemonstrated.Thesensitivityoftheperformanceofthesedevicesonthe

positioninthecouplerwaistwherethegratingiswritten,hasledtothedevelopment

ofanoveltechniqueforthecharacterisationoffibrecouplers.

1.2 Motivation

The main motivation for this research was to develop an understanding of the

aspects related to EDFA gain flattening and routing of signals in WDM optical

communicationsandtodemonstratenoveldevicesormethodsthatmaybeusedin

suchnetworks.Thekeytopicsunderlyingthisworkcanbesummarisedasfollows:

• Todevelopanunderstandingofthedesignandfabricationaspectsrelatedtoadd-

drop multiplexers based on a Bragg grating inscribed in the waist of a fibre-

coupler.


• TodevelopanunderstandingofEDFAgainequalisingfiltersandconfigurations

andtodemonstrateanacousto-optictunablefilterforequalisingtheEDFAgain

spectrumfordifferentamplifiersaturations.

• To demonstrate a compact all fibre add-drop multiplexer with symmetric

operation.

• Todeveloppersonalexperimental,research,engineeringandsoftwareskills.

1.3 MainAchievements

ThisthesisisfocusedmainlyontwoaspectsofWDMopticalcommunications:First

theneedfortheequalisationoftheEDFAgainspectrumandsecondlytheselective

routingofdifferentopticalchannelsbymeansofadd-dropmultiplexers.

Chronologicallytheworkwasinitiatedbydevelopinganacousto-optictunable

filter forequalising theEDFAgainspectrumunderdifferentsaturationconditions.

This device was demonstrated as a simple (easier to reconfigure) although less

flexible alternative to solving the problem. Secondly, a theoretical study of ideal

filters for the EDFA gain equalisation was performed giving an insight into the

possibilities and limitations for extrinsic filters placed either outside or within the

EDFA.

Thesecondaspectoftheworkwasdirectedtowardsthedemonstrationofnovel

add-dropmultiplexerdesignsbasedon inscriptionofgratings in thewaistof fibre

couplers. This project has led to an understanding of aspects related to the

performanceof thesedevicesandhowtheycanbeaddressedpractically.Firstly,a

novelmethodforcharacterisingfibre-couplersbasedonalocalperturbationinduced

by a CO2 laser beam was developed and secondly, a novel add-drop multiplexer

designwasdemonstrated.

DuringmyPhDstudiesnumerousfibre-couplershavebeenfabricatedandthen

inscribed UV induced Bragg-gratings in their waist. The procedure undertaken on


the fibre couplers from the time of fabrication was optimised during the work

according to the facilities available. A considerable amount of time has also been

spent modelling fibre propagation characteristics, add-drop multiplexers based on

fibre couplers with a grating inscribed in the waist, and the local perturbation of

fibrecouplers.

1.4 Summaryofthethesis

This thesis investigates two technologies essential for the deployment of WDM

networks.ThefirstisequalisationoftheEDFAgainspectrum,andthesecondisthe

routingofchannelsthroughall-fibreadd-dropmultiplexerconfigurations.Thethesis

isdividedinfoursections:

-SectionIisanintroductiontothedevicesandtechnologiesinvolvedinthisstudy.

Following this chapter, which puts this thesis into context and outlines the

motivation for the work, chapter 2 introduces the EDFA, and the different issues

relevant to its performance in optical networks. Chapter 3 discusses the add-drop

multiplexer,thedefiningparametersthatareusedtocharacterisetheirperformance

and addresses different configurations used for routing WDM channels. It also

reviews the technologies investigated to date and the advantages or drawbacks or

otherwise of each. Next, in chapter 4, fibre couplers are introduced, where the

understandingandoptimisationofthesedevicesisessentialforoptimisationofadd-

dropmultiplexerconfigurations investigated in this thesis.Couplersaloneare also

important components in WDM networks used to route, split, or combine optical

signalsandthereforeasignificantpartofthisthesisisdedicatedtothem.Theadd-

drop configurations investigated in this work rely on Bragg gratings for filtering

selected wavelengths; chapter 5 provides a brief introduction to these devices and

the subsequent issues related to this thesis. The second and third sections of this

thesisaretheauthor’ scontributiontotheareaofopticalcommunications.


-Section II addresses the equalisationof theEDFAgain spectrum. In chapter6 a

novel technique for tailoring the loss spectrum of an acoustooptic (AO) filter is

proposed. The application of the technique is demonstrated by dynamically

equalising the amplified spontaneous emission (ASE) spectrum of an EDFA for

different saturating input signals. The operation of the device relies on simpler

tuningconditionscomparedtosimilaralternativetechnologies.Chapter7presentsa

theoreticalandnumericalstudyofidealfiltersfortheequalisationoftheEDFAgain

spectrum.Itdiscussesamethodfordeterminingtherequiredidealfiltershapesand

placement position in the amplifier in order to obtain the best performance whilst

equalising the EDFA gain spectrum. It is shown that the optical filter can be

properly designed in order to compensate for its own insertion loss as well as of

otherdevicesincorporatedintheEDFA.

- Section III is dedicated to all-fibre add-drop multiplexer configurations. It

addresses three compact all-fibre configurations basedon the inscriptionof Bragg

gratings in the waist of fibre-couplers. Design and fabrication issues for each of

these configurations are addressed in chapter 8. The need for an experimental

methodforcharacterisingthefibre-couplers,inordertocorrectlypositiontheBragg

gratingswithin thecouplerwaist, led to thedevelopmentofanovel technique for

the non-destructive characterisation of fibre-couplers. This technique is based on

scanninga locally inducedperturbationalong thecouplerwaist toobtain the taper

andwaistprofileanddeterminetheevolutionofpoweralongthecoupler,aswellas,

theshapeofthecouplerwaistandcouplingconstantdistribution.Thisisaddressed

theoreticallyandexperimentallyinchapter9.

- SectionIVdrawsconclusionsregardingtheabove-mentionedtopics,concluding

withpossibledirectionsforthisresearchtocontinue.

2

IntroductiontotheEDFA

A general introduction to the EDFA is presented in this chapter. It starts with an

overview of the implementation of this component in optical communications

networks, thenabriefdescriptionof theamplifieroperationandhowtomodel the

spectral characteristics. Important amplifier parameters such as the optical noise

figure,amplifierbandwidth,andmethodstoachieveequalisationoftheEDFAgain

spectrumareintroduced.

2–IntroductiontotheEDFA 9

2.1 EDFAOverview

TheinventionoftheEDFAinthelateeighties[4,5]wasoneofthemajoreventsin

the history of optical communications. It provided new life to the optical fibre

transmission window centred at 1.55µm and the consequent research into

technologiesthatallowhighbit-ratetransmissionoverlongdistances.Highbit-rates

werealsopossiblewiththeaidofdifferentdispersioncompensationschemes.The

basicconfigurationforincorporatingtheEDFAinanopticalfibrelinkisshownin

Figure 2.1. The signals and pump are combined through a WDM coupler and

launched into an erbium-doped fibre. The amplified output signals can be

transmittedthrough60-100kmbeforefurtheramplificationisrequired.

Figure2.1-BasicconfigurationfortheincorporationofanEDFAinanopticalfibrelink.

IngeneraltheEDFAhasanarrowhighgainpeakcentredcloseto1532nmand

abroadpeakwith lowergain centred at 1550nm.The initialWDMschemesused

fewwavelengths(typically4)acrossthebroadflatamplificationregion.Inorderto

take advantage of the whole amplification band provided by the EDFA gain

spectrumearly equalisation schemeswhere employed [6].However, the useof an

increasednumberofchannelsinthepresentDWDMopticalnetworksrequiresaflat

gain spectrum across the whole usable bandwidth. Different EDFA equalisation

schemesarediscussedinsection2.5.

In order to further increase the capacity of DWDM optical fibre networks,

research efforts have been made to increase the amplified bandwidth either by

shiftingthegainspectrumofconventionalEDFAstolongerwavelengthsorbyusing


newdopantsandglassestoprovideamplificationatdifferentwavelengthbands(see

section2.4)orbyusingRamanamplifiers.

EDFAs have been used successfully in WDM transmission systems as all-

opticallumpedamplifiersatwhichthegainisboostedatapointofthetransmission

line. On the other hand, the fibre amplifiers based on Raman effect also have

attracted huge research attention nowadays due to its tunability of amplification

bandbysimplychangingpumpwavelength,sinceever-increasingdemandofoptical

data transmission capacity expansion in telecommunications has generated

enormous interest inopticalcommunicationbands (S-,L-band) [7,8]outsideofa

conventionalEDFAgainbandwidth(C-band).TheprincipleoftheRamanamplifier

is based on the stimulated emission process associated with Raman scattering in

fibrefortheamplificationofsignals.Theinelasticnon-lineareffectscanberegarded

as scattering of a pump beam off phonon (molecular vibrational state) and the

transfer of energy into a lower energy beam. The Stokes shift corresponds to the

Eigen-energy of an optical phonon, which is approximately 13.2 THz for optical

fibres. InRamanamplifiers,signalwavelengthis longerthanpumpwavelengthby

the equivalent amountof the frequency shift. By usingmultiplepumps across the

targetgainwindow,over100nmbandRamanamplifierscanbeachieved [9].The

majordrawbacksofthistechnologyaretherequirementofhighpumppowerorlong

length of fibre and the related Rayleigh scattering issue. However, availability of

cheapandhighpowerpumplasers,andhighlynon-linearfibresenablesfibreRaman

amplifierstobeapromisingtechnologyfortheincreaseoftransmissioncapacityof

currentandfutureWDMnetworks.

2.2 Theory

2.2.1 Energylevels

TheEDFA absorption andemission cross sections are the signatureof the energy

levelsoftheEr3+ionintheglasshost.Whentheerbiumionisintroducedintoahost


medium the energy levels are modified by local electric fields through Stark-

splitting. These levels are in thermal equilibrium due to rapid nonradiative

transitions between these levels. The amplifier is assumed to have homogeneous

broadeningbutifthelocalelectricfieldisdifferentatvarioussitesalongoracross

thefibreduetoimpurities,clusteringeffects,orotherglassstructuraldisorders,then

inhomogeneous broadening occurs resulting in different electronic transitions at

respectivesites.TheincorporationofanetworkmodifiersuchasAluminium(Al)to

enhance the solubility of the Er3+ ions in the glass structure changes each energy

level’ sStark-splittingandincreasestheinhomogeneityof themedium.Theenergy

transitions typically associatedwithEr3+ ina silicateglassare the 4I11/2,4I13/2, and4I15/2states,andareillustratedinFigure2.2.

Figure2.2–a)Energy leveldiagram forEr3+ ions showing thedominant transitions. b)

Stark-splittingoftheenergylevelsduetothecrystalorglasselectricfield.

W12,W21aretheratesforthestimulatedtransitionswhileA32andA21aretherates

forthespontaneousemission.A32isassumedtobeessentiallynonradiativeandA21

essentially radiative [10].Thesubscripts1,2 and3correspond respectively to the

energylevels4I15/2,4I13/2,and4I11/2.


GenerallytheEDFAispumpedwith980nmradiation,excitingelectronsfrom

the ground state 4I15/2 to level 4I11/2 or at 1480nm by exciting electrons from the

groundstatetoahigh-energyStark-splitsublevelofthe4I13/2manifold.Rigorously

this implies, when pumping the EDFA using a wavelength of 980nm, that the

amplifiercorrespondstoathree-levelsystemwhilewhenusinga1480nmpumpthe

amplifierisaquasithree-levelsystem(aspumpingistoahigher-energyStark-split

statewithin the I13/2manifold).However,bothpumpingschemescanbedescribed

effectivelyintermsofthepopulationsoftwolevels.Thisapproximationisjustified

inthe980nmpumpingcaseduetothenonradiativedecayrateA32beingmuchlarger

thanthestimulatedemissionratefrom3to1,andthereforethepopulationoflevel3

(4I11/2) can be neglected. In the case of 1480nm pumping the two-level system is

justified due to the rapid thermalisation decay that transfers the higher-energy

electronsof the4I13/2manifold to lower-energyStarksublevels.Therateequations

forthepopulationsofatwo-levelsystemarewrittenas:

2212211122 nAnWnW

dtdN −−= (2.1a)

21 nnnt += (2.1b)

wherentistheEr3+iondensityandn1andn2thefractionaldensityofthelowerand

upperexcitedlevelsrespectively.Theseequationsholdevenforthemorecomplex

system where the manifolds are split into Stark sublevels. In this situation the

transitionratescorrespond to thesumoverall thepossible j-k (j,k=1,2) transitions

multiplied by the population weight of the transition, given by the Boltzman

distribution[10]. Inpracticehowever theexactenergy levelscorresponding to the

individual Stark levels are dependent upon the ion distribution and host material.

Thusthepopulationanddecayratesfortheenergylevelsofinterest,typicallyhave

to be determined experimentally through absorption and emission cross section

measurements.


2.2.2 Numericalmodellingofspectralproperties

The wavelength dependent properties of EDFAs can be modelled following the

method proposed by [11] in which the spatial characteristics of the amplifier are

integrated. This model involved dividing the EDFA spectrum into discrete optical

channels of frequency bandwidth, ∆νk, centred at the optical wavelength λk.

Assuming homogeneous broadening and a uniform distribution of the Er3+ ions

across the fibre core, the amplifier can be characterised by introducing four

measurablefibreparameters:Theabsorptionspectrum,αk,thegainspectrumg*k,the

fibresaturationpower,PkSat,andthefibrebackgroundloss,lk,thataregivenby:

tkekk ng Γ= σ* (2.2a)

tkakk nΓ= σα (2.2b)

** )( kk

k

kk

teffkSatk g

hg

nAhP

+=

+=

αξν

ταν

(2.2c)

Where; σak and σek are respectively the wavelength dependent absorption and

emissioncrosssections,ntisthetotalconcentrationoftheerbiumions,ξ=Aeffnt/τis

theratioofthelineardensityoferbiumionstothefluorescencelifetime,Aeff=πb2eff

istheeffectiveareaofthedopedregion,τisthemetastablelevel2lifetime,andΓk

istheoverlapintegralbetweenthedopantandopticalmodedistributionsthatinthe

caseofuniformdopingoftheerbiumions(beff=b)isgivenby:

=Γπ

φφ2

0 0

),(b

kk rdrdrI (2.3)

WherebistheradiusoftheEr3+-dopedregion.Ifthisassumptionisunrealisticthen

modification of the integral is required to include the Er3+ ion distribution. The


aboveoverlapintegraldependsingeneralonthewavelengthchannel,k,forwhichit

iscalculated.Understeady-stateoperation,assumingauniformdistributionfor the

excitedlowerstateandupperstatepopulations(n1andn2respectively),theexcited

upperstatepopulationdensityfortheEDFAisgivenby[11]:

+

+=

kSat

k

k

kSat

k

k

kk

k

t

PzPP

zPg

nn

)(1

)(*

2 αα

(2.4a)

21 nnnt += (2.4b)

Theequationsthatdescribethepropagationofthebeamsofwavelengthλkandthe

pumpthroughthefibreare[10]:

( ) ( )

+−∆++= )()( 2*2* zPlmh

nn

gzPnn

gudzdP

kkkkkt

kkt

kkkk αννα (2.5a)

( ) ( )

+−+= )()(2* zPlzP

nn

gudz

dPpumppumppumppump

tpumppumpk

pump αα (2.5b)

Pk(z) is the signal power at frequency λk at a certain position along the amplifier

length; uk represents the direction of the travelling beam uk=1 for a forward

propagating beam and uk=-1 for backward propagation; the term mhνk∆νk is the

contributionofthespontaneousemissionfromthelocalexcitedstatepopulationn2,

withm=2correspondingtothenumberofpolarisationmodessupportedbythefibre,

andh thePlankconstant; lk is awavelengthdependentbackground loss.Thus the

two-level amplifier system can be fully characterised using equations (2.5a) and

(2.5b)thatdescribethepropagationofthesignal,ASEandpumpalongtheerbium-

dopedfibreandequation(2.4.a)describingthepopulationinversionandsaturation

characteristicsalong theamplifier.Whenusing apumpwavelengthof980nm, the


gaincoefficient isnull 0g*980 = andequation(2.5b)describingthepumpevolution

alongtheEDFcanbesimplified.

Details of the model used herein for numerical simulations of the EDFA

performance are described in Chapter 7. Briefly though it was implemented by

dividingthefullEDFAbandwidth(from1420nmto1620nm)intoequalsegments.

The wavelength dependence of α(λ) and g*(λ) were obtained by digitising

absorptionandgainparametersmeasuredforanactualEDFasillustratedinFigure

2.3.Usingthemeasuredvalueforthefibrebackgroundlosslbgandtheratioofion

density to the fluorescence lifetime ξ, the rate and propagation equations were

solveduntilthespecifiedconvergenceparameterswerereached.

0

1

2

3

4

5

6

7

1420 1470 1520 1570 1620Wavelength(nm)

Abs

orpt

ion/

Gai

n(d

B/m

)

g*(λ)

α(λ)

Figure2.3–Measuredabsorptionandgainparametersforthefibreusedinthenumerical

simulations.

2.3 Noisefigure

The analysis of noise in optical systems is sufficiently complex that it can be

characterised either with simple engineering formulae or by a thorough quantum

theoreticalapproach.Itisnottheaimofthissectiontoprovideadeepintroduction

tonoise inopticalsystems,but rather togive thebasicdefinitions,whichquantify

theopticalnoisegenerationintheEDFA.Thesedefinitionswillbeusedinchapter6


todiscusstheeffectontheEDFAperformance,intermsofanoisefigure,whenthe

concept of gain equalising filters is introduced. The optical noise figure is a

parameter used for quantifying the noise penalty added to a signal due to the

insertionofanopticalamplifier.Thatis,beforelightentersanamplifierthesignalto

noiseratioisSNR(0),afteramplificationitisSNR(z).Thus,opticalnoisefigurecan

bedefinedas:

)()0(

zSNRSNR

NFOpt = (2.6)

Ifthenoisefigureoftheamplifierwere1,thentheinitialsignaltonoiseratiowould

be maintained throughout amplification. However it has been shown that the

quantumlimitforanopticalamplifier[10]is3dB,thereforethesignaltonoiseratio

afteramplificationishalf(50%)oftheoriginalvalue.Forrealopticalamplifiersthe

noise figure can be as high as 6dB whereby the signal quality is sufficiently

deteriorated that the detector’ s ability to discriminate signal from noise is

compromised.

The signal to noise ratio can be described as the ratio between the average

signal intensity and the standard deviation of intensity fluctuations from that

average.Thedefinition follows in termsof theaveragenumberofphotons<n(z)>

andthevarianceσ2=<n(z)2>-<n(z)>2:

)(

)(2

2

z

znSNR

σ= (2.7)

wherezisthepositionalongtheamplifierorfibrelink.Ithasalsobeenshown,[10]

thatthenoisefigureofanopticalamplifiercanbedescribedas:

)(1

)(1)(

2zGzG

zGnNF

kk

kspOpt +−= (2.8)


whereGk(z)istheamplifiergainatagivenposition,z,atawavelengthλkandwhere

nspisthespontaneousemissionfactorthattakestheform:

12

2

NN

Nn

ek

aksp

σσ−

= (2.9)

Here, N1 and N2 are the populations of the ground and excited energy levels

respectively. For a total population inversion N1=0, nsp=1 and therefore the noise

figure is close to 2, which is the quantum limit for the amplifier noise. The

spontaneous emission factor is related to the total power of the amplified

spontaneousemissionPASEwithinthebandwidth,∆νk,bythefollowingexpression

[10]:

( ) kk

ASEsp hG

Pn

νν∆−=

12 (2.10)

2.4 Largerbandwidth

Theusable35nmbandwidthof theEDFAoperating in theConventionalband (C-

Band)enabled fibre communicationsusingWDMandDWDM.Howevergrowing

demand for increased bandwidth and subsequent research have given rise to fibre

amplificationatshorterandlongerwavelengthbands.TheL-bandEDFA,wherethe

EDFA gain is shifted to the longer wavelengths (1560nm-1580nm) [12], in

conjunctionwiththerecentlydemonstratedThulium-dopedfibreamplifieroperating

attheS-band(shortwavelengths)around1490nm[13],providethebasisforfuture

transmission capacity of 10Tbits/s channels multiplexed across the three amplifier

bands [2].Figure2.4 illustrates the threeamplificationbandwidthscoveredby the

threetypesofamplifiers.


1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 1.62Wavelength (µm)

L-Band

S-Band

C-Band

Figure2.4–Wavelengthbandwidthcoveredbytheamplifiers.

2.5 Gainequalisation

Equalisationofanamplifier’ s gain spectrum isessential forbalancing thechannel

powersinordertoachieveerrorfreedetectionofthesignalstransmittedthroughthe

opticalfibrelink.SeveralmethodsforachievingEDFAequalisation,eitherintrinsic

or extrinsic, have been proposed in the literature. Intrinsic methods constitute

changing the spectroscopic properties of the erbium-doped glass absorption and

emission cross sections by co-doping with other ions, different glass matrices or

special fibre designs. Fluoride-based glasses [14, 15] are known to improve the

flatness of the EDFA gain spectrum. Extrinsic methods are based on filtering

devicesthataredesignedwithawavelengthdependentlossspectrum.Severalfilters

have been demonstrated in the literature [16-26]. These can be divided in active

devicesthatarere-configurable,whichmayaccommodatechangesintheamplifier

gain spectrum due to saturation effects, and passive devices that cannot be tuned.

Active devices reported include; acousto-optic tunable filters [16-19], strain-tuned

fibre Bragg gratings [27], micro mechanical filters [24], and a planar integrated

opticalfilter[25].Somepassivedevicesincludelongperiodgratings[21,23],Bragg

gratings[22],andfiltersusingSamariumdopedfibres[20].Allthesedeviceshave


characteristicequalisationpropertiesand insertion losses thatcanbeas lowas the

splicinglossbetweentheEDFfibreandthefilterfibreorashigh8-9dBasreported

in[24,25].

In chapter 6 an acousto-optic tunable filter based on the profile of a multi-

taperedoptical fibre and its spectral transmissionproperties [18,19], isdiscussed.

The amplified spontaneous emission spectrum of an EDFA was equalised for

different saturation levels in order to demonstrate the potential of the device. The

tunable parameters of the device were, the acoustic wave frequency and the filter

lossshape,whichwasdependentuponthetaperedfibreprofile.Althoughthisfilter

lacks the flexibility of reshaping the spectral profile, it is very easy to tune

dependingonlyon2to4parametersasopposedtothe12tuningparametersofother

designs[17].

Figure 2.5 – Basic EDFA gain flattening configurations. Top: Filter placed outside the

amplifier.Bottom:Filterplacedwithintheamplifier.

Determinationof the ideal filtershape inorder toequalise theEDFAisnota

trivialtask.ThelossspectrumoffiltersplacedoutsidetheEDFA(configuration1in

Figure2.5)canbeobtainedbyinvertingtheamplifieroutputgainacrossthedesired

bandwidth.Althoughthelossduetotheinsertionof thefilter,dependentuponthe


type of filter and the fabrication procedure, can be up to 8-9dB [24, 25], and

thereforeanotheramplificationstageisusuallyrequiredafterthefilter.Ifhowever,

thefilterisplacedatacertainpositioninsidetheEDFA(configuration2inFigure

2.5), the penalty in amplifier loss can be reduced but the exact filter shape and

placement is not known. Liaw [20] used the loss spectrum of a samarium-doped

fibreandfoundthebestpositionatwhichitshouldbeplacedinagivenamplifierby

splicingitatdifferentpositionsalongtheamplifier.Acoustooptictunablefilters[26]

have alsobeenused in this configuration andoptimisedby tuning the filter shape

untilthedesiredperformanceisreached.Thisisaniterativeprocessandquitetime

consuming, as the filters may not be placed in the optimum position along the

amplifier. A solution to these problems is proposed in chapter 7, where the

theoreticaldesignofidealfiltersthatinadditiontogainflatteningalsocompensate

forinsertionlosses,andtheirpositionwithinanamplifierforequalisingtheEDFA

gain spectrum is discussed. Performance of the above filter configurations is

compared.

2.6 Summary

AbriefintroductiontotheEDFA,oneofthemostimportantcomponentsinWDM

communications,wasgiveninthisChapter.Startingwithfundamentalprinciplesof

amplifier operation, a well-known model based on a two-level amplifier system

includingthespectralcharacteristicsoftheEDFA,waspresented.Importantissues

relatingtotheamplifierperformance,namelytheopticalnoisefigureandamplified

bandwidthwereintroduced.Finally,thechapterconcludedwithareviewofexisting

technologies utilised for equalising the EDFA gain spectrum. The concepts

introducedinthischapterarefundamentaltosectionIIwheretheequalisationofthe

EDFAgainspectrumisaddressedinmoredetail.

3

IntroductiontoAdd-Drop

Multiplexers

Inthischapterdifferentchannelroutingtechnologiesarereviewed,highlightingthe

advantages and drawbacks of the different devices and configurations. The

parameterstocharacterisetheperformanceoftheadd-dropmultiplexersaredefined.

3-IntroductiontoAdd-DropMultiplexers 22

3.1 OpticalAdd-DropTechnology

Theevolutionofsinglewavelengthpoint-to-pointtransmissionlinestowavelength

division multiplexed optical networks has introduced a demand for wavelength

selective optical add-drop multiplexers (OADM) to separate/route different

wavelengthchannels.Theycanbeusedatdifferentpointsalongtheopticallinkto

insert/remove or route selected channels increasing the network flexibility. This

feature is particularly important in metropolitan WDM lightwave services where

officesorsitescanbeconnectedbydifferentadd-dropchannels,forexampleinan

interofficering.Additionallythereisflexibilityoftransmittingdifferentdataratesin

differentWDMchannelsaccordingtothecapacityneeds.Figure3.1illustrates the

basic operation of an add-drop multiplexer where a stream of 16 channels with

central wavelengths λ1 through λ16 are launched into the input (port 1) and 8

channels are dropped at port 4, the rest go through port 2. Simultaneously, 4

channels are launched into port 3 and added to the signal stream at port 2. The

channelsthatareaddedordroppedatthatnodedependonthenetworkrequirements.

Figure3.1-Basicoperationofanopticaladd-dropmultiplexer.

TherearetwomaintypesofOADMthatcanbeusedinWDMopticalnetworks;

fixed OADMs that are used to drop or add data signals on dedicated WDM

channels,andreconfigurableOADMsthathavetheabilitytoelectronicallyalterthe

selected channel routing through the optical network. The main features of the

second type of OADM is to provide flexibility in rerouting optical streams,


bypassingfaultyconnections,allowingminimalservicedisruptionandtheabilityto

adaptorupgradetheopticalnetworktodifferentWDMtechnologies.

Configurations presented in the literature to perform the required add or drop

functions use both planar and fibre technology. Planar devices [28-36] provide

compact solutionswith thepossibilityofaddingordroppingmanychannelsusing

onlyoneintegratedopticalcircuitusingarrayed-waveguide-grating(AWG)[34]or-

waveguide-grating-router (WGR) technology [35, 36]. The main drawbacks of

planardevicesaretheirhighinsertionloss,whichcanbeashighas7dB,andtheir

polarisation dependence. Alternatively, all-fibre devices [37-47] are attractive

solutionsdue to their low insertion losses,polarisation insensitivity (dependingon

thefibreandconfiguration)andeaseofcouplingbetweendeviceoutputandinputs

oftheopticalnetworkusingsimplesplicesandpigtails.Typically,duetotheirlarger

dimensionsthesedevicesaresensitivetoenvironmentalvariations,dependentupon

the configuration.Devicesbased in free spaceoptics (micromirrors andgratings)

have also been used successfully to perform add-drop operations with good

performance[48].Although,thesedevicesareingeneralmoreexpensiveandhave

relatively high insertion losses. Finally thin film filter devices have been

traditionally used for multiplexing/demultiplexers purposes. Fibre and planar add-

dropconfigurationsandtheirrespectiveperformancearediscussedinthefollowing

section.

3.2 Add-DropConfigurations

Excellent performance and compactness offered by four-port planar-waveguide-

based devices can be rivalled by the simple all-fibre add-drop configuration, as

showninFigure3.2.Itconsistsofa3dBsplitterandagratinginoneoftheoutput

arms; light launched into port1 is split in two, λG is reflected by the grating then

droppedatPort4.Theothercoupleroutputportisimmersedinanindexmatching

fluidsothatthelightisnotreflected.Theselectedsignalemergesatboththeinput


anddropport.Anopticalisolatoratport1protectstheinputnetworkfromtheback-

reflectedsignal.Thedroppedsignalis6dBweakerthantheoriginalinputsignal.In

transmission, a second 3dB coupler splits the signal that was not reflected by the

grating.The add function isperformedby launchinga signal intoport 3which is

reflectedbythegratingandthusaddedtothesignalatport2,asillustratedinFigure

3.2.Anisolator isalsorequiredtoisolatetheAddportfromthesignaltransmitted

fromtheinput.Whenusingthetwoisolators,at theinputandAddports,thisnon-

interferometric configuration provides excellent add-drop performance. In this

configuration thereareno limitationsonthe length,position,orapodisationof the

written grating. Ideal grating filters may be designed using an inverse scattering

method[49,50].Theprimarydrawbackofthisconfigurationistheinsertionlossto

all the channels that is at least 6dB. However, when comparing with planar-

waveguide-baseddevices,ithassimilarinsertionlossesbuthasincreasedflexibility

in writing and tuning ideal gratings. Notwithstanding, planar devices have the

advantageofcompactnessandareeasier tostabilisewithrespect toenvironmental

changes.

Figure3.2–Add-dropmultiplexerconfigurationbasedonagratingandtwo3dBcouplers.

One method to overcome the high insertion loss of the above configuration

requires an additional grating, identical to the first, written in the unused coupler

ports, thus forming a Mach-Zehnder interferometer. Both planar [29-31] and fibre

[40, 44, 46, 47, 51] devices using this configuration have been reported.


Theoreticallythisdeviceissymmetricandcanyieldexcellentperformanceinterms

ofinsertionloss,back-reflectionandcross-talk.

Figure 3.3. illustrates the principle of operation for this configuration: A 3dB

couplersplits light launched intoport1andaspecificwavelength,λG, is reflected

bythetwoidenticalgratings.Thesereflectedsignalsinterfereinthe3dBcouplerin

suchawaythatthesignalisdroppedandtheback-reflectedlightintensityarriving

at port1 is zero, providing the coupler is well matched (50% splitter). The

transmittedwavelengthsaremade to interfere in thesecond3dBcoupler such that

theyarriveattheoutputportwithnoresiduallightattheAddport,againforawell-

matched coupler. This configuration is based on the splitting and interference of

light and is therefore quite sensitive to changes in the signals path length, the

characteristics of the identical gratings, and the matching of the 3dB couplers.

Thereforeenvironmentalstabilisation,UVtrimmingoftheindividualpaths[47]and

identical couplers and gratings are essential for good device performance. The

stability and tolerances for achieving practical WDM performance using this

configuration were analysed by Erdogan [31]. This configuration in planar

technology has shorter path lengths and therefore is easier to stabilise. Also,

identical gratings can be written with one exposure simply by using a small

separationbetweentheinterferometerarms.Alternativeconfigurationsbasedonthe

dual-corefibresthatpresentshorterinterferometerarmsandavoidtheneedforUV

trimminghavebeendemonstratedaspracticaldevicesusingtheMZinterferometer

configuration[40,45].

Figure3.3–Add-dropmultiplexerconfigurationbasedonaMach-Zehnderinterferometer.


Another example of a symmetric four-port add-drop multiplexer is similar to

configuration 1 shown in Figure 3.2, with the 3dB couplers replaced by optical

circulators. Theoretically the operation of this non-interferometric device is ideal:

Thespectralpropertiesdependprincipallyontheperformanceofthegratingthatcan

bedesignedasanidealsquarefilterusinginversescatteringtechniques;theinsertion

loss and cross-talk are mainly dependent on the performance of the optical

circulators. Figure 3.4 illustrates this configuration. Light launched into port 1 is

directed intoa fibreBragggratingwithresonantwavelength,λG, reflectedback to

the circulator and dropped to port 4 with the remaining optical channels being

transmittedtoarriveatport2.Anothersignalofwavelength,λG,islaunchedinport

3,reflectedbythegratingandaddedtotheopticalstreamatport2.

Themaindrawbackof thisconfiguration is thatcirculators are expensiveand

bulky devices. However, with the advent of cheaper circulators and with low

insertionlosses,itwillbeaveryattractiveadd-dropmultiplexersolution,duetoits

inherentstabilityandperformance[51].

Figure3.4–Add-dropmultiplexerconfigurationbasedonagratingandtwocirculators.

Thestabilityof the interferometric add-dropmultiplexer shown inFigure3.3,

configuration 2, can also be improved by using the interference between the

eigenmodesofafibrecoupler.Writingagratinginthewaistofahalf-cycle(100%)

couplerhasbeendemonstratedinbothfibre[41,42]andplanarconfigurationsasa

meansofachievingadd-dropperformance.Thedeviceiscompact,but inprinciple

onlyhasanidealsymmetricperformancewhenthegratingisapoint-likereflector.


Thisisonlypossiblebyusingveryshortandstronggratingsorverylongcouplers.

Figure 3.5 shows schematically this configuration. Light launched into port 1 is

transferredtotheevenandoddeigenmodesofthecoupler.Agratingisplacedatthe

centreofthecouplerwherethephasedifferencebetweentheeigenmodesisπ/4i.e.,

wherelightisequallysplitbetweenthetwocoupledwaveguides(seechapter4for

theeignemodedescriptionofafusedcoupler).Thechannelatthegratingresonance

wavelengthλGisreflectedandtheremainingsignalspropagatethroughthecoupler

arrivingattheoutputport.Inreflection,theeigenmodesreachthebeginningofthe

coupler with a π/2 total phase difference and therefore, the channel is dropped to

port4.Inprinciple,thestabilisationofthisinterferometricdeviceisimprovedwith

respecttotheMach-Zehnder(configuration2)duetothepoint-likereflectionpoint

andtheinterferenceachievedthroughthebeatingbetweenthepropagatingcoupler

eigenmodes. However, limitations in the grating strength and the length of

fabricated couplers compromise the expected performance. Optimisation and

discussionofdifferentschemesusingconfiguration4areaddressedinChapter8.

Figure3.5–Add-dropmultiplexerconfigurationbasedongratinginscribedinthewaistof

acoupler.

3.2.1 ReconfigurableAdd-Drops

Theabilitytoreconfigureanadd-dropmultiplexerbychangingthefilterresonance

wavelengthortoswitchthedeviceonoroffprovidesextraflexibilityinanoptical

network. Compact multi-channel devices using arrayed waveguide grating

technologyhavebeenreportedwithindividualroutingofeachchannel,byswitching


itonoroff[32,34].Eventhoughlowcross-talkisachievablewithmultiplepasses

throughthemultiplexer,thesedeviceshaveunavoidablyhighinsertionlosses.

On theotherhand,all-fibreadd-dropconfigurationshavepotentiallyno cross

talk (dependingon the filter design)withvery low insertion loss.Whenusing the

non-interferometric add-drop configurations 1 or 3, wavelength selection is

achievable by straining [52] or heating [53] the Bragg grating. Whilst using the

interferometric configuration 2, both fibre gratings should be affected equally and

therefore wavelength tuning is not practicable. However, switching is possible by

unbalancingtheinterferometerbystrainingorheatingonlyoneofthearms.

3.3 Add-DropPerformance

The analogue performance of add-drop multiplexers is characterised by using

scatteringparametersSij foreachpairofports [54].Thefirstsubscript, i, refers to

the destination port and the second subscript, j, the input port. Several properties

may be characterised using the scattering parameter namely; the insertion loss,

polarisation dependent loss (PDL), dropped channel isolation, channel uniformity,

frequencyaccuracyandbandwidthconsiderations.InappendixAsystemapplication

characteristicsfortheisolationoftheopticalportsachievablewithcurrent50,100,

200 and 400 GHz channel-spacing technologies as well as, cross-talk, back-

reflectionand insertion loss requirementsaregiven.The remainingparametersare

definedtoin[54].

3.3.1 IsolationandCrosstalk

The two main parameters related to the isolation of channels in an add-drop

multiplexerarethethrough-portisolationofadroppedchannel(S21parameter)and

thedrop-portisolationofthroughchannels(S43parameter).Notethatinasymmetric

device S43=S21. These two parameters represent the sources of the interchannel

crosstalkforthedeviceillustratedinFigure3.6,wheretheS21isolationparameteris


highlighted.Iftheamountofpowerlaunchedintoport1,P1,andthedroppedpower

toport4,P4,theremainingtransmittedpower,P2,emergesatport2asinterchannel

crosstalk.Themeasureofisolationisgivenby-10log(P1/P2).

Figure3.6–ExampleoftheS21isolationofthethroughportofadroppedchannel.

The second kind of crosstalk is due to unwanted signals transferred from

neighbouringchannelstothefilteredone,andisnamedintrachannelcrosstalk[55].

Itcanappearintheinterferometricconfigurationsasaresultofanincorrectsplitting

ratio in the 3dB (50%-50%) couplers. This kind of crosstalk however, has a low

powerpenaltyintheperformanceoftheWDMsystem.

3.3.2 Insertionlosses

Insertion lossesare theattenuation in theopticalpowerof thechannelsdue to the

insertion of the device. The effect of the device insertion loss is schematically

illustratedinFigure3.7whereboththedroppedchannelandtheoutputchannelsare

attenuated.

Figure3.7–Schematicrepresentationoftheinsertionlossofanadd-dropmultiplexer.


Theinsertionloss,linscorrespondingtothetransferefficiencyoflightfromportito

portjaffectsallthechannelsequallyandisdescribedby

=

j

iins P

Pl log10

PiandPjarethepowersofagivensignalchannelattherespectiveportsassuming

thereisnocross-talkorpolarisation-dependentloss(PDL).

3.3.3 Back-reflections

Back-reflectionsaredefinedbythescatteringparametersSii.Thesubscriptiis1or3

correspondingtotheinputoraddportsrespectively.Figure3.8showsschematically

theeffectdescribedbytheseparameters.IfthechannelselectionisbasedonaBragg

gratingwitharesonancewavelengthλG(asinconfigurations1to4),thenwhenthat

channelislaunchedintoeitherport1orport3itwillbereflectedtoeitherthedrop

or out port respectively. However, there is also a percentage of light, which is

reflectedbacktotheoriginalportsP’ 1orP’ 3,thustheSiiback-reflectionparameter

is defined as 10log(Pi/P’ i). The effect of the back-reflections can be avoided by

introducingisolatorsintobothoftheseports(asshowninFigure3.2).However,the

problemcanbeavoidedbyadequateadd-dropmultiplexerbalancing.

Figure3.8–Schematicrepresentationof theS11andS33back-reflectionparametersofan

add-dropmultiplexer.


3.4 Summary

Add-dropmultiplexersaredevicesinhighdemandcompatiblewithbothLANand

longhaulnetworks.DuetothenumberofnodesusedinLANs,andthusthenumber

add-dropmultiplexersrequired,demandforcheapdevicesistheprimarymotivation.

All-fibreadd-dropmultiplexerconfigurationsarepotentialcandidatesforproviding

suchcheapdevices.Thedifferentschemeswillbefurtheraddressedinchapter8.In

summary, this chapter was a review of the existing technologies for routing

wavelength channels,withdiscussion regarding the advantages anddrawbacks for

each. Parameters, which are used to characterise the performance of add-drop

multiplexers, were also introduced. This chapter provides OADM fundamentals

relevant to section III, where the optimisation of three different all-fibre add-drop

multiplexerschemesisdiscussed.

4

IntroductiontoFibre-

Couplers

Theaimofthischapteristoprovideanoverviewoffibrecouplertechnology.The

principles of how fibre couplers exchange power between the two ports are

presented and different methods of fabrication are compared. The information

providedinthischapterintroducestheworkonthecharacterisationoffibrecouplers

(Chapter 9) and is relevant to the optimisation of all-fibre add-drop multiplexers

basedontheinscriptionofgratingsinthecouplerwaist(Chapter8).

4-IntroductiontoFibre-Couplers 33

4.1 CouplerTechnology

Fibre- and integrated-optic couplers are extremely important components in a

number of photonics applications. They are generally four-port devices and their

operation relieson thedistributed couplingbetween two individualwaveguides in

closeproximity,whichresultsinagradualpowertransferbetweenmodessupported

bythetwowaveguides.Thispowertransferandcross-couplingatthecoupleroutput

ports can be viewed also, as a result of the beating between eigenmodes of the

compositetwo-waveguidestructurealongthelengthofthecompositecouplerwaist

[56]. The most common use of fibre- and integrated-optic couplers is as a power

splitter, this is, the fibre-optic equivalentof a free spaceopticbeam-splitter.They

canbeusedtosplittheopticalpowerofanopticalchannel(ofcertainwavelength)

betweentheoutputports[57].Anotherapplicationistocombineorsplitthepower

ofdifferentchannels,correspondingtodifferentwavelengths(wavelength-division-

multiplexing (WDM) splitters/combiners) [58]. Lately fibre- and integrated-optic

couplers,havebeencombinedwithreflectiveBragggratingswrittenintheirwaist,

toprovideselectiveaddinganddroppingofdifferentchannelsinWDMsystems[41,

42].

4.2 TheoreticalCouplerDescription

A fibrecoupler isa four-portdeviceconsistingof two fibres thathavebeen fused

together, etched, or polished over a small interaction region. The mechanism

through which light is exchanged between the two fibres is dependent upon the

fabricationmethod.Whenthefibresareetchedorpolishedandpositionedinclose

proximity, the otherwise insensitive and well confined core modes interact by

exchangingpowerbetweeneach fibre coredue to theoverlapof themodes in the

commoncladding.Thestrengthofthecouplingbetweenthetwomodesisdescribed


by an overlap integral of the fields associated with each of the individual guides.

Fusedcouplersareobtainedbyfusingtogetherandstretchingtwoparalleluncoated

fibres. As the fibres are stretched the core sizes decrease until the modes (at the

wavelength of interest) are no longer guided by the core but by the composite

cladding-airstructure.Ifthetaperisadiabaticonlythetwolowest-ordereigenmodes

of this structure will be excited and the power exchange is due to the beating

betweenthesetwoeigenmodes.Intheworkpresentedhereonlyfusedfibrecouplers

arediscussed.

Figure 4.1 - Four-port coupler schematic showing the coupling region (LC), which is

comprisedoftwotaperregions(LT1,LT2)andthecouplerwaist(LW).

Consider the 2x2 coupler shown schematically in Figure 4.1. When light is

launchedintoport1,thenormalisedfieldamplitudesoftheeven(Ae)andodd(Ao)

eigenmodesatthecouplerinput(z=0)canbeapproximatedby[56]:

2

)0()0()0(;

2)0()0(

)0( 2121 AAA

AAA oe

−=+= (4.1)

whereA1(0)andA2(0)arethenormalisedamplitudesofthefieldslaunchedintothe

twoinputports1and2,respectively.Forsingleportexcitation,A1(0)=1andA2(0)=0

and,throughEquation(4.1),Ae(0)=Ao(0)=1/ 2 .Therefore,lightlaunchedintoone

of the inputportsofa2x2couplerexcitesequally the two lowest-order (evenand

odd) eigenmodes along the coupling region. The two eigenmodes propagate

adiabaticallyalong theentirecouplingregionwithpropagationconstantsβe(z)and

βo(z)respectively.Thebeatingbetweenthesetwomodesthenprovidesthecoupling

ofpoweralongthecoupler.


Even

+ + +

Odd

∆φeo 0 3π/2 2π

P1

P1

P2

P2

ππ/2

Figure4.2-Schematicofevenandoddeigenmodebeatingandtotalpowerevolutionalong

a2x2full-cycle(∆φeo=2π)coupler.

Thepropagatingtotalelectricfieldatanypointalongthecouplerisdescribedby:

+

=+=

−−z

o

z

e di

o

di

eoet ezAezAzEzEzE 00

)()(

)()()()()(ζζβζζβ

(4.2)

During adiabatic propagation, the even and odd eigenmodes retain their

amplitude(Ae(z)=Ae(0)andAo(z)=Ao(0))andchangeonlytheirrelativephase.This

results inspatialbeatingalong thecouplerwaistandpowerredistributionbetween

the two individual waveguides comprising the optical coupler. The peak field

amplitudes for each individual waveguide, along the coupling region, can be

approximatedby[56]:


[ ]

[ ]

−=−=

=+=

+−

+−

z

oe

z

oe

dioe

dioe

ezizEzE

zE

ezzEzE

zE

0

0

)()(21

2

)()(21

1

)(21

sin2

)()()(

)(21

cos2

)()()(

ζζβζβ

ζζβζβ

φ

φ

(4.3)

where [ ] −=∆==z

oe

z

eoeo ddzz00

)()()()()( ζζβζβζζβφφ is the relative

accumulatedphasedifferencebetweentheevenandoddeigenmodes.βeandβoare

the propagation constants of the even and odd eigenmodes, respectively. The

correspondingnormalisedpeakpowercarriedbytheindividualwaveguidesisgiven

byP1(2)=|E1(2)|2,namely

=

=

)(21

sin)(

)(21

cos)(

22

21

zzP

zzP

φ

φ (4.4)

At thepointsalong thecoupler,whereφ iszerooramultipleof2π, the total

powerisconcentratedpredominantlyaroundwaveguide#1(P1=1andP2=0).Atthe

pointsalongthecoupler,whereφismultipleofπ,ontheotherhand,thetotalpower

isconcentratedpredominantlyaroundwaveguide#2(P1=0andP2=1).Finally,atthe

pointswhereφ ismultipleofπ/2, the totalpower isequallysplitbetween the two

waveguides (P1=P2). The even/odd eigenmode beating and total power evolution

alongafull-cyclecoupler(φ=2π)isshownschematicallyinFigure4.2.Thecoupling

coefficient k(z) describing the strength of the interaction between the eigenmodes

andisgivenby:

2)()(

)(zz

zk oe ββ −= (4.5)


The coupler beat length LB is defined as the minimum interaction length the two

eigenmodes,initiallyinphase,musttravelinordertointerfereconstructivelyi.e.,to

beagaininphase:

oeBL

ββπ−

= 2 (4.6)

4.3 FabricationofFusedFibreCouplers

4.3.1 Flame-BrushTechnique

The flame-brush technique for the fabrication of fibre couplers is based on the

scanningofapoint-likeflamewhilepullingthefibres[59].Twofibresareclamped

parallel to each other and the flame is scanned over a given interaction region.

Figure4.3shows theexperimentalconfigurationofsucha rig for fabricatingfibre

tapersorcouplers.

Figure4.3–Flamebrushtechniqueexperimentalsetup

The couplers and tapers fabricated during this work where made using a

configuration similar to thatofFigure4.3.The fibres arepulledby two computer


controlledAerotechstages.TheflameisscannedusingathirdAerotechstage.The

flame gas consists of a mixture of isobutene and oxygen. Both cleaning and

alignmentofthefibresiscrucialforfabricatinguniformtapersorcouplerswithlow

insertion losses. Air draughts or gas pressure variations can severely affect the

quality of the devices, due to variations in the flame temperature and consequent

localnon-uniformitiesalongthetapers/couplers.Duringthepullingofthefibresthe

outputpowerismonitoredandtheprocesshaltedatthedesiredfibreradius(inthe

case of taper fabrication) or extinction ratio (in the case of coupler fabrication).

Figure4.4showsthepoweratboththeoutputports(Port3andPort4)duringthe

pulling process for a half-cycle coupler fabricated using this technique. Coupler

elongation of 46mm represents the point at which coupling of light between the

waveguides starts to occur, corresponding to the monomode regime [60]. As

illustrated, thepoweratport3drops to0Vwhile thepower inport4 increases to

around 7V. The pulling process was halted when Port 3 reached its minimum,

producingthiswayahalf-cyclecoupler.

0

2

4

6

8

46 51 56 61 66 71CouplerElongation(mm)

Cou

pled

pow

er(a

.u.)

Port4

Port3

50%splitter

100%coupler

Figure 4.4 – Power evolution of a coupler fabricated using the flame-brush technique at

λ=1.55µmduringpullingprocess.

The spectral characteristics of the fabricated couplers are determined by

launchingawhitelightsourceintooneoftheportsofthecouplerwhilstmeasuring


theoutputportswithanOpticalSpectrumAnalyser(OSA).Figure4.5illustratesthe

spectral characteristics of a 20mm long full-cycle coupler fabricated using this

technique. It is observed that the extinction ratio was better than 30dB and the

meausurement was noise-limited due to insufficient input power. The pulling

process was halted so that the full-cycle resonance peak was at λ=1.55µm. The

resonanceatλ=1.175µmis thehalf-cycleresonancecorrespondingtoa totalphase

displacementofφ(L)=π.

Disadvantagesofthisfabricationmethodare;thepossiblecontaminationofthe

tapers/couplers by the combustion by-products, the variations of the burner

temperature,andtheflamesize,thatmaynotbeapproximatedtoapointlikesource.

Notwithstanding, throughout thisworkverygoodquality tapersandcouplerswere

obtained.Infact,thequalityofthecouplersproducedwiththerig,asillustratedin

Figure4.5,providedconfidence in theuniformityof the tapersandstabilityof the

flame during the fabrication process. For example, using a standard

telecommunications single mode fibre, typical insertion losses of the fabricated

taperswereonly0.1dB.

-90

-80

-70

-60

-50

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7Wavelength(µµµµm)

Pow

er(d

Bm

)

Port3

Port4

Figure4.5–Spectralresponseofa20mmlongfull-cyclecouplerfabricatedusingtheflame

brushtechnique.


4.3.2 CO2Laser

RecentlyDimmicket.al.[61]reportedthedevelopmentofafused-fibrecouplerand

fibre taper rig that uses a scanning, focused, CO2 laser beam as the heat source,

insteadofthegasburner.Thesetupissimilartothatoftheflame-brushtechnique,

withtwopullingstagesthatstretchthefibreatadesiredspeedwhilsttheCO2laser

radiationisscannedacrossthefibresbyarotatingmirror.Thebeamisfocusedusing

aZnSelenswitha30mmfocallengthgivingaspotsizeof820µm.Anexperimental

setupusedtofabricatefibre-couplersusingaCO2laserisillustratedinFigure4.6.

Figure 4.6 – Experimental setup of the fabrication of fibre tapers/couplers using the

radiationofafocusedCO2laser.

Thissetupprovidesabettercontroloftheshapeofthetaper/couplertaperedregion

duetothesmallerhotspotproducedbythefocusedCO2laserwhencomparedtothe

flame-brush technique. It also allows greater control in producing non-uniform

tapersorcouplersduetothepossibilityofrapidpositioningofthelaserspotandfast

switchingofthelaserbeampowerwithashutter.

However,themaindisadvantageofthistechniqueisthatthetemperatureofthe

heatsourcevariesduring thepullingof thefibre.Heatingofoptical fibresusinga

lasersourcedependsonmanyparameterssuchas;theabsorptioncoefficient(which

varies with temperature and wavelength), the laser power, the fibre-cooling rate


(which depends on the fibre radius and temperature), and the laser spot size. To

overcome this problem the laser power has to be adjusted constantly in order to

maintainaconstant temperatureduring thefibrepulling. Incontrast,whenheating

with a flame burner, the presence or not of the fibre has little or no effect on the

temperatureoftheheatsourceduetothemechanismofheatgeneration.

4.3.3 HeatingOven

Anothertechniqueusedinindustryforfabricatingfibrecouplersandtapersrelieson

heatingthewholeuniformsectionusinganovenorresistiveelectricalheaterwhile

pullingthefibres.Duetothelongheatzonethistechniquehasnocontroloverthe

shape of the tapered region although the sensitivity to environmental factors is

reduced. The quality of the tapers/couplers is essentially dependent on the oven

design,andthetemperatureuniformityalongthelengthofwaistregion.

4.3.4 ShapeoftheTaperedRegion

Accuratecontrolofthetaperedregionshapeofbothfibrecouplersandfibretapers

canbecrucialfortheperformanceofdevicesusingthesecomponents.Forexample,

inchapter5anAOtunablefilterisdiscussed,whichreliesontheaccuratecontrolof

thefibre-tapershapeandlength.Birksetal.[62],usingtheflame-brushtechnique,

produceda longuniformtaperwaist (90mm)withshort transitionregions(35mm)

and very small waist diameters (~2µm), for generating a supercontinuum light

spectrum. Also in fibre couplers, the accurate control of the tapered region is

extremelyimportantforthefabricationofnon-uniformcouplersthatcanbeusedas

an add-drop multiplexer when a grating is inscribed in the waist (chapter 8). In

general, the transition region for both fibre couplers and tapers should obey the

adiabaticcriterion[63],inordertominimiseinsertionlosses.


The shape of fibre tapers/couplers produced by using scanning point-like

heatingsourceshasbeenextensivelystudiedbyBirkselal.[64].Assumingthatthe

localisedheatingofthefibremakestheglasssoftenoughtobestretchedwhilstnot

being so soft that it falls under its own weight, the shape of the tapers can be

calculated without having to recur to fluid mechanics beyond the principle of

conservation of mass. A tapered fibre, at any given time (or elongation) of the

pullingprocess,canbecharacterisedbytheparametersshowninfigure4.7a).ro is

theinitialfibreradiuscorrespondingtoatransitionlength,z0,andr(z)theradiusof

thetapertransitionatagivenpositionz.Thelengthoftheuniformtaperwaistlw(t)

isequaltothelengthofthehot-zoneL(t)atthattime.Thesizeofthehot-zoneL(t)

mayvarywithtimebutissubjecttotheconstraintsL≥0anddL/dx≤1.Thissecond

constraint ensures that the hot-zone does not overtake the pulled transitions. The

time change is proportional to the extension or elongation of the taper i.e., the

pullingspeed isconstant.Figure4.7b)shows theequivalentuntaperedfibrewhere

theinitialhotspotlength(att=0)isL0andxisthetotalpullingextensionatagiven

time.Comparingthetaperedwiththeuntaperedfibreitmaybeobservedthatpoints

AandB are elongated byx. In theparticular casewhere thehot-zone is constant

during thepulling, thewaist length isconstant lw(x)=L0 and the taper transition is

equaltohalfoftheextensionz=x/2.

Figure4.7 -Schematic representationofa fibre taper structure.a)Ata time tduring the

pulling.b)Initialfibrebeforepulling.


From the conservation of mass principle, the following expression can easily be

derived:

Lr

dxdr ww

2−= (4.7)

Secondly, the extension x can be related to the taper transition length z by

comparingtheinitiallengthABatt=0,withthetotaltaperlengthABatanygiven

time:

02 LxLz +=+ (4.8)

The particular case where the hot-zone remains constant during the fibre

extensionhasbeenanalysedby[64-66].InthiscaseL(z)=L0andz=x/2.Integrating

(4.7)givesthewaistshapeforatotalfibreextensionx.

( )00 20

)'('

2/1

0)( LxxLdx

w ererxr

x

−

−

=

= (4.9)

The taper profile is calculated by substituting x=2z in (4.9), resulting in the well-

knownexponentialdecayprofile.Allthetaperandcouplerdevicesdiscussedinthis

thesiswerefabricatedusingaconstanthot-zone,thusexpression(4.9)issufficientto

describe the profiles of the tapered regions. Further examples of interest are

discussedin[64]whereequation(4.7)isdemonstratedaswell.

Inorder tominimizelossesbetweenthefundamentalandthenearestcladding

modes,thetaperangle|dr/dz|hastoobeytheadiabaticcriterion[63].

( ) ( )( )π

ββ2

21 zzrdzdr −≤ (4.10)


Where β1(z) and β2(z) are respectively the local propagation constants of the

fundamental mode and the closest cladding modes, and r is the local core radius.

Experimentally it was observed that intrinsic loss of the fabricated couplers and

tapers using the flame-brush technique were very low and justify the use of the

aboveparametersdescribingsmoothadiabatictransitions.

4.3.5 Effect of the tapered transition on the coupler power

evolution

The long transition regions in couplers fabricatedusing the flame-brush technique

withconstanthotzone,playaroleinthewaythepowerevolvesalongthecoupler.

For a full-cycle coupler with a constant hot zone of L0=30mm fabricated with

standard telecommunications single mode fibre, the evolution of the power at the

outputports is illustrated inFigure4.8. Light fromaDFB-LD at awavelengthof

1.55µmislaunchedinport1andmonitoredatport3andport4duringthepulling

process.Thepowerevolution isonlyplotted fromanextensionofx=47mm(from

x=0tox=47therewasnocoupling)inordertoemphasisethecouplingprocess.

0

1

2

3

4

5

47 52 57 62 67 72 77CouplerElongation(mm)

Mea

sure

do

utpu

t(V

)

Port4

Port3

50%splitter

πcoupler 2π coupler

x1

∆x

x3x2x0 xm xN

........

........ ........

Figure4.8–Measuredpowerevolutionofa30mmlongfull-cyclecouplerataλ=1.55µm

duringthepullingprocess.


Light starts to be coupled between the two fibres for a coupler extension around

x=51mm,thehalf-cyclepointisreachedataroundx=73.5mmwhenallthelightisin

Port4andthepullingprocesswashaltedafteronefull-cycle,i.e.,whenalllightwas

coupledbacktoPort3.UsingtheinformationplottedinFigure4.8andthefactthat

dL/dx=0 (constant hot-zone pulling), an iterative method to extract the coupling

strengthprofileduetothetaperedtransitionregioncanbedeveloped.Afteragiven

extension,x,where coupling starts tooccur, all the interaction isdue to thewaist

section with length L0. The coupling coefficient, k(x), can be evaluated for that

extension(orequivalentlyforthatwaistradius)assumingthatthehot-zonesectionis

uniform and constant during the fabrication process, by solving equation (4.4) in

ordertodetermineφ(x)=∆β(x)L0=2k(x)L0.Nowthephasedisplacementbetweenthe

evenandoddeigenmodescorrespondingtothecoupledpowerP1(x0)atextensionx0

isgivenby:

[ ] 0011

0 )(cos)( LxPx −=∆β (4.11)

andthevalueof∆β(x1)atthenextextensionx1=x0+∆xcanbecalculatediteratively

using;

[ ] 010111

1 )()(cos)( LxxLxPx ∆∆−=∆ − ββ , (4.12)

finallyatthemthsection,xm=x0+m∆x,ityields;

[ ] −

=

− ∆∆−=∆1

0001

1 )()(cos)(m

nnmm x

Lx

LxPx ββ (4.13)

Thereaderisremindedthatz=x/2andtherefore,∆β(z)=∆β(x)/2.Usingthisgeneral

recursive expression and the coupler power evolution Port 3 (blue line in Figure

4.8),thecouplingstrength(solidline)wascalculatedandplottedinFigure4.9thisis


comparedtotheidealcoupler(dashedline)withoutataperedtransitionregion.The

originofthegraphinFigure4.9correspondstoacouplerextensionofx=47mmand

thereforea transition lengthofz=23.5mm.At thisposition thenormalisedcoupler

radiuscanbecalculatedusing(4.9)yieldingr(z=23.5)/r0=exp(-z/L0)≈0.457.

Theidealcouplerhasahighercouplingstrengthalongtheuniformwaist than

the fabricated coupler; although the total coupler phase displacement φ(L)

correspondingtotheintegrationofthecouplingstrengthalongthewholelength,is

the same in both couplers at λ=1.55µm. By comparing the power coupling in the

transition regions with that in the uniform region of the fabricated coupler, it is

realised that22.1%of the totalphasedisplacementalong thecoupler isdue to the

tapered transition regions and 77.9% due to the uniform waist. Therefore, when

optimising add-drop multiplexers based on full-cycle couplers with gratings

inscribed in thewaist,byplacing thembetween theexactpointsalong thecoupler

wherethepowerisequallysplitbetweenthefibres,thecouplertransitionregionhas

to be taken into account. However, the non-destructive coupler characterisation

methodpresentedinChapter9overcomesthisproblem.

0

0.5

1

1.5

0 7.5 15 22.5 30 37.5 45 52.5 60CouplerPosition(mm)

k(z)

(x1

04 m

-1)

Idealcoupler

Realcoupler

Figure4.9–Comparisonofthecouplingstrengthsofanideal(dashedline)andfabricated

(solidline)30mmlongfull-cyclecoupler.


The effect of the tapered transition region on the power evolution along the

coupler length is illustrated directly in Figure 4.10. Both the output coupler ports

(Port3andPort4)are shown.Thedashed line refers to the idealcouplerand the

solidlinetothefabricatedcoupler.Itisobservedthatthefabricatedcouplerislonger

and the coupling smoother corresponding to the transition regions. The coupler

positions where the power is equally distributed in both the waveguides (50-50%

points) are shifted towards the tapered regions. Identification of these coupler

positionsiscriticalfortheoptimisationofadd-dropmultiplexersbasedongratings

inscribedinthecouplerwaistandwillbediscussedinChapter8.

The accuracy of expression (4.13), in determining the coupling strength and

hencethe50-50%pointsofthecoupler,dependsontheuniformityofthehot-zone

lengthandtheadiabaticevolutionofthetaperedtransitionregionduringthepulling

process.Inordertocharacterisethecoupleranddetermineits50-50%pointsanovel

non-destructive characterisation technique for fibre couplerswasdevelopedand is

discussedinChapter9.

0

0.5

1


Nor

mal

ised

Pow

er

P1(z)

P2(z)

Figure 4.10 – Power evolution along the length of an ideal uniform (dashed line) and

fabricated(solidline)30mmlongfull-cyclecoupler.


4.3.6 Couplercrosssection

When fabricating couplers using the flame-brush technique, the degree of fusion

beforepullingthefibresdefinesthecrosssectionalshapeofthecoupler.Thehigher

the degree of fusion the closer the cross section of the fabricated coupler is to a

cylinder.Inthecaseofveryweakfusion,thecouplerhasacharacteristicdumbbell

shapeandforintermediatedegreesoffusion,thecross-sectionhasapproximatelyan

elliptical shape with varying eccentricity [67]. The theoretical description of the

couplerintermsofthecouplereigenmodes,alsoknownassupermodes,isrelatedto

thecouplercross-section,differentapproximationsforcalculatingthesemodeshave

beenaddressedintheliterature.InBureset.al.[68]thefibreswerenotfusedandthe

coreswereneglected;[69]approximatedthecouplercrosssectionusingdifferenta

rectangular cross section, and [70, 71] gives analytical expressions for the two

lowestordermodes,LP01andLP11,fordifferentcouplercross-sections(rectangular,

elliptical,circular)alsoneglectingthefibrecores.

Furtherwork, [67,72]usedaFieldCorrectionMethod toaccuratelycalculate

the coupler eigenmodes, while [73-75] use the rigorous surface integral equation

methoddeterminethecouplercharacteristics.

4.4 Summary

Fibrecouplersare importantcomponentsused inWDMsystems to routeandsplit

signals,monitorthenetwork,orcombinesignalandpumpwavelengthsforfeeding

optical amplifiers. Recently add-drop multiplexer configurations relying on the

inscriptionofBragggratingsinthecouplerwaisthavebeeninvestigated[41,42].In

order to optimise these devices accurate control of the fabrication and suitable

methods of characterisation for the couplers are required. In chapter 9 a non-

destructivemethodforcharacterisingfibre-couplersisdescribed.

In conclusion this chapter gave an introduction to coupler technologies and

described how light is transferred between the two waveguides along the coupler

length. A review of fibre-coupler fabrication technologies, their advantages and


drawbacksforeachwasdiscussed.Finallytheinfluenceofthefibrecouplerstapered

transitionregiononthepowerevolutionalongthecouplerlengthwasdescribed. It

willbeshown(inchapter8) that inorder tooptimiseanadd-dropmultiplexer; the

influenceofthetransitionregionhastobetakenintoaccount.

5

IntroductiontoFibre

BraggGratings

This chapter is a short introduction to fibre Bragg gratings aimed at providing a

fundamental understanding of the spectral properties of the filters designed using

thistechnology.Theconceptspresentedinthischapterareimportantfortheanalysis

andoptimisationofadd-dropmultiplexersbasedongratingsinscribedinthewaistof

fibre-couplers(chapter8).

5-IntroductiontoFibreBraggGratings 51

5.1 PhaseMatchingConditions

Fibregratingsallowthetransferofpowerbetweenmodesofanopticalfibre.Thisis

achievedbyperturbingthephaseofonemodesuchthatitmatchesthephaseofthe

other,“phasematchingcondition”.Fibregratingsareusuallywritteninbarefibres

where theacrylatecoating is removed.Thismeans theoptical fibrebehaves likea

three-layer structurewithdifferent effective refractive indexes in the core, n1, and

thecladding,n2,withafinaloutercladdingbeingair,n3=1.Forasinglemodefibre

withtheseparametersthecore-guidedmodehasapropagationconstantβcogivenby,

12222

nnn coco λπ

λπβ

λπ <=< (5.1)

and the cladding modes that are guided by the cladding-air structure have

propagationconstantsthatfallintherange:

2322

nn cl λπβ

λπ << (5.2)

and finally there are radiation modes that can have propagation constants in the

limit:

32

0 nrad λπβ << (5.3)

With the introductionofaperiodicvariationof theeffective indexalong the fibre

length, the first order phase matching between the fundamental and backward

propagatingfibremodes(fundamentalorcladdingmodes)occurswhen[76]:

Λ=− πββ 2

21 (5.4)


For thecaseofcouplingintothebackwardpropagatingfundamentalmode,β2=-β1

andtheresonanceconditionyields:

Λ= πβ1 (5.5)

Inexpressions(5.4)and(5.5)Λistheperiodoftheeffectiveindexmodulationand

β1,β2arerespectively, thepropagationconstantsof thefundamentalmodeandthe

mode the reflected light is coupled into. Gratings that couple to backward

propagating modes are known as reflection or Bragg gratings. Typically these

devices are based on coupling between the forward and backward fundamental

modes.

Figure5.1–Schematicrepresentationofthemodesexistinginuncoatedsinglemodefibres

andthematchingconditionforthecoremodereflection.

For long period gratings (both β1 and β2 are positive) the phase condition for

forward coupling from the fundamental mode into forward propagating cladding

modesisgivenby:

Λ=− πββ 2

21 (5.6)


5.2 MathematicalDescriptionofBraggGratings

This section describes a simple approach for obtaining the spectral properties of

fibreBragggratings.Foranextensivereviewof the theoryandpropertiesof fibre

Bragggratingsthereferences[77-79]aresuggested.

5.2.1 Coupledmodeequations

Coupledmodetheoryhasbeensuccessfullyusedtodescribethespectralproperties

ofBragggratings[78].RefractiveindexvariationswithaperiodΛalongthelength

ofafibrearegenerallyexpressedas:

( ))(2cos)()( 0 zznnzn θπ +Λ∆+= (5.7)

the functions∆n(z)andθ(z)are slowlyvarying functionscompared to thegrating

period Λ, n0 is the refractive index of the core, and ∆n(z) the envelope of the

refractive indexmodulation.Theparameter,θ(z), defines locally, thephaseof the

effectiveindexmodulation,whichisusedtodescribephaseshiftsorgratingchirp.

For simplicity this introduction will consider unchirped gratings only, therefore

θ(z)=0. Along the grating the forward propagating wave, v1, and backward

propagatingwave,v2,arerelatedbythecoupledmodeequations[80]:

1*

22

211

)(

)(

vziqvidzdv

vziqvidzdv

−+=

+−=

δ

δ (5.8)

where the amplitudes of the waves v1 and v2 are related to the amplitudes of the

forwardandbackwardpropagatingelectricfield,A(z)andB(z)respectively:


zi

zi

evzB

evzAδ

δ

+

−

=

=

2

1

)(

)( , (5.9)

q(z)isthecouplingcoefficientgivenby:

)(2

)(0

znn

zq ∆Λ

= π (5.10)

and δ represents the detuning from the Bragg grating resonance wavelength,

λBragg=2n0Λ,definedas:

Λ−= π

λπδ 0

2n (5.11)

InthecaseofBragggratingswhere∆nvariesalongthegratinglengththespectral

characteristics canbeobtainedby solving thedifferential coupledmode equations

(5.8).Theparticularcaseofauniformgratinghasbeensolvedanalytically[81],the

reflectioncoefficientρ=v1(δ)/v2(δ)andreflectivityR=|ρ|2at thestartof thegrating

(z=0)are:

)cosh()sinh()sinh(

)(LiL

Lqγγγδ

γδρ+

−= (5.12)

222

2

)(cosh)(sinh

)(qL

LR

δγγδ−

= (5.13)

whereγ2=q2−δ2.


Some important features can be inferred from these results. Firstly it can be

demonstrated that the maximum reflectivity Rmax occurs when the resonance

conditionisobserved,i.e.,δ=0andisgivenby

)(tanh2max qLR = (5.14)

and secondly the spectral bandwidth, ∆λzeros, defined as the two first zeros in

reflectivitycalculatedusing(5.13)yielding[78]:

2

0

1

∆+∆=∆

nLnn Braggzeros

λλ

λ (5.15)

For strong gratings where∆nL>>λBragg the normalised bandwidth is approximated

by:

0nnzeros ∆≈∆

λλ

(5.16)

andforweakgratingswhere∆nL<<λBraggthenormalisedbandwidthisapproximated

by:

LnBraggzeros

0

λλ

λ =∆ (5.17)

Whenwritinggratings in fibres, equation(5.15)providesuseful informationabout

the inducedeffective indexchangesimplybymeasuring thespectralbandwidthof

thegrating.Similarlyforuniformgratings,theinducedrefractiveindexchangecan

alsobecalculatedusing(5.14),bymeasuringthemaximumreflectivityattheBragg

wavelength.


To fully understand the dispersive properties of fibre Bragg gratings the

conceptofgrouportimedelaymustbeintroduced.Forauniformgratingthetime

delay can be determined from the phase of the reflection coefficient ρ defined in

(5.12). Ifθρ=phase(ρ), thenthetimedelay,τρ, for lightreflectedfromagratingis

definedas[78]:

λθ

πλ

ωθ

τ ρρρ d

d

cd

d

2

2

−== (5.18)

andtheeffectivelength, leff, that lightataparticularwavelengthtravelswithinthe

gratingbeforeitreturnstotheorigincanbecalculatedfromleff=cτρ/n0. Inuniform

gratings,theminimumtimedelayoccursattheBraggwavelength.Forwavelengths

near the edges of the grating bandwidth and the sidelobes of the reflectivity, the

dispersion is greatest with the time delay varying rapidly with wavelength. Thus,

largetimedelaysarecharacteristicofthisregimeandareduetothesewavelengths

sufferingmultiplereflectionsbeforeexitingthegrating(asinaFabry-Perotcavity).

Figure5.2showsthereflectivityspectrumandthetimedelayforauniformgrating

withastrength,qL=4,andagratinglengthofL=20mm.Themaximumreflectivity,

whichcanbecalculated from(5.14), corresponds to theminimum timedelay.For

wavelengths near the first reflectivity zeros, the time delay is maximum

correspondingtoseveralround-tripsbeforethelightexiststhegrating.


0

100

200

300

400

500

1549.75 1549.85 1549.95 1550.05 1550.15 1550.25Wavelength(nm)

Gro

upd

elay

(ps

)

0

0.2

0.4

0.6

0.8

1

Ref

lect

ivity

Figure 5.2 - Calculated reflection spectra (dotted line) and group delay (solid line) for a

uniformgratingwithqL=4.

5.3 Apodisation

In order to increase side-lobe suppression to achieve the required discrimination

between adjacent wavelength channels (at least 30dB) in WDM systems, fibre

gratings are generally apodised. This is achieved by tapering the refractive index

modulation, ∆n(z), at both ends of the grating structure. The reflectivity of an

apodised grating can be calculated by defining an effective length, Leff, for the

grating calculated using the following expression [79], which describes the

normalisedcouplingstrength.


=L

eff dzzqLq0

max )( (5.19)

ThereflectivityatthegratingresonancewavelengthiscalculatedbysubstitutingLeff

in (5.14) and using q=qmax. When comparing gratings with different apodisations,

the quantity defined by (5.19) must be equal for each. Thus to achieve the same

normalised coupling strength for the same maximum grating refractive index

modulation,∆nmax,orcouplingstrength,qmax,thelengthofthegratingsismultiplied

byL/Leff.

Inter-channel cross-talk of grating based add-drop multiplexers depend upon

side-lobe suppression and the grating spectrum. Ideally a square filter with high

reflectivity and –50dB side-lobes is required. Recently these filters have been

determined using a numerical inverse scattering method [50] and demonstrated

experimentally[82].InOADMsbasedongratingsinscribedinthecouplerwaist,the

fabrication limitations for the grating length play a vital role in the choice of

apodisationandtheconsequentadd-dropperformancediscussedfurtherinchapter8.

Figures 5.3 and 5.4 compare the reflectivity spectrum and penetration depth

respectively, forgratingswith the samenormalisedcouplingstrengthqLeff=4.The

blacklinecorrespondstoauniformgrating,theblueaBlackmanapodisedgrating,

andtheredlinetoasine2apodisedgrating.TheBlackmanapodisedgratingoffers

thebestside-lobesuppressionalthoughithasthehighestpenetrationdepthintothe

grating.Theactual lengthsof thegratings toobtain the samenormalisedcoupling

strength, for each of the apodisations were; Blackman: 47.6mm; sine2: 40mm;

Uniform:20mm.


-80

-60

-40

-20

0

1549.8 1549.9 1550 1550.1 1550.2Wavelength(nm)

Ref

lect

ivity

(dB

)

Blackman

sin2

Uniform

Figure 5.3 – Reflectivity spectrum of gratings with different apodisations. Black line:

Uniformapodisedgrating;Blueline:Blackmanapodisedgrating;Redline:sine2apodised

grating.

0

5

10

15

20

25

30

35

1549.8 1549.9 1550 1550.1 1550.2Wavelength(nm)

Pen

etra

tion

leng

th(m

m)

Blackman

sin2

Uniform

Figure5.4–PenetrationdepthspectrumofthesamegratingsasinFigure5.3.


5.4 TransferMatrix

Formodellingthespectralpropertiesofgratingswitharbitraryapodisationandchirp

profiles, a simple method exists, whereby the grating is described using N sub

matrices representing N uniform sections of the grating; these matrices are then

multiplied to obtain the total grating response [78, 83]. The solution of the

propagationequation(5.8)forauniformmediumoflength∆zandconstantcoupling

coefficientqcanbeexpressedintermsofthewell-knowntransfermatrix[78],MT:

=

∆+∆+

),(),(

),(),(

1

1

2

1

δνδν

δνδν

z

zM

zz

zzT

∆+∆∆

∆∆−∆=

)sinh()cosh()sinh(

)sinh()sinh()cosh(

zss

izszs

zssq

zss

izsMT δ

δ

Where s=|q|2−δ2. The output amplitudes of the entire grating can be found by

multiplyingthetransfermatricescorrespondenttoeachoftheNindividualsections:

=

)0()0(

)()(

1

1

2

1

νν

νν

TML

L; 11 ... T

NT

NTT MMMM ⋅⋅⋅= −

Throughout this thesis the above method, in conjunction with an appropriate

discretisation algorithm [50] was employed to efficiently model the spectral

characteristicsofthegratingsinvestigated.Toincreasethenumericalefficiencyby

reducingthecomputationtime,thematrixMTwasexpressedasaproductofsimpler

matrices[50].Thescatteringprocessisdesbribedasalocalisedeventinthecentre

of each individual grating section. Taking MT in the limit |q|→∞ whilekeeping a

finitproductq∆zwecancalculateasimplifiedmatrix thatdescribes thescattering

processMS(∆z),inthesectionoflength∆z:


∆∆

∆∆=∆

)cosh()sinh(

)sinh()cosh(

)( *

zqzqqq

zqqq

zq

zM S

Thepropagationalongthegratinghasalsotobetakenintoaccount.Thepropagation

matrixMP(∆z,δ),iscalculatedtakingMTinthelimit|q|→0giving:

=∆ ∆+

∆−

zi

zi

P e

ezM δ

δ

δ0

0),(

ThetransfermatrixMTcanbeapproximatedwithanerrorO(∆3) in termsof these

twomatricesas:

( ) .,2

,2

∆∆

∆≈ δδ zMzM

zMM PSPT

5.5 Photosensitivity

To write strong gratings in short fibre lengths, the photosensitivity of the

germanium-dopedfibrecoreshouldbe increased toachieve larger refractive index

changes.This issueisespeciallyimportantwhenwritinggratingsinfibre tapersor

couplerswherethephotosensitiveareaandthereforetheoverlapwiththecoremode

isreduced.Increasedphotosensitivityiscommonlyachievedby;loadingtheoptical

fibres with hydrogen or deuterium under high pressures [84], brushing the optical

fibres/waveguides with an hydrogen flame [85] and, increasing germanium

concentration and adding codopants such as fluorine or boron to reduce NA. The

physicaloriginof thephotosensitivityinopticalfibreisstillamatterforextensive


discussionsandisoutofthecontextofthisthesis.Forgoodreviewsconcerningthe

photosensitizationprocess,readersarereferredto[86-88].

5.6 Summary

AbriefintroductiontofibreBragggratingswaspresentedinthissection.Thephase

matchingbetweenforwardandbackwardpropagatingfundamentalfibremodescan

be achieved with a periodic variation of the effective index. The interaction is

quantifiedusing thewell knowncoupledmode equations.Analytical solutions for

theseequationsexistfor thesimplestcaseofauniformgrating.Forusefuldevices

withlowsidelobes,inordertoaccuratelydiscriminatebetweenadjacentchannels,

differentfibreapodisationsareused.Thespectralresponseofgratingswitharbitrary

apodisations is obtainedby solving the coupled modeequationsusing an efficient

scatteringmatrixmodel.Theconceptoftimedelayandpenetrationdepthoflightin

thegratingwereintroducedaswellinordertooptimisetheperformanceofadd-drop

multiplexersbasedongratingsinscribedinthewaistoffibrecouplers,discussedin

chapter8.

6

Acousto-OpticTunable

FilterDesign

InthischapteramethodfortailoringtheshapeoflossfiltersbasedontheAcousto-

optic(AO)interactionintaperedopticalfibresispresented.Themethodisbasedon

the coupling of light from the fundamental core mode to the fibre taper cladding

modes. The conditions for resonant coupling between the modes in tapered fibres

arecharacterisedandasanapplicationa filter isdesigned todynamicallyequalise

theEDFAgainspectrumwithreducedtuningparameters.

6-Acousto-OpticTunableFilterDesign 64

6.1 Acousto-opticTechnology

Acousto-optic (AO) interaction in optical fibres results from the coupling of light

between the propagation modes of an optical fibre, induced by an acoustic wave.

Frequency shifters, switches, filters, amplitude modulators [16, 17, 26, 89-93, 94,

95-98]areexamplesofpracticaldevicesusingthistechnology.Earlyacousto-optic

devicesreliedonthecouplinginatwo-modefibre[26,99].Inthiscase,aflexural

acoustic wave couples light from the fundamental mode (LP01) to a low-order

claddingmode(LP11).Thesedeviceswereusedasfrequencyshiftersandlossfilters.

Other AO devices were based on the coupling between the modes of a dual-core

opticalfibre[90],orsimilarly,thepolarisationmodesofahighbirefringentoptical

fibre[89].AnewrangeofAOdevicesmakeuseoftaperedfibrestructures[91-93,

100]. Null couplers were used in different configurations giving rise to frequency

shifters, acousticmodulators and switches. In this case, theAO interactionoccurs

betweentheopticalmodessupportedbythethinsilicawaist(witharadiusofafew

microns).Theresidualcoredoesnotconfinethesemodes.Theycanbeconsidered

cladding modes, propagated by the whole silica-air structure. One of the main

advantages of the tapered AO devices is their low power consumption due to the

amplificationoftheacousticwaveinthetaperedregion[64].

Recently,wide interest hasbeendevoted tonovelAO tunable filters [16,17,

26]. Inthesefilters, light iscoupledfromthefundamentalmode(LP01)ofasingle

modeoptical fibretoseveral lowordercladdingmodes(LP11,LP12,LP13) through

anacousticflexuralwave.ThemainfeatureofAOfilterswithrespecttostaticfilters

such as long period gratings [97], is the ability to control dynamically the loss

spectrum in order to compensate for gain saturation effects of optical amplifiers

caused by power fluctuations of the input signal. By changing the acoustic wave

frequencyandamplitudeonecanchangetheresonancewavelengthandstrengthof

the AO interaction, this is to reconfigure the filter loss spectrum to the desired

response.


ThedevicepresentedbyKimetal.[16,17,98]isveryflexibleinreconfiguring

thespectralresponseoftheAOfilter.However,thatflexibilityrelieson6acoustic

wave frequencies with different powers. This means that for tuning the spectral

response of the filter 12 parameters need to be adjusted, which represents a very

complexsystem.

In this chapter an alternativedesign thatusesonly2parameters for adjusting

dynamically the filter response is presented. To demonstrate the operation of the

device,thedesignedfilterwasusedtoflattentheASEspectrumofanerbiumdoped

fibre amplifier (EDFA) for different saturation levels. The effect of a controlled

tapering of a single mode optical fibre on the coupling between the fundamental

modeandlowordercladdingmodesisinvestigated.Thecontroloftheradiusprofile

alonganon-uniformtaperedregionofthefibrecanberegardedasanotherdegreeof

freedomfor tailoringthespectralresponseof theAOfilter.Usingthisapproach,a

singleacousticfrequencycanbeusedwiththespectralcharacteristicsrelyingonthe

complexmulti-taper structure.Consequently there is a reductionof thenumberof

parametersneededtotunethefilterdynamicallybutasacompromise,areductionof

itsflexibility.

The acoustic interaction in tapered fibres has been studied in this work. This

interaction results from the exchange of power between two light modes of the

opticalfibre,duetoaperiodicperturbationintherefractiveindex.Theperturbation

is induced by an acoustic wave that acts like a periodic grating. The resonance

condition for this power exchange requires that the beatlength, LB, of the modes

matchtheacousticwavelength,Λ,intheinteractionregion.Coupledmodeequations

areusedtoquantifytheamountoflightcoupledfromonemodetotheotherdueto

theperturbation.Theamountofopticalpowercoupleddependsontheamplitudeof

theacousticwave,theoverlapbetweentheopticalmodesandtheelasticproperties

ofthefibre.Inthiswork,thecouplingbetweenthefundamentalmodeLP01andthe

lowerordercladdingmodesLP11,LP12andLP13intaperedfibresisstudied.

The effect of tapering a fibre is to change the dispersion relations of the

different optical modes, inducing differences in the beatlength between the

fundamental mode and the cladding modes. These dispersion relations were fully


characterisedboththeoreticallyandexperimentally.Theyarethebasisfordesigning

structures of different tapers and interaction lengths with different spectral

responses.

6.2 Theory

ThissectiongivesthetheoreticalbackgroundforcharacterisingtheAOinteraction

intaperedopticalfibres.Firstthepropagationofacousticwavesinopticalfibresis

analysed.Then,thepropagationoftheopticalmodesintaperedfibresisdescribed.

Finally, the interactionbetween the acousticwaveand theoptical fibremodes are

quantifiedusingcoupledmodetheory.

6.2.1 Propagationoftheacousticwave

Thepropagationofacousticwavesinrodsandcladrodsmadefromisotropiclinear

materials has been extensively studied [101]. Torsional, longitudinal and flexural

wavemotionsarepermittedinthesestructures.Atlowfrequencies,aflexuralwave

propagating along a fibre can be described approximately by the Euler-Bernoulli

theoryofbeams[101].Ifu(z,t)isthetransversedisplacementfromtheequilibrium

axis then the equation describing the propagation of the flexural acoustic wave is

[59]:

0),(1 2

224 =

+− tzuppYIT

p tzz α (6.1)

n

n

xAYI

∂∂== n

x2 pandwith

ρα

WhereTistheaxialtensionappliedtothefibre,YistheYoungModulus,ρisthe

densityofthematerial,Aistheareaofthecrosssection(A=πr2)andIthemoment


ofinertia(I=πr4/4foracylinder).Inthecasewherenotensionisappliedtothesilica

fibre (T=0) then equation (6.1) simplifies and considering solutions of the form

u(z,t)=U(0)e-iKzeiΩtthedispersionrelationscanbefoundtobe:

RVK

ext

Ω=Λ

= 22π (6.2)

WhereKisthepropagationconstantoftheacousticflexuralwave,Ωistheangular

frequency,R is thefibre radiusandVext is thevelocityof theextensionalwave in

silica(Vext=(Y/ρ)1/2=5760ms-1).Atlowfrequencies,thepowerflowoftheflexural

acousticwaveisgivenby[102]:

( ) 20

21

55720

23 2)(2 ufRVufRvP extg πρρπ == (6.3)

Wherevg is thegroupvelocityof thewave(vg=∂Ω/∂K),u0 is theamplitudeof the

acousticwaveandfistheacousticfrequency.

6.2.1.1 Effect of tension applied to the fibre

Iftensionisappliedtothefibrethenthesecondterminequation(6.1)accountsfor

its effect on the propagation of the acoustic wave (T≠0). The roots of the

characteristicequationobtainedbysubstitutingu(z,t)=U(0)e-iKzeiΩtin(6.1)are:

21

21

2

22

Ω+±±=α

γγK (6.4)

Whereγ=T/(2YI).Consideringanappliedtension,theacousticpropagationconstant

willbeperturbedaccordingto,


AIYVT

KdK ext 1

4Ω= (6.5)

AccordingtoHooke’ slawthestrainandtensionappliedtothefibrecanberelated,

forsmallstrains,by:

YAT

S z = (6.6)

6.2.2 Opticalmodesintaperedfibres

A single mode optical fibre is a three-layer structure (core/cladding/air), which

supportsseveralcladdingmodesbesidesthefundamentalmode,guidedbythecore.

Thecladdingmodesareprimarilyguidedbythecladding/airstructure.Bytapering

the fibre, the modes will change their optical properties. If the taper is smooth

enough, the modes will evolve adiabatically, maintaining their identity with

insignificant lossesduring their propagation.The electric fieldof anopticalmode

propagatingalongthetapercanbewrittenas:

( ) ( ) tjdj

mnmnmn

z

mn

eezbrEzAtzrE ωξξβ

φφ

=−

0

)(

)(,,)(,,,

(6.7)

Where βmn(z) is the propagation constant of the mode LPmn, ωmn is its angular

frequency,b(z) the taper radiusatpointz,Emn is thenormalisedfieldpattern,and

Amn(z) is a slowlyvaryingamplitude.βmn(z) andEmn dependon the radiusof the

taper. βmn(z) is considered as a local propagation constant and Emn the local

normalised field pattern. If the taper is smooth enough to allow adiabatic

propagationwithoutlossesandtransferofpowerbetweenthemodes,thenAmn(z)is

constant,independentoftheposition.


LP01(r)LP11(r)∆∆∆∆n(r)

Rad

ius

LP01(r)LP11(r)∆∆∆∆n(r)

Rad

ius

Figure 6.1 – Schematic representation of the field distribution of the fundamental mode

LP01andcladdingmodeLP11aswellastheradialdistributionoftherefractiveindexchange

∆n(r),inducedbytheacousticwave.

Symmetryconsiderationsshowthattheonlycladdingmodesthatareallowedto

couplelightwiththefundamentalmodeareanti-symmetricfieldpatternmodes.This

is due to (6.8) and the anti-symmetric nature of the change in the dielectric

permittivity∆ε inducedbyaflexuralacousticwave,as illustratedschematicallyin

Figure6.1.TheamountoflightcoupledbetweenthefundamentalmodeLP01andan

arbitrarycladdingmodeLPmnisgivenbythecouplingconstant:

∞

∞−

∆= drrzrEzrzrEzk mn2*

01 ),(),(),(4

)( εω (6.8)

ThemodesthatcouplelightwiththefundamentalmodeLP01aretheanti-symmetric

claddingmodesLP1m.Theseopticalmodeschangetheirpropagationconstantsand

field distribution along the taper region. In order to simulate the change in the

propertiesofthemodes,afullyvectorialmodelwascomputed.Theonlyassumption

madewasthattheratiorccbetweenthecoreradius,a,andthecladdingradius,b,is

constant (rcc=4/62.5 for the fibre used). The results, shown in Figure 6.2, use

normalisedUandVparameters (V=2π⋅NA⋅a/λandU2=(2π⋅n0⋅a/λ)2-β2) inorder to

predict theAOresonances in taperedfibres.ThemodesrepresentedbyLP11,LP12


andLP13actuallycorrespondtoasetofthreepolarisationmodeseachofwhichhave

negligiblesplittingamongthem.

0

2

4

6

8

10

12

0 2 4 6 8 10 12Vnumber

Un

umbe

r

LP13

LP12

LP11

LP01

Figure 6.2 – Characterisation of the optical modes LP01, LP11, LP12 and LP13 using

normalisedparametersUandV.

Theresonanceconditionforcouplingbetweentwomodesdependsonthedifference

in propagation constants between the interacting modes. This condition can be

expressed in terms of the normalised effective beatlength, Lm, between the

fundamentalandthecladdingmode(LP01↔LP1m):

mm a

L101

21ββ

π−

= (6.9)

Whereaistheradiusofthecore.Figure6.3showsthedispersionofthebeatlength

LmforthethreeinteractingmodesasafunctionofthenormalisedfrequencyV.


0

500

1000

1500

2000

2500

0 1 2 3 4 5 6

NormalisedFrequencyV

Nor

mal

ised

Bea

tleng

thL

mLP01-LP11

LP01-LP12

LP01-LP13

Figure6.3 -Dispersionof thenormalisedbeatlength for three interactingpairsofmodes

(LP01↔LP1m,m=1..3).

ThebeatlengthissmallforlowandhighvaluesofthenormalisedfrequencyVand

exhibits a maximum for V close to 1. For small values of V, the mode is not

confinedtothecore,behavingasacladdingmode.Thiscanbeconsideredthecase

when a small taper radius is achieved and there is only a residual core. The

maximum beatlength corresponds roughly to the point at which the fundamental

modeLP01beginstobeguidedbythecore.

6.2.3 Acousto-opticinteraction

Considering the case of an acoustic wave, with an acoustic period Λ, travelling

throughasectionof length,Lofasinglemodeopticalfibre, thefundamentalcore

mode LP01 will be coupled with cladding modes LP11, LP12 and LP13. Figure 6.4

represents an acoustic wave propagating in a region of a fibre with radius, R and

length,L.


Figure6.4–Schematicrepresentationofanacousticwavepropagatingalongasectionof

anopticalfibre.Left:Effectindexchangeduetoelasto-opticeffect.Right:Effectiveindex

changeduetothegeometricaleffect.

TheAOinteractionbetweentwomodesofanopticalfibreisduetotwomechanisms

that give opposite contributions [102]. The main contribution is given by the

geometric deformation (Figure 6.4 - right) of the optical fibre induced by the

acousticflexuralwave.Thegeometricdeformationoftheopticalfibregivesriseto

differentopticalpathsexperiencedbylightatdifferentcrosssectionalpositionsand

can be seen as a periodic change in the effective index. The other is due to the

elasto-opticeffectthatresultsinaperiodicchangeofthedielectricpermitivitydue

to local internal stresses causedbyeither compressionor expansionof theoptical

fibre(Figure6.4left).Theeffectivechangeinthedielectricpermitivityofthefibre

incorporatingboththesecontributionscanbewrittenas[102]:

( )χεε −=∆ 12 02

zzSn (6.10)

Where n is the refractive index of silica, ε0 the permittivity in vacuum, Szz the

longitudinalstrainofthefibre,andχaccountsforthechangeintherefractiveindex

due to the stress induced by bending the fibre [102]. At low frequencies, χ has a

valueof0.22,decreasingas thefibrediameter increases.Finally, thestraincanbe

relatedtothedisplacementtotheneutralaxisu(z,t)bythefollowingrelation[100,

103].


2

2

2

2 ),(4Λ

−== ytzuy

dzud

S zzπ

(6.11)

whereyistheradialpositioninthedirectionofthefibredisplacement.

Theresonanceconditionforthecouplingbetweentwoopticalmodesdependsonthe

dispersion relations of the modes as well as the period of the acoustic wave.

Momentum conservation requirements establish the following condition for

resonancecoupling:

Lm(λr,b)=Λ(b,Ω) (6.12)

From equation (6.12) we observe that the beatlength, Lm, between the interacting

modesisequaltotheacousticwavelength,Λ,forresonancecoupling.Theresonance

wavelength,λr,oftheinteractingmodescanbecalculatedasafunctionofthetaper

radius, a, by intersecting the normalised beatlength with the acoustic wavelength.

TheseresultsareshowninFigure6.5.Theyrepresenttheresonancewavelengthof

thethreeAOinteractionsobservedexperimentally,asafunctionofthetaperradius

andforthreeacousticfrequencies1.15MHz,1.25MHzand1.35MHz.


0

1

2

3

5 15 25 35 45 55

TaperRadius(m)

Res

onan

ceW

avel

engt

h(

m)

LP01-LP12LP01-LP11

LP01-LP13

fac=1.35MHz

fac=1.15MHz

Figure 6.5 - Resonance wavelength as a function of the taper radius for the three AO

interactionsandacousticfrequenciesof1.15MHz,1.25MHzand1.35MHz.

From the results shown in Figure 6.5, it can be observed that the resonance

wavelengthisadouble-branchedfunctionofthetaperradiusforeachacousticwave

frequency.TheshortresonancewavelengthbranchcorrespondstohighVnumbers,

wherethefundamentalmodeLP01iswellconfinedinthecore.Incontrast,thelong

resonancewavelengthbranchcorrespondstolowVnumbers,wherethefundamental

modeisguidedmainlybythecladding-airstructure.PracticalAOdevicesthatuse

resonancewavelengths around1.5µmhavebeen implemented.Thesemakeuseof

different branches by using untapered fibres [16, 17, 98] (short resonance

wavelength branch), and very this taper waists [91-93, 100] (long resonance

wavelengthbranch).FromFigure6.5itcanalsobeobservedthatdevicesworkingin

the long-wavelengthbrancharemore sensitive tovariationsof the taper radius,or

dependentparameters,thanthoseworkingintheshort-wavelengthbranchduetothe

steeperslopeofthebranch.

IntheworkdonebyKimetal.[16,17,98],theacousticfrequencyisusedto

tune the resonance wavelength of the AO interaction. This tuning gives opposite

effectsinbothbranchesasobservedinFigure6.5.Forthelong-wavelengthbranch,

an increase in the acoustic frequency will result in a decrease of the resonance


wavelength in contrast to the short-wavelength branch that will translate into an

increase of the resonance wavelength. Applying an axial strain to the fibre also

allows tuning of the resonance wavelength. In this case, it can be inferred from

equation (6.5) and Figure 6.8 that the effect of straining the fibre is to make the

resonance wavelengths of each branch converge. This effect may also be used to

tunetheresonancewavelengthoftheAOinteraction.

The studyof theAO interactionbetween theopticalmodesof a single mode

fibreissimilartothecaseoflongperiodgratings[97].Theamountoflightcoupled

from the fundamental mode to one of the cladding modes is expressed by the

coupledmodeequationsthatcanbewritteninthefollowingform:

−=

−=

Λ⋅+−+

Λ⋅+−−

z

m

z

m

djm

dj

m

ezAzjkdz

zdA

ezAzjkdz

zdA

0

0

00

)/2)()((

0*

)/2)()((0

)()()(

)()()(

ξπξβξβ

ξπξβξβ

(6.13)

A0istheamplitudeofthefundamentalmodeLP01whileAmistheamplitudeofthe

cladding mode LP1m and k(z) is the coupling coefficient. The resolution of this

differential equation for different wavelengths gives the spectral response of a

determined structure. The coupling strength, k(z), depends on the overlap integral

between the interacting modes and the perturbation of the dielectric permittivity

inducedbytheacousticwave[18].

)(~

)1(4)( 2

4

VOIan

zk mΛ−=

λξχπ

(6.14)

whereξistheenvelopeamplitudeoftheacousticwave,a,isthecoreradius,λisthe

optical wavelength and V the normalised frequency. OIm is a normalised overlap

integral between the fundamental mode and the mth interacting cladding mode,

dependingonlyonthenormalisedparameterV:


∞

=0

21

*01 ),(),(

1)( drrVrEVrE

aVOI mm (6.15)

The normalised overlap integral was computed as a function of V and the

resultsare illustrated inFigure6.6.For lowVnumbersandveryhighVnumbers

(corresponding to small fibre radii anduntapered fibres), theLP01-LP11 interaction

has the greatest coupling coefficient. However, for intermediate values of V, the

strengthofthecouplingcoefficientfortheLP01-LP13interactionisthestrongestand

LP01-LP11theweakest.Inthisregion,thefundamentalmodefielddistributionstarts

to expand through the fibre cross section changing the overlap with the cladding

modes. The electric field distribution for the fundamental mode and the three

claddingmodesisshowninFiguresB1andB2inappendixBfordifferentvaluesof

theVnumber.AtV=3,boththefundamentalmodeandtheLP11modeareguidedby

thecoreandastheVnumberdecreases,themodesfielddistributionstarttobeless

affectedbythecoreandthereforeexpandthroughthecladding.

Figure6.6–NormalisedoverlapintegralbetweenthefundamentalmodeLP01andthethree

interactingcladdingmodesasafunctionofthenormalisedfrequencyV.


The accentuated dip in overlap integral between the fundamental mode LP01

and the LP12 claddingmodeobserved inFigure 6.6, is due to theoverlap integral

going from negative values to positive values around V=0.7 and therefore the OI

passesbyanull.

6.3 Experiments

TheprincipleofoperationoftheAOfilterconsistsontheexcitationofthetapered

sectionofasinglemodeopticalfibrebyaflexuralacousticwave.Anacoustichorn,

whichconcentrates theacousticpower in itsapex, isglued to thefibre inorder to

generate the acousticwave.Thehorn is excitedby apiezoelectric element (PZT),

which isdrivenby an amplified radio frequency (RF) signal. Anacousticdamper

limitsthepropagationoftheacousticwavetothetaperedregionofthefibre.Figure

6.7illustratestheexperimentalsetupoftheAOfilter.

RFAMP

RFSynthesizer

CladdingmodesStripper

AcousticHorn+PZT

OpticalSOURCE

OpticalSpectrumAnalyzer

TaperedFibreAcousticdamper

Figure6.7–PrincipleofoperationoftheAOfilter.

TheexperimentalworkperformedinordertostudytheAOdevicewasdivided

in two parts. The first part consisted of the characterisation of the dispersion


relationsintaperedopticalfibres.ThethreeAOinteractionswereidentifiedandthe

double-branched function of the resonance wavelength, shown in Figure 6.5, was

confirmed experimentally for the LP13 mode. The dispersion relations were

measuredbyusing several taperswithdifferent radii.The acoustic frequencywas

variedandtheresonancewavelengthofthemodeswasregistered.

The second part of the experimental work was devoted to designing and

implementinganAOtunablefilter,forflatteningthegainprofileofanEDFA.The

filterwasimplementedbycascadingtwoAOfilters(astheoneshowninFigure6.7)

that consisted of a multi-tapered section of a standard telecommunications single-

mode optical fibre with a numerical aperture of 0.12, a core radius of 4µm and a

claddingradiusof62.5µm.Thefilterwasdrivenbyonlyoneacousticfrequencyand

thentunedbyadjustingboththeacousticwavepowers.

6.3.1 Characterisationofthedispersionrelations

The AO filter dispersion relations were characterised experimentally by the

measurement of the resonance wavelength of each mode. A set of tapers with an

interaction lengthof100mmwas fabricated.The radii of the taper regions ranged

from30µmto50µmwithastepof2.5µm.Thetaperswheremountedunderaslight

tensionandtheacousticwavefrequencywasvariedfrom1MHzto1.8MHz.Figure

6.8 illustrates these experimental results. The theoretical fits assumed a fixed

cladding/coreratio(rcc=4/62.5)andatensionappliedtothefibreof0.9N.


1.1

1.3

1.5

1.7

1 1.2 1.4 1.6 1.8

AcousticFrequency(M Hz)

Res

on

ance

Wav

elen

gth

m

(a)

0.9

1.1

1.3

1.5

0.6 0.9 1.2 1.5 1.8


Res

on

ance

Wav

elen

gth

(

m)

(b)

0.9

1.1

1.3

1.5

0.6 0.9 1.2 1.5 1.8


Res

on

ance

Wav

elen

gth

(

m)

(c)

b=40m

b=30m

b=30m

b=40m

b=30m

b=40m

Figure6.8–DispersionoftheresonancewavelengthoftheLP01-LP11(a),LP01-LP12(b)and

LP01-LP13(c)interactionsfordifferenttaperradii.Thedotsrefertoexperimentalvaluesfor

radiiof30,32.5,35,37.5and40m,andthesolidcurvestothetheoreticalfittings.

TheresonancewavelengthcurvesshowninFigure6.8aredoubled-branchedas

predictedinthenumericalsimulationresultsshowninFigure6.5.Forsmalltapers,

theevolutionof thetwobranchesof theLP01-LP13 interactioncanbeobservedfor

the range of acoustic frequencies used (1MHz-1.8MHz). Figure 6.9 shows the

evolutionofthethreeAOinteractionsofthe32.5µmtaperforacousticfrequencies


of 1.31MHz, 1.30MHz, 1.28MHz and 1.24MHz. The peaks corresponding to the

LP01-LP11 and LP01-LP12 interactions and the double peak for the LP01-LP13

interactioncanbeobserved.

-9

-7

-5

-3

-1

1

1000 1150 1300 1450 1600

W avelength(nm)

No

rmal

ised

res

po

nse

(d

B)

LP11-LP01 LP12-LP01

LP13-LP01

-9

-7

-5

-3

-1

1

1000 1150 1300 1450 1600

Wavelength(nm)

No

rmal

ised

res

po

nse

(d

B)

LP11-LP01 LP12-LP01

LP13-LP01

-9

-7

-5

-3

-1

1

1000 1150 1300 1450 1600

Wavelength(nm)

No

rmal

ised

res

po

nse

(d

B)

LP11-LP01 LP12-LP01 LP13-LP01

-9

-7

-5

-3

-1

1

1000 1150 1300 1450 1600

Wavelength(nm)

No

rmal

ised

res

po

nse

(d

B)

LP11-LP01 LP12-LP01

LP13-LP01

(a)

(c) (d)

(b)

f=1.31MHz f=1.30MHz

f=1.28MHz f=1.24MHz

Figure6.9–EvolutionofthespectralresponseoftheAOinteractionsfordifferentacoustic

frequencies.

ComparingthespectralresponseshowninFigure6.9withthedispersioncurve

corresponding to the 32.5µm taper, the merging and vanishing of both the peaks

corresponding to thebranchesof the LP01-LP13 interaction canbe appreciated.As

frequencyisincreased,bothpeaksmoveinoppositedirectionsinwavelength.Itcan

also be observed that the peak corresponding to the long wavelength branch is

broaderduetothelargerslopeinthedispersionrelations(Figure6.8).Foracoustic

frequencies smaller than 1.24MHz, there is no resonance for the LP01- LP13

interaction.Inthesegraphs,thecouplingefficiencyofthethreeAOinteractionscan

also be compared. The coupling between LP01-LP13 modes is much stronger than

coupling of the LP01-LP11 and LP01-LP12 interactions. This behaviour can be

understoodbycomparingthecomputedresultsfortheoverlapintegralforthethree


interactions, shown in Figure 6.6. The LP01-LP13 interaction was operated at a V

numberof1.25andtheLP01-LP11andLP01-LP12 interactionswereoperatedataV

numberaround1.5withaloweroverlapbetweentheinteractingmodes.Thechange

intheresonancelosspeakswiththefrequencyoftheacousticwaveisrelatedtothe

frequency response of the piezoelectric element, which had a resonance around

1.24MHz.

6.3.2 FlatteningtheEDFAASEspectrum

Withtheinformationprovidedbythedispersionrelationsintaperedfibres,shownin

Figure6.8, thespectral lossshapecanbetailoredfor thedesiredfilterapplication.

Thelengthoftheinteractionregionchangesthebandwidthofthepeakresponseand

thetaperradiuschangestheresonancewavelengthforagivenacousticfrequency.A

simulationprogramofamulti-taperinteractionregionwasimplementedinorderto

designafilterforflatteningthegainprofileofanEDFA.

6.3.2.1 CharacterisationoftheEDFA

TheamplifierusedintheexperimentswasanEDFAwithgermano-silicateglassco-

dopedwithaluminaashostglass.TheEDFAgainandASEspectraconsistedoftwo

mainamplification lobescentredatλ=1532.5nmandλ=1555nm.When thepower

of the input signal is increased, saturation of the gain of the EDFA occurs. The

effectoftheEDFAsaturationvariesnon-uniformlyalongtheASEandgainspectra.

The1532.5nmlobeismoreaffectedbysaturationthanthe1555nmlobeanditcan

also be observed that the centres of both these amplification peaks are slightly

shifted in wavelength with different saturation levels. The EDFA was saturated

usingaDFBlaserdiodeemittingat1548nmasinputsignal.Thepowerofthelaser

diodewassettodifferentlevels(-26dBm,-22dBmand–18.4dBm)andtheEDFA

gainspectraforthedifferentsaturationinputpowersareshowninFigure6.10.


0

5

10

15

20

25

1520 1530 1540 1550 1560 1570 1580Wavelength(nm)

Sig

nalG

ain

(dB

)

PLD=-26dBmPLD=-22dBm

PLD=-18.4dBm

Figure6.10–CharacterisationoftheEDFAfordifferentinputpowersofaDFB-LD.

6.3.2.2 CharacterisationoftheAOfilter:

Todemonstratethepotentialofthismethod,theAOfiltershowninFigure6.11and

Table1wasdesignedtoflattentheASEspectrumoftheEDFA.Thefilterconsists

of two cascaded acousto-optic filters (AOF) relying on the LP01-LP13 interaction.

DuetotheindependentchangeinthelobesoftheEDFAwithsaturation(shownin

Figure 6.10), two independent filters were designed to enable a dynamic

equalisationoftheEDFA.AOF#1compensatestheASEspectrumofthe1532.5nm

lobe and AOF#2 was designed to compensate the 1555nm lobe. Both filters were

driven with a different radio frequency (RF) generator. However, both the filters

weredrivenatthesameacousticfrequencyof1.24MHz.


10 20 60 Length(mm)

40.2 38.4 37.2 Radius(µµµµm)

12 50 Length(mm)

42.0 41.0 Radius(µµµµm)

-6

-4

-2

0

1520 1530 1540 1550 1560 1570 1580

Wavelength(nm )

Att

enu

atio

n(

dB

)_

-8

-6

-4

-2

0

1520 1530 1540 1550 1560 1570 1580

Wavelength(nm )

Att

en

ua

tio

n

(dB

)_

-8

-6

-4

-2

0

1520 1530 1540 1550 1560 1570 1580

Wavelength(nm )

Att

enu

atio

n(

dB

)_

a)

b)

AOF#2AOF#1 c)

AOF#1

AOF#2

AOF#1 AOF#2

Figure6.11–SpectralresponseoftheAOfilter.(a)AOF#1.(b)AOF#2.(c)Bothacousto-

optic filterscascaded.Foreachfilter thespectrumisshownforseveraldifferentRFdrive

powers(solidlines).Thedashedlinecorrespondstotheoreticalsimulations.

Filter Section1 Section2 Section3

AOF#1 L=10mm

R=40.2µm

L=20mm

R=38.4µm

L=60mm

R=37.2µm

AOF#2 L=12mm

R=42µm

L=50mm

R=41µm

Table1–ParametersofthetaperprofileofAOF#1andAOF#2


The taper profile of both AOF is shown in Table 1. The coupling spectrum of

AOF#1 and AOF#2 is seen to be asymmetric due to the non-uniform taper

transition, as shown in Figure 6.11a) and Figure 6.11b) respectively.

Characterisation of the filter change with the acoustic signal power is shown in

Figure6.12where thepeak loss isplottedagainst theRFsynthesiserdrivepower.

ThesignalscomingfromtheRFsynthesiserwasamplifiedusingaRFamplifierbya

factor of 103. The response illustrated in Figure 6.12 corresponds to the first

coupling cycle; with increasing drive powers the peak loss would go through

multiplecycles,withlightcoupledbackandfourthbetweenthecladdingmodeand

the core mode, as happens in directional couplers. The different power behaviour

between the two filters is due the longer length of AOF#1 compared to AOF#2,

therefore,requiredacousticpowertoachieveagivenpeakattenuationislowerand

thepeakisalsonarrower.

-14

-12

-10

-8

-6

-4

-2

0

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35

RFsynthesiserpower(mW)

Pea

klo

ssL

P01

-LP

13(d

B)

AOF#1

AOF#2

Figure6.12–FilterlossatpeakwavelengthfordifferentRFsynthesiseroutputpowers.The

opencirclesrefertoAOF#1andthefilledsquarestoAOF#2.

6.3.2.3 EqualisationoftheASEspectrum:

TheASEspectrumoftheEDFAwasequalisedforseveralpowerlevelsoftheDFB-

LDinputsignal(-26dBm,-22dBmand–18.4dBm).Thefluctuationachievedwas


lessthan1dBforabandwidthofover30nmforallsaturationlevels.TheAOfilter

was tunedby changing only the electricpowersused todrive the acousticwaves.

TheexperimentalresultsobtainedarepresentedinFigure6.13.Theinsertionlossof

thewholefilterwasestimatedtobeless than0.5dB.Thetimeresponseof theAO

filter depends on the length of the device and the group velocity of the acoustic

flexuralwaveandisestimatedtobearound50µs.Thetotalpowerconsumptionof

thefilterwas1W.Thisvaluedidnotcorrespondtotherealcapabilitiesofthefilter

duetoanimpedancemismatchoftheamplifierwiththepiezoelectricelement.The

power consumption for an optimum efficiency is estimated to be of a several of

hundredsofmilliWatts.

TheprocessoftuningthisAOfilterismuchsimplercomparedtoothermulti-

frequency schemes. The filter is tuned by changing only the acoustic wave

amplitudes.Thefilter’ slossspectrumshapeisfixedbythetailoredtaperprofileand

isnotable to react toseverechanges in theamplifiergainspectrum(eachfilter is

designed for a specific amplifier). Reducing the tuning parameters increased the

facility of implementation and tuning the AO filter but, as a compromise, the

flexibility of the filter to accommodate to changes in the amplifier gain spectrum

wasalsoreducedincomparisontopreviousconfigurations.Forincreasedflexibility,

andstillasmallnumberoftuningparameters,thefrequencyoftheacousticwaveof

eachoftheindividualfilterscouldalsobetunedtoachievebetterequalisation.


-30

-25

-20

-15

-10

1520 1530 1540 1550 1560 1570 1580

W avelength(nm)

AS

ES

pec

tru

m(

dB

m/n

m) (b)

-30

-25

-20

-15

-10

1520 1530 1540 1550 1560 1570 1580

W avelength(nm)

AS

ES

pec

tru

m(

dB

m/n

m) (c)

-30

-25

-20

-15

-10

1520 1530 1540 1550 1560 1570 1580

W avelength(nm)

AS

ES

pec

tru

m(

dB

m/n

m) (d)

-30

-25

-20

-15

-10

1520 1530 1540 1550 1560 1570 1580

Wavelength(nm)

AS

ES

pec

tru

m(

dB

m/n

m) (a)

DFBsaturatingSignal

DFBsaturatingSignal

DFBsaturatingSignal

Figure6.13–EqualisationoftheASEspectrumoftheEDFAfordifferentsaturationlevels.

(a) No saturation. (b) Saturation power level of –26.0dBm. (c) Saturation power of –

22.0dBm.(d)Saturationpowerlevelof–18.4dBm.

6.4 Summary

Inthischapter,thepotentialofthecontrolofthetaperprofileinthedesignofAO

filtershasbeenconsidered.ThedependenceofthecharacteristicsoftheAOfilteron

the radius of the taper has been studied theoretically and demonstrated

experimentally. The application of these filters in the equalisation of optical

amplifiershasbeenshownbydesigninganequaliserconsistingoftwocascadedAO

filters to flatten the ASE spectrum of an EDFA. A dynamic equalisation was

achieved for several saturation levelsof theEDFAwitha fluctuationsmaller than

1dBforaspectralbandover30nm.

The present design uses the control of taper profile as another degree of

freedomintailoringthespectralresponseofAOfilters.Thisapproachresultsinthe

reduction of complexity in tuning AO filters compared to other multi-frequency


schemes [16, 17, 26, 98]. However the flexibility of this design is limited to the

shapeofthefiltergivenbythecomplextaperprofile.Combiningbothtechniquesby

the use of several driving acoustic frequencies may allow the use of a minimum

numberofparameterstotunetheAOfilterdynamically.

7

IdealFilterDesignfor

EDFAGainEqualisation

The design of ideal filters to equalise the EDFA gain spectrum is studied using

differentconfigurations.Filtersbasedonanidealwavelength-dependentbackground

lossoftheEDFarepresentedandcomparedwithlumpedfiltersbasedontheinverse

of theamplifiergainspectrum.Special filtersaredesigned inorder tocompensate

for their own insertion losses giving no penalty to the amplifier gain and gain

flatness. With these novel filter designs EDFA equalisation is achieved with no

penaltyintheamplifiergainforinsertionlossesupto8dB.

7-IdealFilterDesignforEDFAGainEqualisation 89

7.1 Introduction

Erbium-doped fibre amplifiers (EDFA) play an important role in wavelength

divisionmultiplexing(WDM)systems.Asthenumberofchannelsinthesesystems

increases, broadband and equalised amplification is required. Several methods of

achievingEDFAequalisation,eitherintrinsicorextrinsic,havebeenproposedinthe

literatureasmentionedpreviouslyinsection2.5.Thesedeviceshavecharacteristic

equalisationpropertiesandinsertionlossesthatcouldbeaslowasthesplicingloss

betweentheEDFfibreandthefilterfibreandashigh8-9dB[24,25].

H. Zech reported a study of a theoretical filter to flatten the EDFA gain

spectrumacrossadesiredbandwidth.Thefilterisproposedintheformofanideal

wavelength-dependent background loss along the EDF length [104]. However, a

practicalimplementationofthistypeoffilteralongtheEDFlengthisverydifficult.

Inthischapter,theinitialtheoryisextendedtocalculatetheidealwavelength-

dependentbackgroundlossbyincludingthefibrebackgroundlosstermoftheEDF.

In the case of a lossy fibre, the ideal filter shape given by Zech [104] should be

corrected to incorporate this wavelength-independent fibre background loss. The

practicalimplementationofthedistributedbackgroundlossbyintegratingitintoone

ortwofiltersinsertedatdifferentpositionsalongtheamplifierisalsostudied.The

optimum placement of these filters within the EDFA is calculated both for a

configuration using one filter and two filters to equalise the gain spectrum.

Furthermore, the incorporation of the filter insertion loss in the filter design is

achievedbytreatingitasawavelength-independentdistributedlossandcorrecting

the filter design for this loss. With this correction, the filter insertion loss can be

compensatedforbyanefficientfilterre-designandoptimumpositioningwithinthe

EDFA. This is especially important for filters with high insertion losses, which

couldgoupto8-9dB[24,25].


7.1.1 TheoreticalModel

Inthisworkaspectralmodel[11]isusedtodescribetheamplifierperformance.The

spatialcharacteristicsoftheEDFareintegratedandtherateequationsareexpressed

asinChapter2.ThemodelisbasedontheknowledgeoftheEDFparametersα(λ),

g*(λ), l(λ) and the ratio of the linear density of erbium ions to the fluorescence

lifetime ξ(λ). Several methods have been studied in the literature to measure the

amplifierlossandgainspectra[105-107].Theα(λ)andg*(λ)usedinthenumerical

simulationsweremeasuredfromareal fibreandare illustrated inFigure2.3,with

ξ=4.7x1015m-1s-1and lBG=4.9dB/kmat1200nm.Theinputsignalwasdividedinto

32 channels on a 100GHz grid from 1532nm to 1557nm. The calculations were

performedby running the rateequations (2.4)and (2.5)backand fourth along the

amplifierlengthuntilalltheparametersmetthegivenconvergencecriteria(shooting

method).The forwardamplified spontaneous emission (ASE)was fixed tozero at

thebeginingoftheamplifierandthebackwardASEwasfixedtozeroattheendof

theamplifier.Thenumericalsimulationsshowthattheconvergenceoftheamplifier

structureisachievedafterafewiterations.

7.2 Theoreticalfilterdesign:

According to Zech [104], in order to equalise the EDFA gain spectrum, an ideal

distributedloss,canbedeterminedanalyticallyifthewavelengthdependenceofthe

modefunctions in thesignal range isneglected.This isa fairassumptionover the

limited C and L EDFA bands. In this case, the spontaneous emission term is

neglectedandtherateequationforthesignalpowercanbewrittenintheform[11]:

( )( )

)(

)()(

2

tkkk

kkkk

nn

zPg

zPldz

zdP

=+

++

α

α (7.1)


Thisisaverygoodapproximationinhigh-gain,saturatedamplifierssuchastheones

used incurrent telecomsystems. In (7.1) thesubscriptk,corresponds toachannel

with a central wavelength λk, and 2n is the local metastable-level average

population concentration, which is wavelength independent under homogeneous

broadening assumption. Thus, equation (7.1) can be written for λk and λi, and

integrated for the EDF length (z=0 to z=L) resulting in the following expression

[104]:

Lba kiikik GG += (7.2)

whereListhefibrelengthandGk,iistheamplifiergainatwavelengthsλkorλi(in

dBs)i.e.,Gk,i=10log(Pk,i(L)/Pk,i(0)).Theparametersak,iandbk,iaregivenby:

( ) ( )[ ] ellab

gg

a

kkiiki

ii

kk

log10ki

ki

+−+=++=

αααα

Equation(7.2)canbeviewedasprovidingthesaturatedgainGkatawavelengthλk

intermsofthegainGiatareferencewavelengthλiandmeasurableparameters,such

as, the gain and absorption coefficients at λk and λi and the corresponding

backgroundloss.Inthemajorityofcases,thefibrebackgroundlossisconsideredto

be wavelength independent. However, it should be stressed that equation (7.2) is

validevenforthegeneralcasewherebackgroundlossiswavelength-dependent.In

the case of a gain flattened EDFA, where Gk=Gi equation (7.2) can be easily

rearrangedtogive:

( ) kiikiiki laG

Lea

l αα −++−=)log(10

1k (7.3)


Expression (7.3) gives the required wavelength-dependent distibuted loss, lk, of a

gain-flattened saturated amplifier. The gain-flattened bandwidth depends on the

choiseofthereferencewavelengthλiandthetargetgainGi.Overthegain-flattened

bandwidth,lk≥0.Outsidethisbandwidth,lk<0,whichimpliesadditionalbackground

gain.Sincethegainprofileisfixedandentirelydeterminedbytherare-earthchoise,

this is an unrealistic requirement. If additional wavelength-independent fibre

background loss, lBG, is considered, it can be shown that (7.2) and (7.3) can be

writtenas:

( ) LlaeLba BGki 1)log(10GG kiikik −++= (7.4)

( ) ( )1)log(10

1k −+−++−= kiBGkiikii

ki allaGLe

al αα (7.5)

Equations(7.4)and(7.5)areobtainedbysimplyreplacinglkwithlk+lBG.However,

theimportanceofthesecorrectedexpressionswillbeevidentwhentheyareusedto

redesignoptical filters inorder to compensate foradditional insertion losses (their

ownorofotherinserteddevicessuchasisolators,taps,etc).

7.2.1 Effectofthefibrebackgroundloss

Inordertoshowthesignificanceofthecorrectionofexpressions(7.2)and(7.3)to

incorporatethefibrebackgroundloss,asetofsimulationswereperformedusingthe

fibreparametersmentionedpreviously.Theidealwavelength-dependentdistributed

loss, lk, was calculated for fibres with different background loss, lBG, using both

expressions(7.3)and(7.5).TheEDFAgainspectrawerecomparedforboth these

cases.InbothsimulationstheamplifierlengthwasL=3mandtheinputsignalswere

dividedinto32channelswith100GHzspacingstartingfrom1532nm.Eachchannel

power was 2.5µW. The EDFA was end pumped in the forward direction with a

50mWpumpat980nm.Theabsorptionandgainparametersofthefibreusedinthe

numerical simulations are illustrated in Figure 2.3. The absorption coefficient at


980nmwas4.5dB/m.Figure7.2showstheEDFAgainspectrumandoutputsignal

fordifferentEDFbackgroundlosses,lBG=0,0.04,0.08,0.12(dB/m).

0.25

1

1.75

2.5

1530 1540 1550 1560Wavelength(nm)

Sig

nal

(m

W)

a)l Bg =0

l Bg =0.1215

20

25

30

1520 1530 1540 1550 1560 1570Wavelength(nm)

Gai

n(d

B)

b)l Bg =0

l Bg =0.12

Figure 7.1 – EDFA performance fordifferent fibrebackground losses, lBG=0, 0.04, 0.08,

0.12dB/m.a)Outputsignal.b)Gain.

From(7.2), theEDFAgainspectrumcanbetotallycharacterisedbyusingthe

gain at a reference wavelength. This is also true in determining the ideal

wavelength-dependentdistributed loss inorder to flatten theEDFAgainspectrum.

The following results were obtained by inserting in the numerical simulations a

wavelength-dependentdistributed loss, lk,givenby(7.3) fordifferent lBG=0,0.04,

0.08,0.12(dB/m).Thewavelength-dependentlosslkandtheequalisedEDFAgain

spectraareshowninFigures7.2a)and7.2b)respectively.

0

0.2

0.4

0.6

1520 1530 1540 1550 1560 1570Wavelength(nm)

Bac

kgro

un

dlo

ss(

dB

/m)

a)

l Bg =0.12

l Bg =0

l Bg =0.12

l Bg =0

15

17.5

20

22.5

25

1520 1530 1540 1550 1560 1570Wavelength(nm)

Sig

nal

gai

n(

dB) b)

l Bg =0.12

l Bg =0

Figure7.2–FlatteningoftheEDFAgainspectrawithawavelength-dependentdistributed

loss, lk, fordifferent fibrebackground losses, lBG=0,0.04,0.08,0.12dB/m.a)Theoretical

wavelength-dependent distributed loss, lk, calculated using (7.3). b) EDFA gain spectra

includingthecalculatedwavelength-dependentloss,lk.


Theidealwavelength-dependentdistributedlossshapechangedforeachvalue

ofthefibrebackgroundloss.Thisisduetochangesinhowtheamplifierwithafixed

lengthL=3msaturates fordifferent fibrebackground losses.Foreachvalueof the

fibre background loss, there is a corresponding amplifier gain at the reference

wavelengthresultingindifferentwavelength-dependentlossshapescalculatedfrom

(7.3)wherethefibrebackgroundlossisconsideredzero(Figure7.2a).Thesefilters

are not suitable to equalise the EDFA gain spectrum when different background

losses are included in the simulations as shown in Figure 7.2b). The higher the

backgroundlossofthefibre,thehigherthepenaltyintheflatnessoftheEDFAgain

spectrum and the error in the calculation of the wavelength-dependent loss using

expression(7.3).

Thesecondsetofnumericalsimulationsconsistsofusingtheexpression(7.5),

which includes the fibre background loss, to calculate the ideal wavelength-

dependentdistributedloss,lk.TheresultsillustratedinFigures7.3a)and7.3b)show

respectively, the ideal distributed-loss spectrum, lk, and the EDFA gain spectrum

equalisedwiththecalculatedfilters.Fromthesefiguresiscanbeconcludedthatfor

a constant fibre background loss across the whole ASE spectrum, the gain profile

shape of the EDFA does not change due to different saturation conditions but is

attenuated equally across the spectrum bandwidth. The result is a universal filter

shape dependent only on the absorption and gain cross sections of the EDF. The

spectralshapeofthecorrectionstoexpression(7.3)duetothefibrebackgroundloss

isalsouniversaltotheamplifieranddependsontheamountofthefibrebackground

loss, lBG,aswellas theabsorptionandgaincrosssections.Figure7.3c) shows the

correctionaddedtoexpression(7.3)duetothecontributionofthefibrebackground

loss.


0

0.2

0.4

0.6

1520 1530 1540 1550 1560 1570Wavelength(nm)

Loss

(dB

/m)

a)

l Bg =0,0.04,0.08,0.12

15

17.5

20

22.5

25

1520 1530 1540 1550 1560 1570Wavelength(nm)

Gai

n(d

B)

b)

l Bg =0.12

l Bg =0

-0.08

-0.04

0

0.04

0.08

1520 1530 1540 1550 1560 1570Wavelength(nm)

Loss

(dB

/m)

c)l Bg =0.12

l Bg =0l Bg =0.12

l Bg =0

Figure7.3–Filters for flattening theEDFAgain spectrumwith awavelength-dependent

distributedlossfordifferentfibrebackgroundlosses,lBG=0,0.04,0.08,0.12(dB/m).a)Ideal

wavelength-dependentdistributedloss,lk,calculatedusing(7.5).b)FlatteningoftheEDFA

gain spectrum with an ideal wavelength-dependent distributed loss for different fibre

backgroundlosses,lBG.c)Correctiontermaddedto(7.3)duetolBG.

The corrected expression for thewavelength-dependent loss allowsan almost

perfect equalisation of the gain spectrum of the EDFA (Figure 7.3b). This result

suggests that theapproximationused inthemodelproposedbyZech is reasonable

and the calculated filters can be used in the more accurate numerical model to

equalisethespectrumofdifferentEDFAs.

In order to quantify the flatness along a certain bandwidth of the EDFA, the

standarddeviationof thegain spectrumacross the filter bandwidthwasused.The

standarddeviationisdefinedas,

( )2/1

1

21..

−= =

n

ii xx

ndevstd (7.6)


wherethesumismadeacrosstheflattenedbandwidth.Thecomparisonbetweenthe

gainspectraobtainedusingboth theexpressioncorrectedfor thefibrebackground

lossand theoneproposedbyZech [104] toequalise theEDFAgain spectrumfor

differentfibrebackgroundlossesisshowninFigure7.4.Thegainstandarddeviation

wascalculatedacrossthecorrespondingfilters’ bandwidth,showninFigures7.2a)

and7.3a).

0

0.1

0.2

0.3

0.4

0.5

0 0.04 0.08 0.12Backgroundloss(dB/m)

Gai

nS

td.D

ev.(

dB)

Usingexpression(7.3)

Usingexpression(7.5)

Figure7.4–StandarddeviationoftheEDFAgainspectrumfordifferentfibrebackground

losses.Thetheoreticaldistributed-lossshapewascalculatedusingbothexpression(7.5)and

expression(7.3).

The greater the fibre background loss, the greater the error introduced when

usingfilterscalculatedwiththeuncorrectedexpression(7.3).Thestandarddeviation

of theEDFAgain spectrumrises from0.03dB to0.45dBwhenchanging the fibre

background loss from0dB/m(no loss) to0.12dB/m(Figure7.4).The filterdesign

should be corrected according to (7.5) in order to achieve a flat EDFA gain

spectrum. This correction becomes especially significant when designing practical

filteringdeviceswithinsertionlossesthatcouldreachupto8-9dB[24,25]aswill

beshowninthefollowingsections.


7.3 Designofpracticalfilters

7.3.1 Idealfilter–Noinsertionloss

Idealfiltersintheformofawavelength-dependentloss,distributedalongtheEDF,

havebeendiscussed.Thesedistributedfiltersareabletoequalisethegainspectrum

oftheEDFAbutarenotpracticaltobeimplementedinrealamplifiers.Inorderto

produce a useful and practical device, this wavelength-dependent loss should be

integrated into a discrete number of filters placed in different positions along the

EDFA. If lk(λ), the ideal filter distributed loss, is tobe incorporated intoN filters

thenthelossspectrumforeachfilterisgivenby:

NLl

N

dzl

F k

L

k

k

)()(

)( 0 λλ

λ ==

(7.7)

In order to achieve inexpensive filtering, the number of filters in the EDFA

should be as low as possible. However it is known that the larger the number of

filtersintheEDFA,thecloseritwillbetotheidealcaseoffilteringbyadistributed

wavelength-dependent loss. Taking these factors into account, in this work two

differentconfigurationswerestudied.Thefirstoneisequalisingtheamplifiergain

spectrumusingone filter (N=1 inequation7.7)placedatdifferentpositionsalong

theamplifier.Thesecondisbasedonusingtwofilters(N=2inequation7.7)thatare

placedsymmetricallyinrelationtothecentreof theEDFA.Againthepositionsof

thefiltersarevariedinordertodeterminetheiroptimumvalues.

7.3.1.1 Onefilterconfiguration

Thisconfigurationisexpectedtobemoresensitivetothepositioningofthefilterin

theEDFAdue toall thedistributed lossbeing incorporated inoneposition.There

should be an optimum position where the filter should be placed in order to


minimize the standard deviation of the EDFA gain spectrum. At this optimum

position the filter loss plus its placement within the EDFA are the best

approximation to the ideal wavelength-dependent distributed loss. However, it is

expected that the standarddeviationof theequalisedEDFAgain spectrum isvery

dependent on the filter position. Figure 7.5 illustrates the EDFA gain flattening

configurationforonefilteringdevice.

Pump980nm

Z1

EDF#1 EDF#2F ( )k λ

GainSpectrum

Figure7.5–One-filterconfigurationfortheEDFAspectrumgainflattening.Thefilter is

positionedatapositionZ1fromthestartoftheamplifier.

In the numerical simulations the amplifier length was divided into M=40

sectionsandthefilterplacedatpositionsZ1=nL/M,n=1..M-1.Theperformanceof

the EDFA was optimised for the flatness of the gain spectrum by varying the

positionofthefilteralongtheamplifier.Figure7.6a)showsthestandarddeviation

oftheEDFAgainspectrumacrossthefilterbandwidthfordifferentpositionsofthe

filter along the amplifier. From this result, the optimum filter placement is at

Z1=1.575m and the standard deviation of the gain spectrum across the filter

bandwidthislessthan0.07dB.Figure7.6b)showstheaveragegainacrossthefilter

bandwidth.Thepositionwherethefiltershouldbeplacedinordertominimisethe

gain penalty should be around Z1=0.5m where the average gain of the amplifier

couldbeupto25.5dB.Howeverifthefilterwereplacedatthisposition,therewould

be a high penalty in the flatness of the gain spectrum. At Z1=1.575m, it can be

observedthattheaveragegainisstillaround24dB.


0.04

0.09

0.14

0.19

0.24

0 1 2 3FilterPosition(m)

Gai

nS

td.D

ev.(

dB) a)

22

23

24

25

26

0 1 2 3FilterPosition(m)

Ave

rage

Gai

n(d

B)

b)

Figure7.6–Performanceoftheequalisedamplifiersfordifferentfilterpositionswithinthe

EDFA.a)Standarddeviationofthegainspectrumacrossthefilterbandwidth.b)Average

EDFAgainacrossthefilterbandwidth.

Theaveragegainisdefinedask

kk )0(P)L(P ,wherethesumisperformedoverthe

filterbandwidth. )L(Pand)0(P kk arerespectively,theinputandoutputsignalpower

atwavelengthλk.Theaveragegainbehaviourfordifferentfilterpositions,shownin

Figure7.6b)isexplainedbythegainrecoverywhenthefilterisplacedattheinput

of the amplifier due to the attenuation of the backward ASE and therefore the

amplifiersaturation.Ontheotherhand,whenthefilterisplacedclosertotheendof

theamplifier,thesignalandforwardASEaremostattenuatedbythefilterandthe

amplifiergainisnotabletorecovertoahighervalue.

Theactualfilterobtainedfromtheintegrationofequation(7.5)andusedinthe

numericalsimulations isshowninFigure7.7a).Aseachwavelengthsaturates ina

differentway,thecorrectfiltershapehastobeplacedatthecorrectpositioninorder

for all wavelengths to reach the same level at the output. Figure 7.7b) shows the

flatteningoftheEDFAgainspectrumfortheoptimisedfilterposition(Z1=1.575m).


-8

-6

-4

-2

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filte

rlo

ss(d

B)

a)

14

18

22

26

30

1520 1530 1540 1550 1560 1570Wavelength(nm)

Sig

nalg

ain

(dB

)

b)UnflattenedEDFA

FlattenedEDFA

3

3.2

3.4

3.6

3.8

4

1520 1530 1540 1550 1560 1570Wavelength(nm)

Noi

seF

igur

e(d

B)

c)

UnflattenedEDFA

FlattenedEDFA

Figure7.7-Flatteningof theEDFAgainspectrumwithonefilterplacedat theoptimum

positionZ1=1.575m.a)Filterlossspectrum.b)EDFAgainspectrumwithandwithoutfilter.

c)Noisefigurewithandwithoutfilter.

InFigure7.7b)itcanbeobservedthattheEDFAgainspectrumbandwidthhas

increasedslightlywhen thefilter is inserted in theamplifier.This isdue to there-

distributionofthepumppoweracrossthebandwidthafterthefilterisinserted.The

wavelengthsthatareoutofthefilterbandwidthandhavenotbeenattenuatedbythe

filteraregainingfromthefactthataftertheinsertionofthefiltertheamplifierisnot

saturated and is absorbing the pump at a high rate. The comparison between the

noisefigureoftheamplifierwithandwithoutthefilterplacedintheEDFAisshown

inFigure7.7c).Itisobservedthatthenoisefigureoftheamplifieracrosstheuseful

bandwidth(1525nmto1565nm)isnearthequantumlimit.Thereisalsonopenalty

inthenoisefigureduetotheinsertionofthefilteratZ1=1.575m.Actually,thenoise

figureimprovesslightlyoutsidethegain-flattenedregionasadirectconsequenceof

the increased gain in this part of the spectrum, as illustrated in Figure 7.7b). The

noise figure for different filter positions is discussed in more detail in section

7.3.1.2.


Figure7.8showsthesignalbuild-upalongtheEDFAlengthforthreedifferent

signalwavelengths.Atλ1=1532.3nmthesignalincreasesveryfastandthereforeitis

attenuatedbyalargeamountinordertogiveaflatsignaloutput.Atλ2=1539.4nm

the signal increases at a slower rate and therefore the loss of the filter at that

wavelength is small in order to achieve the same signal output power as at

λ1=1532.3nm.Thesignalatλ3=1550.7nmistheslowesttoincreaseandattheendof

theEDFAitspowerisstillrising.FortheequalisationoftheEDFAsignalgain,the

correctfilterlosshastobeplacedatthecorrectpositionandthesignalgainateach

wavelengthshouldbethesameattheendoftheamplifierlength.

-25

-15

-5

5

0 1 2 3EDFposition(m)

Po

wer

(dB

m)

λλλλ 3333=1550.7nm

λλλλ 1111=1532.3nm

λλλλ 2222=1539.4nm

Figure 7.8 – Signal build-up along the EDFA length for three different wavelengths.

λ1=1532.3nm;λ2=1539.4nm;λ3=1550.7nm.

7.3.1.2 Twofiltersconfiguration

In this configuration it is expected that the wavelength-dependent distributed loss

alongthefibrebeapproximatedmoreaccuratelybytheinsertionoftwofiltersinthe

EDFA.ThestandarddeviationoftheEDFAgainspectrumshouldbelesssensitive

tothepositionwheretheequalisingfiltersareplaced.However,ifthefiltershapeis

notcorrectthenthisconfigurationshouldnotbeabletoequalisethegainspectrum

of the amplifier wherever they are placed. The filters were calculated using

expression (7.5) and integrated according to (7.7) and the loss was split into two

filters (N=2).Theywereplaced in the amplifier symmetricallywith respect to the


centreoftheEDF.Thisis,forafilterpositionedatZ1,theotherfilterwasplacedat

Z2=L-Z1.Figure7.9showstheconfigurationfortwofiltersplacedintheEDFA.

Pump980nm

Z1 Z2

EDF#1 EDF#2 EDF#3F ( )k λF ( )k λ

Gainspectrum

Figure7.9–Two-filtersconfigurationfortheEDFAspectrumgainflattening.

TheperformanceoftheEDFAwassimulatedfordifferentfilterpositionsalong

theEDFA.HalfoftheamplifierlengthwasdividedintoM=40sectionsandthefilter

placed at positions Z1=nL/2M, n=1..M-1. Figure 7.10a) shows that the standard

deviation of the EDFA gain spectrum across the filter bandwidth for different

positionsofthefiltersalongtheamplifierisaround0.07dB.Theoptimumpositions

for the placement of the filters in order to minimise the standard deviation of the

gain spectrumareZ1=0.038mandZ2=2.962m.However, thepositionof the filters

does not greatly affect the flatness of the gain spectrum and a good performance

would be obtained even if both the filters were placed near the centre of the

amplifier (close to the configurationwhereone filter isplacedat the centreof the

amplifier). Figure 7.10b) shows the average gain across the filter bandwidth. It is

observedthattheEDFAperformanceisnotverysensitivetothepositionwherethe

filtersareplacedandtheaveragegainishigherthan24dB.Thisresultwasexpected

due to this configuration being a closer approximation to a wavelength-dependent

lossdistributedalongtheEDFA.


0.06

0.07

0.08

0.09

0.1

0 0.5 1 1.5FilterPosition(m)

Gai

nS

tdD

ev.(

dB) a)

22

23

24

25

26

0 0.5 1 1.5FilterPosition(m)

Ave

rage

Gai

n(d

B) b)

Figure7.10–Performanceof the amplifiers equalisedusing the two-filters configuration

fordifferent filter positionswithin theEDFA. a)Standarddeviationof thegain spectrum

acrossthefilterbandwidth.b)AverageEDFAgainacrossthefilterbandwidth.

Theactualfilterlossobtainedfromtheintegrationof(7.5)andusedtosimulate

the EDFA performance is shown in Figure 7.11a). When compared to the case

whereonlyonefilterisused(Figure7.7a),itmaybeobservedthatthelossishalved.

Figure7.11b)showstheflatteningoftheEDFAgainspectrumfortheoptimumfilter

positions (Z1=0.038mandZ2=2.962m).Again it is stressed that the flatnessof the

gain spectrum is not affected much by the positioning of the filters in this

configuration and this configuration is a good approximation for an ideal

wavelength-dependentdistributedlossfilter.


-4

-3

-2

-1

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filte

rlo

ss(d

B)

a)

14

18

22

26

30

1520 1530 1540 1550 1560 1570Wavelength(nm)

Sig

nalg

ain

(dB

)

b)

FlattenedEDFA(2filters)

UnflattenedEDFA

3

4

5

6

7

1520 1530 1540 1550 1560 1570Wavelength(nm)

Noi

seF

igur

e(d

B)

c)Z1=0.038mandZ2=1.962m

UnflattenedEDFA

Figure7.11-FlatteningoftheEDFAgainspectrumwithtwo-filtersplacedattheoptimum

positionZ1=0.038mandZ2=2.962m.a)Filter lossspectrum.b)EDFAgainspectrumwith

andwithoutfilter.c)Noisefigurewithandwithoutfilter.

Thecomparisonbetweenthenoisefigureoftheamplifierwithandwithoutthe

filterplacedintheEDFAisshowninFigure7.11c).Inthiscaseitmaybeobserved

thereisasignificantincreaseinthenoisefigureduetotheinsertionoftwofiltersat

Z1=0.038mandZ2=2.962m.Thereasonthenoisefigureishighforthiscaseisdue

to the placement of the first filter near to the start of the amplifier. It will have

practicallyno effect on the forwardASEbuild-upbutwill decrease thegain.The

effectofthesecondfilteronthenoisefigurewillbeverysmallbecauseitattenuates

equallythegainandthealreadybuilt-upforwardASE.Figure7.12givesthepenalty

inthenoisefigureduetotheplacementofanequalisingfilteratthestartoftheEDF.

These simulations correspond to the placement of the filter represented in Figure

7.8a)atdifferentpositionsalongtheamplifier.Aspreviouslyobserved,fortheone

filter configuration, the optimum position of the filter (for equalising the gain

spectrumoftheEDFA)wasZ1=1.575mandtherewasnopenaltyinthenoisefigure


at that position. Figure 7.12 refers to the noise figure at a fixed wavelength

(λ=1532nm).

3

4

5

6

7

8

9

0 0.5 1 1.5 2 2.5 3FilterPosition(m)

Noi

seF

igur

e(d

B)

λλλλ=1532nm

Figure7.12–Noisefigureatλ=1532nmforanEDFAwithoneequalisingfilterplacedat

differentpositionsalongtheamplifier.

Theplacementof an equalising filter close to the start of theEDFAcauses a

highpenaltyintheamplifiernoisefigure.Thereasonistherelativeincreaseofthe

forward ASE build-up compared to the signal due to the placement of the filter

closer to the start of the amplifier. In order to show this effect, the EDFA

performance was simulated for the configuration where two filters are used to

equalisethegainspectrumfordifferentfilterpositions:(Z1=0.038mandZ2=2.962m)

and (Z1=1.425m and Z2=1.575m). As shown in Figure 7.10b), the average gain

across the filterbandwidthdoesnot changeand therefore, thechange in thenoise

figureisduetothedifferentbuild-upsoftheforwardASE.

Figures 7.13a), b) and c) show respectively the build-up of the forward ASE

along the amplifier length at wavelengths λ1=1532.3nm, λ2=1539.4nm and

λ3=1550.7nm.Threedifferentcasesareplotted:TheunfilteredEDFA,theamplifier

flattenedtwowithfiltersat(Z1=0.038mandZ2=2.962m);andtheamplifierflattened

twowithfiltersat(Z1=1.425mandZ2=1.575m).


0

0,004

0,008

0,012

0,016

0,02

0 0,5 1 1,5 2 2,5 3Position(m)

Pow

er(

mW

)a) Z1=0.038m+Z2=2.962m

Z1=1.462m+Z2=1.538m

Unflattened

λλλλ=1532.3nm

0

0.001

0.002

0.003

0.004

0.005

0.006

0 0.5 1 1.5 2 2.5 3Position(m)

Pow

er(m

W)

b)Z1=0.038m+Z2=2.962m

Z1=1.462m+Z2=1.538m

Unflattenedλλλλ=1539.4nm

0

0.001

0.002

0.003

0.004

0.005

0.006

0 0.5 1 1.5 2 2.5 3Position(m)

Pow

er(m

W)

c) Z1=0.038m+Z2=2.962m

Z1=1.462m+Z2=1.538m

Unflattened

λλλλ=1550.7nm

Figure7.13–ForwardASEbuild-upalongtheamplifierlengthfordifferentfilterpositions.

a)ASEpowerat1532.3nm.b)ASEpowerat1539.4nm.c)ASEpowerat1550.7nm.

Asboththecasesofflatteningtheamplifierat(Z1=0.038mandZ2=2.962m)and

(Z1=1.425m and Z2=1.575m) produce the same flat gain spectrum and a similar

averagegain,theincreaseinthenoisefigurecanbeeasilyunderstoodbycomparing

thebehaviouroftheforwardASE:Atallthreewavelengths(Figures7.13a),b)and

c)),theforwardASEishigherinthecasewherethefiltersareplacedat(Z1=0.038m

and Z2=2.962m) compared to the placement of the filters at (Z1=1.425m and

Z2=1.575m)andtheresultisanincreasednoisefigureinthefirstcase.

When comparing the forward ASE build-up of the two cases of the filtered

EDFAit isobservedthatat theendof theamplifier(Z=L=3m), theASEpoweris

always higher (at all three wavelengths) when the filters are positioned at

(Z1=0.038m and Z2=2.962m). Since there is no significant change in the average

gain (Figure 7.10b)), the higher the forward ASE, the higher the amplifier noise

figure.

When comparing the case of the filters placed at Z1=0.038m and Z2=2.962m

with the unfiltered amplifier, it can be observed that the forward ASE build-up is

muchstrongerwhentheEDFAisequalised.Thisisduethefactthattheinputsignal,


which saturates the amplifier, is being filtered out by the first filter placed at

Z1=0.038m.At thispoint theforwardASEispracticallyzeroand the filteraffects

essentiallytheinputsignalpowerandtherefore,theamplifierwillbelesssaturated

bythesignal.TheASEbuild-upinthiscaseismoreeffectiveasshowninFigures

7.13a), b) and c). The noise figure increase can be seen as a direct result of the

decreaseintheaveragegainwhencomparedtotheunfilteredamplifier(seeFigure

7.1a)). In Figure 7.14 the noise figure spectra for these three configurations are

shown (Unfiltered amplifier; (Z1=0.038m and Z2=2.962m) and (Z1=1.425m and

Z2=1.575m)).

3

4

5

6

7

1520 1530 1540 1550 1560 1570Wavelength(nm)

Noi

seF

igur

e(d

B)

Z1=0.038m+Z2=2.962m

Z1=1.462m+Z2=1.538m

Unflattened

Figure 7.14 – Noise figure spectra using the two filters configuration for different filter

positionsintheEDFA.

As expected, the spectral shape of the noise figure with the filters placed at

Z1=0.038mandZ2=2.962missimilartothelossspectrumthatattenuatedthesignal

atthestartoftheamplifier.

7.3.1.3 Conclusionsontheimplementationofpracticalfilters

The possibility of equalising the EDFA gain spectrum based on a theoretical

wavelength-dependentdistributedlossfilter,givenbyequation(7.5),wasstudiedby

considering two different implementation with one filter or two filters inside the

amplifier.Intheconfigurationwhereonefilterisusedtoequalisetheamplifier,the

flatness of the EDFA gain spectrum is very sensitive on the position of the filter

withintheamplifier.Theoptimumpositionisclosetothecentreoftheamplifierand

thereisnopenaltyinthenoisefigureduetotheinsertionofthefilter(theposition


wherethefilterisplacedisfarfromthestartoftheEDF).Intheconfigurationwhere

two filters are used to equalise the amplifier, both the flatness of the EDFA gain

spectrumandtheaveragegainoftheamplifierareveryinsensitivetothepositions

ofthefiltersintheEDFA.Thisisduetothefactthatthetwo-filterconfigurationisa

closerapproximationtothecaseofadistributedlossfilter.However,thecloserthe

first filter is placed to the start of the EDFA, the higher the penalty in the noise

figure will be. On the other hand, the closer to the middle they are inserted, the

closer this is to thecaseof theone-filterconfigurationwiththefilterplacedat the

centre of the amplifier. In either configuration, the accurate filter shape given by

equation(7.5)iscrucialtotheequalisationoftheamplifier.

7.3.2 Inclusionofthefilterinsertionloss

Real filters have insertion losses due to their design, fabrication procedure or the

methodtheyareincorporatedintheamplifier.Deviceswithinsertionlossesaslow

as0.1dBandashighas8dBhavebeenreportedintheliterature.Inordertoequalise

adequately theEDFAgain spectrum, theoptimumpositionof filters including the

insertionlosseshastobedetermined.

The effect of the inclusion of different filter insertion losses in the filters

calculated from equation (7.5) was studied using both one-filter and two-filter

equalising schemes. For the case of one equalising filter, the considered insertion

losses were lins=0, 0.5, 1, 2, 4 and 8dB. For the case of two equalising filters

insertionlossesoflins=0dB,0.5dBand1dBperfilterwereconsidered.Theoriginal

filtersareshowninFigure7.7a)fortheone-filterconfigurationandinFigure7.11a)

forthetwo-filtersconfiguration.

7.3.2.1 One-filterconfiguration

If an insertion loss is incorporated in the filter loss spectrum, the filter shapewill

changeandthereforethepositionwherethefiltershouldbeplacedintheEDFAin

ordertoflattenthegainspectrumhastobedetermined.Duetodifferentwavelengths


saturating at different rates in the EDFA, the inclusion of an insertion loss in the

filterdoesnotonlychangetheaverageEDFAgainbutitalsochangesthesaturation

of the EDFA and consequently, the flatness of the gain spectrum. In order to

compensate the different saturation condition, the filter is placed at different

positionsaccordingtheinsertionloss.Figure7.15a)showsthefiltershapecalculated

from equation (7.5) and Figure 7.15b) the filters including the different insertion

losses.Foreachfiltertheoptimumpositionwasdetermined.

-8

-6

-4

-2

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filte

rlo

ss(d

B)

a)

Ins.Loss=0,0.5,1,2,4,8dB

-16

-12

-8

-4

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filte

rlo

ss(d

B)

b)Ins.Loss=8dB

Ins.Loss=0

0

0.2

0.4

0.6

0.8


Gai

nS

td.D

ev.(

dB)

c)

Ins.Loss=8dB

Ins.Loss=014

18

22

26


Ave

rage

gai

n(d

B)

d)

Ins.Loss=8dB

Ins.Loss=0

10

15

20

25

30

1520 1530 1540 1550 1560 1570Wavelength(nm)

Sig

nalg

ain

(dB

)

e)

Ins.Loss=8dB

Ins.Loss=0Unflattened

3

3.2

3.4

3.6

1520 1530 1540 1550 1560 1570Wavelength(nm)

Noi

sefi

gure

(dB

)

f)

Ins.Loss=8dB

Ins.Loss=0

Figure7.15–Performanceof theEDFAequalisedusing theone-filterconfigurationwith

differentinsertionlosses.Eachfilterwasplacedattheoptimumpositioninordertoflatten

thegainspectrumof theEDFA.a)Filtershapeobtainedfrom(7.5).b)Actual filterswith

insertion lossesof0,0.5,1,2,4and8dB,used in thenumerical simulations.c)Standard

deviationofthegainspectrumacrossthefilterbandwidthfordifferentfilterplacements.d)

Average EDFA gain across the filter bandwidth for different filter placements. e) EDFA

gainspectra.f)EDFAnoisefigure.


For each filter the average and standard deviation of the equalised gain

spectrum for different filter positions was calculated. The results of these

simulationsareshowninFigures7.15d)and7.15c)respectively.Thefilterposition

thatgivestheloweststandarddeviationcorrespondstotheoptimumpositionofthe

filter.Fromtheseresultsitmaybeobservedthatthechangeintheoptimumposition

of the filter due to the insertion loss gives a penalty in the average gain of the

amplifier. However, equalisation of the EDFA gain spectrum is still achieved by

changing the position of the filters. The higher the insertion loss of the filter, the

closertotheendoftheEDFistheoptimumpositionandtheloweristheamplifier

averagegainacrossthefilterbandwidth.Thepenaltyintheaveragegainforfilters

placedattheoptimumpositioncorrespondsroughlytothefilterinsertionloss.Fora

filter with no insertion loss the optimum position is around Z1=1.575m and the

averagegain isclose to24dBandfora filterwith8dBinsertion loss theoptimum

positionisaroundZ1=2.5mandtheaveragegainis16dB.Figure7.15e)showsthe

EDFA gain spectrum for the filters with different insertion losses placed at the

optimumpositions.

The almost linear relation between the amplifier gain reduction and the filter

insertionlosscanbeunderstoodbyobservingthebehaviourofthepumppowerand

populationinversionalongtheEDF.Thepopulationinversionduetotheplacement

of the filter doesnot change significantly and therefore, there is no significant re-

absorptionofthepumpandnosignificantbuild-upoftheASEandsignalafterthe

filter.Figures7.16a)and7.16b)showrespectively,thepumppowerandpopulation

inversionalongtheEDFlengthforfilterswithdifferentinsertionlosses.


0

10

20

30

40

50


Pow

er(m

W)

a)

l Ins =0

l Ins =8dB0.4

0.5

0.6

0.7

0.8

0.9

1


Inve

rsio

nn

2/n

t(a.

u.) b)

l Ins =0

l Ins =8dB

84

0.51

0

2

Figure 7.16 – Pump power and population inversion along the EDFA length for the one

filter configuration.The filtersused to equalise the amplifierhad insertion lossesof0dB,

0.5dB,1dB,2dB,4dBand8dB.a)Pumppower(λ=980nm).b)Populationinversion.

Theperformanceofthefilteralsodependsonthenoisefigurepenaltyduetothe

insertionofthefilterintheEDFA.Figure7.15f)showsthenoisefigurespectrafor

theequalisedEDFAusingfilterswithdifferentinsertionlosses.Itcanbeobserved

that forhigh filter insertion losses, there starts tobea slightpenalty in theEDFA

noisefigure.Thisisdueessentiallytothereductionofthesignalgainandtherefore

theterm1/Ginexpression(2.8)ofthenoisefigurestartstobesignificant.

ThereasonforthesmallpenaltyintheEDFAnoisefigureduetotheplacement

ofthefiltersisthatboththesignalandtheforwardASEareheavilyattenuatedatthe

secondhalfoftheamplifier(whereboththesignalandASEpowersarealmostfully

amplified. Figure C1 in appendix C shows both the forward-backward ASE and

signal gain build-up along the EDFA length for different wavelengths. The ratio

betweentheforwardASEpowerandthesignalgainremainsconstantafterthefilter

due to the small pump re-absorption (Figure7.16a) andconsequently, sodoes the

noisefigure.

7.3.2.2 Two-filtersconfiguration:

As previously mentioned, in this configuration the integrated filters give a good

approximation to a wavelength-dependent distributed loss filter calculated from

equation(7.5).Ifthefilterischangedduetotheinclusionofaninsertionloss,then

theflatnessoftheEDFAgainspectrumwillbedegraded.Asinthisconfiguration,


thegainflatnessisnotsensitivetothepositionsofthefilters,filterrepositioningwill

notbeanextradegreeoffreedomtoequalisetheEDFAandtherefore,adegradation

of the filter performance is expected. The filter spectral shape is shown in Figure

7.17c) and the actual filters that include insertion loss of 0, 0.5dB and 1dB are

showninFigure7.17d).

0.05

0.09

0.13

0.17

0.21

0.25

0 0.25 0.5 0.75 1 1.25 1.5FilterPosition(m)

Gai

nS

td.D

ev.(

dB)

a) 1dB

0.5dB

0dB

22.6

23.4

24.2

25


Ave

rag

eg

ain

(dB

) b)1dB

0.5dB

0dB

-4

-3

-2

-1

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filt

erlo

ss(

dB)

c)

-5

-4

-3

-2

-1

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filt

erlo

ss(d

B)

d)

1dB0.5dB0dB

14

18

22

26

30

1520 1530 1540 1550 1560 1570Wavelength(nm)

Sig

nal

gai

n(

dB)

e)

1dB0.5dB

0dBUnflattened

3

4

5

6

7

8

1520 1530 1540 1550 1560 1570Wavelength(nm)

Noi

seF

igu

re(d

B)

f)

1dB0.5dB

0dBUnflattened

Figure7.17–PerformanceoftheEDFAequalisedusingthetwo-filtersconfigurationwith

differentfilterinsertionlosses.Thefilterswereplacedattheoptimumpositionsinorderto

flattenthegainspectrumoftheEDFA.a)Standarddeviationofthegainspectrumacrossthe

filter bandwidth for different filter positions. b) Average EDFA gain for different filter

positions.c)Filtershapecalculatedfrom(7.5).d)Actualfiltersincludingtheinsertionloss.

e)EDFAgainspectrum.f)EDFAnoisefigure.

TheaverageandstandarddeviationoftheEDFAgainspectrumwerecalculated

fordifferentfilterpositions.Thefilterswereagainplacedsymmetricallyinrelation

tothecentreof theEDF.Aspredicted, theresults illustratedinFigures7.17a)and

7.17b)showthatforfilterswithaninsertionlossas lowas0.5dB,theamplifier is

notequalisedwhereverthefiltersareplaced.AsshowninFigures7.17a)and7.17e),

increasingthefilterinsertionlossresultsinanincreasedpenaltyoftheEDFAgain

flatness. As the two-filter configuration is a good approximation of an ideal

backgroundloss,ifthefiltershapeisnottheidealduetotheinclusionofthefilter

insertionloss,thenequalisationoftheEDFAisnotpossible.Theoptimumposition


of the filters for all the considered insertion losses is Z1=0.038m and Z2=2.962m.

However,aspreviouslymentioned,theinsertionofthefirstfilternearthestartofthe

amplifier causes a high penalty in the amplifier noise figure. The two-filter

configuration does not have the flexibility to accommodate changes in the filter

shape resulting in deterioration of both the gain spectrum flatness and the noise

figure.Figures7.17e)and7.17f)showrespectively,thepenaltyintheamplifiergain

flatnessandnoisefigure.

7.3.2.3 Conclusionsontheinclusionofthefilterinsertionloss

Boththesuggestedconfigurations(one-filterandtwo-filter)offlatteningtheEDFA

gain spectrum using real filters with different insertion losses were analysed. The

two-filterconfiguration is shown tobedisadvantageouscompared to theone-filter

configuration when the filter insertion loss in considered. This is due to the

approximationofthetwo-filtersconfigurationtoadistributedlossandtherefore,itis

veryinsensitivetothepositionofthefiltersintheEDFAandresultsinpenaltiesin

boththeEDFAnoisefigureandthegainflatness(Figure7.17).

Using the one-filter configuration, the optimum position of the filter in the

EDFAchangesaccordingtotheinsertionlossofthefilter.Thehighertheinsertion

loss, the closer to the endof theEDFA is theoptimumposition (Figure7.15).At

these optimum positions the gain flatness is achieved but at the cost of a lower

average gain, due to the placement of the filter nearer to the end of the EDFA

(Figure7.15e). In this configuration theEDFA canbe correctly equalised and the

noise figure penalty is very small even for filters with insertion losses as high as

8dB.

7.3.3 Filterdesignscompensatingthedeviceowninsertionloss

Sofar,ithasbeenshownthathighperformancegain-flatteningopticalfilterscanbe

obtained, by converting an ideal wavelength-dependent distributed loss (equation


(7.5)), intoa lumplossplacedinasingleposition(one-filterconfiguration)or two

symmetricpositions(two-filterconfiguration)alongtheamplifierlength.

Inthissection,thepossibilityofincorporatingthedeviceowninsertionlossinto

thefiltershape,andcompensatingfor it, is investigated.This isaccomplishedina

manneropposite to theone followed in theprevious section.For thispurpose, the

localisedwavelength-independentdeviceinsertionlossisspreaduniformlyalongthe

amplifier length. In this respect, the device insertion loss can be considered as

additionalbackgroundloss,lBG,andequation(7.5)canbenowusedtoprovidethe

correctedgain-flatteningfilterspectrum.Intheone-filterconfigurationtheinsertion

losses of the filters were 0, 0.5, 1, 2, 4 and 8, all in dB units. For the two-filter

configurationeachfilterhadaninsertionlossof0dB,0.5dBand1dB.

7.3.3.1 One-filterconfiguration

ThetreatmentofthefilterinsertionlossasanequivalentEDFbackgroundlosswill

result in different filter shapes in order to equalise the gain spectrum. The

corrections in the filter loss spectrumdue to the inclusionof the fibrebackground

lossareshowninFigure7.3c).Thecorrectedfiltershapesforeachinsertionlossare

illustrated in Figure 7.18a) and the actual filters, including the insertion loss, are

showninFigure7.18b).


-12

-8

-4

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filte

rlo

ss(d

B)

a)

Ins.Loss=8dB

Ins.Loss=0

-20

-15

-10

-5

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filte

rlo

ss(d

B)

b)Ins.Loss=8dB

Ins.Loss=0

0.05

0.15

0.25

0.35

0 0.5 1 1.5 2 2.5 3Filterposition(m)

Gai

nS

td.D

ev.(

dB)

c)

Ins.Loss=0

Ins.Loss=8dB

14

18

22

26

0 0.5 1 1.5 2 2.5 3Filterposition(m)

Ave

rage

gai

n(d

B)

d)

Ins.Loss=8dB

Ins.Loss=0

14

18

22

26

30

1520 1530 1540 1550 1560 1570Wavelength(nm)

Sig

nalg

ain

(dB

)

e)Unflattened

Ins.Loss=0

Ins.Loss=8dB

3

3.5

4

4.5

1520 1530 1540 1550 1560 1570Wavelength(nm)

Noi

sefi

gure

(dB

)

f)

Unflattened

Ins.Loss=8dB

Ins.Loss=0

Figure7.18–PerformanceoftheEDFAequalisedusingfilterscorrectedforbytakinginto

account their insertion losses. Each filter was placed at the optimum position in order to

flattenthegainspectrumoftheEDFA.a)Filtershapeobtainedfrom(7.5).b)Actualfilters

with insertion losses of 0, 0.5, 1, 2, 4 and 8dB, used in the numerical simulations. c)

Standard deviation of the gain spectrum across the filter bandwidth for different filter

placements. d) Average EDFA gain across the filter bandwidth for different filter

placements.e)EDFAgainspectra.f)EDFAnoisefigure.

WhencomparingFigure7.18a)withFigure7.15a)itmaybeobservedthatboth

thefiltershapeandbandwidthchangewhenthefilterinsertionlossisincludedinthe

design.For thesenewfilters, theoptimumpositionwasdeterminedbymonitoring

thegainstandarddeviationandaveragegainacross thefilterbandwidth,shownin

Figures7.18c)and7.18d)respectively.


In contrast with the non-corrected filter results in section 7.3.3, the optimum

filterpositiongetscloser to the startof theEDFwith the increaseof the insertion

loss.Theconsequenceisanincreaseintheaveragegainacrossthefilterbandwidth,

as seen in Figure 7.18d). The average gain builds up to around 24dB for all the

filters,includingthefilterwith8dBofinsertionloss.Figure7.18e)showstheEDFA

gain spectrum using the different filters placed at the optimum position. The

amplifier is equalised and the gain is 24dB for all the filters. As previously

mentioned,thecloserafilterisplacedtothestartoftheEDF,thehigherthepenalty

in the amplifier noise figure. In Figure 7.18f) the noise figure for the filters with

differentinsertionlossescanbecompared.Evenfortheextremecaseofacorrected

filter with 8dB insertion loss, with an optimum position around 1.25m, the

maximumpenaltyinthenoisefigureisbelow1dB.

Tounderstandhowitispossibletorestorethegaintothe24dBgainleveleven

inthecaseoffilterswithinsertionlossesashighas8dB,thepumpandpopulation

inversion along the EDF length are plotted in Figures 7.19a) and 7.19b),

respectively.Thehigherthefilterinsertionloss,theclosertotheoptimumpositionis

totheEDFfrontend(Figure7.18c))andthelargertheamountofrequiredfiltering,

especially around the1532nmgainpeak (Figure7.18a)).This result in substantial

signalandASE(backwardinparticular)powerattenuationandsignificantreduction

in the amplifier saturation. This frees-up a substantial amount of pump-photons,

whicharenowavailableforabsorptionintheremainingEDFlength(Figure7.19a)).

As a result, the population inversion after the location of the filter improves

dramatically, and provides enough extra gain to re-amplify all wavelengths to the

same level (as in the “loss-less” case). The actual power (local gain) evolution at

three different wavelengths is shown in Appendix C (Figure C2). Different

wavelengthsareattenuatedbyadequateamounts(givenbythecorrectfiltershape)

sothattheyendupatthesamelevelattheEDFAoutput.


0

10

20

30

40

50


Pow

er(m

W)

a)

l Ins =0

l Ins =8dB

0.4

0.5

0.6

0.7

0.8

0.9

1


Inve

rsio

nn 2

/nt(

a.u.

) b)

l Ins =0

l Ins =8dB

Figure 7.19 – Pump power and population inversion along the EDFA length. The filters

used to equalise the amplifier were corrected for the insertion loss. a) Pump power

(λ=980nm).b)Populationinversion.

7.3.3.2 Two-filtersconfiguration

The filter shapes corrected for the insertion loss were studied using the two-filter

configuration. The insertion losses for the filters in the configuration were 0dB,

0.5dBand1dB.ThecorrectedfiltershapesareshowninFigure7.20c)andtheactual

filtersincludingtheinsertionlossinFigure7.20d)


0.04

0.08

0.12

0.16


Gai

nS

td.D

ev.(

dB

) a)1dB

0.5dB

0dB

22

23

24

25


Ave

rag

eG

ain

(dB

)

b)1dB

0.5dB

0dB

-5

-4

-3

-2

-1

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filt

erlo

ss(

dB

)

c)1dB

0dB0dB

1dB

-6

-4

-2

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filt

erlo

ss(

dB

)

d)

1dB0.5dB0dB

14

18

22

26

30

1520 1530 1540 1550 1560 1570Wavelength(nm)

Sig

nal

gai

n(

dB

)

e)

1dB0.5dB

0dBUnflattened

3

3.2

3.4

3.6

3.8

4

1520 1530 1540 1550 1560 1570Wavelength(nm)

No

ise

Fig

ure

(dB

)

f)

1dB

0.5dB

0dB

Unflattened

Figure 7.20 – EDFA performance for the amplifier equalised using the two-filters

configurationwithdifferentinsertionlosses.Thefilterdesignwascorrectedfortheinsertion

lossandeachfilterwasplacedattheoptimumpositioninordertoflattenthegainspectrum

of the EDFA. a) Standard deviation of the gain spectrum across the filter bandwidth. b)

Average EDFA gain across the filter bandwidth. c) Filter shape calculated from (7.5). d)

Filterincludingtheinsertionloss.e)EDFAgainspectra.f)EDFAnoisefigure.

Thestandarddeviationofthegainspectrumandtheaveragegainfordifferent

filterpositionswerecalculatedfor thecorrectedfilterprofiles.Theresults indicate

that forall the insertionlosses, thecorrectedfilters,shouldbeplacedatZ1=1.35m

and Z2=1.65m in order to optimise the flatness of the gain spectrum, as seen in

Figure7.20a).Thesepositionsalsocorrespondtomaximaoftheaveragegainacross

eachfilterbandwidth(Figure7.20b).AgoodEDFAequalisationwasnotachieved

withaninsertionlossof1dBperfilter.Thismeansthattheassumptionthatthefilter

insertionlosscanbetreatedasafibrebackgroundlossisnottotallyaccurateandfor

high insertion losses EDFA equalisation is not perfectly achieved. Comparing the

placementofthecorrectedfilterdesignsshowninFigure7.20a),withtheplacement

oftheuncorrectedfilterdesignsshowninFigure7.17a),itisobservedthattheyare

both quite insensitive to the position of the filters. However the corrected filters

shouldbeplacedcloser to thecentreof theEDF toachieveEDFAgain flattening

whiletheun-correctedfiltersshouldbeplacedattheendsoftheEDF.Thepenalty

intheaveragegainacrossthefilterbandwidthissimilarinbothfiltershapesbutthe


noisefigurefortheuncorrectedfiltersismuchhigherduetotheplacementofoneof

thefiltersclosetothestartof theamplifier. Inthecaseof thecorrectedfilters, the

noisefigure(Figure7.20f)islowduetothefiltersbeingplacedneartothemiddleof

theEDF.Itisalsoobservedthattheachievedgainflatteningisalsoimprovedwhen

usingthecorrectedfiltershapes(Figure7.20e).

7.3.3.3 Conclusions on the correction of the filter to compensate for the

deviceinsertionloss

Boththesuggestedconfigurations(one-filterandtwo-filter)offlatteningtheEDFA

gain spectrum using real filters with loss spectra corrected for different insertion

losseswereanalysed.The two-filter configuration is shown tobedisadvantageous

comparedtotheone-filterconfigurationwhenthefilterinsertionlossinconsidered.

However, using the corrected filter shapes there is a slight improvement of the

EDFAgainspectrumflatnessandnoisefigureforthetwo-filterconfiguration.This

configurationcouldbeused in realamplifiersusing filterswithvery lowinsertion

losses. The two-filter configuration also halves the requirements on maximum

filtering loss, which eases the implementation and manufacture of the different

filters.Twolongperiodgratingscouldbeusedoralternatively,onestaticfilterand

an AO tunable filter could be used in order to achieve dynamic equalisation. The

mainadvantageofusingthisconfigurationisthatthefilterpositioningisnotcritical

fortheperformanceoftheamplifierequalisation.

Usingtheone-filterconfiguration,theoptimumpositionofthecorrectedfilters

in the EDFA changes according to the insertion loss of the filter. The higher the

insertionloss,theclosertothestartoftheEDFAlaystheoptimumposition(Figure

7.18). This behaviour is opposite to the one shown by the positioning of the un-

corrected filters in theEDFA (Figure7.15).Using thecorrected filters, theEDFA

gainspectrumwasequalisedwithnopenaltyintheaveragegainevenwithinsertion

lossesupto8dB.Thepenaltyintheamplifiernoisefigureduetotheintroductionof

these filters was bellow 1dB. This configuration however, allows a better gain


equalisation.Ontheotherhand,theperformanceoftheequalisedamplifierdepends

on the exact positioning of the filters and, therefore, may prove more difficult to

implement.

7.3.4 EDFAEqualisationbyusingtheinverseofthegainspectrum

Itiscommonpracticetoplacetheequalisingfilterattheoutputoftheamplifier.The

EDFA gain flattening performance using the one-filter configuration (Figure 7.7)

andusingtheinversegainprofileastheequalisingfilterisshowninFigure7.21.In

order to correctly flatten the EDFA gain spectrum, the inverse shape of the gain

spectrumacrossadesiredbandwidthisused(Figure7.21a).Inordertocharacterise

theperformanceof theequalisationof theEDFAgainspectrumusingfiltersbased

ontheinverseofthegainspectrum,theaverageandstandarddeviationofthegain

spectrum were calculated for different filter placements along the EDF. The filter

loss spectrum was obtained by selecting a bandwidth across the EDFA gain

spectrum(Figure7.21b).Filterinsertionlossesof0,0.5dB,1dB,2dB,4dBand8dB

wereconsideredinthenumericalsimulations(Figure7.21e).Itisexpectedthatthe

optimum positions where the filters should be placed are close to the end of the

EDFA and the standard deviation of the gain spectrum should increase when the

filtersareplacedclosertothestartoftheEDF.


14

18

22

26

30

1520 1530 1540 1550 1560 1570Wavelength(nm)

Sig

nalg

ain

(dB

)a)

Filterbandwidth

-6

-4

-2

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filte

rlo

ss(d

B)

b)

0

1

2

3


Gai

nS

td.D

ev.(

dB)

c)Ins.Loss=8dB

Ins.Loss=0

14

18

22

26


Ave

rage

Gai

n(d

B)

d)

Ins.Loss=8dB

Ins.Loss=0

-16

-12

-8

-4

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filte

rlo

ss(d

B)

e)Ins.Loss=8dB

Ins.Loss=0

5

10

15

20

25

30

1520 1530 1540 1550 1560 1570Wavelength(nm)

Sig

nalg

ain

(dB

)

f)Ins.Loss=0

Ins.Loss=8dB

UnflattenedEDFA

Figure 7.21 – EDFA performance for the amplifier equalised using the one filter

configuration.Thefilterswerecalculatedusingtheinverseofthegainspectrumandseveral

insertion losses were used. a) Selected filter bandwidth for equalising the EDFA gain

spectrum. b) Filter shapes obtained by inverting the EDFA gain spectrum across a given

bandwidth. c) Standard deviation of the gain spectrum across the filter bandwidth for

differentfilterplacements.d)AverageEDFAgainacrossthefilterbandwidthfordifferent

filterplacements.e)Actualfilterswithinsertionlossesof0,0.5,1,2,4and8dB,usedinthe

numericalsimulations.f)EDFAgainspectra.

Asexpected, inorder toequalisethegainspectrumoftheamplifier thefilters

shouldbeplacedattheendoftheamplifierandthepenaltyintheaveragegaindue

to the positioning of the different filters corresponds to the insertion loss of each

filter (Figures 7.21c and 7.21d). For the extreme case of a filter with an insertion


lossof8dBthegaindecreasesfrom24dBtoapproximately16dB.Withtheincrease

of the filter insertion loss thegainof the amplifierdecreases. InFigure7.21f) the

gain spectra for the EDFA equalised with the different filters are illustrated.

However,duetothefilterbeingplacedattheendoftheEDF,thereisfurthersignal

andforwardASEbuild-upandthereforenobroadeningoftheEDFAgainspectrum

duetothedifferentsaturationconditionsaftertheinclusionofthefilter,asseenin

Figures 7.15e) and 7.18e). The noise figure remains unchanged due to the equal

attenuationof the signaland forwardASEspectrumby the filterat the endof the

amplifier.

Theresultsof theequalisationof theEDFAgainspectrumaresimilar inboth

thecasesoffiltersuncorrectedfortheinsertionloss,calculatedfromequation(7.5),

placedwithintheEDFAandthepresentcasewheretheinverseofthegainspectrum

is used to equalise the EDFA gain spectrum. In both cases the EDFA is well

equalised and the penalty in the amplifier gain due to the insertion of filters

correspondstotheirinsertionlosses.

Insummary,using this technique,goodgainflatnesscanbeachievedwithno

penalty in the amplifier noise figure. However, there is no increase in the filter

bandwidthduetore-amplificationwithdifferentsaturationconditionsasseenwhen

thefilterisplacedclosertothecentreoftheamplifier,seeFigures7.15e)and7.18e).

Thepenaltyintheaveragegainacrossthefilterbandwidthduetotheinsertionloss

ofthefiltercorrespondstothedeviceinsertionloss.

7.3.5 Conclusions

AsummaryoftheresultsofthesimulationssofarisshowninFigure7.22wherethe

EDFA noise figure at λ=1532nm, the gain standard deviation and average gain

acrossthefilterbandwidtharecomparedfortheone-filterconfigurationwhenusing

the three types of filters considered: Filters that were corrected for the insertion

losses, theuncorrectedonesandthefiltersbasedontheinverseof thegainprofile

placedattheoutputoftheEDFA.


0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6 7 8Insertionloss(dB)

Gai

nS

td.D

ev.(

dB)

a)

Inversefilter

Correctedfilter

Uncorrectedfilter

15

17

19

21

23

25


Ave

rage

gai

n(d

B)

Inversefilter

Correctedfilter

b)

Uncorrectedfilter

3

3.4

3.8

4.2


NF

at1

532n

m(d

B)

c)

Inversefilter

Correctedfilter

Uncorrectedfilter

Figure7.22–Comparisonof theequalisedEDFAperformanceusing filtersbasedon the

inversegainprofile,filterscorrectedfortheinsertionlossanduncorrectedfiltersintheone-

filterequalisationscheme.a)EDFAgainstandarddeviationacrossthefilterbandwidth.b)

EDFAaveragegainacrossthefilterbandwidth.c)EDFAnoisefigureatλ=1532nm.

The equalisation of the EDFA gain spectrum using a filter based on a

wavelength-dependent distributed loss calculated using equation (7.5) has been

shown to be possible in two different configurations: one-filter configuration and

two-filters configuration. The two-filters configuration is a good approximation to

theidealdistributed-lossfilterandisveryinsensitivetothepositionwherethefilters

areplacedwithintheEDFA.However,forrealfilterswithinsertionlosses,thereisa

penaltyintheflatnessoftheequalisedgainspectrum.Anotherdisadvantageofthis

configurationisthatisreliesontwofiltersandisthereforemoreexpensive.

Using a one-filter configuration is shown to be the more practical due to the

employmentofonlyonefilter toequalise theEDFAgainspectrumanddue to the

extra flexibility in adjusting its relative position. In this configuration different


equalisationschemesfortheEDFAgainspectrumwerecomparedinFigure7.22:1-

Equalisationusingfilterscalculatedfromequation(7.5)thatwerecorrectedfortheir

insertionloss;2-Equalisationusingfilterscalculatedfromequation(7.3)thatwere

not corrected for their insertion losses; 3- equalisation using filters based on the

inverseprofileofthegainspectrumthatwereplacedattheoutputoftheEDFA.The

thirdequalisationschemeis themostcommonlyadopted inequalisingEDFAgain

spectra.Inalloftheschemes,gainflatnessisachievedattheoptimumpositionsfor

allfilters.Boththeuncorrectedfiltersplacedattheoptimumpositionsandthefilters

basedontheinverseofthegainspectrumhavesimilarperformanceintermsofgain

standarddeviation,averagegainandnoise figure.However,whenusing thefilters

correctedfortheinsertionlossitisobservedthatthereisasignificantimprovement

in theEDFA gainwith small penalty in the gain spectrumstandarddeviation and

noisefigure(seeFigures7.22a),b)andc),respectively).Inthisconfiguration,there

isatotalrecoveryintheamplifiergainandlowpenaltyinthenoisefigureevenfor

filters with insertion losses as high as 8dB. The design and positioning of these

novel filters improves the overall performance of equalised EDFAs. Another

advantageofbothschemes1and2,inrelationtoscheme3,istheslightlyincreased

useful bandwidth obtained, as shown in Figures 7.15e) and 7.18e), respectively,

comparedtoscheme3,showninFigure7.21f).

7.4 Gain flattening filters compensating for the insertion

lossesofotherdevices

In the following section, thiswork is extendedeven further to thedesignofgain-

flatteningfiltersthatcancompensateforinsertionlossesofdifferentdevicesplaced

alongtheEDFAaswellastheirowninsertionloss.Firstly,inordertodemonstrate

theprinciple, thecaseofa lumplossof2dBplacedatagivenpositionwithin the

EDFA is considered. This loss could correspond to the insertion loss of a tap or

anotherfilteringdeviceor lossesduetosplicing.Thesefiltercalculationscouldbe


applied generally, to a number of distributed sources of loss within the EDFA.

Secondly, the inclusionofan isolator in theEDFAisconsidered.The isolator isa

special component commonlyused in amplifiers inorder to reduce the amountof

lighttravellinginthereversedirectionandarrivingattheamplifierinput.Thislight

is due mainly to backward propagating amplified spontaneous emission. The

placementofanisolatorintheEDFAisalsoknowntoimprovetheamplifiernoise

figureandsignalgainwhenplacedat theoptimumposition[108].Toequalise the

EDFA+isolatorstructure,theoptimumpositionwheretheisolatorshouldbeplaced

was determined and the equalising filter was designed in order to compensate for

bothitsowninsertionlossaswellastheinsertionlossoftheisolator.

7.4.1 Equalisation of the EDFA with a lump loss positioned at

Z=2m

ApossibleconfigurationforanEDFAcouldincludeatapdeviceformonitoringthe

performanceoftheEDFAatagivenposition.Theinsertionlossofthesedevicesis

in general low (worst case of around 2dB) but will affect, nevertheless, the

saturationandgainoftheamplifier.TheequalisationoftheEDFAwiththelumped

losscanbeachievedbyinvertingthegainspectrumacrossadesiredbandwidthor,

as shown previously in section 7.3, by determining a filter shape from the ideal

background loss and placing it at the correct position in the EDFA. Figure 7.23

shows the EDFA configuration for a device with a certain insertion loss and an

equalising filter. The position where the tap device is placed is Zloss and the

equalisingfilterisplacedatZ1.


Pump980nm

Z1

Zloss

EDF#1 EDF#2 EDF#3

Ins.LossF ( )k λGainspectrum

Figure7.23–ConfigurationoftheequalisationofanEDFAincludingalumplosscaused

bytheinsertionlossofanarbitrarydevice.Z1isthepositionoftheequalisingfilterandZloss

isthepositionofthearbitrarydevice.

InordertoverifythepossibilityofequalisingtheEDFAgainspectruminthis

configuration,theequalisingfilterswerecalculatedusingboththegainspectrumof

theEDFAwithandwithoutthelumploss.Theinsertionlossofeachequalisingfilter

varied(lIns=0,0.5dB,1dB,2dB,4dB,8dB)andthefiltershapeswerecalculatedin

ordertocorrecttheirinsertionlossandtheinsertionlossofthedeviceplacedatZloss

thatwas lloss=2dB.The total loss lloss+lInswascompensatedusingexpression (7.5).

Theinputsignalwasdividedin32channelsplacedwitha100GHzspacingstarting

at 1532nm.Thepower of each channelwasP0=2.5µWand the total lengthof the

amplifier was L=3m. The simulations were performed for a lump loss placed at

Zloss=2mthatcorrespondsto2/3oftheamplifierlength.Theequalisingfilterswere

calculated using both the EDFA gain spectrum with and without the lump loss.

Figure7.23showstheamplifiergainspectrumwithandwithouttheinsertionofthe

lump loss at Zloss=2m. Curve I corresponds to the gain spectrum of the amplifier

without the lump loss while curve II corresponds to the gain spectrum of the

amplifier+lumplossstructurewiththelosspositionedatZloss=2m.


15

20

25

30

1520 1530 1540 1550 1560 1570Wavelength(nm)

Sig

nalg

ain

(dB

)

I-Withoutloss

II-WithlossatZ=2m

Figure 7.24 – EDFA gain spectrum without the lump loss and including a lump loss at

Zloss=2m.

ThegainspectraillustratedinFigure7.24showthattheeffectontheamplifier

gain due to the insertion of the lump loss at Zloss=2m is a penalty in the gain

correspondingapproximatelytotheattenuationofthelumploss.

Thefilterscanbecorrected tocompensate for the insertion lossesofdifferent

loss elements (fibre splices, other filters, etc) distributed by the amplifier length

simplybyaddingalltheinsertionlossesandusingexpression(7.5)tocalculatethe

correctedfilter.Eithertheoriginalamplifiergainspectrum(Figure7.24–curveI)or

thecompositeamplifiergainspectrum(Figure7.24–curveII)isusedtodetermine

the filter shape. The calculated filters are slightly different and therefore have

differentcharacteristicswhenequalisingtheEDFAgainspectrum.Boththesecases

will be discussed for the lump loss positioned at 2.0m due to the different

characteristicsoftheequalisedEDFAresponsewhenusingboththesefilters.

7.4.1.1 Gain spectrum equalisation with filter calculated using the gain

profilewiththeinsertionofthe2dBlumploss(Figure7.24-curveII).

The filter shape used for equalising the EDFA was calculated using the gain

spectrumoftheamplifierincludingthelumploss.Severalfilterinsertionlosseswere

consideredandforeachcase,thefiltershapewascorrectedforbothitsinsertionloss

and the insertion loss of the device (lump loss). The corrected filter shapes are


showninFigure7.25a)andtheactualfiltersincludingtheirinsertionlossareshown

inFigure7.25b).

-20

-16

-12

-8

-4

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filte

rlo

ss(d

B)

a)

Ins.Loss=0

Ins.Loss=8dB

Ins.Loss=0

Ins.Loss=8dB

-28

-24

-20

-16

-12

-8

-4

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filte

rlo

ss(d

B)

b)

Ins.Loss=0

Ins.Loss=8dB

0

0.5

1

1.5

2


Gai

nS

td.D

ev.(

dB)

c)

Ins.Loss=0

Ins.Loss=8dB

6

10

14

18

22

26


Ave

rage

Gai

n(d

B)

d)

Ins.Loss=0

Ins.Loss=8dB

14

18

22

26

30

1520 1530 1540 1550 1560 1570Wavelength(nm)

Sig

nalg

ain

(dB

)

e)I

Ins.Loss=0

Ins.Loss=8dB

II

3

4

5

6

7

1520 1530 1540 1550 1560 1570Wavelength(nm)

Noi

sefi

gure

(dB

)

f)

Unflattened

Ins.Loss=0

Ins.Loss=8dB

Figure7.25–Performanceofthe(amplifier+lumplossatZ=2m)equalisedusingone-filter

configurationwithdifferentinsertionlosses.Eachfilterwasplacedattheoptimumposition

inorder to flatten thegainspectrumof the structure.a)Filter shapecalculatedfrom(7.5)

using the gain spectrum of the EDFA without the lump loss (curve-II). b) Actual filters

includinginsertionlossof0,0.5,1,2,4,8dB,usedinthenumericalsimulations.c)Standard

deviationof thegain spectrumacross the filter bandwidth fordifferent filterpositions. d)

AverageEDFAgainacrossthefilterbandwidthfordifferentfilterpositions.e)EDFAgain

spectra.f)EDFAnoisefigure.


Byvarying thepositionof the filters along the amplifier length, theoptimum

positionswherethefiltersshouldbeplacedweredetermined.Theaveragegainand

standard deviation across the filter bandwidth were determined for different filter

positionsalongtheEDFA.Figures7.25d)and7.25c)showrespectively,theaverage

gainandstandarddeviationof theamplifierfor filterswithinsertionlossesof0to

8dBplacedalongtheamplifierlength.FromFigure7.25c)itmaybeobservedthat

theoptimumpositionsarearoundZ=1.3mor,atthestartoftheamplifier.Itisbest

toplace the filter away from the start of theEDF tominimise the amplifiernoise

figure penalty. Figures 7.25e) and 7.25f) show respectively the amplifier gain

spectrum and noise figure for filters placed at the optimum positions (around

Z=1.3m).

Equalisation of the EDFA gain spectrum was achieved for all the filters.

However, for filters with insertion losses over 4dB, a considerable penalty in the

noisefigureisobserved,duetotheplacementofthesefiltersclosertothestartofthe

amplifier. The insertion loss of each of the filters was compensated as shown in

Figure7.25e).Evenforfilterswithaninsertionlossof8dBthegainspectrumwas

equalised with full recovery of the average gain across the filter bandwidth. The

insertionlossofthelumplosswascompensatedbythefilterdesign.However,there

isapenaltyofroughly2dBcorrespondingtothelumplossthatisnotavoidedwhen

usingthecurrentfilters.

The way these filters achieve spectral equalisation of the amplifier gain is

shown in Figure C3 in appendix C where the signal gain, forward ASE and

backwardASEareplottedalongtheEDFlengthforthreedifferentwavelengths,λ1

=1532.3nm,λ2=1539.7nmandλ3=1550.9nm.Thereisaconstantinsertionlossof

2dBat Z=2mand the equalising filterswith insertion lossesof0, 0.5, 1, 2, 4 and

8dBareplacedat theoptimumpositions(aroundZ=1.3m). Inorder toobtaingain

equalisation across the whole filter bandwidth, the filter loss spectrum and the

positionwhere it isplacedhas tobe adjustedso thateachwavelength reaches the

endoftheamplifierwiththesamegainlevel.InFigureC3,itcanalsobeobserved

that even for filters with high insertion losses the gain can be recovered by the

correctdesignandpositioningofthefilters.


Thephysicalreasonforthesignalrecoveryandgainrestorationacrossthefilter

bandwidthisthesameasinthecasedescribedinsection7.3.3.1.Thesignalpower

andlocalgainevolutionforthreedifferentwavelengthsareshowninFiguresC3a),

b)andc),inAppendixC.Thepumppowerandpopulationinversionre-distribution

(due tochanges inEDFAsaturationcausedby thefiltering)along theEDFlength

areshowninFigureC4.

7.4.1.2 Gain spectrum equalisation with filter calculated using the gain

profilewiththeinsertionofthe2dBlumploss(Figure7.24-curveI).

In the following simulation the gain spectrum of the EDFA plus an insertion loss

placedatZloss=2misequalisedusingafiltercalculatedusingequation(7.5)andthe

amplifier gain spectrum without the lump loss (Curve I in Figure 7.24). Filter

insertion losses of 0, 0.5, 1, 2, 4 and 8dB were considered and for each case, the

filter loss spectrum was corrected for both its own insertion loss as well as the

insertionlossofthelosselementplacedatZloss=2m.Thecorrectedfiltershapesare

showninFigure7.26a)andtheactualFiltersincludingtheirinsertionlossareshown

inFigure7.26b).


-20

-16

-12

-8

-4

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filte

rlo

ss(d

B)

a)

Ins.Loss=0

Ins.Loss=8dB

Ins.Loss=0

Ins.Loss=8dB

-28

-24

-20

-16

-12

-8

-4

0

1520 1530 1540 1550 1560 1570Wavelength(nm)

Filte

rlo

ss(d

B)

b)

Ins.Loss=0

Ins.Loss=8dB

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2FilterPosition(m)

Gai

nS

td.D

ev.(

dB)

c)

Ins.Loss=0

Ins.Loss=8dB84

2

1

00.5

6

10

14

18

22

26


Ave

rage

Gai

n(d

B)

d)Ins.Loss=0

Ins.Loss=8dB

10

15

20

25

30

1520 1530 1540 1550 1560 1570Wavelength(nm)

Sig

nalg

ain

(dB

)

IIns.Loss=0

Ins.Loss=8dB

II

e)

3

4

5

6

7

8

9

1520 1530 1540 1550 1560 1570Wavelength(nm)

Noi

sefi

gure

(dB

)

f)

Unflattened Ins.Loss=0

Ins.Loss=8dB

0.5

0

1

2

48

Figure7.26–Performanceofthe(amplifier+lumplossatZ=2m)equalisedusingone-filter


inorder to flatten thegainspectrumof the structure.a)Filtershapecalculatedfrom(7.5)

using the gain spectrum of the EDFA without the lump loss (curve-I). b) Actual filters

includinginsertionlossof0,0.5,1,2,4,8dB,usedinthenumericalsimulations.c)Standard

deviationof thegain spectrumacross the filter bandwidth fordifferent filter positions. d)

AverageEDFAgainacrossthefilterbandwidthfordifferentfilterpositions.e)EDFAgain

spectra.f)EDFAnoisefigure.

Again, the optimum filter position was determined by minimising the gain

standarddeviation.Thestandarddeviationandaveragegainacrossthebandwidthof

each filter are shown respectively in Figures 7.26c) and 7.26d) for different filter

positions.Theoptimumpositionwherethefiltersshouldbeplacedinordertoflatten


the EDFA gain spectrum is around Z=0.8m. As previously mentioned, the

positioningoftheequalisingfilternearthestartoftheEDFincreasestheamplifier

gain across the filter bandwidth and causes an increased penalty in the amplifier

noise figure. Figures 7.26e) and 7.26f) show respectively the gain spectrum and

noisefigureoftheequalisedamplifierforthefiltersplacedattheoptimumpositions.

The overall behaviour is similar to the previous case, shown in Figure 7.25.

However, because in the current case the target equalisedgain level is higher, the

requiredfilteringisstronger(seeFigures7.25a)and7.26a))andtheoptimumfilter

positionslightlyclosertotheEDFAinputend(seeFigures7.25c)and7.26c)).Asa

result, the noise figure penalty is substantially increased in the current case (see

Figures 7.25f) and 7.26f)). The corresponding signal gain at three different

wavelengths,thepumppowerandmetastable-levelpopulationevolutionsareshown

in Figures C5 and C6 in Appendix C. From Figures 7.25f) and 7.26f), it can be

deducedthatloweringthetargetequalised-gainlevelcanhaveasignificanteffecton

theincurrednoisefigurepenalty.

7.4.2 Equalisationofa(EDFA+isolator)structure:

A common configuration for an EDFA includes an isolator for filtering the

backwardASEthatimprovesthenoisefigureandgainoftheamplifier.Theisolator

isaspecialdevicethatactsasadirection-selectivefilter:forwardpropagatinglight

isunaffectedwhilethebackwardpropagatinglightisattenuated(typicallyby30dB).

The insertion loss of these devices is in general low (maximum of 2dB). The

saturation along the amplifier will be affected by the insertion loss of the isolator

and especially due to the high attenuation of the backward ASE. Due to this

behaviouroftheisolator,itisnotobviousthatthefiltersdesignedusingthismethod

are suitable to equalise the EDFA+isolator structure. The equalisation of the

composite EDFA with the isolator can also be achieved by inverting the gain

spectrumacrossadesiredbandwidthandplacingitattheEDFAoutputor,asshown

here,bydeterminingafilterfromtheidealdistributedlossspectrumandplacingitat

the correct position in the EDFA. Figure 7.27 illustrates the configuration of an


EDFAincludinganisolatorwithagiveninsertionlossandanequalisingfilter.The

positionwhere the isolator isplaced isZisolatorand theequalisingfilter isplacedat

Z1.ThetotallengthoftheEDFAisL=3m.

Pump980nm

Zisolator

Z1

EDF#1 EDF#2 EDF#3F ( )k λ

GainspectrumIsolator

Figure7.27–Configurationof theequalisationofanEDFAincludingan isolatorandan

equalisingfilter.Z1 is thepositionof theequalisingfilterandZisolator is thepositionofthe

isolator.

ThefirststepistooptimisethepositionoftheisolatorintheEDFA.According

to[108,109],theoptimumpositionofanisolatorwithintheEDFAisaround1/3of

theamplifierlength.Atthispositionthenoisefigureisminimumandtheincreasein

theamplifiergainisalmostmaximum.Inthesesimulations,anisolatorwith30dBof

extinctionratioand2dBofinsertionlosswasused.Inthiscase,forwardsignalsare

attenuatedby2dBandbackwardsignalsareattenuatedby32dB.Thepositionofthe

isolatorintheEDFAwasvariedalongtheamplifierlengthandtheperformanceof

theEDFA+isolatorwasmonitoredforawavelengthof1532nm.Figures7.28a)and

7.28b)showrespectivelythenoisefigureandsignalgainatawavelengthλ=1532nm

fordifferentpositionsoftheisolatorwithintheEDFA.


3

3,2

3,4

3,6

3,8

4

0 1 2 3Isolatorposition(m)

Noi

seF

igur

e(d

B)

a)λ=1532λ=1532λ=1532λ=1532 nm

27

28

29

30

31

0 1 2 3Isolatorposition(m)

Sig

nalg

ain

(dB

)

b)λ=1532λ=1532λ=1532λ=1532 nm

Figure 7.28 - Performance of the EDFA+isolator structure for different positions of the

isolatorwithintheamplifier.a)Noisefigureatλ=1532nmversusisolatorposition.b)Signal

gainatλ=1532nmversusisolatorposition.

Thepositionwheretheisolatorshouldbeplacedinorder tooptimisetheamplifier

noisefigureisZisolator=1m.TheamplifierlengthwasL=3mandthereforetheisolator

position corresponds to 1/3 of the amplifier length in agreement with [108, 109].

ThegainspectraoftheEDFAwithandwithouttheisolatorplacedatZisolator=1mis

showninFigure7.29.

15

20

25

30

1520 1530 1540 1550 1560 1570Wavelength(nm)

Sig

nalg

ain

(dB

)

II-WithIsolator

I-WithoutIsolator II

I

Figure7.29-EDFAgainspectrumwithandwithouttheisolatorpositionedattheoptimum

position,Zisolator=1m.

InordertoequalisethegainspectrumoftheEDFAwiththeisolatorplacedat

Zisolator=1m and compensate for the insertion loss of the isolator, the filters where

calculatedusingequation(7.5)and theEDFAgainprofileshowninFigure7.29–


curveI.Filterswithinsertionlossesof0,0.5,1,2and4dB,whichweredesignedto

compensate for theirowninsertionlossaswellas theinsertionlossof theisolator

were considered. These filters are the same as the ones used previously for

compensating for the insertion loss of a 2dB lump loss. The filter shapes and the

actual filters including the insertion loss that were used to equalise the

EDFA+isolator gain spectrum are illustrated respectively in Figures 7.26a) and

7.26b). The optimum position where these filters should be placed in order to

equalisetheamplifiergainspectrumwasdeterminedbyvaryingthepositionofthe

filtersalongtheamplifierlength.Thestandarddeviationandaveragegainacrossthe

filterbandwidthfordifferentfilterpositionsareshowninFigures7.30a)and7.30b),

respectively.

0.04

0.14

0.24

0.34

0 0.5 1 1.5 2FilterPosition(m)

Gai

nS

td.D

ev.(

dB

) a)

l Ins=4dB

l Ins=0l Ins=4dB

l Ins=0

10

14

18

22

26

0 1 2 3Filterposition(m)

Ave

rag

eg

ain

(dB

)

b)l Ins=0

l Ins=4dB

14

18

22

26

30

1520 1530 1540 1550 1560 1570Wavelength(nm)

Sig

nalg

ain

(dB

)

c)

l Ins=4dB

l Ins=0

3

3.5

4

4.5

5

5.5

1520 1530 1540 1550 1560 1570Wavelength(nm)

Noi

sef

igu

re(d

B)

d)

l Ins=4dB

l Ins=0

Figure7.30–Performanceof the (amplifier+isolator atZ=1m)equalisedusingone-filter


in order to flatten the gain spectrum of the structure. a) Standard deviation of the gain

spectrum across the filter bandwidth for different filter positions. b) Average EDFA gain

across the filter bandwidth for different filter positions. c) EDFA gain spectra. d) EDFA

noisefigure.


FromFigure7.30a), theoptimumfilterpositionisaroundZ=1.2m(depending

onthefilterinsertionloss).Thesepositionsareclosetothemiddleoftheamplifier

andthereforeasmallpenaltyinthenoisefigureisobserved.

Theperformanceof the(EDFA+isolator+equalisingfilter)structurefor the

filters placed at the optimum positions is illustrated in Figures 7.30c) and 7.30d)

wheretheamplifiergainspectrumandnoisefigureareplottedrespectively.Forall

the filters, the amplifier gain spectrum is equalised and there is no penalty in the

averagegainasshowninFigure7.30c).Boththeinsertionlossofeachfilterandthe

insertionlossoftheisolatorwerecompensatedforbythefilterdesign.Thepositions

where the filtersareplaced inorder to flatten theamplifiergain spectrumcausea

small penalty in the amplifier noise figure, illustrated in Figure 7.30d). However

when using filtering devices with low insertion losses (below 1dB), a very low

amplifier noise figure can be obtained with gain spectrum equalisation and

compensationfortheisolatorinsertionloss.

Figures C7a), C7b) and C7c) in appendix C show the evolution of the signal

gain along the amplifier length for three different wavelengths, λ1=1532.3nm,

λ2=1539.4nmandλ3=1550.7nmrespectively.Thereisadropof2dBintheamplifier

gain corresponding to the insertion lossof the isolator atZ=1mandawavelength

dependentlossduetothefilterataround1.2m.Theamplifiergainpicksuptovalues

around24dBatallwavelengthsproducinganequalisedgainspectra.

Thewavelength-dependentnoisefigureshowninFigure7.30d)forthevarious

filtersisduetothedifferentevolutionofthesignalandforwardASEalongthefibre

length.TheforwardandbackwardASEpowerevolutionalongtheEDFlengthfor

different wavelengths is also illustrated in Figures C7d)-f). The power of the

backwardASEattheinputendoftheEDFAisverylowduetothepresenceofthe

isolator at Z=1m. Consequently, a lower pump absorption rate at the input of the

EDFA isobserved.Thepumppower andpopulation inversionalong the amplifier

lengthareillustratedinFiguresC8a)andC8b)respectively.Theisolatorpractically

doesnotaffectthepopulationinversionatZ=1mduetoitsrelativelylowinsertion


loss(2dB)butchangestheinitialpopulationinversionandpumpabsorptionatZ=0

duetotheattenuationof32dBinthebackwardASE.

7.4.3 Conclusions

ThedesignoffiltersforequalisingtheEDFAgainspectrumwhilecompensatingfor

their own insertion losses and thatofotherdeviceswas studied.The gainpenalty

duetotheinsertionofanarbitrarylossintheEDFcanbecompensatedforbymeans

of the calculation of the corrected filter shape and determination of the optimum

position.Howeverthereisapenaltyintheamplifiernoisefigureassociatedwiththe

incorporation of these filters in the amplifier. These novel filter designs can be

particularlyuseful incaseswherefilterswith intrinsicallyhigh insertion lossesare

used.

7.5 Summary

An approach for determining ideal wavelength-dependent loss filters and their

optimumpositioninordertoequalisetheEDFAgainspectrumwasdeveloped.One-

filter and two-filter equalisation schemes were compared: the first one is very

sensitive on the position where the filter is placed, and the second gives a good

approximation to a uniformly-distributed wavelength-dependent distributed loss,

insensitive to the exact placement of the filters. Different filter designs with or

without a correction for the device insertion losses, as well as filters obtained by

inverting the EDFA gain spectrum, were compared. Using filters that are not

designedtocompensatefortheinsertionlossesofthedevices,theirperformancein

flatteningtheEDFAgainspectrumissimilartofiltersobtainedusingtheinverseof

the amplifier gain and positioned at the output. When using filters designed to

correct for the respective insertion losses, gain equalisation is achieved with no

penaltyintheamplifiergainforfilterswithupto8dBinsertionlosses.However,in


thiscase,thereisaslightpenaltyintheamplifiernoisefigureduetothefactthatthe

optimum position where these filters are placed is closer to the input of the

amplifier.Thesummaryoftheperformanceofthesedifferentequalisationfiltersis

showninFigure7.22.

Themethodcanbeextendedtodesignfiltersthatcompensateforboththeirown

insertionlossandtheinsertionlossesofdifferentdevicesplacedalongtheamplifier.

Inparticulartwosituationswereaddressed;thefirstoneconsistedofequalisingthe

EDFA gain spectrum that included a loss element placed at Z=2m; the second to

equalisetheEFDAgainspectruminacommonlyusedconfigurationthatincludesan

optical isolator toavoidthebackwardASEbuildupwhile improvingbothitsgain

and noise figure. For both these situations the EDFA gain equalisation was

successful using one equalising filter and the correct filter design and placement

withintheamplifier.

Finally,tunablefilters,similartotheonesdemonstratedinthepreviouschapter,

could be designed using the calculated ideal filter shapes in order to dynamically

equalisetheEDFAgainspectrum.Alternatively,asthistheoryisvalidforarbitrary

EDFA saturations (including pumping conditions and signal power), a range of

filterscouldbedesignedfordifferentsaturationconditionsandatunablefilterused

to scan among the possible configurations in order to equalise the EDFA gain

spectrumwithminimumcomputingtime.

8

All-FibreAdd-Drop

Multiplexers

InthisChapterall-fibreadd-dropmultiplexerconfigurationsbasedontheinscription

of Bragg gratings in the waist of fibre couplers are discussed. The main study is

aroundtheparametersinvolvedintheoptimisationofadesignbasedonahalf-cycle

coupler with a grating in its waist. A solution is presented in the form of a non-

uniform half-cycle coupler. This novel device is demonstrated experimentally and

shown tobe apotential solution for achieving the required specifications for add-

drop multiplexers in WDM systems. A new configuration based on a full-cycle

couplerwithagratinginscribedinitswaistisalsoanalysed.

8-All-FibreAdd-DropMultiplexers 140

8.1 Overview

Wavelength division multiplexing (WDM) is one of the most important means of

obtaining high speed optical communications links. In these links, signals of

selectedwavelengthsneedtobedroppedoraddedtotheopticalstreamatdifferent

pointsalongthenetwork.Severaladd-dropmultiplexerschemeshavebeenreported

in the literature. Planar devices based on gratings in coupler structures present

compact and efficient solutions to the problem. However, these devices have

intrinsicproblemslikepolarisationsensitivityandhighinsertionlosses.Severaladd-

dropmultiplexerdevicesbasedonall-fibreschemeshavebeenextensivelyanalysed

asmentionedinChapter4.

Aparticularfamilyofall-fibreadd-dropmultiplexersisbasedontheinscription

ofBragggratingsinthewaistoffibre-couplers.Theinscriptionofatiltedgratingin

the waist of a null coupler has been demonstrated as suitable for add-drop

performance[39,110]. Inthischapter thefocus isonadd-dropmultiplexersbased

on the inscription of gratings in the waist of half-cycle and full-cycle couplers.

Design considerations and parameters to be optimised are analysed and a novel

design based on a non-uniform half-cycle coupler is presented as an optimised

solution.

8.2 NumericalModel

The performance of add-drop multiplexers based on the inscription of a Bragg

grating in the waist of a fibre coupler can be simulated using the transfer matrix

approachforthewholestructure.Thepropagatingevenandoddcouplereigenmodes

withpropagationconstantsβeandβo,respectively,arereflectedindividuallybythe

grating.Both the reflected lightarrivingat the inputportand the transmitted light

arrivingattheoutputportsaredecomposedintothenormalmodesoftheindividual


waveguides. Given that essentially there are two reflectionpeaksoriginating from

thesamegratingandcorrespondingtothetwoeigenmodes,thelargerthedifference

inpropagationconstantoftheevenandoddmodes,thelargerwillbethedetuning

betweenthecentralwavelengthsofthereflectionpeaksproducinglargedispersions

atthebandedgesandnarrowingthetotalreflectedbandwidth.Thedifferenceofthe

localpropagationconstantsoftheevenandoddsupermodesareproportionaltothe

strengthofthecoupler,k(z),and,ingeneral,variesalongthestructure.Thegrating

responseisdeterminedbyusingatransfermatrixmethodandarbitraryapodisation

profiles[78].ThetotalstructureisshownschematicallyinFigure8.1

Figure8.1–Schematicoftheproposedadd-dropmultiplexers.

LetA1andA2betheamplitudesoftheelectricfieldatthecouplerinputport1and

port2,respectively.Thefieldsaredecomposedintotheevenandoddsupermodesat

thebeginningofthecoupleratpositionA.βeandβoarethepropagationconstantsof

theevenandoddeigenmodes,andAeandAoarethefieldamplitudesapproximated

by(4.1):

2)0( 21 AA

Ae

+= and2

)0( 21 AAAo

−=

FromAtothestartof thegratingatZ=ZG-LG/2(pointBinFigure8.1) themodes

propagate adiabatically along the coupler accumulating a total phase difference

betweentheeigenmodes β−β=β∆z

0 oe dz)( wheretheintegrationisperformedover


the travelled distance. At B the fields of the forward propagating even and odd

eigenmodesaregivenby:

=−

−

−2/

0

,

)0()2/( ,,

GLGZ

oe dzi

oeGGoe eALZEβ

(8.1)

The reflected field for the even and odd eigenmodes is given by multiplying the

gratingreflectioncoefficient,ρe,o(0),bytherespectiveelectricfieldatthestartofthe

grating.DenominatingEe’ andEo’ thefieldsofthebackwardpropagatingevenand

oddeigenmodesrespectively,atthestartofthegratingtheycanbewrittenas

)2/()2/(' ,,, GGoeoeGGoe LZELZE −=− ρ (8.2)

andatthestartofthecoupler(positionA),thefieldsofthebackwardreflectedeven

andoddeigenmodesarewrittenas,

=

−

−2/

0

,2

,,, )0()0('

GLGZ

oe dzi

oeoeoe eAEβ

ρ (8.3)

The phase of the reflected even and odd eigenmodes depends now on both the

propagation along the coupler and the phase due to the corresponding complex

reflectivityof thegrating.The light transmitted through thegrating isobtainedby

multiplying the transmission coefficient, t(LG) by the fields at the input of the

grating.AtpositionCcorresponding toZ=ZG+LG/2, theforwardpropagatingeven

andoddmodefieldscanbewrittenas

=+

−

−2/

0

,

,,, )0()2/(

GLGZ

oe dzi

oeoeGGoe etALZEβ

(8.4)


The expressions for the electric fields of the forward propagating even and odd

eigenmodesattheendofthecoupler(positionD)arewrittenas:

+−

= +

− CZ

GLGZ

oe

GLGZ

oe dzdzi

oeoeCoe etALE 2/

,

2/

0

,

,,, )0()(ββ

(8.5)

The spectral properties of the grating namely the transmission and reflection

coefficients are obtained by using a transfer matrix model [78] by dividing the

grating into N uniform sections and multiplying the transfer functions of each

section.Toobtainthelightpowerarrivingateachoneoftheports,theevenandodd

mode fields are decomposed into the fields of the normal waveguides using

expression (4.1). The response of the grating written within the coupler waist in

general differs from the performance of the device itself due to the difference in

propagationconstantsbetweentheevenandoddeigenmodes.

8.3 Add-DropConfigurations

Devicesthatrelyontheinscriptionofnon-tiltedBragggratingsinthewaistoffibre

couplers are studied. Three different add-drop multiplexer configurations are

analysedandcompared.

The first design relieson agratingwritten in ahalf-cycle fused fibre coupler.

Thisdevicehasbeendemonstrated[42]byplacingagratingatthecentreofafused-

coupler waist and, by writing identical gratings in two separate fibres and then

polishedtogetherformingalongcouplingregion[111].Severalissuesrelatedtothe

optimisationandperformanceofthisdeviceareevaluatedinthischapter.Theeffect

of the difference in propagation constant of the coupler eigenmodes, the grating

apodisationandlength,thepenetrationdepthoftheradiationintheBragggratings

and the position within the couplers where they should be placed are addressed.

Experimentalaspectsthatmaydegradethedeviceperformancearealsodiscussed.


Thesecondisanovelconfigurationthatreliesonwritingagratinginthewaist

ofasymmetricfull-cyclecoupler.Thegratingiswritteninthecentreofthecoupler

waistwithitseffective-reflectionpointscoincidingwiththepositionsinthecoupler

wherethepowerisequallydistributedbetweentheidividual-waveguidemodes(50-

50% point of the coupler). Equivalently, these are the points where the two

eigenmodes are π/2 and 3π/2 out of phase. This gives a symmetric configuration

whereaselectedchanneloflightlaunchedfromport1isdroppedatport2andthe

remaining wavelength channels arrive at port 4 or equally, a selected channel

comingfromport3isdroppedtoport4(seeFigure8.1).Themainadvantageofthis

symmetric configuration is that simultaneous add and drop operation can be

achievedwithonlyonegrating.However,aswillbediscussedinsection8.3.2this

device suffers from intrinsic limitations mainly due to the high dispersion at the

gratingbandedgesandtheneedforexactpositioningofthegratingwithrespectto

the50-50%pointsofthecoupler.Inordertodetermineexperimentallythe50-50%

point of the full-cycle couplers a perturbation method for characterising fibre

couplers(discussedinChapter9)wasdeveloped.

The third device is based on inscribing a grating in a half-cycle coupler

fabricated with a complex coupling profile. This novel device is symmetric and

composedofthreecouplingregionswithdifferentradii:Acentralregionwithavery

low coupling constant that is longer than the two sections on either side that are

stronglycoupled.Thegratingiswritteninthecentralregionwherethepropagation

constants of the coupler eigenmodes are almost the same. This configuration is

equivalenttoameter-longuniformhalf-cyclecouplerandthereforethegratingcan

beconsideredtobeapointreflector.

8.3.1 Grating-baseduniformhalf-cyclefibrecouplerOADM.

The demonstration of add-drop operation based on a Bragg grating written in the

centreofahalf-cyclecouplerhasbeeninitiallydemonstratedbyBakhtietal.[42].

However,theperformancewasnotoptimisedduetotheuncontrolledpositioningof


thegratinginthecoupler.Thegratingwasplacedatthecentreofthecouplerwaist

in order to achieve symmetric operation but the strength, apodisation and length

were not designed so that the effective reflection point, at the grating resonant

wavelength,matchedthecentreofthecouplerwaist.Asaconsequenceanexcessive

cross-talk was observed in the operation of this device. The optimisation of the

grating relative position in the coupler waist is essential for the optimum

performanceof thedevice [41,112]. In this section, several aspects related to the

performance-degradation of this device are discussed and possible solutions are

presented. Figure 8.2 illustrates schematically the principle of operation of this

device.

Figure8.2–Principleofoperationofanadd-dropmultiplexerbasedontheinscriptionofa

Bragggratinginthewaistofahalf-cycle(100%)coupler.a)Devicerepresentation.b)Drop

operation:achannellaunchedinport1isdroppedatport2andtherestofthechannelsare

transmittedtoport4.

8.3.1.1 Optimisingforthepenetrationdepth

Whenagratingisplacedinthecentreofthecouplerwaist,thedeviceissymmetric.

Inthiscasetheperformanceof thedeviceiscompromisedandinorder toachieve

optimumperformancethegratingshouldbepositionedasymmetricallywithrespect


to the centre of the coupler. The grating length, strength and apodisation profile

should be taken into account when correcting the position of the grating. To

demonstrate these issues, simulations of gratings with different apodisations and

lengths are shown in Figures 8.3, 8.4 and 8.5. The maximum refractive index

modulationofthegratingwasassumedtobeconstant,∆n=2x10-4.InFigure8.3,the

transmission (at thegrating resonancewavelength,λG)wasplottedwithrespect to

the grating length for different grating apodisations namely, Blackman and sine2

apodisationsanduniformgratings.For agiven transmissivity, the requiredgrating

lengthassociatedwiththerespectiveapodisationprofilecanbedetermined.

-100

-80

-60

-40

-20

0

0 5 10 15 20 25 30GratingLength(mm)

Tran

smis

sivi

tya

t λλ λλG(d

B)

Uniform

Blackman

Sin2(x)

Figure 8.3 – Grating transmissivity at the resonance wavelength for different grating

lengths and constant index modulation ∆n=2x10-4. Blue line: Blackman apodisation; Red

line:sine2apodisation;Blackline:Uniformgrating.

By choosing the desired grating length, the corresponding correction in the

position of the centre of the grating with respect to the centre of the coupler is

calculatedbydeterminingthepenetrationdepthattheBraggwavelength.Figure8.4

showsthesecorrectionswithrespecttothegratingsplottedabove.Thecorrectionof

thegratingposition,∆Zcorr,wascalculatedbysubtractingthepenetrationdepthatthe

grating resonance wavelength, Zpen(λG), from half the grating length, i.e., ∆Zcorr=


LG/2-Zpen(λG).FromFigure8.4itisobservedthatforauniformgratingwithalength

ofLG=9mmand∆n=2x10-4, theoptimisedpositionisdisplaced+3.27mmfromthe

centre of the coupler and for a raised sinusoid apodisation it is displaced by

+1.35mm.Finally,fortheBlackmanapodisedgratingtheoptimisedpositionofthe

gratingis+1mmfromthecentreofthecoupler.

-2

0

2

4

6

8

10

12

14


Pos

ition

cor

rect

ion

(mm

) Uniform

Blackman

Sin2(x)

Figure 8.4 – Correction of the grating position in order to match the effective reflection

point atλG to the centreof the coupler. Blue line: Blackman apodisation; Red line: sine2

apodisation;Blackline:Uniformgrating.

Additionally, the total length of the coupler, LC, necessary for optimising the

differentgratinglengthsandapodisationscanbecalculatedsimplyby:

∆+= CorrG

C ZL

L2

2 (8.6)

Thisisanimportantparameterduetothelimitationsofcouplerlengthimposed

bythefabricationprocedure.Usingtheflame-brushtechniquedescribed insection

(4.3.1) the limit for fabrication of consistent good quality uniform half-cycle

couplers was about 30mm. For the simulated gratings, the minimum size of the


couplers, inorder tomatch thegrating reflectionpoint to centreof the coupler, is

showninFigure8.5.

0

10

20

30

40

50

60

70


Min

.cou

pler

leng

th(m

m)

Uniform

Blackman

Sin2(x)

Figure 8.5 – Minimum uniform coupler length in order to match the effective reflection

point atλG to the centreof the coupler fordifferentgrating lengths.Blue line:Blackman

apodisation;Redline:Sine2apodisation;Blackline:Uniformgrating.

Theimpactof therelativegratingpositionwithinthecouplerwaistonthespectral

performanceoftheadd-dropmultiplexerisdemonstratedbycomparingFigures8.6

and 8.7. These Figures show the spectral response of an OADM with a sine2

apodised 15mm-long grating written respectively, at the centre of a 30mm long

coupleranddisplacedby∆ZCorr=+3.24mmfromthecentreofthecoupler.Thelabels

PijinFigures8.6and8.7refertothepowerarrivingatportiwhenlaunchedatPortj.

From Figure 8.6 it is observed that there is a significant amount of light back

reflectedtotheinputportattheresonancewavelength.Thisisduetothemismatch

betweenthecentreof thecouplerandtheeffectivereflectionpositionattheBragg

wavelength. There are also significant back-reflections (P11) and leakage through

port3(P13)atthegratingbandedgesduetotheincreasedtimedelayexperiencedat

thosewavelengths,duetomultiplereflectionsinthegrating.Thesecanbereduced

bydecreasingtheseparationbetweenthecouplereigenmodessothatthetimedelay


atthesewavelengthsisthesameforboththemodesandthereforeafteraroundtrip,

theyarriveattheoriginofthecouplerwithatotalphasedifferenceφ=(βe-βo)dz=π.

-60

-50

-40

-30

-20

-10

0

-0.50 -0.25 0.00 0.25 0.50Wavelengthdetuning(nm)

Pow

er(d

B)

P11

P21 P41

P31

Figure 8.6 – Spectral response of the 30mm long half-cycle coupler with a 15mm sine2

apodisedgratingwritteninthecentreofthecouplerwaist.

Theoperationof thisdevice issymmetricwhicheverport the light is launched

into. It can be used in the configuration shown in Figure 3.5 to add and drop the

selected wavelength from the optical stream. However, the back-reflections (S11

parameter)andcrosstalk(S31parameter)arearound–10dBandnotsufficientinreal

applications. Placing two isolators at the input of ports 1 and 3 could solve this

problembutwouldmake thedeviceexpensiveand lesscompetitive.Alternatively,

this add-drop multiplexer could be optimised for either add or drop operation by

placing the grating at the optimum position as shown in Figures 8.7a) and 8.7b).

Anotherissuethatshouldbeaddressedistheisolationofthisdevicethatdependson

thegratingstrength.FromFigure8.3, it isdeduced that foramaximumrefractive

indexmodulationof∆n=2x10-4anda40dBisolation,therequiredgratinglengthis

25mmforasine2apodisationand30mmforaBlackmanapodisationprofile.Inthe

symmetric configuration, the uniform coupler length could be at least the same

lengthasthegrating.


Whenusingan asymmetric configuration,where thegrating isdisplaced from

the centre of the coupler in order to compensate for the penetration depth, the

minimum uniform coupler length is given by equation (8.6) and is illustrated in

Figure8.5forgratingswithdifferentapodisationsandlengths.Thespectralresponse

fortheadd-dropmultiplexerinthisasymmetricoperationisshowninFigures8.7a)

and8.7b)forlightlaunchedinport1(dropoperation)andlightlaunchedinport4

(add operation) respectively. At the grating resonance, the back reflected light is

optimised but due to the different time delays experienced by the even and odd

eigenmodesattheedgesofthegratingbandwidth,theusablebandwidthislimitedto

about20%ofthetotalgratingbandwidth,asillustratedinFigure8.7a).Whenlight

is launched into port 4, in order to add a channel to the optical stream passing

through port 3, the operation is degraded as observed in Figure 8.7b): the back-

reflectionsarehighandthereisaninsertionlossassociatedwiththeaddedchannel.

It should be noted that P41≡P14 and P31≡P24 due to reciprocity in a symmetric

configuration.

-60

-50

-40

-30

-20

-10

0


Pow

er(d

B)

P11

P21 P41

P31

a)

-60

-50

-40

-30

-20

-10

0


Pow

er(d

B)

P44

P34 P14

P24

b)

Figure 8.7 – Spectral response of the 30mm long half-cycle coupler with a 15mm sine2

apodised grating displaced by +3.24mm from the centre of the coupler waist. a) Light

launchedinport1-OptimisedDropoperation.b)Lightlaunchedinport4–DegradedAdd

operation.

ToachieveanOADMwithoptimisedAddandDropoperations, two identical

asymmetric devices should be employed using the configuration shown in Figure

8.8.Theoperationof thedeviceis thefollowing:Anopticalstreamis launchedin


port 1 and the grating written at the optimised position in the coupler drops the

selected channel to port 4. The rest of the channels are transmitted through the

secondcouplerarrivingatport2.Whenanopticalsignal is launchedinPort3the

selectedchannel isaddedtoport2withoptimisedperformance.Theportnumbers

are consistent with the previous nomenclature for add-drop multiplexer ports in

chapter 3: Port1 – Input; Port2 – Output; Port3 – Add and Port4 – Drop. This

numbering may differ from the numbering of the coupler ports depending on the

configurationemployed.

Figure 8.8 – Schematic of a symmetric add-drop operation achieved by cascading two

asymmetricdeviceswithoptimisedaddanddropoperation.

8.3.1.2 Couplerlengthoptimisation.

Theoptimisationof thegratingposition in thewaistof thecouplerdoesnotsolve

the problem of the offset of the reflection peaks of the even and odd eigenmodes

(due to their difference in propagation constant). For a uniform coupler with a

resonance wavelength, λC, the difference between the effective index of the even

andoddeigenmodesisgivenby:

C

eoeoe k

nnnβ∆=−=∆ 0, (8.7)


where∆βeo=βe-βoandkC=2π/λC.ForauniformcoupleroflengthLandwithatotal

phase φ=nπ, n=1,2… (n=1 - half-cycle couplers, n=2 - full-cycle couplers…) the

differencebetweentheevenandoddpropagationconstantsisgivenby∆βeo=φ/L.

Figure 8.9 shows the spectral response of the eigenmodes of a 30mm long

uniformcouplerreflectedoffa15mm-longgratingwithasine2apodisationprofile.

Thedashedlinesrepresentthedifferentgratingresonantwavelengths“seen”byeach

oneoftheeigenmodes.Thetotalresponseoftheadd-dropmultiplexerisdetermined

bytheoverlapbetweenthereflectivitiesofbothmodes.Consequently,anarrowing

oftheoverallfilterbandwidthisexpected.

-60

-50

-40

-30

-20

-10

0


Pow

er(d

B)

Even Odd

Figure8.9–Spectralresponseoftheevenandoddcouplereigenmodesfora15mmlong

gratingwithasine2apodisation inaLC=30mmlongcoupler.Blue lines:Evenmode.Red

lines:Oddmode.Thinline:Transmission.Thinkline:Reflection.

By considering this difference in terms of the time delay or associated

penetrationdepthoftheradiationintothegrating,thereasonforthehighlevelsof

backreflectedlightat theedgesof thegratingbandwidthbecomesclear.Thehigh

dispersion at these points and the slight detuning in the resonance wavelength of

eachoftheeigenmodesgivesrisetodifferenteffectivepathstravelledbyeachone

oftheeigenmodesatthesewavelengths.Consequently,whentheyarriveatthestart


of thecoupler theyhaveacquiredadditionalphasedetuning thepreciseamountof

which, depends on the wavelength. Figure 8.10a) shows the penetration depth for

both theevenandoddeigenmodesfor thesamegratingasused in thesimulations

above.Thedashedlinescorrespondtothewavelengthwherethedifferencebetween

thepenetrationdepthsofeachoneoftheeigenmodesisgreatest.Thetotaldistance

travelledbylightinthegratingisdoublethispenetrationdepth.InFigure8.10b)the

difference between the penetration depths, ∆Zpen, of the coupler eigenmodes is

illustrated. The dashed lines indicate again the wavelength detuning where the

differenceinthepenetrationdepthbetweenthesemodesisgreatest.Atthesepoints

theeffective round-trippath travelledby thecouplereigenmodes from the startof

the coupler is LC+2∆Zpen≈36mm and LC+2∆Zpen≈24mm for the short and long-

wavelengthmaximum,respectively.

4

5

6

7

8

9

10

11


Pen

etra

tion

dept

h(m

m)

Even

Odd

a)-3

-2

-1

0

1

2

3


∆∆ ∆∆Z p

en(m

m)

b)

LC=30mm

LC=50mm

LC=70mm

Figure8.10–a)Penetrationdepthoftheevenandoddeigenmodesofa30mmlongcoupler

ina15mmlonggratingwithraisedsinusoidapodisation.Blueline:Evenmode.Redline:

Odd mode. b) Spectral difference between the penetration depths of the even and odd

eigenmodes for coupler lengths of 30mm (Black line), 50mm (red line) and 70mm (blue

line).

As shown inFigure8.10b), increasing thecouplerphysical length reduces the

effect of the difference in the eigenmodes propagation constants on the add-drop

performance, according to equation (8.7). Numerical simulations of the spectral

response of an add-drop multiplexer based on a 15mm long grating with sine2

apodisation written in the centre of uniform couplers with different lengths are


shown in Figure 8.11. The lengths of the couplers used in the simulations were

30mm,50mmand70mm.Itisobservedthatwiththeincreasingcouplerlengthand

consequentdecreaseinthedifferenceinthepropagationconstantsboththeamount

back-reflectedlighttoport1(P11)andlightleakingthroughport3(P31)attheedges

of the grating stop-band, are reduced and the overall spectral performance of the

deviceisimproved.

-60

-50

-40

-30

-20

-10

0


Pow

er(d

B)

P11

P21 a)

-60

-50

-40

-30

-20

-10

0


Pow

er(d

B) P31

P21 b)

Figure 8.11 - Numerical simulations of the spectral response of an add-drop multiplexer

basedona15mmlonggratingwithsine2apodisationwritteninthecentreofauniformhalf-

cyclecouplerwithdifferentlengths,LC.a)Droppedport(P21)andback-reflectedlight(P11).

b)Droppedport and leakage throughport 3.Black line:LC=30mm.Red line:LC=50mm.

Blueline:LC=70mm.


Even for a coupler length of LC=70mm, the amount of back-reflections and light

leakingthroughport3attheedgesofthegratingbandwidtharestillaround-20dB.

Tofurtherincreasetheperformanceofthisdevice,thecouplerlengthshouldbethe

longestpossible.However,thecompactnessofthedeviceiscompromisedwhenthe

coupler length is increased and could give rise to stabilisation and packaging

problems. Furthermore, the fabrication of fused fibre-couplers with long uniform

waists isnot trivialand therefore this isaserious limitation to theperformanceof

this device. However, this device could be suitable for 50GHz channel spacing

across thegratingbandwidthwith thereducedrequirementof–20dBfordrop-port

and add-port isolations (see table A1 in appendix A). In order to overcome these

limitations a novel add-drop design was developed based on a non-uniform half-

cyclecoupler(section8.3.2).

8.3.1.3 Experiments

Experimentally,theprincipleofoptimisingoneoftheoperations(AddorDrop)by

displacing the grating from the centreof the coupler is shown.Thegrating length

was4mm-longandwithnoapodisation(uniform).Thegratingwaswrittenusingan

excimer laseroperating at193nm.TheUVbeamwasbroadenough to expose the

whole cross section of the coupler, which was oriented perpendicularly to the

incidentbeamsothatbothphotosensitiveareaswereequallyexposed.Thecoupler

was fabricated using a fibre with boron co-doped core and high NA. The coupler

was then loaded with hydrogen at 160bar for 15 days in order to increase the

photosensitivity to the UV radiation [84]. The coupler’ s spectral response was

measured both before loading with hydrogen and after the writing of the grating.

The reason for the choice of a boron co-doped fibre instead of fibres with a

photosensitivecladding,thatareusuallyusedtoincreasethephotosensitiveareawas

the poor coupler extinction ratios obtained when fabricating couplers with these

fibres[113].


The grating was displaced by approximately 2mm from the centre of the

coupler.Inordertooptimisetheplacementofthegratingwithinthecoupler,the50-

50% point of the coupler should have been determined experimentally. The non-

destructive characterisation method developed in Chapter 9 could be used to

determinethispositioninthecasewhenthecouplerisasymmetricduetofabrication

errors, or when there is no clear marker of the start and end of the couplers and

thereforethecentrecannotbedetermined.However,intheseexperimentsitwasnot

possible to integrate the coupler characterisation system with the grating writing

system.After15daysunderhigh-pressurehydrogenandbeforewritingthegrating,

thecouplerresonancewavelengthwasdetunedbymorethan100nm.Afterwriting

the grating and letting the remaining hydrogen out-diffuse, the coupler remained

permanently detuned by 80nm. The original coupler resonance wavelength was

λC=1605nmandafterwritingthegratingitwasλC=1685nmasillustratedinFigure

8.12 where the coupler output at port 3 was measured for the initial fabricated

coupler (blue line), the hydrogen loaded coupler (black line) and the exposed

coupler(redline).

-90

-80

-70

-60

-50

1100 1200 1300 1400 1500 1600 1700Wavelength(nm)

Por

t3P

ower

(dB

)

Fabricated

ExposedH 2 loaded

Figure8.12–Measuredpowerattheoutputport3forthefabricatedcoupler(blueline),H2

loadedcoupler(blackline)andexposedcoupler(redline).

The degradation of the coupler due to the writing of the grating in its waist is

discussed inmoredetail insection8.3.3.Thespectralcharacteristicsof thedevice


weredeterminedbylaunchinglightintothecouplerports1and4andmeasuringthe

power received at each one of the output ports. Figures 8.13a) and 8.13b) show

respectively the experimental results and numerical fits when light is launched in

port1.Thetheoreticalfitsshowexcellentagreementwiththeexperimentaldata.For

optimumfittingoftheexperimentalresultsthegratingwasassumedtobedisplaced

-1.5mmrelativetothecentreofthecouplerandtheeffectiveindexmodulationwas

∆n=1.5x10-4,calculatedbymeasuringthereflectivityofthegratingattheresonance

wavelengthandusingexpression(5.13).Theerrorof0.5mminthelocationof the

grating is a reasonable experimental error taking into account the position of the

centreofthecouplerisdeterminedbyrudimentarymeans.

The experimental results (Figure 8.13a)) show that a 5dB grating with a

resonancewavelength1539.87nmwaswritteninthecouplerwaist.Therestof the

structureisduetothepoorbeamquality,characteristicoftheUVlaserusedforthis

exposure.Excitationofhigher-ordercladdingmodessupportedbythecoupler-waist

also contributes to these spectral features. These characteristics can be improved

considerably by proper choice of photosensitive fibres and coupler design. The

measured coupler extinction ratio is 5dB, consistent with the coupler spectral

responseshowninFigure8.12–redline.Theamountoflightback-reflectedtoport

1(P11)wasapproximately-10dB.ThenumericalsimulationresultsshowninFigure

8.13b)areinqualitativeagreementwiththemeasureddata.


-40

-30

-20

-10

0

1538 1539 1540 1541 1542Wavelength(nm)

Pow

er(d

Bm

) P41 P31P21

P11

a)

-40

-30

-20

-10

0

1538 1539 1540 1541 1542Wavelength(nm)

Pow

er(d

B) P41

P31

P21

P11

b)

Figure 8.13 – Spectral performance of the fabricated half-cycle coupler with a 4mm

uniform grating written at -2mm off the centre of the coupler when launching light from

port1.a)Experimentalresults.b)Numericalsimulations.

Toshowtheasymmetricoperationofthisdevicewhenlaunchinglightfrom

port3andtheimprovedperformance,thedevicewascharacterisedlaunchinglight

from the respectiveport.The experimental resultsare shown inFigure8.14a)and

the results from the numerical simulations are shown in Figure 8.14b). In the

simulations, the sameparameterswereusedas thosewhen launching fromport1.

The experimental results (Figure 8.14a)) show that the amount of light back-

reflected to port 1 (P11) was approximately -15dB which is an improvement

compared to the previous case. The remaining port responses were practically the


same as before. The numerical simulation results shown in Figure 8.14b) are in

qualitativeagreementwiththeexperiments.

-35

-25

-15

-5

5

1538 1539 1540 1541 1542Wavelength(nm)

Pow

er(d

Bm

)

P13

P23

P43

P33

a)

-35

-25

-15

-5

5

1538 1539 1540 1541 1542Wavelength(nm)

Pow

er(d

B)

P13

P23

P43

P33

b)

Figure 8.14 – Spectral performance of the fabricated half-cycle coupler with a 4mm

uniform grating written at -2mm off the centre of the coupler when launching light from

port3.a)Experimentalresults.b)Numericalsimulations.

8.3.1.4 Conclusions

Design considerations for add-drop multiplexers based on the writing of a Bragg

grating in the waist of a half-cycle coupler were discussed. The placement of the

grating at the centre of the coupler waist gives a device with compromised


performance as the effective reflection point at the resonance wavelength is

displacedfromthecentreofthecoupler.Thisdevicecanpotentiallyprovidecloseto

idealoperationwhenverylongcouplerlengthsareused.Suchlongdevicesarenot

practicaltofabricateandwouldhavestabilisationproblems.

Alternatively, Add or Drop operation can be optimised by placing the grating

displaced from the centre of the coupler by an appropriate amount, given by the

grating reflectivity and apodisation profile. In this asymmetric configuration, the

usable optimised bandwidth is reduced and to perform both Add and Drop

operationstwoidenticaldevicesarecascaded,asshowninFigure8.8.

TheexperimentalresultsshowninFigures8.13a)and8.14a)supportthedesign

considerations. The numerical simulations matched the experimental results and

gaveconfidencetothemodelused.Themaximumindexmodulationobtainedusing

this was 1.5x10-4, which is relatively low for practical applications. However it

could be improved by using fibres with larger photosensitive areas, i.e., with a

photosensitive cladding. However, the penalty would be the poor quality of the

fabricated couplers [113]. An additional factor that should be taken into account

experimentallywhenwritingthegratingsisthatthephasemaskandthecouplerare

aligned so that the grating is not tilted. The effect of the grating tilt is the

degradationofthespectralcharacteristicsofthegratingduetocouplingwithhigher

ordermodesasdiscussedinreference[114].

8.3.2 Grating-baseduniformfull-cyclefibrecouplerOADM.

This is a novel symmetric design that that relies on the positioning of the grating

between the two 50%-50% points of a full-cycle (2π) coupler. The theoretical

performanceanddesignparametersofthisdeviceareanalysedinthissection.Figure

8.15 shows schematically the principle of operation of this device. A full-cycle

coupler is obtained when the total phase difference between the coupler even and

oddeigenmodesalongthecoupleris2π.Thepowerevolutionalongthelengthofa

full-cyclecoupler, illustratedschematically inFigure8.15b),has twopointswhere


thepowerisequallydistributedbetweentheindividualwaveguides(50-50%points)

orequivalentlywherethetotalphasedifferencebetweenthecouplereigenmodesis

π/2 and 3π/2 respectively. In a symmetric coupler these points are located

symmetricallyrelativelytothecentreofthecouplerandthedistancebetweenthem

isdesignatedL3dB.Thegratingiswrittensymmetricallyalongthecouplerwaistand

its length, LG, should be optimised so that the reflection points in the grating

coincide with the 50-50% points of the coupler and therefore it is necessary that

LG=L3dB+2Zpen. It sould be pointed out that L3dB=LC/2 for ideal couplers with no

taperedregionsandconstantcouplingstrength.

Figure 8.15 – Principle of operation of a symmetric add-drop multiplexer based on the

inscription of a Bragg grating in the waist of a full-cycle (2π) coupler. a) Device

representation.b)Dropoperation:achannellaunchedinport1isdroppedtoport2andthe

restofthechannelsaretransmittedtoport3.

8.3.2.1 Optimisingforthepenetrationdepth

Theoperationofthisdevicedependscriticallyontheexactplacementofthegrating

so that its two effective reflection points coincide with the 50-50% points of the

coupler.ForthesamegratingsasillustratedinFigure8.3,thegratinglengththatis

requiredformatchingthereflectionpointswiththe50-50%pointsofthecoupleris

showninFigure8.16,fordifferentdistancesbetweenthecouplerreflectionpoints.


Forasymmetric30mmlonguniformfull-cyclecouplerthedistancebetweenthetwo

50-50%pointsofthecouplerishalfthecouplerlength(15mm)andforaneffective

index modulation of ∆n=2x10-4 the required grating lengths are LG≈17.6mm and

LG≈27.3mm,foruniformandsine2apodisationprofiles,respectively(representedin

Figure 8.3 by the dashed lines). The required length for a Blackman apodised

grating isgreater than the coupler length and thereforenotpossiblewith an index

modulationof∆n=2x10-4.

0

5

10

15

20

25

30

0 5 10 15 20 25 30Reflectionpointsdifference,L3dB(mm)

Gra

ting

leng

th,L

G(m

m)

Uniform

BlackmanSin2(x)

Figure 8.16 – Grating length, LG, required for different distances between the reflection

points of the coupler, L3dB. Black line: Uniform apodisation. Blue line: Blackman

apodisation.Redline:Sine2apodisation.

Thespectralresponsesofa30mmlongfull-cyclecoupler,withbotha27.3mm

longsine2apodisedgratinganda17.6mmlonguniformgratingwritteninthewaist,

areshowninFigures8.17a)and8.17b),respectively.Lightislaunchedinport1and

thepowerarrivingateachoneof theports iscalculated.Theportsarerepresented

by:P11–Thinredline;P21–Thickredline;P31–Thickblackline;P41–Thinblack

line. The second subscript in Pij refers to the input port (j) and the first subscript

referstotheoutputport(i).


-60

-50

-40

-30

-20

-10

0


Pow

er(d

B)

P11

P21

P31

P41

L C =30mm

a)

-60

-50

-40

-30

-20

-10

0


Pow

er(d

B) P11

P21 P31

P41

L C =30mm b)

Figure8.17–Spectralresponseofauniform30mmlongfull-cyclecouplerwithagrating

lengthoptimisedforthepenetrationdepthinscribedinitswaist.a)LG=27.3sine2apodised

grating.b)LG=17.6mmuniformgrating.

Foroptimisedsymmetricoperation, theperformanceof thisdevice is shown tobe

poorintermsofoutofbandback-reflectionandlightleakagethroughthedropport.

As shown previously, this effect is due to the high dispersion at the edges of the

grating bandwidth. For DWDM networks, these high cross-talk values could be

suppressedbyusingtwoisolatorsplacedateachtheinputanddropports.However

this requirement reduces the cost effectiveness of the device. Alternatively, this

devicecouldbeemployedtoroutesignalswithlargechannelspacingwheretheout

ofbandcross-talk is small.Due to the largedifference in the coupler eigenmodes

(double the case of a similar length half-cycle coupler), the overlap between the

individualgratingsaffectingtheevenandoddeigenmodesissmallergivingriseto

shorteravailablebandwidths.Thiseffectismorepronouncedforapodisedgratings

wherethebandwidthisreduced,asshowninFigure8.17a).Themainadvantagesof

this device are the large grating lengths required for optimising the couplers that

allowforhigherdeviceisolationduetothelargereflectivityofthegratingandthe

symmetricoperation(underoptimisedconditions),incontrastwiththepreviouscase

of the half-cycle coupler. Variation of the coupler length and maximum effective

index modulation do not improve further the performance of the device. There is

always a compromise between the optimisation of the penetration depth at the

resonant wavelength and the high penetration depths at the edges of the grating

bandwidth.Theincreasedcouplerlengthrequiresalongergratingand,therefore,the

deviceperformanceisscaledaccordingly.


8.3.2.2 Sensitivitytothedeterminationofthecoupler50-50%points

Theamountofback-reflectedpowerdue to the incorrectdeterminationof the50-

50%pointsofthecouplerisanalysedforauniformgratingplacedinthewaistofa

uniform full-cycle coupler. Generally, it is assumed that the positions of these

optimumreflectionpointsof thecouplerareatLC/4and3LC/4,which is trueonly

forthecaseofuniformcouplers.Thetaperingofthecouplerwaistduetofabrication

irregularities and especially the tapered coupler region at both ends, influence the

locationofthesepointswithinthecoupler,asshowninFigure4.10.Infact,forthe

couplerprofileillustratedinFigure4.9,thatwasdeterminedbymeasuringthepower

evolution during the coupler fabrication, and using expression (4.13), the distance

betweenthe50-50%couplerpointsisL3dB=18.6mmincreasingby3.6mmrelativeto

thecaseofauniformfull-cyclecoupler.Theeffectoftheerrorinthegratinglength

ontheamountofback-reflectedlightatthecentrewavelengthisshowninFigureD1

inappendixD.Optimallytheerrorinthegratinglengthshouldbelessthan1mmfor

less than -20dB of back-reflected light to be achieved and therefore, the

determinationof theexact50-50%pointsof thecoupler iscritical.This issuewas

tackledbydevelopinganon-destructivemethodforcharacterisingfibrecouplersin

Chapter9 thatallows thedeterminationof the50-50%pointswithanaccuracyof

less than 1mm, which could be further improved by suitable optimisation. It is

shown that for a full-cycle coupler the 50-50% points are identified even for

couplerspresentingtaperedwaistsandlongtransitionregions.

8.3.2.3 Conclusions

Theoperationof thisdevice is symmetricbut theout-of-bandcrosstalk andback-

reflections are very high. Its employment in DWDM systems where the optical

channelsaretightlyspacedwouldonlybepossiblewiththeuseofopticalisolators

at the input and add ports that would increase its total cost. However, this device

couldbeusedtoroutechannelswithlargespacingbetweenthemwithout theneed


for optical isolators. The performance of this device relies on the correct grating

lengthanddeterminationofthe50-50%positionsinthecoupler.

8.3.3 Grating-basednon-uniformfibrecouplerOADM.

Thethirdconfigurationisanoveldesignthatisbasedonanon-uniformhalf-cycle

couplerwithagratingwritten in itswaist.Thecouplercomprises twoendregions

withhighcouplingconstantformingtwocloseto3dBcouplersand,acentralregion

thatisweaklycoupledandwheretheevenandoddmodespropagationconstantsare

almost equal, βe≈βo. Figure 8.18 illustrates the proposed configuration. The two

laterallengthsofthecouplerL1formtwo3dBsplittersandthecentrallengthL2is

veryweaklycoupledasshowninFigure8.18a).Thewholestructureformsahalf-

cyclecoupleroflengthLC,i.e. π=β∆CL

0dz)z( .Figure8.18b)showstheevolution

ofpowerbetweentheindividualcouplerwaveguidesandthedropoperationof the

device.Light launchedfromport1isalmostequallysplitat theendofL1andthe

“50-50%point”ofthecouplerisnowexpandedthroughtheentireweaklycoupled

centralregion.Thewavelengthselectedbythegratingisdroppedtoport2andthe

remainingchannelsaretransmittedthroughport4.


Figure 8.18 - Representation of an add/drop multiplexer configuration based on a non-

uniform half cycle coupler with a grating in the waist. b) Drop operation: a channel

launchedinport1isdroppedtoport2andtherestofthechannelsaretransmittedtoport4.

Thisdesigncanbeviewedasanoptimisedhalf-cyclecouplerwheretheproblemof

theseparationbetweenthecouplereigenmodesissolvedbybringingthemtogether

andexpanding the50-50% regionof the coupler.This effectively is equivalent to

inscribing a grating in the waist of a very long coupler where the grating can be

consideredasapointreflector.

8.3.3.1 Numericalsimulations

The performance of this device was simulated based on parameters used

experimentally in the fabrication of the non-uniform couplers. The total coupler

length was LC=30mm and all the three sections were L1=L2=10mm. The total

coupling due to the lateral sections was 49.5% and the intermediate region was

responsible for 1% of the total phase detuning between the even and odd

eigenmodes. The coupling strength profile, k(z) of the coupler used in the

simulationsisshowninFigure8.19.


0.E+00

2.E-05

4.E-05

6.E-05

8.E-05

0 5 10 15 20 25 30Couplerposition(mm)

Cou

plin

gst

reng

th,k

(z)(

µµ µµm

-1)

49.5% 49.5%

1%

Figure8.19–Coupling strengthprofileof anon-uniformhalf-cycle couplerwith lengths

L1=10mm,andL2=10mm.

Thegratingwritteninthewaistofthenon-uniformcouplerwas9mmlongwith

an effective index modulation of ∆n=4x10-4 and a sine2 apodisation profile. The

spectralresponseofthisdevice,showninFigure8.20a),wascomparedwiththecase

ofa1mlonguniformhalf-cyclecouplerwiththesamegratingplacedatthecentreof

thecoupler,showninFigure8.20b).

-80

-60

-40

-20

0

-0.5 -0.25 0 0.25 0.5WavelengthDetuning(nm)

Nor

mal

ised

res

pons

e(d

B)

P11

P21P41

P31

a)

-80

-60

-40

-20

0

-0.5 -0.25 0 0.25 0.5WavelengthDetuning(nm)

Nor

mal

ised

res

pons

e(d

B)

P11

P21P41

P31

b)

Figure8.20–Spectralresponseofahalf-cyclecoupleroptimisedforadd-dropperformance

witha9mmlongsine2apodisedgratingwith∆n=4x10-4.a)Gratingwritteninanon-uniform

(L1=10mm+L2=10mm+L1=10mm)coupler.b)Gratingwrittenina1mlongcoupler.

As shown in the simulations above, thisdevice has excellentperformance. Its

symmetric configuration allows for both add and drop operations to be achieved


simultaneously in one compact device. The amount of back-reflected light and

cross-talk are very low, around –40dB, and therefore this device is suitable for

DWDMsystems (according to the specifications shown inappendixA).The filter

resonant wavelength, bandwidth and isolation are grating design parameters

depending the grating period, apodisation and index modulation. The spectral

characteristicsofthisdesignareidenticaltothecaseofusinga1mlonghalf-cycle

coupler.Howeverithashugepracticaladvantagesduetoitsfeasibilityandcompact

size. The reason both these devices are equivalent is understood by observing the

behaviourof thepowerdistributionalong the lengthsofboth thenon-uniformand

1m long half-cycle couplers, illustrated in Figure 8.21. The red lines refer to the

non-uniformhalf-cyclecouplerused in theprevioussimulationsand theblue lines

refer to the meter-long uniform half-cycle coupler. In Figure 8.21a) the expanded

viewofthewholemeter-longcouplerisillustratedandinFigure8.21b)theregion

correspondingtothenon-uniformcouplerisenhanced.

0

0.25

0.5

0.75

1

-500 -250 0 250 500CouplerPosition(mm)

Nor

mal

ised

Pow

er

P2

P1

a)

0

0.25

0.5

0.75

1

-15 -10 -5 0 5 10 15CouplerPosition(mm)

Nor

mal

ised

Pow

er

P2

P1b)

Figure 8.21 – Comparison between the power evolution along the waist of a meter-long

uniformhalf-cyclecoupler(blueline)anda30mmlongnon-uniformhalf-cyclecoupler(red

line).a)Expandedview.b)Magnificationofnon-uniformcoupler.

Thesesimulationsshowthatintheregionfrom–5mmto5mmthepowerevolution

alongboththecouplersisthesame.Thisregioncorrespondstotheweaklycoupling

region of the non-uniform coupler, where the grating is written. Therefore, the

response of the gratings will be similar due to the same difference between the


coupler eigenmodes in both couplers. The rest of the couplers to the left of the

gratingandrightofthegratingcanbeconsideredasalmost3dBsplitters.

The main advantages of this non-uniform coupler design compared to the

traditionalMach-Zehnderinterferometer[47]arethatboththephotosensitiveareas

are incloseproximitywhenwriting thegrating and therefore, thegratingswillbe

identical without the need for post-trimming. Furthermore, as the arms of the

interferometer are short and both modes are slightly coupled, it is much less

sensitive to environmental variations and vibrations than the traditional

interferometer.

8.3.3.2 Experiments

The feasibility of this device was demonstrated experimentally. Non-uniform

couplersweresuccessfullyfabricatedusingtheflame-brushtechniquedescribedin

section4.3.1.Couplerswithdifferentend-regionlengthsandweakly-coupledcentral

regions were fabricated. In order to optimise the length of the central region,

couplerswithL1=5mmandL2=20mmwerefabricatedwithsuccess.Theuniformity

of the central coupler region is critical for the adiabaticpropagationof themodes

whenwritingthegrating.Experimentalresultsofanon-uniformcoupler(L1=6mm

andL2=18mm)witha8mmlonguniformgratingwritteninthewaistareshownin

Figures8.22and8.23wherethedeviceischaracterisedbylaunchinglightinports1

and 3, respectively. The numerical simulations obtained for this device are also

illustratedforcomparison.Theparametersusedinthenumericalsimulationswereas

follows: the grating refractive index modulation was ∆n=1.05x10-4, obtained by

measuringthetransmissivityoftheuniformgratingandthegratinglengthwas8mm;

thenon-uniformhalf-cyclecouplerstrengthprofilewassimilartotheoneillustrated

inFigure8.19withL1=6mmandL2=18mm.Thecentralregionwasassumedtobe

responsible for 1% of the coupling and coupler resonance wavelength to be

displacedby80nm.TheexperimentaldataillustratedinFigure8.22a)showthatthe

gratingwritteninthecouplerwaistwaspoor.Thiswasduetothebadqualityofthe

excimerlaserbeam(duetotheirregularitiesinthebeamprofile)ornon-uniformities


inthecoupler-waistcausingnon-uniformexposuresalongthegratinglengthand,a

possible tilt between the coupler waist and the phase mask [114]. The operating

wavelengthoftheexcimerlaserwas193nm.Thecouplerwasexposedatarepetition

rateof20Hzfor90seconds.Thefluenceperpulsewas0.5Jcm-2resultinginatotal

fluenceof0.9KJcm-2.Thecouplerdropportwasmonitoredduringexposureandthe

couplerexposurewasstoppedwhenthegratingreflectivitystartedtodecrease.The

fibreusedtofabricatethecouplerwasboron-co-dopedwithahighconcentrationof

Germania and without a photosensitive cladding. It was loaded in deuterium at a

pressureof100barfor10daysbeforetheexposure.

-35

-25

-15

-5

5

-1.5 -1 -0.5 0 0.5 1 1.5Wavelengthdetuning(nm)

Pow

er(d

Bm

)

P11

P31

P41 P21

a)

-35

-25

-15

-5

5


Pow

er(d

Bm

)

P11

P31

P41 P21

b)

Figure8.22–Spectralperformanceofthenon-uniformcouplerwitha8mmlonggratingin

the waist when launching light from port 1. a) Experimental measurement. b) Numerical

simulations.


Inordertoshowthatthedeviceissymmetricitwascharacterisedbylaunchinglight

intoport3andmeasuringthepowerarrivingateachoneoftheports.Theresultsare

shown inFigure8.23a) where it is observed that theperformanceof thedevice is

identicaltothepreviouscase,withlightlaunchedinport1.

-35

-25

-15

-5

5


Pow

er(d

Bm

)

P33

P13

P23 P43

a)

-35

-25

-15

-5

5


Pow

er(d

Bm

)

P33

P13

P23 P43

b)

Figure8.23–Spectralperformanceofthenon-uniformcouplerwitha8mmlonggratingin

the waist when launching light from port 3. a) Experimental measurement. b) Numerical

simulations.

ComparingFigure8.22a)with8.23a)itisconcludedthattheoperationofthisdevice

is symmetric. However, due to the cross-talk of this particular device is not low

enough for use in DWDM systems. The -10dB level of back-reflected light and

powerleakingthroughtheaddportwerecausedbythedetuningofthecouplerafter


loadingthecouplerwithdeuteriumplusexposingtotheUVbeam.Theseshiftshave

beenreportedintheliterature[115,116]andshouldbeappropriatelycompensated

for. They can be understood by considering the overlap between the coupler

eigenmodes and the photosensitive regions. The overlap between the refractive

indexchangeandtheoddeigenmodepowerdistributionisgreaterthantheoverlap

with the power distribution of the even eigenmode and therefore, the coupler

resonancewavelengthchanges.Theabsolutevalueof thedifferencebetweeneven

andoddeigenmodeoverlapintegralsisillustratedinFigureD2,inappendixD,for

differentcoupler radiiandaphotosensitiveareawith1.5µmradius.Thechange in

the eigenmode propagation constants is proportional to the overlap integrals and

therefore, for a refractive index modulation of ∆n=2x10-4 and a coupler radius of

16µm the difference between the coupler eigenmodes due to the exposure can be

calculatedtobe∆neo=∆ne-∆no=8.3x10-6.InFiguresD3a),b)andc)theevenandodd

eigenmodepowerdistributionacross thewaistandcouplercross section including

the photosensitive areas can be visualised, respectively. The detuning in the

resonance wavelength of a 30mm long half-cycle non-uniform (L1=6mm and

L2=18mm)couplerdue to adifference in the evenandoddeigenmodes along the

8mmlengthofthegratingisshowninFigureD4,inappendixD,fordifferent∆neo.

For ∆neo=8.3x10-6 the coupler was calculated to be detuned by 42nm. Comparing

thisvaluewiththeexperimentalresponseofthecouplermeasuredbeforeandafter

the writing of the grating, shown in Figures D5a) and b), respectively, the

wavelength shift was measured to be approximately 44nm showing a good

agreement.

8.3.3.3 Fabricationissues

Owingtothelimitedrefractiveindexmodulationachievablebecauseofthereduced

sizeofthephotosensitiveareasinthecouplerwaist,inordertoachievethedesired

isolation requirements (e.g. 40dB for 200GHz channel spacing), it is necessary to

increase the lengthof thegratingwritten in thecouplerwaist.However,when the


writing isovershot and agrating iswritten in the tapered regions itwasobserved

experimentally that the coupler performance was degraded and huge losses were

apparent.Hence,thelengthofthenon-uniformregionshouldbeoptimisedinorder

to achieve long grating lengths without degradation of the coupler spectral

performance. The following experiments illustrate the consequences of writing a

grating in the tapered regionof thecoupler.Anon-uniformcouplerwithL1=7mm

and L2=14mm was fabricated and a uniform grating with a length of 4mm was

writtenafewmillimetresoffthecentreofthecouplerasshowninFigure8.23.

Figure8.23–Non-uniformcouplerwithagratingwrittenoffthecentreofthewaist.

Thespectralresponseof thecouplerwasmeasuredusingawhite lightsourceboth

after fabrication of the coupler and after exposure of the grating. The results,

illustrated inFigure8.24, show that the effect of exposing the tapered region is a

degradation of the coupler performance. The black line represents the original

couplerperformanceandthegreenlinetheperformanceoftheexposedcoupler.The

isolation of the coupler changes from 25dB at the resonance wavelength to

approximately10dBandtheresonancewavelengthisalsoshifted.Thelossisdueto

non-adiabaticpropagationoftheeigenmodesinthatregionandcanbeunderstoodif

a differential loss is induced to the coupler eigenmodes. The shift in the coupler

resonancewavelengthcanbeexplainedaspreviouslymentionedbythedifferential

increaseintheeffectiveindexoftheeigenmodesalongthe4mmgrating.


-80

-70

-60

-50

1100 1300 1500 1700Wavelength(nm)

Pow

er(d

Bm

)

Port3 Port4

Figure8.24–Measurementofthecouplerspectralresponseusingawhitelightsource.The

blacklinesrefertotheoriginalunexposedcouplerandthegreenlinesrefertotheexposed

coupler.

Thefibreusedtofabricatethecouplerhadaphotosensitivecladdingringaround

the core in order to increase the overlap between the coupler eigenmodes power

distribution and the photosensitive areas. After fabrication the coupler was

hydrogen-loadedatapressureof160barfor15daysatroomtemperatureinorderto

increase its photosensitivity. The grating was exposed using an excimer laser

operatingatawavelengthof193nm.ThefluenceoftheUVbeamwas0.5Jcm-2per

pulseandthecouplerswereexposedatarepetitionrateof20Hzfor140syieldinga

totalfluenceof1.4kJcm-2.Theachievedeffectiveindexmodulationwasdetermined

bymeasuringthereflectivityof thegratingyielding∆n≈3x10-4,whichisshownto

be an improvement relatively to the previous written couplers. The spectral

characteristicsofthedeviceweremeasuredbylaunchinglightbothfromport1and

port3illustratedinFigures8.25a)andb),respectively.


-35

-25

-15

-5

5

1543 1544 1545 1546 1547 1548Wavelength(nm)

Pow

er(d

Bm

)

P33

P23

P13

P43

a)

-35

-25

-15

-5

5

1543 1544 1545 1546 1547 1548Wavelength(nm)

Pow

er(d

Bm

)

P11

P31

P41

P21b)

Figure8.25–Measurementof the spectralperformanceofanon-uniformcouplerwitha

4mmlonggratingwritteninitswaist.a)Lightlaunchedinport3.b)Lightlaunchedinport

1.

In transmission, the out-of-band characteristics of this OADM are dictated by the

degradedcouplerperformance,whichhadonly6dBisolationatthegratingresonant

wavelength.However,thisdeviceisshowntobeasymmetric:Whenlaunchingfrom

port1,adegradedcouplerresponseisobservedandlightisreflectedequallytothe

dropportand the inputport.When launching fromport3however, the coupler is

intactandthereforealllightisdroppedtoport4andtheleveloftheback-reflected

light is –15dB corresponding to the original coupler extinction ratio at that

wavelength.


8.3.3.4 Conclusions

An improved add-drop performance can be achieved by using a complex coupler

design that is based on a non-uniform coupler. Theoretically this device is

equivalent to a uniform half-cycle configuration with a very long coupler length

yieldingoptimumcross-talkcharacteristics.However,as inall thesecoupler-based

devices, efforts shouldbemade to increase the grating strengthby increasing: the

fibre photosensitivity; the overlap between the photosensitive areas and the field

distributionofthecouplereigenmodes;andthelengthofthenon-uniformregionof

the coupler. A refractive index modulation of ∆n=3x10-4 that could yield an

extinction ratio of –29dB using a 16mm long grating with sine2 apodisation was

experimentallyachieved.Thecorrectpositionof thegratinginthecouplerwaistis

criticalfortheoperationofthedevicedependingontheexactdeterminationofthe

uniform region in the centre of the coupler. The method for characterising fibre

couplers developed in Chapter 9 is suitable for the experimental determination of

theseregions.

8.4 Summary

Limitationsandoptimisationofdifferentadd-dropmultiplexerconfigurationswere

discussed.Firstly,aconfigurationusingauniformhalf-cyclecouplerwithagrating

inscribedinthewaistwasanalysed.Itcanbeoptimisedbycascadingtwoidentical

asymmetricdeviceswherethegratingisdisplacedfromthecentreofthecoupleror,

by fabricating extremely long couplers where the grating can be considered as a

pointreflector.However, thegratingstrength,apodisationandidentificationof the

position within the coupler, where it should be placed and the difficulty of

fabricating long uniform couplers are serious problems that compromise its

performance.Additionally,theirreducedphotosensitivityandtheconsequentlimited

channelisolationofthegratingfiltersisanotherlimitationthathastobeovercome.

Todetermineexperimentallythepositionofthecentreofthehalf-cyclecoupleror,


optimise their fabrication procedure, a non-destructive method for characterising

fibrecouplerswasdevelopedandisdiscussedinChapter9.

Secondly,aconfigurationbasedonafull-cyclecouplerwithagratinginscribed

initswaist,placedbetweenitstwo50-50%points,wasstudied.Thisdevicesuffers

fromintrinsichighout-of-bandback-reflectionsandcross-talkwhenitsisolationis

optimised. However, the device isolation of the grating channel depends on the

usingtheexactgratinglengthbydeterminingthepositionof the50-50%pointsof

the coupler. These can again be experimentally determined by the coupler

characterisationmethoddevelopedinthefollowingchapter.

Finally, a device based on a non-uniform half-cycle coupler that theoretically

yields an optimum add-drop multiplexer performance was demonstrated to be

suitableforDWDM.ItissymmetricachievingbothAddandDropfunctionsinone

compact device. The grating, written in the slightly-coupled central region, is

position independent as longas the tapered regions arenot exposed.Asymmetric

devicewasexperimentallydemonstratedsuggestingthatthetwoendregionsofthe

couplerwerewellmatched.Maindrawbacksforthisdeviceisthelimitedsizeofthe

central regionhence thegrating length.Procedures for fabricating longer couplers

with a better control over the central region shape, length should be investigated.

The accurate placement of the grating within the coupler can be achieved by

characterisingitusingthemethodexploitedinChapter9.

9

CharacterisationofFibre-

Couplers

Anovelnon-destructive technique for characterising couplersbymeans of a local

perturbation is described. The method is studied theoretically and verified

experimentallybycharacterisingdifferent typesof fused fibre-couplers.Using this

technique,boththeinformationofthepowerdistributionandcouplingprofilealong

thecouplerwaistareobtained.

9-CharacterisationofFibre-Couplers 179

9.1 Introduction

The performance of couplers and coupler-based devices depends on the coupling-

constant and/or power distribution along the coupling region. The response of

coupler-based OADMs involving Bragg gratings, for example, is critically

dependentontheexactpositioningofthegratingwithrespecttothepointsinsidethe

couplerwaistwherethepoweroneachindividualcoreisequallysplitasmentioned

in chapter 8, or equivalently, where the phase difference between the two waist

eigenmodes is multiple of π/2. Development of non-destructive coupler

characterisationtechniques,inordertodeterminethepowerevolutionandcoupling

constant distribution along the coupler length, is, therefore, of paramount

importanceindevelopingcouplersforhighperformanceapplications.

Various methods for determining different parameters of uniform directional

couplershavebeenreportedintheliterature[117,118].Bourbinetal.[118]reported

amethodforcharacterisingcouplersinplanarwaveguides.Themethodisbasedon

inducing a small differential loss in one of the coupled waveguides. In order to

localisethelossperturbationinoneofthewaveguidesonly,theotherwaveguideis

covered with aprotective resist film.Gnewuchet al. [117] reported an alternative

local-perturbation method for measuring the beatlength of uniform couplers in

buried planar-waveguide geometry. The method consists of inducing a local

perturbation in one of the waveguides by heating it with an incident 980 nm

semiconductor laser diode. To facilitate the 980 nm laser absorption by the

otherwise transparent waveguides and achieve local heating, a 1µm-thick layer of

absorptiveblackinkwasspin-coatedontothecouplersurface.Themethoddidnot

giveanyresultswhenthecouplerwasperturbedsymmetrically(laserdiodefocused

at the centre between the two waveguides). It should be stressed that the two

reported methods require some degree of post-fabrication coupler treatment (e.g.,

applicationofresistfilminoneofthewaveguides[118]andspin-coatedabsorptive

thin-layer[117])inordertoachievetherequireddifferentialperturbation.Although

such steps and processes can be acceptable in planar waveguide geometries, they

cannot be applied or should be avoided in fused fibre coupler geometries. This is


due to the fact that the very small waist diameters involved are quite fragile and

pronetopost-fabrication-treatmentfailures.

Inthischapter,anewnon-destructivemethodforfullcouplercharacterisationis

described.Themethoddoesnotinvolveanypost-fabricationtreatmentand/orextra

coupler preparation. Firstly, by applying an asymmetric perturbation between the

two lowest-orderwaist eigenmodes, the complexpower evolution along the entire

coupling region can be measured non-destructively. Furthermore, in the particular

case of a 100% coupler, the asymmetric perturbation of the coupler provides a

markerforthepositionalongthecouplerwherethepowerisequallysplitbetween

both the waveguides (50-50% point) independently of the wavelength of the light

used to monitor the coupler. Secondly, by applying a symmetric perturbation

between the two lowest-orderwaisteigenmodes, thecoupling-constantdistribution

alongtheentirecouplingregioncanbemeasuredwithoutdamagingthecoupler.

9.2 Local Perturbation Coupler Characterisation

Technique

9.2.1 GeneralDescriptionoftheProposedMethod

As already mentioned, optical couplers are formed by bringing two or more

waveguides(planar,ridge,diffusedwaveguidesorfibres)incloseproximitysothat

they exchange power through evanescent field interaction. In four-port (2x2)

couplers,shownschematicallyinFigure4.1,twowaveguidesexchangepowersover

acoupling region (LC),whichcomprises thecouplerwaist (LW)and the two taper

regions (LT1,LT2)on either side.The taper regionsareadiabatic inorder toavoid

higher-order, as well as, radiation mode excitation that contribute to losses. The

coupling process along the taper lengths is non-uniform, described by a varying

couplingconstant,andaccountsforasubstantialpartofthetotalexchangedpower.

These regions should be taken into account when considering practical coupled


devices.Thewaistregion,ontheotherhand,inmostofthecasesissupposedtobe

uniform and is described by a fixed coupling constant. However, in practise,

dependingonthefabricationprocess,thewaistshowssizeablenon-uniformitiesthat

should be properly accounted for, in order to describe accurately the device

performance. This is particularly important in more complex devices, such as

OADMs,thatcombinecouplerswithgratingsintheirwaistsasdiscussedinchapter

8.

Figure 9.1a) illustrates the principle of operation of the proposed technique.

Lightof theappropriatewavelength is launched intooneof the inputports (#1or

#2). The coupler characterisation method consists of inducing a local perturbation

along its coupling region (taper + waist) and monitoring the change in power (or

phase) at oneor twoof theoutputports (#3 and#4).The localperturbation is, in

general, induced non-destructively by a temperature gradient across the coupler

waist,asshownschematicallyinFigure9.1.Theperturbation(shownbytheshaded

area)canbeasymmetric(Figure9.1b)-top)orsymmetric(Figure9.1b)-bottom)with

respecttothepowerdistributionofevenandoddeigenmodes.Asitwillbeshown

theoretically and confirmed experimentally in subsequent sections, the type of the

applied perturbation can provide information about different coupler parameters.

The temperature gradients were induced by two different techniques, involving

differentheatsources.Thefirstonewasaheatedwireandthesecondoneapower-

controlled CO2 laser. The CO2 laser radiation is highly absorbed by fused silica

(typical absorption length of ~5µm [119]) and provides the required perturbation

gradientwithouttheneedforapplicationofextraabsorbinglayers(asinRef.[117]).


Perturbingelement

P1

P4++++ ∆∆∆∆P4

P3++++ ∆∆∆∆P3

Localisedperturbation

(b)

(a)

Perturbingelement

P1

P4++++ ∆∆∆∆P4

P3++++ ∆∆∆∆P3

Localisedperturbation

(b)

(a)

Figure 9.1 - a) Principle of operation of the coupler characterisation technique. b)

Schematic of the coupler-waist perturbation using an asymmetric (top) or symmetric

(bottom)configuration.

Themethodhasbeenfirststudiedtheoreticallyusingcoupledmodetheory,and

thendemonstratedexperimentally,showingexcellentagreement.Furthermore,ithas

been successfully applied to a number of different coupled structures, such as

standardfibrefusedcouplersofdifferentlengths,aswellas,complexnon-uniform

coupledfibrestructures.Themethodcanprovideboththepowerevolutionalongthe

couplerwaistandthedistributionofthecorrespondingcouplingconstant.

9.3 TheoreticalModel

9.3.1 CouplerDescription

Thecouplerisdescribedbythebeatingbetweenthetwopropagatingevenandodd

eignemodes (see chapter 4). Denominating βe andβo the propagation constants of

the even and odd eigenmodes respectively and φ(z) the accumulated phase

difference between the eigenmodes from the start of the coupler until a given


positionz,thepowerevolutionoftheunperturbedcoupleralongthewaistiswritten

as:

=

=

)(21

sin)(

)(21

cos)(

22

21

zzP

zzP

φ

φ (9.1)

[ ] −=∆==z

oe

z

eoeo ddzz00

)()()()()( ζζβζβζζβφφ

9.3.2 EffectofExternalPerturbation

Inthepresenceofalocalnon-adiabatic(symmetric/asymmetric)externallyinduced

refractive indexperturbation,atagivendistancez0, theotherwiseuncoupledeven

andoddeigenmodesscatterlightintoeachotherandperturbtheiramplitudesAeand

Ao.Theinteractionbetweenthetwopropagatingeigenmodescanbedescribedbythe

followingcoupled-modeequations:

zieoeooo

o

zioeoeee

e

eAikAikdzdA

eAikAikdzdA

⋅∆−

⋅∆

−−=

−−=

β

β

(9.2)

where∆β=βe-βo.Theoverallcouplingprocess ischaracterisedby fourparameters,

namelykee,koo,keoandkoe.Theparameterskeeandkooareself-couplingcoefficients,

describingthescatteringofeachmodeintoitself,andresultinamodificationofthe

modepropagationconstant locally. Theparameterskeoandkoe,ontheotherhand,

arecross-couplingcoefficients,describingthescatteringofeachmodeintotheother,

andgivetheinteractionandpowerexchangebetweentheevenandoddmodes.The

scatteringprocessandcouplingmechanisminducedbytheexternalrefractiveindex

perturbation∆n(markedbytheshadedarea),isshownschematicallyinFigure9.2.


Evenkee

koo

keo

koe

Odd

z0

∆z

z + z0 ∆

n n+ n∆ n

Figure 9.2 - Schematic of even and odd eigenmode self-coupling (kee, koo) and cross-

coupling(keo,koe)inducedbytheexternalperturbation.Theshadedareamarkstheexternal

perturbation∆n.

Thecouplingcoefficientscanbeexpressedas:

dxdyyxEyxEzyxzk

dxdyyxEyxEzyxzk

dxdyyxEyxEzyxzk

eooeoeeo

oooo

eeee

∆=

∆=

∆=

),(),(),,(4

)(

),(),(),,(4

)(

),(),(),,(4

)(

)(*

)()(

*

*

εω

εω

εω

(9.3)

where ∆ε≈2ε0n∆n is the dielectric permittivity perturbation. When the refractive

index perturbation is uniform across the waist cross-section or symmetric with

respect to the waist centre, the cross-coupling coefficients are zero (keo=koe=0).

When the refractive index perturbation is antisymmetric with respect to the waist

centre, theself-couplingcoefficientsarezero(kee=koo=0). Inthegeneralcaseofan

asymmetric perturbation, all coupling coefficients are non-zero. Solving the

coupled-mode equations along the local perturbation length ∆z, the following

expressions for the amplitudes of the perturbed even and odd mode fields are

obtained:


zi

oeeo

o

zi

oeo

ee

ezAzss

izszAzs

sik

zzA

ezAzss

ikzAzs

si

zszzA

∆′′∆−

∆′∆

∆+∆+∆−=∆+

∆−

∆−∆=∆+

2000

2000

)()sin()cos()()sin()(

)()sin()()sin()cos()(

β

β

σ

σ

(9.4)

where,

( )oe

ooeeooeediffdiffeo

kk

kkk

kkkkks

βββββββ

βσσ

−=∆+∆

=′′∆

−∆

=′∆

+=

−=+

∆=+=

,22

,22

2,

2,

2,

2/122

Thepropagationalonganunperturbedcouplerregion,extendedbetweenz1andz2,

canbedescribedby:

⋅

=

)()(

),(00),(

)()(

1

1

21

21

2

2

zE

zE

zz

zzzE

zE

o

e

o

e

o

e

αα

(9.5)

with,

=

−2

1

)( )(

21)( ),(

z

z

oe dzzi

oe ezzβ

α (9.6)

FromEquation(9.4),ontheotherhand,thepropagationalongtheperturbedregion

canbewritteninatransfermatrixformas:

⋅

⋅=

∆+∆+

)()(

)()(

0

0

2221

1211

0

0zE

zE

TT

TTzzE

zzE

o

e

o

e (9.7)

where


zi

zieo

zi

ezss

izsT

ezss

kiTT

ezss

izsT

∆−

∆−

∆−

∆+∆=

∆−==

∆−∆=

β

β

β

σ

σ

)sin()cos(

)sin(

)sin()cos(

22

2112

11

(9.8)

where22

ooeeoe kk ++

+=

βββ is the average of the two perturbed propagation

constants.Theeven-andodd-modefieldsat thecoupleroutput (z=L)Ee(L,z0)and

Eo(L,z0),respectively,withtheperturbationappliedatz=z0,areobtainedintermsof

the input fields Ee(0)= Ae(0) and Eo(0)= Ao(0) by multiplying the three pertinent

propagationmatricesandcanbeexpressedas:

⋅

⋅

⋅

∆+∆+

=

)0()0(

),0(00),0(

),(00),(

),(),(

0

0

2221

1211

0

0

0

0

o

e

o

e

o

e

o

e

A

A

zz

TT

TT

LzzLzz

zLE

zLE

αα

αα (9.9)

Thetransfermatrix[T]oftheperturbationcanbefurthersimplifiedbydisentangling

the coupling event from the propagation process over the perturbation length ∆z

[120]. The perturbation transfer matrix is then expressed as the product of a

localised and instantaneous coupling matrix and a simple propagation matrix as

follows:

=

∆+−

∆+−

zki

zki

ooo

eee

e

eCC

CC

TT

TT)(

)(

2221

1211

2221

1211

00

β

β

(9.10)

where

( )zkCC eo ∆== cos2211 and ( )zkiCC eo ∆−== sin2112


Theerrorinvolvedintheapproximation(9.10)isO(∆3)andisnegligiblewhenthe

perturbation length ∆z is very small. Substituting (9.10) into (9.9) the perturbed

fieldsEe(L,z0)andEo(L,z0)oftheevenandoddmodes,respectively,atthecoupler

output can be calculated with the perturbation at z0. Using expression (9.1) the

fields of the outputs of the corresponding individual waveguides E1(L,z0) and

E2(L,z0)canbecalculated.Aftersimplemathematicmanipulations,thepoweratthe

outputsofthecorrespondingindividualwaveguidesP1(2)(L,z0)=|E1(2)(L,z0)|2andare

expressedas:

−∆+∆

= peoeop zkzkzLP φφφ21

cos)|(|sin)|(|cos21

cos),( 12222

01 (9.11a)

−∆+∆

= peoeop zkzkzLP φφφ21

sin)|(|sin)|(|cos21

sin),( 12222

02 (9.11b)

whereφp=φ(L)+∆φp is thetotalperturbedphasedifferencebetweenevenandodd

modes,expressedasthesumofthetotalphasedifferencebetweentheevenandodd

modesoftheunperturbedcoupler ∆=L

dzzL0

)()( βφ andperturbationterm∆φp=(kee-

koo)∆z. The term ∆=0

01 )(

z

dzzβφ is the accumulated phase difference up to the

perturbationpointanditisthereforeafunctionofz0. Forauniformcoupler,φ1is

the only z0-dependent term. Monitoring the power variation as the perturbation is

scanned along the coupler length, extremely useful information about the coupler

waist characteristics and the power evolution along the coupling region can be

extracted.Twodifferenttypesofperturbationcanbeconsidered,namely:

a)Symmetrictypes,wheretheperturbationisappliedsymmetricallywithrespectto

power distribution of the even and odd eigenmodes. Figure 9.1b)-bottom shows a

specific arrangement of symmetric perturbation. From Equations (9.3), it can be


easily deduced that in this case only the self-coupling coefficients kee and koo are

non-zerowhilethecross-couplingcoefficientskeoandkoearezero.

b)Asymmetrictypes,wheretheperturbationisappliedasymmetricallywithrespect

to power distribution of the even and odd eigenmodes. Figure 9.1b)-top shows a

specificarrangementofasymmetricperturbation.Inthiscase,boththeself-coupling

andcross-couplingcoefficientsarenon-zero.

9.3.2.1 Symmetricperturbation(kee≠≠≠≠koo≠≠≠≠0,keo=koe=0)

Undersymmetric-perturbationconditions,Equations(9.11)become:

[ ]

[ ]

∆+=

=

∆+=

=

pp

pp

LLLP

LLLP

φφφ

φφφ

)(21

sin)(21

sin)(

)(21

cos)(21

cos)(

222

221

(9.12)

For an ideal multiple-cycle coupler of length L0, the unperturbed total phase

difference φ(L0) is givenby φ(L0)=mπ,m=1,2,3,… Inpractice,however, couplers

areslightlydetunedfromtheideallength(L≠L0and|L-L0|<<1).Theunperturbed

total phasedifference φ(L), in this case, is givenby φ(L)= φ(L0)+∆φL=mπ+∆φL,

m=1,2,3,… and πβφ <<∆=∆ L

LL dzz

0

)( .Formultiplefull-cyclecouplers(meven),

inthelimitofsmallperturbation[(kee-koo)∆z<<1],equations(9.12)become:

zkkLP

zkkLP

ooeeLLpL

ooeeLLpL

∆−∆+∆≈

∆+∆≈

∆−∆−∆−≈

∆+∆−≈

)(21

41

2)(

)(21

41

12

1)(

2

2

2

2

2

1

φφφφ

φφφφ

(9.13)


For multiple half-cycle couplers (m odd), the expressions for P1(L) and P2(L) are

interchanged. From Equations (9.13), it can be observed that, in the case of

symmetric perturbation, the power leakage at the null port (P2) has two

contributions. In addition to the initial residual power, due to manufacturing

tolerancesanderrorsresultinginasmalldetuning∆φL≠0,thereexistsanotherterm

that depends on the difference between the perturbation-induced self-coupling

coefficients, i.e., ∆φP≠0. Although the first contribution is fixed and perturbation

independent, thesecondone,asdiscussedextensively insection9.4.1,dependson

the overlap between the perturbation profiles induced by the heating element

(heatingwire,CO2laser radiation,etc)andtheevenandoddmodesof thecoupler

waist. This overlap is shown to depend on the coupler-waist radius and the

perturbation penetration depth. Under symmetric perturbation, the power variation

on either output port can be used to map the coupling-region outer diameter

variation.Itcan,therefore,provideusefulinformationaboutthetaper-regionshape

and waist uniformity. In the case of non-uniform couplers (see section 9.4.2.4), it

canalsoprovidetheexactprofileoftheentirecouplingregion.Incaseofaperfect

coupler (∆φL=0), the information by the symmetric perturbation is given by the

quadraticterm[(kee-koo)∆z]2.

9.3.2.2 Asymmetricperturbation(kee≠≠≠≠0,koo≠≠≠≠0,keo≠≠≠≠0,koe≠≠≠≠0)

In the general case, all coupling coefficients arenon-zero. For a slightly detuned

couplerwithmeven,andanasymmetricperturbationappliedatapositionz0along

thecouplingregion,Equations(9.11)taketheform:

( )

( )

∆−∆+∆

∆=

∆−∆+∆

∆=

φφφ

φφφ

21

)(sin)|(|sin||cos21

sin),(

21

)(cos)|(|sin||cos21

cos),(

012222

02

012222

01

zzkzkLzP

zzkzkLzP

eoeo

eoeo

(9.14)


where ∆φ=(∆φL+∆φp) is the total detuning due to the length mismatch and the

perturbation.P2istheperturbedpowerleakingatthenullport(outputport#2)and,

for small total detuning (∆φ<<π) and a small perturbation (|keo|∆z≈0), can be

approximatedby:

( )

∆−∆+

∆≈2

)(sin||2

),( 0122

2

02φφφ

zzkLzP eo (9.15)

The first termof equation (9.15) is the residualpower atoutputport#2due to the

small total phase detuning and the non-zero difference between the symmetric

perturbation coefficients (kee-koo) (see Figure 9.4). This term is similar to the one

appearingunderthesymmetricperturbationofthecouplerinequation(9.13).The

second term depends on the relative position of the applied perturbation (through

φ1(z0))andthesquareofperturbationstrength(through(|keo|∆z)2).From(9.15)itis

alsoobservedthatforasmallphasedetuningthepowerevolutionalongthecoupler

is followed. It can be easily shown that the leaking power P2 acquires maximum

valuesatpositionsz0nalongthecouplingregion,forwhich:

1,2...,0n2

)12(21

)( 01 =++∆= πφφ nz n (9.16)

Thetotalnumberofsuccessivemaximaisdeterminedbytherelation0≤φ1(z0n)≤mπ

where m=2,4,6… Equation (9.16) is also valid for multiple half-cycle couplers

wheremisoddnumber.Inthiscase,however,theexpressionsforoutputpowersP1

and P2 in Equations (9.14) are interchanged. For the related ideal coupler (where

∆φ=0), the corresponding P2 maxima positions z’ 0n fulfil the relation

φ1(z’ 0n)=(2n+1)π/2.Itcanbeeasilyshownthatatthesepositionsthetotalpoweris

split equally between P1 and P2 (50-50% points). The leaking power acquires

minimum values at the points where the perturbation term in (9.15) vanishes, i.e.

when:

1,2...,0n21

)( 01 =+∆= πφφ nz (9.17)


Againfortheidealcoupler(∆φ=0),atthesepointsthepowerisconcentratedatonly

oneofthewaveguides(0-100%points).

9.3.3 Asymmetricperturbationsofnon-idealcouplers

From equation (9.16) it is deduced that the presence of a finite phase detuning

(∆φ≠0)introducesanerrorinthedeterminationofthe50-50%points.Thedetuning

ofthecouplermaybecausedbythefabricationprocessoritscharacterisationusing

awavelengthdifferentfromitsresonantwavelength.

9.3.3.1 Maintainingthecouplerstrengthandvaryingthecouplerlength:

For uniform couplers the error in the determination of the 50-50% points of the

coupler (at the resonance) due to a phase detuning ∆φ originated by varying the

couplerlengthtoL+∆Lwhilemaintainingthestrengthofthecouplerisgivenby:

βφφ

βφ

∆∆+∆

=∆∆=−=∆

22'00

pLnnn zzz (9.18)

Wherez0naretheactual50-50%pointsoftheidealcouplerandz’0narethemaxima

ofthenon-idealasymmetricperturbation.Thiserrorcanbeminimisedbylaunching

lightwithawavelengthclosetotheresonancewavelengthofthecouplerandusinga

very small perturbation.For a full-cycle coupler (m=2)with20dBextinction ratio

(∆φL=0.2) and a length of 30mm, the error in the 50-50% point positions is ≈-

0.5mm.

9.3.3.2 Varyingthecouplerstrengthandmaintainingthecouplerlength:

Thissituationariseswhencharacterisingthecoupleratadifferentwavelength(test

wavelength, λt) than the resonance wavelength, λ0. For full-cycle couplers, at the


testwavelength,λt, thedifferencebetweentheevenandoddpropagationconstants

is∆βt=2π(ne-no)/λtwhileattheresonancewavelength,λ0, itis∆β0=2π(ne-no)/λ0.It

is assumed thatλt is very close to λ0 and therefore (ne-no) is considered constant.

The coupler phase displacement from the resonance is given by ∆φ=(∆βt-∆β0)L

whereListhelengthoftheuniformcoupler.Foratestwavelengthofλt<λ0ityields

∆φ>0andwhenλt>λ0ityields∆φ<0.Ifthecouplerischaracterisedattheresonance

wavelength then λt=λ0 and ∆φ=0. It can be easily shown that, for a uniform full-

cyclecouplertheerrorinthe50-50%pointsduetoaphasedetuning∆φisgivenby:

tnnn

tnnn

zzz

zzz

βφβφ

∆∆+=−=∆

∆∆−=−=∆

===

===

4'

4'

)1(0)1(0)1(

)0(0)0(0)0(

(9.19)

Wheren=0,1correspondtothefirstandsecond50-50%pointrespectivelyandZ0n

correspondstothepositionofthe50-50%pointoftheidealcouplerandZ’ 0narethe

maximaofthenon-idealasymmetricperturbation.Itisinterestingtonotethatthe(0-

100%) point of the coupler corresponds to the minimum of the perturbation

independently of the phase detuning ∆φ. When calculating the error between the

local minimum of the asymmetric perturbation given by Equation (9.17) and the

positionofthe(0-100%)pointofthefull-cyclecouplerityields:

0' 10101 =−=∆ === nnn zzz (9.20)

For a uniform half-cycle coupler the error in the 50-50% points due to a phase

detuning∆φisgivenby:

0' 00000 =−=∆ === nnn zzz (9.21)


Therefore, for a half-cycle coupler the maximum of the leaking power due to an

asymmetric perturbation is a marker of the 50-50% point of the coupler

independently of the phase detuning of the coupler i.e., independent of the test

wavelength.

9.3.4 OutputRelativePhaseMeasurements

Theasymmetricperturbationofthecouplerwillalsoaffecttheelectricfieldphaseat

the output ports. The phase of the output light of the perturbed coupler will vary

withtheperturbationpositionalongthecouplerwaist.Theoutputphaseisgivenby

( ))Re()Im(arctan iii AA=θ ,whereAi (i=1,2) is the fieldamplitudeat theoutput

port#1 or port#2. Solving (9.7) for a perfect full-cycle coupler (m=2, ∆φ=0) the

phasechangeattheoutputportinrelationtotheunperturbedcouplerisgivenby:

[ ] )(cos)|tan(|arctan)( 0101 zzkz eo φθ ⋅∆−= (9.22)

Forsmallperturbations(keo∆z≈0)thephasedifferenceisapproximatedby:

zkzzkz eoeo ∆+

∆−≈ ||)(21

cos||2)( 012

01 φθ (9.23)

From Equations (9.1) it is then deduced that, with the perturbation applied at

position z0, the relative phase change of the field amplitude at output port#1 is

proportionaltotheindividual-waveguidepowerP1(z0).Therefore,thechangeinthe

relativephaseof the fieldat thecoupleroutputmapsdirectly thepowerevolution

along the corresponding individual waveguide. This information can be used to

calculate the coupling constant distribution k(z) along the coupling region. For a

perfectfull-cyclecoupler(∆φ=0)nolightarrivesatport#2andtherefore thephase

displacement cannot be measured at that port. In the case of non-ideal full-cycle


couplerswithaslightphasedetuning(m=2,∆φ≠0)phasechangeattheoutputport

induetotheasymmetricperturbationofthecouplerisgivenby:

∆+∆≈

∆−∆+∆−≈

))(sin(||1

12

)(

)(21

cos||2||2

)(

0102

012

01

zzkz

zzkzkz

eo

eoeo

φφθ

φφθ (9.24)

For full-cycle couplers with a small phase detuning, the phase change at output

port#1continuestomapthepowerevolutionalongthecoupler.However,thephase

changeatoutputport#2doesnotprovideadirectmeasurementofthecouplerpower

evolution,asshownin(9.24).

9.4 NumericalSimulations

9.4.1 Overlap integrals between the coupler eigenmodes and the

perturbationprofile.

Characterisationofcouplersusingasymmetricandasymmetricperturbationallows

thelocationofthe50-50%powerpointsofthecouplerandmeasurementofthebeat

lengthaswell radiusnon-uniformities inthetaperprofile.Theperturbationcanbe

inducedbyanumberoflocalisedheatsources,suchasexternalheatingelementsor

illuminationbylightsources(whitelight,CO2laser,He-Nelaser,laserdiodes,etc).

The various sources will induce different perturbation profiles and therefore will

haveadifferentoveralleffect.

In order to investigate the effectiveness of the perturbation we consider a

simplifiedphenomenologicalmodel inorder tocalculate the relativemagnitudeof

the coupling coefficients kij (i,j=e,o). The highly fused coupler waist is first

approximated by a circular cross-section glass structure with negligible core. The


couplermodesareapproximatedbythelowestordermodes(LP01andLP11)ofthis

multimode cladding-air structure [70, 121]. The coupler is perturbed locally by

radiation incident from side xx (symmetric perturbation) and side yy (asymmetric

perturbation), as shown in Figure 9.1. The absorption of the radiation generates

instantaneous heating of the structure that follows an exponential decay (~e-αx)

acrossthewaist.Thisresultsinalocalchangeoftherefractiveindexofthestructure

by TTn

n ∆∂∂=∆ . According to [56], for fused silica, the coefficient

)(K101.1 1-5−⋅≈∂∂Tn

.FortheCO2radiation,typicalvalueforthepenetrationlength

is 1/α≈1µm-6µm [119]. Figure 9.3 illustrates the symmetric and asymmetric

perturbationofacouplerwitharadiusof30µmandaradiationpenetrationlengthof

16µm.Thepenetrationdepthwaschosenforabettervisualisationofthetemperature

gradientthroughthecouplercrosssection.


Figure9.3-Perturbationofa30µmcouplerbyCO2radiation.Left:Symmetricperturbation

configuration; Right: Asymmetric perturbation configuration. Top: Even mode profile,

middle:Oddmodeprofile,bottom:temperaturedistributionf(x,y)acrossthefibre.

The perturbation is quantified by calculating the overlap integrals OIij (i,j=e,o)

between the temperature distribution and the mode profiles. The overlap integrals

aredefinedby

oejidAyxfEEOIA

jiij ,,,),( ==

where f(x,y) is the normalised temperature profile. The distribution f(x,y) is

proportional totheperturbedindexprofileand, therefore, theoverlapintegralsOIij

(i,j=e,o)areproportionaltothecouplingcoefficientskij(i,j=e,o).


Firstly,theeffectoftheradiationpenetrationdepthonthecouplingcoefficient

magnitude for both a symmetric and asymmetric perturbation is considered. The

couplerwaist radius is considered tobe16µm,which is typicalof thedeviceswe

routinelyfabricateusingtheflamebrushtechnique(seechapter4).Figure9.4shows

the relative variation (in arbitrary units) of the coupling constant keo and the

corresponding difference kee-koo, under symmetric (dashed lines) and asymmetric

perturbations (solid lines), for different radiation absorption lengths. It should be

reminded that under pure symmetric perturbation (Section 9.3.2.1), the perturbed

outputpowerisproportionaltothedifference(kee-koo),asinequation(9.13),while

under pure asymmetric perturbation (Section 9.3.2.2), the perturbed power is

proportionalto 2eok ,asshowninexpression(9.15).FromFigure9.4itisrealisedthat

both asymmetric-perturbation keo and symmetric-perturbation (kee-koo) are

maximised for a range of absorption lengths between 10µm and 17µm, i.e., the

proposed perturbation method is optimised for radiation absorption lengths

comparabletothecouplerwaistradius.Italsoshowsthatasymmetricperturbations

result in finite kee-koo, which nevertheless, is appreciably smaller than the

accompanyingkeo.Undersymmetricperturbation,asexpected,keo isnegligible for

everyabsorption length. Finally,as theabsorption length is increasedappreciably

theperturbationbecomesincreasinglyuniformacrosstheentirecouplerwaistcross-

sectionandalltheparameterstendtozero,undereitherperturbation.Thissuggests

that theproposednon-destructiveperturbationmethodwouldnotwork incase the

perturbing radiation was provided by a He-Ne laser radiation of λ=633nm

(absorption length in silica ~1m) or any other visible or near-infrared laser (with

absorption lengthswell above thewaistdiameter).Theuseof radiationwith large

absorptionlengthwouldhaverequiredapplicationofanextrahighlyabsorbinglayer

(asin[117]),whichisnotnecessaryusingtheCO2laserradiation.


-0.5

0

0.5

1

1.5

2

5 20 35 50 65 80 95Absorptionlength(µµµµm)

Cou

plin

gC

oeff

cien

ts(

a.u.

)

(kee-koo)sym

|keo|2asym

|keo|2sym

(kee-koo)asym

Coupler radius=16 µµµµm

Figure 9.4 - Coupling coefficient variation with the absorption length of the incident

radiationforacouplerwaist radiusof16µm.Dashed lines:symmetricperturbation.Solid

lines:asymmetricperturbation.

Next,inFigure9.5,thevariationofthecouplingcoefficientskeoandthedifferences

(kee-koo)fordifferentcoupler-waistradii,underCO2lasersymmetric(dashedlines)

andasymmetric(solidlines)side-perturbationisconsidered.Forthecalculations,a

typical absorption length of 5µm was assumed. As before, the asymmetric-

perturbation keo and symmetric-perturbation (kee-koo) are maximised for a coupler

waistofabout5µm,i.e.,comparabletotheradiationabsorptionlength.FromFigure

9.5 it is also realised that for small coupler-waist radii, asymmetric perturbations

resultin(kee-koo)appreciablysmallerthantheaccompanyingkeo.However,forlarger

coupler-waist radii, the difference (kee-koo) becomes comparable with and finally

equaltokeoandthesimpleanalyticformula(9.15)isnolongervalid.Inthiscase,

the power perturbation at the coupler output ports should be calculated using

equations(9.11).Again,undersymmetricperturbation,keoiszeroindependentlyof

thecoupler-waistradius.


-0.5

0

0.5

1

1.5

2

5 10 15 20 25 30Couplerradius(µµµµm)

Cou

plin

gC

oeff

icie

nts

(a.u

.)

(kee-koo)sym

|keo|2asym

|keo|2sym

(kee-koo)asym

Absorption length=5 µµµµm

Figure 9.5 - Coupling coefficient variation with coupler-waist radius. The perturbing

radiation absorption length was 5µm (typical of CO2 laser). Dashed lines: symmetric

perturbation.Solidlines:asymmetricperturbation.

Under symmetric perturbation, the difference (kee-koo) changes quasi-linearly with

thecoupler-waistradius.FromEquation(9.13),itisthendeducedthatoutputpower

perturbationwillfollowcloselythecoupler-waistouterdiameterastheCO2laseris

scannedalong thecoupling region.Theoutputpowervariationcan thenprovidea

reliablemappingof theentirecouplingregiongivinganaccurateestimationof the

coupleruniformity.

Under asymmetric perturbation, the coupling coefficient keo changes

appreciablywiththecoupler-waistradius.FromEquation(9.15),itisthendeduced

thatastheperturbationisscannedalongthevaryingcouplingregion,inadditionto

theexpression inparenthesesof the second term, theperturbationoutputpower is

appropriatelyweightedbythevaryingkeo2coefficient.Additionally,ifthe(kee-koo)

islargerorcomparabletokeo2(forlargecoupler-waistradiiorunderweakCO2laser

power), the significant (kee-koo) term in Equation (9.15) should also be taken into

account.

InFigure9.6theeffectofdifferentincidentradiationpowersonthemagnitude

ofthecoefficients(kee-koo)andkeo2underasymmetricperturbationisconsidered.It

isassumedthatthereisalineardependenceoftherefractiveindexwiththepowerof

the incident radiation and therefore, the coupling coefficients (kee-koo) and keo are


proportional to the power of the incident radiation. The absorption length of the

incident radiationwas5µm(CO2 laser radiation)and thecouplerwaist radiuswas

16µm.ForhighpowersoftheCO2laser(region3inFigure9.6),(kee-koo)<<keo2and

theasymmetricperturbationofthecouplercanbeusedtolocatethe50-50%points

ofthecoupler.ForsmallvaluesoftheCO2laserpowerwhere(kee-koo)>>keoor(kee-

koo)≈keo(regions1and2inFigure9.6respectively)thefirstterminEquation(9.15)

shouldbetakenintoaccount.

0

1

2

3

4

5

0 5 10 15 20 25 30CO2laserPower(a.u.)

Cou

plin

gC

oeff

cien

ts(

a.u.

)

|keo|2

|kee-koo|

1

3

2

Figure 9.6 - Coupling coefficient variation with the power of the incident CO2 laser

radiation under asymmetric perturbation. The perturbing radiation absorption length was

5µmandthecouplerwaistradiuswas16µm.

9.4.2 CouplerPerturbationResults

Inorder toverifythevalidityof theapproximateexpressions(9.13)and(9.15),an

exact model based on the transfer-matrix method was implemented. The entire

coupler structure was divided in M uniform sections and the transfer matrices

corresponding to each section were calculated using equations (9.5) to (9.7). The

transfer matrix of the entire coupler is then easily calculated by multiplying the

individual transfer matrices. No simplifications to the perturbation matrix were

made. In thismodel, anarbitrarycouplingprofilek(z)canbe introducedandboth


thesymmetricandasymmetricperturbationscanbeaccountedforbymodifyingthe

valuesof thecouplingcoefficientskeo ,kee andkoo.Anumberofdifferentcoupler

configurations were considered with coupling coefficient profiles of varying

complexity.Theyareintendedtoprovethatforallcouplingcoefficientgeometries,

anasymmetricperturbationscannedalongthecouplingregionalwaysprovidesthe

50-50%powerpoints. In the followingsimulations idealasymmetricperturbations

areconsideredwithkee=koo=0and∆φ≠0.

9.4.2.1 UniformCoupler

The first simulation refers to an ideal uniform coupler with constant coupling

coefficientthroughoutthecouplingregion.ThetotalcouplerlengthisL=30mm.The

total phase difference between the even and odd eigenmodes was φ(L)=2π (full-

cyclecoupler).Figure9.7showsthenormalisedpowerevolutionP1(z)andP2(z)of

each “individual” waveguide (dashed lines), as well as, the output power

perturbation∆P2(L)(solidline)asafunctionoftheperturbationpositionalongthe

coupling region. The coupling coefficient profile is also superimposed for better

visualisation.

0

0.25

0.5

0.75

1

1.25


Nor

mal

ised

Pow

er(

a.u.

)

P1(z)

P2(z) ∆∆∆∆P2(L,z0)(x103)

k(z)(x104µµµµm-1)

Figure9.7-Normalisedpowerevolutionalongeach“individual”waveguide(dashedlines),

aswellas,outputpowerperturbation(solidline)asafunctionoftheperturbationposition

alongthecouplingregionofanidealuniformcoupler. Thecouplingcoefficientprofileis

alsosuperimposedforbettervisualisation.


Theseresultsillustratethatthepositionsinthecouplerwheretheoutputpower

perturbation is maximum correspond to the points where the power is equally

distributedbetweenthetwo“ individual” waveguidesP1(z)=P2(z)=0.5.Foranideal

uniform coupler of length L, these points are situated at L/4 and 3L/4. The

simulationresultsshowthatthe50-50%pointsareat7.5mmfromthecentreofthe

coupler,asexpected.

9.4.2.2 UniformCouplerwithTwoTaperedRegions

Thesecondsimulationreferstoamorerealisticcouplerprofilewithonetaperregion

oneithersideoftheuniformcouplerwaist.Eachtaperedregionisconsidered10mm

long and the uniform waist region is 30mm long. The total coupler length is

therefore L=50mm. Again, the total phase difference between the even and odd

eigenmodes was φ(L)=2π (full-cycle coupler). This coupling profile is typical of

couplers fabricated with the flame brush technique. The results of the simulation,

illustrated in Figure 9.8, show that the effect of the taper region on the power

distributionalongthecoupleristomovethe50-50%pointsawayfromthecentreof

thecouplerdue to somecouplingbetween themodes in the transition region.The

resultsalsoillustratethatthemaximaoftheoutputperturbationpowercoincidewith

the50-50%points,whichareplaced9.5mmawayfromthecentreofthecoupler.


0

0.25

0.5

0.75

1


Nor

mal

ised

Pow

er(

a.u.

)∆∆∆∆P2(L,z0)(x103)

P1(z)

P2(z)


Figure9.8-Normalisedpowerevolutionalongeach“ individual” waveguide(dashedlines),


along the coupling region of a uniform coupler with two tapered regions. The coupling

coefficientprofileisalsosuperimposedforbettervisualisation.

9.4.2.3 UniformlyTaperedCoupler

Next, some examples of non-uniform couplers are considered. First, a uniformly

tapered coupling coefficient profile with small taper ratio is simulated. These

profiles canbe encountered in real fused couplers andmaybedue to temperature

non-uniformities along the fused waist or other experimental inaccuracies. The

resultsof thesimulationareshowninFigure9.9.Figure9.10showsthesimulated

perturbationresultsofauniformlytaperedcouplerwithextremetaperratio.Inboth

cases,thetotalcouplerlengthwasL=30mmandthetotalphasedifferencebetween

theevenandoddeigenmodeswasφ(L)=2π(full-cyclecoupler).


0

0.2

0.4

0.6

0.8

1

1.2


Nor

mal

ised

Pow

er(

a.u.

)

P1(z)

P2(z)

∆∆∆∆P2(L,z0)(x103)


Figure9.9-Normalisedpowerevolutionalongeach“ individual” waveguide(dashedlines),


alongthecouplingregionofauniformly-taperedcoupler(smalltaperratio).Thecoupling

coefficientprofileisalsosuperimposedforbettervisualisation.

0

0.5

1

1.5

2


Nor

mal

ised

Pow

er(

a.u.

)

∆∆∆∆P2(L,z0)(x103)

P1(z) P2(z)


Figure 9.10 - Normalised power evolution along each “ individual” waveguide (dashed

lines), as well as, output power perturbation (solid line) as a function of the perturbation

positionalongthecouplingregionofauniformly-taperedcoupler(extremetaperratio).The

couplingcoefficientprofileisalsosuperimposedforbettervisualisation.


Despitethedifferentindividualpowerdistributions,inbothcases,theoutputpower

perturbationmaximacoincidewiththepointsalongthecouplerwherethepoweris

splitequallybetweenthetwo“ individual” waveguidesP1(z)=P2(z)=0.5.

9.4.2.4 Non-UniformCoupler(Mach-ZenhderInterferometer)

Thefinalsimulationconcernsacomplexnon-uniformcouplingstructureconstituted

bytwoweakly-coupledregionsandanintermediateuncoupledregion. Thelength

ofeachweakly-coupledregionisL0=10mmandthetotalcouplerlengthLc=30mm.

The phase difference between the even and odd eigenmodes along each weakly-

coupledregionis2

)(0

0

πβ =∆L

dzz .Thetotalphasedifferencebetweentheevenand

oddeigenmodes, in thiscase, is πβφ =∆= 0

0

)()(L

C dzzL (half-cyclecoupler).Since

the coupler is half-cycle long, the perturbation is measured at the output of

waveguide#1.Figure9.11showsthenormalisedpowerevolutionP1(z)andP2(z)of

each “ individual” waveguide (dashed lines), as well as, the output-power

perturbation∆P1(L)(solidline)asafunctionoftheperturbationpositionalongthe

coupling region. The coupling coefficient profile is also superimposed for better

visualisation.


0

0.25

0.5

0.75

1

0 5 10 15 20 25 30 35 40CouplerPosition(mm)

Nor

mal

ised

Pow

er(

a.u.

) ∆∆∆∆P2(L,z0)(x103) P2(z)P1(z)


Figure 9.11 - Normalised power evolution along each “ individual” waveguide (dashed

lines), as well as, output power perturbation (solid line) as a function of the perturbation

position along the coupling region of a non-uniform coupler (Mach-Zenhder

interferometer).Thecouplingcoefficientprofileissuperimposedforbettervisualisation.

At the end of the first weakly-coupled region, the power is equally split

between the “ individual” waveguides #1 and #2 (P1=P2). The powers remain

unchangedoverthecentraluncoupledregionandcross-couplecompletelyattheend

ofsecondweakly-coupledregion.Theoutput-powerperturbation∆P1(L)(solidline)

maps exactly this power evolution. It is shown that ∆P1(L) reaches a maximum

valuewhentheperturbationreachestheendofthefirstweakly-coupledregionand

retains it over the entire uncoupled central region. It is easily realised that this

complex coupled structure corresponds to a Mach-Zenhder interferometer if the

centralregionistotallyuncoupled.

9.4.3 Perturbationsofnon-idealcouplers

Asalreadymentioned in section9.3.3, in thepresenceof a finitedetuning∆φ the

perturbationpowermaximaaredisplacedfromtheactual50-50%powerpointsby

anamountgivenbyEquation(9.18)orEquation(9.19)dependingonthenatureof

thephasedetuning.


9.4.3.1 Maintainingthecouplerstrengthandvaryingthecouplerlength

Figure9.12 shows the simulationof the asymmetricperturbationof couplerswith

different phase displacements from the optimum point, ∆φL=0, ±0.21 (∆φp is

considered 0). The thick solid lines show the power evolution along the coupler

length. The dashed line shows the asymmetric perturbation of the ideal coupler,

while the thin solid lines show the corresponding perturbations of the detuned

couplers. The shifts in the perturbation maxima from the ideal case, given by

expression (9.19), are clearly shown. In these simulations, the coupling strength

remained constant and the phase displacement, ∆φL, was achieved by varying the

coupler length by ∆Lcoupler=∆φL/∆β=±1.0mm. The length of the ideal coupler

(∆φL=0)wasL=30mmandthecouplingstrengthofallcouplerswas∆β=2⋅π/L.The

cross-couplingcoefficientremainedconstant,keo∆z=0.22and(kee-koo)=0.

0

0.5

1

0 7.5 15 22.5 30Couplerposition(mm)

Nor

mal

ised

pow

er(a

.u.)

∆φ∆φ∆φ∆φ====−−−−0.210.210.210.21 ∆φ∆φ∆φ∆φ=0.21=0.21=0.21=0.21

∆φ∆φ∆φ∆φ=0=0=0=0

P2(z)P1(z)

0

0.5

1

0 7.5 15 22.5 30Couplerposition(mm)

Nor

mal

ised

pow

er(a

.u.)

∆φ∆φ∆φ∆φ====−−−−0.210.210.210.21 ∆φ∆φ∆φ∆φ=0.21=0.21=0.21=0.21

∆φ∆φ∆φ∆φ=0=0=0=0

P2(z)P1(z)

Figure9.12-Asymmetricperturbationoffull-cyclecouplersfordifferentdetuningvalues

(∆φ=0, ±0.21) achieved by using different lengths for each coupler. The asymmetric

couplingcontributionremainedconstant,keo∆z=0.22.Thicklines:Powerdistributionalong

thecoupler.Thinlines:Perturbationof thedetunedcouplers.Dashed line:Perturbationof

theidealcoupler.(Theperturbationpowerismultipliedbyafactorof10).

Forauniform2π couplerwithacouplingstrengthof∆β=2π/LwhereL=30mmis

the optimum coupler length and for a phase displacement of ∆φL=±0.21, the


correction to the perturbation maxima positions, in order to obtain the 50-50%

pointsofthecouplerisgivenby(9.18), mmL

L pert 5.04

±≈⋅∆=∆π

φ.

9.4.3.2 Varyingthecouplerstrengthandmaintainingthecouplerlength

In the followingsimulations, thecoupling length remainedconstant and thephase

displacement, ∆φ, was achieved by varying the difference between the coupler

eigenmodesby∆φ/L.Asalreadymentioned,thisphasedetuningcouldbeachieved

by characterising the coupler at a wavelength different from its resonance

wavelength.Figure9.13shows the simulationof theasymmetricperturbationofa

uniformfull-cyclecouplerwithdifferentphasedisplacements.Figure9.13a)shows

the power evolution (solid lines) and asymmetric perturbation (dashed line) of an

idealcoupler(∆φ=0)testedattheresonancewavelength(λt=λ0).Theverticaldashed

linesshowthepositionsof theasymmetric-perturbationmaximathatcoincidewith

theactual50-50%pointsofthecoupler(shownbythearrows).Figures9.13b)and

9.13c) show the corresponding power evolution (solid lines) and asymmetric

perturbation(dashedlines)ofthecouplertestedatthewavelengthsλt<λ0(∆φ=0.3)

andλt>λ0(∆φ=-0.3)respectively.Theverticaldashedlinesshowthecorresponding

asymmetricperturbationmaximawhichnowdifferfromtheactual50-50%pointsof

theidealcoupler(markedbythearrows).Inaccordancewithexpression(9.19)the

perturbationmaxima in thesecasesare shifted inside (∆φ>0)oroutside (∆φ<0)of

theactual50-50%points.Boththemagnitudeandthedirectionofthisshiftshould

be accounted correctly in order for the actual 50-50% points to be retrieved. It

shouldalsobestressedthatinallcasesthe(0-100%)point(givenbytheasymmetric

perturbation minimum) remains fixed as the theory predicts (expression 9.20). In

thesesimulationsthedifferencebetweentheeigenmodesoftheidealcoupler(∆φ=0)

was∆β=2⋅π/LandthelengthofallcouplerswasL=30mm.Thedifferencebetween

the eigenmodes of the detuned couplers was ∆β’=(2⋅π+∆φ)/L. The cross-coupling

coefficientremainedconstant,keo∆z=0.22and(kee-koo)=0.


0

0.5

1

0 7.5 15 22.5 30

Nor

mal

ised

pow

er(a

.u.)

P2(z,λλλλt)P1(z,λλλλt)

b)∆∆∆∆P2(L,Z0)∆φ∆φ∆φ∆φ = 0.3= 0.3= 0.3= 0.3

λλλλ t < λ< λ< λ< λ0

0

0.5

1

0 7.5 15 22.5 30

Nor

mal

ised

pow

er(a

.u.)

P1(z,λλλλt) P2(z,λλλλt)

a)∆∆∆∆P2(L,Z0)

∆φ∆φ∆φ∆φ = 0= 0= 0= 0

λλλλ t = λ= λ= λ= λ0

0

0.5

1

0 7.5 15 22.5 30CouplerPosition(mm)

Nor

mal

ised

pow

er(a

.u.)

P1(z,λλλλ0) P2(z,λλλλ0)

c)

∆φ∆φ∆φ∆φ = = = = −−−−0.30.30.30.3

λλλλ t > λ> λ> λ> λ0

∆∆∆∆P2(L,Z0)

0

0.5

1

0 7.5 15 22.5 30

Nor

mal

ised

pow

er(a

.u.)


b)∆∆∆∆P2(L,Z0)∆φ∆φ∆φ∆φ = 0.3= 0.3= 0.3= 0.3


0

0.5

1

0 7.5 15 22.5 30

Nor

mal

ised

pow

er(a

.u.)


b)∆∆∆∆P2(L,Z0)∆φ∆φ∆φ∆φ = 0.3= 0.3= 0.3= 0.3


∆φ∆φ∆φ∆φ = 0.3= 0.3= 0.3= 0.3


0

0.5

1

0 7.5 15 22.5 30

Nor

mal

ised

pow

er(a

.u.)


a)∆∆∆∆P2(L,Z0)

∆φ∆φ∆φ∆φ = 0= 0= 0= 0


0

0.5

1

0 7.5 15 22.5 30

Nor

mal

ised

pow

er(a

.u.)


a)∆∆∆∆P2(L,Z0)

∆φ∆φ∆φ∆φ = 0= 0= 0= 0


∆φ∆φ∆φ∆φ = 0= 0= 0= 0


0

0.5

1


Nor

mal

ised

pow

er(a

.u.)


c)

∆φ∆φ∆φ∆φ = = = = −−−−0.30.30.30.3


∆∆∆∆P2(L,Z0)

0

0.5

1


Nor

mal

ised

pow

er(a

.u.)


c)

∆φ∆φ∆φ∆φ = = = = −−−−0.30.30.30.3


∆φ∆φ∆φ∆φ = = = = −−−−0.30.30.30.3


∆∆∆∆P2(L,Z0)

Figure9.13-Asymmetricperturbationoffull-cyclecouplersfordifferentdetuningvalues

(∆φ=0, ±0.3) achieved by using different coupling strengths for each coupler. The arrow

markers correspond to the50-50%pointsof the ideal coupler and thedashed lines to the

maxima of the asymmetric perturbation. a) Power evolution and perturbation of the ideal

coupler(∆φ=0).b)Powerevolutionandperturbationofacouplerdetunedby∆φ=+0.3.c)

Powerevolutionandperturbationofacouplerdetunedby∆φ=-0.3.(Theperturbationpower

ismultipliedbyafactorof10).


For a uniform 2π coupler with a length L=30mm, where ∆β=2π/L is the

optimumcouplingstrengthandforaphasedisplacementof∆φ=±0.3,thecorrection

to the perturbation maxima positions, in order to obtain the 50-50% points of the

idealcoupleraregivenby,)2(41 φπ

φ∆+⋅

⋅∆−=∆ LL and

)2(42 φπφ

∆+⋅⋅∆+=∆ L

L .Itcan

be seen that the corrections are different for ∆φ=+0.3 (∆L1≈-0.34mm and

∆L2≈+0.34mm)and∆φ=-0.3(∆L1≈+0.38mmand∆L2≈-0.38mm).

Figure 9.14 illustrates the simulation of the asymmetric perturbation of a

uniformhalf-cyclecouplerwithdifferentphasedisplacements.Figure9.14a)shows

the power evolution (solid lines) and asymmetric perturbation (dashed line) of an

idealcoupler(∆φ=0)testedattheresonancewavelength(λt=λ0).Theverticaldashed

lineshowsthepositionoftheasymmetric-perturbationmaximumthatcoincideswith

the actual 50-50% point of the coupler (shown by the arrow). Figures 9.14b) and

9.14c) show the corresponding power evolution (solid lines) and asymmetric

perturbation(dashedlines)ofthecouplertestedatthewavelengthsλt<λ0(∆φ=0.2)

andλt>λ0(∆φ=-0.2)respectively.Theverticaldashedlinesshowthecorresponding

asymmetricperturbationmaximumthatstillcoincidewiththeactual50-50%points

of theidealcoupler (markedbythearrows)aspredictedby thetheory(expression

9.21). In these simulations the difference between the eigenmodes of the ideal

coupler (∆φ=0) was ∆β=π/L and the length of all couplers was L=30mm. The

differencebetweentheeigenmodesofthedetunedcouplerswas∆β’=(π+∆φ)/L.The

cross-couplingcoefficientremainedconstant,keo∆z=0.22and(kee-koo)=0.


0

0.5

1

0 7.5 15 22.5 30

Nor

mal

ised

pow

er(a

.u.)


b)∆∆∆∆P1(L,Z0)

∆φ∆φ∆φ∆φ = 0.2= 0.2= 0.2= 0.2λλλλ t < λ< λ< λ< λ0

0

0.5

1

0 7.5 15 22.5 30

Nor

mal

ised

pow

er(a

.u.)


a)∆∆∆∆P1(L,Z0)

∆φ∆φ∆φ∆φ = 0= 0= 0= 0λλλλ t = λ= λ= λ= λ0

0

0.5

1


Nor

mal

ised

pow

er(a

.u.)

P2(z,λλλλ0)P1(z,λλλλ0)

c)∆φ∆φ∆φ∆φ = = = = −−−−0.20.20.20.2λλλλ t > λ> λ> λ> λ0

∆∆∆∆P1(L,Z0)

0

0.5

1

0 7.5 15 22.5 30

Nor

mal

ised

pow

er(a

.u.)


b)∆∆∆∆P1(L,Z0)

∆φ∆φ∆φ∆φ = 0.2= 0.2= 0.2= 0.2λλλλ t < λ< λ< λ< λ0

0

0.5

1

0 7.5 15 22.5 30

Nor

mal

ised

pow

er(a

.u.)


b)∆∆∆∆P1(L,Z0)

∆φ∆φ∆φ∆φ = 0.2= 0.2= 0.2= 0.2λλλλ t < λ< λ< λ< λ0

∆φ∆φ∆φ∆φ = 0.2= 0.2= 0.2= 0.2λλλλ t < λ< λ< λ< λ0

0

0.5

1

0 7.5 15 22.5 30

Nor

mal

ised

pow

er(a

.u.)


a)∆∆∆∆P1(L,Z0)

∆φ∆φ∆φ∆φ = 0= 0= 0= 0λλλλ t = λ= λ= λ= λ0

0

0.5

1

0 7.5 15 22.5 30

Nor

mal

ised

pow

er(a

.u.)


a)∆∆∆∆P1(L,Z0)

∆φ∆φ∆φ∆φ = 0= 0= 0= 0λλλλ t = λ= λ= λ= λ0

∆φ∆φ∆φ∆φ = 0= 0= 0= 0λλλλ t = λ= λ= λ= λ0

0

0.5

1


Nor

mal

ised

pow

er(a

.u.)


c)∆φ∆φ∆φ∆φ = = = = −−−−0.20.20.20.2λλλλ t > λ> λ> λ> λ0

∆∆∆∆P1(L,Z0)

0

0.5

1


Nor

mal

ised

pow

er(a

.u.)


c)∆φ∆φ∆φ∆φ = = = = −−−−0.20.20.20.2λλλλ t > λ> λ> λ> λ0

∆φ∆φ∆φ∆φ = = = = −−−−0.20.20.20.2λλλλ t > λ> λ> λ> λ0

∆∆∆∆P1(L,Z0)

Figure9.14-Asymmetricperturbationofhalf-cyclecouplersfordifferentdetuningvalues

(∆φ=0, ±0.2) achieved by using different coupling strengths for each coupler. The arrow

markerscorrespondtothe50-50%pointofthecouplerandthedashedlinestothemaxima

of the asymmetric perturbation. a) Power evolution and perturbation of the ideal coupler

(∆φ=0).b)Powerevolutionandperturbationofthecouplerdetunedby∆φ=+0.2.c)Power

evolutionandperturbationof thecouplerdetunedby∆φ=-0.2. (Theperturbationpower is

multipliedbyafactorof10).


Thecorrectiontothemaximumoftheasymmetricperturbationofahalf-cycle

couplergivenby(9.21)iszeroandthereforeit isamarker tothe50-50%pointof

the half-cycle coupler independently of the coupling strength of the coupler or

equivalently,independentlyofthewavelengthatwhichthecouplerischaracterised

aslongas∆φ=<<π.

Finally,itshouldbestressedthatinthecasethatthecouplerwaististwisted,as

theperturbingelementisscannedalongthecouplerlengthresultsinbothsymmetric

and asymmetric perturbation with mixed results that do not provide any useful

information.

9.4.4 OutputPhasePerturbation

In section 9.3.4 it was shown analytically that for a perfect coupler under pure

asymmetric perturbation (∆φ=0), the phase of the electric field at the output port

with non-null power given by Equation (9.22) is proportional to the power of the

corresponding “ individual” waveguide at the point of the perturbation. Therefore,

the output phase variation maps directly the power evolution along the

corresponding “ individual” waveguide. The phase change due to an asymmetric

perturbationwassimulatedforanidealuniformfull-cyclecoupler(∆φL=0)byusing

expression(9.9).Theasymmetriccross-perturbationcoefficientwaskeo∆z=0.07and

the self-perturbation coefficientswere considered zero (kee=koo=0).The resultsof

thesimulationinFigure9.15showthatthephasevariationθ1oftheelectricfieldat

theoutputof the “ individual” waveguide#1 (solid line) follows indeed closely the

powerevolutionalongthecorresponding“ individual” waveguide.Foranoptimum

cyclecoupler(∆φL=0)withlightlaunchedinport1,thecouplingprofilek(z)canbe

obtainedbymeasuringtheoutputphaseatthesameport.Theoutputphasechanges

canbemeasuredbyusingaphasesensitive(interferometric)technique.


-0.2

0

0.2

0.4

0.6

0.8

1


Out

put

phas

e(a

.u)

P1(z)

P2(z) θθθθ1(z0)

Figure 9.15 - Simulation of the phase displacement at port#1 due to an asymmetric

perturbationofafull-cyclecoupler.Thedashedlinesrefertothepowerdistributionalong

thecoupler.Thesolidlinereferstothephaseshiftrelativetotheunperturbedcase.

9.5 ExperimentalResults

To demonstrate the proposed method experimentally, two different external, non-

destructiveperturbationtechniqueswereused.Initiallytheperturbationwasinduced

byscanningacrossthecouplerwaista100µmouterdiameterheatingelectricwire.

The temperature of the wire could be controlled by varying the applied electric

current. This method, however, introduced large errors due to oscillations in the

electric current as well as heat convection losses that influenced significantly the

temperature of the wire and therefore, the induced symmetric and asymmetric

perturbations. Subsequently, the perturbation was induced by scanning across the

couplerwaisttheoutputofaCO2laserat10.6µm.Thistechniqueprovedtobemuch

more stable, repeatable and accurate. In order to reduce the noise of the

measurement,thelaseroutputwasmodulatedandthepoweroscillationsduetothe

perturbation where detected and amplified using a lock-in amplifier. The

experimentalsetupisillustratedinFigure9.16.


Figure9.16 -Experimental setup for thecharacterisationofcouplersusingaperturbation

inducedbyaCO2laser.

Lightislaunchedinthecouplerthroughport1at1.55µmusingaDFB-LDand

the light arriving at port3 and port4 is detected and amplified using a lock-in

amplifier.AmirrorismountedonatranslationstageinordertoscantheCO2laser

across the coupler waist. A symmetric perturbation is induced to the coupler by

shining the CO2 laser perpendicularly to the two cores (see Figure 9.1b). An

asymmetricperturbationof thecoupler isaccomplishedby rotating thecouplerby

90°arounditsaxis.

Several experiments were performed in order to prove the theoretical

predictions mentioned before. Three different couplers where fabricated and

characterisedusing the perturbationmethod: ahalf-cycle coupler (φ(L)=π), a full-

cycle coupler (φ(L)=2π) and a complex non-uniform coupler. The length for all

thesecouplerswas30mm,however, theywere all approximately twice that length

due to a long transition region. Both the symmetric perturbation and asymmetric

perturbationwereusedtocharacterisethecouplers.


9.5.1 Characterisationofahalf-cyclecoupler[φ(L)=π]

Thesecouplerstransferlightfromonefibretotheother(lightthatislaunchedinto

port1 exits at port4) has one point where the power is equally distributed in both

fibres that should be localised in the centre of the coupler. Under asymmetric

perturbation,theperturbedpowerwillpeakonceatthe50-50%point.Theresultsof

the characterisation of a π coupler are shown in Figure 9.17. The asymmetric

perturbationthepowerdistributionalongthecouplerandthesymmetricperturbation

follows thecouplingprofile.Thesymmetricperturbationwasnormalised toπ and

usedasthecouplingprofiletofittheoreticallytheasymmetricperturbation(Figure

9.17–theoreticalFit2).Althoughthesymmetricperturbationfollowsthedifference

between the self-coupling perturbation coefficients, (kee-koo), it will match closely

the coupling profile, k(z) of the measured coupler differing mainly in the tapered

regions. Additionally the asymmetric perturbation was fitted using the coupling

strength profile calculated by equation (4.13) and measuring the power evolution

duringthefabricationprocess(Figure9.17–theoreticalFit1).

0

1

2

3

4

5

6


Nor

mal

ised

pow

erx

10-3

Asym.Perturbationexperimentaldata TheoreticalFit2

Sym.PerturbationexperimentaldataTheoreticalFit1

Figure 9.17 - Characterisation of a π coupler using the symmetric and asymmetric

perturbation. The asymmetric perturbation was fitted using both the coupling profile

retrievedfromthesymmetricperturbationdata(fit2)andcalculatedfromthemeasurement

ofthepowerevolutionduringtheelongationprocess(fit1).


From this Figure it was realised that the experimental coupling strength obtained

using the symmetric perturbation of the coupler was better suited than the one

calculated from (4.13), to fit the asymmetric perturbation data. This is due to

equation (4.13)beingan approximation to the casewhere the sizeof the flame is

infinitesimalandinrealityitisapproximately4mmwideresultinginanactualwaist

radiusdifferingfromtheburnertravel.

Insections9.3.3itwasmentionedthatthemaximumofthepowerchange,due

to an asymmetric perturbation, is a marker for the 50-50% points of the coupler

independentlyofthesmallphasedetuningofthecoupler(eitherduetostraininthe

mountingof thecoupleror thecharacterisation atawavelengthdifferent from the

coupler resonance wavelength). This information is very useful since the 50-50%

pointsofhalf-cyclecouplerscanbealwaysobtainedwithusinganormallaserdiode

tocharacterisethecouplerandwithouttheneedofatunablelasersettothecoupler

exact resonance wavelength. In Figure 9.18 experimental results of the

characterisation of a half-cycle coupler at different wavelengths are shown. A

tunablelaserwasusedtolaunchlightinthecouplerport#1insteadoftheDFB-LD

as shown in experimental setup (Figure 9.14). Three different test wavelengths

where used: λ0=1510nm, λ1=1550nm (coupler resonance wavelength) and

λ2=1590nm. The power of the CO2 laser was the same for all the experiments

(100mWthrougha2mmpinhole).


0

0.025

0.05

0.075

0.1

0.125

0 20 40 60 80 100Couplerposition(mm)

Per

turb

atio

n(V

)λλλλ=1550nm

λλλλ=1510nm

λλλλ=1590nm-90

-80

-70

-60

1300 1500 1700Wavelength(nm)

Pow

er(

dBm

)

1590nm1510nm

1550nm

0

0.025

0.05

0.075

0.1

0.125

0 20 40 60 80 100Couplerposition(mm)

Per

turb

atio

n(V

)λλλλ=1550nm

λλλλ=1510nm

λλλλ=1590nm-90

-80

-70

-60

1300 1500 1700Wavelength(nm)

Pow

er(

dBm

)

1590nm1510nm

1550nm

Figure 9.18 - Characterisation of a half-cycle coupler a three different wavelengths

(λ0=1510nm, λ1=1550nm and λ2=1590nm) using an asymmetric perturbation. The inset

represents the measured spectral response of the characterised coupler and the markers

correspondtothecharacterisedwavelengths.

FromthisFigureitcanberealisedthatforahalf-cyclecouplerthemaximumofthe

powerchange inPort#2due toanasymmetricperturbationof thecoupler remains

the same for different test wavelengths. The difference in the magnitude of the

perturbation at the different wavelengths is due to differences in the tunable laser

outputpoweratthethreewavelengths.

9.5.2 Characterisationofafull-cyclecoupler[φ(L)=2π]

AsshowninFigure9.8,theasymmetricperturbationofafull-cyclecouplerhastwo

maximathatcorrespondtothe50-50%powerpointsofthecoupler.Thefabricated

coupler was characterised using both the symmetric perturbation and asymmetric

perturbation.Asforthecaseofthehalf-cyclecoupler,thecouplingprofileobtained

from the symmetric perturbation was used to fit theoretically the asymmetric

perturbation response. The experimental and theoretical results are in good

agreement (Figure 9.19). The symmetric perturbation resulted in a very weak and

thereforenoisyafteramplificationsignal.Theexperimentalasymmetricperturbation

hastwopointswherethepoweroftheperturbationisamaximum.However,thereis


a slight difference in the height of the two peaks accompanied by a variation of

symmetric-perturbation signal.This canbedue to a small variationof the coupler

waist outer diameter or a slight waist twist. A small misalignment between the

coupler waist and the scanning CO2 could also produce similar asymmetries. The

asymmetric perturbation was fitted assuming a linear variation of 5% in the

asymmetriccouplingcoefficientfromwaistendtoend.Themeanvalueisassumed

tobekeo=2.3×10-4µm-1.

-1

-0.5

0

0.5

1

1.5

2

-48 -32 -16 0 16 32 48Couplerposition(mm)

Per

turb

atio

n(V

)

Sym.perturbationexperimentaldata

Asym.perturbationexperimentaldata

Asym.perturbationtheoreticalfit

Figure 9.19 - Characterisation of a 2π coupler using the symmetric and asymmetric

perturbation. The asymmetric perturbation was fitted using the coupling profile retrieved

fromthesymmetricperturbationdata.

In order to prove that asymmetric perturbation of the coupler follows the

coupling profile for weak perturbations (keo very small) and follows the power

distributioninthecouplerforlargekeo,asmentionedinsection9.4.1,a2πcoupler

wascharacterisedusingdifferentCO2laserpowers(Figure9.20).Thelaseroutput

powersusedwere30mW,42mWand96mW.Theactualpowerthathitsthefibreis

much lower, given approximately by the ratio (≈7.5×10-3) of outer waist diameter

(≈30µm)overtheunfocusedlaserspotsize(≈4mm).Toreducethespotsizeofthe

CO2 laser and increase the resolution of the method, a 1mm aperture was used,

reducingthepowerhittingthecouplerto1.87×10-3oftheoutputpower.


FromFigure9.20,foraCO2laserpowerof30mWtheasymmetricperturbation

seemstofollowthecouplingprofileofthestructureandnomaxima(50-50%points)

areobserved.Thissituationcorrespondstoregion-1inFigure9.6.Byincreasingthe

power to42mW,an intermediate response isobservedwhere the twoperturbation

maximastartarisingandthecouplingprofileeffectisstrongerduetotheincreaseof

the(kee-koo)coefficientaswell.Atthispowerthemagnitudeofthecoefficients(kee-

koo) andkeo2 is comparable (corresponding to region-2ofFigure9.6).For slightly

largerpowersof theCO2 laser(96mW),thekeocoefficient ispredominantandthe

powerdistributioninthecouplerisfollowed(region-3ofFigure9.6).Thecorrection

inthepositionofthe50-50%pointsofthecouplerinrelationtothemaximaofthe

asymmetric perturbation due to a phase detuning of the coupler and the (kee-koo)

coefficientcanbedeterminedusingexpression(9.19).

7

10

13

16

0 10 20 30 40 50 60 70CouplerPosition(mm)

Per

turb

atio

n(m

V)

CO2power42mW

CO2power96mW

CO2power30mW

Figure 9.20 - Characterisation of a 2π coupler using the asymmetric perturbation for

differentpowersoftheCO2laser.

Itisalsorealisedthatusingtheasymmetricconfiguration,thereisathresholdin

the CO2 laser power in order to track the power distribution of the coupler and

identifythe50-50%positions.


9.5.3 Characterisationofacomplexnon-uniformπcoupler

Acomplexnon-uniformcouplerwithathreeinteractionregionswithlength10mm

each was fabricated using the flame brush technique. The theoretical coupling

profileofthestructureissimilartotheoneshowninFigure9.11.However,thereal

couplerhastransitiontapersbetweeneachofthethreeregionsandthewidthofthe

burnerflame(approximately4mm)wouldhavesomeinfluenceontheshapeofthe

real structure averaging out the profile. Both symmetric and asymmetric coupler

perturbations were carried out. The power oscillations due to the symmetric

perturbation are very weak giving a very noisy signal. However the result for the

symmetricperturbation(Figure9.21)followsthecouplingprofileofthetheoretical

structurewithtwocouplingregionsandaregionwithlowcouplingstrengthinthe

middle.Theprofilemaybedistortedduetoaveragingoftheidealprofilebythesize

oftheflame,bynoisewhilecharacterisingthestructureandalsobyamisalignment

of theCO2 laserpositionalong thecoupler.Theasymmetricperturbationwasalso

characterisedbyrotatingthefibreby90o.TheresultillustratedinFigure9.21shows

anincreaseoftheperturbationuntiltheuncoupledregionandthenadecreaseinthe

second coupling region. The slight tilt in the perturbation is probably due to a

changeinkeoalongthecoupler.However,whencomparedtothetheoreticalresults

showninFigure9.11,theexperimentaldataareinverygoodagreement.

-0.5

-0.3

-0.1

0.1

0.3

0 12.5 25 37.5 50 62.5 75Couplerposition(mm)

Per

turb

atio

n(V

)

Symmetric

Asymmetric

Figure 9.21 - Experimental characterisation of a complexnon-uniformπ coupler using a

symmetricandasymmetricperturbation.


It is thereforeconcludedthatboththeasymmetricandsymmetricperturbation

configurations can be used to identify the central region of the non-uniform

couplers. This method can be employed when writing gratings in non-uniform

couplers so that thegrating iswritten in the correctposition, avoiding the tapered

regionsthatdegradetheperformanceofthesedevicesasdiscussedinchapter8.

9.6 Summary

Afulldescriptionofamethodofnon-destructivelycharacterisinguniformandnon-

uniform fibre couplerswasdescribed.Themethodconsists inperturbing locally a

fibrecouplerusingaCO2laserradiationorotherradiationwithapenetrationlength

closetothecouplerdiameter.Byinducingasymmetricperturbationwithrespectto

the two lowest order waist eigenmodes, useful information about the taper profile

and uniformity of the coupler waist can be obtained. By inducing an asymmetric

perturbation,ontheotherhand,thepowerevolutionalongtheentirecouplingregion

canbefollowed.Additional informationmaybeobtainedbymeasuring theoutput

electricfieldphaseinthecaseoftheasymmetricandsymmetricperturbations.The

method can used for the optimisation of add/drop multiplexers based on different

couplerstructureswithinscribedgratings.Itcanalsobeusedinindustrialfacilities

fortheidentificationoferrorsandoptimisationofthefabricationprocedureoffibre

couplers (powersplittersorWDMcouplers)by thesuitablecharacterisationof the

devices. This method was initially developed with the purpose of optimising the

performance of the add-drop multiplexers discussed in chapter 8. Experimentally,

the CO2 laser was integrated with the grating writing system in order to first

characterise the couplers and subsequently inscribe the gratings at the correct

positioninordertooptimisetheadd-dropperformance.However,hydrogenloading

canchange thecoupler resonancewavelengthbymore than100nmandwhen it is

characterised themeasured50-50%positions shouldbe corrected.For an accurate

correctionof thesepositions, theknowledgeof theexactcouplerprofile shouldbe


known.Anotherdrawbackistheacceleratedhydrogenout-diffusingduetotheheat

generatedbytheincidentCO2radiationthatinducesfurtherchangestothecoupler

andmakingthemeasurementlessreliable.Therefore,ideally,thecouplersshouldbe

characterisedbeforetheprocessofincreasingitsphotosensitivity.

10

Summary

Theincreasingdemandforhigh-speedcommunicationshasledtothewidespreadof

WDMLAN,METRO,andlong-haulnetworks.InLANandMETROinparticular,

duetothenumberofopticalnodesinvolved,cost-effectivesolutionsareimportant

toassurecompetitive services.Keycomponentsof theseopticalnetworks include;

equalised EDFA that amplify uniformly the optical signals, add-drop multiplexers

thatareusedtorouteselectedchannelstodifferentlocationsatstrategicpointsalong

thenetwork, and fibre couplers that areused to monitor thenetwork, split optical

signals or provide pump/signal discriminators when launching into EDFAs. The

contentofthisthesiswasaimedmainlyatthesetwotechnologies,essentialforthe

deploymentofWDMnetworks.

10–Summary 225

10.1 EDFAgainequalisation

Design of ideal filters for EDFA gain equalisation can reduce the number of

amplifiers needed in the optical network by: compensating for insertion losses,

increasing theamplifiergain,and increasing theamplifierbandwidth. In thiswork

theoreticaldesignandnumericalmodellingofidealfiltersfortheequalisationofthe

EDFAgainspectrumwerestudied,givingamethodofdetermining the ideal filter

shapesandoptimumpositionintheamplifierinordertoobtainthebestperformance

whileequalisingtheEDFAgainspectrum.Thesefiltersareoriginallydeterminedas

an ideal wavelength-dependent background loss and are integrated in one or two

filters, placed within the EDFA. The performance of both these configurations is

compared.Furthermore,theperformanceoffilters,designedtobeplacedbothinside

andoutsidetheEDFA,wasalsoconsideredandcompared.Itwasalsodemonstrated

thatfollowingourdesignprocedure,thegain-flatteningfiltercouldberedesignedto

compensateforitsown,aswellas,otherdevices’ insertionloss.Withthismethodit

wasshownthatfilterswithinsertionlossupto8dBcouldbeusedwithoutpenaltyin

the amplifier gain. Such filter designs can have huge impact on the number of

amplifiersusedinalinkandthesystempowerbudget.

The saturation of the EDFA varies with the several parameters including the

input signal. Dynamic equalisation is needed to adjust the filter shape in order to

equalise the EDFA gain spectrum for different saturations. An alternative AO

tunable filter design was demonstrated as a means of achieving dynamic gain

equalisation with reduced tuning parameters comparing to previous design. The

controlofthewaistradiusofafibre-taperwasdemonstratedtobeanewmeansof

tailoring the loss spectrum of AO filters. Tunable filters can be designed to have

faster reshaping algorithms and ideal spectral shapes using the theoretical method

fordesigningidealfilters,inordertoequalisetheEDFAgainspectrumwiththebest

performance.

10–Summary 226

10.2 Add-dropmultiplexers

The ability to route selected wavelength channels at different locations along the

opticallinkisessentialforthedesignofefficientWDMnetworks.Thedemandfor

cheapadd-dropmultiplexerstoperformtheseoperationshasledtotheinvestigation

ofcompactall-fibreconfigurationsbasedonBragggratingsinscribedinthewaistof

fibre couplers. Optimisation parameters of three different designs are discussed in

this work with emphasis on a novel device based on a non-uniform half-cycle

coupler with a grating inscribed in its waist. This device is demonstrated

experimentallytopotentiallysatisfyDWDMstandardswhenfullyoptimised.

Thesuitablecharacterisationoffibrecouplersisimportantforthedetermination

ofthecorrectposition,wherethegrating,shouldbeinscribed,withinthecoupler.A

novel non-destructive coupler characterisation technique was developed in this

work.Allthreeadd-dropconfigurationsanalysedinthisworkcanbenefitfromthis

techniquebothbydeterminationofthecouplerstrengthprofileandpowerevolution

andbyusing themethodasanassessment tool tooptimise thecoupler fabrication

procedure.Finally,onitsownthistechniqueisanimportanttooltoassesserrorsin

couplerfabricationprocessesingeneral.

10.3 FutureWork

The work on the design of ideal filters for EDFA gain equalisation could be

followedbyexperimentalverification.Thiswould involve themeasurementof the

EDFparametersandthedesignoffiltersforthegivenamplifierusingshort-aswell

as long-period grating technology or AO technology. The impact of the EDFA

inhomogeneousbroadeningontheperformanceofthesefilterscouldbeaddressedas

well.

Following the work on the add-drop multiplexer it will be interesting to

integratethecouplercharacterisationtechniquewithagoodqualitygratingwriting

setupinordertohavefullcontrolonthepositionthegratingsareplacedwithinthe

10–Summary 227

couplers. Optimisation of the fabrication of non-uniform couplers to increase the

centrallengthandhavebettercontrolonthetaperedregionshapecouldbeachieved

byfabricatingthecouplersusingafocusedCO2laserbeam.

APPENDIXA

Add-droprequirements

Thetablepresentedinthisappendixrefertochapters3and8.Itgivesvaluesforthe

performancerequirementsofadd-dropmultiplexerdevices.

AppendixA 229

ThescatteringparameterSijrepresentsthespecifiedoperationwherethesubscriptj

referstotheportwherelightislaunchedanditotheportwhereitisreceived.The

designationsoftheadd-dropportsare:Inputport–1;Outputport–2;Addport–3;

Dropport–4.

Valuesforthedeviceisolation[54]:

Type Parameterspecification 50GHzSpacing

100GHzSpacing

200GHzSpacing

S21 Through-portisolationofdrop-channel(dB) 20 30 40

S43 Drop-portisolationofaddchannels(dB) 20 30 40

TableA1–Add-dropmultiplexercharacteristicsforsystemswithdifferentchannelspacing

usingcurrentfiltertechnology.

Valuesthedevicebackreflections:

S11back-reflection<-20dB

S33back-reflection<-20dB

Valueforthedeviceinsertionloss:<0.5dB

Valueforthedevicecross-talk:<-20dB

AppendixB

Fielddistributions

Thefiguresillustratedinthisappendixrefertochapter6.

AppendixB 231

-0.02

0

0.02

0.04

0.06

0.08

0 20 40 60Radius(µm)

Fie

lda

mpl

itude

(a.

u.) LP01

LP11LP12LP13

V=0.1

-0.02

0

0.02

0.04

0.06

0.08


Fiel

dam

plitu

de(a

.u.) LP01

LP11LP12LP13

V=1

-0.02

0

0.02

0.04

0.06

0.08


Fiel

dam

plitu

de(a

.u.) LP01

LP11LP12LP13

V=2

-0.04

-0.02

0

0.02

0.04

0.06

0.08


Fiel

dam

plitu

de(a

.u.) LP01

LP11LP12LP13

V=3

Figure B1 – Normalised field distribution of the fundamental mode LP01 and three low-

ordercladdingmodesLP11,LP12andLP13fordifferentvaluesoftheVnumber.

0

0.02

0.04

0.06

0.08


Fiel

dam

plitu

de(a

.u.) V=0.1

V=1V=2V=3

LP01

0

0.02

0.04

0.06


Fiel

dam

plitu

de(a

.u.) V=0.1

V=1V=2V=3

LP11

-0.02

0

0.02

0.04


Fiel

dam

plitu

de(a

.u.) V=0.1

V=1V=2V=3

LP12

-0.04

-0.02

0

0.02

0.04


Fiel

dam

plitu

de(a

.u.)

V=0.1V=1V=2V=3

LP13

FigureB2–Evolutionof fielddistributionsof the fundamentalmodeLP01 and theLP11,

LP12andLP13claddingmodeswiththeVparameter.

AppendixC

NumericalSimulations

The figures illustrated in this appendix are simulations concerning the design of

idealfiltersfortheEDFAgainequalisation,studiedinchapter7.

AppendixC 233

-60

-45

-30

-15

0 0.5 1 1.5 2 2.5 3EDFposition(m)

Pow

er(d

Bm

)

a)

λλλλ=1532.3nm

ForwardASEBackwardASE

Ins.Loss=8dBIns.Loss=0

0

5

10

15

20

25

30

35

0 0.5 1 1.5 2 2.5 3EDFposition(m)

Sig

nalg

ain

(dB

)

d)

Ins.Loss=8dB

Ins.Loss=0

0.5 1 2 4 80

λλλλ=1532.3nm

-60

-45

-30

-15

0 0.5 1 1.5 2 2.5 3EDFposition(m)

Pow

er(d

Bm

)

b)

λλλλ=1539.4nm



0

5

10

15

20

25

30

35

0 0.5 1 1.5 2 2.5 3EDFposition(m)

Sig

nalg

ain

(dB

)

e)

Ins.Loss=8dB

Ins.Loss=0

λλλλ=1539.4nm

-60

-45

-30

-15

0 0.5 1 1.5 2 2.5 3EDFposition(m)

Pow

er(d

Bm

)

c)

λλλλ=1550.7nm



0

5

10

15

20

25

30

35

0 0.5 1 1.5 2 2.5 3EDFposition(m)

Sig

nalg

ain

(dB

)

f)


λλλλ=1550.7nm

FigureC1–EDFAsignalgain and forwardASEbuild-upalong the amplifier length for

different filter insertion losses. The filters were placed at the optimum position. a) ASE

powerat1532.3nm.b)ASEpowerat1539.4nm.c)ASEpowerat1550.7nm.d)EDFAgain

at1532.3nm.e)EDFAgainat1539.4nm.f)EDFAgainat1550.7nm.

AppendixC 234

0

5

10

15

20

25

30


Gai

n(d

B)

a)

Ins.Loss=8dB

Ins.Loss=0

λλλλ=1532.3nm

0

5

10

15

20

25

30


Gai

n(d

B)

λλλλ=1539.4nm

b) Ins.Loss=0

Ins.Loss=8dB

0

5

10

15

20

25

30


Gai

n(d

B)

λλλλ=1550.7nm

c)

Ins.Loss=8dB

Ins.Loss=0

FigureC2–EDFAsignalgainbuild-upalongtheamplifierlengthforfilterinsertionlosses

of0dB,0.5dB,1dB,2dB,4dB,8dB.Thefilterscorrectedfortheinsertionlossandplacedat

the optimum position. a) Signal gain at 1532.3nm. b) Signal gain at 1539.4nm. c) Signal

gainat1550.7nm.

AppendixC 235

0

5

10

15

20

25

30


Sig

nalg

ain

(dB

)a)

Ins.Loss=8dB

Ins.Loss=0λλλλ=1532.3nm

-55

-50

-45

-40

-35

-30

-25

-20

-15

0 1 2 3

d)

l Ins=8dB

l Ins=0

λλλλ=1532.3nm

l Ins=0

ASE_F

ASE_B

ASE_B

ASE_F

0

5

10

15

20

25

30


Sig

nalg

ain

(dB

)

b)

Ins.Loss=8dB

Ins.Loss=0

λλλλ=1539.4nm

-55

-50

-45

-40

-35

-30

-25

-20


Pow

er(d

Bm

)

e)

l Ins=8dB

l Ins=0

λλλλ=1539.4nm

l Ins=0

ASE_F

ASE_B

ASE_B

ASE_F

0

5

10

15

20

25

30


Sig

nalg

ain

(dB

)

c)

Ins.Loss=8dB

Ins.Loss=0

λλλλ=1550.7nm

-55

-50

-45

-40

-35

-30

-25

-20


Pow

er(d

Bm

)

f)

l Ins=8dB

l Ins=0

λλλλ=1550.7nm

l Ins=0

ASE_F

ASE_B

ASE_B

ASE_F

Figure C3 – Signal gain spectra along the EDFA length for different wavelengths. The

equalisingfilterswithinsertionlossesof0,0.5,1,2,4,and8dBwereplacedattheoptimum

positionsanda2dB lump losswas insertedatZ=2.0m.Left: a)Gainatλ1=1532.3nm.b)

Gainatλ2=1539.4nm.c)Gainatλ3=1550.7nm.Right:d)ASEpoweratλ1=1532.3nm.e)

ASEpoweratλ2=1539.4nm.f)ASEpoweratλ3=1550.7nm.

AppendixC 236

0

10

20

30

40

50


Po

wer

(m

W)

a)

l Ins =8dB

l Ins =0

0.4

0.5

0.6

0.7

0.8

0.9

1


Inve

rsio

nn

2/n

t(a.

u.) b)

l Ins =8dB

l Ins =0

FigureC4–a)980nmPumppoweralongtheEDFAlength.b)Populationinversionalong

the EDFA length. Equalising filters with different insertion losses where placed at the

optimumposition,aroundZ=1.2mandalumplossof2dBwasinsertedatZ=2.0m.

0

5

10

15

20

25


Sig

nalg

ain

(dB

)

a)

Ins.Loss=8dB

Ins.Loss=0

λλλλ=1532.3nm-55

-50

-45

-40

-35

-30

-25

-20

-15

0 1 2 3EDFPosition(m)

Pow

er(d

Bm

)

d)

l Ins=8dB

l Ins=0

λλλλ=1532.3nm

l Ins=0

ASE_F

ASE_B

ASE_B

ASE_F

0

5

10

15

20

25


Sig

nalg

ain

(dB

)

b)

Ins.Loss=8dB

Ins.Loss=0

λλλλ=1539.4nm-55

-50

-45

-40

-35

-30

-25

-20


Pow

er(d

Bm

)

e)

l Ins=8dB

l Ins=0

λλλλ=1539.4nm

l Ins=0

ASE_F

ASE_B

ASE_B

ASE_F

AppendixC 237

0

5

10

15

20

25


Sig

nalg

ain

(dB

)c)

Ins.Loss=8dB

Ins.Loss=0

λλλλ=1550.7nm-55

-50

-45

-40

-35

-30

-25

-20


Pow

er(d

Bm

)

f)

l Ins=8dB

l Ins=0

λλλλ=1550.7nm

l Ins=0

ASE_F

ASE_B

ASE_B

ASE_F

Figure C5 – Signal gain spectra along the EDFA length for different wavelengths. The

equalisingfilterswereplacedattheoptimumpositionsanda2dBlumplosswasinsertedat

Z=2.0m.Left:a)Gainatλ1=1532.3nm.b)Gainatλ2=1539.4nm.c)Gainatλ3=1550.7nm.

Right: d) ASE power at λ1=1532.3nm. e) ASE power at λ2=1539.4nm. f) ASE power at

λ3=1550.7nm.

0

10

20

30

40

50


Po

wer

(m

W)

a)

l Ins =8dB

l Ins =00.4

0.5

0.6

0.7

0.8

0.9

1


Inve

rsio

nn

2/n

t(a.

u.) b)

l Ins =8dB

l Ins =0



optimumposition,aroundZ=0.8mandalumplossof2dBwasinsertedatZ=2.0m.

AppendixC 238

0

5

10

15

20

25


Sig

nalg

ain

(dB

)a)

Ins.Loss=4dB

Ins.Loss=0

λλλλ=1532.3nm-55

-50

-45

-40

-35

-30

-25

-20


Pow

er(d

Bm

)

d)

l Ins=4dB

l Ins=0

λλλλ=1532.3nm

l Ins=0

ASE_F

ASE_B

ASE_B

ASE_F

0

5

10

15

20

25


Sig

nalg

ain

(dB

)

b)

Ins.Loss=4dB

Ins.Loss=0

λλλλ=1539.4nm-55

-50

-45

-40

-35

-30

-25

-20


Pow

er(d

Bm

)

e)

l Ins=4dB

l Ins=0

λλλλ=1539.4nm

l Ins=0

ASE_F

ASE_B

ASE_B

ASE_F

0

5

10

15

20

25


Sig

nalg

ain

(dB

)

c)

Ins.Loss=4dB

Ins.Loss=0

λλλλ=1550.7nm-55

-50

-45

-40

-35

-30

-25

-20


Pow

er(d

Bm

)

f)

l Ins=4dB

l Ins=0

λλλλ=1550.7nm

l Ins=0

ASE_F

ASE_B

ASE_B

ASE_F

FigureC7–Signalgain,backwardASEandForwardASEspectraalongtheEDFAlength

fordifferentwavelengths.Theequalisingfilterswereplacedat theoptimumpositionsand

an isolator with 2dB of insertion loss and 30dB of isolation inserted at Z=1.0m. Left: a)

Gain at λ1=1532.3nm. b) Gain at λ2=1539.4nm. c) Gain at λ3=1550.7nm. Right: d) ASE

poweratλ1=1532.3nm.e)ASEpoweratλ2=1539.4nm.f)ASEpoweratλ3=1550.7nm.

AppendixC 239

0

10

20

30

40

50


Pow

er(m

W)

a)

l Ins =4dB

l Ins =0

0.4

0.5

0.6

0.7

0.8

0.9

1


Inve

rsio

nn

2/n

t(a.

u.) b)

l Ins =4dB

l Ins =0



optimumposition,aroundZ=1.2mandan isolatorwith2dBof insertion lossand30dBof

isolationwasplacedatZ=1.0m.

AppendixD

Add-Dropsimulations

Thefiguresillustratedinthisappendixrefertochapter8.

AppendixD 241

-60

-40

-20

0

0 0.5 1 1.5 2 2.5Gratinglengtherror(mm)

Bac

k-R

efle

ctio

n(d

B)

Figure D1 – Back-reflected light at the centre wavelength in a 30mm long full-cycle

coupler with a uniform grating in its waist as a function of the error in the used grating

length.

0

0.02

0.04

0.06

0.08

0.1

10 12 14 16 18 20CouplerRadius(µµµµm)

Ove

rlap

diff

eren

ce(a

.u.)

FigureD2–a)Absolutevalueofthedifferencebetween theoverlap integralbetween the

evenandoddcouplereigenmodeselectric-fieldpowerandthephotosensitiveregionsofthe

coupler.

AppendixD 242

FigureD3–Crosssectionandpowerdistributionoftheevenandoddcouplereigenmodes

for a radius of 16µm and a photosensitive area of radius 1.5µm. a) Even mode. b) Odd

mode. c) Cross section of the coupler. The red areas indicate the photosensitive residual

cores.

AppendixD 243

0

20

40

60

0.E+0 2.E-6 4.E-6 6.E-6 8.E-6 1.E-5Effectiveindexdifference(∆∆∆∆neffe-∆∆∆∆neffo)

Wav

elen

gth

detu

ning

(nm

)

FigureD4-Detuningoftheresonancewavelengthofa30mmlonghalf-cyclenon-uniform

couplerduetothedifferencebetweentheevenandoddeigenmodeseffectiveindexchange

duetotheinscriptionofa8mmlonggratinginthecouplerwaist.

Figure D5 – Measurement of the coupler spectral response with a white light source. a)

Afterfabrication.b)Afterhydrogenloading+exposuretoUVradiation.

References 244

References

[1] S.S.Wagner andK.L.Lemberg, "Technologyand system issues foraWDM-based

fiberloop,"JournalofLightwaveTechnology,vol.7,pp.1759-1768,1989.

[2] T.Ono,"40Gb/s-basedlargecapacityDWDMtransmission,"presentedatOFC,2001.

[3] M. Nakazawa, "Solitons for breaking barriers to terabit/second WDM and OTDM

transmission in the next millennium," IEEE SelectedTopics in Quantum Electronics,

vol.6,pp.1332-1343,2000.

[4] E. Desurvire, J. R. Simpson, and P. C. Becker, "High-gain erbium-doped traveling-

wavefiberamplifier,"OpticsLetters,vol.12,pp.888-890,1987.

[5] R.J.Mears,L.Reekie,M.Jauncey,andD.N.Payne,"Low-noiseerbium-dopedfibre

amplifieroperatingat1.54µm,"ElectronicsLetters,vol.26,pp.1026-1028,1987.

[6] M. Tachibana, R. I. Laming, P. R. Morkel, and D. N. Payne, "Erbium-doped fiber

amplifier with flattened gain spectrum," IEEE Photonics Technology Letters, vol. 3,

1991.

[7] T.Sakamoto,"S-bandfiberopticamplifier,"presentedatOFC,2001.

[8] P. C. Reeves-Hall, D. A. Chestnut, C. J. S. D. Matos, and J. R. Taylor, "Dual

wavelengthpumpedL-andU-bandRamanamplifier,"ElectronicsLetters,vol.37,pp.

883-884,2001.

[9] S. Namiki and Y. Emori, "Ultrabroad-Band Raman Amplifiers Pumped and Gain-

Equalized by Wavelength-Divison-Multiplexed High-Power Laser Diodes," IEEE

SelectedTopicsinQuantumElectronics,vol.7,pp.3-16,2001.

[10]E.Desurvire,Erbium-dopedfiberamplifiers:Principlesandapplication.NewYork:J.

Wiley&Sons,1994.

[11]C. R. Giles and E. Desurvire, "Modeling erbium-doped fiber amplifiers," Journal of

LightwaveTechnology,vol.9,pp.271-283,1991.

[12]A. Buxens, H. N. Poulsen, A. T. Clausen, and P. Jeppesen, "Gain flattened L-band

EDFA based on upgraded C-band EDFA using forward ASE pumping in an EDF

section,"ElectronicsLetters,vol.36,pp.821-823,2000.

[13]T.KasamatsuandY.Yano,"Tm-dopedfiberamplifiersforS-band,"presentedatOFC,

2001.

References 245

[14]M. Yamada, T. Kanamori, Y. Terunuma, K. Oikawa, M. Shimizu, S. Sudo, and K.

Sagawa, "Fluoride-Based Erbium-Doped Fiber Amplifier with Inherently Flat Gain

Spectrum,"IEEEPhotonicsTechnologyLetters,vol.8,pp.882-884,1996.

[15]D.Bayart,B.Clesca,L.Hamon,andJ.L.Beylat,"Experimental Investigationof the

GainFlatnessCharacteristics for1.55mmErbium-DopedFluorideFiberAmplifiers,"

IEEEPhotonicsTechnologyLetters,vol.6,pp.613-615,1994.

[16]H.S.Kim,S.H.Yun,H.K.Kim,N.Park,andB.Y.Kim,"Dynamicgainequalization

oferbium-dopedfiberamplifierwithall-fiberacousto-optic tunablefilters,"presented

atOFCtechnicaldigest,1998.

[17]H. S. Kim, S. H. Yun, H. K. Park, and B. Y. Kim, "Actively gain-flattened erbium-

dopedfiberamplifierover35nmbyusingall-fiberacousto-optictunablefilters,"IEEE

PhotonicsTechnologyLetters,vol.10,pp.790-792,1998.

[18]R. Feced, C. Alegria, R. I. Laming, and M. N. Zervas, "Acoustooptic Attenuation

Filters Based on Tapered Optical Fibers," IEEE Selected Topics in Quantum

Electronics,vol.5,pp.1278-1288,1999.

[19]C.Alegria,R.Feced,M.N.Zervas,R. I.Laming,andS.G.Farwell,"Acousto-optic

filtersbasedonmulti-tapered fibre structures,"ElectronicsLetters,vol.35,pp.1006-

1007,1999.

[20]S.-K.LiawandY.-K.Chen,"PassiveGain-EqualisedWide-BandErbium-DopedFiber

AmplifierUsingSamarium-DopedFiber,"IEEEPhotonicsTechnologyLetters,vol.8,

pp.879-881,1996.

[21]P. D. Greene and H. N. Rourke, "Tailoring long period optical fibre gratings for

flatteningEDFAgainspectra,"ElectronicsLetters,vol.35,pp.1373-1374,1999.

[22]M.Rochette,M.Guy,S.Larochelle,J.Lauzon,andF.Trépanier,"Gainequalizationof

EDFA's with Bragg gratings," IEEE Photonics Technology Letters, vol. 11, pp. 536-

538,1999.

[23]P. F. Wysocki, J. B. Judkins, R. P. Espindola, M. Andrejco, and A. M. Vengsarkar,

"Broad-band erbium-doped fiber amplifier flattened beyond 40nm using long-period

gratingfilter,"IEEEPhotonicsTechnologyLetters,vol.9,pp.1343-1345,1997.

[24]J.E.FordandJ.A.Walker,"Dynamicspectralpowerequalizationusingmicro-opto-

mechanics,"IEEEPhotonicsTechnologyLetters,vol.10,pp.1440-1442,1998.

[25]P. M. Schiffer, C. R. Doerr, L. W. Stulz, M. A. Cappuzzo, E. J. Laskowski, A.

Paunescu,L.T.Gomez,andJ.V.Gates,"Smartdynamicwavelengthequalizerbased

References 246

onanintegratedplanaropticalcircuitforuseinthe1550-nmregion,"IEEEPhotonics

TechnologyLetters,vol.11,pp.1150-1152,1999.

[26]S. H. Yun, B. W. Lee, H. K. Kim, and B. Y. Kim, "Dynamic erbium-doped fiber

amplifierbasedonactivegainflatteningwithfiberacoustooptictunablefilters,"IEEE

PhotonicsTechnologyLetters,vol.11,pp.1229-1231,1999.

[27]S.K.Liaw,K.P.Ho,andS.Chi,"Dynamicpower-equalizedEDFAmodulebasedon

strain tunable fiberBragggratings," IEEEPhotonicsTechnologyLetters, vol. 11, pp.

797-799,1999.

[28]J.-M.Jouanno,D.Zauner,andM.Kristensen,"LowCrosstalkPlanarOpticalAdd-Drop

MultiplexerFabricatedwithUV-InducedBraggGratings,"ElectronicsLetters,vol.33,

pp.2120-2121,1997.

[29]J.Albert,F.Bilodeau,D.C.Johnson,K.O.Hill,K.Hattori,T.Kitagawa,Y.Hibino,

andM.Abe,"Low-LossPlanarLightwaveCircuitOADMwithHighIsolationandNo

Polarisation Dependence," IEEE Photonics Technology Letters, vol. 11, pp. 346-348,

1999.

[30]Y.Hibino,T.Kitagawa,K.O.Hill,F.Bilodeau,B.Malo,J.Albert,andD.C.Johnson,

"WavelengthDivisionMultiplexerwithPhotoinducedBraggGratingsFabricated in a

Planar-Lightwave-Circuit-Type Asymmetric Mach-Zehnder Interferometer on Si,"

IEEEPhotonicsTechnologyLetters,vol.8,pp.84-86,1996.

[31]T.Erdogan,T.A.Strasser,M.A.Milbrodt,E.J.Laskowski,C.H.Henry,andG.E.

Kohnke, "Integrated-opticalMach-Zehnderadd-drop filter fabricatedbya singleUV-

inducedgratingexposure,"AppliedOptics,vol.36,pp.7838-7845,1997.

[32]W.D.Zhong,S.Dods,J.P.R.Lacey,andR.S.Tucker,"Reconfigurablemultichannel

add-drop multiplexer with improved performance," Electronics Letters, vol. 32, pp.

1477-1478,1996.

[33]H. Takahashi, S. Susuki, K. Kato, and I. Nishi, "Arrayed-Waveguide Grating for

Wavelength Division Multi/Demultiplexer with Nanometre Resolution," Electronics

Letters,vol.26,pp.87-88,1990.

[34]K. Okamoto, M. Okuno, A. Himeno, and Y. Ohmori, "16-channel optical add/drop

multiplexer consisting of arrayed-waveguide gratings and double-gate switches,"

ElectronicsLetters,vol.32,pp.1471-1472,1996.

[35]C. R. Doerr, L. W. Stulz, J. Gates, M. Cappuzzo, E. Laskowski, L. Gomez, A.

Paunescu,A.White,andC.Narayanan,"ArrayedWaveguideLensWavelengthAdd-

DropinSilica,"IEEEPhotonicsTechnologyLetters,vol.11,pp.557-559,1999.

References 247

[36]C.R.Doerr,L.W.Stulz,M.Cappuzzo,E.Laskowski,A.Paunescu,L.Gomez,J.V.

Gates,S.Shunk,andA.E.White,"40-WavelengthAdd-DropFilter,"IEEEPhotonics

TechnologyLetters,vol.11,pp.1437-1439,1999.

[37]T. Mizuochi, T. Kitayama, K. Shimizu, and K. Ito, "Interferometric Crosstalk-Free

OpticalAdd/DropMultiplexerUsingMach-Zehnder-BasedFiberGratings,"Journalof

LightwaveTechnology,vol.16,pp.265-276,1998.

[38]L.Dong,P.Hua,T.A.Birks,L.Reekie,andP.S.J.Russell,"NovelAdd/DropFilters

for Wavelength-Division-Multiplexing Optical Fiber Systems Using a Bragg Grating

AssistedMismatchedCoupler,"IEEEPhotonicsTechnologyLetters,vol.8,pp.1656-

1658,1996.

[39]A.S.Kewitsch,G.A.Rakuljic,P.A.Willems,andA.Yariv,"All-fiberzero-insertion-

lossadd-dropfilterforwavelength-divisionmultiplexeing,"OpticsLetters,vol.23,pp.

106-108,1998.

[40]B.Ortega,L.Dong,andL.Reekie,"All-fiberopticaladd-dropmultiplexerbasedona

selectivefusedcouplerandasinglefiberBragggrating,"AppliedOptics,vol.37,pp.

7712-7717,1998.

[41]F. Bakhti, X. Daxhelet, P. Sansonetti, and S. Lacroix, "Influence of Bragg grating

locationinfused100%couplerforaddanddropmultiplexerrealization,"presentedat

OFC,1998.

[42]F.Bakhti,P.Sansonetti,C.Sinet,L.Gasca,L.Martineau,S.Lacroix,X.Daxhelet,and

F. Gonthier, "Optical add/drop multiplexer based on UV-written Bragg grating in a

fused100%coupler,"ElectronicsLetters,vol.33,pp.803-804,1997.

[43]S.Y.Kim,S.B.Lee,S.W.Kwon,S.S.Choi,andJ.Jeong,"Channel-switchingactive

add/dropmultiplexerwithtunablegratings,"ElectronicsLetters,vol.34,pp.104-105,

1998.

[44]J.T.Ahn,H.K.Lee,K.H.Kim,M.-Y.Jeon,D.S.Lim,andE.-H.Lee,"Astabilised

fibre-opticMach-Zehnder interferometer filterusingan independantstabilisation light

source,"OpticsCommunications,vol.157,pp.62-66,1998.

[45]S. Bethuys, L. Lablonde, L. Rivoallan, J. F. Bayon, L. Brilland, and E. Delavaque,

"Optical Add/Drop Multiplexer Based on UV-Written Bragg Gratings in Twincore

FibreMach-ZehnderInterferometer,"ElectronicsLetters,vol.34,pp.1250-1252,1998.

[46]S.Y.Kim,S.B.Lee,J.Chung,S.Y.Kim,I.J.Park,J.Jeong,andS.S.Choi,"Highly

Stable Optical Add/Drop Multiplexer Using Polarization Beam Splitters and Fiber

BraggGratings,"IEEEPhotonicsTechnologyLetters,vol.9,pp.1119-1121,1997.

References 248

[47]F.Bilodeau,D.C.Johnson,S.Thériault,B.Malo,J.Albert,andK.O.Hill,"AnAll-

Fiber Dense-Wavelength-Division Multiplexer/Demultiplexer Using Photoimprinted

BraggGratings,"IEEEPhotonicsTechnologyLetters,vol.7,pp.388-390,1995.

[48]R. Watanabe,K. Nosu, and Y. Fujii, "Optical Grating Multiplexer in the 1.1-1.5 µm

WavelengthRegion,"ElectronicsLetters,vol.16,pp.108-109,1980.

[49]J.SkaarandK.M.Risvik,"AGeneticAlgorithmfortheInverseProbleminSynthesis

ofFiberGratings,"JournalofLightwaveTechnology,vol.16,pp.1928-1932,1998.

[50]R.Feced,M.N.Zervas,andM.A.Muriel,"AnEfficientInverseScatteringAlgorithm

for the Design of Nonuniform Fiber Bragg Gratings," IEEE Journal of Quantum


[51]T. Erdogan, "Optical add-drop multiplexer based on an asymmetric Bragg coupler,"

OpticsCommunications,vol.157,pp.249-264,1998.

[52]A. Iocco, H. G. Limberger, R. P. Salathé, L. A. Everall, K. E. Chisholm, J. A. R.

Williams,andI.Bennion,"BraggGratingFastTunableFilterforWavelengthDivision

Multiplexing,"JournalofLightwaveTechnology,vol.17,pp.1217-1221,1999.

[53]B.J.Eggleton,"RecentAdvancesofTunableFiberGratingTechnologies,"presentedat

BGPP,2001.

[54]C. R. Giles and M. Spector, "The Wavelength Add/Drop Multiplexer for Lightwave

Communication Networks," The Bell System Technical Journal, vol. January-March,

pp.207-229,1999.

[55]G. E. Keiser, "A Review of WDM technologies and applications," Optical Fiber

Technology,vol.5,pp.3-39,1999.

[56]D.Marcuse,Theoryofdielectricopticalwaveguides,2ndEditioned:AcademicPress,

1991.

[57]A.W.Snyder,"Polarisingbeamsplitterfromfused-tapercouplers,"ElectronicsLetters,

vol.21,pp.623-625,1985.

[58]A. Lord, I. J. Wilkinson, A. Ellis, D. Cleland, R. A. Garnham, and W. A. Stallard,

"Comparison of WDM coupler technologies for use in erbium-doped fibre amplifier

systems,"ElectronicsLetters,vol.26,pp.900-901,1990.

[59]S. G. Farwell, "Fused Tapered Fibre Optic Devices," in PhD Thesis, Universtity of

Southampton,1997.

[60]J. Bures, S. Lacroix, C. Veilleux, and J. Lapierre, "Some particular properties of

monomodefusedfibercouplers,"AppliedOptics,vol.23,pp.968-969,1984.

References 249

[61]T.E.Dimmick,G.Kakarantzas,T.A.Birks,andP.S.J.Russell,"Carbondioxidelaser

fabricationoffused-fibercouplersandtapers,"AppliedOptics,vol.38,pp.6845-6848,

1999.

[62]T. A. Birks, W. J. Wadsworth, and P. S. J. Russell, "Supercontinuum generation in

taperedfibers,"OpticsLetters,vol.25,pp.1415-1417,2000.

[63]J. D. Love, M. W. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier,

"Teperedsingle-modefibresanddevicesPart1:Adiabaticitycriteria,"IEEProceedings-

J,vol.138,pp.343-354,1991.

[64]T. A. Birks and Y. W. Li, "The Shape of Fiber Tapers," Journal of Lightwave


[65]W. K. Burns, M. Abebe, and C. A. Villarruel, "Parabolic model for shape of fiber

taper,"AppliedOptics,vol.24,pp.2753-2755,1985.

[66]R.P.Kenny,T.A.Birks,andK.P.Oakley,"ControlofOpticalFibreTaperShape,"


[67]S. Lacroix, F. Gonthier, and J. Bures, "Modeling of symmstric 2 x 2 fused-fiber

couplers,"AppliedOptics,vol.33,pp.8361-8369,1994.

[68]J. Bures, S. Lacroix, and J. Lapierre, "Analyse d'un coupleur bidirectionnel à fibres

optiquesmonomodesfusionnées,"AppliedOptics,vol.22,pp.1918-1922,1983.

[69]F. P. Payne, C. D. Hussey, and M. S. Yataki, "Modelling Fused Single-Mode-Fibre

Couplers,"ElectronicsLetters,vol.21,pp.461-462,1985.

[70]A.W.SnyderandX.-H.Zheng,"Opticalfibersofarbitrarycrosssections,"Journalof

theOpticalSocietyofAmericaA,vol.3,pp.600-609,1986.

[71]A. W. Snyder and X.-H. Zheng, "Fused Couplers of Arbitrary Cross-Section,"


[72]F. Gonthier, S. Lacroix, and J. Bures, "Numerical Calculations of Modes of Optical

Waveguides with Two-Dimensional Refractive Index Profiles by a Field Correction

Method,"OpticalandQuantumElectronics,vol.26,pp.S135-S149,1994.

[73]C.-C. Su, "A Surface Integral Equations Method Homogeneous Optical Fibers and

CoupledImageLinesofArbitraryCrossSections,"IEEETransactionsonMicrowave

TheoryanTechniques,vol.MTT-33,pp.1114-1119,1985.

[74]T.-L. Wu and H.-C. Chang, "Rigorous Analysis of Form Birefringence of Weakly

FusedFiber-OpticCouplers,"JournalofLightwaveTechnology,vol.13,pp.687-691,

1995.

References 250

[75]S.-W. Yang and H.-C. Chang, "Numerical Modeling of Weakly Fused Fiber-Optic

Polarization Beamsplitters-Part I: Accurate Calculation of Coupling Coefficients and

FormBirefringence,"JournalofLightwaveTechnology,vol.16,pp.685-690,1998.

[76]T.Erdogan,"Cladding-moderesonancesinshort-andlong-periodfibergratingfilters,"

JournaloftheOpticalSocietyofAmericaA,vol.14,pp.1760-1773,1997.

[77]C.R.Giles,"LightwaveApplicationsofFiberBraggGratings,"JournalofLightwave


[78]T. Erdogan, "Fiber Grating Spectra," Journal of Lightwave Technology, vol. 15, pp.

1277-1294,1997.

[79]H. Kogelnik, "Filter Response of Nonuniform Almost-Periodic Structures," The Bell

SystemTechnicalJournal,vol.55,pp.109-126,1976.

[80]A.YarivandP.Yeh,Opticalwavesincrystals:JohnWiley&SonsInc,1984.

[81]A.Yariv,"Coupled-ModeTheoryforGuided-WaveOptics,"IEEEJournalofQuantum

Electronics,vol.QE-9,pp.919-933,1973.

[82]M.Ibsen,R.Feced,P.Petropoulos,andM.N.Zervas,"99.9%Reflectivitydispersion-

lesssquare-filter fibreBragggratings forhighspeedDWDMnetworks,"presentedat

OFC,Baltimore,2000.

[83]J.Skaar,"SynthesisandCharacterizationofFibreBraggGratings,"PhDThesis,2000.

[84]P.J.Lemaire,R.M.Atkins,V.Mizrahi,andW.A.Reed,"HighPressureH2Loadingas

aTechnique forAchievingUltrahighUVPhotosensitivity andThermalSensitivity in

GeO2DopedOpticalFibres,"ElectronicsLetters,vol.29,pp.1191-1193,1993.

[85]F.Bilodeau,B.Malo,J.Albert,D.C.Johnson,K.O.Hill,Y.Hibino,M.Abe,andM.

Kawachi, "Photosensitization of Optical Fiber and Silica-on-Silicon/Silica

Waveguides,"OpticsLetters,vol.18,pp.953-955,1993.

[86]J. Stone, "Interactions of Hydrogen and Deuterium with Silica Optical Fibers: A

Review,"JournalofLightwaveTechnology,vol.LT-5,pp.712-733,1987.

[87]A. Iino,M.Kuwabara,andK.Kokura,"MechanismsofHydrogen-InducedLosses in

Silica-BasedOpticalFibers,"JournalofLightwaveTechnology,vol.8,pp.1675-1679,

1990.

[88]J.Canning,"PhotosensitizationandPhotostabilizationofLaser-InducedIndexChanges

inOpticalFibres,"OpticalFiberTechnology,vol.6,pp.275-289,2000.

[89]M. Berwick, C. N. Pannel, P. S. J. Russell, and D. A. Jackson, "Demonstration of

birefringent optical fibre frequency shifter employing torsional acoustic waves,"


References 251

[90]H. Sabert, L. Dong, and P. S. J. Russell, "Versatile acousto-optic flexural wave

modulator, filter and frequency shifter in dual-core fiber," International Journal of

Optoelectronics,vol.7,pp.189-194,1992.

[91]T.A.Birks,D.O.Culverhouse,S.G.Farwell,andP.S.J.Russell,"2X2Single-mode

fiberroutingswitch,"OpticsLetters,vol.21,pp.722-724,1996.

[92]T. A. Birks, S. G. Farwell, P. S. J. Russell, and C. N. Pannell, "Four-port fiber

frequency shifter with a null taper coupler," Optics Letters, vol. 19, pp. 1964-1966,

1994.

[93]T.A.Birks,P.S.J.Russell,andC.N.Pannell,"Lowpoweracousto-opticdevicebased

ona taperedsingle-modefiber,"IEEEPhotonicsTechnologyLetters,vol.6,pp.725-

727,1994.

[94]W.P.Risk,G.S.Kino,andH.J.Shaw,"Fiber-opticfrequencyshifterusingasurface

acousticwaveincidentatanobliqueangle,"OpticsLetters,vol.11,pp.115-117,1986.

[95]D.O.Culverhouse,S.G.Farwell,T.A.Birks,andP.S. J.Russell,"Four-port fused

taper acousto-optic devices using standard singlemode Telecomunications fibre,"


[96]S.G.Farwell,D.O.Culverhouse,T.A.Birks,andP.S.J.Russell,"Low-lossall-fibre

amplitudemodulatorat1.55mm,"ElectronicsLetters,vol.32,pp.577-578,1996.

[97]A.M.Vengsarkar,J.R.Pedrazzani,J.B.Judkins,P.J.Lemaire,N.S.Bergano,andC.

R.Davidson,"Long-periodfiber-grating-basedgainequalisers,"OpticsLetters,vol.21,

pp.336-338,1996.

[98]H.S.Kim,S.H.Yun, I.K.Kwang,andB.Y.Kim,"All-fiberacousto-optic tunable

notchfilterwithelectronicallycontrollablespectralprofile,"OpticsLetters,vol.22,pp.

1476-1478,1997.

[99]H. E. Engan, B. Y. Kim, J. N. Blake, and H. J. Shaw, "Propagation and optical

interactionofguidedacousticwavesintwo-modeopticalfibres,"JournalofLightwave


[100] T.A.Birks,P.S.J.Russell,andD.O.Culverhouse,"Theacousto-opticeffectin

single-modefibertapersandcouplers,"JournalofLightwaveTechnology,vol.14,pp.

2519-2529,1996.

[101] R.N.Thurston, "Elasticwaves in rodsandclad rods,"Journalof theAcoustical

SocietyofAmerica,vol.64,pp.1-37,1978.

References 252

[102] J. N. Blake, B. Y. Kim, H. E. Engan, and H. J. Shaw, "Analysis of intermodal

coupling in a two-mode fibre with periodic microbends," Optics Letters, vol. 12, pp.

281-283,1987.

[103] H. F. Taylor, "Bending Effects in Optical Fibers," Journal of Lightwave

Technology,vol.LT-2,pp.617-628,1984.

[104] H. Zech, "Theoretical investigation of the gain profile of erbium-doped fiber

amplifiers,"OpticalFiberTechnology,vol.1,pp.327-330,1995.

[105] H.Zech,"MeasurementTechniquefortheQuotientofCrossSectionsσe(λs)/σa(λs)

of Erbium-Doped Fibers," IEEE Photonics Technology Letters, vol. 7, pp. 986-988,

1995.

[106] W. L. Barnes, R. I. Laming, E. J. Tarbox, and P. R. Morkel, "Absorption and

Emission Cross Section of Er3+ Doped Silica Fibers," IEEE Journal of Quantum


[107] R. E. Tench and M. Shimizu, "Fluorescence-Based Measurement of g*(l) for

Erbium-DopedFluorideFiberAmplifiers,"JournalofLightwaveTechnology,vol.15,

pp.1559-1564,1997.

[108] M. N. Zervas, R. I. Laming, and D. N. Payne, "Efficient Erbium-doped fiber

amplifiers incorporating an optical isolator," IEEE Journal of Quantum Electronics,

vol.11,pp.472-480,1995.

[109] J.H.Povlsen,A.Bjarklev,O.Lumholt,H.Vendeltorp-Pommer,K.Rottwitt,and

T.Rasmussen,"OptimizinggainandnoiseperformanceofEDFA'swithinsertionofa

filteroranisolator,"SPIEFiberLaserSourcesandAmplifiersIII,vol.1581,pp.107-

113,1991.

[110] C. Riziotis and M. N. Zervas, "Design Considerations in Optical Add/Drop

Multiplexers Based on Grating-Assited Null Couplers," Journal of Lightwave


[111] I.Baumann,J.Seifert,W.Nowak,andM.Sauer,"CompactAll-FiberAdd-Drop-

MultiplexerUsingFiberBraggGratings,"IEEEPhotonicsTechnologyLetters,vol.8,

pp.1331-1333,1996.

[112] C. Riziotis, P. G. R. Smith, and M. N. Zervas, "Performance characteristics of

interferrometric Bragg grating based OADMs in WDM transmission systems,"

presentedatBGPP,Stresa,Italy,2001.

[113] L.MartineauandS.Lacroix,"FabricationofHighlyGe-dopedFibreCouplersby

Fusing-TaperingTechnique,"ElectronicsLetters,vol.33,pp.798-800,1997.

References 253

[114] E.Marin,R.Ghosh,J.-P.Meunier,X.Daxhelet,andS.Lacroix,"BraggGratings

in 2 x 2 Symmetric Fused Fiber Couplers: Influence of the Tilt on the Wavelength

Response,"IEEEPhotonicsTechnologyLetters,vol.11,pp.1434-1436,1999.

[115] R.J.Orazi,S.D.Russell,T.T.Vu,andP.K.L.Yu,"UVfinetuningofnarrow

channel fused fibre wavelength division multiplexing couplers," Electronics Letters,

vol.33,pp.154-155,1997.

[116] S. J. Ashby, R. B. Charters, J. D. Love, F. Ladouceur, and M. C. Elias, "Large

wavelength shifts in UV-exposed 3dB and WDM fused taper fibre couplers,"


[117] H.Gnewuch,J.E.Roman,M.Hempstead,J.S.Wilkinson,andR.Ulrich,"Beat-

length measurement in directional couplers by thermo-optic modulation," Optics

Letters,vol.21,pp.1189-1191,1996.

[118] Y. Bourbin, A. Enard, M. Papuchon, and K. Thyagarajan, "The local absorption

thechnique: A straightforward characterisation method for many optical devices,"

JournalofLightwaveTechnology,vol.LT-5,pp.684-687,1987.

[119] N. Sahba and T. J. Rocket, "Infrared absorption coefficients of silica glasses,"

JournalAmericanCeramicSociety,vol.75,pp.209-212,1992.

[120] R.FecedandM.N.Zervas,"Efficientinversescatteringalgorithmforthedesignof

grating-assisted codirectional mode couplers," Journal of the Optical Society of

AmericaA,vol.17,pp.1573-1581,2000.

[121] A.SnyderandJ.D.Love,OpticalWaveguideTheory:ChapmanandHall,1983.

ListofPublications

ConferencePublications:

C Alegria, R Feced, M N Zervas and R I Laming, “ Acousto-optic effect in optical fibre

taperedstructuresforthedesignoffilters” ,IEEColloquium:Newdevelopmentsinoptical

amplifiers,London,November2nd1998.

C.Alegria,R.Feced,M.N.Zervas,R.I.Laming,“ Dynamicacousto-opticfiltersforgain

flattening of optical amplifiers” , Proc. II conferência de telecomunicações, Sesimbra-

Portugal,15-16April1999.

F.Ghiringhelli,C.Alegria,M.N.Zervas,“ Effectofphaseshiftperturbationsandcomplex

localtimedelayinfiberBragggratings” ,Proc.BGPP2001,StresaCongressCenter,Stresa-

Italy,4-6July2001.

C. Alegria, F. Ghiringhelli, M. N. Zervas, “ Non-Destructive Characterisation of Fibre

Couplers” , ECOC 2001, Rai Congress Center, Amsterdam-Holand, 30 September to 4

October2001.–Invitedpaper

C.Alegria,M.N.Zervas,R.Feced “ GratingAdd-DropMultiplexerbasedonaCompact

Non-UniformFusedFiberCoupler” ,acceptedtoOFC2002,Anaheim,USA.

JournalPublications:

R.Feced,C.Alegria,M.N.Zervas,R. I.Laming,"Acoustoopticattenuationfiltersbased

ontaperedopticalfibres",IEEEJournalofSelectedTopicsinQuantumElectronics,Vol.5,

No.5,pp.(1999).

C. Alegria, R. Feced, M. N. Zervas, R. I. Laming, S. G. Farwell, "Acousto-optic filters

basedonmulti-taperedfibrestructures",ElectronicsLetters,Vol.35,No.12,pp.(1999).

C. Alegria, M. N. Zervas, “ Non-destructive Coupler Characterisation Technique” ,

submittedtoJournalofLightwaveTechnology.

Patentapplications:

C. Alegria, M. N. Zervas, University of Southampton, “ Methods and apparatus for

analysingwaveguidecouplers” ,Filed14thSeptember2001.EuropeanPatentapplicationNo.

01306893.7-1236.

university of southampton research repository eprints soton › 30238 › 1 ›...

Documents