Thomas Szkopek

A tliesis siibrni t ted in conformity witli the requirements for the degree of EvIaster's of Applied Science

Graduate Departinent of Edward S. Rogers Senior Departnient of Electrical and Compu ter Engineering

University of Toronto

Copyright @ 2001 by Thomas Szkopek

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Abstract

Esperinient and Theory of a Novel. Multiple Wavelengtli. Erbium-Doped Fiber Laser

Thomas Szkopek

A,Iaster's of Applied Science

Graduate Department of Edward S. Rogers Senior Department of Electrical and

Cornputer Engineering

University of Toronto

2001

For the first tinle. four-wave niising and intracavity spectral slicing is sliown to pro-

duce siiilultaneous miilt iple wavelengtli laser oscillation in an erbium doped fiber laser.

A key application is multiple optical carrier generation for wavelengt h encoded. 1ight.-

wave c.ommunication systenis. The esperimental prototype used active rnodelocking for

gca tc r peak powers and tlius iiicreased four-wave mising witliin the cavity W report

herc the esperimental deinonstration of simultaneous oscillation of G wavelengtli cliannels

ivitii i5 d B variation in powver and 45 GHz cliannel spacing.

Kiinierical simulations based on a split-step Fourier metliod are presented, agreeirig

qiialitat ively wit li esperinient. Simulations indicate we may espect improved temporal

piilsc profiles wi tli the insertion of an all-fiber. iilt rafast saturable absorber.

Pour le Parcier et x td t re de I'Eglise de l'Art de Jésus Conducteur.

Contents

1 Introduction 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 hlotivation 1

. . . . . . . . . . . . . . . . . . . . . . . 1.2 Erbium Dopecl Fiber Amplifiers 3

. . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 EDFL Developnient 6

. . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Cont. ributions of this Thesis 6

. . . . . . . . . . . . . . . . . . . . . . . . . . 1 .1 Organization of tliis Tliesis 7

2 Erbium Doped Fiber Lasers 8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 1ntroduct.ion 8

. . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Multiple Wavelengtli Lasing 8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Sat. uration 9

. . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Stability Criterion 12

. . . . . . . . . . . . . . . . . . . . . . 2.3 Multiple Wavelengtli Fiber Lasers 13

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 ProposedTeclmique 18

. . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Four-Wave Mixiiig 19

. . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Active Modelocking 20

. . . . . . . . . . . . . . . . . . . . . . 2.4.3 Intracavi ty Spectral Slicing 21

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Laser Operation 24

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Summary 26

3 Experimental Demonst ration 27

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 11it. rodrrct ion 27

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . i3.2 Laser Implementat ion 27

3.3 Lasers versus Spont. aneous Emission Sources . . . . . . . . . . . . . . . . 29

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Power Spectral Density 31

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Temporal Characterization 32

. . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Autocorrelation Traces 37

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Statist icaI h~leasurements 42

3.6.1 Pulse Enerm Correlation . . . . . . . . . . . . . . . . . . . . . . . 42

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Conclusions 44

4 Numerical Simulation 46

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction 46

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Theory 46

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Xunlerical h4ethod 51

4.4 Sirnulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.5 Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Sum~nary 64

5 Concluding Remarks

A Inhomogeneously Broadened Media Under Saturation 70

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Introduction 70

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Voigt Profile 71

. . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Susceptibility Saturation 71

B Theory of the Lyot Filter 74

B. l Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

B.2 Birefringent Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . 75

. . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Poincar i. Sphere Formalism 76

. . . . . . . . . . . . . . . . . . . . . . B.4 Frequency Tra~lsmission Function 79

B.5 Sumniary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5

C Phase-Matching for Second Harmonic Generation 86

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 Introduction 86

. . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Phase-Matching Condition 86;

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Angle-Tuning 88

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4 External Angles 88

D Derivation of the Ginzburg-Landau Equation 91

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . I 1nt.roduction 91

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2 Bloch Equations 91

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3 h4ilswell's Equations 94

. . . . . . . . . . . . . . . . . . . . . . D.4 First-Order Pert. urbation Theory 95

Bibliography 100

List of Tables

2.1 Cornparison of technicliles for multiple wavelength generation in EDFL's

3.1 Parameters of the fiber laser iised to obtairi spectral. temporal and statis-

tical nieasurenients. The R F repetition rate was tuned for resonance with

the fast, asis patli. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1 Laser simiilatiori paraineters aricl tlieir respective values taken to match

thosc in esperiiiient iinless otherwise notecl in the test. . . . . . . . . . . 5.1

vii

List of Figures

( A ) A siiiiplified scliematic of an erbium doped fiber amplifier in a counter-

propagatiiig punipsignal configuration. (B) Most significant energy levels

. . . . . . . . and rates for trivalent erbiiim ions in a silicate glass liost. 5

The saturateci gain. norrnalized to an unsaturated ilnit peak. is plotted

above for an irilioniogeneously broadened medium. Tlie frequency is 11or-

nidized t O tlic widtii of the assiirned Gaussian inhoniogeneoiis distribu-

tion. Ttie tiomogeneous widtli is Au = 0.01 and tlie saturating frequency

is u;, = -0.1. Tlie plots are labelled witli normalized saturating powers

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I i / I s a , . 11

. . . . . . . A sclieniatic of the laser demonstrated in this body of work. 18

Polarization rotation in a birefringent fiber is sliown above. after Agrawal.

The polarization evolves witli a spatial period known as the beat lengtli. 22

The time and iingular frequency representation of an incident pulse is illus-

trated in ( A ) and (B). respectively. Repeated spectral slicing, represented

witli an effective transfer function (C): leads to a sliced spectrum (D) and

. . . . . . . . . . . . . . . . . . . . . pulse profile with substructure (E). 25

The double-pass EDFA constructed for initial

t iple wavelengt h laser. . . . . . . . . . . . .

experiments witli the mul-

. . . . . . . . . . . . . . . . 30

viii

The oiitput powr versus the pump power for the proposed fiber laser witli

u low gai11 (-15 dB) EDFA. Note tlie tliresliold indicative of the onset of

laser oscillat ion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Out put power spectral densities measiired witli an optical spectriim ana-

Iyzer \vit h a rcsoliit ion banclwidt li of 10 pm. Tlie abscissae are wavelengtl~s

in units of niii and the ordinates are optical powers in units of dBni. The

peak intracavity pulse energies are: A. 40 pJ: B. 255 pJ: C. 510 pJ: D. 989

p.J: E. 1368 pJ: and F. 1902 pJ. . . . . . . . . . . . . . . . . . . . . . . . 33

.A sdicmatic of tlie aiitocorrclator designed and built to cliaracterize the

pulse siibstruct tire of the esperimental fiber laser. . . . . . . . . . . . . . 35

Autocorrelation traces observed as a funetion of peak intracavity pulse

cners , The abscissae are delay times T in units of ps and the ordinates

are secoiid liarnionic iiitcnsities nornialized to peak values. The pcak in-

tracavity pulse energies arc: A. 40 p.J: B. 255 pJ: C. 510 pJ; D. 989 pJ: E.

. . . . . . . . . . . . . . . . . . . . . . . . . . 126Sp.J:andF. 1902pJ. 39

Ilinimuni piilse substriict ure widtli infered froni the central autocorrela-

tion peak iinder the assuniption of negligible cliirp and sech2(j pulse sliape. 40

Ratios of central peak SHG to pedestal SHG as a fuiiction of peak intra-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . cavity pulse eiiergy.

Scat ter plots of iiornialized pulse energies over 10 000 collected series. The

abscissae are first pulse energies E,, and the ordinates are: A, second pulse

eriergies E,,+l: B. fourth pulse energies Elif3: C. sixth pulse energies En+5;

and D. sewntli pulse energies E7t+6. . . . . . . . . . . . . . . . . . . . .

Tlie correlation behveen energy measurements of pulses separated by the

indicated number ~f round trips. . . . . . . . . . . . . . . . . . . . . . .

An illustration of t lie t heoret ical mode1 used for numerical simulations of

the fiber laser experimentally demonstrat,ed. . . . . . . . . . . . . . . . .

The siniulated tirne average spectrum of the fiber laser assumirig slight

detiining froni the slow asis. Tlie spectrum is norrnalized to unit peak. .

TIic siniulated tinie average aiitocorrelâtion trace of tlie fiber laser assum-

ing sliglit det~ining from the slow mis. . . . . . . . . . . . . . . . . . . .

Instantaneous pulse power indicat.ed dark shading versiis time refer-

cnced to peak rnodulator trarisniission on the abscissa. The evoliition of

pulse profile is shown with corresponding round trip numbers along the

ordiiiatc. The modulator wiiidow is assumed t o be a 400 ps FWHM sii-

pergaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Pulse energy transmission tlirougti various cavity elements are plot ted

above as a function of round trip number: Tl Lyot filter. T2 electro-optic

rriod~ilator. EDFA and T., undoped fiber. . . . . . . . . . . . . . . . .

The niodified laser cavity here includes an al1 fiber saturable absorber

consist ing of two polarization controllers. a lengt h of low birefringence

. . . . . . . . . . . . . . . . . . . . . . fiber ancl a polarizatiori analyzcr.

Sinitilated p o w r spectral density versus optical frequency for the modified

laser cavi t y including an ultrafast saturable absorber. Tlie spectrurn is

normalized to unit peak. . . . . . . . . . . . . . . . . . . . . . . . . . .

The simiilated tirne average aiitocorrelation trace of the fiber laser includ-

. . . . . . . . . . . . . . . . . . . . . ing an ultrafast satiirable absorber.

Simulated inst.antaneous pulse power indicated by dark shading versus

. . . . . . . . . time. An ult,rafast saturable absorber lias been assurned.

The Poincaré spliere C illustrated with two co-ordinate systems: A: the

. . . . . . . . conventional $ and X; B. the alternative system @ and 8.

The arripiitude t,ransmission function with 4, = 4, as the labelled param-

eter. The trajectories in the cornples plane are clockwise with increasing

opt ical frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B.3 The p o w r t ransinission function witli 9, = O,.. One abscissa is the optical

frecyuency nornialized to the Lyot FSR and referenced t o a bias freqriency.

The otlier abscissa is the angle O, normalizecl to ~ / 2 . . . . . . . . . . . . 82

B.4 The amplitude transmission f~inction witli O, = n / 4 and O, as the iabelled

paraiiietcr. Tlie traject ories in t lie comples plane are clockwise wit h in-

creasirig opt ical frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . 53

B.5 Tlie p o w r transmission function witli 0, = ~ 1 - l . One abscissa is the

optical frecliieiicy nornializeci to the L o t FSR and referenced to a. bias

fwqucncy. Tlie other abscissa is the angle du normalized to ri /2. . . . . . $4

C.1 Tlie pliase ~iiatcliing condit.ion for two ordinary fundamental beams with

wavevectors k,* and an estraordinary second harmonic beam with urcwevec-

. . . . . . . . tor kSL. Tlieopt~ic~xiscoftliecrystalisalsoillustrated. 87

C.2 The refractiori at the air-crystal interface for tlie two fu~idarnental beams

and t lie second liarinonic beani are illiist rated. The sliaded region repre-

. . . . . . . . . . . selits t lie cryst,aI: tlie unstiaded region represents air. 89

C.3 ( A ) The pliase matclied. second harmonic esternal angle (' as a function

of the esternal f~indamental beam spread A<. (B) The mean fiindamental

esterriai angle 1/2(cf + <-) versus tlie esternal fundamental beam spread

A<. The crystal axis is 30" with respect to normal. as in the crystal used. 90

Chapter 1

Introduction

1.1 Motivation

TLic first ~pe ra t~ ing fiber optic conimunication link was installed in a police station in

the city of Boiirnemoutli. United Iiingdoni in 1975 as a liglitning resistant replacement

for a two-way radio [l]. Out of siicli liumble beginnings. the fiber optics industry has

burgcoiied into t lie j uggernaut behirid the internet and the unprecedented econorny of

long distance teleplio~iy. Data transfer among internet users lias on average doubled every

year sinre 1996 [2]. Bit rates supported by fiber optic trunks are puslied t o multiples of

terabits per second to accomodate the growtli in data trafic.

Over a quartcr century since tlie Boiirnemouth Iink. the fiber optics industry has

nia tured under t lie nioniker of pliotonics. Functional. reliable. commercial components

are available for the construction of a photonic information infrastructure incorporating

long haiil triinks. met ropolitan area networks (h/IAN!s) and local area networks ( LAN's) .

Wavelengt h division mult iplesing ( WDM) is the prevailing multiplexing scheme in long

iiaul trunks and MAN'S. Data channeis are simply alloted slots in tlie optical frequency

domain in WDM schemes. Al1 optical carriers are intensity modulated with their respec-

tive data. A key benefit of WDM is that a large aggregate bit rate can be supported

n-hile the bit ratcs of eacli charinel are low enough to allow for niodiilation with electro-

optic clcvices (typicaily 10 Gbit/s to 40 Gbit/s). Photonic coniponents are now readily

availablc to gcnerate. modulate. niultiples. transport. amplify. route. demultiples and

dctcct opt ical signais as required in WDh4 networks.

At present . seniiconcluctor clist ribrited feedback lasers ( DFB's) perform well as optical

sources in \;Z7D31 nctworks. Hoivever. each WDM cliannel requires an individual temper-

atrirc stabilized DFB. resulting iii higlilÿ comples and costly systenis [3]. It is tlesirable to

reduce bot li cos t and coniplcsity of opt ical sources for sniall-scale (-1 0 cliannel) W D 3 1

nctivorks. sudi as t liose current ly used for met ropolitan eschange or t liose proposed for

local acccss [il]. Doing so iticreases tlie economic viability of MrDRI teclinology. thus

aiigniciitiiig efforts towards proliferation of photonic infrastructure.

A rolmst multiple ivavelength soiirce is ideal as an alternative to an array of DFB's.

TIic desirecl propert ies of sucli a mult iplc wavelengtli source are:

the capabi1it.y to sim~iltuneously generate niiiltiple wavelengtli carriers u.it.11 siiit-

able cliannel spacing. A paradigrnatic WDh4 systein employs a cllannel grid witli

nanometer to siib-nanonlet.er spacing. tliiis giving a sirnilar requirement for any

proposed source.

0 t lie iiicorporation of wavclengt li stabilisation for t lie entire cliannel grid. Ideally.

wc wisli to maintain control over only turo variables. tJhe regular cliannel spacing

anci tlie absolute freqiiency offset of tlie cliannel source with tlie network channcl

grid.

simple fabrication procediires. Altliough it is difficult to quantify a complexity

nietric. the multiple wavelength source sliould be cornpetitive with DFB rack arrays

considering resources in i ts manufacture. installation and online operation.

Ii iere are numerous candidate WDM sources tliat are curre~itly being investigated

hy the research community. A short list includes solid state lasers, integrated DFB's,

irit egrat ecl vertical cavity surface emit t ing lasers. semiconductor optical amplifier lasers

and erbium doped fiber lasers ( EDFL's).

Not al1 candidates are equally attractive hotvever. Solid st ate lasers. recently demon-

strated witli suitable wavelengt h spacing in Ckforsterite [5]. are bulky and typically very

sensitive to the environment. Altbough capable of supplying more power than any other

sclienie. integration witli fiber optic networks is not trivial. Researcli work has been

fervent witli compact. integrated DFB arrays but wavelength stabilisation must still be

pcrforniecl on a per clianncl basis [6. 71. Integrated vertical cavity surface emitting lasers

arc ais0 beirig actively piirsued. but there is considerable effort required in their fabri-

cat ioii [8. 91. Semiconductor opt ical amplifier lasers have been demonstrated witli bulk

comporients [ IO. 111 and with al1 fiber comporients [12. 131. The limitations are again

size and sensitivity for biilk optic lasers. Al1 fiber semiconductor optical amplifier lasers

are perliaps the most proniising multiple wavelength sources. altliough tliere is presently

liniited conimcrcial prodiiction of seniiconductor optical amplifiers.

EDFL's are tlic carididates investigated furt her in tliis tliesis. Constructed from off

t lie slielf fiber optic components. EDFL's are obviously the most cornpat ible teclinology

witli fibcr optic networks. It is the purpose of this thcsis to demonstrate a novel multiple

\vaveleiigtli EDFL wliicli satisfies tlie above defiiied criteria. The on-goiiig search for a

compact. robust. ccononiic. source of multiple carrier frequencies is the motivation for

tliis tliesis. We briefly review EDFA's here to faniiliarize readers witli the technology

dcalt witb through out this thesis.

1.2 Erbium Doped Fiber Amplifiers

The first fiber laser and amplifier were demonstrated by Koester et al. [14] in 1964. The

elegant design consisted of a neodymium doped silica g l a s fiber wound into a coi1 about

a Rash lamp. Further development of fiber lasers \vas hampered by difficulties in the

relicible doping of optical fibers.

Tlie b reak lirougli came wit 11 t lie irivent ion of an estended niodified clieniical vapour

deposition (AICVD) teclinique by Poole et al. 1151. wliicli is still in use today for tlie

fabrication of erbium doped silica fiber. The estended MCVD technique opened the way

for reliable fabrication of silica doped wi t li rare-eartlis [IG] . Tlie most important resiil t

for EDFL's was tlic new found ability to form low loss fibers witli Er& concentrations

of up to > 1 wt %. acliieved \vit11 A1203 codoping [li]. Rapid development followed in

erbium doped fiber amplifier (EDFA) technology. Integrated modules pumped with 980

niii laser diodes. the indostry standard today. already began to appeur by 1990 [18. 191.

Gaiiis frorri 30 dB to 46 dB at 1.53 prn were readily provided [20. 211. as were 3 dB

spectral gain I~andwiclths of 35 nni [22].

WC esplain here the basic opcration of an EDFA's. and refer the reader to the coni-

prelicnsivc treatnieiit of Desiirvire [23] for advanced topics. Tlie prototypical EDFA is

illustrated iii Fig. l . l ( A ) . A piimp bearii. comrnonly a 980 nrn laser diode source. is

ciirected througti erbium doped fiber witli a wavelength dependent fiber coupler. The

pump liglit produces an inversion in the electronic states of Er" ions wliich varies across

t lie lcngtli of tlic amplifier.

Tlic rclevant cnergv structure and transition rates of an Er3+ ion witliin a g l a s matrix

arc illustrated in Fig. 1 . l ( B ) . T!ie energy structure and transition rates are in fact critical

ta t,lie success of EDFA teclinolog~.. An inversion can be readily estabIislied in an EDFA

first bccause of tlie relatively quick decay from the pump manifold 'Il 1/2 to the escited

manifold t llrougli ni~lltiple plionon collisions [24]. Secondly. tlie lifetime of the 4113/2

manifold is relatively long (- 10 nis). Radiative dipole transitions a t 1550 nm are in fact

liiglily irnprobably for erbium ions [25] counter intuitive perhaps for an optical amplifier

operating at 1550 nrn. Once inversion is established, gain is provided by stimulated

ernission fronl tlie erbium ions over a wavelength range spanning tens of nanometers.

There are inany possible configurations of pump and signal directions giving different

980 nm Laser Diode I

Wavelength Selective /- =T

output signal

input + P

j multiple 7 phonon 1 f decay i I

411 u2

-1 550 nm -1550 nm spontaneous stimulated transitions transitions

980 nm PumP absorption

Figure 1.1: ( A ) A siniplified sclieinatic of an erbium doped fiher amplifier in a counter-

propagati~ig punip-signal configuratioii. (B) 3lost significant energy levels and rates for

trivalent erbiuni ions in a si1icat.e glass liost.

inversion profiles and tliiis different maximum gains. saturation powers and noise figures

The widt 11 of the gain spectruni is due to nunierous factors. Tlie greatest contribution

to tlie spectral widtli of tlie EDFA gain is the manifold nature of the levels involved in

laser transitions. The erbium ions find themselves in a siiica glass liost wliich does not

possess long range order. but does possess short range order. The local electric field

produces ii splitting in energy levels referred to as Stark splitting. giving rise to the

observed manifold structure. The Stark splitting gives an approximate spectral width

of 50 nm [26]. However! further complications arise because tlie manifolds are in fact

statistical distributions of energy levels due to variations in the local environment of each

erbium ion within tlie glass matrix. Prediction of spectral shape from first principles

is an untractable probieni. and thus EDFA spectral properties are most successfully

cliaracterized wi t li various empirical rnodels [23].

1.2.1 EDFL Development

Tiie rapid cvolution of EDFA tedinology resiilted in a concomitant development of

EDFL's. Bot h cont iniioiis wave (CW) and Q-switcliing modes of laser operation were

investigated for lixicar ciivities in early years [27. 28. 291. The availability of 30 dB EDF.4

gaiil and greater fibcr optic componerit f~inctionality led to a cornucopia of EDFL de-

s ign~. '\Iost notably. active niodelocking and soiiton pulse shaping were dernonstrated

[30. 311. T h liigli peak powers achievaiile witli sliort optical pulses (sub 10 ps) resulted

in significant self phase niocliilation due to the Kerr nonlinearity of the silica fiber itself.

Single longitudiiial niode EDF ring lasers have been dernonstrated with lincwidtlis

as narrolv as 10 kHz [32]. Siidi lasers take advantage of the very liigli quality factors

wchievablc in dl-fiber ring resonators. 0 1 1 ttic otlier liaiid. EDF can provide gairi over ii

35 nni I~andwid t li. and t hiis sliort piilse sources can be developed. Passive modelocking

in t lie forni of additive pulse modelocking lias beeii successfully implemented in various

nays [33]. The principle behind additive pulse niodelocking is iionlinear interferometry

providiiig lowr losses to higher energy piilses. The technique llas proven successful and

has lecl to the generation of pulses as short as 100 fs [34]. Tliere are many configurations

wit il performance ranging between t lie estremes ment ioned.

1.3 Contributions of this Thesis

We piirsiie here an esperi~nental and theoretical investigation of a novel multiple wave-

lengtli EDFL. The main cont.ributions of the work here are:

0 a. novel concept for generating multiple wavelengths within an EDFL is proposed.

The novel feature of the technique is the combination of intracavity spectral slicing

and foiir-wâve mising.

the results of a detailed esperimental study of the novel multiple wavelengtli EDFL

is perforrned. We present a full cliaracterization of the EDFL. noting that pulse

encrgy fltict uat ion linlits the immediate utility of the configuration denionstrated

tiere.

a refined laser design lias been proposed using an iiltrafast saturable absorber. NU-

riierical simulatioris indicate a superior pulse profile niay be espected in esperiment.

1.4 Organization of this Thesis

Tliis thesis is organized into five cliapters followed by an extensive appendix. Tlie present

chapter. being t lie first . is the prolegonienon for the work that follows. Tlie second chapter

piits the present work into contcxt among conteniporaneoiis research efforts. Discussions

of tlie proposed techiiique. tlie principle of operation and physical implementation are

found in the second chapter. The third diapter gives a detailed account of the esperi-

mental results from t Iie laboratory demonst ration. including tinie average nieasurements

and dyriariiical measurements. Thc fourt 11 chapter presents t lie met hods adopted for

niinierical siniulation of the proposed EDFL. Discussions regarding a refined laser cavity

and correspoiiding sinii11ation results are presented as well. FinaIly. the fifth chapter

bririgs the main body of the thesis to a close with concluding remarks regarding future

prospects for tlie denionstrated multiple wavelength EDFL. Wit hin the appendices, one

will find dctails which the author believes would only detract from the main arguments

of the tliesis with tedious longueurs.

Chapter 2

Erbium Doped Fiber Lasers

2.1 Introduction

Tlie focus of this cliapter is erbium doped fiber lasers (EDFL's) designed for operation at

iiiiiltiple ~vavelengtlis. We procced to examine a general t l~eory regarding simiiltaneous

laser oscillation of multiple wavelengtlis. arriving at the required criteria for stable opera-

tion. The tedinical difficulty of acliieving stable multiple wavelength opcration are made

apparent. To put the work here in contest. we provide a survey of techniques previously

employed to generate multiple wavelengt lis in EDFL's. Finally. t lie technique proposed

and dernonstrated in this tliesis is described.

2.2 Multiple Wavelength Lasing

Development of rnult iple wavelengt h EDFL's has been comparatively slow compared to

t hat of other EDFL categories. The retarded development is due to the inherent difficulty

in adiieving stable multiple wavelength laser operation. Soon after the discovery of the

laser itself: the broadening of spectral gain bandwidths was clizssified as homogeneous,

inhomogeneous or a combinstion of both. As will be shown, the nature of the spectral

broadening for gain media plays a key role in the stability of multiple wavelength lasers.

Horriogeneous hroadening is causeci by eacli individual atonl. ion or molecule liaviiig

the sanie finit e spectral widt h for a given optical transition. Contributing niechanisms

iiiclude dipole interactions witli the vacuum field1 (351 and phonon stimulated broaden-

ing (361. Siicli nieclianisms are more generally refered to as lifetime broadening. Each

radiatiiig entity is spectrally identical. Iieiice the name homogeneous. Also. note tliat

lioniogeileous broadening is always present altliough it may not be the dominant contri-

t~ution to the spectral width.

Inhoriiogcneous broadening is caused bj. a redistribution of resonance frequencies

arnong a collection of atoms. ions or molecules. Each entity interacting with the electro-

niagnetic field is spectrally different. 111 solids. inhomogeneous broadening is caused by

the inevitable variations in tlie host rnaterial striictiire a t ttie local site of each ion. atom

or molecole [36]. The importance in the distinction between the two broadening meclia-

riisms is crucial because of ttie different satiiration beliaviour that results from eacli. as

disciissed f~irtiier below. The saturation beliwioiir of an ampliQing medium determines

t lie relative ease with which stable miilt iple wavelength operation can be achieved.

2.2.1 Saturation

First. let us consider a purely homogeneously broadened medium represented as a col-

lection of two level systems puniped esternally iii an unspecified way. We assume a ho-

mogeneous linewidt li Au. a resonant frequeiicy u~~ and effective transit ion cross-section

o. T lie comples suscept ibility ~ ( w ) relating tlie material polarization phasor P ( w ) to tlie

electric field phasor E ( w ) takes tlie following form2

where c is the vacuum speed of light and N = IVmcitd - Nground is the population inversion

of tlie active medium. The optical gain provided by our prototypical amplifier is simply

'spontaneous emission in the semichssicat theory of photons and atoms 'refer t o the Appendix D for a derivation of this result from the Bloch equations

t 11e iriiaginary part of the faniiliar coniplex Lorentzian of Eq. 2.1. Saturation of the gain

nieclium is the result of a reduction of tlie inversion JV according to [37].

ivtierc !Veq is the inversion due to esternal pu~riping in the absence of the resonant signal

of iiitensity I and Is,, is the saturation intensity of the transition. Obviously. saturation

of a Iioniogeneoiisly broadened niedium results in the reduction of gain across the entire

spect r d widtli of the transition.

Let us now consider saturation of a medium eshibiting inhomogeneous broadening as

~ ~ 1 1 . So te that lifctime broadening will always be non-zero. and thus some honiogeneous

spectral broadeiiing is always present . Tlie statist ical distribution of resonances. denoted

<(Y-.) Iiere. is assunied nornialized sucli tliat J' C ( ~ T . ) & ~ = 1. The susceptibility z ( w )

iirider sat iirittion 1- a11 optical sigiial of frequency w and intensity I I can be written in

the foriii [37].

wliere the saturation fact'or S(w,) is giveii by,

aiid wliere t lie deptii of the saturation is determined by.

The deptli of saturation increases witli saturating intensity and proximity to the satu-

r a t h g signal frequency. as one would espect.

Inliomogeneous broadening leads to a frequency dependent saturation3. Plots of the

norinalized gain spectra under varying saturation conditions are illustrated below in Fig.

2.1. Spectral Iiole-btirning is clearly observed, a phenornenon unobserved in purely l i e

mogeneously broadened gain media. Tlie greater the rat,io of inhomogeneous broadening

"refer to Appendk A for an analytic method to calculate the saturated susceptibility f ( w )

to homogeneous broadcning. the narrower the spec t ral hole. In the s t rongly inhomoge-

rieous limit. the spectral Iioles are twice the liomogeneous linewidth for weak saturation

conditions. The spectral Iiole burning effect is reduced as the medium becomes more

iioiiiogeneous. The broadening mechanisms of amplifying media must be understood in

c l t t ail for any prediction of saturation behaviour.

O -1.5 -1 -0.5 O 0.5 1 1.5

Nomalited Frequency

Figure 2.1: The saturated gain. normalized to an unsaturated unit peak. is plotted

above for an inhornogeneously broadened medium. The freqiiency is normalized to the

widt il of t lie assunied Gàussian inhomogeneous distribution. The hoinogeneous widt h

is A ~ J = 0.01 and the saturating frequency is wl = -0.1. The plots are Iabelled with

norrnalized saturating powers I I / I,,, .

It is worth discussing the nature of spectral broadening in EDF. The homogeneous

linewidth of typical germano-alumina-silica based EDF a t room temperature is approxi-

niately - 10 nm [38]. Alt liough moderately well approximated as a homogeneous medium

in opt.ica1 signal amplification applications [39]: deep saturation resuhs in complex spec-

tral Iiole I~iirning belial-iour [do. 41. 12. 13. -141. Our two-level assumption is a simplifi-

cation of the case for EDF. As noted earlier. there is a rnultiplicity of electronic energy

levels giviiig a ground state manifold and escited state manifold. To date. there does

not esist a tractable tlieory allowing one to cakulate tlie contributions to the suscepti-

t~ility froni eacli transition aniong the manifolds. even for crystalline hosts. Interaction

between cIect roi1 witvef~iriction configurations are forbiding due to t lie slieer size of tlie

niany-electron atom problem [45. 461. An empirical mode1 was developed by Desurvire

1471 to overronie the niathematical difficulties of a first principles treatment. The requi-

site Stark split eriergy levels have been measured t.hrough low temperature spectroscopy

126. 181. but the requisite plienonienological dipole moments of eacli transition have y t

t O \le measiired. Present ly. t lie Iiomogeneously broadened two-level approximation is tlie

best available for EDFL simiilations.

2.2.2 Stability Criterion

Tlie conditions under wliicli stable laser oscillation are expected to ensile were investi-

giitecl early in tlie developrnent of lasers by Lamb [dg]. The crit.erion for stable dual

wavelengtli. continuous lvave oscillation was determined by Lamb througli linearization

of the gain saturation in the rate equations for a laser cavity witli two modes?

wtiere t is t,ime. I I and I2 are the optical intensities a t two different frequencies. gj are the

optical gains. aj are the optical loss. and K j k are the gain saturation coeffients. Neglecting

wavelength dependent loss (al = oz). the criterion for stable laser oscillation was found

to be that of weak beam coupling [49]:

In ot lier words. t lie cross saturation of gain must be less than the self saturation of gain

in order to achieve stable oscillation. In view of the previous discussion pertaining to ho-

iriogeneous and inliomogeiieoiis gain hroadening. one readily concludes that the spectral

holc hirniiig effect results in weak coupling and tlius supports stable multiple wavelengtli

oscillation. The widtti of a spectral hole. proportional to the Iiomogeneoiis linewidth to

first order. deterniines the order of magnitude of the minimum spacing of stable multi-

ple wavelengtli oscillations. Oncc wit hin a homogeneous linewidtli. gain cross saturation

approaclies gain self saturatiori for the two oscillating frequencies. Instability arises be-

cause ail>- wavclength dependent loss can upset the fine balance achieved between the

two wa\-~lengths competing for gain tlirough cross saturation. A completely homoge-

iieous gain mediuni will not support stable multiple wavelengtli operation. However. any

niectianisili wtiich reduces pet gain cross saturation below net self saturation can induce

st ubility. Thiis. altliougli the liomogencous Iinewidt h imposes a limit on tlie niinirnuni

separat ion for stable oscillation. this can be overconie wit li esternai niet hods to reduce

cffect ive gain cross saturation.

2.3 Multiple Wavelength Fiber Lasers

Tlie 10 nm homogeneous linewidth of EDF and a desired channel spacing of less tlian

10 nrn necessitates tlie use of anotlier pliysical niedianisrn for t,he reduction of net gain

cross saturation t o allo\v stable operat.ion. Researcli in the field is fervid and numerous

stratagems Iiave been proposed and demonstrated with varying degress of success. Tlie

tecliniques previously employed can be categorised roughly as follows ':

0 gain equalization

cooling

'we do not report here methods obtaining simultaneous laser oscillation with 8 nm channel spaciiig or greater

spatial nitil t iplesing

0 polarization niult iplesing

stinliilated Brillouin scat tering

0 frecluency sliift ing

0 estracavity spectral slicing

In the following paragraplis. we briefly describe the principle used in eaclï technique to

overcome the limit imposed by a 10 nm room temperature homogeneous liiiewidth. A

suniriiary of esperinientally demonstrated results is given.

Gain Equalization

AIultiple frequencies will conipete for gain in the erbium gain medium. Ericli frequency

sat urat es t lie gain. wit h t.lw resiilt t liat t,lie freqiiency wi t li t lie liighest unsatiirated net

gain (gain rniiiiis loss) will esperience the great.est aniplification and eventually saturate

tlie gain below tlie loss level for al1 other frequencies. This plienomenon is referred to

as gain clampilig. Gain ecliialization is the technique by wliicli one adjusts the loss of

eacli cliannel sucli t liat no single frecluency will esperience preferentiâl net amplification

per round trip. As with al1 otlier techniclues below. the cliannel spacing is determined

bjr an iiitracavit.y filter uiiless otlierwise specified. Esperimental dernonstration of this

teclinique stiowecl a mininiuni 4.8 niii cliannel spacing coold be obtained [50. 51. 521. Tlie

gain ecliialization teclinique is sensitive to cavit,y perturbations because only a slight wave-

lengt li dependent loss can upset t lie balanced amplification of each channel undergoing

laser oscillation.

Cooling

Tlie homogeneous linewidtli of erbium doped fiber exhibits an approximate temper-

ature dependence d o m t.o 20 K [38]. Cooling erbium doped fiber to 77 K with readily

availa ble licpid nit rogen reduces t lie homogeneoiis linewidt h to approxiniately 1 nm. Es-

periiiiental denionstration of tlie technique [53. 511 lias shown its utility in increasing tlie

stability of multiple waveIengtli EDFL's. in fact. cooling is often used in conjuction with

ot lier tedinicpes listed Iiere t O increase output power stability. However. the use of liquid

nit rogen limits t lie pract ical use of t his technique outside research laboratories.

Spatial Multiplexing

Spatial miiltiplesing includes al1 schemes in wliicti different freqiiency ctisnnels access

spatially distinct erbium ions for gain. Tlie rnost mundane and costly approach is to

force each freqüency to access separate EDFA's for individual amplification 155. 561. An

elegant impleiiientation of the technique is tlie use of dual core erbium doped fiber [57].

Tlie oscillation of optical power between tlie cores of the fiber is freqiiency dependent.

TIius the overlap of optical signals witli disparate frecluencies is reduced in the amplifying

cores. -4 siiiiilar reduction in overlap can be acliieved tlirough the use of a miniature,

liiglily dopcd fibe? as a standing wave resonator 1581. Again. disparate frequencies

esperience a rediiced overlizp within tlie ampiifying medium and thus a reduction in

gain cornpetition is cffected. An 0.4 nm wavelengtli separation can be achieved. The

primary limitation of this technique is the required fabrication of novel erbium doped

fiber structures.

Polarizat ion Mult iplexing

Polarizat ion mu

cross saturation

saturation by a

ltiplexing is based upon tlie phenomenon of polarization dependent gain

. often referred t o as polarization Iiole burning [59]. Following amplifier

pulse of definite polarization, a probe pulse will experience more gain

when orthogonally polarized. This phenomenon was used to reduce gain cornpetition --

%he high doping concentration was achieved by CO-doping with ytterbium, which did not dramatically change the homogeneous linewidtli

betweeii freqiiencies by wrapping erbium doped fiber around a spool to induce birefrin-

gciice: a 1.1 nni clianriel spacing was obtained [GO] . Other mechanisnis sucli as spectral

holc biirnirig due t,o gain inlioniogenei ty were siiggested to aid the multiple wavelengt h

operation.

Stimulated Brillouin Scattering

Spontaneous Brillouin scattering is the process by whicli light is back scattered

acoustic wave witliin film- [Cl] . Stiniulüted Brillouin scattering (SBS) occurs wlien si

an

lffi-

ciciit Brillouin scat tcred liglit int erferes wit 11 t lie incident light . resulting in the generat ion

of niore acotist ic waves and thiis increased Brillotiin scattering. SBS results in a frequency

sliift clcpeiident ripoii tlie frequency of tiie acoustic wave. Typical freqiiency shifts are on

tlie ordcr of 10 GHz [Cil]. Ernploying two EDFA's. 53 cliannels witli a 0.08 nni cliannel

spacing wcrc generated tlirougli feedback of suc-cessive signals generated by SBS [G2].

Tlic problcni of gain competitioii is avoided Iiere becailse of the amplification provided

tliroiigli tlic SBS process witliin iindoped fiber. Of coiirse. SBS provides no control over

t lie rlianiiel spaciiig. siiice it is deterinined by acoustic wave frequencies witliin the fiber.

Frequency Shifting

Frecpency shifting t ~ y ari esternally driven acousto-optic rnodulator finds application

in generutirig miilti ple wavelengtlis from a single EDFL. Esperiniental demonstrations

have iitilized a frequency shift below that of tlie channel spacing by a factor of at. least

1000 [63. 641. Wit li subcliannel frequeiicy shifting. the laser is operating in a dynamic

eqiiilibriiini. Optical energy is continiiously being sliifted in one frequency direction

among the cavity modes, preventing any single channel from clâmping the gain for al1

other channels. The spectral width of each channel was verified to be less tlian tha t of the

defining intracavity spectral filter? indicative of laser oscillation. Ho~vever~ a quantitative

theoretical esplanation of this technique is yet t o be given.

Extracavity Spectral Slicing

Est racavity spectral slicing is the tedinique in wliich broadband optical power is spec-

trally filtered esternal to the laser cavity. A pulsed mode of operation is required to

geiierate the several nanonieters of bandwidtli required througli nonlinear effects within

fibcr. Pulses from sub-picosecond fiber lasers have been filtered with wavelength demul-

t iplesers for mdt iple wavelengt h output [65]. Nonlinear pulse compression and super-

continuum gcricration have also been used to generate siifficient bandwidth for spectral

slicing [66. 671. The contrat between peak to trougli in the output power spectral den-

sity is liniited by tliut of the estracavity filter. Furtlierrnore. estracavity spectral slicing

rcjccts a significant fraction of the output power. reducing the final powr efficiency of

t lie source.

Summary of Techniques

A coniparisori of the aforementioned techniques is found iri Table 2.1. The minimum

diannel spacing and corresponding number of channels are indicated.

1 Methoci 1 Cliannel Spacing 1 Nurnber of Cliannels 1 cw / pulsed 1

Cooling 1531

Spatial Mult iplexing 1571

1 Frequency Sliifting [63] 1 0.8 nm 1 14 1 CW 1

Polarization Multiplexing [GO]

1 SBS [G2]

1 Extracavity Spectral Slicing [66] 1 0.08 nrn 1 150 ( pulsed 1

0.65 nm

0.5 nrn

--

Table 2.1: Cornparison of techniques for multiple wavelengtli generation in EDFL's

1.1 nm

0.08 nrn

11

8

cw

cmr

7

53

cw

cw

CHAPTER 3. ERBICM DOPED FIBER LASERS

2.4 Proposed Technique

iCé wish to combine the desirable properties of previously nientioned tecliniqiies. nainely

the environnienta1 stability of estracavity spectral slicing and tlie liigh peak to trough

cont rast of t lie ot her techiques witli intracavity filters. The stratagem proposed liere is

to iise four-wave misirig (F\VAtI) to couple optical power 11etwvee11 multiple wavelengths

dcfined 11'- an int racavity spectral filter. A scliernat ic of the pulsed laser demonst rated

in tliis wvork is illustrated Selon. in Fig. 2.2. ive first discuss how FWM. intracavity

spcctrwi slicing arid active nioclelocking wvork togetlier in the demonstrated laser to achieve

niiiltipie wavelength oscillatiori. We finally discuss the propagation of a pulse around the

k c r cavity to illiistrat e t lie sigrtificant effects at play.

RF Source

I Output

/-

I

EO Modulator 10:90 Coupler EDFA

Polarization Highly Polarization Controller Birefringent Controller

Fiber

Figure 2.2: A schematic of the laser demonstrated in this body of work.

2.4.1 Four-Wave Mixing

A classical mode1 of FU91 is the niising of three electromagnetic wavcs in a medium hy

virtue of a tliird order sucseptibility tensor 1(3) according to.

ivliere E" is the vacuum perniittivity. P,vL is the nonlinear material polarization and E

tlie electric field. The resultiiig niaterial polarization oscillates at frequencies given by al1

possible signed sums of the incident frequencies to drive a fourtli electroniagnetic wave

[(is]. A special case of degenerate (or nearly degenerate) FWM is the optical Iierr effect.

riamcly an intensity dependcnt refractive indes. The optical Iierr effect results in what

is knotin as self p h i ~ e modiilation for obvious reasons. Third order nonlinearity typically

arises from the nonresonant. optical driving of elect,roris in anharmonic potential tvells

!vit hin the niediuni of iriterest.

Vie illiistratc licre how FWAI acts to reduce effective cross-saturation with a simple

esamplc. Let us consiclcr a laser witli a ho~nogeneously broadened gain medium operating

at a single frequency &,, duc to gain clumping. We defirie the respective electric field.

wliere e is a unit wctor defining the polarizat,ion state and Ep is the comples electric

field phasor including a11 spatial dependericy. In the absence of any nonlinearity otlier

tlian gain saturation. the electric field would continue to oscillate at a single fsequency.

If now the cavity is assunied to possess a X(3) nonlinearity. FWM will occur. Let us

call our original single frequency electric field tlie "pump" . A full quantum mechani-

cal treatment is reqiiired for us to observe how light at other frequencies will build up.

We take the result, here tliat spontaneous emission from the gain medium and vacuum

field fluctuations tliemselves will result in broadband optical noise [35]. Let us con-

sider noise Ec a t frequency w, = w, + 6. whicli we will call the "conjugate'' field. The

rionlinearity witliiii tlie cavity will resiilt in a niaterial polarization with many terms

of different ficquency dependence. We are interested in polarization terms of tlie form

col(") Ep Ep E; esp(-i;,t). wliere ds = ,> - 6. Tlie oscillating material polarization will

tlius drive a .bsignal" electric field Es a t frequency The signal and conjugate are in

fact intercliangeable liere. and eacli will lead to the growtli of the ot lier. One rnay tliink

of FWII as tlie annihilation of two punip photons leading to the creation of a signal

plioton and a conjugate photon.

Tlic effect of FI431 is non. clear. A wavelength pump cliannel will divert power to

adjacent wavelengt li clianncls via FWM. The gain mediuni is saturated by the pump

so as to provide insiifficient gain to other wavelengths for laser oscillation. but gain

proport ional t O piimp power is provided t lirough the FW3 1 meclianism. Tlius. FWRI

reduccs t lie effective gain cross sat iiration and aids us in aciiieving multiple wavelength

laser oscillation. Furtliermore. a regiilar clianncl spacing enforced with an intrâcavity

filter will facilitate tlic FWXI process. Anÿ cliannel witli suficierit power will serve as a

pump bcam for adjciccnt signal and conjugate cliannels.

Tlie ~ncdiiim that rnediûtes tlie FWhI process witliin the demonstrated laser cavity

is thc silica fibcr itself. Pulscd opcration and propagation over esterided lengths can be

iised to increase tlie total FWM generated. Fiber nonlinearity (A(") = G x l~ -~ ' c rn"e r~

[GS]) is sufficiently large tliat F\.CTM can provide gain over a 200 GHz bandwidtli witli a

1 \Ar continuoiis wave punip in standard teleconimunications fiber. resulting in an effect

knoan as modulation instability [69].

2.4.2 Active Modelocking

To obtain the greatest nonlinearity possible. and thus maximize FWM. the demonstrated

laser is operated in pulsed niode. Tlie pulse propagation direction is L ~ e d to be clockwise

tlirougli the use of an isolator within the EDFA. With the same average power provided

by tlie EDFA to the laser cavity, the peak power of a puise increases in proportion

to Tl&. wlicre T is the round trip time and At tlie pulse width. To achieve pulsed

operation. an active niodelocking teclinique is employed here.

The airi pli tutle modulator rcsponsible for modelocking tlie EDFL is a LiN bOJ electro-

opt ic (EO) niodiilator in a Mach-Zelinder interferometer configuration. An applied volt-

age controls tlie optical transmission through the modulator by inducing a phase shift

in one arni of a LIacli-Zelinder interferometer via the electro-optic effect. To obtain a

single pulse circulating witliin tlie cavity a t a time. we apply a voltage pulse to the EO

niodulator witli a repetition rate equal to the fiindamental cavity resonance frequency

(tlic frequenry witli wliicli an optical pulse traverses the laser cavity once). A transmis-

sion wiridow is opened at tlie precise tinie an optical piilse arrives a t the EO modulator

e d i round trip. Tlie process is self-starting because noise wliich passes tlirough tlie

t rarisniission winclow will esperience less loss t lian noise wliicli does not. The circulatirig

piilse csperiences niiniriiiini loss and will tlius be the preferred mode of operation. Tlic

forriiatioii of a pulse train periodic in tinie witliin tlie cavity is equivalent to locking tlie

ptiase bctween t lie longitudinal cavity modes. lience the name niodelocking.

2.4.3 Intracavity Spectral Slicing

We liave discussed Iiow FWbi will provide gain tlirougli fiber nonlinearity to counter

gain cross-satuiat,ion. Howver. an iiitracavity spectral slicing filter is required to define

tlie waveleiigtli cliannels. The filter ive ernploy is a Lyot filter coniposed of highly bire-

fringent fiber. polarization controllers and the electro-optic (EO) modiilator previously

discussed. We use tlie rnodiilator here as a polarization state analyzer. The modulator

is impleinented witli planar waveguides which are designed to allow the propagation of a

single polarizat ion only.

The Lyot filter operates as follows. Linearly polarized light from the modulator

irnpinges on a polarization controller. This polarization controller, which can perform

arbitrary polarizat,ion transformations. is adjusted to launch light with linear polarization

i i / 4 radians froni tlie principal axes of the liighly birefringent fiber. As liglit propagates

dov-n tlic birefringent fiber. the polarization state evolves as in Fig. 2.3.

c, FAST WOE

Figure 2.3: Polarizatiorl rotation in a birefringent fiber is shown above. after Agrawal-

Tlie polarization cvolves witli a spatial period known as tlie beat length.

Thc rate at whicli polarization state evolves is deterniined by the accumulation of

pliase difference O = &AnL/c . wliere is tlie optical radian frequericy. An is the b i r c

fringencr of tlie fi ber. c is t lie viiciiiini speed of liglit and L is tlie fiber lengtli. Tlie

polariziitioii rotation is frequeiicy dependent. and tliiis ive can perform spectral slicing

witli a polarizütion analyzer at tlie terminal end of tlie birefringent fiber. A polarization

controller at the terminal end is t h adjusted to set the polarization state wliich will

esperiencc mininial lûss tlirougli our analyzer. the EO modulator. Tlie polarization con-

trollers are used in the laser as a nicans to acliieve the desired alignment between the

pririciple axes of the Iiirefiingent fibcr and the transniitting mis of the EO niodulator.

A detailed analysis of the Lyot filter with arhitrary polarization controller settings

is presented in Appendis B. We present some general resirlts liere. and leave the reader

to tlie appendis for details. The transmission function for the electric field phasor of an

optical signal a t carrier frequency w is6.

'we clenote tlie transmission fuiictiori l ~ y H ( u ) rather than t liere to avoid confusion with the time variable

wtiere a.. 6 and <f> are constants determined by polarization controller setting. The unde-

sirabie situation of frequency independent transmission (a = 6 = 1 or a = 6 = 0). occurs

when liglit is launciied or analÿzed along a principle a..is of the birefringent fiber. The

free spectral range (FSR) of tlie Lyot filter is given in Hz by.

Tlie tinle-doniain impulse response of the L o t filter is given by.

ivhere d(t) is tlie Dirac delta function. An incident. pulse thus produces two oiitpiit pulses

with a d e l q T = l/Au in between tliem. Tlie first (second) pulse corresponds to the

portion of tlie pulse whicli traversed tlie fast (slow) xxis of the birefringent fiber. We

discuss tlie iniportance of spectral-temporal relationsliips furtlier below.

Tliere are several attractive features of tlie Lyot filter as a spectral slicing filter. The

FSR is sliglit ly acljiistable t liroiigli fiber birefringence and lengt li. and can be finelv tuned

beiiding the fiber for stress induced birefrigence [70]. For a typical 10 m fiber section.

tlic FSR can be tuncd by approximately 3 GHz with a birefringence tuning of 1 0 - ~ . as

coiild be ind~iced witli a bend radius of about 3 cm for a typical fiber [70Iï. Anotlier

desirable feature is that the FSR of the Lyot filter eshibits negligible frequency depen-

dcnce over the wavelengt 11 span of interes t,. The FSR depends upon t lie birefringence.

and t,lius oniy dispersion of tlie birefringence itself contributes to a variation in FSR witli

frequency The high birefringence of the fiber used in this work. a so called "bow-tie"

fiber. is due to a permaiient stress induced birefringence within the fiber core [71]. The

frequency dependence of silica photoelasticity over a 10 nm bandwidth is negligible. One

can readily keep the FSR variation to below 500 MHz over a 10 nm bandwidth, that is,

1 part in 100 for a prototypical WDM channei spacing of 50 GHz.

'we assume here a permanent fiber birefringence of approximately 1 0 - ~ , a representative value for t h work here

Given the uniforrnity of the Lyot filter response. one would only be recliiired to monitor

and coiit roi two degrees of freedom for wavelengtli stabilisation in a system application:

cliannel grid spacing and cliannel grid bias. The bias point and contrast ratio of the

Lyot filter can be adjusted tlirougli tlie polarization controllers. Tlie Lyot filter is tlius

excellent for irit racavi t .~ spect rai slicing.

2.4.4 Laser Operation

WC trace a piilse propagating tlirough tlie cavity t o give furtlier insight into tlie operation

of tlie tfenionst rated laser. Uk begin with a linearly polarized pulse entering the EDFA.

Mé assiinie tliat nunieroiis wavelengtli dianncls add to give oiir single pulse.

Upon propagatiiig tlirougli the EDFA. the wavelengtli channel a t the gain peak of the

EDF.4 will i x aniplified more tlian adjacent cliarinels. A fraction of the piilse energv is

cliverteci to the outpiit of the laser by an al1 fiber coiipler arid the remainder is lwpt witliiii

tlie cavit~.. \\~liile traversiiig the fiber ring. FWA4 occiirs witliin the fiber to produce two

cffects. First. FWiI provides gaiil to lower power channels as described carlier. Second.

piilsecl opcrat ion iiiiplios u finite spectral widtli for each wavelength cliannel and t hus

FMrA4 acts to broaden the spectriini of each cliannel itself. In the absence of spect ral

filtcring. we would obtain a piilsed broadband output from the EDFL.

The piilse tlien propagates througli the Lyot filter elements. As described. the Lyot

filter produces two wciglited replicûs of tlie incident pulse with a dcii~y T = I/Av between

tliem. Tlie Lyot filter narrows tlie widtli of each wavelength cliannel. Tlie result of

repeated spectral filtering of a short pulse. as would be incurred througli several round

trips. is illustrated in Fig. 2.4. One can readily see that spectral broadenixlg by FWM will

be balanced by the spectral filtering of the Lyot filter. The periodic temporal structure

enforced by repeated filtering is seen in the esperimental and numerical work reported

later in this thesis.

Finally. the pulse must pass through the EO modulator. It will be noticed that

Repeated Spectral

4 F iltering

A At.

1, l

-: ;* * I I

'+ -2d9

I I t

Figure 2.4: Tlie time and angiilar frequency representation of an incident pulse is il-

liist rateci in ( A ) and ( B) . respect ively. Repeated spectral slicing, represented with an

effective transfer function (C). leads to a sliced spectrum (D) and pulse profile with

substructiire (E) .

we have in fact two round trip times througli the ring cavity corresponding to tlie fast

and slow pat lis t lirough the birefringent fiber. When t lie modulator driving voltage

resonates wit li the fast axis. a portion of t lie pulse will always arrive late for the modulator

transmission wiiidow. Tlie pulse eilergy will thus accumulate at. tlie faliing edge of the

transrriission window. If resonating with the slow axis: pulse energy d l accumulate on

t Iie rising edgc. Tlie attenuation of the portion of tlic pulse wliicli takes the "wrong" patli

tlirough t,lie cavity increases the effective cavity loss. Furthermore. t.he accumulation of

pulse energy at the edge of the modulator window brings about sensitivity to timing jitter

and pulse shape in the voltage signal driving the EO modulator. Discussions regarding

these issues are presented with the experimental and numerical results of Chapters 3 and

4: respectively. The pulse, once shapcd by the modulator transmission window, emerges

with linear polarization a t the incident side of the EDFA. We have followed a pulse over

a round trip.

2.5 Summary

\\é have given a bricf tlieoretical survey of relevant pliysical effects in multiple wavelength

lasers. In particular. we have reviewed tlie nature of gain saturation and the stability

criterion for multiple wavelengtli laser oscillation. M;t3 have also presented a brief review of

tccliiiiques den-ionstrated experinientally by other groiips to acliieve multiple wavelengtli

operation of EDFL's. No single technique presented itself as an outstanding practical

solution for multiple wavclengtli gcneration for an WDM optical network.

l i é have proposed a novel technique to induce stable multiple ivavelengtli oscillation in

an EDFL. Tlie role of FU'M in rediicing gaiii cross-saturation was esplained in detail. A

description of tlic operation of the proposed EDFL structure was given. higlilighting thc

importarice of FWlI. active niodelocki~ig and intracavity spectral slicing. The nuniber of

cffects at play and the iionlinear nature of EDFA gain saturation and FWM motivates tlie

~iririicrical simulations prescntcd in Chaptcr 4 for a qiiantitative theoretical treat ~ilcnt .

Chapter 3

Experimental Demonst ration

3.1 Introduction

Tlie novel fiber laser was constructed and characterized experimentally in tivo corifigu-

rations. 1% present tlic time average spectral and temporal cliaracterization of the laser

pulses iinder differerit operating conditions. The time average spectra of the pulsed laser

oiitpiit are encouraging. hleasiirement of the pulse substructure agrees well with that

espec teci from t lie spec t ra. al t tiougli imperfect modelocking was found. Finally. s t a t is-

tical measiirernents of the output pulse energy are reported to quantiS. the wave noise

inlierent. to the laser.

3.2 Laser Implementat ion

Two lasers were constructed according to Fig. 2.2 for the experiments reported here.

Both cavities were identical apart from the EDFA used. The details regarding common

cavity elements are described first .

A11 fiber components were connected to eacli other with angle cleaved, physical contact

connectors where possible to avoid back reflections. The exceptions were a t the input

and output ends of the EDFA's, where standard physical contact connectors were used.

The output coupler \vas a commercial (EsB) 10:90 fused-fiber coupler. Both polarization

cont rollers wcre based on commercial rotation paddles (Thor Labs FPC560) wound wit h

loops of industry standard single-mode fiber (Corning ShIF-28). Tlie three paddles of

eadi controller were wound witli 2. 4 and 2 loops of fiber to best approsiniate the 1/4

uxvcplate. 1/2 waveplate and 1/4 waveplate combination that can perform arbitrary

polarizat ion t ransforniat ions.

Tlie liiglily birefringent fiber used wrts a 12.4 ni segment of commercial "bow-tie" fiber

( Sewport F-SPPC- 15) fusion spliced to SMF-28 segments witli angle cleaved. physical

contact coiinectors. The birefringence was found by constriict ing a Lyot filtcr with the

fiber segment and fiber coupled polarizers. The Lyot. filter transmission was measured

with an optical spectrum analyzer and broadband spontaneoiis emissiori source. The fiber

birefringence corresponding to the Lyot filter FSR of 15.2 iz 0.5 GHz is 5.35 x IO-"

0.01 x 10-".

The cornmercial EO rriodulstor iised was a LiNbOs Mach-Zehnder interferometer

iiripleriiented witli planar waveguides (Unipliase S5-150-1-l-C2-Pl-AP). The insertion

loss at rnaximiini transmission was measiired to be approximately 6 dB. The on/off

estinction ratio was specified as 24.5 dB with ait RF electrode V7 of 3.0 V. The 3 dB

bandw-idth of the RF modiilation response was specified as 3.4 GHz. A DC supply was

used to bias the EO modulator to nul1 transmission. The RF source used to drive the EO

modulator coiisistecl of two parts. A frequency syntliesizer (Hewlet t-Packard 8656B) was

used t O t rigger a short pulse generator (Avtech AVMN- 1-PS-UTI) capable of gcnerating

sub-picosecond pulses. The pulse generator was specified with a trigger to output timing

ji t t er of 15 ps RMS. a rise time of less t han 100 ps and a fa11 time of less than 200 ps. For

the esperimental results reported k r e : an approximately square pulse of duration 1.01

ns and peak amplitude 4.48 V was used. Direct measurernent with a digital sampling

oscilloscope (Tektronix TIC1 1805) was used to adjust the pulse source to give the shortest

pulse possible with a symmetric square shape.

As nicntioned previoiisly. two different EDFA's were used in the laser cavity for the

espcrinients reported lierc. The first EDFA was used only for the test between laser

oscillatior~ and spontaneous eniission. This EDFA was constriicted within the laboratory

according t,o the diagram in Fig. 3.1. An optical circulator (JDS Fitel CR5500-3P)

clefiiied the direction for the signal to traverse t lie EDF.4. The EDF was a 7.5 m segment

froni Institute National d'Optique witli a doping of approximately 790 ppm by weight

of Er203. a niimerical aperatiire of 0.22 and an effective confinement factor of 0.5. A

wavelengt li dependent coupler (.JDS Fi tel CVD 915-T4-A) directed pump light from a

980.30 nni seniicondiictor laser diode (SDL 2564-145-BN) into the EDF. A loop mirror

was constriicted from a third polarization controller and a 50:50 fused fiber coupler

( EsB). Tlie loop mirror witli polarization controller served to reflect the signal for a

second pass tlirougli tlie amplifier. Tlie esceedingly short strarid of EDF did not provide

sufficient gain witli a single pass. and tliiis the double pass configuration was employed.

Tlie iiiiswturated gairi was rneasiired to be 15 d B wlien purnped with 74 mW froni the

semiconductor laser diode. Apart. from pliysical contact connectors a t the 50:50 coupler.

al1 fiber coiinections were niade witli fusion splices.

Tlie EDFA used for the reiriaiiiing experirnents reported here was a commercial unit

(Corning FGR1-S-045-01). An tinsatiirated gain of 28.2 dB at a wavelengtli of 1535 nni

was provided. The gain spectrum was limited by a 5 nni FWHM filter within the EDFA.

3.3 Lasers versus Spontaneous Emission Sources

We consider this question to distinguish the behaviour of our source from otliers [63]

wliere the distinction between laser oscillation and amplification of spontaneous emission

\.as less clear and the subject of debate. To determine experimentally whether a source

is a laser or a spontaneous emission source, one need only observe the presence of a

t hresliold optical pump power. Below t hreshold, the spontaneous emission is so low that

Diode Wavelength Dependent

Input - Output

Figure 3.1 : Tlic double-pass EDFA constructed for initial esperiments witli the multiple

wavelengt li laser.

t lie t liere is riegligible probabili ty of photon feedback and stimulated emission. Above

tlireshold. the sporitaneous emissiori rate is great enough thât there is an increase of

plioton density and st iniulated eniission eventuslly becomes significant . The stimulated

emissio~i rate will dominate over spontaneous eniission and the optical gain will finally

satiirate to equal the photon loss rate. Detailed treatments of the onset of laser oscillation

are niany. and Loudon's [35] is but one excellent treatinent of this fundamental subject.

A plot of output power versus EDFA pump power from the 980.30 nm semiconductor

laser diode is given in Fig. 3.2. Tlie pump power was determined by noting the input

pump current and using a supplied calibration curve for the laser diode. Cavity losses were

not minimized in the laser, but a threshold is clearly visible. The multiple wavelength

source is indeed a laser.

O6 20 40 60 80 100 1 Pump Power (mW1

Figure 3.2: The output power versus the pump power for the proposed fiber laser with a

lo~v gain (-15 dB) EDFA. Note the threshold indicative of the onset of laser oscillation.

3.4 Power Spectral Density

Tlic tinie average power spectral density of the pulsed laser output was investigated.

The FWM nieclianism for spectral broadening is power dependent and thus spectra

were observed wit,li different peak intracavity pulse energies. The intracavity power

waç viriried by controlling the EDFA punlp power and thus the EDFA saturation power.

Tlie pulse eriergy was det,ermined tlirough measurement of the output power with an Si

pliotodetector (Newport 818-IR). A simple calculation requiring knowledge of the output

couplirig and pulse repetition rate gives the peak intracavity pulse energy. A summary

of some laser parameters is presented in Table 3.1.

Power spectral densi ty measurements of t lie laser output were performed wi th an

AND0 AQû317 optical spectrum analyzer. The spectra are illustrated below in Fig.

3.1 for sis different intracavity pulse energies. The channel spacing was measured to be

36Of 5 pm (45.Of 0.6 GHz). The change in spectral shape with peak intracavity pulse

enerw is that expected from fiber nonlinearity. The triangular shape of the spectrum on

a logarithmic scale is coiisistent witli a secli2 function. The sech function is self Fourier

( Lyot filter FSR 1 45.2 GHz

Parameter

Unsaturated EDFA gain

1 Output coupling loss 1 0.5 dB

Value

28.2 dB

1 RF pulse widtli

- - -

EO modulator l o s ~

Connec t ion losses

-- 1 RF repetition rat,e 1 2.9015 MHz

6 dB

0.5 dB

Table 3.1: Parameters of the fiber laser used to obtain spectral. temporal and statistical

iiieasurenients. The R F repetition rate was tuned for resonance with the fast a i s patli.

triinsforniing [72]. irnplying a Iiyperbolic secant shape in field envelope. Sucli a sliape is

tliat of pulses esperiencing dispersion and self phase modulation. the omnipresent solitons

144th 40 p.] of peak intracavity piilse energy. the laser is operating just. above thresliold

and the spect rum is the narrowst observed. Wi t 11 increasing pulse energies. two trends

are observable. The envelope of the spect rum broadens and eacli ciiannel undergoes

spectral broacle~iiiig. in accordance witli tlie discussion of laser operation in Chapter 2.

With 2902 pJ peak pulse energies, the cliannel 3 dB widtlis were 6Of 5 pm. Allowing for

a 5 dB wriation i r i power. we observe G wavelerigth channels oscillating simultaneously.

3.5 Temporal C haracterization

From spectral measurements we may conclude that fine substructure is present in the

ptilsed output, as espected from our djscussion in Chapter 2. A total spectral width ex-

ceeding 2 nm corresponds to a sub-picosecond pulse width assuming a Fourier transform-

limited time-bandwidth product [74]. In order to resolve pulse substructure on tlie pi-

Figure 3.3: Output power spectral derisities measured wit h an optical spectrum analyzer

with a resolution bandwidth of 10 pm. The abscissae are wavelengths in units of nm and

the ordinates are optical powers in units of dBm. The peak intracavity pulse energies

are: A. 40 pJ: B? 255 pJ: C. 510 pJ; DI 989 pJ; El 1268 pJ: and F, 1902 pJ.

cosecoiid scak. the autocorrelation technique is used. We briefly discuss the technique

and ive report key features of the autocorrelat.or constriicted. We then discuss the ob-

served aut~ocorrelat ion traces.

Tlic purpose of the autocorrelation technique is to allow the resolution of temporal

features of opt ical pulses t liat are beyond the liniits of direct photodetection. Higli-speed

photodetectors are typically liniited to response times of -100 ps due to photocarrier

t.rarisit t imes (7.51. The tecliniq~ic used liere is t hat of background free. non-interferonietric

second liarmonic gcnerwtion (SHG). LVe briefly discuss this technique here: a more general

account of aitocorrelation is given by lppen et al. [74].

Autocorrelation by Second Harmonic Generation

A scliematic of the autocorrelator coiistructed is illustrated in Fig. 3.5. The principle of

operatiori is as follows. An incident pulse train is split irito t,wo equal intensity trains. An

optical trorihone is iised to varj- the delay between the two pulse trains as tliey are fo-

ciissed ont0 a crystal possessirig large second order nonlinearity characterized by a second

order soscept i hili ty X('). Tliree second liarmonic barns are generated wi t h comparable

efficiency if the incident beanis are aligned witli the crystal to match wavevector pliase.

The details of phase matcliing are given in Appendis C. Two second liarnionic beams

are generated the material polarization escited by each beani individua.11~. The ot.her

second tiarmonic beam is the result of material polarization escited by the interaction of

the two incident beams in the crystal. Using ail aperature located appropriately, tlie

SHG resulti~ig from the overlap of bot11 incident beams is selected and detected with a

liighly sensitive photodetector such as a photomultiplier tube.

T here are four notable festures of the autocorrelator construc ted for fiber laser char-

ac terization:

0 tlie scanning length of the optical trombone is approximately 10 cm: which allows

for a delay range of approximately k300 ps. The trombone consists of a retrore-

Figure 3.4: A sctiematic of the autocorrelator designed and built to ctiaracterize the pulse

substructure of the experimental fiber laser.

flector rnoiintecl upon a translation stage with stepping motor control. A minimum

10 p m resolution in position gives a minimuni 66 fs resolution in the autocorrela-

tion trace. The trombone \vas chosen with paranieters to allow for measurenient of

siib-picosecond struct.iire on pulses tliat were nleasured to be somewhat wider than

100 ps via direct pliotodetection.

the crystal used for SHG is commercially available LiI03 (Femtochrome) witli an

anti-refiection coating. The lmm thick crystal was cut witli a 30° angle between

crystal asis and normal. Type 1 [FI] phase matching was employed. for wliicli a de-

t ailed caiciilat ion appears in appendis C. Phase matciiing is polarizat ion sensitive.

and thus a polarization controller ivas required at the entrance to the autocorrela-

tor to enstire correct linear polarizat ion of the fundanient al beams witliin the M O s

crystal.

un opt ical cliopper and lock-in amplifier (S tanford Research Systenis 83ODS P) are

present. The total pliotocurrent shotnoise is proportional to the square root of tlie

electrical measurement bandwidtli [76]. The optical cliopper and lock-in ampli-

fier w r e iised to reduce the electrical measurement bândwidth to 0.3 Hz, thereby

significantly increasing tlie electrical signal to noise ratio.

the pliotomultiplier tube is a Hamamatsii mode1 R928. wliicli has a spectral re-

sponse estended to the near infra-red. Tlie quantum efficiency a t tlie second iiar-

nionic wavelength of interest (765 nm) is fi. while a t the fundamental (1530nm)

t here is a negligi ble quantum efficiency.

We now give a quantitative description of the SHG detected. referred to as the a u t e

correlation trace. The greater the pulse coincidence within the $') crystal, the greater

the second Iiarmonic. The second Iiarmonic generated in the present scheme will be of a

relative intensity S(T) given by (741.

n-liere r is t lie time del- between the fundaniental signais at the x(2) cristal. Tlie first

terni contributing to the second liarnionic is the second order temporal coberence.

wtiere E ( t ) is tlie comples electric field at time t and the integrat ion time is deterniined

by that of tlie measurement system bandwidth. The second term of Eq. 3.1 is a rapidly

oscillating function of delay time r due to interference of the two fundamental beams.

Only the average of r ( r ) . ivliicli is zero. is actually detected because the interference

pattern of the two fundaniental beams is spatially averaged over the entire volume of

non-colinear overlàp within the crystal. Lock-in aniplification gives a rnesurenient in-

tegration time of 300 ms resulting in an averaging of over one million pulses. Thus. the

autocorrelation trace will be proportional to the ensemble average of the second order

t,cmporal coherence integrated over a single output pulse:

Tlie pulse sliape can not be extracted from the autocorrelation trace because al1 field

phase inforrnat ion is lost . althougll some information about pulse subst ructure can be

detcrrnined.

3.5.1 Autocorrelat ion Traces

Having discussed the autocorrelation technique employed. we now present and discuss

experiment ally observed nutocorrelat ion traces. A commercial ED FA (Calmar Optcom

EDFA-02) was used to amplify the EDFL output to several mMr for increased SHG within

the aut,ocorrelator. The measurements are shown consistent with the observed spectra

and the contribution of FWh3 t,o laser operation is clearly shown.

Witli the same laser parameters and powers as used to obtain the results in Fig. 3.4,

autocorrelation traces were obtained and are presented below. A number of featmes

arc ivortliy of notc. Each autocorrelation trace consists of a number of peaks situated

iipoii a broad pedestal. Tlie delay between adjacent peaks is 22.1i10.1 ps. indicating

t h presence of same such periodicity in pulse substructure. Tlie inverse of the peak

separatiori is 45.2h0.2 GHz. agreeing ive11 to the periodicity of the frequency spectra

observed.

There is a rioticeable narrowing of the autocorrelation peaks with the increase of peak

pulse energ'- from 40 pJ to 255 pJ. with no noticeable change for further increases. The

narrowing of the autocorrelat ioii pcaks is the result of pulse substructure sbortening by

t lie combinat ion of optical Kerr effect and anomaloiis dispersion wit hin the fiber ring.

Su b-pulses of short duration accurriulate pliase by bot li t lie Kerr effect and anomalous

clispersioii. Tlie sub-pulses inay thiis take on a soliton-like cliaracter. In fact. it is a

property of nieclia witli arion~aloiis dispersion and Iierr nonlinearity that solitons typically

forni alter tlie propagation of an arbitrary pulse sliape of sufficient peak powr [73].

Solitons. as nonlinear pherioiiieiia. esliibit the property that tlieir duration is inverse11

proportional to tlieir energy (691. CVe tlius expect to see a reduction in tlie soliton-like

sub-piilsc duration as the energy of tlie total pulse increases.

Higher resolution scaiis (66 fs st,epping resolution) were used to probe more accurately

tlie cent ral peak of t lie autocorrehtion traces. Illustrated below are pulse substructure

widt lis infered from t lie airtocorrelatioii traces as a function of peak intracavity power.

A hyperbolic sccant. trarisform-limit,ed sub-pulse sliape was assumed to obtain the pulse

FWHM from the autocorrelation trace FWHM. It is emphasized that since the rneasured

SHG is in fact the ensemble average of the second order temporal coherence function. we

are in fact probiiig an average substructure width. The minimum feature width obtained

here is 590 fs. whicli corresponds to a spectral width of 4.3 nm. It has already been noted

however. tliat the potver spectraI density eiievelope had an approximate 2 nm FWHh4.

The apparent discrepancy is easily explained by the fact that oniy i3 fraction of the pulse

eilergy is in the form of sub-picosecond structures, and thus contributes to the average

Figure 3.5: Autocorrelation traces observed as a function of peak intracavity pulse e n e r w

The abscissae are delay tinies T in units of ps and the ordinates are second harmonic

intensities normalized to peak values. The peak intracavity pulse energies are: A! 40 pJ;

B. 255 pJ: CI 510 pJ: Dt 989 pJ; E. 1268 pJ; and F. 1902 pJ.

- . !

O ' O 400 800 1200 1600

Peak Pulse Energy (pJ]

Figure 3.6: Mininium pulse substructure width infered froni the cent.ra1 âritocorrelation

peak iinder tlie assuniption of iiegligi ble cliirp and ~ e c h , ~ () pulse shape.

potver spectral density proportionately. The large widtli pedestal clearly indicatcs a large

fraction of pulse c~icrgy is not found in the sub-picosecond structure.

\Ve now consider the ratio of tlie central peak lieigtit to the pedestal background. as

is prescntcd in Fig. 3.5.1 for various intracavity pulse energies. The background autocor-

rclation Level was taken to be tliat midway betwcen the central peak and adjacent peaks.

Tlie ratio rapidly increases froni approximately 2 at low pulse energies to 4 a t higli pulse

energies. We wish to assign nieaning to tlie observed ratios. Hence. we consider tlie tlieo-

ret ically espectcd second order temporal coherence for cliaotic light burst s with Gaussian

p o w r spectral density. partially modelocked pulses and also perfectly modelocked pulses.

It was sliown by Grutter et al. [77] that for chaotic liglit bursts witli Gaussian pourer

spectral densi t.y. the second order colierence func t ion sat isfies the condition g(2 ) (0) = 2

and g(2) ( T ) = 1 for T greater tlian the colierence time of the chaotic light. Thus. we have

a ratio of 2 between central peak and pedestal. A perfectly modelocked pulse train is

predicted to satisfy 9(2) (0 ) T/2At where At is the pulse width and T is the pulse train

repetition rate. The factor in an exact relation depends upon pulse shape. Furthermore,

1 ,:

2b 800 1200 1600 Peak Pulse Energy [pl]

Figure 3.7: Ratios of central peak SHG to pedestal SHG as a function of peak intracavity

piilse energy.

t k r c is prcdicted to be zero pedestal for perfect modelocking. Partial rnodelocking results

iii a reductioii of g ( 2 ) ( 0 ) . and an increase in backgroiind pedestal. Calculation of accurate

peak to pedestal ratios is dificult for intermediate degrees of temporal coherence and

results Vary depending upon the fornialisni one aclopts to mode1 partial modelocking

[TT. 781.

Looking I~ack upori the observed autocorrelation traces. we can see that the low

pulse enerm traces correspond well to that espected for Gaussisn noise bursts. witli a

colierence time rouglily equal to G ps. With increasing pulse energy, the ratio of peak to

pedestal reaclies approxiinately 4. There is a reduction in phase noise among the modes

of the cavity. meaning t lie cavi ty modes have become part ially modelocked. However.

tlie peak to pedestal ratio is still far below wliat is expected for a perfectly modelocked

source. From the autocorrelation ratios, Ive sec that partial modelocking was irnproved

by increasing the pules energies, but there was st.ill significant phase noise present even

at the highest energies.

3.6 Statist ical Measurements

Tlie power spectral densities indicate muitiple wavelength operation was achieved. al-

tliougli perfect modelocking was not achieved. Tlie statist ical distribution of output

piilse energy was measiired in order to deterniine tlie intrinsic noise of the laser wit h the

saine laser parameters as in Table 3.1. The esperiinental setup. illustrated below. was

used to nieasurc pulse energy. Tlie operating peak pulse energy was chosen to be 1430

p J . tlic cnergy a t wliicli we observe close to iniiliniiim pulse substructure widtii. The

ptiotocurrcnt from a 125 AIHz bandwidtli InGaAs p-i-n photodiode (New FOCUS 181 1)

was nicasurecl witli a digital oscilloscope of 100 MHz bandwidtli and transferred to a

coiiipiiter for processing. The oscilloscope window encompassed seven pulses which iwre

cacli iritcgx-ated numerically to determine the relative pulse energies. The apparatus \vas

used to collect 10 000 series. eacii series corisisting of seven pulse energies.

3.6.1 Pulse Energy Correlation

Uk qiiu~itified the inlierent noise of the laser t,lirougli analysis of the statistical distri-

butioris of tlie pulse eiiergies. The fluctuation of pulse energy was significant. with an

obscrved staiidard deviation eclual to 0.395 of tlie niean piilse energy. Scatter plots of

iiorinalized pulse energies against eacli otlier are ill~st~rated in Fig. 3.8. For instance,

grapli A records the second pulse energy âgainst tlie first pulse enerar for each series. A

distribution concentrated dong the line =En indicates a liigh correlation between

first and second pulse energies. For a given pulse, tlie energy of the following pulse will

most often be almost equal to the given pulse eiiergy. As the number of round trips be-

tween observed pulses increases. a greater deviation in energy is indicated by espansion

of the distribution outwards from the equal energy a i s .

The correlation functiori of pulse energy rneaçurements quantifies the correlation of

measurents between pulses sorne nurnber of round trips apart [Tg]. A value of zero indi-

Figure 3.8: Scatter plots of normalized pulse energies over 10 000 collected series. Tlie

abscissae are first pulse energies En and the ordinates are: A, second pulse energies

B. fourth pulse energies En+3: C, sisth pulse energies and D. seventh puIse energies

O O 1 2 3 4 5 6

Round Trips

Figiire 3.9: Tlie correlation between energy nleasurements of piilses separated by the

indicated number of round trips.

cates no correlation and statistical independence. while indicates perfect correlation

aiid anticorreiation. respectively. The correlat.ion f~inctioii is plotted in Fig. 3.9. Six

round trips are sufficient to bring tlie pulse energies close to statistical independence.

3.7 Conclusions

Wc have presented here nurnerous experiments characterizing our fiber laser source. A

brief suriirnary of t,lie key experirnental results is.

A tlireshold in output power versus optical pumping power was observed. indicating

the onset of laser oscillation.

m A timc average power spectral density with a 45,Of 0.6 GHz channel spacing was

measured. corresponding well to tlie Lyot filter FSR. Spectral broadening of u p was

observed wit h a features characteristic of that due to Kerr nonlinearity within fiber.

Simiilt.arieous Iasing at multiple wûvelengt lis was thus sirccessf~~lly demonstrated

with 6 adjacent channels eshibiting less than 5 dB powr variation.

a Arltocorrelation by SHG \vas used t,o cliaracterize pulse substructure. Autocorre-

lation peaks were observed at 23.1 ps in te~vds as espected from the Lyot filter

FSR. A 4:l peak to pedestal ratio was observed. indicating partial niodelocking

\vas sichieved.

hleasiirenient of pulse energies by direct ptiotodetection revealed fi uctuations with

staridard deviat ion 0.395 of tlie mean pulse energy

Chapter 4

Numerical Simulation

4.1 Introduction

\Ve present hcre nunierical simulations whicli clualitatively predict the experimentally

observed laser perforniance. Nunierical simulations give us it means to investigate origins

of instability. Tlie tlieoretical niodel appiied to describe the operation of the fiber laser

is describeci first. witli care taken to describe the approximations made. The numerical

tecliriicpes used to irnplenient t,lie theorctical niodel of the EDFL are briefly reviewed

due to tlieir importance in obtaining meaningful results.

The nunierical simulation resul ts are presented for t lie EDFL configuration inves-

t igated in tliis tliesis. Agreement with esperirnental results are good. l~owever sonle

discrepiincies appear and ;ire discossed. Finally the issue of stabiiity is tackled witli tlie

proposeci insertion of a fast saturable absorber. Tlie reasoning, modelling and simulation

results of tliis solution are discussed.

4.2 Theory

The theoretical mode1 we employ for the EDFL demonstrated is based upon tlie propa-

gation of a broadband (several nanometers) pulse tlirough the laser cavity. We describe

t iie pulse circulating tlirougli the ring cavity with a single carrier and a single envelope

t h t c m possess fine su bst ruct ure.

The laser was constructed in single niode fiber and thus the electric field cân be

described byl.

1 E(r. t ) = - ( é F ( x . g)A(r. t) exp [i(poz - dot)] + CL)

3 (4.1)

where ê is a polarization unit vector. F ( x . g ) is tlie mode field distribution. A(=. t ) is the

pulse envelope. do is the carrier propagation constant and ~ J O is the carrier freqiiency

The puise erivelope gives 110th amplitude and phase. thus completely characterizing the

piilse. The Fourier transforni of the envelope is here defined as.

For the propagation of the enwlope through fiber. it is required that tlie envelope be

a basebarid funct.ion. In other words. the spectral components of A(z . w) are possibly

noiizero strictly for « WO. CVe disciiss each non-trivial elenient within the laser cavit,y.

supplying t lie equat,ions used to transform tlie envelope appropriately. A schernat ic of

the cavity element niodels and the propagat.ion of A(:. t ) is presented below in Fig. 4.1.

Standard Single-Mode Fiber

Tlie propagation of a pulse th-ougli single mode fiber is described by a nonlinear Schrodinger

(NLS) eqiiation [69].

wliere T = t - z/,u, is the time in a reference frame moving with the group velocity

eg of tlie pulse. & = #/3/C)w2 is the group velocity dispersion and -, is a nonlinea,rity

coefficient given by.

'refer to appendis D for a description of this expression

Gaussian White Noise 1

Round Trip Modulation

Nonlinear Schrodinger Equation (4.3)

Transmission Equation (4.7)

Transmission Equation (4.8)

Doped Fiber Propagation

Ginzburg-Landau Equation (4.lO,4.ll)

Figure 4.1: Ari illustration of tlie tlieoretical mode1 used for iiumerical simulations of tlie

fiber laser esperinientally demoristrated.

wliere ri? is tlie nonlinear Kerr index of the glass and ..leff is aii effective area related to

t lie mode field by.

111 tlic expressions abow. it is assumed tliat IA(2. T)12

(4.5)

is normalized to instant aiieous

power w i t h t.he pulse. It. is of interest to look at the form of the NLS equation in the

Fourier dornain to see more clearly the effect of propagation through fiber.

The above equation clearly illustrates the quadratic phase accumulation of the pulse due

to group velocity dispersioli. Also. tlie optical Kerr nonlinearity takes the form of an

iritegration over al1 possible contributions from FWM. We see e~plicit~y the FWM within

the envelope that will result iii both the coupling between wavelength channels aiid the

spectral broacieiiing of eacli ch ine l .

Lyot Filter

Nonlincar polarization rotation effects are assumed negligible within the Lyot filter. The

large teiilporal walk-off due to group velocity niismatcli does not allow nonlinear Kerr

induced phase to accumulate between the two polarization cornponents within the bire-

fringent fiber [69]. The Lyot filter tlius linearlv filters the pulse envelopes according to

the amplitude transmission equation derived in Appendix B? rewritten in the form.

wliere a. b and Q> are constants. For the simulations: we assume a = b = 1 / fi and = 0.

Tliis corresponds to the physical situation in which light is launched and analyzed at a/4

radians to the principle axes of the birefringent fiber.

?in the context hem, wve use H ( w ) rather than t t o avoid confusion with the time variable

Electro-optic Modulation

Tlie electro-optic rnodulator is a simple L~lacli-Zehnder interferorneter witli a variable

refractive index in one ami. Application of a voltage to the single arm effects refractive

indes cllanges via the electro-optic effect. Neglecting the linear wavelengtli dependence

of tlic liàlf-NYLW voltage k, reqnircd to switch tlie modulator. the temporal transmission

fiinction is [YO].

- wliere Jz is tlie peak niodulat.or transmission. V ( t ) is tlie voltage appearing across

the intcrferonieter ami. Tlie forni of the voltage applied is assuniecl t,o be a ratlier general

r q t ) - = 1 - exp 11;

wliere at is tlie FWH h I of the voltage functioii. Tlie bias is present to effect higli transmis-

sion owr tlic FWIlhI of t,lict voltage hnction. Tlie negative sign preceeding the Gaussiaii

could just. as tvell have been cliosen t.o be positive. the only effect being the sign of the

indiiced cliirp on the enwlope a t the edges of the modulator wiridow.

EDFA

The e~ivelope is propagated through the EDFA with a NLS equation including gain and

gain dispersion. Tlie modified NLS equation is in fact a Ginzburg-Landau (GL) equation

of tlie forni [69j.

where g is the gain of the amplifier per unit length of fiber and T2 = 2/Aw is tlie

liomogeneous lifetime inversely proportional to the homogeneous linewidth Au. The gain

term ciearly leads to amplification of tlie envelope. The gain dispersion term is included

to incroporate the finite gain bandwidth of the EDFA. The optical Kerr nonlinearity and

the fiber dispersion are included in the (GL) equation. The inversion of the erbium ions,

and t hiis the gain g. was assumed constant dong tlie lengt h of tlie fiber amplifier for each

roii~id trip. From the tirrie dependence of tlie inversion derived in Appendis D. we find

tlie gain g is governed by the eqiiation.

d i e r e geQ is the unsaturated gain. Tl is the population relaxation time. I is the average

optical intensity over tlie fiber area and ISat is tlie saturation intensityaveraged over

tlie fiber mode area. The population relaxation time is 10 ms. while a round trip is

approxixiiately 350 ns. Ué tllus assume a constant EDFA gain per round trip.

4.3 Numerical Met hod

Tlie analysis of the propagation of the enve!ope A(- . T) is trivial for the Lyot filter.

oii t put coupler and elect ro-optic niodulator. We treat t hem liere as lumped elenients

acting upon the envelope in either the time or frequency doniain. However. analysis of

t.lic propagation of the envelope through the doped and undoped fiber requires integration

of Eqs. 4.3 and 1.10. Tlie metliod cliosen tiere is tiie split-step Fourier metliod [69]. We

brieffy consider the method liere in the contest of the NLS eqiiation. Tlie form of the

NLS eqiiatioii is that of propagation by the suni of a linear operator D and a nonlinear

operator N . where.

The NLS equation takes the simple form 8 A / & = (b + N I A . The split-çtep Fourier

method consists of taking the step size Az = h for numerical integration sufficiently small

that we may assunie the operators D and N cornmute. Under said assumption. step wise

integration may be perfornied with the forma1 symnietric recipe,

(2 ' ) (JI'"' f i ( 2 ) d ~ ' ) exp (:fi) A(io T) A(z + h.. T) = exp -D exp

The linear esponential operators are performed in tlie hequency domain. We take ad-

vmtage of t,lie Fourier operator eqiiivalence a/t9'T @ -id in applÿing esp(h.b/2). Tlie

linear operat or is convenient ly diagonal in the frecpency domairi. The nonlinear oper-

ation is perforniecl in the time doinain. wl~ere in tliis case ive use a simple first order

approsirnat ion.

\fi' take advantage of tlie siniplicity of tlie forni of tlie optical Kerr nonlinearity ïepre-

sciitecl in the tirne tloniain. Tlie present compiitational ease of applying Fourier trans-

fornis rcndcrs t lie split-step Fourier met liod attractive.

Tlie nuiiicrical algoritlirn was impleniented in ii AzIATLAB environment. In order to

niiriiinize coniputation time. adaptive step sizing was used. It is important to choose

tlie step size siifficiently small so tliat the phase of the erivelope changes gradually with

eacli step. This is a reqiiirement for accurate sirriirlation witli t tie spli t-step rnet liod.

LfTe clioosc liere to keep tlie pliase c h i g e below ~ / 2 for eacli step. Altliougli the pliase

cfiange due to dispersion is negligible over a round trip. tlie Kerr nonlinear pliase shift

rriay not Iw. Estimation of tlie masimiim phase accumulation in the envelope for a single

step leads us to tlie condition.

wlicre tlie riunierical factor IVES cllosen as a comproinise between cornputational time

and accuracy. For t lie simulations presented here. we chose r ) = 0.1. The value was based

upon tlie acciiri-icy of simulation results for the propagation of a 1 ps soliton over many

soliton periods. Tlie step size was iipdated once per round trip.

lntegration of the GL equation 4.10 was perfomed also with a split-step Fourier

metliod. Tlie first modification required is the generalization of the linear operator D to

incorporate tlie gain and gain dispersion of Eq. 4.10. The second modification is in the

clioice of step size. Not only may the phase of the envelope change significantly with a

siiigle step. but tlie amplitude will change as well in the presence of gain. The step size

is chosen to 11e the niiniinum of that determined IF Eq. 4.15 or h g < qt. wliere again qt

is a nunierical factor cliosen as a compromise between computation time and accurary.

For the siniiilations reported Iiere. ive liave chosen q' = 0.2. This ensiires the envelope

does not grow by more than a factor of 1.2 per step. Again. the step size was updated

once per round trip.

4.4 Simulation Results

,\;unierical simulation of tlie demonstrated laser was undertaken according to the schenie

described. The pulse envelope was propagated t hrougli the laser cavity with Gaussian

wliitc noise as the initial conditions. Tlie pulse envelope was discretized iiito 8192 Sam-

ples ilcross a total time window of 800 ps. The relevant laser parameters iised in the

simulations are presented below in Table 4.1. rnatcliing tliose from experiment. We

have assunied = -20 ps' km-L and = 3 rad km-' as for standard SMF-28 fiber.

Wc assunie tliese same vztliies for the tiighly birefringent fiber and the EDF within the

coiiiiiiercial EDFA because these parameters are of approximately the same order of mag-

iiitiide as in SMF-28. The EDF length was estimated knowing the experimental cavity

resonance freqilency and direct measiirement of al1 other fiber lengtlis. The unsaturated

gaiil per iinit length was thus estimated. The rernainiiig erbium ion parameters were

tüken to be tliose typically found (811. The modulator pulse width was assumed t o be

40% of that in experiment to reduce the time required for simulations. The modulator

window widtli is less important tlian the window shape because the pulse energy typically

bunches up dong a window edge.

Tlie last important parameter still not mentioned is that of the modulator pulse rep-

etition rate. which is not necessarily exactly equal to the fast (slow) cavity round trip

time. The repetition rate was chosen to be that which gave the best correspondance

1 Undoped Fiber Lengtli 1 Lu 1 38 m

Parameter

1 Doped Fiber Lengtli 1 L d 1 30 m

S y mbol Value

Unsaturated Gain

1 Dipole Depliasing Time 1 T2 1 500 fs

Sat iirat ion Power

1 Population Relasation T i m 1 T l 1 10 ms

g

1 Group Velocity Dispersion 1 A 1 -20 ps2 km-'

0.23 ni-'

Ps,~ 250 /LW

1 klodiilator SuperGaussian Order 1 ni 1 4

Kerr Xonlinearity Coefficient

Lyot Filter FSR

Peak h~IocIiilator Transmission

Table 4.1: Laser simulation parameters and their respect ive values taken to match those

in esperirnent iinless otlierwise noted in the text .

with the csperiinentally observed data. A 2 ps increniental (decremental) detuning per

round trip from the slow (fast) a i s repetition rate was chosen. This detuning resulted

iii nuinerically siniiilated beliaviour wliicli qualitatively best matched the time average

esperir~ient~al resiiltjs reported i r i Chapter 3. Al1 the experimental results qiioted were per-

formed following no less tlian 600 round trips with Gaussian white noise initial conditions

for the pulse envelope. The population response of the EDF was assumed instantaneous

during the initial 600 round trips so tliat convergence towards the average inversion would

be accelerated.

i

Au

TEOM

The simulated time average output spectrum and autocorrelation trace are iliustrated

below in Figs. 4.2 and 4.3. The simulated peak intracavity pulse energy was an aver-

-

3 rad km-'

45 GHz

0.20

Frequency (THz]

Figure 4.2: Tlie simulated t h e average spectrum of the fiber laser assuming slight de-

timing from the slow asis. Tlie spectriim is normalized to unit peak.

age 1390 pJ. in accordance witli esperinlent where the energ?; was varied from 40 pJ

to 1920 pJ. Tlie tiieoretical spectrum eshibits the 45 GHz cliannel spacing as espected

and approsiniatelj~ liiiear roll-off of t,lie spectral envelope. Bot,li featiires agree well with

the esperimental data of Fig. 3.4. The average autocorrelation trace corresponds qual-

itat ively wi tli t lie esperin~entally observed traces of Fig. 3.5.1. Autocorrelation peaks

separated by 33.1 ps are readily observed iipon a broad pedestal. The 3:1 peak to pedestal

ratio of the numeric-ally siniiilated trace indicates poorer modelocking than the esperi-

inentally observed 4: 1 ratio. Also. the simulated pedestal width is a factor of 2 wider

t han t lie experimentally observed pedestal. In general \ve have qualitative agreement,

indicating our model may at l e s t be useful for qualitative predictions.

There are two siniplifications in the model which may affect the agreement with

experirnentâl results. First? the nonidealities of the modulator transmission have been

omitted. No timing jitter in the transmission window has been assumed, although a

nlaximum jitter of 100 ps was observed in the electrical pulses driving the EO modulator.

'\

O I 1 I I

-400 -300 -200 -100 O L

1 0 0 200 300 .! Time [ps]

Figure 4.3: Tlie siniulated tirne average autocorrelation trace of the fiber laser assuniing

slight detuning froni t.he slow axis.

We have also neglected pulse distortion due to tlie capacitance of the electrodes and sligtit

inipedsince rnisiilatch wi tli tlie pulse generator. Botli effects make an esact niodelling of

t lie trans~iiission window sliape estrcmely difficiilt at best. As will be discussed further.

t lie modula tor window sliape may be significant in deterrniniiig laser dy narnics.

Tlie second simplification was done with regards to the EDFA. A gain response based

iipon a siniple. liornogeneously broadened two level systein was used in tlie rnodel. We

Iiavc neglected siicli difficiilt topics as spectral hole burning of deeply saturated amplifiers

resulting from gain inhomogeneity. We have also ignored the manifold nature of the

ground and escited states of erbium ions with silica fiber. A niore rigourous treatrnent

of tlie EDFA rcquires more detailed knowledge of the EDF than is currently available in

t Iie literature. In particular. the transition wavelengt hs, the inliomogeneous broadexiing

of each transition. and relative strengths of each transition involved are required. The

detailed niodellirig of deeply saturated amplifiers thus rests upon new approaches to

spect roscopic s t udies and t heoret ical modelling of EDF.

Nonetheless. our numerical mode1 allows us to probe the qualitative nature of tlie

dyiianiics of the fiber laser. giving insight into the physical mechanisms involved. The

instantarieous p o w r of the simulated output pulses is illustrated in Fig. 4.4 as a function

of round trip number. A number of interesting feat ures are apparent. We see the pulse

cnerg. gatlier towards the rising edge of tlie modulator window. With each round trip.

t hc portion of the pulse which travels along the fast axis leads aliead by 22.1 ps. An

increase in effective loss is created by the continuous shifting of the pulses traversing tlie

fast path of tlie Lyot filter. The dope of the edge of the modulator window determines

the rate at which eriergy is lost due to t his shifting. A closer examination of the striations

in ttic profile evolution plot reveaIs that a 2 ps per round trip shift can be observed. as

obviously espected due to the detunirig frorn resonance witli the slow axis.

Rapid fluctuations in pulse sliape are visible. mucii like pulse break up espected

due to Kerr indoced modiilûtion instability [82]. Nonlineariy-induced variations in the

rcfract ive index lead to ins t abili t ies t liat typically result in sub-picosecond formations. An

ecliiivalcrit interpret ation is spect r d broadening due to F Wh4 and the resiilting beating

betweeii tiie various spectral components gives rise to a rapidly varying temporal pulse

profile.

We take advantage of the numerical simulations in anotlier respect by probing the

clynaniics of tiie total pulse energy as it propagates through the laser cavity. The pulse

energy t ransmissioii t.hroiig1i tlie Lyot filter. electro-optic modulator, EDFA and undoped

fiber are individually plotted for 800 round trips in Fig. 4.5. From the pulse energy

transmission plot we can discern the elements giving rise to the output pulse energy

fluctuations observed. Pulse energy transmission t hrough the spectral slicing filter is

observed t.o fluctuate by as much as 25%. The variation in transmission is caused by the

nonlinear generation of spectral components by FWM which are removed by the Lyot

filter. In particular, broadening of the individual channels through FWM contributes to

a nonlinear loss through the Lyot filter. Similarly, variations in EDFA gain are caused

-200 -100 O 1 00 200 300 400 Time [ps]

Figurc 4.4: Iristatitaiieous pulse power indicated by cidark sliading versus tinie referenced

to peak modolator transmission on the abscissa. The evolution of pulse profile is shown

witli correspondiiig round trip nunibers dong the ordinate. The modulator window is

assumed to be a 400 ps FWHM supergaussian.

filtering of spectral component s lying outside t lie gain bandwidtli [83] determined

t lirough tlie dipole depliasing tinie (recall Au; = 2/T2 for a homogeneous transition). The

nonlincar generatioii of spectral components tlirough FWM outside tlie gain bandwidth

varies froni round trip to round trip. Transmission variation through the modulator is

siniply il result of the sliifting of energy out of the transmission window. Successive round

trips give different pulse energy distributions at the modulator window edge and thus a

variation in loss following propagation throught tlie Lyot filter. Transmission througli

undoped fiber iinsurprisingly gives no variation in transmission. The conclusion that

caii be drawn froni the above analysis is tliat spectral broadening by FWM gives rise

to cavity losses whicli var i significaiitly froin round trip to round trip. Although FLVM

coiintcracts gain competition to allow multiple wavelengtli laser oscillation. undesired

spectral broadeniiig of eacli chaniiel leads to pulse energy instability.

Tlie iiiagnitude of t lie siniulated pulse energy Hiictuations is less tlian tliat observed

in esperiment. Tlie standard deviation of the ouptiit pulse energy was simulated to

be a fi-actio~i 0.208 of the niean pulse energy. The corresponding ratio is 0.395 for the

esperiniental results. We tlius conjecture t hat tlie esperimental pulse energy fluctuation

would be reduced witli a decrease in tiie RF source timing jitter. The distribut ion of pulse

energy dong thc EO modulator window cdge results in extrenie sensitivity to any jitter

in tlie n~odulator window. A single instance of a 100 ps jitter is sufficient to completely

nul1 the transmission of a 100 ps pulse that would normally pass the EO modulator on

the edge of the window.

4.5 Pulse Shaping

In the previous section we have shown how the numerical mode1 we have employed

predict s similar bel-iaviour to the laser esperiment ally realized. An undesirable feature

of the output pulses inferred from the numerical simulations is the randomly fluctuating

1 # 1 I t 1 1 O0 200 300 400 500 600

Round Trips

Figiire 4.5: Pulse eiiergy transmission through various cavity elemerits are plotted above

as a functiori of roiind trip number: TI Lyot filter. T2 electro-optic modulator. Tg EDFA

and T.l undoped fiber.

pulse profilc. We propose here a method for pulse stiaping wit.hin the laser cavity witli

t lie intent t O create a more uniform t rain of pulse substructiire.

Before iritroducing the method and esperimental results. we consider in a bit more

detail tlie nature of the fluctuations. Short pulse substriictures in tlie simulations are

typically subpicosecond in width and multiple Watts in peak power. The first order

soliton p o w r for the fiber used gives a peak power of 20 W assuming 1 ps pulse widtli3.

The pulse substructiires are thiis not far from that of low order solitons. Considering the

siib pulses as a collection of low order solitons allows us to use the conclusions drawn from

pert urbat ive t reat ments of interact ing solitons [84. 8.51. The principal result of interest

to us is tJhat regularly spaced solitons are only very weakly bound to each other. and are

tlius very emily perturbed. We thiis can not expect the Lyot filter to prodiice a pulse

substriict tire \vit li perfect periodicity. Modulational instability inevitably results in the

randoni fliict iiat ions reported liere.

We show liere tliat pulse quality is dramatically iniproved throiigh the introduction

of a nonlinear pulse shaping element. In the language of lasers. an additive pulse mod-

clocking technique is being proposed. An illustration of the proposed structure is given

in Fig. 4.6. An ultrafast saturable absorber based upon nonlinear polarization rotation

lias been iiicorporated into the laser. Demonstrated esperimentally (861 and analyzed

tlieoretically (87. 58. 891. nonlinear polarization rotation nom- often finds application as a

saturable absorber for passively modelocked lasers (33. 901.

Nonlinear polarization rotation is simply the result of an intensity dependent. index

of refraction. A pulse experiences an intensity dependent polarization rotation since the

indices of refraction dong the fast and slow axes change according to the power launched

along each mis4. A polarization analyzer at the terminal end of tlie birefringent section

of fiber acts to project the pulses from each axis to a common polarization. We thus

%alculated using the fornitda quoted in (691 "a detailed analysis [87, 891 reveals subtleties that we need riot consider Iiere.

RF Source Polarization

1 Controller Polarization Controller

-. - *--. * .- .. -.c,

Low

EO Modulator Birefringence EDFA Fiber Polarization Analyzer

I

Polarization Controller Birefringence Controller

Fiber

Figure 4.6: The rnoclified laser cavity here incliides an al1 fiber saturable absorber consist-

ing of two polarization controllers. a lengtli of low birefringence fiber and a polarization

analyzer .

liaw a rioiiliiiear iiiterferonieter of the gerieric sort used in additive pulse niodelocking.

Appropriate adjiistment of tlie polarization analyzer will give a transmission increasing

with intensity. a saturable absorher in other words.

We reiterute a n important point liere. Tlie Lyot filter can be approximated as a

linear interferorneter because of the great disparity in group velocities along each mis.

The sat urable absorber eshibits its nonlinear beliaviour because the birefringence of the

fiber segnient is siifficiently low tliat piilses travel togetlier. allowing significant nonlinear

interaction.

Insertion of saturable absorption into a fiber laser cavity is a simple and common

technique to achieve modelocking. High intensitÿ peaks. experiencing less loss. are more

energetically favourable within a laser cavity and hence short pulse operation ensues. In-

cident pulses are shortened with each pass through the saturable absorber. The ultimate

width of the pulse is usually lirnited not by the ultrafast response times of the optical

Kerr effect (100 fs [Gl]). l>ut by the finite gain bandwidth of the EDFA within the cavity.

The piilse sliaping mechanisln introduces greater nonliriearity into the laser cavity.

The favourable reduction iii loss for higli peak power subpulses makes a regular train of

pulses cnergetically niore favourable tlian the fluctuating pulse structures of Fig. 4.4.

The Lyot filter acts to introduce temporal coherence with periodicity on the order of

22.1 ps ttirougli coupling of pulse coinponents 22.1 ps apart. The combined effect of the

Lyot filter and saturablc absorber effectively increase the binding energy of tlie pulses.

iiiaking tlie desired pulse substriicture more resilient against perturbations due to EDFA

spectral filtering and temporal modulation.

The arguments above were tested wit h a numerical simulation of the new cavity struc-

tiirc in Fig. 4.6. The ultrafast saturable absorber was assumed to have instantaneous

responsc. since WC have noted tliut the EDFA gain bandwidth d l be the limiting band-

width and not tlie nonliiiear resporise of optical fiber. The absorption \vas thus assumed

to take the form.

wliere a. is tlie unsaturated absorption and P,,, is the saturation poiver. Typically

available valiies of cru = 0.9 and P,,, = 5 Mi were cliosen liere [86]. The peak transn~ission

throiigh the iiiodiilator was assumed to be an optimistic TEohr = 0.9 for the simulations

wit li saturable absorbers.

T h results of nuinerical siniulations are plotted below in Figs. 4.7. 4.8 and 4.9.

First note that multiple wavelength oscillation is still occuring, although t.he contrast

between the channel power spectral density and background spectral density has been

reduced. The change in pulse structure is more dramatic. No longer a rapidly fluctuating

profile, the pulse has taken on a much less random and more periodic structure. The

quality of the modelocking is superior because of the reduction in noise among cavity

modes as indicated by the vastly irnproved peak to pedestal ratio. This sliould not corne

as a surprise since lasers modelocked with saturable absorbers based upon nonlinear

polariziit ion rotation are known t O eshibit escept ional pulse uni formi ty [CIO].

-40 1 !

- 1 -0.5 O 0.5 1 Frequency [THz]

Figiirc 4.7: Simulated p o w r spectral density versus opticai frequency for the inodified

laser cavity iiicludiiig an iiltrafast sat,urable absorber. The spect rum is normalized to

unit peak.

4.6 Summary

This cliapter Ilas presented t lie t heoretical mode1 used to perforrn nunierical simulations

of the proposcd fiber laser. Tlie mode1 did not incoporate jitter iri the EO modulator

transiiiissiori window and the EDFA was approsimated with homogeneoiisly broadened

two level systems. but qualitative agreement between experirnent and numerical simula-

tion was found. Variation in loss from round trip t o round trip was predicteci. although

riot to the estent observed in experiment. Spectral components generated each round trip

by FWhl were found to be filtered by the Lyot filter and EDFA in a nonlinear manner.

Furt herniore. t lie simulations indicate the intrinsic pulse energy fluctuation from the laser

is less than that observed. We conjecture that pulse energy fluctuation can be reduced by

Figure 4.8: The siniulated time average autocorrelation trace of the fiber laser including

an uit.rafast. saturable absorber.

t h reduction of timing jitter in the RF pulse source driving tlie EO modulator. Finally.

ail ultrafast saturable absorber was found to dramatically iniprove tlie pulse profile in

niixnerical simulations.

. . '. ; :. O - , - - -400 -300 -200 -100 O 100 200 300 400

Time [ps]

Figure 4.9: Simulated instantaneous puise power indicated by dark sliading versus time.

Ar1 ultrafast saturable absorber has been assumed.

Chapter 5

Concluding Remarks

\I7c have int roduced in t his t liesis a novel technique for siniultaneoiis miilt iple wavelength

generation in an EDFL. striving to satisfy the criteria oiitlined in Chapter 1 for practical

application in WDhll networks. The novelty liere lies in the simultaneous use of FWhI

aiid intracavity spectral slicing to acliieve miiltiple wavelength laser oscillation. Following

an introduction to tlie cliallenge in developing stable. multiple wavelength EDFL's. a

brief siirvey of previo~is tectiniqiies demonst rat.ed was given in Chapter 2. The principle

and esperiinent al irnpIenient ation of the laser proposed liere was t hen introduced. A

sumniary of tlie esperimental setup was given in Chapter 3. Temporal. spectral and

statist ical characterization of tlie output laser pulses were presented. The time average

spectra indicated niultiple wavelength oscillation was acliieved. but fluctuation in pulse

energy was significant . In Chapter 4. LW introduced tlie theoretical mode1 wliich formed

the basis for numerical si~nulations. Alttiough a number of approximations w r e made,

qualitative agreement was found between experiment and numerical simulation. Finally.

we proposed a modified laser structure including an al1 fiber ultrafast saturable absorber.

Nunierical simulations indicate the expected pulse shaping yields improved pulse profiles.

The important dernonstrations of this thesis include,

0 the combination of FViW and intracavity spectral slicing has been shown as a

met hod for niiilt iple wavelerigt h laser oscillation in EDFLms

t,irne average spectra of G channels with 5 dB power variation were denionstrated

with a 45 GHz channel spacing

nunierical siniulat ions of a laser incorporat ing an ult rafast saturable absorber indi-

cate iniproved pulse qiiali ty niay be espected in esperiment

The work perfornied liere provides avenues for fruit, ful researcli into multiple wave-

lerigt li sources.

m espcrimental demonstration of the EDFL with a tuneable R F source with less

t iniing jit ter sliould be perfornied. Timing jitter is believed to be a lirniting factor

i r i the EDFL performance.

espcririient al denionstrat ion of t lie niodifiect E DFL incorporat ing a saturable a b

sorber sliould be performed. The espected iniprovement in pulse cpality could

easi ly bc evinced froni ail au tocorrelat ion rneasurement .

rn tlie nimierical siiiiulations of the laser. although giving qualitatmive agreement with

csperirnent. reqiiire a niodel tliat can incorporate inhoniogeneous effects tliat be-

corrie important in strongly saturated EDFA's. A comprehensive spectroscopic

st,udy of EDF to estract t,he plienornenological parameters required for the more

coniplete cinpirical mode1 of Desurvire [26] would

wwelengt h EDFL modelling.

O the incorporation of a semicondiictor optical ampli

enable more accurate. multiple

fier (SOA), in lieu of an EDFA.

should be demonstrated as well. The success of the proposed technique for EDFL7s

indicates it sliould easily be applied to SOA lasers since FWM occurs within SOA's

tliemselves and the SOA can be used as a modelocking element itself due to much

quicker gain dynamics.

Ii i conclusiori. a riem teclinique for multiple wavelengtli generation witliin EDFL's has

bcen demonst rated. The results of the proof of principle demonst ration are encouraging.

Cont inuing development of t lie t eclinique proposed. including an experimental invest iga-

t ion of t lie niodified laser st riict iire proposed. stiould be purstied for ultiinate application

t O U'D A I liglitwave sys tems.

Appendix A

Inhomogeneously Broadened Media

Under Saturat ion

A. 1 Introduction

The susceptibility %(u) of a n inhoinogeneously broadened medium is given by an integra-

tion over the resonance frequency distribut ion <(da) wit h the homogeneous susceptibility

,y"(~;) and a saturation factor S(du). In general. we may assume a Gaussian resonance

distribution [Xi].

wliere is the mean resonance frequency and Au, is the FWHM of the resonancy

frequency distribut ion. Even with this cornmon distribution, the saturated susceptibility

does not lend itself to a tractable analytic expression in closed form. In order to avoid

numerical integration. we provide bere a derivation of an analytic expression for the

saturated susceptibility.

A.2 Voigt Profile

First . we consider tlie unsaturated susceptibility. The susceptibility is given by [37]

wlierc 8 = - V e ~ ~ o L l p ( 2 / 3 c ~ o A ~ denotes the strength of the transition in ternis of

tlie dipole iiionieilt p. Lorentz correction factor L and unsaturated inversion IVeq- The

lirieshape is t lie ubiquitous Voigt profile. the convolution of a Gaussian distribut ion and

a Lorentzian lineshape. A conipact expression for the above is.

n-Liere w ( t ) is the Faddeeva fiinction defined by.

witli z = 3' + i.1 and t given by.

Efficient algoritlinis esist for calculation of tlie Faddeeva function (91. 921. and it is ttius

as accessible as esp() for esample. The unsaturated susceptibility is thus known in terms

of a well cliaractcrized analytic function.

A.3 Susceptibility Saturation

In order to include saturation. we follow Siegman [37] and consider the change in suscep

tibility due to saturation by a signal of intensity I I and frequency wl,

d i e r e the saturation fact,or S(k1,) is given by.

and wtlere the deptli of the saturation is determined by.

The reason for considering the change in susceptibility rather than the susceptibility

as a wliole is tliat 1 - S(&,) is a simpier espression ttian S(ua) and the unsaturated

susceptibility is e a i l y calculated.

The ~ i m t lieniatics is simplifed when we transform to normalized variables according

to t lie scheme.

Our susccpt i bili ty change t hus niay be wri t ten.

The integrand does not lend itself to any simple integration technique. An approximation

is tlierefore made. Using exp(o(u - r) 2 , = 1 + a(u - r ) + a(u - r)"2 for sufficiently small

lu - rl. we can approximate the Gaussian distribution with an "extended" Lorentzian,

-4s - 1.1 increases. the approsiniation does not fail catastrophically becaiise the expan-

sion is found in the denominator. The integral thus becomes.

Coniples contour integration is readily applicable to the above integral. The poles of the

integrand in the cornples u plane are al1 simple and are given by

u,,, E { r d= 1 / da esp(i3ir/8). r I 1 / \la & exp(i5r/8). i z i JG. u + i} (A. 13)

We now consider the clocktvise complex contour integral dong path -1 consisting of the

real asis and an arc in the lower half plane.

14;e can easily verify tliat integration dong an arc about the centre of the complex plane

can be made vanishingly sinall. This. ive can apply Caiiclv's residue theorem for comples

contour integration to obtain.

1 62,;) = -2.i"' 1 resiciue ($ . 'u = u,, u - ' U ,

11

Esplicit ly caiculating the residues leaves us with the following analytical formula.

+ (A. 1G) (& - &) (1 + p + ( r - (V - + i + &/\/O)

where QI = exp(i3n/8) / 2 ' / h d 4 2 = exp( i5~ /8 ) /2 ' /~ . Ait liough the analytic formula

does not give one insight into the nature of the spectrum saturation, i t allows for quick

numerical calculations of saturated susceptibili ties. The method can also be applied with

higher order expansions, albeit at the expense of factoring higher order polynomials.

Appendix B

Theory of the Lyot Filter

B. 1 Introduction

The de~iioristrated niiiltiple wavelengtli laser eniploys a Lyot filt.er for intracavity spectral

slicing. Based on linear polarization rotation. the Lyot filter is a simple device. Howvever.

the liriear tlieory of Lyot filter operation is described here in detail because it is not

as iibiqiiitous as otlier freqiiency comb filters such as dielectxic stacks and Fabry-Perot

resoriators. Konlinear polariza t ion rotation is neglected because tlie 11igli birefringence of

the fiber used in the Lyot filter produces a large velocity mismatcli between the polariza-

tion coniponents of a piilse. Thus. the temporal walk-off intiibits nonlinear interaction

between eacli polarization cornponent. A second reason to study the linear theory of a

Lyot filter is to ascert.ain what parameters are critical to the Lyot filter characteristics,

such as free spectral range (FSR) and extinction rat.io between pass and stop bands.

We begin this appendix chapter with a description of birefringence in optical fibers.

We tlien describe Poincaré sphere formalism, which will be useful in our description of

polarization rotation. An analytical expression for the transmission function of a Lyot

filter is gjven and we provide illuminating examples.

B.2 Birefringent Optical Fibers

A single niode optical fiber. witii perfect axial symnletry. in fact supports tlie propaga-

tion of two CO-propagating modes a t a given optical frequency. The doubly degenerate

nioclc conies about diie to the polarization degree of freedom. analagous to the double

degeneracy associated with CO-propagating plane waves in vacuum. Any HEiI mode field

distribution and a duplicate rotated through an angle ?i/2 about the symmetry mis serve

as a b a i s for al1 possible guided optical excitations. The niode fields approsirnate two

lincar polarizations. and the literatiire often speaks of both modes as siicli.

The introduction of fiber asymmetry about a plane through the asis destroys the

degcricracy of botli HEll modes. Al1 giiided optical escitations are now described by a

iiniclue basis const it u ted by two modes. Eacli niode will possess a different longitudinal

propagation coi~starit and a different mode field distribution. If we again identify eacli

mode witli linear polarizatiori. ive have an obvious analogue to the propagation of linearly

polarized planc waves in a birefringent crystal [93]. Birefringent fiber designs employ

geonietric core asymrnetry or stress induced index birefringence. As in the case of a

bircfringciit crystal. we identify a fast asis and a slow mis tvith the fiber. althougli we

are strictly speaking of a fast niode (high phase velocity, low effective index) and a slow

modc (low phase velocity. liigli effective index).

The rilodes of a birefringent fiber are not linearly polarizcd. but it is convenient to

speak of tliern in this way for two reasons:

the differerice is immaterial except a t tlie entrance and exit of the birefringent

fiber. The exact electromagnetic field distribution does not mat ter escept when we

consider the launcliing of light into and out of the birefringent fiber.

a the modes ARE very close to being lineary polarized, especially for weakly guiding

fibers.

For the above reasons, we will use the language of plane wave polarization to elucidate

APPENDIX B. THEORY OF T H E LYOT FILTER

the operation of the fiber Lyot filter.

B.3 Poincaré Sphere Formalism

In order to fiilly characterize the operation of the Lyot filter. we present a Poincaré sphere

formalism. Alt liough it m a i seem to the reader as thougli we are cracking the proverbial

wlniit witli a sledgehammer. the Poincaré sphere provides a clear visualization of the

Lyot filter operat ion giving further meaning to the mat hematics. The assumptions we

niake ticre are tliat negligible loss and negligible mode coupling are induced over the

lerigth of the birefringent fiber. We begin by espressing the electric field within the fiber

as a siiperposition of modes at a point 20 along the fiber asis.

wliere el and e2 are tlie slow and fast mode field distributions respectively. Similarly,

,$1 aiid are t.lie slow and fast longitudinal propagation constants. It is convenient tao

writc tlie superposition of modes in a normalized vector forin as in the calculus of Jones.

Xote that wc oiily require two real variables d and 0 to describe the polarization statc

since absolute magnitude and absoliite phase are irrelevant.

The Poincaré formalism makes use of the Stokes parameters [94]' defined as follows.

APPENDIX B. THEORY OF T H E LYOT FILTER -- 1 (

It cati easily be sliown tliat Si = Sf + S.: + Si. the equation of a spliere. The Poincaré

spliere X is the set of al1 possible polarization states in the Cartesian coordinate system

{S ) = {(S1. 4.4) ) E Hg3. We assume So = 1 for convenience because the polarization

state of the elect romagnet ic field is independent of amplitude. The conventional spherical

CO-ordinate system describing C is.

where the angles tb arid ,x are given by.

The plysical interpretation of .L!I and ly are worth rioting. For a general elliptical polâriza-

tion state. the electric field vector will describe a lociis a t fised 2. The major axis of the

polarizatiori ellipse is an angle ,@ away from the a i s defined by el. The tangent of t,lie

angle is the ratio of major axis to minor a i s of the polarization ellipse. An illustration

of the standard coordinate system is given in Fig. B.l (A) . With the angles defined

as above. ive c m easily associate points on C with definite polarization states. Linear

polarizat,ion states are located on the SI - Ss plane. Linear polarization states on the

fast and slow axes are located a t (- 1.0.0) and (1.0. O ) , respectively. Polarization states

at +7r/4 and -7r/4 with respect to the slow axis are located at (0.1,O) and (0, - 1.0).

Left- and right-lianded polarizations are located a t ( O , O. - 1) and (0, OJ).

For our purposes. however. we use the less coriventional angles @ and B to describe

t.he polarization state on C as iliustrated in Fig. B.l (B). The Stokes parameters can be

Figure B.1: Tlie Poincaré spliere C illustrated with two CO-ordinate systenis: A. the

coriventioilal c': and 1: B. the alternative system o and 8.

written in terms of tlie angles H and (3 <W.

Tlie angle 0 is sirnply tlie phase of t tie ratio El / E2. The cotangent of 4 is the amplitude

ratio 1 El I/1 E21. Tlie utility of these angles will sooii be apparent.

Having introdiiced tlie Poincaré spliere formslism. we now introdiice three useful

properties of C .

1. For every polarization state S' tliere is a unique polarization state S orthogonal to

S . The state s is located dianietrically opposite to S on C.

2. The evoliition of a polarization state is described by a unitary transformation U that

corresponds to rotation of C about the mis described by the orthogonal polarization

eigenvectors of t lie transformation U.

APPENDIX B. THEORY OF T H E LYOT FILTER 79

3. Tlie inner product of two fields described by polarization states on C has a magni-

tude equal to cos(iP). wliere 2Q is the minimum angle subtending both polarization

states.

Proofs of the above are outlined in [93]. Witli the above arsenal. we are now able to

describe the operation of the Lyot. filter mathematically and graphically.

B .4 Frequency Transmission Function

\Ve iiow consicter tlie Lyot filter structure. The incident polarization analyzer and po-

larization cont roller launch liglit into the birefringent fiber with a polarization state Su.

Propagation t lirough t lie birefringent fiber gives a wavelengt h dependent polarization

state Sul. The output polarization analyzer and polarization controller are set to pass a

polarization state S , witiiout loss.

Assuniing ive have a n initial polarizatiori state Su at 2 = 0. the Jones vector will be.

Propagation 11 a distance 2 tlirough the birefringent fiber is given by a diagonal unitary

transformation in the basis of tlie fast and slow modes. Tlie result is a polarization state

SUI with .Jones vector.

cos(&) esp(iPl t )

sin(qbu) exp(i(& t - I I , ) ) 1 We factor out an irrelevant global phase of Pl: radians to obtain,

APPENDIX B. THEORY OF THE LYOT FILTER 80

wliere An = n l - ri--. We see that tlie angle 0 increases by Anwzlc radians upon

propagation t hrougli t lie birefringent medium. On the Poincaré sphere. ive have an

ecpivalent rot,ation aboiit the SI axis. as given by tlie two polarization eigenvectors

correspondirig to iinear polarizat ion along the fast and slow axes of the fiber. For a given

lengtli L of fiber. ive t h s have an optical freqiiency periodicity Au in polarization state

given b>-.

We takc t lie iriner prodiict. of the polarizatiori states S, and Sut to calculate the amplitude

transmission fiinction t . LVe do this esplicitly in the Jones calculus.

= cos(o,.) cos(b,,) + sin(&) sin(&,) esp(-i(2ïrv/Av + O,, - H u ) )

The transmit tancc T = t't is readily obtained.

W e iniiiiediat,cly see tliût the angle 0,, -O,! simply represents a bias phase. shifting tlie en-

tire transnlission spectrurn. The angles d,, and +, determine the maximum and minimum

The transmission amplitude loci and t ransmittances are illustrated beiow in Figs.

B.2.B.4 and B.3.B.5 respectively for two cases. In the first case. 4, = 4 , a t a number

of different values. We observe a peak unity transmission and a minimum dependent on

the value of 4, = 9,. Launching optical power dong fast or slow axes results in unity

transmission for al1 frequencies. while equal optical power in fast and slow axes gives

transmission minima of zero. On the Poincaré sphere, the loci are circles of intersection

between C and planes with the normal SI. Transmission maxima occur with the coin-

cidence of Sul and Sv. Transmission minima occur when Sul is maximally distant from

Figure B.2: The riniplit.~ide transmission function with d, = Qu as the labelled parameter.

Tlie t rajectories in t lie comples plane are clockwise with increasing optical frequency.

S , on C. Nulls in transmission will only occur if the locus of polarization state includes

the state 8 , opposite Su. Only equal excitation of botli fast and slow ases allows the

polarization t,o evolve tlirough orthogonal states. such as opposite circular polarizations

or linear polarizations witii 9 = &7ï/4.

The second case considered is that of fixed polarization analysis with 4, = ~ r / 4 and

variable 9,. Tlie maxi~num transmittance is unity only for the case of 4, = 7r/4: as

is made obvious through consideration of polarization state loci on C . Furthermore, by

considering the loci for 0, # n/4, as illustrated below, we see the transmission nulls occur

only for 4, = 7r/4 also. Tlie present case corresponds to launching elliptical polarizations

of light into the birefringent fiber and analyzing with a linear polarizer 7r/4 radians to the

APPENDLS B. THEORY OF THE LYOT FILTER

Figure B.3: The power transmission furictioii with 4, = 9.. One abscissa is t h e optical

frequency norrnalized to the Lyot FSR and referenced to a bias frequency. The otlier

abscissa is the angle 4 , norinalized to ~ / 2 .

APPENDIX B. THEORY OF T H E LYOT FILTER

Figure B..?: Tlie amplitude transmission fiinction with 4 . = 7r/4 and pu as the labelled

parameter. Tlie t rajectories in the coniples plane are clockwise wit li iricreasing opt ical

frequency.

slow asis. Mmima (minima) occur wlicn the output polarization ellipticity is greatest

and the major a i s is î ~ / 4 ( - ~ / 4 ) radians to the slow mis.

The above analysis gives us the esplicit form of the transmittance of the Lyot filter.

The periodicity of the transmittance in the frequency domain is determined by tlie total

length of fiber and size of the index difference between slow and fast axes. We can also

see that to obtain unity maxima and nul1 minima, we are required to set the polarization

controllers and polarization analyzers sucli that 4, = 9, = 7r/4. In other words? we must

excite equally the fast and slow axes of the birefringent fiber. We must also analyze a

polarization corresponding to equal excitation of the fast and slow axes of the optical

Figure B-5: The power transmission function with 9, = ~ / 4 . One abscissa is the optical

frequency riormalized to the Lyot FSR and referenced to a bias frequency. The otlier

abscissa is the angle 4, norinalized to ~ / 2 .

APPENDIX B. THEORY OF THE LYOT FILTER 85

film-. Finally. LW note tliat sliifting the transmision spectrum is done readily by varying

t lie pliase difference between tlie fields esciting the fast and slow axes of the fiber (or

varyiiig the analyzed polarization state in a like wise nianner).

B.5 Summary

Iri coiiclusion. we have given a complete tlieoretical descript ion of tlie linear operation of

a Lyot filter. hlatlieniat ical atid gapliical descriptions were provided to elucidate Lyot

filter opel-at ion and f'îîcilit,ate Lyot filter design.

Appendix C

Phase-Matching for Second

Harmonic Generat ion

C. 1 Introduction

The autocorrelation teclinique c~nployed Iiere was based upon second harmonic generation

(SHG) in birefringeiit LiI03. The cfficiency of tlie SHG process is strongly dependent on

satisfaction of t lie pliase rnatcliing condition [Gl] . We present here calcuiat ions based on

a Type 1 ariglc tuning approacli to achieve pliase rnatcliing. Two fundamental. ordinary

t~eanls are corribineci to create a second liarmonic. estraordinary beam.

C .2 P hase-Matching Condition

The rion-interferometric. background fiee autocorrelation technique requires the efficient

generat,ion of SHG from two, non-colinear fundamental beams. The wavevectors of the

fiindamental beams are liere denoted 14, and k,-. We let the wavevect.or of the desired

SHG beam be ksb. The phase matching condition is simply.

APPENDIX C . PHASE-MATCHING FOR SECOND HARMONIC GENERATION 87

The pliase-inatching condition is illustrated below in Fig. C. 1. togetlier wit 11 the optic

axis c of t lie crystal. The polarizations of botli fundanienta: beams are out of the page.

corresponding to ordinary beams. The second liarmonic beam is a n estraordinary beani.

The polarization arrangement liere is according to the Type 1 phase matching scheme.

Siihst it iit iiig vacoiini wavelengtlis and indices of refraction for t lie wavevectors in Eq.

C.1. one rcadily obtains the relation.

wlierc ,- is tlie angle betwveen the second harmnnic and the optic axis. 2+ is tlie angular

spread of the fundamental beams. 6 ,*2 i r . e (~) is the +-dependent refractive indes of the

estraordinary second tiarmonic beani and n.,., is the ordinary refractive index of t.lie

fiiridainental beams.

Figure C.l: The phase matching condition for two ordinary fundamental bearns with

wavevectors kcJf and an extraordinary second harmonic beam with wavevector k&. The

optic mis c of the crystal is also illustrated.

APPENDIX C . P I - 1 . 4 ~ ~ - A T N G FOR SECOND HARMONIC GENERATION 88

C .3 Angle-Tuning

In order to satisfy Ecl. C.2. u-e note tlie refractive index of tlie extraordinary beam can

be represented as [!95].

wliere nL., and n L - , are the ordinary and extraordinary refract ive indices wit liin the

crystal. hlanip~ilation of tlie above and substitution of Eq. C.2 gives us the relation.

Tlie recluired angle of the second Iiarmonic to tlie optic axis can tli~is be determined from

tlie arigular spread of the fundamental beams in the crystal. The indices of refraction for

LiI03 cari l x determined witli the Sellmeir formulae (961.

wliere X is tlie vacuum wavelengtli in micrometers. Here. we use X = 1 . 5 3 0 p ~ tlie

approxirriatc centre wavelength of fiber laser pulses we wish to characterize.

C.4 External Angles

Mie are iinterested in tlie esterilal angles of the beams wlien aligning tlie crystal within

tlie aiitocorrelator. The refraction of each beam is illustrated in Fig. C.2 below. The

optir asis angle in the crystal used here is 0 = 30°.

Snellk law is applied with the angles as illustrated above to yield the following,

sin(<+) = n,,, sin(@ - v - $J)

sin(<-) = nu,, sin(8 - 9 + v )

APPENDIS C . PHASE-~IATCI~INC FOR SECOND HARMONIC GENERATION

Figiire C.2: The refràctiori at the air-crystal interface for the two fundamental beams

aild t lie second liarmonic beam are illustrated. The shaded region represents the crystal;

the unsliaded region represents air.

Put ting togctlier Eqs. C.6. C.5 and C.4. the esternal second Iiarnionic angle < can be

plotteci versus the external fiindamental beam spread A< = <- - E+. The result is given

in Fig. C.3 (A) below.

APPENDIX C. PHASE-MATCHING FOR SECOND HARMONIC GENERATION 90

A 5 [degrees]

A 6 [degrees]

Figure C.3: ( A ) The phase matclied. second liar~nonic esternal angle < as a fiiriction of

the esterniil f~indaiiiental beairi spread Ac. (B) Tlie niean fundamental external angle

1/2(EÇ + E - ) vci.jus tlie estcrnal fundamental beam spread A<. Tlie crystal avis is 30°

with respect to normal. as in the crystal used.

Appendix D

Derivat ion of the Ginzburg-Landau

Equat ion

D. 1 Introduction

The tlieoretical treatment we give liere is based upon tliat first given by Agrawal [8l. 691.

but \ve provide here corrections and further cletails as is seen to be helpful. Notes regard-

iiig the validity of various approximations are presented as the approximations are niade.

We begin witli the Bloch equations for the interaction of optical fields with a collection

of noninterac ting rare-eartli ions. We then proceed to apply Ma-xwell's equations to a

lossless fiber. wi t 11 the rare-earth ion interaction with the optical field and Kerr nonlin-

esrity treated as first order perturbations. Tlie final equation of motion derived is shown

to be of Ginzburg-Landau form.

D.2 Bloch Equations

We begin first by considering the interaction of an optical electromagnetic field with a

collection of rare earth ions sitting in bulk silica glass. The relevant energy structure

of erbium ions is not a simple three-level system, but is in fact three manifolds due to

APPENDIX D. DERIVATION O F THE GINZBURG-LANDAU EQUATION 92

Stark splitting and iiihornogeneity in ion sites witliin the glass matris [25] . However.

esperiments 1% Desurvire et al. (391 liave sliown tliat EDFA's betiave similnrly to strictly

lioniogeneoiisly t~roadened ne dia at roorn temperature. Thus. we consider the erbium

ions 21s simple. liomogeneously broadened three level systems.

We will notv apply the Bloch equations of motion for tlie ions interacting with an

electroniagiiet.ic field. Tlie cliaracteristic time Tl for population relk~at ion from the

escited state to tlie ground state is on the order of 10 ms [97]. Tlie dipole depliasing time

G. or tlie inverse of the Iiomogeneous linewidth. is on the order of 100 fs (381. We are

coricerned with pulses tliat possess widttis greater than 5 but less tlian T l . We assume

a niaterial inversion chaiiging in time on scales of T l . since colierent effects such as Rabi

oscillations d l last for times coniparable to fi and will generally be unobservable at

room temperature. Tlie Bloch equations take the following form [(il].

wherc P is t lie rare-eartli polarization. E the electric field. time derivatives are indicated

by raised dots. tinie averages are given by angled brackets. €0 is the vacuum permittivity.

kta is t lie resonance frequencv of tlie ions: IpI is the dipole moment moment of the transi-

tion. N = & - Ni is tlie population inversion. Neq is t.he equilibrium population inversion

in tlie alxence of tlie resonant, optical field and L is the Lorentz correction factor relating

niacrosopic fields to microscopie fields. Note tliat a factor of 1/3 is present to account for

random oricntûtions of the dipoles within tlie glass host. The polarization of the electric

field and rare-eartli polarization are assumed constant and aligned. Since tliere is only

residual birefriiigence within tlie EDFA. this is an appropriate approximation.

Introducing tlie Fourier transforms of P and E as follows,

APPENDIX D. DERIVATION OF THE GINZBURG-LANDAU EQUATION

ive caii arrive at a susceptibility ~ ( L J ) satistking the relation.

Applying Fourier transforms to the Bloch equations of Eq. D.1. we obtain

Since we are concerned wit li

%(u/ - d(,) and tlius rewrite

optical frequencies w. - ua. we can approximate w2

ut) in the form

ivliere c is tlie vacuum speed of light and ive have introduced the cross-section a =

T2ir., LI I L l2/3cTiO. Tlie reason for the introduction of a defined in the above mannes will

become apparent as will its interpretation as a cross-section. The form of ~ ( w ) is tlie

'miiliar cornples Lorentzian as expect,ed for resonant interactions of light with at,oms or

ions.

Befose we continiie in deriving an equation of motion for optical pulse propagation in

fiber. we fiirtlier specify N by sirnplificat ion of tlie second of Eq. D.1. Tlie final result.

wliose teclinical derivation is giveii by Siegmaii [37]. can be written in the form

wliere I is tlie optical intensity averaged over a tinie rnuch greater than T2 but shorter

tlian Tl . The saturation intensity is given by ISat = h/20T1. The expression above is

strictly valid for a iiarrow optical spectrum witli zero detuning from resonance. but. the

reduction in saturation due to the violation of these restrictions is negligible in cases of

interest to us here. It is edifying to consider the forrn of N a t steady state?

The cause of homogeneous gaiii saturation is clearly

medium inversion.

(D.7)

seen to be the reduction of the gain

APPENDIX D. DERIV.V~ION OF THE GINZBURG-LANDAU EQUATION

D. 3 Maxwell's Equat ions

11-e non- t urri our attention to hfaswell's equations. beginiiing by considering an undoped.

lossless. linear fiber first. We follow the metliod of Argawal (691. beginning wit h hhxwell's

vcctor equat ion for the elect ric field.

For w-eakly guiding fil~ers. as is the case of interest. tlie semi-vectorial approsimation

applies.

, \ I i ~ ~ \ ~ e l l ' s watre ecluat ion t liiis takes t lie famiiiar form.

Taking Fourier transforms in time. we r an use the relation P(r. w ) = eoxo(r. w)E(r. w)

w1iei.e ,yo(r. w) is t lie susçept ibility of t,lie silica corist i tuents of the fiber.

+ xo(r. w ' ) ) E(r. uf) = O (D. 11)

The z-asis is cliosen as the fiber a i s and we define a relative permittivity dependent

solely on transverse coordinates E , (T. g. i ~ ) = 1 i xo(r. LJ). In deterinining the fiber modes.

WC assunie a solution of the forni E(r. &) = e F ( x . jy.+xp(i&+). Siniply. e is a m i t

direction vector. F ( x . y . ~ ) is the mode profile for Our single mode fiber and O(w) is

t lie longitudinal propagation constant. The propagation constant and mode profile are

determined tlirough substitut ion into Eq. D. l l .

W 2

er(x.y. w)-F(z . y. W ) = f i ( ~ ) ~ F ( x . Y.W) (D.12) c2

For a given relative permittivity. the above equat.ion can be solved by numerous methods.

Analytic solutions esist orily for certain index profiles. More complex profiles require the

use of numerical methods such as the finite element method. which is a robust and

efficient method widely employed [98]. The mode profile and propagation constant are

assiimed to be known from here onwards.

APPENDIX D. DERIV.~TION OF T H E GINZBURC-LANDAU EQUATION

D.4 First-Order Perturbation Theory

\4é are now in a position to consider perturbations to the propagation of liglit in a linear.

lossless fiber. Following Agrawal [69]. tlie relative permit tivity is perturbed to give the

follow-ing to first order.

wliere n(k1) = Dl(&)/m is tlie effective niode index o is the fiber loss. ~ ( w ) is the

susceptibility of the rare-earth ions and An' ( U I ) is tlie change in the square of effective

index due to t lie optical Kerr nonlinearity of the silica glass. Esplicitly: the optical Kerr

nonlinearity is the intensity dependence of refractive index

n j = no + n2 I ( x . y) (D. 14)

u-i-iiex-e n l is the silica refractive indes. no is tlie linear refractive index. n2 is the Kerr

coefficient and I ( x . y ) is tlie local light intensity.

The pert iirbat ions to the relative permit tivity may be considered individually. and

are takcn into accoiint by considering Eq. D.12 t o determine the perturbation t o tlie

propagation constant O(&). Let t ing 3 be tlie perturbed propagation constant. we have

Assiiming a siiiall perturbation as is reqiiired in our first order approach. we can ap-

prosiniate ,$ - 13' z 20 (0 - ,O). Thus. the perturbation to the propagation constant

is. w JJ F'Ar, Fdxdy

"-"=?,<,>, JJP'Fdrdp (D. 16)

\.Ve nonr apply the above formula to calculate the propagation constant perturbation

from loss, gain

the transverse

and optical Kerr nonlineari ty respec tively. We assume uniform loss across

plane and obtain the simple result.

(D. 17)

APPENDIX D. DERIVATION O F THE GINZBC~RG-LANDAU EQUATION 96

The perturbation due to gain requires niore careful t reat.ment since the rare-eart h ions

will bc confined to a volume wit hin the fiber core. We tiius assume a transverse density

profile of ions p ( x . y). normalized such tliat Jy p(x. .g)d:rdq = 1. The rare-earth ions will

t lius cont ribiite.

wliere the overlap factor i? is given by

The Kerr nonlinearity reqiiires a similar t reatrnent.

(D.19)

Using the approximation An' - 2.n ( ~ * ) n ~ l ( x . y). ive again arrive a t a simple expression for the 13 pertiirbation.

(D. 18)

wliere fil is the total optical power aiid = n n ~ / c A e I f . Tlie effective area A e f j is defined

as.

Adding t>he cotitribut.ions of Eqs. D.lî.D.18.D.20. we arrive at the total perturbation.

\4? t h tlie abow perturbation. we cari proceed to derive an equation of motion for optical

pulses.

As \vas iiientioned earlier. we assume piilses of widtli greater tlian T2, but less tlian

T l . In tliis regime. an envelope approximation is appropriate.

1 E(r. t ) = - ( ê F ( z . y)A(z. t ) exp [i(Poz - u0t)] + c.c.)

2

Mie will also make use of the Fourier transform of the envelope,

(D. 24)

APPENDIX D. DERIVATION OF THE GINZBL~RG-LANDAU EQUATION 97

The corripIes envelope A( t . t ) t hus carries the information of pulse amplitude and phase at

a carrier optical frecluency wi t h corresponding propagation constant o0. Furthermore.

under our assumptions about piilse width. we espect A(z. t ) t.o have significant spectral

coiiipoiients only for frecpencies Iw( « wo. An equat,ion of motion for A(--. t ) will now

be determined. U'e take the Fourier transform of Eq. D.23 with the intent to substit.ute

t lie result into Eq. D. 11.

The approximation ~ v e liave made above. based upon the assumption that A(z. t ) h;is

conlponents only for frequencies less than the optical carrier, restricts us to forward

( posit i1.e z ) propagating pulses only. To siniplify the continuing analysis. we use the

sliifted frecluericy 2 = &* - S~~bstittition of the above result for E(r, u) into Eq. D.11

yields the following after sonie algebraic gymnastics.

wliere /3(&) is determincd from Eq. D.12. The slowly varying envelope approsiniation

(SVEA) caii be used for pulse envelopes whicli change negligibly over distances on the

order of optical wavelengths. We are considering such pulses and tlius the SVEA allows

us to iieglect #A(-. ;>)/a-' relative to the otlier terms appearing in Eq. D.26. Using

,6"(&) - 0: ;= 2&(D(w) - 130). we arrive at the peculiarly elegant equation of motion for

A(2.G).

The above equation is quite general, and numerous physical effects may be taken into

account through appropriate modification of P(o ) . We proceed to derive the promised

Ginzburg-Landau equation by including the physical effects claimed to be relevant. Prior

APPEND~X D. DERIVATION OF T H E GINZBURG-LANDAU EQUATION 98

to corisidering the perturbations given in Eq. D.22. ive espand /3(w) in a Taylor series.

Tlie clepcndcnce of 13 upon u: corresponds to tlie combination of rnaterial and waveguide

dispersion witliin the fiber. In the work considered here. rJ is sufficiently srnall that we

l i e d only consicler terins up to second order in 2. Inspection of the terms in the above

expansion allows one to make the following pliysical connections. The phase velocity is

triviallÿ givcn by 11, = as expected with our assumption about carrier velocity.

Tlie teriii dl = X?/& is tlie inverse of tlie groiip velocity i7, = &/a/jl. Tlie second

order coeficicnt /3? is referred to as the group velocity dispersion (GVD) parameter. The

greater tlie GVD paranieter. tlie greater the ciirvatiire of (3 with LJ. The physical result is

a ciifference in groiip velocities of constituent spectral components of a puise. resulting in

chirping and broadcriing with propagation. The above expansion indudes effects which

preserve piilse energy, Ilut result in pliase accumulation and possible pulse spreading or

compression.

Prior to deriving t.l~e specific eqiiation of motion from Eq. D.27. ~ v e rnake one final

Taylor expansion. The susceptibilitÿ ~ ( w ) of Ecl. D.5 gives the detailed frequency de-

pendence of the rare-eartli response. As with tlie material and waveguide dispersion of

the optical fiber. ive are interested in the material response over a selected frequency

range. The work here is concerned with laser oscillations? and we thus expect a carrier

fiequency for our pulses to correspond to the gain peak in the absence of cavity losses

forcing oscillation elsewhere. Thus, we set wo = u, and expand the susceptibility up to

second order. Forseeing that we will be summing the rare-earth perturbation to P(w),

we give the niodified coefficients of expansion of LU.).

Tlie piiysical nieaning behind the modifications of the B(w) are easily seen. The imaginary

contribution to gelf is tlie gain provided by the rare-earth ions. proportional to the

inversion iV. The real contribution to @ y f f is the result of the group delay at the linecentre

of the Lore~it~ziaii respoiise. Tlie second order contribution to @ z f f is iniaginary and is the

curvature of tlic Loreritzian gain profile. The curvature inodels the limited bandwidth of

the gain provided by the erbium ions.

Iilcorporating the expansion coefficients of B(u) in Eq. D.29 and including the per-

turbation teriiis of Eqs. D.17. D.20. the equation of motion D.27 becomes the following

upon inverse Fourier t ransforination.

where we have defiiied the gain g = l?o-V/,n(wo) and we Iiave assumed tlie pulse envelope

A(:. t ) is scaled t,o root power. rneanirig Po = IA(s. t ) 1'. The transformation to a moving

refereiice franie witli T = t - 4;jf z gives us our heralded equation of motion.

The above equation is in the promised form of a Ginzburg-Landau equation.

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