Numerical design of an optical solid-state amplifier

by

Gysbert Johannes van der Westhuizen

Thesis presented inpartial fulllment of the requirements

for the degreeof

Master of Science

atthe University of Stellenbosch

Supervisor:

Dr. J.P. Burger

Co-supervisor:

Declaration

I,theundersigned,herebydeclarethattheworkcontainedin thisthesisismyownoriginalwork

andthatIhavenotpreviouslyinitsentiretyorinpartsubmitteditatanyuniversityforadegree.

...

Signature

...

Date

Copyright

c 2007StellenboschUniversity Allrightsreserved

Abstract

Theoretical models fulll an important role in the eld of laser research as a resultof the time

andmoneyrequiredto performanexperiment. Inparticular, theoreticalmodelsmaybe usedas

toolsinthedesignofmoreecientlasersystemsandcontributeto abetterunderstandingofthe

physicalprocessesinuencinglasermedia. Intheworkpresented,asolid-stateamplierismodeled

accordingtowellknowntheoryforthepurposeofdesigninganamplierstage,tobeusedaspart

ofahybridbre-bulklasersystem.

Variousprocesseshaveaninuenceonthequalityofalasermaterialandonlyanumberofthem

canrealisticallybeincludedin amodel. Twoprocesseswerespecicallyinvestigatedinthiswork.

Therstprocessisthatofampliedspontaneousemission(ASE),wherebythepopulationinversion

ofalasermaterial isdecreased. Thesecondprocessinvolvesthenatureofbeampropagationand

isofimportance fortheprecisemodelingoflight-matterinteraction. Thebackgroundtheoryand

proceduresfollowedtotakeaccountoftheabovementionedprocessesarediscussed.

The model equations were numerically solved for the case of continuous wave (CW) probe

amplication as well as the case of pulsed probe amplication. An experiment was conducted,

using readily available equipment, for the purpose of verifying the CW model, whilst the pulse

model was validated by means of ananalytical approach. In both casesthe agreementbetween

the simulated results and those obtained from experiment or theory was acceptable. Possible

shortcomingswithinthemodelsareidentied.

Finally the models were used for the design of an Yb:YAG amplier. The optimum crystal

characteristics were identied and the thermal properties of the crystal were evaluated. With

limitedequipment, aqualitativeanalysis of the optimumcrystal wasperformed bymeans of an

experimentalinvestigation. TheinvestigationconrmedtheCWmodelasapotentialdesigntool.

Opsomming

Teoretiesemodellevervul'nbelangrikerolindieveldvanlasernavorsingasgevolgvandietyden

geldwatbenodigwordvirdieuitvoervan'neksperiment. Indiebesonderkanteoretiesemodelle

gebruikwordasgereedskapin dieontwerpvanmeerdoeltreendelasersistemeenkanookbydra

tot 'n beter begrip van die siese prosesse wat inwerk op 'n laser medium. Die navorsing wat

hier voorgedraword, behelsdie modelering van'n vaste-toestand, optieseversterkervolgenswel

bekende teorie met die doel om 'n versterker wat deel vorm van 'n hibried vesel-vaste-toestand

lasersisteemte ontwerp.

Verskeie prosesse beïnvloed die kwaliteit van 'n laser materiaal en slegs sekere hiervan kan

realistiesin 'n model ingesluitword. Twee spesiekeprosesse wasasprioriteit beskouin hierdie

navorsing.Dieeersteprosesgenaamd,versterktespontaneemissie(ASE),is'nproseswatleitotdie

afnameindiepopulasieinversievandielasermateriaal. Dietweedeprosesbehelsdievoortplantings

kenmerkevan'nligbundelenisvanbelangvirdieakkuratemodeleringvandieinteraksietussen

ligen'nvoorwerp. Dieteorieendiemaatre

¨e

lswatgevolgisombeidebo-genoemdeprosesseinte reken,wordbespreek.

Die model vergelykings is numeriesopgelos vir die versterking van 'n kontinue intree bundel

(CW) en vir die versterking van 'n gepulsde intree bundel. 'n Eksperiment met die doel op

die verikasie van die CW model is uitgevoerdeur gebruik te maak van beskikbare toerusting,

terwyl 'n analitiese benadering gevolg is om die puls model te ondersoek. In beide gevalle is

die ooreenstemming tussen die gesimuleerde resultate en die resultate wat deur middel van die

eksperimentofanalitiesebenaderingverkryis,bevredigend.Moontliketekortkomingeindiemodel

isgeïdentiseer.

Die modelle is ten slotte gebruik in die ontwerp van 'n Yb:YAG versterker. Die optimale

kristal karakteristieke is geïdentiseer, waarna die kristal se termiese eienskappe ontleed is. 'n

Kwalitatiewe ondersoek na die optimale kristal is deur middel van 'n eksperimentele opstelling

bewerkstellig. Dieondersoekhetdie CW model as'n potensi

¨e

le gereedskapstuk, in die ontwerp vanversterkers,bevestig. Verbeteringeaan dieCW, asook diepulsmodel wordvoorgestel enkan

Acknowledgments

Iwouldliketothankthefollowingpeople:

•

Dr. J.P.Burgerforhissupervisionofthisworkandcountless usefuldiscussions.

•

Prof. E.G.Rohwerforcontributinghis knowledgeandexperiencetotheproject.

•

Dr. J-NMaranforhisguidanceandinspiringapproachto research.

•

Mr. A.S.BothaandMr. E. Shieldsfortheirtechnicalassistance.

•

Mr. D.Spangenbergforthesourcecodeofaraytracingalgorithmusedin thiswork.

•

Myfellow students,especially Gurthwin Bosman and Pieter Neethling, wholistened to all myproblems.

•

MissF.H.MountfortandMissM.VanZylwhotookthetimetoreadthisthesis.

•

Myparentsfortheirsupportand patience.

•

Last,butnotleast,Margueriteforherloveandcountlessairplanetickets.

Contents

Introduction 1

1 The semi-classicalapproachto light-matterinteraction 3

1.1 Thewaveequation . . . 3

1.2 Classicalpictureofabsorption. . . 4

1.3 Quantummechanicaldescriptionofmatter . . . 4

1.3.1 Thetime-dependentSchrödingerequation . . . 5

1.3.2 A two-statequantumsystem . . . 6

1.3.3 Thedensitymatrix. . . 8

1.3.4 Elastic andinelasticcollisions . . . 9

1.3.5 Thepopulationrateequations . . . 10

1.4 Thecontinuityequation . . . 11

2 Numerical models 13 2.1 Adjustingthepopulationrateequations . . . 13

2.1.1 Assumptions . . . 13

2.1.2 Focusingeects . . . 16

2.1.3 Amplied spontaneousemission. . . 18

2.2 Continuouswavemodel . . . 20

2.2.1 Simulationalgorithm . . . 21

2.2.2 Typicaloutput . . . 21

2.2.3 Addition oftransversecoordinates . . . 28

2.3 Time-dependentmodel . . . 35

2.3.1 Adjustingthesimulationalgorithm . . . 35

2.3.2 Typicaloutput . . . 37

2.3.3 Dual-passmodel . . . 39

3 Vericationof the models 43 3.1 Experimentalvalidation ofcontinuouswavemodel . . . 43

3.1.1 Material propertiesofNd:YVO 4 . . . 43

3.1.2 Designofexperimental setup . . . 46

CONTENTS ii

3.1.4 Comparison ofresults . . . 57

3.2 Theoreticalvalidation oftime-dependentmodel . . . 62

3.2.1 TheFrantz-Nodvik model . . . 62

3.2.2 Comparison ofresults . . . 64

4 The Yb:YAGexperiment 67 4.1 DesignofYb:YAGcrystals . . . 67

4.1.1 Material properties . . . 67

4.1.2 Implications onthepopulationrateequations . . . 69

4.1.3 Resultsfromthecontinuouswavemodel . . . 70

4.1.4 Resultsfromthetime-dependentmodel . . . 74

4.2 Thermalconsiderations . . . 77

4.2.1 Solvingtheheatequation . . . 77

4.2.2 Eectongain. . . 81

4.2.3 Theoryofelasticity. . . 83

4.2.4 Possibilityofthermalfracture . . . 86

4.3 Experimentalsetup . . . 87

4.3.1 Optical designforpumpbeam . . . 88

4.3.2 Characterization . . . 89

4.4 Resultsanddiscussion . . . 91

4.4.1 Finalexperimentalsetup . . . 92

4.4.2 Discussion . . . 95

5 Conclusions 102

Appendix: Specications of equipmentused in experiments 103

List of Figures

1.1 Theprocessesofabsorptionandemissionforatwo-stateatomhaving

∆E = ~ω

. . 6 2.1 Afour-statesystemshowingthemechanismsresponsibleforlaseraction.. . . 14

2.2 Simplied"four-state"systemasaresultofassumingfastnon-radiativetransitions. 15

2.3 Denition of the relevant quantities of a Gaussian beam diverging away from its

waist[18, 19]. . . 18

2.4 Thesolidanglewithinwhichspontaneousemissionisamplied(solidline),compared

totheevolutionin thespotsize ofaGaussianpump beam(dashedline). . . 20

2.5 Adiagrammaticgraphshowingthesimulationalgorithm. . . 22

2.6 Illustration of the consequences of not having transverse coordinates in the CW

model. Thepumpandprobebeamsaremodeledwithatophattransverseintensity

distribution (left-hand side). The probe waist radius and the input probe power

havetobeadjustedtocompensateforthemodel'sinabilitytomodelpartialspatial

overlapbetweenthebeams(right-handside). . . 23

2.7 Probeintensityat therearofthecrystalconvergestoanumericalsteadystate. . 25

2.8 Decreaseinpumppowerduetoabsorptionoverthelengthofthecrystal. Thepump

beamisfocusedin thecenterofthecrystal. . . 25

2.9 Upperstatepopulationdensityasafunctionofdistancealongtheopticalaxis. The

characteristicshapeis determined bythe intensities ofthe pump, probeand ASE

atthecorrespondingcoordinate. . . 26

2.10 Thecontributionoverallwavelengths totheforwardpropagatingASE(solidline)

and backward propagating ASE (dashed line) with respect to position along the

opticalaxis. . . 26

2.11 Increase in probe power due to amplication over the length of the crystal. The

probebeamisfocusedinthecenter ofthecrystal. . . 27

2.12 Gain experienced by the input probe power as a function of distance along the

opticalaxis. Theinputprobepowerof2.62

µW

isampliedtoanoutputpowerof 7.8

mW

overthelengthofthecrystal. . . 27

LISTOFFIGURES iv

2.13 Procedure fortheinclusion of transverse coordinates. ThenormalCW model can

onlysimulate uniform transverseintensitydistributions and therefore isunable to

accountfornon-uniform transversegainsaturation(left-hand side). Theimproved

CW model includes transversecoordinates by arrangingtophat beamsaccording

toanynon-uniformdistribution. . . 28

2.14 ComparisonbetweentheGaussianintensitydistributionattwodierentlongitudinal

positionswithin thecrystal. . . 29

2.15 Relativeradialamplicationattwodierentlongitudinalpositionswithinthecrystal

asaresultofnon-uniformtransverseamplication. . . 30

2.16 Thedivergingeect ofnon-uniformtransverseamplicationonaprobebeam(gain

distortion). . . 31

2.17 Top hat pump intensity as a function of longitudinal and transverse coordinates.

Thepump beamisfocusedonthefrontsurfaceofthecrystal. . . 33

2.18 Upperstatepopulationdensityat allpositions withinthecrystaloverlappingwith

theGaussian intensitydistributedprobebeam. . . 33

2.19 ContributionoverallwavelengthstotheforwardpropagatingASEasafunction of

position withinthecrystal. . . 34

2.20 Contributionoverallwavelengths tothebackwardpropagatingASEasafunction

ofposition withinthecrystal. . . 34

2.21 Procedure onwhichtime-dependentmodel relies. Thesquarepulse isdividedinto

discretepowerswhicharepropagatedthroughthegainmediumindividually,starting

fromtime

t

1

andendingattime

t

N

. Theampliedpowersarere-combinedtoform theoutputpulse. . . 36

2.22 Gaussianinputpulse(dashedline)givingrisetoanon-Gaussianoutputpulse(solid

line),shiftedwithrespecttothetemporalcentreoftheinputpulse. Theupperstate

populationdynamicswithrespecttotimeisalsoindicated(dashed-dottedline). . 38

2.23 Top hat inputpulse (dashed line) givingrise to a severely distorted output pulse

(solid line). The upper state population dynamics with respect to time is also

indicated(dashed-dottedline). . . 39

2.24 (a)Gaussian inputpulse(dashedline)beingampliedintherstpassthroughthe

amplier. (b) Theoutputpulse from (a)is usedasthe inputpulse forthesecond

passthroughtheamplier. . . 41

2.25 Upper state population density as a function of iterations. The rst dotted line

representsthetransitionfromthephasewhere thenumericalprocessesconvergeto

thephasewheretherst passiscalculated. Theseconddotted line representsthe

transitionbetweenthecalculationoftherstandsecond pass. . . 42

3.1 EnergyleveldiagramforNd:YVO

4

. . . 44

3.2 Absorption cross section spectrum for the

π

-polarization of Nd:YVO 4

[34]. The

LISTOFFIGURES v

3.3 Emission cross section spectrum for the

π

-polarization of Nd:YVO 4

[34]. The

strongestfeatureisfoundat1064

nm

. . . 45 3.4 Imageshowingan aerial viewof a top hatintensity distribution. The right-hand

siderepresentsaquarterofthetophatinpolarcoordinatesandisusedtocalculate

thevarianceviaintegration. . . 47

3.5 Opticaldesignforthepumpbeam. Thedashedlinesindicatethepositionandradius

oftheconvexlenses. Thewaistradiusisobtainedas200

µm

. . . 50 3.6 Experimentalsetupforsimpleamplier. . . 51

3.7 Electro-opticalpropertiesofpumpdiodesat

±

23

o

_C

. . . 52

3.8 Powerdependenceofwavelengthforpump diodesat

±

23

o

_C

. . . 52

3.9 Measuredpump beampropagationpathusedfor

M

2

p

-calculation. . . 53 3.10 Top hatproleof pump beamat waist. 60%ofthe poweriscontainedin anarea

witharadiusof200

µm

. . . 54 3.11 Electro-opticalpropertiesoftheprobelaserat20

o

_C

. . . 55

3.12 Measuredprobebeampropagationpathusedfor

M

2

s

-calculation. Theexperimental setupusedwasbuildespeciallyforthepurposeofmeasuringthebeamqualityand

shouldnotbemistakenfortheactualbeampropagationpath. . . 55

3.13 DependenceofgainonpumppowerinaNd:YVO

4

amplier. Thesolidcircles

repre-sentthenumericalresultsandtheblankcirclesthoseobtainedfromtheexperiment. 58

3.14 SaturationcharacteristicsoftheNd:YVO

4

amplier. Thesolidcirclesrepresentthe

numericalresultsandtheblankcirclesthoseobtainedfromtheexperiment. . . . 58

3.15 SchematicrepresentationofupconversionmechanismsinNd:YVO

4

[35]. . . 59

3.16 TheinuenceofASEat (a)lowgainand(b)high gain. Forbothguresadashed

line is used to indicate results simulatedwithout theeect ofASE, whilst a solid

lineisusedfortheresultsthataccountforASE. . . 61

3.17 Normalized photon density at the exit surface from the ruby crystal at dierent

times,normalizedtothepulsewidth (

η = 4 × 10

18

_{photons cm}

−2

). . . 65

3.18 Normalized photon density at the exit surface from the ruby crystal at dierent

times,normalizedtothepulsewidth (

η = 4 × 10

19

_{photons cm}

−2

). . . 65

4.1 EnergyleveldiagramforYb:YAG[46]. . . 68

4.2 Absorption cross section spectrum of Yb:YAG. The strongest feature is found at

941

nm

[47]. . . 68 4.3 Emission cross section spectrum of Yb:YAG. The strongest feature is found at

1030

nm

[48]. . . 69 4.4 TheinuenceofASEathighgain. Adashedlineisusedtoindicateresultssimulated

withoutthe eect of ASE included, whilst asolid line is used for the resultsthat

accountforASEas well. . . 70

4.5 Gain dependence on pump waist radius for (a)

N = 9 × 10

26

_m

−3

and (b)

N =

LISTOFFIGURES vi

4.6 Resultsfrom the CW simulation foran Yb:YAG crystalwith

N = 9 × 10

26

_m

−3

.

(a)Gradientofpump powerabsorption. (b)Evolutionofpowergainalong optical

axis. . . 73

4.7 Resultsfrom the CW simulationfor an Yb:YAG crystalwith

N = 7 × 10

26

_m

−3

.

(a)Gradientofpump powerabsorption. (b)Evolutionofpowergainalong optical

axis. . . 74

4.8 Resultsfromthe dual-passnumericalmodel foranYb:YAGcrystalwith

N = 7 ×

10

26

_m

−3

. (a) Energy gain for the rst pass: 10.83

dB

. (b) Energy gain for the secondpass: 10.79

dB

. . . 75 4.9 Resultfrom the dual-pass numericalmodel for an Yb:YAG crystalwith

N = 7 ×

10

26

_m

−3

. Theupperstatepopulationdensityasafunction ofiterationsindicates

almostnosaturationduringtherstpassandonlyaslightamountofsaturationas

aresultofthesecond pass. . . 76

4.10 Saturation characteristic of the Yb:YAG crystal with

N = 7 × 10

26

_m

−3

,

calcu-lated with the time-dependent model. The theoretical saturation input energy is

approximately0.25

mJ

. . . 76 4.11 Graphicalpresentationofheatgenerationwithin anYb:YAGcrystal. . . 78

4.12 GraphicalpresentationofFTCS-method,usedtosolvethe2Dheat equation. . . 79

4.13 Temperature distribution on the front surfaceof the crystal. The pump beam is

focused 2

mm

into the crystal, resultingin a pump spot radius of 266

µm

on the frontsurface. Themaximumtemperatureof 70.4

o

Cis locatedat thecentreof the

crystal. . . 81

4.14 Magnied 2

F

7/2

manifold of Yb:YAG. The process whereby an increase in

tem-perature inuences the level population densitiesare indicated. The energy level

associatedwithre-absorptionislabeledRA, whilePdenotesthelowerpumplevel. 82

4.15 Stresscomponentsat radialpositionsacrosstheshortest dimensionofthe crystal.

Thecrystalissubjecttoextremecompressivestresses. . . 87

4.16 Resultantstressvalue,normalizedtothesurfacefracturestrengthofthecrystal,as

afunctionof radialposition. . . 88

4.17 Designofpump beampropagationpathobtainedfrom geometricalraytracing. . 89

4.18 PreliminaryexperimentalsetupwithYb:YAGasamplifyingmedium. . . 90

4.19 Electro-opticalpropertiesofpumpdiodesat24 o

C

. . . 90 4.20 Diodewavelengthdependenceonpumppowerandtemperature. . . 91

4.21 Finalexperimentalsetupusedtoinvestigatethepropertiesofthespontaneous

emis-sionfrom a5at.%Yb:YAGamplifyingmedium. . . 92

4.22 Spontaneous emission image obtained from a CCD camera at a 90 o

angle with

respecttothesideoftheYb:YAGcrystal. Thepumpbeamisfocusedinthecentre

ofthecrystalatfullpower. . . 93

4.23 Spot radii along the optical axis according to the experimental data as well as

LISTOFFIGURES vii

4.24 Normalizedexperimental result forintensityalong the optical axis. Theintensity

is proportional to the pump intensity within the crystal at the corresponding

z

position. Thepumpbeamisfocusedatalongitudinalpositionof 1.3

mm

. . . 96 4.25 Numericalresultfor normalizedpump intensityalong theoptical axis. Thepump

beamisfocusedatalongitudinalposition of1.3

mm

.. . . 97 4.26 Normalizedexperimental result forintensityalong the optical axis. Theintensity

is proportional to the pump intensity within the crystal at the corresponding

z

position.The pumpbeamisfocusedin thecentreofthecrystal. . . 97

4.27 Numericalresultfor normalizedpump intensityalong theoptical axis. Thepump

beamisfocusedin thecentreofthecrystal. . . 98

4.28 Comparisonbetweenthenormalizedpump intensityobtainedfrom the simulation

andthenormalizedexperimentallymeasuredintensity. Thefullpowerpumpbeam

isfocusedinthecentreofthecrystal. . . 98

4.29 Comparison between the Experimental and Numerical results for the normalized

ASEintensities. Themeasurementisperformedbehindthecrystal. . . 99

4.30 Measuredresultfornormalizedspontaneousemission(SE)intensity. The

measure-mentisperformedatthesideofthecrystal. . . 100

4.31 Comparisonbetweenpump light transmissionpredictedby simulationand

experi-ment. . . 101

4.32 Diodewavelengthspectrumat 24 o

List of Tables

2.1 ParametersusedintheCWsimulation. ANd:YVO

4

crystaland

π

-polarizedbeams areusedforthepurposeofillustratingtypicalresults. . . 24

2.2 Parameters used in the CW simulation incorporating transverse coordinates. A

Nd:YVO

4

crystal and

π

-polarized beams are used for the purpose of illustrating typicalresults. . . 32

2.3 Parametersused inthetime-dependentsimulation. AnYb:YAGcrystalisused for

thepurposeof illustratingtypicalresults. . . 38

3.1 ParametersfortheNd:YVO

4

simulation. Thevaluesindicatedforthepump

wave-length,pumpabsorptioncrosssectionandinitialpumppowerwereusedtogenerate

Figure3.14,whilethevaluefortheinitialprobepowerisspecictoFigure3.13. . 57

4.1 ConstantparametersforalltheYb:YAGdesigns. . . 72

4.2 ImportantparametersforthetwooptimumYb:YAGdesigns. . . 73

4.3 Parametersusedforthecalculationofthethermalstresstensorcomponentsonthe

Introduction

The continuous search for better materials and a more comprehensive understanding of the

underlyingprinciples intheeldoflasersandopticsreliesontwomethods. Therstand

histori-callyprimary methodofevaluatingamaterialand itspropertiesis anexperiment. Experimental

investigationsareverygoodinthesensethattheygiveanexactaccountofthephysicalprocesses

governingthematerial. Oftenitisdiculttoidentifytherelativesignicanceoftheprocesses

in-uencingaspecic experiment. Inaddition,theexperimentalevaluationofanumberofmaterials

isnotalwayspracticalduetothehighcostofsuchmaterialsandthetimeittakestomanufacture

them. As aresult of such diculties,the second method of evaluating a material, a theoretical

model, playsan important role in the research environment. Apart from the fact that

theoret-ical models can be implemented as tools in the design of laser materials for specic purposes,

the models also lead to an improvedunderstanding of the processesgoverningthe usefulness of

a material. The use of theoretical models is common in the literature and nds application in

various branchesof laser physics. Optical ampliers specicallyhave received much attentionin

theeldsofsemiconductorscience[1,2,3,4],breoptics[5]andsolid-state(SS)laserengineering,

in both the single [6, 7] and multi-pass [8] regimes. As far as lasers are concerned, theoretical

modelssimilarlycontinuetocontributetosemiconductorscience[9]aswellasSSengineering[10].

Althoughnumericalmethodsareoftenusedto solvetheequationsgoverningthedynamicsinside

thelasermaterials,analyticalsolutionsareavailableundercertainassumptions[11,12].

Theworkpresentedinthisthesisformspartofaprojectdedicatedtothedevelopmentofalaser

systeminthepulsedregime. Thelasersystemis requiredto deliver

ps

-pulses withanenergyper pulseinthe

mJ

-rangeandarepetitionrateofatleast1

kHz

. Thepulsedprobebeamisprovided byabreoscillator (Yb:glass). Fibrelasers haveanumberof advantages overconventionalbulk

SSlasers. Due to alarge surface-to-volume ratio, bre lasers dissipateheat veryeectivelyand

requireno externalcooling. Inaddition, brelasers are compact with high mechanicalstability

and produce outputs with exceptionalbeam quality. The disadvantages of bre lasers are high

non-lineareects and thepossibility of optical damage, arisingfrom the large intensities within

the small bre core [13]. It was decided to use a SS amplier (Yb:YAG) to scale the output

fromthebreoscillator upto the

mJ

-range. Solid-stateampliershaveenergystoragecapacities superiortothatofbrelasersandarefrequentlyusedtoachievehighenergypulses[13]. Yb:YAG

ispreferredaslasermaterialduetoitspropertiescontributingtohighpulseenergiesandhasbeen

LISTOFTABLES 2

The excellent thermalproperties and bandwidthof Yb:YAGenables thegeneration of

ps

-pulses at highrepetition rates. Althoughmaterials such asNd:glassorTi:Sapphire are widelyused for

short pulse generation, the low thermalconductivity associatedwith Nd:glass and the low laser

eciencyofTi:Sapphiredeemsitnecessarytoinvestigateothermaterials.

Incontributiontothedevelopmentofthishybridbre-bulklasersystem,theresearchpresented

hereisaimedatthedesignofaSSamplier.

Thedesign is basedon anumerical approach, whereby the population rateequations are solved

usingaRunga-Kuttamethod. ThenumericalcalculationsareexecutedinMATLAB.Althoughthe

continuouswave(CW)scenariohasbeenapriority,solutionstothetime-dependentequationsare

obtainedbymeansofanitedierencemethod. Thepurposeofthenumericaldesignistwo-fold.

Therstpurpose isto developadesigntoolcapable ofpredictingthegainthat canbeexpected

from acertain SSamplier under givenboundaryconditions, whilst thesecond is to investigate

theinuenceofampliedspontaneousemission(ASE) athighgain.

InChapter1thetheoryleadinguptothepopulationrateequationsisdiscussedwithemphasis

onthe assumptionsused in thederivation. Chapter 2showsthe manipulationof thepopulation

rate equations to account for additional eects and also describes the algorithms on which the

various models are based. The validity of the models are evaluated in Chapter 3, making use

of experimental procedures as well as known analytical solutions. The models are applied to

thedesign of an optimumYb:YAGcrystal in Chapter 4,also serving as nal verication of the

Chapter 1

The semi-classical approach to

light-matter interaction

Thedesignofanylaseroramplierrequiresafundamentalunderstandingofthewayinwhich

lightandmatter interacts. Thepurposeofthischapter istobrieyillustratethemostimportant

aspects of the theory eventually leading to the population rate equations. The light-matter

in-teractionwill be explainedatthehand of thesemi-classicalapproach. Thismeansthat thelight

(electromagneticwave)isdescribedclassically(Maxwell),whilstthematterisdescribedquantum

mechanically. Forthemostpart,theapproachofMilonniandEberly[18]shallbefollowed.

1.1 The wave equation

Whendealingwithelectromagneticelds,themostfundamentaldescriptionisgivenbyMaxwell's

equations. Byusingsimplevectorcalculus,theinhomogeneouswaveequationresultsfrom

rewrit-ing Maxwell's equations for a neutraldielectric medium and assuming that the medium is

non-magnetic.

∇

2

E −

~

_c

1

2

∂

2

E

~

∂t

2

=

1



0

c

2

∂

2

P

~

∂t

2

(1.1)

~

E

and

P

~

denotes theelectric eld and thepolarization (also called the dipole momentdensity) respectively,whilst

c

isthevelocityoflightin vacuum.

The inhomogeneous wave equation is satised by transverse elds (elds transverse to the

propagationdirection). An exampleofatransverseeld isamonochromaticplanewaveofsingle

frequency

ω

and amplitude

E

0

which are independent of

t

or

z

(time and space coordinate in propagationdirection).

~

E

=

εE

ˆ

0

cos ω



t − n (ω)

z

c



(1.2)

1.2.CLASSICALPICTUREOF ABSORPTION 4

Theunitvector

ε

ˆ

indicatestheorientationofthepolarizationforthismonochromaticplanewave and

n (ω)

istherefractiveindexofthemediumatthefrequencyspeciedby

ω

.

1.2 Classical picture of absorption

Priorto the developmentof quantum mechanics, the theory that wasused to predict anatom's

interaction with an external electromagnetic eld is called the electron oscillator model. The

electron oscillator model (Lorentz model) makes use of the dipole approximation and basically

comprisesofNewton'sequationofmotionfortherelativeelectron-nucleusdisplacementcoordinate

(

~x

)subjecttoabindingforce

F

~

en

(exertedbythepresumablystationarynucleusontheelectron) as well as the Lorentz force

F = e ~

~

E

(describing the interaction of the charged electron with theexternal eld). Asserting that thebinding force may be modeled accordingto aspring with

constant

k

s

,theLorentz-modelequationisgivenby:

m

d

2

_~x

dt

2

= e ~

E

 ~

R, t



− k

s

~x

(1.3)

The parameters

m

and

R

~

denote the reduced mass and centre of mass of the electron-nucleus system,respectively.

When the equation of motion, given in equation 1.3, is subject to a frictional force, caused

by inter-atomic elastic collisions, it is found that the dipoles (electron-nucleus pairs) experience

aphase lag relative to that of the driving eld. The phaselag eectivelydampens the dipoles'

oscillations. SolvingtheLorentz-modelequationincludingdamping,fornearlyresonantradiation

(the eld frequency

ω

corresponds to a natural oscillation frequency of the atom

ω

0

), leads to theclassicaltheoryofabsorption. Accordingto theclassicaltheoryofabsorptiontheintensityof

the external eld exponentiallydecays to zero with propagation distance at a rate proportional

to theabsorption coecient. Although theclassicaltheory of absorption correctlydescribesthe

lineshapebroadeningwithrespectto theabsorptioncoecient'sdependenceonfrequency(inthe

presence of damping), aparameter

f

, called the oscillator strength, has to be assigned to each naturaloscillationfrequency of themedium to compensatefor discrepancies betweenthe theory

andexperiment.

1.3 Quantum mechanical description of matter

The purpose of this section is to describe a theory able to predict how matter responds to an

external electromagnetic eld. Specically of interest is the transitions an atomic medium can

undergowhensubjecttosuchanexternaleld. Duetothestatisticalnatureofquantummechanics,

one can only assign a probability or expectation value for a system to be found in a discrete

energystate. Usingthetime-dependentSchr

¨o

dingerequationasstartingpoint,themathematical structure and assumptions necessaryto arriveat the probabilityamplitudes areexplained. The

1.3.QUANTUMMECHANICALDESCRIPTIONOFMATTER 5

1.3.1 The time-dependent Schrödinger equation

TheSchr

¨

o

dingerequationforaparticleinatime-independentpotential

V (~r)

isgivenby

i~

∂Ψ

∂t

=



−

~

2

2m

∇

2

_{+ V (~r)}



Ψ

(1.4)

where

~

= h/2π

isPlanck'sconstantandthewavefunctionislabeled

Ψ

. Thesolutionstoequation 1.4areoftheform

Ψ (~r, t) = Φ (~r) e

−iEt/~

(1.5)

wherethespacedependentfunctions,

Φ (~r)

,arecalled thestationarystates. Thescalar

E

simply representstheallowed energystatesofthe particle. Thestationary statesmustsatisfythe

time-independentSchr

¨o

dingerequation, with

H

ˆ

denotingthe Hamiltonian (operator representingthe kineticandpotentialenergyoftheparticle).

ˆ

HΦ (~r) = EΦ (~r)

(1.6)

Everystationarystateisassociatedwithaspecicenergyandaspecictheprobabilitydensityof

theparticleinspace.

Whenaparticlendsitselfin anexternaltime-dependentpotential(

V

ext

(~r, t)

),anew Hamil-tonianmaybedened as,

ˆ

H

=

H

ˆ

a

+ V

ext

(~r, t)

(1.7)

where

H

ˆ

a

is the atomic (unperturbed particle's) Hamiltonian. For a Hamiltonian dened by equation1.7thetime-dependentSchr

¨o

dingerequationisgivenby:

i~

∂Ψ

∂t

=



−

~

2

2m

∇

2

_{+ V (~r) + V}

ext

(~r, t)



Ψ

(1.8)

Due to the completeness of the functions

Φ

n

(~r)

, a solution to the wavefunction satisfying the time-dependentSchr

¨o

dingerequation maybeexpressedasalinearcombinationof thestationary states.

Ψ (~r, t) =

X

n

1.3.QUANTUMMECHANICALDESCRIPTIONOFMATTER 6

Figure1.1: Theprocessesofabsorptionandemissionforatwo-stateatomhaving

∆E = ~ω

.

Equation1.9requiresthesolutionstothetime-dependentcoecients

a

n

(t)

,knownasprobability amplitudes.

1.3.2 A two-state quantum system

Whenamonochromatic planewaveof frequency

ω

is incidentuponanatom with twoelectronic energylevelssuch that

∆E = E

2

− E

1

= ~ω

,aphotonwill beabsorbed. Conversely, ifforsome reasontheatom's energyisdecreasedfrom

E

2

to

E

1

,aphotonofenergy

E = ~ω

willbeemitted (Figure1.1).

Since only two states are assumed, the probability of theatom beingin any one of the two

statesshould beunity:

|a

1

(t)|

2

+ |a

2

(t)|

2

=

1

(1.10)

Fromequation1.9thewavefunction forthistwo-statesystemisgivenby:

Ψ (~r, t) = a

1

(t) Φ

1

(~r) + a

2

(t) Φ

2

(~r)

(1.11)

1.3.QUANTUMMECHANICALDESCRIPTIONOFMATTER 7

Afterusing thetime-independent Schr

¨o

dingerequation togetherwith thefact that thefunctions

Φ

n

(~r)

areorthogonal,atime-dependentSchr

¨o

dingerequationforeachof thestatesarises 1

:

i~ ˙a

1

(t) = E

1

a

1

(t) + V

12

a

2

(t)

i~ ˙a

2

(t) = E

2

a

2

(t) + V

21

a

1

(t)

(1.12)

The symbols

V

12

and

V

21

denote the matrix elements of the external potential arisingfrom the electromagneticeld. Althoughtherearetwomorematrixelementsforthissystem(

V

11

and

V

22

), theyarerequiredtobezerobytheparityselectionrule. Intheelectricdipoleapproximationevery

matrixelement

V

ab

(t)

isrelatedtotheelectriceld(incomplexnotation)by 2

V

ab

(t) = −e~r

ab

·

1

2

εE

ˆ

0

e

−iωt

₊

complexconjugate

(1.13)

where

~r

ab

isknownasthecoordinatematrixelementfortheelectrondisplacement(see

~x

inLorentz model). Theequationsin 1.12maybethereforeberewrittenas

3 :

i ˙a

1

(t) =

−

1

2

χ

12

e

−iωt

_{+ χ}

∗

21

e

iωt

a

2

(t)

i ˙a

2

(t) =

ω

21

a

2

(t) −

1

2

χ

21

e

−iωt

_{+ χ}

∗

12

e

iωt

a

1

(t)

(1.14)

The frequency corresponding to the energy of the upper state has been dened as

ω

21

= E

2

/~

(assuming

E

1

tobethegroundstate),while

χ

ab

= e (~r

ab

· ˆε) E

0

/~

isdenedastheRabifrequency. Due to the fact that absorption takes place whenever the external eld is resonant to the

naturaloscillationfrequency, thecaseforwhich

ω ≈ ω

21

isofspecicinterest. Thetrialsolutions

a

1

(t) = c

1

(t)

and

a

2

(t) = c

2

(t) e

−iωt

allow the time-dependent Schr

¨

o

dinger equations to be rewrittenas:

i ˙c

1

(t) = −

1

2

χ

12

e

−2iωt

_{+ χ}

∗

21

c

2

(t)

i ˙c

2

(t) = (ω

21

− ω) c

2

(t) −

1

2

χ

21

+ χ

∗

12

e

2iωt

c

1

(t)

(1.15)

Therotating-waveapproximationstatesthat thetermsdescribedby

e

±2iωt

maynowbetakento

averagetozero,sincetheirvariationwithtimeissorapidincomparisontothatoftheotherterms.

1

Themathematicalnotationofadotisusedtoindicatethetimederivativeofthequantity

a

. 2

Thesubscripts

a

and

b

referto1and2orviceversa. 3

1.3.QUANTUMMECHANICALDESCRIPTIONOFMATTER 8

If one takes

χ = χ

21

and dene the detuning from resonance as

∆ = ω

21

− ω

, the nal form ofthetime-dependentSchr

¨o

dingerequationsaregivenby:

i ˙c

1

(t) = −

1

2

χ

∗

_c

2

(t)

i ˙c

2

(t) = ∆c

2

−

1

2

χc

1

(t)

(1.16)

1.3.3 The density matrix

Using the denition of the expectation value and the solution to the two-state system's wave

function,thequantummechanicalelectrondisplacementisfoundtobe:

h~ri =

Z

allspace

Ψ

?

(~r, t) ~rΨ (~r, t) d

3

r

=

Z

allspace

(a

∗

1

Φ

∗

1

+ a

∗

2

Φ

2

∗

) ~r (a

1

Φ

1

+ a

2

Φ

2

) d

3

r

= ~r

12

a

∗

1

(t) a

2

(t) +

complexconjugate (1.17)

Analogousto theLorentz model, anequation ofmotionforthe expectation valueoftheelectron

displacementmay now be derived using the time-dependent Schr

¨

o

dinger equations of 1.12. For linearlypolarizedlightandarealvalued

~r

12

,theequationofmotionmaybewritten as:

 d

2

dt

2

+ ω

2

0



h~ri =

2eω

_~

0

~r

12



~r

21

· ~

E

 

|a

1

(t)|

2

− |a

2

(t)|

2



(1.18)

Under the assumption that the excited stateoccupation probability is verysmall, thequantum

mechanical equation of motion (1.18) reduces to theLorentz-model equation of 1.3 and enables

fundamental atomic parametersto be assigned to the oscillator strength. Comparing equations

1.18and1.3,thenecessityfortheparameter

f

becomesapparent. Equation1.3showsanelectron responding to an electric eld independent of orientation, while equation 1.18 uses the scalar

product to determine the degree to which the dipole is aligned with the external eld, thereby

assigningastrengthto theoscillators.

Thepreceding discussion regardingthequantum mechanicalequivalentto theLorentz model

isextremelyusefulsinceitprovidestheincentive(viathepositionexpectationvalue)toworkwith

theproducts

c

∗

1

c

2

and

c

1

c

∗

1.3.QUANTUMMECHANICALDESCRIPTIONOFMATTER 9

Forthisreason,thedensitymatrixelementsof atwo-statesystemmaybedenedas:

ρ

12

=

c

1

c

∗

2

ρ

21

=

c

2

c

∗

1

ρ

11

=

c

1

c

∗

1

= |c

1

(t)|

2

ρ

22

=

c

2

c

∗

2

= |c

2

(t)|

2

(1.19)

The rst twoequations of 1.19 represent the complex amplitudes of the expectation value

h~ri

, while the last two equations are simply the occupation probabilities of the states. Writing the

time-dependentSchr

¨o

dingerequationsof1.16intermsofthedensitymatrixelementsofequations 1.19,thefollowingequationsofmotionareobtained:

˙ρ

12

= i∆ρ

12

+ i

χ

∗

2

(ρ

22

− ρ

11

)

˙ρ

21

= −i∆ρ

21

− i

χ

2

(ρ

22

− ρ

11

)

˙ρ

11

= −

i

2

(χρ

12

− χ

∗

_ρ

21

)

˙ρ

22

=

i

2

(χρ

12

− χ

∗

_ρ

21

)

(1.20)

1.3.4 Elastic and inelastic collisions

Theequationsof1.20mustnowbemodiedtoaccommodateprocessessuchasinelasticandelastic

collisions(aswasdiscussedinSection1.2). Althoughproperstatisticalmathematicsmaybeused

to derive the extra terms due to collisions, these terms will simply be introduced along with a

discussion oftheir meanings. It is howeverimportantto note that from this point onwards, the

equationsofmotionforthe

ρ

'srepresentanaverageatom.

As the name"inelasticcollision" suggests,it is accompaniedby energyloss. The energyloss

translatesintochangesintheoccupationprobabilities

ρ

11

and

ρ

22

:

˙ρ

11

=

−Γ

1

ρ

11

+ A

21

ρ

22

−

i

2

(χρ

12

− χ

∗

_ρ

21

)

˙ρ

22

=

− (Γ

2

+ A

21

) ρ

22

+

i

2

(χρ

12

− χ

∗

_ρ

21

)

(1.21)

Theterms

−Γ

1

ρ

11

and

−Γ

2

ρ

22

arearesultofinelasticcollisionsleadingtoanexponentialdecayof theoccupationprobabilityfromthelowerandupperenergylevelsrespectively,into otherenergy

levelsoftheatom. Similarly,

±A

21

ρ

22

denotesspontaneousemissionandsigniesexponential de-cayfrom theupperenergylevelinto thelowerlevel,whilstemittingaphoton. Thethirdtermin

1.3.QUANTUMMECHANICALDESCRIPTIONOFMATTER 10

In contrast to inelastic collisions, elastic collisions conserve the energy involved, thereby not

inuencing the occupation probabilities. Elastic collisions do however lead to a phase lag, as

waspreviouslydiscussed, and forthis reasonhavean eecton thequantum mechanicalelectron

displacement (

ρ

12

and

ρ

21

). Although elastic collisions do not inuence

ρ

11

and

ρ

22

, inelastic collisionsalsoaect

ρ

12

and

ρ

21

:

˙ρ

12

=

− (β − i∆) ρ

12

+ i

χ

∗

2

(ρ

22

− ρ

11

)

˙ρ

21

=

− (β + i∆) ρ

21

− i

χ

2

(ρ

22

− ρ

11

)

(1.22)

Intheequationsof1.22,

β

denotesthetotalrelaxationrate,givenby:

β =

1

τ

+

1

2

(Γ

1

+ Γ

2

+ A

21

)

. Theterm

−

1

τ

ρ

21

givestherateatwhichelasticcollisionsoccur,whilsttheinelastictermsincluded in

β

arisefrom thefact that the density matrixelement

ρ

12

(forexample) isrelated toelements

ρ

11

and

ρ

22

by:

|ρ

12

| =

√

ρ

11

ρ

22

(1.23)

Elasticcollisionswilloftendominateandfortheremainderofthisdocument

β ≈

1

τ

isapproximated.

1.3.5 The population rate equations

In the case where the total relaxation rate (

β

) is dominated by the elastic collision rate

1

τ

, it is found that the atom's response to theexternal eld is no longer oscillatory. In fact, external

forcesonlyproduceaquasisteady-stateincreaseordecreaseintheoccupationprobabilities. This

increase or decrease corresponds to the terms stimulated emission and absorption respectively.

Although the expressions for the absorption and stimulated emission coecients as well as the

spontaneousemissioncoecient(rstintroducedbyEinstein)shallnotbederivedhere,theorigins

ofthecoecientswill beexplainedat thehand oftheequationsofmotionforthedensitymatrix

elements.

Theelasticcollisionsresponsibleforthefastdecayintheatom'sresponsetoanexternaleldis

onlylocatedintheequationsgivenby1.22. Thismeansthattheelements

ρ

12

and

ρ

21

veryquickly reachaquasisteady-stateandtheirtime-derivativesmaybesetto zero,yieldingthefollowing:

ρ

12

=

iχ

∗

_/2

β − i∆

(ρ

22

− ρ

11

)

ρ

21

=

−iχ/2

1.4.THECONTINUITYEQUATION 11

Substituting the expressions for

ρ

12

and

ρ

21

(equations 1.24) into the equations of 1.21, gives thepopulationrateequations:

˙ρ

11

= −Γ

1

ρ

11

+ A

21

ρ

22

+

|χ|

2

β/2

∆

2

_{+ β}

2

(ρ

22

− ρ

11

)

˙ρ

22

= − (Γ

2

+ A

21

) ρ

22

−

|χ|

2

β/2

∆

2

_{+ β}

2

(ρ

22

− ρ

11

)

(1.25)

Thersttwotermsinbothexpressionsof1.25havealreadybeendiscussedintheprevioussection,

whilethelasttwotermsindicatestimulatedemissionandabsorptionrespectively. Sincethelasttwo

termsareequivalentexceptforachangeinsign,thecoecient

|χ|

2

β/2

∆

2

+β

2

iswrittenmorecompactlyas

R = σφ

,where

σ (∆)

iseithertheemissionorabsorptioncrosssectionand

φ = I/~ω

iscalledthe photonux. Multiplyingtheoccupationprobabilitiesby

N

,thedensityofatoms,anddeningthe gaincoecient

g (∆) = σ (∆) (N

2

− N

1

)

,thenalformofthepopulationrateequationsbecomes:

˙

N

1

= −Γ

1

N

1

+ A

21

N

2

+ gφ

˙

N

2

= −Γ

2

N

2

− A

21

N

2

− gφ + K

(1.26)

ThesymbolKintheupper-staterateequationrepresentsthepumpingrate. Thepumping

mecha-nismisresponsibleforcreatingapopulationinversion

(N

2

− N

1

)

andwillbediscussedin Chapter 2.

1.4 The continuity equation

Thepurpose ofthis sectionisto obtainanequationdescribingtheamplicationoflight. For

this,useshallbemadeofPoynting'stheoreminonedimension. Theenergydensitywithinalight

waveis denedastheintensity(modulus squaredoftheelectriceld's amplitude)dividedbythe

velocityofthewave,

u (z) = I/c

.

Theelectromagneticpowertransportedbyaplanewaveacrosssomeplaneofarea

A

atposition

z

,isgivenby

I (z) A

. Similarly,thepowertransportedacrossaplaneatposition

z + ∆z

isgivenby

I (z + ∆z) A

. Accordingto thedenition of thederivative,thechange in thepowerwithrespect tothe

z

-coordinateis:

∂

∂z

(IA)

= [I (z + ∆z) − I (z)] A/∆z

∂

1.4.THECONTINUITYEQUATION 12

Thechangeinpowershould howeverbeconsistentwiththerateatwhich electromagneticenergy

leavesthevolume

A∆z

:

−

_∂t

∂

(uA∆z) =

∂

∂z

(IA) ∆z

(1.28)

Rewritingequation1.28,theequationofcontinuityresults:

1

c

∂

∂t

I

+

∂

∂z

I = 0

(1.29)

Althoughtheright-handsideofequation1.29correspondstoavacuum,itmayeasilybereplaced

by rates responsible for the change in electromagnetic energy in a so-called gain medium

(two-statesystemforexample). Whenamonochromatic planewaveoffrequency

ω

propagatesin the

z

-direction throughamedium with aresonancefrequency at

ω

, the absorption of radiationwill create a population inversion. The upper state population may be stimulated by an external

radiationsourcetoemit photonsandthenetresultisamplicationoftheradiationasquantied

bythegaincoecient,

g (∆)

. Theequationofcontinuitythus becomes:

1

v

∂

∂t

I

+

∂

∂z

I = g (∆) I

(1.30)

The parameter

v

replaces

c

and denotes the velocity of the electromagnetic wavein a medium (

v ≤ c

).

Equation1.30,togetherwiththepopulationrateequationsof1.26formsthebasisofthetheory

oflight-matterinteractioninthesemi-classicalapproachandwillbediscussedtodetailinthenext

Chapter 2

Numerical models

Thepopulationrateequationsandthedierentialequationdescribingthedynamicsoftheeld

intensity are coupled by the population densities

N

1

and

N

2

. Manytextbooks show how these coupled equations can be solved analytically under certain (often crude) approximations. The

fact of the matter remains that these coupled equations can notbe solved analytically without

such approximationsand therefore the use of anumerical approach is required. In this chapter

anumerical model capable of predictingthe gain that can be expected from acontinuous wave

(CW)amplieraswellasapulseamplierwillbedescribed. Sincenotalleectscanrealistically

beaccountedfor,the modelis basedonanumberofapproximationsandcertainphenomena are

madeapriority.

2.1 Adjusting the population rate equations

2.1.1 Assumptions

The equations of 1.26 and 1.30 were derived for a hypothetical two-state system. In such

a system the radiation absorbed from an external electromagnetic wave, corresponding to the

resonance frequency of the medium, also leads to stimulated emission of radiation at the same

frequency. In addition, the eect of spontaneous emission and other possible inelastic collisions

contributetothede-populationoftheupperstate,makingapositivepopulationinversion(

N

2

−N

1

) impossibletoobtain. Theuseofgainmediaconsistingofthreeormoreactivestatesaretherefore

requiredto obtainlaseraction[18].

Figure2.1showsatypicalfour-statesystem. Forafour-statesystemapump(previouslylabeled

K

)withradiation atfrequency

ω

p

isused toexcite atomsfrom energylevel

|1i

to level

|4i

. The energyofthesystemisthenloweredfrom

|4i

to

|3i

bywhatisassumedtobeveryfastnon-radiative transitions. The mechanism offast non-radiativetransitions populates

|3i

. In the presenceof a probewithradiationatfrequency

ω

s

,correspondingtotheenergydierence

∆E

23

= ~ω

s

between

|2i

and

|3i

,energylevel

|3i

ispartiallydepopulatedbystimulatedemission(

ω

s

)andspontaneous emission,leadingto amplicationoftheprobewave.

2.1.ADJUSTINGTHEPOPULATION RATEEQUATIONS 14

Figure2.1: A four-statesystemshowingthemechanismsresponsibleforlaseraction.

Although the process whereby amplication takes place results in the atom lowering its energy

from

|3i

to

|2i

,fastnon-radiativetransitionsonceagaindecreasestheatom'senergyto

|1i

[18,19]. The eciency of alaser scheme depends heavily on therate at which thefast non-radiative

transitionsdepopulate

|4i

and

|2i

. If thede-populationratesare sucientlyfast,thepump may betreatedas thoughit excitesatomsto

|3i

insteadof

|4i

, whilethestimulatedand spontaneous emissionmaybetreatedasthoughitde-excitesatomsdirectlyto

|1i

. Sincenoneoftheatomsget trappedin

|4i

and

|2i

, therapidde-populationeectivelypreventsthepumpand probewavesof actingin reverse to repopulate

|1i

and

|3i

respectively. Thefour-statesystemof Figure 2.1may thereforebeviewedasatwo-statesystemconsisting oflevels

|1i

and

|3i

,whilst keepingin mind that the frequenciesof thepump andthe probe are dierentfrom each other (Figure 2.2). The

approximation of a two-state system does not imply a lossof information, since the multi-level

structureof theenergydiagram iscontainedwithin the respectiveabsorptionandemission cross

sectionsof agiven transition. Forsimplicity thelevels previouslylabeled as

|1i

and

|3i

are now denotedaslevelsas

|1i

and

|2i

,respectively.

2.1.ADJUSTINGTHEPOPULATION RATEEQUATIONS 15

Figure 2.2: Simplied"four-state"systemasaresultofassumingfastnon-radiativetransitions.

With the simplications as illustrated in Figure 2.2, the appropriaterate equations are very

similar to that of equation 1.26. Inelastic collisions may be neglected asa result of its relative

magnitude, withrespect to thestimulatedprocessesand spontaneousemission, yielding therate

equationsgivenby:

˙

N

1

= A

21

N

2

+ (σ

21

p

φ

p

+ σ

21

s

φ

s

) N

2

− (σ

p

12

φ

p

+ σ

12

s

φ

s

) N

1

˙

N

2

= −A

21

N

2

− (σ

p

21

φ

p

+ σ

s

21

φ

s

) N

2

+ (σ

12

p

φ

p

+ σ

s

12

φ

s

) N

1

(2.1)

A comparisonbetween thesets of equationsgivenby 1.25,1.26and 2.1 showsthe way in which

thegaincoecient nowconsists of contributions from four dierentcross sections(

σ

p

21

,

σ

s

21

,

σ

p

12

and

σ

s

12

). Furthermore, theequations of 2.1is subjectto twodistinct photonux contributions (

φ

p

and

φ

s

). Thecrosssection subscript

21

denotesstimulated emissionfrom

|2i

to

|1i

, while

12

denotesabsorptionfrom

|1i

to

|2i

. A

p

-superscriptforthecrosssectionsandphotonuxrefersto radiation at thepump frequency (

ω

p

) and an

s

-superscript to that at the probe frequency(

ω

s

). Addingthetwoequationsof2.1,itisfoundthatthetotalpopulationdensityistime-independent

andthatthesumofthelevelpopulationdensitiesthereforeremainsconstant.

N

1

+ N

2

=

N

(2.2)

Thesymbol

N

waspreviouslydenedasthedensityofatoms. Fromtherelationbetween

N

1

and

N

2

(2.2),thetwoequationsof2.1mayconvenientlybeexpressedonlyintermsoftheupper-state populationrateequationandthedensityofatoms:

˙

N

2

=

−A

21

N

2

− (σ

p

21

φ

p

_{+ σ}

s

21

φ

s

) N

2

+ (σ

p

12

φ

p

_{+ σ}

s

12

φ

s

) (N − N

2

)

(2.3)

2.1.ADJUSTINGTHEPOPULATION RATEEQUATIONS 16

For the purpose of simulating the gain dynamics of an amplier in CW mode, a temporal

steady-stateisassumed. Atemporal steady-stateimpliesthat thetime derivativeofequation 2.3

is takenaszero. Applying therelation betweenthe photonux and intensityand rewriting the

frequencyas

ν = ω/2π

,thesolutionto thesteadystaterateequationbecomes:

N

2

=



_σ

p

12

I

p

hν

p

+

σ

s

12

I

s

hν

s



N

h

_σ

p

12

I

p

hν

p

+

σ

s

12

I

s

hν

s



+



σ

21

p

I

p

hν

p

+

σ

s

21

I

s

hν

s



+

1

_τ

i

(2.4) Theterm

1

τ

,with

τ

beingthespontaneousemissionlifetime,replaces

A

21

.

Since the system is subject to two radiation elds (pump and probe), the dynamics of the

intensityofboththeradiationeldsmustbedescribedbyitsowndierentialequationinanalogy

with equation 1.30. Again, the gain coecient asdened at the end of Chapter 1is no longer

applicableandisreplacedbytheappropriatecrosssectionterms. Usingthetemporalsteady-state,

thedierentialequationsfortheradiationeldsaregivenintermsofintensitybythefollowing:

∂I

p

∂z

=

[σ

p

21

N

2

− σ

12

p

(N − N

2

)] I

p

∂I

s

∂z

=

[σ

s

21

N

2

− σ

12

s

(N − N

2

)] I

s

(2.5)

The equations of 2.5, together with equation 2.4, are the most general form of the equations

that shall be used to describe the CW mode amplicationof light. Section 2.1.2 dealswith the

modicationofthesegeneralequationsto alsoincludeotheraspectsofampliertheory.

2.1.2 Focusing eects

WhendealingwithSSampliers,diodepumpingisoftenpreferredduetoitsrelativesimplicity

andhighoutputpower. AswillbediscussedindetailinChapter3,thesediodemodules arebre

coupled and havea multiple transverse mode output. A multi-mode output is characterized by

anon-ideal beam quality factor (

M

2

_{> 1}

), resultingin anumber of complications with respect

tooptical design[19,20]. The beamquality factorisessentiallyameasureof howtightlyalaser

beamcan befocusedunder certainconditionsandgivesanideaoftheextenttowhichthebeam

isdiractionlimited[21]. Toappreciatetheimplicationsofsuchanon-idealbeam,itisnecessary

totakealookat thephysical propertiesofGaussianbeams.

AGaussianbeampropagatinginthe

z

-direction,isdenedbyanintensityprolegivenby[18]

I (x, y, z) =

|A|

2

e

−2

(

x

2

+y

2

)

/w

2

(2.6)

where

A

isthecomplexscalarwaveamplitude,

x

and

y

thetransversepositioncoordinatesand

w

thespot size radiusofthebeam(lateralposition fromthe

z

-axiswheretheintensity decreaseby afactorof

1

e

2

).

2.1.ADJUSTINGTHEPOPULATION RATEEQUATIONS 17

Theamplitude,

A

, maybeexpressedin termsofthecomplexradiusofcurvature, givenby:

1

q (z)

=

1

R (z)

+

iλ

πw

2

_(z)

(2.7)

R (z)

istheradiusofcurvatureofthewavefront. Theparameter

q (z)

hastheadditionalproperty that

q (z) = q

0

+ z

,where

q

0

= q (0) =

iλ

πw

2

0

,since

R

0

= R (0) = ∞

(plane).

w

0

= w (0)

isdened asthebeamwaist,sinceitrepresentstheminimumspotsize radius alongthebeampropagation

path(Figure 2.3). Inthecase of

M

2

_{6= 1}

, thetwoexpressionsfor

q (z)

maybe equatedand the realandimaginarypartsrespectivelyyield:

R (z) =

z +

z

2

0

(M

2

₎

2

_z

(2.8)

w (z) =

w

0

s

1 +

(M

2

₎

2

_z

2

z

2

0

(2.9)

The newparameter,

z

0

=

πw

2

0

λ

, is dened as the Rayleighrange and is a measure of the length overwhich thebeammaybeassumedtobecollimated aboutthewaist region(Figure2.3). With

thefactor

M

2

2

included,thespotsize radiuswill divergeataratelinearlyproportionalto the

beamqualityfactor(

M

2

)and willalsobelargerforasmallerbeamwaist.

Whenconsideringthederivativeoftheintensitywithrespecttothepositioncoordinatealong

theopticalaxis(say

z

),asin equations2.5,thefollowingisfound:

dI

dz

=

d

dz

 P

A



=

1

A

dP

dz

+ P

d

dz

 1

A



=

1

A

dP

dz

+

2Iπw

A

dw

dz

(2.10)

The rstterm represents so-calledpowergainand givesthe change in intensity asa resultof a

changein power,with

A = πw

2

labelingthecross-sectionalareaofthebeam. Thesecondtermis

denotedthegeometricalgain,sinceitgivesthechangeinintensityresultingfrom achangein the

spot sizeradiusofthebeam(focusingorde-focusing).

The dierential equations describing the dynamics of the intensities of the pump and probe

beams (equations2.5) onlytakeaccountof intensitychangesdue to changes in power. Thefact

that theintensity at agiven

z

-positionis dependentonthe spot size radius ofthebeamat that position,motivatestheinclusionforatermdescribingthebeampropagationdynamics,especially

intheeventofahigh

M

2

2.1.ADJUSTINGTHEPOPULATION RATEEQUATIONS 18

Figure2.3: DenitionoftherelevantquantitiesofaGaussianbeamdivergingawayfromitswaist

[18,19].

Theequationsin 2.5maythenberewrittenas:

∂I

p

∂z

= [σ

p

21

N

2

− σ

12

p

(N − N

2

)] I

p

−

2I

p

M

p

4

λ

2

p

z

πw

4

0p

+ M

p

4

λ

2

p

z

2

(2.11)

∂I

s

∂z

=

[σ

s

21

N

2

− σ

s

12

(N − N

2

)] I

s

−

2I

s

M

4

s

λ

2

s

z

(πw

4

0s

+ M

s

4

λ

2

s

z

2

)

(2.12)

Onceagainasubscript

p

or

s

indicate thequantities relatedtothepumporsignalbeams respec-tively.

2.1.3 Amplied spontaneous emission

Thenalphenomenonthatwillbeincludedintherateequationapproach,usedinthenumerical

modeltofollow,isthatofampliedspontaneousemission(ASE).Asthenamesuggests,ASEisthe

mechanismwherebythespontaneousemission,resultingfromthespontaneousde-excitationofthe

2.1.ADJUSTINGTHEPOPULATION RATEEQUATIONS 19

Itis importantto considerASEsince, especially athigh gain,ithas thepotentialto deplete the

excitedstatepopulationsubstantially,leadingtoseveregainsaturation.

Amplication ofspontaneousemissiontakesplace overthefullemissionspectrumandin

ran-domdirections,dueto thenature ofspontaneousemission[18]. It howeverfollowsthatonlyASE

componentsparallel to theopticalaxis(forwardandbackward)are ampliedtoasignicant

de-gree. ForsimplicitytheemissioncrosssectionsofthemediaaremodeledaccordingtoaLorentzian

lineshape,centered aboutthepeakemissionwavelength.

σ

ASE

21

(λ)

= σ

21

s

(λ

0

)

1

2

Γ

π (λ − λ

0

)

2

+

1

₂

Γ

2

(2.13)

In equation 2.13,

λ

0

denotes the peak emission wavelength, while

Γ

is the full width at half maximum (FWHM) of the Lorentzian prole. For the purpose of the numerical calculation the

wavelength band is divided into a number of discrete intervals, each corresponding to a certain

stimulatedemissioncrosssectionvalue.

Propagationequations, describingtheintensityevolutionfortheforwardand backward

prop-agatingASE,canbederivedfrom equation1.30asweredonefortheintensitiesofthepumpand

probeinequations2.5.

∂I

ASE,f

(λ

i

)

∂z

=

σ

ASE

21

(λ

i

) N

2

I

ASE,f

(λ

i

) +

2N

2

hν

ASE

(λ

i

)

τ

A

1

4πL

2

ν

4

ASE

(λ

i

) σ

21

ASE

(λ

i

)

P

λ

i

ν

4

ASE

(λ

i

) σ

21

ASE

(λ

i

)

(2.14)

∂I

ASE,b

(λ

i

)

∂z

=

σ

ASE

21

(λ

i

) N

2

I

ASE,b

(λ

i

) +

2N

2

hν

ASE

(λ

i

)

τ

A

2

4πL

2

ν

4

ASE

(λ

i

) σ

21

ASE

(λ

i

)

P

λ

i

ν

4

ASE

(λ

i

) σ

21

ASE

(λ

i

)

(2.15)

Subscripts

f

and

b

are used to denote theforwardand backwardpropagating ASEcomponents, respectively,whilsttheindex

i

simplyreferstotherelevantwavelengthintervalforwhichtheASE componentsaresolved. Thersttermontheright-handsideofequations2.14and2.15represents

amplication of the ASEintensities at the various wavelength intervals, whilst the second term

givesthecontributiontotheASEintensitiesduetospontaneousemission[22]. Therstfactorin

thesecondterm,

N

2

hν

ASE

(λ

i

) /τ

,hasunitsofpowerperunitvolumeandthereforerepresentsthe energyperunit time,perunit volume,spontaneously emittedfromthe excitedstatein theform

ofphotons. Thefractionof thespontaneouslyemittedphotonsthat willbeampliedis givenby

thesolidangle

A

1,2

/4πL

2

2.2.CONTINUOUSWAVEMODEL 20

Figure2.4: Thesolidanglewithinwhichspontaneousemissionisamplied(solidline),compared

totheevolutioninthespot sizeofaGaussian pumpbeam(dashedline).

The cross sectional areas

A

1

and

A

2

are those at the rear and front surface of the amplica-tion medium respectively and correspond to the pump beam, since the pump is responsible for

creatingthepopulationinversionleadingtospontaneousemission(Figure2.4)[18,19]. Thesolid

angleused in equations 2.14and2.15is arathercrudeapproximation,but is sucientasarst

approach despite the fact that more sophisticated methods of modeling the ASE fractions are

available[23, 24]. Thenalfactorinthesecondtermofequations2.14and2.15isobtainedfrom

theFuchtbauer-Ladenburgequation [25,22,26]andsimplygivesthecontributionofspontaneous

emissionfromagivenwavelengthinterval.

TheadditionoftheASEpropagationequationsbringsaboutachangetothe

N

2

-rateequation, duetothefactthat ASEdepletestheexcitedstateatoms. Thenalformofthe

N

2

-rateequation isgivenbythefollowing:

N

2

=

N

 σ

p

12

I

p

hν

p

+

σ

s

12

I

s

hν

s



/

 σ

p

12

I

p

hν

p

+

σ

s

12

I

s

hν

s

+

σ

p

₂₁

I

p

hν

p

+

σ

s

21

I

s

hν

s

+

X

λ

i

σ

ASE

21

(λ

i

) I

ASE,f

(λ

i

)

hν

ASE

(λ

i

)

+

X

λ

i

σ

ASE

21

(λ

i

) I

ASE,b

(λ

i

)

hν

ASE

(λ

i

)

+

1

τ

#

(2.16)

2.2 Continuous wave model

Theequationsdescribingthepropagationoftheintensitiesforthepump,probe,forwardASE

andbackwardASE(equations2.11,2.12,2.14,2.15),togetherwiththerateequationfortheupper

statepopulationdensity(equation2.16)forms theheartof theCWnumericalmodel. InSection

2.2,thebasicalgorithmuponwhichthesimulationis basedaswellassomeofthetypicalresults