Intense lasers: high peak power Part 1: amplificationoldilil.ino.it › Capri2017 › Slides › Capri_2017_blg_part1.pdf · 2017-07-10 · Bruno Le Garrec LPA schoolCapri 2017 page

pageBrunoLeGarrec 1

Intense lasers: high peak powerPart 1: amplification

BrunoLeGarrecDirecteurdesTechnologiesLasersduLULILULI/EcolePolytechnique,routedeSaclay

91128Palaiseaucedex,France

[email protected]

31/08/2016

LPA school Capri 2017

pageBrunoLeGarrec 2

What do you need for building a laser

:An amplifying medium = an energy converterAn electromagnetic radiation = an electromagnetic wave that propagatesA resonant cavity = a set of mirrors facing each other = a « Fabry-Perot » cavity

That’s true for both an oscillator and an amplifier.For many reasons, a Master Oscillator Power Amplifier (MOPA) is the most commonly used

SOURCE Pre-- amplification

AMPLIFICATIONFREQUENCYCONVERSION

10-9 J < 1 J 15 to 20 kJ 7,5 to 10 kJ

pageBrunoLeGarrec 3LPA school Capri 2017

Laser-matterinteraction:theBlackbodyradiation

Wavelength

What is the power spectral density ofthe radiation emitted by a gas inside a box at a given temperature ?


Laser-matter interaction:theBlackbodyradiationtheory

• Semi-classicalmodelbetweenelectromagneticradiationandapopulationofatoms:– Thespectralenergydensityisdefinedasρ (ν) =U(ν)dN(ν)/Vthe

productoftheaveragenumberofphotonsperenergymodetimesthephotonenergytimesthemodesdensitybetweenν andν+dν:

– Photonenergy

– Averagenumberofphotonspermode

– Photonsdensitybetweenν andν+dν– That’sthePlanckformulafromtheBlackbodyradiationtheory

νπνννννρ d

ckThhd 8

1)exp(1 )( 3

2

−=

ννπνννρ d

ch

kThd 8

1)exp(1 )( 3

3

−=


Laser-matterinteraction:theatomicsystemismadeofmanylevels

Any2,3or4levelsystemcanbeseenasa2levelsystem:– Degeneracy g=2m+1(relatedtothenumberofsub-levelsofagivenkineticmomentum:orbital L,mL,totalJ=L+S,J,mJ,F=J+I,F,mF)

– Homogeneousbroadeningrelatedtothelifetimeoftheatomicsystem(freeatoms,electronsinthecrystalfield,molecules):– Lorentzian

– Inhomogeneousbroadening(Dopplereffectofmovingatomsandmolecules)– Gaussian

Δ istheFullWidthatHalfMaximum(FWHM)ofthelineshapewhenΔ =1/Trad+1/Tnon rad and

⎥⎦⎤

⎢⎣⎡ Δ+−

Δ=

42)( 22

)( 0ωωπωg

( ))2ln4(2ln2)(2

0

Δ−−

Δ= ωω

πω Expg

∫ =1)( ωω dg


Emission,Fluorescence,Phosphorescence

White light


Einstein’scoefficients(1)

Atthermodynamicequilibrium,eachprocessgoing“down”mustbebalancedexactlybythatgoing“up”andthetransitionprobabilitycanbewritten:

• Nj thepopulation(orpopulationdensity)ofleveli• N= N0+Nj =csteAbsorption• δN0

a=- N0 B0j ρ(ν) δtStimulatedorinducedemission• δN0

sti=+Nj Bj0 ρ(ν) δtSpontaneousemission• δN0

spont=+Nj Aj0 δt

• The balanceΔN0=δN0a +δN0

sti +δN0spont =[(Nj Bj0 - N0 B0j )ρ(ν) +Nj Aj0 ]δt

• ΔNj =-ΔN0=[(N0 B0j -Nj Bj0)ρ(ν) - Nj Aj0 ]δt• CommonlywrittendNj /dt,dN0 /dt

tBP jj )(00 νρ=→

Level 0, g0

Level j, gj

Abso

rptio

n Spontaneousemission


Einstein’scoefficients(2)Relationship between thecoefficients :• BecauseNj ∝ gj exp-Ej /kT andabsorption =sumofemissions• g0 B0j= gj Bj0

• Aj0/Bj0 =8πhν3/c3

• Nj isthepopulation (ornj thepopulation density) of leveli

ρ (ν) dν =ρ (ω) dω thenρ (ω) =ρ (ν) /2π

• Bj0 ρ (ν) /Aj0 =n(ν) isthenumberofphotons permode ρ (ν) =8πν2/c3 hν n(ν) withthermal radiation included thenn(ν) =1/Exp(hν/kT)-1) onefindshν >>kT intheopticaldomain and

stimulated/spontaneous ≈ Exp-hν/kT whileinthethermaldomain hν <<kT andstimulated/spontaneous ≈ kT/hν

Whentherefraction indexisn,v=c/n andonedefines the(laser) intensityas

Iν=cρ (ν) /n

• dNj /dt =(N0 B0j -Nj Bj0)ρ(ν) - Nj Aj0

• dNj /dt =(nIν /c)Aj0 (c3/n3)/8πhν3(N0 gj /g0 -Nj)- Nj Aj0

• dNj /dt =- Aj0 (λ2/8πn2)(Iν /hν )(Nj -N0 gj /g0)- Nj Aj0


Einstein’sapproach(3)

Transition(lifetime)broadening:• Generalcaseρ(ν) and g(ν):• AbecomesA’=Ag(ν) andBbecomesB’=Bg(ν)• Differentcasestobeconsidered:• Narrowtransitiong(ν) <<ρ(ν) theng(ν) =δ(ν−ν0)

• Broadtransitiong(ν) >>ρ(ν) then ρ(ν) = Iν /c= I0 g(ν −ν0)/c

Relationshipbetweenthecoefficients:• Spontaneousemissionisotropicandun-polarized• Stimulatedemission:transmittedwavehasthesamefrequency,isinthesame

direction,andhasthesamepolarizationastheincidentwave.• Thereisarelationbetweengainandintensity


Amplification(1)

OnewritestheintensitybalanceΔI =Itransmitted +Ispontaneous- Iincident asafunctionoftheEinstein’scoefficientsinatwo-levelatomicsystem(1,2)foragivenmediumthicknessΔz

ΔI =hν B21 Iν /cg(ν)N2 Δz- hν B12 Iν /cg(ν)N1 Δz+hν A21 Δν g(ν)N2 ΔzdΩ /4π onedirection acceptancecone

ΔI /Δz =hν B21 (N2 –N1 g2 /g1)g(ν)Iν +hν A21 Δν g(ν)ΔzdΩ /4π

Thereisgainif:• N2 >N1 g2 /g1

Witha« noise »contributionevenwithoutincidentlight

Δz

Polarizer filter

Incident intensity Transmitted intensity with dΩ

Level 1, g1

Level 2, g2


Amplification(2)

Thegainfactor reads:dIν/dz =A21(λ2/ 8πn2)g(ν)(N2 –N1 g2 /g1)Iν =γ0(ν) IνThisγ0(ν) org0(ν) isthesmall signal gainwhen Iincident issmall compared toa

so-called« saturation »value Isaturation .ThefirstpartofdIν/dz isthetransition cross section.Thereisadifference

betweenstimulated emission crosssection andabsorption cross section.γ0(ν) =A21(λ2/ 8πn2)g(ν)(N2 –N1g2 /g1)σse =A21(λ2/ 8πn2)g(ν)σab =A21(λ2/ 8πn2)g(ν) g2 /g1ΔN=N2 –N1g2 /g1, sofar :γ0(ν)=σse ΔN

WhendIν/dz canbeintegrated over zthen:Iν(z) =Iν(0) Exp[γ0(ν) z]G0(ν) =Exp[γ0(ν) z]=Iν(z)/ Iν(0) isthegain.Another veryimportant factor isthesaturation fluence :Fsat=hν/σ


Populationinversion(1)

Assoonas:N2 >N1 g2 /g1or γ0(ν)>0,thereispopulationinversionorpopulationsaresaidtobe“inverted”

WhentherearerelationsbetweenEinstein’scoefficientsor rateequationsPopulation inversion⇔ amplification

Atthermodynamicequilibrium,levelpopulationsaregivenbytheMaxwell-Boltzmannrelationship:Nj∝ gj exp-Ej/kT

N/g

Level 1, g1

Level 2, g2

E If E2 > E1, then N2/g2< N1/g1

So far:• N2/g2> N1/g1 is an abnormal state of affairs.This state has to be sustained to compensate for emission losses.• The extracted energy E = ΔN hν tells us that anytime 1 photon is emitted ⇔the atom “goes” from E2 to E1


Populationinversion(2):2-levelsystem

• Ina2-levelsystem,population inversion isimpossible

• Theprobability toempty level2isalwaysgreater than thattoemptylevel1.Inthe« open »case,leupper levelwillbeprogressively drainedtothemeta-stable leveland thelower levelwillbe“depleted”: thisprocess iscalled« opticalpumping ».

1

2

«closed» 2-level

1

2

«open» 2-level

m


Populationinversion(2):3-leveland4-levelsystems

• According toselection rulesbetween levels(parity,ΔL,ΔJ,ΔF=0,±1),absorption, spontaneous orstimulated emission areorarenotpossiblebetweenanysetof2levels.

• Nonradiative transitions arepossible: collisions (gas), crystalvibrations.Thesetransitions canallowfastpopulation transfers between neighborlevels.

0

12

3 levels1

23

04 levels


Thelaserwasbornin1960,May16th .

• Maiman hasusedaflashlamp(GEFT-506model) insideasimplealuminumtube.

• Therodhasa0.95cmdiameter(3/8inch)anda1.9cmlength (3/4inch)withendfacescoatedwithsilver.

• Ononeface,thecentralpartof thesilvercoatingisremoved inordertolettheradiationescapefromtherod.


3-levelsystem

• dN2 /dt =(N0 B02 –N2 B20)ρP(ν) – N2 A20- N2 A21

• dN1 /dt =(N0 B01 –N1 B10)ρL(ν) – N1 A10+N2 A21

• dN0 /dt =(-N0 B01 +N1 B10)ρL(ν) +N1 A10+ (-N0B02 +N2 B20)ρP(ν) +N2 A20

• d(N0+N1+N2)/dt=0

• ToachieveN1>N0 (N1>50%Nt):– dN2/dt=0– A21 greaterthanA20andB20 ρP(ν)

• HighρP(ν),meanshighpumping intensity• 3levels:Al203 ,Rubylaser

0

12

3 levels

ρP(ν)ρL(ν) A21


4-levelsystem

• dN3 /dt =(N0 B03 –N3 B30)ρP(ν) –N3 A30-N3 A32

• dN2 /dt =(N1 B12 –N2 B21)ρL(ν) –N2 A21+N3 A32

• dN1 /dt =(-N1 B12 +N2 B21)ρL(ν) +N2 A21- N1 A10

• dN0 /dt =(-N0 B03 +N3 B30)ρP(ν) +N3 A30+N1 A10

• d(N0+N1+N2+N3)/dt=0• att=0,N1=N2=0• IfA32 ismuchgreaterthanA30andB30 ρP(ν) ,as

soonasN3 ≠ 0,thendN2 /dt =N3 A32,soN2 ≠ 0• ThereisalwaysgainwhenN2>N1

• Inordertosustainthecycle,A10 mustbelargeenough,otherwisea« bottleneck»effectcanoccur.

• 4levels :Nd3+ :Y3Al5012 ,YAGlaser(garnet)

1

23

04 levels

ρP(ν) ρL(ν)

A32

A10


Quasi3-levelsystem

• dN3 /dt =(N0 B03 –N3 B30)ρP(ν) – N3 A30-N3A31-N3 A32

• dN2 /dt =(N1 B12 –N2 B21)ρL(ν) – N2 A21– N2A20+N3 A32

• dN1 /dt =(-N1 B12 +N2 B21)ρL(ν) +N2 A21+N3A31- N1 A10

• dN0 /dt =(-N0 B03 +N3 B30)ρP(ν) +N3 A30+N2A20+N1 A10

• d(N0+N1+N2+N3)/dt=0• IfA32 greaterthanA30andB30 ρP(ν) ,assoon

asN3 ≠ 0, thendN2 /dt =N3 A32and N2 ≠ 0• thereisnogainsinceN2<N1becauseN1

related totemperature• Inorder tosustain thegaincycle, thepump

intensitymustbegreater thanathresholdvalue.

• YtterbiumionYb3+ canbepumpedeither1-6or1-5

1

23

04 levels

ρP(ν) ρL(ν)

A32

A10

E

2F5/2

2F7/2l1

l2

l3

l4

u1

u2

u3E

2F5/2

2F7/2l1

l2

l3

l4

u1

u2

u3 765

4321

E

2F5/2

2F7/2l1

l2

l3

l4

u1

u2

u3E

2F5/2

2F7/2l1

l2

l3

l4

u1

u2

u3 765

4321


Amplification:intensityand/orfluence

00,51

1,52

2,53

3,54

4,5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

n° de tranche

Flue

nce

J/cm

2

• Thereisgain« gorγ =σΔn»

IinIoutIntensity :

Iout = Iin Exp(g.l)FluenceFout = Fin Exp(g.l)

What’s going on if the thickness increases indefinitely ?


Amplification:intensityand/orfluence

0

0,05

0,1

0,15

0,2

0,25

0,3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

n° de tranche

Gai

nIinIout

L. Frantz & J. Nodvik : « Theory of pulse propagation in a laser amplifier », J. Appl. Phys. 34,8, 2346 (1963)

Themediumissplitin«n»slices(or«n»amplifiersarelined)Thebluecurveistheinitialgain,therosecurveisthegainafteronepassSince1photonisemitted⇔ the atom“goes”fromE2toE1andbecause γ =σΔn,theextractedenergyisΔE=ΔNhνThemoretheamplification,themorethegaindecreasesSamephenomenononthetemporalside

Slice n°


RateequationsfromFrantzandNodvik*

• StartingfromMaxwellequations,onegetsapropagationequation«Helmoltz type»andasetofpopulationequations(oneequationperlevel)thatcanbereducedto:

• ThisistheFrantzetNodvikmodelthatisafunctionofonesingle«z»coordinate,andafunctionoftime«t»

• Thissystemhasnoanalyticalsolution• Thissystemcanbeintegratedformallyasafunctionoftimetoleadtoan

energyrelation(orfluence)inwhichthesaturationfluence appearsFsat=hν/σ

NhI

tNNIz

I Δ−=∂Δ∂Δ=

∂∂

νσσ 2et


FrantzandNodvik

Starting from theintegrated formula (Fisthe fluence=energy E/surfaceΔS) :

Itisstraightforward toshowthat:

with

Foragivenamplification slice,onecomputes theresidual gainafteramplification:

AnEXCELfilecanbemadeeasily.

])1)/()[(1( −+= satinsatout FFExpglExpLnFF

]1)/()[(1)/( −=− satinsatout FFExpglExpFFExp

))](1))(/((1[ 1)1( lgExpFFExpLnlg isatiini −− −−−−−=

)(/ : glExpFFFF inoutsatin =<<SglFglFFFFF satsatinoutsatin Δ=Δ=−>> Eou :

L. Frantz & J. Nodvik : « Theory of pulse propagation in a laser amplifier », J. Appl. Phys. 34,8, 2346 (1963)


FrantzetNodvik:2ratesorregimes

LinearbehaviorwhenFin<<Fsat,thenFout/Fin =Exp(g*l)

Example:– l=5cm,– g=0,0461cm-1

– (G=10)– Fsat =4,5J/cm2

SaturatedbehaviorwhenFin>>Fsat,thenFout=Fin +g*l*Fsat

Greencurveapproximatedsolution, redcurve exactsolution.

0.05 0.1 0.15 0.2 0.25 0.3Fin J/cm2

0.5

11.5

2

2.5

3FoutJ/cm2

5 10 15 20Fin J/cm2

510

15

2025

30FoutJ/cm2


Temporalvariations(1)

• Atagivenzcoordinate,forathinslideofmedium,att=0,thegainequalsg0.

• Atanytimet>0,thegainissmallerthang0 becauseIhaveusedsomepartofthepopulationinversion

• Thereforeasquareinputpulsewillbechangedintoadecreasingexponentialshape

• Inversely,foragivensquareoutputpulse,Ihavetogenerateanincreasingexponentialinput

• InthecaseofaLorentzian oraGaussianshape,therisingedgeismoreamplifiedthantheleadingedge.Itseemsthatthepulseisgoingforwardsteeperandsteeper

t

P or I

t


Exactsolution:anamplifierwithoutloss

5 10 15 20 25 30t*3.10̂8

0.1

0.2

0.3

0.4

0.5

0.6

0.7

I*0.5410̂ - 8 W/cm2 et g en cm̂ - 1

• ExactsolutionfromMathematica• Gain=bluecurveandintensity=redcurve


Laseroscillation

R1,R2 aretheintensityreflexion coefficientsofthemirrorsg isthesmallsignalgainα arethelosses,Lc thecavitylength,L thegainmediumlengthTheoscillationconditionisgain=losses onasingleround-trip:R1R2 Exp (2gL-2αLc)=1Linearbehaviour abovethreshold

R1 I

R2 I

I ExpgL

I ExpgL

P out

P pumpP thresholdP pump

P out


Exactsolution:cavitywithlosses

• Cavitylossesconsideredasathresholdforgain:R1R2 Exp(2gL-2αLc)=1

• Whenthisgainratioequalsorisgreaterthan1,intensityincreasesuntilgainratiogoesbackto1,thenintensitydecreases

100 200 300 400 500

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

100 200 300 400 500

0.5

0.75

1.25

1.5

1.75

2

Gain ratio = gain/losses

Intensity

Time

Time

pageBrunoLeGarrec 28

Single pass amplification

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

n° de tranche

Flue

nce

J/cm

2

0,0000,0200,0400,0600,0800,1000,1200,1400,1600,180

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

n° de trancheG

ain

])1)/()[(1( −+= satinsatout FFExpglExpLnFF

I ExpgL

))](1))(/((1[ 1)1( lgExpFFExpLnlg isatiini −− −−−−−=

I ExpgL

Slice number Slice number


Double pass amplification

R I

I ExpgL

I ExpgL

0

0,02

0,04

0,06

0,08

0,1

0,12

1 3 5 7 9 11 13 15 17

Slab number

Res

idua

l gai

n First passSecond pass

0

0,5

1

1,5

2

2,5

1 3 5 7 9 11 13 15 17

n° de plaque

fluen

ce d

e so

rtie

Slice number Slice number

Flue

nce

J/cm

2


Doublepassamplifier

• Angularmultiplexing

• Polarizationmultiplexing

1st passMirror Amplifier Lens Pinhole

2nd pass

Polarizer

1st pass« p » polarization

2nd pass« s » polarization

Mirror Amplifier

Quarter wave plate


4passamplifier

• Quarterwaveplate=4pass• Pockels celltoswitch« on »and« off »

MirrorAmplifier

Polarizer

Quarter wave plate

1st passPolarization « s »

2nd passPolarization « p » 3rd pass

Polarization « p »

4th passPolarization « s »


npassamplifier

• Phasedifference:π/2(λ/4)betweenthe2nd andthe3rd pass• Outputpossibleatthe2(k+1)passifthephasedifference-π/2(-λ/4)

betweenthe2kandthe(2k+1)pass

MirrorAmplifier

Polarizer

Quarter wave plate

Pockels cell

2(k+1) passPolarization « s »

3ème passagePolarisation « p »

2ème passagePolarisation « p »

1st passPolarization « s »

2k passPolarization « p » (2k+1) pass

Polarization « p »


Optimization:1ω outputenergyasafunctionofinjectedenergyinafourpassamplifierwith14,16or18slabs

Injected energy in Joules

Out

put e

nerg

y in

Jou

les

18 slabs16 slabs14 slabs


End of part 1LasersLaser electronics : Joseph T. Verdeyen, Prentice-Hall International, Inc. (1989)Lasers : Anthony E. Siegman, University Science Books (1986) Principles Of Optics Electromagnetic, Theory of Propagation Interference and Diffraction of Light : M.

Born, E. Wolf, 6th ed. Pergamon Press (1980) Solid-state laser engineering : Walter Koechner, Springer-Verlag (1976)Principles of Lasers : Orazio Svelto, 2nd ed. Springer (1982)

AmplificationAlbert Einstein, "Zur Quantentheorie der Strahlung”, Physika Zeitschrift, 18, 121-128 (1917)Rudolpf Ladenburg, Review of Modern Physics 5, 243 (1933)Theodore Maiman, “Stimulated optical radiation in ruby”, Nature, 493-494 (August 6,1960)L. Frantz & J. Nodvik, « Theory of pulse propagation in a laser amplifier », J. Appl. Phys.

34,8, 2346 (1963)J. E. Murray & W. H. Lowdermilk, “The multipass amplifier: theory and numerical analysis”, J. Appl.

Phys. 51, 5 (1980); “Nd:YAG regenerative amplifier”, J. Appl. Phys. 51, 7 (1980)

The Laser OdysseyTheodore. H. Maiman, “The laser Odyssey”, Laser Press (2000)Jeff Hecht, “BEAM, the race to make the laser”, Oxford University Press (2005)Andrew H. Rawicz, “Theodore Maiman and the invention of the laser”, SPIE 7138, 713802 (2008)

Documents

Intense lasers: high peak power Part 1: amplificationoldilil.ino.it › Capri2017 › Slides › Capri_2017_blg_part1.pdf · 2017-07-10 · Bruno Le Garrec LPA schoolCapri 2017 page