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pageBrunoLeGarrec 1
Intense lasers: high peak powerPart 1: amplification
BrunoLeGarrecDirecteurdesTechnologiesLasersduLULILULI/EcolePolytechnique,routedeSaclay
91128Palaiseaucedex,France
bruno.le-garrec@polytechnique.edu
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 ?
pageBrunoLeGarrec 4LPA school Capri 2017
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
−=
pageBrunoLeGarrec 5LPA school Capri 2017
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
pageBrunoLeGarrec 6LPA school Capri 2017
Emission,Fluorescence,Phosphorescence
White light
pageBrunoLeGarrec 7LPA school Capri 2017
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
pageBrunoLeGarrec 8LPA school Capri 2017
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
pageBrunoLeGarrec 9LPA school Capri 2017
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
pageBrunoLeGarrec 10LPA school Capri 2017
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
pageBrunoLeGarrec 11LPA school Capri 2017
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
pageBrunoLeGarrec 12LPA school Capri 2017
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
pageBrunoLeGarrec 13LPA school Capri 2017
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
pageBrunoLeGarrec 14LPA school Capri 2017
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
pageBrunoLeGarrec 15LPA school Capri 2017
Thelaserwasbornin1960,May16th .
• Maiman hasusedaflashlamp(GEFT-506model) insideasimplealuminumtube.
• Therodhasa0.95cmdiameter(3/8inch)anda1.9cmlength (3/4inch)withendfacescoatedwithsilver.
• Ononeface,thecentralpartof thesilvercoatingisremoved inordertolettheradiationescapefromtherod.
pageBrunoLeGarrec 16LPA school Capri 2017
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
pageBrunoLeGarrec 17LPA school Capri 2017
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
pageBrunoLeGarrec 18LPA school Capri 2017
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
pageBrunoLeGarrec 19LPA school Capri 2017
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 ?
pageBrunoLeGarrec 20LPA school Capri 2017
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°
pageBrunoLeGarrec 21LPA school Capri 2017
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
pageBrunoLeGarrec 22LPA school Capri 2017
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)
pageBrunoLeGarrec 23LPA school Capri 2017
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
pageBrunoLeGarrec 24LPA school Capri 2017
Temporalvariations(1)
• Atagivenzcoordinate,forathinslideofmedium,att=0,thegainequalsg0.
• Atanytimet>0,thegainissmallerthang0 becauseIhaveusedsomepartofthepopulationinversion
• Thereforeasquareinputpulsewillbechangedintoadecreasingexponentialshape
• Inversely,foragivensquareoutputpulse,Ihavetogenerateanincreasingexponentialinput
• InthecaseofaLorentzian oraGaussianshape,therisingedgeismoreamplifiedthantheleadingedge.Itseemsthatthepulseisgoingforwardsteeperandsteeper
t
P or I
t
pageBrunoLeGarrec 25LPA school Capri 2017
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
pageBrunoLeGarrec 26LPA school Capri 2017
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
pageBrunoLeGarrec 27LPA school Capri 2017
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
pageBrunoLeGarrec 29
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
pageBrunoLeGarrec 30LPA school Capri 2017
Doublepassamplifier
• Angularmultiplexing
• Polarizationmultiplexing
1st passMirror Amplifier Lens Pinhole
2nd pass
Polarizer
1st pass« p » polarization
2nd pass« s » polarization
Mirror Amplifier
Quarter wave plate
pageBrunoLeGarrec 31LPA school Capri 2017
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 »
pageBrunoLeGarrec 32LPA school Capri 2017
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 »
pageBrunoLeGarrec 33LPA school Capri 2017
Optimization:1ω outputenergyasafunctionofinjectedenergyinafourpassamplifierwith14,16or18slabs
Injected energy in Joules
Out
put e
nerg
y in
Jou
les
18 slabs16 slabs14 slabs
pageBrunoLeGarrec 34
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)
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