Terahertz and higher-order Brunel harmonics: from tunnel to multiphoton ionization regime in tailored fields

0.42 MB·

by  I. Babushkin

Terahertzandhigher-orderBrunelharmonics: fromtunneltomultiphotonionizationregimein
tailoredfields
I. Babushkin,1,2 C. Bre´e,3 C. M. Dietrich,1 A. Demircan,1,4 U. Morgner,1,4 and A. Husakou2
1InstituteforQuantumOptics,LeibnizUniversita¨tHannover,Welfengarten1,30167Hannover,Germany
2Max Born Institute, Max Born Str. 2a, 12489 Berlin, Germany
3Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany
4Hannover Centre for Optical Technologies, Nienburgerstr. 17, 30167 Hannover, Germany
Brunel radiation appears as a result of a two-step process of photo-ionization and subsequent acceleration
ofelectron, withouttheneedofelectronrecollision. WeshowthatforgenerationofBrunelharmonicsatall
frequenciesthesubcycleionizationdynamicsisofcriticalimportance. Namely, suchharmonicsdisappearat
7
1 lowpumpintensitieswhentheionizationdynamicsdependsonlyontheslowenvelope(socalledmultiphoton
0 ionizationregime)andnotontheinstantaneousfield.Nevertheless,ifthepumppulsecontainsincommensurate
2 frequencies,Brunelmechanismdoesgeneratenewfrequencieseveninthemultiphotonionizationregime.

n
a
I. INTRODUCTION ofplasma-inducedrectificationandleadstogenerationoffre-
J
quenciesinterahertz(THz)range[16–22],orthe“zero-order”
0
Brunelharmonic. Higherorderharmonicsarealsogenerated,
3 The study of the dynamics of ionization process attracted
forbothsingle-andmulti-colorfields,althoughwithdecreas-
significantattentionlastyears,becauselasersourcesallowing
] intenseultra-shortpulseswiththecontroloverthefastoptical ingefficiency. Brunelharmonicsaretobedistinguishedfrom
h theonesappearingduetodynamicsoftheboundedelectrons
p waveform shape became available. Such studies gained im-
(socalledfour-wave-mixingprocess). Ingeneral,experimen-
- portance also because generation of shortest possible pulses
m taldifferentiationbetweenthetwoisarathercomplicatedtask
isbasedheavilyonhighharmonicgeneration,whichexploits
[23–27].
s the ionization dynamics. Although the ionization process in
a In this article we show that, looking from the perspective
most simple single-electron systems such as noble gases is
l oftheclassicalelectronmotioninthecontinuum,theBrunel
p studied for several decades, its details are still far from be-
. ingfullyclarified, thusbeinginafocusofrecentstudies[1– harmonics appear to be extremely sensitive to the subcycle
s
step-wisecharacteroftheionizationprocess. Iftheionization
c 11]. One of the most known peculiarities of the ionization
i is“fastenough”(tunnelionizationregime),itgivesrisetothe
s processisitsstronglynonlineardependenceontheinputfield
y strength. Forhighenoughintensitiesionizationhaspredomi- newharmonicswhereasiftheionization“slow”(multiphoton
h nantlyacharacterofaninstantaneoustunneling,whereasfor ionizationregime),Brunelharmonicsdonotappear.Weshow
p thetransitionbetweenthesetworegimesforthepumppulses
lowerintensitiesitisbetterdescribedasratherslowmultipho-
[ containingonlyonefrequency. Wealsoprovethatthisisthe
tonprocess. Theboundaryisnotexactandwasseveraltime
case for the THz harmonics in two-color case. Furthermore,
1 revisited[12,13],but,roughlyspeaking,itdependsonthera-
v tio of the ionization time τ and the optical oscillation period we show that the situation may change if the two-color field
4 T known as Keldysh parameter γ = 2πτ/T. The ionization contains incommensurate harmonics, that is harmonics with
7 thenon-integerfrequencyratio. Suchharmonicsleadtobeat-
process above this threshold γ (cid:28) 1 is nearly instantaneous
8 ings in the envelope which can be slow enough to switch on
and depends on the current field strength. This leads to an
8
theBrunelmechanismagain,eveniftheionizationprocessis
0 essentially step-wise dynamics of ionization rate, with steps
slow.
. localizednearextremaoftheelectricfield. Ontheotherhand,
1 below this threshold γ (cid:29) 1, ionization depends more on the The article is organized as follows: in Sec. II we make a
0
short introduction into the mechanism of Brunel harmonics
7 cycle-averagedfieldintensityratherthanontheinstantaneous
generationanddescribethedetailsofionizationprocessrele-
1 field. It should be noted also that for multicolor fields the
vant to our consideration. In Sec. III we analyze the Brunel
: Keldyshparameterscannotbedefinedanymoreinauniversal
v harmonics in tunnel and multiphoton ionization regime in
way.
i
X single-color pump whereas Sec. IV and Sec. V are devoted
Creationoffreeelectronsduetoionizationleadstomodifi-
totwo-colorpulses,withcommensurateandincommensurate
r cationofthematerial’srefractiveindex,thusmakingthelater
a frequencies,respectively. Theanalyticaldescriptionisdevel-
field-dependent and therefore introducing nonlinearity. The
opedanddiscussedinSec.6followedbyconcludingremarks
presence of nonlinearity leads to generation of harmonics of
Sec.VII.
thepumpfield,whicharesometimescalledBrunelharmonics
[14, 15]. We remark that this mechanism does not requires
returnoftheelectronbacktothecoreandthereforeinvolves
theharmonicsofrelativelyloworder(incontrasttohighhar- II. ELECTRONTRAJECTORIES,MACROSCOPIC
monics generation). In particular, if the pump field is also PLASMACURRENT,ANDIONIZATION
asymmetric in time, this may create also a net macroscopic
asymmetric current which persists after the pulse and thus Brunelradiationcanbedescribedclassicallyasaradiation
givesrisetoalow-frequencyharmonic. Thisprocessisatype created by a current of all free electrons. In particular, as-
2
suming a small volume of gas (smaller than the considered thenetcurrentinthecaseofatwo-colorfieldinFig.1(b)pos-
wavelengths) ionized in a pump field E(t), the Brunel radia- sessesaspatialasymmetry,whichisabsentinthesingle-color
tion created by the free electron current J(t) in this volume case. Therefore, the net current J(t) is also very different in
willbe: both cases. Already this simple example shows that the net
current J(t) and thus the Brunel radiation E can be signifi-
dJ(t) Br
E (t)=g , (1) cantlymodifiedbythefielddynamicsonthesubcyclescale.
Br dt
Herewewillshowthatthedynamicsofionizationprocess
itself can be also of critical importance for the macroscopic
where g is a factor depending on the observation point. If
current buildup. As far as we know, there is no universally
delays in light propagation can be neglected but still the dis-
acceptablemodelwhichdescribesionizationdynamicsforall
tance to the observer r is large enough, is given as [22]:
g=δV/(4πε c2r),whereδV isthevolumeoftheplasmaspot. intensities on all relevant timescales which is also valid for
0
multicolorfield. Asitwasmentionedbefore,forhighintensi-
Inthecurrentarticleweassumeargonasaworkinggas.
TheexpressionforthecurrentisgivenbyJ(t)=−eρ(t)v(t), ties(γ (cid:28) 1)theionizationtakesplaceonthesubcyclescale.
Inthiscase,theionizationratedependingontheinstantaneous
wheree,ρ(t),v(t)aretheelectroncharge,densityofelectrons
fieldstrength,describingtunnelingprocessinquasi-staticap-
andtheiraveragevelocity,respectively.Thechangeofcurrent
δJaftertimemomentδtcanbeexpressedasδJ =−e(ρ(t)δv+ proximation,isacceptableforarbitrarypulseshapesandthus
arbitrarynumberoffrequenciesinthepulse. Weuseherethe
v(t)δρ). Whencalculatingthechangeofaveragevelocityδv,
statictunnelingapproximationformulaintheform[28]:
we take into account that the newly ionized electrons have
zero average velocity, thus reducing v: δv = −eE(t)δt/me − α (cid:40) β (cid:41)
aδρrve(mt)a/rρk(at)b,lywshiemreplmeeeixsptrheesseiloenctronmass. Asaresult,weget WST(t)= E˜(t)exp −E˜(t) , (6)
EBr(t)=gdJd(tt) = gme2E(t)ρ(t). (2) 5w.h1e4re×W10S1T1(tV)mis−t1hiesitohneizfiaetlidonunraittei,nE˜a(tto)m=ic|sEy(stt)e|/mEao,fEuanit≈s
e
(a.u.),coefficientsα = 4ω r5/2 andβ = (2/3)r3/2 aredefined
a H H
The electron density ρ(t) is related to the ionization rate throughtheratioofionizationpotentialsofargonandhydro-
W(t)as genatomsr = U /U ,U = 13.6eV,U = 15.6eV,and
H Ar H H Ar
ω ≈4.13×1016s−1.
a
∂ρ(t)
=(ρ −ρ(t))W(t), (3) Forlowerintensitiesγ (cid:29) 1,whenionizationprobabilityis
∂t 0 muchlessthenanopticalcycle,ionizationratedependsmore
onthecycle-averagedintensity-thatis,onthefieldenvelope
where ρ is the density of non-ionized atoms before interac-
0 rather than instantaneous field strength. The tunnel ioniza-
tionwiththepulse. Ionizationratedynamicsisimportantand
tion formula Eq. (6) does not catch this dynamics anymore.
isdescribedbelow.
To calculate corresponding ionization rate for both single-
InFig.1onecanseeanexemplaryprocessoffreeelectron
and two-color fields we used the generalization of the well
creation and their dynamics for single- and two-color pump
known Keldysh-Faisal-Reistheory [29–31] for thefield with
field. WeconsidertheelectricfieldE(t)intheform:
arbitrarynumberofharmonics,whichwasdevelopedbyGuo,
E(t)= F (t)cosω t+F (t)cos(2ω t+φ (t)), (4) Aberg and Crasemann [32] and which we abbreviate here as
1 0 2 0 12
GACmodel. InApp.Athecorrespondingionizationratefor
where φ (t) is the phase which we allow to vary slowly in the case where the field contains two colors ω0, 2ω0. This
12
model provides us the dependence of the ionization rate on
time,and
bothintensitiesI (t),I (t)andtherelativephaseφ inexplicit
(cid:112) (cid:112) 1 2 12
Fi(t)= 2Ii(t)/(c(cid:15)0), Ii(t)= Ii0e−t2/Ti2, F0i = 2I0i/(c(cid:15)0). (thoughcumbersome)formalism:
(5)
areslowlyvaryingfieldamplitudesandintensitieswhichde- WGAC(t)=WGAC(I1(t),I2(t),φ12), (7)
scribethecorrespondingpulseenvelope,T isthepulsedura-
i
tion,andφ12istherelativephase. whereWGAC isgivenbyEq.(A10)anddescribedindetailsin
In Fig. 1, several exemplary trajectories have been gener- AppendixA.Importantly,nowionizationdoesnotdependon
ated with the electron ionized randomly with the probability theinstantaneousfieldbutratherontheslowenvelopesofthe
corresponding to the tunnel ionization rate as discussed be- harmonicsaswellasontheirphaseφ12.
low,forthecasesof I = 0inFig.1(a)and I = 20TW/cm2 Finally, for the case of a single-color field there is a rela-
2 2
inFig.1(b). InbothcasesweassumedI =100TW/cm2,the tivelysimpleformuladerivedbyIvanovandYudin[1],which
1
frequency ω corresponding to 2 µm wavelength, and pulse correctlydescribesbothcycle-averagedionizationrateandits
0
duration of 50 fs. For the parameters chosen here, the elec- slow and subcycle dynamics in both tunnel and multiphoton
tronsareionizedwithhighestprobabilityclosetothemaxima regimes. WewillrefertothismodelasIYone,andthecorre-
ofelectricfieldandbysubsequentaccelerationmove,inaver- spondingionizationrateisgivenby:
age,awayfromtheorigin. Neitherrecollisionnocorepoten-
(cid:16) (cid:17)
tialweretakenintoaccountinoursimulations. Remarkably, W (t)=αexp −β(cid:48)F(t)2−β(cid:48)(cid:48)E(t)2 , (8)
IY
3
1.0
(a) (b)
u.
a. 0.5
x(t),
0, 0.0
1
×
E(t) 0.5
1.0
40 20 0 20 40 40 20 0 20 40
t, fs t, fs
FIG.1.Trajectoriesx(t)(redlines)ofsampleelectronsionizedintheelectricfieldE(t)(blueline)ofthestrongone-color(a)andtwo-color(b)
opticalpulses(parametersaregiveninthetext).In(b),becauseofpresenceoftwocolors,thefieldshapeisslightlyasymmetricwhichleadsto
anasymmetrybetweenup-anddown-directionsintheelectrontrajectoriesandthustosignificantchangeoftheresultingmacroscopiccurrent
J(t)andthustheBrunelradiationE (t).Thisillustratestheimportanceofsubcycledynamicsoftheionizationandmovement.
Br
wherethefieldisassumedtobeintheformEq.(4)withF = in Fig. 2(c). Namely, one can see that the Brunel harmonics
2
0,thusF(t)≡ F (t)/E istheslowfieldenvelopeinEq.(5)in atthefrequenciesω = nω , n = 3,5,7...indeedariseifthe
1 a 0
theunitsofE ,andthecoefficientsaredefinedas: ionizationisinthetunnelregime(redlines)anddisappearsif
a
multiphotonionizationtakesplace(bluelines). Asitisshown
α= πCk2 (cid:32)2κ3(cid:33)2Q/κ, β(cid:48) = 2σ0+σ(cid:48)0(cid:48),β(cid:48)(cid:48) = σ(cid:48)0(cid:48) , (9) inthenextsection,thisisbecausethefieldofBrunelharmon-
τ F 2ω3 2ω3 ics appears as a convolution of the fast optical field with the
0 0 electron density, and this convolution has no new harmonics
1 1 γ
σ = (G2− )lnC−G , (10) iftheρ(t)changesslowlyintime,asitisthecaseforthemul-
0
2 2 2 tiphotonregime. Notethattheharmonicatω = ω isrelated
(cid:113) 0
γ
σ(cid:48)(cid:48) =lnC−2 , G = 1+γ2, C =2G2−1+2Gγ. (11) relatedtothefreeelectronmovement(incontrasttoelectron
0 G birth)andisnotinterestingforusinthepresentarticle.
√
Hereκ = 2r (r isdefinedbelowEq.(6)),τ = κ/F isthe
H H UsingIYformulaitispossibletoobservethetransitionbe-
tunneling time, γ = ω τ is the Keldysh parameter, Q is the
0 tween the two types of dynamics. Such transition is shown
effectivechargeoftheioniccore(Q = 1fornoblegases),C
k in Fig. 3, where the energies of the 3th and 5th Brunel har-
isrelatedtotheradialasysmptoteofthequantumorbitaland
monics are shown (solid black and blue lines, respectively)
in our case also C ≈ 1. This formula is not applicable for
k in dependence on the peak amplitude F of the single-color
01
multicolorfields,butforsingle-coloronesitprovidesapossi-
field with the parameters as in Fig. 1(a). As the field am-
bilitytoinspectthetransitionbetweentunnelandmultiphoton
plitudedecreases,theintensitiesoftheBrunelharmonicsde-
regimeindetails,whichisdoneinthenextsection. creaseratherdramatically. Therearetwoeffectsinthis,oneis
connectedtotheoveralldensityoffreeelectrons,whichalso
decreases, the other is connected to the ionization dynamics
III. BRUNELHARMONICSINSINGLE-COLORFIELDS onitsown. Todemonstrateclearlythelatereffect,weplotted
the energies rescaled per single ionized atom (technically, to
We start our consideration of Brunel radiation from dis- obtainsuchquantitieswerescaletheionizationcurrent J and
cussingthesingle-colorfield(I02 =0)inEq.(5),Eq.(4). We thus EBr toρ(t = ∞)/ρ0). Theresultingquantitiesareshown
useEq.(2)togetherwithequationforthefreeelectrondensity inFig.3bydashedlines.
Eq.(3)andionizationcalculatedusingIYformulaW = W
IY
Eq. (8). In Fig. 2 the dynamics of the electric field and ion- IftheBrunelharmonicenergiesdependedonlyonthenum-
izationaswellastheBrunelradiationisshownfortheparam- berofionizedelectronsandnotonthecorresponding(subcy-
eters as in Fig. 1(a) but with two different intensities corre- cle)ionizationdynamics,suchrescaledenergieswouldnotde-
spondingtothetunnel[Fig.2(a)]andmultiphoton[Fig.2(b)] pendonF. However,thisisnotthecase.Asonecansee,they
regime. Of course, the bare ionization rate W in the case of also decay quickly with decreasing F (although with much
theweakfieldismanyordersofmagnitudelowerthaninthe lower rate), demonstrating the importance of the fast tunnel
tunnelone.Weareinterestedinthedynamics,thustheioniza- ionizationdynamics.Thisdecayisalsomuchfasterforthe5th
tionrateaswellasBrunelradiationE (t)isrenormalizedto harmonicsthanforthe3thone,indicatingstrongnonlinearity
Br
theirmaximumvaluesineverycase.Theionizationrateinthe of the process. As it will be shown in Sec. 6,the appearance
tunnel regime demonstrates strong subcycle-scale dynamics of 3th and 5th harmonics would be impossible if the ioniza-
whereasinthemultiphotonregimeitchangesonmuchlonger tionrateweretrulyslow(containsnoharmonicsatthepump
timescale of the slow field envelope. The Brunel radiation frequencyandhigher). Thisshowsalsothatthefasttunneling
E (t)looksnotverydifferentinthetimedomain,butsignifi- componentinionizationdynamicsisstillpresentalsoforvery
Br
cantdifferencecanbeobservedinthefrequencyrangeshown lowfields.
4
1.0 1.0
max (a) max (b) 100 (c)
Br, Br, 10-1
E E 10-2
/ /
WE(t), maxBr 00..05 WE(t), maxBr 00..05 arb. units 11110000----6543
W(t)/ W(t)/ (ω), Br 1100--87
, max 0.5 , max 0.5 E 10-9
E E 10-10
/ /
E(t) E(t) 10-11
1.0 1.0 10-12
40 200 20 40 60 80100 40 200 20 40 60 80100 0 1 2 3 4 5
t, fs t, fs ω/ω0
FIG.2.Brunelradiationinthetunnelandmultiphotonregimesinthecaseofsinglecolorpulse.In(a)and(b),thefield(blueline),ionization
(blackline)and E (t)(redline)areshownforF = 0.053a.u. (a)andF = 2.6×10−4 (200timesless)(b). Theotherparametersasin
Br 01 01
Fig.1(a).In(c),thespectraofBrunelradiationE (ω)for(a)(redline)and(b)(bluedashedline)areshown.
Br
photonregimeaswedidthisinFig.3. Nevertheless,wecan
100
consider these two cases separately, using tunnel ionization
10-2 rate W (Eq. (6)) for higher intensities and GAC ionization
ST
10-4 rateWGAC forloweronesasdiscussedinSec.II.
uts 10-6 Thecorrespondingfigurefortwo-colorcaseispresentedin
n
gy, arb. u1100-1-80 FFcaiiggse..14i(n.b)FT.ighIn.e2pp,aarbrtauicmtunelaoterw,rswIae02rhea(cid:44)tvhe0et,asnkaaemmneeφlaysI=i0n2π/t/Ih02e1ws=ihnig0clh.e2-dcaeosllivoin-r
er 12
en10-12 ersthemaximalenergyfortheTHzharmonic[33]. Onecan
ctral 10-14 seethatthesamebehaviorisobserved:inthetunnelionization
pe regimetheBrunelharmonicsatω = nω arevisiblewhereas
s10-16 0
in the multiphoton regime they disappear. This behavior is
10-18 completelyanalogoustothesingle-colorcaseandtakesplace
10-20 becauseinthemultiphotonregimeionizationis“tooslow”to
10-22 createharmonicsasdiscussedfurtherinSec.6.
10-3 10-2 10-1
F, a.u.
FIG. 3. The energy of Brunel harmonics in dependence on the
V. BRUNELHARMONICSINTWO-COLORFIELDS:
peak amplitude F of a single-color pulse with the parameters as
01 INCOMMENSURATECASE
inFig.1(a). Black(blue)linesshowtheenergyofthe3th(5th)har-
monics. Theabsoluteenergiesareshowninsolidlineswhereasthe
dashedlinesrepresenttheenergiespersingleionizedatom. Letusnowtrytwo-colorpumpfieldswithincommensurate
frequencies. By incommensurate frequencies we assume the
oneswiththeratiobetweenthembeingnot-integer. Forthis,
IV. BRUNELHARMONICSINTWO-COLORFIELDS: weslightlydetunethesecondharmonic2ω0to2ω0+∆ωwith
COMMENSURATECASE ∆ω(cid:28)ω0. Thatis,theelectricfieldweconsideris:
(cid:112) (cid:112)
Nowletusreturntothecasewhenthepumppulseconsists E(t)= 2I1(t)/(c(cid:15)0)cosω0t+ 2I2(t)/(c(cid:15)0)cos(2ω0t+∆ωt+φ12).
of the fundamental frequency ω and its second harmonics (12)
0
2ω , that is, Eq. (4), Eq. (5) with I (cid:44) 0, φ = const. In The idea behind consideration of such field shape is the fol-
0 02 12
this case, both odd and even harmonics can appear. That is, lowing:slightdetuning∆ω(cid:28)ω0leadsinsomeslowbeatings
Brunelharmonicsarelocatednearfrequenciesω = nω with inthefieldamplitude,andinthiscaseeven“slow”ionization
0
n = 0,2,3,4.... In particular, a zero one, which is located inmultiphotonregimemay“see”thesebeatingsandthusgen-
typicallyinrealisticsetupsatTHzfrequency,isofspecialim- eratesomeharmonics.
portance because THz radiation is difficult to generate. The For the high intensity in the tunnel ionization regime we
appearanceofevenharmonicsisrelatedtotheasymmetryof can still use the tunnel formula Eq. (6). On the other hand,
thefieldasshowninFig.1(b).Forthetwocolorcase,asitwas the GAC formalism is valid for commensurate frequencies.
mentioned before, there is no general formula for ionization However, wenotethatwiththecondition∆ω (cid:28) ω , wecan
0
processallowingtotrackthetransitionfromtunneltomulti- interpretthefrequencyshiftasadditionalrelativephasewhich
5
1.0 1.0
max (a) max (b) 100 (c)
Br, Br, 10-1
E E 10-2
/ /
WE(t), maxBr 00..05 WE(t), maxBr 00..05 arb. units 11110000----6543
W(t)/ W(t)/ (ω), Br 1100--87
, max 0.5 , max 0.5 E 10-9
E E 10-10
/ /
E(t) E(t) 10-11
1.0 1.0 10-12
40 200 20 40 60 80100 40 200 20 40 60 80100 0 1 2 3 4 5
t, fs t, fs ω/ω0
FIG.4. Brunelradiationinthetunnelandmultiphotonregimesinthecaseoftwo-colorpulse. In(a)and(b),thefield(blueline),ionization
(blackline)andE (t)(redline)areshownforF =0.053a.u. (a)andF =2.6×10−4a.u. (200timesless)(b). Theotherparametersasin
Br 01 01
Fig.1(b).In(c),thespectraofBrunelradiationE (ω)for(a)(redline)and(b)(bluedashedline)areshown.
Br
1.0 1.5 0
(a) (b) s (c)
W(t), arb. units00000.....56789 W(t)E(t), , arb. units 0001....5050 E(ω)g(), arb. unitBr||11505
o
l
0.4 1.0 20
0 π 2π 150 100 50 0 50 100 150 0 1 2 3 4 5
φ t, fs ω/ω0
FIG. 5. Brunel harmonics for incommensurate frequencies. (a) Two color multiphoton ionization rate W (I ,I ,φ) (normalized to the
GAC 1 2
maximumoverallφ)dependingonphasesbetweentwoharmonicsφforI = I = 20TW/cm2 (blueline)andI = I = 10TW/cm2 (green
1 2 1 2
line). (b) The field (black line), tunnel ionization rate (red line) and multiphoton ionization rate (blue line) for a 200 fs pulse containing
frequenciesω and2.2ω withI =I =20TW/cm2.(c)Brunelharmonics(persingleionizedelectron)duetotunnelionization(redline)and
0 0 1 2
multiphotonionization(blueline).
isslowlyvaryingintime: The spectrum of the radiation generated by the Brunel
mechanism for ∆ω = 0.2ω is presented in Fig. 5(c). One
φ12(t)→φ12+∆ωt (13) can see the generation of ne0w frequencies in the vicinity of
thepumpfrequenciesintheformofacomb-likepatternwith
andthusconsiderthefieldintheformEq.(4)withφ (t)given
12 spacingofroughly0.4∆ωbothinthecaseoftunnelandmul-
by Eq. (13). We then use the GAC formalism for two-color
tiphoton ionization regimes. This is in contrast to the pump
pumpwithfrequenciesω ,2ω andthecorrespondingioniza-
0 0 pulse which have peaks only at ω , 2ω +3∆ω. In the mul-
tionrate 0 0
tiphotonregime,onlyfewsuchfrequenciesisgenerated[blue
W(t)=WGAC(I1(t),I2(t),φ12+∆ωt), (14) l2iωne+in∆Fωig±.35∆(cω)]a.ppEeffareicntivtheilsy,caosnel.yInfrceoqnuteranscti,eisnωth0e±tun∆nωel,
0
withW giveninEq.(A10). ionizationregime,thiscombcontainsmuchmorefrequencies.
GAC
As one can see in Fig. 5(a), where the dependence of the
multiphoton ionization rate on the phase between pulses for
exemplaryintensitiesI ,I isshown,thechangeoftheeffec-
1 2
tive phase effective phase ∆ωt+φ lead to noticeable mod- VI. ANALYTICALCONSIDERATIONSOFBRUNEL
12
HARMONICS
ification of multiphoton ionization rate. Indeed, as one can
see in the Fig. 5(b), not only tunnel ionization rate but also
multiphotononevariesacrossthepulse.Forthiscaseweused To get a further insight into the processes governing the
longerpulses(T = 200fs)tomakeslowbeatingspossible, generation of new frequencies by Brunel mechanism, we
1,2
withtheintensitiesI = I =100TW/cm2 forthetunnelcase present the quantities E (t), E(t), and ρ(t) as Fourier series
1 2 Br
andI = I =20TW/cm2forthemultiphotonone. nearthemaximumofthepumppulse. Thismethodwasfirst
1 2
6
proposedin[14]andfurtherdevelopedin[22,34,35]. Here ∆ωanditsmultiples:
we neglect the slow amplitude changes due to Gaussian en-
(cid:88) (cid:88)
velopes,thatis,weconsiderpulsesformallyinfiniteintimeto EBr(t)= EBr,n,meit(nω0+m∆ω), E(t)= En,meit(nω0+m∆ω),
keepconsiderationsimple. Underthiscondition,wehavefor n,m n,m
thepumppulseswithcommensuratefrequencies: (21)
(cid:88) (cid:88)
EBr(t)=(cid:88)EBr,neinω0t, E(t)=(cid:88)Eneinω0t, ρ(t)= n,m ρn,meit(nω0+m∆ω),W(t)= n,m Wn,meit(nω0+m∆ω). (22)
n n
(cid:88) (cid:88)
ρ(t)= ρneinω0t,W(t)= Wneinω0t. (15) Substituting into Eq. (2) and multiplying by e−it(n(cid:48)(cid:48)ω0+m(cid:48)(cid:48)∆ω),
wewillhave:
n n
(cid:88)
Ifρ(cid:28)ρ0wecanalsoassume∂tρ≈ρ0W,thatis: EBr,n,meit(ω0(n−n(cid:48)(cid:48))+∆ω(m−m(cid:48)(cid:48))) =
n,m
ρ (cid:88)
ρn = iω0Wn. (16) = En,mρn(cid:48),m(cid:48)eit(ω0(n+n(cid:48)−n(cid:48)(cid:48))+∆ω(m+m(cid:48)−m(cid:48)(cid:48))), (23)
n,m,n(cid:48),m(cid:48)
BysubstitutingtheseriesEq.(VI)intoEq. (2),multiplicat-
ing the result by e−in(cid:48)ω0t and integrating over time (assuming We can now integrate over time as we did by deriva-
dthealtta(cid:82))−,+∞w∞eegitωe0t(:n−n(cid:48))dt = 2πδn,n(cid:48), where δn,n(cid:48) is the Kronecker t(cid:82)Mi−o+∞n∞theoiatf(tω0E(Nnq−ω.n(cid:48)0(cid:48))(+1+∆7ω)(.Mm−∆m(cid:48)Iω(cid:48)n))dtth==e2cπ0aδsnew,n(cid:48)eδifmc,mtah(cid:48)n(cid:48)e,rtehuuseissotbnhtoeainsriuenlcgah:tioNn:,
ge2 (cid:88)
EBr,n = me j Ejρn−j. (17) EBr,n,m = gmee2 (cid:88)j,i Ej,iρn−j,m−i. (24)
First, consider the situation arising when we are in multi- This equation is relatively good applicable in our situation
photon ionization regime. In this case, ρ contains only the since we need large N, M to get Nω + M∆ω = 0, which
0
componentρ0. Thatis, wouldrequirelargenumberofharmonicsinthespectrum.
Inthemultiphotonregime,becausemultiphotonionization
ρ(t)=ρ0. (18) isfastenoughtoresolvebeatingswithfrequency∆ωbutstill
slowonthescaleoffundamentalfrequencyω ,wehave:
0
Inthiscase, theharmonicspresentin E coincidewiththe
Br,n
(cid:88)
onesinthepumpEn: ρ(t)= ρ eim∆ωt, (25)
0,m
m
ge2 (cid:88)
EBr,n = m ρ0En, EBr(t)= EBr,neinω0t. (19) which is to be compared to the multiphoton commensurate
e n case Eq. (18). Because of this, instead of Eq. (19) we will
have:
One can see that the slow dynamics of the electron density
does not lead to generation of new harmonics, but rather to ge2 (cid:88)
modification of the existing ones. This is valid for arbitrary EBr,n,m(ω)= m ρ0,jEn,m−j. (26)
pumpspectra(withcommensuratefrequencies). e j
In contrast, in the case of tunneling ionization, ρ(t) in-
As follows from the symmetry considerations and as one
creases steeply each half optical period, and contains strong
components with n (cid:44) 0. In this case, the Brunel harmonics can see in Fig. 5, only even harmonics of ∆ω occur in W(t)
and in ρ. This leads, according to Eq. (24), to generation of
appear. In particular, if E contains only a single frequency
(thatis,onlyE ,E (cid:44)0),thenfromEq.(17)wehave: newfrequenciesω0+(2j+1)∆ωand2ω0+(2j+2)∆ω, j =
1 −1 0,±1,±2,...,byBrunelmechanism,asshowninFig.5.Inthe
  tunnelnon-commensuratecase, thegeneralformulaEq.(24)
EBr,n = gme2 E1(cid:88)ρn−1+E−1(cid:88)ρn+1. (20) applies.
e j j
In this case, only even components ρ , n = 2N are present, VII. DISCUSSIONANDCONCLUSION
n
which results, according to Eq. (17), in appearance of odd
harmonicsin E (t). Incontrast,forthetwo-colorpump(we In conclusion, we have considered generation of the so-
Br
donotwritetheexplicitformulaforthiscasehere),allcompo- called Brunel harmonics in one- and two-color fields, com-
nentsarepresentinρ ,andaccordinglyalsoinE ,including paringthesituationswhentheionizationisfastsubcycletun-
n Br,n
thosewithn=0. nelingprocessdependingonlyontheinstantaneousfield(tun-
In the case of non-commensurate frequencies, the series nelionizationregimevalidforhighfields),andthecasewhen
(VI)mustbeextendedtoincludecomponentswithfrequency it has “memory” and depends on the cycle-average intensity
7
(multiphotonregimevalidforlowerfields).Weusedapproach AppendixA:Two-colormultiphotonionization
ofdirectmodelingoftheclassicalnetcurrentproducedbythe
ionized electrons and have shown that, at least in this classi-
cal approximation, in the multiphoton ionization the Brunel
harmonicsvanishinthemultiphotonionizationregime. This Fordescriptionofmultiphotonionizationweadoptherethe
showsthatBrunelharmonicsareverysensitivetotheinternal Guo, AbergandCrasemann(GAC)model[32], whichisthe
dynamicsoftheionizationprocess.Forthesingle-colorfields, generalizationofthewellknownKeldysh-Faisal-Reistheory
wewereabletoshowthetransitionbetweenthesetwolimits [29–31] for the field of sufficiently different frequencies and
usingIvanov-Yudinapproachwhichgivescorrectlybothion- whichwasusedinanopticalsetup[36,37]. TheGACmodel
izationrateandionizationdynamics. Weconfirmedthatthis is based on the scattering-matrix formalism, providing non-
behaviorisalsovalidfortwo-colorpumppulsescontaininga perturbativetreatmentofelectroninelectromagneticfield. In
fundamentalfrequencyanditssecondharmonics. Suchpump this section the units are used in which (cid:126) = 1, c = 1. It
pulsesareespeciallyinterestingbecausetheyallowtogener- provides the cycle-average ionization rate for all intensities
ate THz frequencies, which are otherwise difficult to access butfailstoresolvethefastsubcycleionizationratedynamics
by other optical methods. In this case, we considered only in the tunnel ionization regime. In our case, we have two-
limitingcasesofpurelytunnelorpurelymultiphotonregime. color field (for instance ω+2ω) field. The Volkov state of
Nevertheless, for the two-color fields with the incommen- theelectroninbichromaticlaserfieldwithaquantizedvector
suratefrequencies(ω andω +∆ω)andlongenoughpulses, potentialA:
0 0
theBrunelharmonicsoftheformω +j∆ω(forinteger j)can
0
arisealsoinmultiphotonregime,becauseoftheslowbeatings
between two harmonics on the time scale ∼ 1/∆ω. In gen-
eral, bychangingofthisbeatfrequency, onecanthusswitch
from“typicaltunnel”to“typicalmultiphoton”regime,giving (cid:88)2
us a new tool to study the transition region between the two A(−kr)= gi((cid:126)(cid:15)jeikjraj+(cid:126)(cid:15)j∗eikjra†) (A1)
ionizationregimeswithoutchangingtheintensitywhichisan j=1
interestingdirectionforthefuturestudies.
FUNDING
with polarization(cid:126)(cid:15) = [(cid:126)(cid:15) cos(ξ /2)+i(cid:126)(cid:15) sin(ξ /2)]exp(iφ ),
j x j y j j
We gratefully acknowledge support by the DFG BA definedbyellipticityξ andphaseφ forthemodes j=ω,2ω,
j j
4156/4-1,MO850/20-1andNieders. VorabZN3061. canbewrittenas[32,38]
(cid:12) (cid:69) (cid:88)
(cid:12)(cid:12)Ψµ =Ve−1/2 exp{i[Pe+(up1− j1)k1+(up2− j2)k2]}Xj1,j2(z)|n1+ j1,n2+ j2(cid:105), (A2)
j1,j2
whereV isanintegrationvolume,n isabackgroundphoton where i = 1,2, ω = ω, ω = 2ω, 2Λ is an amplitude of
e i 1 2 i
number in the ith mode, j is the number of transferred pho- theithmode,obtainedfromaquantizedfieldEq.(A1)inthe
i √
tons, P usthemomentumoftheelectroninthefield, k , k largephotonlimitg n →Λ.
e 1 2 i i i
are the wavevectors of the corresponding field components,
andu ,u aretheponderomotiveparametersoftheelectron
p1 p2
inthefieldsωand2ωcorrespondingly:
u = e2Λ2i , (A3) [38X,3j19,j2](z)arethegeneralizedphasedBesselfunctions(GPB)
pi m ω
e i
 
Xj1,j2(z)= (cid:88)∞ X−j1+2q1+q3+q4(ζ1)X−j2+2q2+q3−q4(ζ2)(cid:89)4 X−qi(zi), (A4)
q1...q4=−∞ i=2
where X (z) = J (|z|)exp{inarg(z)} is a phased Bessel func- arguments
n n
tion[39],calculatedthroughtheusualBesselfunctionJ with
n 1
ζ = 2|e|ΛiP(cid:126)(cid:15), z = u (cid:126)(cid:15)(cid:126)(cid:15), (i=1,2) (A5)
i meωi i i 2 pi i i
2e2Λ Λ (cid:126)(cid:15) (cid:15)∗
z = 2e2Λ1Λ2(cid:126)(cid:15)1(cid:15)2,z = 1 2 1 2. (A6)
3 (ω1+ω2)me 4 (ω1−ω2)me
8
Using the interaction Hamiltonian of the electron and the onecanobtainthetransitionamplitudefromthegroundstate
quantizedfieldA: inside an atom, described by a wavefunction Φ(r) and the
i
energy E, and l , l photons in the corresponding modes of
i 1 2
thefield,toanelectronfreestatewithawavefunctionφ (r)=
−e e2A2 f
V(r)= 2m [(−i∇)A+A(−i∇)]+ 2m , (A7) eiPefrandtheenergyEf,andm1, m2photonsinthefield[32].
e e
(cid:88) (cid:68) (cid:69)(cid:68) (cid:69)
Tfi =Ve−1/2 φf(r);m1,m2|Ψµ Ψµ|V(r)|Φi(r);l1,l2 =Ve−1/2Bq(ω1,1q,2ω2)(Pef), (A8)
εµ=εi
whereq =l −m,andB(ω1,ω2)(P )isan(unnormalized)elec-
i i i q1,q2 ef
trontransitionamplitudetothefinalstateφ (r),accomplished
f
byabsorptionofexactlyq photonsfromthei-thfieldmode:
i
(cid:88)(cid:104) (cid:105)
B(qω1,1q,2ω2)(Pef)=Φi(Pef) (up1− j1)ω1+(up2− j2)ω2 Xj1−q1,j2−q2(zf)X∗j1,j2(zf), (A9)
j1,j2
withΦ(P )beingaFouriertransformofΦ(r). “biphotons”q=q +2q leadingfinallyto:
i ef i 1 2
(cid:90) (cid:90)
V (cid:110) (cid:111)
W = e dΩd|P | |P |2δ(E −E )|T |2 =
GAC (2π)2 ef ef i f fi
(cid:112)2m3ω(cid:88) (cid:112) (cid:90) (cid:12)(cid:12)(cid:12)(cid:88) (cid:12)(cid:12)(cid:12)2
4π2e q−B/ω dΩ(cid:12)(cid:12)(cid:12)(cid:12) B(qω−,22qω2),q2(pef)(cid:12)(cid:12)(cid:12)(cid:12) . (A10)
q q2
The total transition rate is then given by integration over Here, B is an ionization potential, Ω is a solid angle (dΩ =
(cid:112)
athlleppoasrstiiballeperolebcatbroilnitimesomofeanbtasoarnpdtioisneoxfppreasrstiecdulaasransuummbeorvoerf usinnitθpveefcdtθopre,fdpoφipnefti)n,ganindtpheefdi=recntiΩond2emfienωe(dqb−yBth/eωa),ngnlΩeΩis.a
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