Novel aspects of radiation reaction in the classical and the quantum regime

0.21 MB·

by  Norman Neitz

4
Novel aspects of radiation reaction in the classical and the
1
0
2 quantum regime

n
a
J NormanNeitz,NaveenKumar,FelixMackenroth,KarenZ.Hatsagortsyan,

3 ChristophH.KeitelandAntoninoDiPiazza
1
Max-Planck-Institutfu¨rKernphysik,Saupfercheckweg1,69117Heidelberg,Germany

]
h E-mail:[email protected]
p
-
p Abstract.
e Thisworkisdedicatedtothestudyofradiationreactionsignaturesintheframeworkofclassicaland
h quantum electrodynamics. Sincetherehasbeennodistinct experimental validationof radiationreaction
[ anditsunderlyingequationssofaranditsimpactisexpectedtobesubstantialfortheconstructionofnew

experimentaldevices,e.g.,quantumx-freeelectronlasers,aprofoundunderstandingofradiationreaction
1
effects is of special interest. Here, we describe how the inclusion of quantum radiation reaction effects
v
9 changesthedynamicsofultra-relativisticelectronbeamscollidingwithintenselaserpulsessignificantly.
0 Thereafter, the angular distribution of emitted radiation is demonstrated to be strongly altered in the
7 quantum framework, if in addition to single photon emission also higher order photon emissions are
3 considered. Furthermore, stimulated Raman scattering of an ultra-intense laser pulse in plasmas is
. examinedandforwardRamanscatteringisfoundtobesignificantlyincreasedbytheinclusionofradiation
1
0 reactioneffectsintheclassicalregime. Thenumericalsimulationsinthisworkshowthefeasibilityofan
4 experimentalverificationofthepredictedeffectswithpresentlyavailablelasersandelectronaccelerators.
1
:
v
i
X
r 1. Introduction
a
If a charged particle, an electron for definiteness, is exposed to an electromagnetic background field,
it will be accelerated and subsequently emit radiation. Although this process is fundamental in
electrodynamics, theusualclassicaltreatmentisinsufficient,sinceitdoesnottakeintoaccountradiation
reaction (RR), i.e., the back reaction of the emitted radiation on the charged particle itself [1, 2]. In
this work high intensity plane wavelaser fields ofelectric fieldamplitude , central angular frequency
0
E
ω , central wavelength λ and propagation direction n are investigated, in order to theoretically probe
0 0 0
the parameter regime relevant for RR. The study of electron-laser interactions in classical as well as
quantum electrodynamics (QED) is of special interest, as they are expected to have impact on various
fields like accelerators, quantum x-free electron lasers [3, 4, 5] or the production of multi-GeV photon
beams[6]. However,RReffectsarealsoofpuretheoreticalinterest,asevenintheframeworkofclassical
electrodynamics theoretical methods such as renormalization are necessary to describe the self-coupled
dynamics[7].
Recently it has been reported, that the so-called Landau-Lifshitz (LL) equation is in theory the
accurate equation of motion for an electron of mass m and charge e < 0 in the framework of classical
electrodynamics [1, 2, 7, 8, 9, 10, 11]. (Units with ~ = c = 1 are used.) In case of an electromagnetic
plane wave the LL equation allows for an analytical solution [12] and it was demonstrated that if the
parameter R = αχ ξ is of order of unity, the dynamics of an electron with initial momentum pµ
c 0 0 i
colliding with a plane wave laser field is significantly altered by RR effects. Here, we introduced
the classical and quantum nonlinearity parameters ξ = e /mω and χ = ((n p )/m) /E ,
0 0 0 0 0 i 0 cr
respectively, and defined nµ = kµ/ω = (1,n ), where we|m|Eade use of the abbreviation (ab)E= a bµ
0 0 0 0 µ
denoting the product of two four-vectors aµ and bµ. Furthermore, α = e2 is the fine structure constant
and E = m2/e = 1.3 1016V/cm is the critical field of QED. In addition, in a bichromatic laser
cr
| | ×
pulse the RR force was shown to modify the trajectory of an electron also in the case of R 1 [13]
c
≪
(see also [14]). The head-on collision of a laser field with an ultra-relativistic electron of initial energy
ε yields R = 3.2ε [GeV]I [1023 W/cm2]/ω [eV], where I = 2/4π is the laser peak intensity.
i c i 0 0 0 E0
Thus, the experimental challenges in observing RR effects can be understood, since this expression is
usually very small. Nevertheless, an alternative method has been proposed to measure RR effects also
atmoderatelaserintensities andlargerpulsedurations [15]. However,forupcoming ultra-high intensity
laserfacilitiesitwascalculatedthatforlaserintensitiesexceedingI > 2 1023W/cm2 RReffectshave
0
×
tobetaken into account andforI > 4 1024W/cm2 quantum effects willbecome important [16]and
0
×
leadtoastrongalteration oftheparticle’s dynamics[17].
In addition, the consistency of QEDand the diverse classical approaches was examined [18]. In the
quantum description a laser field is depicted as a stream of photons and the scattering process with an
electronleadstotheabsorption ofmanyphotonsfromthefieldandtothesubsequent emissionofoneor
more photons. In fact, the emission of a single photon by an electron in strong laser pulses (nonlinear
single Compton scattering (NSCS)) has been studied thoroughly [19, 20, 21, 22, 23] and the classical
spectra without RR [24] were shown to coincide with the NSCS spectra for χ 1. Hence, in the
0
≪
ultra-relativistic regimeand for negligible pair creation the quantum analogue of RRcan beunderstood
as the emission of a higher number of photons [25]. Providing the first results for such higher order
photonemission,theemissionoftwophotonsbyanelectroninaplanewave(nonlineardoubleCompton
scattering (NDCS))wasconsidered recently [26,27,28,29].
The influence of the RR force on the collective particle dynamics of a system can be considerably
different from the single particle dynamics as the collective energy loss of the particles due to RR can
give rise to unexpected physical phenomena in a medium. In the classical electrodynamics regime, an
analysis of the influence of the Landau-Lifshitz RR force [7] on the collective plasma dynamics of the
particles has been carried out recently, where it was found that the inclusion of RR counterintuitively
strongly enhances the forward Raman scattering (FRS) of the laser radiation in plasmas [30]. This
growth enhancement is attributed to the nonlinear mixing of the two Raman sidebands mediated by the
RRforce.
In the sections concerning quantum RR (sections 2 and 3), we will employ light-cone coordinates,
which for a four-vector aµ = (a0,a) are defined as aµ = (a+,a−,a ), where a = a0 a , with
⊥ ± k
±
a = k a/ω anda = a a k /ω .
k 0 0 ⊥ k 0 0
· −
2. Kineticapproachtoquantumradiationreaction
Inthissection,weinvestigateRReffectsinthecollisionofanintenselaserpulsewithanultra-relativistic
electron beam. In classical electrodynamics RR was shown to reduce the energy width of electron [31]
and ion [32, 33, 34, 35] bunches. However, in the quantum regime we find that RR has the opposite
tendency and leads to a broadening in the width of the energy distribution describing the electron
beam. Thedifference between theclassical and the quantum regime can beexplained by theincreasing
importanceofthestochasticnatureofphotonemissioninthequantumregime. TheclassicalLLequation
ignores the stochasticity of photon emission and even for small χ a correct treatment of RR requires
0
an additional stochastic term. In fact, a Langevin-like equation can be employed for not too large χ ’s,
0
though in the full quantum regime at χ 1 this approximated description is not valid anymore. The
0
∼
broadening in the energy distribution of the electron beam displayed by our numerical simulations is
expectedtobedetectable withnowadaysavailable electronaccelerators andstronglaserfields.
An exact treatment of RR in the realm of strong-field QED would in principal result in the
determination of the full S-matrix, taking into account multiple photon emission, radiative corrections
andpaircreation following photon emission [1,25]. However,RRmainlystemsfromincoherent multi-
photon emission in the so-called “nonlinear moderately-quantum” regime ξ 1, χ . 1 [25], where
0 0
≫
nonlinear effects in the laser field are considered to be large and quantum effects are already important
but pair production can still be neglected. In order to investigate RR in this regime we apply a kinetic
approach [36, 37, 38] and in turn characterize electrons and photons by distribution functions. Due
to the neglect of pair production the distribution function of positrons is considered to vanish and the
kinetic equation of the electrons is decoupled from that of the photons [36, 37, 38]. If the average
energy of the electron beam ε fulfills the constraint ε mξ , the transverse momentum of the
∗ ∗ 0
≫
electronscanbedisregarded,sincethroughoutthewholeinteractionitwillremainmuchsmallerthanthe
longitudinal momentum [7]. Assumingthe collision ofpresently available optical (ω = 1.55 eV)laser
0
fields of intensity 1022 W/cm2 [39] with electron bunches with typical energies of ε = 1 GeV yields
∗
mξ = 25 MeV and allows us to treat the present problem as an one-dimensional one. Considering a
0
linearly polarized plane wave propagating along the positive y direction, we introduce the electric field
of the laser field by E(η) = g(η)zˆ depending on the laser phase η = ω (t y) via a pulse-shape
0 0
E −
function with g(η) 1. Since the ultra-relativistic regime ξ 1 is investigated, the emission
max 0
| | ≤ ≫
in a plane wave field of a photon with four-momentum kµ = (ω,k) by an electron with initial four-
momentum pµ = (ε,p) can be characterized by adopting the well-known single photon differential
emissionprobability dPp− perunitphaseandperunitu= k−/(p− k−)[40]
−
dPp− α m2 1 1 2u ∞
= 1+u+ K dxK (x) , (1)
2 1
dηdu √3πω0p−(1+u)2 "(cid:18) 1+u(cid:19) 3 (cid:18)3χ(cid:19)−Z32χu 3 #
where K () is the modified Bessel function of νthorder. The quantum nonlinearity parameter is given
ν
·
by χ χ(η,p ) = (p /m) (η)/E and is now depending on the laser phase due to the oscillating
− − cr
≡ |E |
electric field (η) = g(η). Thefact that Eq. (1) only depends onthe variables η and p allows usto
0 −
E E
employanelectrondistribution f (η,p )forthedescription ofanelectronbeamanditsphaseevolution
e −
isdetermined bythekineticequation (seeRef. [36])
∂fe ∞ dPp′− p− dPp−
= dp f (η,p ) f (η,p ) dk (2)
′− e ′− e − −
∂η Zp− dηdp− − Z0 dηdk−
with
dPp′− = p′− dPp′− , (3)
dηdp− (p−)2 dηdu (cid:12)u=p′−p−−p−
dPp− = p− dPp(cid:12)(cid:12)− . (4)
dηdk− (p−−k−)2 dηdu(cid:12)u=p−k−−k−
(cid:12)
(cid:12)
Eq. (2)isnon-local inthemomentum p since itisanintegro-differential equation. Hence, anelectron
−
withinitialmomentump iscoupledtotheelectronwithmomentump k ,wherethemomentumof
−0 −0 − −
theemittedphotonk variesbetween0andp . Inturn,theevolutionoff (η,p )isnotonlyinfluenced
− −0 e −
bythe neighborhood ofp but byallpossible values of p . Inorder tostudy the classical limit ofRR,
− ′−
weexpandEq. (2)uptotheorderofχ3(η,p )
−
∂f ∂ 1 ∂2
e
= [A(η,p )f ]+ B(η,p )f , (5)
∂η −∂p − e 2∂(p )2 − e
− −
(cid:2) (cid:3)
which is a Fokker-Planck-like equation [38, 41, 42]. Here, we introduced the “drift” coefficient
A(η,p ) = 2αm2χ2(η,p )[1 55√3χ(η,p )] and the “diffusion” coefficient B(η,p ) =
− − 3ω0 − − 16 − −
αm2 55 p χ3(η,p ). Eq. (5)isnolongeranintegro-differential equationandtheevolutionoff (η,p )
3ω0 8√3 − − e −
Figure 1. (color online) Phase evolution of the electron distribution for a 7-cycle sin2-like laser pulse
accordingtoEq. (2)(parta)),totheclassicalkineticequationwithout(partb))andwiththereplacement
I (η,p ) I (η,p ) (part c)). The laser and the initial electron distribution parameters are given in
cl − q −
→
thetext.
isdeterminedbythemomentap closetop ,duetothelocality inp . Ifalsohigher-order corrections
′− − −
inχ(η,p )are considered, the expansion of Eq. (2) leads to terms with higher derivatives off (η,p )
− e −
withrespecttop .
−
By expanding the kinetic equation (2) we obtain two quantum corrections to the classical kinetic
equation ∂f /∂η = ∂/∂p (f dp /dη), with dp /dη = I (η,p )/ω and the classical radiation
e − e − − cl − 0
− −
intensity I (η,p ) = (2/3)αm2χ2(η,p ). The first correction modifies the drift coefficient A(η,p ),
cl − − −
but does not affect the analytical structure of the classical equation. Therefore the phase evolution is
purely deterministic [43] and this correction coincides with the well-known leading-order correction to
the total intensity of radiation [36, 40]. As the correction term is negative, we expect the reduction of
the energy width to be smaller than in classical electrodynamics. However, if the classical radiation
intensity I (η,p ) is substituted by the corresponding quantum one I (η,p ) (see, e.g., [40]), the
cl − q −
corresponding Liouville equation still predicts a decrease in the energy width, due to the fact that
electrons with higher energy on average will emit more radiation. On the other hand, the second
quantum correction introduces the diffusion coefficient B(η,p ) and transforms the classical kinetic
−
equationintoaFokker-Planck-likeequation,whichisequivalenttothesingle-particlestochasticequation
dp = A(η,p )dη + B(η,p )dW, with dW an infinitesimal stochastic function [41]. Reflecting
− − −
−
thestochasticityofphotonemission,thisequationisnolongerdeterministicandthestochasticevolution
p
ofthesystemcausesthebroadening oftheenergydistribution [43].
We solved Eq. (2) numerically by employing a finite difference method and we ensured that for
χ = (p , /m) /E 1, where p , isthe average momentum, ournumerical simulations coincide
∗ ∗− 0 cr ∗−
E ≪
with classical results, as well as with the results in the quantum regime [25]. We now consider a laser
pulsewithshapefunctiong(η) = sin2(η/2N )sin(η),whereN isthenumberoflasercyclesandwith
L L
ω = 1.55 eV, to collide head-on with an initial Gaussian electron distribution that is normalized to
0
unity. Further, we assume the laser peak intensity I = 2.5 1022 W/cm2, the average momentum
0
×
p∗,− = 1.8 GeV (ε∗ 900 MeV) and the initial width of the electron distribution σp− = 0.18 GeV
≈
corresponding to χ = 0.8, and N = 7 corresponding to about 21 fs. The results of our numerical
∗ L
simulations areshowninFig. 1. Asexpected, Fig. 1a)displays abroadening oftheelectron distribution
in the quantum regime. However, if the full kinetic Eq. (2) is not applied but the quantum intensity
I (η,p ) (see, e.g., [36, 40]) is set into the classical kinetic equation, the photon emission still reduces
q −
the energy spread of the electron distribution (Fig. 1c)) as in the classical case (Fig. 1b)). This clearly
indicatesthatthebroadeningoftheelectrondistributionisinducedbytheimportanceofthestochasticity
ofphotonemissioninthefullquantumregime.
3. NonlineardoubleComptonscattering
In this section, we are going to demonstrate how in the ultra-relativistic quantum regime, distinguished
bytheconditions ξ 1,χ & 1,theNDCSsignalcanbespatiallyseparatedfromthedominantNSCS
0 0
≫
signal. Theinvestigation ofthegivenparameterregimeistimelyasitisabouttocomeintoexperimental
reach. To clearly interpret the results of this analysis and to establish an intuitive understanding of the
underlying physics we are going to work out a semi-classical picture of two smoothly joined classical
electron trajectories from which the two separate photons are consecutively emitted. These trajectories
arefoundtofeatureadiscontinuity onlyintheelectron’senergy, whichcanbeattributedtotheemission
of a photon of finite energy. We are going to connect this picture of a discontinuous change of the
trajectorytotheclassicalaccountofRRwhichleadstoasmoothchangeoftheelectron’strajectory[25].
Finally, we are going to demonstrate that the discussed effect is likely to be observable with already
available lasers, featuring intensities beyond 1022 W/cm2 [39], and electron acceleration technology
eitherfromconventional accelerators [44]ormodernplasma-based laseraccelerators [45,46].
As in the regime mξ ε , with the electron’s initial energy ε , the radius of the laser’s focal
0 i i
≪
volume routinely exceeds the electron’s perpendicular excursion, which is on the order of λ (mξ /ε )
0 0 i
it is justified to approximate the laser field by a plane wave. In the present study we are thus going
to model the laser pulse by a plane wave field of arbitrary temporal shape Aµ(η) = ( /ω )ǫµψ(η)
depending on the space-time coordinates only via the invariant phase η = kµx0. Here ǫµE0is th0e w0ave’s
0 µ 0
polarizationfour-vectorandtheshapefunctionψ(η)describesthelaserpulse’sarbitrarytemporalshape.
Sincein theregime ξ 1nonlinear effects have to betaken into account exactly wewillperform our
0
≫
calculations intheso-called Furrypicture ofquantum dynamics [47,48]. Theessence ofthisprocedure
istoattributetermsintheQEDLagrangiandescribing thestrongbackground field,i.e.thelaserpulsein
thiscase,tothefreeLagrangianandtosubsequentlyquantizethechargedfermionicfieldsinthepresence
of this strong background. Technically this amounts to a replacement of the vacuum wave function of
an electron with momentum pµ and spin quantum number σ by a solution of the Dirac equation in the
presence ofthestrong plane wavebackground, knownasVolkov wave function Ψ (x)[49]. Itisthen
p,σ
straightforward to obtain an expression for the scattering matrix element S of an electron with initial
fi
(final) four-momentum pµ = (ε ,p ) (pµ = (ε ,p )) and spin quantum number σ (σ ) emitting two
i i i f f f i f
photons with wave vectors kµ and kµ and polarization four-vectors ǫµ and ǫµ , respectively. The
1 2 k1,λ1 k2,λ2
(1) (2)
resultingexpression canbewrittenasS +S with
fi fi
S(1) = e2 d4xd4y Ψ (y)/ǫ eik2yG(y,x)/ǫ eik1xΨ (x) (6)
fi − pf,σf ∗k2,λ2 ∗k1,λ1 pi,σi
R
and S(2) = S(1)(1 2). Here /a = γµa is the common Feynman slash notation with the four-
fi fi ↔ µ
vector of the Dirac matrices γµ. Furthermore, we introduced the Dirac conjugate wave function
Ψp,σ(x) = Ψ†p,σ(x)γ0 and made use of the laser dressed electron propagator G(y,x) [40]. Due to
(1)
symmetryreasonsitissufficienttoonlycomputethequantityS ,whencethecross-channel amplitude
fi
(2)
S can be obtained by the exchange of indices (1 2) in the final expression. According to earlier
fi ↔
work[28,29]thispartialamplitudenaturally splitsupintotwocontributions inthefollowingway
2
S(1) = (2π)3 (a f δ +b f )δ(p k k p )δ(2)(p k k p ). (7)
fi r r r,s r,s r,s −i − 1− − 2−− −f ⊥i − 1⊥− 2⊥− ⊥f
r,s=0
X
Thematrixcoefficients a andb arerather involved butnotneeded hereandthus notgivenexplicitly.
r r,s
Anyway, the important information on the dynamics of the two-photon emission process is encoded in
thedynamicintegrals [28,29]
f = dηψr(η)exp i[S (η)+S (η)] , (8a)
r x y
{− }
Z
f = dη dη Θ(η η )ψs(η )ψr(η )exp i[S (η )+S (η )] , (8b)
r,s x y y x x y x x y y
− {− }
Z
where S (η) = ηdη [α ψ(η ) + β ψ2(η ) + γ ], with α = mξ[(p ǫ )/(k p )
x/y 0 ′ x/y ′ x/y ′ x/y x − i 0 0 i −
(p ǫ )/(k p )], β = m2ξ2(k k )/2(k p )(k p ), γ = (k p )/(k p ). The parameters α , β
antd0γ are0otbtainexd frRom−α ,β 0and1 γ ,re0sptecti0veily, suxbstitu−ting1pµi p0µ,tpµ pµ andkµ kyµ. Iny
y x x x t → f i → t 1 → 2
theseexpressionswedefinedthetransitionalelectronmomentumpµ,distinguishedbythefourconditions
t
p = p k , p = p k and p2 = m2. From the above expressions one can read off that the
−t −i − 1− ⊥t ⊥i − 1⊥ t
integrals f , f and f are divergent. These divergences, however, can be analytically regularized
0 0,s r,0
by an integration by parts technique [28, 29]. Carrying out the resulting replacements in the scattering
matrix element, taking the modulus square and summing overthe discrete andcontinuous phase spaces
of all particles involved in the scattering the final result for the differential energy spectrum is found to
be
ω +ω d3p 2 d3k 2
1 2 f i (1) (2)
dE = S +S , (9)
2 (2π)3 (2π)3 fi fi
Yi=1 {Xσ,λ}(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12) (1,2) (cid:12)
where σ,λ σ ,σ ,λ ,λ . The δ-functions contained in S serve to fix the final electron
{ } ≡ i f 1 2 fi
momentumpµ whenceweonlyhavetointegrateoverthephasespacesoftheemittedphotons.
f
AsanextstepwewishtogiveanexemplarynumericalstudyofEq.(9)todemonstrateitstractability
aswellastohighlighttheaforementionedspatialseparationoftheNSCSandtheNDCSemissionsignals.
For the numerical evaluation we assume the laser’s shape function to be ψ(η) = sin4(η/4)sin(η) for
η [0,4π] and zero elsewhere with a central frequency ω = 1.55 eV, corresponding to an optical
0
∈
laser pulse of approximately 5 fs duration. We consider the case of an ultra-intense laser pulse of peak
intensity I = 1022W/cm2 corresponding toξ 48, asisalready available atnowadays workinglaser
0 0
≈
facilities suchastheHercules laser [39]. Furthermore, westudy thecaseofanultra-relativistic electron
of initial energy ε = 5 GeV colliding head on with the specified laser pulse, resulting in a quantum
i
nonlinearity parameter χ = 2.8, indicating theimportance ofquantum effects. Next, wewishtorecall
0
several properties of the NSCSsignal, that are to be expected for the given experimental setup [50]: At
ξ 1 the single photon signal is limited to a polar angular range π θ ψ θ , where the typical
0 0 0
≫ − ≤
opening angle of the single photon emission cone is given by θ = mξ /ε . The maximum value of
0 0 i
the shape function specified above is ψ = max(ψ(η)) = 0.8. Any emission predicted towards polar
0
| |
anglesoutsidethisemissionconewillbeclearlyseparated fromthedominant NSCSsignal.
As our reference frame we choose the coordinate frame in which the laser pulse is polarized along
the x-direction and propagates towards the positive z-axis. This axis we also choose as polar axis, i.e.
θ = π corresponds to the electron’s initial propagation direction opposite to the laser’s propagation
direction at θ = 0. To stress that two-photon emission will be important for the electron’s radiation
pattern giventhespecified experimental parameters weestimate theaverage numberofemittedphotons
tobe = 1.6[40].
NSCS
N
In the given reference frame we consider an experiment where one photon detector observes any
photon emitted towards the direction (θ = π θ /2,φ = π) and a second detector to trace
1 0 1
−
photons emitted towards the two different directions (θ = π θ /2,φ = 0) (see fig. 2a)) and
2 0 2
−
(θ = π θ ,φ =0)(seefig.2b)). WhilethefirstdetectorispositionedinsidetheNSCSemissioncone
2 0 2
−
thechoice forthesecond observation direction corresponds tophoton detection inside (see fig.2a)) and
outside(seefig.2b))thiscone,respectively. Theazimuthalanglesφ = 0,πarechosenforobservation
1,2
oftheemittedphotons withinthelaser’splaneofpolarization, wheremostradiation isemitted.
In fig. 2a) we wish to highlight that the cutoff for the emitted photon’s frequencies (white line in
fig. 2a)), set by energy-momentum-conservation, is closely approached, whence we infer that quantum
effects indeed are non-negligible. Comparing now this figure to Fig. 2b), which is plotted in the
same color-scale, we immediately conclude that there is a considerable amount of radiation emitted
to directions outside the NSCS emission cone (recall that this latter figure corresponds to the detection
of a photon outside this cone). We can thus conclude a clear spatial separation between the NSCS
signal, confined exclusively to the angular range π θ 0.8θ , and the NDCS signal, which is also
0
− ≤
Figure2. (color online) Two-photon energy emission spectra dE/Π2 dω dΩ [eV 1 sr 2]atχ 2.8
i=1 i i − − 0 ≈
observedatθ = π θ /2,andatθ = θ (parta))andatθ = π θ (partb)),withθ = 5 10 3 rad.
1 0 2 1 2 0 0 −
− − ×
Other numerical parameters are given in the text. The solid white lines correspond to the cutoff-energy
ω + ω = ε (part a)) and the threshold frequency ω (part b)), respectively. Part c): Two classical
1 2 i 1∗
electron trajectories with initial electron momentum p (solid line) and p (dashed line). In color-code:
i t
Actualelectrontrajectory foraphotonwithenergyω = 2GeVemittedatη¯ towards(θ ,φ ).
1 x,1 1 1
detectable under π θ = θ . The classical analog of this quantum result is an effect attributed to
0
−
RR changing the angular distribution of the radiation emitted by an electron [14]. For a qualitative
interpretation of the presented separation of the single and two photon signals we take advantage of a
stationary-phase analysis that was recently developed for the analysis of NDCS spectra [29]. It was
shownthat itwasonly necessary toanalyze thepartofthescattering matrix elementproportional tothe
bivariatedynamicintegralsf asthispartialamplitudelargelydominatesthescatteringamplitude. The
r,s
corresponding dynamicintegralswereapproximatedas[29]f Θ(η¯ η¯ )fy(η¯ )fx(η¯ )
r,s ≈ l,n y,n− x,l r y,n s x,l
with fx/y = dηψr(η) exp[ iS (η)] with the exponential phases S (η) defined above. The sum
r x/y P x/y
−
over the indices l and n is a sum over all stationary points in whose vicinities the only non-negligible
R
contributions tothe dynamic integrals are formed. These stationary points are found as solutions of the
equations ψ(η¯ ) = 1/2 and ψ(η¯ ) = ∆ϑ ω /ε (1/2+∆ϑ ) with ∆ϑ = (π θ )/θ . In
x,l y,n 2 1 i 2 2 2 0
− − −
these expressions the fixed values of θ ,φ and φ were already inserted and only θ was left variable.
1 1 2 2
Thegivenequationsformallycorrespondtothestationarypointconditionsoftwoseparatesinglephoton
emissions,wherefirstaphotonwithwavevectorkµ isemittedbyanelectronwithinitialmomentumpµ,
1 i
whereasthesecondphotonwithwavevectorkµ isemittedbyanelectronwithinitialmomentumpµ. We
2 t
can consequently interpret two-photon emission in the regime ξ 1as the sequential emission of the
0
≫
firstandsecondphotonemittedfromtheclassicaltrajectory anelectronwouldtakeifitenteredthelaser
fieldwithaninitialfour-momentum pµ andpµ,respectively. Forthesakeofcontinuity thesetrajectories
i t
naturallyhavetobejoinedattheemissionpointofthefirstphotonkµ. However,theenergiesaclassical
1
electronwillhavewhenfollowingthesetwoseparatetrajectorieswillbediscontinuous attheconnection
point, reflecting the finite loss of energy and momentum due to the quantum photon emission. We note
that, of course, there are several solutions for the stationary points η¯ and hence several combinations
x,l
ofclassicaltrajectoriesforeachchoiceofobservationdirectionsfortheemittedphotons. Oneexemplary
combination of two such classical trajectories computed for an electron scattered from a laser pulse of
thegivenparameters isshowninfig.2c). Anelectron withinitial momentum pµ wouldfollow thesolid
i
trajectory,whereas,ontheotherhand,anelectronofinitialelectronmomentumpµwouldtakethedashed
t
trajectory. In the case shown here the transitional momentum pµ is computed from pµ by assuming the
t i
emissionofaphotonofenergyω = 2GeVintothedirection(θ ,φ ). Sincethiscorrespondstotheloss
1 1 1
ofasignificantportionofitsinitialenergybytheelectronthemomentumpµsignificantlydiffersfrompµ
t i
and thus the two corresponding trajectories are clearly distinguishable. This feature then also explains
the considerable spatial separation of the NSCS and the NDCS signal in the regime χ 1. We note,
0
∼
however, that the tangent vectors of the two shown trajectories are parallel at the junction, as they have
to be for an ultra-relativistic electron emits photons almost exclusively into its instantaneous direction
of propagation and thus can lose momentum only in this direction. The aforementioned loss of energy,
however, can be read off from the color-coding of the two trajectories and is clearly discontinuous at
the point of emission of the first photon. It is also this significantly decreased energy of the electron
that leads to a stronger deflection of the second trajectory inside the laser field. To find an analytical
prediction of the energy threshold the first photon has to carry away to render this deflection strong
enoughtofacilitateemissiontowardsthechosenobservationdirectionθ = π θ wesolvethedefining
2 0
−
equation of the stationary point η¯ for the first emitted photon’s frequency ω . From this procedure
y,n 1
we find the threshold ω = ε (1 0.8)/(1 + 1/2) 660 MeV, which is well confirmed in fig. 2b).
1∗ i − ≈
Furthermore,wehavetostressthatintheoverallscattering amplitude wealsohavetoincludethecross-
channeltermS(2) wherethephotons’wavevectorskµ andkµ areexchanged. Theinterpretation forthis
fi 1 2
cross-channel is in terms of classical trajectories is analogous to the above given arguments, however,
withtheorderoftheemissionofthetwophotons exchanged aswell.
4. InfluenceofRRontheparametricinstabilities inplasmas
Parametric instabilities of a laser pulse in a plasma are important due to their applications in the
area of laser-driven fusion, laser wakefield acceleration, and have been investigated for decades
[51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62]. The FRS—a scattering process belonging to the
general Stimulated Raman scattering (SRS) processes in plasmas—is one of the prominent examples
of parametric instabilities in plasmas. In the FRS, the incident pump laser decays into two forward
movingdaughterelectromagnetic waves,andaplasmawave. Thedaughterwaveshavetheirfrequencies
upshifted (anti-Stokes waves) and downshifted (Stokes wave) from the pump laser by the magnitude
whichequalstheexcitedplasmawavefrequency.
Though, at high laser intensities I 1019W/cm2, the growth rate of the parametric instabilities
0
≥
becomessmallerduetotherelativistic Lorentzfactor [57],theroleofRRforcebecomes alsoimportant
especially atultra-high laser intensities, I 1022W/cm2[1,33,34, 14,63, 64,65]. Such ultra-intense
0
≥
lasersystemsareexpected tobeavailable inanearfuture afterthecommissioning oftheExtremeLight
Infrastructure (ELI)project in Europe [66]. Due to the RRforce, the laser pulse suffers damping while
propagating in a plasma. As the laser loses energy due to RR force it facilitates, apart from the usual
parametric decay processes, the availability of an additional source of free energy for perturbations to
grow in the plasma. Its effective intensity also decreases which lowers the relativistic Lorentz factor.
Moreover, thephaseshift, caused bytheRRforce, inthenonlinear current densities causes polarization
rotations of the scattered daughter electromagnetic waves. This necessitates to include the effect of RR
forceinthetheoretical formalismoftheparametric instabilities intheplasma.
We study the FRS of an ultra-intense laser pulse in a plasma including the RR force effects in
the classical electrodynamics regime where quantum effects arising due to photon recoil and spin are
negligible [1]. This approach is valid if the wavelength and magnitude of the external electromagnetic
fieldintheinstantaneous restframeoftheelectronsatisfyλ λ , E ,whereλ = 3.9 10 11
C cr C −
≫ E ≪ ×
cmisthe Compton wavelength and E isthe critical fieldofthe quantum electrodynamics [1]. Forthe
cr
laserintensities planned intheELIproject I 1022 23W/cm2 [66],thesetwocriteria canbefulfilled.
0 −
∼
In the classical electrodynamics regime, the Landau-Lifshitz RR force [7] correctly accounts for the
radiationemittedbyarelativisticchargedparticle[1]. Considerthepropagation ofacircularlypolarized
(CP) pump laser along the zˆ direction in an underdense plasma with uniform plasma electron density
n . Ions areassumed tobeatrest. Eq. ofmotion foranelectron including theleading order termofthe
e
Landau-Lifshitz RRforceinthelaserfieldis
∂p 2e4
+υ p = e E +υ B γ2υ E +υ B 2 υ E 2 , (10)
∂t ·∇ − × − 3m2 × − ·
h i
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
where γ = 1/√1 υ2 and the velocity of the light in vacuum c = 1. The other terms of the
−
Landau-Lifshitz RR force are ignored as they are 1/γ times smaller than the leading order term
[7]. We first ignore the RR term and express the electric and magnetic fields in potentials as E =
φ ∂A/∂t, B = A. Ina1Dapproximationvalidwhenr λ (wherer isthespot-sizeand
0 0 0
−∇ − ∇× ≫
λ isthewavelengthofthepumplaserpulse),thetransversemomentumandz-componentofmotionare
0
p = eA, and ∂υ /∂t = e φ/(mγ ) e2 A2/(2m2γ2), where A = A eiη0/2+c.c, A =
σ⊥A ,σ = (xˆ +iyˆ)/√z2,γ = (∇1+ξ2/20)1/−2,ξ ∇=|eA| /m,η 0= k z ω t,and0ω , k arethecar0rier
0 0 0 0 0 0 0 − 0 0 0
frequency and wavevector of the pump laser respectively [57, 67]. A plane monochromatic CP light
doesn’t cause any charge separation since for it A 2 = 0 and consequently there is no component
0
∇| |
of velocity in the zˆ direction. This solution is known as the Akhiezer-Polovin solution for a purely
transverse monochromatic CPlightinplasmas[57,68,69]. Thescattering ofthelaserpulseresults into
thetotalvectorpotentialoftheformA =[A0eiη0 +δA+eik⊥.x⊥eiη+ +δA∗e−ik⊥.x⊥e−iη−∗]/2+c.c.,
where δA = σδA , δA = σδA , δA and δA represent the anti-St−okes and the Stokes waves
+ + ∗ ∗ +
respectively,η = (k +k −)z (ω+ω− )t, η = (k −k )z (ω ω )t[57,67]. BeatingoftheStokes
+ z 0 0 ∗ z 0 ∗ 0
andtheanti-Stokes waveswith−thepumplase−rleadst−othep−lasma−waveexcitation δn/n ,whichcanbe
e
estimatedfromtheequationofcontinuity,Poissonequation,andthez-componentofequationofmotion,
yielding δn˜ = e2k2/2m2γ2D (A δA +A δA ), where D = ω2 ω′2, ω′2 = ω2/γ , ω2 =
4πn e2/m, δn/n =zδn˜eiηe0ik⊥e.x⊥/2∗0+c+.c,and0η −η η ηe +η −= kpz pωt[57,p67].0Plapsma
e (cid:0)e (cid:1) ≡ +− 0 ≡ − 0 z −
waveoscillation causesanaxialcomponentofvelocityandmomentumυ 1,p p .
z z
On using the above solutions for transverse and longitudinal compone≪nts of m≪ome⊥nta to simplify
the RR term in Eq.(10), the full equation of motion after expressing the CP laser pulse as A =
A⊥(x ,z,t)eiη0/2+c.c., withitsamplitude varying slowlyi.e. ∂A /∂t ω0A , ∂A /∂z
k A ⊥,and φ A ,ω2/γω2 1,andγ = (1+e2 A 2/m2)1|/2,y⊥ields|,≪ | ⊥| | ⊥ | ≪
| 0 ⊥| | | ≪ | | p 0 ≪ | |
∂
(p eA) = eµω Aγ A 2(1 2υ ), (11)
0 z
∂t ⊥− − | | −
whereµ = 2e4ω /3m3,υ = (ω/k )δn˜eik⊥.x⊥eiη/2+c.c.,andwehaveassumed µγ A 2 1,valid
0 z z
| | | | ≪
for laser intensity I 1023W/cm2. Since φ A and the RR effects in the case of the collinear
0
≤ | | ≪ | |
movement of plasma electrons and the plasma wave are negligible, we don’t consider the effect of RR
on plasma oscillations. One can solve Eq.(11) by expressing the transverse component of the quiver
momentum as e.g. p = [p0eiη0 + p+eik⊥.x⊥eiη+ +p∗e−ik⊥.x⊥e−iη−∗]/2 +c.c., where p+ and p
have similar polarizat⊥ions as the anti-Stokes and the Sto−kes modes. The wave equation for the vecto−r
potentialafterthedensityperturbation n = n +δnreadsas
e
∂2A ω2 δn p
2A = p 1+ ⊥. (12)
∇ − ∂t2 γ n e
(cid:18) e(cid:19)
Thedispersion relation fortheequilibrium vectorpotential canbeobtained aftercollecting theterms
containing eiη0, and it reads as ω2 = k2 + ω′2 1 iµ A 2γ /2 , implying that the RR term causes
0 0 p − | 0| 0
dampingofthepumplaserfield. Weincorporate thisdampingbydefiningafrequencyorawavenumber
(cid:0) (cid:1)
shift in the pump laser 1. On writing ω = ω iδω , δω ω we get the frequency shift δω as
0 0r 0 0 0r 0
δω = ω′2ε γ ξ2/2ω , where ε = r ω /3−, r = e2/m≪is the classical radius of the electron and
0 p rad 0 0 0r rad e 0r e
without the loss of generality we have assumed ξ = ξ . For the SRS growth to occur, this frequency
0 0∗
shiftmustbelessthanthegrowthrate. Oncollectingthetermscontainingeiη±eik⊥.x⊥ inEq.(12),weget
D δA = R (δA +δA )andD δA = R (δA +δA )whichyieldsthedispersion relation
+ + + + +
− − − − −
R R
+
+ − = 1, (13)
D D
(cid:18) + −(cid:19)
1 OnecanalsoincorporatetheRRtermbyappropriatelymodifyingtheplasmafrequency,whichessentiallyimplieschangein
thelaserpumpwavevectorarisingduetotheit’sdispersionintheplasma.
where
D = (ω ω )2 ω′2 1 iεradξ02γ0ω0 [(k k )2+k2],
± ± 0 − p (cid:18) − ω±ω0 (cid:19)− z ± 0 ⊥
ω2ξ2 k2 2iε ξ2γ ωω ω ω
R = p 0 z 1 iε ξ2γ + rad 0 0 0 1 iε ξ2γ +4iε γ3 0 .
± 4γ03 "De (cid:18) ∓ rad 0 0 kz ω±ω0(cid:19)−(cid:18) ∓ rad 0 0ω±ω0 rad 0ω±ω0(cid:19)#
TheRRtermmodifiesthecoupling betweentheStokesandtheanti-Stokes modes(R = R ),andthe
+
6 −
formofdispersion relationfromthedispersion relation derivedbefore[51,53,57,67].
′
FortheestimationoftheFRSgrowthrateinalow-densityplasma,ω ω ,boththeStokesandthe
p ≪ 0r
anti-Stokes modes have to be taken into account [30]. Substituting the pump laser frequency shift δω
0
andignoring the finitek gives D = (ω ω )2 ω′2 (k k )2. Onexpressing ω = ω′ +iΓ ,
⊥ ± ± 0r − p − z ± 0 ′ p′ frs
whereΓ isthegrowth rate ofthe FRSinstability, yields D 2iΓ (ω ω ), D 2iω Γ . On
assumingfrsk2 ω′2, ω′2 ω2 ω2 , we get, in the wea±kly≈-coupflresd rpeg±ime0rΓ e ≈ω′, thpe gfrrsowth
z ≈ p p − 0r ≈ − 0r frs ≪ p
rateoftheFRSas
Γ = ωp2εradξ02 ωp2ξ0cos(θ/2) 4 (1+2ε2 ξ2γ4)2+ εradξ02γ0ω0r 2,
frs − 2ω0r ± √8γ02ω0r s rad 0 0 (cid:18) ωp′ (cid:19)
ε ξ2γ (ω /ω′)
tanθ = − rad 0 0 0r p . (14)
(1+2ε2 ξ2γ4)
rad 0 0 !
WithouttheRRforceε = 0,onerecovers therelativistic growthrateoftheFRSinstability asderived
rad
before [57, 67]. Fig.3 shows the growth rate ratio of the FRS with (Γ δω ) and without (Γ ) RR
frs 0 0
−
force. It is evident that the RR force strongly enhances the growth rate of the FRS at lower plasma
′
densities ω /ω 1 and higher laser amplitude ξ 1, which is also apparent from Eq.(14). The
p 0r ≪ 0 ≫
strong growth enhancement due to the RR force is counterintuitive as the later is generally considered
as a damping force similar to collisions in plasmas. One can attribute this enhancement in the growth
rate of the FRS due to the mixing between the Stokes and the anti-Stokes modes mediated by the RR
force. Without the RR force, nonlinear currents driving the Stokes and the anti-Stokes modes have
opposite polarizations. Since the phase shift induced by the RR force is polarization dependent, it is
opposite for the Stokes and the anti-Stokes modes. The opposite phase shifts, consequently, lead to the
interaction between the nonlinear current terms and phase shifts accumulation in Eq.(13). This phase
shift accumulation is termed as the manifestation of the nonlinear mixing of the two modes, and it is
responsible fortheenhanced growthrateoftheFRSinstability. Intuitivelythisgrowthenhancement can
beimaginedtooccurduetotheavailability ofanadditional channelofthelaserenergydecayduetothe
RRforceinduced dampinganditssubsequent utilization byboththeStokesandtheanti-Stokes modes.
Since,theresonant excitation ofboththeStokesandtheanti-Stokes modesistheessential condition
for the growth enhancement of the FRS, let us estimate the conditions under which both the modes
are excited. Resonant excitation of the Stokes modes (D = 0) is always possible due to kinematical
−
considerations. However, the simultaneous resonant excitation of both the modes is only possible in a
′
tenuous plasma (ω ω ). The resonant excitation of the Stokes mode leads to frequency mismatch
p ≪ 0r
for the anti-Stokes mode defined as ∆ω = ω′ + ω [ω′2 + (k′ + k )2 + D ]1/2, and it reads
m p 0r − p p 0 +
as ∆ω = ω′3/ω2 +9ω′4/4ω3 . As shown in Ref. [30], this frequency mismatch is smaller than
m − p 0r p 0r
the actual growth rate Γ δω of the FRS instability. This makes the inclusion of both the modes
frs 0
−
importantwhilederivingthegrowthrateoftheFRS.TheRRforceonlymarginallyenhancesthegrowth
rate of the FRS, if only the Stokes mode is resonantly excited in the plasma. This can be understood
easilyasthenonlinear mixingofthetwoRamansidebands isabsentinthiscase. Thephaseshiftcaused
bythe RR force maintains the laser energy transfer to the Stokes mode for alonger timecausing minor
enhancement inthegrowthrateoftheFRS.