The breather like penetration of the ultra-short linearly polarized laser into over-dense plasmas

0.98 MB·English

by  Dong Wu

The breather like penetration of the ultra-short linearly polarized laser into
over-dense plasmas
Dong Wu,1,2,a) C. Y. Zheng,1,2,3,b) and X. T. He1,2,3,c)
1)Center for Applied Physics and Technology,
Peking University, Beijing, 100871, China.
2)Key Laboratory of High Energy Density Physics Simulation,
Ministry of Education, Peking University, Beijing, 100871, China.
3)Institute of Applied Physics and Computational Mathematics,
Beijing, 100088, China.
3
1 (Dated: 11 January 2013)
0
Relativistic electromagnetic penetration behavior is reexamined in the relativistic transparency region. The
2
interaction is modeled by the relativistic hydrodynamic equations coupled with the full system of Maxwell

n equations,whicharesolvedbyafullyimplicitenergy-conservingnumericalscheme. Forthefirsttime,wehave
a
studied the penetration behavior through ultra-short circularly polarized and linearly polarized laser pulses
J
interactionwithover-denseplasmas. Itisshownthatfortheultra-shortcircularlypolarizedlaserpenetration
0
occurs through a soliton like behavior, which is quite consistent with the existing studies. However, we have
1
found that the ultra-short linearly polarized laser penetrates through a breather like behavior, and there is

] an energy transition mechanism with a frequency 2ω0 between the penetrated breather like structure and
h the background plasmas. A qualitative interpretation has been given to describe this mechanism of energy
p
exchange.
-
m
PACS numbers: 52.38.Kd,41.75.Jv, 52.35.Mw, 52.59.-f
s
a
l
p The dynamics of the penetration of ultra-intense laser should be emphasized that their is no theoretical analy-
.
s irradiation into classically-forbidden plasmas has at- sis on the penetration into over-dense plasmas with lin-
c tracted wide attention for its distinct prospective appli- early polarized (LP) laser, because the second-harmonic
i
s cations. The penetration of intense laser radiation into oscillating component of the ponderomotive force could
y
over-dense plasma core would be crucial in the concept disturb the background plasma density and make the
h
of the fast ignition fusion1, plasma lens for ultra-intense problem even more complicated. The difficulty of the
p
laser focusing2and betatron oscillations for high intense fluid-MaxwellsimulationsinstudyingLPsolitonsisthat
[
electron acceleration3,4. Kaw et al. are the first to pre- the explicitnumericalscheme willbreakdownin dealing

2 dict that relativistically strong laser can reduce the ef- with density ripples caused by second-harmonic oscillat-
v fectiveplasmafrequencybelowthefrequencyofthelaser ing ponderomotive force25. However, the crucial issue,
4
radiation, allowing the laser to penetrate through the which is of fundamental influence to the fluid-Maxwell
2
over-dense plasmas5. This problem has been greatly en- simulations, is the vacuum heating and J × B heat-
3
0 richedbya seriesoftheoreticalandparticle-in-cellsimu- ing mechanisms on the vacuum-plasma interface. These
. lationstudies6–21. Tushentsovetal. arethefirsttostudy heatings are totally kinetic mechanisms which can not
2
the irradiation penetration of the circularly polarized be described by a fluid treatment. Whereas in our work,
1
2 (CP) laser pulse through fluid-Maxwell simulations23, these straitproblemsaresolvedin twoways. Afully im-
1 but there is a serious shortcoming: the Maxwell equa- plicit energy-conserving numerical scheme is applied to
: tions are solved within the framework of parabolic ap- well deal with the density ripples caused by the oscillat-
v
proximation, which will certainly break down for ultra- ing term. Abandoning the parabolic approximation24,25,
i
X short laser pulse. Afterward, Berezhiani et al. abandon wesolvethe fullsystemofrelativistichydrodynamicand
r the parabolic approximation and solve the full system Maxwellequations, and focus on ultra-shortlaser irradi-
a of relativistic hydrodynamic and Maxwell equations24. ation with full-width-half-maximum 5T . In such cases
0
However in their results, we have not found the intrinsic the vacuumheating andJ×B heating canbe neglected
application to the study of ultra-short laser penetration considering the ultra-short interaction time between the
into over-dense plasmas. In their fluid-Maxwell simula- laser and plasmas on vacuum-plasma interface.
tions, the laser duration and solitons formation time are
over 100T . Their simulation results just reconfirm that In this paper, the interaction between the ultra-short
0
of Tushentsov et al. with parabolic approximation. It laser irradiation and the plasmas is modeled by the rel-
ativistic hydrodynamic equations coupled with the full
systemofMaxwellequations,whicharesolvedbya fully
implicit energy-conservingnumerical scheme. The pene-
a)Electronicmail: [email protected] tration of the ultra-short CP irradiation occurs through
b)Electronicmail: zheng [email protected] a soliton like behavior which is consistent with previous
c)Electronicmail: [email protected] theoretical and fluid-Maxwell simulation studies23,24,29,
2
anditspropagationinthehomogeneousbackgroundplas- impliestheaccumulationofamacroscopicnumberofpar-
mas is quasi-static. However, it is shown that, for the ticles within the interest region, and it seems that the
ultra-short CP laser, the penetration occurs through a fluid treatment of the regionof zero plasma density may
breaker like behavior, and there is an energy transition result in unphysical outcome. When numerically inte-
mechanismwithafrequency2ω betweenthepenetrated grating the equation of motion Eqs. (5) and (6) within
0
breather like structure andthe backgroundplasmas. We thevacuum-likeregion,thelackofthebalanceforcefrom
have proposed a qualitative interpretation to describe the electrostatic field may result in the unphysical x di-
this mechanism of energy exchange. rectionspeedingoftheelectronsandionsunderthelaser
The governing equations are the relativistic hydrody- ponderomotive force. However, the plasma density is
namic fluid equations coupled with the full system of zero, and the total x direction current is zero too, which
Maxwell equations. As we are considering the relativis- doesnotcontributetothesolvingofthecoupledMaxwell
tic intense laser irradiation and the electron quiver ve- equationsandmakestheglobalfluid-Maxwellsimulation
locity under the laser electric field is much greater than applicable24.
the thermal velocity, we treat the plasmas as cold liq- To ensure the accuracy of the numerical scheme, we
uid. Below are the group of normalized full system of follow the energy of the system at each time step during
equations24,29: our simulations. The total energy can be separated into
four components:
∂2a ∂2a n n
y − y +( e +Z2 i )a =0, (1)
∂t2 ∂x2 γe mγi y ǫ = 1 dx[(∂ay)2+(∂az)2+(∂ay)2+(∂az)2](,10)
∂2a ∂2a n n l 2Z ∂t ∂t ∂x ∂x
z − z +( e +Z2 i )a =0, (2)
∂t2 ∂x2 γ mγ z 1 ∂φ
e i ǫ = dx[( )2], (11)
∂n ∂ n p p 2Z ∂x
e e e
+ ( )=0, (3)
∂t ∂x γ
e ǫ = dx[(γ −1)n ], (12)
∂n ∂ n p e Z e e
i i i
+ ( )=0, (4)
∂t ∂x γ
i ǫ = dx[m(γ −1)n ], (13)
∂p ∂γ ∂φ i Z i i
e + e − =0, (5)
∂t ∂x ∂x
whereǫ isthelaserenergy,ǫ istheenergyoftheplasma
∂p ∂γ Z ∂φ l p
i + i + =0, (6) wave, ǫe is the electron kinetic energy and ǫi is the ion
∂t ∂x m∂x
kinetic energy. In Eq. (10), we can see that the laser en-
∂2φ
=(n −n ), (7) ergy consists of four parts: y component electric energy
∂x2 e i [(∂a /∂t)2], y component magnetic energy [(∂a /∂x)2],
y z
γe =[1+a2y+a2z+p2e]1/2, (8) z component electric energy [(∂az/∂t)2] and z compo-
nent magnetic energy [(∂a /∂x)2]. It should be empha-
γ =[1+(a /m)2+(a /m)2+p2]1/2. (9) y
i y z i sizedthatthisfieldisplaguedbysubstandardsimulation
Thegroupofequationsarenormalizedintermsofdimen- methodology, which has led to bold conclusions being
sionless quantities t = ω t, x = ω x/c, a = eA /m c2, drawn from what amounts to numerical artifacts. The
0 0 y y e
a = eA /m c2, n = n /n , n = n /n , p = p /m c, energyconservingofthesimulationsconvinceusthatthe
z y e e e c i i c e e e
p =p /m c, m=m /m and φ=eφ/m c2, where ω is new physics we report is not from numerical effect.
i i i i e e 0
the laser frequency, a and a are the normalized ampli- We first study the problem of an ultra-short CP laser
y z
tudeofthelaserirradiation,n andn arethenormalized pulse penetration into over-dense plasmas. The parame-
e i
plasma density for electrons and ions, p and p are the ters of the incident CP laser pulse are: normalized am-
e i
normalizedmomentumalongthelaserpropagationdirec- plitude a = 0.71, a = 0.71, [sin(πt/2τ)2] time profile
y z
tionforelectronsandions,misthenormalizedionmass, with τ = 5.0T and laser frequency ω = 0.88. The
0 0
φ is the normalized electrostatic potential and n is the background plasma density is set to be n = 1.0, cor-
c e
critical density. responding to n = 1.3n , in which n = m ω2/4πe2.
e c c e 0
Tosolvethegroupofequations,afullyimplicitenergy- The vacuum-plasma is modeled with [sin(πx/2s)2] den-
conversing numerical scheme has been applied26, which sity profile, where s is the shape factor to describe the
can well deal with the density ripples and restrain over- steepness of the vacuum-plasma interface.
shootingofthenumericalscheme. Herewehaveassumed In Fig. 1, the space-time dynamics of the penetration
that the boundary conditions [a =0, a =0 and φ=0 process is demonstrated when there is a sharp density
y z
on both sides of the simulation box] are always satisfied slope at the plasma boundary. A relativistic electromag-
during the simulation. At t = 0.0T the laser pulse is neticsolitonisgeneratedduringthepenetrationprocess,
0
apart from plasmas, and at later time, the laser irradi- whichisinagreementwiththetheoreticalpredictionthat
ates on the plasmas. Our simulation is a global fluid- the penetration occurs through soliton-like structure29.
Maxwell simulation, which is quite different from pre- Compared with the incident pulse, the penetrated soli-
vious studies that the soliton structure is put in as an ton owns a sharply narrow width which is less than one
initial condition15,16,27,28. The exact definition of a fluid laser wavelength and the amplitude is greatly enhanced.
3
t=0T t=1T t=2T t=3T t=4T
t=5T t=6T t=7T t=8T t=9T
FIG. 1. (color online) The formation process of CP solitons: red line represents the normalized laser intensity and the black
line is for theplasma density. Here, ay =0.71, az =0.71, τ =5T0, ω0 =0.88, ne =1.0 and s=0.4λ0.
FIG.2. (coloronline)Thespace-timedistributionofthenor-
malized laser intensity: (a) is for CP laser and (b) for LP
laser. Here the simulation parameters for CP laser pulse are
thesameasthatofFig.1,andthesimulation parametersfor
LP laser pulse are ay = 1.00, az = 0.0, τ = 5T0, ω0 = 0.88,
ne =1.0 and s=0.1λ0.
FIG.3. (coloronline)Theenergyhistoryofthefluid-Maxwell
simulations: backlinerepresentsthenormalizedenergyofthe
CP laser, the red line represents the plasma energy (mainly
After the formation, the soliton propagates into the uni-
theelectronkineticenergy)drivenbyCPlaser,thegreenline
form plasmas with a constant velocity v = 0.172, which
is for energy of the LP laser and blue line is for the plasma
is shown in Fig. 2 (a). From Fig. 3, the energy history
energy by LP laser. Here the simulation parameters are the
demonstrates that during the soliton formation stage,
same as that of Fig. 2.
within the early 7.0T , there is an energy transfer from
0
the incident laser to the background plasmas. It is easy
to understand this energy transfer mechanism: the pon-
deromotive force drives the plasmas inward in the laser steep interface the penetrated CP soliton is quite weak,
propagationdirection,duringthisprocessworkhasbeen and the penetration ratio is rather low. In contrast, the
donebythelaserpulse. Howeverduringthe propagation rathersmoothinterfaceallowsastrongpenetration. Fig.
processofthe soliton,we see fromFig.3 thatthe energy 3 (e) shows the penetration ratio for different shape fac-
ofthelaserpulseandbackgroundplasmasdonotchange tor s, demonstrating the generalpenetration ratio range
anymore,whichdemonstratesthatthe CP solitonprop- 10%∼60%.
agation is quasi-static. Fig. 5 shows the spectral intensity of the initially in-
Considerable part of the incident CP laser pulse is cident CP irradiation, the reflected part and the pene-
transformed into the sharply narrow and relativistic in- trated soliton. In the soliton formation stage, the pon-
tense electromagnetic soliton. Our simulations demon- deromotiveforcepushesthe plasmasinwardinthedirec-
strate that there is an obvious relationbetween the pen- tionoftheincidentpulsepropagation. Asredlineshows,
etration ratio and the steepness of the vacuum-plasma thereflectedradiationfromthevacuum-plasmainterface
interface. As is shown in Fig. 3 (a)-(d), for a relatively is mostly red-shifted. The frequency of the penetrated
4
20% penetration (a) (a) (b)
(e)
s=0.1
28% penetration (b)
s=0.2
(c) (d)
FIG. 5. (color online) The spectral intensity of the incident
Penetrated
56% penetration 57% penetration CPlaserpulse,reflectedpartandpenetratedpart. Theblack
line is for initially incident CP laser pulse, red line is for re-
flected and green line for penetrated. (a) is the case s = 0.1
s=0.4 s=0.8
Reflected and (b) is the case s = 0.4. Here the remaining simulation
parameters are thesame as that of Fig. 1.
FIG. 4. (color online) (a)-(d) demonstrate the reflected and
penetrated CP laser pulse structure under different vacuum- boundary. We change the incident laser amplitude (the
plasmasteepness,wheretheredlineisthelaserintensityand same intensity with that of CP) to a =1.0 and a =0,
y z
blacklineistheplasmaelectron density. (e)Thepenetration while keeping other parameters the same as that of CP
ratio of theincident laser pulsevs. shape factor s, where the laser. Similar to the CP case, during the early 7.0T
0
black triangle is from thefluid-Maxwell simulation data. interaction time, the energy is transfered from the inci-
dent LP laser pulse into backgroundplasmas in one way
although there is some small scale perturbations. Fig. 2
CP soliton is blue-shifted for s=0.1 [subplot (a) in Fig.
(b)showsthatthepropagationvelocityofthepenetrated
4]and red-shifted for s=0.4 [subplot(c) in Fig.4]. The
electromagnetic structure is almost the same as that of
behaviorofthe weakpenetratedsoliton[s=0.1,subplot
CPsolitonwhichisv =0.172. However,wefindthatun-
(a) in Fig. 4] can be well described by the known finite
liketheCPsoliton,duringitspropagation,thereexistan
soliton velocity solutions, which are based on the weak
energytransitionmechanismbetweenthe LPpenetrated
density response30
structure and the background plasmas with a frequency
(1−v2−η2)1/2 ηv t about double the incident laser frequency. The energy
ay =amsech[ 1−v2 (x−vt)]×cos[1−v2(x− v)],exchange is even more clearly indicated from the energy
history map shown in Fig. 3, where the laser energy and
where
(1−v2)(1−v2−η2) the background plasma energy coupled together and ex-
a =4[ ]1/2 change with each other with a frequency 2ω .
m 4η2−(1−v2)(3+v2) 0
However, from Fig. 6 (c) and (d) for the LP penetra-
istheamplitudeoftheCPsoliton,andη isthefrequency tioncase,wecanseethatthepenetratedelectromagnetic
parameter with η = ω(1 −v2). The carrier frequency structure disappears at t = 16.00T and suddenly reap-
0
of the weak penetrated soliton can be derived from the pearsatt=16.25T ,withaperiodT /2. Att=16.00T
0 0 0
frequency-amplitude-velocity relation, almostallthepenetratedelectromagneticenergyistrans-
formed into background plasmas. While at t = 16.25T
[16(1−v2)2+a2 (1−v2)(3+v2)]1/2 0
ω = m . (14) theelectromagneticstructurehasreturnedbackfromthe
(1−v2)[4a2 +16(1−v2)]1/2
m backgroundplasmas. Itbehaveslikeablinkingbreather,
For the weak penetrated soliton [s = 0.1, subplot (a) in which is quite different from the quasi-stable soliton as
Fig. 4] with a = 0.482 and v = 0.172, Eq. (14) gives shown in Figs. 6 (a) and (b).
m
ω = 1.008 = 1.145ω , which is quite consistent with the Here,weprovideaqualitativeinterpretationfortheen-
0
one 1.142ω obtained from the fluid-Maxwell simulation ergytransitionmechanismwith2ω betweentheLPpen-
0 0
in Fig. 5 (a). etrated structure and the background plasmas. During
To study the penetration process of the LP laser, we our simulations, we follow the kinetic energy of the elec-
should be very careful. The density ripples caused by trons,thekineticenergyoftheions,thelaserenergyand
thesecond-harmonicoscillatingponderomotiveforcewill the plasma wave energy. As the electron kinetic energy
break down the simulation process, while an fully im- [ǫ ]occupiesthemainpartofthebackgroundplasmaen-
e
plicit energy-conserving numerical scheme is applied to ergy[ǫ +ǫ +ǫ ],theplasmawaveenergydoesnotdom-
i e p
well deal with this problem. In addition, in order to inate during these process,thus we candropthe Poisson
avoidorneglectthevacuum-heatingandtheJ×B effect Equation[Eq.(7)] duringour analysis. We now focus on
on the vacuum-plasma interface, the incident laser pulse theEq.(1). Therefractiontermfromthebackgroundion
should be ultra-shortto shortenthe interaction time be- motions canbe neglected due to the high ion to electron
tween the incident laser and the sharp vacuum-plasma mass ratio and we set the electron momentum along the
5
When the background plasma density n is greater
e
(a) (b) than 1.5nc, the interaction results in the generation of
plasma field structure consisting of alternating plasma
and vacuum regions with the CP laser penetrating into
the over-denseplasmasoverafinite length,whichiscon-
sistentwiththepreviousstudies23,24. Becausetheforma-
tiontimeofthealternatingplasmafieldstructureisquite
long which is about hundreds of laser period, it call for
quite long incident laser pulse. As analyzed, it beyonds
(c) (d)
the ability of fluid-Maxwell simulations in studying such
long pulse LP laser interaction with over-dense plasmas.
In summary, the relativistic electromagnetic penetra-
tion behavior is reexamined in the relativistic trans-
parency region, which is modeled by the relativistic hy-
drodynamic equations coupled with the full system of
Maxwellequations. We havestudied the penetrationbe-
haviorthroughultra-shortCP andLP laserpulses inter-
FIG. 6. (color online) The distribution of laser intensity action with over-dense plasmas. It is shown that for the
and plasma density: (a) and (b) is for CP laser pulse at
ultra-shortCPlaserpenetrationoccursthroughasoliton
t = 16.00T and t = 16.25T ; (c) and (d) for LP laser pulse
0 0 like behavior, which is quite consistent with the existing
at t = 16.00T and t = 16.25T . The black line represents
0 0 studies. Thepenetrationratiooftheultra-shortCPirra-
theplasma electron density,redlineis forlaser intensityand
diation is systematically studied with different steepness
green line for electron kinetic energy density. Here the simu-
of the vacuum-plasma interface, which indicates that a
lation parameters are the same as that of Fig. 2.
rather smooth vacuum-plasma interface allows a strong
penetration. The propagation of the CP soliton in ho-
laser propagation direction to p = 0, then Eq. (1) can mogeneousbackgroundplasmasisquasi-static. However,
e
be rewritten as we have found that the ultra-short LP laser penetrates
through a breather like behavior, and there is an energy
∂2a ∂2a n transition mechanism with a frequency 2ω between the
y − y + e a =0. (15) 0
∂t2 ∂x2 (1+a2)1/2 y penetrated breather like structure and the background
y
plasmas. A qualitative interpretation has been given to
describe this mechanism of energy exchange.
As a = a sin(ω t), the refraction term n /(1 +
y m 0 e
a2)1/2 which describes the effective plasma density un-
y
der the LP laser irradiation becomes n /[1 + a2 /2 −
e m
a2 /2cos(2ω t)]1/2. In our fluid-Maxwell simulations, ACKNOWLEDGMENTS
m 0
the laser frequency is ω = 0.88 which corresponds to
0
nc =0.76andthebackgroundplasmadensityisne =1.0 This work was supported by the National Natural
which corresponds to ne = 1.3nc. Unlike the soliton Science Foundation of China (Grant Nos. 11075025,
structure in the CP laser case where there is a density 10835003, 10905004, 11025523, and 10935002) and the
cavity for the background plasmas, in the LP laser case, Ministry ofScience andTechnologyofChina (GrantNo.
the background density is almost unchanged, ne ∼ 1.0, 2011GB105000).
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