Excitation of lower hybrid and magneto-sonic perturbations in laser plasmainteraction

Ayushi Vashistha1,2,∗ Devshree Mandal1,2, and Amita Das3†1 Institute for Plasma Research, HBNI, Bhat, Gandhinagar - 382428, India

2 Homi Bhabha National Institute, Mumbai, 400094 and3 Physics Department, Indian Institute of Technology Delhi, Hauz Khas, New Delhi - 110016, India

Lower hybrid (LH) and magneto-sonic (MS) waves are well known modes of magnetized plasma.These modes play important roles in many phenomena. The lower hybrid wave is often employed inmagnetic confinement fusion experiments for current drive and heating purposes. Both LH and MSwaves are observed in various astrophysical and space plasma observations. These waves involve ionmotion and have not therefore been considered in high power pulsed laser experiments. This papershows, with the help of Particle - In - Cell (PIC) simulations, a simple mechanism for excitation oflower hybrid and magnetosonic excitations in the context of laser plasma interaction. A detailedstudy characterising the formation and propagation of these modes have been provided. The schemefor generating these perturbations relies on the application of a strong magnetic field in the plasmato constrain the motion of lighter electron species in the laser electric field. The magnetic fieldstrength is chosen so as to leave the heavier ions un-magnetized at the laser frequency. This helpsin the excitation of the LH waves. On the other hand at the slower time scale associated with thelaser pulse duration, even the ions show a magnetized response and magnetosonic excitations areobserved to get excited.

I. INTRODUCTION

The magnetized plasma modes, like lower hybrid (LH)and other modes (e.g. electron and ion cyclotron modes)have found an important place in the context of mag-netic confinement fusion studies. They are traditionallyemployed for the purpose of current drive and heatingof fusion plasma in magnetic confinement devices. Lowerhybrid waves have been extensively studied in a variety ofcontexts in laboratory and space plasmas [1–4]. The LHwaves can impart their energy to plasma species throughvarious mechanisms like electron heating via breaking oflower hybrid wave [3], providing energy to electrons viawave-particle interaction. An application of the latter isfound in toroidal tokamak devices where landau dampingof externally launched lower hybrid wave helps in drivingplasma current required for plasma confinement [5, 6].Furthermore, it is very common to observe lower hybridwaves and several interesting phenomena due to theirbreaking or turbulence in space plasmas [7, 8]. Localisedlower hybrid turbulence is observed in density depletionregions in the early phases of an intense magnetic storm[9]. Many studies in the auroral region report observationof lower hybrid emissions, giving rise to solitary struc-tures and/or ion-heating [7, 10]. The LH mode is uniquein the sense that the dynamical response of both electronand ions are relevant for this mode. Basically, the magne-tized electron response couples with the un-magnetizedion dynamics for this particular mode to get excited.

The linear modes for magnetized plasma have, how-ever, not been explored in the context of laser plasma

∗ayushivashistha@gmail.com†amitadas3@yahoo.com

interaction studies. With recent technological progress,quasi-static magnetic field of the order of 1.2 kilo Tesla[11] can be produced in laboratory. With rapid advance-ments in technology it is quite likely that the regime ofmagnetized plasma response at laser frequencies can bewithin reach of laboratory experiments. In the contextof CO2 lasers the magnetic field requirement to observemagnetized electron response (i.e. ωce > ωl) turns outto be around 1.2 kilo Tesla. In view of this our grouphas been engaged in investigating this particular regimewith the help of particle - in - cell (PIC) simulations. Re-cently, we illustrated a possible new mechanism of direction heating by laser pulse [12]. For relativistically intenselasers we had also demonstrated the formation of mag-netosonic solitons [13]. In this paper we show the directcoupling of the laser energy to electrostatic lower hybridfluctuations in plasma. In addition the magnetosonic per-turbations are also observed in simulation. A detailedparametric study for the frequency regime of the excita-tions of these modes has been carried out. This paper hasbeen organized as follows. In Sec.II, we have provided thedetails of the simulation configuration. Sec.III containsour numerical observations and analysis. Finally, Sec.IVprovides the summary of the work.

II. SIMULATION DETAILS

We have carried out two dimensional PIC simulationsusing OSIRIS-4.0 [14–16]. A schematic of the simulationgeometry has been shown in Fig.1. A rectangular boxin x-y plane of dimensions Lx = 3000c/ωpe and Ly =100c/ωpe has been chosen for simulation. Here ωpe =√

4πn0e2/me is the plasma frequency corresponding tothe plasma density n0. The left side of the box uptox = 500c/ωpe is vacuum. Thereafter, a uniform density

arX

iv:2

008.

1028

6v1

[ph

ysic

s.pl

asm

-ph]

24

Aug

202

0

2

TABLE I: Values of simulation parameters in normalised andcorresponding standard units for mi=25

Parameter Normalised Value Value in standardunit

Plasma Parameters

no 1 3 × 1020 cm−3

ωpe 1 1015Hz

ωpi 0.2 (for M/m =25)

0.2 × 1015Hz

Laser Parameters

ωl 0.2ω−1pe 0.2 × 1015Hz

λl 31.4c/ωpe 9.42µm

Intensity a0 = 0.5 3.5 × 1015W/cm2

External Magnetic Field Parameters

Bz 2.5 14.14KT

plasma has been placed. A p-polarized plane laser pulseis incident normally from the left side. We have chosenthe parameters associated with CO2 laser pulse having awavelength of 10µm for our studies. This choice reducesthe requirement on the applied external magnetic field bytypically 10 times compared to the conventional laserswith a wavelength close to 1µm. This is because theexternal magnetic field has been chosen so as to have thelighter electron species magnetized and the ions to remainun-magnetized at the laser frequency. This translatesinto the requirement of ωce > ωl > ωci where ωl is thelaser frequency. To satisfy this condition, we have chosenexternal magnetic field such that ωce = 2.5ωpe = 12.5ωl.The motion of both electrons and ions are tracked in thesimulation. A reduced ion to electron mass ratio of 25 hasbeen considered. In some cases the mass ratio of 100 hasalso been chosen, which has been explicitly mentionedwhile discussing those results. The small mass ratio hasbeen chosen to expedite the simulations. The boundarycondition along y-axis for particles as well as fields hasbeen taken to be periodic. However, along x-axis theabsorbing boundary condition has been implemented.

In Table I we list out the main simulation parametersfor a quick reference. We have also varied some parame-ters like magnetic field and the laser frequency in a cou-ple of our simulation runs for exploring the conditionsof the excitation of these modes. These specific choiceshave been specified explicitly where the results concern-ing them are discussed.

III. OBSERVATIONS AND ANALYSIS

From the laser and plasma parameters of Table-I, thelaser frequency is ωl = 0.2ωpe. The plasma is overdensefor this laser frequency. The laser propagates along xaxis and is incident normally on the plasma target. It islinearly polarised with electric field of the laser pointingalong y. As expected, in the absence of any applied ex-

ternal magnetic field, the simulation shows that the laserpulse gets reflected from the plasma target and no distur-bance gets excited in the plasma medium. We then car-ried out simulations in the presence of an external mag-netic field pointing along z direction. The magnetic fieldwas chosen to satisfy the condition ωce > ωl > ωci. Thisensures that at the laser oscillation time period, the elec-trons would exhibit a magnetized response and the ionson the contrary will remain unmagnetized. The resultsof these simulations have been presented in this section.We observe that with the addition of external magneticfield, a part of laser energy gets absorbed by the plasmamedium, despite it being overdense. This is clearly evi-dent from Fig. 2 where the excited disturbances in theplasma medium have been shown at various times. Theplasma surface starts from x = 500. The color plot showsthe charge density in 2-D. The blue and green lines de-note the y and x component of electric field (Ey and Ex)respectively. At t = 0 the plasma is undisturbed and nocharge density fluctuations can be seen in the medium.The x component of the electric field (Ex) is also zeroinitially. One observes only the y component of electricfield (Ey) associated with the laser pulse in the vacuumregion at t = 0. As the laser hits the plasma surface, theplasma medium gets disturbed by it. The effect can beseen in the subplots of Fig.2 where the plots are shownat times t = 500 and t = 1000. The charge densitydisturbances are evident in the zoomed plots at thesetimes. Presence of both components of electric fieldsEx and Ey in the plasma medium can be seen. Withtime, these disturbances propagate inwards, towards thedeeper region of the plasma medium (comparison of plotsat t = 500 and t = 1000 illustrates this). These dis-turbances are not random fluctuations but appear to bequite regular. In Fig. 3(a), we have plotted the evolution

of I1 =∫| ∇ · ~E | dxdy and I2 =

∫| ∇ × ~E | dxdy (inte-

grated over the bulk region of the plasma) to determinethe electrostatic/electromagnetic character of these dis-turbances. It can be observed that though both I1 andI2 start with zero initially, they increase with time. How-ever, I1 increases much more rapidly and clearly domi-nates over I2. This suggests that the fluctuations have adominating electrostatic character. This is further borneout from Fig. 3(b) where we show directional distribu-

tion of ~E in the plasma region. It demonstrates that the

electric field ~E is dominantly along x. We have also eval-uated the spatial FFT of Ex in bulk plasma region att = 1000 (when the laser has been reflected back fromthe system) and observe that the spectrum peaks at aparticular value of kx = 0.73 (fig. 4). The scale length ofthe fluctuations appear to be longer in the deeper regionof the target as can be observed in Fig.2.

At a significantly later time (shown at t = 4000 andt = 6000 in Fig. 5) formation of another structure quitedistinct from the oscillations discussed so far can be seenclearly in the plot. This structure is significantly longercompared to the oscillations observed at earlier times(Fig. 5). We now make an attempt at understanding

3

the excitation and detail characterization of these twokind of structures in the subsections below.

A. Identification of short scale fluctuation as lowerhybrid mode

We now show that the short scale electrostatic distur-bance generated in the plasma is essentially the lowerhybrid mode. We also illustrate the physical mechanismand the condition that need to be satisfied for such exci-tations.

It should be noted that in the presence external mag-netic field and the oscillating electric field of the laser,

both ions and electrons would experience ~E × ~B drift

along x. Here ~E is the electric field of the laser pulse

along y and ~B is the applied external magnetic field.Since the electric field of the laser is oscillating in timethe magnitude of this drift is dependent on the chargespecies and takes the following form [17]

~Vs, ~E× ~B(t) =ω2cs

ω2cs − ω2

l

~E(t)× ~B

B2(1)

Here, the suffix s = e, i stand for electron and ion speciesrespectively. Thus, ωce and ωci denote the electron andion cyclotron frequency respectively and ωl is the laserfrequency. We have maintained the condition of ωce >ωl > ωci in all our simulation studies here. The differencein this drift speed between the two species

gives rise to a current in the plasma medium, viz.,~J = en0(~Vi, ~E× ~B(t) − ~Ve, ~E× ~B(t)). This current is di-

rected along x. The spatial variation of laser electric field

along x at laser wavelength makes the ~∇ · ~J(= ∂Jx/∂x)finite. The finite divergence of the current leads to spacecharge fluctuation from the continuity equation. Thischarge density fluctuation is responsible for the creationof the electrostatic field Ex. Once Ex sets up the twocharge species also respond to the force (qEx) along thex. However, the electrons being magnetized, they do notaccelerate freely by Ex. On the other hand ions beingun-magnetized get directly accelerated by this field. It isobserved in the simulations that dominant x componentof the velocity for electron is provided by the expression

of ~E× ~B drift given in Eq.(1) in the laser electric field, forthe ions on the other had the velocity gained by directacceleration through Ex dominates. The plot in Fig.6shows the spatial variation of the x component of velocityof the two species obtained from simulations. The twospecies move together, however, the amplitude of theirvelocity differs, with the ions showing higher amplitude.

The frequency spectrum of the observed oscillationhas been shown in the subplot alongside. The spectrumpeaks at the frequency of ω = 0.176 which differs fromthe laser frequency of 0.2. It matches closely with thelower hybrid frequency of 0.186 evaluated from the ana-

lytical expression of

ωLH =

(1

ω2pi

+1

ωceωci

)−1/2(2)

It is thus clear that the laser excites an electrostatic modein the medium in which both electrons and ions havesignificant roles to play.

B. Necessary condition for the generation of LHmode

We have shown that the generation of the electrostatic

field is linked to the difference in the ~E × ~B field of thetwo species in the presence of external magnetic fieldand the oscillating transverse electric field of the laser.Clearly, this requires that the laser field should penetratethe plasma. It should be noted that we have chosen theplasma medium to be overdense for the chosen laser fre-quency in our simulations here. Thus the laser field inthe normal course would not have penetrated the plasmaregion.

However, the laser radiation is able to penetrate theplasma despite having frequency lower than the electronplasma frequency due to the presence of applied externalmagnetic field. The oscillatory electric field of the elec-tromagnetic field is directed orthogonal to the appliedmagnetic field. This suggests that the X mode is therelevant mode for our simulation geometry. The disper-sion curve of the X mode has the characteristics shownin Fig.7 (adapted from Boyd and Sanderson [18]). Thefigure shows the presence of a stop band between ωLH

and ωL indicated by the shaded region, which has beendenoted as Region II in the figure. Region I in frequencyband ranges from 0 to ωLH and is the pass band, so isRegion III from ωL to ωUH for the incoming electromag-netic wave. Here, ωLH is the lower hybrid frequency and

has been defined in equation 2 , ωUH =√ω2pe + ω2

ce is

the upper hybrid oscillation frequency.The frequencies ωL and ωR shown in Fig. 7 corre-

sponds to the left and right hand cut off defined by thefollowing expressions

ωL = [ω2pe +ω2

pi + (ωci +ωce)2/4]1/2− (ωci−ωce)/2 (3)

ωR = [ω2pe +ω2

pi + (ωci +ωce)2/4]1/2 + (ωci−ωce)/2 (4)

On the basis of this distinction between the three re-gions we now identify the necessary condition for theexcitation of the lower hybrid mode. We choose threedifferent values of the laser frequency corresponding tothese three regions. To have a better distinction betweenthe three regions for these simulation runs, we have cho-sen the ratio of ion to electron mass to be 100. Thishelps in having the three frequencies ωl, ωce and ωci tobe well separated. Since, the physics that needs to be

4

addressed can be explored in a simple 1-D geometry wehave chosen the same for these set of runs. We have al-ready, in all our earlier studies, shown that 2-D effectsdo not alter any physics associated with the main themeof the investigation [12]. The simulation box in 1-D ge-ometry is chosen to be larger with Lx = 4000c/ωpe andthe simulation duration was also increased. We choseto give simulation runs for three different values of laserfrequency corresponding to the three different regions inthe X mode dispersion curve (marked as region I, IIand III in fig. 7). The three values of frequencies areωl1 = 0.08ωpe, ωl2 = 0.16ωpe and ωl3 = 0.5ωpe fallingin region I, II and III respectively. For our given simu-lation parameters ωL, ωLH and ωpi are 0.376, 0.093 and0.1 respectively. The stop-band for propagation of theX mode electromagnetic wave in plasma lies between ωL

and ωLH . Thus we expect that while the laser wouldpropagate inside plasma for runs with frequency ωl1(passband) and ωl3(pass band), there should be total reflectionof the laser for frequency lying in ωl2(stop band). This isindeed what we observe in the simulations. Since thereis no propagation of the laser field at ωl2 the laser doesnot interact with plasma and it is not possible to excitelower hybrid oscillations in this particular case. On theother hand, the other two frequencies, ωl1 and ωl3 lie inthe pass-band and hence are expected to interact withthe plasma.

We, however, observe that the laser energy gets cou-pled into plasma only when the laser frequency lies inregion I, where the electrostatic lower hybrid wave getexcited by the mechanism discussed above. This is be-cause the lower hybrid frequency is nearby and thereforegets excited. In region III where the laser frequency ismuch higher than the lower hybrid frequency the cou-pling of laser energy is found to be very weak with theplasma. The laser simply gets transmitted in this partic-ular case. This can be clearly seen from the evolution ofkinetic and field energies shown in Fig. 8 for frequencieslying in the three regions.

It thus appears that to excite the lower hybrid wave weneed to choose the frequency of the driving laser pulse tobe in region I, i.e. lower than the lower hybrid resonanceto satisfy the pass band criteria. Furthermore, it shouldalso not be in Region III which satisfies the pass bandcriteria but the laser frequency is much higher than thelower hybrid wave frequency to couple with it.

C. Identification of long scale disturbances asMagnetosonic excitation

We now discuss the other long scale disturbances thatget generated in the plasma by the laser, which becomeevident very clearly at a later time (See Fig. 5). Thesedisturbances have electromagnetic character as the per-turbations in the z component of magnetic field has alsobeen observed. It can be observed from Fig. 9 that whileit is observed in the perturbed magnetic field Bz (the

TABLE II: Frequencies and their respective normalised valuesfor mi = 100 described in for section C of results

.

Frequency Normalisedvalue

Observation

ωl1 (Re-gion I)

0.08 absorption via ex-citation of LH

ωl2 (Re-gion II)

0.16 stop-band, hence,no absorption

ωl3 (Re-gion III)

0.5 pass band butno coupling oflaser energy intoplasma

applied field has been subtracted) and Ey, no formationof such a structure is observed in Ex.

Furthermore, the ion and electron density perturba-tions both show this structure. These structures appearas a result of the ponderomotive force that acts on theplasma due to the finite longitudinal laser pulse width.The envelope of the laser pulse pushes the surface ofthe plasma target, creating a magnetosonic perturbationwhich propagates inside the plasma. The frequency as-sociated with the envelope of the laser pulse is slow sothat both ions and electrons display a magnetized re-sponse and hence it excites magnetosonic perturbation.The single hump of the disturbance testifies that it hasbeen excited by the laser envelope. We carried out simu-lations with different pulse width of the laser and observethat the width of these structures scale linearly with thelaser pulse duration as can be seen from Fig. 10. Wealso made two successive laser pulse of different durationto fall on the plasma target. In this case as expectedtwo distinct long scale structures get formed indicatingclearly that the temporal profile of the laser pulse is re-sponsible for this.

It is also interesting to note that since these structuresare essentially excited by the envelope of the laser pulse,they are independent of the laser frequency. Thus evenwhen the laser frequency lies in the stop band of RegionII, we observe the formation of these structures. Thishas been shown in Fig. 5 where we compare the for-mation of these long scale structures for both the casesof laser frequencies lying in region I and region II andhaving the same pulse width. It should be noted that inthe former case the structures form along with the shortscale lower hybrid (LH) excitations. However, when thelaser frequency lies in region II, the laser is unable topenetrate the plasma to excite the short scale LH fluc-tuations are absent but the long scale structure contin-ues to be present. It is yet another demonstration ofthe fact that the ponderomotive force due to the finitelaser pulse induces the formation of this long scale struc-tures. The velocity of this long scale structure is foundto be 0.24c which matches closely with the Alfven speed(vA = (me/mi)

1/2 × ωce/ωpe) of the medium, which is0.25c for our choice of parameters.

5

At higher intensity of the laser field the pondero-motive force is higher and increases the amplitude ofthese magnetosonic perturbations driving them in non-linear regime. Such a disturbance then forms magne-tosonic solitons. The properties of this structure thenmatches with KdV magnetosonic soliton as has been re-ported earlier [13]. This study can help in estimating theelectromagnetic wave frequency in astrophysical obser-vations where solitary structures and/or ion heating areobserved.

IV. CONCLUSION

To summarize, we have shown with the help of PICsimulations, the possibility of exciting electrostatic lowerhybrid perturbations in plasma with the help of a laser inthe presence of an external magnetic field. The plasmaprofile was chosen to be sharp and the external magneticfield was chosen to be normal to both the laser propa-gation direction and the oscillatory electric field. It wasshown that the necessary condition for the LH excita-tion is that the laser frequency should lie in the lowerpass band of the X mode. For this case the electromag-netic field of the laser gets partially transmitted insidethe plasma and then drives the lower hybrid oscillationsby an interesting mechanism arising due to the differ-

ence between the ~E × ~B drift of the two species of ionand electrons in the oscillatory laser electric field and theapplied external magnetic field. We have also demon-strated that the finite extent of the laser pulse exciteslong scale magnetosonic perturbations in the medium.These excitations are independent on the laser frequencyand depend merely on pulse width as they get driven by

the ponderomotive pressure of the laser pulse.

Recently, there has been a lot of technological progressleading to the development of low frequency short pulselasers such as CO2 lasers and also the magnetic fields ofthe order of 1.2 kilo Tesla have already been produced inthe laboratory. This would, therefore, soon open up thepossibility of investigating magnetized plasma responsein laser plasma related experiments, a regime we haveconsidered in this paper. The importance of LH modeis well known used in the context of magnetic confine-ment fusion studies where it is used for the purpose ofcurrent drive and heating. Here, we have shown the pos-sibility of exciting this particular mode in the contextof laser plasma interaction. The laser experiments withmagnetized response may thus have important implica-tions for several frontier experiments which rely on laserplasma interactions. For instance, it opens up a possibil-ity of designing smarter fusion experiments which coulduse the best of both inertial and magnetic confinementprinciples.

Acknowledgements

The authors would like to acknowledge the OSIRISConsortium, consisting of UCLA ans IST(Lisbon, Por-tugal) for providing access to the OSIRIS4.0 frameworkwhich is the work supported by NSF ACI-1339893. ADwould like to acknowledge her J. C. Bose fellowshipgrant JCB/2017/000055 and the CRG/2018/000624grant of DST for the work. The simulations for the workdescribed in this paper were performed on Uday, an IPRLinux cluster. AV and DM would like to thank Mr.Omstavan Samant for constructive discussions and ideas.

References:

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7

Laser

Plasma

Lower hybrid

(Laser Frequency < Lower hybrid frequency)

Vacuum

0 500 3000

100

X

Y

K

E

laser

laser

Blaser

B0 ,

FIG. 1: Schematic showing laser energy being coupled into plasma via excitation of lower hybrid oscillations in the system.Our simulation geometry is in X-Y plane with a planar laser being incident on plasma along +x. An external magnetic fieldhas been applied in X-mode configuration for the incoming laser. The external magentic field is high enough to magnetise theelectrons, keeping ions unmagnetised. For the sake of simplicity, the transverse extent of laser has been kept infinite.

t=500

0 500 1000 1500x(c/ω

pe)

0

20

40

60

80

100

y(c

/ωp

e)

-0.1

-0.05

0

0.05

0.1

Ex

, E

y(m

c ω

pe/e

)

-0.01 0 0.01 0.02 0.03 0.04 0.05

Ex

Ey

t=1000

0 500 1000 1500x(c/ω

pe)

0

20

40

60

80

100

y(c

/ωp

e)

-0.05

0

0.05

Ex, E

y(m

cω

pe/e

)

-0.01 0 0.01 0.02 0.03 0.04 0.05

Ex

Ey

t=0

0 500 1000 1500

x(c/ωpe

)

0

20

40

60

80

100

y(c

/ ω

pe)

-0.1

-0.05

0

0.05

0.1

Ex

, E

y(m

c ω

pe/e

)Ex

Ey

-0.01 0 0.01 0.02 0.03 0.04 0.05

t=500

490 500 510 520 530 540 550 560

x(c/ωpe

)

0

20

40

60

80

100

y(c

/ωpe)

-0.1

-0.05

0

0.05

0.1

EX,

Ey

(mcω

pe/e

)

-0.01 0 0.01 0.02 0.03 0.04 0.05

t=1000

500 550 600 650x(c/ω

pe)

0

20

40

60

80

100

y(c/ω

pe)

-0.05

0

0.05

Ex,

Ey(m

cωp

e/e

)

-0.01 0 0.01 0.02 0.03 0.04 0.05

FIG. 2: Plot of electric field components (Ex and Ey) superposed over 2-D color plot of charge density. At t = 0, only Ey

is present in the system which is due to laser. As laser interacts with plasma (t = 500), Ex is generated in bulk plasma andspatial variation of Ex is found to be in accordance with that of charge density (zoomed plot of t = 500 and t = 1000). Thissuggests that density perturbations along x have generated Ex in bulk plasma.

8

time( ωpe

-1)

0 200 400 600 800 1000

|∇.E

| , |∇

× E

|

0

10

20

30

40

50

(a)|∇ × E|

|∇.E|

0.01

0.02

0.03

0.04

30

210

60

240

90

270

120

300

150

330

180 0

(b) Resultant Electric field

FIG. 3: (a) Temporal Variation of electrostatic and electromagnetic component of electric field in bulk plasma exhibiting thatelectrostatic part dominates in bulk plasma. (b) Direction of resultant electric field in bulk plasma at t = 1000 which is pointingdominantly along x.

0 1 2 3 4 5

kx

0.5

1

FF

T(E

x)

t=1000X: 0.733

Y: 1.185

FIG. 4: FFT of Ex in bulk plasma along x after laser has been reflected back from the system.

9

Laser frequency lying in Region I

x(c/ ωpe

)1000 1500 2000 2500

Bz(m

cω

pe/e

)

×10-3

-1

-0.5

0

0.5

1t=4000

x(c/ ωpe

)

1000 1500 2000 2500

Bz(m

cω

pe/e

)

×10-3

-1

-0.5

0

0.5

1t=6000

Laser frequency lying in Region II

1000 1500 2000 2500

Bz(m

cω

pe/e

)

×10-3

-5

0

5 t=4000

1000 1500 2000 2500

Bz(m

cω

pe/e

)

×10-3

-5

0

5 t=6000

FIG. 5: Spatial variation of Bz at different times. We observe ion heating as well as a structure ahead of the oscillations forlaser frequency lying in Region I of frequency band, on the other hand, only a structure for laser frequency lying in Region II.

x(c/ ωpe

)

500 600 700

Vx

-0.01

0

0.01

ions

electrons

(a)

ω(ωpe

)

0 0.2 0.4 0.6 0.8 1

FF

T(V

x)

×10-3

0

0.5

1

1.5

2ions

electrons(b)

FIG. 6: (a) Spatial variation of Vx for both the charged species showing that the magnitude of Vx for ions is more than thatfor electrons but their motion is in correspondance with each other. (b) FFT of Vx of both the charged species with time againconfirms that the frequency of their motion is same i.e they move together, indicating presence of a hybrid mode in the system.

10

Region I

2.85

2.69

Region III

0 38

upper hybrid

lower hybrid

k

FIG. 7: Dispersion curve for X-mode indicating different stop band and pass band frequencies (Fig. courtesy [18]). Value ofdifferent frequencies (normalised to ωpe) corresponding to mi = 100 and Bz = 2.5 is shown at right in blue.

time( ωpe

-1)

0 500 1000 1500 2000 2500En

erg

y D

en

sity (

m2 c

2 ω

pe

2 e

-2) ×10

-5

0

2

4

6

Region I

(Kinetic Energy)elec

(Kinetic Energy)ion

Total Energy

EMF Energy

time( ωpe

-1)

0 500 1000 1500 2000 2500

En

erg

y D

en

sity

×10-5

0

2

4

6

Region II

(KinE)elec

(KinE)ion

Total E

EMF E

time( ωpe

-1)

0 500 1000 1500 2000 2500

En

erg

y D

en

sity

×10-5

0

2

4

6

Region III

(KinE)elec

(KinE)ion

Total E

EMF E

FIG. 8: Absorption of laser energy into plasma when laser frequency falls into different regions of frequency spectrum. It canbe observed that laser energy is coupled into plasma only in Region I of laser frequency. Laser frequency in Region II falls instop-band and hence, is not able to interact with plasma. On the other hand, laser frequency in Region III lies in pass-bandand hence could interact with plasma but could not impart its energy into plasma species. Plasma species acquire energy onlytill the time laser is present in this case, their energy goes off after that.

11

Laser frequency lying in Region II

x(c/ ωpe

)

1000 1500 2000 2500

Bz (

mc ω

pe /e)

×10-3

-5

0

5t=1200

t=2400

t=3600

t=4800

t=6000

x(c/ ωpe

)

1000 1500 2000 2500

Ion

De

nsity

0.999

1

1.001

t=1200

t=2400

t=3600

t=4800

t=6000

x(c/ ωpe

)

1000 1500 2000 2500

Ele

ctr

on

De

nsity

0.999

1

1.001

t=1200

t=2400

t=3600

t=4800

t=6000

FIG. 9: Spatial variation Bz , ion density and electron density of the strucutre has been plotted at different times. Laserfrequency lies in Region II for this case. The perturbation shows similar structure in Bz as well as density of the chargedspecies. Also, the structure is maintaining shape and moving with a velocity close to the Alfven velocity of the medium for ourchoice of simulation parameters.

12

Laser Pulse Width (c/ ωpe

)

500 1000 1500 2000 2500 3000 3500

Str

uctu

re w

idth

(c/

ωp

e)

100

200

300

400

500

600

700

FIG. 10: Formation of solitary peak for three different laser pulse widths lying in Region II of frequency band, keeping otherparameters same. We observe that the width of the structure increases upon increasing pulse width.

0 2000 4000 6000 8000

x(c/ ωpe

)

-0.1

0

0.1

Bz (

mcω

pe/e

)

t=0

4000 4500 5000 5500

x(c/ ωpe

)

-5

0

5

Bz(m

c ω

pe/e

)

×10-3 t=4000

4500 5000 5500 6000

x(c/ ωpe

)

-5

0

5

Bz(m

c ω

pe /

e)

×10-3 t=6000

FIG. 11: Spatial variation of Bz at different times. At t=0, we show two laser pulses of different widths propagating along x.The red dotted line shows plasma vaccuum interface. At subsequent times, we observe formation of two structures of differentwidths as a result of interaction of the laser pulses with plasma. Note that both the structures vary in their width dependingon the initial pulse width of the laser. Both these structures move close to Alfven velocity of the medium (0.25c for our chosenparameters).