Generation of higher harmonic by laser interacting with magnetized plasma
Srimanta Maity, Devshree Mandal, Ayushi Vashistha, Laxman Prasad Goswami, Amita Das
aa r X i v : . [ phy s i c s . p l a s m - ph ] F e b Generation of higher harmonic by laser interacting with magnetized plasma
Srimanta Maity, ∗ Devshree Mandal,
2, 3
Ayushi Vashistha,
2, 3
Laxman Prasad Goswami, and Amita Das Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India Institute for Plasma Research, HBNI, Bhat, Gandhinagar 382428, India Homi Bhabha National Institute, Mumbai, 400094, India
The nonlinearity of the plasma medium provides for an interesting prospect for High HarmonicGeneration (HHG) of electromagnetic (EM) wave. One can tune the characteristic features ofharmonics by varying the plasma as well as the EM wave properties. It is shown here that anapplication of an external magnetic field adds to the richness of the HHG generation mechanisms inthe plasma medium. The plasma dispersion properties get modified in the presence of a magneticfield and it opens various possibilities for HHG both in O-mode and X-mode configurations. Thishas been shown here with the help of Particle-In-Cell (PIC) simulations of a moderately intense laser(to avoid complications arising from relativistic nonlinearities) of sub-picoseconds duration fallingon a plasma with an applied external magnetic field. The observations have been analyzed andan understanding of the mechanism has been provided. A discussion of the appropriate parameterregime and pertinent conditions has also been provided. I. INTRODUCTION
High Harmonic Generation (HHG) of electromagneticradiation in plasma has been an important area of re-search for decades now [1]. It was first theoreticallypredicted by Margenau and Hartman [2] using time de-pendence of the various parts of the electron velocitydistribution function in the presence of an alternatingelectric field and has been studied extensively since thenfor many different plasma scenarios ranging from lab tospace [3–10]. The observation of electromagnetic radi-ation at the fundamental and higher harmonics of theplasma frequency have also been reported in space plas-mas which helps in the detection of plasma parameters[11]. The HHG observations in laboratory plasmas haveopened up a wide range of applications. It is consideredas one of the most efficient techniques known to obtainelectromagnetic wave of higher frequency in a controlledmanner [12, 13]. HHG has been used as a distant probefor the detection of turbulence in a toroidal magneticallyconfined plasma [14]. Polarization measurement of theharmonic has also been used to detect the poloidal mag-netic field profile in Tokamak devices [15]. Recently, thesecond harmonic radiation generated in an underdenseplasma has been used to experimentally verify some ofthe fundamental properties of photons including the con-servation of total angular momentum [16].Owing to the wide range of applications, HHG has beenextensively studied for magnetized as well as unmagne-tized plasmas. The high harmonics and low-frequencygeneration from an overdense unmagnetized plasma sur-face was reported in ref. [17]. It has been understoodthat in this case, the high harmonics have been gener-ated because of the oscillations of the electron density atthe vacuum plasma interface, known as oscillating mir-ror . The oscillating mirror model has also been developed ∗ Electronic address: [email protected] and used to explain the higher harmonic generation in ref.[18] for an unmagnetized overdense plasma. The secondharmonic generation in a uniform magnetized plasma hasbeen studied by Jha et al., [19], where they considered anintense linearly polarized laser pulse propagation throughplasma medium. Harmonic generation in the interactionof laser fields with a magnetized plasma having a densitybelow the critical density has been reported in ref. [20].These studies are mostly analytical involving approxi-mations and do not deal with the dynamics. We havecarried out PIC simulations here for a finite laser pulsefalling on a magnetized plasma target and study the pro-cess of evolution leading to the harmonic generation. Wehave focused here on the generation of higher harmonicsin a magnetized plasma for both O and X-mode config-urations. We show that the general perception that O -mode configuration, in general, has similar traits as thatof an unmagnetized plasma behavior does not hold asfar as the generation of harmonics is concerned. Further-more, we have also addressed in detail the following issuesin this work: (i) the frequency spectrum that has beenobserved, (ii) polarization of the higher harmonic radia-tion, and (iii) forbidden frequencies for given plasma andEM wave parameters.In our study, we have concentrated on the generationof higher harmonics via moderately intense ( a = 0 . .
35% for the second harmonic.These harmonics propagate inside the plasma and theyalso are observed to move towards the vacuum from thevacuum plasma interface.The paper has been organized as follows. Section IIdescribes the simulation setup. Section III describes re-sults and in various subsections generation and charac-terization of harmonics have been discussed. Section IVprovides a summary of this work.
II.
SIMULATION DETAILS
We have carried out one-dimensional Particle-In-CellSimulations using OSIRIS 4.0 framework [25–27] for ourstudy. Our simulation geometry has been shown in Fig.1. It has a longitudinal extent of 3000 d e with plasmaboundary starting from x = 1000 d e . Here, d e is theelectron skin depth c/ω pe and c is the speed of light invacuum. We have chosen 60000 grid points, which cor-responds to dx = 0 .
05. The number of particles percell has been chosen to be 8. Time has been normalizedby t N = ω − pe , where ω pe is the plasma frequency corre-sponding to the density n . The length is normalized by x N = c/ω pe = d e , and fields by B N = E N = mcω pe /e ,where m and e represent the mass of an electron and themagnitude of an electronic charge, respectively. The ex-ternal magnetic ( B = 2 .
5) field has been applied alongthe ˆ z direction.Table I presents laser and plasma parameters in nor-malized units and a possible set of values in the stan-dard unit. Both electron and ion( M = 100 m ) dynamicshave been considered in the simulation. A laser pulsewith intensity ≈ W/cm ( a = 0 .
5) and frequency ω l = 0 . ω pe is normally incident from the left side ofplasma. The electric field of the laser is chosen to bealong ˆ z for the O-mode configuration and along ˆ y for theX-mode configuration. In Fig. 1 the O-mode configu-ration has been depicted in a schematic representation.The longitudinal profile of laser pulse is a polynomialfunction with rise and fall time of 100 ω − pe which trans-lates to 200 f s and it starts from x = 950 d e . The value ofthe external magnetic field is chosen to elicit magnetizeresponse of electrons while ions remain unmagnetized atthe laser frequency i.e. ω ce > ω l > ω ci , where ω ci , ω ce are cyclotron frequencies of ion and electron respectivelywhile ω l is laser frequency. Absorbing boundary con-ditions are used for fields and particles. We have cho-sen the plasma to be overdense. For an unmagnetizedplasma (i.e. in the absence of an external magnetic field)the laser is unable to penetrate inside the plasma. In TABLE I: Simulation parameters: In normalized units andpossible values in standard units.Parameters Normalized Value A possible valuein standard unitLaser ParametersFrequency ( ω l ) 0 . ω pe . × HzWavelength 15 . c/ω pe . µm Intensity a = 0 . . × W/cm Plasma ParametersNumberdensity( n ) 1 3 . × cm − Electron Plasmafrequency ( ω pe ) 1 1 × HzElectron skindepth ( c/ω pe ) 1 0 . µm External FieldsMagnetic Field( B ) 2 . ≈ kT the X-mode configuration even in the overdense case, ifthe laser frequency lies in the pass band of magnetizedplasma it propagates inside and generates Lower hybridand magnetosonic excitation as has been illustrated insome of the earlier works [22–24]. However, for the O-mode configuration, the laser cannot propagate inside theplasma if it is overdense. In this case, the laser light onlypenetrates the plasma up to a skin depth. We observethat this is sufficient for the generation of harmonics inthe O-mode configuration. For the higher frequency ofharmonics that get generated the plasma could be un-derdense. In such a situation the generated harmonicradiation is free to propagate inside the plasma. III.
RESULTS AND DISCUSSION
When a laser pulse incident on an overdense plasmasurface, it will interact with the plasma particles onlywithin a skin depth. In the X-mode configuration, theEM radiation of the laser penetrates the plasma in therespective permitted pass bands even at frequencies lowerthan the plasma frequency. In this case, even the bulkplasma is able to interact with the incident radiation. Inthe O-mode, however, for the EM wave frequency smallerthan the plasma frequency the interaction is confinedonly to the skin depth layer. The plasma responds tothe EM wave electric and magnetic fields and the ap-plied external magnetic field through the Lorentz force.The spatial profile of the laser pulse is also responsible forproviding a ponderomotive force to the plasma medium.We have chosen to work in the frequency domain shownin in the table I. The condition ω ci < ω l < ω ce is cho-sen to be satisfied. Here, ω l defines the laser frequency,and ω ce and ω ci represent the electron and ion cyclotronfrequency, respectively. Thus, in the time period corre-sponding to a laser cycle, electrons would exhibit mag-netized response whereas ions remain unmagnetized. Invarious subsections below we provide the details of theobservations made by the simulations. A. HHG in O-mode configuration ( E l k B ) We first consider the case when the frequency of the in-cident laser pulse was chosen to be 0 . ω pe and the polar-ization of the laser fields were considered to be in O-modeconfiguration i.e., E l k B (in ˆ z direction). Here, E l isthe laser electric field and B is the externally appliedmagnetic field. The transverse y and z components ofthe magnetic field, B y and B z have been shown at a par-ticular instant of time t = 1000 in subplots (a) and (b) ofFig. 2, respectively. It is to be noted that at time t = 0,laser pulse with the electromagnetic fields B y and E z wasset to propagate along positive ˆ x direction from the lo-cation x = 950. Thus, the structure in B y present in thevacuum region ( x ≈ t = 1000, as can be seen fromsubplot (a), is associated with the reflected part of theincident laser propagating along − ˆ x direction. A smallfraction of B y is also present inside the bulk plasma, ashas been depicted in the zoomed scale in subplot (a1). Inthe consecutive section, we will identify this structure asthe third harmonic radiation. In the subplot (b) of Fig.2, it is seen that the z − component of the oscillating mag-netic field B z which was not present before the laser hitsthe plasma surface, has been produced at a later time andexists in both vacuum and bulk plasma. There are twotypes of disturbances that are observed inside the bulkplasma, as can be seen from the subplot (b). One is thelarge scale disturbance near the plasma surface, whichwill be identified as the magnetosonic perturbation. Andanother disturbance moving with the faster group veloc-ity. In the following discussion, we will show that thisis the second harmonic radiation generated due to theinteraction of laser pulse with the plasma particles.The FFT (Fast Fourier Transform) of these reflectedand transmitted radiations in time have been shownin Fig. 3. The FFTs of E z and B y at the location x = 500 (vacuum) show two distinct peaks at the fre-quency ω ≈ . ω pe and ω ≈ . ω pe , as can be seenfrom the subplot (a) of Fig. 3. It is important to re-call that at t = 0, the incident laser pulse was locatedin between x = 750 and 950. The first peak with higherpower at the location ω ≈ . ω pe is essentially the orig-inal laser pulse which has got reflected from the plasmasurface, as plasma is overdense. The second peak locatedat ω ≈ . ω pe is the third harmonic radiation. The thirdharmonic is also present inside the bulk plasma and hasbeen demonstrated by carrying out the FFT in time forthe E z and B y signals at the location x = 2000 (plasma),as has been shown in subplot (b) of Fig. 3. Thus, it isnow clear that the small disturbance in B y present in-side the plasma, as shown in subplot (a1) of Fig. 2 isessentially associated with the third harmonic radiation.The FFT of E y and B z in time at the locations x = 500and 2000 have been shown in subplots (c) and (d) of TABLE II: Conversion efficiencies of harmonics when S-polarized light (O-mode configuration) is incident on plasma a ω l a ω l η nd ( ref ) η nd ( trans ) η rd ( trans )0.5 0.7 0.35 0.346 0.209 0.03420.5 0.6 0.30 0.264 0.173 0.0240.5 0.5 0.25 0.193 0.128 0.0160.5 0.4 0.20 0.130 0.085 0.00940.4 0.5 0.20 0.125 0.0833 0.009 Fig. 3, respectively. The FFT at the location x = 2000(plasma), in subplot (d), has been evaluated within atime window t = 1000 to 3000. This is just to eliminatethe slowly moving magnetosonic disturbance as shownin the subplot (b) of Fig. 2. It is seen that the fre-quency spectrum of E y and B z has a distinct peak atthe location ω ≈ . ω pe in both the cases (vacuum andplasma). This ensures that the second harmonic radia-tion has been generated and travels in both vacuum andplasma medium. It is interesting to note that in eachsubplot, we have shown the FFTs of transverse electricand magnetic field components as a pair. This is done toshow the electromagnetic nature of harmonic radiations.It is also important to observe that the polarization of thethird harmonic radiation is the same as the incident laserpulse. Whereas, the polarization of the second harmonicradiation is different from that of the incident laser pulse.The reason behind this will be discussed later as whenwe analyze the origin of the higher harmonic radiationsin subsequent subsections.The conversion efficiencies of second and third harmon-ics have been provided in table II for different values of a (= eE/mω l c ) and laser frequency ω l . We have ob-served that the conversion efficiency solely depends onthe strength of the laser fields. Keeping a constant whenwe increase ω l , field strength increases and so the efficien-cies increase for both the harmonics. On the other hand,as we keep the value of ( a ω l ) constant for a differentcombination of a and ω l , the field strength of the inci-dent laser pulse remains the same and so does the con-version efficiency. This has been clearly shown in tableII. B. HHG in X-mode configuration ( E l ⊥ B ) The higher harmonics can also be observed for the casewhen the polarization of the incident laser pulse is cho-sen to be in X-mode configuration i.e., E l ⊥ B . Thishas been clearly illustrated in Fig. 4. Laser pulse withfrequency 0 . ω pe was set up initially ( t = 0) to propa-gate in ˆ x direction from the location x = 950. In thiscase also the external magnetic field B has been cho-sen to be 2 . z direction. It is tobe noticed that for our chosen values of system param-eters, the frequency of the incident laser pulse lies inbetween left-hand cutoff ( ω L = 0 . X K laser E l 0 PlasmaVacuum
FIG. 1: A summary of the observations of this study have been shown in this schematic. We have performed 1D PIC simulation(along ˆ x ) with a laser being incident on the plasma surface at x = 1000. The external magnetic field B has been appliedalong z − direction. The polarization of the incident laser has been chosen in O-mode configuration in this schematic, i.e., theelectric field of the incident laser pulse is parallel to the external magnetic field B . It is observed that as laser interacts withthe plasma surface, it generates higher harmonics with different polarizations in the reflected and transmitted radiations, ashas been shown in the schematic. The magnetosonic disturbance has also been observed in these interactions. brid frequency ( ω UH = 2 . z − component of the transverse magnetic field B z asa function of x at a particular instant of time t = 1000.It is also seen that a part of the incident pulse gets re-flected from the vacuum-plasma interface and propagatesin − ˆ x direction in vacuum. The other part of the inci-dent laser penetrates the plasma surface and propagatesthrough the medium. It can be observed that a small dis-turbance, as highlighted by the dotted rectangular boxin subplot (a), is also present which moves with a highergroup velocity in the plasma medium. These are essen-tially the higher harmonics generated by the plasma. TheFFTs in time for this signal of the transverse fields E y and B z in subplots (b) and (c) of Fig. 4 corroborate this.The FFTs of E y and B z inside the plasma ( x = 2500), ashave been shown in subplot (c), evaluated within a timewindow t = 200 to 2200. This choice eliminates the origi-nal transmitted laser pulse with frequency 0 . ω pe (havinghigher power) from the frequency spectrum. Two distinctpeaks observed at ω ≈ . ω pe and 1 . ω pe in both thesubplots correspond to the second and third harmonic,respectively. It is interesting to notice that unlike theprevious case (O-mode configuration), here the polariza-tion of the higher harmonics, both second and third, isthe same as the incident laser pulse. C. Mechanism of HHG in a magnetized plasma
Let us now try to understand the mechanism of theharmonic generation. When a laser pulse with O-modeconfiguration, i.e., E l k B is incident on the vacuum-plasma interface, plasma particles will experience a force( ∝ e v z e B l exp( i ω l t )) due to the Lorentz force ( v × B l )along ˆ x direction. Here, v is the particle’s quiver veloc-ity initiated due to the laser electric field E l in ˆ z . Asa result, plasma electrons wiggle forming an oscillatingcurrent at the surface of the plasma in ± ˆ x direction witha frequency twice the incident laser frequency. This hasbeen clearly shown in subplots (a1) and (a2) of Fig. 5.The electrons being magnetized in the presence of theexternal magnetic field B ˆ z gyrate in the x − y plane.Thus, the oscillatory motion of electrons along x gener-ated by the process discussed above is coupled with itsmotion along ˆ y having the same amplitude and similarfrequency. Consequently, an oscillating current with afrequency twice the laser frequency is produced in the ˆ y direction, as have been shown in subplots (b1) and (b2)of Fig. 5. This acts as an oscillating current sheet an-tenna and radiates electromagnetic wave with the fields e E y and e B z propagating along both ± ˆ x directions. Theseare the second harmonics that have been captured in oursimulations in both vacuum and plasma regime. It iseasy to understand that the oscillating current sheets inˆ x and ˆ y directions might have all the even higher har-monics depending upon the strength of the nonlinearityinvolved in the electron dynamics. In our simulations,we have captured up to sixth harmonic and can be easily -3 (a1) (a)(b) t = 1000t = 1000 ω l = 0.4 ω pe ω l = 0.4 ω pe E l ~ B ( ) FIG. 2: Transverse time varying magnetic fields B y and B z with respect to x have been shown at a particular instant of time t = 1000 (when laser already gets reflected back from the system) in subplots (a) and (b), respectively. In subplot (a1), B y which exist inside the plasma has been in different scale. Here, the colored dotted line at x = 1000 represents the plasmasurface. It is to be noted that the electromagnetic fields e B l and e E l of the incident laser pulse were along ˆ y and ˆ z directions,respectively. X Y X Y X Y (b) (c) (d) X Y (a) X Y In vacuum (x = 500) In plasma (x = 2000)In plasma (x = 2000) (t = 1000 - 3000)(t = 0 - 1000) (t = 1000 - 3000)
In vacuum (x = 500) (t = 0 - 1000)
FIG. 3: Fourier transform of electromagnetic fields with time after laser reflected back from the plasma surface. In subplot (a)and (b), the FFT of E z and B y with time in vacuum ( x = 500) and the bulk plasma ( x = 2000) have been shown, respectively.The same have been shown for the fields ( E y , B z ) in subplots (c) and (d), respectively. From these FFTs, it is a clear indicationthat the higher harmonics have been generated and they are present in both vacuum and the bulk plasma. seen in the FFTs of J ex and J ey in subplots (a2) and (b2) of Fig. 5. X Y X Y X Y X Y X Y Higher harmonics t = 1000
In vacuum (x = 500) (t = 0 - 2000)
In plasma (x = 2500) (t = 200 - 2200) (a)(b) (c) ω l = 0.4 ω pe E l ~ B ( ) FIG. 4: The generation of higher harmonics has been depicted here for the case where the polarization of incident laser hasbeen chosen to be in X-mode configuration, i.e., e E l ⊥ B . The electromagnetic part of the magnetic field along ˆ z , B z has beenshown in subplot (a) at a particular instant of time t = 1000. In the subplot (b), the FFT of E y and B z at the location x = 500(vacuum) has been sown. It is clearly seen that in addition to the original reflected laser field ( ω ≈ . ω ≈ . , .
2) are also present in the reflected radiation. The existence of these higher harmonics inside the bulk plasma hasbeen depicted in subplot(c), where the FFTs have been performed at the location x = 2500. -3 -3 X Y X Y X Y X Y X Y X Y X Y (a1) (b1) (c1)(a2) (b2) (c2) FIG. 5: Figure shows the time evolution of (a1) x -component, (b1) y -component, and (c1) z -component of electron currents J ex , J ey , and J ez at the vacuum plasma interface ( x = 1000), respectively. In subplots (a2), (b2), and (c2), the FFTs of J ex , J ey , and J ez have been shown, respectively. In addition, the electron motion along ˆ x will couple tothe laser magnetic field e B l (which is along ˆ y for O-modeconfiguration) resulting in an oscillating current sheetalong ˆ z with the frequency thrice of laser frequency andalso at higher odd harmonic values, as have been shownin subplots (c1) and (c2) of Fig. 5. Thus, another radi-ation will be produced propagating in ± x directions butin this case, the electromagnetic fields associated withthis radiation are e E z and e B y . It is to be noticed thatideally, all the odd higher harmonics might be present inthe J ez . In our simulations, we have identified up to fifth harmonic and shown in subplot (c2) of Fig .5.For the laser polarization in the X-mode configuration,the magnetic field e B l of the laser is parallel to B (alongˆ z ), accordingly for this case using similar arguments bothsecond and third harmonics (in fact, all the high harmon-ics) will be generated for which the electromagnetic fieldsare e E y and e B z , i.e., the generated HHG radiation also hasthe X-mode configuration. This is exactly what we haveobserved in our simulations, as have been shown in Fig.4. D. Characterization of higher harmonics
We now analyze in further detail and characterize thehigh harmonic radiations. As have been shown previ-ously, these higher harmonics can be observed for thelaser polarization in both O-mode ( E l k B ) and X-mode ( E l ⊥ B ) configurations. We consider here theobservations corresponding to the O-mode configuration.As discussed in the section III A, the higher harmonicradiations are electromagnetic in nature. Thus, in thevacuum these radiations will travel with the speed oflight, i.e., the frequency and wavenumber (in normalizedunits) will have the same values. On the other hand,when they propagate through the plasma medium, theywill have to follow certain plasma dispersion relationsto sustain inside the medium. Thus, the group veloc-ity, as well as the phase velocity of these radiations, willhave different values depending upon the nature of thedispersion relation they are following. These propertieshave been clearly shown in Fig. 6. The spatial FFTsof the transverse fields ( E z , B y ) for vacuum and bulkplasma have been shown in subplots (a) and (b) of Fig.6, respectively. It is clearly seen from the subplot (a)that, as expected, the reflected laser pulse ( ω l = 0 .
4) andthe third harmonic ( ω ≈ .
2) radiation are associatedwith the wavenumbers k x ≈ . k x ≈ .
2, respec-tively, as they travel in vacuum. On the other hand, thespectrum of the spatial FFTs of E z and B y in the bulkplasma region shows a distinct peak at a particular valueof k x ≈ .
67. It is interesting to realize that the third har-monic radiation is associated with the transverse electricfield ( E z ) parallel to the external magnetic field B andtraveling perpendicular to B . Thus, it matches the con-dition of plasma ordinary ( O ) mode. We have evaluatedthe theoretical dispersion curve for the O mode for ourchosen values of system parameters and has been shownin the subplot (a) of Fig. 7. It is seen that the valueof wavenumber corresponding to the frequency ω = 1 . .
66. Thus, It matches wellwith the properties of third harmonic radiation observedin our simulation inside the bulk plasma region.The FFT of transverse fields E y and B z in space as-sociated with the second harmonic radiation have beendepicted in subplots (c) and (d) of Fig. 6 for vacuumand bulk plasma, respectively. The FFT spectrum re-veals that in the vacuum the second harmonic radiation( ω ≈ .
8) propagates with a finite wavenumber k x ≈ . k x ≈ .
76. Thetheoretical model analysis again affirms that this secondharmonic radiation matches the condition for plasma ex-traordinary ( X ) mode and propagates through the passband (region III) lying in between left-hand cutoff ( ω L )and upper hybrid frequency ( ω UH ) of the X mode dis-persion curve. This has been clearly demonstrated in thesubplot (b) of Fig. 7.The mode analysis of harmonic radiations has a direct significance in the sense that we can now have controlover the excitation of these radiations inside the plasmamedium. For example, if the value ω l or B is changed,the frequency of the higher harmonics as well as the dis-persion curves of the plasma modes (Fig. 7) will bemodified accordingly. Thus, there might be some situ-ations where these harmonics will not be allowed to passthrough the plasma medium. For instance, we have con-sidered a particular case where the frequency of the in-cident laser pulse is chosen to be 0 . ω pe and all othersystem parameters have been kept the same as the previ-ous case. It has been observed that the second harmonicradiations initiated at the vacuum-plasma interface aretraveling in both vacuum and plasma. This has beenclearly demonstrated in the subplot (a) of Fig. 8. It isto be noticed that the frequency of the second harmonicradiation which is happened to be 0 . ω pe in this case(subplots (b1) & (b2)), is still higher than the left-handcutoff ω L = 0 .
376 and thus, lies within the pass bandregion in between ω L and ω UH . Hence, the second har-monic radiation is allowed to pass through the plasma.On the other hand, the third harmonic radiation whichhas the O-mode characteristics will have the frequency0 . ω pe . Thus, in this particular case, the third harmonicradiation is lying below the cutoff ( ω = ω pe ) of the O-mode dispersion curve and is forbidden to propagate in-side the plasma. This has been clearly illustrated in Fig.9. In subplot (a) and (b) of Fig. 9, we have shown they-component of the transverse magnetic field e B y at a par-ticular instant of time t = 1000 for two different incidentlaser frequencies ω l = 0 . ω pe and 0 . ω pe , respectively. Itis seen that for ω l = 0 . ω pe (subplot (a)), electromagneticfield e B y of the incident laser pulse has been reflected fromthe vacuum-plasma interface and no signal of e B y existsinside the plasma. Whereas, for ω l = 0 . ω pe (subplot(b)), a part of e B y is also present inside the plasma andwhich is associated with the third harmonic radiation, asalso has been demonstrated in the section III A.It is straightforward that in the presence of an externalmagnetic field B , a plasma wave with the electric field E ⊥ B and the propagation vector k ⊥ B always tendto be elliptically polarized instead of plane-polarized [29].That is, as such a wave propagates through the plasma,an electric field component parallel to the propagationdirection will also be present. The wave, therefore, hasboth electromagnetic and electrostatic features. In ourstudy, the observed second harmonic radiation is associ-ated with an electric field perpendicular to B and prop-agates along ˆ x , as has been shown in section III A. Thus,an electric field component along x , e E x is also expectedto be present and will be traveling along with the secondharmonic EM radiation. This has been clearly depictedin the subplot (a) of Fig. 10. In this subplot, the E x fieldprofile is clearly seen to be associated with the secondharmonic radiation. The Fourier spectra of this partic-ular profile in both ω and k -space also show the samecharacteristic properties as the electromagnetic second X Y X Y X Y (b)(c) (d) In plasma (x = 1500 - 3000) t =
In vacuum (x = 0 - 1000) t = 600 In plasma (x = 1500 - 3000) t = 1600 X Y In vacuum (x = 0 - 1000) t = 600 (a) X Y FIG. 6: Fourier transforms in space of the electromagnetic fields after laser reflected from the vacuum-plasma interface. Insubplots (a) and (b), the FFT of ( E z , B y ) along ˆ x at time t = 600 and 1600 have been shown for vacuum and bulk plasma,respectively. On the other hand, the same have been shown for the fields ( E y , B z ) in subplots (c) and (d), respectively. X Y III R egionRegion II Region I LH (cid:1) L (cid:0) UH (cid:2) R (cid:3) X Y (b)(a) FIG. 7: (a) Dispersion curves [28] of (a) Ordinary ( O ) mode and (b) extraordinary ( X ) mode for the chosen values of thesystem parameters of this study. harmonic radiation, as have been shown in subplots (b1)and (b2) of Fig. 10.We also observe from the subplot (a) of Fig. 10 that alarge scale disturbance is present in E x near the plasma-vacuum interface. Such a disturbance has also been ob-served in the transverse fields E y and B z , as has beenshown in subplot (b) of Fig. 2. This fluctuation is elec-tromagnetic in nature but becomes elliptically polarizedin the presence of an external magnetic field B , as dis-cussed earlier. It is also observed that this disturbancetravels with a much slower velocity compared to thehigher harmonic radiation present in the system. Sucha disturbance has also been observed in both electronand ion density profile, as can be seen in Fig. 11. Thenormalized density profiles of electrons and ions at a par-ticular instant of time t = 1000 have been shown by thered solid line and blue dotted line in Fig. 11, respectively.A spatial electron density profile travels along with thegenerated second harmonic pulsed structure wave and isthus associated with it. As the frequency of the secondharmonic is much higher than the ion response time scale,this structure appears only in the electron density pro- file, not in ions. On the other hand, near the plasmasurface, the fluctuations are present in both electron andion density profiles and they are almost identical. Thesefluctuations are associated with the low-frequency mag-netosonic perturbations initiated due to the ponderomo-tive force associated with the finite pulse width of thelaser. Such a disturbance has also been reported in arecent study by Vashistha et al. [22] for X-mode con-figuration. We confirm that this is also present in theO-mode configuration. IV.
SUMMARY
We have demonstrated the generation of higher har-monics in a magnetized plasma by following the dynam-ics of a laser pulse interacting with a plasma. One di-mensional PIC simulations using OSIRIS-4.0 has beenused for this purpose. A laser pulse coming from vac-uum is chosen to fall on an overdense plasma medium( ω l < ω pe ) in the presence of an externally applied mag-netic field. The dynamical mechanisms leading to higher X Y X Y (a)(b1) (b2) t = 1000 In vacuum (x = 500) In plasma (x = 2000) (t = 0 - 1000) (t = 1000 - 3000) ω l (cid:4)(cid:5)(cid:6) ω pe FIG. 8: The z-component of the magnetic field B z (electromagnetic) with respect to x at time t = 1000 has been shown insubplot (a). The FFTs of ( E y , B z ) at the locations x = 500 and x = 2000 in the frequency domain, as have been shownin subplots (b1) and (b2), clearly demonstrate that second harmonic is present in both reflected and transmitted radiation,respectively. -3 -3 (a)(b) t = 1000t = 1000 ω l (cid:7) (cid:8)(cid:9)(cid:10) ω pe ω l (cid:11) (cid:12)(cid:13)(cid:14) ω pe FIG. 9: The y-component of magnetic field B y (electromagnetic) with respect to x at a particular instant of time t = 1000 hasbeen shown for incident laser frequency (a) ω l = 0 . ω l = 0 .
4. In both the cases, the polarization of the incident laserhas been chosen to be in O-mode configuration, i.e., e E l k B (along ˆ z ). harmonic generation in both O-mode ( E l k B ) and X-mode configurations ( E l ⊥ B ) have been demonstratedand analysed. The required conditions for the propaga-tion of the harmonic radiation inside the plasma havebeen identified. Our study also demonstrates a conver-sion efficiency of about 0 .
35% for second harmonic for alaser intensity of a = 0 .
5, which is typically in a similar range as some of the analytical studies have reported incontext to magnetized plasma. We feel that this can beimproved further by appropriate tailoring of the plasmaand magnetic field profiles. We believe that this studywill make a significant impact in Astrophysical as well aslaboratory fusion plasma scenarios.0 -3 X Y X Y (a) (b1) (b2) x = 2000t = 1000 - 3000 x = 1500 - 3000 t = 1600 t = 1000 ω l (cid:15) (cid:16)(cid:17)(cid:18) ω pe FIG. 10: The longitudinal electric field E x with respect to x has been shown in the subplot (a) at a particular instant of time t = 1000. In the subplots (b1) and (b2), the FFTs of E x in frequency and k − space have been shown, respectively. t = 1000 ω l (cid:19) (cid:20)(cid:21)(cid:22) ω pe FIG. 11: The electrons and ions density fluctuations have been shown by red solid and blue dotted lines, respectively. V. ACKNOWLEDGMENT
The authors would like to acknowledge the OSIRISConsortium, consisting of UCLA and IST (Lisbon, Por-tugal) for providing access to the OSIRIS 4.0 frame- work which is the work supported by NSF ACI-1339893.This research work has been supported by the J. C.Bose fellowship grant of AD (JCB/2017/000055) and theCRG/2018/000624 grant of DST. The authors thank IITDelhi HPC facility for computational resources. [1] M. S. Sodha and P. K. Kaw (Academic Press, 1970),vol. 27 of
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