Self-Focusing/Defocusing of Chirped Gaussian Laser Beam in Collisional Plasma with Linear Absorption∗

Manzoor Ahmad Wani and Niti Kant

Department of Physics,Lovely Professional University,Phagwara-144411,Punjab,India

1 Introduction

The interaction of high power laser beams with plasmas occupies a unique place in the field of research due to its various applications such as laser-driven fusion,charged particle accelerators,x-ray lasers etc.[1−3]In these applications,it is necessary for a high power laser beam to propagate over extended distances without loss of energy.When such a laser beam interacts with the plasma,various nonlinear phenomena’s(self-focusing,harmonic generation,electron acceleration in vacuum etc.)are likely to occur.Among,these phenomena’s self-focusing is an important nonlinear phenomenon in which the wave front of laser acquires a curvature and laser tends to focus.In general,there are two types of self-focusing viz.,relativistic self-focusing[4]and ponderomotive self-focusing[5]and many papers have been published in achieving the selffocusing of laser beams in plasmas.[6−10]The self-focusing decreases with increase in intensity of the beam due to dominance of diffraction effect at high intensity.[11]Gillet al.[12]used the higher order paraxial theory to study the relativistic self-focusing of super Gaussian laser beam in plasma and reported that the inclusion of higher order terms of dielectric function affects the behavior of beam width parameter significantly and the magnetic field improves the self-focusing of laser beam in plasma.[13−14]Recently,Habibi and Ghamari[15]have extended the same theory for the focusing of a cosh-Gaussian laser beam in quantum plasma.By using more effective decentered parameter,better self-focusing is observed for cosh-Gaussian laser beams as compared to Gaussian beams.

The plasma density ramp is considered to be important in laser plasma interactions,particularly for the selffocusing of short pulse laser in an under dense plasma.Increase in initial density and ramp slope decreases the minimum spot size of the laser beam.The laser and plasma parameters are crucial for self-focusing of laser beam in plasma as it is enhanced with optimized laser and plasma parameters.[16−20]Kant and Wani[21]studied the density transition based self-focusing of laser beam in plasma with linear absorption.The absorption weakens the self-focusing effect and density transition sets an earlier and stronger self-focusing of laser beam.Guptaet al.[22]considered the relativistic ponderomotive nonlinearity and found that the ion temperature causes the thermal self-focusing and has a serious influence on the evolution of laser beam in collisional plasma.The higher order axial electron temperature decreases the influence of collisional nonlinearity.It changes the electron density distribution and increases the dielectric constant therefore,leads to fast divergence of the laser beam.[23]Sprangleet al.[24]investigated the relativistic self-focusing of short-pulse radiation beams in plasmas and reported that for self-focusing to occur,the laser power exceeds a critical value.Again,Wanget al.[25]found that there is an upper limit of the laser power for self-focusing in plasma beyond which the laser pulse is defocused due to the ponderomotive force.It is further,observed that there is a lower limit of the plasma density for self-focusing below which self-focusing does not occur for any laser powers and a new ponderomotive defocusing occurs.

The chirp was used to study the electron acceleration in vacuum.It increases the electron energy and hence momentum so that the electron escapes from the laser beam.The value of chirp parameter decreases with laser intensity and initial electron energy.It further,increases the amplitude of wake wave generated in the plasma by an electromagnetic pulse.[26−27]Ghotra and Kant[28]used the chirped laser pulse to study the electron acceleration in vacuum in presence of azimuthal magnetic field.The chirp increases the duration of interaction of laser beam with electron and strengths the resonance for longer duration.Further,the magnetic field improves the electron acceleration to high energy of the order of GeV.Wanget al.[29]showed that the laser pulse experiences self-focusing and the THz pulse spreads very quickly due to small spot size comparable to THz wavelength.The experimental observations of enhanced acceleration of background electrons in a laser wake field accelerator have been studied by using nonlinearly chirped laser pulses.The effect of nonlinear chirp is to cause pulse shape asymmetries with fast rises,which are more unstable to self-modulation and Raman forward scattering(RFS)instability.It has also been found that pulse shaping is more effective in controlling plasma instabilities and in optimizing the performance of laser-plasma devices.[30]

The laser pulses are very useful in studying the mechanism of powerful terahertz radiation generation from gas targets.The quasi-static transverse currents created by laser field ionization in plasmas are responsible for the THz emission.The chirped laser pulses are used to generate strong THz pulses with amplitudes scaling linearly with the laser amplitude.[31]Since,the propagation properties of cosh-Gaussian laser beams are important technological issues as these beams posses high efficient power.Therefore,the THz radiation is investigated by beating of two cosh-Gaussian lasers with spatial envelops.The lasers exert a nonlinear ponderomotive force along the transverse direction which imparts an oscillatory velocity to electrons that couples with the density ripple to generate a stronger THz radiation.[32]Kumaret al.[33]investigated the effect of self-focusing and defocusing on terahertz(THz)generation by amplitude–modulated Gaussian laser beam in rippled density plasma.It is observed that the amplitude–modulated laser beam self-focuses and defocuses when it propagates in the plasma.The focusing changes with time and THz amplitude get enhanced.Further,the THz generation by self-focusing of amplitude–modulated Gaussian laser beam is a potential scheme for the efficient generation of the radiation.

In the present communication,we analyzed the effect of chirp on the self-focusing/defocusing of Gaussian laser beam propagating in collisional plasma with linear absorption.effects of chirp parameter,collision frequency and other laser plasma parameters are seen on the selffocusing/defocusing of laser beam in plasma.Although,without chirp,the laser beam shows self-focusing but,as the propagation distance increases,it starts to experience defocusing.To reduce this defocusing,the effect of chirp is considered.The chirp parameter minimizes the defocusing and increases the ability of self-focusing of laser beam in plasma.Further,the amplitude of oscillations decreases with the distance of propagation so that an earlier and stronger self-focusing of laser beam is achieved.The paper is organized as follows:in Sec.2 the nonlinear dielectric constant and the equation governing the behaviour of beam width parameter with the distance of propagation is presented.Section 3 is devoted to results and discussions.Finally,the conclusion is given in Sec.4.

2 Self-Focusing of Chirped Gaussian Laser Beam

Consider the propagation of a Gaussian laser beam in plasma along thez-axis.Its initial intensity distribution is given by

where,is the electric vector andr0is the spot size of laser beam.The wave equation governing the propagation of laser beam may be written as

The last term on left hand side of Eq.(2)can be neglected providedk−2∇2(lnε)≪1,wherekrepresents the wave number of the laser beam.Thus,

The effective dielectric constant of the plasma can be expressed as

where,ε0= 1−and Φ are the linear and nonlinear parts of dielectric constant respectively,εi=()(ν/ω)takes care of linear absorption(εi≪ε0),νis the collision frequency,ω=ω0(1+b(ω0t−ω0z/c))is the angular frequency of chirped Gaussian laser beam,ω0is the angular frequency of incident laser beam,bis the chirp parameter,cis the velocity of light,ωpis the plasma frequency given by=4πn0e2/m,where,mis the rest mass of electron,eis the charge of electron,andn0is the equilibrium electron density.Following Ref.[34],Φ(EE∗)can be expressed as:

where,s0is a parameter characterizing the nature of collisions,α=e2M/6m2ω20kBT,Mis the mass of scaterrer in the plasma,Tis the equilibrium plasma temperature,andkBis the Boltzmann constant.Now,introducing,E=A(r,z)exp[i(ωt−kz)],where,A(r,z)is the complex amplitude and employing the WKB approximation,Eq.(3)becomes as:

To solve Eq.(6),we expressAas

where,A0andSare real functions ofrandz(Sbeing the eikonal of the laser beam).Substituting Eq.(7)in Eq.(6)and separating real and imaginary parts,one can obtain

Following Akhmanovet al.[7]and Sodhaet al.,[34−35]we can write as follows

where,τis the dimensionless retarded time,ki=kεi/2ε0is the absorption coefficient withk=/candβ(z)=(1/f)(∂f/∂z),β−1is interrupted as the radius of curvature of the laser beam andf(z)is the dimensionless beam width parameter.Substituting Eq.(10)and Eq.(11)in Eq.(8),the differential equation for beam width parameter is obtained as:

where,ξ=z/Rdis the normalized distance of propagation,Rd=represents the diffraction length,ρ0=r0ω0/cis the equilibrium beam radius,is the normalized absorption coefficient.Equation(12)represents the spot size variation of laser beam with the distance of propagation.

3 Results and Discussion

Equation(12)is the second order nonlinear differential equation governing the behavior of beam width parameter of chirped Gaussian laser beam in collisional plasma with linear absorption.We have solved Eq.(12)numerically by applying the initial condition atξ=0,f=1,(∂f/∂ξ)=0 and(∂2f/∂ξ2)=0 with the following set of typical parameters;ω0=1.778×1014rad/sec,laser beam radius 20µm and equilibrium plasma densityn0=4×1019cm−3.By optimizing suitable laser and plasma parameters,we have investigated the self-focusing/defocusing of chirped Gaussian laser beam in collisional plasma.

Fig.1 Variation of beam width parameter f with the normalized propagation distance ξ for different values of ν/ω0. The other parameters are: ωp/ω0= 0.4,α =0.4,and b=0.

Figure 1 shows the variation of beam width parameterfwith the normalized propagation distanceξfor different values ofν/ω0.The other parameters are:ωp/ω0=0.4,α=0.4,andb=0.It is observed that in the absence of chirp,the laser beam shows defocusing character.The defocusing of laser beam increases with increase in the values ofν/ω0.It is due to the fact that the absorption(corresponding to collision frequency termν/ω0)becomes significant and the laser beam shows fast divergence.The amplitude of oscillations of beam width parameter becomes too large,there by the beam width parameter diverges continuously.In other words,the laser beam shows self-focusing up to a certain critical value and then defocusing character is observed.This is because,for the self-focusing to occur the laser power should exceed a critical value as presented by Sprangleet al.[24]and Wanget al.[29]Further,the results of present analysis can be compared with those of Navareet al.,[36]wherein increase in collision frequency is subjected to increase in oscillation amplitude of beam width parameter.Moreover,for higher values ofν/ω0,the absorption is more significant and overcomes the self-focusing effect.Again,Jafari Milaniet al.[37]investigated the ponderomotive self-focusing of Gaussian laser beam in warm collisional plasma and reported that the collision frequency at first causes selffocusing and then defocusing of laser beam takes place.

Fig.2 Variation of beam width parameter f with the normalized propagation distance ξ for different values of b for(a)positive chirp(b)negative chirp.

Now,in order to account for the defocusing of laser beam in plasma,the effect of chirp is considered.For investigating the effect of chirp parameter(b)on the propagation of laser beam in collisional plasma,various values ofbare considered.Figure 2(a)illustrates the behaviour of beam width parameterfwith the normalized propagation distanceξfor different values of positive chirp parameterband the other parameters are same as taken in Fig.1.It is observed from Fig.2(a)that as soon as the chirp parameter is increased,the amplitude of oscillations of the laser beam decreases with the distance of propagation.Further,with the passage of laser beam in plasma,the angular frequency of the laser beam increases with the result,the dielectric constant of the plasma decreases.The decrease in dielectric constant reduces the amplitude of spot size of laser beam close to the propagation axis.Consequently,the beam width parameter attains a minimum value for further distance of propagation.The effect of negative chirp on the self-focusing or defocusing of laser beam is shown in Fig.2(b)which represents the variation of beam width parameterfwith the normalized propagation distanceξfor different values of negative chirp.From Fig.2(b),it is clear that on increasing the values of negative chirp,the self-focusing at first is strengthened and after attaining a critical value the laser beam defocuses.This is because the frequency of a linear and negative chirped laser beam changes during the propagation in the plasma.Therefore,the spot size of laser beam depends onξand at propagation distances much greater than the Rayleigh length the temporal shape of the chirped laser beam will be changed.Therefore,the defocusing of laser beam is weakened and there by the self-focusing effect is strengthened by using chirp.Hence,the chirp parameter is important for minimizing the defocusing and increasing the ability of self-focusing of laser beam in collisional plasma.

Fig.3 Variation of beam width parameter f with the normalized propagation distance ξ for different values of ωp/ω0.The other parameters are: ν/ω0=0.002,αE20=0.4.and b=0.002.

Fig.4 Variation of beam width parameter f with the normalized propagation distance ξ for different values of α The other parameters are: ν/ω0=0.002,ωp/ω0=0.4,and b=0.002.

Fig.5 Variation of beam width parameter f with the normalized propagation distance ξ for different values of intensity.The other parameters are: ν/ω0=0.002,ωp/ω0=0.6,and b=0.002.

Figure 3 presents the variation of beam width parameterfwith the normalized propagation distanceξfor different values ofωp/ω0.The other parameters are:ν/ω0=0.002,α=0.4,andb=0.002.It is evident from Fig.3 that with increase inωp/ω0,the nonlinearity of plasma medium increases,with the result,the amplitude of oscillations decreases further close to the propagation axis.Consequently,fminshifts towards lower value ofξ=0.4.Therefore,the self-focusing of laser beam occurs earlier and thus supports the results.[36,38]Figure 4 illustrates the behaviour of beam width parameterfwith the normalized propagation distanceξfor different values ofα.The relative plasma density is fixed atωp/ω0=0.4 and the other parameters are same as mentioned in Fig.3.The curves demonstrate that with increase inαof the beam,the laser spot size and hence the self-focusing length decreases.Again,increase in laser intensity results in increasing the nonlinearity which is responsible for the selffocusing of laser beam in plasma.Consequently,the laser beam bends more towards the focusing mode for moderately high intensity values of the laser beam.Furthermore,at higher intensity and for higher plasma density,a beam with more electrons travels with the laser beam and generates a higher current.Consequently,a higher quasistationary magnetic field is generated,which reduces the focusing length and hence adds to self-focusing.Again,taking into account the laser intensities(1020W/cm2)closer the realistic values,the variation of beam width parameter with the dimensionless distance of propagation is shown in Fig.5.From Fig.5,it is observed that at higher intensities,the oscillatory behavior of beam width parameter is destroyed during propagation in plasma and the laser beam undergoes defocusing.In other words,the selffocusing of laser beam disappears with very high intensity and ponderomotive defocusing occurs.This is due to dominance of diffraction effect at high intensity.Further,the frequency of a chirped laser beam changes during the propagation in plasma.As the spot size of laser beam depends onξand at propagation distances much greater than the Rayleigh length,the temporal shape of the chirped laser beam will be changed.However,for propagation distances less than the Rayleigh length,the change in laser pulse shape is not considerable.

4 Conclusion

In the present communication,we have investigated the self-focusing/defocusing of chirped Gaussian laser beam in collisional plasma with linear absorption.We have derived the differential equation for the beam width parameter by using the WKB and paraxial ray approximations and investigated the impression of laser and plasma parameters on the self-focusing/defocusing of laser beam in collisional plasma. From the results,one can conclude that the chirp parameter is important for the selffocusing/defocusing of laser beam in plasma and plays a vital role in laser plasma interaction.The laser beam is defocused due to strong diffraction and absorption effects at higher oscillation frequencies.It is further,revealed that initially the amplitude of beam width parameter is too large and continuously diverges in the collisional plasma.The chirp parameter minimizes the divergence and consequently,an earlier self-focusing of laser beam is observed.Thus,apart from electron acceleration,the chirp can also be used to study the self-focusing/defocusing of laser beam in plasma.The results of present investigation may be useful in laser—driven fusion and laser plasma based accelerators.

[1]P.Sprangle,E.Esarey,and J.Krall,Phys.Plasmas 3(1996)2183.

[2]D.N.Gupta,M.S.Hur,and H.Suk,J.Appl.Phys.100(2006)103101.

[3]D.Umstadter,Phys.Plasmas 8(2001)1774.

[4]A.R.Niknam,M.Hameshemzadeh,and B.Shokri,Phys.Plasmas 16(2009)033105.

[5]Bruce I.Cohen,Barbara F.Barbara,A.Bruce Langdon,and Jullian C.Cummings,Phys.Fluids B 3(1991)766.

[6]H.Hora,Z.Physik 226(1969)156.

[7]S.A.Akhmanov,A.P.Sukhorukov,and R.V.Khokhlov,Sov.Phys.Usp.10(1968)619.

[8]N.Kant,S.Saralch,and H.Singh,Nukleonika 56(2011)149.

[9]D.N.Gupta,M.S.Hur,H.Hwang,and H.Suk,J.Opt.Soc.Am.B 24(2007)1155.

[10]S.D.Patil and M.V.Takale,Phys.Plasmas 20(2013)083101.

[11]S.D.Patil,M.V.Takale,S.T.Navare,M.B.Dongare,and V.J.Fulari,Optik 124(2013)180.

[12]T.S.Gill,R.Mahajan,R.Kaur,and S.Gupta,Laser and particle Beams 30(2012)509.

[13]M.Aggarwal,S.Vij,and N.Kant,Optik 126(2015)5710

[14]K.I.Hassoon,A.K.Sharma,and R.A.Khamis,Phys.Scr.81(2010)025505.

[15]M.Habibi and F.Ghamari,IEEE Trans.Plasma Sci.43(2015)2160.

[16]N.Kant,M.A.Wani,and A.Kumar,Opt.Commun.285(2012)4483.

[17]V.Nanda and N.Kant,Phys.Plasmas 21(2014)042101.

[18]S.Zare,E.Yazdani,S.Razaee,A.Anvari,and R.Sadighi-Bonabi,Phys.Rev.ST Accel.Beams 18(2015)041301.

[19]M.Aggarwal,H.Kumar,and N.Kant,Optik 127(2015)2212.

[20]V.Nanda,N.Kant,and M.A.Wani,Phys.Plasmas 20(2013)113109.

[21]N.Kant and M.A.Wani,Commun.Theor.Phys.64(2015)103.

[22]D.N.Gupta,M.R.Islam,D.G.Jang,H.Suk,and D.A.Jaroszynski,Phys.Plasmas 20(2013)123103.

[23]XIA Xiong-Ping and YI Lin,Commun.Theor.Phys.57(2012)873.

[24]P.Sprangle,C.M.Tang,and E.Esarey,IEEE Trans.Plasma Sci.15(1987)145.

[25]W.M.Wang,Z.M.Sheng,M.Zeng,et al.,Appl.Phys.Lett.101(2012)184104.

[26]K.P.Singh,Appl.Phys.Lett.87(2005)254102.

[27]A.G.Khachatryan,F.A.Van Goor,J.W.J.Verschuur,and K.J.Boller,Phys.Plasmas 12(2005)062116.

[28]H.S.Ghotra and N.Kant,Appl.Phys.B 120(2015)141.[29]W.M.Wang and C.Y.Zheng,Phys.Plasmas 13(2006)053112.

[30]W.P.Leemans,P.Catravas,E.Esarey,et al.,Phys.Rev.Lett.89(2002)174802.

[31]W.M.Wang,Z.M.Sheng,H.C.Wu,M.Chen,C.Li,J.Zhang,and K.Mima,Opt.Express 16(2008)16999.

[32]M.Singh,R.K.Singh,and R.P.Sharma,Frontiers Phys.104(2013)35002.

[33]S.Kumar,R.K.Singh,M.Singh,and R.P.Sharma,Laser and Particle Beams 33(2015)257.

[34]M.S.Sodha,A.K.Ghatak,and V.K.Tripathi,Prog.Opt.13(1976)169.

[35]M.S.Sodha,A.K.Ghatak,and V.K.Tripathi,Self-Focusing of Laser Beams in Dielectric,Plasmas and Semiconductors,Tata McGraw-Hill,Delhi(1974).

[36]S.T.Navare,M.V.Takale,S.D.Patil,V.J.Fulari,and M.B.Dongare,Opt.and Lasers in Engineering 50(2012)1316.

[37]M.R.Jafari Milani,A.R.Niknam,and A.H.Farahbod,Phys.Plasmas 21(2014)063107.

[38]S.D.Patil,M.V.Takale,V.J.Fulari,D.N.Gupta,and H.Suk,Appl.Phys.B 111(2013)1.