Keshav Walia

Department of Physics,DAV University Jalandhar,India

Abstract The present work explores the propagation characteristics of high-power beams in weakly relativistic-ponderomotive thermal quantum plasma.A q-Gaussian laser beam is taken in the present investigation.The quasi-optics equation obtained in the present study is solved through a well-established Wentzel–Kramers–Brillouin approximation and paraxial theory approach for obtaining the second-order differential equation describing the behavior of beam width of the laser beam.Further,a numerical simulation of this second-order differential equation is carried out for determining the behavior of the beam width with dimensionless distance for established laser–plasma parameters.The comparison of the present study is made with ordinary quantum plasma and classical relativistic plasma cases.

Keywords: q-Faussian beam,WKB approximation,thermal quantum plasma,weakly relativisticponderomotive force,relativistic regime

1.Introduction

Recent advances in technology have led to the construction of lasers with intensities of the order of 1018W cm?2.Interaction of such lasers with plasma causes the generation of numerous instabilities including self-focusing,filamentation,generation of harmonics,self-phase modulation,and scattering instabilities [1–13].The investigation of these nonlinear phenomena is one of the hot topics amongst several experimental/theoretical researchers as a result of their manifold applications including laser-driven fusion,ionospheric modification,and accelerations of plasma particles [14–21].The success of these applications largely depends on the much deeper penetration of laser beams through plasma.The transition of laser beams through plasmas is greatly affected due to these nonlinear phenomena.So,it becomes important to have in-depth information on these nonlinear phenomena to keep these nonlinear phenomena at a low level.Moreover,it will also help researchers improve the coupling efficiency of the laser–plasma interaction.

Out of the various laser–plasma instabilities discussed above,the phenomenon of self-focusing occupies a major role as a result of the direct connection with several other instabilities.[22–29].The self-focusing phenomenon was initially given by Askar’yan in 1962 [30].The plasma medium has a nonlinear response to the incoming beam in the self-focusing of the beam.The refractive index profile across the beam’s cross-section is created in the plasma medium as a result of the laser beam transition through it.The refractive index profile starts increasing along the beam irradiance thereby producing self-focusing.Self-focusing has a major impact on other instabilities also.Relativistic self-focusing(RSF) in plasma medium is the most promising research topic.RSF arises on account of enhancement in electronic mass,whenever electrons start traveling with a velocity equivalent to the velocity of light.When power associated with a given beam is much greater than critical power,then electrons start traveling with a velocity the same as that of light thereby causing a change in the effective dielectric function of plasma and hence causing the focusing of the beam [31,32].In ponderomotive self-focusing (PSF),the electrons are expelled from the high field portion to the low field portion due to nonlinear ponderomotive force.The plasma’s dielectric function is modified thereby producing self-focusing of the beam [33,34].The phenomenon of RSF occurs almost instantaneously,whereas PSF takes a finite time as a result of the expulsion of plasma electrons from regions with high irradiance to regions with low irradiance.PSF only is added to RSF and does not obstruct RSF.Early work on the self-focusing phenomenon is carried out in the classical regime of plasma.High temperature and less number density are characteristics connected with classical plasma.But,the properties linked with quantum plasmas are low temperature and high number density.The quantum contribution can be well understood with the help of the parameterχ=.(TFandTcorrespond to Fermi temperature and temperature of plasma).Forχ≥1,the quantum contribution becomes superior.The complete statistical description for classical plasma is given with the help of Maxwell–Boltzmann (MB) statistics,whereas the complete statistical description for quantum plasmas is given with the help of Fermi–Dirac (FD) statistics.Further,the de-Broglie wavelength for plasma particles is very small for the classical regime.So,plasma particles are usually considered point-like.But,the de-Broglie wavelength for plasma particles is the same as that of the inter-particle distance for the quantum regime.The quantum effects become important with an increase in number density and also with a decrease in plasma temperature.The interaction of intense lasers with quantum plasma is an active research area amongst distinct experimental/theoretical research groups due to its importance in manifold applications such as the laser–matter interaction,fusion science,and astrophysical systems etc [35–39].The nonlinear effects become predominant in quantum plasma rather than classical plasma.The beam spot size oscillates with larger frequencies and lesser amplitudes for the case of quantum plasmas thereby enhancing the beam’s focusing behavior [40–45].Several research groups have already explored the interaction of intense lasers with quantum plasmas in the past [46–50].But,these investigations are carried out by taking Gaussian beams with cylindrical cross-sections.In recent years,there has been interest from researchers to explore a new class of laser systems known as q-Gaussian beams.The distribution of intensity for such lasers is of the formf(r)=f(0).One can change the q-Gaussian beams into ordinary Gaussian beams by consideringq→∞.Moreover,the power associated with q-Gaussian beams is much lesser than cylindrical Gaussian beams.So,the motivation of the current work is to explore propagation characteristics of q-Gaussian beam thermal quantum plasma(TQP) under relativistic-ponderomotive nonlinearities.In section 2,the well-established Wentzel–Kramers–Brillouin(WKB) approximation and paraxial theory are taken for obtaining a nonlinear differential equation representing the change in beam width with dimensionless propagation distance.In section 3,the results obtained through numerical simulation of nonlinear differential equations are discussed.Finally,the conclusion of the present work is discussed in section 4.

2.Evolution of spot size of laser beam

The transition of the high-power q-Gaussian beam in TQP along the z direction is considered in the present investigation.The present research problem is carried out under the combined action of relativistic-ponderomotive forces.The initial irradiance distribution of such beams at z=0 is expressed as

In equation (1),the initial beam radius,axial field amplitude,and complex field amplitude are represented as r0,E00and E0,respectively.In the above equation,the q-parameter is really helpful in describing deviation in irradiance distribution of the q-Gaussian beam from ordinary Gaussian beams.Irradiance distribution associated with the q-Gaussian beam is converted to ordinary Gaussian beams asq→∞.Forz＞ 0,distribution of irradiance for the q-Gaussian beam is best described by

For an isotropic non-conducting medium,the differential forms of Ampere’s law and Faraday’s law take the form

In equations (3)and (4),D=εEis the electric displacement vector.Further E and B correspond to electric and magnetic field vectors.Equations (3) and (4) can be solved together to get a wave equation for electric field vector E as

The effective dielectric function for TQP considering the Fermi pressure,quantum contribution,and Bohm potential may be represented as [51,52]

One can express general dielectric function for TQP medium as

Following [54–56],one can express the solution for equation (6) as

In equation(13),‘S’represents the eikonal for the beam.The phase shift of a given beam is denoted by Φ0(z).Since,we are stressing on beam irradiance in the present investigation rather than its phase.So,the Φ0(z)term will not be needed in further analysis.′f′ denotes beam waist associated with q-Gaussian beam and satisfies the following second-order differential equation

3.Discussion

Figure 3 denotes the change in beam widthfwith dimensionless propagation distanceηat different Fermi temperaturesTF(TF=107K,108K,109K)with other parameters kept fixed.The black curve,red curve,and green curve correspond toTF=107K,108K and 109K.The beam widthfis shifted towards lesserηvalues with increments inTFvalues.In other words,the focusing character of the beam is enhanced with an increase inTFvalues.This is because the converging term starts dominating over the diffractive term at largerTFvalues.Hence,there is an increase in focusing behavior at larger Fermi temperatureTF.

Figure 4 denotes the change in beam widthfwith dimensionless propagation distanceηat different ‘q’ values(q=1,2,3) with other parameters kept fixed.The black curve,red curve,and green curve correspond toq=1,2 and 3.The beam widthfgets shifted towards lesserηvalues with increment in q values i.e.focusing character of the beam gets improved with increments inqvalues.This is due to the shrinking of beam irradiance for a given class of laser beam towards the axial portion.Since focusing happens fast for rays along the axis rather than off-axial rays.Hence,there is an enhancement in focusing behavior at largerqvalues.

Figure 5 denotes the change in beam widthfwith dimensionless propagation distanceηin different plasma environments.The black curve,green curve,blue curve,and red curve correspond to relativistic-ponderomotive thermal quantum plasma (RPTQP),thermal quantum plasma (TQP),cold quantum plasma(CQP),and classical relativistic plasma(CRP).The beam width is shifted towards smallerηvalues in the RPTQP case in contrast with other cases of plasma environments such as TQP,CQP,and CRP cases.In other words,there is more focusing in the RPTQP case instead of other plasma cases.If we compare remaining plasma cases such as TQP,CQP,and CRP.It is found that the TQP case is found to have more focusing in contrast with CQP and CRP cases.So,it is observed that beam focusing is enhanced including quantum contribution and ponderomotive effects.

Figure 3.Denotes change in beam width f with dimensionless propagation distanceη at different Fermi temperatures TF(TF=107 K,108 K,109 K)with other parameters kept fixed.The black curve,red curve,and green curve are for TF=107 K,108 K and 109 K,respectively.

Fig.5.Denotes change in beam width f with dimensionless propagation distanceη in different plasma environments.The black curve,green curve,blue curve,and red curve correspond to RPTQP,TQP,CQP,and CRP cases.

4.Conclusion

In the present investigation,propagation characteristics of a high-power beam in TQP under the combined action of relativistic-ponderomotive nonlinearities are explored by adopting a well-established WKB approximation and paraxial theory approach.The q-Gaussian beam is taken in the present investigation.The important results concluded from the current research are as follows;

1) The decrease in focusing behavior is found with an increase invalues.

2) The focusing ability of the beam is enhanced with the rise in plasma density,q values,and Fermi temperature.

3) The focusing ability of the beam is improved by including ponderomotive effects and quantum effects.The present results are very useful in the inertial con-finement fusion scheme.