Keshav Walia,Kulkaran Singh and Deepak Tripathi

1 Department of Physics,DAV University,Jalandhar,India

2 Department of Physics,AIAS,Amity University,Noida,India

Abstract The purpose of this study is to explore the second harmonic generation(SHG)of a high power Cosh-Gaussian beam in cold collisionless plasma.The ponderomotive force causes carrier redistribution from high field to low field region in presence of a Cosh-Gaussian beam thereby producing density gradients in the transverse direction.The density gradients so produced the results in electron plasma wave(EPW)generation at the frequency of the input beam.The EPW interacts with the input beam resulting in the production of 2nd harmonics.WKB and paraxial approximations are employed for obtaining the 2nd order differential equation describing the behavior of the beam’s spot size against normalized distance.The impact of well-established laser-plasma parameters on the behavior of the beam’s spot size and SHG yield are also analyzed.The focusing behavior of the beam and SHG yield is enhanced with an increase in the density of plasma,the radius of the beam and the decentred parameter,and with a decrease in the intensity of the beam.The results of the current problem are really helpful for complete information of laser-plasma interaction physics.

Keywords:second harmonic generation,cold collisionless plasma,ponderomotive force,electron plasma wave,Cosh-Gaussian beam

1.Introduction

Several theoretical and experimental research groups are interested in exploring laser-plasma interaction physics as a result of its connection with a variety of applications including laser-driven fusion,plasma-based accelerators and higher harmonic generation[1–8].One can achieve success in the above-mentioned applications through much deeper transition of laser beam inside plasma and acquiring minimum spot size so that maximum energy from the laser beam to the system could be transferred.Several nonlinear phenomena such as harmonic generation,scattering instabilities,self-focusing etc are produced on intense laser interaction with plasma[9–21].Researchers are exploring these instabilities theoretically as well as experimentally for detailed information of intense laser interaction with plasma[22–27].Amongst these nonlinear phenomena,the phenomenon of self-focusing occupies a distinctive place.This phenomenon was first time discovered by Askaryan in 1962[28].The selffocusing phenomenon is receiving major attention of many researchers on account of its direct relevance to other nonlinear phenomena.This phenomenon arises on account of a change in the plasma’s overall dielectric function.The overall plasma’s dielectric function can change as a result of three main mechanisms namely relativistic effects,collisions and ponderomotive force.

The most important research area in the laser-plasma interaction process is the production of harmonics.In fact,plasma is the most promising medium for the production of harmonics.It results in the conversion of the laser beam fundamental frequency into several harmonics.Harmonic production strongly influences laser beam transition through plasma medium.Generation of harmonics helps in finding several plasma parameters including local electron density,opacity and electrical conductivity[29,30].The transit of beam through plasma medium can be easily tracked through second harmonic generation(SHG).Earlier work on SHG was carried out by Sodha and Kaw[31].In the field of spectroscopy,harmonic radiations play a commanding role[32–35].The production of harmonics can be done through several mechanisms including plasma wave excitation electron plasma wave(EPW),plasma instabilities and resonant absorption[8,36–39].However,the commonly used method for harmonics production is EPW excitation.In this method,EPW generated at input wave frequency interacts nonlinearly with input wave resulting in the production of SHG.Major work on this mechanism has been done by various theoretical and experimental research groups in the past[40–51].The commanding role is played by plasma wave production in laser-induced fusion due to the generation of high velocity electrons.The plasma wave interacts with plasma particles and transfers energy to particles further causing particle acceleration[52].The past research work confirms that the majority of research on SHG is carried out either through uniform beam or through Gaussian beam.SHG of intense Cosh-Gaussian beam in cold collisionless plasma is explored in the present problem.The 2nd order differential equation reporting the behavior of laser beam spot size is set up in section 2.The 2nd harmonic source term and expression for second harmonic yield is derived in sections 3 and 4 respectively.The discussion of findings obtained is given in section 5.

2.Evolution of laser beam spot size

3.EPW excitation mechanism

We know that the Cosh-Gaussian beam is self-focused on account of collisionless plasma if the power of the beam gets larger than the critical beam power.Moreover,the production of density gradients in the perpendicular direction takes place.There is excitation of EPW at input beam frequency due to these density gradients.EPW excitation mechanism can be understood through below mentioned set of equations:

By solving the above set of equations and further using linear perturbation theory,we have

One can solve equation(18)in order to have the source equation of SHG as

4.Second harmonic generation

Excited EPW interacts nonlinearly with the input beam thereby producing the 2nd harmonics.One can begin from the well known Maxwell’s equations for obtaining the 2nd harmonic field equationE2as

Figure 1.Dependence of beam waist f onη at distinct values of E 00(i .e.E 00 = 3.0 × 109 Vm -1,4.0 × 109 V m-1,5.0 ×10 9 V m-1)at fxied values of other parameters.Green,red and black curves correspond to E 00 = 3.0 × 109 V m -1,4.0 × 109 V m-1 and 5.0 ×10 9 V m-1 respectively.

5.Discussion

The dependence of beam widthfand yield of SHGY2against normalized distance in cold collisionless plasma is represented by equations(13)and(24).Since the analytical solution of equations(13)and(24)cannot be obtained.Hence the numerical solution of these equations is carried out through the RK4 method for well-established laser and plasma parameters;

Figure 2.Dependence of beam waist f onη at distinct values of at fixed values of other parameters.Green,red and black curves correspond to and 0.06respectively.

Figure 3.Dependence of beam waist f onη at distinct values of beam radius w0(i .e.w0 =30 μm,40 μm,50 μm)at fixed values of other parameters.Green,red and black curves correspond tow0 =30 μm,40 μm and 50 μm respectively.

Figure 4.Dependence of beam waist f onη at distinct values of decentred parameter′b′(i.e.b=0,1,2)at fixed values of other parameters.Green,red and black curves correspond tob =0,1 and 2 respectively.

Figure 5.The dependence of SHG yieldY2 againstη at distinct values ofE00(i.e.E00=3.0 ×10 9V m-1,5.0 ×109 V m-1)at fixed values of other parameters.Green and red curves correspond toE00=3.0 ×10 9V m -1and 5.0 ×10 9V m-1 respectively.

Figure 6.The dependence of SHG yieldY2 againstη at distinct values of at fixed values of other parameters.Green and red curves correspond to and 0.06respectively.

Figure 7.The dependence of SHG yieldY2 againstη at distinct values of beam radius w0(i .e.w0 =30 μm ,50 μm)at fixed values of other parameters.Green and red curves correspond tow0 =30 μm and 50 μm respectively.

Figure 8.The dependence of SHG yieldY2 againstη at distinct values of the decentred parameter b(i .e.b =0 and 2)at fixed values of other parameters.Green and red curves correspond to b =0 and 2 respectively.

6.Conclusions

SHG of high power Cosh-Gaussian beam in cold collisionless plasma is explored in the present work with the help of WKB and paraxial theory approach by considering ponderomotive nonlinearity.The outcome of the current research work is mentioned below:

(1)The focusing behavior of the beam is enhanced with an increase in density of plasma,the radius of the beam and decentred parameter,and with a decrease in the intensity of the beam.

(2)SHG yield is enhanced with an increase in density of plasma,the radius of the beam and decentred parameter,and with a decrease in intensity of the beam.

The results of the current problem are really helpful for complete information of laser-plasma interaction physics.