Effect of Plasma Density Ramp and Axial Electron Temperature on Self-focusing / Defocusing of a Laser Beam in Plasma Manzoor Ahmad Wani

Effect of Plasma Density Ramp and Axial Electron Temperature on Self-focusing / Defocusing of a Laser Beam in Plasma
Manzoor Ahmad Wani, Vishal Thakur, Harjit Singh Ghotra and Niti Kant*
Department of Physics,
Lovely Professional University,
Phagwara-144411, Punjab
*E-mail: [email protected] Abstract
In the current paper, we investigated the self-focusing / defocusing of an intense laser beam in the plasma under combined effect of density ramp and higher order axial electron temperature. By following WKB and paraxial approximations the differential equation for beam width parameter is obtained and then solved numerically. The equation governing the behavior of beam width parameter for propagation distance is extensively investigated for different values of optimized parameters and is strongly influenced by the higher order axial electron temperature. Further, results are discussed and presented graphically. It is seen that the electron temperature component is not trivial than and . Under, the influence of plasma density ramp, higher order axial electron temperature on the one way reduces defocusing in course of steady divergence and on the other way; it animates the self-focusing in the course of steady convergence, especially for larger propagation distance. Furthermore, increase in the density ramp results more reduction in the laser beam spot size and consequently increases rate of phenomenon of self-focusing.

Keywords: Laser Beam; Plasma Density Ramp; Self-focusing / Defocusing; Axial electron temperature.

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Introduction
The non-linear interaction of highly powerful laser beam with plasma and phenomena corresponding to self-focusing of an intense laser beam has become a matter of significant interest from last three decades because of various wide ranging applications such as laser-driven acceleration, x-ray lasers, harmonic generation etc. 1-5. For the success of these applications, high intensity laser beam should propagate with high precession to directionality over long distance. When laser beam interacts with a nonlinear medium like plasma, it provides an oscillatory velocity to the electron and hence, changes the dielectric constant related to plasma 6. The process of self-focusing of laser beam in the plasma has been studied by various researchers 7-9 and is important for the above mentioned applications. Describing the plasma as a nonlinear medium, the process of self-focusing of electromagnetic beams was initiated and the effective dielectric constant was supposed to consisting effect of the electric vector amplitude 10. Later on expression for the nonlinear dielectric constant was proposed and considered in laser plasma interactions 11-13 and it resulted in to three distinct regions viz steady divergence, self-focusing and oscillatory divergence. Thereafter, the phenomenon of self-focusing of a laser beam in plasma has extended over to the fields of magnetoplasma 14-17, ramped density plasma 17-24, rippled density plasma 25-30.

Wang and Zhou 31 considered the ponderomotive nonlinearity and found that the plasma density is excelled radially from the central axis and results in plasma cavitation at lower laser temperature. Liu and Tripathi 32 proposed a non-paraxial theory and expanded the eikonal in powers of for the self-focusing of electromagnetic beams. They found that because of strong self-convergence of marginal rays, the ring shaped laser intensity profiles are formed within a very few Rayleigh lengths. Later, Amrita and Sharma 33 used the same theory in studying the thermal self-focusing of the laser beyond near axis approximation in collisional plasma. They reported that the extent of self-focusing increases with enhancement in laser power. Further, the temperature dependence of plasma ionization and recombination processes influences the propagation dynamics of laser beam. Due to increase in recombination and ionization, the electron heating gets reduced which in turn reduces the focusing effect 34.However, for hot magnetized plasmas increase in the strength of wiggler magnetic field leads the laser spot size to decrease and vice versa for increase in temperature. With the result the self-focusing becomes more focused for right-hand polarization and vice versa 35.Furthermore, decrease in plasma temperature and increase in oblique magnetic field increases the quality of self-focusing for right-hand polarization and increases defocusing for left-handed polarization 36.Again, Wani et al. 37 reported that the quality of self-focusing is improved in semiconductor quantum plasma (ScQp). The HchG beam gives freedom to decentered parameter which changes the self-focusing nature of laser beam significantly.

Now, keeping in view the ongoing development of high intense laser beams, current work is aimed to examine combined effect of the plasma density transition and higher order axial electron temperature on the self-focusing / defocusing of laser beam propagating in plasma. The effect of optimized parameters pertaining to laser and plasma is seen on self-focusing of the laser beam in plasma under density ramp. Results so obtained reveal that the higher order axial electron temperature is highly useful for self-focusing than the research of neglecting its effect. The paper is organized as: in section 2, electron temperature and nonlinear dielectric constant is presented. In section 3, the basic formulation and the differential equation defining the behavior of beam width parameter is given. Section 4 pertains to results and discussions. Finally, in section 5, the conclusion is added.

Electron Temperature and Dielectric constant
The electron temperature of for a beam of finite size by considering the effect of higher powers of axial electron temperature, can be expressed as
, (1)
where, , and are the components of electron temperature. The higher powers of axial electron temperature are taken in order to investigate how it affects self-focusing of the laser beam in plasma under ramped density. Further, steady state equation can be taken as:
,(2)
where, andare the electron densities in the presence and absence of laser beam respectively, , and are the components of electron temperature in presence of the laser beam, represents temperature of the plasma in the absence of laser beam. Using Eq. (2), effective dielectric function related to plasma can be stated in the paraxial approximation as:
, (3)
where, the first term on the right hand side represents the linear part and the second and third terms and represent the nonlinear parts of the dielectric function. Their dependence on the electron temperature under plasma density ramp is shown by the following equations:
, ,
, (4)
where,, ,, is the electron plasma frequency in the absence of beam, and are the rest mass and electron charge, is the angular frequency of the incident laser, is plasma frequency, is equilibrium electron density, , represents the diffraction length and is normalized distance of propagation. Substituting for Eq. (4) in Eq. (3), we can have equation of the dielectric function as:
. (5)
Self-focusing
Let us take Gaussian laser beam propagating in plasma along the -axis. The wave equation related can be given as
. (6)
In case of transverse electric field (such as linearly polarized), the term has been neglected as long as ?? 1.

Now, introducing,, where, is the complex amplitude,is wave number corresponding to laser beam. Using WKB approximation, Eq. (6) becomes as:
. (7)
Further, for the laser beam propagation in plasma, we express as
, (8)
where, and are real functions of and (represents amplitude of the electric field andis the eikonal of the laser beam). Using Eq. (8) in Eq. (7) and after separating real and imaginary parts, we can have
Real Part Equation
, (9)
And imaginary part equation

, (10)
In order to solve the equations (9) and (10), we assume in the paraxial region as and the eikonal , where, is dimensionless beam width parameter and , is radius of curvature of the laser beam. Substituting the above values in Eq. (9) and after equating coefficients of on both sides of equation so obtained, the differential equation for the beam width parameter is presented as:

,(11)
where, represents equilibrium beam radius. Eq. (11) represents the spot size variation of laser beam with the distance of propagation in the paraxial region.

Results and Discussion
Eq. (11) represents a nonlinear differential equation which governs the behavior of beam width parameter as a function of propagation distance under the influence of higher order axial electron temperature and slowly varying plasma density ramp. Eq. (11) is solved with initial condition at ,,and for the following set of parameters; , and equilibrium plasma density cm-3. By considering optimized values of laser and plasma parameters, laser beam dynamics in plasma under combined influence of plasma density transition and higher order axial electron temperature have been investigated.

Fig. 1 exhibits graph of beam width parameter with propagation distance ? for various values of . The other parameters taken are, , , and . It is noticed that the laser beam indicates fast divergence in absence of axial electron temperature. However, the beam width parameter increases with increase in showing that defocusing (steady divergence) of laser beam takes place and is minimum at . This is because of the fact that the electron temperature variation is less in initial retarded time and the electron temperature then quickly rises and lastly reduces and tends to a steady state. Therefore, by increasing the temperature component , defocusing (steady divergence) of laser beam increases which can be minimized by increasing quadratic component of axial electron temperature as is depicted in fig. 2 which exhibits the graph of as a function of ? for various values of . Rests of other parameter are same as taken in figure 1. It is clear from fig. 2 that beam width parameter decreases with increase in there by showing that the steady divergence (defocusing) decreases. In other words, the steady convergence (self-focusing) increases and the region for self-focusing gets larger. Therefore, the quadratic temperature component is more sensitive for self-focusing than the component . This is considered to be due to the fact that collisional nonlinearity begins to play its influence and competes with the beam diffraction and leads to beam self-focusing in plasma. Xiong-Piong and Lin 38 studied the self-focusing under the effect of higher order axial electron temperature and their results show that laser beam suffers steady divergence (defocusing) in some regions and oscillatory convergence in other regions. But, in the current work oscillatory behavior of laser beam is completely destroyed and the steady convergence (self-focusing) is enhanced due to the combined effect of density ramp and axial electron temperature.

Figure 3 exhibits the graph of with ? for various values of . Taking the other parameters as , , , and . It is observed from fig. 3 that the steady divergence (defocusing) becomes weaker in absence of axial electron temperature. In other words, the phenomenon of self-focusing gets strengthened for a long propagation distance. This is because of the higher order axial electron temperature that changes the radial distribution of electron density and increases effective dielectric constant. Further, the temperature component decreases the effect of collisional nonlinearity in plasmas. Thus, introduction of higher powers of are more useful than and . Hence, on the one hand reduces defocusing and on the other hand, it quickens the self-focusing. Figure 4 exhibits the graph of with ? for various values of . Other parameters are, , , and . It is observed from fig. 4 that in the absence of slowly varying plasma density transition, the laser beam shows fast divergence (defocusing). However, with increase in , the nonlinearity of medium increases and the amplitude of oscillations decreases more close to the propagation axis than off axis. Consequently, shifts towards larger values of . This is because of the fact that the nonlinearity arising by variation of electron mass due to high intense laser and to the change in the electron density on account of wake field generation. Therefore, self-focusing is noticed at larger values and thus supports the results 39, 40. Hence, under the combined influence of plasma density ramp and axial electron temperature, the quality of self-focusing increases in the focused zone.

Fig. 1: Dependence of on ? for various values of . Other parameters taken are, , , and

Fig. 2: Dependence of on ? for various values of . Other parameters taken are, , , and

Fig. 3: Dependence of on ? for various values of . Other parameters taken are, , , and

Fig. 4: Dependence of on ? for various values of . Other parameters taken are, , and
Conclusion
In the current communication, we observed self-focusing / defocusing of a laser beam in plasma under combined effect of higher order axial electron temperature and plasma density ramp at various optimized laser and plasma parameters. Differential equation for the beam width parameter has been obtained. The results have been plotted and discussed. The results revealed that the higher powers of axial electron temperature are more useful for self-focusing of laser beam in plasma than the research of omitting its effect. Under, the effect of plasma density transition, temperature component on the one hand reduces defocusing in the course of steady divergence and on the other hand, it animates the self-focusing in the course of steady convergence. Further, the higher powers of are more sensitive than and which can change the radial redistribution of electron density and increase effective dielectric constant. Also, after the initial focusing of laser beam, the relativistic mass effect is more prominent in the high plasma density region. Hence, plasma density transition increases the self-focusing effect to a greater extent. The outcomes obtained are expected to add additional information which can be useful in plasma based accelerators and laser driven fusion.

References
1H. Y. Niu, X. T. He, B. Qiao, C. T. Zhou, Resonant acceleration of electrons by intense circularly polarized Gaussian laser pulses, Laser and Particle Beams 26 (2008) 51.

2 P. Sprangle, E. Esarey, J. Krall, Laser driven electron acceleration in vacuum, gases and plasmas, Phys. Plasmas3(1996) 2183.

3 W. Yu, M. Y. Yu, J. Zhang, Z. Xu, Harmonic generation by relativistic electrons during irradiance of a solid target by a short pulse ultra-intense laser, Phys. Rev. E, 57 (1998) 2531
4 P. Sprangle, E. Esarey, Interaction of ultrahigh laser fields with beams and plasmas, Physics Fluids B 4 (1992) 2241.

5 D. Umstadter, Review of physics and applications of relativistic plasmas driven by ultra-intense lasers, Phys. Plasmas 8(2001) 1774.

6 H. Hora, Theory of relativistic self-focusing of laser radiations in plasmas, J. Opt. Soc. Am. 65(1975) 882.7 M. S. Sodha, L. A. Patel, S. C. Kaushik, Self-focusing of a laser beam in an inhomogeneous plasma, Phys. Plasmas 21(1979) 1.8 H. S. Brandi, C. Manus, G. Mainfray, T. Lehner, G. Bonnaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. 1. Paraxial approximation, Phys. Fluids B 5(1993) 3539.

9 A. R. Niknam, M. Hashemzadeh, M. M. Montazeri, Numerical, investigation of ponderomotive force effect in an underdense plasma with a linear density profile. IEEE Trans. Plasma Sci. 38(2010) 2390.10 S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov, Self-focusing and diffraction of light in a nonlinear medium, Sov. Phys. Usp. 10(1968) 609.11M. S. Sodha, A. K. Ghatak, V. K. Tripathi, Self-focusing of lasers in plasmas and semiconductors, Prog. Opt. 13(1976) 169.12 M. S. Sodha, A. K. Ghatak, V. K. Tripathi, Self-focusing of laser beams in dielectrics, Plasmas and semiconductors, Tata McGraw-Hill, Delhi, 1974.

13M. S. Sodha, V. K. Tripathi, Nonlinear penetration of an inhomogeneous laser beam in an overdense plasma, Phys. Rev. A 16 (1977) 2101.

14 T. S. Gill, R. Mahajan, R. Kaur, S. Gupta, Relativistic self-focusing of super-Gaussian laser beam in plasma with transverse magnetic field, Laser and particle Beams30(2012)509.

15 S. D. Patil, M. V. Takale, S. T. Navare, M. B. Dongare, Focusing of Hermite-cosh-Gaussian laser beams in collision-less magneto plasma, Laser and Particle Beams28(2010) 343.

16 H. Xiong, S. Liu, J. Liao, X. Liu, Self-focusing of laser pulse propagating in magnetized plasma, Optik121(2010) 1680.

17 V. Nanda, N. Kant, M. A. Wani, Self-focusing of Hermite-cosh Gaussian laser beam in a magnetoplasma with a ramp density profile, Phys. Plasmas 20 (2013) 113109.18 M. A. Wani, N. Kant, Nonlinear propagation of Gaussian laser beam in an inhomogeneous Plasma under plasma density ramp, Optik 127 (2016) 6710.

19 N. Kant, M. A. Wani, A. Kumar,Self-focusing of Hermite- Gaussian laser beams in plasma under plasma density ramp, Opt. Commun. 285(2012)4483.20 S. D. Patil, M. V. Takale, Self-focusing of Gaussian laser beam in weakly relativistic and ponderomotive regime using upward ramp of plasma density, Phys. Plasmas 20(2013) 083101.

21 V. Nanda, N. Kant, Enhanced relativistic self-focusing of Hermite- cosh Gaussian laser beam in plasma under density transition, Phys. Plasmas 21(2014) 042101.

22 S. Zare, E. Yazdani, S. Razaee, A. Anvari, R. Sadighi-Bonabi, Relativistic self-focusing of intense laser beam in thermal collisionless quantum plasma with ramped density profile, Phys. Rev. ST Accel. Beams 18(2015)041301.23 M. Aggarwal, H. Kumar, N. Kant, Propagation of Gaussian laser beam through magnetized cold plasma with increasing density ramp, Optik 127(2015) 2212.

24 N. Kant, M. A. Wani, Density transition based self-focusing of cosh-Gaussian laser beams in plasma with linear absorption, Commun. Theor. Phys.64(2015)103.

25 S. Kaur, A. K. Sharma, C. M. Ryu, Nonlinear propagation of a short-pulse laser in a plasma with density ripple, Korean Phys. Soc. 53(2008)3768.

26 M. Aggarwal, S. Vij, N. Kant, Propagation of cosh-Gaussian laser beam in plasma with density ripple in relativistic-ponderomotive regime, Optik 125(2014)5081.

27 S. Kaur, A. K. Sharma, Self-focusing of a laser pulse in plasma with periodic density ripple, Laser and particle Beams 27(2009)193.28 M. A. Wani, N. Kant, Self-focusing of a laser beam in the rippled density magnetoplasma, Optik 128 (2017) 1.29 H. C. Barr, F. F. Chen, Raman scattering in a nearly resonant density ripple, Phys. Fluids 30(1987) 1180.
30 D. Dahiya, V. Sajal, A. K. Sharma, Phase-matched second and third harmonic generation in plasmas with density ripple, Physics of Plasmas 14 (2007) 123104.

31 Y. Wang, Z. Zhou, Propagation characteristics of Gaussian laser beams in collisionless plasma: Effect of plasma temperature, Phys. Plasmas 18 (2011)043101
32 C. S. Liu, V. K. Tripathi, Laser frequency upshift, self-defocusing and ring formation in tunnel ionizing gases and plasmas,Phys. Plasmas 7 (2000)4360.

33 Amrita, A. K. Sharma, Thermal self-focusing of a laser in collisional plasma: beyond the near axis approximation, Phys. Scr. 74 (2006) 128.

34 S. Misra, M. S. Sodha, S. K. Mishra, Self-focusing of coaxial electromagnetic beams in a plasma with electron temperature dependent electron– ion recombination coefficient,Opt. Commun. 385(2017)71-77.

35 M. A. Varaki and S. Jafari, Relativistic self-focusing of an intense laser pulse with hot magnetized plasma in the presence of a helical magnetostatic wiggler,Phys. Plasmas 24 (2017)082309.

36 M. A. Varaki and S. Jafari, Nonlinear interaction of intense left and right hand polarized laser pulse with hot magnetized plasma, J. Plasma Phys., 83 (2017) 655830401.

37 M. A. Wani, H. S. Ghotra,N. Kant, Self-focusing of Hermite-cosh-Gaussian laser beam in semiconductor quantum plasma, Optik 154 (2018) 497-502.

38 X. Xiong-Piong, Y. Lin, Effect of higher order axial electron temperature on self-focusing of electromagnetic pulsed beam in collisional plasma, Commun. Theor. Phys. 57 (2012) 873.39 S. T. Navare, M. V. Takale, S. D. Patil, V. J. Fulari, M. B. Dongare, Impact of linear absorption on self-focusing of Gaussian laser beam in collisional plasma, Opt. and Lasers in Engineering 50 (2012) 1316.

40 S. D. Patil, M. V. Takale, V. J. Fulari, D. N. Gupta, H. Suk, Combined effect of ponderomotive and relativistic self-focusing on laser beam propagation in a plasma, Appl. Phys. B 111 (2013) 1.