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.
Keshav Walia. Propagation characteristics of a high-power beam in weakly relativistic-ponderomotive thermal quantum plasma[J]. Communications in Theoretical Physics, 2023, 75(9): 095501. DOI: 10.1088/1572-9494/accf82
1. Introduction
Recent advances in technology have led to the construction of lasers with intensities of the order of 1018 W 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 parameter $\chi =\tfrac{{T}_{{\rm{F}}}}{T}.$ (${T}_{{\rm{F}}}\,{\rm{and}}\,T$ correspond to Fermi temperature and temperature of plasma). For $\chi \geqslant 1,$ 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 form $f\left(r\right)=f\left(0\right){\left(1+\tfrac{{r}^{2}}{q{r}_{0}^{2}}\right)}^{-q}.$ One can change the q-Gaussian beams into ordinary Gaussian beams by considering $q\,\to \,\infty .$ 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, E00 and 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 as $q\to \,\infty .$ For $z\gt 0,$ distribution of irradiance for the q-Gaussian beam is best described by
In equations (3) and (4), $D=\varepsilon E$ is 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
In equation (5), one can ignore the polarization term ${\rm{\nabla }}\left({\rm{\nabla }}\cdot E\right)$ considering the r.m.s beam radius to be more than the wavelength in vacuum. Further, ${\rm{\nabla }}\left({\rm{\nabla }}\cdot E\right)$ can also be ignored on the assumption that $\tfrac{1}{{k}^{2}}\left|{{\rm{\nabla }}}^{2}\,\mathrm{ln}\,\in \right|\ll 1,$ where, $k,\,\omega $ and $\varepsilon $ correspond to the wave vector, frequency of laser beam, and effective dielectric function of medium, respectively. So, equation (5) reduces to
In equation (7), ${v}_{{\rm{f}}}=\sqrt{\tfrac{2{K}_{{\rm{B}}}{T}_{{\rm{f}}}}{m}}$ is known as the Fermi speed, relativistic Lorentz factor $\gamma ={\left(1+\alpha E{E}^{* }\right)}^{1/2}$ with $\alpha =\tfrac{{e}^{2}}{{m}_{0}^{2}{c}^{2}{\omega }^{2}}$ as the nonlinear coefficient and $\delta q=\tfrac{4{\pi }^{4}{h}^{2}}{{m}^{2}{\omega }^{2}{\lambda }^{4}}.$ If the Fermi temperature (${T}_{{\rm{f}}}$) is set equal to zero, then the above expression for the dielectric function for TQP becomes an expression for cold quantum plasma (CQP). Further, if both ${T}_{{\rm{f}}}$ and $\tfrac{h}{2\pi }$ are set equal to zero, then equation (7) becomes the simple case of classical relativistic plasma (CRP). The plasma frequency ${\omega }_{{\rm{p}}}$ is represented as ${\omega }_{{\rm{p}}}=\sqrt{\tfrac{4\pi n{e}^{2}}{m}}.$ The number density of electrons is changed due to the action of the nonlinear ponderomotive force. The changed number density for electrons can be expressed as [53]
In equation (9), the linear and nonlinear parts connected with a dielectric function can be represented as ${\varepsilon }_{0}=1-\tfrac{{\omega }_{{\rm{p}}}^{2}}{{\omega }_{0}^{2}}$ and $\varphi \left(E{E}_{0}^{* }\right).$ In the present case, we are adopting relativistic and ponderomotive nonlinearities collectively. So, under the combined action of relativistic-ponderomotive nonlinearity in TQP, the expression for dielectric function becomes
In equation (13), ‘S’ represents the eikonal for the beam. The phase shift of a given beam is denoted by ${{\rm{\Phi }}}_{0}\left(z\right).$ Since, we are stressing on beam irradiance in the present investigation rather than its phase. So, the ${{\rm{\Phi }}}_{0}\left(z\right)$ term will not be needed in further analysis. $^{\prime} f^{\prime} $ denotes beam waist associated with q-Gaussian beam and satisfies the following second-order differential equation
In equation (15), η denotes the dimensionless propagation distance. We need to have some initial boundary conditions for obtaining numerical solutions of equation (15). We adopt specific boundary conditions as = 1, $\tfrac{{\rm{d}}f}{{\rm{d}}\eta }=0$ at η = 0.
3. Discussion
The initial boundary condition for solving the second-order differential equation represented by equation (15) is f = 1 and $\tfrac{{\rm{d}}f}{{\rm{d}}\eta }=0$ at η = 0. From equation (15), it is clear that there are two terms on RHS. The first term is a diffractive term, whereas the second term arising due to a nonlinear medium is a converging term. We have taken plasma temperature and number density in such a manner that the condition of quantum plasma $n{\lambda }_{B}^{3}\geqslant 1$ is fulfilled. During laser beam transition through plasma, the overall converging/diverging tendency is governed by the comparative magnitude of both diffractive and converging terms. The analytic solution of equation (15) is not possible, so this equation is solved numerically. The following laser–plasma parameters are chosen for carrying out the numerical simulation of equation (15);
Figure 1 denotes the change in beam width $f$ with dimensionless propagation distance $\eta $ at different beam intensities $\alpha {E}_{00}^{2}(\alpha {E}_{00}^{2}=2.0,\,3.0,\,4.0)$ with other parameters kept fixed. The black curve, red curve, and green curve are for $\alpha {E}_{00}^{2}=2.0,\,3.0\,{\rm{and}}\,4.0$ respectively. The shift in beam width $f$ towards larger $\eta $ values is found with an increase in $\alpha {E}_{00}^{2}$ values. In other words, the focusing character of the beam is reduced with an increase in $\alpha {E}_{00}^{2}$ values. This is because the diffractive term starts dominating over the converging term at larger $\alpha {E}_{00}^{2}$ values. Hence, there is a decrease in focusing behavior at larger $\alpha {E}_{00}^{2}.$
Figure 1. Denotes change in beam width $f$ with dimensionless propagation distance $\eta $ at different beam intensities $\alpha {E}_{00}^{2}\left(\alpha {E}_{00}^{2}=2.0,\,3.0,\,4.0\right)$ with other parameters kept fixed. The black curve, red curve, and green curve are for $\alpha {E}_{00}^{2}=2.0,\,3.0\,{\rm{and}}\,4.0$, respectively.
Figure 2 denotes a change in beam width $f$ with dimensionless propagation distance $\eta $ at different plasma densities $\tfrac{{\omega }_{{\rm{p}}}^{2}}{{\omega }^{2}}\left(\tfrac{{\omega }_{{\rm{p}}}^{2}}{{\omega }^{2}}=0.3,\,0.4,\,0.5\right)$ with other parameters kept fixed. The black curve, red curve, and green curve are for $\tfrac{{\omega }_{{\rm{p}}}^{2}}{{\omega }^{2}}=0.3,\,0.4\,{\rm{and}}\,0.5$, respectively. The shift in beam width $f$ towards smaller $\eta $ values is found with an increase in $\tfrac{{\omega }_{{\rm{p}}}^{2}}{{\omega }^{2}}$ values. In other words, the focusing character of the beam is enhanced with an increase in $\tfrac{{\omega }_{{\rm{p}}}^{2}}{{\omega }^{2}}$ values. This is because the converging term starts dominating over the diffractive term at larger $\tfrac{{\omega }_{{\rm{p}}}^{2}}{{\omega }^{2}}$ values. Hence, there is an increase in focusing behavior at larger $\tfrac{{\omega }_{{\rm{p}}}^{2}}{{\omega }^{2}}.$
Figure 2. Denotes change in beam width $f$ with dimensionless propagation distance $\eta $ at different plasma densities $\tfrac{{\omega }_{{\rm{p}}}^{2}}{{\omega }^{2}}\left(\tfrac{{\omega }_{{\rm{p}}}^{2}}{{\omega }^{2}}=0.3,\,0.4,\,0.5\right)$ with other parameters kept fixed. The black curve, red curve, and green curve are for $\tfrac{{\omega }_{{\rm{p}}}^{2}}{{\omega }^{2}}=0.3,\,0.4,\,0.5=0.3,\,0.4\,{\rm{and}}\,0.5$ respectively.
Figure 3 denotes the change in beam width $f$ with dimensionless propagation distance $\eta $ at different Fermi temperatures ${T}_{{\rm{F}}}({T}_{{\rm{F}}}={10}^{7}\,{\rm{K}},\,{10}^{8}\,{\rm{K}},\,{10}^{9}\,{\rm{K}})$ with other parameters kept fixed. The black curve, red curve, and green curve correspond to ${T}_{{\rm{F}}}={10}^{7}\,{\rm{K}},\,{10}^{8}\,{\rm{K}}\,{\rm{and}}\,{10}^{9}\,{\rm{K}}.$ The beam width $f$ is shifted towards lesser $\eta $ values with increments in ${T}_{{\rm{F}}}$ values. In other words, the focusing character of the beam is enhanced with an increase in ${T}_{{\rm{F}}}$ values. This is because the converging term starts dominating over the diffractive term at larger ${T}_{{\rm{F}}}$ values. Hence, there is an increase in focusing behavior at larger Fermi temperature ${T}_{{\rm{F}}}.$
Figure 3. Denotes change in beam width $f$ with dimensionless propagation distance $\eta $ at different Fermi temperatures ${T}_{{\rm{F}}}\left({T}_{{\rm{F}}}={10}^{7}\,{\rm{K}},\,{10}^{8}\,{\rm{K}},\,{10}^{9}\,{\rm{K}}\right)$ with other parameters kept fixed. The black curve, red curve, and green curve are for ${T}_{{\rm{F}}}={10}^{7}\,{\rm{K}},{10}^{8}\,{\rm{K}}\,{\rm{and}}\,{10}^{9}\,{\rm{K}}$, respectively.
Figure 4 denotes the change in beam width $f$ with dimensionless propagation distance $\eta $ at different ‘q’ values $(q=1,\,2,\,3)$ with other parameters kept fixed. The black curve, red curve, and green curve correspond to $q=1,\,2\,{\rm{and}}\,3.$ The beam width $f$ gets shifted towards lesser $\eta $ values with increment in q values i.e. focusing character of the beam gets improved with increments in $q\,$values. 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 larger $q$ values.
Figure 4. Denotes change in beam width $f$ with dimensionless propagation distance $\eta $ at different ‘q’ values $(q=1,\,2,\,3)$ with other parameters kept fixed. The black curve, red curve, and green curve correspond to $q=1,\,2\,{\rm{and}}\,3.$
Figure 5 denotes the change in beam width $f$ with dimensionless propagation distance $\eta $ in 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 smaller $\eta $ values 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 5. Denotes change in beam width $f$ with dimensionless propagation distance $\eta $ 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 in $\alpha {E}_{00}^{2}$ values.
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 confinement fusion scheme.
LemoffB EYinG YGordonIIIC LBartyC P JHarrisS E1995 Demonstration of a 10-Hz femtosecond-pulse-driven XUV Laser at 41.8 nm in Xe IX Phys. Rev. Lett.74 1574
DeutschCBretAFirpoM CGremilletLLefebvreELifschitzA2008 Onset of coherent electromagnetic structures in the relativistic electron beam deuterium-tritium fuel interaction of fast ignition concern Laser Part. Beams26 157
BhatiaAWaliaKSinghA2021 Second harmonic generation of intense Laguerre–Gaussian beam in relativistic plasma having an exponential density transition Optik244 167608
WaliaK2020 Stimulated Brillouin scattering of high power beam in unmagnetized plasma: effect of relativistic and ponderomotive nonlinearities Optik221 165365
WaliaKVermaR KSinghA2021 Second harmonic generation of laser beam in quantum plasma under collective influence of relativistic-ponderomotive nonlinearities Optik255 165745
24
WaliaKSharmaPSinghA2021 Second harmonic generation of Cosh-Gaussian beam in unmagnetized plasmas: effect of relativistic-ponderomotive force Optik245 167627
BhatiaAWaliaKSinghA2021 Influence of self-focused Laguerre–Gaussian laser beam on second harmonic generation in collisionless plasma having density transition Optik245 167747
AndreevA V2000 Self-consistent equations for the interaction of an atom with an electromagnetic field of arbitrary intensity J. Exp. Theor. Phys. Lett.72 238
WaliaKSinghA2021 Non-linear interaction of Cosh-Gaussian beam in thermal quantum plasma under combined influence of relativistic-ponderomotive force Optik247 167867
ZareSYazdaniERezaeeSAnvariASadighi-BonabiR2015 Relativistic self-focusing of intense laser beam in thermal collisionless quantum plasma with ramped density profile Phys. Rev. ST Accel. Beams18 041301
50
KumarHAggarwalMRichaGillT S2016 Combined effect of relativistic and ponderomotive nonlinearity on self-focusing of Gaussian laser beam in a cold quantum plasma Laser Part. Beams34 426
NiknamA RHashemzadehMShokriB2009 Weakly relativistic and ponderomotive effects on the density steepening in the interaction of an intense laser pulse with an underdense plasma Phys. Plasmas16 033105