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Generation of terahertz radiation by a Hermite–Gaussian laser beam inside magnetoplasma with a density ramp

Published online by Cambridge University Press:  11 April 2023

Proxy Kad ,
Vidisha Rana  and
Arvinder Singh Open the ORCID record for Arvinder Singh [Opens in a new window]
Proxy Kad
Affiliation:
Dr. B.R. Ambedkar National Institute of Technology, Jalandhar 144011, Punjab, India
Vidisha Rana
Affiliation:
Dr. B.R. Ambedkar National Institute of Technology, Jalandhar 144011, Punjab, India
Arvinder Singh*
Affiliation:
Dr. B.R. Ambedkar National Institute of Technology, Jalandhar 144011, Punjab, India
*
†Email address for correspondence: [email protected]
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Abstract

In the present scheme of work, the Hermite–Gaussian (HG) laser beam dynamics has been investigated under the influence of an upward density ramp inside magnetized plasma, where both relativistic and ponderomotive nonlinearities are operative. One can achieve self-focusing of laser beam due to the change in the medium's dielectric function, which comes into operation due to the expulsion of plasma electrons from the high intensity to the low-intensity region by ponderomotive force and their motion at relativistic speeds. The dynamics of the laser beam and terahertz generation have been investigated by using the moment theory approach. It has been observed from the present analysis that the dynamics of the laser beam and the production of terahertz radiations strongly depends upon the HG laser beam and plasma parameters. In addition to this, the effect of density ramp and magnetic field has also been investigated on the efficiency of terahertz generation. It has been observed that higher-order modes of the HG laser beam play a dominant role in the production of terahertz radiations.

Keywords

plasma nonlinear phenomenaplasma waves
Type
Research Article
Information
Journal of Plasma Physics , Volume 89 , Issue 2 , April 2023 , 905890207
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press

1. Introduction

Terahertz radiations refer to the high-frequency radiations lying in the range of $(0.1$–$1)\times 10^{12}$ Hz. These radiations find a wide range of applications in the area of communication (Song & Nagatsuma Reference Song and Nagatsuma2011), spectroscopy (Beard, Turner & Schmuttenmaer Reference Beard, Turner and Schmuttenmaer2002), medical imaging (Han Reference Han, Park, Kim, Han, Han, Ahn, Son, Park and Jeong2012), security (Kemp et al. Reference Kemp, Taday, Cole, Cluff, Fitzgerald and Tribe2003), etc., due to their ability to penetrate any medium without causing any ionization. Terahertz generation using terawatt-level lasers was first proposed and exhibited by Hamster et al. (Reference Hamster, Sullivan, Gordon, White and Falcone1993). Some methods to produce terahertz radiations involve using nonlinear crystals by optical rectification (Singh et al. Reference Singh, Singh, Rajouria and Sharma2017), photo-conduction processes (Cai et al. Reference Cai, Brener, Lopata, Wynn, Pfeiffer and Federici1997) and by coupling of lasers with anharmonic carbon nanotubes (CNTs) (Kumar et al. Reference Kumar, Vij, Kant and Thakur2022a,Reference Kumar, Vij, Kant and Thakurb, Reference Kumar, Vij, Kant and Thakur2023). Using nonlinear crystals, it has been possible to go up to a laser intensity of the order of $10^{10-11}\,{\rm W}\,{\rm cm}^{-2}$. Other methods involve using accelerator-based sources which can produce terahertz radiations with high repetition rates, brightness and power. However, the limited accessibility and availability of accelerators and the breakdown of nonlinear crystals at high-intensity electric field cause a limitation in the production of high power and efficient terahertz radiation. An alternate way to produce efficient terahertz radiation involves using laser–plasma interaction (Hassan et al. Reference Hassan, Al-Janabi, Singh and Sharma2012).

Laser–plasma interaction has led to many developments in terahertz generation (Sobhani, Dadar & Feili Reference Sobhani, Dadar and Feili2017; Sun, Wang & Zhang Reference Sun, Wang and Zhang2022), second-harmonic generation (Upadhyay & Tripathi Reference Upadhyay and Tripathi2005; Sharma, Thakur & Kant Reference Sharma, Thakur and Kant2020), plasma-wakefield excitation for particle acceleration (Kim et al. Reference Kim, Pathak, Hojbota, Mirzaie, Pae, Kim, Yoon, Sung and Lee2021; Kad & Singh Reference Kad and Singh2022b), etc. Plasma being a nonlinear medium has an extraordinarily high damage threshold, and is thus capable of sustaining a high-intensity electric field (Hassan et al. Reference Hassan, Al-Janabi, Singh and Sharma2012). An electromagnetic wave can travel only small distances (called Rayleigh length) due to its tendency to diffract in any medium. But for efficient terahertz generation, the laser-medium interaction time should be long enough. Thus, using plasma as the medium, one can overcome these restrictions. The laser beam on interacting with the plasma medium undergoes nonlinear phenomena such as self-focusing (Mori et al. Reference Mori, Joshi, Dawson, Forslund and Kindel1988), self-trapping (Singh & Walia Reference Singh and Walia2010), self-compression (Saedjalil & Jafari Reference Saedjalil and Jafari2016), etc. Self-focusing helps to balance out the diffraction of the laser beam in plasma by acting as a waveguide.

Much work has already been done on terahertz generation by either coupling two different laser beams (Malik, Malik & Nishida Reference Malik, Malik and Nishida2011) or by using a single laser beam. The efficiency of terahertz generation can be increased by introducing a density ramp, thereby taking into account the non-homogeneity of plasma. The effect of introducing a density ramp on terahertz radiation has been illustrated by Miao, Palastro & Antonsen (Reference Miao, Palastro and Antonsen2016). Singh & Sharma (Reference Singh and Sharma2013) presented terahertz generation in a rippled density plasma. Niknam et al. (Reference Niknam, Banjafar, Jahangiri, Barzegar and Massudi2016) has investigated the generation of terahertz radiation in inhomogeneous collisional plasma by the interaction of two laser beams. Furthermore, the magnetic field also affects the generation of terahertz radiation by affecting the extent of self-focusing of laser beams. The effect of magnetized plasma in nonlinear plasma has been depicted by Sharma et al. (Reference Sharma, Monika, Sharma, Chauhan and Ji2010). Strong terahertz production under the non-relativistic ponderomotive regime in magnetized collisional plasma has been demonstrated by Varshney et al. (Reference Varshney, Upadhayay, Madhubabu, Sajal and Chakera2018). Gupta & Jain (Reference Gupta and Jain2021) have used a super-Gaussian laser pulse to produce terahertz radiation in magnetized plasma.

From the literature review, it has been concluded that most of the previous research work in the field of terahertz generation has been carried out by the interaction of Gaussian profile beams with a collisionless plasma medium using the paraxial theory approach. The present work demonstrates the scheme of generation of terahertz radiation using a laser beam with Hermite–Gaussian (HG) profile in magnetized plasma with a density ramp under a relativistic-ponderomotive regime. To the best of the authors’ knowledge, no earlier theoretical investigation using the moment theory approach for terahertz generation has been carried out by the higher-order modes of a HG laser beam in a relativistic-ponderomotive magnetoplasma having an exponential density ramp. Ponderomotive self-focusing takes place when the electrons are expelled from the high- to low-intensity region due to the ponderomotive force which is associated with a spatial gradient in the laser beam intensity. This, along with relativistic effects, changes the refractive index and dielectric properties of the plasma medium due to the motion of plasma electrons at relativistic speeds. The paper is aimed at investigating the influence of higher-order modes of a HG laser beam, its intensity and the slope of density ramp and magnetic field on the efficiency of terahertz generation. Unlike the laser beams with Gaussian profiles which are studied by paraxial theory, its interaction with plasma is studied through the method of moments. This is because for laser beams with super-Gaussian profile such as Laguerre–Gaussian (Kad & Singh Reference Kad and Singh2022a,Reference Kad and Singh2022c), Bessel–Gaussian (Kad et al. Reference Kad, Choudhary, Bhatia, Walia and Singh2022), HG (Wadhwa & Singh Reference Wadhwa and Singh2020), etc., one needs to take into account the intensity of the off-axial parts along with the axial parts. In paraxial theory, only the axial part of the laser intensity is taken into consideration and the off-axial parts are neglected. The structure of the paper is as follows. In § 2, the HG laser beam profile is depicted. Sections 3 and 4 illustrate the modification of the dielectric function of plasma and laser dynamics inside the plasma, respectively. In § 5, plasma wave excitation and generation of terahertz radiation are discussed. The results are discussed in § 6, followed by the conclusion of the obtained results in § 7.

2. HG laser beam profile

For a laser beam propagating in plasma, the electric field vector along the $z$ axis is given by

(2.1)\begin{equation} E(x,y,z) = \varPsi(x,y,z) {\rm e}^{{\rm i}\{\omega t - kz\}}, \end{equation}

where, $\omega$ and k represents the angular frequency and the wavevector, respectively. Here $\varPsi$ denotes the complex amplitude of HG laser beam's electric field and the equation that governs its intensity distribution is given as

(2.2)\begin{equation} I = \varPsi \varPsi^*, \end{equation}

where

(2.3)\begin{equation} \varPsi \varPsi^* = \frac{E_{00}^2}{f_x f_y} \exp \left(-\left(\frac{x^2}{x_0^2 f_x^2} + \frac{y^2}{y_0^2 f_y^2}\right)\right) H_m^2 \left(\frac{x}{x_0 f_x}\right) H_n^2 \left(\frac{y}{y_0 f_y}\right). \end{equation}

Here, $x_0$ and $y_0$ denote the HG laser beam's spot size before entering the plasma along transverse axes $x$ and $y$. Here $E_{00}$ is the maximum electric field amplitude along the axis and $f_x$ and $f_y$ represents the beam width parameters along the transverse axes, respectively. Here $m$ and $n$ correspondingly depict the number of nodes along the $x$ and $y$ directions as well as the degrees of Hermite polynomials $H_m$ and $H_n$. The intensity distribution plots for transverse electromagnetic (TEM) modes (0,0), (0,1) and (0,2) of the HG laser beam have been depicted by figures 1(a), 1(b) and 1(c), respectively.

Figure 1. Three-dimensional normalized intensity distribution plots for various TEM modes (0,0), (0,1) and (0,2) depicted by panels (a), (b) and (c) respectively.

3. Nonlinear dielectric function

Due to nonlinear laser–plasma interaction, a relativistic ponderomotive force acts upon the electrons in the plasma which causes a relativistic variation in its mass which implies that the rest mass of an electron ($m_0$) gets modified by a factor of $\gamma$ (i.e. $m_0 \rightarrow m_0\gamma$). Due to this, the refractive index and the dielectric function gets modified, which now comprises of a linear $(\epsilon _0)$ and a nonlinear term $\chi (EE^*)$. Under the combined effect of magnetic field, density ramp and relativistic-ponderomotive nonlinearity, the dielectric function is obtained as follows:

(3.1)\begin{equation} \epsilon = \epsilon_0 + \chi(EE^*), \end{equation}

where

(3.2)\begin{gather} \epsilon = 1 - \frac{\omega_{p}^2}{\gamma \omega^2}\exp \left(-\frac{m_0c^2}{T_e}(\gamma - 1)\right), \end{gather}
(3.3)\begin{gather} \epsilon_0 = 1 - \frac{\omega_{p}^2}{\omega^2}. \end{gather}

Where, $\omega _p$ is the non-homogeneous plasma frequency in the absence of laser beam which increases exponentially with the distance of propagation $z$ and is given by

(3.4)\begin{equation} \omega_p^2 = \omega_{p0}^2 \exp\left(\frac{z/kx_0^2}{d}\right), \end{equation}

where $d$ is the slope of density ramp, $T_e$ is the temperature in energy units and $\gamma$ is the Lorentz factor given by

(3.5)\begin{equation} \gamma = (1 + \beta^\prime EE^*)^{{1}/{2}}, \end{equation}

and

(3.6)\begin{equation} \beta^\prime = \frac{\beta}{\left(1 - \dfrac{\omega_c}{\omega}\right)}, \end{equation}

is the coefficient of nonlinearity which takes into account the effect of the plasma being magnetized.

Using (2.2), (3.2)–(3.5) and substituting in (3.1), the nonlinear dielectric function is obtained as

(3.7)\begin{equation} \chi(EE^*) = \frac{\omega_{p}^2}{\omega^2} \left[1 - \frac{1}{\gamma} \exp \left(\frac{-m_0c^2}{T_e}(\gamma - 1)\right)\right]. \end{equation}

4. HG laser beam dynamics

The dynamical behaviour of the laser beam while travelling through plasma is governed by the following Maxwell's equations:

(4.1)\begin{gather} \boldsymbol{\nabla} \times{\boldsymbol{E}} ={-}\frac{1}{c} \frac{\partial \boldsymbol{B} }{\partial t}, \end{gather}
(4.2)\begin{gather} \boldsymbol{\nabla} \times{\boldsymbol{B}} ={-} \frac{\epsilon}{c} \frac{\partial \boldsymbol{E}}{\partial t}. \end{gather}

On using (4.1) and (4.2) one can obtain the wave equation as

(4.3)\begin{equation} \nabla^2 \boldsymbol{E} - \frac{\epsilon}{c^2} \frac{\partial^2 \boldsymbol{E}}{\partial t^2} = 0. \end{equation}

Assuming that the modifications in the transverse directions are much faster than those in the $z$ direction and using (2.1), (2.2) in (4.3), one can obtain the nonlinear Schrodinger wave equation as follows:

(4.4)\begin{equation} \iota\frac{{\rm d}\varPsi}{{\rm d}z} = \frac{1}{4k}\left(1 + \frac{\epsilon_{0+}}{\epsilon_{0_{zz}}}\right)\boldsymbol{\nabla}_\perp^2\varPsi + \frac{k}{2\epsilon_0}\chi(EE^*) \varPsi. \end{equation}

Using the method of moments, given by Vlasov, Petrishchev & Talanov (Reference Vlasov, Petrishchev and Talanov1971), one can find the root mean square width of a laser beam as

(4.5)\begin{equation} a_{rms} = \frac{\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}(x^2 + y^2) \varPsi \varPsi^*\,{{\rm d} x}\,{{\rm d} y}}{\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \varPsi \varPsi^*\,{{\rm d} x}\,{{\rm d} y}}. \end{equation}

On differentiating equation (4.5) twice with respect to $z$ and using (4.4), we get

(4.6)\begin{gather} \frac{{\rm d}^2 a_{rms}}{{\rm d}z^2} = \frac{2}{k \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \varPsi \varPsi^*\,{{\rm d} x}\,{{\rm d}y}} \left\{\frac{1}{4k}\left(1+ \frac{\epsilon_{0+}}{\epsilon_{0zz}}\right)^2 \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} |\boldsymbol{\nabla}_\perp \varPsi|^2 \,{{\rm d} x}\,{{\rm d} y} \right.\nonumber\\ \left.+\frac{k}{4\epsilon_0} \left(1+\frac{\epsilon_{0+}}{\epsilon_{0zz}}\right) \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} |\varPsi|^2 \left(x\frac{\partial \chi(EE^*) }{\partial x} + y\frac{\partial \chi(EE^*)}{\partial y}\right) \,{{\rm d} x}\,{{\rm d} y}\right\}. \end{gather}

On using (2.2) in (4.5), we obtain the value of $a_{rms}$ as

(4.7)\begin{equation} a_{rms} = x_0^2f_x^2 \left(m + \frac{1}{2}\right) + y_0^2f_y^2\left(n+\frac{1}{2}\right). \end{equation}

On differentiating equation (4.7) with respect to $z$ and using (2.2), (3.7) and (4.6), we get the two coupled second-order differential equations as

(4.8)\begin{gather} \dfrac{{\rm d}^2f_x}{{\rm d}\zeta^2} + \dfrac{1}{f_x}\left(\dfrac{{\rm d} f_x}{{\rm d}\zeta}\right)^2 = \dfrac{1}{4}\left(1 + \dfrac{\epsilon_{0+}}{\epsilon_{0zz}}\right)^2\dfrac{1}{f_x^3} + \dfrac{1}{2}\left(1 + \dfrac{\epsilon_{0+}}{\epsilon_{0zz}}\right)\dfrac{\beta^\prime E_{00}^2}{{\rm \pi} f_x^2f_y}\varPhi\nonumber\\ \dfrac{I_1}{\left(1 - \dfrac{\omega_c}{\omega}\right)(m + 1/2) 2^{m + n + 1} m! n!} , \end{gather}
(4.9)\begin{gather}\dfrac{{\rm d}^2f_y}{{\rm d}\zeta^2} + \dfrac{1}{f_y}\left(\dfrac{{\rm d} f_y}{{\rm d}\zeta}\right)^2 = \dfrac{1}{4}\left(1 + \dfrac{\epsilon_{0+}}{\epsilon_{0zz}}\right)^2\dfrac{(x_0/y_0)^4}{f_y^3} + (x_0/y_0)^4\dfrac{1}{2}\left(1 + \dfrac{\epsilon_{0+}}{\epsilon_{0zz}}\right)\dfrac{\beta^\prime E_{00}^2}{{\rm \pi} f_x f_y^2}\nonumber\\ \varPhi\dfrac{I_2}{\left(1 - \dfrac{\omega_c}{\omega}\right)(n + 1/2) 2^{m + n + 1} m! n!}, \end{gather}

where

(4.10)\begin{gather} I_1 = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x^\prime \exp\left({-2\left({x^\prime}^2 + {y^\prime}^2 + \dfrac{m_0c^2(J(x^\prime,y^\prime)^{1/2} - 1)}{2T_e}\right)}\right)\nonumber\\ \times H_m^3(x^\prime)H_n^4(y^\prime)\dfrac{(x^\prime H_m(x^\prime)-H_{m+1}(x^\prime))}{J(x^\prime,y^\prime)} \left(\dfrac{1}{J(x^\prime,y^\prime)^{1/2}} - \dfrac{m_0c^2}{T_e}\right) \,{{\rm d} x}^\prime\,{{\rm d} y}^\prime, \end{gather}
(4.11)\begin{gather}I_2= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} y^\prime \exp\left({-2\left({x^\prime}^2 + {y^\prime}^2 + \dfrac{m_0c^2(J^{1/2} - 1)}{2T_e}\right)}\right)\nonumber\\ \times H_m^4(x^\prime)H_n^3(y^\prime)\dfrac{(y^\prime H_n(y^\prime)-H_{n+1}(y^\prime))}{J(x^\prime,y^\prime)} \left(\dfrac{1}{J(x^\prime,y^\prime)^{1/2}} - \dfrac{m_0c^2}{T_e}\right)\,{{\rm d} x}^\prime\,{{\rm d} y}^\prime, \end{gather}

and where

(4.12)\begin{gather} J(x^\prime,y^\prime) = \left(1 + \dfrac{\beta^\prime \psi_{00}^2}{f_xf_y} H_m^2(x^\prime) H_n^2(y^\prime) \exp({-({x^\prime}^2 + {y^\prime}^2)})\right), \end{gather}
(4.13)\begin{gather}\varPhi = \left(\dfrac{\omega_{p0}^2{x_0}^2}{c^2}\right) \exp\left(\dfrac{\zeta}{d}\right), \end{gather}
(4.14)\begin{gather}x^\prime = \frac{x}{x_0f_x}, \end{gather}
(4.15)\begin{gather}y^\prime = \frac{y}{y_0f_y}, \end{gather}
(4.16)\begin{gather}\zeta = \frac{z}{kx_0^2} \end{gather}

is the dimensionless distance of propagation.

On solving (4.8) and (4.9) numerically and subjecting them to boundary conditions $f_{x,y} = 1$ and ${f^\prime }_{x,y} = 0$ dictates the spot size variation of HG laser beam along the transverse $x$ and $y$ directions, respectively.

5. Electron plasma wave excitation and terahertz generation

Poisson's equation, the adiabatic equation of state, the equation of motion and the continuity equation are the four main equations that govern the excitation of the electron plasma wave. The relativistic-ponderomotive force experienced by the electrons is given by

(5.1)\begin{equation} F_{R-P} ={-}m_0c^2 \boldsymbol{\nabla} (\gamma - 1), \end{equation}

and following Wadhwa & Singh (Reference Wadhwa and Singh2020), the expression for perturbed electron density is given as

(5.2)\begin{gather} n_1 ={-} \frac{{e} n_0}{m_0} \frac{E_{00}}{\sqrt{f_xf_y}} \exp\left(-\left(\frac{x^2}{x_0^2f_x^2} + \frac{y^2}{y_0^2f_y^2}\right)\right) \left\{ H_m \left(\frac{x}{x_0f_x}\right) H_n \left(\frac{y}{y_0f_y}\right) \left(\frac{x}{x_0^2f_x^2} + \frac{y}{y_0^2f_y^2}\right) \right.\nonumber\\ \left.-\frac{1}{x_0f_x}H_{m+1}\left(\frac{x}{x_0f_x}\right)H_n\left(\frac{y}{y_0f_y}\right) - \frac{1}{y_0f_y}H_{m}\left(\frac{x}{x_0f_x}\right)H_{n+1} \left(\frac{y}{y_0f_y}\right) \right\}\nonumber\\ \frac{1}{\left(\omega^2 - k^2 v_{T}^2 -\frac{\omega_{p}^2}{\gamma} \exp \left(\dfrac{-m_0c^2}{T_e}(\gamma - 1)\right)\right)}. \end{gather}

The wave equation governing terahertz generation is given as

(5.3)\begin{equation} \nabla^2 E_{Th} + \frac{\omega_{Th}^2}{c^2}\epsilon_{Th} (\omega_{Th})E_{Th} = \frac{\omega_p^2}{c^2} \left( \frac{n_1}{n_0}\right) E, \end{equation}

where, $\epsilon _{Th}$ $(=k_{Th}^2c^2/\omega _{Th}^2)$ is the dielectric constant at terahertz frequency and $\omega _{Th} = \omega - \omega _{ep}$ is the terahertz radiation frequency. Taking $E_{Th} = A_{Th}$ $\exp ({{\rm i}(w_{Th}t-k_{Th}z)})$ where $A_{Th}$ is the amplitude of the generated terahertz radiation and considering that the change in the $z$ direction is much greater than in the transverse directions (i.e. $\partial E_{Th}/\partial z > \partial E_{Th}/\partial r$), (5.3) can be written as

(5.4)\begin{equation} 2{\rm i}k_{Th}\frac{\partial A_{Th}}{\partial z} \approx \frac{\omega_{p}^2}{c^2} \frac{n_1}{n_0} \varPsi, \end{equation}

where the magnitude of wavevector of terahertz generation is

(5.5)\begin{equation} k_{Th} = \frac{\omega}{c} \left(1 - \frac{\omega_{p}^2}{\omega^2}\right)^{{1}/{2}}. \end{equation}

The yield $\eta$ of terahertz radiation is evaluated as

(5.6)\begin{equation} \eta = \frac{P_{Th}}{P_0}, \end{equation}

where, $P_{Th}$ is the power of the generated terahertz radiation and $P_{0}$ is the power of incident laser beam. Since power is directly proportional to the intensity, it can be calculated (Sodha, Ghatak & Tripathi Reference Sodha, Ghatak, Tripathi and Wolf1976) as

(5.7)\begin{gather} P_{Th} =\frac{c}{8{\rm \pi}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}A_{Th} A_{Th}^*\,{\rm d}x\,{\rm d}y, \end{gather}
(5.8)\begin{gather} P_0 =\frac{c}{8{\rm \pi}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\varPsi \varPsi^*\,{\rm d}x\,{\rm d}y. \end{gather}

Using (5.7) and (5.8) in (5.6), we get

(5.9)\begin{equation} \eta = \frac{ \beta^\prime E_{00}^2}{f_xf_y} \varPhi \frac{I_3}{2^{m+n} {\rm \pi}m! n! }, \end{equation}

where,

(5.10)\begin{gather} I_3 = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \exp({-2({x^\prime}^2 + {y^\prime}^2)}) H_m^2(x^\prime) H_n^2(y^\prime) \left(H_m(x^\prime) H_n(y^\prime) \left(\dfrac{x^\prime}{f_x} + \dfrac{x_0}{y_0} \dfrac{y^\prime}{f_y}\right) \right.\nonumber\\ \left.-\dfrac{1}{f_x} H_{m+1}(x^\prime) H_n(y^\prime) - \dfrac{x_0}{y_0 f_y} H_m(x^\prime) H_{n+1}(y^\prime) \right)^2\nonumber\\ \dfrac{1}{\left(\dfrac{\omega^2x_0^2}{c^2} - \left(\dfrac{\omega^2x_0^2}{c^2} - \varPhi\right) \dfrac{v_{T}^2}{ c^2} - \dfrac{\varPhi}{\gamma} \left(\dfrac{-m_0c^2}{T_e}(\gamma - 1)\right)\right)^2} \,{{\rm d} x}^\prime \,{{\rm d} y}^\prime. \end{gather}

6. Results and discussion

Self-focusing of the HG laser beam and the variation of the beam width variables $f_x$ and $f_y$ with the dimensionless distance of propagation $\zeta$ under the effect of magnetic field and upward density ramp has been analysed by solving (4.8) and (4.9) simultaneously. Also, the effect of different parameters such as density ramp slope, static magnetic field, etc., on the efficiency of terahertz generation has been studied. Simulation of the mentioned work has been carried out using the following set of laser variables: $\omega = 1.78 \times {10^{15}}\,{\rm rad}\,{\rm s}^{-1}$ (Nd: YAG laser beam) and $x_0 = 15\,\mathrm {\mu }{\rm m}$.

Figure 2(a,b) represent the change in $f_x$ and $f_y$ for different modes with respect to $\zeta$ at a fixed value of ‘d’ = 10 and $\omega _c = 0.1\omega$. These depict the oscillatory nature of $f_x$ and $f_y$ which correspond to the beam's convergence and divergence while propagating inside the plasma. The application of a magnetic field enhances self-focusing in both transverse directions. Due to the coupling of $f_x$ and $f_y$, maximum focusing is achieved for the TEM$_{02}$ mode as compared with TEM$_{00}$ and TEM$_{01}$. This is due to the symmetric distribution of intensity along the $y$ axis as depicted in figure 1(c) with the intensity being maximum in the off-axial parts. Also, due to the presence of a density ramp, the self-focusing of the laser beam increases with $\zeta$ as the diffraction effects are minimized. This is because the density ramp profile acts as a slowly narrowing waveguide for the laser beam propagating inside the plasma. The increase in laser intensity leads to an increase in the nonlinearity of the medium, which then results in a high refractive index and thus better focusing in the transverse directions as shown by figure 3(a,b). This increase in self-focusing leads to an increase in the efficiency of generated terahertz radiation (figure 4).

Figure 2. Change in $f_x$ and $f_y$ with $\zeta$ for higher-order modes at specified values of $d = 10$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $\omega _c = 0.1 \omega$, normalized laser intensity $\beta \psi _{00}^2 = 2$, $T_e = 10 \,{\rm KeV}$ and $x_0/y_0$ = 1 as depicted by panels (a) and (b), respectively.

Figure 3. Change in $f_x$ and $f_y$ with $\zeta$ for various normalized intensities at specified values of $d = 10$, TEM mode $(m,n) = (0,2)$, $\omega _c = 0.1 \omega$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $T_e = 10\,{\rm KeV}$ and $x_0/y_0 = 1$ as depicted by panels (a) and (b), respectively.

Figure 4. Modification in efficiency $\eta$ of generated terahertz radiation with $\zeta$ for various normalized intensities at specified values of $d = 10$, TEM mode $(m,n) = (0,2)$, $\omega _c = 0.1 \omega$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $T_e = 10\,{\rm KeV}$ and $x_0/y_0$ = 1.

Figure 5. Change in $f_x$ and $f_y$ with $\zeta$ for various values of d at specified values of normalized laser intensity $\beta \psi _{00}^2 = 2$, TEM mode $(m,n) = (0,2)$, $\omega _c = 0.1 \omega$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $T_e = 10 \,{\rm KeV}$ and $x_0/y_0 = 1$ as depicted by panels (a) and (b), respectively.

Figure 6. Modification in efficiency $\eta$ of generated terahertz radiation with $\zeta$ for various $d$ at specified values of normalized laser intensity $\beta \psi _{00}^2 = 2$, TEM mode $(m,n) = (0,2)$, $\omega _c = 0.1 \omega$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $T_e = 10 \,{\rm KeV}$ and $x_0/y_0$ = 1.

Figure 7. Change in $f_x$ and $f_y$ with $\zeta$ for various values of $\omega _c$ at specified values of normalized laser intensity $\beta \psi _{00}^2 = 2$, TEM mode $(m,n) = (0,2)$, $d = 10$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $T_e = 10\,{\rm KeV}$ and $x_0/y_0 = 1$ as depicted by panels (a) and (b), respectively.

Figure 8. Modification in efficiency $\eta$ of generated terahertz radiation with $\zeta$ for various $\omega _c$ at specified values of $d = 10$, normalized laser intensity $\beta \psi _{00}^2 = 2$, TEM mode $(m,n) = (0,2)$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $T_e = 10 \,{\rm KeV}$ and $x_0/y_0 = 1$.

Figure 9. Modification in efficiency $\eta$ of generated terahertz radiation with $\zeta$ for various higher-order modes at specified values of $d = 10$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $\omega _c = 0.1 \omega$, normalized laser intensity $\beta \psi _{00}^2 = 2$, $T_e = 10\,{\rm KeV}$ and $x_0/y_0 = 1$.

Figure 5(a,b) illustrate the effect of the slope of density ramp ‘$d$’ on the focusing of the laser beam. It is observed from the figures that there is more self-focusing as the value of ‘$d$’ decreases. This is due to the dominance of the nonlinear refractive term over the diffractive term as the value of ‘$d$’ decreases. As the value of ‘$d$’ increases, the density variation/transition goes on decreasing (figure 5a,b). Therefore, the efficiency of generated terahertz radiation also increases with a decrease in the value of ‘$d$’ (figure 6). Figure 7(a,b) show that the increase in the applied magnetic field results in deeper self-focusing and hence greater efficiency ($\eta$) of terahertz radiation (figure 8). Figure 8 depicts the efficiency variation with $\zeta$ at different values of $\omega _c = 0.1, 0.2$ and 0.3. An increase in the value of the magnetic field leads to more convergence and hence more efficiency.

Figure 9 represents efficiency variation of generated terahertz radiation with $\zeta$ for different TEM modes. Efficiency is better for TEM$_{02}$ as compared with TEM$_{00}$ and TEM$_{01}$ because of its better self-focusing as compared with other modes.

7. Conclusions

The present work demonstrates the HG laser beam dynamics and terahertz generation while propagating through plasma under the effect of an exponential density ramp and a static magnetic field in a relativistic-ponderomotive regime. It has been observed that self-focusing is more for TEM$_{02}$ as compared with other modes and it also increases with an increase in normalized laser intensity, applied magnetic field and with a decrease in the slope of density ramp ‘$d$’. It is concluded from the present investigation that maximum terahertz efficiency is observed for TEM$_{02}$ and further efficiency is also enhanced at higher values of normalized laser intensity, applied magnetic field and with a decrease in the density ramp slope. The results of the present investigation may be useful for the experimentalist working in the field of terahertz generation.

Acknowledgements

Editor V. Malka thanks the referees for their advice in evaluating this article.

Funding

The authors express their gratitude towards the Ministry of Education, India, for providing financial support to carry out this research work.

Declaration of interests

The authors report no conflict of interest.

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Figure 0

Figure 1. Three-dimensional normalized intensity distribution plots for various TEM modes (0,0), (0,1) and (0,2) depicted by panels (a), (b) and (c) respectively.

Figure 1

Figure 2. Change in $f_x$ and $f_y$ with $\zeta$ for higher-order modes at specified values of $d = 10$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $\omega _c = 0.1 \omega$, normalized laser intensity $\beta \psi _{00}^2 = 2$, $T_e = 10 \,{\rm KeV}$ and $x_0/y_0$ = 1 as depicted by panels (a) and (b), respectively.

Figure 2

Figure 3. Change in $f_x$ and $f_y$ with $\zeta$ for various normalized intensities at specified values of $d = 10$, TEM mode $(m,n) = (0,2)$, $\omega _c = 0.1 \omega$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $T_e = 10\,{\rm KeV}$ and $x_0/y_0 = 1$ as depicted by panels (a) and (b), respectively.

Figure 3

Figure 4. Modification in efficiency $\eta$ of generated terahertz radiation with $\zeta$ for various normalized intensities at specified values of $d = 10$, TEM mode $(m,n) = (0,2)$, $\omega _c = 0.1 \omega$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $T_e = 10\,{\rm KeV}$ and $x_0/y_0$ = 1.

Figure 4

Figure 5. Change in $f_x$ and $f_y$ with $\zeta$ for various values of d at specified values of normalized laser intensity $\beta \psi _{00}^2 = 2$, TEM mode $(m,n) = (0,2)$, $\omega _c = 0.1 \omega$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $T_e = 10 \,{\rm KeV}$ and $x_0/y_0 = 1$ as depicted by panels (a) and (b), respectively.

Figure 5

Figure 6. Modification in efficiency $\eta$ of generated terahertz radiation with $\zeta$ for various $d$ at specified values of normalized laser intensity $\beta \psi _{00}^2 = 2$, TEM mode $(m,n) = (0,2)$, $\omega _c = 0.1 \omega$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $T_e = 10 \,{\rm KeV}$ and $x_0/y_0$ = 1.

Figure 6

Figure 7. Change in $f_x$ and $f_y$ with $\zeta$ for various values of $\omega _c$ at specified values of normalized laser intensity $\beta \psi _{00}^2 = 2$, TEM mode $(m,n) = (0,2)$, $d = 10$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $T_e = 10\,{\rm KeV}$ and $x_0/y_0 = 1$ as depicted by panels (a) and (b), respectively.

Figure 7

Figure 8. Modification in efficiency $\eta$ of generated terahertz radiation with $\zeta$ for various $\omega _c$ at specified values of $d = 10$, normalized laser intensity $\beta \psi _{00}^2 = 2$, TEM mode $(m,n) = (0,2)$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $T_e = 10 \,{\rm KeV}$ and $x_0/y_0 = 1$.

Figure 8

Figure 9. Modification in efficiency $\eta$ of generated terahertz radiation with $\zeta$ for various higher-order modes at specified values of $d = 10$, normalized plasma density $\omega _{p0}^2x_0^2/c^2 = 12$, $\omega _c = 0.1 \omega$, normalized laser intensity $\beta \psi _{00}^2 = 2$, $T_e = 10\,{\rm KeV}$ and $x_0/y_0 = 1$.