# On validity of paraxial theory for super-Gaussian laser beams propagating in a plasma

## Abstract

In the present paper, we have investigated a situation where a high intensity laser beam passes through a gas and ionizes this gas by tunnel ionization. Here the electric field of the laser provides a sufficient velocity to the electrons to surpass the Coulomb barrier of the atom. Owing to the ionization the plasma density enhances which affects the laser beam propagation. The use of paraxial ray approximation theory for the present case of super-Gaussian lasers reveals the self-focusing of the beams and frequency upshifting. The predicted self-focusing of the laser beams is contrary to the expected outcome of defocusing of these beams in the plasma, indicating that the paraxial theory may not be valid for the case of super-Gaussian lasers even for the inclusion of most of the near axis region in the theory.

## Keywords

Tunnel ionization Self-focusing Super-Gaussian lasers Frequency upshift Paraxial ray approximation## Introduction

The interaction of high power laser pulse with gases and plasmas is a major field of research due to its applications in laser electron accelerator, laser driven fusion, and super-continuum generation [1, 2, 3, 4]. For many of these applications, it is highly desirable that the laser pulse propagate extended distances (many Rayleigh lengths) at high intensity. The problem of spot size evolution, self-focusing and defocusing of the laser pulse in plasma are of great interest. Self-focusing is a non-linear optical process induced by a change in refractive index of material exposed to intense laser radiation. A medium whose refractive index increases with electric field intensity acts as focusing lens for a laser beam characterized by an initial transverse intensity gradient. There are studies where people have tried to enhance the Rayleigh length. Most of the studies have been conducted for the laser beams having Gaussian profile using paraxial ray approximation [5, 6, 7, 8].

Sodha et al. [9] reviewed the self-focusing of electromagnetic beams in plasmas, referring primarily to steady state self-focusing. Esarey et al. [10] gave an extensive review of paraxial ray theory of self-focusing and self-guiding of short laser pulses in ionizing gases and plasmas. Fedosejevs et al. [11] reported experimental results on the competing processes of ionization-induced refraction and relativistic self-focusing. Liu and Tripathi [7] developed a theoretical framework to study the combined effects of the laser-frequency upshift, self-defocusing, and ring formation. The self-defocusing of the laser prepulse and laser guiding of the second laser pulse in an axially nonuniform plasma channel has also been studied by them [12]. The Ohmic and the ponderomotive nonlinearities at sub-picosecond pulse lengths are weak, as the time scale for ambipolar plasma diffusion is much longer. However, two other major nonlinearities are important. These arise due to tunnel ionization of atoms in the optical field of the laser and due to the relativistic dependence of electron mass on its velocity. The former nonlinearity is important when the laser field is comparable with the atomic Coulomb field.

However, the super-Gaussian lasers have recently attracted the researchers all over the world [13, 14, 15, 16, 17] for their better extraction from saturated laser amplifiers, better non-linear conversion for same peak intensity, and top-hat beams have a parabolic thermal lens which does not decrease beam quality. Due to this reason, it becomes vital to investigate the profile of super-Gaussian lasers when they tunnel ionize the gas and produce the plasma. Hence, in the present article, we have derived the coupled equations for the amplitude and phase of super-Gaussian laser beam and solve it numerically using initial conditions.

## Coupled equations for amplitude and phase

Let us consider the propagation of a circularly polarized laser beam in a gas jet target. At *z* = 0, the laser field is given by

*t*> 0 and

*E*

_{0}= 0 for

*t*< 0.

*n*

_{p}and plasma frequency

*ω*

_{p}, given by:

*E*, \(\omega_{\text{N}}^{2} = 4\pi n_{\text{N}} e^{2} /m,\)

*n*

_{N}is the initial density of neutral atoms, \(E_{\text{a}} = (4/3)\sqrt {2m} \left( {I_{0} } \right)^{3/2} /e\hbar\) is the characteristic atomic field,

*I*

_{0}is the ionization potential, and

*h*= 2

*π*\(\hbar\) is the Planck’s constant.

*z*> 0), \(\vec{E} = (\hat{x} + i\hat{y})A(t,z,r)\exp ( - i\varphi ),\) where

*A*is the slowly varying complex amplitude and

*φ*(

*t*,

*z*) is the fast phase of the wave. We define

*ω*= ∂

*φ*/∂

*t*,

*k*= − ∂

*φ*/∂

*z*, \(\omega_{\text{P0}}^{2} = \omega_{\text{P}}^{2} \left( {t,z,r = 0} \right).\) The wave equation for the laser pulse is written as:

*t*, using ∂

*k*/∂

*t*= − ∂

*ω*/∂

*z*, and defining \(v_{\text{g}} = c\left( { 1 - \omega_{{{\text{P}}0}}^{2} /\omega^{ 2} } \right)\), we obtain

*A*

_{F}= (

*ω*/

*ω*

_{0})

^{1/2}

*A*,

*t*′ =

*t*−

*z*/

*c*,

*z*′ =

*z*, and assuming \(\omega_{\text{P}}^{2} /\omega^{ 2} \ll 1\), we can rewrite Eqs. (6) and (7) simply as

*γ*

_{T0}=

*γ*

_{T}(

*r*= 0). We consider cylindrically symmetric propagation and write

*A*

_{F}=

*A*

_{F0}exp (

*iQ*), where

*A*

_{F0}(

*t*′,

*z*′,

*r*) and

*Q*(

*t*′,

*z*′,

*r*) are real. Separating the real and imaginary parts of Eq. (8), we obtain:

*Q*,

*γ*

_{T}in powers of

*r*

^{2}, as:

*r*

^{6}) so that most of the region can be included. Keeping the analysis up to

*r*

^{4}order terms, a significant error shall occur. However, the higher order terms shall have negligible impact in the case of Gaussian laser beams.

*γ*

_{T}can be determined using Eq. (3). On comparing the successive powers of \(r^{2}\), we obtain the various equations:

The above coupled equations are written in the dimensionless quantities ς = *z*′/*R* _{d}, *τ* = *γ* _{T00} *t*′, *ω* _{SF} = *ω*/*ω* _{0}, and *R* _{d} = (*ω* _{0}/*c*)\(r_{0}^{ 2}\) is Rayleigh length.

## Results and discussion

The coupled equations are solved by applying boundary conditions for an initially plane wave front. The initial and boundary conditions are at \(\tau = 0, \omega_{\text{P0}}^{2} ,\omega_{\text{P2}}^{2} , \omega_{\text{P4}}^{2} ,\omega_{\text{P6}}^{2} = 0\) for all ς and at ς = 0, \(f_{\text{B}} = 1, \frac{{\partial f_{\text{B}} }}{\partial \varsigma }\) = 0, \(\omega_{\text{SF}} = 1, a_{2} , a_{4} , a_{6} = 0, Q_{4} ,Q_{6} = 0\) for all *τ*. Since the Keldysh parameter *γ* decides the nature of ionization and *γ* ≪ 1 for the tunnel ionization, we select the laser and plasma parameters such that this condition is satisfied. The Keldysh parameter is given by \(\gamma = \sqrt {\frac{{E_{\text{I}} }}{{2U_{\text{P}} }}} = \sqrt {\frac{{\varepsilon_{0} m_{\text{e}} cE_{\text{I}} \omega^{2} }}{{q^{2} I}}}\), where *E* _{I} is the ionization potential of atom, *q* is charge on the electron, *I* is the intensity of the laser beam and other symbols have their usual meanings.

The theory used in the present article also predicts the frequency shifting of the lasers when they tunnel ionize the gas and propagate in the formed plasma. The change in the frequency with various parameters is shown in the forthcoming figures.

## Conclusions

We have considered super-Gaussian beam profile having maximum intensity on the axis and minimum intensity away from the axis. As the laser beam propagates in the gas to form the plasma, the plasma density should stay maximum on the axis which should result in minimum refractive index on the axis. Since the phase velocity of the lasers depends inversely on the refractive index, the phase velocity becomes maximum on the axis and falls down sharply on the edges. This would lead to a diverging wavefront, causing the laser beam to defocus. Kumar and Tripathi [6] observed the self-defocusing of Gaussian laser pulse based on the paraxial ray theory in tunnel ionized helium plasma. Liu and Tripathi [7] also observed the self-defocusing of Gaussian beam based on the paraxial ray theory in tunnel ionized neon plasma. However, we point out that the use of this theory in the case of super-Gaussian lasers beams provides an unphysical result of self-focusing of these beams. Hence, the paraxial ray approximation theory does not seem to be authentic in the case of super-Gaussian laser beams.

## Notes

### Acknowledgements

One of the authors, Lalita Devi, would like to thank CSIR, Govt. of India for providing the financial support for this work.

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