Intense Laser Plasma Interaction Studies with Density Ramp Profile

Meenu Dinesh Varshney1,a), Surbhi Bidawat1, Brahm Prakash1, Aditi Varshney and Sonu Sen2

1School of Physics, Vigyan Bhavan, Devi Ahilya University, Khandwa road Campus, Indore - 452001, India 2School of Automobile and Manufacturing Engineering, Symbiosis University of Applied Sciences, Bada Bangarda,

Super corridor, Indore - 453112, India a)Corresponding author: meenuv13@rediffmail.com

Abstract.An intense short pulse laser through dense plasma expels electrons and ions from the focal spot by the ponderomotive pressure and forms a channel, which acts as a propagation guide for the laser beam. Transport properties of electrons has been observed in the direction of laser propagation for different ramped profile plasma. Based on this analogy in the present paper we make an analytical investigation of intense laser plasma interaction studies with dense plasma. For the given values of plasma frequency and laser frequency, propagation of laser beam are characterized by dimensionless beam width parameter. The variation of beam width with distance of propagation has been obtained for typical values of parameters in ramped density plasma.

1. INTRODUCTION

With the modern innovations in laser technology beam power can be thousand times larger than the critical value. A new generation of lasers are equipped for accomplishing high electric field and consequently plasma can continue greatly extreme electric fields in which the plasma electron wavers at relativistic speeds and thus, the relativistic impact observed at such a high power of the beam.In the past few decades, relativistic interaction of a laser beam withplasma has been studied theoretically and experimentally by many authors. Relativistic laser–plasma interaction are of much attention for their applications to inertial confinement fusion [1-4], X-ray lasers [5], harmonic generation [6, 7], and laser electron acceleration [8-11]. In these studies, the laser must propagate up to some Rayleigh range for maintaining an effective interaction with the plasmas. Thus, self-focusing and propagation of a high-intensity laser beam up to a feasible extent Rayleigh range in plasma is a very significant research issue.

If the beam power with respect to the radiant energy are in the limit where the dependency of the dielectric function on radiant energy is articulated, the radial outward deviation of the dielectric function may be strongly adequate to cause convergence, which overcomes the divergence due to diffraction. Soon the laser beam converges to a particular minimum value where the divergence on account of the dependence of the dielectric function on the irradiance; this process remains until the laser beam attains its actual thickness, subsequently the sequence becomes repeated. This condition refers to oscillatory convergence or self-focusing. After the self-focusing effect, the nonlinear refraction gradually starts fading and the radius of the laser beam increases, performing oscillatory nature with propagation. Further, it was observed that the effects of plasma density function profile become more effective in order toovercome the diffraction and have stronger self-focusing. In presence of density function profile, the mutual result of ponderomotive and relativistic nonlinearities are very important for the penetration of intense laser pulses into the plasmas for self-focusing phenomena. A predefined plasma density function profile can avoid the laser beam diffraction and allows the propagation of an intense laser beam up to certain Rayleigh range.

In the present paper we studied the intense laser plasma interaction with density ramp profile. In section 2 we have made an analytical investigation and derived the equation of the beam width parameter for intense laser beam

in presence of density profile plasma followed by numerical calculations. Numerical results and discussions are made in section 3.

2. PROPAGATION EQUATION FOR DENSITY RAMP PROFILE PLASMA

Consider the propagation of a Gaussian laser beam of frequency ‘’ along the z-direction, at z= 0 the intensity distribution of the beam in this situation is expressed as )/exp( 2

022

0* rrEEE (1)

If initial electron density of plasma is n0 then due to propagation of this intense Gaussian laser beam the dielectric constant for cold quantum plasma given by [12]

10

22 /1)/)(/(1 qnnep (2)

where, )/4( 20

2 menp is the plasma frequency, 2/1*2 ])/(1[ EEcme is the relativistic Lorentz factor,

42222 / mq , here, m is the rest mass of electron, e is electronic charge and ne is the modified electron density due to the ponderomotive force. Using poisson’s equation one may write the modified electron density as )4/( 2

0 enne (3) If we employ a Gaussian constant shape ansatz for amplitude [13]

22

0

2

2

202 exp

frr

fq

q (4)

where, f is the dimensionless beamwidth parameter which is unity at z = 0 and *

020 EEq , )/( 2222

0 cme , then we obtain the modified electron density,

2

2

220

22

220

2

2

04/111

q

frrq

frcnn

pe

(5) Now from Equation (2) dielectric function is a function of irradiance EE* of a Gaussian beam and hence function of r2, therefore in the paraxial approximation can be expanded in power of r2. Expanding dielectric constant around r = 0 by Taylor expansion, one can write

2

0

2'

0 rr

(6)

where,

'

0

220

220

2

2

'0

22

0/1

/1/

1

fqfr

cq p

p

and

'

0

220

220

2

2

3'0

420

2

2' /8

1/

4

fqfr

cfq

p

p

(7)

here, 2/1220

'0 /1 fq (8)

The wave equation governing the electric vector of the beam in plasmas with the effective dielectric constant given by Equation (7) can be written as 0)/( 222 EE c (9)

Keeping in mind that (c2/2) ln << 1, within the WKB approximation, we neglect the term (E) while writing Equation (9). We further express r and z dependence of E following our earlier analysis as [14],

z

dzc

tiz

zrAzrE0

2/104/1

0

4/10 )(exp

)()0(

),(),(

(10) substituting for E and from equation (10) and (6), neglecting (2A/z2) we obtain

012

20

2'

2

2

2

22/1

0

Ar

rcr

Arr

AzA

ci

(11) Separating real and imaginary parts of A, we follow

),(exp),( 2/1

00 zrSc

izrAA

(12) in equation (11) and get

2

0

2

0

'0

20

2

002

20

0

2 112rr

rA

rrA

AcS

zrS

zS

(13)

and

01

2

220

20

2

20

rS

rrSA

rA

rS

zA

(14) S(r, z) is called the eikonal and related to the curvature of the wave front. Following Akhmanov et.al.[13], the solution of (13) and (14), for an initially Gaussian beam (1), can be written as

),()(2

2zzrS

),/exp()/( 220

2220

20 frrfEA )/)(/1( dzdff (15)

Here in the above set of Equation’s (15), ‘β’ represents the inverse of the radius of curvature of the wave front and (r0f)is the width of the beam. In the geometrical optics approximation, r = r0f (z) represents a ray in a plane containing the z-axis. We substitute for S and A0 from Equation (15) in Equation (13) and making use of the paraxial ray approximation i.e., [(r/r0f)4 <<1]. Finally we equate the coefficients of r2 on both sides of the resulting equation, and can now be reduced to a rather simple form by transforming the coordinate z and the initial beam width r0 to dimensionless forms: )/( 2

0 rzc and )/( 00 cr . We get the characteristic beam propagation equation as,

f

ddf

dd

fdfd

0

'200

03

02

2

211

(16) In axial inhomogeneous plasmathe electron density is written as: n() = n(0) R() (17)

where n(0) is initial density of plasmaand R() is the density ramp profile. Choosing “increasing parabolic”, “Sec” and “Cosh” density ramp functions as (i) RPar() = 1 + (B)2 (18) (ii) RSec() = Sec(B) (19) (iii) RCosh() = Cosh(B) (20)

here B is a constant which determines the slope and is adjustable. Considering the propagation Eq. (16) of laser beam, the modified plasma frequency is )(2

02 RppR (21)

hence, in presence of density ramp propagation equation gets modified as

fff

ddf

ddk

kfkcdfd

R

R 20'

0

1322

2

2

2

)()(1

(22)

After initial focusing of the laser, the relativistic mass effect will be much more pronounced in the region of increasing plasma density. Therefore, the laser focuses more during propagation in a plasma density profile.

3. RESULTS AND DISCUSSION

Figures shows variation of beamwidth parameter (f) with dimensionless distance of propagation (). In the area of low plasma density, the electrons are repelled from the high-intensity region by ponderomotive force; the nonlinearity in a plasma comes by electron mass variation, which is due to intense laser and change in electron density. If there is no density profile, the beamwidth parameter decreases because of the nonlinear effects. As the diffraction effect become predominant the beamwidth parameter increases after attaining a minimum value and the laser beam starts diverging owing to saturation of nonlinearity. Hence, the laser becomes focused and defocused and shows an oscillatory behavior. If there is a increasing density profile, the beamwidth parameter decreases up to a Rayleigh length and does not increase much, as in case without any density profile, for suitable parameters.

The saturation behavior of the beamwidth parameter shows the strong self-focusing of the laser in a plasma with a density increasing density profiles. Careful observation of these results from Figures indicates that for parabolic, Sec and Cosh axially increasing density profile plasma, minimum value of the dimensionless beamwidth parameter for the second and higher orders decreases continuously, as compared to homogeneous plasma. This is because on increasing electron density, nonlinearity increases. These curves also indicate that for axial distance, Sec density profile plasma has low value of f as compared to the value obtained for homogeneous plasma, parabolic and Cosh density profile plasma.

(a) (b)

(c) (d)

FIGURE: Variation of beamwidth parameter f with normalized distance of propagation corresponds to (q0, 0) = (3,

20). Here, (a). )/( 22 p = 0.1, B = 0.25 (b). )/( 22 p = 0.1, B = 0.2 (c). )/( 22 p = 0.5, B = 0.25 and (d)

)/( 22 p = 0.5, B = 0.2.

0 1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

1.2 Ho mogene ous plasm a Parabolic profile Sec fun ction Profi le Co sh function Profile

Beam

wid

th p

aram

eter

(f)

0 1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

1.2

Beam

wid

th p

aram

eter

(f)

H omogeneous plasma Parabolic profile Sec function Pro file C osh function Profile

0 1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

1.2 H om ogeneous plasm a Parabolic profile Sec function Profile C osh fun ction Profile

Beam

wid

th p

aram

eter

(f)

0 1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

1.2

Beam

wid

th p

aram

eter

(f)

H omogene ous plasm a Parabolic profile Sec function Profile C osh function Profile

Norm alized dis tance of propagation ()

Substantially, the analytical and numerical work presented here shows reduction in defocusing effect with the introduction of plasma density profiles. Thus, density profile functions significantly increase relativistic self-focusing of a laser beam in plasma producing ultra-high laser irradiance over distances much greater than the Rayleigh length which can be used for various applications towards plasma-based accelerators and Inertial Confinement Fusion.

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