View of Influence of External Magnetic Field Alignment on Terahertz Generation through laser plasma nonlinear interaction in Nonparaxial Region

14496 http://annalsofrscb.ro

Influence of External Magnetic Field Alignment on Terahertz Generation through laser plasma nonlinear interaction in Nonparaxial Region

Munther B. Hassan

^1,(a)

and Lina Mohammed Hatem

^2,(b)

(a) ¹ Professor, Department of physics, faculty of science, university of Kufa.

(b)

Department of physics, faculty of science, university of Kufa.

(a)[email protected],^(b)[email protected]

Abstract

This paper presents a theoretical study to explain the nonlinear self-focusing of Gaussian laser beam through magnetized plasma under the non- paraxial ray approximation where the Relativistic nonlinearity is operated.

The theoretical analysis and numerical calculations show that in both cases (longitudinal and transverse magnetic fields), the increasing of external magnetic field values will lead to increase and enhance the nonlinear self-focusing of Gaussian laser beam, therefore the intensity will reach in the order of 10¹⁸ W.cm^-2 or more which is enough to excite a terahertz radiation by relativistic nonlinearity. The terahertz radiation generation is depending on the down-conversion of three wave mixing technique and on the fulfilling of the phase matching conditions between the pump wave (laser beam), plasma wave and generated terahertz wave. The results showed the self-focused was stronger in the case of the longitudinalmagnetic fieldsthan in thecase of the transverse magnetic fields and the product THZ was also higher in longitudinal than in the case of the transverse magnetic fields in non- paraxial region .

Keywords : laser– plasma interaction, self-focusing, Terahertz (THz) radiation,Plasmafrequency, non- paraxial,Relativistic.

1. Introduction

Among many nonlinear optical effects related to laser– plasma interaction, the phenomenon of self-focusing has played significant role[1-5]. In 2014 Munther B. Hassan et al . [6]. study the impact of external magnetic fields for laser beams on the self-focused laser beam through the plasma. They find that the nonlinear dielectric constant related to the high-intensity Gaussian beam is related to the relativistic electron motion resulting from it. Theoretical and numerical calculations in 2020 AbdulkareemTh.jabbar [7] show that the geometric of applied Initial laser Intensity for both parallel and transversal cases leading to change the self-focusing in non- paraxial region leads to the laser beam self-focusing faster and stronger. Currently the terahertz physics, abbreviated (THz), has been of much interest to researchers for its vast dominance of applications in fields such as security identification, medical imaging and spectroscopy of time domain [8-10]. Laser with ultra-short pulse was employed by Monika Singh et al [11]. Laser pulse reactions are being studied worldwide such as communications, medical applications, and clean energy sources[12-15]. In these applications, the laser pulse should spread across multiple Rayleigh lengths while maintaining effective interaction with the laser beam and plasma. self-focus should play a signinificant role to ensure good compatibility through laser and plasma

14497 http://annalsofrscb.ro

interactions [16,17]. One of the main goals of this study is to derive convenient equations to investigate the nonlinear behavior of laser beam self-focusing inside magnetized plasma and in order to contribute to terahertz radiation generation. Thus we try to enhance the terahertz field stability by using laser parameters (intensity and external magnetic field, in both cases longitudinal and transverse).

In sections (2) the appropriate expressions of the non-paraxial laser beam self- focusing via magnetized plasma have been derived for Initial Laser Intensity.in sections (3) derive equations production the THZ inside magnetic plasma. To more understand of the self-focusing phenomenon and production the THZ so deep discussion of numerical results and the most distinguished conclusions have been introduced in section (4) and section (5) respectively.

2. Self-focusing of non-paraxial laser beam inside plasma

In this section, the self-focusing equations of laser beam propagation inside plasma will be derived. Two alignments of external magnetic fields (with respect of laser beam propagation) will be introduced. First in presence of longitudinal magnetic field and second in presence of transversely magnetic field

.

2.1 external longitudinal magnetic field

Postulating a Gaussian laser beam is propagating along an external magnetic field B₀ in z-direction. Hence the electric field( 𝐸 ₀₊ ) of laser ray may be given as

𝐸 0+= 𝐴 0+𝑒𝑥𝑝 𝑒𝑥𝑝𝑖 𝜔0𝑡 − 𝑘0+𝑧 (1)

where𝐴 ₀₊= 𝐸 _𝑥+ 𝑖𝐸 _𝑦 is the electric field amplitude,

𝜔

𝑜is theangular frequency and 𝑘₀₊ is the wave vector.

To govern the laser beam propagation via magnetized plasma, one may introduce the wave equation as following:

𝛻²𝐸 ₀₊− 𝛻 𝛻 ∙ 𝐸 ₀₊ +𝑐²

𝜔₀²𝜀₊𝐸 ₀₊= 0 (2)

The relativistic dielectric tensor 𝜀+ may be evaluated according upon Munther et. al. technique to become [6]

𝜀

+= 𝜀0++ 𝜀2+𝐴 0+𝐴 0+∗ = 1 −

𝜔 𝑝𝑒 𝜔 0 2

1−^{𝜔 𝑐𝑒} 𝜔 0

+

𝜔 𝑝𝑒 𝜔 0 2

1−^{𝜔 𝑐𝑒} 𝜔 0

𝑎𝑟+𝐴 0+𝐴 0+∗ (3)

It is worth noting that the dielectric tensor consists of a linear (

𝜀

0+=1 −

𝜔 𝑝𝑒 𝜔 0

2

1−^{𝜔 𝑐𝑒} 𝜔 0

)and a nonlinear (

𝜀

2+𝐴 0+𝐴 ∗0+ =

𝜔 𝑝𝑒 𝜔 0

2

1−^{𝜔 𝑐𝑒} 𝜔 0

𝑎𝑟+𝐴 0+𝐴 ∗0+) parts.

where

2 2 2 2 0 0 2

0

1

2 (1 )

r

ce

e

 m c  





 



is the relativistic nonlinearity factor in presence of longitudinal magnetic field.

The final equations of laser beam self-focusing in non-paraxial region can be written as:

14498 http://annalsofrscb.ro

𝑑²𝑓₀₊

𝑑𝑧² =

1 +^𝜖0+

𝜖_𝑧𝑧 2

4𝑘₀₊² 𝑥₀⁴ 1

𝑓₀₊³ 8𝛼₀₂− 3𝛼002 − 2𝛼00+ 1 −

1 +^𝜖0+

𝜖_𝑧𝑧

2𝜖0+

(4)

𝜕

𝑆+

𝜕𝑧 = 𝜖𝑟+ 1 − 𝛼₀₀+ 𝛼02 𝐸₀₀² 2𝜖₀₊ƒ₀₊⁶ −

1 + ^𝜖0+

𝜖𝑜_𝑧𝑧

2𝑥₀²𝑘₀₊²ƒ₀₊⁶ 𝛼₀₀²− 7𝛼₀₀𝛼₀₂+ 𝛼₀₀³− 2𝛼₀₂ −4𝑆2+

ƒ₀₊

𝑑ƒ0+

𝑑𝑧 (5)

𝜕𝛼

00

𝜕𝑧 =−8ƒ₀₊²𝑆₂₊

𝑥𝑜2 1 + 𝜖_𝑜+

𝜖𝑜𝑧𝑧

(6)

𝜕𝛼

02

𝜕𝑧 = 4ƒ0+2

𝑆2+

𝑥₀² −12𝛼00ƒ0+2

𝑆2+

𝑥₀² 1 + 𝜖0+

𝜖_0𝑧𝑧 (7)

The first term in the right hand side of the Eq.4 is referring to the natural diffraction effect while the second term of it is representing the nonlinear self-focusing effect. It is more importance to mention that the oscillation pattern of the laser wave inside plasma is appearing as a result of the rivalry between these two effects (i.e.

diffraction effect and self-focusing).

2.2 external transverse magnetic field

In presence of the transversal magnetic field along z- axis, the electric field of the laser wave along the x- axis may be written as :

𝐸 = 𝐴 𝑒𝑥𝑝 𝑒𝑥𝑝𝑖 𝜔₀𝑡 − 𝑘₀𝑥 (8) where the laser wave amplitude 𝐴 = 𝑥 𝐴_𝑥+ 𝑦 𝐴_𝑦is complex function of space which is written as:

𝐴 =𝐴𝑜𝑒𝑥𝑝𝑖 𝑘𝑜𝑆 (9)

𝐴_𝑜and S are representing the real and the phase functions of the laser ray amplitude . The wave equation decidingthe laser ray propagation through plasma is written as

𝛻

²𝐸 = 𝛻 𝛻 ∙ 𝐸 −𝜔𝑜2

𝑐² 𝜀_𝑟. 𝐸 10

The dielectric tensor𝜀_𝑟 of magnetized plasma is responsible the nonlinear behavior of laser wave. Motivating the relativistic nonlinearity and following Monika et. al.[11], one may introduce as

2 2 2 2

2 2

2 2 2

2 2

0 0 0

0 0

0 2 2 2 2

2 2

0 0

1 1

1 ,

1 1

pe pe ce

pe pe

r y y r y y

u u

E E E E

  

 

  

 

   

 

 

 

     

          

     

     

     

    

       

     

     

 

(11)

14499 http://annalsofrscb.ro

This equation contains a linear nonrelativistic term

2 2

2 2

0 0

0 2

2 0

1 1

1

pe pe

u

 

 

 



 

  

 

 

   

  

 

and nonlinear relativistic term

2 2 2 2

2 2 2

0 0 0

2 2 2

2 0

1

1

pe pe ce

y y r y y

u

E E E E

  

  

 





 

     

         

       

 

    

    

 

 

 

.

where

2 2 2 2

0 0 0 0 0

1 1 3 4

4

ce ce pe

r

e m c

  

    

 

       

 

          

 

         

is the relativistic nonlinearity factor in presence of transverse magnetic field.

As in section 2.1 and taken in our account the influence of transverse magnetic field on laser beam propagation via plasma, the final equations related with the self-focusing in non-paraxial region will become

𝜕^ƒ2_𝑜

𝜕𝑥²= ¹

𝑘_𝑜²𝑥_𝑜⁴ 1

ƒ_𝑜³ (1-2

𝛼

𝑜𝑜 − 3𝛼_𝑜𝑜²+ 8𝛼_𝑜2)-(1-

𝛼

𝑜) ^{𝜖𝑟 𝐸𝑜𝑜2}

𝜖₀𝑥_𝑜² 1

ƒ_𝑜² (12)

𝜕𝑆

𝜕𝑥= 𝜖𝑟 𝐸𝑜𝑜2

2𝜖_𝑜ƒ_𝑜⁶ 1 − 𝛼₀₀+ 𝛼₀₂ − 𝛼₀₀²− 7𝛼₀₀𝛼₀₂+ 𝛼₀₀³− 2𝛼₀₂

1 + ^𝜖0

𝜖𝑜_𝑧𝑧

2𝑥_𝑜²𝑘_𝑜²ƒ_𝑜⁶ −4𝑆2

ƒ_𝑜 𝑑ƒ𝑜

𝑑𝑥 (13)

𝜕𝛼

00

𝜕𝑥 =−8ƒ𝑜2

𝑆2

𝑥_𝑜² 1 + 𝜖𝑜

𝜖_𝑜𝑥𝑥 (14)

𝜕𝛼

02

𝜕𝑥 = 4ƒ_𝑜²𝑆₂

𝑥𝑜2 −12𝛼_𝑜ƒ_𝑜²𝑆₂

𝑥𝑜2 1 + 𝜖₀ 𝜖0𝑥𝑥

(15)

Both set of the Eqs. (4-7) and (12-15) for longitudinal and transverse magnetic field respectively have been numerically solved to understand the geometric arrangement influence on laser beam self-focusing inside plasma.

3. Technique of Terahertz (THz) generation

To investigate the THz wave generation, one may use the final equations of laser beam self-focusing first in presence of longitudinal magnetic field (equation 4 ) and other in presence of transversely magnetic field (equation 12).

3.1 in

external longitudinal magnetic field

Now we find the technique terahertz ( 𝐸 _𝑡+, 𝜔_𝑡, 𝑘 _𝑡+) with the effect of relativity by nonlinear coupling between rippled density plasma (𝐸 ₁, 𝜔₁, 𝑘 ₁) with high intense laser beam (𝐸 ₀₊, 𝜔₀, 𝑘 ₀₊ ). The phase-matching conditions

14500 http://annalsofrscb.ro

there are (

𝜔

0= 𝜔₁+ 𝜔_𝑡 ) , ( 𝑘 ₀₊= 𝑘 ₁+ 𝑘 _𝑡+ ) , where E₁ is the electric fields to plasma , k₁is the wave vectors for plasma wave ,𝐸𝑡+is the electric fields of THz wave ,𝐾𝑡+ is the wave vectors of THz wave so the waves electric fields:

𝐸 1= 𝑧 𝐸1𝑒𝑥𝑝 𝑒𝑥𝑝𝑖 𝜔1𝑡 − 𝑘1𝑧 16 𝐸 _𝑡+= 𝐴 _𝑡+𝑒𝑥𝑝 𝑒𝑥𝑝𝑖 𝜔_𝑡𝑡 − 𝑘_𝑡+𝑧 17 Where 𝐴 _𝑡+ is the amplitude for the right polarized circular field to THz.

𝑣 ₁= − ^{𝑖𝑒 𝐸 1}

𝑚_𝑒𝛾𝜔₁ Plasma wave field at z-direction produces in the relativistic oscillating velocity associated ,with ripple plasma of density where 𝑣₁=^𝜔1

𝑘₁

𝜇

and 𝜇 =^{𝑛 𝑝}

𝑛₀ the normalized amplitude for ripple density symbolizing ratio between density perturbed 𝑛 _𝑝 and density of background plasma 𝑛₀ and 𝑛_𝑝 = 𝑛₀+ 𝑛 _𝑝 𝑒𝑥𝑝 𝑒𝑥𝑝𝑖(𝜔₁𝑡 − 𝐾₁𝑧) .

We are using the equations the nonlinear interaction between the electromagnetic field and plasma wave field can be formulated:

−𝑒𝐸 − ^𝑒

𝑐 (𝑣_𝑗× (𝐵 + 𝐵 ₀)) = 𝑚_𝑗𝛾 =^{𝜕𝑣 𝑗}

𝜕𝑡 + 𝑚_𝑗 𝑣 _𝑗 . 𝛻 𝑣 _𝑗is the momentum equation and continuity equation

𝜕𝑛_𝑗

𝜕𝑡 = −𝛻 . 𝑛𝑗𝑣 𝑗

𝑊ℎ𝑒𝑟𝑒 𝑚𝑗 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑎𝑠𝑠 , 𝑛𝑗 𝑡ℎ𝑒 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 , 𝑎𝑛𝑑 𝑣𝑗is the species velocity 𝑗 = 𝑖,e ; 𝐸 , 𝐵 they are electric fields with self-consistency related for the wave.

In the magnetoplasma atmosphere, the electric vector wave equation (𝐸 _𝑡+) with magnetoplasma will formulated as [18]:

4𝜋 𝑐²

𝜕𝐽 𝑡+

𝜕𝑡 = 𝛻²𝐸 _𝑡+− 1 𝑐²

𝜕²𝐸 𝑡+

𝜕𝑡² 18

Where 𝐽 _𝑡+is the total vector for current density with lower electric field to frequency, 𝐸 _𝑡+and is formulated : 𝐽 ₁₊+ 𝐽 ₂₊= 𝐽 _𝑡+ 19

The current linear density 𝐽 ₁₊ and current nonlinear density 𝐽 ₂₊ are produced by:

𝐽 ₁₊= −𝑒𝑛₀𝑣 ₁₊^𝑒 + 𝑒𝑛₀𝑣 ₁₊^𝑖 20 𝐽 ₂₊= −𝑒𝑛 _𝑝^∗𝑣 ₀₊− 𝑒𝑛₀𝑣 ₂₊^𝑒 21

The ion𝑣 1+𝑖 and the electron linear velocities 𝑣 1+𝑒 , can be found by solving, the momentum equation as:

𝑣 ₁₊^𝑒 = 𝑖𝑒𝐸 𝑡+

𝑚₀ 𝛾𝜔_𝑡− 𝜔_𝑐𝑒 22 𝑣 ₁₊^𝑖 = −𝑖𝑒𝐸 _𝑡+

𝑚0 𝛾𝜔_𝑡+ 𝜔𝑐𝑖

23 The ion–cyclotron frequency given by

𝜔

𝑐𝑖 =^𝑒𝐵0

𝑚_𝑖𝑐 .Presenting 𝐵 = ^{𝑐𝐾 0+}

𝜔₀ × 𝐸 0+ for the equation to momentum, 𝑣 2+𝑒 that is nonlinear velocity interacting for ripple density and electric field to the magnetoplasma laser beam:

𝑣 ₂₊^𝑒 = −𝑖𝑒𝐸 0+𝜔𝑐𝑒𝑘0+𝑣1∗

2𝑚₀𝜔₀ 𝜔₀𝛾 − 𝜔_𝑐𝑒 𝜔_𝑡𝛾 − 𝜔_𝑐𝑒 (24) 𝐽 ₁₊=−𝑖𝜔𝑝2

4𝜋

𝛾𝜔𝑡𝐸 𝑡+

𝛾𝜔_𝑡− 𝜔_𝑐𝑒 𝛾𝜔_𝑡+ 𝜔_𝑐𝑖 25 The quiver electron velocity at laser field is :

𝑣 0+𝑒 = 𝑖𝑒𝐸 ₀₊

𝑚0(𝛾𝜔0− 𝜔𝑐𝑒) 26

14501 http://annalsofrscb.ro

Substituting (22) and (23) in (20) and (29), (31) in (26), the expressions for densities of nonlinear with linear current are:

𝐽 ₂₊= −𝑖𝜔_𝑝² 4𝜋

𝜇^∗

𝛾𝜔₀− 𝜔_𝑐𝑒 1 − 𝜔₁𝑘₀₊𝜔_𝑐𝑒

2𝜔₀𝑘₁₊ 𝛾𝜔_𝑡− 𝜔_𝑐𝑒 𝐸 ₀₊ 27

The contribution at nonlinear interaction has been ignored to its immobility. Placing (25) and 27) in (28) the following equation is given :

𝜕

²𝐸

_𝑡+

𝜕𝑧² +𝜔_𝑡²

𝑐² 1 − 𝛾𝜔_𝑝²

𝛾𝜔_𝑡− 𝜔_𝑐𝑒

𝛾𝜔_𝑡+ 𝜔_𝑐𝑖 𝐸

_𝑡+ = 𝜇^∗𝜔₀

𝛾𝜔₀− 𝜔_𝑐𝑒 1 − 𝜔₁𝑘₀₊𝜔_𝑐𝑒

2𝜔₀𝑘₁₊ 𝛾𝜔_𝑡− 𝜔_𝑐𝑒 𝐸

₀₊ 28 By a similar technique, we calculate relativistic factor

𝛾

as in Sec. 2, (28) by the following equation :

𝜕

²𝐸

_𝑡+

𝜕𝑧² + 𝜔_𝑡²

𝑐² 1 − 𝜔_𝑝𝑒²

𝜔_𝑡− 𝜔_𝑐𝑒

𝜔_𝑡+ 𝜔_𝑐𝑖 + 𝛼_𝑡+ 𝐸

_𝑡+

= 𝜔_𝑝² 𝑐²

𝜇^∗𝜔₀

𝜔₀− 𝜔_𝑐𝑒 1 − 𝜔₁𝑘₀₊𝜔_𝑐𝑒

2𝜔₀𝑘₁₊ 𝜔_𝑡− 𝜔_𝑐𝑒 + 𝛼_𝑡𝑡+ 𝐸

₀₊ 29

The relativistic growing mass contribution terms can symbolized by

𝛼

𝑡+ 𝑎𝑛𝑑 𝛼𝑡𝑡 + , and following equation that produced:

𝛼

𝑡+= 𝜔₀𝜔_𝑝𝑒²

𝑐² 𝜔₀− 𝜔_𝑐𝑒 ²𝛼₊𝐸₀₊𝐸₀₊^∗ 30

𝛼

𝑡𝑡 +=𝜔𝑡𝜇^∗

𝑐²

[ 𝜔

𝑝𝑒2 2𝜔_𝑐𝑒 − 𝜔0

𝜔₀ 𝜔₀− 𝜔_𝑐𝑒 ² − 𝜔0𝑘1+𝜔𝐶𝑒𝜔𝑝𝑒2

𝜔₁𝑘₀ 𝜔_𝑐𝑒− 𝜔_𝑡 𝜔₀− 𝜔_𝑐𝑒

× 𝜔

_𝐶𝑒

𝜔_𝑐𝑒− 𝜔_𝑡 + 𝜔_𝐶𝑒

𝜔₀− 𝜔_𝑐𝑒 − 2

]𝛼

+𝐸₀₊𝐸₀₊^∗ 31

It is evident that the non-relativistic issue can be satisfied at

𝛼

𝑡+ and

𝛼

𝑡𝑡 + have disappeared, and to THz generation intensity; (29) was numerically solved. The function of z is 𝑓₊ it is governed by (4). The field for THz produced by this the technique can numerically found also straightforward by the equation (29) , and appropriate limited conditions at the parameter width for laser beam alters and propagation distance given by (4).

3.2in external transverse magnetic field

The mechanism of the THz wave

 ^{E , }

^t

^

^t

^{, k }

^t



production depends upon the nonlinear interaction between X-

mode high intense laser beam

 ^{E , }

⁰

^

⁰

^{, k }

⁰



and the UHW

 ^{E , }

¹

^

¹

^{, k }

¹



introducing the momentum phase matching

k 

t

 k 

0

 k 

1

and energy phase matching



t

  

0



1 conditions. The variation of the electric field of UHW and THz wave can be written as

 

1

ˆ

1

exp

1 1

,

E   xE  i  t  k x

(32)

 

ˆ exp ,

t t t t

E   yE  i  t  k x

(33)

Where

 ^k

¹

^  ^

¹²

^{ }

^u2

 ^v

^2th



is the propagation constant of UHW and

 ^

^th

^{ k T}

^{B e}

^m

⁰



is electron thermal velocity.

The essential equations which govern the interactions between above electric fields inside plasma are

14502 http://annalsofrscb.ro

(a) The momentum equation

  

⁰



0

1 ,

j B j j

j j j j j

k T n

m e E B B

t c n

   

 

          

      

 

      

(34)

(b) The continuity equation

  ^0,

j

j j

n n

t 

    





(35)

(c) Faraday equation

1 B ,

E c t

    



 

(36)

(d) Ampere equation

4 1

E ,

B J

c c t

 

   



  

(37)

where m_j,



^j,T_j,n_j are the mass, the fluid velocity, the temperature and the particle density of species

j  e i ,

respectively.

k

B_and

c

are Boltzmann constant and light velocity in vacuum.

B 

0

is the external static magnetic field whereas

E 

and

B 

are the self-consistent electric and magnetic fields which corresponded with waves.

In the presence of the electric field of THz wave

E

t, the total current density J^t may be governed by the wave equation taking the curl of the Faraday equation and using Ampere equationas [19]

2 2

2 2 2 2

4 1

t ,

t J t

E E

x c t c t



^

 

 



  (38) the total current density

J 

t

is given by

 

^{* y}

0 0 0 0

0

( ) v v 1 v ,

2

i e i e u

t ty ty t nl y t nl y

J en e n n e n n

 

^{ }_{  }^_

n

^_

   

    

  

(39)

wheren^u represents the complex conjugate of the fluctuation density

n

^u in UHW. In low frequency field, using Eq.

(34), the ion and electron velocities

i

 ^

ty _and

 ^

_ty^e may be given as

2 1

2 2 2 2

0

1 ,

i ci ci B i t

ty ty x

t i i

t t

k T n

ie E

m m n

 

      



 

   

 

   

     

   

    



(40)

14503 http://annalsofrscb.ro

2 1

2 2 2 2

0 0 0

1 ,

e ce ce B e t

ty ty x

t

t t

k T n

ie E

m m n

 

      



 

    

                 

 

(41) where

n

tis the fluctuation density in THz field.

The term

e n 

0

v

_{t nl y}ⁱ

 n

0

v

^e_{t nl y}



represents the nonlinear contribution of the species by the ponderomotive force

[ (v. )v

^e

(v )]

e e

F m q B

    c 

     

see Eq. (34).

Using the continuity equation, the fluctuation density

n

u may be written as:

1

0 1

1 u

n n k 

 

here

^

¹

^{ m}

⁰

 ^{ieE }

¹²¹

^{ }

¹

^

^ce2



 

is the electron velocity in UHW field.

Substituting equation of fluctuation density

n

uand Eqs. (40 and 41 ) in Eq.(39) , one can rewrite Eq. (38) to be

𝜕

²𝐸_𝑡𝑦

𝜕𝑥² − 1 𝑐²

𝜕²𝐸_𝑡𝑦

𝜕𝑡² − 1 𝑣_𝐴²

𝜕²𝐸_𝑡𝑦

𝜕𝑡² = −4𝜋𝑛0𝑇𝑒

𝐵₀²

𝜕²𝐸_𝑡𝑦

𝜕𝑥² +4𝜋𝑛0𝑒

𝑐² −𝑖𝜔𝑡𝐹𝑒𝑥

𝜔_𝐶𝑒𝑚₀− 𝜔_𝑡²𝐹_𝑒𝑦 𝛾𝜔_𝐶𝑒² 𝑚0

+ 𝑖𝜔_𝑡4𝜋𝑛0𝑒

2𝑐² 𝜇^∗𝑣_𝑥 42

where ⁰

n

u

  n

is normalized fluctuation density of UHW to the nonfluctuation plasma density

n

0_,



^ _{is a}

complex conjugate of



_and

V

_A

 B

0²

 4   n

0

m

_i



is the velocity of Alfvén wave.

The components of ponderomotive force may written as



¹



ex 2 2

0

F 1 ,

2

ce

y u

i e   E

  

 



      

(43)

 

2

ey 2 2

0

F 1 ,

2

ce

y u

e  E

  

 



     

(44) Where E_ywhich represents the electric field of laser in non- paraxial region.

It is important to mention that Eq.(42) has been written by introducing a quasi-neutrality condition

 n

i

 n

e

 n

0



and cold ion plasma

 T

i

 ⁰ 

. It is also written by assuming

𝜔

𝑡2≪ 𝑘_𝑡²𝑣_𝑡ℎ² and

𝜔

𝑡2 ≪ 𝜔_𝐶𝑒² . Substituting Eqs. (43 and 44) in Eq.(42) one can obtain

𝜕

²𝐸_𝑡𝑦

𝜕𝑥² + 𝜔_𝑡²

𝑐² −𝜔_𝑝𝑒² 𝑘_𝑡² 𝑐²𝜔_𝐶𝑒²

𝑘_𝐵𝑇_𝑒 𝑚₀ +𝜔_𝑡²

𝑣_𝐴² 𝐸_𝑡𝑦 =𝜔_𝑡𝜇^∗ 2𝑐²

𝜔_𝑝𝑒² 𝜔_𝑡− 𝜔₁ 𝛾 𝜔₀²− ^{𝛾−1𝜔𝑝𝑒}

2+𝜔_𝑢² 𝛾²

− 𝑖 𝜔_𝑝𝑒²𝜔₀ 𝛾 𝜔₀²− ^{𝛾−1𝜔𝑝𝑒}

2+𝜔_𝑢² 𝛾²

𝐸_𝑦 45

Using similar procedure in Sec.2.2 [20] to derive the relativistic factor



, Eq. (45) will take the following form

14504 http://annalsofrscb.ro

𝜕

²𝐸_𝑡𝑦

𝜕𝑥² + 𝜔_𝑡²

𝑐² −𝜔_𝑝𝑒² 𝑘_𝑡² 𝑐²𝜔_𝐶𝑒²

𝑘_𝐵𝑇_𝑒 𝑚₀ +𝜔_𝑡²

𝑣_𝐴² 𝐸_𝑡𝑦= 𝜔_𝑡𝜇^∗ 2𝑐²

𝜔_𝑝𝑒² 𝜔_𝑡− 𝜔₁

𝜔₀²− 𝜔_𝑢² 1 + 𝛼₁ − 𝑖 𝜔_𝑝𝑒² 𝜔₀

𝜔₀²− 𝜔_𝑢² 1 + 𝛼₂

𝐸_𝑦 46

where

 

1

and 

2



are nonlinear factors related with the relativistic nonlinearity



,see Eqs. (* and **), as follows

𝛾 = 1 +1 4

𝑒 𝑚₀𝜔₀𝑐

2

1 + 3 𝜔𝐶𝑒

𝜔₀

2

+ 4 𝜔𝐶𝑒

𝜔₀

2 𝜔𝑝𝑒

𝜔₀

2

+ 2 𝜔𝐶𝑒

𝜔₀

4

+ 5 𝜔𝐶𝑒

𝜔₀

2 𝜔𝑝𝑒

𝜔₀

4

+ 2 𝜔𝐶𝑒

𝜔₀

2 𝜔𝑝𝑒

𝜔₀

6

+ 13 𝜔_𝐶𝑒 𝜔0

4 𝜔𝑝𝑒

𝜔0 4

+ 8 𝜔_𝐶𝑒 𝜔0

6 𝜔𝑝𝑒

𝜔0 2

+ 6 𝜔_𝐶𝑒 𝜔0

4 𝜔𝑝𝑒

𝜔0 6

+ 8 𝜔_𝐶𝑒 𝜔0

6 𝜔𝑝𝑒

𝜔0 4

+ 4 𝜔𝐶𝑒

𝜔₀

6 𝜔𝑝𝑒

𝜔₀

6

+ 4 𝜔𝐶𝑒

𝜔₀

8 𝜔𝑝𝑒

𝜔₀

4

𝐸_𝑦𝐸_𝑦^∗ ∗

It is important to mention that we have neglected the terms of higher orders than

E E

^{y y} _order.

The relativistic factor



may be written as follows [20],

𝛾 ≅ 1 + 𝛼𝐸_𝑦𝐸_𝑦^∗ ∗∗

Where:

     

 

2 2 2 2 2 2

0 1

2 2

1 0

2

1 2 2 2 1

0

2

2 ,

2

u ce t ce pe ce

u t pe

y y

u

E E c

       

  

    

 



 

 

 

   

   

 

   

 

 



2

2 2 2 2 2 2

0 0

3 2 4 2 2 2 2 2 2

0 0 0 0 0

2 3

0 0 2 2 2 2

0 0

4 3

3

2

4

,

t pe

u ce

ce ce pe pe ce

y y ce

u ce

c

E E

  

   

         

   

   

  

  

  

  

   

   

   

 

    

    

    

 



    

 

  

 

In nonrelativistic regime



1

and 

2 will vanish, thus Eq. (46) will reduce to non relativistic regime.

Eq.(46) tell us that the beating between high intense laser beam and UHW will lead to generate magnetosonic wave (MSW) at a THz frequency as long as the phase matching conditions are satisfied.

4. Results and discussion

Figure (1) shows that the laser wave propagation through plasma in non- paraxial region in presence the external magnetic fields.it has been observed that in presence of LMF, the laser beam self-focusing is quicker and stronger comparing with TMF. Same behavior of laser beam self-focusing has been recorded in paraxial region (see Figure (2)). By comparing of two figures (Fig. 1 and Fig. 2), one may note that the laser beam self-focusing in nonparaxial region is showing asymmetrical pattern through its propagation via plasma. In opposite, the laser beam self-focusing in paraxial region is showing symmetrical pattern through its propagation via plasma.

14505 http://annalsofrscb.ro

If the phase matching conditions between three waves, namely laser wave, plasma wave and THz wave, have been satisfied thus an excited wave in THz frequency range may be generated. Figure (3) and figure (4) explain THz generation in non-paraxial and paraxial regions respectively. In paraxial region, the normalized THz field amplitude (𝑬^𝒕⁄𝑬^𝟎𝟎) is showing higher value in comparing with nonparaxial region. by comparing between figures 1 and 2 with figures 3 and 4 respectively , it is observed that whenever the laser beam self-focusing is stronger so the excited terahertz field is also higher.

Figure 1

. Variation Comparison of LMF and TMF Magnetic Field Self-focused Laser Beam Inside Magnetized Plasma owing to Relativistic Laser Strength Parameter in Non-paraxial region .

Figure 2.

Variation Comparison of LMF and TMF Magnetic Field Self-focused Laser Beam inside Magnetized Plasma owing to Relativistic Laser Strength Parameter in Paraxial region .

14506 http://annalsofrscb.ro

Figure 3.

Variations of the normalized THz field amplitude (𝑬^𝒕⁄𝑬^𝟎𝟎) along normalized propagation distance ( ξ ) comparison between terahertz wave generation at non-paraxial region in presence of LMF and TMF(normalized vector potential α0+=1.4) .

Figure 4.

Variations of the normalized THz field amplitude (𝑬𝒕⁄𝑬𝟎𝟎) along normalized propagation distance ( ξ ) comparison between terahertz wave generation at paraxial region in presence of LMF and TMF(normalized vector potentialα0+=1.4) .

5. Conclusions

In this paper the external magnetic fields have a considerable role in enhancement of the self-focusing phenomenon in both LMF case and TMF case but in LMF case it has more influence.

14507 http://annalsofrscb.ro

Along the distance of laser beam propagation through plasma, the THz amplitudes are corresponding to the positions of stronger self-focusing. This may be understood because of higher laser beam intensities at those positions.

This THz wave generation mechanism has excited a THz field power at range of GW and frequency

 ^

^t

^{  1 10}

¹²

^{rad .sec}

^1

 _s

o it is consider a prefer source of THz wave generation.

6.Acknowledgment

This work was supported by Ministry of Higher Education and Scientific Research, Government of Iraq. The authors thank the Department of Physics, Faculty of Science, University of Kufa, Najaf, Iraq, for helpful work.

References

[1] P.K. Kaw, Nonlinear laser–plasma interactions. Rev. Mod. Plasma Phys. 1, 2 (2017)

[2] Rawat, P., &Purohit, G. "Self-focusing of a cosh-Gaussian laser beam in magnetized plasma under relativistic-ponderomotive regime. Contributions to Plasma Physics", Vol. 59.2,(2018).

[3] Munther B. Hassan, IbtisamJaaferAbd-Ali, AhmedObaidSoary "The filamentation instability of nonparaxial laser beam inside magnetized Plasma (2019).

[4] Nidhi Pathak1 Manpreet Kaur1,2 Sukhdeep Kaur1 T. S. Gill1 "Non-paraxial theory of self- focusing/defocusing of HermitecoshGaussian laser beam in rippled density plasmas",(2019)"

[5] Malik, H.K., Devi, L.," Relativistic Self Focusing and Frequency Shift of superGaussian Laser Beam in Plasma", Results in Physics (2020),17

[6] Hassan, Munther B., and Ahmed O. Soary. "The effects of external magnetic field on laser beam self- focusing through plasma." Journal of Kufa-Physics 6.1 (2014).‏

[7] Jabbar, A. T., “The Geometrical Effect of External Magnetic Field on Non-Paraxial Laser Beam Interaction with Plasma”, M. Sc. Thesis, University of Kufa, (2020).

[8] Moldosanov, K. A., Postnikov, A. V., Lelevkin, V. M. &Kairyev, N. J. Terahertz imaging technique for cancer diagnostics using frequency conversion by gold nano-objects. Ferroelectrics 509, 158–166 (2017).

[9] Tóth, G., Tibai, Z., Sharma, A., Fülöp, J. A. &Hebling, J. Single-cycle attosecond pulses by thomson backscattering of terahertz pulses. J. Opt. Soc. Am. B 35, A103–A109 (2018).

[10] Baierl, S. et al. Nonlinear spin control by terahertz-driven anisotropy fields. Nat. Photonics 10, 715–718 (2016).

[11] Monika Singha, Munther B. Hassan b, A. H. Al- Janabi)c .R. P. Sharma)d Narender Kumar)e" Study of Terahertz Generation in Magnetized Plasma Via Self Focused Ultra-Relativistic Laser Beam)e"(2016).

[12] Aggarwal, Munish, ShivaniVij, and Niti Kant. "Self-Focusing of Quadruple Gaussian Laser Beam in an InhomogenousMagnetized Plasma with Ponderomotive Non-Linearity: Effect of Linear Absorption."

Communications in Theoretical Physics 64.5: 565. (2015)

[13] Zhang P, Thomas AGR." Enhancement of high-order harmonic generation in intense laser interactions with solid density plasma by multiple reflections and harmonic amplification". ApplPhys Lett;106:131102.(

2015)

[14] R. Kaur, T.S. Gill, R. Mahajan," Relativistic effects on evolution of a q-Gaussian laser beam in magnetoplasma: application of higher order corrections", Phys. Plasmas 053105 -053105 (2017)

14508 http://annalsofrscb.ro

[15] Nanni, E. A. et al. Terahertz-driven linear electron acceleration. Nat. Commun. 6, 8486 (2015).

[16] Jin, Q., Yiwen, E., Williams, K., Dai, J. & Zhang, X.-C. Observation of broadband terahertz wave generation from liquid water. Appl. Phys. Lett. 111, 071103 (2017).

[17] Kuk, D. et al. Generation of scalable terahertz radiation from cylindrically focused two-color laser pulses in air. Appl. Phys. Lett. 108, 121106 (2016).

[18] Hassan M. B., "Terahertz radiation generation by nonlinear interaction of intense Laser beam with magnetoplasma", PhD Thesis, University of Baghdad, Iraq, (2012).

[19] P. K. Shukla and R. P. Sharma, "Alfvén-wave generation in a beam-plasmasystem", Phys. Rev. A, 25, 2816, (1982).

[20] H. A. Salih and R. P. Sharma, "Plasma wave and second-harmonic generation of intense laser beams due to relativistic effects", Phys. Plasmas,11, 6, (2004).