Second harmonic generation of high power laser beam in cold quantum plasma

Journal Pre-proof Second harmonic generation of high power laser beam in cold quantum plasma Keshav Walia, Vinit Kakkar, Deepak Tripathi

PII:

S0030-4026(19)32049-2

DOI:

https://doi.org/10.1016/j.ijleo.2019.164150

Reference:

IJLEO 164150

To appear in:

Optik

Received Date:

2 December 2019

Accepted Date:

27 December 2019

Please cite this article as: Walia K, Kakkar V, Tripathi D, Second harmonic generation of high power laser beam in cold quantum plasma, Optik (2019), doi: https://doi.org/10.1016/j.ijleo.2019.164150

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Second harmonic generation of high power laser beam in cold quantum plasma

Keshav Walia1, Vinit Kakkar2 , Deepak Tripathi2

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(1) Department of Physics, DAV University Jalandhar, India (2) Department of Physics, AIAS, Amity University Noida, India

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Email: [email protected]

Abstract

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Second harmonic generation of high power laser beam in cold quantum plasma (CQP) is investigated in present communication. In order to derive 2nd order differential equations

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governing the behavior of laser beam against normalized distance, paraxial theory and WKB approximation are adopted. There is self-focusing of beam on account of relativistic increase in

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mass of electron, which further results in production of intensity gradients in transverse direction. The electron plasma wave (EPW) gets excited due to these intensity gradients at pump wave frequency. Further, there is an interaction of EPW with pump beam thereby producing to

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second harmonics. In order to study the effect of various parameters of laser and plasma medium on self-focusing of pump beam and second harmonic yield, we have carried out numerical simulations. The comparison of present results is made with classical relativistic plasma (CRP).

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Keywords: Self-Focusing, Cold quantum plasma, Electron plasma wave, Second harmonic yield.

PACS: 52.38.Hb, 52.35.Mw, 52.38.Dx

1. Introduction Interaction of lasers with plasma medium is gaining importance among several research groups due to its diverse applications including laser induced fusion, laser driven particle accelerators, harmonic generation, super continuum generation[1-12]. In order to have success in above mentioned applications, deeper penetration of lasers in plasma is preferable. Several nonlinear phenomena such as filamentation, self-focusing, scattering instabilities, self-phase modulation etc. comes in to picture during penetration of lasers through plasma [13-27]. So, investigation of

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some of these nonlinear effects is extremely important for much deeper understanding of laserplasma interaction physics.

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These days, construction of lasers having intensities in the range of 1018 𝑤/𝑐𝑚2 is made

possible on account of inertial confinement fusion programs. During propagation of intense

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lasers through plasmas, quiver electron velocity becomes equivalent to light’s velocity, thereby triggering notable increase in electron’s mass. This variation of mass will result in modification

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of dielectric function of plasma. As a result, plasma behave as convex lens resulting in selffocusing of beam. Most demanding work in laser-plasma interaction physics is generation of

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harmonics [28-31]. There is vigorous impact on propagation of laser beams through plasmas on account of harmonic generation. The information regarding various plasma parameters including electrical conductivity, electron density can be easily obtained from harmonic generation.

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Plasmas are considered as most promising medium for production of higher harmonics. This medium is contained in ionized state and can handle intense electric fields. In material and biological imaging, harmonic radiations are useful on account of higher penetrating power. The production of harmonics has been studied in the past by theoretical as well as experimental

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researchers [32-38].

Among several mechanisms of generating harmonics in plasmas, the most common include resonance absorption, parameteric instabilities, transverse density gradients, excitation of EPW and photon acceleration [39-46]. Excitation of EPW at pump wave frequency is most common mechanism of generating second harmonic in plasmas. These days, theoretical as well as experimental work on interaction of laser beam with quantum plasmas is going on due to its several applications. In quantum plasmas, density of plasma is

high whereas its temperature is low. Quantum plasmas are also gaining importance due to their relevance to fusion science, laser-solid interaction, nanotechnology and quantum dots etc.[4754]. Fermi dirac distribution is employed in quantum plasmas, whereas Boltzman-Maxwell distribution is employed in classical plasmas. Classical plasma treats every particle as point like, because de-Broglie wavelength linked with it very small. In quantum plasmas, de-Broglie wavelength linked with particle is of the order of average inter-particle distance. Literature survey on second harmonic reveals that past work on second harmonics is done in classical

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plasmas and with uniform laser beam. But, beams with Gaussian intensity distribution have nonuniform intensity distribution. Focusing/de-focusing behavior is associated with such beams. Self-focusing phenomena has strong influence on yield of second harmonic generation (SHG).

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So, motivation of present work is to study impact of self-focused beam on SHG in CQP by

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incorporating relativistic nonlinearity.

In section 2, 2nd order differential equation for beam width of pump beam has been derived. In

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section 3, source term of SHG is derived. Expression for yield of SHG is derived in section 4. Detailed discussion of computational results and conclusion are discussed in section 5 and

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section 6.

2. Evolution of beam width parameter

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Consider the propagation of gaussian beam in CQP along Z-axis. The distribution of electric field for such beam is given by

𝐸(𝑟, 𝑧) =

𝐸0 𝑟2 𝑒𝑥𝑝 (− ) 𝑓 2𝑟02 𝑓2

(1)

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Here, 𝐸0 represents amplitude of beam, ‘𝑟0 ′ and ‘f’ represents spot size and beam width of the beam. The high power beam propagating through cold quantum plasma spreads oscillatory velocity to electrons, 𝑒𝐸 . 0 𝜔𝛾

whose magnitude is given by 𝑣 = 𝑚

Here, electronic charge, rest mass of electrons and frequency

associated with incoming laser beam is represented by 𝑒, 𝑚0 and 𝜔 respectively. The Lorentz relativistic factor is represented by 1⁄ 2

𝛾 = (1 + 𝛼𝐸𝐸 ∗ )

(2)

2

Where 𝛼 = 𝑒 ⁄ 2 2 2 is the nonlinear coefficient. One may obtain the dielectric function for nonlinear 𝑚0 𝜔0 𝑐 medium by following approach of Sodha et al.[55-56] and may be expressed as 𝜀 = 𝜀0 + 𝜑(𝐸𝐸0∗ ) Here 𝜀0 = 1 −

2 𝜔𝑝

𝜔02

(3)

represents linear part of dielectric function. 𝜔𝑝 = √

4𝜋𝑛0 𝑒 2 𝑚0

is known as plasma

frequency. The unmagnetized dielectric function for CQP is expressed as[57] 2 𝜔𝑝

𝜀 = 1 − 𝛾𝜔2 (1 − 0

4𝜋4 ℎ 2 , 𝑚2 𝜔02 𝜆4

(4)

′ℎ′ and ‘λ’ denotes Planck’s constant and wavelength of beam. By setting 𝛿𝑞−>

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Where, 𝛿𝑞 =

𝛿𝑞 −1 ) 𝛾

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0, one can easily obtain dielectric function of CRP. Following wave equation is satisfied by electric field E of laser beam, 𝜔02 𝜀𝐸 𝐶2

=0

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∇2 𝐸 − ∇(∇. 𝐸) +

one can get 𝜔2

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By WKB approximation, second term of above equation can be ignored, provided

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∇2 𝐸 + 𝐶 20 𝜀𝐸 = 0 The solution of Eq. (6) is

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𝐸 = 𝐴(𝑟, 𝑧)exp ι(𝜔0 𝑡 − 𝑘0 𝑧) On substituting Eq.(7) in Eq.(6) and neglecting 2𝜄𝑘0

𝜕𝐴 𝜕𝑧

𝜕2 𝐴 , 𝜕𝑧 2

= ∇2⊥ 𝐴 +

𝑐2 1 2 | ∇ 𝑙𝑛 𝜔02 ∈

(5) ∈| ≪ 1, so

(6)

(7)

one obtains quasi optic equation represented by

𝜔02 𝜑(𝐸𝐸0∗ )𝐴 𝑐2

(9)

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Here 𝜑(𝐸𝐸0∗ ) is nonlinear part of dielectric function represented by 𝜑(𝐸𝐸0∗ ) =

2 𝜔𝑝

𝜔02

[1 − (1 −

𝛿𝑞 −1 ) ] 𝛾

(10)

Now, further assuming variation of A(r, z) as 𝐴 = 𝐴0 (𝑟, 𝑧)exp(−𝜄𝑘0 𝑆)

(11)

Where, 𝐴0 (𝑟, 𝑧) and S are the functions of r and z. Now, on substituting Eq.(11) in Eq.(9) and equating real part and imaginary part, one can get

𝜕𝑆 2

𝜕𝑆

1

𝜑

2 (𝜕𝑧) + (𝜕𝑟) = 𝑘 2 𝐴 ∇2⊥ 𝐴0 + 𝜀 0 0

(12)

0

and 𝜕𝐴2

𝜕𝐴2

𝜕𝑆

( 𝜕𝑧0 ) + (𝜕𝑟 ) ( 𝜕𝑟0 ) + 𝐴20 ∇2⊥ 𝑆 = 0

(13)

Following [55-56], One can write the solutions of Eqs.(12) and (13) as 1

𝑆 = 2 𝑟 2 𝛽(𝑧) + 𝛷(𝑧) 𝐸02 𝑟2 𝑒𝑥𝑝 (− ) 𝑓2 𝑟02 𝑓2

of

𝐴20 =

(14)

1 𝑑𝑓

(15)

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Where, 𝛽(𝑧) = 𝑓 𝑑𝑧 , 𝛷(𝑧) represents phase shift of beam. On substituting Eqs. (14) and (15) in Eq.(12) and using paraxial approach, differential equation for beam width ‘f’ of beam can be obtained as

𝑑2𝑓 𝑑𝜂2

=

1 𝑓3

𝜔𝑝 𝑟0 2 𝛼𝐸02

−(

𝑐

)

2𝑓

3 (1 +

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−2

−3/2 𝛼𝐸02 ) 𝑓2

1−

𝛼𝐸2 √1+ 20 𝑓

(16) )

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(

𝛿𝑞

3 Excitation of EPW

For exciting EPW, we need to consider only plasma electrons. One may ignore contribution of

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ions, since they are heavy and immobile. There is modification in plasma density via variation in relativistic electron mass. So, amplitude of EPW interacts strongly with input beam. The EPW generated through input beam is governed by following equations: (1) Equation of continuity 𝜕𝑁𝑒 𝜕𝑡

+ ∇ ∙ (𝑁𝑒 𝑉) = 0

(17)

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(2) Poisson’s equation

∇ ∙ 𝐸 = 4𝜋(𝑍𝑁𝑜𝑖 − 𝑁𝑒 )𝑒

(18)

(3) Equation of state 𝑃 𝛾

𝑁𝑒

= 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡

(19)

(4) Equation of motion 𝜕𝑉

1

𝛾

𝑚 [ 𝜕𝑡 + (𝑉 ∙ ∇)V] = −𝑒 [𝐸 + 𝑐 𝑉 × 𝐵] − 2𝛤𝑚𝑉 − 𝑁 ∇𝑃𝑒 𝑒

(20)

With the help of linear perturbation theory, equation of EPW can be as follows 2 𝛿𝑞 −1

2 2 −𝜔02 𝑛1 − 𝑣𝑡ℎ ∇ 𝑛1 + 2𝜄𝛤𝜔0 𝑛1 + 𝜔𝑝2 [(1 − 𝛾 ) ]

𝑒 𝑛1 ≅ 𝑚 (𝑛0 ⃗⃗⃗ ∇. 𝐸⃗ )

(21)

Now, On introducing value of E in Eq.(21), one can obtain source equation for second harmonic as 1 −1 2 −𝜔2 ((1−𝛿𝑞) ) {𝜔02 −𝑘 2 𝑣𝑡ℎ 𝑝 𝛾

(22)

2

}

of

𝑒𝑛0 𝐸0 𝑟2 𝑟 exp (− 2 2 ) { 2 2 } 𝑚 𝑓 2𝑟0 𝑓 𝑟0 𝑓

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𝑛1 =

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4 Second harmonic Generation

Second harmonic is produced on account of an interaction between excited EPW and pump

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beam. Through Maxwell’s equations, one can obtain wave representing electric field 𝐸2 of second harmonic as

∇2 𝐸2 +

𝜔22 𝑐2

𝜀2 (𝜔2 )𝐸2 =

2 𝜔𝑝 𝑛1

𝑐 2 𝑛0

𝐸0

(23)

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Where, frequency of second harmonic is denoted by 𝜔2 = 2𝜔0 , effective dielectric function of plasma at 𝜔2 is denoted by 𝜀2 (𝜔2 ). One can get expression for magnitude of 𝐸2 as 𝐸2 =

2 𝜔𝑝 𝑛1 𝐸0

𝑐2

𝑛0 𝑓

exp (−

𝑟2 2𝑟20 𝑓2

1

) (𝑘 2 −4𝑘 2 ) 2

0

(24)

Second harmonic yield Y2 is defined as ratio of power of second harmonic to power of input beam and its expression can be expressed as

𝑒 2 𝐸02 𝜔𝑝4 1 𝑌2 = 4 2 2 4 2 4𝑓 𝑟0 𝑚 𝑐 (𝑘2 − 4𝑘02 )2

1 −1

𝜔20 − 𝑘2 𝑣2𝑡ℎ − 𝜔2𝑝

2

𝛿𝑞

1−

√1 +

((

{

2

𝛼𝐸200 𝑓2 ) ) }

(25)

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3. Discussion

The behavior of ‘𝑓’ 𝑎𝑛𝑑 ‘𝑌2 ’ against the normalized distance 𝞰 in CQP is represented by Eqs.(16)

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and (25). These equations are to be solved numerically, because it is not possible to have

and (25): 2 𝜔𝑝

𝜔02

= 0.2, 0.4,0.6,

𝑟0 = 3𝜇𝑚, 4𝜇𝑚, 5𝜇𝑚

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𝛼𝐸02 = 2, 4, 6,

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analytical solutions of these equations. Following set of parameters are used for solving Eqs. (16)

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On RHS of Eq. (16), there are several terms. There is some physical mechanism behind each term. First term has origin in Laplacian in Quasi optics equation (9). This term is accountable for diffraction divergence of beam. Second term is due to nonlinearity, is accountable for refraction

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of laser beam. There is competition between these two terms, while transit of laser through plasma. Overall focusing/defocusing behavior of beam is mainly decided by relative magnitude of two terms.

The plot of beam width ‘f’ against at distinct beam (𝛼𝐸02 = 2, 4, 6) is depicted in figure 1.

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Again other parameters of laser and plasma medium are kept constant. Analysis from the figure shows that beam’s tendency to converge is moved towards larger values of 𝞰 with increasing value of laser intensity. This is further due to weakening of refractive term in contrast to diffractive term as laser intensity is increased. 2 𝜔𝑝

The plot of beam width ‘f’ against 𝞰 at distinct plasma density (𝜔2 = 0.2, 0.4,0.6) is represented 0

by figure 2. Again other parameters of laser and plasma medium are kept constant. Analysis

from the figure shows that beam’s tendency to converge is moved towards lesser values of 𝞰 as plasma density is increased. As we increase plasma density, there is production of more electrons at relativistic speeds, which improves the self-focusing effect. The plot of beam width ‘f’ against 𝞰 at distinct beam radius (𝑟0 = 3𝜇𝑚, 4𝜇𝑚, 5𝜇𝑚) is represented by figure 1. Here, other parameters of laser and plasma medium are kept constant. The beam’s tendency to converge is moved towards lesser values of 𝞰 as the value of beam radius is increased. This is mainly because of comparative superiority of refraction term over

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diffraction term as beam radius is increased.

The plot of beam width ‘f’ against 𝞰 for two plasma cases is represented in figure 4. Analysis

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from the figure shows that beam’s tendency to converge is moved towards lesser values of 𝞰 in CQP as compared to CRP. This is again because of reason that additional self-focusing effect is

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added due to quantum contribution, which was absent in classical relativistic case. The plot of second harmonic yield 𝑌2 against 𝞰 at distinct values of 𝛼𝐸02 (𝛼𝐸02 = 2, 4) is depicted in

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figure 5. Analysis from figure depicts that with increase in 𝛼𝐸02 , 𝑌2 decreases. This is due to the reason that focusing extent associated with beam has strong impact on 𝑌2 . Since, with increase in

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𝛼𝐸02 , there is decrease in self-focusing. So, lesser will be generation of transverse intensity gradients thereby leading to decrease in amplitude of EPW and hence 𝑌2 .

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The plot of second harmonic yield 𝑌2 against 𝞰 at distinct values of

2 𝜔𝑝

𝜔02

2 𝜔𝑝

(𝜔2 = 0.2, 0.4) is represented 0

by figure 6. Analysis from the figure shows that increase in density of plasma electrons results in increase in 𝑌2 . This is again due to sensitiveness of 𝑌2 on self-focusing behavior associated with beam. As self-focusing effect becomes more pronounced, more will be generation of transverse

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intensity gradients resulting in increase in 𝑌2 . The plot of second harmonic yield 𝑌2 against 𝞰 at distinct values of 𝑟0 (𝑟0 = 3𝜇𝑚, 4𝜇𝑚) is represented

by figure 7. Increase in 𝑌2 is observed as radius of beam in increased. This is due to fact that 𝑌2 is highly sensitive to extent of self-focusing associated with beam. More the self-focusing behavior of beam, more will be the generation of transverse intensity gradients and hence more will be second harmonic yield.

The plot of second harmonic yield 𝑌2 against 𝞰 for two plasma cases is represented in figure 8. The comparison between 𝑌2 of CRP and CQP is studied. Analysis from figure shows that 𝑌2 is found to be more for CQP as compared to CRP. This is due to fact that self-focusing behavior of beam is more pronounced in CQP.

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4. Conclusion Second harmonic generation of laser beam in CQP is studied by using WKB and paraxial ray

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approximation in the present communication. Following results were concluded:-

(1) There is shifting of beam’s tendency to converge towards lower η values as beam radius and

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plasma density is increased.

(2) Focusing extent of the beam decreases as intensity is increased.

(3) With increase of beam radius and plasma density, second harmonic yield increases.

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(4) There is decrease in second harmonic yield with increase in intensity. (5) On incorporating quantum effects, there is improvement in the self-focusing of pump beam

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and second harmonic yield in CQP in comparison to CRP. These present results are helpful in understanding laser plasma interaction physics. For

guide.

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experimental researchers working in laser-plasma interaction, present results may serve as a

Decleration of interest statement

Please find enclosed the manuscript entitled “Second harmonic generation of high power laser

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beam in cold quantum plasma” authored by Keshav Walia, Vinit Kakkar and Deepak Tripathi for publication in your esteemed journal. This manuscript is original work and is not submitted elsewhere.

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Black line-->2.0 Red line---->4.0 Green line-->6.0

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1.0

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0.8

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f

0.6

0.2

0.5

1.0

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0.0

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0.4

1.5

2.0

2.5

3.0



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Figure 1: The plot of beam width ‘f’ against at distinct beam intensity (𝛼𝐸02 = 2, 4, 6)

Black line-->0.2 Red line---->0.4 Green line-->0.6 1.0

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0.8

f

ro

0.6

re

-p

0.4

0.0 0.5

1.0

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0.0

lP

0.2

1.5

2.0

2.5

3.0

 2 𝜔𝑝

Jo

Figure 2: The plot of beam width ‘f’ against 𝞰 at distinct plasma density (𝜔2 = 0.2, 0.4,0.6) 0

Black line-->3 Red line---->4 Green line-->5 1.0

of

0.8

ro

f

0.6

-p

0.4

0.0 0.5

1.0

1.5

lP

0.0

re

0.2

2.0

2.5

3.0



Jo

ur na

Figure 3: The plot of beam width ‘f’ against 𝞰 at distinct beam radius (𝑟0 = 3𝜇𝑚, 4𝜇𝑚, 5𝜇𝑚)

Black line--->CQP Red line----->CRP 1.0

of

0.8

f

ro

0.6

0.2

0.5

1.0

1.5

lP

0.0

re

-p

0.4

2.0

2.5

3.0

ur na



Jo

Figure 4: The plot of beam width ‘f’ against 𝞰 for two plasma cases (CQP vs CRP)

Red line---> 2.0 Black line-->4.0 0.020 0.018 0.016

of

0.014 0.012

ro

Y2

0.010 0.008

-p

0.006 0.004

re

0.002

-0.002

0.5

ur na

0.0

lP

0.000

1.0

1.5

2.0

2.5



Jo

Figure5: The plot of second harmonic yield 𝑌2 against 𝞰 at distinct beam intensity (𝛼𝐸02 = 2, 4)

Black line---> 0.2 Red line ---> 0.4

0.020 0.018 0.016 0.014

of

0.012

ro

Y2

0.010 0.008

-p

0.006 0.004

re

0.002 0.000

0.0

0.5

lP

-0.002 1.0

1.5

2.0

2.5

3.0

ur na



Jo

Figure6: The plot of second harmonic yield 𝑌2 against 𝞰 at distinct plasma density (

2 𝜔𝑝

𝜔02

= 0.2, 0.4)

Red Line--> 4m Black line-->3m

0.040 0.035 0.030

of

0.025

ro

Y2

0.020 0.015

-p

0.010

0.000 -0.005 1.0

lP

0.5

ur na

0.0

re

0.005

1.5

2.0

2.5

3.0



Jo

Figure 7: The plot of second harmonic yield 𝑌2 against 𝞰 at distinct beam radius(𝑟0 = 3𝜇𝑚, 4𝜇𝑚)

Red line--> CRP Black Line-->CQP 0.016 0.014 0.012 0.010

Y2

of

0.008

ro

0.006 0.004

-p

0.002

-0.002 0.0

0.5

1.0

re

0.000

1.5

2.0

2.5

3.0

lP



Jo

ur na

Figure 8: The plot of second harmonic yield 𝑌2 against 𝞰 for two plasma cases (CQP vs CRP)