Second harmonic generation of Cosh-Gaussian laser beam in collisional plasma with nonlinear absorption

Optics Communications 381 (2016) 180–188

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Second harmonic generation of Cosh-Gaussian laser beam in collisional plasma with nonlinear absorption Navpreet Singh a, Naveen Gupta b, Arvinder Singh b,n a b

Guru Nanak Dev University College, Kapurthala, Punjab, India Department of Physics, National Institute of Technology, Jalandhar, India

art ic l e i nf o

a b s t r a c t

Article history: Received 9 February 2016 Received in revised form 14 June 2016 Accepted 16 June 2016

This paper investigates second harmonic generation (SHG) of an intense Cosh-Gaussian (ChG) laser beam propagating through a preformed underdense collisional plasma with nonlinear absorption. Nonuniform heating of plasma electrons takes place due to the nonuniform irradiance of intensity along the wavefront of laser beam. This nonuniform heating of plasma leads to the self-focusing of the laser beam and thus produces strong density gradients in the transverse direction. The density gradients so generated excite an electron plasma wave (EPW) at pump frequency that interacts with the pump beam to produce its second harmonics. To envision the propagation dynamics of the ChG laser beam, moment theory in Wentzel–Kramers–Brillouin (W.K.B) approximation has been invoked. The effects of nonlinear absorption on self-focusing of the laser beam as well as on the conversion efficiency of its second harmonics have been theoretically investigated. & 2016 Published by Elsevier B.V.

Keywords: Self-Focusing Cosh-Gaussian Plasma Waves Second Harmonic

1. Introduction Invention of laser [1] in 1960 opened a completely new area of laser-plasma interactions, which is extremely complex, fascinating, challenging, and rich in physics. Over the last 45 years, these laser– plasma interactions are at the vanguard of theoretical as well as experimental investigations. The research in this field has given birth to many exciting applications. Some of the applications such as guiding microwaves [2], lightning protection [3], triggering rain and snow [4], and inertial confinement fusion (ICF) [5–7] have obvious impact on the real world. Other applications such as generation of terahertz radiations [8–10], electron acceleration [11–14], X-ray lasers [15,16], etc., are though subtler, yet no less important to the scientific community and for the development of new technologies. Most of these applications require efficient coupling of laser energy with plasmas. In the absence of an optical guiding mechanism, diffraction broadening of the laser beams negates the efficiency of laser–plasma coupling and thus jeopardizes the feature of the above mentioned applications. Therefore, there have been ongoing efforts to explore the methods or processes that may help in increasing the efficiency of laser– plasma coupling. Self-focusing is such a nonlinear phenomenon that averts the diffraction broadening of the laser beam [17]. It n

Corresponding author. E-mail addresses: [email protected] (N. Singh), [email protected] (N. Gupta), [email protected] (A. Singh). http://dx.doi.org/10.1016/j.optcom.2016.06.047 0030-4018/& 2016 Published by Elsevier B.V.

arises on account of nonlinear response of material medium to the field of incident laser beam leading to the modification of its refractive properties in such a way that medium starts behaving like a converging lens. In collisional plasmas, Ohmic heating of plasma electrons due to nonuniform irradiance along the wavefront of the laser beam modifies the refractive properties of plasma. Several processes like parametric instabilities (stimulated Raman scattering [18–20], stimulated Brillouin scattering [21–23], etc.), higher harmonic generation (HHG) [24–26], etc., can compete to determine the propagation dynamics of laser beams through plasmas. In laser produced plasmas, HHG is an important nonlinear phenomenon and has become an important field of research. Harmonic generation has strong influence on the nature of laser propagation through plasmas. It is an important diagnostic tool for obtaining information of various plasma parameters without requiring a probe into the plasma [27]. An important example is SHG, which helps in detecting the passage of intense laser pulses through underdense plasmas [28]. Ultra short pulse duration, and good spatial and temporal coherence of harmonic radiations make them good candidates for applications in the vacuum ultraviolet (VUV) or extreme ultraviolet (XUV) region [29]. This includes investigations of ultra fast ionization of atoms, molecules, and clusters in strong fields of short wavelengths, ultra fast spatial interferometry in the XUV region, and ultrafast holography. Several mechanisms (resonance absorption [30], parametric instabilities [31], photon acceleration [32], ionization fronts [33], transverse density gradients associated with light filaments [28], and excitation of EPW [34–37]) are responsible for generation of

N. Singh et al. / Optics Communications 381 (2016) 180–188

higher harmonics of laser beams in plasmas. The excitation of EPW is particularly important because it provides a potent mechanism for the generation of higher harmonics of laser beams in plasmas. Harmonic generation of electromagnetic radiations in plasmas has been the subject of extensive study for quite sometime [38– 41]. Hora and Ghatak [42] derived and evaluated second harmonic resonance for a perpendicular incidence at four times the critical density. Kant et al. [43] investigated the effect of pulse slippage on SHG of short laser pulse in plasma. Sukhdeep et al. [44] investigated resonant SHG of Gaussian laser beam in collisional magneto plasma. The presence of magnetic field significantly enhances the second harmonic yield. Resonant SHG of a millimetre wave in the presence of magnetic wiggler has been studied by Aggarwal et al. [45]. The effects of self-focusing of Gaussian laser beam on SHG in collisional [36], collisionless [37], and relativistic [46] plasmas have been investigated by Singh and Walia with the help of moment theory approach [47]. SHG of p-polarized laser beam in underdense plasma has been investigated by Jha and Agrawal [48]. The literature review reveals the fact that the main thrust of theoretical and experimental investigations on SHG is so far directed towards the revelations of propagation characteristics of Gaussian laser beams. Keeping in view the intensity profile of the Vulcan Petawatt laser [49] at Rutherford Appleton laboratory, Gupta et al. [50] reported SHG by q-Gaussian laser beam in collisional plasma channel. The intensity profile of the q-Gaussian laser beam is just a deviation from Gaussian profile. However, in the past few years, a significant interest has been gained by ChG laser beams, a new class of laser beams that possess high power and low divergence than that of Gaussian as well as q-Gaussian laser beams [51–54]. The intensity profile of ChG laser beam resembles with Gaussian, flat-topped, and dark hollow laser beam with a suitable choice of decentered parameter. This means that Gaussian, flat-topped, and dark hollow spot can be derived from a single profile. Hence, ChG laser beams are more versatile than both Gaussian and q-Gaussian laser beams. Thus, such lasers can be utilized to achieve large and uniform energy transport in ICF. To the best of authors' knowledge, no theoretical investigation on S.H. G in collisional plasmas with nonlinear absorption has been earlier carried out for ChG irradiance along the wavefront of laser beam. This article for the first time aims to delineate the effects of selffocusing and nonlinear absorption of ChG laser beam on SHG in collisional plasmas. The systematic organization of this paper is as follows: In Sections 2 and 3, the nonlinear dielectric function and nonlinear attenuation function of the plasma have been derived. In Section 4, moment theory has been invoked to derive differential equation governing the propagation dynamics of the laser beam. The source term for S.H.G has been obtained in Section 5. In Section 6, an expression for second harmonic yield has been derived. The detailed discussion and conclusions drawn from the present investigation have been summarized in Sections 7 and 8 respectively.

181

where E0 is the slowly varying complex amplitude of the field of laser beam and r0 is the spot size of the laser beam at plane of incidence i.e., at z¼ 0. The parameter b is known as decentered parameter of the laser beam. For 0 ≤ b ≤ 1, the intensity profile of the laser beam resembles with that of flat topped laser beam, and for b > 1, the intensity profile resembles with that of beams with central shadow. For z > 0, the energy conserving Cosh-Gaussian ansatz for the laser beam is

E0E0⋆ =

2 E00

f2

2 ⎛ − r r02f 2 cosh2⎜

⎞ b ⎟ −2Kabz r e ⎜ r0f ⎟ ⎝ ⎠

e

(2)

where, E00 is the axial amplitude of the field of laser beam and r0f is the instantaneous spot size of the laser beam. Hence, the function f is termed as dimensionless beam width parameter, which is a measure of both axial intensity and spot size of the laser beam. The function Kab is known as attenuation function of the plasma averaged over the cross section of the laser beam. The dielectric function of the plasma can be written as

ϵ = ϵ r + ϵi

(3)

where ϵr and ϵi are the real and imaginary parts of the dielectric function respectively. The real part is responsible for the nonlinear refraction of the laser beam; whereas imaginary part is responsible for its attenuation.

ϵr = 1 −

ωp2 ω02

(4)

4π e 2 ne m

(5)

where

ωp2 =

is the plasma frequency in the presence of laser beam, and e and m are electronic charge and mass respectively. The nonuniform intensity distribution of the laser beam along its wavefront produces nonuniform heating of the plasma electrons. Consequently, redistribution of electrons takes place. The resultant distribution of electrons is given by [57] s

⎛ 2T ⎞1 − 2 0 ne = n 0 ⎜ ⎟ ⎝ T0 + Te ⎠

(6)

where, T0 is the equilibrium temperature of the plasma channel. Te is the temperature in the presence of laser beam, which is related to electric field of laser beam by

⎛ ⎞ r2 βE 2 − 2 2 b ⎟ −2Kabz T = 1 + 200 e r0 f cosh2⎜ r e ⎜ r0 f ⎟ T0 f ⎝ ⎠ where β =

e2 M 6K0T0m2ω02

(7)

is the coefficient of collisional nonlinearity.

Using Eqs. (5)–(7) in (4), one obtains s

2. Nonlinear dielectric response of the plasma Consider the propagation of a linearly polarized laser beam with electric field vector E(r , z, t ) through an underdense plasma of equilibrium density n0. The initial intensity distribution of the laser beam along its wavefront at z = 0 is assumed to be CoshGaussian and is given by[54–56] 2

E0E0⋆

= z= 0

⎞ b ⎟ r ⎟ ⎝ r0 ⎠

4πe

(8)

2

where ωp20 = m n0 is the plasma frequency in the absence of the laser beam. The parameter s describes the nature of collisions and can be defined through the dependence of collision frequency ν on v2

⎛

−r 2 2 E00 e r0 cosh2⎜ ⎜

⎫2 − 1 ⎧ ⎛ ⎞ r2 ⎪ ωp20 ⎪ 1 βE002 − r 2 f 2 2⎜ b ⎟ −2Kabz ⎬ e 0 cosh r e ϵr = 1 − 2 ⎨ 1 + 2 ⎜ ⎟ r f 2 f ⎪ ω0 ⎪ ⎝ 0 ⎠ ⎭ ⎩

electron's random velocity v and temperature Te as ν ∝ ( T )s . For e

(1)

velocity independent collisions s¼0, for collisions between electrons and diatomic molecules s ¼2, and for electron–ion collisions

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N. Singh et al. / Optics Communications 381 (2016) 180–188

s¼ -3. Taking

ϵ r = ϵ 0 + ϕ(E0E0⋆)

(9)

where ϵ0 and are the linear and nonlinear parts of the dielectric function of the channel, respectively, we get

ϕ(E0E0⋆ )

ϵ0 = 1 −

∇2E +

ωp20 ω02

(10)

and s ⎛ ⎞ ⎫ ⎧ 2 ⎞ 2 − 1⎪ ⎛ ⎛ ⎞ 2 − r ⎜ ⎟ ωp20 ⎪ β E 1 00 r 2f 2 ⎜ 2⎜ b ⎟ −2Kabz ⎟ ⋆ ⎬ ϕ⎜ E0E0 ⎟ = 2 ⎨ 1 − ⎜ 1 + e 0 cosh r e ⎟ ⎪ ⎜ r0f ⎟ 2 f2 ω0 ⎪ ⎜⎜ ⎟⎟ ⎝ ⎠ ⎠ ⎪ ⎝ ⎪ ⎝ ⎠ ⎭ ⎩

(11)

⎛ ⎞ r2 2 − ∞ E00 ⎜ b ⎟ 2 2 ϵi 2 e r0 f cosh2⎜ r f r ⎟rdr 0 0 f

∫ k 0c 2

⎝ ⎠ ⎛ ⎞ 2 ∞ E2 ∫0 002 e r02 f cosh2⎜⎜ r b f r ⎟⎟rdr f 0 ⎝ ⎠ 2 − r

(12)

s ⎛ 2 ⎧ ⎛ ⎞⎫ 2 2 − r ⎪ s + 5) ⎜ ωp20 ν0 ⎪ ( βE00 4 b 2 2 2 ⎜ ⎟ ⎜ ⎨ 1 + 2 e r0 f cosh ϵi = πΓ r ⎬ ⎜ r0 f ⎟⎪ 3 2 ⎜⎜ ω02 ω0 ⎪ f ⎝ ⎠⎭ ⎩ ⎝

(s + 5) 2 eb

2

(13)

∫0

(20)

Using Eq. (20) in (19), one obtains

ω02 c2

ϕ(E0E0⋆)E0 − ιKabE0 − 2ιk 0

where, ∇⊥2 =

∂2 ∂r 2

+

1 ∂ r ∂r

∂E0 =0 ∂z

(21)

is the Laplacian along transverse direction.

The first term on the left-hand side of Eq. (21) is known as spatial dispersion term that spreads the beam in transverse direction. The second term on the left-hand side of Eq. (21) arises due to collisional nonlinearity of the plasma and is responsible for self-contraction of the transverse dimensions of the laser beam. In deriv∂ 2E0 ∂z 2

has been neglected under the as-

sumption that the wave–amplitude scale length along the z-axis is much larger as compared to the characteristic scale in the transverse direction. The solution to Eq. (21) is obtained by taking spatial moments of a given set of trial function. In moment theory approach, the average value of a physical quantity A(z ) is defined as

A(z ) =

1 I0

∫ E0⋆A(z)E0d2r

(22)

where

1 Iab k 0r02

(14)

I0 =

∫ E0E0⋆d2r

(23)

and

where

Iab =

(19)

Since we are interested in global beam behavior, it makes sense to separate here the rapid variations in the laser field phase, Ψ = ω0t − k0z , from its amplitude E0(r , z ) by applying the slowly varying envelope approximation [59].

Using Eq. (12) in (13), we get

Γ

ϵE = 0

ing Eq. (21) , the term

Following Sodha et al. [57]

4 Kab = π 3

c2

∇2⊥E0 +

The nonlinear attenuation function of the plasma averaged over the cross section of the laser beam is defined as [58]

Kab =

ω02

E = E0eιΨ e x

3. Attenuation function of the plasma

ω02

The polarization term ∇(∇ . E) of wave equation (18) can be neglected, provided r . m . s beam radius is much greater than the vacuum wavelength [47]. We need to stress only that Eq. (18) is a nonlinear wave equation, since ϵ depends on the field amplitude E0 via Eq. (8). Under this approximation, Eq. (18) reduces to

∞⎧

⎨1 + ⎩

2 βE00

⎪

f2

−x 2

e

s ⎫2

−x 2

cosh (bx)⎬ e ⎭ 2

⎪

2

cosh (bx)xdxx =

d2r = rdrdθ

r r0 f

(15)

The propagation of an electromagnetic beam in an isotropic and nonconducting medium ( J = 0, ρ = 0, μ = 1) is described by Maxwell's equations

1 ∂D c ∂t

1 I0

σ 2(z ) =

4. Propagation dynamics of laser beam

∇×B=

The quantity of particular importance from the standpoint of selffocusing is mean square radius <σ 2(z )>, which is defined as

(16)

∫ E0⋆r 2E0d2r

(24)

Following the procedure of Lam and Lippmann [47,60], we get following quasi-optic equation governing the evolution of mean square radius of the laser beam with distance of propagation

I d2 H < σ 2(z ) > = 4 2 + 4 2 I0 I0 dz

(25)

where

1 ∂B ∇×E=− c ∂t

(17)

∫ 21k 2 (|∇⊥E0|2 − F )d2r 0

where E and B are electric and magnetic fields respectively, associated with the laser beam and D = ϵ(E0E0⋆ )E is the electric displacement vector. By eliminating B from Eqs. (16) and (17), it can be shown that the electric field vector E(r , z, t ) of the laser beam satisfies the wave equation

∇2E − ∇(∇. E) +

I2 =

ω02 c2

ϵE = 0

(18)

E0E0⋆

F (E0E0⋆) =

1 2ϵ 0

H = 2F −

1 E0E0⋆ϕ(E0E0⋆) 2ϵ 0

∫0

ϕ(E0E0⋆)d(E0E0⋆)

(26)

(27)

(28)

N. Singh et al. / Optics Communications 381 (2016) 180–188

Using Eqs. (2), (23)–(24), (26)–(28) in (25), we get 2

d f dξ 2

+

2 1 ⎛ df ⎞ ⎜ ⎟ = f ⎝ dξ ⎠

1 + e−b

2⎞ ⎜1 − b ⎟ ⎝ ⎠

(

f

3

+

∇E T = − 4πeN

where N is the total electron density, E T is the sum of electric fields of laser beam and EPW

)

2

⎛ −b2 ⎞ ωp20 r02 ⎛ s ⎞ βE 2 ⎟(T − bT ) ′ ξ⎜ e ⎜ − 1⎟ 300 e−Kab 2 2 ⎝ 2 2⎟ 1 ⎜ ⎠ c f ⎝1+b ⎠

N = n0 + n′E T = E + E′ (29)

is the oscillatory velocity of electrons. Using linear perturbation theory, Eqs. (30)–(32) reduce to

z k 0r02

∂n′ = ∇(n0 ve) = 0 ∂t

is the dimensionless distance of propagation and s

T1 =

∫0

∞

2 ⎧ ⎫2 − 2 2 1 βE00 −x 2 −2Kab 2 ′ ξ⎬ ( ) x 3⎨ 1 + e bx e e−2x cosh4 (bx)dxT2 cosh 2 f2 ⎩ ⎭ ⎪

∫0

∞

m

2 ⎧ ⎫2 − 2 2 1 βE00 −x 2 −2Kab 2 ′ ξ⎬ ( ) x 2⎨ 1 + e bx e e−2x cosh3(bx) cosh 2 2 f ⎩ ⎭ ⎪

(33)

⎪

s

=

and

v = ve

where

ξ=

(32)

2⎛

2 1+b 1

183

d ve KT = − eE′ − 3 0 0 ∇n′ dt m

(34)

⎪

sinh(bx)dx

(35)

∇E′ = − 4πen′

and

Combining Eqs. (33)– (35), we get

(s + 5) Γ 4 2 I ′ = Kab π ab 2 3 eb

∂ 2n′ e 2 2 − vth ∇ n′ + ωp2n′ = n0∇E m ∂t 2

is the dimensionless attenuation function of the plasma. For an initially plane wavefront, Eq. (29) is subjected to boundary condf ditions f = 1, dξ = 0 at ξ ¼0. There are two terms both on the right- left-hand sides of Eq. (29). As the laser beam propagates through the plasma, the evolution of its spot size is governed by the relative competition between these two terms. The first term on the right-hand side has its origin in the Laplacian ( ∇⊥2 ) in the wave equation (21) and is known as spatial dispersion term that spreads the beam in transverse directions. Whereas, the second term comes into picture under the effect of collisional nonlinearity and is responsible for the contraction of transverse dimensions of the laser beam. Depending on the numerical values of these two terms, one can observe focusing/defocusing of the laser beam. The first term on the left-hand side determines the evolution of spot size of the laser beam with distance of propagation. The second term on the left-hand side, which is zero at ξ = 0, evolves with distance of propagation and contracts the diffraction divergence of the laser beam.

(36)

Taking

n′ = neι(ω0t − k 0z ) where n is the amplitude of density perturbation associated with EPW, we get 2 −ω02n′ + k 02vth n′ + ωp2n′ =

e n0∇E m

(37)

Using Eq. (2) in Eq. (37), we get the following source term for S. H.G of Cosh-Gaussian laser beam in collisional plasma. −

r2

2 2 en0 e 2r0 f e−Kabz n= ⎛ ⎞ s 1 m 2 ⎜ ω02 − k 02vth − ωp20(1 + βE0E0⋆)2 − 1⎟ ⎝ ⎠ 2

⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ ⎜ r − b tanh⎜ b r ⎟⎟cosh2⎜ b r ⎟ ⎜ ⎟ ⎜ r02 f 2 ⎜ r0 f ⎟ r0 f ⎝ r0 f ⎠⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

(38)

5. Excitation of electron plasma wave The background plasma density is modified via nonuniform irradiance of intensity along the wavefront of the laser beam. Therefore, the amplitude of EPW that depends on the background electron density, gets strongly coupled with the laser beam. The contribution of the ions is negligible due to their heavier mass and hence, provides only a positive background in the dynamics of excitation of EPW. The generated plasma wave is governed by

 equation of continuity ∂N + ∇(Nv) = 0 ∂t

KT dv = − eE T − 3 0 0 ∇N dt N

 Poisson's equation

The generated plasma wave interacts with the incident pump beam to produce second harmonic frequency ω2. The wave equation for the second harmonic field E2 can be deduced from Maxwell's equations

∇2E2 +

ω22 c2

ϵ2(ω2)E 2 = −

8π ιω2 JNL 2 c2

where ω2 = 2ω0 is the second harmonic frequency,

(30)

 momentum balance equation m

6. Second harmonic generation

ϵ2 is the ef-

fective dielectric constant at second harmonic frequency, and J2NL is the nonlinear second harmonic current density and is given by

JNL = − ne v2 2 (31)

(39)

(40)

where v2 is the oscillatory velocity of the electrons at second harmonic frequency ω2, and is given by

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N. Singh et al. / Optics Communications 381 (2016) 180–188

v2 = − n

e2 E 0 mιω2

(41)

Using Eqs. (40)–(41) in (39), we get

∇2E 2 +

ω22 c

2

ϵ2(ω2)E 2 =

ωp2 n E0 c 2 n0

(42)

From the above equation, we get expression for field E2 of 2nd harmonic as

ωp2 n E0 2 n 2 c 0 (k2 − 4k 02)

E2 =

Now the 2

P2 =

nd

(43)

harmonic power can be written as

∫ E2E2⋆d2r

(44)

also, the power of initial pump beam is given by

P0 =

∫ E0E0⋆d2r

(45)

Defining 2nd harmonic yield as

Y2 =

Fig. 1. Variation of beam width parameter f with normalized distance of propagation ξ for different values of decentered parameter b viz., b¼ 0, 0.25, 0.50, 1, 1.15, ωp0r0 2 ν 2 1.30 and at fixed values of βE00 ¼ 3.0, ( ) = 12, 0 = 0.05, s = − 3.

P2 P0

c

we get 2 2 4 m ⎛ K0T0 ⎞ βE00 ⎜⎜ ⎟⎟ Y2 = e−b H 3 M ⎝ mc 2 ⎠ f 4

(46)

where H=

∞

∫0

e−2x

( )

(x − btanh( bx ))2cosh4 bx

2

s ⎫2 ⎧ ⎞ 2 − 1⎪ ⎛ ⎞ 2 ω 2r 2 2 ω p20r 02 ⎛ ⎪ ω 2r 2 vth βE 00 2 1 2 ξ − 0 0 0 0 K 2 − x ′ ⎜1 + ⎬ ⎨ e cosh ⎜⎜ bx ⎟⎟e − ϵ0 − ab ⎟ ⎟ ⎜ 2 f2 ⎪ ⎪ c2 c2 c2 c2 ⎝ ⎝ ⎠ ⎠ ⎭ ⎩

dx

7. Discussion The evolution of spot size and second harmonic yield of CoshGaussian laser beam with dimensionless distance of propagation ξ in a collisional plasma with nonlinear absorption is governed by Eqs. (29) and (46), respectively. These equations have been solved numerically for the following set of laser as well as plasma parameters

ω0 = 1.78 × 1015 rad/sλ = 1.06 μmr0 = 15 μmT0 = 106 K to investigate the effect of propagation dynamics of Cosh-Gaussian laser beam on generation of its second harmonics in collisional plasma by including the effect of nonlinear absorption. Fig. 1 illustrates the effect of decentered parameter of the ChG laser beam on its focusing/defocusing behavior. It is observed that the spot size of the laser beam decreases monotonically with distance of propagation as it propagates through the plasma. Beyond the focus, the diffraction effects become predominant. Hence, the beam width keeps on increasing monotonically. This happens due to the damping of the laser energy as a consequence of nonlinear absorption of the laser beam. The plots in Fig. 1 also depict that there is increase in the extent of self-focusing of the laser beam with increase in the value of decentered parameter of the laser beam for 0 ≤ b ≤ 1. The underlying physics behind this fact is that the intensity profile of the ChG laser beam resembles with that of flat topped laser beams for 0 ≤ b ≤ 1, and the degree of flatness increases with the increase in the value of b. Hence, for

ω0

0 ≤ b ≤ 1, the intensity of off axial rays become equal to that of axial rays. As a result, there is increase in the extent of self-focusing of the laser beam with increase in the value of b for 0 ≤ b ≤ 1. It is observed from Fig. 1 that there is decrease in the extent of self-focusing of the laser beam with increase in the value of decentered parameter of the laser beam for b > 1. This is due to the fact that the intensity maxima of the laser beam for b > 1 appears in the off axial part of the laser beam rather than in axial part. As a result, the axial part of the laser beam becomes weaker as compared to the off axial part with increase in the value of b beyond 1, which in turn, leads to reduced focusing of the laser beam with increase in the value of b for b > 1. Fig. 2 illustrates the effect of peak intensity of the laser beam on its focusing/defocusing. It is observed that there is decrease in the extent of rate of increase of the spot size of the laser beam beyond the focus with increase in its intensity. This is due to the fact that increase in the intensity of the laser beam leads to weakening of its nonlinear absorption. Fig. 3 illustrates the effect of plasma density on the focusing/ defocusing of the laser beam. The stronger and earlier focusing of the laser beam is observed with increase in the density of the plasma. The stronger focusing of the laser beam is due to the fact that the number of electrons contributing to the collisional nonlinearity also increases with increase in the density of plasma. Whereas, earlier focusing is due to the increase in the magnitude of nonlinear refractive term in Eq. (29) with increase in the plasma density. Fig. 4 illustrates the effect of collisional frequency on focusing/ defocusing of the laser beam. It is observed that there is decrease in the extent of self-focusing of the laser beam with increase in the collisional frequency. This is due to the fact that increase in collisional frequency leads to enhanced damping of the laser energy. As a result, the magnitude of the refractive term in Eq. (29) decreases. Hence, increase in collisional frequency leads to reduced focusing of the laser beam. Fig. 5 illustrates the effect of nature of collisions on focusing/ defocusing of the laser beam. It is observed that no self-focusing of the laser beam takes place in plasmas dominant with collisions between electrons and diatomic molecules. This is due to the fact that s ¼2 corresponds to the situation, where ne ¼n0 ¼n0i and

N. Singh et al. / Optics Communications 381 (2016) 180–188

Fig. 2. Variation of beam width parameter f with normalized distance of propa2 2 gation ξ for different values of laser intensity βE00 viz., βE00 = 3.0, 3.50, 4.0 and at ωp0r0 2 ν fixed values of b ¼ 0.25, ( ) = 12, 0 = 0.05, s¼  3. c

ω0

Fig. 4. Variation of beam width parameter f with normalized distance of propaν ν gation ξ for different values of collisional frequency 0 viz., 0 ¼0, 0.05, 0.10 and at ωp0r0 2 ω0 ω0 2 fixed values of b ¼0.25, βE00 = 3.0 , ( ) ¼12, s ¼  3. c

Fig. 3. Variation of beam width parameter f with normalized distance of propaωp0r0 2 gation ξ for different values of normalized plasma density ( ) viz., c ωp0r0 2 ν0 2 ¼ 0.05, s ¼  3. ( ) = 12, 14, 16 and at fixed values of b¼ 0.25, βE00 = 3.0 , c

185

ω0

hence, no redistribution of electrons takes place. Thus, the magnitude of nonlinear refractive term in Eq. (29) becomes zero. The diffraction divergence is a dominating mechanism in this case. Fig. 6 illustrates the effect of decentered parameter b on attenuation of the laser beam. It is observed that the extent of nonlinear absorption of the laser beam decreases as the decentered parameter of the laser beam increases towards b¼ 1. So, it is concluded that flat topped laser beams are more suitable for extended propagation of laser beams through plasmas. Fig. 7 illustrates the effect of decentered parameter b on yield Y2 of second harmonics of the incident laser beam. It is observed that maximum yield of second harmonics occurs at the focal spots of the laser beam. This is due to the fact that the focal spots of the laser beam are the regions of very high intensity and thus act as a source of plasma wave at pump frequency. The plasma wave so generated interacts with the pump beam to produce its second harmonics. Thus, maximum yield of second harmonics occurs at focal spots of the laser beam. The plots in Fig. 7 also depict that there is increase in the second harmonic yield with increase in the value of decentered

Fig. 5. Variation of beam width parameter f with normalized distance of propagation ξ for different values of s viz., s¼ -3, 0, 2 and at fixed values of b¼ 0.25, ωp0r0 2 ν 2 ¼ 3.0, ( βE00 ) = 12, 0 = 0.05. c

ω0

parameter for 0 ≤ b ≤ 1. However, there is decrease in the second harmonic yield with further increase in the value of decentered parameter for b > 1. This is due to the fact that there is one-to-one correspondence between second harmonic yield and extent of self-focusing of the laser beam. Greater is the extent of self-focusing of the laser beam, higher will be the yield of second harmonics. It may be seen from Fig. 1 that there is increase in the extent of self-focusing of the laser beam with increase in the value of its decentered parameter for 0 ≤ b ≤ 1. There is decrease in the extent of self-focusing of the laser beam with increase in the value of its decentered parameter for b > 1. Hence, increase in the value of its decentered parameter for 0 ≤ b ≤ 1 leads to increase in the second harmonic yield whereas; for b > 1, there is decrease in the yield of second harmonics. Figs. 8 and 9 illustrate the effects of peak intensity of the laser beam and plasma density on second harmonic yield Y2 respectively. It is observed that there is substantial increase in the yield of second harmonics with increase in laser intensity or plasma density. This is due to increase in the extent of self-focusing of the

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′ with normalized distance of propaFig. 6. Variation of attenuation function Kab gation ξ for different values of decentered parameter b viz., b¼ 0, 0.50, 1.0 and at ωp0r0 2 ν0 2 fixed values of βE00 = 3.0 , ( ) = 12, = 0.05, s¼  3. c

ω0

Fig. 8. Variation of second harmonic yield Y2 with normalized distance of propa2 2 gation ξ for different values of laser intensity βE00 viz., βE00 = 3.0, 3.50, 4.0 and at ωp0r0 2 ν fixed values of b¼ 0.25, ( ) = 12, 0 = 0.05, s ¼  3. c

Fig. 7. Variation of second harmonic yield Y2 with normalized distance of propagation ξ for different values of decentered parameter b viz., b ¼0, 0.25, 0.50, 1, 1.15, ωp0r0 2 ν 2 1.30 and at fixed values of βE00 = 3.0 , ( ) = 12, 0 ¼0.05, s ¼  3. c

ω0

Fig. 9. Variation of second harmonic yield Y2 with normalized distance of propaωp0r0 2 gation ξ for different values of normalized plasma density ( ) viz., c ωp0r0 2 ν0 2 ( ) = 12, 14, 16 and at fixed values of b ¼0.25, βE00 = 3.0 , = 0.05, s ¼  3. c

ω0

ω0

laser beam with increase in laser intensity or plasma density. Fig. 10 illustrates the effect of collision frequency on second harmonic yield Y2. It is observed that there is decrease in the yield Y2 of second harmonics with increase in the value of collision frequency. This is due to the fact that increase in the collision frequency leads to enhanced damping of the laser energy, which in turn, leads to reduction in the yield of second harmonics. Fig. 11 illustrates the effect of nature of collisions on second harmonic yield Y2. It is observed that there is no SHG in plasmas dominant with collisions between electrons and diatomic molecules. This is due to the fact that self-focusing of the laser beam does not occur in plasmas dominant with collisions between electrons and diatomic molecules.

8. Conclusions In the present work, the authors have investigated the effects of self-focusing and nonlinear absorption of Cosh-Gaussian laser beam on second harmonic generation in collisional plasma. It is

Fig. 10. Variation of second harmonic yield Y2 with normalized distance of proν ν pagation ξ for different values of collisional frequency 0 viz., 0 = 0, 0.05, 0.10 and ω0 ωp0r0 2 ω0 2 at fixed values of b¼ 0.25, βE00 = 3.0 , ( ) = 12, s ¼  3. c

N. Singh et al. / Optics Communications 381 (2016) 180–188

Fig. 11. Variation of second harmonic yield Y2 with normalized distance of propagation ξ for different values of s viz., s ¼  3, 0, 2 and at fixed values of b ¼0.25, ωp0r0 2 ν 2 βE00 = 3.0 , ( ) = 12, 0 = 0.05. c

ω0

observed from present analysis that there is increase in the extent of self-focusing of the laser beam for 0 ≤ b ≤ 1 whereas; for b > 1, there is decrease in the extent of self-focusing. Also, harmonic yield increases with increase in decentered parameter of the laser beam for 0 ≤ b ≤ 1. It is further observed that nonlinear absorption of the laser beam decreases with increase in the decentered parameter of the laser beam. Therefore, ChG laser beams are more suitable than Gaussian beams to suppress nonlinear absorption, and to enhance the harmonic yield. The present results may be useful for the scientists working in the field of laser-plasma interactions.

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