2. EXCITATION OF PLASMA WAVE IN PRESENCE OF HGLB

To study Raman scattering of a HGLB propagating in collisionless plasma, first the excitation process of EPW through the same is to be studied, since the amount of scattered light is proportional to the plasma wave's amplitude. Therefore, in this section first the propagation of high power HGLB through the plasma is being studied and hence the excitation process of plasma wave is presented considering relativistic nonlinearity. For this purpose, a high power HGLB of frequency ω₀ and the wave vector k ₀ is considered to be propagating in hot, collisionless, and homogeneous plasma in the z-direction. The irradiance distribution (z = 0) of the incident beam is

(1)$${{\bf E}_{\bf 0}}{\bf \cdot E}_{\bf 0}^ * \left\vert {_{z = 0}} \right. = E_{00}^2 {({r^2}/2r_0^2 )^{2m}}\exp ( - {r^2}/r_0^2 )$$

where the direction of propagation has been assumed to be along the z-axis and r refers to radial coordinate of the cylindrical coordinate system, r ₀ is the spot size of the beam, respectively, E ₀ is a real constant characterizing the amplitude of the HGLB, respectively, E ₀₀ refers to the complex amplitude of the beam which denotes the electric field maximum at ${r_{\max}} = {r_0}\sqrt {2m} $, corresponding to z = 0. The positive integer number m decides order of HGLB which characterizes the shape of the HGLB and position of its irradiance maximum. Equation (1) represents fundamental Gaussian beam of width r ₀ for m = 0. We can write the following equation for the field inside the plasma

(2)$${\nabla ^2}{{\bf E}_{\bf 0}} + \displaystyle{{{\rm \omega} _0^2} \over {{c^2}}}\left( {1 - \displaystyle{{{\rm \omega} _{\rm p}^2} \over {{\rm \omega} _0^2}}} \right){{\bf E}_{\bf 0}} = 0$$

where ω_p is the plasma frequency. The wave equation governing the electric vector of the HGLB in plasma can be written as assuming the variation of electric field in terms of vector potential as

(3)$${{\bf E}_{\bf 0}} = {{\bf A}_0}(x,y,z)\,{e^{ - i{k_0}z}}.$$

For the considered HGLB (in a steady state) the vector potential A₀ satisfies the wave equation as

(4)$$2i{k_0}\displaystyle{{{\partial} {{\bf A}_0}} \over {{\partial} z}} + i{{\bf A}_0}\displaystyle{{{\partial} {k_0}} \over {{\partial} z}} = \left( {\displaystyle{{{{\partial} ^2}{{\bf A}_{\rm 0}}} \over {{\partial} {r^2}}} + \displaystyle{1 \over r}\displaystyle{{{\partial} {{\bf A}_0}} \over {{\partial} r}}} \right) + \displaystyle{{{\rm \omega} _0^2} \over {{k_0}{c^2}}}({\rm \varepsilon} - {{\rm \varepsilon} _0})$$

where ${k_0}(z) = ({{\rm \omega} _0}/c)\sqrt {{{\rm \varepsilon} _0}(z)} $ is the wave number of the system, c is the speed of light in vacuum, and ε₀ is the linear part of the plasma dielectric constant given by

(5)$${{\rm \varepsilon} _0} = 1 - \displaystyle{{{\rm \omega} _{{\rm pe}}^2} \over {{\rm \omega} _0^2}}$$

where, ω_pe = (4πnN ₀e ²/m _c)^1/2 is the plasma frequency. Here N ₀ is the electron density in absence of laser beam; e and m _c are the charge and relativistic mass of the electron, respectively. Substituting the relativistic mass (m _c = γm ₀) where m ₀ is the electron rest mass, and ω_p = (4πN ₀e ²/m ₀)^1/2 we have,

(6)$${\rm \varepsilon} = 1 - \displaystyle{{{\rm \omega} _{\rm p}^2} \over {{\rm \gamma} \,{\rm \omega} _0^2}} $$

where the relativistic factor γ is given by,

(7)$${\rm \gamma} = {\left[ {1 + \displaystyle{{{e^2}} \over {{c^2}m_0^2 {\rm \omega} _0^2}} {{\bf E}_{\bf 0}}{\bf\cdot E}_{\bf 0}^ *} \right]^{{1 / 2}}}$$

Further, the complex amplitude A₀(r, z) may be expressed as, A₀(r, z) = A₀(r, z)exp[−ik ₀(z)S ₀(r, z)] where S ₀(r, z) is the eikonal associated with the HGLB. Now putting A₀ in Eq. (4) and segregating the real and imaginary parts from the resulting equation, we get the following set of equations as

(8)$$2\displaystyle{{{\partial} {S_0}} \over {{\partial} z}} + {\left( {\displaystyle{{{\partial} {S_0}} \over {{\partial} z}}} \right)^2} = \displaystyle{{{\rm \omega} _0^2 {\rm \varepsilon}} \over {{c^2}k_0^2}} + \displaystyle{1 \over {k_0^2 {{\bf A}_0}}}\left( {\displaystyle{{{{\partial} ^2}{{\bf A}_0}} \over {{\partial} {r^2}}} + \displaystyle{1 \over r}\displaystyle{{{\partial} {{\bf A}_0}} \over {{\partial} r}}} \right)$$

(9)$$\displaystyle{{{\partial} A_0^2} \over {{\partial} z}} + \displaystyle{{{\partial} {S_0}} \over {{\partial} r}}\displaystyle{{{\partial} {\bf A}_0^2} \over {{\partial} r}} + {\bf A}_0^2 \left( {\displaystyle{{{{\partial} ^2}{S_0}} \over {{\partial} {r^2}}} + \displaystyle{1 \over r}\displaystyle{{{\partial} {S_0}} \over {{\partial} r}}} \right) = 0$$

Moreover, in Gaussian beams, we expand all the parameters around central point where the intensity is maximum that is, r = 0. However, in case of HGLB, the irradiance is not maximum at the center, thus it is better to define a point along the radial distance of the beam at which all the power of the beam is supposed to be concentrated, let say $r = {r_0}{\,f_0}(z)\sqrt {2m} $ which indeed justified in the paraxial like approximation, where f ₀(z) is the beam width parameter for the HGLB. We expand all the parameters around this point and define this point η as

(10)$${\rm \eta} = \Big[(r/{r_0}{\,f_0}) - \sqrt {2m} \Big]$$

Here, the condition ${\rm \eta}\!\ll\! \sqrt {2m} $ is appropriate like the case of paraxial theory. The conversion indicates as

(11)$$\displaystyle{{\partial} \over {{\partial} z}} = \,\displaystyle{{\partial} \over {{\partial} z}} - \displaystyle{{\Big(\sqrt {2m} + {\rm \eta} \Big)} \over {{\,f_0}}}\displaystyle{{d{\,f_0}} \over {d{\rm \eta}}} \displaystyle{{\partial} \over {{\partial} {\rm \eta}}} $$

(12)$$\displaystyle{{\partial} \over {{\partial} r}} = \displaystyle{1 \over {{r_0}{\,f_0}}}\displaystyle{{\partial} \over {{\partial} {\rm \eta}}} $$

The appropriate parameters like eikonal, ε, and irradiance may be expanded around the maximum of the HGLB. So,

(13)$${\rm \varepsilon} ({\rm \eta}, z) = {{\rm \varepsilon} _0}(z) + {\rm \phi} (E.{E^*})$$

where ε₀(z) is given by Eq. (5) and ϕ(E.E ^*) is given as

(14)$${\rm \phi} (E.{E^*}) = \displaystyle{{{\rm \omega} _{\rm p}^2} \over {{\rm \omega} _0^2}} \,\,\left\{ {1 - \displaystyle{1 \over {\left[ {1 + \left( {{{{e^2}} / {{c^2}m_0^2 {\rm \omega} _0^2}}} \right)\,{E_0}.E_0^*} \right]}}} \right\}$$

Moreover, expanding the effective dielectric function using Taylor series around the irradiance maximum η = 0 of the HGLB as

(15)$${\rm \varepsilon} ({\rm \eta}, z) = {{\rm \varepsilon} _f}({\rm \eta} = 0) + {{\rm \eta} ^2}{{\rm \varepsilon} _2}({\rm \eta} = 0)$$

where ε₂ is defined as ∂ε/∂η². The expressions for these coefficients ε_f and ε₂ have been derived later. Substitution for ε(η, z) from Eq. (15) in real and imaginary part equations that is, Eqs. (8) and (9) leads to

(16)$$\eqalign {& \displaystyle{1 \over {r_0^2 f_0^2}} {\left( {\displaystyle{{{\partial} {S_0}} \over {{\partial} {\rm \eta}}}} \right)^2} + \displaystyle{{2{\partial} {S_0}} \over {{\partial} z}} = \displaystyle{1 \over {k_0^2 {{\bf A}_0}r_0^2 f_0^2}} \left[ {\displaystyle{1 \over {(\sqrt {2m} + {\rm \eta} )}}\displaystyle{{{\partial} {{\bf A}_0}} \over {{\partial} {\rm \eta}}} + \displaystyle{{{{\partial} ^2}{{\bf A}_0}} \over {{\partial} {{\rm \eta} ^2}}}} \right] \cr & \quad + {{\rm \eta} ^2}\displaystyle{{{\rm \omega} _0^2 ({{\rm \varepsilon} _2})} \over {{c^2}k_0^2}} }$$

(17)$$\displaystyle{{{\partial} {\bf A}_0^2} \over {{\partial} z}} + \displaystyle{{{\bf A}_0^2} \over {r_0^2 f_0^2}} \left[ {\displaystyle{{{{\partial} ^2}{S_0}} \over {{\partial} {{\rm \eta} ^2}}} + \displaystyle{1 \over {(\sqrt {2m} + {\rm \eta} )}}\displaystyle{{{\partial} {S_0}} \over {{\partial} {\rm \eta}}}} \right] + \displaystyle{1 \over {r_0^2 f_0^2}} \displaystyle{{{\partial} {\bf A}_0^2} \over {{\partial} {\rm \eta}}} \displaystyle{{{\partial} {S_0}} \over {{\partial} {\rm \eta}}} = 0$$

One can express solution of Eq. (16), with the paraxial approximation ${\rm \eta} \!\ll\! \sqrt {2m} $ as

(18)$${\bf A}_0^2 = \displaystyle{{E_0^2} \over {{2^{2m}}f_0^2}} {\Big(\sqrt {2m} + {\rm \eta} \Big)^{4m}}\exp \Big( \!\!- {\Big(\sqrt {2m} + {\rm \eta} \Big)^2}\Big)$$

with

$${S_0}({\rm \eta}, z) = \displaystyle{{{{\Big(\sqrt {2m} + {\rm \eta} \Big)}^2}} \over 2}{\rm \beta} (z) + {\rm \varphi} (z)$$

where

(19)$${\rm \beta} \left( z \right) = r_0^2 {\,f_0}\displaystyle{{d\,{\,f_0}} \over {d\,z}}$$

Here φ(z) is an arbitrary function of z. As we assumed that nearly all the power of the beam is concentrated in the region around η = 0, there is certainly some power of the beam beyond this limitation, which is accounted for in an approximate manner by Eq. (18).

Further, the present study considers a plasma, characterized by relativistic nonlinearity caused by the relativistic change in the mass of electron. To derive the value of the coefficients ε_f and ε₂ given in Eq. (15), it is convenient to expand the solution for ${\bf A}_0^2 $ as a polynomial in η²; for further algebraic analysis,

(20)$$A_0^2 = {d_0} + {d_2}{{\rm \eta} ^2}$$

where d ₀ and d ₂ are expressed as

(21)$$d_0^2 = \displaystyle{{E_0^2} \over {\,f_0^2}} {m^{2m}}\exp ( - 2m)$$

(22)$$d_2^2 = \displaystyle{{2E_0^2} \over {\,f_0^2}} {m^{2m}}\exp ( - 2m)$$

Following the paraxial like approximation one can expand the dielectric function in axial and radial parts around the maximum of the HGLB. Thus, from the set of Eqs. (6, 7) and (20–22), one finds

(23)$${{\rm \varepsilon} _f}({\rm \eta} = 0) = 1 - \displaystyle{{{\rm \omega} _{\rm p}^2 } \over {{\rm \omega} _0^2 }}\displaystyle{1 \over {{{(1 + {d_0})}^{1/2}}}}$$

(24)$${{\rm \varepsilon} _2}({\rm \eta} = 0) = - \displaystyle{{{\rm \omega} _{\rm p}^2} \over {2{\rm \omega} _0^2}} \displaystyle{{{d_2}} \over {{{(1 + {d_0})}^{3/2}}}}$$

The dimensionless beam width parameter f ₀ can be obtained using the boundary conditions at z = 0 as f ₀ = 1 and df ₀/dz = 0 (Akhmanov et al., Reference Akhmanov, Sukhorukov and Khokhlov1968) as

(25)$$\displaystyle{{{{\partial} ^2}{\,f_0}} \over {{\partial} {{\rm \xi} ^2}}} = \displaystyle{4 \over {\,f_0^3}} - \displaystyle{{R_{d0}^2 {\rm \omega} _{\rm p}^2 {d_2}} \over {2{\rm \omega} _0^2 k_0^2 {{\rm \varepsilon} _0}{\,f_0}{{(1 + {d_0})}^{3/2}}}}$$

where ${R_{d0}} = \,{k_0}r_0^2 $, ξ = z/R _d0. The above Eq. (25) gives the variation of beam width parameter with the normalized distance and Eq. (18) provides the intensity profile of the laser beam in the plasma along with the radial direction when relativistic nonlinearity is operative. We perform numerical computation of Eqs. (18) with (25). The equation has been solved for an initial plain wave front and the boundary conditions for HGB, and the results are presented in Section 4.

Further, the propagation of the laser beam through the plasma leads the variation of density of the plasma channel. The relativistic mass change of electron is responsible for the change in the density of the plasma and hence changes in the refractive index of the medium. As a result the laser beam self focuses in the plasma. Here, it is important to mention that the seed EPW exists in the plasma nonlinearly interacts with the propagating laser beam and as a result the plasma wave gets excited. The excitation process of EPW in the presence of relativistic nonlinearity can be described with the equations presented as follows.

1. (a) Momentum equation

   (26)$$\eqalign{{m_c}\left[ {\displaystyle{{{\partial} V} \over {{\partial} t}} + (V\cdot\nabla )V} \right] = & - eE - \displaystyle{e \over c}V \times B - 2{\Gamma _e}{m_c}V \cr & - \displaystyle{{3{K_0}{T_e}} \over N}\nabla N}$$

where the Landau damping factor is

(27)$$2{\Gamma _e} \simeq \,\sqrt {\displaystyle{{\rm \pi} \over 8}\,} \displaystyle{{{{\rm \omega} _{\rm p}}} \over {{k^3}{\rm \lambda} _d^3}} \exp \left( { - \displaystyle{3 \over 2} - \displaystyle{1 \over {2{k^2}{\rm \lambda} _d^2}}} \right)$$

where λ_d = (k _βT ₀/4πN ₀e ²)^1/2 is the Debye length V is the velocity of electron fluid, N is the instantaneous electron density, E the electric and B the magnetic field vectors, and k is the wave number of the electrostatic wave.

1. (b) Equation of continuity

   (28)$$\displaystyle{{{\partial} N} \over {{\partial} t}} + \nabla. (NV) = 0\comma$$

2. (c) Poisson's equation for electric field

   (29)$$\nabla\cdot E = - 4{\rm \pi} eN.$$

Using the customary techniques (Akhmanov et al., Reference Akhmanov, Sukhorukov and Khokhlov1968) and factorizing the density into equilibrium and perturbed part as N = N _0e + n _e0 the equation describing the electron density variation as

(30)$$\displaystyle{{{{\partial} ^2}{n_{e0}}} \over {{\partial} {t^2}}} - v_{{\rm th}}^2 {\nabla ^2}{n_{e0}} + 2{\Gamma _e}\displaystyle{{{\partial} {n_{e0}}} \over {{\partial} t}} + \displaystyle{{{\rm \omega} _{\rm p}^2} \over {\rm \gamma}} {n_{e0}} = 0$$

The solution of the form n _e0 = n _e00(r, z)exp(iωt − ikz) satisfies the dispersion relation

(31)$${{\rm \omega} ^2} = \displaystyle{{{\rm \omega} _{\rm p}^2} \over {\rm \gamma}} + {k^2}v_{{\rm th}}^2 $$

Further, assuming n _e00(r, z) in the form of exp(−ikS) in Eq. (8) and splitting the real and imaginary quantities, we get the real part equation as

(32)$$2\left( {\displaystyle{{{\partial} S} \over {{\partial} z}}} \right) + {\left( {\displaystyle{{{\partial} S} \over {{\partial} r}}} \right)^2} = \displaystyle{1 \over {{k^2}{n_{e00}}}}\left( {\displaystyle{{{{\partial} ^2}{n_{e00}}} \over {{\partial} {r^2}}} + \displaystyle{1 \over r}\displaystyle{{{\partial} {n_{e00}}} \over {{\partial} r}}} \right) + \displaystyle{{{\rm \omega} _{\rm p}^2} \over {{k^2}v_{{\rm th}}^2}} \left( {1 - \displaystyle{1 \over {\rm \gamma}}} \right)$$

and imaginary part equation as

(33)$$\displaystyle{\partial n_{e00}^2 \over \partial z} + \displaystyle{\partial S \over \partial r} \displaystyle{\partial n_{e00}^2 \over \partial r} + \displaystyle{n_{e00}^2 \over k^2} \left(\displaystyle{\partial^2 S \over \partial r^2} + \displaystyle{1 \over r} \displaystyle{\partial S \over \partial r} \right) + 2\Gamma_e \displaystyle{{\rm \omega} \,n_{e00}^2 \over kv_{{\rm th}}^2} = 0$$

The term S describes the eikonal for plasma wave. Here, to deal the HGLB the transformation of (r, z) coordinate into (η, z) coordinate is required. So, using the Eq. (10) we made the transformation. However, using paraxial like approximation Eqs. (32) and (33) can be solved and for that we assume the initial radial variation of density perturbation and the solution as

(34)$$n_{e00}^2 = \displaystyle{{N_{e00}^2} \over {{2^{2m}}f_e^2}} {\Big(\sqrt {2m} + {\rm \eta} \Big)^{4m}}{\left( {\displaystyle{{{r_0}{\,f_0}} \over {a{\,f_e}}}} \right)^{4m}}\exp \Big[\!\! - {\Big(\sqrt {2m} + {\rm \eta} \Big)^2}\Big]\exp [ - {k_i}(z)]$$

(35)$$S({\rm \eta}, z) = \displaystyle{{{{\Big(\sqrt {2m} + {\rm \eta} \Big)}^2}} \over 2}r_0^2 f_0^2 {\rm \beta} (z) + {\rm \phi} (z),\quad {\rm \beta} (z) = \displaystyle{1 \over {{\,f_e}}}\displaystyle{{d{\,f_e}} \over {dz}}$$

where f _e is a dimensionless parameter of the electron plasma wave, a is the initial beam width of plasma wave, ${k_i} = 2{\Gamma _e}{\rm \omega} /kV_{{\rm th}}^2 $ is the damping factor and $N_{e00}^2 $ is the magnitude of the excited plasma wave. Further, we have used the following boundary conditions (for an initially plane wave front): At df _e/dz = 0 and f _e = 1 and S = 0 z = 0. Substituting, Eqs. (34) and (35) in Eq. (32) and equating the coefficients of η² on both sides, we get the equation for f _e as

(36)$$\displaystyle{{{{\partial} ^2}{\,f_e}} \over {{\partial} {{\rm \xi} ^2}}} = \displaystyle{{R_{d0}^2 {\,f_e}} \over {\,f_0^2}} \left\{ {\displaystyle{1 \over {{k^2}r_0^4 f_0^2}} \left[ {3 + {{\left( {\displaystyle{{{r_0}{\,f_0}} \over {a{\,f_e}}}} \right)}^4}} \right]} \right\} - \displaystyle{{R_{d0}^2 {\,f_e}} \over {2{\rm \omega} _0^2 f_0^2}} \displaystyle{{{\rm \omega} _{\rm p}^4} \over {{k^2}V_{{\rm th}}^2}} \displaystyle{{{d_2}} \over {{{(1 + {d_0})}^{3/2}}}}$$

where R _d = ka ², Further, the amplitude of the density perturbation at finite z is to be obtained, for that purpose, Eq. (34) has to be solved numerically with Eq. (36) and results are given in Section 4.

3. SRS

SRS governs the amount of laser energy that can be propagated over long distances through plasma without being lost to scattering and electron heating. SRS is the process in which light from an incident pump pulse is scattered by the electron density perturbations of a plasma wave. If the wave has frequency ${\rm \omega} _{\rm p}^2 = 4{\rm \pi} {n_0}{e^2}/m$, with electron charge e, mass m and number density n ₀ and the incident light's frequency is ω₀, then light will be scattered from noise to frequencies ω₁ = ω₀ − ω_p and ω₁ = ω₀ + ω_p, which are called the Stokes and anti-Stokes lines. The beating between the ω₀ light and that scattered to ω₁ resonantly drive a plasma wave, which creates a feedback loop, as the amount of scattered energy is proportional to the amplitude of excited plasma wave. To study the scattering of HGLB, first we split the total electric field of the pump wave into two parts scattered and remaining, and further only the scattered part of the laser beam is treated to observe the beam width parameter and reflectivity of the same. Thus, the high frequency pump wave is propagating through the plasma is considered to have total electric field E _T as

(37)$${E_{\rm T}} = {E_0}\exp \,(i{{\rm \omega} _0}t) + {E_{\rm s}}\exp \,(i{{\rm \omega} _{\rm s}}t)$$

where E ₀ is the electric field of the pump laser beam and E _S, electric field of the scattered wave. The electric field E _S originates owing to scattering of the pump beam off the plasma wave. Thus, the wave equation

(38)$${\nabla ^2}{E_{\rm T}} - \nabla \Big(\nabla\cdot {E_T}\Big) = \displaystyle{1 \over {{c^2}}}\displaystyle{{{{\partial} ^2}{E_{\rm T}}} \over {\partial {t^2}}} + \displaystyle{{4{\rm \pi}} \over {{c^2}}}\displaystyle{{{\partial} {J_{\rm T}}} \over {{\partial} t}}$$

where J _T is the total current density vector corresponding to E _T. Comparing the zeroth order terms in above equation, one gets Eq. (2) and equating the terms at scattered frequency. Further, separating the terms at different frequencies, one obtains

(39)$${\nabla ^2}{E_{\rm s}} + \displaystyle{{{\rm \omega} _{\rm s}^2} \over {{c^2}}}\left( {1 - \displaystyle{{{\rm \omega} _{\rm p}^2} \over {{\rm \omega} _{\rm s}^2}} \displaystyle{1 \over {\rm \gamma}}} \right){E_{\rm s}} = \displaystyle{1 \over 2}\displaystyle{{{\rm \omega} _{\rm p}^2} \over {{c^2}}}\displaystyle{{{{\rm \omega} _{\rm s}}} \over {{{\rm \omega} _0}}}\displaystyle{{{n^*}} \over {{N_0}}}{E_0}$$

Further substituting E _s as

(40)$${E_{\rm s}} = {E_{{\rm s}0}}(r,z)\exp (i{k_{{\rm s0}}}t) + {E_{{\rm s1}}}(r,z)\exp ( - i{k_{{\rm s}1}}t)$$

where

(41)$$k_{{\rm s}0}^2 = \displaystyle{{{\rm \omega} _{\rm s}^2} \over {{c^2}}}{\left( {1 - \displaystyle{{{\rm \omega} _{\rm p}^2} \over {{\rm \omega} _{\rm s}^2}}} \right)^{{1 / 2}}} = \displaystyle{{{\rm \omega} _{\rm s}^2} \over {{c^2}}}\sqrt {{{\rm \varepsilon} _{{\rm s}0}}} $$

and k _s1 and ω_s follows

$${{\rm \omega} _{\rm s}} = {{\rm \omega} _0} - {\rm \omega}, \quad {k_{{\rm s1}}} = {k_0} - k$$

Substituting Eq. (40) in (39), one gets

(42)$$\eqalign{ - k_{{\rm s}0}^2 {E_{{\rm s0}}} + 2i{k_{{\rm s0}}}\displaystyle{{{\partial} {E_{{\rm s}0}}} \over {{\partial} z}} + \left( {\displaystyle{{{{\partial} ^2}} \over {{\partial} {r^2}}} + \displaystyle{1 \over r}\displaystyle{{\partial} \over {{\partial} r}}} \right){E_{{\rm s}0}} \cr+ \displaystyle{{{\rm \omega} _{\rm s}^2 } \over {{c^2}}}\left[ {{{\rm \varepsilon} _{{\rm s0}}} + \displaystyle{{{\rm \omega} _{\rm p}^2 } \over {{\rm \omega} _{\rm s}^2 }}\left( {1 - \displaystyle{1 \over {\rm \gamma} }} \right)} \right]{E_{{\rm s0}}} = 0}$$

(43)$$\eqalign { & - k_{{\rm s}1}^2 {E_{{\rm s}1}} - 2i{k_{{\rm s1}}}\displaystyle{{\partial {E_{{\rm s}1}}} \over {\partial z}} + \left( {\displaystyle{{{\partial ^2}} \over {\partial {r^2}}} + \displaystyle{1 \over r}\displaystyle{\partial \over {\partial r}}} \right){E_{{\rm s1}}} \cr & \quad \quad \quad \quad + \displaystyle{{{\rm \omega} _{\rm s}^2 } \over {{c^2}}}\left\{ {{{\rm \varepsilon} _{s0}} + \displaystyle{{{\rm \omega} _{\rm p}^2 } \over {{\rm \omega} _{\rm s}^2 }}\left( {1 - \displaystyle{1 \over {\rm \gamma} }} \right)} \right\} \cr & \quad \quad \quad \quad{E_{{\rm s}1}} = \displaystyle{{{\rm \omega} _{\rm p}^2 {{\rm \omega} _{\rm s}}{n^*}{E_0}} \over {2{c^2}{{\rm \omega} _0}{N_0}}}{e^{( - i{k_0}{S_0})}}}$$

Moreover, using ${E_{{\rm s}0}} = {E_{{\rm s}00}}(r,z)\,{e^{ + ik{S_c}}}$ in Eq. (42) and separating the real and imaginary parts, we get the two equations as

(44)$$\displaystyle{{2{\partial} {S_c}} \over {{\partial} z}} + {\left( {\displaystyle{{{\partial} {S_c}} \over {{\partial} r}}} \right)^2} = \displaystyle{1 \over {k_{{\rm s}0}^2 {E_{{\rm s}00}}}}\left( {\displaystyle{{{{\partial} ^2}} \over {{\partial} {r^2}}} + \displaystyle{1 \over r}\displaystyle{{\partial} \over {{\partial} r}}} \right){E_{{\rm s}00}} + \displaystyle{{{\rm \omega} _{\rm p}^2} \over {{{\rm \varepsilon} _{{\rm s}0}}{\rm \omega} _{\rm s}^2}} \left( {1 - \displaystyle{1 \over {\rm \gamma}}} \right)$$

(45)$$\displaystyle{{{\partial} E_{{\rm s}00}^2} \over {{\partial} z}} + E_{{\rm s}00}^2 \left( {\displaystyle{{{{\partial} ^2}{S_c}} \over {{\partial} {r^2}}} + \displaystyle{1 \over r}\displaystyle{{{\partial} {S_c}} \over {{\partial} r}}} \right) + \displaystyle{{{\partial} {S_c}} \over {{\partial} r}}\displaystyle{{{\partial} E_{{\rm s}00}^2} \over {{\partial} r}} = 0$$

The solution of these equations can be written as

(46)$$E_{{\rm s}00}^2 = \displaystyle{{B_1^2} \over {{2^{2m}}f_{\rm s}^2}} {\Big(\sqrt {2m} + {\rm \eta} \Big)^{4m}}{\left( {\displaystyle{{{r_0}{\,f_0}} \over {{b_0}{\,f_{\rm s}}}}} \right)^{4m}}\exp \Big[ - {(\sqrt {2m} + {\rm \eta} )^2}\Big]$$

(47)$${S_c}({\rm \eta}, z) = \displaystyle{{{{\Big(\sqrt {2n} + {\rm \eta} \Big)}^2}} \over 2}r_0^2 f_0^2 {\rm \beta} (z) + {\rm \phi} (z),\quad {\rm \beta} (z) = \displaystyle{1 \over {{\,f_{\rm s}}}}\displaystyle{{d{\,f_{\rm s}}} \over {dz}}$$

where B ₁ is the constant determined by the boundary conditions, f _s is a dimensionless parameter of the backscattered beam, S _c is the eikonal of the scattered beam and b ₀ is the initial beam width of scattered beam. In the presence of relativistic nonlinearity the dielectric constant at scattered frequency

(48)$${{\rm \varepsilon} _{\rm s}}({\rm \eta}, z) = {{\rm \varepsilon} _{{\,f_{\rm s}}}}({\rm \eta} = 0) + {{\rm \eta} ^2}{{\rm \varepsilon} _{2{\rm s}}}({\rm \eta} = 0)$$

Next, using the paraxial like approximation the dielectric function can be expanded around the maximum of the HGLB. Thus, we get

(49)$${{\rm \varepsilon} _{{\,f_{\rm s}}}}({\rm \eta} = 0) = \,1 - \displaystyle{{{\rm \omega} _{\rm p}^2} \over {{\rm \omega} _{\rm s}^2}} \displaystyle{1 \over {{{(1 + {d_0})}^{1/2}}}}$$

(50)$${{\rm \varepsilon} _2}({\rm \eta} = 0) = - \displaystyle{{{\rm \omega} _{\rm p}^2} \over {2{\rm \omega} _{\rm s}^2}} \displaystyle{{{d_2}} \over {{{(1 + {d_0})}^{3/2}}}}$$

The boundary conditions df _s/dz = 0, f _s = 1 and S _c = 0 at z = 0 is used for initially plane wave front. Using Eq. (44) and comparing the coefficients of η² one gets

(51)$$\eqalign{\displaystyle{{{{\partial} ^2}{\,f_{\rm s}}} \over {{\partial} {{\rm \xi} ^2}}} &= \displaystyle{{R_{d0}^2 {\,f_{\rm s}}} \over {\,f_0^2}} \left[ {\displaystyle{1 \over {k_{{\rm s}0}^2 r_0^4 f_0^2}} \left\{ {3 + {{\left( {\displaystyle{{{r_0}{\,f_0}} \over {{b_0}{\,f_e}}}} \right)}^4}} \right\}} \right] \cr&\quad- \displaystyle{{R_{d0}^2 {\,f_{\rm s}}} \over {2{\rm \omega} _{\rm s}^2 f_0^2}} \displaystyle{{{\rm \omega} _{\rm p}^4} \over {k_{{\rm s}0}^2 {c^2}}}\displaystyle{{{d_2}} \over {{{(1 + {d_0})}^{3/2}}}}}$$

where $R_{{\rm ds}}^2 = k_{{\rm s}0}^2 \,b_0^2 $ is the diffraction length of the scattered radiation. It is clear from Eq. (34) that the plasma wave is damped as it propagates along the z-axis. Consequently the scattered wave amplitude should also decrease with increasing z. Thus, the appropriate boundary condition would be

(52)$${E_{\rm S}} = {E_{{\rm S}0}}(r,z){e^{ + i{k_{{\rm s}0}}z}} + {E_{{\rm S}1}}(r,z){e^{ - i{k_{{\rm s}1}}z}} = 0\,\,{\rm at}\,\,Z = {Z_c}$$

If L is the interaction length then length Z _c is chosen (Z _c = L − z) sufficiently large such that n _e00 is nearly zero. Substituting Z _c in Eq. (40), one gets

(53)$$\displaystyle{1 \over {b_0^2 {\,f_{\rm s}}({Z_c})}} = \displaystyle{1 \over {r_0^2 f_0^2 ({Z_c})}} + \displaystyle{1 \over {{a^2}f_e^2 ({Z_c})}}$$

and

(54)$$\eqalign{& {B_1} \approx - \displaystyle{1 \over 2}\displaystyle{{{\rm {\rm \omega}} _{\rm p}^2 } \over {{c^2}}}\displaystyle{{{{\rm \omega} _{\rm s}}} \over {{{\rm \omega} _0}}}\displaystyle{{\,f_{\rm s}^2 } \over {\,f_e^2 }}{\left( {\displaystyle{{{b_0} {\,f_{\rm s}}} \over {a {\,f_e}}}} \right)^{4m}}\left( {\displaystyle{{N_{e00}^2 } \over {{N_0}}}} \right) \cr & \quad \times \displaystyle{\matrix{{E_{00}}\exp (i{\rm \omega} t - kz - S) \exp \cr \left[ { - {{\Big(\sqrt {2m} + {\rm \eta} \Big)}^2} - 2{k_i} z} \right] \exp \left[ { - (i{k_0}{S_0} + i{k_{{\rm s}1}}{Z_c})} \right]} \over \matrix{\left\{ {k_{{\rm s}1}^2 - k_{{\rm s}0}^2 - ({\rm \omega} _{\rm p}^2 /{c^2})\left[ {1 - (1/{\rm \gamma} )} \right]} \right\} \exp \cr \left\{ { - {{\Big(\sqrt {2m} + {\rm \eta} \Big)}^2}\left[ {\Big(r_0^2 f_0^2 \Big)/\Big(b_0^2 f_{\rm s}^2 \Big)} \right]} \right\} \exp \left( {i{k_{{\rm s}0}}{S_c} + i{k_{{\rm s}0}}{Z_c}} \right) } }} $$

The back reflectivity R = |E _S| ²/|E ₀₀| ² is calculated by using the above set of equations

(55)$$R = \displaystyle{\eqalign{& \left\{ {({B_1}/{2^{4m}}f_{\rm s}^2 ){{(\sqrt {2m} + {\rm \eta} )}^{4m}}{{\left( {{r_0}{f_0}/{b_0} {f_{\rm s}}} \right)}^{4m}}} \right. \cr & \quad \exp \left[ { - {{(\sqrt {2m} + {\rm \eta} )}^2}\left( {r_0^2 f_0^2 /b_0^2 f_{\rm s}^2 } \right)} \right] \cr & \quad \exp \left[ { - (i{k_{s0}}{S_0} + i{k_{s0}}{Z_c})} \right] \cr & + (1/2)({\rm \omega} _{\rm p}^2 /{c^2})({{\rm \omega} _s}f_s^2 /{{\rm \omega} _0}f_e^2 ){({b_0}{f_s}/a {f_e})^{2m}} \cr & \quad \left\{ {[{E_{00}}\exp (i{\rm \omega} t - kz - S)]/[(k_{{\rm s}1}^2 - k_{{\rm s}0}^2 ) - ({\rm \omega} _p^2 /{c^2})} \right. \cr & \quad {\left. {(1 - 1/{\rm \gamma} )]\} exp[ - (i{k_0}{S_c} + i{k_{{\rm s}1}}{Z_c})]({n^{\ast}}/{n_0})} \right\}^2}} \over {\left( {E_0^2 /{2^{2m}}f_0^2 } \right){{(\sqrt {2m} + {\rm \eta} )}^{4m}}\exp [ - {{(\sqrt {2m} + {\rm \eta} )}^2}]}} $$

The above equation expresses the reflectivity of back scattered beam. The SRS reflectivity can be calculated numerically using the relevant above set of equations. It is clear from the above equation that the reflectivity R depends upon the beam width parameter of incident pump wave, EPW, and scattered wave.

4. NUMERICAL RESULTS AND DISCUSSION

In the present work, the phenomena of SRS of HGLB have been studied considering relativistic nonlinearity in collisionless plasma. In the process the weak EPW propagating in the z-direction is observed to be nonlinearly coupled with the laser beam. The nonlinear coupling between these two waves leads to excitation of the EPW. This pump beam scattered by the EPW and back reflectivity has been calculated. In this process, first we see the propagation of HGLB and the focusing of beams depends upon dielectric function corresponding to the relativistic nonlinearity. The dielectric function is modified due to the presence of high power HGLB in the plasma. In the analysis all the relevant parameters have been expanded around the axis of maximum irradiance of the HGLB in paraxial like approach. Equations (18) and (19) show the intensity profile of HGLB in plasma along the radial direction in the presence of relativistic nonlinearity, while Eq. (25) determines the focusing/defocusing of the HGBs, along the distance of propagation in the plasma. The numerical calculation have been done with using following typical laser beam parameters: The HGLB power flux (10²² W cm⁻²), r ₀ = 15 μm, ω₀ = 1.776 × 10¹⁴ rad s⁻¹, and plasma density n = 0.35 n _cr., the electron thermal speed v _th = 0.05c. The boundary condition used here for an initial plane wave front of the laser beams, f ₀ = 1, df ₀/dξ = 0, and S ₀ = 0 at ξ = 0.

Further, Eq. (25) explains the beam dynamics in plasma with relativistic nonlinearity taken into account and decides the propagation behavior of HGLB inside the plasma. Equation (25) is nonlinear second order differential equations governing the normalized beam width parameter f ₀ in the plasma. The first term on the right-hand side of Eq. (25) represents spatial dispersion and is responsible for diffractional divergence. On the other hand, second term is nonlinear in nature leading to self-focusing of the beam. Analytical solutions to these equations are not possible. Thus, we go for the numerical computational techniques to study beam dynamics. The variation of initial intensity of the HGLB for the order m = 2, has been shown in Figure 1a at ξ = 0. It is obvious from the figure that in paraxial regime the intensity of laser beam is maximum at η = 0. Similarly the laser profile has also been shown to the first focal point for ξ = 0 and the results is shown in the form of Figure 1b. Figure 2a illustrates the focusing of HGLB for the purely Gaussian mode (m = 0) of HGLB when relativistic nonlinearity is operative in the system. Figure 2a clearly displays the oscillatory self-focusing in the presence of relativistic nonlinearity, while Figure 2b gives the effect of relativistic nonlinearity on the propagation of HGLB for m = 1 mode. Figure 2c depicts the dynamics of laser beam for m = 2 mode and Figure 2d give the same for m = 3 mode of propagation of HGLB.

Fig. 1. Normalized intensity of HGLB for laser power ${\rm \alpha} E_{00}^2 = 1.6$ when only relativistic nonlinearity is operative; (a) at ξ = 0, (b) at the first focal point.

Fig. 2. Normalized intensity of HGLB with normalized distance (ξ = z/R _d0) for various order of propagation for laser power ${\rm \alpha} E_{00}^2 = 1.6$ when only relativistic nonlinearity is operative; (a) m = 0, (b) m = 1, (c) m = 2, and (d) m = 3.

To observe the consequences of the propagation of laser beam and numerical appreciation of them, the excitation of EPW has been studied. It is clearly seen that the EPW excited due to the coupling between seed plasma wave and pump beam. Since the propagation of high power HGLB leads to the modification of electron density in the plasma. In the present formulation the density profile of the EPW is governed by Eq. (34) with Eq. (35). The variation of initial intensity of the HGLB for the order m = 2 has been shown in Figure 3a at ξ = 0. It is obvious from the figure that in paraxial regime the intensity of laser beam is maximum at η = 0. Figure 3b expresses the EPW at first focal point. It is clear from Eq. (36) that the dynamics of EPW is influenced by the coupling term. The beam width parameter of EPW controls the intensity profile of the excited EPW. Therefore, Eq. (36) has been numerically calculated to observe the dynamics of the EPW for various orders of HGLB for the same chosen parameters as previously. The beam width parameter f _e of the EPW as a function of dimensionless distance of propagation has been computed with relativistic nonlinearity and the results has been presented in the form of Figure 4. Figure 4b–4d depicts the excited EPW for m = 1, 2, 3 modes of HGLB, respectively.

Fig. 3. Normalized intensity of EPW when relativistic nonlinearity is operative; (a) at ξ = 0, (b) at the first focal point.

Fig. 4. Normalized intensity of EPW with normalized distance (ξ = z/R _d0) for various order of propagation for laser power ${\rm \alpha} E_{00}^2 = 1.6$ when relativistic nonlinearity is operative; (a) m = 0, (b) m = 1, (c) m = 2, and (d) m = 3.

In addition, Eq. (51) shows that the beam width parameter of scattered HGLB depends upon the beam width parameter of EPW and incident HGLB. The beam width parameter f _s of the scattered beam as a function of dimensionless distance of propagation have been shown with relativistic nonlinearity by Eq. (51). Moreover, Eq. (55) gives the reflectivity against the distance of propagation. We have numerically solved Eq. (55) and the results are given in the form of Figure 5. Figure 5 clearly depicts the change in the back reflectivity with normalized distance for different modes of propagation of HGLB. The back reflectivity is displayed with normalized distance ξ for different order of HGLB. It is found that the reflectivity is higher at the points where the HGLB focuses. It is also shown that as we increase the order of the HGLBs the reflectivity decreases because the focusing of HGLBs decreases with increasing order of the beam.

Fig. 5. Variation in normalized back scattered reflectivity of HGLB with normalized distance for various orders of propagation for ${\rm \alpha} E_{00}^2 = 1.6$ when relativistic nonlinearity is operative.

In conclusion, the SRS of HGLB has been presented in the present work. It is shown that all the modes of propagation of HGLB contribute to the reflectivity of the propagating pump HGLB. In the scattering process of HGLB when the relativistic nonlinearity is taken into consideration the intensity profile is modified due to the presence of higher modes of propagation.