1. INTRODUCTION
The propagation of high power laser of finite spot size in plasmas is influenced by diffraction and self focusing effects. The former causes divergence of the beam on the scale of a Rayleigh length R d = kr 02, where k is the wave number of the laser and r 0 is its spot size. The self focusing arises due to nonlinear permittivity that may be caused by a variety of sources, e.g., ohmic nonlinearity, ponderomotive nonlinearity and relativistic mass nonlinearity (Sodha et al., Reference Sodha, Khanna and Tripathi1974a, Reference Sodha, Mittal, Kumar and Tripathi1974b, Reference Sodha, Ghatak and Tripathi1976; Gill et al., Reference Gill, Mahajan and Kaur2010; Saini & Gill, Reference Saini and Gill2004; Singh & Tripathi, Reference Singh and Tripathi2009; Verma & Sharma, Reference Verma and Sharma2011). The theories of self focusing are mostly limited to paraxial ray approximation where eikonal of the wave is expanded up to the second power of r, the radial (transverse) coordinate. Sodha et al. (Reference Sodha, Khanna and Tripathi1974a, Reference Sodha, Mittal, Kumar and Tripathi1974b) have surveyed the early studies on self focusing and filamentation in unmagnetized plasmas with primary focus on long time scale Ohmic and ponderomotive nonlinearities. Esaray et al. (1997) have studied the short-pulse laser propagation in gases undergoing ionization. They also observed optical guiding of a laser beam in plasmas by taking relativistic effects, ponderomotive effects, and preformed density channels. Kurki-Suonio et al. (Reference Kurki-suonio, Morrison and Tajima1989) have studied the propagation of a circularly polarized laser in underdense plasma, including relativistic and ponderomotive nonlinearities, and showed that it remains intact as it propagates inside the plasma. Liu and Tripathi (Reference Liu and Tripathi2000) have developed a nonparaxial theory of laser self defocusing in a tunnel ionizing gas by expanding the eikonal up to fourth power in r.
Pathak and Tripathi (Reference Pathak and Tripathi2006) have developed a formalism to study the nonlinear Eigen modes of plasma without resorting to any expansion of the eikonal. The wave equation is solved numerically. The laser intensity profile in an unmagnetized plasma under relativistic and ponderomotive nonlinearity, turns out to be close to a Lorentzian. The Eigen modes in high density plasmas are stable to stimulated Raman scattering (SRS). However, in low density plasmas, laser Eigen modes are unstable to SRS. This study is limited to unmagnetized plasma.
In this paper, we study the nonlinear Eigen modes of a laser in a self created magnetized plasma channel (Verma & Sharma, Reference Verma and Sharma2009a; Panwar & Sharma, Reference Panwar and Sharma2009), under the combined effects of ponderomotive and relativistic mass nonlinearities, and examine their stability to stimulated Raman scattering. This study is relevant to intense short pulse laser produced plasmas where multi mega-gauss magnetic fields have been observed (Cai et al., Reference Cai, Yu, Zhu and Zhou2007; Borghesi et al, Reference Borghesi, MacKinnon, Barringer, Gaillard, Gizzi, Meyer, Willi, Pukhov and Meyer-ter-Vehn1997; Verma & Sharma, Reference Verma and Sharma2009b). Horovitz et al. (Reference Horovitz, Eliezer, Henis, Paiss, Moshe, Ludmirsky, Werdiger, Arad and Zigler1998) experimentally measured the generation of axial magnetic fields. They performed experiment with 1.06 µm laser of 7 ns pulse duration and 109 to1014 W/cm2 intensity range. They observed axial fields from 500 G to 2.17 MG. Naseri et al. (Reference Naseri, Bychenkov and Rozmus2010) reported that inverse Faraday Effect is the main mechanism of axial magnetic field generation in inhomogeneous plasmas. Pukhov and Meyer-ter-Vehn (Reference Pukhov and Meyer-ter-Vehn1998) in their three-dimensional particle-in-cell simulations of intense short pulse laser gas jet interaction have observed 100 MG magnetic fields in one-tenth critical density plasmas at ~1019 W/cm2 intensity at 0.8 µm wavelength. Frolov (Reference Frolov2009) observed quasi-static magnetic fields generated in low-density plasma in the interaction of counter propagating laser pulses. The magnetic field introduces anisotropy and cyclotron resonance effects in the propagation of electromagnetic waves and opens up a rich variety of electrostatic modes. Parametric coupling (Liu & Tripathi, Reference Liu and Tripathi1986; Porkolab & Chang, Reference Porkolab and Chang1972; Short & Simon Reference Short and Simon2004) between these modes has been a fascinating subject of extensive study for quiet some time. Clark and Fisch (Reference Clark and Fisch2005) proposed a scheme that utilizes the stimulated Raman backscattering (SRBS) in plasma to amplify a short and frequency downshifted seed pulse. Sharma and Kourakis (Reference Sharma and Kourakis2009) used SRBS to investigate the pulse compression and pulse amplification mechanisms. Trines et al. (Reference Trines, Kamp, Schep, Leemans, Esarey and Sluijter2004) did PIC simulation and show that suppression of SRBS increases the high energy electron yield. Sajal et al. (Reference Sajal and Tripathi2004, Reference Sajal, Panwar and Tripathi2006) have studied relativistic stimulated Raman forward scattering (SRFS) and shown that the Langmuir wave in the presence of a strong azimuthal magnetic field goes over to a localized lower hybrid wave, and the growth rate of the stimulated Raman backward scattering process is decreased. Liu and Tripathi (Reference Liu and Tripathi1996) have studied the stimulated forward Raman scattering of a laser in the channel for none relativistic case. The localization effect reduces the growth rate of Raman process. Liu and Tripathi (Reference Liu and Tripathi2001) also studied the relativistic laser guiding in a strong azimuthal magnetic field in plasma in the paraxial ray approximation. A linearly polarized laser with circular spot size is seen to acquire an elliptical spot size as it is self focused (Liu & Tripathi, Reference Liu and Tripathi1994).
In Section 2, we deduce the nonlinear wave equation governing the propagation of a right circularly polarized mode in plasma in the presence of axial magnetic field, allowing for a transverse amplitude variation. The nonlinearity is taken to arise due to relativistic mass effect and ponderomotive force induced electron density redistribution. The ions are taken to be stationary. The wave equation is solved numerically to obtain the nonlinear Eigen modes. In Section 3, we study the stimulated Raman backscattering of a large amplitude Eigen mode. In Section 4, we discuss the results.
2. NONLINEAR ELECTROMAGNETIC PLASMA EIGEN MODE IN PRESENCE OF AXIAL MAGNETIC FIELD
Consider pre-formed plasma of electron density n 0 embedded in a static magnetic field 
. A circularly polarized electromagnetic mode propagates through it with t, z variation of the electric field as
where A 0 has a slow dependence on x. The magnetic field of laser is
The laser imparts an oscillatory velocity to electrons,
where ωc = eB s/m, −e and m are the electron charge and mass and 
is the gamma factor. For |v0|2/c 2 ≪ 1, simplifies to
where a 02 = e 2A 02/m 2ω02c 2.
The laser exerts a direct current ponderomotive force on electrons,
Using identities 
and 
, we get 
. But we assumed ωc/ω0γ0 < 1, hence one can neglect the second term from above equation and hence we get
hence
The ponderomotive force pushes the electrons away from the laser axis, creating a low electron density channel near the laser axis. For laser pulse duration shorter than the ion response time, the ions may be taken to be stationary. The ion space charge field 
pulls the electrons back. In a time on the order of ωp−1, where ωp2 = n 0e 2/mɛ0, a quasi-steady-state is realized, when the ponderomotive potential balances the space charge potential and further lowering of electron density at the axis stops. The time scale of plasma channel formation is of the order of electron plasma period (~1/ωp). Hence in the quasi-steady-state
Using ϕs in Poisson's equation, ∇2ϕs = e(n e − n 0)/ɛ0, we obtained the modified electron density
With the relativistic modification of electron mass and ponderomotive modification of density, the non-zero component of the conductivity tensor of plasma can be written as
Using third and fourth Maxwell's equations, we deduce the wave equation as
Effective dielectric constant for right circularly polarized extraordinary mode in magnetized plasma can be written as
where
Using 
, and following Sodha et al. (Reference Sodha, Khanna and Tripathi1974a, Reference Sodha, Mittal, Kumar and Tripathi1974b, Reference Sodha, Ghatak and Tripathi1976) the wave equation governing the propagation of right circularly polarized wave can be written as
when ωc ≪ ω0, then 
and hence the above equation reduces to well-known result as derived by Liu and Tripathi (Reference Liu and Tripathi1994). Here A + = E x + iE y is the extraordinary mode of the laser. Let A + = A +0 (x)e ik 0z, where A +0 = 2A 0. In an under dense plasma ωp2 ≪ ω02 and 
. Using these values, Eq (13) becomes
Put the value of modified electron density from Eq. (8), ɛ_ can be written as
Put the value of ɛ_ in the above equation, it becomes
![\eqalign{ \displaystyle{{d^2 a_0^2 } \over {dX^2 }} &=\displaystyle{1 \over {2a_0^2 \lpar 1 - 2{\rm \alpha} a_0^2 \rpar }}\left({\displaystyle{{da_0^2 } \over {dX}}} \right)^2\cr &\quad +\displaystyle{{2a_0^2 } \over {\lpar 1 - 2{\rm \alpha} a_0^2 \rpar }}\left[{\left({\displaystyle{{k_0 c} \over {{\rm \omega} _0 }}} \right)^2 - 1} \right]\cr &\quad +\displaystyle{{2\lpar {\rm \omega} _p^2 /{\rm \omega} _0^2 \rpar a_0^2 } \over {\lpar 1 - 2{\rm \alpha} a_0^2 \rpar \lpar {\rm \gamma} _0+{\rm \omega} _c /{\rm \omega} _0 \rpar }}} \eqno\lpar 16\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021065705771-0980:S0263034611000619_eqn16.gif?pub-status=live)
where a 0 = eA 0/mω0c, X = xω0/c and
We have solved Eq. (16) numerically to obtain the mode structure of the Eigen mode for parameters: ωp/ω0 = 0.4, k 0c/ω0 = 0.91652 and ωc/ω0 = 0.4.
Figure 1 shows the mode structure of a laser in the presence of magnetic field. It resembles Gaussian profile with half width x 0 = 0.436c/ω0, a 0 = a 00ψ(x), 
. This is different from the Pathak and Tripathi, Lorentzian profile that have obtained in unmagnetized plasma. Figure 2 shows the variation of ω0/ωp with k 0c/ωp for different values of magnetic fields. The value of ω0/ωp is minimum at minimum k0c/ωp and increases monotonically with increasing k 0c/ωp. The minimum value of ω0/ωp increases with increasing the value of ωc/ωp.
Fig. 1. (Color online) Electromagnetic mode amplitude square as a function of normalized transverse distance for ωp2/ω02 = 0.25, a 00 = 0.4, and ωc/ω0 = 0.4.
Fig. 2. (Color online) Variation of normalized frequency ω0/ωp with normalized wave vector k 0c/ωp for a high power laser in plasmas. The other parameters are: a 00 = 0.4, ωc/ωp = 0.2 and 0.4.
3. STIMULATED RAMAN BACK SCATTERING
Now we examine the parametric decay of a large amplitude Eigen mode into a reactive quasi-mode of potential
and a backscattered electromagnetic wave with electric and magnetic fields
where ω1 = ω−ω0, k 1 = k − k 0. In an underdense plasma, ω ≪ ω0 hence |ω1| ≈ ω0, |k 1| ≈ k 0 and k ≈ 2k 0. The scattered wave imparts oscillatory velocity to electrons
The pump and scattered waves exert a ponderomotive force on electrons at 
, 
, where
![{\rm \phi} _{\,p{\rm \omega} }=\displaystyle{e \over {m{\rm \gamma} _0\, k}}\left[{\displaystyle{{k_1 {\rm \omega} _0 \lpar {\rm \omega} _1+{\rm \omega} _c /{\rm \gamma} _0 \rpar +k_0 {\rm \omega} _1 \lpar {\rm \omega} _0 - {\rm \omega} _c /{\rm \gamma} _0 \rpar } \over {{\rm \omega} _0 {\rm \omega} _1 \lpar {\rm \omega} _0 - {\rm \omega} _c /{\rm \gamma} _0 \rpar \lpar {\rm \omega} _1+{\rm \omega} _c /{\rm \gamma} _0 \rpar }}} \right]A_0 A_1 .\eqno\lpar 21\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021065705771-0980:S0263034611000619_eqn21.gif?pub-status=live){kind=small}
The electron response to ϕω and ϕpω can be obtained by solving the equation of motion,
using this value in the continuity equation we get the modified density of electrons
Using Poisson's equation ∇2ϕω = en ω/ɛ0, and defining eϕω/mc 2 = φω, we get
![\eqalign{& \displaystyle{{\partial ^2 {\rm \varphi} _{\rm \omega} } \over {\partial x^2 }}\left[{1 - \displaystyle{{{\rm \omega} _p^2 } \over {{\rm \gamma} _0 \lpar {\rm \omega} ^2 - {\rm \omega} _c^2 /{\rm \gamma} _0^2 \rpar }}} \right]- k_z^2 \lpar 1 - {\rm \omega} _p^2 /{\rm \omega} ^2 {\rm \gamma} _0 \rpar {\rm \varphi} _{\rm \omega}\cr &\quad= - \, \, \displaystyle{{{\rm \omega} _p^2\, k_z {\rm \omega} _0^2 a_0 a_1 } \over {{\rm \omega} ^2 {\rm \gamma} _0^2 }}\cr &\qquad\times\left[{\displaystyle{{k_1 {\rm \omega} _0 \lpar {\rm \omega} _1+{\rm \omega} _c /{\rm \gamma} _0 \rpar +k_0 {\rm \omega} _1 \lpar {\rm \omega} _0 - {\rm \omega} _c /{\rm \gamma} _0 \rpar } \over {{\rm \omega} _0 {\rm \omega} _1 \lpar {\rm \omega} _1+{\rm \omega} _c /{\rm \gamma} _0 \rpar \lpar {\rm \omega} _0 - {\rm \omega} _c /{\rm \gamma} _0 \rpar }}} \right]\comma} \eqno\lpar 23\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021065705771-0980:S0263034611000619_eqn23.gif?pub-status=live)
Where a 1 = eA 1/mω0c. For ω > ωc, ωp, Eq. (23) can be written as
![\eqalign{\displaystyle{{\partial ^2 {\rm \varphi} _{\rm \omega} } \over {\partial x^2 }} - k_z^2 {\rm \varphi} _{\rm \omega}&=- \displaystyle{{{\rm \omega} _p^2\, k_z {\rm \omega} _0^2 a_0 a_1 } \over {{\rm \omega} ^2 {\rm \gamma} _0^2 \lpar 1 - {\rm \omega} _p^2 /{\rm \omega} ^2 {\rm \gamma} _0 \rpar }}\cr&\quad \times \left[{\displaystyle{{k_1 {\rm \omega} _0 \lpar {\rm \omega} _1+{\rm \omega} _c /{\rm \gamma} _0 \rpar +k_0 {\rm \omega} _1 \lpar {\rm \omega} _0 - {\rm \omega} _c /{\rm \gamma} _0 \rpar } \over {{\rm \omega} _0 {\rm \omega} _1 \lpar {\rm \omega} _1+{\rm \omega} _c /{\rm \gamma} _0 \rpar \lpar {\rm \omega} _0 - {\rm \omega} _c /{\rm \gamma} _0 \rpar }}} \right]\comma \; \eqno\lpar 24\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021065705771-0980:S0263034611000619_eqn24.gif?pub-status=live)
the current density at the side band 
can be written as
The normalized wave equation for the side band, on assuming 
, can be written as
![\eqalign{& \displaystyle{{d^2 a_1 } \over {dx^2 }}+\Bigg[ \[\displaystyle{{{\rm \omega} _1^2 } \over {c^2 }} - k_1^2 - \displaystyle{{{\rm \omega} _p^2 } \over {c^2 {\rm \gamma} _0 \lpar 1+{\rm \omega} _c /{\rm \omega} _1 {\rm \gamma} _0 \rpar }} \cr &\quad - \displaystyle{{\lpar 2 - {\rm \omega} _c /{\rm \omega} _0 {\rm \gamma} _0 \rpar \lpar \partial ^2 a_0^2 /\partial x^2 \rpar } \over {4{\rm \gamma} _0^2 \lpar 1+{\rm \omega} _c /{\rm \omega} _0 {\rm \gamma} _0 \rpar \lpar 1 - {\rm \omega} _c /{\rm \omega} _0 {\rm \gamma} _0 \rpar ^2 }} \Bigg ] a_1 \cr &\quad =- \displaystyle{{n_{\rm \omega} e^2 {\rm \omega} _1 a_0^{\ast}} \over {2{\rm \varepsilon} _0 mc^2 {\rm \gamma} _0 \lpar {\rm \omega} _0 - {\rm \omega} _c /{\rm \gamma} _0 \rpar }}.} \eqno\lpar 26\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021065705771-0980:S0263034611000619_eqn26.gif?pub-status=live)
The left-hand side of this equation is the same as that of Eq. (14) with ω0 replaced by 
. Hence, in the absence of the parametric coupling the mode structure of a 1 is the same as that of the pump Eigen mode,
In low density plasma when the pump amplitude is large, the growth rate of plasma wave may become larger than the plasma frequency, ω2≫ωp2 turning into a reactive quasi-mode. Substituting for a 1 in Eq. (26), multiplying the resulting equation by ψdx′, x′ = x/x 0 , and integrating over x′ from −∞ to +∞ we obtain
![\eqalign{& {\rm \omega} _1^2 - k_1^2 c^2=\displaystyle{{c^2 } \over {x_0^2 }}\left[1 - \mathop{\scale190%\vint}_\limits{ - \infty }_^\infty {x^{\prime 2} } {\rm \psi} ^2 dx^{\prime} \right]\cr &\quad - \displaystyle{{{\rm \omega} _p^2 {\rm \omega} _1\, k^2 c^2 \left\vert {a_{00} } \right\vert ^2 } \over {2{\rm \omega} ^2 {\rm \omega} _0 }}\mathop{\scale190%\vint}\limits_{ - \infty }^\infty {\displaystyle{{{\rm \psi} ^4 dx^{\prime}} \over {{\rm \gamma} _0^3 \lpar 1 - {\rm \omega} _c /{\rm \omega} _0 {\rm \gamma} _0 \rpar ^2 }}} \cr &\quad +\mathop{\scale190%\vint}_{ - \infty }^\infty \left[\displaystyle{{{\rm \omega} _p^2 } \over {{\rm \gamma} _0 \lpar 1+{\rm \omega} _c /{\rm \omega} _1 {\rm \gamma} _0 \rpar }}\right. \cr &\quad \left. +\displaystyle{{\lpar 2 - {\rm \omega} _c /{\rm \omega} _0 {\rm \gamma} _0 \rpar c^2 } \over {4{\rm \gamma} _0^2 \lpar 1+{\rm \omega} _c /{\rm \omega} _1 {\rm \gamma} _0 \rpar \lpar 1 - {\rm \omega} _c /{\rm \omega} _1 {\rm \gamma} _0 \rpar ^2 }}\displaystyle{{\partial ^2 } \over {\partial x^2 }}\left\vert {a_0 } \right\vert ^2 \right]{\rm \psi} ^2 dx^{\prime}.}\eqno\lpar 28\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021065705771-0980:S0263034611000619_eqn28.gif?pub-status=live)
One may approximate the left-hand side as 2ωω0. Since ω3 < ω2, we can neglect ω3 terms. Hence Eq. (28) can be written as
where
other normalized quantities are as follows, Ωp = ωp/ω0, Ωc = ωc/ω0, Ω = ω/ω0, 
, 
, x′ = x/x 0 and 
. Solving Eq. (29) numerically, one obtains the growth rate Γ = Imω.
Figure 3 shows the schematic diagram of the laser plasma interaction. Figure 4 shows the variation of normalized growth rate as a function of normalized axial laser amplitude a 00 for ωc/ω 0 = 0.4 and ωp/ω0 = 0.1, 0.2 and 0.4. As ω0 is a constant, different value of ωp/ω0 imply different value of background electron density. Therefore, Figure 4 shows the importance of background electron density and showed that as the plasma frequency (density of electrons) increases, Γ/ω0 increases slowly. For larger values of laser intensity, the SRBS instability grows more rapidly. This shows that SRBS is significantly influenced by the plasma frequency. Figure 5 shows the variation of growth rate with axial laser amplitude. As a 00 increases from 0 to 0.4, Γ/ω0 varies from 0.001 to 0.024 for ωc/ω0 = 0.1. But as ωc/ω0 increases from 0.1 to 0.4, the value of Γ/ω0 increases from 0.024 to 0.035. This shows that magnetic field plays an important role in the growth of SRBS, and instability become stronger with increasing the magnetic field.
Fig. 3. (Color online) Schematic diagram of laser plasma interaction.
Fig. 4. (Color online) Normalized growth rate Γ/ω0 as a function of normalized axial laser amplitude a 00 for ωc/ω0 = 0.4 and ωp/ω0 = 0.1, 0.2 and 0.4.
Fig. 5. (Color online) Normalized growth rate as a function of normalized axial laser amplitude for a 00 for ωp/ω0 = 0.4 and ωc/ω0 = 0.1,0.2 and 0.4.
4. DISCUSSION
The presence of magnetic field modifies the Eigen frequency of the fundamental laser Eigen modes. The mode structure resembles a Gaussian distribution. This is different from unmagnetized plasma where Pathak and Tripathi obtain the mode structure resembling a Lorentizian profile. The growth rate of stimulated Raman backscattering in the reactive quasi-mode regime, relevant at high intensity, increases with the magnetic field. This is caused due to the cyclotron resonance effect though the theory is not applicable close to cyclotron resonance. It also increases with plasma density. As the laser intensity increases the width of nonlinear Eigen modes decreases. This is due to the fact that at higher laser intensity nonlinear self focusing effect becomes stronger, hence to compensate for it stronger diffraction effects, caused by reduced spot size, are required. Our theory is limited to a 0 ≤ 0.5 i.e., for weakly relativistic cases.
ACKNOWLEDGMENTS
The authors are grateful to Prof. V. K. Tripathi, IIT Delhi for fruitful discussions. The author is very thankful to the Govt. Degree College Bilaspur, Rampur, U.P. for encouragement and financial support.
