2. NONLINEAR PERMITTIVITY

Consider a right circularly polarized Gaussian laser beam of spot size r ₀, propagating through uniform plasma of density n ₀ in the presence of guide magnetic field B _s ẑ. The electric and magnetic fields of the laser field are given by

(1)$$\vec{E} = \lpar \hat{x} + i \hat{y} \rpar A \lpar r\comma \; z \rpar \exp \left[-i \lpar {\rm {\rm {\rm \omega}}} t - k z \rpar \right] \comma $$$\overrightarrow{E}=(\hat{x}+i\hat{y})A(r,z)exp[-i(\text{ω}t-kz\left)\right],$

(2)$$\vec{B} = {\vec{k} \times \vec{E} \over {\rm {\rm {\rm \omega}}} } - {i\lpar \nabla A \times \lpar \hat{x} + i \hat{y}\rpar \rpar \over {\rm {\rm {\rm \omega}}}} \exp \lsqb \!\!- i\lpar {\rm {\rm {\rm \omega}}} t - kz\rpar \rsqb . $$$\overrightarrow{B}=\frac{\overrightarrow{k}\times \overrightarrow{E}}{\text{ω}}-\frac{i(\nabla A\times (\hat{x}+i\hat{y}\left)\right)}{\text{ω}}exp[-i(\text{ω}t-kz\left)\right].$

where k = (ω/c) (1 − (ω²_p/(ω(γω − ω_c)))^1/2 , ω_P=(4πn ₀e ²/m)^1/2 are the plasma frequency, ω_c = eB _s/mc is the electron cyclotron frequency, −e and m are the electronic charge and mass, and c is the velocity of light. At z = 0 the laser field amplitude is given by

(3)$$A^2=A^2_0 {\exp}\left[-r^2/r^2_0\right].$$$A^2=A_0^{2}exp[-r^2/r_0^2].$

The laser imparts an oscillatory velocity to electrons given by

(4)$$\vec{v} = {eA\lpar \hat{x} + i \hat{y}\rpar \over mi\lpar {\rm {\rm {\rm \gamma}}} {\rm {\rm {\rm \omega}}} - {\rm {\rm {\rm \omega}}}_c \rpar } \exp \Big[ \!\!-\! i\lpar {\rm {\rm {\rm \omega}}} t - kz\rpar \Big] \comma $$$\overrightarrow{v}=\frac{eA(\hat{x}+i\hat{y})}{mi(\text{γ ω}-{\text{ω}}_c)}exp[-i(\text{ω}t-kz)],$

and exerts ponderomotive force given by

(5)$$\eqalign{{\vec {F_p}} & = - \displaystyle{m \over 2}\left( {\vec {v}^{\ast}} .\nabla {\gamma} {\vec {v}} \right) - \displaystyle{e \over 2}\left( {\vec {v}^{\ast}} \times {\rm {\vec {B}}} \right), \cr & =- \hat r\displaystyle{{e^2 } \over {4m{\rm \omega} ({\rm \gamma} {\rm \omega} - {\rm \omega} _c )}}\left[ \displaystyle{{2 - {{{\rm \omega} _c } / {{\rm \gamma} {\rm \omega} }}} \over {1 - {{{\rm \omega} _c } / {{\rm \gamma} {\rm \omega} }}}}\displaystyle{{\partial (AA^{\ast} )} \over {\partial r}} - \displaystyle{{{\rm \omega} _c AA^ {\ast } \over {{\rm \omega} {\rm \gamma} ^2 (1 - {{{\rm \omega} _c } / {{\rm \gamma} {\rm \omega} }})^2 }}\displaystyle{{\partial {\rm \gamma} } \over {\partial r}}} \right].}$$$\begin{array}{cc}\overrightarrow{F_p}& =-\frac{m}{2}({\overrightarrow{v}}^{*}.\nabla \gamma \overrightarrow{v})-\frac{e}{2}({\overrightarrow{v}}^{*}\times \overrightarrow{\text{B}}),\\ & =-\hat{r}\frac{e^2}{4m\text{ω}(\text{γ ω}-{\text{ω}}_c)}[\frac{2-{\text{ω}}_c/\text{γ ω}}{1-{\text{ω}}_c/\text{γ ω}}\frac{\partial \left({AA}^{*}\right)}{\partial r}-\frac{{\text{ω}}_c{AA}^{*}}{\text{ω}{\text{γ}}^2(1-{\text{ω}}_c/\text{γ ω}{)}^2}\frac{\partial \text{γ}}{\partial r}].\end{array}$

Here γ is the relativistic factor deduced from the following relation

(6)$${\rm {\rm {\rm \gamma}}} = \left(1 + \displaystyle{{e^2 AA^{\ast} } \over m^2 {\rm {\rm {\rm \omega}}} ^2 \lpar 1 - {\rm {\rm {\rm \omega}}}_{c} /{{\rm {\rm {\rm \gamma}}} {\rm {\rm {\rm \omega}}} } \rpar ^2 c^2 } \right)^{1/2} . $$$\text{γ}={(1+\frac{e^2{AA}^{*}}{m^2{\text{ω}}^2(1-{\text{ω}}_c/\text{γ ω}{)}^2{c}^2})}^{1/2}.$

This is a transcendental equation that can be solved numerically. The expression for γ can be written as fourth order differential equation in γ. Using Eq. (6) in Eq. (5) we get

(7)$$\vec{F}_p = - {\hat{r} \lpar mc^2\rpar \over 4{\rm {\rm {\rm \gamma}}} \left(1 - {\rm {\rm {\rm \omega}}}_c / {\rm {\rm {\rm \gamma}}} \right)^2} \left[2 - {{\rm {\rm {\rm \omega}}}_c \over {\rm {\rm {\rm \omega}}} {\rm {\rm {\rm \gamma}}}} - {{\rm {\rm {\rm \omega}}}_c / {\rm {\rm {\rm \omega}}} \lcub {\rm {\rm {\rm \gamma}}}^2 - 1\rcub \over 2 \lcub {\rm {\rm {\rm \gamma}}}^3 - {\rm {\rm {\rm \omega}}}_c / {\rm {\rm {\rm \omega}}} \rcub } \right]{\partial aa^{\ast} \over \partial r}. $$${\overrightarrow{F}}_p=-\frac{\hat{r}\left({mc}^2\right)}{4\text{γ}{(1-{\text{ω}}_c/\text{γ})}^2}[2-\frac{{\text{ω}}_c}{\text{ω γ}}-\frac{{\text{ω}}_c/\text{ω}\{{\text{γ}}^2-1\}}{2\{{\text{γ}}^3-{\text{ω}}_c/\text{ω}\}}]\frac{\partial {aa}^{*}}{\partial r}.$

For ω_c → 0, this reduces to the usual expression

(8)$$\vec{F}_p = e\nabla {\rm {\rm {\rm \phi}}}_p\comma \; \quad {\rm {\rm {\rm \phi}}}_{\rm p} = - \displaystyle{{mc^2 } \over e} \left[\lpar 1 + a^2 \rpar ^{1/2} - 1 \right]\comma \; a = \displaystyle{{eA \over m{\rm {\rm {\rm \omega}}} c}}. $$${\overrightarrow{F}}_p=e\nabla {\text{ϕ}}_p,{\text{ϕ}}_{\text{p}}=-\frac{{mc}^2}{e}\left[\right(1+a^2{)}^{1/2}-1],a=\frac{eA}{m\text{ω}c}.$

In the non-relativistic limit it reduces to (Sodha et al., Reference Sodha, Ghatak and Tripathi1976)

(9)$${\rm {\rm {\rm \phi}}}_{\rm p} = - {mc^2 \over e} {a^2 \over 4} {2 - {\rm {\rm {\rm \omega}}}_c / {\rm {\rm {\rm \omega}}} \over \lpar 1 - {\rm {\rm {\rm \omega}}}_c / {\rm {\rm {\rm \omega}}} \rpar ^2}. $$${\text{ϕ}}_{\text{p}}=-\frac{{mc}^2}{e}\frac{a^2}{4}\frac{2-{\text{ω}}_c/\text{ω}}{(1-{\text{ω}}_c/\text{ω}{)}^2}.$

One may note that $\vec{F}_p$${\overrightarrow{F}}_p$ changes sign at ω ~ ω_c/2γ. The sign reversal implies that the ponderomotive force, instead of expelling electrons, attracts them when ω ≤ ω_c/2γ. As we are considering ω ≥ ω_c/2γ, the ponderomotive force expels electrons in our case.

The modified density is given by

(10)$${n_e \over n_0} = 1 - {\nabla .\vec{F}_p \over 4{\rm {\rm {\rm \pi}}} n_o e^2}. $$$\frac{n_e}{n_0}=1-\frac{\nabla .{\overrightarrow{F}}_p}{4\text{π}{n}_o{e}^2}.$

The effective plasma permittivity for the right circularly polarized wave (ɛ₊ = ɛ_xx + iɛ_xy where ɛ_xx and ɛ_xy are the components of plasma permittivity tensor) is given by

(11)$${\rm {\rm {\rm \varepsilon}}}_{+} = 1 - {{\rm {\rm {\rm \omega}}}_p^2 n_e / n_0 \over {\rm {\rm {\rm \omega}}}^2 \lpar {\rm {\rm {\rm \gamma}}} - {\rm {\rm {\rm \omega}}}_c / {\rm {\rm {\rm \omega}\rpar}}}. $$${\text{ɛ}}_{+}=1-\frac{{\text{ω}}_p^2{n}_e/n_0}{{\text{ω}}^2(\text{γ}-{\text{ω}}_c/\text{ω})}.$

In order to explicitly write ɛ₊ in the paraxial approximation (r ²/r ₀²f ²), we expand γ as

(12)$${\rm {\rm {\rm \gamma}}} = {\rm {\rm {\rm \gamma}}}_0 + {{\rm {\rm {\rm \gamma}}}_1 r^2 \over f^2 r_0^2}. $$$\text{γ}={\text{γ}}_0+\frac{{\text{γ}}_1{r}^2}{f^2{r}_0^2}.$

Eq. (6) gives

(13)$${\rm {\rm {\rm \gamma}}}_0 = \left(1 + {a_0^2 {\rm {\rm {\rm \gamma}}}_0^2 \over f^2 \lpar {\rm {\rm {\rm \gamma}}}_0 - \Omega_c\rpar ^2} \right)^{1/2}\semicolon \; \quad \Omega_c = {{\rm {\rm {\rm \omega}}}_c \over {\rm {\rm {\rm \omega}}}}\comma \; $$${\text{γ}}_0={(1+\frac{a_0^2{\text{γ}}_0^2}{f^2({\text{γ}}_0-{\Omega }_c{)}^2})}^{1/2};{\Omega }_c=\frac{{\text{ω}}_c}{\text{ω}},$

(14)$${\rm {\rm {\rm \gamma}}}_1 = - {a_0^2 {\rm {\rm {\rm \gamma}}}_0 \over 2f^2 \lpar {\rm {\rm {\rm \gamma}}}_0 - \Omega_c\rpar ^2 \left(1 + {a_0^2 \Omega_c \over f^2 \lpar {\rm {\rm {\rm \gamma}}}_0 - \Omega_c\rpar ^3} \right)}. $$${\text{γ}}_1=-\frac{a_0^2{\text{γ}}_0}{2f^2({\text{γ}}_0-{\Omega }_c{)}^2(1+\frac{a_0^2{\Omega }_c}{f^2({\text{γ}}_0-{\Omega }_c{)}^3})}.$

Substituting for F _p in Eq. (10) and using Eq. (12) we get the normalized electron density

(15)$${n_e \over n_0} = 1 - {\rm {\rm {\rm \delta}}} n_0 - {\rm {\rm {\rm \delta}}} n_1 {r^2 \over r_0^2}. $$$\frac{n_e}{n_0}=1-\text{δ}{n}_0-\text{δ}{n}_1\frac{r^2}{r_0^2}.$

Where

(16)$$\eqalign{{\rm \delta} n_0 &= c^2 \left({a_0^2 \left({4{\rm \gamma} _0^4 - 3{\rm \gamma} _0 \Omega _c - 3{\rm \gamma} _0^3 \Omega _c + 2\Omega _c^2 } \right)} \right)\cr & \qquad/\left({2{\rm \omega} _p^2 r_0^2 f^4 \left({{\rm \gamma} _0 - \Omega _c } \right)^2 } {\left({{\rm \gamma} _0^3 - {\rm \omega} _c } \right)} \right)} $$$\begin{array}{cc}\text{δ}{n}_0& =c^2\left(a_0^2(4{\text{γ}}_0^4-3{\text{γ}}_0{\Omega }_c-3{\text{γ}}_0^3{\Omega }_c+2{\Omega }_c^2)\right)\\ & /\left(2{\text{ω}}_p^2{r}_0^2{f}^4{({\text{γ}}_0-{\Omega }_c)}^2({\text{γ}}_0^3-{\text{ω}}_c)\right)\end{array}$

(17)$$\eqalign{ &{\rm \delta} n_1 = a_0^2 c^2 \left({a_0^2 \left({4{\rm \gamma} _0^7 - 4{\rm \gamma} _0^4 \Omega _c - 6{\rm \gamma} _0^6 \Omega _c + 3{\rm \gamma} _c \Omega _c^2 + 7{\rm \gamma} _0^3 \Omega _c^2 - 4\Omega _c^3 } \right)} \right. \cr & \qquad \left. { - 2f^2 \left({{\rm \gamma} _0 - \Omega _c } \right)^2 \left({4{\rm \gamma} _0^7 - 7{\rm \gamma} _0^4 \Omega _c - 3{\rm \gamma} _0^6 \Omega _c + 3{\rm \gamma} _0 \Omega _c^2 + 5{\rm \gamma} _0^3 \Omega _c^2 - 2\Omega _c^3 } \right)} \right) \cr & \qquad \Big / \left({2\Omega _p^2 r_0^2 f^6 \left({{\rm \gamma} _0 - \Omega _c } \right) \left({{\rm \gamma} _0^3 - \Omega _c } \right)^2 \left({f^2 \left({{\rm \gamma} _0 - \Omega _c } \right)^3 + a_0^2 \Omega _c } \right)} \right.} $$$\begin{array}{cc}& \text{δ}{n}_1=a_0^2{c}^2(a_0^2(4{\text{γ}}_0^7-4{\text{γ}}_0^4{\Omega }_c-6{\text{γ}}_0^6{\Omega }_c+3{\text{γ}}_c{\Omega }_c^2+7{\text{γ}}_0^3{\Omega }_c^2-4{\Omega }_c^3)\\ & -2f^2{({\text{γ}}_0-{\Omega }_c)}^2(4{\text{γ}}_0^7-7{\text{γ}}_0^4{\Omega }_c-3{\text{γ}}_0^6{\Omega }_c+3{\text{γ}}_0{\Omega }_c^2+5{\text{γ}}_0^3{\Omega }_c^2-2{\Omega }_c^3))\\ & /(2{\Omega }_p^2{r}_0^2{f}^6({\text{γ}}_0-{\Omega }_c){({\text{γ}}_0^3-{\Omega }_c)}^2(f^2{({\text{γ}}_0-{\Omega }_c)}^3+a_0^2{\Omega }_c)\end{array}$

In Figure 1, we have plotted electron density (using Eq. (15)) on the axis i.e., at r = 0 as a function of normalized magnetic field (Ω_c) for different values of Ω_p and normalized laser spot size. The other parameters are a ₀ = 1 and f = 1 At Ω_p = 0,1 and r ₀/λ = 3 the electron cavitations is around 60% at normalized amplitude a ₀ = 1. The electron expulsion increases with increase in magnetic field due to increase in ponderomotive force. For fixed laser parameters and guide magnetic field, the electron expulsion in radial direction decreases with increase in plasma density. The electron expulsion also decreases with the increase in ωr ₀/c. The laser field strength required for complete electron evacuation decreases with increase in magnetic field but increases rapidly with increase in plasma density for a given magnetic field (see Figs. 2 and 3). Using Eq. (15) in the expression for the plasma permittivity we get

(18)$${\rm {\rm {\rm \varepsilon}}}_{+} = {\rm {\rm {\rm \varepsilon}}}_0 - {\rm {\rm {\rm \Phi}}} {r^2 \over r_0^2}\comma \; $$${\text{ɛ}}_{+}={\text{ɛ}}_0-\text{Φ}\frac{r^2}{r_0^2},$

(19)$${\rm {\rm {\rm \varepsilon}}}_0 = 1 - {{\rm {\rm {\rm \omega}}}_p^2 \lpar 1 - {\rm {\rm {\rm \delta}}} n_0\rpar \over {\rm {\rm {\rm \omega}}}_0^2 \lpar {\rm {\rm {\rm \gamma}}}_0 - \Omega_c\rpar }\comma \; $$${\text{ɛ}}_0=1-\frac{{\text{ω}}_p^2(1-\text{δ}{n}_0)}{{\text{ω}}_0^2({\text{γ}}_0-{\Omega }_c)},$

(20)$${\rm {\rm {\rm \phi}}} = {\Omega_p^2 \lpar {\rm {\rm {\rm \gamma}}}_1 \lpar 1 - {\rm {\rm {\rm \delta}}} n_0\rpar + f^2 \lpar {\rm {\rm {\rm \gamma}}}_0 - \Omega_c\rpar {\rm {\rm {\rm \delta}}} n_1\rpar \over \lpar {\rm {\rm {\rm \gamma}}}_0 - \Omega_c\rpar ^2 f^2}. $$$\text{ϕ}=\frac{{\Omega }_p^2\left({\text{γ}}_1\right(1-\text{δ}{n}_0)+f^2({\text{γ}}_0-{\Omega }_c\left)\text{δ}{n}_1\right)}{({\text{γ}}_0-{\Omega }_c{)}^2{f}^2}.$

For a constant magnetic field, ɛ₀ and ϕ increases with a ₀ However, ϕ decreases with the increase in magnetic field for constant laser field strength.

Fig. 1. Variation of normalized charge density at r = 0 with normalized magnetic field for different values of laser beam radius and plasma density. The other parameters are f = 1 and a ₀ = 1.

Fig. 2. Variation of laser field strength required for complete electron cavitation on the axis (a ₀(cavitation)) with normalized magnetic field (Ω_c). The other parameters are r ₀/λ = 3 and Ω_p = 0.2.

Fig. 3. Variation of laser field strength required for complete electron cavitation on the axis (a ₀(cavitation)) with normalized plasma density (Ω_p). The other parameters are r ₀/λ = 3 and Ω_c = 0.9.

3. SELF-FOCUSING

The wave equation governing A is given as (Sodha et al., Reference Sodha, Ghatak and Tripathi1976)

(21)$$2ik {\partial A \over \partial z} + {1 \over 2} \left(1 + {{\rm {\rm {\rm \varepsilon}}}_0 \over {\rm {\rm {\rm \varepsilon}}}_p} \right)\nabla_{\perp}^2 A + \left({{\rm {\rm {\rm \omega}}}^2 \over c^2} {\rm {\rm {\rm \varepsilon}}}_0 - k^2 \right)A=0.$$$2ik\frac{\partial A}{\partial z}+\frac{1}{2}(1+\frac{{\text{ɛ}}_0}{{\text{ɛ}}_p}){\nabla }_{\perp }^{2}A+(\frac{{\text{ω}}^2}{c^2}{\text{ɛ}}_0-k^2)A=0.$

where

$${\rm {\rm {\rm \varepsilon}}}_p = 1-({ {\rm {\rm {\rm \omega}}} _p^2 / {\rm {\rm {\rm \gamma}}} {\rm {\rm {\rm \omega}}}^2} ).$$${\text{ɛ}}_p=1-({\text{ω}}_p^2/\text{γ}{\text{ω}}^2).$

Following Sodha et al. (1976) we introduce an eikonal, A = A ₀ (r, z) exp [ikS (r, z)], expand S in the paraxial region as S = S ₀(z) + βr ², and introduce a function f such that β = (2/(f(1 + ɛ/ɛ_p) ∂f/(∂z)).

Equating coefficients of different powers of r, we obtain the normalized laser intensity distribution for z > 0 as

(22)$$a^2 = {{a_0^2 } \over {\,f^2 }}\exp \Big[ - r^2 /\Big( r_0^2 \, f^2 \Big) \Big]\comma \; $$$a^2=\frac{a_0^2}{f^2}exp[-r^2/(r_0^2{f}^2)],$

where f is the beam width parameter (f = 1 at z = 0) and a ₀ = eA ₀/mωc. The equation governing beam width parameter f turns out to be

(23)$${\partial^2 f \over \partial z^2} = {1 \over R_d^2 f^3} - {{\rm {\rm {\rm \Phi}}} \over r_0^2 {\rm {\rm {\rm \varepsilon}}}_0} f. $$$\frac{{\partial }^{2}f}{\partial z^2}=\frac{1}{R_d^2{f}^3}-\frac{\text{Φ}}{r_0^2{\text{ɛ}}_0}f.$

here $R_d = {2kr_0^2 \over 1 + {\rm {\rm {\rm \varepsilon}}}_0 /{\rm {\rm {\rm \varepsilon}}}_p}$$R_d=\frac{2{kr}_0^2}{1+{\text{ɛ}}_0/{\text{ɛ}}_p}$ is Rayleigh diffraction length. Substituting z = ξ(ωr ₀²/c) in Eq. (21) we get

(24)$${\partial^2 f \over \partial {\rm {\rm {\rm \xi}}}^2} = {1 \over 4} \left(1 + {{\rm {\rm {\rm \varepsilon}}}_0 \over {\rm {\rm {\rm \varepsilon}}}_p} \right)^2 {1 \over {\rm {\rm {\rm \varepsilon}}}_0 f^3} - {1 \over 2} \left(1 + {{\rm {\rm {\rm \varepsilon}}}_0 \over {\rm {\rm {\rm \varepsilon}}}_p} \right)\left({r_0^2 {\rm {\rm {\rm \omega}}}^2 \over c^2} \right){\Phi \over {\rm {\rm {\rm \varepsilon}}}_0} f. $$$\frac{{\partial }^{2}f}{\partial {\text{ξ}}^2}=\frac{1}{4}{(1+\frac{{\text{ɛ}}_0}{{\text{ɛ}}_p})}^2\frac{1}{{\text{ɛ}}_0{f}^3}-\frac{1}{2}(1+\frac{{\text{ɛ}}_0}{{\text{ɛ}}_p})\left(\frac{r_0^2{\text{ω}}^2}{c^2}\right)\frac{\Phi }{{\text{ɛ}}_0}f.$

In Figure 4, numerical results are presented to show self-focusing action as a consequence of (1) purely relativistic effect (relativistic mass increase of the plasma electrons) and (2) the combined effect of relativistic effect and radial ponderomotive expulsion of electrons. We have plotted f as a function of ξ for different values of normalized magnetic field (Ω_c) for both the cases. The other parameters are a₀ = 1 and Ω_p = 0.2. We have ${\rm {\rm {\rm \omega}}} \;r_{0}/c= {2{\rm {\rm {\rm \pi}}} r_0 \over {\rm {\rm {\rm \lambda}}}} = 18.84$$\text{ω}{r}_0/c=\frac{2\text{π}{r}_0}{\text{λ}}=18.84$ that corresponds to normalized spot size r ₀/λ = 3. We have chosen this value as a lower limit on spot size. For Ω_p = 0.1 and r ₀/λ = 3, we have $r_{0} = 1.88 {c \over {\rm {\rm {\rm \omega}}}_p}$$r_0=1.88\frac{c}{{\text{ω}}_p}$. When only relativistic effects are considered the plasma permittivity is obtained by substituting n_e = n₀ in Eq. (11) and on using ε₀=1−(Ω_p²/(γ₀−Ω_c)) and Φ=γ₁Ω²_p/f ²(γ₀−Ω_c)² in Eq. (24) we obtain f as a function of ξ (dashed line). The periodic focusing and defocusing occur due to relative dominance of relativistic nonlinearity and diffraction effects. As the laser beam acquires smaller spot size, the diffraction divergence increases and takes over the convergence due to nonlinear refraction. This occurs in a cyclic manner. The relativistic focusing is enhanced by increasing the magnetic field. For the case when both relativistic focusing and ponderomotive effects are present (bold line), it is seen that the beam continues to focus until complete electron evacuation occurs on the axis (beyond this point theory is not valid). The distance at which the complete electron evacuation occurs (referred to as cavitation length) is an important parameter. The cavitation length can be increased by decreasing plasma density, normalized laser spot size r ₀/λ and guide magnetic field. The self-focusing of the laser beam is also a sensitive function of normalized beam radius ωr ₀/c as seen in Figure 5 where we have plotted f versus Ω_c at a fixed ξ. Ponderomotive effects are significant at lower value of ωr ₀/c.

Fig. 4. The variation of beam width parameter (f) with normalized distance (ξ = z/(ωr ₀²/c)) for different values of magnetic field (Ω_c = 0, 0.5, and 0.9) for two cases namely (1) when combined effect of relativistic effect and radial ponderomotive expulsion of electrons is considered (solid line) and (2) only relativistic effect is included (dashed line). The other parameters are a ₀ = 1, r ₀/λ = 3 and Ω_p = 0.2.

Fig. 5. The variation of beam width parameter (f) with normalized magnetic field (Ω_c) for different values of normalized laser spot size (r ₀/λ = 3 and 6, λ = 1 μm) at fixed axial distance z = 22.4 μm for two cases namely (1) when combined effect of relativistic effect and radial ponderomotive expulsion of electrons is considered (solid line) and (2) only relativistic effect is included (dashed line). The other parameters are a ₀ = 1 and Ω_p = 0.2.

The self-focusing increases with laser field strength a ₀. An upper limit on a ₀ is at values at which complete electron evacuation occurs. In Eq. (23) the two terms on the right-hand side denote the diffraction divergence and the self-focusing effect (due to ponderomotive and relativistic effects) respectively. For self sustained Gaussian beam, these two terms balance each other i.e., Φ=r ²₀/R ²_d. From this value of Φ the threshold laser field strength is deduced. In Figure 6, we have plotted the normalized radius of the Gaussian beam ω_pr ₀/c as a function of threshold laser field strength a _cr for Ω_c = 0.3 and 0.5. The threshold of laser intensity for self-ducting of laser beam depends on beam radius and the applied magnetic field. It increases with the decreasing radius of the self sustained laser beam. For a fixed beam radius the threshold of laser intensity decreases with increase in magnetic field.

Fig. 6. Normalized beam radius (ω_↓pr _↓0/c) as a function of threshold laser field strength (a _cr) for different values of magnetic fields (Ω_c = 0.3 and 0.5).

When the laser intensity is very high (a ₀² > 5), the relativistic gamma factor is numerically deduced by plotting γ versus a ₀² for different values of magnetic field (see Fig. 7). Beyond a ₀² > 5, at any given a ₀² the values of γ differs by nearly the same constant amount (vertical separation of the curves ~0.8–1Ω_c depending on the choice of a ₀². Hence (Eq. (6)) can be approximately modeled as ${\rm {\rm {\rm \gamma}}} = .8\Omega_c + \sqrt{1 + a_0^2}$$\text{γ}=.8{\Omega }_c+\sqrt{1+a_0^2}$. For applications involving propagation of laser in an overdense plasma where a ₀ ≥ 10 it is noted that for Ω_c ≤ 1 the effect of guide magnetic field is not very prominent.

Fig. 7. Relativistic gamma factor versus a ₀² for different values of magnetic field (Ω_c = 0, 0.4 and 0.8).