II. PROPAGATION OF LASER THROUGH A NON-UNIFORM PLASMA CHANNEL

Consider a linearly polarized laser beam propagating along the $\hat z$-direction through a non-uniform plasma channel. Density non-uniformity of plasma channel,

(1)$$n(r) = {n_{{\rm e0}}}\left( {1 + \displaystyle{{{n_2}} \over {{n_{{\rm e0}}}}}\displaystyle{{{r^2}} \over {r_2^2}}} \right),$$

can be developed by two laser pulses [Kumar et al. (Reference Kumar, Dahiya and Sharma2011)], where n _e0 is electron density on channel axis. The plasma channel non-uniformity n ₂/n _e0 and spot size r ₂ are related to the properties of the channel. Then the plasma frequency profile becomes,

(2)$${\rm \omega} _{\rm p}^2 (r) = {\rm \omega} _{{\rm p}0}^2 \left( {1 + \displaystyle{{{n_2}} \over {{n_{{\rm e0}}}}}\displaystyle{{{r^2}} \over {r_2^2}}} \right),$$

where ${\rm \omega} _{{\rm p}0}^2 = n_{{\rm e0}}^{\rm 2} {m_{\rm e}}/{{\rm \varepsilon} _0}$ is the electron plasma frequency, m _e and ε₀ are the electron mass and electric permittivity, respectively. The electric field of a laser pulse is given by,

(3)$${\vec E_0} = \hat x{A_0}(r,z,t){e^{ - i({{\rm \omega} _0}t - {k_0}z)}},$$

where k ₀ = (ω₀/c)η₀, η₀ is the refractive index of the plasma, and A ₀(r, z, t) is a slowly varying complex amplitude of the laser. The intensity distribution of this mode at “z = 0” is given by,

(4)$${A_0} \vert _{z = 0}^2 = E_{00}^2 {e^{ - {r^2}/r_0^2}} {g_0}(t),$$

where g ₀(t) is the temporal shape of the main pulse. And at z > 0, the intensity distribution is modified by the beam width parameter “f”,

(5)$$ \vert {A_0}{ \vert ^2} = \displaystyle{{E_{00}^2} \over {{\,f^2}}}{e^{ - {r^2}/r_0^2 {\,f^2}}}{g_0}(t - z/c).$$

The main laser pulse imparts oscillatory velocity to electrons,

(6)$${\vec v_0} = \displaystyle{{e{{\vec E}_0}} \over {{m_{\rm e}}i{{\rm \omega} _0}{\rm \gamma}}}, $$

where –e is the electron charge. Here we have neglected $({\vec v_0} \times {\vec B_0})$, because of its twice high frequency. In the paraxial approximation, relativistic gamma factor can be expanded by using Taylor's expansion,

(7)$${\displaystyle{1 \over {{\rm \gamma} (r)}} = \displaystyle{1 \over {{{\rm \gamma} _0}}}\left( {1 + \displaystyle{{a_{00}^2 {r^2}/4r_0^2 {\,f^4}} \over {1 + a_{00}^2 /2{\,f^2}}}} \right),} $$

where $a_{00}^2 = e{A_{00}}/{m_{\rm e}}{{\rm \omega} _0}c\vert{^2}$ is the normalized laser amplitude and ${\rm \gamma}_0 = (1 + a_{00}^2 {/2)^{1/2}}$ is the value of γ on the axis. The wave equation governing the electric field of the pump laser beam in a plasma can be written as,

(8)$${\nabla ^2}{\vec E_0} - \nabla (\nabla \cdot {\vec E_0}) + \displaystyle{{{\rm \omega} _0^2} \over {{c^2}}}{{\rm \varepsilon} _{{\rm eff}}}{\vec E_0} = 0,$$

where the effective dielectric permittivity becomes in terms of z and r as,

(9)$${{\rm \varepsilon} _{{\rm eff}}}(r,z) = {{\rm \varepsilon} _{{\rm eff}0}} + {{\rm \varepsilon} _2}{r^2},$$

where

(10)$${{\rm \varepsilon} _{{\rm eff}0}} \equiv 1 - \displaystyle{{{\rm \omega} _{{\rm p}0}^2} \over {{\rm \omega} _0^2 {{\rm \gamma} _0}}},\quad {{\rm \varepsilon} _2} \equiv - \displaystyle{{{\rm \omega} _{{\rm p}0}^2 \Phi} \over {{\rm \omega} _0^2 {{\rm \gamma} _0}r_0^2 {\,f^2}}},$$

(11)$${\rm and}\quad \Phi \equiv \displaystyle{{{n_2}} \over {{n_{{\rm eo}}}}}\displaystyle{{r_0^2 {\,f^2}} \over {r_2^2}} + \displaystyle{{a_{00}^2} \over {4{\,f^2}(1 + a_{00}^2 /2{\,f^2})}}.$$

Electromagnetic wave satisfies the condition, $\nabla \cdot {\vec E_0} = 0$. In terms of a slowly varying laser amplitude, one can neglect (∂²A ₀/∂z ²) by using the WKB approximation,

(12)$$2i{k_0}\displaystyle{{\partial {A_0}} \over {\partial z^{\prime}}} + \nabla _ \bot ^2 {A_0} - k_0^2 {A_0} + \displaystyle{{{\rm \omega} _0^2} \over {{c^2}}}\left( {{{\rm \varepsilon} _{{\rm eff}0}} + {{\rm \varepsilon} _2}{r^2}} \right){A_0} = 0,$$

where the old variables z and t are transformed to z′ = z and t′ = (t − z/v _g), respectively. Now, following Akhmanov et al. (Reference Akhmanov, Sukhorukov and Khokhlov1966) and Sodha et al. (Reference Sodha, Ghatak and Tripathi1974) to evaluate the pulse evolution we introduce an eikonal approximation,

(13)$${\vec A_0} = {\vec A_{00}}(r){e^{i{k_0}S(r,z^{\prime})}},$$

and separate the real and imaginary parts. This results in the following equations,

(14)$$\eqalign{ - 2{k_0}&\left( {\displaystyle{{\partial S} \over {\partial z^{\prime}}}{A_{00}}} \right) + \left[ {\displaystyle{1 \over r}\left( {\displaystyle{{\partial {A_{00}}} \over {\partial r}}} \right) + \displaystyle{{{\partial ^2}{A_{00}}} \over {\partial {r^2}}} - {A_{00}}{{\left( {\displaystyle{{\partial S} \over {\partial r}}} \right)}^2}} \right] \cr &+ \displaystyle{{{\rm \omega} _0^2} \over {{c^2}}}{{\rm \varepsilon} _2}{r^2}{A_{00}} = 0,} $$

and

(15)$${k_0}\displaystyle{{\partial A_{00}^2} \over {\partial z^{\prime}}} + \left( {\displaystyle{{{\partial ^2}S} \over {\partial {r^2}}} + \displaystyle{1 \over r}\displaystyle{{\partial S} \over {\partial r}}} \right)A_{00}^2 + \displaystyle{{\partial A_{00}^2} \over {\partial r}}\displaystyle{{\partial S} \over {\partial r}} = 0.$$

The coupled set of equations for $A_{00}^2 $ and S are solved in the paraxial approximation by expanding S as,

(16)$$S = {\rm \phi} (z^{\prime}) + {\rm \beta} \displaystyle{{{r^2}} \over 2},$$

where

(17)$${\rm \beta} = \displaystyle{1 \over f}\displaystyle{{df} \over {dz^{\prime}}},$$

and ϕ(z′) is the axial phase. After solving the Eqs. (14) and (15), we get the single equation for laser beam width parameter,

(18)$$\displaystyle{1 \over f}\displaystyle{{{\partial ^2}f} \over {\partial {{z^{\prime}}^2}}} = \displaystyle{1 \over {{\,f^4}}}\left( {\displaystyle{1 \over {R_d^2}} - \displaystyle{{{\rm \omega} _{\,p0}^2 \Phi} \over {{\rm \omega} _0^2 {{\rm \gamma} _0}r_0^2}} {\,f^2}} \right),$$

where ${R_{\rm d}} = {k_0}r_0^2 $ is the diffraction length. The second term on the right hand side of the above Eq. (18) contributes to non-linear self-focusing length. The laser beam width parameter governs the equation through the dimensionless distance ξ = z/R _d,

(19)$$\displaystyle{{{\partial ^2}f} \over {\partial {{\rm \xi} ^2}}} = \displaystyle{1 \over {{\,f^3}}} - \displaystyle{{{\rm \omega} _{{\rm p}0}^2 \Phi} \over {{\rm \omega} _0^2 {{\rm \gamma} _0}}}\displaystyle{{R_{\rm d}^2} \over {r_0^2 f}}.$$

The boundary conditions at ξ = 0 for an initially plane wavefront are f = 1, ∂f/∂ξ = 0, and S ₀ = 0. We observed the variation of a normalized pump intensity distribution $I = (a_{00}^2 /{\,f^2})exp( - {r^2}/r_0^2 {\,f^2})$ with respect to the propagation distance ξ for different values of plasma channel non-uniformities n ₂/n _e0 are plotted in Figure 1. Larger values of plasma channel non-uniformity induce more focusing of laser pulse, which further increase the normalized laser pulse intensity. Normalized pump intensity distribution through the deep plasma channel n ₂/n _e0 = 0.3 for different normalized laser amplitude a ₀₀ are given in Figure 2. In a deep plasma channel, the self-focusing is only due to the relativistic mass correction. When both the terms on the right hand side of Eq. (18) are equal, then the beam width parameter (f = 1) becomes unity for all the values of z. Thus, the Gaussian intensity profile given in Eq. (4) of the laser would remain the same for z > 0. We obtain the equation for the laser spot size,

(20)$$\displaystyle{{r_{00}^2} \over {{{\rm \gamma} _0}}}\left( {\displaystyle{{{n_2}} \over {{n_{{\rm e0}}}}}\displaystyle{{r_{00}^2} \over {r_{22}^2}} + \displaystyle{{a_{00}^2} \over {4(1 + a_{00}^2 /2)}}} \right) = 1,$$

where r ₀₀ = r ₀ω_p0/c and r ₂₂ = r ₂ω_p0/c. In Figure 3, we plotted the variation of the normalized pump intensity a ₀₀ with the plasma channel non-uniformity for the fixed normalized spot size r ₂₂. The normalized pump amplitude a ₀₀ must decrease with the increase in plasma channel non-uniformity to maintain the constant shape of the pulse.

Fig. 1.

Variation of a normalized pump intensity distribution $I = (a_{00}^2 /{\,f^2})exp( - {r^2}/r_0^2 {\,f^2})$ with respect to the normalized distance of propagation ξ and pulse with r/r ₀ for plasma channels with non-uniformities, n ₂/n _e0 = 0.0 (1a), n ₂/n _e0 = 0.1 (1b), n ₂/n _e0 = 0.2 (1c), and n ₂/n _e0 = 0.3 (1d) and normalized laser intensities is a ₀₀ = 0.57.

Fig. 2.

Variation of normalized pump intensity distribution $I = (a_{00}^2 /{\,f^2})exp( - {r^2}/r_0^2 {\,f^2})$ with respect to the normalized distance of propagation ξ and pulse with r/r ₀ through a deep plasma channel with non-uniformity n ₂/n _e0 = 0.3 for normalized laser intensities a ₀₀ = 0.65 (2a), a ₀₀ = 0.75 (2b), a ₀₀ = 0.85 (2c), and a ₀₀ = 0.95 (2e).

Fig. 3.

Variation of a normalized pump amplitude a ₀₀ of main laser pulse with respect to plasma channel non-uniformities n ₂/n _e0 for r ₀₀ = 3 (red-dashed line) and r ₀₀ = 5 (blue-line).

III. MODULATIONAL INSTABILITY

A laser pulse can exerts a ponderomotive force on the electrons, ${\vec F_{\rm p}} = e\nabla {{\rm \phi} _{\rm p}}$, where

(21)$${{\rm \phi} _{\rm p}} \approx ({m_{\rm e}}{c^2}/e)a_{00}^2 \big[(1/2{\,f^2})(1 + {r^2}/r_0^2 {\,f^2})\big].$$

We perturb the equilibrium by a plasma wave of potential ϕ with the density perturbation n _p and the velocity perturbation $\vec v$. The plasma wave modifies the beam width parameter to f = 1 + Δf, and the ponderomotive potential to ϕ′_p,

(22)$${\phi ^{\prime}_{\rm p}} \approx \displaystyle{1 \over 2}\displaystyle{{{m_{\rm e}}{c^2}} \over e}a_{00}^2 \left[ {\displaystyle{1 \over 2}(1 - 2\Delta f)\left( {1 + (1 - 2\Delta f)\displaystyle{{{r^2}} \over {r_0^2}}} \right)} \right],$$

where Δf ≪ 1. The equations governing $\vec v$ and n _p are,

(23)$$\displaystyle{{\partial \vec v} \over {\partial t}} = \displaystyle{e \over {{m_{\rm e}}}}\nabla ({\rm \phi} + {{\rm \phi} ^{\prime}_{\rm p}}),$$

and

(24)$$\displaystyle{{\partial {n_{\rm p}}} \over {\partial t}} + \nabla \cdot ({n_0}\vec v) = 0.$$

Taking ∂n _p/∂t of Eq. (24) and using Eq. (23) and Poisson's equation ${\nabla ^2}{\rm \phi} = 4{\rm \pi} {n_{\rm p}}e$, we obtain

(25)$$\displaystyle{{{\partial ^2}{n_{\rm p}}} \over {\partial {t^2}}} + {\rm \omega} _{{\rm p0}}^2 {n_{\rm p}} = - \displaystyle{{{n_{{\rm e0}}}e} \over {{m_{\rm e}}}}\displaystyle{{{\partial ^2}{{{\rm \phi} ^{\prime}}_{\rm p}}} \over {\partial {z^2}}},$$

writing,

(26)$${n_{\rm p}} = {n_{{\rm p0}}} + {n_{{\rm p2}}}{r^2}/r_0^2, $$

Eq. (25) can split into,

(27)$$\displaystyle{{{\partial ^2}{n_{{\rm p0}}}} \over {\partial {{t^{\prime}}^2}}} + {\rm \omega} _{{\rm p0}}^2 {n_{{\rm p0}}} = {n_{{\rm e0}}}{c^2}a_{00}^2 \displaystyle{{{\partial ^2}\Delta f} \over {\partial {z^2}}},$$

and

(28)$$\displaystyle{{{\partial ^2}{n_{{\rm p2}}}} \over {\partial {{t^{\prime}}^2}}} + {\rm \omega} _{{\rm p0}}^2 {n_{{\rm p2}}} = 2{n_{{\rm e0}}}{c^2}a_{00}^2 \displaystyle{{{\partial ^2}\Delta f} \over {\partial {z^2}}},$$

where ∂/∂z = ∂/∂z − (1/c)∂/∂t′. The relevant combination of the permittivity becomes,

(29)$${{\rm \varepsilon} _{{\rm eff}}}(r,z) = {{\rm \varepsilon} _{{\rm eff0}}} + {{\rm \varepsilon} ^{\prime}_2}{r^2},$$

where

(30)$${{\rm \varepsilon} ^{\prime}_2} \equiv - \displaystyle{{{\rm \omega} _{{\rm p0}}^2} \over {{\rm \omega} _0^2 {{\rm \gamma} _0}r_0^2 {\,f^2}}}\Phi - \displaystyle{{{\rm \omega} _0^2} \over {{\rm \omega} _{{\rm p0}}^2 r_0^2}} \displaystyle{{{n_{{\rm p2}}}} \over {{n_{{\rm e0}}}}}.$$

Thus, one can obtain the following perturbation equation in terms of the beam width parameter,

(31)$$\displaystyle{{{\partial ^2}\Delta f} \over {\partial z^{\prime}}} + \left( {\displaystyle{3 \over {R_{\rm d}^2}} - \displaystyle{{{\rm \omega} _{{\rm p0}}^2} \over {{\rm \omega} _0^2 {{\rm \gamma} _0}}}\displaystyle{{{n_2}} \over {{n_{{\rm e0}}}}}\displaystyle{3 \over {r_2^2}}} \right)\Delta f = - \displaystyle{{{\rm \omega} _{{\rm p0}}^2} \over {{\rm \omega} _0^2}} \displaystyle{{{n_{{\rm p2}}}} \over {{n_{{\rm e0}}}r_0^2}}, $$

and ignoring the product of perturbed quantities Δf and n _p2/n _e0. Letting,

(32)$$\Delta f = {a_{\rm f}}exp[ - i({\rm \omega} t - qz^{\prime})],$$

and

(33)$$\displaystyle{{{n_{{\rm p2}}}} \over {{n_{{\rm e0}}}}} = {a_{\rm n}}exp[ - i({\rm \omega} t - qz^{\prime})],$$

and looking for the solutions for which q is complex, we obtain,

(34)$$\left( {{q^2} - \left( {\displaystyle{3 \over {R_{\rm d}^2}} - 3\displaystyle{{{\rm \omega} _{\,p0}^2} \over {{\rm \omega} _0^2 {{\rm \gamma}_0}r_0^2}} \displaystyle{{{n_2}} \over {{n_{{\rm e0}}}}}\displaystyle{{r_0^2} \over {r_2^2}}} \right)} \right){a_f} = \displaystyle{{{\rm \omega} _{{\rm p0}}^2} \over {r_0^2 \omega _0^2}} {a_{\rm n}},$$

(35)$$\left( {{{\rm \omega} ^2} - {\rm \omega} _{{\rm p0}}^2} \right){a_{\rm n}} = - 2a_0^2 {\left( {{\rm \omega} + qc} \right)^2}{a_{\rm f}},$$

which yields the following dispersion relation for the modulational instability of the laser,

(36)$$\eqalign{& \left( {{q^2} - \left( {\displaystyle{3 \over {R_{\rm d}^2}} - 3\displaystyle{{{\rm \omega} _{\,p0}^2} \over {{\rm \omega} _0^2 {{\rm \gamma} _0}r_0^2}} \displaystyle{{{n_2}} \over {{n_{{\rm e0}}}}}\displaystyle{{r_0^2} \over {r_2^2}}} \right)} \right)\left( {{{\rm \omega} ^2} - {\rm \omega} _{{\rm p0}}^2} \right) \cr & = - {{\rm \alpha}[ ^2}{\left( {{\rm \omega} + qc} \right)^2},} $$

where,

(37)$${{\rm \alpha} ^2} = 2\displaystyle{{a_{00}^2 {\rm \omega} _{{\rm p0}}^2} \over {r_0^2 {\rm \omega} _0^2}}. $$

An instability occurs when the quantity within the square root is negative. The spatial growth rate becomes,

(38)$$\Gamma = \displaystyle{{{{\left[ {\left( {{\Omega ^2} - 1 + {{{\rm \alpha} ^{\prime}}^2}} \right)\left( {{{{\rm \alpha} ^{\prime}}^2}{\Omega ^2} - {{{\rm \beta} ^{\prime}}^2}} \right) - {{{\rm \alpha} ^{\prime}}^4}{\Omega ^2}} \right]}^{1/2}}} \over {\left( {{\Omega ^2} - 1 + {{{\rm \alpha} ^{\prime}}^2}} \right)}},$$

where

(39)$${{\rm \alpha} ^{\prime 2} = 2\displaystyle{{a_{00}^2} \over {R_0^2 \Omega _0^2}},} $$

and

(40)$${{\rm \beta} ^{\prime ^2} = \left( {\displaystyle{3 \over {R_{\rm d}^2}} - \displaystyle{3 \over {{{\rm \gamma} _0}\Omega _0^2}} \displaystyle{{{n_2}} \over {{n_{{\rm e0}}}}}\displaystyle{1 \over {R_2^2}}} \right)\left( {{\Omega ^2} - 1} \right).}$$

Here, we non-dimensionalized the above equations with Γ = γc/ω_p0, Ω = ω/ω_p0, Ω₀ = ω₀/ω_p0, and R ₀ = r ₀ω_p0/c, R ₂ = r ₂ω_p0/c. The normalized growth rates of the modulational instability of a main pulse are plotted for shallow (Figs 4a and 4b) and deep plasma channels (Figs 5a and 5b). In both the cases, the growth rate increases very sharply at Ω = 1 and sharply drops to zero on further increase in Ω. For a small laser pump amplitude the growth rate increases with the increase in plasma channel non-uniformity (Fig. 4b). In a deep plasma channel with non-uniformity 0.3, the growth rate of the modulational instability also increases with the normalized laser amplitude (Fig. 5b).

Fig. 4.

The normalized growth rate Γ = γ ₀ω_p0/c of modulational instability as a function of a normalized frequency of the plasma wave Ω = ω/ω_p0 for normalized laser intensity a ₀₀ = 0.3 through different channel non-uniformities n ₂/n _e0 = 0.0 (red-line), n ₂/n _e0 = 0.2 (blue-dashed line), and n ₂/n _e0 = 0.3 (green-long-dashed line). And Figure 4b is the zoom in portion of Figure 4a near the maximum growth rate.

Fig. 5.

The normalized growth rate Γ = γ ₀ω_p0/c of modulational instability as a function of normalized frequency of the plasma wave Ω = ω/ω_p0 through a deep channel with non-uniformity 0.3 for normalized laser intensity a ₀₀ = 0.5 (red-line), a ₀₀ = 0.65 (blue-dashed line), and a ₀₀ = 0.75 (green-line). And Figure 5b is the zoom in portion of Figure 5a near the maximum growth rate.

IV. RESULTS AND DISCUSSSION

We have carried out an analysis of the plasma channel non-uniformity effect on the modulational instability, and found that self-focusing is induced by the relativistic mass non-linearity and the plasma channel non-uniformity. The normalized laser pulse intensity increases with the increase in non-uniformity of the plasma channel such as n ₂/n _e0 = 0.1, 0.2, and 0.3 in Figure 1. However, in deep plasma channels with non-uniformity n ₂/n _e0 = 0.3, the laser pulse intensity increases due to the relativistic mass non-linearity as shown in Figure 2. To maintain the constant shape of the laser pulse, the plasma channel non-uniformity must decrease with the increase of the laser pulse amplitude (Fig. 3). A co-propagating plasma wave perturbation with the laser group velocity produces the axial intensity gradient. Further, the laser pulse derives a plasma wave by exerting a ponderomotive force on the electrons, and develops the modulational instability. The normalized growth rate of the modulation for the main pulse is modified by the plasma channel non-uniformity as well as by the laser intensity. In Figures 4a and 4b, it is shown that in the presence of a plasma channel the growth rate increases with non-uniformity n ₂/n _e0 = 0.0, 0.2, and 0.3. In the presence of deep cavitation n ₂/n _e0 = 0.3, the normalized growth rate also increases with the intensity of the main laser pulse as shown in Figures 5a and 5b. However, the growth rate peak shifts towards the larger value of normalized plasma wave frequency.