2. LASER PULSE COMPRESSION AND SELF-FOCUSING THROUGH A PRE-FORMED RIPPLED PLASMA CHANNEL

Consider a plasma channel with density ripple $ n_q = n_{q0}e^{ik_{q}z}$ (Shibu & Tripathi, Reference Shibu and Tripathi1998), the equilibrium electron density becomes,

(1)$$n\lpar r\comma \; z\rpar = \lpar n_{e0} + n_{q0} e^{ik_q z} \rpar \left({1 + \displaystyle{{n_2 } \over {n_{e0} }}\displaystyle{{r^2 } \over {r{}_2^2 }}} \right)\comma \;$$

where the electron density on channel axis n _e0 ≫ n _q0, plasma channel non-uniformity n ₂/n _e0, and spot size r ₂ are related to the properties of channel (Ashok et al., 2011). The laser pulse propagates through rippled plasma channel with electric field,

(2)$$\vec E _0 = \widehat x A_0 \lpar r\comma \; z\comma \; t\rpar e^{ - i\lpar {\rm \omega} _0 t - k_0 z\rpar } \comma \;$$

where k ₀ = (ω₀/c)η₀, η₀ is the refractive index of the plasma, and A ₀(r, z, t) is the slowly varying complex amplitude of the laser. At z = 0,

(3)$$A_0 \vert _{z = 0}^2 = A_{00}^2 e^{ - r^2 /r_0^2 } e^{ - t^2 /{\rm \tau} _0^2 } \comma \; $$

where r ₀ and τ₀ are the beam radius (in cylindrical coordinates) and pulse width (in time), respectively. And A ₀₀ is the amplitude of the beam at axis. At z > 0, laser beam amplitude is modified as,

(4)$$\vert A_0 \vert ^2 = \displaystyle{{A_{00}^2 } \over {g{\rm }f^2 }}e^{ - r^2 /r_0^2 f^2 } e^{ - t^2 /{\rm \tau} _0^2 g^2 } \comma \;$$

where f and g represent the beam width parameter (in space) and pulse width parameter (in time), respectively. The electron response to laser pulse from momentum equation becomes,

(5)$$m_e {\rm \gamma} \displaystyle{{\partial \vec {\rm \upsilon} _0 } \over {\partial t}} = - e\vec E _0 \comma \; $$

where −e and m _e are electronic charge and mass, respectively, and relativistic gamma factor $ \gamma=(1 + (a^{2}_{00}e^{-r^{2}/r_{0}^{2}\,f^{\,2}/}e^{-t^{2}/\tau_{0}^{2}g^{2}})/2g\,f^{2})^{1/2} $ is described in terms of noramalized laser amplitude a ₀₀² = |e A ₀₀/m _eω₀c|². Here we also have neglected $\lpar \vec {\rm \upsilon} _{0} \times \vec B _{0} \rpar $, because of its twice high frequency. The electron current density at the main laser frequency $\vec J_0 = -n\lpar r\rpar e\vec {\rm \upsilon} _0$ gives,

(6)$$\partial _t \vec J _0 = - n\lpar r\rpar e\partial _t \vec {\rm \upsilon} _0 = n\lpar r\comma \; z\rpar \displaystyle{{e^2 \vec E _0 } \over {m_e {\rm \gamma} }}.$$

The wave equation governing the electric field of the pump laser beam in plasma can be drived from Maxwell's equations. Using Maxwell's equations in MKS units,

(7)$$\nabla ^2 \vec E _0 - \nabla \lpar \nabla \cdot \vec E _0 \rpar - {\rm \mu} _0 \displaystyle{{\partial \vec J _0 } \over {\partial t}} - \displaystyle{1 \over {c^2 }}\displaystyle{{\partial ^2 \vec E _0 } \over {\partial t^2 }} = 0 $$

where $c = 1/\sqrt {{\rm \mu} _0 {\rm \varepsilon} _0 } $ is the speed of light in free space. For a transverse electromagnetic fields (such as linear polarized) $\nabla \cdot \vec E _0 = 0$. Using Eq. (1) and Eq. (6) into wave equation Eq. (7).

(8)$$\eqalign{& 2ik_0\displaystyle{\partial \over {\partial z^\prime }}A_0 + \nabla _ \bot ^2 A_0 + {\rm \beta} \displaystyle{\partial \over {\partial {\rm \tau} ^2 }}A_0 \cr &\quad - \left[{k_0^2 - \displaystyle{{{\rm \omega} _0^2 } \over {c^2 }}\left({1 - \displaystyle{{{\rm \omega} _{\,p0}^2 /{\rm \gamma} } \over {{\rm \omega} _0^2 }}\left({1 + \displaystyle{{n_2 } \over {n_{e0} }}\displaystyle{{r^2 } \over {r_2^2 }}} \right)} \right)} \right]A_0 = 0\comma} $$

where ${\rm \beta} = \left({1/c^2 } \right)\sqrt {1 - {\rm \varepsilon} _{e f f0}/{\rm \varepsilon} _{e f f0}} $, and transforming the variables z and t to z = z and τ = (t − z/υ_g), respectively. Here the parallel variation of the amplitude A ₀ in terms of z is assumed to be slower than the transverse one (in r), therefore terms in ∂²/∂z′∂τ and ∂²/∂z′² can be neglected. The third term in Eq. (8) is known as the group velocity dispersion (GVD) term and it results in pulse compression when combined with (balanced by) nonlinearity. One can expand γ by using Taylor's expansion,

(9)$$\displaystyle{1 \over {\rm \gamma} }\lpar r\comma \; t\rpar = \displaystyle{1 \over {{\rm \gamma} _0 }}\left({1 + \displaystyle{{a_{00}^2 r^2 /4r_0^2 g{\rm }f^{\rm 4} } \over {1 + a_{00}^2 /2g{\rm }f^{\rm 2} }} + \displaystyle{{a_{00}^2 t^2 /4{\rm \tau} _0^2 g^3 {\rm }f^{\rm 2} } \over {1 + a_{00}^2 /2g{\rm }f^{\rm 2} }}} \right)\comma \; $$

where γ₀ = (1 + a ₀₀²/2)^1/2 is the value of γ on the axis. We get the total dielectric constatnt,

(10)$${\rm \varepsilon} \lpar r\comma \; z\comma \; t\rpar = {\rm \varepsilon} _{ef\,{\rm }f0} + {\rm \varepsilon} _{2r} r^2 + {\rm \varepsilon} _{2t} t^2 \comma \;$$

where

(11)$$\eqalign{& {\rm \varepsilon} _{2r} = \displaystyle{{ - {\rm \omega} _p^2 } \over {{\rm \omega} _0^2 r_0^2 {\rm \gamma} _0 }}\Phi _{2r} \comma \; \Phi _{2r} = \left({\displaystyle{{n_2 } \over {n_{e0} }}\displaystyle{{r_0^2 } \over {r_2^2 }} + \displaystyle{{a_{00}^2 } \over {4g{\rm }f^4 \left({1 + a_{00}^2 /2g{\rm }f^{\rm 2} } \right)}}} \right)\comma \; \cr & {\rm \varepsilon} _{2t} = - \displaystyle{{{\rm \omega} _p^2 } \over {{\rm \omega} _0^2 {\rm \tau} _0^2 {\rm \gamma} _0 }}\Phi _{2t} \comma \; \, \, \, \Phi _{2t} = \displaystyle{{a_0^2 } \over {4g^3 {\rm }f^2 \left({1 + a_{00}^2 /2g{\rm }f^{\rm 2} } \right)}}.} $$

and

And using $k_0 = \left({{\rm \omega} _0 /c} \right)\sqrt {{\rm \varepsilon} _{ef f 0}} $, we get the wave equation Eq. (9),

(12)$$\left({2ik_0 \displaystyle{\partial \over {\partial z^\prime }} + \nabla _ \bot ^2 + {\rm \beta} \displaystyle{{\partial ^2 } \over {\partial {\rm \tau} ^2 }}} \right)A_0 + \displaystyle{{{\rm \omega} _0^2 } \over {c^2 }}\left({{\rm \varepsilon} _{2r} r^2 + {\rm \varepsilon} _{2t} t^2 } \right)A_0 = 0.$$

This equation describes the propagation of laser pulse through the plasma with relativistic mass nonlinearity and plasma channel non-uniformity. In the simplified analytical model, we take into account the nonlinearity by relativistic mass and plasma channel non-uniformity, and neglect electron density perturbation. Eq, (12) is a nonlinear Schrödinger equation, where third term in the first bracket representing the dispersion broadening and second bracket term representing the nonlinear compression. When the two effects balance each other, one obtains a coherent structure in the form of a solitary wave. Now, following Akhmanov et al. (1996) and Sodha et al. (Reference Sodha, Ghatak and Tripathi1974) to evaluate the pulse evolution we introduce an eikonal approximation, A ₀ = A ₀₀exp(iS), where A ₀₀(r, z′, τ) and S(r, z′, τ) are real functions and separate out the real and imaginary parts of the resulting equations,

(13)$$\eqalign{& - 2k_0 \left({\displaystyle{{\partial S} \over {\partial z^\prime }}A_{00} } \right)+ \left[{\displaystyle{1 \over r}\left({\displaystyle{{\partial A_{00} } \over {\partial r}}} \right)+ \displaystyle{{\partial ^2 A_{00} } \over {\partial r^2 }} - A_{00} \left({\displaystyle{{\partial S} \over {\partial r}}} \right)^2 } \right]\cr & + {\rm \beta} \left[{\displaystyle{{\partial ^2 A_{00} } \over {\partial {\rm \tau} ^2 }} - A_{00} \left({\displaystyle{{\partial S} \over {\partial {\rm \tau} }}} \right)^2 } \right]+ \displaystyle{{{\rm \omega} _0^2 } \over {c^2 }}\left({{\rm \varepsilon} _{2r} r^2 + {\rm \varepsilon} _{2{\rm \tau} } {\rm \tau} ^2 } \right)A_{00} = 0\comma \; }$$

and

(14)$$\eqalign{& k_0 \displaystyle{{\partial {\rm }A_{00}^2 } \over {\partial z^\prime }} + \left({\displaystyle{{\partial ^2 S} \over {\partial r^2 }} + \displaystyle{1 \over r}\displaystyle{{\partial S} \over {\partial r}}} \right)A_{00}^2 + \displaystyle{{\partial A_{00}^2 } \over {\partial r}}\displaystyle{{\partial S} \over {\partial r}} \cr & + {\rm \beta} \left[{\displaystyle{{\partial A_{00}^2 } \over {\partial {\rm \tau} }}\displaystyle{{\partial S} \over {\partial {\rm \tau} }} + A_{00}^2 \displaystyle{{\partial ^2 S} \over {\partial r^2 }}} \right]= 0.} $$

The coupled set of equations for A ₀₀² and S are solved in the paraxial approximation by expanding S only to the first power of r ² and r ² such as,

(15)$$S\lpar r\comma \; z^\prime \comma \; {\rm \tau} \rpar = S_0 \lpar z^\prime \rpar + S_{2r} \lpar z^\prime \rpar \displaystyle{{r^2 } \over {r_0^2 }} + S_{2r} \lpar z^\prime \rpar \displaystyle{{{\rm \tau} ^2 } \over {{\rm \tau} _0^2 }}\comma \; $$

where function f(z′) and g(z′) are,

(16)$$S_{2r} \lpar z^\prime \rpar = \displaystyle{1 \over {2f}}\displaystyle{{\partial f} \over {\partial z^\prime }}\quad {\rm and}\quad S_{2r} \lpar z^\prime \rpar = \displaystyle{1 \over {2g}}\displaystyle{{\partial f} \over {\partial z^\prime }}\comma \; $$

and S ₀(z′) is the axial phase. We obtain the beam width and pulse width parameter equations by collecting coefficient of r ² and r ² in terms of dimensionless distance ξ = z/R _d,

(17)$$\displaystyle{{\partial ^2 f} \over {\partial {\rm \xi} ^2 }}{\rm = }\displaystyle{1 \over {\,f^3 }} - \displaystyle{{{\rm \omega} _{\,p0}^2 \Phi _{2r} } \over {{\rm \gamma} _0 {\rm \omega} _0^2 }}\displaystyle{{R_d^2 } \over {r_0^2 }} f\comma \; $$

and

(18)$$\displaystyle{{\partial ^2 g} \over {\partial {\rm \xi} ^2 }}{\rm = }\displaystyle{{{\rm \beta} ^2 } \over {g^3 }}\displaystyle{{r_0^4 } \over {r_0^4 }} - {\rm \beta} \displaystyle{{{\rm \omega} _{\,p0}^2 \Phi _{2 t} } \over {{\rm \gamma} _0 {\rm \omega} _0^2 }}\displaystyle{{R_d^2 } \over {r_0^2 }}g\comma \;$$

where R _d = k ₀r ₀² is the diffraction length, the first and second term in the right-hand side of above Eq. (17) contribute to the the diffraction divergence and nonlinear self-focusing, respectively. Coupled equations contain nonlinear corrections by relativistic mass and plasma channel non-uniformity. So the self-focusing and compression of main pulse can be controlled by the plasma channel non-uniformity and laser intensity. The boundary conditions at ξ = z′ = 0 for an initially plane wavefront are f = g = 1, ∂f/∂ξ = ∂g/∂ξ = 0 and S ₀ = 0. In Figures 1 and 2, we plotted the variation of f and g with respect to ξ for main laser pulse frequency ω₀ = 2.8195 × 10¹⁴/s.⁻¹, spot size r ₀ = 14 µm, τ₀ = 10 ps and normalized pump amplitude a ₀₀ = e |E00|/m _eω₀c = 0.3. The other parameters are n _e0 = 0.85 × 10²⁴ m⁻³, ω_p0/ω₀ = 0.18, and the spot size of second laser beam, r ₂(>r ₀) = 16 µm. The main laser pulse shows the periodic focusing (Fig. 1) and compression (Fig. 2) inside the plasma channels, it increases with increasing density non-uniformity of the plasma channels with n ₂/n _e0 = 0.1, 0.2, and 0.3. For the main laser pulse with different intensities, a ₀₀ = 0.65, 0.75, 0.85, and 0.95 through a deep plasma channel with non-uniformity n ₂/n _e0 = 0.3, the variation of f and g with respect to ξ is plotted in Figures 3 and 4, and other plasma paremeters are the same as (Fig. 1). The Self-focusing behavior for different intensities show a similarity due to the ineffectiveness of intensity dependent nonlinearity (Fig. 3), however, pulse compression increases with the main laser intensity (Fig. 4).

Fig. 1.

(Color online) Variation of beam width parameter f with respect to the normalized propagation distance ξ = z′/R _d for a normalized laser amplitude a ₀₀ = 0.3 through different plasma channels with non-uniformities n ₂/n _e0 = 0, 0.1, 0.2 and 0.3. The other parameters are ω₀ = 2.8195 × 10¹⁴ s.⁻¹, r ₀ = 14 µm, n _e0 = 0.85 × 10²⁴ m⁻³, τ₀ = 10 ps and r ₂ = 16 μm.

Fig. 2.

(Color online) Variation of temporal parameter g with the normalized propagation distance ξ = z/R _d for a normalized laser amplitude a ₀₀ = 0.3 in different plasma channels for the parameters same as in Figure 1.

Fig. 3.

(Color online) Variation of beam width parameter f with the normalized propagation distance ξ = z/R _d for different laser amplitudes a ₀₀ = 0.65, 0.75, 0.85, and 0.95 through a deep plasmas channel with non-uniformity n ₂/n _e0 = 0.3. The other parameters are ω₀ = 2.8195 × 10¹⁴ s.⁻¹, r ₀ = 14 µm, n _e0 = 0.85 × 10²⁴ m⁻³, τ₀ = 10ps and r ₂ = 16 µm.

Fig. 4.

(Color online) Variation of temporal parameter g with the normalized propagation distance ξ = z/R _d for different laser amplitudes a ₀₀ = 0.65, 0.75, 0.85, and 0.95through a deep plasma channel with non-uniformity n ₂/n _e0 = 0.3 for the same parameters same as in Figure 3.

3. THIRD HARMONIC GENERATION

Laser pulse exerts a pondermotive force on electrons at second harmonic (2ω₀, 2k ₀),

(19)$$\vec F _{2{\rm \omega} _0 \comma 2k_0} = - \displaystyle{e \over 2}\left({\vec {\rm \upsilon} _0 \times \vec {\rm \beta} _0 } \right)= \displaystyle{{e^2 } \over {2m_e i{\rm \gamma} {\rm \omega} _0 c}}\hat z {\rm \eta} _0 A_0^2 \left({r\comma \; z\comma \; t} \right)e^{ - 2i\Psi } \comma \; $$

where Ψ = (ω₀t − k ₀z). Ponderomotive force $\vec F _{2{\rm \omega} _0 \comma 2k_0} $ is primarily along $\hat z$ direction and its electron response will be only in the z-direction. The electron oscillatory velocity due to ponderomotive force becomes,

(20)$$\vec {\rm \upsilon} _{_{2{\rm \omega} _0 } }^{NL} = \displaystyle{{a_{00}^2 c} \over {4{\rm \gamma} }}\hat z {\rm \eta} _0 e^{ - 2i\Psi } .$$

This oscillatory velocity couples with ripple density n _q, through the equation of continuity to produce a density perturbation at (2ω, 2k ₀ + k _q),

(21)$$n_{2{\rm \omega} _0 \comma 2k_0 + k_q } = \displaystyle{1 \over 2}n\lpar r\rpar \lpar 2\vec k _0 + k_q \rpar \cdot \displaystyle{{\vec {\rm \upsilon} _{2{\rm \omega} _0 \comma 2k_0 } } \over {2{\rm \omega} _0 }} $$

(22)$$\approx - n_{q0} \left({1 + \displaystyle{{n_2 } \over {n_{e0} }}\displaystyle{{r^2 } \over {r_2^2 }}} \right)\displaystyle{{a_{00}^2 } \over {8{\rm \gamma} }}{\rm \eta} _0 e^{ - i\lsqb 2{\rm \omega} _0 t - \lpar 2k_0 + k_q \rpar z\rsqb } .$$

The perturbation Eq. (22) beats with electron velocity Eq. (5) to give the third harmonic nonlinear current density at (3ω₀, 3k ₀ + k _q),

(23)$$\eqalign{ \vec J _{3{\rm \omega} _0 \comma 3k_0 + k_q }^{NL} & = - \displaystyle{e \over 2}n_{2{\rm \omega} _0 \comma 2k_0 + k_q } \vec {\rm \upsilon} _{{\rm \omega} _0 \comma k_0 } \cr & = - \hat x \displaystyle{i \over {16{\rm \gamma} ^2 }}n_{q0} eca_{00}^3 {\rm \eta} _0 e^{ - i\lpar 3{\rm \omega} _0 t - \lpar 3k{}_0 + k_q \rpar z\rpar } .} $$

The linear current density due to the self-consistent second harmonic field, $\vec E _{3{\rm \omega} _0 } $ is

(24)$$\vec J _{3{\rm \omega} _0 }^L = - \displaystyle{{n\lpar r\rpar e^2 \vec E _{3{\rm \omega}_{0} } } \over {3im_e {\rm \omega} _0 }} \approx - n_{e0} \left({1 + \displaystyle{{n_2 } \over {n_{e0} }}\displaystyle{{r^2 } \over {r_2^2 }}} \right)\displaystyle{{e^2 \vec E _{3{\rm \omega} _0 } } \over {3im_e {\rm \omega} _0 }}\comma \;$$

and the wave equation for third harmonic field,

(25)$$\nabla ^2 \vec E _{3{\rm \omega} _0 } - \nabla \left({\nabla \cdot \vec E _{3{\rm \omega} _0 } } \right)- {\rm \mu} _0 \displaystyle{{\partial \vec J _{_{3{\rm \omega} _0 } }^L } \over {\partial t}} - \displaystyle{1 \over {c^2 }}\displaystyle{{\partial ^2 \vec E _{3{\rm \omega} _0 } } \over {\partial t^2 }} = {\rm \mu} _0 \displaystyle{{\partial \vec J _{_{3{\rm \omega} _0 } }^{NL} } \over {\partial t}}. $$

Using Eq. (22) and Eq. (24) into Eq. (22) wave equation becomes,

(26)$$\eqalign{& \nabla ^2 \vec E _{3{\rm \omega} _0 } + \displaystyle{{{\rm \omega} _{\,p0}^2 } \over {c^2 }}\left({1 + \displaystyle{{n_2 } \over {n_{e0} }}\displaystyle{{r^2 } \over {r_2^2 }}} \right)\vec E _{3{\rm \omega} _0 } - \displaystyle{1 \over {c^2 }}\displaystyle{{\partial ^2 \vec E _{3{\rm \omega} _0 } } \over {\partial t^2 }} \cr &\quad = \widehat x S_3 e^{ - i\lpar 3{\rm \omega} _0 t - \lpar 3k_0 + k_q \rpar z\rpar } \comma \; }$$

where S ₃ = −3n _q0eω₀a ₀₀³η₀/16ɛ₀cγ². We may write $\vec E _{3{\rm \omega} _0 } = \widehat x A_{3{\rm \omega} _0 } \lpar r\comma \; z\comma \; t\rpar e^{ - \lpar 3{\rm \omega} _0 t - k_3 z\rpar } $ and in terms of slowly varying laser amplitude by using the WKB approximation. Transforming the variables z and t to z and t′ = (t − z/υ_g3ω0), respectively, and using third harmonic group velocity, υ_g3ω0 = k ₃c ²/3ω₀,

(27)$$\eqalign{& 2ik_3 \displaystyle{{\partial A_{3{\rm \omega} _0 } } \over {\partial z^\prime }} + \nabla _{^ \bot }^2 A_{3{\rm \omega} _0 } + \left({\displaystyle{{9{\rm \omega} _{^0 }^2 - {\rm \omega} _{^{\,p0} }^2 /{\rm \gamma} _0 } \over {c^2 }} - k_3^2 } \right)A_{3{\rm \omega} _0 } \cr & - \displaystyle{{{\rm \omega} _{^{\,p0} }^2 } \over {c^2 }}\displaystyle{{n_2 } \over {n_{e0} }}\displaystyle{{r^2 } \over {r_2^2 }}A_{3{\rm \omega} _0 } = S_3 e^{ - i\Delta z^\prime } \comma \; } $$

where Δ = (k ₃ − 3k ₀ − k _q). We can also write the above equation as,

(28)$$2ik_3 r_3^2 \displaystyle{{\partial A_{3{\rm \omega} _0 } } \over {\partial z^{\prime} }} + \displaystyle{{\partial ^2 A_{3{\rm \omega} _0 } } \over {\partial {\rm \rho} _3^2 }} + \displaystyle{1 \over {{\rm \rho} _3 }}\displaystyle{{\partial A_{3{\rm \omega} _0 } } \over {\partial {\rm \rho} _3 }} + \left({{\rm \lambda} _3 - {\rm \rho} _3^2 } \right)A_{3{\rm \omega} _0 } = S_3 r_3^2 e^{ - i\Delta z^\prime } \comma \; $$

where r ₃² = ((c ²r ₂²n _e0)/n ₂ω_p0²)^1/2 and ρ₃ = r/r ₃. To solve the above equation, we express A _3ω0 as,

(29)$$A_{3{\rm \omega} _0 } = A_{3{\rm \omega} _0 } \lpar z^\prime \rpar \Psi _n \lpar {\rm \rho} _3 \rpar \comma \; $$

where Ψ_n (ρ₃) are the eigen-functions satisfying,

(30)$$\displaystyle{{\partial ^2 \Psi _n } \over {\partial {\rm \rho} _3^2 }} + \displaystyle{1 \over {{\rm \rho} {}_3}}\displaystyle{{\partial \Psi _n } \over {\partial {\rm \rho} _3 }} + \left({{\rm \lambda} _3 - {\rm \rho} _3^2 } \right)\Psi _n = 0\comma \; $$

with λ₃ = (2n + 1). This equation gives an eigen function, $ A_{{3\omega}_{0}} = \rho_3^n e^{- \rho_3^{2/2}}$. For n = 0 mode, the eigen function $ \Psi_n (\rho_3) = e^{- \rho_3^{2/2}}$ is Gaussian and eigen value λ₃ = 1,

(31)$$k_3^2 = \displaystyle{1 \over {c^2 }}\left({9{\rm \omega} _0^2 - \displaystyle{{{\rm \omega} _{\,p0}^2 } \over {{\rm \gamma} _0 }} - \displaystyle{{c^2 } \over {r_{{}_3}^2 }}} \right).$$

Now substituting the above solution to the third harmonic wave equation and multiplying the resulting equation by Ψ* and integrating from −∞ to ∞ one obtains,

(32)$$\displaystyle{{\partial A_{3{\rm \omega} _0 } } \over {\partial z^\prime }} = \displaystyle{{S_3 } \over {2ik_3 }}e^{ - i\Delta z^\prime } . $$

For a Gaussian pulse A ₀² = (A ₀₀²/2g f ²)e ^{−r 2/r 2₀f 2}e ^{−τ2/τ₀2g 2} and γ = [1 + (a ₀₀²/2g f ²)e ^{−r 2/r ₀2f 2}e ^{−τ2/τ₀2g 2}]^1/2 are functions of r and t. For a given k _q, one cannot have phase matching (Δ = 0) for third harmonic generation all r and t (Liu & Tripathi, Reference Liu and Tripathi2008). If one matches the ripple wave number at the peak of the laser pulse at ‘r = 0’ and ‘t = 0’ i.e.,

(33)$$k_q = \lpar k_3 - 3k_0 \rpar = \displaystyle{4 \over 3}\displaystyle{{{\rm \omega} _{\,p0}^2 } \over {{\rm \omega} _0^2 {\rm \gamma} _0 }}\displaystyle{{{\rm \omega} _0 } \over c}\comma \; $$

where γ₀ = (1 + a ₀₀²/2)^1/2, then at other r and t,

(34)$$\Delta = k_q \left({\displaystyle{{{\rm \gamma} _0 } \over {{\rm \gamma} \lpar r\comma \; t\rpar }}\displaystyle{{{\rm \omega} _p^2 \lpar r\comma \; t\rpar } \over {{\rm \omega} _{\,p0}^2 }} - 1} \right). $$

Using Eq. (34) into Eq. (32) and after integrating with perfect to z′, we get a third harmonic amplitude,

(35)$$\eqalign{\vert a_{3{\rm \omega} _0 } \vert ^2 & = \left({\displaystyle{1 \over {32}}\displaystyle{{n_{q0} } \over {n_{e0} }}\displaystyle{{{\rm \omega} _0 } \over {k_3 c}}\displaystyle{{{\rm \omega} _{\,p0}^2 } \over {{\rm \omega} _0^2 }}\displaystyle{{a_{00}^3 } \over {{\rm \gamma} ^2 }}} \right)^2 k_0^2 \cr &\quad \times \displaystyle{{Sin^2 \left[{\displaystyle{{k_q } \over 2}\left({\displaystyle{{r_0 } \over {r\lpar r\comma \; t\rpar }}\displaystyle{{{\rm \omega} _p^2 \lpar r\comma \; t\rpar } \over {{\rm \omega} _{\,p0}^2 }} - 1} \right)z^\prime} \right]} \over {\left[{\displaystyle{{k_q } \over 2}\left({\displaystyle{{r_0 } \over {r\lpar r\comma \; t\rpar }}\displaystyle{{{\rm \omega} _p^2 \lpar r\comma \; t\rpar } \over {{\rm \omega} _{\,p0}^2 }} - 1} \right)} \right]^2 }}\comma \; } $$

where the third harmonic normalized amplitude is normalized as, |a _3ω0|² = |eA _3ω0/3m _eω₀c|². For

(36)$$z^\prime < \displaystyle{2 \over {k_q \left({\displaystyle{{{\rm \gamma} _0 } \over {{\rm \gamma} \lpar r\comma \; t\rpar }}\displaystyle{{{\rm \omega} _p^2 \lpar r\comma \; t\rpar } \over {{\rm \omega} _{\,p0}^2 }} - 1} \right)}}\comma \;$$

we get the normalized third harmonic amplitude,

(37)$$\vert a_{3{\rm \omega} _0 } \vert ^2 \approx \left({\displaystyle{1 \over {32}}\displaystyle{{n_{q0} } \over {n_{e0} }}\displaystyle{{{\rm \omega} _0 } \over {k_3 c}}\displaystyle{{{\rm \omega} _{\,p0}^2 } \over {{\rm \omega} _0^2 }}k_0 z^\prime} \right)^2 \displaystyle{{a_{00}^6 } \over {{\rm \gamma} ^4 }}. $$

The total power P _3ω0 of the third harmonic across the transverse cross section at any value of z′ can be obtained by integrating a _3ω0a _3ω0* over r from ‘0’ to ∞ (Sodha et al., Reference Sodha, Faisal and Verma2009) and t from −∞ to ∞ (Liu & Tripathi, Reference Liu and Tripathi2008) i.e.,

(38)$$P_{3{\rm \omega} _0 } = \displaystyle{{c{\rm \eta} _{3{\rm \omega} _0 } {\rm \varepsilon} _0 } \over 2}\vint_0^\infty {} \vint_{ - \infty }^\infty {} a_{3{\rm \omega} _{\rm 0} } a_{_{3{\rm \omega} _{\rm 0} } }^{\ast} 2{\rm \pi} rdrdt\comma \;$$

and initialy pump power at z = 0 and t = 0, P _ω0 = (η_ω0/2)πr ₀²a ₀₀². So we get the conversion efficiency of third harmonic across the transverse cross section,

(39)$$\eqalign{ \displaystyle{{P_{3{\rm \omega} _0 } } \over {P_{{\rm \omega} _0 } }} & = \left({\displaystyle{1 \over {96}}\displaystyle{{n_{q0} } \over {n_{e0} }}\displaystyle{{{\rm \omega} _{\,p0}^2 } \over {{\rm \omega} _0^2 }}k_0 R_d } \right)^2 a_{00}^4 {\rm \xi} ^2 \cr &\quad \times \left({\displaystyle{1 \over {{\rm \gamma} _0^4 }} + \displaystyle{{a_{00}^2 /4g{\rm }f^4 } \over {{\rm \gamma} _0 \left({1 + a_{00}^2 /2g{\rm }f^2 } \right)^{5/2} }} + \displaystyle{{a_{00}^2 /8g^5 {\rm }f^2 } \over {{\rm \gamma} _0 \left({1 + a_{00}^2 /2g{\rm }f^2 } \right)^{5/2} }}} \right).}$$

One can straight forwardly obtain from the dependence of the third harmonic power on beam width parameter f, pulse width parameter g and intensity of the fundamental laser beam, and other plasma parameters.

We numerically plotted the energy conversion ratio P _3ω0/P _1ω0 with respect to the normalized propagation distance ξ = z′/R _d in different plasma channels in Figure 5. The presence of plasma channel nor-uniformity guide the main laser pulse and increase the self-focusing in Figure 1 and compression Figure 2, which further increase the energy conversion ratio P _3ω0/P _1ω0 with distances of propagation. In Figure 6, the presense of a nonuniform deep plasma channel n ₂/n _e0 = 0.3 focusing is independent of the laser intensity. However, the pulse compression and the energy conversion ratio increase with the intensity of the main laser pulse.

Fig. 5.

(Color online) Variation of energy conversion ratio ‘η = P _3ω0/P _1ω0’ with respect to the normalized propagation distance ‘ξ = z′/R _d’ for a normalized laser amplitude a ₀₀ = 0.3 through different plasma channels with non-uniformities n ₂/n _e0 = 0, 0.2 and 0.3. The other parameters are ω₀ = 2.8195 × 10¹⁴Sec⁻¹, r ₀ = 14μm, n _e0 = 0.85 × 10²⁴ m⁻³, τ₀ = 10ps and r ₂ = 16μm.

Fig. 6.

(Color online) Variation of energy conversion ratio ‘P _3ω0/P _1ω0’ with respect to the normalized propagation distance ‘ξ = z′/R _d’ for different laser amplitudes a ₀₀ = 0.65, 0.85 and 0.95 through a deep plasma channels with non-uniformity n ₂/n _e0 = 0.3. The other parameters are ω₀ = 2.8195 × 10¹⁴Sec⁻¹, r ₀ = 14μm, n _e0 = 0.85 × 10²⁴ m⁻³, τ₀ = 10ps and r ₂ = 16μm.