Introduction
The interaction of the high-power laser beams with plasma gives rise to a number of non-linear effects (Ganeev et al., Reference Ganeev, Hutchison, Witting, Frank, Okell, Zair, Weber, Redkin, Lei, Roschuk, Maier, L'Opez-Quintas, Martin, Castillejo, Tisch and Marangos2012). Some of these non-linear phenomena includes THz radiation generation (Kumar et al., Reference Kumar, Pandey, Cui and Nahata2011; Vij et al., Reference Vij, Kant and Thakur2019), wakefield acceleration (Ibbotson et al., Reference Ibbotson, Bourgeois, Rowlands-Rees, Caballero, Bajlekov, Walker, Kneip, Mangles, Nagel, Palmer, Delerue, Doucas, Urner, Chekhlov, Clarke, Divall, Ertel, Foster, Hawkes, Hooker, Parry, Rajeev, Streeter and Hooker2010), self-focusing (Aggarwal et al., Reference Aggarwal, Kumar and Kant2016; Kumar et al., Reference Kumar, Aggarwal, Richa and Gill2018; Thakur et al., Reference Thakur, Wani and Kant2019), and harmonic generation (Tripathi et al., Reference Tripathi, Liu, Shao, Eliasson and Sagdeev2009; Vij et al., Reference Vij, Aggarwal and Kant2017; Sharma et al., Reference Sharma, Thakur and Kant2019). The phenomenon of harmonic generation is noteworthy in terms of laser–plasma interaction and has brought remarkable notice due to its various applications. Over the last few years, a great progress in the laser technology has been witnessed in which intensity of the laser beam has been amplified progressively. As the electric field of such lasers is relatively high and this forces electrons in the plasma channel to oscillate with the relativistic energy.
A Gaussian laser expels the electrons radially away from the central axis which is referred to as the electron cavitation. This charge displacement because of the expelled electrons removes ions and forms a channel having density depression on the axis. Hence, under the influence of ultra-intense Gaussian laser pulse, the plasma channel gets depleted from the region of high field to low field, which results in the establishment of the transverse density gradient. Sharma and Sharma (Reference Sharma and Sharma2012) investigated the second-harmonic generation in the presence of the static magnetic field with the extended paraxial ray approximation. They considered both the relativistic and ponderomotive non-linearity simultaneously and concluded that when a laser beam incident on the inhomogeneous magneto-plasma, a density gradient is created which is parallel to a static magnetic field. Further, the plasma wave at the pump wave frequency is caused which leads to the generation of second harmonic.
In most of the laser interactions with homogeneous plasma, it is observed that odd harmonics with laser frequency are produced (Mori et al., Reference Mori, Decker and Leemans1993; Zeng et al., Reference Zeng, Shen, Yu and Xu1996). However, in the presence of density gradient, second harmonics have been noticed. This is because of the laser-induced quiver motion of the electrons across the density gradient, which, in turn, gives rise to the perturbation in electron density. Density perturbation couples with quiver motion of electrons, which produces the source current at the second-harmonic frequency.
In this paper, for the first time, we point out the probability of the second-harmonic generation when the laser beam propagates in the homogeneous plasma under the influence of a planar magnetostatic wiggler with the exponential density profile. With the increase in the value of the positive chirp parameter, the second-harmonic generation enhances. Influence of cyclotron frequency associated with the plasma electrons is also seen in the enhancement of second-harmonic generation. It is due to the fact that dynamics of the oscillating electrons is altered due to the Lorentz force which, in turn, modifies the plasma wave and, hence, results in the efficient second-harmonic generation. As we know that the exponential plasma density ramp is crucial for stronger self-focusing which further enhances the second-harmonic generation. In addition to this chirped pulse laser is considered as it increases the laser–plasma interaction for a longer duration. Also, wiggler magnetic field is introduced here, as it provides the necessary phase-matching condition. Therefore, the combined influence of exponential density ramp, wiggler magnetic field, and chirped pulse laser are considered here for an efficient second-harmonic generation. The construction of the manuscript is as follows: In the section “Theoretical considerations”, non-linear current density and the dispersion relations for the fundamental/pump and the second-harmonic frequencies have been observed. In the section “Second-harmonic generation”, the normalized wave amplitude of second harmonic and its conversion efficiency are derived. Numerical results and conclusions are given in the sections “Result and discussion” and “Conclusion”.
Theoretical considerations
Consider a linearly polarized chirped pulse laser propagating along the z direction under the influence of planar magnetostatic wiggler field 
$\vec{B}_w.$ The wiggler field and vector potential of chirped pulse laser can be written as follows (Esmaeildoost et al., Reference Esmaeildoost, Zolghadr and Jafari2017):
$$\vec{B}_w = B_0(\hat{e}_y\sin (k_wz)),$$
$$\vec{A} = \hat{e}_xA(z,t)\sin (k_1z-{\rm \omega} _1)t,$$where B 0 is the amplitude of the magnetic field, k w = 2π/λ w represents the wave number of the wiggler field, ω1 = ω0(1 + b(ω0t − ω0z/c)) is the frequency of chirped pulse laser, b represents the chirp parameter, c represents the velocity of the light in the vacuum, ω0 is the laser frequency, k 1 = ω0(1 + b(ω0t − ω0z/c))μ1/c is the propagation vectors at frequencies ω0, and μ1 is the refractive index for fundamental wave. Further, following Jha et al. (Reference Jha, Mishra, Raj and Upadhyay2007), penetration of laser beam through the transversely magnetized plasma generates transverse current density at the frequency 2ω0 and acts as the source for the generation of second harmonic. As a result, for frequencies ω0 and 2ω0, the vector potential can be given as follows:
$$\vec{A}_1 = \hat{e}_xA_1\sin (k_1z-{\rm \omega} _1t),$$
$$\vec{A}_2 = \hat{e}_xA_2\sin \lpar {k_2z-2{\rm \omega}_1t} \rpar,$$where A 1 and A 2 are amplitudes of the laser beam and its second harmonic, respectively, k 2 = 2ω0(1 + b(ω0t − ω0z/c))μ2/c is the propagation vector at the frequency 2ω0, and μ2 is the refractive index for the second-harmonic wave.
Further, using Maxwell's equations, we obtain the wave equation as follows:
$$\left( {-\nabla^2 + \displaystyle{1 \over {c^2}}\displaystyle{{\partial^2} \over {\partial t^2}}} \right)A = \displaystyle{{4{\rm \pi}} \over c}J,$$
$$J = -n_eev_e,$$where J is the current density of plasma electrons, n e is the electron density, e is the magnitude of electron charge, and v e represents the electron velocity.
Also, the relativistic equation of motion for the electrons is written as follows:
$$\displaystyle{{\partial ({\rm \gamma} m_{0e}v_e)} \over {dt}} = -e\left[ {E + \displaystyle{1 \over c}v_e \times B_w} \right]-\displaystyle{1 \over {n_e}}\nabla p_e,$$and the continuity equation is
$$\displaystyle{{\partial n} \over {\partial t}} + \vec{\nabla}. (n_e\vec{v}_e) = 0,$$where p e represents the pressure of electrons, and m 0e is electron's rest mass. Further, the continuity equation is 
$\partial n/\partial t + \vec{\nabla}. (n_e\vec{v}_e) = 0,$ where 
${\rm \gamma} = \sqrt {1 + p_e^2 /m_{0e}^2 c^2}$ is the relativistic factor. By substituting Eq. (1) into Eq. (7), velocity in x- and z-directions are given by 
$\partial v_x^{(1)} /\partial t = -e\;E_1/m_{0e} + v_z^{(1)} {\rm \omega} _w\sin (k_wz)$ and 
$\partial v_z^{(1)} /\partial t = -e\;E_z^{(1)} /m_{0e}-v_x^{(1)} {\rm \omega} _w\sin (k_wz)$, respectively, where 
$E_z^{(1)}, \;E_1 = -\partial A_1/\partial t,{\rm \omega} _w = eB_0/m_{0e}c$ are first-order perturbation for the polarization field, the amplitude of the laser field, and the wiggler frequency, respectively. Further, first-order transverse and longitudinal velocities can be given as follows:
$$\eqalign{v_x^{(1)} = &\displaystyle{{ca_1({\rm \omega} _0^2 -{\rm \omega} _p^2 )} \over {({\rm \omega} _0^2 -{\rm \omega} _p^2 -{\rm \omega} _w^2 {\sin} ^2(k_wz))}} \cr &\times \sin (k_wz)\cos (k_1z-{\rm \omega} _0(1 + b({\rm \omega} _0t-{\rm \omega} _0z/c))t),} $$
$$\eqalign{v_z^{(1)} = &-\displaystyle{{ca_1{\rm \omega} _w{\rm \omega} _0} \over {({\rm \omega} _0^2 -{\rm \omega} _p^2 -{\rm \omega} _w^2 {\sin} ^2(k_wz))}} \cr &\times \sin (k_wz)\cos (k_1z-{\rm \omega} _0(1 + b({\rm \omega} _0t-{\rm \omega} _0z/c))t),} $$where a 1 = eA 1/mcω0 is the normalized amplitude of the pump wave, ωP = (4πn(z′)e 2/m 0e)1/2 represents plasma frequency, 
${\rm \omega} _{p0} = 4{\rm \pi}n_0e^2/m_{0e}$, z′ = ω0z/c is the normalized distance of propagation, 
${k}^{\prime}_w = \;k_wc/{\rm \omega}$ is the normalized wiggler wave number. The plasma density profile is considered as n(z′) = n 0exp(z′/d), where d is the adjustable constant, n 0 is the equilibrium electron density, and e is the charge of the electron.
$$\eqalign{J_x^{(1)} ({\rm \omega} _0) = &\displaystyle{{a_1en_0({\rm \omega} _0^2 -{\rm \omega} _{\,p0}^2 \exp ({z}^{\prime}/d))} \over {({\rm \omega} _0^2 -{\rm \omega} _{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar -{\rm \omega} _w^2 {\sin} ^2({{k}^{\prime}}_w{z}^{\prime}))}} \cr &\times \sin ({{k}^{\prime}}_w{z}^{\prime})\sin (k_1z-{\rm \omega} _0(1 + b({\rm \omega} _0t-{z}^{\prime}))t).} $$By using Eqs (5) and (11), non-linear dispersion relation can be obtained as follows:
$$c^2k_1^2 = {\rm \omega} _0^2 -\displaystyle{{({\rm \omega} _0^2 -{\rm \omega} _{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar ){\rm \omega} _{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {({\rm \omega} _0^2 -{\rm \omega} _{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar -{\rm \omega} _w^2 {\sin} ^2({{k}^{\prime}}_wz^{\prime}))}}.$$Further, in the absence of the wiggler field, Eq. (12) reduces to dispersion relation in the linear limit. The refractive index associated with the frequency ω0 can be written as follows:
$$n_1 = \left( {1-\displaystyle{{({\rm \omega}_0^2 -{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar ){\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 ({\rm \omega}_0^2 -{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar -{\rm \omega}_w^2 {\sin}^2({{k}^{\prime}}_w{z}^{\prime}))}}} \right)^{1/2}.$$Similarly, for the frequency 2ω0, we have
$$n_2 = \left( {1-\displaystyle{{({\rm \omega}_0^2 -{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar ){\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {4{\rm \omega}_0^2 (4{\rm \omega}_0^2 -{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar -{\rm \omega}_w^2 {\sin}^2({{k}^{\prime}}_w{z}^{\prime}))}}} \right)^{1/2}.$$By employing perturbation theory and Eq. (8), we can derive the first-order plasma electron density as follows:
$$n^{(1)} = \displaystyle{{a_1cn_0k_1{\rm \omega} _w\sin ({{k}^{\prime}}_wz^{\prime})\cos (k_1z-{\rm \omega} _0(1 + b({\rm \omega} _0t-{z}^{\prime}))t)} \over {({\rm \omega} _0^2 -{\rm \omega} _{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar -{\rm \omega} _w^2 {\sin} ^2({{k}^{\prime}}_w{z}^{\prime}))}}.$$In a similar way, the second-order velocity component can be given as follows:
$$\eqalign{v_x^{(2)} = &\displaystyle{{c^2a_1^2 k_1{\rm \omega} _w\lsqb {{({\rm \omega}_0^2 -{\rm \omega}_p^2 )}^2-\lpar {{\rm \omega}_w^2 {\sin}^2\lpar {k_wz} \rpar } \rpar \lpar {(4{\rm \omega}_0^2 -{\rm \omega}_p^2 )} \rpar } \rsqb } \over {2({\rm \omega} _0^2 -{\rm \omega} _p^2 -{\rm \omega} _w^2 {\sin} ^2(k_wz))(4{\rm \omega} _0^2 -{\rm \omega} _p^2 -{\rm \omega} _w^2 {\sin} ^2(k_wz))}} \cr & \times \sin (k_wz)\sin 2(k_1z-{\rm \omega} _0(1 + b({\rm \omega} _0t-{\rm \omega} _0z/c))t).} $$Following the same steps to calculate the first order n (1), the second-order plasma electron density can be given as follows:
$$\eqalign{n^{(2)} = &\left[ {-\displaystyle{\matrix{a_1^2 c^2n_0k_1^2 [({\rm \omega}_0^2 -{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar )^2- \hfill \cr {\rm \omega}_w^2 \sin^2({{k}^{\prime}}_wz^{\prime})(4{\rm \omega}_0^2 -{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar )] \hfill} \over \matrix{({\rm \omega}_0^2 -{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar -{\rm \omega}_w^2 \sin^2({{k}^{\prime}}_wz^{\prime}))^2(4{\rm \omega}_0^2 \hfill \cr -{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar -{\rm \omega}_w^2 \sin^2({{k}^{\prime}}_wz^{\prime})) \hfill}}} \right] \cr& \times \cos 2(k_1z-{\rm \omega} _0(1 + b({\rm \omega} _0t-{z}^{\prime})).} $$As we know harmonics are driven by the non-linear current density; hence, we can derive second-order equations of the non-linear current density for second harmonic by using 
$J_x^{\lpar 2 \rpar } = -e\lpar {n_0v_x^{\lpar 2 \rpar } + n^{\lpar 1 \rpar }v_x^{\lpar 1 \rpar }} \rpar$ and Eq. (16). As a result, one can have
$$\eqalign{J_x^{(2)} (2{\rm \omega} _0) = &\left[ {\displaystyle{\matrix{3a_1^2 c^2en_0k_1{\rm \omega}_w{\rm \omega}_0^2 ({\rm \omega}_0^2 \hfill \cr -{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar + {\rm \omega}_w^2 \sin^2({{k}^{\prime}}_wz^{\prime})) \hfill} \over \matrix{2({\rm \omega}_0^2 -{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar -{\rm \omega}_w^2 \sin^2({{k}^{\prime}}_wz^{\prime})) \hfill \cr \times (4{\rm \omega}_0^2 -{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar -{\rm \omega}_w^2 \sin^2({{k}^{\prime}}_wz^{\prime})) \hfill}}} \right] \cr &\times \sin ({{k}^{\prime}}_wz^{\prime})\sin 2(k_1z-{\rm \omega} _0(1 + b({\rm \omega} _0t-{z}^{\prime}))t).} $$Second-harmonic generation
By using Eqs (5) and (18), one can derive the expression for the amplitude of the second-harmonic field. We are considering that distance over which ∂a 2(z)/∂z changes considerably is large as compared with the wavelength (∂2a 2(z)/∂z 2 <<k 2∂a 2(z)/∂z) and that A 1 reduces very slightly. Here, a 2(z) represents the normalized amplitude of the second-harmonic wave. The quantity 
$a_1^2$ can be assumed to be independent of z and growth of the second harmonic is given as follows:
$$\eqalign{& a_2(z) = \displaystyle{\matrix{3a_1^2 {\rm \omega} _w\sin ({{k}^{\prime}}_w{z}^{\prime}){\rm \omega} _{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar \left[ \matrix{1-\displaystyle{{\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}\left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)} \over {\left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}-\displaystyle{{{\rm \omega}_w^2 {\sin}^2({{k}^{\prime}}_w{z}^{\prime})} \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)}} \hfill \cr \hfill} \right]^{1/2} \hfill \cr \times \left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}} + \displaystyle{{{\rm \omega}_w^2 {\sin}^2({{k}^{\prime}}_w{z}^{\prime})} \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)\exp \left[ {\displaystyle{{i\Delta kz} \over 2}} \right]\displaystyle{{\sin \left( {\displaystyle{{\Delta kz} \over 2}} \right)} \over {\Delta k}} \hfill} \over \matrix{16c{\rm \omega} _0^2 \left[ \matrix{1-\displaystyle{{\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}\left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)} \over {\left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}-\displaystyle{{{\rm \omega}_w^2 {\sin}^2({{k}^{\prime}}_w{z}^{\prime})} \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)}} \hfill \cr \hfill} \right]^{1/2} \times \left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}-\displaystyle{{{\rm \omega}_w^2 {\sin}^2({{k}^{\prime}}_w{z}^{\prime})} \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)^2 \hfill \cr \times \left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}-\displaystyle{{{\rm \omega}_w^2 {\sin}^2({{k}^{\prime}}_wz^{\prime})} \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right) \hfill}}, \cr &} $$where Δk = 2k 1 − k 2. Further, the second-harmonic conversion efficiency η2(z) can be given as follows:
$${\rm \eta} _2(z) = \displaystyle{{n_1} \over {n_2}}\displaystyle{{{\left \vert {\displaystyle{{\partial a_2} \over {\partial t}}} \right \vert} ^2} \over {{\left \vert {\displaystyle{{\partial a_1} \over {\partial t}}} \right \vert} ^2}}.$$By using Eqs (19) and (20), we obtain the following equation:
$$\eqalign{& {\rm \eta} _2(z) = \displaystyle{\matrix{9a_1^2 {\rm \omega} _w^2 \sin ^2({{k}^{\prime}}_w{z}^{\prime}){\rm \omega} _{\,p0}^4 \exp \lpar {{z}^{\prime}/d} \rpar \left[ {1-\displaystyle{{\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}\left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)} \over {\left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}-\displaystyle{{{\rm \omega}_w^2 {\sin}^2({{k}^{\prime}}_w{z}^{\prime})} \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)}}} \right]^{1/2} \hfill \cr \times \left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}} + \displaystyle{{{\rm \omega}_w^2 {\sin}^2({{k}^{\prime}}_w{z}^{\prime})} \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)\left( {\displaystyle{{\sin \left( {\displaystyle{{\Delta kz} \over 2}} \right)} \over {\Delta k}}} \right)^2 \hfill} \over \matrix{128c^2{\rm \omega} _0^4 \left[ {1-\displaystyle{{\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}\left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)} \over {\left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}-\displaystyle{{{\rm \omega}_w^2 {\sin}^2({{k}^{\prime}}_w{z}^{\prime})} \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)}}} \right]^{1/2} \hfill \cr \times \left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}-\displaystyle{{{\rm \omega}_w^2 {\sin}^2({{k}^{\prime}}_w{z}^{\prime})} \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)^4 \times \left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}-\displaystyle{{{\rm \omega}_w^2 {\sin}^2({{k}^{\prime}}_w{z}^{\prime})} \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)^2 \hfill}} \cr & \cr &}. $$The minimum value of the z for which the value of η is found to be maximum can be given as follows:
$$z = l_c = \displaystyle{\pi \over {\Delta k}},$$where l c represents the plasma length. Further, the maximum second harmonic efficiency is written as follows:
$$\eqalign{& {\rm \eta} _{\max} = \displaystyle{\matrix{9a_1^2 {\rm \omega} _w^2 \sin ^2({{k}^{\prime}}_w{z}^{\prime}){\rm \omega} _{\,p0}^4 \exp \lpar {{z}^{\prime}/d} \rpar \left[ {1-\displaystyle{{\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}\left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)} \over {\left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}-\displaystyle{{{\rm \omega}_w^2 {\sin}^2({{k}^{\prime}}_w{z}^{\prime})} \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)}}} \right]^{1/2} \hfill \cr \times \left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}} + \displaystyle{{{\rm \omega}_w^2 {\sin}^2({{k}^{\prime}}_w{z}^{\prime})} \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)\left( {\displaystyle{1 \over {\Delta k}}} \right)^2 \hfill} \over \matrix{128c^2{\rm \omega} _0^4 \left[ {1-\displaystyle{{\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}\left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)} \over {\left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}-\displaystyle{{{\rm \omega}_w^2 {\sin}^2(k_wz)} \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)}}} \right]^{1/2} \times \left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}-\displaystyle{{{\rm \omega}_w^2 {\sin}^2({{k}^{\prime}}_w{z}^{\prime})} \over {{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)^4 \hfill \cr \times \left( {1-\displaystyle{{{\rm \omega}_{\,p0}^2 \exp \lpar {{z}^{\prime}/d} \rpar } \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}-\displaystyle{{{\rm \omega}_w^2 {\sin}^2(k_w^{\prime} z^{\prime})} \over {4{\rm \omega}_0^2 {(1 + b({\rm \omega}_0t-{z}^{\prime}))}^2}}} \right)^2 \hfill}}. \cr & \cr &} $$Result and discussion
In order to obtain conversion efficiency of the second-harmonic generation, Eq. (21) is solved numerically. Consider plasma irradiated by a 1.06 μm Nd:YAG laser (intensity I 0 ≈ 5 × 1017 W/cm2). Figure 1 depicts the dependence of conversion efficiency (η) upon the normalized distance of propagation z′ for different values of 
${\rm \omega} _{p0}/{\rm \omega} _0 = 0.4,\;0.6,\, {\rm and} \;0.8$ (corresponding electron density values are 1.58 × 1019, 3.01 × 1019, and 6.05 × 1019 cm−3, respectively). Rest of the values are d = 10, b = 0.7, and ωw/ω0 = 0.06. The maximum value of η is found to be 
$0.08\%$ for 
${\rm \omega} _{p0}^{} /{\rm \omega} _0 = 0.8$ at normalized distance z′ = 0.6. It is because, with the density of the plasma channel, the second-harmonic generation shows an oscillatory behavior and peaks at certain plasma density. Lorentz force influences the dynamics of oscillating electrons which, in turn, alters the plasma wave. This, in turn, enhances the second harmonic significantly. Further, it is observed that cyclotron frequency related to the plasma electrons in the presence of the exponential density ramp profile plays a key role in the enhancement of the harmonic generation of the second order. Similar effects were obtained previously in our work (Thakur et al., Reference Thakur, Vij, Sharma and Kant2018) and reported that the conversion efficiency of second-harmonic generation depends on the laser and plasma frequencies.
Dependence of the conversion efficiency (η2) on the normalized propagation distance z′ for different values of ωp0/ω0. Other parameters are d = 10, b = 0.7, and ωw/ω0 = 0.06.
Figure 2 shows the variation of the conversion efficiency (η) with z′ for different values of 
${\rm \omega} _w/{\rm \omega} _0 = 0.02,\;0.04,\,{\rm and} \;0.06.$ Rest of the parameters are same as detailed in Figure 1. It is studied that with an increase in the transverse magnetic field efficiency of the second-harmonic generation enhances considerably. Cyclotron frequency related to plasma electrons under exponential density transition plays a key role in enhancement of the harmonic generation of the second order. Figure 3 shows the conversion efficiency (η) for the various values of the chirped parameter 
$b = 0.3,\;0.5,\;{\rm and} \;0.7.\;$ Rest of parameters are same as taken in Figure 1. The sharp rise in the conversion efficiency 
$\lpar {\approx 0.08\%} \rpar$ of the second-harmonic amplitude is noticed for b = 0.7. This is because of the efficient electron heating by the shorter wavelength component of the chirped pulses which results in higher second-order susceptibility. The results are in good agreement with Jha et al. (Reference Jha, Malviya and Upadhyay2009). Equations describing the evolution of the phase shift and laser spot size were derived to analyze the propagation of the chirped laser pulses in a plasma channel. The enhancement in the conversion efficiency is observed with an increase in the chirp parameter value for positive chirp as shown in Figure 3. The physics behind this is that chirped laser pulse either positive or negative causes in the heating of the electrons which results in the enhancement of the second-harmonic generation as studied by Guo et al. (Reference Guo, Rodriguez and Taylor2001). Further, the influence of the exponential plasma density ramp is observed in making the self-focusing of the laser beam efficient which, in turn, enhances the second-harmonic generation.
Dependence of the conversion efficiency (η2) on the normalized propagation distance z′ for different values of ωw/ω0. Rest of the parameters are same as taken in Figure 1.
Dependence of the conversion efficiency (η2) on the normalized propagation distance z′ for different values of the chirped parameter b. Rest of the parameters are same as taken in Figure 1.
Conclusion
In present work, we investigate the second-harmonic generation of the relativistic self-focusing chirped pulse laser under the influence of the exponential density transition with a planar magnetostatic wiggler. It is observed that conversion efficiency for the second harmonic increases as the wave progresses along the z direction under the exponential density transition. Under the influence of exponential density transition, fundamental laser beam propagates up to larger distance without getting so much divergence, and hence, plasma density ramp plays a major role in enhancing self-focusing. Also, with the increase in the chirp parameter value in the positive direction, the enhancement in the second-harmonic generation is observed. Further, the magnetostatic wiggler helps in enhancing the harmonic generation of the second order. This is due to the fact that dynamics of the oscillating electrons is changed due to the Lorentz force which, in turn, modifies the plasma wave and, hence, results in the efficient second-harmonic generation.
Financial support
This work is supported by the TARE Scheme (Grant No. TAR/2018/000916) of SERB, DST, New Delhi, India.
