Analytical formulation

Consider two right circularly polarized CGBs with different frequencies (ω_j+, j = 1, 2) and wave vectors (k _j+) propagating in the collisionless magnetoplasma. We consider that the external applied static magnetic field (B ₀) is along to the propagation direction of the beams, i.e., in the z-direction. The complex amplitude of the laser beams is given by the following equation:

(1)$$(A_{\,j^{+}} )_{z = 0} = \displaystyle{{E_{0j^{+}}} \over 2}\exp \left( {\displaystyle{{b^2} \over 4}} \right)\left[ {\exp \left\{ {-{\left( {\displaystyle{r \over {r_{\,j^{+}}}} + \displaystyle{b \over 2}} \right)}^2} \right\} + \exp \left\{ {-{\left( {\displaystyle{r \over {r_{\,j^{+}}}} -\displaystyle{b \over 2}} \right)}^2} \right\}} \right]$$

where “+” sign denotes the right circular mode of propagation, E _0j+ is the initial amplitude of the CGBs, r is the radial coordinate of the cylindrical system, r _j+ is the initial width of the laser beams, and b is the decentred parameter. The intensity profile of a CGLB is shown in Figure 1. Such profile of laser beam having specific features, namely steep gradient in the distribution of laser intensity and a wider cross-section relative to the other laser beam profiles, which leads to stronger pondermotive force and higher power THz radiation.

Fig. 1.

Intensity profile of a CGLB with normalized transverse distance (r/r ₀) for different values of b. Here, b = 0 corresponds to the Gaussian laser beam.

Effective dielectric constant of the plasma

The dielectric constant $({{\rm \varepsilon} _{0j+}})$ of plasma is given by the following equation:

(2)$${\rm \varepsilon} _{0j+} = \left[ {1-\displaystyle{{{\rm \omega} _{\,p0}^2} \over {{\rm \omega} _{\,j+}^2 (1-\Omega _{\,j+} )}}} \right]\comma \;\;\; \Omega _{\,j+} = \displaystyle{{{\rm \omega} _{ce}} \over {{\rm \omega} _{\,j+}}} $$

where ω_p0 = (4πn ₀e ²/m ₀)^1/2 and ω_ce = (eB ₀/m ₀c) are the plasma frequency and cyclotron frequency, respectively, e is the electronic charge, m ₀ is the rest mass of the electron, and n ₀ is the electron density of the plasma in the absence of laser beam. The laser fields impart oscillatory velocity $(\vec{\rm \upsilon} _{j +} )$ to plasma electrons at the beat frequency (ω₁ − ω₂) and wave number (k ₁ − k ₂) given by the following expression:

(3)$$\vec{\rm \upsilon} _{\,j+} = \vec{\rm \upsilon} _{\,jx} + i\vec{\rm \upsilon} _{\,jy} = \displaystyle{{ie} \over {m_0{\rm \omega} _{\,j+} {\rm \gamma} (1-\Omega _{\,j+} /{\rm \gamma} )}}\vec{E}_{\,j+} $$

The laser beams exert ponderomotive force on the electrons in the presence of relativistic nonlinearity. The relativistic–ponderomotive force by laser beams in magnetized plasma is given by the following equation (Sharma and Tripathi, Reference Sharma and Tripathi2012):

(4)$$\eqalign{{\vec{F}}_{\rm RP} &= -\displaystyle{{m_0} \over 2}\lpar {(\vec{\rm \upsilon}_{\,j+}^{^{\vskip -1.5pt \ast}} \cdot \nabla ){\rm \gamma} {\vec{\rm \upsilon}}_{\,j+}} \rpar -\displaystyle{e \over 2}(\vec{\rm \upsilon} _{\,j+} ^{^{\vskip -1.5pt \ast}} \times {\vec{B}}_{\,j+} ) \cr &= -\hat{r}\displaystyle{{e^2} \over {4m_0{\rm \omega} _{\,j +} ({\rm \gamma \omega} _{\,j+} -{\rm \omega} _{ce})}}\left[{\displaystyle{{2-(\Omega_{\,j+} /{\rm \gamma} )} \over {\lpar {1-(\Omega_{\,j+} /{\rm \gamma} )} \rpar }}} \displaystyle{{\partial (A_{\,j+} A_{\,j+} ^{^\ast} )} \over {\partial r}} \right.\cr &\quad -\left. {\displaystyle{{{\rm \omega} _{ce}A_{\,j+} A_{\,j+}^{^\ast}} \over {{\rm \omega} _{\,j+}^2 {\rm \gamma}^2{\lpar {1-(\Omega_{\,j+} /{\rm \gamma} )} \rpar }^2}}\displaystyle{{\partial {\rm \gamma}} \over {\partial r}}} \right]} $$

where $\vec{B}_{j +} = \lpar {ic(\nabla \times {\vec{E}}_{j+} )/\omega_{j+}} \rpar$ is the magnetic field of respective laser beams, and γ is the relativistic factor given by the following equation:

(5)$${\rm \gamma} = \left( {1 + \sum\limits_j {\displaystyle{{a_{0j}A_{\,j+}^{} A_{\,j+}^{^\ast}} \over {{\lpar {1-(\Omega_{\,j+} /{\rm \gamma} )} \rpar }^2}}}} \right)^{1/2}$$

where $a_{0j} = e^2/m_0^2 c^2{\rm \omega} _{j+} ^2$. The expression for γ can be written as the fourth order differential equation in γ. Using Eq. (5) in Eq. (4), we get the following equation:

(6)$$\eqalign{\vec{F}_{\rm RP} &= -\hat{r}\displaystyle{{m_0c^2} \over {4{\rm \gamma} {\lpar {1-(\Omega_{\,j+} /{\rm \gamma} )} \rpar }^2}}\left[ {2-\displaystyle{{{\rm \omega}_{ce}} \over {{\rm \gamma \omega}_{\,j+}}} -\displaystyle{1 \over 2}\displaystyle{{{\rm \omega} _{ce}} \over {{\rm \omega} _{\,j+}}} \displaystyle{{({\rm \gamma}^2-1)} \over {({\rm \gamma}^3-\Omega_{\,j+} )}}} \right] \cr &\quad\times\displaystyle{\partial \over {\partial r}}a_{\,j+} a_{\,j+} ^{^\ast}} $$

where a _j+ = eA _j+/m ₀cω_j+. The modified electron density of the plasma due to this relativistic–ponderomotive force can be written as follows:

(7)$$\displaystyle{n \over {n_0}} = 1-\sum\limits_j {\displaystyle{{\nabla \cdot {\vec{F}}_{\rm RP}} \over {4{\rm \pi}n_0e^2}}} $$

The effective dielectric constant $({\rm \varepsilon} _{j+})$ of the plasma for the right circularly polarized beams is given by the following equation:

(8)$${\rm \varepsilon} _{\,j+} = {\rm \varepsilon} _{xxj}-i{\rm \varepsilon} _{xyj} = 1-\displaystyle{{{\rm \omega} _{\,p0}^2} \over {{\rm \omega} _{\,j+} ^2 {\rm \gamma} \lpar {1-(\Omega_{\,j+} /{\rm \gamma} )} \rpar }}\displaystyle{n \over {n_0}}$$

where

$${\rm \varepsilon} _{xxj} = {\rm \varepsilon} _{yyj} = 1-\displaystyle{{{\rm \omega} _{\,p0}^2} \over {{\rm \omega} _{\,j+} ^2 {\rm \gamma} \lpar {1-(\Omega_{\,j+}^2 /{\rm \gamma}^2)} \rpar }}\displaystyle{n \over {n_0}}$$

$${\rm \varepsilon} _{xyj} = {\rm \varepsilon} _{yxj} = \displaystyle{{-i\lpar {{\rm \omega} _{\,p0}^2 /{\rm \gamma \omega} _{\,j+}^2} \rpar \lpar {\Omega_{\,j+} /{\rm \gamma}} \rpar } \over {\lpar {1-(\Omega_{\,j+}^2 /{\rm \gamma}^2)} \rpar }}\displaystyle{n \over {n_0}}$$

and   

$${\rm \varepsilon} _{zzj} = 1-\displaystyle{{{\rm \omega} _{\,p0}^2} \over {{\rm \omega} _{\,j+} ^2}} \displaystyle{n \over {n_0{\rm \gamma}}}$$

In order to explicitly write ${\rm \varepsilon} _{j+}$ in the non-paraxial region, γ can be expended as follows:

(9)$${\rm \gamma} = {\rm \gamma} _0 + r^2{\rm \gamma} _2 + r^4{\rm \gamma} _4$$

One can obtain from Eq. (11):

(10)$${\rm \gamma} _0 = 1 + \sum\limits_j {{\rm \gamma} _{0j}} + \sum\limits_{\,j \ne k} {{\rm \gamma} _{0jk}} = \sum\limits_j {\left( {\displaystyle{{a_{0j}^2} \over {2f_{\,j+}^2}} X_j-\displaystyle{{a_{0j}^4} \over {8f_{\,j+}^4}} Y_j-\displaystyle{1 \over 8}\sum\limits_k {\displaystyle{{a_{0j}^2} \over {\,f_{\,j+}^2}} \displaystyle{{a_{0k}^2} \over {2f_{k+}^2}}} Z_{\,jk}} \right)} $$

(11)$$\eqalign{{\rm \gamma} _2 &= \sum\limits_j {{\rm \gamma} _{2j}} + \sum\limits_{\,j \ne k} {{\rm \gamma} _{2jk}} = \sum\limits_j \left\{ \displaystyle{{a_{0j}^2} \over {2r_{\,j+}^2 f_{\,j+}^2}} (b_j^2 -{\rm \alpha} _{2j}-2)X_j \right.\cr &\quad +\left.-\displaystyle{{a_{0j}^4} \over {8r_{\,j+}^2 f_{\,j +}^6}} (b_j^2 -{\rm \alpha}_{2j}-2)Y_j \right. \cr &\quad -\displaystyle{1 \over 8}\sum\limits_k {\displaystyle{{a_{0j}^2} \over {\,f_{\,j+} ^2}} \displaystyle{{a_{0k}^2} \over {\,f_{k+} ^2}}} \left. {\left[ {\displaystyle{{(b_j^2 -{\rm \alpha}_{2j}-2)} \over {r_{\,j+}^2 f_{\,j+}^2}} + \displaystyle{{(b_k^2 -{\rm \alpha} _{2k}-2)} \over {r_k^2 f_{k+}^2}}} \right]Z_{\,jk}} \right\}} $$

and

(12)$$\eqalign{{\rm \gamma} _4 &= \sum\limits_j {{\rm \gamma} _{4j}} + \sum\limits_{\,j\ne k} {{\rm \gamma} _{4jk}} = \sum\limits_j \left\{ \displaystyle{{a_{0j}^2} \over {2r_{\,j+}^4 f_{\,j+}^2}} A_{1j}X_j \right.\cr &\quad -\displaystyle{{a_{0j}^4} \over {8r_{\,j+}^4 f_{\,j+}^6}} A_{2j}Y_j -\sum\limits_k {\displaystyle{{a_{0j}^2} \over {\,f_{\,j+} ^2}} \displaystyle{{a_{0k}^2} \over {\,f_{k+} ^2}}} \cr &\quad \times \left. {\left[ {\displaystyle{{{(b_j^2 -{\rm \alpha} _{2j}-2)}^2} \over {r_{\,j+}^4 f_{\,j+}^4}} + \displaystyle{{(b_j^2 -{\rm \alpha} _{2j}-2)} \over {r_{\,j +}^2 f_{\,j+}^2}} \displaystyle{{(b_k^2 -{\rm \alpha} _{2k}-2)} \over {r_k^2 f_{k+}^2}}} \right]Z_{\,jk}} \right\}} $$

where

$$X_j = 1 + 2\displaystyle{{{\rm \omega} _{ce}} \over {{\rm \omega} _{\,j+}}} + 3\displaystyle{{{\rm \omega} _{ce}^2} \over {{\rm \omega} _{\,j+} ^2}}$$

$$Y_j = 1 + 8\displaystyle{{{\rm \omega} _{ce}} \over {{\rm \omega} _{\,j+}}} + 22\displaystyle{{{\rm \omega} _{ce}^2} \over {{\rm \omega} _{\,j+} ^2}} + 9\displaystyle{{{\rm \omega} _{ce}^3} \over {{\rm \omega} _{\,j+} ^3}} + 9\displaystyle{{{\rm \omega} _{ce}^4} \over {{\rm \omega} _{\,j+} ^4}}$$

$$Z_{\,jk} = 1 + 4\left( {\displaystyle{{{\rm \omega} _{ce}} \over {{\rm \omega} _{\,j+}}} + \displaystyle{{{\rm \omega} _{ce}} \over {{\rm \omega} _{k+}}}} \right) + \left( {9\displaystyle{{{\rm \omega} _{ce}^2} \over {{\rm \omega} _{\,j+}^2}} + 9\displaystyle{{{\rm \omega} _{ce}^2} \over {{\rm \omega} _{k+}^2}} + \displaystyle{{{\rm \omega} _{ce}^2} \over {{\rm \omega} _{\,j+} {\rm \omega} _{k+}}}} \right) + 6\displaystyle{{{\rm \omega} _{ce}^3} \over {{\rm \omega} _{\,j+} {\rm \omega} _{k+}}} \left( {\displaystyle{1 \over {{\rm \omega} _{\,j+}}} + \displaystyle{1 \over {{\rm \omega} _{k+}}}} \right) + 9\displaystyle{{{\rm \omega} _{ce}^4} \over {{\rm \omega} _{\,j+} ^2 {\rm \omega} _{k+} ^2}}$$

$$A_{1j} = \displaystyle{{b_j^4} \over 3}-b_j^2 + b_j^2 {\rm \alpha} _{2j}-2{\rm \alpha} _{2j} + {\rm \alpha} _{4j} + 2$$

$$A_{2j} = \displaystyle{{5b_j^4} \over 3}-8b_j^2 + 2b_j^2 {\rm \alpha} _{2j} + 2b_j {\rm \alpha} _{2j} + {\rm \alpha} _{2j}^2 -8{\rm \alpha} _{2j} + 2{\rm \alpha} _{4j} + 8$$

Substituting $\vec{F}_{\rm RP}$ from Eq. (6) in Eq. (7) and using Eq. (9), we get the normalized electron density as follows:

(13)$$\displaystyle{n \over {n_0}} = 1-\left( {{\rm \delta}n_{0j} + \displaystyle{{r^2} \over {r_{\,j+}^2}} {\rm \delta}n_{2j} + \displaystyle{{r^4} \over {r_{\,j+}^4}} {\rm \delta}n_{4j}} \right)$$

where

$${\rm \delta}n_{0j} = \displaystyle{{-c^2a_{0j}^2} \over {2{\rm \omega} _p^2 r_{\,j+} ^2 f_{\,j+} ^4}} \times \displaystyle{{(b_j^2 + {\rm \alpha} _{2j}-2)} \over {{({\rm \gamma} _0-\Omega _{\,j+} )}^2({\rm \gamma} _0^3 -\Omega _{\,j+} )}} [{B_{1j}}]$$

$$\eqalign{{\rm \delta}n_{2j} &= \displaystyle{{-c^2a_{0j}^2} \over {{\rm \omega} _p^2 r_{\,j+} ^2 f_{\,j+} ^6 {({\rm \gamma} _0-\Omega _{\,j+} )}^4{({\rm \gamma} _0^3 -\Omega _{\,j+} )}^3}} \cr &\quad\times\left\{ {8B_{1j}A_{1j}{({\rm \gamma} _0-\Omega_{\,j+} )}^2{({\rm \gamma} _0^3 -\Omega_{\,j+} )}^2 + \displaystyle{{a_{0j}^2} \over {\,f_{\,j+}^2}} {\rm \gamma}_0^2 A_{3j}} \right. \cr &\quad\times \left.\vphantom{\displaystyle{{a_{0j}^2} \over {\,f_{\,j+}^2}}} {\left[ {B_{2j}({\rm \gamma} _0-\Omega_{\,j+} )\displaystyle{{{\rm \omega} _{ce}} \over {{\rm \omega} _{\,j+}}} -B_{1j}({\rm \gamma} _0 + \Omega_{\,j+} )({\rm \gamma} _0^3 -\Omega_{\,j+} )} \right]} \right\}}$$

and

$$\eqalign{{\rm \delta}n_{4j} &= \displaystyle{{-c^2a_{0j}^2} \over {2{\rm \omega} _p^2 r_{\,j+} ^2 f_{\,j+} ^8 {({\rm \gamma} _0-\Omega _{\,j+} )}^4{({\rm \gamma} _0^3 -\Omega _{\,j+} )}^3}}\cr &\quad\times\left\{ { 9{\rm \gamma}_0A_{4j}B_{1j}{({\rm \gamma} _0-\Omega_{\,j+} )}^2{({\rm \gamma} _0^3 -\Omega_{\,j+} )}^2 + 2\displaystyle{{a_{0j}^2} \over {\,f_{\,j+}^2}} {\rm \gamma} _0^2 A_{5j}} \right. \cr &\quad\times \left.\vphantom{\displaystyle{{a_{0j}^2} \over {\,f_{\,j+}^2}}} {\left[ {B_{2j}({\rm \gamma} _0-\Omega_{\,j+} )\displaystyle{{{\rm \omega} _{ce}} \over {{\rm \omega} _{\,j+}}} -B_{1j}({\rm \gamma} _0 + \Omega_{\,j+} )({\rm \gamma} _0^3 -\Omega_{\,j+} )} \right]} \right\}}$$

with

$$B_{1j} = 4{\rm \gamma} _0^4 -3{\rm \gamma} _0\Omega _{\,j+} -3{\rm \gamma} _0^3 \Omega _{\,j+} + 2\Omega _{\,j+} ^2$$

$$B_{2j} = 3{\rm \gamma} _0^6 -3{\rm \gamma} _0^4 -2{\rm \gamma} _0^3 \Omega _{\,j+} + 2\Omega _{\,j+} ^2$$

$$A_{3j} = b_j^4 -4b_j^2 + 2b_j^2 {\rm \alpha} _{2j}-4{\rm \alpha} _{2j} + {\rm \alpha} _{2j}^2 + 4$$

$$A_{4j} = b_j^6 /24 - 2{\rm \alpha} _{2j}b_j^2 + {\rm \alpha} _4b_j^2 + (a_{2j}b_j^4 - 2b_j^4 )/3 + 2{\rm \alpha} _{2j} - 2{\rm \alpha} _{4j} + 2b_j^2 - 1$$

$$A_{5j} = -2{\rm \alpha} _{2j}^2 + {\rm \alpha} _{2j}{\rm \alpha} _{4j} + 6b_j^2 + 6{\rm \alpha} _{2j}-4 + (4{\rm \alpha} _{2j}b_j^4 -8b_j^4 + b_j^6 )/3-6{\rm \alpha} _{2j}b_j^2 -2{\rm \alpha} _{4j} + {\rm \alpha} _{4j}b_j^2 + {\rm \alpha} _{2j}^2 b^2 + b_j^2$$

The effective dielectric constant of the plasma can be expressed as follows:

(14)$${\rm \varepsilon} _{\,j+} (r,z) = {\rm \varepsilon} _{0j+} (z)-\displaystyle{{r^2} \over {r_{\,j+} ^2}} {\rm \varepsilon} _{2j+} (z)-\displaystyle{{r^4} \over {r_{\,j+} ^4}} {\rm \varepsilon} _{4j+} (z)$$

where ${\rm \varepsilon}_{0j+} (z), \, {\rm \varepsilon} _{2j+} (z), \, {\rm {and}} \, {\rm \varepsilon} _{4j+} (z)$ are the coefficients associated with r ⁰, r ², and r ⁴, respectively, in the expansion of ${\rm \varepsilon} _+ (r,z)$ given by the following equation:

$${\rm \varepsilon} _{0j+} (z) = 1-\displaystyle{{{\rm \omega} _p^2} \over {{\rm \omega} _{\,j +} ^2}} \displaystyle{{(1-{\rm \delta} n_{0j})} \over {({\rm \gamma} _0-\Omega _{\,j +} )}}$$

$${\rm \varepsilon} _{2j+} (z) = \displaystyle{{{\rm \omega} _p^2} \over {{\rm \omega} _{\,j+} ^2}} \left\{ {\displaystyle{{({\rm \gamma} _2(1-{\rm \delta}n_{0j}) + f_{\,j+}^2 ({\rm \gamma} _0-\Omega_{\,j+} ){\rm \delta}n_{2j})} \over {r_{\,j+}^2 f_{\,j+}^2 {({\rm \gamma} _0-\Omega_{\,j+} )}^2}}} \right\}$$

and

$${\rm \varepsilon} _{4j+} (z) = {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}2\displaystyle{{{\rm \omega} _p^2} \over {{\rm \omega} _{\,j+} ^2}} \left\{ {\displaystyle{\matrix{(1-{\rm \delta}n_{0j}) [ {{\rm \gamma}_4({\rm \gamma} _0-\Omega_{\,j+} )-{\rm \gamma} _2^2} ]\cr + f_{\,j+}^2 ({\rm \gamma} _0-\Omega_{\,j+} ) [ {\,f_{\,j+}^2 ({\rm \gamma} _0-\Omega_{\,j+} ){\rm \delta}n_{4j}-{\rm \gamma}_2{\rm \delta}n_{2j}} ] {\kern 1pt} {\kern 1pt} {\kern 1pt}} \over {r_{\,j+}^4 f_{\,j+}^4 {({\rm \gamma}_0-\Omega_{\,j+} )}^3}}} \right\}$$

Propagation of CGLBs in magnetized plasma

The wave equation for the beams propagating through in collisionless magnetized plasma can be written as follows:

(15)$$\nabla ^2\vec{E}_{\,j+} -\nabla \cdot (\nabla \cdot \vec{E}_{\,j+} ) + {\rm \varepsilon} _{\,j+} (r,z)\displaystyle{{{\rm \omega} _{\,j+} ^2} \over {c^2}}\vec{E}_{\,j+} = 0$$

The electric fields of the laser beams are given by the following equation:

(16)$$\vec{E}_{\,j+} (r,z) = \hat{r}\,A_{\,j+} (r,z)\exp [i({\rm \omega} _{\,j+} t-k_{\,j+} z)]$$

where $\hat{r} = (\hat{x} + i\,\hat{y})$ and $k_{j+} = \lpar {{\rm \omega} _{j+} /c} \rpar \sqrt {{\rm \varepsilon} _{0j+} (z)}$. The maximum amplitude of the laser beams is defined as follows:

(17)$$A_{\,j+} = A_{0j+} (r,z)\exp \{ -ik_{\,j+} S_{\,j+} (r,z)\} $$

where A _0j+ is a complex function of space, and S _j+(r, z) is the eikonal associated with the CGLBs. Substituting Eq. (17) into Eq. (15) and separating real and imaginary parts of the resulting equation, the following set of equations is obtained:

(18)$$\eqalign{ & 2\left( {\displaystyle{{\partial S_{\,j+}} \over {\partial z}}} \right) + \displaystyle{1 \over 2}\left( {1 + \displaystyle{{{\rm \varepsilon} _{0j+}} \over {{\rm \varepsilon} _{0zzj}}}} \right)\left( {\displaystyle{{\partial S_{\,j+}} \over {\partial r}}} \right)^2 \cr &\quad = \displaystyle{1 \over {2k_{\,j+} ^2 A_{0j+}}} \left( {1 + \displaystyle{{{\rm \varepsilon} _{0j+}} \over {{\rm \varepsilon} _{0zzj}}}} \right)\left( {\displaystyle{{\partial^2A_{0j+}} \over {\partial r^2}} + \displaystyle{1 \over r}\displaystyle{{\partial A_{0j+}} \over {\partial r}}} \right) \cr &\qquad-\displaystyle{{r^2} \over {r_{\,j+} ^2}} \displaystyle{{{\rm \varepsilon} _{2j+} (z)} \over {{\rm \varepsilon} _{0j+} (z)}}-\displaystyle{{r^4} \over {r_{\,j+} ^4}} \displaystyle{{{\rm \varepsilon} _{4j}(z)} \over {{\rm \varepsilon} _{0j+} (z)}}} $$

and

(19)$$\displaystyle{{\partial A_{0j+} ^2} \over {\partial z}} + \displaystyle{1 \over 2}\left( {1 + \displaystyle{{{\rm \varepsilon} _{0j+} (z)} \over {{\rm \varepsilon} _{0zzj}}}} \right)\left\{ {A_{0j+}^2 \left( {\displaystyle{{\partial^2S_{\,j+}} \over {\partial r^2}} + \displaystyle{1 \over r}\displaystyle{{\partial S_{\,j+}} \over {\partial r}}} \right) + \displaystyle{{\partial A_{0j+}^2} \over {\partial r}}\displaystyle{{\partial S_{\,j+}} \over {\partial r}}} \right\} = 0$$

The solution of Eqs (18) and (19) can be written as follows (Liu and Tripathi, Reference Liu and Tripathi2001; Sodha and Faisal, Reference Sodha and Faisal2008):

(20)$$\eqalign{A_{0j+} ^2 (r,z) &= \displaystyle{{E_{\,j+} ^2} \over {4f_{\,j+} ^2}} \exp \left( {\displaystyle{{b^2} \over 2}} \right) \times \left( {1 + \displaystyle{{r^2} \over {r_{\,j+}^2 f_{\,j+}^2 (z)}}{\rm \alpha} _{2j}(z) + \displaystyle{{r^4} \over {r_{\,j+}^4 f_{\,j+}^4 (z)}}{\rm \alpha} _{4j}(z)} \right) \cr &\quad\times \left( {\exp \left[ {-{\left( {\displaystyle{r \over {r_{\,j+} f_{\,j+} (z)}} + \displaystyle{b \over 2}} \right)}^2} \right] + \exp \left[ {-{\left( {\displaystyle{r \over {r_{\,j+} f_{\,j+} (z)}}-\displaystyle{b \over 2}} \right)}^2} \right]} \right)^2} $$

The eikonal S _j+(r, z) of the pump beams is given as follows:

(21)$$S_{\,j+} (r,z) = S_{0j+} (z) + \displaystyle{{r^2} \over {r_{\,j+} ^2}} S_{2j+} (z) + \displaystyle{{r^4} \over {r_{\,j+} ^4}} S_{4j+} (z)$$

where f _j+ is the dimensionless beam width parameters for beam 1 and 2, respectively, S _0j(z) is the axial phase shift,

$$S_{2j}(z) = \displaystyle{{r_j^2} \over {2f_{\,j+}}} \displaystyle{{df_{\,j+}} \over {dz}}\displaystyle{1 \over {(1 + {\rm \varepsilon} _{0j+} /{\rm \varepsilon} _{0zzj})}}$$

where S _2j(z) is indicative of the spherical curvature of the wavefront, and S _4j(z) corresponding to its departure from the spherical nature.

Substituting $A_{0j +} ^2$ and S _j+ from Eqs (20) and (21) into Eqs (18) and (19) and equating the coefficients of r ⁰, r ², and r ⁴ in the resulting equation on both sides of the resulting equation, following equations are obtained for f _j+, ${S}^{\prime}_{4j+}$, α_2j, and α_4j:

(22)$$\displaystyle{{d^2f_{\,j+}} \over {d{\rm \xi} _j^2}} = \left( {1 + \displaystyle{{{\rm \varepsilon}_{0j +}} \over {{\rm \varepsilon}_{0zzj}}}} \right)^2\displaystyle{{{\rm \chi} _{2j}} \over {{\rm \varepsilon} _{0j+} f_{\,j+} ^3}} -\left( {1 + \displaystyle{{{\rm \varepsilon} _{0j+}} \over {{\rm \varepsilon} _{0zzj}}}} \right)\displaystyle{{2{\rm \varepsilon} _{2j+} f_{\,j+}} \over {{\rm \varepsilon} _{0j+}}} {\rm \rho} _j^2 $$

(23)$$\displaystyle{{d{{S}^{\prime}}_{4j+}} \over {d{\rm \xi}}} = \displaystyle{1 \over 4}\left( {1 + \displaystyle{{{\rm \varepsilon} _{0j+}} \over {{\rm \varepsilon} _{0zzj}}}} \right)\displaystyle{{{\rm \chi} _{4j}} \over {{\rm \varepsilon} _{0j+} f_{\,j+} ^6}} -\displaystyle{{{\rm \varepsilon} _{4j+} {\rm \rho} _j^2} \over {2{\rm \varepsilon} _{0j+}}} -\displaystyle{{2{{S}^{\prime}}_{4j+}} \over {\,f_{\,j+}}} \displaystyle{{df_{0+}} \over {d{\rm \xi} _j}}$$

(24)$$\displaystyle{{\partial {\rm \alpha} _{2j}} \over {\partial {\rm \xi} _j}} = -8\left( {1 + \displaystyle{{{\rm \varepsilon} _{0j+}} \over {{\rm \varepsilon} _{0zzj}}}} \right){S}^{\prime}_{4j +} f_{\,j+} ^2$$

(25)$$\displaystyle{{\partial {\rm \alpha} _{4j}} \over {\partial {\rm \xi} _j}} = -4\left( {1 + \displaystyle{{{\rm \varepsilon} _{0j+}} \over {{\rm \varepsilon} _{0zzj}}}} \right){S}^{\prime}_{4j+} f_{\,j+} ^2 (b_j^2 + 3{\rm \alpha} _{2j}-2)$$

where

$${S}^{\prime}_{4j+} = S_{4j+} \displaystyle{{{\rm \omega} _{\,j+}} \over c}$$

Eliminating ${S}^{\,\prime}_{4j +}$ from Eqs (24) and (25) and integrating the resulting equation with the initial conditions α_4j = 0 and α_2j = 0 at ξ_j = 0, we obtain the following equation:

(26)$${\rm \alpha} _{4j} = \displaystyle{{(3{\rm \alpha} _{2j}^2 -4{\rm \alpha} _{2j})} \over 4}$$

where

$${\rm \chi} _{2j} = -\displaystyle{{b_j^4} \over 3}-4b_j^2 + 2{\rm \alpha} _2b_j^2 -3{\rm \alpha} _{2j}^2 -4{\rm \alpha} _{2j} + 8{\rm \alpha} _{4j} + 4$$

$${\rm \chi} _{4j} = {\rm \alpha} _{2j}^3 + 4{\rm \alpha} _{2j}^2 + 2(2b_j^4 -{\rm \alpha} _{2j}b_j^4 )/3-5b_j^6 /12 + 4{\rm \alpha} _{4j}b_j^2 -8{\rm \alpha} _{4j} + 7/2{\rm \alpha} _{2j}{\rm \alpha} _{4j}-2a_{2j}^2 b_j^2 -4$$

Here, ${\rm \xi} _j = \lpar {(c/r_{j +}^2 {\rm \omega}_{j+} )z} \rpar$ and ρ_j = (r _j+ω_j+/c) describe the dimensionless distance of propagation and the dimensionless initial width of the beam, respectively. Equation (22) describes the laser beams propagation in magnetized plasma.

Formalism of generation of THz radiation

In order to generate THz radiation by beating/cross-focusing of two intense cosh-Gaussian laser beams, we have taken rippled density plasmas as n = n ₀ + n _q, where n _q[ = n _q0 exp (iqz)], here, n ₀ is the electron plasma density without a ripple, n _q0 is the amplitude of the density ripple, q (= 2π/λ) is the ripple wave vector, and λ is the wavelength of the ripple. The rippled plasma density provides phase matching condition for the generation of THz radiation. Such ripple density plasmas can be produced by various schemes (Pai et al., Reference Pai, Huang, Kuo and Lin2005; Kuo et al., Reference Kuo, Pai, Lin, Lee, Lin, Wang and Chen2007; Layer et al., Reference Layer, York, Antonson, Varma, Chen, Lengand and Milchberg2007), namely a machining beam, a patterned mask, and transmission ring grating. The coupling between density ripples with the density perturbations provided by pondermotive force give rise to nonlinear current, which is responsible for THz generation. Figure 2 represents schematic representation for the generation of THz radiation in the presence of an external applied static magnetic field in the ripple density plasma.

Fig. 2.

Schematic representation for THz radiation generation by beating of two laser beams in the presence of external static magnetic field and ripple density plasma.

The laser beams impart a nonlinear ponderomotive force to the plasma electrons at the beat frequency ω_T+ = ω₁ − ω₂ and wave number k _T+ = k ₁ − k ₂. The range of THz radiation occurs in the beat frequency or frequency difference of two laser beams. The ponderomotive force exerted by laser beams at a beat frequency can be written as follows:

(27)$$\scale90%{\eqalign {&\vec {{F}_{\,p+} ^{{\rm \omega} _{T+}}} = -\displaystyle{{m_0} \over 2}(\vec{\rm \upsilon} _1^{^\ast} \cdot \nabla ){\rm \gamma} {\vec{\rm \upsilon}} _2-\displaystyle{e \over 2}(\vec{\rm \upsilon} _1^{^\ast} \times {\vec{B}}_2)-\displaystyle{{m_0} \over 2}(\vec{\rm \upsilon} _2^{^\ast} \cdot \nabla ){\rm \gamma}{\vec{\rm \upsilon}} _1- \displaystyle{e \over 2}(\vec{\rm \upsilon} _2^{^\ast} \times {\vec{B}}_1) \cr &= \displaystyle{{-\hat{r}e^2} \over {4{\rm \gamma}m_0}}\displaystyle{{E_{01}E_{02}} \over {{\rm \omega} _1{\rm \omega} _2f_1f_2\lpar {1-({\rm \omega}_{ce}/{\rm \gamma \omega} _1)} \rpar \lpar {1-({\rm \omega}_{ce}/{\rm \gamma \omega}_2)} \rpar }} \left[ 4-\displaystyle{{{\rm \omega}_{ce}} \over {{\rm \gamma \omega}_1}} -\displaystyle{{{\rm \omega}_{ce}} \over {{\rm \gamma \omega}_2}} \right.\cr &\quad \left.-\displaystyle{1 \over 2} \displaystyle{{\lpar {({\rm \omega}_{ce}/{\rm \omega}_1) + ({\rm \omega}_{ce}/{\rm \omega}_2)-2({\rm \omega}_{ce}^2 /{\rm \omega}_1{\rm \omega}_2)} \rpar \lpar {{\rm \gamma}^2-1} \rpar } \over {[ {\lpar {1-({\rm \omega}_{ce}/{\rm \gamma \omega}_1)} \rpar \lpar {{\rm \gamma}^3-({\rm \omega}_{ce}/{\rm \gamma \omega}_2)} \rpar + \lpar {1-({\rm \omega}_{ce}/{\rm \gamma \omega}_2)} \rpar \lpar {{\rm \gamma}^3-({\rm \omega}_{ce}/{\rm \gamma \omega} _1)} \rpar } ] }} \right] \cr &\quad\times \left\{ \cosh \left( {\displaystyle{{rb_1} \over {r_1f_1}}} \right)\cosh \left( {\displaystyle{{rb_2} \over {r_2f_2}}} \right)\left( {\displaystyle{r \over {r_1^2 f_1^2}} + \displaystyle{r \over {r_2^2 f_2^2}}} \right) \right.\cr &\quad \left.+ \left[ \cosh \left( {\displaystyle{{rb_1} \over {r_1f_1}}} \right)\sinh \left( {\displaystyle{{rb_2} \over {r_2f_2}}} \right)\left( {\displaystyle{{b_2} \over {r_2f_2}}} \right) \right.\right.\cr &\quad \left.\left.+ \sinh \left( {\displaystyle{{rb_1} \over {r_1f_1}}} \right)\cosh \left( {\displaystyle{{rb_2} \over {r_2f_2}}} \right) \left( {\displaystyle{{b_1} \over {r_1f_1}}} \right) \right] \right\} \cr &\quad\times \exp \left[ {-\displaystyle{1 \over 2}\left( {\displaystyle{{r^2} \over {r_1^2 f_1^2}} + \displaystyle{{r^2} \over {r_2^2 f_2^2}}} \right)-i({\rm \omega}_{T +} t-k_{T +} z)} \right]}} $$

This nonlinear ponderomotive force causes perturbations in the density of the plasma and imparts nonlinear oscillatory velocity to the electrons. The nonlinear oscillatory velocity of electrons at beat wave frequency can be obtained from the equation of motion, i.e.,

(28)$$\displaystyle{\partial \over {\partial t}}\lpar {m_0{\rm \gamma} \vec {{\rm \upsilon}_ +^{\rm NL}}} \rpar = \mathop {\vec {F_{\,p+} ^{\omega _{T+}}} -\displaystyle{e \over c}} \,(\vec {{\rm \upsilon} _ + ^{\rm NL}} \times \vec{B}_0)$$

Fourier transform of Eq. (28) can be written as follows:

(29)$$ \vec {{\rm \upsilon} _ + ^{\rm NL}} = \displaystyle{1 \over {-im_0{\rm \gamma \omega} _{T+} \lpar {1 + ({\rm \omega} _{ce}/{\rm \gamma \omega} _{T+} )} \rpar }} \vec {F_{\,p+} ^{{\rm \omega} _{T+}}} $$

This nonlinear velocity couples with the density ripple and gives rise to the nonlinear current density. The nonlinear current density is given as follows:

(30)$$ \vec {J_ + ^{\rm NL}} = -\displaystyle{1 \over 2}en_{q0}e^{iqz} \vec {{\rm \upsilon} _ + ^{\rm NL}} $$

This current is responsible for the generation of THz radiation at the beat frequency ω_T+ = ω₁ − ω₂ and wave number k _T+ = k ₁ − k ₂ + q.

The wave equation for the propagation of the THz wave in magnetized plasma can be written as follows:

(31)$$-\nabla ^2\vec{E}_{T+} + \nabla \, (\nabla \cdot \vec{E}_{T+} ) = -\displaystyle{{4{\rm \pi}i{\rm \omega} _{T+}} \over {c^2}} \vec {J_ + ^{\rm NL}} + \displaystyle{{{\rm \omega} _{T+} ^2} \over {c^2}}{\rm \varepsilon} _ + \vec{E} _{T +} $$

where $\vec{E}_{T+} [ { = \hat{r}E_{T0 +} (r,z)\exp [ {-i({\rm \omega}_{T +} t-k_{T+} z)} ] } ]$ is the electric field associated with the THz radiation, and

$${\rm \varepsilon} _ + = {\rm \varepsilon} _{xx}-i{\rm \varepsilon} _{xy} = 1-\displaystyle{{{\rm \omega} _p^2} \over {{\rm \omega} _{T+} ^2 {\rm \gamma} \lpar {1-{\rm \omega}_{ce}/{\rm \gamma \omega}_{T+}} \rpar }}\displaystyle{n \over {n_0}}$$

where ${\rm \varepsilon} _+$ is the dielectric constant of the plasma at the THz frequency (ω_T+).

Substituting the values of $ \vec {J_ + ^{\rm NL}}$ and ${\rm \varepsilon} _+$ into Eq. (31), we get THz electric field as follows:

(32)$$\eqalign{\left\vert \displaystyle{E_{T0 +} \over E_{10}} \right\vert &= \left\vert \displaystyle{z \over 16} \left(\displaystyle{{\rm \omega}_p^2 \over k_{T+}} \right) \left(\displaystyle{n_{q0} \over n_0} \right) \left(-\displaystyle{ieE_{20} \over c^2m_0} \right) \right.\cr &\quad \left.\times\displaystyle{1 \over {\rm \gamma} _0 (1 + ({\rm \omega}_c/{\rm \gamma}_0{\rm \omega}_T))} [X] [Y] \exp \left[-\left(\displaystyle{r^2 \over r_1^2 f_1} + \displaystyle{r^2 \over r_2^2 f_2} \right) \right] \right \Bigg \vert} $$

where

$$\eqalign{&[ X ] = \displaystyle{1 \over {\lpar {{\rm \gamma}_0 {\rm \omega}_1-{\rm \omega}_{ce}} \rpar \lpar {{\rm \gamma}_0 {\rm \omega}_2-{\rm \omega}_{ce}} \rpar }} \left[ 4-\displaystyle{{{\rm \omega}_{ce}} \over {{\rm \gamma}_0 {\rm \omega}_1}}-\displaystyle{{{\rm \omega}_{ce}} \over {{\rm \gamma}_0 {\rm \omega}_2}} \right.\cr &\quad \left.-\displaystyle{1 \over 2}\displaystyle{{\lpar {({\rm \omega}_{ce}/{\rm \omega}_1) + ({\rm \omega}_{ce}/{\rm \omega}_2)-2({\rm \omega}_{ce}^2 /{\rm \omega}_1{\rm \omega}_2)} \rpar \lpar {{\rm \gamma} _0^2 -1} \rpar } \over {[ \matrix{\lpar {1-({\rm \omega}_{ce}/{\rm \gamma}_0 {\rm \omega}_1)} \rpar \lpar {{\rm \gamma}_0^3 -({\rm \omega}_{ce}/{\rm \gamma}_0 {\rm \omega}_2)} \rpar\cr + \lpar {1-({\rm \omega} _{ce}/{\rm \gamma} _0 {\rm \omega} _2)} \rpar \lpar {{\rm \gamma}_0^3 -({\rm \omega}_{ce}/{\rm \gamma} _0 {\rm \omega}_1)} \rpar } ] }} \right]}$$

and

$$\eqalign{[Y] &= \left\{\cosh \, \left(\displaystyle{rb_1 \over r_1f_1} \right) \cosh \, \left(\displaystyle{rb_2 \over r_2f_2} \right)\left(\displaystyle{r \over r_1^2 f_1^2} + \displaystyle{r \over r_2^2 f_2^2} \right) \right.\cr &\quad \left.+ \left[\cosh \, \left(\displaystyle{rb_1 \over r_1f_1} \right) \sinh \, \left(\displaystyle{rb_2 \over r_2f_2} \right)\left(\displaystyle{b_2 \over r_2f_2} \right) \right. \right. \cr &\quad \left. \left. + \sinh \, \left(\displaystyle{rb_1 \over r_1f_1} \right) \cosh \, \left(\displaystyle{rb_2 \over r_2f_2} \right) \left(\displaystyle{b_1 \over r_1f_1} \right) \right] \right\}}$$

The efficiency of the emitted THz radiation is given by the ratio of the THz radiation energy to the energy of incident laser beams. The energy per unit area of the incident lasers is given in the following equation (Singh et al., Reference Singh, Singh and Sharma2013):

(33)$$W_{\rm pump} = \displaystyle{{{\rm \varepsilon} _0r_1{\vert {E_{10}} \vert } ^2} \over 4}\sqrt {\displaystyle{{\rm \pi} \over 2}} \left\{ {1 + \exp \left( {\displaystyle{{b^2} \over 2}} \right)} \right\}$$

and the energy per unit area of the THz radiation is given as follows:

(34)$$\eqalign{W_{\rm THz} &= \displaystyle{{{\rm \varepsilon} _0r_1^3 {\vert {E_{10}} \vert } ^2} \over 4}\sqrt {\rm \pi} \left \vert \displaystyle{z \over {16}}\left( {\displaystyle{{{\rm \omega}_p^2} \over {k_{T+}}}} \right)\left( {\displaystyle{{n_{q0}} \over {n_0}}} \right) \right.\cr &\quad \left.\times\left( {-\displaystyle{{ieE_{20}} \over {c^2m_0}}} \right)\displaystyle{1 \over {{\rm \gamma}_0 \lpar {1 + ({\rm \omega}_c/{\rm \gamma}_0{\rm \omega}_T)} \rpar }}[ X ] \right \vert ^2 \cr &\quad \times \left[ {1 + \exp \left( {\displaystyle{{b^2} \over 4}} \right)} \right] \left[ {3-2b^2 + \exp \left( {\displaystyle{{b^2} \over 4}} \right) + \exp \left( {\displaystyle{{3b^2} \over 4}} \right)} \right]} $$

The normalized efficiency of THz radiation generation is given by the following equation:

(35)$$\eqalign{{\rm \eta} &= \displaystyle{{W_{\rm THz}} \over {W_{\rm laser}}} = \sqrt 2 r_1^2 \left \vert \displaystyle{z \over {16}}\left( {\displaystyle{{{\rm \omega}_p^2} \over {k_{T+}}}} \right)\left( {\displaystyle{{n_{q0}} \over {n_0}}} \right)\left( {-\displaystyle{{ieE_{20}} \over {c^2m_0}}} \right) \right.\cr &\quad \left.\times\displaystyle{1 \over {{\rm \gamma}_0 \lpar {1 + ({\rm \omega}_{ce}/{\rm \gamma}_0{\rm \omega}_T)} \rpar }}[ X ] \right \vert ^2 \displaystyle{1 \over {[ {1 + \exp \lpar {b^2/2} \rpar }] }} \cr &\quad \times \left[ {1 + \exp \left( {\displaystyle{{b^2} \over 4}} \right)} \right] \left[ {3-2b^2 + \exp \left( {\displaystyle{{b^2} \over 4}} \right) + \exp \left( {\displaystyle{{3b^2} \over 4}} \right)} \right]} $$

Numerical result and discussion

This section presents a numerical study for THz radiation by two intense cross-focused CGLBs in the rippled density magnetized plasma under the relativistic–ponderomotive regime. The following set of laser and plasma parameters has been used in numerical calculations: ω₁ = 1.776 × 10¹⁵ rad/s, ω₂ = 1.706 × 10¹⁵ rad/s, intensity of laser beams a ₁ = 1.21 × 10¹⁸ W/cm², a ₂ = 4.84 × 10¹⁸ W/cm², initial beam waist width of laser beams r _j (r ₁ = r ₂) = 20 µm, ω_p0 = 0.02ω₁, n ₀ = 1.56 × 10¹⁷/cm³, n _q0 = 0.1n ₀, ω_ce = 0, 0.2ω_p0, 0.4ω_p0, and 0.6ω_p0, b = 0, 0.4, 0.8, and 1. We have used the initial conditions: f_j = 1; df_j/dz = 0; and E _T = 0 at z = 0.

At first, we investigated the effects of relativistic and ponderomotive nonlinearities on the cross-focusing of CGLBs in magnetized plasma. Due to mutual interaction of two laser beams, focusing of one laser beam is affected by the presence of second laser beam, and hence, cross-focusing of laser beams takes place in the plasma. When two intense laser beams propagate through the magnetized plasma, the charge carriers are redistributed due to ponderomotive and relativistic nonlinearities. The dielectric constant of the plasma becomes modified due to redistribution of charge carriers. Equation (20) describes the intensity profile of laser beams in magnetized plasma along with the radial direction. Equation (22) determines the converging/diverging behavior of the laser beams in the magnetized plasma, where the first and second term on the right-hand side show divergence (diffraction) and the convergence (nonlinear coupling term) of the beams in propagation direction, respectively. It is obvious from Eq. (20) that the intensity profile of both laser beams depends on the dimensionless beam width parameter f _j and the coefficients (α_2j and α_4j) of r ² and r ⁴ in the non-paraxial region. Equations (22) and (20) have been solved with the help of Eqs (23)–(26) for cross-focused CGLBs in magnetized plasma, and the numerical results are presented in Figures 3 and 4.

Fig. 3.

Variation of beam width parameters (f _1,2) with normalized distance (ξ) in the direction of propagation of CGLBs for different values of decentered parameter b. The dashed and solid lines are for the first and second laser beams, respectively.

Fig. 4.

Variation of intensities of CGLBs with normalized transverse distance (r/r ₀) in the direction of propagation for different values of decentered parameter b. The dashed and solid lines are for the first and second laser beams, respectively.

Figures 3 and 4 reveal the variation of dimensionless beam width parameters f ₁ and f ₂ and normalized intensity profiles of CGLBs with the normalized distance of propagation (ξ) for different values of the decentred parameter (b), when relativistic and ponderomotive nonlinearities are simultaneously operative. Focusing of both laser beams depends upon the initial powers of laser beams, initial beam radii r _j+ of the laser beams, and plasma density (ω_p0/ω_j+). In this study, we have fixed the initial powers, beam radii of the laser beams, and cyclotron frequency (ω_ce) and considered the variation in decentred parameter (b). It is obvious from Figure 3 that with the increase in the value of b, the beam width parameter of both beams decreases. In general, the self-focusing ability of laser beams enhances the higher values of b (Gill et al., Reference Gill, Mahajan and Kaur2011; Purohit et al., Reference Purohit, Gaur and Rawat2016). In our case, focusing on both laser beams decreases with increase in the values of b. This may be due to the contribution of the coefficients α_2j and α_4j of r ² and r ⁴ in the non-paraxial-ray approximation. It is interesting to note that the amplitude and efficiency of THz radiation increase for higher values of b. The normalized intensity profile of the cosh-Gaussian beams for different values of b is plotted in Figure 4. The intensity of both laser beams in magnetized plasma is increasing with the increases of b at the focused position. It is also clear from Figure 4 that the cosh-Gaussian beam is much squarer than fundamental Gaussian beam (b = 0).

When CGLBs propagate in rippled density magnetized plasma, the motion of electrons responds at the beat pumps wave frequency. The rippled density couples with the density perturbations induced by relativistic–pondermotive force of the laser beams and give rise to nonlinear current, which leads to generating THz radiation. It is important to mention here that ripples in plasma density play a very important role to enhance the amplitude of THz radiation because huge numbers of electrons participate for the generation of nonlinear current and the amplitude of THz field increases. For the resonant excitation of the THz radiation, the phase-matching condition (k _T+ = k ₁ − k ₂ + q) should be appropriately satisfied. The ripple parameters (n _q0 and q) are the decisive parameter for the exact phase matching. Due to the exact phase matching, energy transfer from the pump laser to the generated THz waves will be maximum, which leads to a considerable enhancement in the efficiency of THz radiation. In addition, the rippled density of plasma increases with the increasing the strength of the magnetic field; therefore, a static magnetic field can be used to enhance the THz radiation.

In order to investigate the resonant THz radiation from rippled density magnetized plasma irradiated by two cross-focused CGLBs, Eq. (32) has been solved numerically and the numerical results are presented in Figures 5a–5c. The maximum value of THz amplitude found at a resonance condition [i.e., ${\rm \omega} _T\approx ({\rm \omega} _{ce}^2 + {\rm \omega} _{p0}^2 )^{1/2}$] irrespective of the laser profile. Figure 5a shows the variation of the normalized amplitude of THz radiation with the normalized transverse distance (r/r ₀) at different values of decentred parameter b. It is seen that, with the increasing value of the decentred parameter, the amplitude of generated THz radiation increases significantly, and its distribution takes place in a wider area in comparison to the Gaussian laser beam (b = 0). The normalized amplitude of THz radiation increases with increasing b because the THz amplitude is directly proportional to the gradient in the intensity of the pump laser beams. Figure 5b presents the normalized amplitude of the THz radiation field as a function of transverse distance (r/r ₀) for different values of magnetic field ω_ce (cyclotron frequency). It is observed that the normalized amplitude of the THz radiation field increases with the increasing strength of the magnetic field (ω_ce). It is also seen that, in magnetized plasma, the magnitude of emitted THz radiation field increases significantly in comparison to non-magnetized plasma. The change in magnitude is of two to three orders from non-magnetized plasma. Therefore, THz wave radiation enhances to a great extent in magnetized plasma. Figure 5c illustrates the effect of density ripple (n _q0) on the amplitude of THz radiation. The amplitude of the emitted THz radiation increases with the increase in the value of n _q0 because the number of electrons involved in the generation of oscillatory nonlinear current increases on increasing ripple amplitude. The magnitude of the generated THz field strongly depends on the decentred parameter, cross-focusing of the laser beams, relativistic–ponderomotive force, amplitude of the density ripple, and applied external magnetic field.

Fig. 5.

(a) Variation of normalized emitted THz amplitude with normalized transverse distance (r/r ₀) for different values of decentered parameter b. (b) Normalized THz amplitude variation with normalized transverse distance (r/r ₀) for different values of ω_ce. (c) Normalized THz amplitude variation with normalized transverse distance (r/r ₀) for different values of the amplitude of density ripples.

The effect of the decentred parameter (b), cyclotron frequency (ω_ce/ω_p0), and density ripple (n _q0) on the efficiency of emitted THz radiation is shown in Figures 6a–6c, respectively. The efficiency of THz radiation depends on the quality of the laser beams, which is directly proportional to the product of the beam divergence and beam waist size of the laser beams, and the amplitude of the generated THz field. Figure 6a shows the efficiency of THz radiation (ɳ) versus the normalized beating frequency (ω_T/ω_p0) for different values of decentred parameter b. There is an enhancement in the efficiency of THz radiation by increasing the value of b. The effect of cyclotron frequency (ω_ce/ω_p0) in the efficiency of THz radiation generation is depicted in Figure 6b. It is evident that the efficiency of THz radiation increases with increasing cyclotron frequency. Moreover, the efficiency in the non-magnetized plasma (i.e., ω_ce/ω_p0 = 0) is very small as compared to magnetized plasma, i.e., the efficiency of THz radiation increases several orders of magnitude in the presence of the magnetic field. Figure 6c depicts the efficiency of THz radiation generation with normalized beating frequency for different values of normalized ripple amplitudes. It is shown that by increasing value of n _q0 from 0.1 to 0.4, the efficiency of THz radiation increases significantly.

Fig. 6.

(a) Variation of efficiency of THz radiation generation with normalized beating frequency (ω_T/ω_p0) for different values of decentered parameter b. Here, ω_p0 = 0.02ω₁, ω_ce = 0.4ω_p0, and n _q0 = 0.1n ₀. (b) Efficiency of the THz radiation generation with normalized beating frequency (ω_T/ω_p0) for different values of ω_ce. Here, b = 0.4, ω_p0 = 0.02ω₁, and n _q0 = 0.1n ₀. (c) Efficiency of the THz radiation generation with normalized beating frequency (ω_T/ω_p0) for different values of the amplitude of density ripples. Here, b = 0.4, ω_ce = 0.4ω_p0, and ω_p0 = 0.02ω₁.

The THz radiation efficiency of the present scheme is of the orders of ~7 × 10⁻⁹, 4.2 × 10⁻⁶, and 1.3 × 10⁻⁵ for optimized values of b, ω_ce, and n _q0, respectively, comparable to that of several other types of schemes. For example, Singh et al. (Reference Singh, Singh and Sharma2013) have reported the conversion efficiency of THz radiation is ~4.5 × 10⁻⁶ by beating of CGLBs in spatially periodic density plasma in the presence of nonlinear ponderomotive force. Mann et al. (Reference Mann, Sajal, Varshney and Sharma2017) observed that the THz radiation efficiency reaches about 10⁻⁴ (for ~10¹⁵ W/cm² laser intensity) by the nonlinear beating of two cosh-Gaussian laser pulses propagating in a corrugated plasma. Varshney et al. (Reference Varshney, Upadhayay, Madhubabu, Sajal and Chakera2018) obtained the efficiency of THz generation up to 15% by CGLBs in axially magnetized collisional plasma under the non-relativistic–ponderomotive regime. THz radiation efficiency up to 8.3% have been obtained by beating of two flat-topped laser beams in a magnetized plasma (Bakhtiari et al., Reference Bakhtiari, Esmaeilzadeh and Ghafary2017). The conversion efficiency of THz radiation of the order of 10⁻³ has been achieved by the beating of two focused Gaussian laser beams in rippled density plasma when both ponderomotive and relativistic nonlinearities are operative (Kumar et al., Reference Kumar, Singh and Sharma2015). Thus, the proposed scheme is quite effective along with the use of two CGLBs, and the THz radiation efficiency from the present scheme is significantly enhanced relative to the efficiency obtained in the previously studied schemes.