2. BASIC EQUATIONS FOR EVALUATION OF NONLINEAR CURRENT DENSITY

We consider two cosh-Gaussian lasers with different frequencies (ω₁ and ω₂) and wave numbers ( $\vec k_1 $ and $\vec k_2 $ ) co-propagating in a magnetized plasma $(B_0 \,\hat z)$ having corrugated plasma density given by n = n ₀ + n′,  n′ = n _α0 e ^iαx , where n _α0 is the amplitude of ripple and α is the repetition factor for corrugation ripples. These density ripples may be produced using various techniques involving transmissive ring grating and a patterned mask (Hazra et al., Reference Hazra, Chini, Sanyal, Grenzer and Pietsch2004; Malik et al., Reference Malik, Singh and Sajal2014; Malik & Malik, Reference Malik and Malik2011, Reference Malik and Malik2012; Kuo et al., Reference Kuo, Pai, Lin, Lee, Lin, Wang and Chen2007). Here we can control ripple parameter by changing the groove period, groove structure, and duty cycle in such a grating and by adjusting the period and size of the masks. The profile of laser electric fields is given as

(1) $$\vec E_j = \hat yE_0 \cosh \left( {\displaystyle{{yb} \over {a_0}}} \right)e^{ - {{y^2} / {a_0^2}}} e^{ - i({\rm \omega} _j t - k_j x)}, $$

where j = 1,2 for two lasers, a ₀ is the initial beam width, and b is the decentered parameter. The electric field profile given by Eq. (1) is plotted in Figure 1. One can notice that as a decentered parameter of the cosh-Gaussian laser beam changes from b = 0 to 5, the profile changes its shape in the following sequences: (1) Gaussian (b = 0), (ii) cosh-Gaussian (b = 1), (iii) flat-top (b = 1.45), (iv) ring shape (b = 2), and (v) hollow-Gaussian (b = 5).

Fig. 1. Normalized cosh-Gaussian laser field amplitude as a function of transverse distance y for de-centered parameters 0 < b ≤ 5. Other laser parameters are a ₀ = 0.05 mm,  ω₁ = 2.4 × 10¹⁴ rad/s.

Beating lasers exert a nonlinear ponderomotive force $\vec F_{\rm p} ( = F_{\,px} \hat x + F_{\,py} \hat y)$ on plasma electron having oscillatory velocity (due to the electric field of beating lasers) $\vec v_j = e[ - {\rm \omega} _{\rm c} \hat x + {\rm \omega} _j \hat y]/ m({\rm \omega} _j^2 - {\rm \omega} _{\rm c}^2 )\,E_{\,jy} $ at frequency ω = ω₁ − ω₂ and wave vector $\vec k'' = \vec k_1 - \vec k_2 $ . The components of the ponderomotive force F _px and F _py are as follows:

(2a) $$F_{{\rm p}x} = \displaystyle{{e^2 E_0^2} \over {2m}}\,\left[ {{\rm \delta} _{11} - {\rm \delta} _{12} {\rm \Delta}} \right]\cosh ^2 \left( {\displaystyle{{yb} \over {a_0}}} \right)e^{ - 2{{y^2} / {a_0^2}}},$$

(2b) $$F_{{\rm p}y} = \displaystyle{{e^2 E_0^2} \over {2m}}\,\left[ {{\rm \delta} _{21} - {\rm \delta} _{22} {\rm \Delta}} \right]\cosh ^2 \left( {\displaystyle{{yb} \over {a_0}}} \right)e^{ - 2{{y^2} / {a_0^2}}},$$

where ${\rm \delta} _{12} = [i{\rm \omega} _{\rm c} {\rm \omega} /({\rm \omega} _1^2 - {\rm \omega} _{\rm c}^2 )({\rm \omega} _2^2 - {\rm \omega} _{\rm c}^2 )]$ ,

$$\eqalign{{\rm \delta} _{11} = \left[ {\displaystyle{{i{\rm \omega} _1 k_2} \over {{\rm \omega} _2 ({\rm \omega} _1^2 - {\rm \omega} _{\rm c}^2 )}} - \displaystyle{{i{\rm \omega} _2 k_1} \over {{\rm \omega} _1 ({\rm \omega} _2^2 - {\rm \omega} _{\rm c}^2 )}}} \right. \left. { - \displaystyle{{i{\rm \omega} _{\rm c}^2 k^{\prime \prime}} \over {({\rm \omega} _1^2 - {\rm \omega} _{\rm c}^2 )({\rm \omega} _2^2 - {\rm \omega} _{\rm c}^2 )}}} \right],} $$

$$\eqalign{{\rm \delta} _{21} &= \left[ {\displaystyle{{ - {\rm \omega} _{\rm c} k_1} \over {{\rm \omega} _1 ({\rm \omega} _2^2 - {\rm \omega} _{\rm c}^2 )}} - \displaystyle{{{\rm \omega} _{\rm c} k_2} \over {{\rm \omega} _2 ({\rm \omega} _1^2 - {\rm \omega} _{\rm c}^2 )}}} \right. \cr & \left. {\quad - \displaystyle{{{\rm \omega} _{\rm c} {\rm \omega} _2 k_2} \over {({\rm \omega} _2^2 - {\rm \omega} _{\rm c}^2 )({\rm \omega} _1^2 - {\rm \omega} _{\rm c}^2 )}} - \displaystyle{{{\rm \omega} _{\rm c} {\rm \omega} _1 k_1} \over {({\rm \omega} _2^2 - {\rm \omega} _{\rm c}^2 )({\rm \omega} _1^2 - {\rm \omega} _{\rm c}^2 )}}} \right],} $$

$${\rm \delta} _{22} = \left[ {\displaystyle{{2i{\rm \omega} _1 {\rm \omega} _2} \over {({\rm \omega} _1^2 - {\rm \omega} _{\rm c}^2 )\,({\rm \omega} _2^2 - {\rm \omega} _{\rm c}^2 )}}} \right],$$

$${\rm \Delta} = - \displaystyle{{2y} \over {a_0^2}} + \displaystyle{b \over {a_0}} \tanh \left( {\displaystyle{{yb} \over {a_0}}} \right),$$

where ω_c = eB ₀/m

Figure 2 shows the variation of ponderomotive force as a function of decentered parameter b. The ponderomotive force couples with pre-existing space-charge mode (of negligible amplitude) having potential φ at frequency ω = ω₁ − ω₂ and wave number $\vec k'' = \vec k_1 - \vec k_2 $ , to provide oscillatory velocity to plasma electrons in the presence of transverse magnetic field, given by

(3a) $$v_x = \displaystyle{1 \over {m({\rm \omega} ^2 - {\rm \omega} _{\rm c}^2 )}}\left[ {ie{\rm \omega} \nabla {\rm \varphi} + {\rm \omega} _{\rm c} F_{{\rm p}y} + i{\rm \omega} F_{{\rm p}x}} \right],$$

(3b) $$v_y = \displaystyle{1 \over {m({\rm \omega} ^2 - {\rm \omega} _{\rm c}^2 )}}\left[ { - e{\rm \omega} _{\rm c} (\nabla {\rm \varphi} ) - {\rm \omega} _{\rm c} F_{{\rm p}x} + i{\rm \omega} F_{{\rm p}y}} \right].$$

The oscillations of plasma electron perturb the equilibrium of plasma and, density perturbation due to this can be calculated by substituting Eq. (3) into continuity equation. The density perturbation n = n ^NL + n ^L consists of both linear part (due to space-charge mode) and nonlinear part (due to the ponderomotive force). Here, density perturbation is assumed to be small as compared with the density ripple. Substituting n = n ^L + n ^NL in the Poisson's equation $\nabla ^2 {\rm \varphi} = 4{\rm \pi} ne$ , we obtain

(4) $${\rm \varphi} = - \displaystyle{{4{\rm \pi} e} \over {k^2}} \displaystyle{{n^{{\rm NL}}} \over {\rm \varepsilon}}, $$

where ε = 1 + χ and ${\rm \chi} = - {\rm \omega} _{\rm p}^2 /({\rm \omega} ^2 - {\rm \omega} _{\rm c}^2 )$ . On rearranging Eqs. (3) and (4), we obtain the oscillatory velocity components of plasma electrons:

(5a) $$v_y = \displaystyle{1 \over {m({\rm \omega} ^2 - {\rm \omega} _{\rm h}^{\rm 2} )}}\left[ {{\rm \omega} _{\rm c} F_{{\rm p}y} + i{\rm \omega} F_{{\rm p}x}} \right],$$

(5b) $$v_z = \displaystyle{{ - {\rm \omega} _{\rm c}} \over {m({\rm \omega} ^2 - {\rm \omega} _{\rm h}^2 )}}F_{px} + i\displaystyle{{({\rm \omega} ^2 - {\rm \omega} _{\rm p}^2 )} \over {m({\rm \omega} ^2 - {\rm \omega} _{\rm h}^2 ){\rm \omega}}} F_{{\rm p}y},$$

where ${\rm \omega} _{\rm h} = \sqrt {{\rm \omega} _{\rm c}^2 + {\rm \omega} _{\rm p}^2} $ . In the presence of density ripple n _α0 e ^iαx , the oscillatory velocity excites a nonlinear current density at $({\rm \omega}, \,\vec k_1 - \vec k_2 + \vec {\rm \alpha} ),$ which can be written as

(6) $${\vec J}\,^{{\rm NL}} = - \displaystyle{1 \over 2}n_{{\rm \alpha} 0} e{\vec v}\,_{\rm \omega} ^{{\rm NL}} e^{i{\rm \alpha} x}. $$

It can be seen that the nonlinear current oscillates at frequency difference ω = ω₁ − ω₂ similar to the ponderomotive force, but its wave number $\vec k( = \vec k_1 - \vec k_2 + \vec {\rm \alpha} )$ is different from the ponderomotive force.

Fig. 2. Plot of normalized ponderomotive force F _p/eE ₁ as a function of decentered parameter b.

3. THz AMPLITUDE

Now, we solve the wave equation to find the amplitude of the THz wave

(7) $$\nabla ^2 \vec E - \vec \nabla (\vec \nabla. \vec E) + \displaystyle{{{\rm \omega} ^2} \over {c^2}} (\overline{\overline \varepsilon } \vec E) = - \displaystyle{{4{\rm \pi} i{\rm \omega}} \over {c^2}} \vec J,$$

where $\overline{\overline \varepsilon } $ is the plasma permittivity tensor at ω. We can separate out the coupled $\hat x$ and $\hat y$ components of $\vec E$ . On neglecting second-order term $\hat y$ component of THz radiation is given by

(8) $$\eqalign{& - \displaystyle{{2ikc^2} \over {{\rm \omega} ^2}} \displaystyle{{\partial E_y} \over {\partial z}} + \left( {\displaystyle{{k^2 c^2} \over {{\rm \omega} ^2}} - {\rm \varepsilon} _{yy} + \displaystyle{{{\rm \varepsilon} _{yx} {\rm \varepsilon} _{xy}} \over {{\rm \varepsilon} _{xx}}}} \right) \cr & \quad E_y = \displaystyle{{4{\rm \pi} i} \over {\rm \omega}} J_y^{{\rm NL}} - \displaystyle{{{\rm \varepsilon} _{yx}} \over {{\rm \varepsilon} _{xx}}} \displaystyle{{4{\rm \pi} i} \over {\rm \omega}} J_x^{{\rm NL}},} $$

where ${\rm \varepsilon} _{yy} \!=\! [1 \!\!-\! ({\rm \omega} _{\rm p}^2 /{\rm \omega} _{}^2 \!\!-\! {\rm \omega} _{\rm c}^2 )]$ and ${\rm \varepsilon} _{xy} \!=\! - i[{\rm \omega} _{\rm c} {\rm \omega} _{\rm p}^2 /({\rm \omega} /{\rm \omega} _{}^2 \!-\! {\rm \omega} _{\rm c}^2 )]$ are components of the dielectric tensor. One can conclude from Eq. (8) that THz radiation generation has to satisfy the following dispersion relation for exact phase-matching condition in the corrugated magnetized plasma, which includes the corrugation factor and suggests that the maximum energy transfer from beating lasers to oscillating electrons will take place at resonance condition leading to the maximum THz radiation

(9a) $$\displaystyle{{k^2 c^2} \over {{\rm \omega} ^2}} = \left \vert {{\rm \varepsilon} _{yy} - \displaystyle{{{\rm \varepsilon} _{xy} {\rm \varepsilon} _{yx}} \over {{\rm \varepsilon} _{xx}}}} \right \vert.$$

The corrugation factor (periodicity of density ripples) required for the fine tuning of maximum energy transfer is calculated utilizing Eq. (9a) along with resonance conditions $\vec k = \vec k_1 - \vec k_2 + {\rm \alpha} $ and ω = ω₁ − ω₂ ≈ ω_h. The corrugation factor comes out as follows:

(9b) $${\rm \lambda} _{\rm c} = \displaystyle{{2{\rm \pi}} \over {\rm \alpha}} = \displaystyle{{2{\rm \pi} \,c} \over {{\rm \omega} [ \vert ({\rm \varepsilon} _{yy} - {{{\rm \varepsilon} _{xy} {\rm \varepsilon} _{yx}} / {{\rm \varepsilon} _{xx}}} )^{1/2} - 1 \vert ]}}\,\,.$$

Varshney et al. (Reference Varshney, Sajal, Singh, Kumar and Sharma2013, Reference Varshney, Sajal, Sharma, Chauhan and Kumar2015) have shown that the wave number of density ripples (α) increases with THz frequency (ω), attains maximum value at ${\rm \omega} \sim {\rm \omega} _{\rm h} $ (resonance), and then starts decreasing. On combining Eqs. (8) and (9), we obtain the phase-matched normalized amplitude of THz radiation as follows:

(10) $$\eqalign{& \left \vert {\displaystyle{{E_y} \over {E_0}}} \right \vert = \displaystyle{1 \over {4k^{\prime}}}\displaystyle{{n_{q0}} \over {n_0}} \displaystyle{1 \over {{\rm \omega} ^{\prime}({\rm \omega} ^{\prime 2} - {\rm \omega}_{\rm h}^{\prime 2} )}} \cr & \quad x^{\prime}\left[ {\left \vert {\left( {\displaystyle{{{\rm \omega} ^{\prime 2} - 1} \over {{\rm \omega} ^{\prime}}} - \left \vert {\displaystyle{{{\rm \varepsilon} _{xy}} \over {{\rm \varepsilon} _{xx}}}} \right \vert {\rm \omega} ^{\prime}_{\rm c}} \right)F_{{\rm p}y} + \left( {{\rm \omega} ^{\prime}_{\rm c} - {\rm \omega} ^{\prime}\left \vert {\displaystyle{{{\rm \varepsilon} _{yx}} \over {{\rm \varepsilon} _{xx}}}} \right \vert} \right)F_{{\rm p}x}} \right \vert} \right],} $$

where k′ = kc/ω_p, ω′ = ω/ω_p, ω′_c = ω_c/ω_p, and x′ = xω _p/c. Substituting the ponderomotive force into Eq. (10), we obtain normalized amplitude of THz wave as follows:

(11) $$\eqalign{\left \vert {\displaystyle{{E_y} \over {E_0}}} \right \vert &=\! \left[ \matrix{{\rm \xi}_1^{\prime} {\rm \delta}_{11}^{\prime} +\! \displaystyle{{2y^{\prime}{\rm \xi}_1^{\prime} {\rm \delta}_{12}^{\prime}} \over {a_0^{\prime 2}}} - \displaystyle{{b{\rm \xi}_1^{\prime} {\rm \delta}_{12}^{\prime}} \over {a_0^{\prime}}} \tanh \left( {\displaystyle{{y^{\prime}b} \over {a_0^{\prime}}}} \right) \hfill +\! {\rm \xi}_2^{\prime} {\rm \delta}_{21}^{\prime} \!+ \displaystyle{{4y^{\prime}{\rm \xi}_2^{\prime} {\rm \delta}_{22}^{\prime}} \over {a_0^{\prime 2}}} \hfill \cr - \displaystyle{{b{\rm \xi}_2^{\prime} {\rm \delta}_{22}^{\prime}} \over {a_0^{\prime}}} \tanh \left( {\displaystyle{{yb} \over {a_0^{\prime}}}} \right) \hfill} \right] \cr & \quad \cosh ^2 \left( {\displaystyle{{y^{\prime}b} \over {a_0^{\prime}}}} \right)e^{ - 2\left( {{{y^{\prime 2}} / {a_0^{\prime 2}}}} \right)},} $$

where y′ = yω _p/c, a′₀ = a ₀ω_p/c, ${\rm \delta} '_{12} = {\rm \delta} _{12} /{\rm \omega} _{\rm p}^{\rm 2} $ , δ′₁₁ = δ₁₁/cω _p, δ′₂₁ = δ₂₁/cω _p, ${\rm \delta} '_{22} = {\rm \delta} _{22} /{\rm \omega} _{\rm p}^2 $ ,

$${\rm \xi}_1^{\prime} = c{\rm \omega} _{\rm p} \left[ {\displaystyle{1 \over {8k^{\prime}}}\displaystyle{{n_{{\rm \alpha} 0}} \over {n_0}} \left( {\displaystyle{{{\rm \omega} ^{\prime}{\rm \omega}_2^{\prime}} \over {{\rm \omega} ^{\prime 2} - {\rm \omega}_{\rm h}^{\prime 2}}} x^{\prime}v_2^{\prime} \left\{ {{\rm \omega} ^{\prime} + {\rm \omega}_{\rm c}^{\prime} \left \vert {\displaystyle{{{\rm \varepsilon} _{xy}} \over {{\rm \varepsilon} _{xx}}}} \right \vert} \right\}} \right)} \right],$$

$${\rm \xi}_2^{\prime} = c{\rm \omega} _{\rm p} \left[ {\displaystyle{1 \over {8k^{\prime}}}\displaystyle{{n_{{\rm \alpha} 0}} \over {n_0}} \left( {\displaystyle{{{\rm \omega} ^{\prime}{\rm \omega}_2^{\prime}} \over {{\rm \omega} ^{\prime 2} - {\rm \omega}_{\rm h}^{\prime 2}}} x^{\prime}v_2^{\prime} \left\{ {{\rm \omega}_{\rm c}^{\prime} + \displaystyle{{{\rm \omega} ^{\prime 2} - 1} \over {{\rm \omega} ^{\prime}}}\left \vert {\displaystyle{{{\rm \varepsilon} _{yx}} \over {{\rm \varepsilon} _{xx}}}} \right \vert} \right\}} \right)} \right].$$

Equation (10) exhibits that greater the value of corrugation amplitude (n _α0/n ₀), greater will be the amplitude of excited THz radiation due to larger number of oscillating electrons involved in the process of nonlinear current generation at $({\rm \omega}, \,\vec k)$ . In Figure 3a and 3b, the normalized THz amplitude is plotted as a function of normalized THz frequency (ω/ω_p) for different laser profiles (corresponding to decentralized parameter b = 0, 0.5, 1.0, 1.4, 3, 4, 5) respectively at ω_c/ω_p = 0.2 (B ₀ = 71 kG) and ω_c/ω_p = 0.3 (B ₀ = 107 kG). The maximum THz amplitude is obtained at resonance condition ω ≈ ω_h irrespective of laser profiles. It can be attributed to the factor $({\rm \omega} ^2 - {\rm \omega} _{\rm h}^2 )$ present in the expression of nonlinear current $\vec J_{{\rm NL}} $ given by Eq. (6) (in ${\vec v}\,_{\rm \omega}^{{\rm NL}} $ ). This factor is introduced in Eq. (4) (potential φ of space charge mode of plasma) due to the presence of perpendicular magnetic field. In the presence of magnetic field, the potential of space-charge mode will achieve its peak value at ω ≅ ω_h and energy transfer in beat wave process will be maximal. This maximally developed space-charge mode along with density ripple gives rise to strong nonlinear current responsible for the maximum THz amplitude. The amplitude of THz wave decreases as one moves away from the resonance position. At the same time α has maximum value (or λ_c is small) at resonance ${\rm \omega} \sim {\rm \omega} _{\rm h} ;$ thus, density ripples with small corrugation factor (λ_c) are best suited for efficient THz excitation. This result matches with the observation of Antonsen et al. (Reference Antonsen, Palastra and Milchberg2007), who proposed that the phase-matching requirements for efficient energy transfer from laser pulse to THz can be matched in a parabolic plasma channel (in radius) by z sequence of δ-function peaks with period d and strength triangle Δ, in axial distance, that is, $n_0 (r,z) = n_{00} [1 + {{r^2} / {2r_{{\rm ch}}^2}} + {\rm \Delta} \sum\limits_{l = - \infty} ^\infty {{\rm \delta} (z - ld)} $ .

Fig. 3. (a) Plot of normalized THz amplitude (E _y /E ₁) as a function of normalized THz frequency (ω/ω_p) for different values of decentered parameter 0 < b ≤ 1.4. All other parameters are same as Figure 1. (b) Plot of normalized THz amplitude (E _y /E ₁) as a function of normalized THz frequency (ω/ω_p) for different values of decentered parameter 1.5 < b ≤ 5. All plasma parameters are same as Figure 1.

Here, δ function-type axial coagulated plasma density acts as ripples. This inhomogeneity couples with the density perturbation provided by ponderomotive force and gives rise to a nonlinear current responsible for THz generation. Figure 3a shows that the THz amplitude increases with b for b < 1.5. The maximum amplitude (E _y ) is obtained for Gaussian laser profile (b = 0) at resonance condition ω ≈ ω_h. For higher value of decenterd parameter (1.5 ≤ b ≤ 5), the normalized THz amplitude increases with b and the maximum amplitude is again achieved at THz frequency ${\rm \omega} \sim {\rm \omega} _{\rm h} $ as shown in Figure 3b. THz amplitude (E _y ) achieves maximum value for hollow-Gaussian beating lasers (with b = 5). The variation of THz amplitude with b can be attributed to the variation of nonlinear ponderomotive force, which depends upon the shape of beating lasers beams as shown in Figure 2. Figure 3 also exhibits that the normalized THz field amplitude decreases and peak shifts toward higher value of THz due to new resonance frequency $\sqrt {{\rm \omega} _{\rm p}^2 + {\rm \omega} _{\rm c}^2} $ by varying applied magnetic field from B ₀ = 71 kG (ω_c/ω_p = 0.2) to B ₀ = 107 kG (ω_c/ω_p = 0.3). Thus, THz amplitude (at a particular THz frequency) can be tuned by changing dc magnetic field.

Finally, we calculate the efficiency (η) of the present scheme. Average EM energy stored per unit volume in electric and magnetic fields are given by the relations (Malik et al., Reference Malik, Malik and Nishida2011, Reference Malik, Malik and Stroth2012b ; Singh et al., Reference Singh, Singh and Sharma2013; Varshney et al., Reference Varshney, Sajal, Singh, Kumar and Sharma2013, Reference Varshney, Sajal, Sharma, Chauhan and Kumar2015) $\left\langle {W_{E{\rm i}}} \right\rangle = \displaystyle{1 \over {8{\rm \pi}}} {\rm \varepsilon} _0 \displaystyle{\partial \over {\partial {\rm \omega} _{\rm i}}} \left[ {{\rm \omega} _{\rm i} \left( {1 - {{{\rm \omega} _{\rm p}^2} / {{\rm \omega} _{\rm i}^2}}} \right)} \right]\left\langle {\left\vert {E_{\rm i}} \right\vert^2} \right\rangle $ and ${\rm } \,\left\langle {W_{B{\rm i}}} \right\rangle = {{\left\langle {\left\vert {B_{\rm i} ^2} \right\vert} \right\rangle} / {8{\rm \pi} {\rm \mu} _0}}, $ respectively; where 〈B _i〉 = k〈E _i〉/ω_i. The energy densities of the incident lasers (W _pump) and emitted radiation (W _THz) are as follows:

(12) $$W_{{\rm pump}} = \displaystyle{{{\rm \varepsilon} _0 a_0} \over 2}\left \vert {E_0} \right \vert ^2 \sqrt {\displaystyle{{\rm \pi} \over 2}} \left\{ {1 + \exp \left[ {\displaystyle{{b^2} \over 2}} \right]} \right\}$$

and

(13) $$W_{{\rm THz}} = {\rm \varepsilon} _0 \left \vert {E_0} \right \vert ^2 \left[ \matrix{\displaystyle{1 \over 2}\sqrt {\displaystyle{{\rm \pi} \over 2}} \left( {{\rm \xi} _1^2 {\rm \delta} _{11}^2 + {\rm \xi} _2^2 {\rm \delta} _{21}^2} \right)a_0 \{ 1 + e^{b^2 /2} \} \hfill \cr + \displaystyle{1 \over 2}\sqrt {\displaystyle{{\rm \pi} \over 2}} \left( {{\rm \xi} _1^2 {\rm \delta} _{12}^2 + 4{\rm \xi} _2^2 {\rm \delta} _{22}^2} \right)a_0 \{ 1 + (1 + b^2 )e^{b^2 /2} \} \hfill \cr - \displaystyle{{b^2} \over {a_0^{}}} \sqrt {\displaystyle{{\rm \pi} \over 2}} \left( {{\rm \xi} _1^2 {\rm \delta} _{12}^2 + {\rm \xi} _2^2 {\rm \delta} _{22}^2} \right)\{ - 1 + e^{b^2 /2} \} \hfill} \right].$$

The normalized efficiency of THz radiation generation is evaluated

(14) $$\eqalign{& {\rm \eta} = \displaystyle{{W_{{\rm THz}}} \over {W_{{\rm pump}}}} = \left[ {\left( {{\rm \xi}_1^{\prime 2} {\rm \delta}_{11}^{\prime 2} + {\rm \xi}_2^{\prime 2} {\rm \delta}_{21}^{\prime 2}} \right) + \displaystyle{1 \over {a_0^{\prime 2}}} \left( {{\rm \xi}_1^{\prime 2} {\rm \delta}_{12}^{\prime 2} + 4{\rm \xi}_2^{\prime 2} {\rm \delta}_{22}^{\prime 2}} \right)} \vphantom{\quad \displaystyle{{\{ 1 + (1 + b^2 )e^{b^2 /2} \}} \over {\{ 1 + e^{b^2 /2} \}}} + \displaystyle{{b^2} \over {a_0^{\prime}}} \left( {{\rm \xi}_1^{\prime 2} {\rm \delta}_{12}^{\prime 2} + {\rm \xi}_2^{\prime 2} {\rm \delta}_{22}^{\prime 2}} \right)\displaystyle{{\{ 1 - e^{b^2 /2} \}} \over {\{ 1 + e^{b^2 /2} \}}}} \right.. \cr & \left. {\quad \displaystyle{{\{ 1 + (1 + b^2 )e^{b^2 /2} \}} \over {\{ 1 + e^{b^2 /2} \}}} + \displaystyle{{b^2} \over {a_0^{\prime}}} \left( {{\rm \xi}_1^{\prime 2} {\rm \delta}_{12}^{\prime 2} + {\rm \xi}_2^{\prime 2} {\rm \delta}_{22}^{\prime 2}} \right)\displaystyle{{\{ 1 - e^{b^2 /2} \}} \over {\{ 1 + e^{b^2 /2} \}}}} \right].} $$

Figure 4 shows that the efficiency η of the present scheme increases with decentered parameter (b) and decrease with laser beam width (a ₀). The efficiency can be optimized with an applied static magnetic field corresponding to THz frequency. An efficiency ~20% is achieved by frequency mixing of two hollow-Gaussian laser beams at beam width $a_0 \sim 3c/5{\rm \omega} _{\rm p} $ and dc magnetic field $B_0 \sim 107\,{\rm kG}$ . Wu et al. (Reference Wu, Sheng and Zhang2008) achieved energy conversion efficiency ~0.005 in inhomogeneous plasma at peak laser intensity 5.48 × 10¹² W/cm², which is much lower than the present model. Malik et al. (Reference Malik, Malik and Nishida2011, Reference Malik, Malik and Stroth2012b ) have reported the conversion efficiency ~0.002 and ~0.006 by beating of two spatial-Gaussian lasers and two super Gaussian lasers, respectively. Varshney et al. (Reference Varshney, Sajal, Singh, Kumar and Sharma2013) have reported the conversion efficiency $\sim\!10^{ - 3} $ by beating of two planer x-mode lasers in a magnetized plasma. Further Varshney et al. (Reference Varshney, Sajal, Sharma, Chauhan and Kumar2015) improved one order of conversion efficiency and achieved ~10⁻² by photomixing of two super-Gaussian laser. Malik et al. (Reference Malik, Singh and Sajal2014) obtain efficiency about 15% by frequency mixing of two top-hat lasers in the magnetized plasma.

Fig. 4. Plot of THz efficiency (η) as a function of normalized laser beam width a _oω_p/c for different values of b and ω_c/ω_p. Other normalized parameters are same as in Figure 1.