1 Introduction

Radiation of terahertz electromagnetic waves, as an important research tool in chemistry, biology and medicine, has stirred much interest in the scientific community [1,2,3]. Various approaches in nonlinear optics, solid-state electronics and laser-plasma interaction are presented and examined for generation of high-power THz radiation. Quantum cascade lasers, uni-travelling-carrier photodiode, photoactivated semiconductor switches, free electron lasers, and p-Germanium laser are examples of today’s THz sources [1, 4, 5].

Various mechanisms responsible for THz radiation in ionized gases are proposed where most of them are based on interaction of laser and plasma. Generation of terahertz radiation through the interaction of laser and plasma has been observed in various studies [6,7,8,9,10,11,12,13]. Emission of THz waves via the interaction of single-laser beam and plasma was first demonstrated by Hamster and coworkers [6]. The efficiency of THz generation in plasma produced by single-color laser pulses is quite low. This efficiency can be enhanced by adding a static electric field across the produced plasma. Dai et al. investigated the polarization properties of the generated THz waves through the interaction of two-color lasers and plasma [7]. Berge et al. studied the generation of THz waves generated by two-color laser filamentation [8]. The effect of the polarization states of ionizing femtosecond two-color pulses on the emitted terahertz radiation in gases was investigated both theoretically, as well as experimentally by Tailliez and his coworkers [9]. Kim, et al. interpreted the generation of THz waves in interaction of two-color lasers and plasma as the result of the nonzero drift velocity of ionized electrons [10]. Hussain et al. studied the generation of high-power THz waves in the interaction of two super-Gaussian laser beams and plasma where plasma was immersed in a DC electric field [11].

Beating of two laser beams was considered as the reason of THz generation in Refs. [12, 13] which are focused on emission of THz waves in the interaction of two-color lasers and magnetized plasma. Mann et al. investigated THz generation through beating of two cosh-Gaussian laser beams propagating in a corrugated magnetized plasma where the effect of pulse slippage was considered in the analysis [12]. Varshney, et al. investigated the emission of terahertz waves in the interaction of two X-mode spatially triangular lasers and magnetized plasma [13]. Malik et al. considered excitation of terahertz radiation based on the beating of two spatial-Gaussian lasers having different frequencies and wave numbers in a spatially periodic density plasma in the presence of an applied static magnetic field [14]. In yet another study, Malik et al. investigated high-intensity terahertz wave generation by frequency-mixing of two super-Gaussian lasers in a corrugated plasma in the presence of external static magnetic field [15]. Singh and Sharma presented a theoretical model for terahertz (THz) radiation generation by two cross-focused Gaussian laser beams in a collision-less magneto-plasma [16]. Their study points to initiation of nonlinearity in current density as a result of ponderomotive nonlinearity leading to THz radiation generation.

The present article focuses on generation of THz waves through the interaction of two-color X-mode lasers and magnetized plasma in weakly relativistic regime where the induction of nonlinear current density is considered as the physical mechanism of THz waves. This mechanism is proposed by Pennano, et al. which is based on solving cold-plasma fluid equations through a perturbative method where the equations are solved to the third order in the vector potentials [17]. This approach is capable of explaining the radiation properties of the generated THz waves and reproducing a number of their radiation properties observed in experimental studies [17]. The motivation of the present work is to use the perturbation technique, among the different mechanisms presented for generation of THz waves, to investigate the interaction of two-color lasers in x-mode with collision-less magnetized plasma for the first time. The mechanisms used in Refs. [11,12,13,14,15,16] and other references are based on the assumption that the interaction of two laser beams with frequencies \({\omega }_{1}\) and \({\omega }_{2}\) generates a THz radiation with frequency \({\omega }_{1}-{\omega }_{2}\). However, this mechanism cannot be applied to the case of two-color lasers. Furthermore, ease and practicality of a two-color laser set-up over two different laser systems is of utmost interest in most experimental studies. The focus is on x-mode beams instead of circular polarized lasers that was undertaken in earlier study [20].

In the present study, emission of THz waves through the interaction of two-color X-mode lasers and magnetized plasma is investigated via the nonlinear coupling of two-color lasers for the first time. Plasma is embedded in a constant magnetic field where the propagation vector of lasers is perpendicular to the direction of the magnetic field. Five different profiles of Gaussian, cosh-Gaussian, flat-top, ring shape and hollow Gaussian are considered as the profile of the propagating lasers. Presence of DC magnetic fields and consideration of different laser profiles affect the radiation properties of the generated THz waves in plasmas.

The structure of the manuscript is as follows: In “Derivation of current density”, generation of THz waves in magnetized plasma is investigated in the context of plasma fluid model with consideration of ponderomotive effect in weakly relativistic approximation. Radiation properties of THz radiation are investigated in “THz radiation characteristics” through solution of electromagnetic wave equations in far-field approximation. The next section (“Discussion”) is devoted to the analysis of properties of THz radiation and discussion of results. Conclusions are drawn in the last section (“Conclusion”).

2 Derivation of current density

The electric fields of two X-mode lasers propagate in z-direction can be written as:The electric fields of two X-mode lasers propagate in z-direction can be written as:

$$\vec{E}_{1} = \frac{1}{2}E_{1} \left( {\hat{x} + {\text{i}}\alpha_{1} \hat{z}} \right){\text{exp}}\{ ik_{1} z - \omega_{0} t)\} + C.C.$$
(1)
$$\vec{E}_{2} = \frac{1}{2}E_{2} \left( {\hat{x} + {\text{i}}\alpha_{2} \hat{z}} \right){\text{exp}}\{ ik_{2} z - 2\omega_{0} t)\} + C.C.$$
(2)

Here, \({k}_{1}\) and \({k}_{2}\) are the wave numbers of laser beams, respectively. The parameters \(\alpha_{1}\) and \(\alpha_{2}\) are defined as:

$$\alpha_{1} = \frac{{\omega_{c} }}{{\omega_{0} }}\frac{{\omega_{p}^{2} }}{{\omega_{0}^{2} - \omega_{c}^{2} - \omega_{p}^{2} }}$$
(3)
$$\alpha_{2} = \frac{{\omega_{c} }}{{2\omega_{0} }}\frac{{\omega_{p}^{2} }}{{4\omega_{0}^{2} - \omega_{c}^{2} - \omega_{p}^{2} }}$$
(4)

where \({\omega }_{p}\) and \({\omega }_{c}\) are plasma and cyclotron frequencies of plasma electrons. These laser beams with frequencies \({\omega }_{0}\) and \({2\omega }_{0}\) propagate in a magnetized plasma which is immersed in a constant magnetic field \({\overrightarrow{B}}_{0}={\widehat{y}B}_{0}\). Interaction of femtosecond laser beams with plasma guides electrons to move in the direction of polarization of the laser electric field and the electron motion induces electron current density. The spatial variation of vector potential in transverse direction modifies density profile of plasma and changes electrons velocity. The coupling of transverse velocity and modulated electron density induces a nonlinear electron current density that is responsible for THz radiation. Considering the phase matching condition, the coupling of second order modified electron density profile (\({n}^{\left(2\right)}\)) and first order modified transverse velocity (\({\overrightarrow{v}}^{(1)}\)) is accounted for THz radiation. The superscripts denote the order of the quantity with respect to the laser amplitude. The current density corresponding to THz radiation in magnetized plasma is given as:

$${\overrightarrow{J}}_{THz}=-e{n}^{\left(2\right)}{\overrightarrow{v}}^{(1)}$$
(5)

where \({n}^{(2)}\) and \({\overrightarrow{v}}^{(1)}\) are the second-order density and first-order velocity of electrons. In order to evaluate the perturbed quantities, one can start from momentum equation:

$$\frac{{d\vec{P}}}{dt} = - e\left[ {\vec{E}_{L} + \frac{{\vec{\nu }}}{c} \times (\vec{B}_{L} + \vec{B}_{0} )} \right]$$
(6)

where \(\overrightarrow{P}=\gamma m\overrightarrow{v}\) and \(\gamma\) is Lorentz factor. It is assumed that the laser intensity is not high; therefore, the relativistic effects are not considered and the Lorentz factor γ = 1. Here, \({\overrightarrow{E}}_{L}\) and \({\overrightarrow{B}}_{L}\) are total electric and magnetic field of two-color lasers. Solution of Eq. (6) yields the electron velocity:

$$\overrightarrow{v}=\frac{1}{2}\left({\overrightarrow{v}}_{1}{e}^{i{k}_{1}z-i{\omega }_{0}t}+{\overrightarrow{v}}_{2}{e}^{i{k}_{2}z-2i{\omega }_{0}t}\right)+c.c.$$
(7)

where

$$\vec{v}_{1} = \left[ {\frac{{ - ieE_{1} \left( {1 + \frac{{\omega_{c} \alpha_{1} }}{{\omega_{0} }}} \right)}}{{m\omega_{0} \left( {1 - \frac{{\omega_{c}^{2} }}{{\omega_{0}^{2} }}} \right)}}} \right] \, \hat{x} + \left[ {\frac{{eE_{1} \left( { - \alpha_{1} - \frac{{\omega_{c} }}{{\omega_{0} }}} \right)}}{{m\omega_{0} \left( {1 - \frac{{\omega_{c}^{2} }}{{\omega_{0}^{2} }}} \right)}}} \right] \, \hat{z}$$
(8)
$$\vec{v}_{2} = \left[ {\frac{{ - ieE_{2} \left( {1 + \frac{{\omega_{c} \alpha_{2} }}{{2\omega_{0} }}} \right)}}{{2m\omega_{0} \left( {1 - \frac{{\omega_{c}^{2} }}{{4\omega_{0}^{2} }}} \right)}}} \right] \, \hat{x} + \left[ {\frac{{eE_{2} \left( { - \alpha_{2} - \frac{{\omega_{c} }}{{2\omega_{0} }}} \right)}}{{2m\omega_{0} \left( {1 - \frac{{\omega_{c}^{2} }}{{4\omega_{0}^{2} }}} \right)}}} \right] \, \hat{z}$$
(9)

Here, \({\upomega }_{\mathrm{c}}= \frac{{e\mathrm{B}}_{0}}{{\mathrm{m}}_{\mathrm{e}}c}\) is the electron cyclotron frequency. Combination of continuity and Poisson equations along with Eq. (6) leads to the following equation for \({n}^{(2)}\):

$$\left( {\frac{{\partial^{2} }}{{\partial t^{2} }} + \omega_{p}^{2} } \right)\frac{{n^{(2)} }}{{n_{0} }} = \frac{e}{mc}\vec{\nabla }.\left( {\vec{\nu } \times \vec{B}_{L} } \right)$$
(10)

Explicit form of \({n}^{(2)}\) can be derived by the application of phase-matching technique as:

$$n^{\left( 2 \right)} = \frac{{n_{0} c^{2} }}{2}\left\{ {\frac{{k_{1}^{2} a_{1}^{2} \left( {1 + \frac{{\omega_{c} \alpha_{1} }}{{\omega_{0} }}} \right)e^{{2i\psi_{1} }} }}{{\left( {4\omega_{0}^{2} - \omega_{p}^{2} } \right)\left( {1 - \frac{{\omega_{c}^{2} }}{{\omega_{0}^{2} }}} \right)}} + \frac{{a_{1}^{*} a_{2} \left( {k_{2} - k_{1} } \right)e^{{i\left( {\psi_{2} - \psi_{1} } \right)}} }}{{2\left( {\omega_{0}^{2} - \omega_{p}^{2} } \right)}} + \left( { - \frac{{k_{1} 1 + \frac{{\omega_{c} \alpha_{1} }}{{\omega_{0} }})}}{{\left( {1 - \frac{{\omega_{c}^{2} }}{{\omega_{0}^{2} }}} \right)}} + \frac{{k_{2} \left( {1 + \frac{{\omega_{c} \alpha_{2} }}{{2\omega_{0} }}} \right)}}{{\left( {1 - \frac{{\omega_{c}^{2} }}{{4\omega_{0}^{2} }}} \right)}}} \right) + c.c.} \right\}$$
(11)

where \({\psi }_{1}={k}_{1}z-{\omega }_{0}t\) and\({\psi }_{2}={k}_{2}z-{2\omega }_{0}t\). The normalized vector potentials are defined as \({\overrightarrow{a}}_{1}=-\frac{e {\overrightarrow{A}}_{1}}{m{c}^{2}}\) and \({\overrightarrow{a}}_{2}=-\frac{e {\overrightarrow{A}}_{2}}{m{c}^{2}}\) where \(\overrightarrow{E}=-\frac{1}{c}\frac{\partial \overrightarrow{A}}{\partial t}-\nabla \phi\). Therefore, Eqs. (7) and (11) lead to the following equation for current density of THz radiation:

$$\vec{J}_{THz} = \frac{nec}{8}a_{1}^{{{*}^{2} }} a_{2} \left( {i{\Gamma }\hat{x} + \beta \hat{z}} \right)\exp \left( {i\left( {k_{2} - 2k_{1} } \right)z} \right) + c.c.$$
(12)

where

$${\Gamma } = \frac{{c^{2} \left( {k_{2} - k_{1} } \right)\left( {1 + \frac{{\omega_{c} \alpha_{1} }}{{\omega_{0} }}} \right)}}{{2\left( {\omega_{0}^{2} - \omega_{p}^{2} } \right)\left( {1 - \frac{{\omega_{c}^{2} }}{{\omega_{0}^{2} }}} \right)}}\left( { - \frac{{k_{1} \left( {1 + \frac{{\omega_{c} \alpha_{1} }}{{\omega_{0} }}} \right)}}{{\left( {1 - \frac{{\omega_{c}^{2} }}{{\omega_{0}^{2} }}} \right)}} + \frac{{k_{2} \left( {1 + \frac{{\omega_{c} \alpha_{2} }}{{2\omega_{0} }}} \right)}}{{\left( {1 - \frac{{\omega_{c}^{2} }}{{4\omega_{0}^{2} }}} \right)}}} \right) + \frac{{c^{2} k_{1}^{2} \left( {1 + \frac{{\omega_{c} \alpha_{1} }}{{\omega_{0} }}} \right)\left( {1 + \frac{{\omega_{c} \alpha_{2} }}{{2\omega_{0} }}} \right)}}{{\left( {4\omega_{0}^{2} - \omega_{p}^{2} } \right)\left( {1 - \frac{{\omega_{c}^{2} }}{{\omega_{0}^{2} }}} \right)\left( {1 - \frac{{\omega_{c}^{2} }}{{4\omega_{0}^{2} }}} \right)}}$$
(13)
$$\beta = \frac{{c^{2} \left( {k_{2} - k_{1} } \right)\left( {\alpha_{1} + \frac{{\omega_{c} }}{{\omega_{0} }}} \right)}}{{2\left( {\omega_{0}^{2} - \omega_{p}^{2} } \right)\left( {1 - \frac{{\omega_{c}^{2} }}{{\omega_{0}^{2} }}} \right)}}\left( { - \frac{{k_{1} \left( {1 + \frac{{\omega_{c} \alpha_{1} }}{{\omega_{0} }}} \right)}}{{\left( {1 - \frac{{\omega_{c}^{2} }}{{\omega_{0}^{2} }}} \right)}} + \frac{{k_{2} \left( {1 + \frac{{\omega_{c} \alpha_{2} }}{{2\omega_{0} }}} \right)}}{{\left( {1 - \frac{{\omega_{c}^{2} }}{{4\omega_{0}^{2} }}} \right)}}} \right) - \frac{{c^{2} k_{1}^{2} \left( {1 + \frac{{\omega_{c} \alpha_{1} }}{{\omega_{0} }}} \right)\left( {\alpha_{1} + \frac{{\omega_{c} }}{{\omega_{0} }}} \right)}}{{\left( {4\omega_{0}^{2} - \omega_{p}^{2} } \right)\left( {1 - \frac{{\omega_{c}^{2} }}{{\omega_{0}^{2} }}} \right)\left( {1 - \frac{{\omega_{c}^{2} }}{{4\omega_{0}^{2} }}} \right)}}$$
(14)

Equation (12) can be recast in the following form:

$$\vec{J}_{THz} = \frac{nec}{{8\alpha }}a_{1}^{{{*}^{2} }} a_{2} {\text{J}}_{THz} {\text{ cosh}}\left( {\frac{{{\text{zb}}}}{L}} \right)\exp \left( {\begin{array}{*{20}c} { - \frac{{\left( {t - \frac{z}{{v_{g} }}} \right)^{2} }}{2}} \\ \tau \\ \end{array} } \right)\exp \left( {i\left( {k_{2} - 2k_{1} } \right)z} \right)\hat{e}_{ \bot } + c.c.$$
(15)

where L is the interaction length and \({v}_{g}\) is the group velocity of THz wave. The parameter b can take values from 0 to 5, which is the representative of the following profiles: (1) Gaussian (b = 0), (2) cosh-Gaussian (b = 1), (3) flat-top (b = 1.45), (4) ring shape (b = 2), and (5) hollow Gaussian (b = 5) [12]. The parameter \(\alpha\) is a normalization factor which will be selected accordingly for each profile.

3 THz radiation characteristics

The terahertz generation is characterized by angular distribution of radiation pattern. The electric field of THz waves is obtained by solution of wave equation:

$${\nabla }^{2}\overrightarrow{E}-\frac{1}{{c}^{2}}\frac{{\partial }^{2}\overrightarrow{E}}{\partial {t}^{2}}=\frac{4\pi }{{c}^{2}}\frac{\partial \overrightarrow{J}}{\partial t}$$
(16)

Integration of Eq. (16) provides the explicit form of radiation electric field:

$${E}_{THz}=-\frac{1}{{c}^{2}}\iiint d{x}^{^{\prime}}d{y}^{^{\prime}}d{z}^{^{\prime}}\frac{{e}^{ik\sqrt{{\left(x-{x}^{^{\prime}}\right)}^{2}+{\left(y-{y}^{^{\prime}}\right)}^{2}+{\left(z-{z}^{^{\prime}}\right)}^{2}}}\frac{\partial {J}_{THz}}{\partial t}}{\sqrt{{\left(x-{x}^{^{\prime}}\right)}^{2}+{\left(y-{y}^{^{\prime}}\right)}^{2}+{\left(z-{z}^{^{\prime}}\right)}^{2}}}$$
(17)

Through application of Fourier transform and the use of far-field approximation, the Fourier transform of \({E}_{THz}\) (\({\tilde{E }}_{THz}\)) can be expressed in spherical coordinates:

$${\tilde{E }}_{THz}=\frac{\pi {r}_{0}^{2}{L}_{0}{J}_{THz}}{\sqrt{2}{c}^{2}\alpha }\frac{{e}^{\frac{i\omega R}{c}}}{R}G\left(\theta ,\omega \right)$$
(18)

\(G\left(\theta ,\omega \right)\) is defined as:

$$G\left( {\theta ,\omega } \right) = i\frac{{J_{1} \left( {\frac{\omega }{c}r_{0} sin\theta } \right)}}{{\frac{{r_{0} }}{c\tau }sin\theta }}\frac{{\omega {\uptau }}}{4}H\left( {\theta ,\omega } \right)e{\text{xp}}\left( { - \frac{{{\upomega }^{2} \tau^{2} }}{4}} \right)$$
(19)

where \({J}_{1}\) is the Bessel function of the first kind of order one and

$$H(\theta ,\omega )=\frac{1}{(f+ig-ih)}{(e}^{i\frac{\left(f+ig-ih\right)L}{2}}-{e}^{-i\frac{\left(f+ig-ih\right)L}{2}})+\frac{1}{\left(-f+ig-ih\right)}{(e}^{i\left(-f+ig-ih\right)}-{e}^{-i\left(-f+ig-ih\right)})+\frac{1}{\left(f-ig-ih\right)}{(e}^{i\left(f-ig-ih\right)}-{e}^{-i\left(f-ig-ih\right)})+\frac{1}{(f+ig+ih)}{(e}^{i(f+ig+ih)}-{e}^{-i(f+ig+ih)})$$
(20)
$$f=\frac{b}{L}$$
(21)
$$g=\Delta k$$
(22)
$$h = \frac{\omega }{c}\left( {cos\theta - \frac{c}{{v_{g} }}} \right)$$
(23)

The magnetic component of THz radiation can be readily evaluated by means of Maxwell’s equation; \(\overrightarrow{\nabla }\times \overrightarrow{E}=-\frac{1}{c}\frac{\partial \overrightarrow{B}}{\partial t}\). According to the energy flow equation of Poynting’s theorem:

$$\int dt \overrightarrow{S}.\widehat{n} =\frac{i{c}^{2}}{4\pi }\int \frac{d\omega }{\omega }\left(\overrightarrow{E}\times \overrightarrow{\nabla }\times \overrightarrow{{E}^{*}}-\overrightarrow{{E}^{*}}\times \overrightarrow{\nabla }\times \overrightarrow{E}\right).\widehat{n}$$

Therefore, the total energy radiated by terahertz waves can be calculated by the following integral

$${W}_{THz}={R}^{2}\int d\Omega \int dt \overrightarrow{S}.\widehat{n}$$
(24)

where \(\overrightarrow{S}\) and \(\widehat{n}\) are Poynting and normal vector, respectively. The time integration of the Poynting vector is attained [12]:

$${\int }_{-\infty }^{\infty }dt \overrightarrow{S}.\widehat{n}=\frac{c}{2\pi }(1-{\left(\widehat{n}.{\widehat{e}}_{\perp }\right)}^{2}){\int }_{0}^{\infty }d\upomega {\left|{\tilde{E }}_{THz}\right|}^{2}$$
(25)

Thus, the total energy of THz radiation per solid angle per frequency interval \(\frac{{{\partial }^{2}W}_{THz}}{\partial\Omega \partial\upomega }\) is obtained by comparison of Eqs. (24) and (25):

$$\frac{{{\partial }^{2}W}_{THz}}{\partial\Omega \partial\upomega }=\frac{c{R}^{2}}{2\pi }{\left|{\tilde{E }}_{THz}\right|}^{2}(1-{\mathrm{sin}}^{2}\theta {\mathrm{sin}}^{2}\phi )$$
(26)

In interaction of two-color lasers and magnetized plasma, the quantity \(\frac{{{\partial }^{2}{\varvec{W}}}_{{\varvec{T}}{\varvec{H}}{\varvec{z}}}}{\partial{\varvec{\Omega}}\partial {\varvec{\upomega}}}\) provides insight into radiation angular distribution and it also relates to the terahertz radiated power.

4 Discussion

The quest for high peak power THz waves has been the subject of much interest over the last decade. Two-color lasers have been employed to generate high-intensity THz waves. In the present study, the interaction of two-color lasers and magnetized plasma leading to generation of THz waves is investigated analytically where X-mode lasers propagate perpendicular to the direction of DC magnetic field. Generation of THz waves is considered as a nonlinear coupling process [17]. Perturbative solution of cold-plasma fluids along with solution of electromagnetic wave equations in far-field approximation leads to the derivation of the quantity \(\frac{{{\partial }^{2}W}_{THz}}{\partial\Omega \partial\upomega }\) which indicates the angular and spectral dependence of radiation. This quantity is dependent upon density and cyclotron frequency of electrons, as well as, laser profile and its polarization. A literature survey indicates that a limited number of articles have analyzed the radiation pattern of THz waves generated in the interaction of laser and plasma. Furthermore, their analysis is limited to filamentation of single-laser in plasma [18, 19]. Previous studies on the interaction of two-color lasers and plasma (which were reviewed in “Introduction”) were focused on the derivation of analytic results for electric field of THz waves through the solution of plasma fluid equations. The analysis does not provide any insight on the angular dependence of generated THz waves [11,12,13]. To qualify the theoretical predictions, in the present study, plots of the radiation patterns of THz waves are presented for various laser and plasma parameters. In order to compare the findings of the present study with those of Ref. [16], the laser parameters were selected in accordance with those of Ref. [14], namely interaction length L = 5 mm, laser spot diameter \({r}_{0}\)=14 \(\mu m\), laser pulse duration \({\tau }_{L}\)=1 ps. The plots of the quantity \(\frac{{{\partial }^{2}W}_{THz}}{\partial\Omega \partial\upomega }\) are shown in Figs. 1, 2, 3 and 4 for various plasma densities, cyclotron frequencies and laser profiles. Comparison of radiation patterns in different conditions will provide sufficient information on the orientation of radiation patterns in specific directions.

Fig. 1
figure 1

Angular distribution of THz radiation for Gaussian profile (b = 0) and different plasma frequencies for \({\lambda }_{0}=\) 800 nm, \({a}_{1}={10}^{-2}\), \({a}_{2}={5\times 10}^{-3}\), \(\omega\)=1 THz, and \(\frac{{\omega }_{c}}{{\omega }_{0}}=0.1\). (a) \({\omega }_{p}=3.05 THz\), (b) \({\omega }_{p}=9.66 THz\), (c) \({\omega }_{p}=71.6 THz\), (d) \({\omega }_{p}=79.6THz\), and (e) \({\omega }_{p}=178.1 THz\)

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Fig. 2
figure 2

Angular distribution of THz radiation for hollow Gaussian profile (b = 5) and different plasma frequencies for \({\lambda }_{0}=\) 800 nm, \({a}_{1}={10}^{-2}\), \({a}_{2}={5\times 10}^{-3}\), \(\omega\)=1 THz, and \(\frac{{\omega }_{c}}{{\omega }_{0}}=0.1\). (a) \({\omega }_{p}=3.05 THz\), (b) \({\omega }_{p}=9.66 THz\), (c) \({\omega }_{p}=71.6 THz\), (d) \({\omega }_{p}=79.6 THz\), and (e) \({\omega }_{p}=178.1 THz\)

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Fig. 3
figure 3

Angular distribution of THz radiation for Gaussian profile (b = 0) and different cyclotron frequencies for \({\lambda }_{0}=\) 800 nm, \({a}_{1}={10}^{-2}\), \({a}_{2}={5\times 10}^{-3}\), \(\omega\)=1 THz, and \({\omega }_{p}=79.6 THz\). (a) \(\frac{{\omega }_{c}}{{\omega }_{0}}=0.1\), (b) \(\frac{{\omega }_{c}}{{\omega }_{0}}=0.12\), and (c) \(\frac{{\omega }_{c}}{{\omega }_{0}}=0.07\)

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Fig. 4
figure 4

Angular distribution of THz radiation for different profiles for \({\lambda }_{0}=\) 800 nm, \({a}_{1}={10}^{-2}\), \({a}_{2}={5\times 10}^{-3}\), \(\omega\)=1 THz, \(\frac{{\omega }_{c}}{{\omega }_{0}}=0.1\), and \({\omega }_{p}=79.6 THz\). a Gaussian profile (b = 0), b cosh-Gaussian (b = 1), c flat-top (b = 1.45), d ring- shape profile (b = 2), and e Hollow Gaussian profile (b = 5)

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The effect of plasma density on radiation pattern of THz waves is examined in Fig. (1), where a Gaussian laser beam with wavelength \({\lambda }_{0}=\) 800 nm along with its frequency-doubled counterpart propagates in a magnetized plasma and radiates 1 THz waves. Figure 1a–e present the radiation patterns for five different selected plasma frequencies 3.05, 9.66, 71.6, 79.6, 178.1 THz, respectively. Different shapes of radiation patterns in plots of Fig. 1 indicate the capability of generation of THz waves in different directions (including backward and forward directions) through the interaction of two-color lasers and magnetized plasma. The wave directionality depends sensitively on plasma density which is consistent with numerical studies of Ref. [17, 20]. For \({\omega }_{p}=79.6 THz\), radiation power is maximized in the backward direction (Fig. 1d); however, its concentration in forward direction is realized for lower plasma frequencies \({\omega }_{p}=\) 3.05 THz and 9.06 THz (Fig. 1a and b). The radiation pattern for plasma frequency \({\omega }_{p}=71.6 THz\) is presented in Fig. 1c, which consists of two lobes in the backward direction. This plot reveals the capability of production of two-lobe pattern in the backward direction by two-laser method, in contrast to laser filamentation method which generates two-lobe pattern in forward direction [18, 19].

Plots of Fig. 2 correspond to radiation pattern of two-color lasers with fundamental wavelength \({\lambda }_{0}=\) 800 nm. The considered plasma densities are the same as those chosen for plots of Fig. 1. Hollow Gaussian profile, which can be produced and controlled by optical techniques [21], is considered for the two-color laser beams. Comparison of Figs. 1 and 2 indicates that the change of laser profile modifies the shape of radiation patterns and induces new lobes in directivity for plasma frequencies of 71.6, and 79.6 THz while no change is noticed for other plasma frequencies. Furthermore, selection of Hollow Gaussian profile leads to reduction of radiation power. The maximum THz power is obtained at \({\omega }_{p}=79.6 THz\) while its minimum is observed at plasma frequency \({\omega }_{p}=3.05THz\).

Cyclotron frequency is another parameter which affects the radiation pattern of THz waves where its dependence is investigated in plots of Fig. 3. The increase in cyclotron frequency leads to the rise in electrons velocity and their current density which amplifies the radiation power, as is evident in Fig. 3a–c. Therefore, the increase in DC magnetic field, which can be achieved by utilization of non-destructive magnets or irradiation of high-power lasers to plasma medium [22], leads to generation of more intense THz waves in the interaction of two-color lasers and plasmas.

Figure 4a–e present the radiation patterns of THz waves generated in the interaction of plasma and two-color lasers with Gaussian, cosh-Gaussian, flat-top, ring shape and hollow Gaussian profiles, respectively. The fundamental laser wavelength \({\lambda }_{0}=\) 800 nm and plasma frequency \({\omega }_{p}=79.6 THz\) were considered for all five profiles. Inspection of the radiation patterns for all profile types indicates that all are concentrated in the backward direction. Therefore, the change of laser profile will not change the general behavior of radiation pattern for this specific plasma density while it will change the magnitude of radiation power. The selection of flat-top profile leads to maximization of radiation power, while hollow Gaussian profile has the opposite effect (the radiation power is minimized). Figure 5 shows the plot of the efficiency of terahertz radiation versus normalized cyclotron frequency (different magnetic field magnitudes). In order to calculate radiation efficiency, the flat-top laser profile was considered at fundamental wavelength \({\lambda }_{0}=\) 800 nm and plasma density \(n=2.7\times {10}^{19}c{m}^{-3}\). The plot clearly indicates that the radiation efficiency increases as the DC magnetic field gets larger. This is due to the fact that the larger magnetic field leads to larger longitudinal electric field components which in turn provide higher radiation power. The spatial distribution of the emitted THz energy integrated over all frequencies is plotted in Fig. 6. The radial extent of the surface in a given direction specifies the amplitude of this quantity. This plot shows that terahertz energy is largest in the backward direction, consistent with the previous findings for radiation patterns.

Fig. 5
figure 5

Efficiency of THz waves versus normalized cyclotron frequency (\(\frac{{\omega }_{c}}{{\omega }_{0}}\)) for flat-top profile; \({a}_{1}=0.01, {a}_{2}=0.005, n=2.7\times {10}^{19}c{m}^{-3},{\lambda }_{0}=800nm\)

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Fig. 6
figure 6

Spatial distribution of the emitted terahertz energy integrated over all frequencies for \({a}_{1}=0.01, {a}_{2}=0.005, n=2.7\times {10}^{19}c{m}^{-3},{\lambda }_{0}=800nm\)

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Plots of Figs. 1, 2, 3 and 4 indicate that the change of plasma density, cyclotron frequency and laser profiles change the radiation patterns of THz waves which shows the sensitivity of directivity to the parameters of lasers and plasmas. These results provide the insight on the appropriate conditions for enhancement of radiation power in specific directions.

5 Conclusion

Interaction of two-color lasers in X-mode and magnetized plasmas leading to THz generation is investigated by consideration of ponderomotive effects in weakly relativistic regime. Perturbative solution of cold-plasma fluid equations to the third order in the vector potentials results in the current density of THz waves. The corresponding electric field is evaluated by solution of wave equations in far-field approximation. The analysis leads to the evaluation of radiation pattern which specifies the frequency and angular dependence of radiated power. The effect of plasma density, cyclotron frequency and laser profile on the directivity of THz waves is investigated through the study of radiation patterns for different laser and plasma parameters. Different shapes of radiation patterns indicate that the directivity of THz waves depends sensitively on laser and plasma parameters. The analysis also shows that radiation power is maximized in the backward direction for a specific range of plasma densities. Increase in DC magnetic field leads to enhancement of radiated power. Examination of different laser profiles indicated that flat-top is the best choice leading to the maximum radiation power while the output is minimum for hollow Gaussian profile. Comparison of radiation patterns in different conditions will be helpful in design of experimental setups.