Terahertz wave excitation by nonlinear coupling of intense laser field with magnetized plasma

In this article, the plasma density, the laser intensity and the external magnetic field are playing vast roles firstly to generate a coherent THz wave then to enhance the stability of generated THz. Due to the relativistic increase of electron mass, the relativistic self-focusing of a right circular polarized (RCP) laser beam inside magnetized plasma will occur which leads to raising the laser power to enough limits for exciting the terahertz wave. By fulfilling the energy–momentum conservation conditions, a terahertz wave frequency at the difference between the laser pump wave frequency and plasma wave frequency is obtained. More stabilization and higher power (reaching to tens of gigawatts) of terahertz field amplitude have been observed whenever the plasma density, the laser intensity, and the external magnetic field are increased. Better results are recorded at high THz frequency (5 THz) compared with low THz frequency (1 THz).


Introduction
Recently terahertz science has attracted more attention from researchers due to its wide usage in different scientific fields such as medical, biological, chemical applications, and others (Cherkasova et al. 2016;Zheng et al. 2006;Ferguson and Zhang 2002;Shen et al. 2005). Due to the unique properties of THz wave, medical imaging has to earn nowadays expand interests. As long as the THz radiation considers nonionizing radiation, so using THz rays in body human imaging is safer compared with X-rays imaging (Yan et al. 2022;Zhang et al. 2021). The THz radiation has different penetration abilities depending upon the types of materials. For example, metals have a very short penetration depth and high reflectivity in the THz region, hence the metallic objects completely block the THz radiation, and in contrast, the plastic objects are partially transparent. Hence the detection technologies of concealed weapons, hidden explosives, and biological terror materials have been improved greatly by using THz imaging (Zhang et al. 2018;Wang et al. 2017). Depending on nonlinear laser plasma interaction to excite an electromagnetic wave in terahertz scope, many methods have been suggested. Some of these methods have operated the ponderomotive nonlinear force exerted by an intense laser field on electrons Malik and Singh 2015;Singh and Malik 2015;Gupta and Kumar 2020a). Other methods have exploited the relativistic movement of electrons due to high intense laser field which leads to an increase in the electron effective mass hence to modifying the plasma refractive index (Hussain et al. 2016(Hussain et al. , 2014Liao et al. 2017;Gupta and Kumar 2020b;Abdullah et al. 2015). Coherent, high power and stable terahertz wave are significant requirements for most terahertz applications so many efforts have been done to enhance it (Hassan et al. 2019;Ge et al. 2014;Singh and Sharma 2013;Priyanka Rawat1 2017). Motivating the ponderomotive nonlinearity, Sharma et al.  have obtained terahertz in the gigawatt range by increasing the external longitudinal magnetic field. A. Hematizadeh et al. (Hematizadeha et al. 2016) have proposed a scheme to generate terahertz waves by beating two laser beams inside the plasma with a spatially periodic density taking into account the electron-neutral collisions. Munther et. al. (Hassan and Sharif 2017) have enhanced terahertz waves by controlling the laser beam diameter and laser beam intensity in a relativistic nonlinearity regime. N. Gupta and S. Kumar (Gupta and Kumar 2021a), reported a theoretical study on optical self-action effects of intense q-Gaussian laser beams interacting with collision less plasma with axial density ramp. As well as the generation of second harmonics of relativistically self-focused q-Gaussian laser beams in under dense plasma with axial density ramp was investigated by N. Gupta and S. Kumar (Gupta and Kumar 2021b). Bakhtiari et al. (Bakhtiari et al. 2017) proposed to generate a terahertz (THZ) radiation in plasma by striking two laser beams with present of a static magnetic field which applied to the plasma in a parallel or perpendicular direction. Where they showed that the presence of a magnetic field enhances the power and efficiency of THz radiation, and the enhancement of the perpendicular magnetic field is greater than that of the parallel one. Whereas Purohit et al. (Purohit et al. 2019) studied the generation of (THz) radiation by knocking off two cosh-Gaussian laser beams in rippled density magnetized plasma under the relativistic-ponderomotive system. The amplitude, and efficiency of the generated terahertz radiation, were found to increase significantly with the increase in the values of the decentered parameter, magnetic field, and density ripple. The THz wave production is widely used in several application which need high THz wave field stability, especially in imaging, medical diagnosis, and chemical and biological identification (Amini et al. 2021;Gill et al. 2018).
In this study, the plasma density has a vast effect to generate and enhance the terahertz wave considering the nonlinear relativistic force exerted by the laser field on electrons inside the plasma medium.
In Sects. 2 and 3 the plasma dielectric tensor and laser beam self-focusing have been derived respectively taking into our account the relativistic nonlinearity. Section 4 has focus on extracting suitable expressions for terahertz wave generation. The discussion of numerical results and the most prominent conclusions have been introduced in Sects. 5 and 6 respectively.

Plasma dielectric tensor in relativistic nonlinearity
Proposing a propagation right circular polarized (RCP) laser beam inside a uniform magnetized plasma in presence of the external applied magnetic field B 0 . The laser beam and external applied magnetic field are both aligned along the z-direction. The electric field vector ⇀ E 0 of the RCP laser beam is given by Ginzburg (1964): where 0 and k 0 are the angular frequency and wave vector of RCP laser beam respectively, here The equation of the relativistic motion of the electron in presence of high intense RCP laser beam is given by Sharma et al. (2010): where the relativistic factor = 1 √ 1 − 2 0 c 2 and the oscillation velocity imparted from RCP laser beam By following Munther et al. procedure (Hassan et al. 2012), the relativistic factor will become as: where r is the relativistic nonlinearity coefficient given by: The propagating equation of (RCP) laser beam through magnetized plasma can be given as (Sharma 1978): The components of the relativistic dielectric tensor corresponding to (RCP) laser beam are taking the following form (Hassan and Sharif 2017): (1) where as the effective dielectric constant eff will become: where pe = 4 n e e 2 m e 1 2 is the electron plasma frequency.
Putting relativistic factor of Eq. (3) in Eq. (6) so the effective dielectric constant eff may be written as: It is clear that the effective dielectric constant eff includes a linear part 0 and a nonlinear part 2 A 0 A * 0 which may be written as follows: The nonlinear part 2 A 0 A * 0 of the effective dielectric constant is obtaining due to the high intensity of the laser beam field so it will vanish at low intensity.

Relativistic self focusing of RCP laser beam
Since the RCP laser field is varying along external magnetic field (i.e. z-direction) larger than its variation via wave front plane (i.e. x-y direction) so one may postulate that the electromagnetic wave inside magnetized plasma is transverse wave thus no space charge occur therefore (Sodha et al. 1974): Using Eq. (10) and Eq. (5) so one may obtain: Equation (11) is a zero-order approximation that may be exploited to solve (Eq. 4) to be (Hasson et al. 2010): where the product of nonlinear parts with is a complex amplitude thus Eq. (12) may become as: Now assuming a two dimensional Gaussian beam (ie y = 0 ) and introducing an eikonal A � 0 = A 0 0 exp ik 0 S , where A 0 0 and S are real and phase functions of RCP laser beam inside the magnetic plasma, hence Eq. (13) can be written as (Sodha et al. 1974): It is important to mention that the two above equations (Eq. 14 and Eq. 15) have been written after the separation of real and imaginary parts.
Introducing the paraxial ray approximation theory, where, x 2 << x 2 0 f 2 0+ , the real function A 0 0 and phase function S of RCP laser beam can be written as: where f 0+ and x 0 represent the beam width parameter and initial diameter of the laser beam respectively. By substituting Eq. (16) in Eq. (14), so (z) will become as (Sodha et al. 1980): Now putting Eq. (17) and Eq. (18) in Eq. (15) and assuming that ( f 0+ = 1 and df 0+ dz = 0 at z = 0 ) so one may obtain: Equation (19) represents the variation of the beam width parameter f 0+ of RCP laser beam through it is propagating inside plasma. This equation illustrates the competition between the natural diffraction effect (first term on the right hand side of Eq. (19)) and nonlinear self-focusing effect (second term on the right hand side of Eq. (19)) respectively.

Terahertz generation theory
To excite a terahertz frequency E T , T , k T by the nonlinear interaction between a laser beam E 0 , 0 , k 0 and a rippled plasma wave E RP , RP , k RP , the phase-matching conditions between the above three waves should be satisfied. These conditions are the energy conservation condition 0 = RP + T and the momentum conservation condition The electric field of the rippled plasma wave E RP and the terahertz electric field E T can be written as: where A T (x, y, z) = E Tx + iE Ty and A RP (z) are the amplitudes of the terahertz wave and the rippled plasma wave respectively.
Due to the plasma wave field E RP , the plasma electrons will gain an oscillating velocity RP given as follows: So the primary plasma density n 0 will modify to become where ñ RP is the amplitude of the electron rippled density. Now firstly introduce the continuity and the momentum equations as follows  where m j , n j , j are the mass, the particle density, and the velocity of species j = i, e respectively, and secondly following Munther et al. procedure (Hassan et al. 2012) thus the final equation of terahertz field excitation can be written as: To extract the relativistic factor so a similar technique in Sect. 2 will be used therefore Eq. (26) may become as: The nonlinear self-focusing of laser beam and terahertz field excitation through magnetized plasma have been studied by computing Eq. (19) and Eq. (27) numerically.

The numerical results and discussion
This study is based on the nonlinear interaction between magnetized plasma and the fundamental mode of Carbon Dioxide (CO 2 ) pulsed laser. The typical parameters which have been used are: • The angular frequency of CO 2 laser 0 = 1.778 × 10 14 rad. sec −1 corresponding to the frequency f = 2.83 × 10 13 Hz • The initial laser beam intensities are I ≈ (5, 7, 9) × 10 18 W∕cm 2 corresponding to normalized vector potential 0 = (0.7, 0.8, 0.9) where a 0 = 0.85 × 10 −9 √ I[W∕cm 2 ] × [ m] (Umstadter 2003): • The initial laser beam radius is x 0 = 30 m . • The plasma densities are n e = (6, 7, 8) × 10 18 cm −3 corresponding to the plasma frequencies pe = 0.7 0 , pe = 0.8 0 , pe = 0.9 0 respectively. • The external longitudinal magnetic fields are in order of (B ≈ (50, 100, 150) kG) corresponding to the cyclotron frequencies ce = 0.005 0 , ce = 0.015 0 , ce = 0.02 0 The RCP laser beam will show a relativistic self-focusing if and only if the initial incident power of the laser beam is greater than the critical power P cr ≈ 17 × 10 9 × 0 pe 2 (Sprangle et al. 1987). Figure 1 demonstrates an oscillating self-focusing behavior of a laser beam through its progressing inside the plasma medium. The laser beam self-focusing is appearing stronger as long as the plasma frequency is increased. This is because the plasma frequency increase will lead to a rise in the nonlinear part of the dielectric constant 2 (Eq. 9) thus increase the nonlinear self-focusing effect (second term on the right-hand side of Eq. 19). At high plasma frequency magnitude ( i. e. pe = 0.9 0 ), the RCP laser beam will nearly propagate in a constant diameter. In general, similar behavior of laser beam self-focusing has been observed when the laser intensity and external magnetic fields are increased (Figs. 2 and 3). The terahertz wave, that generated due to the interaction of high power RCP laser beam with magnetized plasma, will be increased with the increase of plasma density Fig. 4. This may be understood as following: the increasing of plasma density will impose more nonlinear interacted electrons which it in the role will contribute to increasing the THz wave field. The terahertz wave field is exhibiting the best stability at higher plasma frequency pe = 0.9 0 due to the higher and best stability of laser beam self-focusing pe = 0.9 0 (Fig. 1).
both the increment of the initial laser intensities and the increment of the applied magnetic fields result in to increase the self-focusing (Figs. 5 and 6 respectively), hence an increase in the laser beam intensity through plasma which in turn leads to increasing of terahertz field amplitude. It has been noticed that the high THz wave stability are corresponding to high values of the initial laser intensities and the increment of the applied magnetic fields.
By satisfying the phase matching conditions of the interacted waves namely RCP laser beam, plasma wave, and terahertz wave, one may tune selected terahertz frequencies. In this article, two distinguished terahertz frequencies have been excited at T∕ 2 = 1 THz and T∕ 2 = 5 THz . Figure 7 Illustrates that the terahertz wave field at T∕ 2 = 5 THz is higher with the best stability compared with that at T∕ 2 = 1 THz . This may be reasonable as follows: higher terahertz frequency is corresponding to longer penetration distance through plasma thus higher excitation of a terahertz wave field.

Conclusions
The RCP laser beam is undergoing an oscillating self-focusing phenomenon when the nonlinear divergent term is competing the natural diffraction term (Eq. 19). The excitation of a terahertz wave is more expected due to the laser beam power density be highest which corresponds to the self-focusing state (Eq. 27).
One may deduce that the generated terahertz wave field will become highest with more stabilization when the plasma density, the laser intensity and the external magnetic field are increased. This may be understood because the increase of the plasma density, the laser intensity, and the external magnetic field are corresponding to the highest and more stabilization of laser beam self-focusing. During only a few diffraction length distances, the power of an excited terahertz wave field is in order of gigawatts with high stability. This will be noted largely with the best stability at T∕ 2 = 5 THz comparing with that at T∕ 2 = 1 THz . This mechanism of THz wave production is widely used in several application which need high THz wave field stability, especially in imaging, medical diagnosis, and chemical and biological identification. It is important to mention that nonlinear interaction between highly intense laser and plasma will give high THz power without of ionization breakdown limitation since the plasma is an ionization gas.