E�cient THz Generation by Nonlinear Interaction of Gaussian Laser Beam With the Anharmonic and Rippled CNTs Aligned Vertically in the Array

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The Gaussian laser beam passing through the CNTs exerts a nonlinear ponderomotive force on the electrons of CNTs and provides them resonant nonlinear transverse velocity. This produces the nonlinear current which is further responsible for the generation of THz radiation. The anharmonicity plays a vital role in the efficient generation of THz radiation. The anharmonicity arises due to the nonlinear variation of restoration force on the various electrons of CNTs. This anharmonicity in the electrons of CNTs helps in broadening the resonance peak. We have observed that externally applied static magnetic field (110 kG to 330 kG) also paves the way for the enhancement of the normalized THz amplitude.

E-mail:vishal20india@yahoo.co.in I. Introduction
One of the main tasks in the modern technical world is to develop compact and highly efficient terahertz generation sources. Such THz generation sources can bring revolutionary changes in many fields of science and technology, a few of these are THz-spectroscopy, security systems, medical and health sectors, etc. [1][2][3][4][5]. In the present century, researchers have provided variety of schemes for the efficient generation of THz radiations. In these schemes, THz radiations are generated using various methods, for example by employing laser coupling to anharmonic CNTs [6], by using a wiggler magnetic field on vertically aligned carbon nanotubes (VA-CNTs) [7], buy using anharmonic CNTs in the presence of static magnetic field [8], by using laser filaments in the presence of static electric and magnetic fields [9], etc. According to Batrakov et al. [10], THz radiations can be generated through CNTs by using a static electric field. Parashar and Sharma [11] have applied optical rectification in CNTs to generate THz radiations. The CNTs, when irradiated with laser beams provide one of the promising ways for the creation of compact and efficient sources of THz radiations. The extraordinary electrical conductivity, thermal conductivity, and compact dimensional characteristics of CNTs make them a front runner in the field of THz generation [12][13][14][15][16].
In the present paper, we develop an analytical model for the efficient THz generation by the interaction of Gaussian laser beam with the array of VA-CNTs in the presence of static magnetic field applied perpendicular to the direction of propagation of laser and length of CNTs. In 2 nd segment of the paper, we have derived the relation for the nonlinear current density, which is further responsible for the generation of THz radiation. In 3 rd segment of the paper, we have provided the THz wave dynamics to calculate the normalized THz electric field. The discussion of results and conclusion has been provided in the last segment of this paper.

II. Evaluation of nonlinear current density
Consider a vertical array of single-walled anharmonic CNTs nested in dielectric surface (glass) as shown in Fig. 1. To magnetize these CNTs, a static magnetic field is applied along the y-direction transverse to the direction of laser propagation (z-direction) and the longitudinal axis of CNTs (xdirection). The microwave plasma-enhanced chemical vapor deposition synthesizing (MPECVD) technique can be engineered to obtain the deformed VA-CNTs [17][18][19][20]. The CNTs obtained by the above mentioned technique are known as rippled VA-CNTs. These are very softer and more prone to ripples as compared to crystalline CNTs obtained by other available methods like the arc discharge technique [21][22][23][24]. In the above mentioned MPECVD technique, the dimensions and alignment of SWCNTs can be easily controlled [25,26]. In this way, we can produce density ripples of desired period and size in the CNTs. The number of CNTs per unit area in the array is n q and corresponding modulated density of CNTs is n q = n 0q e iqx , here the term n 0q is the amplitude and q is the wavenumber of density ripples produced in CNTs. The free electron density of each CNT is n 0 . Each CNT is characterized by the inner radius a, outer radius b, and length L. Each SWCNT is normally shaped as a hollow cylinder of compact dimensions to determine the electrical conductivity [27]. As far as we are concerned with the response of SWCNTs to the transverse electric and magnetic fields of the laser beam, these nanotubes as solid cylindrical tubes [28,29]. The amplitude modulated Gaussian laser beam of angular frequency ω and wavenumber k, having non-uniform intensity distribution propagates through VA-CNTs with electric field profile where μ is the modulation depth, Ω is the modulation frequency in the THz range, c is the speed of light and x is the unit vector. The intensity profile of the incident Gaussian laser beam is represented by the relation E 0 2 = E 00 2 e −x 2 r 0 2 ⁄ , here r 0 is initial beam radius. (2) By using the above equation (2), one can calculate x and z components of the nonlinear ponderomotive force in the exponential form. These components are acting perpendicular to the direction of the applied magnetic field (y-direction). These components are responsible for THz generation at resonance.
The Thus the net electric field at the point (r, φ, z) can be written as E ⃗ ⃗ = E ⃗ ⃗ + + E ⃗ ⃗ − .
where, ϵ = ϵ 0 ϵ r is known as the electric permittivity of the medium. As explained above, the xcomponent of the ponderomotive force F x is responsible for the oscillatory motion of the electrons of CNTs along the x-direction, thus the expression for the corresponding x-component of the spacecharge electric field can be derived from Eq. (5).
The restoring force for the electrons of CNTs along can be obtained by using the relation, F Rx = −eE x , ) ∫ ( By using standard integrals, the above linear and nonlinear components can be simplified to get net average restoration force where, ω P = [n 0 e 2 mϵ 0 ⁄ ] As the static magnetic field B ⃗ ⃗ is applied along the y-axis, therefore, magnetic force can be resolved into their x and z components, F Bx = −ev z B c ⁄ and F Bz = ev x B c ⁄ respectively. Under the influence of electric fields of the lasers, an external static magnetic field B ⃗ ⃗ and space charge electric field, the displacement of electrons in CNTs can be controlled by the following set of equations where, represents electron-neutral collision frequency, which is lesser than ω.

Fig. 2 shifting of the electron cylinder by displacement ∆ ⃗ ⃗ concerning the ion cylinder along the x-direction
On solving the equations (9) and (10), we obtain the x and z components for the displacement of the electrons of CNTs in the array where, ω c = eB m ⁄ is known as cyclotron frequency of the electrons in CNTs.
With the help of the above equations (11) & (12) and by using the fundamental relation of the velocity v = dΔ dt ⁄ we can determine the corresponding nonlinear velocity components of the electrons of One can use the relation J ω NL = −e n 0 v to calculate the nonlinear current density of the electrons of CNTs. The value of nonlinear current density is non-zero over the cross-sectional area of CNTs in the array. At the same time, its value is zero for the area lying in between the CNTs. Therefore we have calculated the average nonlinear current density of the array of CNTs by using the equation (3) & (4) and the relation J av.ω NL = −e n 0 v (n q * π(b 2 − a 2 )).
However, with the ripples produced in CNTs have wave number q, there occurs matching of phase.
Under this condition, resonant excitation of THz radiation can be realized. The nonlinear current density terms make a significant contribution to the THz field (E TH ) and one can observe this contribution from the THz wave propagation equation.

III. THz wave dynamics
The wave equation is derived by using Maxwell's equations and describes the propagation of terahertz waves through the array of VA-CNTs.
In the presence of an external static magnetic field applied along the y-axis in the collisional plasma of CNTs, the electric permittivity assumes the form of an anisotropic tensor. ϵ = | ϵ xx ϵ xy ϵ xz ϵ yx ϵ yy ϵ yz ϵ zx ϵ zy ϵ zz |.
With the use of the above components of dielectric tensor, Eq. (17) can be modified as: By using the phase-matching condition one can write With the help of the above equations (20) and (21)

IV. Results & Discussion
To perform numerical calculations, we have used a carbon dioxide laser beam with the following specified parameters. The angular frequency and wavelength of the laser beam is ω = 1.78 ×   presented by Nemilentsau et al. [31]. Watanabe et al. [32] have also shown THz electric field variation with radii of CNTs in their experimental work. From figure 4, it is clear that in each curve normalized THz amplitude has its peak value at the surface plasmon resonance point. Moreover, as the surface plasmon resonance condition depends upon the characteristic parameter β, therefore with the increase in the value of β, the surface plasmon resonance point slips towards the right side of the normalized THz frequency. This increase in the number density of CNTs results in the increase of the nonlinearities of the array of VA-CNTs. Vij et al. [7] has shown the similar results in their theoretical study of THz generation by using CNTs under the effect of a wiggler magnetic field. Also, the surface plasmon resonance condition ω = ω p [(1 + β) 2ϵ r ⁄ + ω α 2 ] 1 2 ⁄ is independent of the inter-tube separation distance d, therefore in this graph surface plasmon resonance point remains same for the three curves.  figure 6, the surface plasmon resonance point shifts towards a higher value. This is because of the increase in the value of the static magnetic field as explained above. Figure 6, also explains the importance of modulation index, for the enhancement of normalized THz amplitude. Such dependence has also been explained by Kumar et al. [33] in their theoretical study THz generation by amplitude-modulated laser beam in ripple density plasma. So, the output of THz wave generation can be tuned by using appropriate values of the applied static magnetic field strength and modulation index.  However by applying a suitable magnetic field, one can overcome this loss up to some extent. Similar results have been shown by Singh and Malik [35] in their theoretical study of enhanced THz generation in magnetized collisional plasma.

Conclusion
The Gaussian laser beam makes a nonlinear interaction with VA-CNTs to resonantly excite THz radiation at the modulation frequency under the influence of an external magnetic field. This is responsible for the production of the nonlinear current which results in THz generation. The anharmonicity in the electrons of VA-CNTs helps in broadening the surface resonance peak. This also results in the further enhancement of THz generation. The surface plasmon resonance condition ω = ω p [(1 + β) 2ϵ r ⁄ + ω α 2 ] 1 2 ⁄ can be altered by varying the dimensions of CNTs and the strength of the applied static magnetic field.

Author Contribution
Sandeep Kumar: derivation, methodology and analytical modeling Shivani Vij: graph plotting and writing Niti Kant : numerical analysis and result discussion Vishal Thakur: supervision, reviewing and editing

Data Availability
The data that supports the findings of this study are available insise the manuscript

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