Generation of terahertz radiation by beating of two color lasers in hot clustered plasma with step density profile

The terahertz region of electromagnetic radiation spectrum has been an important area of research for the last two decades. THz radiation sources have a number of applications in remote sensing, biological and chemical imaging [1, 2], non destructive testing, high-field condensed-matter studies [3], short distance wireless communications and sensing [4], explosives detections [5] etc. In view of above mentioned applications, a variety of new schemes are proposed by various researchers to generate THz radiation. Laser based THz emitters involve wide band gap semiconductors, electro-optic crystals and photoconductive antennas [6–9]. However, all these media have the limitation of material breakdown and can not handle very high potential gradients. To overcome this limitation, plasma can be used as an interactive medium, which being in ionized state can handle very high potential gradients. Hence, plasma based THz radiation sources have the great potential for generating the broadband high-power THz pulses and have become the topic of attention for many researchers [10–19]. Bhasin and Tripathi [17] have presented a model of THz generation via optical rectification of a short pulse laser in rippled density plasma in the presence of an external magnetic field. They observed the power conversion efficiency of 0.04% of a laser of intensity 3 × 10¹⁵ W cm⁻² in a 0.01% of critical density plasma with 30 kG. Kumar and Tripathi [18] have theoretically studied the process of THz generation by nonlinear mixing of two laser pulses propagating in a plasma at an angle to the density ripple wave vector. They observed the direct dependence of THz amplitude on the ripple orientation angle. Kumar et al [19] have reported THz radiation generation by beating of two laser pulses in a hot plasma with step density profile. The amplitude of generated THz radiation is increased due to coupling with plasma wave. They reported THz power of the order of 90MW with a laser of intensity 3 × 10¹⁶ W cm⁻², wavelength 1 μm and beam radius 100 μm.

Clusters are also attractive contenders for the generation of THz radiation because of their peculiar properties. They are vanderwalls bonded aggregates of atoms or molecules that combine advantages of both solid and gaseous media. They inherit high local electron density leading to high value of nonlinear response and hence a number of applications [20–25] in the interaction of high-power laser pulses with gas-cluster. The efficient absorption of laser energy by clusters (up to 95%) has strengthened the motivation for studying nonlinear phenomena in clustered plasma by various researchers [26–29]. Using cluster as interacting medium, the THz radiation of high power can be generated due to surface plasmon resonance associated with nanoparticles of the cluster medium. Authors of [26] have experimentally studied THz generation from argon clusters using a weakly relativistic laser pulse. In their report, the enhancement in the average THz power by forty times is observed from cluster targets as compared to unclustered targets. THz generation via optical rectification of an intense laser pulse in a magnetized clustered plasma has been reported by Bhasin et al [30]. In order to excite THz wave resonantly, the phase matching is provided by applying density ripple on nano clusters. Recently Malik et al [31] have presented a scheme of THz generation by the interaction of two color lasers in plasma embedded with clusters. The ripple on cluster density providing the requisite phase matching and cluster plasmon resonance significantly enhances the THz amplitude when laser frequency approaches ${\omega }_{{pe}}/\sqrt{3}$.

In the present paper, we study the terahertz generation via interaction of two color laser beams with plasma embedded with clusters having step density profile. The nonlinear mechanisms of generation of THz waves could be attributed to bound state nonlinearities within a neutral cluster and nonlinearities due to plasma electrons within an electronically non neutral cluster. The primary mechanism of THz generation can be explained as follows: an intense short laser pulse at $({\omega }_{1},{\vec{k}}_{1})$ is mixed with its frequency shifted second harmonic $({\omega }_{2},{\vec{k}}_{2})$ (where ω₂ = 2ω₁ + ω and ω is THz frequency in a clustered plasma). The cluster electrons experience ponderomotive force by the incident lasers. This force drives density perturbations at $(2{\omega }_{1},2{\vec{k}}_{1})$ and $({\omega }_{2}-{\omega }_{1},{\vec{k}}_{2}-{\vec{k}}_{1})$. Using equation of motion, one can find the oscillatory velocities of cluster as well as plasma electrons which produce a nonlinear current density at $({\omega }_{2}-2{\omega }_{1},{\vec{k}}_{2}-2{\vec{k}}_{1})$ to generate THz radiation. Self-generated plasma wave enhances the conversion efficiency of THz radiation. Hence, the plasmsa electrons are considered as a nonlinear medium while cluster electrons are considered to analyze the effect of surface plasmons on THz generation. The expressions for the nonlinear current density and the THz wave amplitude are derived in sections 2 and 3 respectively. The normalized power of THz wave emitting as an antenna is also derived in section 3. The results of the analysis are plotted and discussed in section 4 and in the last section, the investigation is concluded.

Consider a clustered gas jet of argon emerging from a structured nozzle which produces nanoclusters of radius r with step density profile (of cluster density)

$$\begin{eqnarray}\begin{array}{rcl}{n}_{c} & = & {n}_{{co}},\quad \mathrm{for}\ {\rm{x}}{\text{}}\gt 0\\ & = & 0\quad \quad \mathrm{for}\ {\rm{x}}{\text{}}\lt 0\end{array}\end{eqnarray} \tag{ 1 }$$

where n_co is the initial cluster density. By using a focusing device on a supersonic cluster beam that produces a sudden turn in the flow passage prior to the free jet and diverts the clusters from their streamlines toward the nozzle axis, one can control the density of cluster to get different density profiles [32]. The electron density inside each clustered ball is n_e. Two plane polarized lasers of frequency ω₁ and ω₂ are obliquely incident on vacuum-cluster interface at an angle θ (refer to Fig. 1). The electric and magnetic fields of the beams are

$$\begin{eqnarray}&&{\vec{E}}_{j}=(-\sin \theta \hat{x}+\cos \theta \hat{z}){A}_{j}{e}^{-{\text{}}i({\omega }_{j}t-{k}_{{jx}}x-{k}_{{jz}}z)}\end{eqnarray} \tag{ 2 }$$

and

$$\begin{eqnarray}&&{\vec{B}}_{j}=\displaystyle \frac{{\vec{k}}_{j}\times {\vec{E}}_{j}}{{\omega }_{j}},\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray}&&{k}_{j}=\displaystyle \frac{{\omega }_{j}}{c}{\left[1-\displaystyle \frac{{\omega }_{p}^{2}}{{\omega }_{j}^{2}}-\displaystyle \frac{4\pi {n}_{c}{r}^{3}{\omega }_{{pe}}^{2}}{3({\omega }_{j}^{2}-{\omega }_{{pe}}^{2}/3)}\right]}^{1/2},\end{eqnarray} \tag{ 4 }$$

${k}_{{jx}}={k}_{j}\cos \theta$, ${k}_{{jz}}={k}_{j}\sin \theta$, j = 1, 2 and A_j is the amplitude of laser pulse. Here ${\omega }_{p}=\sqrt{4\pi {n}_{o}{e}^{2}/m}$ is the plasma frequency, ${\omega }_{{pe}}=\sqrt{4\pi {n}_{e}{e}^{2}/m}$ is the plasma frequency inside the cluster. The plasma electron density outside the cluster is

$$\begin{eqnarray}\begin{array}{rcl}{n}_{o} & = & {n}_{{oo}},\quad \mathrm{for}\ {\rm{x}}\gt 0\\ & = & 0\quad \quad \mathrm{for}\ {\rm{x}}\lt 0\end{array}\end{eqnarray} \tag{ 5 }$$

where n_oo is the initial plasma electron density.

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e and m are respectively the charge and rest mass of an electron.

The electric field of lasers exert an electric force on cluster electrons and they start oscillating in x-z plane. The ionization of atoms inside the cluster leads to their conversion into plasma balls. Here, the nanoplasma model given by Ditmire et al [33] is used, in which the cluster in the presence of strong laser field is treated as a spherical ball of nanoplasma with uniform density and temperature. By using, equation of motion for cluster electrons, their oscillatory velocity v_ωj with its components can be written as,

$$\begin{eqnarray}&&{\vec{v}}_{\omega j}=\displaystyle \frac{e{\omega }_{j}{\vec{E}}_{j}}{m\iota ({\omega }_{j}^{2}-{\omega }_{{pe}}^{2}/3)},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{v}_{\omega {jx}}=\displaystyle \frac{-e{\omega }_{j}{A}_{j}\sin \theta (\hat{z}-\hat{x}\tan \theta ){e}^{-{\text{}}i({\omega }_{j}t-{k}_{{jx}}x-{k}_{{jz}}z)}}{m\iota ({\omega }_{j}^{2}-{\omega }_{{pe}}^{2}/3)},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{v}_{\omega {jz}}=\displaystyle \frac{e{\omega }_{j}{A}_{j}\cos \theta (\hat{z}-\hat{x}\tan \theta ){e}^{-{\text{}}i({\omega }_{j}t-{k}_{{jx}}x-{k}_{{jz}}z)}}{m\iota ({\omega }_{j}^{2}-{\omega }_{{pe}}^{2}/3)}.\end{eqnarray} \tag{ 8 }$$

The lasers couple nonlinearly at difference frequency ω₂–ω₁ and 2ω₁ to exert a ponderomotive force on cluster electrons which is given as

$$\begin{eqnarray}&&{F}_{p({\omega }_{2}-{\omega }_{1})}=\displaystyle \frac{{e}^{2}}{2m\iota }{\vec{E}}_{1}^{\star }.{\vec{E}}_{2}\left(\displaystyle \frac{{\omega }_{1}{k}_{2}}{{\omega }_{2}({\omega }_{1}^{2}-{\omega }_{{pe}}^{2}/3)}-\displaystyle \frac{{\omega }_{2}{k}_{1}}{{\omega }_{1}({\omega }_{2}^{2}-{\omega }_{{pe}}^{2}/3)}\right)(\cos \theta \hat{x}+\sin \theta \hat{z}),\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{F}_{p(2{\omega }_{1})}=\displaystyle \frac{{e}^{2}{k}_{1}{E}_{1}^{2}}{2m\iota ({\omega }_{1}^{2}-{\omega }_{{pe}}^{2}/3)}(\cos \theta \hat{x}+\sin \theta \hat{z}).\end{eqnarray} \tag{ 10 }$$

The behavior of electron in the influence of this ponderomotive force is governed by the equation of motion

$$\begin{eqnarray}&&m\displaystyle \frac{{d}^{2}\vec{{\rm{\Delta }}}}{{{dt}}^{2}}={\vec{F}}_{p({\omega }_{2}-{\omega }_{1})}-m\displaystyle \frac{{\omega }_{{pe}}^{2}}{3}\vec{{\rm{\Delta }}}\end{eqnarray} \tag{ 11 }$$

where $\vec{{\rm{\Delta }}}$ is the displacement due to electron excursion and $m{\omega }_{{pe}}^{2}\vec{{\rm{\Delta }}}/3$ is the restoring force experienced by the cluster electrons when the electron sphere of a cluster is displaced with respect to the ion sphere. This equation gives nonlinear oscillatory velocity at ω₂–ω₁ and 2ω₁ as

$$\begin{eqnarray}&&{\vec{v}}_{c}^{{NL}}({\omega }_{2}-{\omega }_{1})=\displaystyle \frac{-{\vec{F}}_{p({\omega }_{2}-{\omega }_{1})}({\omega }_{2}-{\omega }_{1})}{m\iota ({\left({\omega }_{2}-{\omega }_{1}\right)}^{2}-{\omega }_{{pe}}^{2}/3)}\end{eqnarray} \tag{ 12 }$$

and

$$\begin{eqnarray}&&{\vec{v}}_{c}^{{NL}}(2{\omega }_{1})=\displaystyle \frac{-\vec{F}p(2{\omega }_{1})2{\omega }_{1}}{m\iota (4{\omega }_{1}^{2}-{\omega }_{{pe}}^{2}/3)}\end{eqnarray} \tag{ 13 }$$

respectively. These longitudinal velocities generate density oscillations at (ω₂−ω₁, k₂−k₁) and (2ω₁, 2k₁) as

$$\begin{eqnarray}&&{n}_{c({\omega }_{2}-{\omega }_{1})}^{{NL}}=\displaystyle \frac{{n}_{a}{e}^{2}({k}_{2}-{k}_{1})\vec{{E}_{1}^{* }}.\vec{{E}_{2}}}{2{m}^{2}({\left({\omega }_{2}-{\omega }_{1}\right)}^{2}-{\omega }_{{pe}}^{2}/3)}\left(\displaystyle \frac{{\omega }_{1}{k}_{2}}{{\omega }_{2}({\omega }_{1}^{2}-{\omega }_{{pe}}^{2}/3)}-\displaystyle \frac{{\omega }_{2}{k}_{1}}{{\omega }_{1}({\omega }_{2}^{2}-{\omega }_{{pe}}^{2}/3)}\right),\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{n}_{c(2{\omega }_{1})}^{{NL}}=\displaystyle \frac{{n}_{a}{e}^{2}{k}_{1}^{2}{E}_{1}^{2}}{{m}^{2}({\omega }_{1}^{2}-{\omega }_{{pe}}^{2}/3)(4{\omega }_{1}^{2}-{\omega }_{{pe}}^{2}/3)}\end{eqnarray} \tag{ 15 }$$

respectively, where ${n}_{a}=(4\pi /3){n}_{e}{n}_{c}{r}^{3}$ is the average number of cluster electrons per unit volume. Similarly for plasma electrons, density perturbation is

$$\begin{eqnarray}&&{n}_{p({\omega }_{2}-{\omega }_{1})}^{{NL}}=\displaystyle \frac{{n}_{o}{e}^{2}{\left({k}_{2}-{k}_{1}\right)}^{2}\vec{{E}_{1}^{* }}.\vec{{E}_{2}}}{2{m}^{2}{\omega }_{1}{\omega }_{2}{\left({\omega }_{2}-{\omega }_{1}\right)}^{2}}\end{eqnarray} \tag{ 16 }$$

and

$$\begin{eqnarray}&&{n}_{p(2{\omega }_{1})}^{{NL}}=\displaystyle \frac{{n}_{o}{e}^{2}{k}_{1}^{2}{E}_{1}^{2}}{4{m}^{2}{\omega }_{1}^{4}}.\end{eqnarray} \tag{ 17 }$$

This nonlinear density perturbations at (ω₂ − ω₁, k₂−k₁) perturb the electrostatic potential ϕ_s which varies as ${e}^{-\iota [({\omega }_{2}-{\omega }_{1})t-({k}_{2}-{k}_{1})(z\cos \theta +x\sin \theta )]}$ to produce a self-consistent space charge field $\vec{{E}_{s}}=-\vec{{\rm{\nabla }}}{\phi }_{s}$. The space charge field causes linear electron density perturbation in clusters which can be wriiten as,

$$\begin{eqnarray}&&{n}_{c({\omega }_{2}-{\omega }_{1})}^{L}=\displaystyle \frac{{\left({k}_{2}-{k}_{1}\right)}^{2}{\chi }_{c}{\phi }_{s}}{4\pi e},\end{eqnarray} \tag{ 18 }$$

where χ_c = $-(4\pi /3){n}_{c}{r}^{3}{\omega }_{{pe}}^{2}/({\left({\omega }_{2}-{\omega }_{1}\right)}^{2}-{\omega }_{{pe}}^{2}/3)$ is the susceptibility of the cluster electrons.

Also, one can deduce the linear perturbed density of plasma electrons as,

$$\begin{eqnarray}&&{n}_{p({\omega }_{2}-{\omega }_{1})}^{L}=\displaystyle \frac{{\left({k}_{2}-{k}_{1}\right)}^{2}{\chi }_{p}{\phi }_{s}}{4\pi e},\end{eqnarray} \tag{ 19 }$$

where ${\chi }_{p}=-{\omega }_{p}^{2}/{\left({\omega }_{2}-{\omega }_{1}\right)}^{2}$ is the susceptibility of the plasma electrons.

Now from Poisson's equation one can get,

$$\begin{eqnarray}&&{\phi }_{s}=\displaystyle \frac{-4\pi e({n}_{c({\omega }_{2}-{\omega }_{1})}^{{NL}}+{n}_{p({\omega }_{2}-{\omega }_{1})}^{{NL}})}{{\left({k}_{2}-{k}_{1}\right)}^{2}(1+{\chi }_{c}+{\chi }_{p})}.\end{eqnarray} \tag{ 20 }$$

The net density of cluster electrons can be obtained as,

$$\begin{eqnarray}&&{n}_{c({\omega }_{2}-{\omega }_{1})}={n}_{c({\omega }_{2}-{\omega }_{1})}^{{NL}}\left(1-\displaystyle \frac{{\omega }_{p}^{2}/}{{\left({\omega }_{2}-{\omega }_{1}\right)}^{2}}\right)\displaystyle \frac{1}{{\epsilon }_{1}^{{\prime} }}+{n}_{p({\omega }_{2}-{\omega }_{1})}^{{NL}}\displaystyle \frac{(4\pi /3){n}_{c}{r}^{3}{\omega }_{{pe}}^{2}}{({\left({\omega }_{2}-{\omega }_{1}\right)}^{2}-{\omega }_{{pe}}^{2}/3){\epsilon }_{1}^{{\prime} }},\end{eqnarray} \tag{ 21 }$$

where ${\epsilon }_{1}^{{\prime} }=1+{\chi }_{c}+{\chi }_{p}$.

Similarly, for plasma electrons, the net density can be written as,

$$\begin{eqnarray}&&{n}_{p({\omega }_{2}-{\omega }_{1})}={n}_{p({\omega }_{2}-{\omega }_{1})}^{{NL}}\left(1+\displaystyle \frac{{\omega }_{p}^{2}}{{\left({\omega }_{2}-{\omega }_{1}\right)}^{2}{\epsilon }_{1}^{{\prime} }}\right)+{n}_{c({\omega }_{2}-{\omega }_{1})}^{{NL}}\left(\displaystyle \frac{{\omega }_{p}^{2}}{{\left({\omega }_{2}-{\omega }_{1}\right)}^{2}{\epsilon }_{1}^{{\prime} }}\right).\end{eqnarray} \tag{ 22 }$$

The net density of cluster electrons at (2ω₁)is

$$\begin{eqnarray}&&{n}_{c(2{\omega }_{1})}={n}_{c(2{\omega }_{1})}^{{NL}}\displaystyle \frac{(1-{\omega }_{p}^{2}/4{\omega }_{1}^{2})}{{\epsilon }_{1}}+{n}_{p(2{\omega }_{1})}^{{NL}}\displaystyle \frac{(4\pi /3){n}_{c}{r}^{3}{\omega }_{{pe}}^{2}}{(4{\omega }_{1}^{2}-{\omega }_{{pe}}^{2}/3){\epsilon }_{1}},\end{eqnarray} \tag{ 23 }$$

where ₁ = $1-{\omega }_{p}^{2}/4{\omega }_{1}^{2}-(4\pi /3){n}_{c}{r}^{3}{\omega }_{{pe}}^{2}/(4{\omega }_{1}^{2}-{\omega }_{{pe}}^{2}/3)$ and the the net density of plasma electrons at (2ω₁) is

$$\begin{eqnarray}&&{n}_{p(2{\omega }_{1})}={n}_{c(2{\omega }_{1})}^{{NL}}\displaystyle \frac{{\omega }_{p}^{2}}{4{\omega }_{1}^{2}{\epsilon }_{1}}+{n}_{p(2{\omega }_{1})}^{{NL}}\left(1+\displaystyle \frac{{\omega }_{p}^{2}}{4{\omega }_{1}^{2}{\epsilon }_{1}}\right).\end{eqnarray} \tag{ 24 }$$

The conjugate of the oscillatory velocity at ω₁ beats with the perturbed density at ω₂–ω₁ to generate nonlinear current density at ω = ω₂ –2ω₁ and k = k₂–2k₁. Also, the oscillatory velocity at ω₂ beats with the conjugate of the perturbed electron density at 2ω₁ to generate nonlinear current density at ω, k.

Thus, the expression for the genarated nonlinear current density at THz frequency ω = ω₂ –2ω₁ with wave number k = k₂–2k₁ can be derived as

$$\begin{eqnarray}\begin{array}{rcl}{\vec{J}}_{c\omega }^{{NL}} & = & -\displaystyle \frac{e}{2}{n}_{c({\omega }_{2}-{\omega }_{1})}{\vec{v}}_{{\omega }_{1}}^{\star }-\displaystyle \frac{e}{2}{n}_{c(2{\omega }_{1})}^{\star }{\vec{v}}_{{\omega }_{2}}\\ & = & \left[{\left(\displaystyle \frac{{k}_{1}c}{{\omega }_{2}}\right)}^{2}\displaystyle \frac{{a}_{1}^{\star }{a}_{2}{n}_{c}{r}^{3}{\omega }_{{pe}}}{12\iota {\epsilon }_{1}^{{\prime} }}\left(\displaystyle \frac{{\omega }_{{pe}}}{{\omega }_{2}}\right)\displaystyle \frac{({k}_{2}/{k}_{1}-1)}{(1-{\omega }_{{pe}}^{2}/3{\omega }_{1}^{2})}\displaystyle \frac{1}{{\left(1-{\omega }_{1}/{\omega }_{2}\right)}^{2}(1-{\omega }_{{pe}}^{2}/3{\left({\omega }_{2}-{\omega }_{1}\right)}^{2})}\right.\\ & & \times \left\{\displaystyle \frac{{\omega }_{2}}{{\omega }_{1}}\left(1-\displaystyle \frac{{\omega }_{p}^{2}}{{\left({\omega }_{2}-{\omega }_{1}\right)}^{2}}\right)\left(\displaystyle \frac{{k}_{2}/{k}_{1}}{(1-{\omega }_{{pe}}^{2}/3{\omega }_{1}^{2})}-\displaystyle \frac{1}{(1-{\omega }_{{pe}}^{2}/3{\omega }_{2}^{2})}\right)\right.\\ & & \left.+\left(\displaystyle \frac{{k}_{2}}{{k}_{1}}-1\right)\left(\displaystyle \frac{{\omega }_{p}}{{\omega }_{1}}\right)\left(\displaystyle \frac{{\omega }_{p}}{{\omega }_{2}}\right)\displaystyle \frac{1}{{\left(1-{\omega }_{1}/{\omega }_{2}\right)}^{2}}\right\}-{\left(\displaystyle \frac{{k}_{1}c}{{\omega }_{1}}\right)}^{2}\displaystyle \frac{{a}_{1}^{\star }{a}_{2}{n}_{c}{r}^{3}{\omega }_{{pe}}}{24\iota {\epsilon }_{1}(1-{\omega }_{{pe}}^{2}/3{\omega }_{2}^{2})(1-{\omega }_{{pe}}^{2}/12{\omega }_{1}^{2})}\\ & & \left.\times \left(\displaystyle \frac{{\omega }_{{pe}}}{{\omega }_{1}}\right)\left(\displaystyle \frac{1-{\omega }_{p}^{2}/4{\omega }_{1}^{2}}{(1-{\omega }_{{pe}}^{2}/3{\omega }_{1}^{2})}+\displaystyle \frac{{\omega }_{p}^{2}}{4{\omega }_{1}^{2}}\right)\right]({\cos }^{2}\theta -{\sin }^{2}\theta )\times (\cos \theta \hat{z}-\sin \theta \hat{x}){A}_{1}^{\star }\\ & & \times {e}^{-{\text{}}i(\omega t-{k}_{x}x-{k}_{z}z)}.\end{array}\end{eqnarray} \tag{ 25 }$$

Similarly

$$\begin{eqnarray}\begin{array}{rcl}{\vec{J}}_{p\omega }^{{NL}} & = & -\displaystyle \frac{e}{2}{n}_{p({\omega }_{2}-{\omega }_{1})}{\vec{v}}_{{\omega }_{1}}^{\star }-\displaystyle \frac{e}{2}{n}_{p(2{\omega }_{1})}^{\star }{\vec{v}}_{{\omega }_{2}}={\left(\displaystyle \frac{{k}_{1}c}{{\omega }_{2}}\right)}^{2}\displaystyle \frac{{a}_{1}^{\star }{a}_{2}{\omega }_{p}}{16\pi \iota {\epsilon }_{1}^{{\prime} }}\left(\displaystyle \frac{{\omega }_{p}}{{\omega }_{2}}\right)\displaystyle \frac{{\left({k}_{2}/{k}_{1}-1\right)}^{2}}{{\left(1-{\omega }_{1}/{\omega }_{2}\right)}^{2}}\\ & & \times \left[\displaystyle \frac{{\omega }_{2}}{{\omega }_{1}}\left(1-\displaystyle \frac{{\omega }_{p}^{2}}{{\left({\omega }_{2}-{\omega }_{1}\right)}^{2}}\right)+\left(\displaystyle \frac{{\omega }_{p}}{{\omega }_{1}}\right)\left(\displaystyle \frac{{\omega }_{p}}{{\omega }_{2}}\right)\displaystyle \frac{1}{{\left(1-{\omega }_{1}/{\omega }_{2}\right)}^{2}}-{\left(\displaystyle \frac{{k}_{1}c}{{\omega }_{1}}\right)}^{2}\displaystyle \frac{{a}_{1}^{\star }{a}_{2}{\omega }_{p}}{32\pi \iota {\epsilon }_{1}}\left(\displaystyle \frac{{\omega }_{p}}{{\omega }_{1}}\right)\right]\\ & & \times ({\cos }^{2}\theta -{\sin }^{2}\theta )(\cos \theta \hat{z}-\sin \theta \hat{x}){A}_{1}^{\star }{e}^{-{\text{}}i(\omega t-{k}_{x}x-{k}_{z}z)},\end{array}\end{eqnarray} \tag{ 26 }$$

where ${a}_{1}^{\star }={{eA}}_{1}^{\star }/m{\omega }_{1}c$, a₂ = eA₂/mω₂c. The linear current density due to self-consistent terahertz field at frequency ω is

$$\begin{eqnarray}&&{\vec{J}}_{p\omega }^{L}=\displaystyle \frac{-{n}_{o}{e}^{2}{\vec{E}}_{\omega }}{m\iota \omega }+\displaystyle \frac{{v}_{{th}}^{2}{\epsilon }_{o}}{\iota \omega }\vec{{\rm{\nabla }}}(\vec{{\rm{\nabla }}}.{\vec{E}}_{\omega })\end{eqnarray} \tag{ 27 }$$

where _o and v_th are the permittivity of the free space and electron thermal velocity respectively. Hence net current density becomes, ${\vec{J}}_{\omega }={\vec{J}}_{c\omega }^{{NL}}$ + ${\vec{J}}_{p\omega }^{{NL}}$ + ${\vec{J}}_{p\omega }^{L}$.

Using Maxwell's equations, the wave equation for THz field ${\vec{E}}_{\omega }$ can be derived as,

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}{\vec{E}}_{\omega }-\left(1-\displaystyle \frac{{v}_{{th}}^{2}}{{c}^{2}}\right)\vec{{\rm{\nabla }}}(\vec{{\rm{\nabla }}}.{\vec{E}}_{\omega })+\displaystyle \frac{({\omega }^{2}-{\omega }_{p}^{2})}{{c}^{2}}{\vec{E}}_{\omega }=-\displaystyle \frac{\iota \omega }{{c}^{2}{\epsilon }_{o}}{\vec{J}}_{\omega }^{{NL}}.\end{eqnarray} \tag{ 28 }$$

In the absence of nonlinear source, this equation provides two distinct solutions. One is an EM wave with $\vec{{\rm{\nabla }}}.{\vec{E}}_{m}=0$ as,

$$\begin{eqnarray}&&{\vec{E}}_{m}={A}_{m}\left(\hat{x}-\displaystyle \frac{{k}_{{mx}}}{{k}_{z}}\hat{z}\right){e}^{{\text{}}{{ik}}_{{mx}}x}{e}^{-{\text{}}i(\omega t-{k}_{z}z)},\end{eqnarray} \tag{ 29 }$$

and a Langumuir wave with $\vec{{\rm{\nabla }}}\times {\vec{E}}_{s}=0$ as,

$$\begin{eqnarray}&&{\vec{E}}_{s}={A}_{s}\left(\hat{x}+\displaystyle \frac{{k}_{z}}{{k}_{{sx}}\hat{z}}\right){e}^{{\text{}}{{ik}}_{{sx}}x}{e}^{-{\text{}}i(\omega t-{k}_{z}z)},\end{eqnarray} \tag{ 30 }$$

where ${k}_{{mx}}={\left({\omega }^{2}-{\omega }_{p}^{2}-{k}_{z}^{2}{c}^{2}-\tfrac{4\pi {n}_{c}{r}^{3}{\omega }_{{pe}}^{2}}{3(1-{\omega }_{{pe}}^{2}/3{\omega }^{2})}\right)}^{1/2}$/c, ${k}_{{sx}}={\left({\omega }^{2}-{\omega }_{p}^{2}-{k}_{z}^{2}{v}_{{th}}^{2}-\tfrac{4\pi {n}_{c}{r}^{3}{\omega }_{{pe}}^{2}}{3(1-{\omega }_{{pe}}^{2}/3{\omega }^{2})}\right)}^{1/2}$/${v}_{{th}}$, ${k}_{z}=\omega \sin \theta$/c. In the presence of nonlinear term, considering the variation of ${\vec{J}}_{\omega }^{{NL}}$ as ${e}^{-\iota (\omega t-\vec{k}.\vec{r})}$, the solution of equation (28) becomes,

$$\begin{eqnarray}&&{\vec{E}}_{p}=\displaystyle \frac{-\iota \omega {\vec{J}}_{\omega }^{{NL}}}{{\epsilon }_{o}\left({\omega }^{2}-{\omega }_{p}^{2}-{k}^{2}{v}_{{th}}^{2}-\tfrac{4\pi {n}_{c}{r}^{3}{\omega }_{{pe}}^{2}}{3(1-{\omega }_{{pe}}^{2}/3{\omega }^{2})}\right)}.\end{eqnarray} \tag{ 31 }$$

Since E_s and E_p both are irrotational fields, so no magnetic field is associated with them. The magnetic field associated with ${\vec{E}}_{m}$ is

$$\begin{eqnarray}&&\vec{H}=\displaystyle \frac{{\vec{k}}_{m}\times {\vec{E}}_{m}}{{\mu }_{o}\omega }=\hat{y}\displaystyle \frac{{k}_{{mx}}^{2}+{k}_{z}^{2}}{{\mu }_{o}\omega {k}_{z}}{A}_{m}{e}^{{\text{}}{{ik}}_{{mx}}x}{e}^{-{\text{}}i(\omega t-{k}_{z}z)}.\end{eqnarray} \tag{ 32 }$$

The THz wave exists only in the reflection side. So, for x < 0, the electric and magnetic field components of THz wave are

$$\begin{eqnarray}&&{\vec{E}}_{r}={A}_{r}(\hat{x}+{k}_{{rx}}/{k}_{z}\hat{z}){e}^{{\text{}}-{{ik}}_{{rx}}x}{e}^{-{\text{}}i(\omega t-{k}_{z}z)}\end{eqnarray} \tag{ 33 }$$

and

$$\begin{eqnarray}&&{\vec{H}}_{r}={A}_{r}\hat{y}\displaystyle \frac{({k}_{{rx}}^{2}+{k}_{z}^{2})}{{\mu }_{o}\omega {k}_{z}}{e}^{{\text{}}-{{ik}}_{{rx}}x}{e}^{-{\text{}}i(\omega t-{k}_{z}z)}\end{eqnarray} \tag{ 34 }$$

where ${k}_{{rx}}=\omega \cos \theta /c$.

Applying boundary condition that the z-component of electric field vector should be continuous at cluster-vacuum interface, x = 0 i.e.

${\vec{E}}_{{mz}}$ + ${\vec{E}}_{{sz}}$ +${\vec{E}}_{{pz}}$ = ${\vec{E}}_{{rz}}{| }_{x=0}$, we get

$$\begin{eqnarray}&&\displaystyle \frac{-{k}_{{mx}}}{{k}_{z}}{A}_{m}+\displaystyle \frac{{k}_{z}}{{k}_{{sx}}}{A}_{s}-\displaystyle \frac{i\omega {J}_{\omega z}^{{NL}}}{{\epsilon }_{o}\left({\omega }^{2}-{\omega }_{p}^{2}-{k}^{2}{v}_{{th}}^{2}-\tfrac{4\pi {n}_{c}{r}^{3}{\omega }_{{pe}}^{2}}{3(1-{\omega }_{{pe}}^{2}/3{\omega }^{2})}\right)}=\displaystyle \frac{{k}_{x}}{{k}_{z}}{A}_{r}.\end{eqnarray} \tag{ 35 }$$

Applying second boundary condition that the y-component of magnetic field vector should be continuous at x = 0 i.e. ${\vec{H}}_{{my}}={\vec{H}}_{{ry}}{| }_{x=0}$, we get

$$\begin{eqnarray}&&{A}_{m}=\left(\displaystyle \frac{{k}_{{rx}}^{2}+{k}_{z}^{2}}{{k}_{{mx}}^{2}+{k}_{z}^{2}}\right){A}_{r}.\end{eqnarray} \tag{ 36 }$$

For third boundary condition, one can integrate the z-component of equation (28) over x from 0⁻ to 0⁺, to get

$$\begin{eqnarray}&&\left({k}_{{mx}}^{2}+{k}_{z}^{2}(1-\displaystyle \frac{{v}_{{th}}^{2}}{{c}^{2}})\right){A}_{m}-{k}_{z}^{2}\displaystyle \frac{{v}_{{th}}^{2}}{{c}^{2}}{A}_{s}-\displaystyle \frac{i\omega {k}_{{rx}}{k}_{z}{v}_{{th}}^{2}{J}_{\omega z}^{{NL}}}{{\epsilon }_{o}{c}^{2}\left({\omega }^{2}-{\omega }_{p}^{2}-{k}^{2}{v}_{{th}}^{2}-\tfrac{4\pi {n}_{c}{r}^{3}{\omega }_{{pe}}^{2}}{3(1-{\omega }_{{pe}}^{2}/3{\omega }^{2})}\right)}=({k}_{{rx}}^{2}+{k}_{z}^{2}){A}_{r}.\end{eqnarray} \tag{ 37 }$$

On solving equations (35)–(37) we get the amplitude of the reflected THz wave denoted as A_ω as,

$$\begin{eqnarray}&&{A}_{\omega }=\displaystyle \frac{i\omega (1+{k}_{{sx}}/{k}_{x}){J}_{\omega z}^{{NL}}}{{\epsilon }_{o}{\omega }_{p}^{2}\left[1-\tfrac{{\omega }^{2}}{{\omega }_{p}^{2}}+\tfrac{4\pi {n}_{c}{r}^{3}{\omega }_{{pe}}^{2}}{3{\omega }_{p}^{2}(1-{\omega }_{{pe}}^{2}/3{\omega }^{2})}+\tfrac{{k}^{2}{v}_{{th}}^{2}}{{\omega }_{p}^{2}}\right]}\displaystyle \frac{1}{\left[\tfrac{{k}_{{mx}}{k}_{{sx}}/{k}_{x}{k}_{z}+{k}_{z}/{k}_{x}}{(1-{\omega }_{p}^{2}/{\omega }^{2})}+\tfrac{{k}_{{sx}}}{{k}_{z}}\right]}.\end{eqnarray} \tag{ 38 }$$

The retarded vector potential at a far point $\vec{r^{\prime} }(r,{\theta }_{1},z)$ due to the current density of clustered plasma electrons in the filament of L of radius r_f over its entire length and cross-section is given as,

$$\begin{eqnarray}&&\vec{A}(r,t)=\displaystyle \frac{{\mu }_{o}}{4\pi }\int \int \int \displaystyle \frac{{J}_{\omega }^{{NL}}(\vec{{r}^{{\prime} }},t-R/c)}{R}{d}^{3}{r}^{{\prime} },\end{eqnarray} \tag{ 39 }$$

where $R=| r-{r}^{{\prime} }|$. For small cross sectional beam $R\simeq r-{z}^{{\prime} }\cos {\theta }_{1}$, where θ₁ is angle between r and z^' axis and k = k₂ − 2k₁  ≈ ω/c(1 + 3ω_p/4ω).

$$\begin{eqnarray}&&\vec{A}(r,t)=\displaystyle \frac{{\mu }_{o}}{4\pi }\left({\vec{J}}_{1c\omega }^{{NL}}+{\vec{J}}_{1p\omega }^{{NL}}\right){\int }_{0}^{L}\displaystyle \frac{{e}^{-\iota [\omega (t-(r-{z}^{{\prime} }{\cos }{\theta }_{1})/c)]}{e}^{\iota [\omega (1+3{\omega }_{p}/4{\omega }_{1})/c]}{{dz}}^{{\prime} }}{r-{z}^{{\prime} }{\cos }{\theta }_{1}}{\int }_{0}^{2\pi }d{\phi }^{{\prime} }{\int }_{0}^{{r}_{f}}{\rho }^{{\prime} }d{\rho }^{{\prime} }\end{eqnarray} \tag{ 40 }$$

where $({\vec{J}}_{1c\omega }^{{NL}}$ = $({\vec{J}}_{c\omega }^{{NL}}{e}^{{\text{}}i(\omega t-k(z\cos \theta +x\sin \theta ))}$ and ${\vec{J}}_{1p\omega }^{{NL}}$=${\vec{J}}_{p\omega }^{{NL}}{e}^{{\text{}}i(\omega t-k(z\cos \theta +x\sin \theta ))}$. At larger distance, the vector potential A becomes,

$$\begin{eqnarray}&&\vec{A}(r,t)=\displaystyle \frac{{\mu }_{o}{r}_{f}^{2}}{4r}\left({\vec{J}}_{1c\omega }^{{NL}}+{\vec{J}}_{1p\omega }^{{NL}}\right)\displaystyle \frac{{e}^{-\iota \left[\omega (t-r/c)\right]}\left({e}^{\tfrac{\iota \omega }{c}((1+3{\omega }_{p}/4{\omega }_{1})\cos \theta -\cos {\theta }_{1})L}-1\right)}{\tfrac{\iota \omega }{c}((1+3{\omega }_{p}/4{\omega }_{1})\cos \theta -\cos {\theta }_{1})}.\end{eqnarray} \tag{ 41 }$$

The magnetic field of the THz wave can be derived using the relation, $\vec{B}=\vec{{\rm{\nabla }}}\times \vec{A}$ and the time averaged Poynting vector of the THz wave can be written as,

$$\begin{eqnarray}&&{\vec{S}}_{{av}}=\hat{r}\displaystyle \frac{c}{2{\mu }_{o}}| B{| }^{2}=\hat{r}\displaystyle \frac{c{\mu }_{o}{r}_{f}^{4}{L}^{2}{\omega }^{2}}{32{r}^{4}{c}^{2}}{\left({J}_{1c\omega }^{{NL}}+{J}_{1p\omega }^{{NL}}\right)}^{2}\displaystyle \frac{{\sin }^{2}{\theta }^{{\prime} }}{{\theta }^{{\prime} 2}}\end{eqnarray} \tag{ 42 }$$

where $\theta ^{\prime} =\tfrac{\omega L}{2c}\left((1+\tfrac{3\omega }{4{\omega }_{1}})\cos \theta -\cos {\theta }_{1}\right)$.

The normalized power of the radiated field in the forward direction (normalized with ${P}_{1}=({{cB}}_{o}^{2})/(2{\mu }_{o}\pi {r}_{o}^{2}$) is

$$\begin{eqnarray}&&\displaystyle \frac{{P}_{{THz}}}{{P}_{1}}=\displaystyle \frac{\hat{r}}{32\pi }\displaystyle \frac{{r}_{f}^{2}}{{r}_{o}^{2}}\displaystyle \frac{{\mu }_{o}^{2}{r}_{f}^{2}{c}^{2}{\left({J}_{1c\omega }^{{NL}}+{J}_{1p\omega }^{{NL}}\right)}^{2}}{{A}_{1}^{2}}\displaystyle \frac{{L}^{2}{\omega }^{2}}{{c}^{2}}\displaystyle \frac{{\sin }^{2}{\theta }^{{\prime} }}{{\theta }^{{\prime} 2}}.\end{eqnarray} \tag{ 43 }$$

Equation (38) is solved numerically by choosing suitable laser, plasma and cluster parameters. As excitation source radiation, Ti:sapphire laser of intensity 7 × 10¹⁴ W cm⁻² with fundamental frequency ω₁ = 2.126 × 10¹⁵ rad s⁻¹ and slightly shifted second harmonic frequency ω₂ = 4.270 × 10¹⁵ rad s⁻¹ is chosen, such that difference ω₂–2ω₁ lies in THz range. Wavelengths of two beams are λ₁ = 0.89 μm and λ₂ = 0.44 μm. The cluster electron plasma frequency ω_pe = 3.8 × 10¹⁵ rad s⁻¹, plasma frequency ω_p = 1.78 × 10¹³ rad s⁻¹, n_e = 1.8 × 10²² cm⁻³, n_oo = 10¹⁷ cm⁻³ and v = 0.03 c. Ar cluster of radius r = 10 nm with cluster density n_co = 2 × 10¹³ cm⁻³ is considered.

Figure 2 represents the 3D plot of normalized amplitude of THz wave A_ω/A₁ with angle of incidence θ and normalized THz frequency ω/ω_p. It can be clearly observed from this figure that the THz wave amplitude shows maxima at particular values of θ. The first maxima is observed at θ = 22° and the second one is observed at θ = 68° , with slightly lesser amplitude. So, θ = 22° is the optimized value of angle of incidence at which the lasers should be incident at the vacuum-cluster interface so that the maximum amplitude of THz wave can be achieved. In this plot, the THz wave amplitude decreases with the increase in THz frequency and is maximum when ω ∼ ω_p. This is because, when plasma density is close to critical density corresponding to the beat frequency ω (THz frequency), strong coupling of Langmuir wave with THz wave occurs. Thus the THz amplitude decreases with THz frequency due to the weakening of this coupling.

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In figure 3(a), we have presented the variation of the normalized THz wave amplitude as a function of normalized cluster electron frequency ω_pe/ω₁. In this plot, a sharp increase in the THz amplitude is observed as laser frequency approaches ${\omega }_{{pe}}/\sqrt{3}$. This is because, at this frequency, response of cluster electrons to the laser is resonantly enhanced. If we plot the variation of THz wave amplitude with cluster electron frequency normalizing it with THz frequency (i.e. A_ω/A₁ versus ω_pe/ω), again the peak in the plot is observed when ω_pe approaches $\sqrt{3}$ times the THz frequency ω (figure 3(b)).

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The plot of the normalized THz wave amplitude with normalized THz frequency is presented in figure 4 for different values of intensity of incident lasers I = 3 × 10¹⁴, 5.6 × 10¹⁴ and 7 × 10¹⁴. The amplitude of THz generation increases with the increase of intensity of laser beam. The reason of this enhancement can be explained as follows: with the increase of laser intensity, the ponderomotive force strengthens leading to enhanced outer ionization of cluster atoms. As the result, higher charge states are generated, converting maximum energy of the incident laser to generated THz radiation.

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Figure 5 demonstrates dependencies of THz amplitude from Argon gas cluster jet as a function of THz frequency for different values of electron thermal velocity v = 0.02c and 0.03c. v = 0.03c corresponds to electron temperature of 0.5 keV. It can be well observed that as the thermal velocity of electrons is increased, there is significant enhancement in the THz amplitude.

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In figure 6, the effect of cluster radius on the THz yield is observed. Argon cluster of three different radii r = 5 nm, 8 nm and 10 nm is considered. Cluster of bigger radius generates THz radiation of larger amplitude. This is because, by increasing cluster size, ionization ignition [34] and outer ionization of cluster atoms increase [35]. Hence, cluster of larger size would create high charge states and the THz radiation of higher amplitude is generated. The magnitude enhancement in the amplitude of THz radiation due to surface plasmon resonance in clustered plasma is seen as compared to unclustered plasma. The same observation was made experimentally by authors of [27].

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Figure 7 shows the variation of normalized THz power with conical angle θ₁ for different filament lengths L = 1 mm, 2 mm and 3 mm at an initial laser spot size r_o = 6.4 μm and r_f = 16 μm. The power shows the maxima at θ₁ = 21.2°. It is observed that, the THz power is more for the filament of larger length.

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In summary, we presented the analytical model of the generation of THz radiation via nonlinear mixing of obliquely incident two color lasers in hot clustered nano-plasma. The lasers exert ponderomotive force on the cluster electrons. The cluster electrons start oscillating in the plane of incidence and as the result space charge electric field is generated to maintain the plasma neutrality. The ponderomotive force and space charge field drive a nonlinear surface current which generates electromagnetic wave of THz frequency on the reflection side. The strong coupling between this generated electromagnetic wave and plasma wave is observed when the plasma density is near to the critical density of the beat frequency (ω₂ –2 ω₁). When the plasma frequency of cluster electrons ω_pe approaches $\sqrt{3}$ times the laser frequency, the electron response to the laser is resonantly enhanced and sharp increase in THz amplitude is observed. On further expansion of clusters, as ω_pe approaches $\sqrt{3}$ times the terahertz frequency ω, again the enhancement in THz amplitude is observed. If there would be no step density profile (density gradient), i.e., for a homogeneous medium, the curl of nonlinear current density will be zero inside the uniform density region. As a result, only electrostatic field is produced. There will be no transverse component of nonlinear current density and it could not give rise to an electromagnetic radiation (THz radiation). This scheme for generation of THz is attractive because (i) there is no problem of material breakdown (ii) high conversion efficiency can be achieved because of the presence of the cluster as an interactive medium. The power conversion efficiency 1.4 × 10⁻⁴ is observed for the above mentioned parameters of our analysis.