2 Theoretical analysis

Consider the propagation of two coaxial high power vortex laser beams along the $z$ -direction in the nonlinear plasma medium. It is supposed that the lasers electric field is polarized along the $x$ -axis, with frequencies $\unicode[STIX]{x1D714}_{1}$ , $\unicode[STIX]{x1D714}_{2}$ and wavenumbers $k_{1}$ , $k_{2}$ . The laser beams with LG distribution can be described by the phase singularity on axis with strength $l$ that is called the optical vortex charge number, and by the radial index $p$ . Here, the beam width evolution is ignored and the distribution of the laser beams is considered doughnut shaped ( $p=0$ ) as:

While

where $j=1,2$ . In (2.1), $E_{0}$ is the initial electric field amplitude, $r_{0}$ is the beam width, $\unicode[STIX]{x1D6FD}(z)$ is related to the curvature of wave front and $\unicode[STIX]{x1D713}_{0,l}(z)$ is the Gouy phase. The laser beams provide a nonlinear oscillatory velocity to the electrons due to the ponderomotive force, at the beat wave frequency $\tilde{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}_{1}-\unicode[STIX]{x1D714}_{2}$ and wavenumber $\tilde{k}=k_{1}-k_{2}$ . This oscillation is perpendicular to the propagation direction of the lasers. Further, a space-periodically modulated plasma, $n=n_{0}+n^{\prime }$ is considered, where $n_{0}$ is background electron density. Also $n^{\prime }=n_{q}\text{e}^{\text{i}qz}$ , where $n_{q}$ and $q$ are the amplitude and wavenumber of the density ripple, respectively. The density ripple can be created by various techniques (Hazra et al. Reference Hazra, Chini, Sanyal, Grenzer and Pietsch2004; Antonsen Jr, Palastro & Milchberg Reference Antonsen, Palastro and Milchberg2007; Kuo et al. Reference Kuo, Pai, Lin, Lee, Lin, Wang and Chen2007), which operate as a slow wave structure that guides the phase matching condition and induces resonant excitation for generating higher THz radiation. The laser fields impart oscillatory velocities $\boldsymbol{v}_{\boldsymbol{j}}=e\boldsymbol{E}_{j}/\text{i}m\unicode[STIX]{x1D714}_{j}$ to the electrons, and furthermore exert a ponderomotive force, $\boldsymbol{F}^{NL}=m/2\unicode[STIX]{x1D735}\boldsymbol{v}_{1}\boldsymbol{\cdot }\boldsymbol{v}_{2}^{\ast }$ , at the beat wave frequency $\tilde{\unicode[STIX]{x1D714}}$ and wavenumber $\tilde{k}$ . Considering equal beam width, the $x$ -component of the nonlinear ponderomotive force due to beating of two LG lasers is obtained as

where $\tilde{l}=l_{1}-l_{2}$ and $z_{R}$ is the Rayleigh length. The ponderomotive force is a superposition of two OAM. This ponderomotive force gives rise to the nonlinear perturbation, $n^{NL}$ , in the electron density, that produces a space charge field. This space charge field self-consistently induces another type of density perturbation, $n^{L}$ . Using the equations of motion and continuity of electrons we obtain $n^{L}$ . By employed Poisson’s equation, the potential of the space charge field, with the combined effects of both the perturbation in electron density, $n^{L}+n^{NL}$ , can be derived. The electrostatic potential, $\unicode[STIX]{x1D719}$ , as the main source of perturbation, is obtained (Malik, Malik & Nishida Reference Malik, Malik and Nishida2011a )

Using the equation of motion, by taking into account the contribution of the self-consistent field, $e\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ , and the ponderomotive force, $\boldsymbol{F}^{NL}$ , the response of the electrons turns out to be

The intensity variation in the transverse direction produces a large change in the ponderomotive force that gives rise to a component of current oscillating at the beat frequency of the lasers. In the presence of ripple plasma density, the nonlinear current density is obtained as

Here the density perturbation is assumed to be small enough so that $n^{L}\ll n_{q}$ . As is clear from (2.6), the current density changes in accordance with the nonlinear ponderomotive force, $\boldsymbol{F}^{NL}\sim \text{e}^{\text{i}\tilde{k}z-i\tilde{\unicode[STIX]{x1D714}}t}$ . Since $\boldsymbol{J}^{NL}$ is responsible for the generation of THz radiation, the THz field will vary as $\text{e}^{\text{i}\tilde{k}z-i\tilde{\unicode[STIX]{x1D714}}t}$ . Using Maxwell’s equations, the THz wave equation is obtained as follows

Taking the $x$ -component of (2.7), the emitted THz radiation, in response to the nonlinear current, is a superposition of two OAM. By employing a suitable plasma density distribution, one of the OAM is eliminated. So THz radiation having one OAM can be generated. Here, the plasma density distribution is considered as $\unicode[STIX]{x1D714}_{p}^{2}=\unicode[STIX]{x1D714}_{p0}^{2}[1+b(l_{T})+r^{2}/a^{2}]$ , where $\unicode[STIX]{x1D714}_{p0}=(4\unicode[STIX]{x03C0}e^{2}n_{0}^{0}/m)^{1/2}$ , $b(l_{T})=-(c^{2}/\unicode[STIX]{x1D714}_{p0}^{2})(4(|l_{T}|+1)/w_{0}^{2})$ and $a=w_{0}^{2}\unicode[STIX]{x1D714}_{p0}/2c$ . This electron density distribution can be produced by different sources of energy (Kaur, Sharma & Salih Reference Kaur, Sharma and Salih2009; Sobhani et al. Reference Sobhani, Vaziri, Rooholamininejad and Bahrampour2016c ,Reference Sobhani, Vaziri, Rooholamininejad and Bahrampour d ). The $x$ -component of (2.7) can be deduced as

where $\unicode[STIX]{x1D700}=1-\unicode[STIX]{x1D714}_{p}^{2}/\tilde{\unicode[STIX]{x1D714}}^{2}$ is the permittivity of the plasma medium at the THz frequency $\tilde{\unicode[STIX]{x1D714}}$ . By choosing suitable plasma parameters $a$ and $b$ , the first three terms on the left-hand side of (2.8) vanish, so the beam width evolution of THz field can be ignored and the vortex charge number of the generated twisted THz radiation is predicted. Using the separation of variables method, a homogeneous solution of (2.8) in the absence of the nonlinear current density, is given as (Sobhani et al. Reference Sobhani, Rooholamininejad and Bahrampour2016b )

where the amplitude of the wave, $A^{l_{T}}$ is a constant. It is expected on physical grounds that, when the nonlinear source term is not too large, the solution to (2.8) will still be of the form of (2.9), except that $A^{l_{T}}$ will be a slowly varying function of  $z$ . So the solution may be written as

This solution has a doughnut LG form in which the beam width, $w_{0}$ is related to the parameters of plasma density. So the plasma density distribution plays the fundamental role in the distribution of the emitted THz radiation. Also the charge number of the emitted THz radiation is determined by the parameters of plasma density. The fast phase variations in $E_{T}$ are taken as $\text{e}^{\text{i}k_{T}z-i\tilde{\unicode[STIX]{x1D714}}t}$ , and the phase matching condition demands $k_{T}=k_{1}-k_{2}+q$ . Substituting (2.10) into (2.8), one obtains

The beam width evolution of the THz radiation, $w_{0}$ , in the nonlinear plasma is ignored. Using the definition of $k_{T}^{2}=(\tilde{\unicode[STIX]{x1D714}}^{2}/c^{2})\unicode[STIX]{x1D700}$ , the second term on the left-hand side of (2.11) vanishes. Under the resonant excitation of the THz radiation, the wavenumber of the rippled plasma density is given as $q=\tilde{\unicode[STIX]{x1D714}}/c[(1-\unicode[STIX]{x1D714}_{p}^{2}/\tilde{\unicode[STIX]{x1D714}}^{2})^{1/2}-1]$ . The normalized period of the density ripple structure, $qc/\unicode[STIX]{x1D714}_{p}$ , plays an important role in the phase matching and generation of efficient vortex THz radiation. Equation (2.11) is multiplied by $\text{e}^{-\text{i}l^{\prime }\unicode[STIX]{x1D719}}D_{p^{\prime }}^{l^{\prime }\ast }(\unicode[STIX]{x1D709})$ , where $\unicode[STIX]{x1D709}=2r^{2}/w_{0}^{2}$ , and integrated over $r$ and $\unicode[STIX]{x1D719}$ . Using the orthogonality relations of $D_{p}^{l}(\unicode[STIX]{x1D709})$ and $\text{e}^{\text{i}l\unicode[STIX]{x1D719}}$ , the value of the THz radiation amplitude is zero, except for two indexes. Therefore one can obtain

While

The plasma parameters, $w_{0}$ and $b(l_{T})$ , determine the characteristics of the THz radiation such as the beam width and the topological charge. By beating the two LG lasers having relative topological charge, $\tilde{l}$ , in a plasma medium characterized by parameter $b(l_{T})$ , the THz amplitude is non-zero when $l_{T}=\tilde{l}\pm 1$ . Equation (2.12) illustrates that the suitable plasma distribution, $b(\tilde{l}\pm 1)$ , is emphasized that vortex THz radiation with only one OAM can be produced. The amplitude of the THz radiation generated can be written as

The features of the THz field amplitude due to the beating of the two lasers are solved numerically. Due to the relative Gouy phase, defined as the difference of the Gouy phases of the input lasers, the relative phase between the lasers plays an important role in the generation of the THz radiation. The relative Gouy phase induces a decrease in the amplitude of the THz radiation generation, as it is obvious from (2.14). If the topological numbers of the lasers are equal, $\tilde{l}=0$ , the Gouy phase shift of each laser is removed others and the amplitude of the THz radiation generation is higher.

3 Results and discussion

Recently, the generation of THz radiation by the beating of laser beams has attracted a great deal of attention related to research and technological applications. The interest in vortex waves in the THz domain can also be applied to data transmission (Wang et al. Reference Wang, Yang, Fazal, Ahmed, Yan, Huang, Ren, Yue, Dolinar and Tur2012), communications (Zhu et al. Reference Zhu, Wei, Wang, Zhang, Li, Zhang, Li, Wang and Liu2014) and active THz holography (Xie, Wang & Zhang Reference Xie, Wang and Zhang2013). Also, the THz frequency region includes the eigenfrequencies of molecules so THz imaging has been applied to biomedicine, security, and spectroscopy (Markelz, Roitberg & Heilweil Reference Markelz, Roitberg and Heilweil2000; Ferguson & Zhang Reference Ferguson and Zhang2002; Kawase et al. Reference Kawase, Ogawa, Watanabe and Inoue2003; Chen et al. Reference Chen, O’Hara, Azad, Taylor, Averitt, Shrekenhamer and Padilla2008; Zhang et al. Reference Zhang, Park, Li, Lu, Zhang and Zhang2009; Ohno et al. Reference Ohno, Hamano, Miyamoto, Suzuki and Ito2009, Reference Ohno, Miyamoto, Minamide and Ito2010). Here (2.14) is solved to study the features of the THz field amplitude for the following laser and plasma parameters; the frequencies of the laser beams are $2.4\times 10^{14}~\text{rad}~\text{s}^{-1}$ and $2.1\times 10^{14}~\text{rad}~\text{s}^{-1}$ , also the initial beam width of both lasers is $50~\unicode[STIX]{x03BC}\text{m}$ and the plasma frequency is $2.75\times 10^{14}~\text{rad}~\text{s}^{-1}$ . The plasma frequency should be selected to be lower than $\tilde{\unicode[STIX]{x1D714}}$ so that the THz wave can propagate in the plasma medium. The value of the normalized ripple amplitude is $n_{q}/n_{0}^{0}=0.1$ and the parameter of plasma density, related to the THz radiation beam width, $w_{0}$ , is $75~\unicode[STIX]{x03BC}\text{m}$ . When an intense laser encounters plasma, a ponderomotive force pushes the electrons out of strong power region, decreasing the local electron density, which leads to the further growth of the plasma dielectric function and consequently, an even stronger self-focusing of laser take places. The diffraction causes defocusing of the laser beams. Under the competition between the ponderomotive force effect and beam diffraction, periodic self-focusing and defocusing occurs and a density-modulated filament is created. Therefore intensity variation of LG lasers in the transverse direction is strong, and the plasma current or laser ponderomotive force can generate a THz wave at the beating frequency (Sobhani et al. Reference Sobhani, Rooholamininejad and Bahrampour2016b ).

Figure 1. Normalized THz amplitude versus normalized propagation distance. The cases shown are, (a) $l_{1}=1$ , $l_{2}=1$ , for $l_{T}=1$ (solid line), $l_{T}=-1$ (dashed line), (b) $l_{1}=2$ , $l_{2}=0$ , for $l_{T}=3$ (solid line), $l_{T}=1$ (dashed line), (c) $l_{1}=2$ , $l_{2}=0$ , for $l_{T}=3$ (solid line), $l_{T}=1$ (dashed line) and (d) $l_{1}=3$ , $l_{2}=2$ , for $l_{T}=2$ (solid line), $l_{T}=0$ (dashed line), in b ( $l_{T}$ ) plasma distribution.

Figure 2. Normalized THz amplitude versus normalized propagation distance by beating of lasers having $l_{1}=2$ , $l_{2}=1$ , when the normalized ripple amplitude of $n_{q}=0.1n_{0}$ (solid line) and $n_{q}=0.2n_{0}$ (dashed line), for (a) $l_{T}=2$ , (b) $l_{T}=0$ in corresponded plasma distribution.

Figure 3. Profile of the input lasers and generated THz radiation: the topological charge numbers of the lasers are $l_{1}=2$ (a) and $l_{2}=1$ (b), and the topological charge numbers of the generated THz radiation are $l_{T}=2$ (c) and (d) $l_{T}=0$ , in corresponded plasma distribution.

Figure 4. Normalized THz amplitude versus normalized propagation distance by beating of lasers having $l_{1}=2$ , $l_{2}=1$ , when $\unicode[STIX]{x1D714}_{p}=2.7\times 10^{13}~\text{rad}~\text{s}^{-1}$ (solid line) and $\unicode[STIX]{x1D714}_{p}=2.5\times 10^{13}~\text{rad}~\text{s}^{-1}$ (dashed line) for (a) $l_{T}=2$ , (b) $l_{T}=0$ in corresponded plasma distribution.

For studying the effect of the topological charge of the incident lasers, the variation of the normalized THz field amplitude versus normalized propagation distance, $Z=z/z_{R}$ , for different values of topological charge of input lasers is presented in figure 1. As seen, the normalized field amplitude of the emitted THz radiation for different OAM is non-similar, through the plasma density characterized by parameter $b(\tilde{l}\pm 1)$ . One of these plasma parameters is such that the electron density is higher and the number of charge carriers involved in the generation of the oscillatory current is greater. Higher oscillatory current results in a higher vortex THz radiation. Because the pumps depletion effect is ignored, the THz amplitude is a rising function of $z$ , and in the presence of the pump depletion effect it is expected that the output THz amplitude approaches a saturation value. The presence of the pump depletion effect is under study by our group and results will appear in the near future. Also, it is obvious from figure 1 that the THz field amplitude generated is higher when a lower vortex charge number is chosen. With the lower vortex charge number, the beam focusing–defocusing oscillations occur strongly; the density-modulated filament induces hardly any intensity variation in the transverse direction, and the ponderomotive force can generate higher THz radiation at the beat frequency. This fact is clear from (2.14).

Figure 5. Normalized THz amplitude versus normalized distance by beating of lasers having $l_{1}=2$ , $l_{2}=1$ , under the phase matching condition, $k_{T}=k_{1}-k_{2}+q$ (solid line) and $k_{T}=k_{1}-k_{2}-\tilde{l}/z\tan ^{-1}(z/z_{R})+q$ (dashed line) for (a) $l_{T}=2$ , and (b) $l_{T}=0$ in corresponded plasma density.

For instance, by beating two vortex lasers having charge numbers $l_{1}=2$ and $l_{2}=1$ , THz radiation with OAM of $l_{T}=2,0$ is generated. The variation of the normalized THz field amplitude versus normalized propagation distance for different values of the normalized ripple amplitude is presented in figure 2. As seen in this figure, the THz field amplitude generated is higher when a larger value of the normalized ripple amplitude is selected. For clarification, a larger value of the normalized ripple amplitude is acquired for better phase matching and maximum energy transfer to the THz radiation. The growth in the THz field amplitude with the ripple amplitude is appreciable as larger numbers of electrons take part (in phase) in the oscillating nonlinear current. In figure 3, the intensity profiles of the input lasers (having charge numbers 2 and 1) and generated vortex THz, having charge numbers 2 and 0, are depicted.

The evolution of the normalized THz field amplitude as a function of the normalized propagation distance for different background plasma frequencies, $\unicode[STIX]{x1D714}_{0}^{0}$ , is shown in figure 4. Similar to the effect of ripple amplitude, the enhanced field with the growth in background plasma frequency is plausible as more electrons take part in the oscillating nonlinear current. For generating strong THz radiation, the plasma frequency should be near to the value of the beat wave frequency, $\unicode[STIX]{x1D714}_{p}\approx \unicode[STIX]{x1D714}$ , this corresponds to the resonance condition so maximum energy transfer will occur.

Figure 6. Normalized THz amplitude versus normalized propagation distance by beating of lasers having $l_{1}=2$ , $l_{2}=1$ and $r_{0}=50~\unicode[STIX]{x03BC}\text{m}$ (solid line), $r_{0}=40~\unicode[STIX]{x03BC}\text{m}$ (dashed line) for (a) $l_{T}=2$ and (b) $l_{T}=0$ in corresponded plasma distribution.

Figure 7. Normalized THz amplitude generated by beating of the lasers having $l_{1}=1$ , $l_{2}=1$ , versus normalized propagation distance for THz radiation beam width $w_{0}=1.5r_{0}$ (solid line) and $w_{0}=1.4r_{0}$ (dashed line), viz., (a) $l_{T}=2$ , (b) $l_{T}=0$ in corresponded plasma distribution.

Another parameter for generating strong THz radiation is the wavenumber of the density ripple. The expression in (2.14) predicts a dramatic decrease in the efficiency of the THz radiation generation process when the relative Gouy phase of the incident lasers is non-zero. By using the ripple wavenumber, considered in the previous section, perfect phase matching is not provided. Consequently, the THz radiation generated will be weak. By considering a new ripple wavenumber, $q=\tilde{\unicode[STIX]{x1D714}}/c[(1-\unicode[STIX]{x1D714}_{p}^{2}/\tilde{\unicode[STIX]{x1D714}}^{2})^{1/2}-1]+\tilde{l}/z\tan ^{-1}(z/z_{R})$ , which can be created by various techniques (Kaur et al. Reference Kaur, Sharma and Salih2009; Sobhani et al. Reference Sobhani, Vaziri, Rooholamininejad and Bahrampour2016c ,Reference Sobhani, Vaziri, Rooholamininejad and Bahrampour d ), one may define the perfect phase matching condition as $k_{T}=k_{1}-k_{2}-(\tilde{l}/z)\tan ^{-1}(z/z_{R})+q$ . In the presence of this ripple wavenumber, the effect of the relative Gouy phases of the input lasers is removed and perfect phase matching is achieved. With this condition, the interaction length is infinite and the energy transfer from the pump laser to the emitted THz radiation will be maximized. A comparison of the generation of the THz radiation between the two phase matching conditions is presented in figure 5. This new ripple wavenumber is constructed at smaller distances and maximum energy transfer will happen. So efficient THz radiation generation will result.

In order to uncover the role of the beam width value of the incident lasers, the variation of the normalized THz field amplitude versus normalized propagation distance is illustrated in figure 6. As shown, the THz field amplitude decreases for the greater beam width. This observation is clear with regards to the beam width in the distribution of the LG beam. The ratio of the beam width of the input lasers $r_{0}$ to the parameter of plasma density, related to the beam width of the THz radiation $w_{0}$ , is found to play an important role to the mechanism of generation of the THz radiation. Figure 7 gives the variation of the normalized THz amplitude as a function of the normalized propagation for different $r_{0}/w_{0}$ . For $w_{0}=1.4r_{0}$ , the slope of plasma density is greater than that for $w_{0}=1.5r_{0}$ ; so the transverse variation and ponderomotive force of the input lasers for the first case is greater and the THz radiation efficiency is better.