Generation of Terahertz Radiation under Interaction of Counterpropagating Laser Pulses in Underdense Plasma

Abstract

Generation of terahertz waves under interaction of two counterpropagating laser pulses with different frequencies in underdense plasma is analyzed. Spectral, angular, and energy characteristics of terahertz radiation are investigated as functions of the frequency difference of the laser pulses. It is shown that, due to the frequency difference, in addition to the line at the doubled plasma frequency [L.M. Gorbunov, A.A. Frolov, JETP 98, 527 (2004)], a peak near the plasma frequency appears in the emission spectrum. The total energy of the terahertz signal is calculated, and the condition under which emission at the plasma frequency dominates is found. It is shown that the energy of radiation at the plasma frequency reaches its maximum value when the frequency difference of the laser pulses is close to the plasma frequency.

1 INTRODUCTION

Generation and detection of terahertz (THz) radiation attracts considerable interest of many research groups due to the possibilities of its use in science, technology, and practical applications. Generation of THz radiation under laser action on solid and gaseous targets has been discovered experimentally relatively long ago, as early as 1990s [1, 2]. Under the conditions of modern experiments, THz pulses with an energy reaching several hundred microjoules were obtained upon laser irradiation of organic crystals [3, 4]. In this case, the conversion efficiency of the laser energy into the energy of the THz wave can reach 3%. The problems related to the generation of THz radiation under laser–matter interaction were analyzed theoretically in [5–8] for the case of underdense plasma formed as a result of ionization of gas jets, while the cases of supercritical plasma and conductive media of a solid-state density were analyzed in [9–12]. Relatively recently, THz radiation was experimentally detected also under interaction of laser radiation with clusters [13–15] formed under injection of a high-pressure noble gas jet into a vacuum chamber. Interaction of THz waves under laser–pulse interaction with clusters was analyzed theoretically in [16–18], where the possibility of generation of high-power THz pulses with relatively high conversion efficiency was demonstrated.

Here, we propose an efficient scheme for generation of THz waves that is based on the interaction of two counterpropagating laser pulses in underdense plasma and develop the relevant theory. The earlier developed theory of radiation generation at the doubled plasma frequency in the interaction of pulses with equal frequencies [7] is generalized for the case of laser pulses with different frequencies. Similar to [7], low-frequency electromagnetic radiation in the proposed scheme is generated due to the excitation of small-scale wake plasma fields and their interaction in the region where the laser pulses overlap with one another. The electromagnetic wave thus appears as a result of fusion of two plasma waves, or, in other words, low-frequency radiation is generated due to an elementary nonlinear process of fusion of two plasmons accompanied by creation of a photon. The dependences of spectral, angular, and energy characteristics of the THz pulse, as well as of its time profile, on the frequency difference of the laser pulses are investigated. It is shown that, in addition to the line at the doubled plasma frequency, a peak at the plasma frequency appears in the spectrum, which is related to the difference in the frequencies of the laser pulses. An inequality is derived under which THz radiation is generated mainly at the plasma frequency. It is shown that, under the conditions in which the plasma frequency dominates, the energy of the THz signal reaches its maximum value when the frequency difference of the laser pulses is equal to the plasma frequency.

The paper is organized as follows. In Section 2, we present the main equations describing generation of low-frequency electromagnetic fields in plasma under the action of the ponderomotive forces of laser radiation. In Section 3, these equations are solved using the perturbation theory and expressions for the components of the THz electromagnetic field in the wave zone far from the region of interaction of the laser pulses are derived. Spectral, angular, and energy c-haracteristics of the THz waves are investigated in Section 4. Analysis of the spectrum of THz radiation shows that it critically depends on the frequency difference of the laser pulses. It is demonstrated that a peak near the plasma frequency appears in addition to that at the doubled plasma frequency even at a small difference in the frequencies of the laser pulses. The amplitude of this peak increases with increasing frequency difference, reaching its maximum value when the frequency difference is equal to the plasma frequency. The angular distribution of the THz waves under the conditions in which radiation at the plasma frequency dominates is investigated. It is shown that, in the case of tight focusing of the laser pulses, radiation is emitted in transverse direction relative to the propagation direction of the laser pulses. As the focal spot size increases, the emitted radiation gradually concentrates closer and closer to the axis of propagation of the laser pulses and becomes directed nearly along this axis for vary broad pulses. The total energy of the THz pulse is calculated, and the condition under which radiation at the plasma frequency dominates is found. It is demonstrated that the energy of radiation at the plasma frequency reaches its maximum value when the frequency difference of the laser pulses coincides with the plasma frequency. The time profile of the THz pulse in the far zone at long distances from the region of pulse interaction is investigated. It is shown that, when the resonance condition is satisfied, the field of the THz pulse oscillates at the plasma frequency and contains many periods, so that its duration is substantially longer than that of the laser pulses. Finally, in the Conclusions, we discuss the applicability conditions of the theory and estimate the parameters of the THz signal.

2 BASIC EQUATIONS

Let two laser pulses with slightly different frequencies ${{\omega }_{1}} = {{\omega }_{0}} + {{\Delta {{\omega }_{0}}} \mathord{\left/ {\vphantom {{\Delta {{\omega }_{0}}} 2}} \right. \kern-0em} 2}$, ${{\omega }_{2}} = {{\omega }_{0}} - {{\Delta {{\omega }_{0}}} \mathord{\left/ {\vphantom {{\Delta {{\omega }_{0}}} 2}} \right. \kern-0em} 2}$ ($\Delta {{\omega }_{0}} > 0$, $\Delta {{\omega }_{0}} \ll {{\omega }_{1}},{{\omega }_{2}}$), and duration τ be counterpropagating in an completely ionized underdense plasma with the electron density ${{N}_{{0e}}}$, which is substantially lower than the critical value ${{N}_{{{\text{cr}}}}} \simeq {{{{m}_{e}}\omega _{0}^{2}} \mathord{\left/ {\vphantom {{{{m}_{e}}\omega _{0}^{2}} {\left( {4\pi {{e}^{2}}} \right)}}} \right. \kern-0em} {\left( {4\pi {{e}^{2}}} \right)}}$, where e is the electron charge, ${{m}_{e}}$ is the electron mass, $\Delta {{\omega }_{0}} =$${{\omega }_{1}} - \;{{\omega }_{2}}$, and ${{\omega }_{0}} = {{\left( {{{\omega }_{1}} + \,{{\omega }_{2}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{\omega }_{1}} + \,{{\omega }_{2}}} \right)} 2}} \right. \kern-0em} 2}$. If the laser pulses propagate along the z axis, their electric field can be expressed in the form

$$\begin{gathered} {{{\mathbf{E}}}_{L}}({\mathbf{r}},t) = \frac{1}{2}{{{\mathbf{E}}}_{1}}\left( {{\mathbf{r}},t} \right)\exp \left( { - i{{\omega }_{1}}t + i{{k}_{1}}z} \right) \\ + \;\frac{1}{2}{{{\mathbf{E}}}_{2}}\left( {{\mathbf{r}},t} \right)\exp \left( { - i{{\omega }_{2}}t - i{{k}_{2}}z} \right) + {\text{c}}{\text{.c}}., \\ \end{gathered} $$

((2.1))

where ${{E}_{{1,2}}}\left( {{\mathbf{r}},t} \right)$ are the amplitudes of the electric field of the corresponding pulses slowly varying with time and in space on the scales ${1 \mathord{\left/ {\vphantom {1 {{{\omega }_{{1,2}}}}}} \right. \kern-0em} {{{\omega }_{{1,2}}}}}$ and ${1 \mathord{\left/ {\vphantom {1 {{{k}_{{1,2}}}}}} \right. \kern-0em} {{{k}_{{1,2}}}}}$, respectively; ${{k}_{{1,2}}} = {{\sqrt {\omega _{{1,2}}^{2} - \omega _{p}^{2}} } \mathord{\left/ {\vphantom {{\sqrt {\omega _{{1,2}}^{2} - \omega _{p}^{2}} } c}} \right. \kern-0em} c}$ are their wavenumbers; ${{\omega }_{p}}$ is the plasma frequency, which, in an underdense plasma, is much lower than the frequencies of the laser pulses (${{\omega }_{p}} \ll {{\omega }_{1}},{{\omega }_{2}}$); and c is the speed of light. According to expression (2.1), the pulse with the amplitude ${{E}_{1}}\left( {{\mathbf{r}},t} \right)$ and frequency ${{\omega }_{1}}$ propagates in the positive z direction, while the pulse with the amplitude ${{E}_{2}}\left( {{\mathbf{r}},t} \right)$ and frequency ${{\omega }_{2}}$ propagates in the opposite direction. The reference frame is chosen in such a way that the laser pulses meet one another at the coordinate origin $z = 0$ at time $t = 0$.

The ponderomotive forces of laser radiation (2.1) induce low-frequency electromagnetic fields in underdense plasma. For their description, we will use the following set of time-averaged hydrodynamic equations and Maxwell’s equations for the electron velocity ${\mathbf{V}}\left( {{\mathbf{r}},t} \right)$, electron density perturbation $\delta {{N}_{e}}\left( {{\mathbf{r}},t} \right)$, electric field ${\mathbf{E}}\left( {{\mathbf{r}},t} \right)$, and magnetic field ${\mathbf{B}}\left( {{\mathbf{r}},t} \right)$, which are slowly varying over the laser period (see, e.g., [8]):

$$\frac{\partial }{{\partial t}}\delta {{N}_{e}}\left( {{\mathbf{r}},t} \right) + \nabla \cdot \left\{ {\left[ {{{N}_{{0e}}} + \delta {{N}_{e}}\left( {{\mathbf{r}},t} \right)} \right]{\mathbf{V}}\left( {{\mathbf{r}},t} \right)} \right\} = 0,$$

((2.2))

$$\left( {\frac{\partial }{{\partial t}} + {{\nu }_{{ei}}}} \right){\mathbf{V}}\left( {{\mathbf{r}},t} \right) = \frac{e}{{{{m}_{e}}}}\left[ {{\mathbf{E}}\left( {{\mathbf{r}},t} \right) - \nabla \Phi \left( {{\mathbf{r}},t} \right)} \right],$$

((2.3))

$$\nabla \times {\mathbf{E}}\left( {{\mathbf{r}},t} \right) = - \frac{1}{c}\frac{\partial }{{\partial t}}{\mathbf{B}}\left( {{\mathbf{r}},t} \right),$$

((2.4))

$$\begin{gathered} \nabla \times {\mathbf{B}}\left( {{\mathbf{r}},t} \right) = \frac{1}{c}\frac{\partial }{{\partial t}}{\mathbf{E}}\left( {{\mathbf{r}},t} \right) \\ + \;\frac{{4\pi e}}{c}\left[ {{{N}_{{0e}}} + \delta {{N}_{e}}\left( {{\mathbf{r}},t} \right)} \right]{\mathbf{V}}\left( {{\mathbf{r}},t} \right), \\ \end{gathered} $$

((2.5))

where ${{\nu }_{{ei}}}$ is the electron–ion collision frequency. The potential of the ponderomotive forces $\Phi \left( {{\mathbf{r}},t} \right)$ has the form

$$\begin{gathered} \Phi \left( {{\mathbf{r}},t} \right) = \frac{e}{{4{{m}_{e}}\omega _{1}^{2}}}{{\left| {{{{\mathbf{E}}}_{1}}\left( {{\mathbf{r}},t} \right)} \right|}^{2}} \\ + \;\frac{e}{{4{{m}_{e}}\omega _{2}^{2}}}{{\left| {{{{\mathbf{E}}}_{2}}\left( {{\mathbf{r}},t} \right)} \right|}^{2}} + \frac{e}{{4{{m}_{e}}{{\omega }_{1}}{{\omega }_{2}}}} \\ \times \;\left\{ {{{{\mathbf{E}}}_{1}}\left( {{\mathbf{r}},t} \right) \cdot {\mathbf{E}}_{2}^{*}\left( {{\mathbf{r}},t} \right)\exp \left( { - i\Delta {{\omega }_{0}}t + 2i{{k}_{0}}z} \right) + {\text{c}}{\text{.c}}{\text{.}}} \right\}, \\ \end{gathered} $$

((2.6))

where we introduced the notation ${{k}_{0}} = {{\left( {{{k}_{1}} + {{k}_{2}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{k}_{1}} + {{k}_{2}}} \right)} 2}} \right. \kern-0em} 2}$. The third term in expression (2.6) for the ponderomotive potential describes the contribution from pulse interaction. Note that thermal motion, electron heating, and relativistic effects in Eq. (2.3) for the velocity of slow electron motion are neglected. This is possible if the electron velocity in the laser field ${{V}_{L}}$ is much lower than the speed of light but is higher than the electron thermal velocity ${{V}_{T}}$. The thermal motion of electrons and the effects of heating can be neglected if $\tau \ll {1 \mathord{\left/ {\vphantom {1 {{{\nu }_{{ei}}}}}} \right. \kern-0em} {{{\nu }_{{ei}}}}}$, i.e., if the laser pulse duration τ is much shorter than the electron–ion collision time.

3 EXCITATION OF LOW-FREQUENCY ELECTROMAGNETIC FIELDS UNDER INTERACTION OF COUNTERPROPAGATING LASER PULSES

Let us solve set of equations (2.2)–(2.5) by using the perturbation theory, taking into account that the electron velocity in the laser field ${{V}_{L}}$ is much smaller than the speed of light. To this end, let us represent all quantities in the form of power series expansions in the small parameter ${{V_{L}^{2}} \mathord{\left/ {\vphantom {{V_{L}^{2}} {{{c}^{2}}}}} \right. \kern-0em} {{{c}^{2}}}} \ll 1$ (e.g., ${\mathbf{E}} = {{{\mathbf{E}}}^{{\left( 1 \right)}}} + {{{\mathbf{E}}}^{{\left( 2 \right)}}} + ...$ for the electric field). In the linear approximation, we obtain the following equations:

$$\left( {\frac{{{{\partial }^{2}}}}{{\partial {{t}^{2}}}} + {{\nu }_{{ei}}}\frac{\partial }{{\partial t}} + \omega _{p}^{2}} \right)\frac{{\delta N_{e}^{{\left( 1 \right)}}\left( {{\mathbf{r}},t} \right)}}{{{{N}_{{0e}}}}} = \frac{e}{{{{m}_{e}}}}\Delta \Phi \left( {{\mathbf{r}},t} \right),$$

((3.1))

$$\left( {\frac{{{{\partial }^{2}}}}{{\partial {{t}^{2}}}} + {{\nu }_{{ei}}}\frac{\partial }{{\partial t}} + \omega _{p}^{2}} \right){{{\mathbf{V}}}^{{\left( 1 \right)}}}\left( {{\mathbf{r}},t} \right) = - \frac{e}{{{{m}_{e}}}}\frac{\partial }{{\partial t}}\nabla \Phi \left( {{\mathbf{r}},t} \right),$$

((3.2))

$$\left( {\frac{{{{\partial }^{2}}}}{{\partial {{t}^{2}}}} + {{\nu }_{{ei}}}\frac{\partial }{{\partial t}} + \omega _{p}^{2}} \right){{{\mathbf{E}}}^{{\left( 1 \right)}}}\left( {{\mathbf{r}},t} \right) = \omega _{p}^{2}\nabla \Phi \left( {{\mathbf{r}},t} \right),$$

((3.3))

which describe excitation of only potential currents and fields.

Low-frequency electromagnetic fields are generated due to nonlinear interaction of the longitudinal plasma fields. The corresponding equations for the fields in the quadratic approximation with respect to the small parameter ${{V_{L}^{2}} \mathord{\left/ {\vphantom {{V_{L}^{2}} {{{c}^{2}}}}} \right. \kern-0em} {{{c}^{2}}}} \ll 1$ can be obtained from Eqs. (2.2)–(2.5) and have the form

$$\begin{gathered} \frac{\partial }{{\partial t}}\delta N_{e}^{{\left( 2 \right)}}\left( {{\mathbf{r}},t} \right) \\ + \;\nabla \cdot \left[ {{{N}_{{0e}}}{{{\mathbf{V}}}^{{\left( 2 \right)}}}\left( {{\mathbf{r}},t} \right) + \delta N_{e}^{{\left( 1 \right)}}\left( {{\mathbf{r}},t} \right){{{\mathbf{V}}}^{{\left( 1 \right)}}}\left( {{\mathbf{r}},t} \right)} \right] = 0, \\ \end{gathered} $$

((3.4))

$$\left( {\frac{\partial }{{\partial t}} + {{\nu }_{{ei}}}} \right){{{\mathbf{V}}}^{{\left( 2 \right)}}}\left( {{\mathbf{r}},t} \right) = \frac{e}{{{{m}_{e}}}}{{{\mathbf{E}}}^{{\left( 2 \right)}}}\left( {{\mathbf{r}},t} \right),$$

((3.5))

$$\nabla \times {{{\mathbf{E}}}^{{\left( 2 \right)}}}\left( {{\mathbf{r}},t} \right) = - \frac{1}{c}\frac{\partial }{{\partial t}}{{{\mathbf{B}}}^{{\left( 2 \right)}}}\left( {{\mathbf{r}},t} \right),$$

((3.6))

$$\begin{gathered} \nabla \times {{{\mathbf{B}}}^{{\left( 2 \right)}}}\left( {{\mathbf{r}},t} \right) = \frac{1}{c}\frac{\partial }{{\partial t}}{{{\mathbf{E}}}^{{\left( 2 \right)}}}\left( {{\mathbf{r}},t} \right) \\ + \;\frac{{4\pi e}}{c}\left[ {{{N}_{{0e}}}{{{\mathbf{V}}}^{{\left( 2 \right)}}}\left( {{\mathbf{r}},t} \right) + \delta N_{e}^{{\left( 1 \right)}}\left( {{\mathbf{r}},t} \right){{{\mathbf{V}}}^{{\left( 1 \right)}}}\left( {{\mathbf{r}},t} \right)} \right]. \\ \end{gathered} $$

((3.7))

To solve set of equations (3.4)–(3.7), we apply Fourier transformation with respect to time. As a result, Eqs. (3.5)–(3.7) yield the following equations for the Fourier transform of the low-frequency magnetic field:

$$\begin{gathered} \Delta {{{\mathbf{B}}}^{{\left( 2 \right)}}}\left( {\omega ,{\mathbf{r}}} \right) + \frac{{{{\omega }^{2}}}}{{{{c}^{2}}}}\varepsilon \left( \omega \right){{{\mathbf{B}}}^{{\left( 2 \right)}}}\left( {\omega ,{\mathbf{r}}} \right) \\ = \; - \frac{{4\pi }}{c}\nabla \times {{{\mathbf{j}}}^{{\left( 2 \right)}}}\left( {\omega ,{\mathbf{r}}} \right), \\ \end{gathered} $$

((3.8))

where ${{{\mathbf{j}}}^{{\left( 2 \right)}}}\left( {\omega ,{\mathbf{r}}} \right)$ is the Fourier component of the nonlinear current ${{{\mathbf{j}}}^{{\left( 2 \right)}}}\left( {{\mathbf{r}},t} \right) = e{\kern 1pt} \delta N_{e}^{{\left( 1 \right)}}\left( {{\mathbf{r}},t} \right){{{\mathbf{V}}}^{{\left( 1 \right)}}}\left( {{\mathbf{r}},t} \right)$, which is responsible for the excitation of electromagnetic fields at the frequency $\omega$ that is much lower than the laser frequency ${{\omega }_{0}}$, and $\varepsilon \left( \omega \right) = 1 - {{\omega _{p}^{2}} \mathord{\left/ {\vphantom {{\omega _{p}^{2}} {\left[ {\omega \left( {\omega + i{{\nu }_{{ei}}}} \right)} \right]}}} \right. \kern-0em} {\left[ {\omega \left( {\omega + i{{\nu }_{{ei}}}} \right)} \right]}}$ is the low-frequency plasma permittivity. Using first-order equations (3.1) and (3.2), we find an expression for the Fourier transform of the nonlinear current in the form

$$\begin{gathered} {{{\mathbf{j}}}^{{\left( 2 \right)}}}\left( {\omega ,{\mathbf{r}}} \right) = \frac{{ie{\kern 1pt} \omega _{p}^{2}}}{{4\pi {\kern 1pt} {{m}_{e}}}}\int\limits_{ - \infty }^{ + \infty } {\frac{{d\omega '}}{{2\pi }}} \\ \times \;\frac{{\Delta \Phi \left( {\omega ',{\mathbf{r}}} \right)\nabla \Phi \left( {\omega - \omega ',{\mathbf{r}}} \right)}}{{\omega '\left( {\omega ' + i{{\nu }_{{ei}}}} \right)\varepsilon \left( {\omega '} \right)\left( {\omega - \omega ' + i{{\nu }_{{ei}}}} \right)\varepsilon \left( {\omega - \omega '} \right)}}, \\ \end{gathered} $$

((3.9))

where $\Phi \left( {\omega ,{\mathbf{r}}} \right)$ is the Fourier component of ponderomotive potential (2.6). Applying Fourier transformation with respect to the spatial coordinates and taking into account expression (3.9) for the nonlinear current, the solution to Eq. (3.8) can be written in the form

$$\begin{gathered} {{{\mathbf{B}}}^{{\left( 2 \right)}}}\left( {\omega ,{\mathbf{r}}} \right) = \frac{{ie{\kern 1pt} \omega _{p}^{2}}}{{{{m}_{e}}c{\kern 1pt} }}\int {\frac{{d{\mathbf{k}}}}{{{{{\left( {2\pi } \right)}}^{3}}}}} \frac{{\exp \left( {i{\mathbf{k}} \cdot {\mathbf{r}}} \right)}}{{{{k}^{2}} - \left( {{{{{\omega }^{2}}} \mathord{\left/ {\vphantom {{{{\omega }^{2}}} {{{c}^{2}}}}} \right. \kern-0em} {{{c}^{2}}}}} \right)\varepsilon \left( \omega \right)}} \\ \times \;\int\limits_{ - \infty }^{ + \infty } {\frac{{d\omega '}}{{2\pi }}\int {\frac{{d{\mathbf{k}}'}}{{{{{\left( {2\pi } \right)}}^{3}}}}} } \\ \times \;\frac{{{\mathbf{k}} \times \left( {{\mathbf{k}} - {\mathbf{k}}'} \right){{k}^{{'2}}}\Phi \left( {\omega ',{\mathbf{k}}'} \right)\Phi \left( {\omega - \omega ',{\mathbf{k}} - {\mathbf{k}}'} \right)}}{{\omega '\left( {\omega ' + i{{\nu }_{{ei}}}} \right)\varepsilon \left( {\omega '} \right)\left( {\omega - \omega ' + i{{\nu }_{{ei}}}} \right)\varepsilon \left( {\omega - \omega '} \right)}}. \\ \end{gathered} $$

((3.10))

To calculate integrals in expression (3.10), the spatiotemporal structure of the field of laser pulses should be specified explicitly. We assume that the group velocities ${{V}_{1}}$ and ${{V}_{2}}$ of the linearly polarized counterpropagating pulses are close to the speed of light c and that the pulses have equal electric field amplitudes ${{E}_{{0L}}}$ and equal longitudinal and transverse dimensions, $L \simeq c\tau$ and R. The electric field of Gaussian laser pulses can be expressed in the form

$$\begin{gathered} {{{\mathbf{E}}}_{1}}\left( {{\mathbf{r}},t} \right) = {{{\mathbf{e}}}_{x}}{{E}_{{0L}}}\exp \left[ { - \frac{{{{{\left( {z - ct} \right)}}^{2}}}}{{2{{L}^{2}}}} - \frac{{{{\rho }^{2}}}}{{2{{R}^{2}}}}} \right], \\ {{{\mathbf{E}}}_{2}}\left( {{\mathbf{r}},t} \right) = {{{\mathbf{e}}}_{x}}{{E}_{{0L}}}\exp \left[ { - \frac{{{{{\left( {z + ct} \right)}}^{2}}}}{{2{{L}^{2}}}} - \frac{{{{\rho }^{2}}}}{{2{{R}^{2}}}}} \right], \\ \end{gathered} $$

((3.11))

where ${{{\mathbf{e}}}_{x}}$ is the basis vector along the x axis and $\rho = \sqrt {{{x}^{2}} + {{y}^{2}}}$. Henceforth, we will be interested in investigation of the low-frequency waves emitted from the laser pulse interaction region and will retain only the third term in expression (2.6) for the ponderomotive potential. It follows from expression (3.9) for the nonlinear current that it is this term that makes the main contribution to the excitation of the low-frequency electromagnetic fields due to the large value of the derivative with respect to the spatial coordinate z, which is proportional to the doubled wavenumber $2{{k}_{0}}$. In view of the conditions ${{k}_{0}}R \gg 1$ and ${{k}_{0}}L \gg 1$, this derivative much larger than the derivatives of the first two terms in expression (2.6). Taking into account formulas (3.11), expression (2.6) for the ponderomotive potential and its Fourier transform take the form

$$\begin{gathered} \Phi \left( {{\mathbf{r}},t} \right) = \frac{e}{{2{{m}_{e}}\omega _{0}^{2}}}E_{{0L}}^{2}\exp \left( { - \frac{{{{t}^{2}}}}{{{{\tau }^{2}}}} - \frac{{{{z}^{2}}}}{{{{L}^{2}}}} - \frac{{{{\rho }^{2}}}}{{{{R}^{2}}}}} \right) \\ \times \;\cos \left( {\Delta {{\omega }_{0}}t - 2{{k}_{0}}z} \right), \\ \end{gathered} $$

((3.12))

$$\begin{gathered} \Phi \left( {\omega ,{\mathbf{k}}} \right) = \frac{{e{\kern 1pt} {{\pi }^{2}}{{R}^{2}}L\tau }}{{4{{m}_{e}}\omega _{0}^{2}}}E_{{0L}}^{2}\exp \left( { - \frac{{k_{ \bot }^{2}{{R}^{2}}}}{4}} \right) \\ \times \;\left\{ {\exp \left[ { - \frac{{{{{\left( {\omega - \Delta {{\omega }_{0}}} \right)}}^{2}}{{\tau }^{2}}}}{4} - \frac{{{{{\left( {{{k}_{z}} - 2{{k}_{0}}} \right)}}^{2}}{{L}^{2}}}}{4}} \right]} \right. \\ + \;\left. {\exp \left[ { - \frac{{{{{\left( {\omega + \Delta {{\omega }_{0}}} \right)}}^{2}}{{\tau }^{2}}}}{4} - \frac{{{{{\left( {{{k}_{z}} + 2{{k}_{0}}} \right)}}^{2}}{{L}^{2}}}}{4}} \right]} \right\}, \\ \end{gathered} $$

((3.13))

where $k_{ \bot }^{2} = k_{x}^{2} + k_{y}^{2}$ and we took into account that ${{\omega }_{1}}{\kern 1pt} {{\omega }_{2}} \approx \omega _{0}^{2}$. After performing a relatively cumbersome but not too complicated integration, we find from Eqs. (3.10) and (3.13) that the low-frequency magnetic field has only the azimuthal component, which is described by expression

$$\begin{gathered} B_{\varphi }^{{\left( 2 \right)}}\left( {\omega ,{\mathbf{r}}} \right) = - \frac{{i{{e}^{3}}E_{{0L}}^{4}{{\pi }^{{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2}}}}\omega _{p}^{2}{{\tau }^{2}}k_{0}^{2}{{R}^{2}}{{k}_{0}}L}}{{16\sqrt 2 m_{e}^{3}c\omega _{0}^{4}{{\omega }_{p}}\left( {\omega + i{{\nu }_{{ei}}}} \right)}} \\ \times \;\exp \left( { - \frac{{{{\omega }^{2}}{{\tau }^{2}}}}{8}} \right)\left\{ {{{W}_{1}}\left( \omega \right) - \frac{{i\omega }}{{2{{k}_{0}}}}} \right. \\ \end{gathered} $$

$$ \times \;\left. {\left[ {\frac{{{{W}_{2}}\left( \omega \right)}}{{\omega - 2{{\omega }_{p}} + i{{\nu }_{{ei}}}}} - \frac{{{{W}_{{ - 2}}}\left( \omega \right)}}{{\omega + 2{{\omega }_{p}} + i{{\nu }_{{ei}}}}}} \right]\frac{\partial }{{\partial z}}} \right\}$$

((3.14))

$$\begin{gathered} \times \;\frac{\partial }{{\partial \rho }}\int {\frac{{d{\mathbf{k}}}}{{{{{\left( {2\pi } \right)}}^{3}}}}} \frac{1}{{{{k}^{2}} - \frac{{{{\omega }^{2}}}}{{{{c}^{2}}}}\varepsilon \left( \omega \right)}} \\ \times \;\exp \left( {i{\mathbf{k}} \cdot {\mathbf{r}} - \frac{{k_{ \bot }^{2}{{R}^{2}}}}{8} - \frac{{k_{z}^{2}{{L}^{2}}}}{8}} \right), \\ \end{gathered} $$

where

$$\begin{gathered} {{W}_{1}}\left( \omega \right) = W\left( {\frac{{\omega - 2{{\omega }_{p}} + 2\Delta {{\omega }_{0}}}}{{2\sqrt 2 }}\tau } \right) \\ - \;W\left( {\frac{{\omega - 2{{\omega }_{p}} - 2\Delta {{\omega }_{0}}}}{{2\sqrt 2 }}\tau } \right) \\ + \;W\left( {\frac{{\omega + 2{{\omega }_{p}} - 2\Delta {{\omega }_{0}}}}{{2\sqrt 2 }}\tau } \right) \\ \end{gathered} $$

$$ - \;W\left( {\frac{{\omega + 2{{\omega }_{p}} + 2\Delta {{\omega }_{0}}}}{{2\sqrt 2 }}\tau } \right),$$

((3.15))

$$\begin{gathered} {{W}_{2}}\left( \omega \right) = W\left( {\frac{{\omega - 2{{\omega }_{p}} + 2\Delta {{\omega }_{0}}}}{{2\sqrt 2 }}\tau } \right) \\ + \;W\left( {\frac{{\omega - 2{{\omega }_{p}} - 2\Delta {{\omega }_{0}}}}{{2\sqrt 2 }}\tau } \right), \\ {{W}_{{ - 2}}}\left( \omega \right) = W\left( {\frac{{\omega + 2{{\omega }_{p}} - 2\Delta {{\omega }_{0}}}}{{2\sqrt 2 }}\tau } \right) \\ + \;W\left( {\frac{{\omega + 2{{\omega }_{p}} + 2\Delta {{\omega }_{0}}}}{{2\sqrt 2 }}\tau } \right), \\ \end{gathered} $$

while the function $W(x)$, which was studied in detail and tabulated in [14], has the form

$$\begin{gathered} W(x) = \exp \left( { - {{x}^{2}}} \right)\left[ {1 + {\text{erf}}\left( {ix} \right)} \right], \\ {\text{erf}}(x) = \left( {{2 \mathord{\left/ {\vphantom {2 {\sqrt \pi }}} \right. \kern-0em} {\sqrt \pi }}} \right)\int\limits_0^x {dt} \exp \left( { - {{t}^{2}}} \right). \\ \end{gathered} $$

((3.16))

Here, ${\text{erf}}\left( x \right)$ is the probability integral. Note that, when deriving expression (3.14), we used the inequalities ${{k}_{0}}R \gg 1$ and ${{k}_{0}}L \gg 1$, which mean that the longitudinal and transverse dimensions of the laser pulses are much larger than their wavelength. Since we are interested in the radiation field, the integral with respect to wave vectors in Eq. (3.14) should be calculated in the wave zone at long distances, $r \gg R,L,{c \mathord{\left/ {\vphantom {c \omega }} \right. \kern-0em} \omega }$, from the region of interaction of the laser pulses. Under these conditions, the following asymptotic expression is valid for the integral [19]:

$$\begin{gathered} \int {\frac{{d{\mathbf{k}}}}{{{{{\left( {2\pi } \right)}}^{3}}}}} \frac{1}{{{{k}^{2}} - \frac{{{{\omega }^{2}}}}{{{{c}^{2}}}}\varepsilon \left( \omega \right)}}\exp \left( {i{\mathbf{k}} \cdot {\mathbf{r}} - \frac{{k_{ \bot }^{2}{{R}^{2}}}}{8} - \frac{{k_{z}^{2}{{L}^{2}}}}{8}} \right) \\ \approx \frac{1}{{4\pi {\kern 1pt} r}}\exp \left\{ {i\frac{\omega }{c}\sqrt {\varepsilon \left( \omega \right)} r} \right. \\ \left. { - \;\frac{{{{\omega }^{2}}}}{{{{c}^{2}}}}\varepsilon \left( \omega \right)\frac{{{{R}^{2}}{{{\sin }}^{2}}\theta + {{L}^{2}}{{{\cos }}^{2}}\theta }}{8}} \right\}, \\ \end{gathered} $$

((3.17))

where θ is the angle between the positive z direction and the direction of observation. Taking into account Eqs. (3.14), (3.17), and (3.7), we find that the low-frequency electromagnetic fields in the wave zone are described by the expressions

$$\begin{gathered} B_{\varphi }^{{\left( 2 \right)}}\left( {\omega ,{\mathbf{r}}} \right) = \sqrt {\varepsilon \left( \omega \right)} {\kern 1pt} {\kern 1pt} E_{\theta }^{{\left( 2 \right)}}\left( {\omega ,{\mathbf{r}}} \right) \\ = \;\frac{{{{\pi }^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}\omega _{p}^{2}{{\tau }^{2}}k_{0}^{2}{{R}^{2}}{{k}_{0}}L}}{{\sqrt 2 {\kern 1pt} {{\omega }_{0}}}}{{\left( {\frac{{{{V}_{E}}}}{{4c}}} \right)}^{3}}{{E}_{{0L}}}\frac{{\omega \sqrt {\varepsilon \left( \omega \right)} }}{{\omega + i{{\nu }_{{ei}}}}} \\ \times \;\frac{{\sin \theta }}{{{{k}_{p}}r}}\exp \left( {i\frac{\omega }{c}\sqrt {\varepsilon \left( \omega \right)} {\kern 1pt} r - \frac{{{{\omega }^{2}}{{\tau }^{2}}}}{8}} \right. \\ \left. { - \;\frac{{{{\omega }^{2}}}}{{{{c}^{2}}}}\varepsilon \left( \omega \right)\frac{{{{R}^{2}}{{{\sin }}^{2}}\theta + {{L}^{2}}{{{\cos }}^{2}}\theta }}{8}} \right)F\left( {\omega ,\cos \theta } \right), \\ \end{gathered} $$

((3.18))

where $F\left( {\omega ,\cos \theta } \right)$ is expressed in terms of the functions ${{W}_{1}}\left( \omega \right)$, ${{W}_{2}}\left( \omega \right)$, and ${{W}_{{ - 2}}}\left( \omega \right)$ from formulas (3.15),

$$\begin{gathered} F\left( {\omega ,\cos \theta } \right) = {{W}_{1}}\left( \omega \right) + \frac{{{{\omega }^{2}}}}{{2{{\omega }_{0}}}}\sqrt {\varepsilon \left( \omega \right)} \\ \times \;\cos \theta \left[ {\frac{{{{W}_{2}}\left( \omega \right)}}{{\omega - 2{{\omega }_{p}} + i{{\nu }_{{ei}}}}} - \frac{{{{W}_{{ - 2}}}\left( \omega \right)}}{{\omega + 2{{\omega }_{p}} + i{{\nu }_{{ei}}}}}} \right]. \\ \end{gathered} $$

((3.19))

According to Eqs. (3.18) and (3.19), the electromagnetic field in the far zone $r \gg R,L,{c \mathord{\left/ {\vphantom {c \omega }} \right. \kern-0em} \omega }$ represents a spherical wave with the frequency ω and wave vector ${{{\mathbf{e}}}_{r}}{{\omega \sqrt {\varepsilon \left( \omega \right)} } \mathord{\left/ {\vphantom {{\omega \sqrt {\varepsilon \left( \omega \right)} } c}} \right. \kern-0em} c}$, which propagates from the region of interaction of the laser pulses in the radial direction. It has the azimuthal component of magnetic field $B_{\varphi }^{{\left( 2 \right)}}\left( {\omega ,{\mathbf{r}}} \right)$ and the meridional component of electric field ${\kern 1pt} {\kern 1pt} E_{\theta }^{{\left( 2 \right)}}\left( {\omega ,{\mathbf{r}}} \right)$, where ${{{\mathbf{e}}}_{r}}$ is the basis vector of the spherical coordinate system in the direction of the radius vector.

It should be noted that the function ${{W}_{1}}\left( \omega \right)$ vanishes in the case of pulses with equal frequencies, ${{\omega }_{1}} = {{\omega }_{2}} = {{\omega }_{0}}$, $\Delta {{\omega }_{0}} = 0$, while the expression for the magnetic field with allowance for Eqs. (3.15), (3.18), and (3.19) takes the form

$$\begin{gathered} B_{\varphi }^{{\left( 2 \right)}}\left( {\omega ,{\mathbf{r}}} \right) = \sqrt {\varepsilon \left( \omega \right)} E_{\theta }^{{\left( 2 \right)}}\left( {\omega ,{\mathbf{r}}} \right) \\ = \;\frac{{{{\pi }^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}\omega _{p}^{2}{{\tau }^{2}}k_{0}^{2}{{R}^{2}}{{k}_{0}}L}}{{\sqrt 2 \omega _{0}^{2}}}{{\left( {\frac{{{{V}_{E}}}}{{4c}}} \right)}^{3}}{{E}_{{0L}}}\frac{{{{\omega }^{3}}\varepsilon \left( \omega \right)}}{{\omega + i{{\nu }_{{ei}}}}}\frac{{\sin \theta \cos \theta }}{{{{k}_{p}}r}} \\ \end{gathered} $$

$$\begin{gathered} \times \;\exp \left( {i\frac{\omega }{c}\sqrt {\varepsilon \left( \omega \right)} r - \frac{{{{\omega }^{2}}{{\tau }^{2}}}}{8}} \right. \\ \left. { - \;\frac{{{{\omega }^{2}}}}{{{{c}^{2}}}}\varepsilon \left( \omega \right)\frac{{{{R}^{2}}{{{\sin }}^{2}}\theta + {{L}^{2}}{{{\cos }}^{2}}\theta }}{8}} \right) \\ \end{gathered} $$

((3.20))

$$\begin{gathered} \times \;\left\{ {\frac{1}{{\omega - 2{{\omega }_{p}} + i{{\nu }_{{ei}}}}}W\left( {\frac{{\omega - 2{{\omega }_{p}}}}{{2\sqrt 2 }}\tau } \right)} \right. \\ \left. { - \;\frac{1}{{\omega + 2{{\omega }_{p}} + i{{\nu }_{{ei}}}}}W\left( {\frac{{\omega + 2{{\omega }_{p}}}}{{2\sqrt 2 }}\tau } \right)} \right\}. \\ \end{gathered} $$

Comparison of expression (3.20) with formula (3.6) from [7] shows that the results are similar, in principle. The only difference is that, in the present work, the decay of plasma oscillations is related to electron–ion collisions within the hydrodynamic model, while the kinetic equation was used in [7] to describe collisionless Cherenkov damping of these oscillations.

The obtained expressions (3.18) and (3.19) for electromagnetic fields make it possible to investigate spectral, angular, and energy characteristics of the low-frequency wave fields generated under interaction of counterpropagating laser pulses with different frequencies in underdense plasma.

4 PHYSICAL CHARACTERISTICS OF TERAHERTZ RADIATION

To calculate the radiated energy, the Poynting v-ector

$${\mathbf{S}}\left( {{\mathbf{r}},t} \right) = \frac{c}{{4\pi }}{{{\mathbf{E}}}^{{\left( 2 \right)}}}\left( {{\mathbf{r}},t} \right) \times {{{\mathbf{B}}}^{{\left( 2 \right)}}}\left( {{\mathbf{r}},t} \right)$$

((4.1))

should be integrated over time and spherical surface of a large radius. As a result, we obtain the following expression for the energy $d{{W}_{{{\text{rad}}}}}\left( {\omega ,\theta } \right)$ emitted in a unit frequency interval $d\omega$ into a unit solid angle $dO = 2\pi \sin \theta d\theta$ for positive values of the frequency ($\omega > 0$):

$$d{{W}_{{{\text{rad}}}}}\left( {\omega ,\theta } \right) = \frac{{c{{r}^{2}}}}{{4{{\pi }^{2}}}}\frac{{\operatorname{Re} \left( {\sqrt {\varepsilon \left( \omega \right)} } \right)}}{{\left| {\varepsilon \left( \omega \right)} \right|}}{{\left| {B_{\varphi }^{{\left( 2 \right)}}\left( {\omega ,{\mathbf{r}}} \right)} \right|}^{2}}d\omega {\kern 1pt} dO.$$

((4.2))

Performing integration over the solid angle in Eq. (4.2) with allowance for formulas (3.18) and (3.19), we find the dependence of the THz radiation energy on the dimensionless frequency $\Omega = {\omega \mathord{\left/ {\vphantom {\omega {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}}$ in the form

$$\begin{gathered} \frac{{d{{W}_{{{\text{rad}}}}}\left( \Omega \right)}}{{d\Omega }} = \frac{{{{\pi }^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}\omega _{0}^{2}{{\tau }^{2}}k_{0}^{2}{{R}^{2}}}}{4}{{\left( {\frac{{V_{E}^{2}}}{{4{{c}^{2}}}}} \right)}^{3}}{{W}_{L}}I\left( \Omega \right),\quad \\ \Omega > 0, \\ \end{gathered} $$

((4.3))

where the function $I\left( \Omega \right)$ is given by

$$\begin{gathered} I\left( \Omega \right) = {{\left( {\frac{{{{\omega }_{p}}\tau }}{2}} \right)}^{3}}\operatorname{Re} \left( {\sqrt {\frac{{{{\Omega }^{2}} - 1}}{{{{\Omega }^{2}}}}} } \right) \\ \times \;\exp \left[ { - {{{\left( {\frac{{{{\omega }_{p}}\tau }}{2}} \right)}}^{2}}{{\Omega }^{2}}} \right]\int\limits_{ - 1}^1 {d\left( {\cos \theta } \right)} {{\sin }^{2}}\theta \\ \times \;\exp \left\{ { - \left( {{{\Omega }^{2}} - 1} \right)\left[ {{{{\left( {\frac{{{{k}_{p}}R}}{2}} \right)}}^{2}}{{{\sin }}^{2}}\theta } \right.} \right. \\ + \;\left. {\left. {{{{\left( {\frac{{{{\omega }_{p}}\tau }}{2}} \right)}}^{2}}{{{\cos }}^{2}}\theta } \right]} \right\}{{\left| {F\left( {\Omega ,\cos \theta } \right)} \right|}^{2}}. \\ \end{gathered} $$

((4.4))

Here, ${{k}_{p}} = {{{{\omega }_{p}}} \mathord{\left/ {\vphantom {{{{\omega }_{p}}} c}} \right. \kern-0em} c}$, ${{V}_{E}} = {{e{{E}_{{0L}}}} \mathord{\left/ {\vphantom {{e{{E}_{{0L}}}} {\left( {{{m}_{e}}{{\omega }_{0}}} \right)}}} \right. \kern-0em} {\left( {{{m}_{e}}{{\omega }_{0}}} \right)}}$ is the electron oscillation velocity in the laser field; ${{W}_{L}} = {{E_{{0L}}^{2}\sqrt \pi {{R}^{2}}L} \mathord{\left/ {\vphantom {{E_{{0L}}^{2}\sqrt \pi {{R}^{2}}L} 8}} \right. \kern-0em} 8}$ is the energy of a single laser pulse; and the dependence $F\left( {\Omega ,\cos \theta } \right)$, according to expression (3.19), has the form

$$\begin{gathered} F\left( {\Omega ,\cos \theta } \right) = {{W}_{1}}\left( \Omega \right) + \frac{{{{\omega }_{p}}}}{{2{{\omega }_{0}}}}{\kern 1pt} \Omega \sqrt {{{\Omega }^{2}} - 1} \\ \times \;\cos \theta \left[ {\frac{{{{W}_{2}}\left( \Omega \right)}}{{\Omega - 2 + i\gamma }} - \frac{{{{W}_{{ - 2}}}\left( \Omega \right)}}{{\Omega + 2 + i\gamma }}} \right], \\ \end{gathered} $$

((4.5))

where $\gamma = {{{{\nu }_{{ei}}}} \mathord{\left/ {\vphantom {{{{\nu }_{{ei}}}} {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}}$, while the functions ${{W}_{1}}\left( \omega \right)$, ${{W}_{2}}\left( \omega \right)$, and ${{W}_{{ - 2}}}\left( \omega \right)$, according to formulas (3.15), are described by the expressions

$$\begin{gathered} {{W}_{1}}\left( \Omega \right) = W\left( {{{\omega }_{p}}\tau \frac{{\Omega - 2 + 2\delta }}{{2\sqrt 2 }}} \right) \\ - \;W\left( {{{\omega }_{p}}\tau \frac{{\Omega - 2 - 2\delta }}{{2\sqrt 2 }}} \right) + W\left( {{{\omega }_{p}}\tau \frac{{\Omega + 2 - 2\delta }}{{2\sqrt 2 }}} \right) \\ - \;W\left( {{{\omega }_{p}}\tau \frac{{\Omega + 2 + 2\delta }}{{2\sqrt 2 }}} \right), \\ \end{gathered} $$

$${{W}_{2}}\left( \Omega \right) = W\left( {{{\omega }_{p}}\tau \frac{{\Omega - 2 - 2\delta }}{{2\sqrt 2 }}} \right)$$

((4.6))

$$\begin{gathered} + \;W\left( {{{\omega }_{p}}\tau \frac{{\Omega - 2 + 2\delta }}{{2\sqrt 2 }}} \right), \\ {{W}_{{ - 2}}}\left( \Omega \right) = W\left( {{{\omega }_{p}}\tau \frac{{\Omega + 2 - 2\delta }}{{2\sqrt 2 }}} \right) \\ + \;W\left( {{{\omega }_{p}}\tau \frac{{\Omega + 2 + 2\delta }}{{2\sqrt 2 }}} \right), \\ \end{gathered} $$

where $\delta = {{\Delta {{\omega }_{0}}} \mathord{\left/ {\vphantom {{\Delta {{\omega }_{0}}} {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}}$ is the dimensionless frequency difference.

It follows from Eqs. (4.3)–(4.6) that only a single line at the doubled plasma frequency appears in the spectrum in the case of equal frequencies of the laser pulses, which agrees with the results reported in [7]. For a nonzero frequency difference, $\Delta {{\omega }_{0}} \ne 0$, in addition to the spectral line at the frequency $2{{\omega }_{p}}$, a new maximum appears near the plasma frequency (see Fig. 1). The height of this maximum increases with increasing frequency difference, and, at a certain value of $\Delta {{\omega }_{0}}$, becomes higher than the spectral line at the doubled plasma frequency (dash-dotted line in Fig. 1). It can be seen from Fig. 1 that the radiation energy at the frequency ${{\omega }_{p}}$ becomes higher than that at the frequency $2{{\omega }_{p}}$ at an even smaller value of the frequency difference, which is due to a significant width of the peak at the plasma frequency. For typical parameters of laser–plasma interaction, when the frequency difference of the laser pulses becomes equal to the plasma frequency, the peak at the plasma frequency prevails in the spectrum of radiation, which is illustrated in Fig. 2. In this case, the height of the spectral line reaches its maximum value when the frequency difference of the laser pulses becomes equal to the plasma frequency. The radiation spectra for several values of the parameter ${{\omega }_{p}}\tau$ under the condition $\Delta {{\omega }_{0}} = {{\omega }_{p}}$ are presented in Fig. 3. It can be seen that the height of the spectral line is maximal when ${{{{\omega }_{p}}\tau } \mathord{\left/ {\vphantom {{{{\omega }_{p}}\tau } 2}} \right. \kern-0em} 2} = 1$. The latter condition corresponds to efficient excitation of plasma fields in the region of interaction of the laser pulses.

Fig. 1.

Normalized energy $I\left( \Omega \right)$ (4.4) of the THz radiation emitted from the region of interaction of counterpropagating laser pulses as a function of the dimensionless frequency $\Omega = {\omega \mathord{\left/ {\vphantom {\omega {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}}$ for ${{{{\omega }_{p}}\tau } \mathord{\left/ {\vphantom {{{{\omega }_{p}}\tau } 2}} \right. \kern-0em} 2} = 1$, ${{{{k}_{p}}R} \mathord{\left/ {\vphantom {{{{k}_{p}}R} 2}} \right. \kern-0em} 2} = 1$, ${{{{\omega }_{p}}} \mathord{\left/ {\vphantom {{{{\omega }_{p}}} {{{\omega }_{0}}}}} \right. \kern-0em} {{{\omega }_{0}}}}$ = 0.01, and $\gamma = {{{{\nu }_{{ei}}}} \mathord{\left/ {\vphantom {{{{\nu }_{{ei}}}} {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}} = 0.01$. The dashed, solid, and dash-dotted curves correspond to the dimensionless frequency difference $\delta = {{\Delta {{\omega }_{0}}} \mathord{\left/ {\vphantom {{\Delta {{\omega }_{0}}} {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}}$ equal to 0.01, 0.025, and 0.05, respectively.

Fig. 2.

Normalized energy $I\left( \Omega \right)$ of the THz radiation emitted from the region of interaction of counterpropagating laser pulses for the same parameters as in Fig. 1. The dashed, solid, and dash-dotted curves correspond to the dimensionless frequency difference $\delta = {{\Delta {{\omega }_{0}}} \mathord{\left/ {\vphantom {{\Delta {{\omega }_{0}}} {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}}$ equal to 0.8, 1, and 1.2, respectively.

Fig. 3.

Normalized energy $I\left( \Omega \right)$ (4.4) of the THz radiation emitted from the region of interaction of counterpropagating laser pulses as a function of dimensionless frequency $\Omega = {\omega \mathord{\left/ {\vphantom {\omega {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}}$ for $\delta = {{\Delta {{\omega }_{0}}} \mathord{\left/ {\vphantom {{\Delta {{\omega }_{0}}} {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}} = 1$, ${{{{k}_{p}}R} \mathord{\left/ {\vphantom {{{{k}_{p}}R} 2}} \right. \kern-0em} 2} = 1$, ${{{{\omega }_{p}}} \mathord{\left/ {\vphantom {{{{\omega }_{p}}} {{{\omega }_{0}}}}} \right. \kern-0em} {{{\omega }_{0}}}}$ = 0.01, and $\gamma = {{{{\nu }_{{ei}}}} \mathord{\left/ {\vphantom {{{{\nu }_{{ei}}}} {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}} = 0.01$. The dashed, solid, and dash-dotted curves correspond to the dimensionless pulse width ${{{{\omega }_{p}}\tau } \mathord{\left/ {\vphantom {{{{\omega }_{p}}\tau } 2}} \right. \kern-0em} 2}$ equal to 0.8, 1, and 1.2, respectively.

The appearance of the peak at the plasma frequency in the radiation spectrum can be explained as being due to the excitation of currents and electromagnetic fields at both the doubled plasma frequency and at the zeroth harmonic of the plasma frequency as a result of interaction of the wake waves of the counterpropagating laser pulses. The fields at the zeroth harmonic consist of a quasi-static magnetic field that remains in the region of interaction of the laser pulses [20] and a low-frequency electromagnetic wave (3.18) propagating from this region. The latter is described by the first term in Eq. (3.19). It follows from Eqs. (3.18), (3.19), and (3.15), along with relations (2.24) and (2.25) from [20], that the electromagnetic fields at the zero frequency are driven only when the frequencies of the laser pulses are different, i.e., $\Delta {{\omega }_{0}} \ne 0$, and the excitation is the most efficient when $\Delta {{\omega }_{0}} = {{\omega }_{p}}$, i.e., when the difference frequency is equal to the plasma frequency. It should be noted that the line at the zeroth harmonic of the plasma frequency (see expressions (4.4), (4.5)) has a relatively large half-width of about ${1 \mathord{\left/ {\vphantom {1 \tau }} \right. \kern-0em} \tau }$, i.e., about the reciprocal of the laser pulse duration. Since only electromagnetic waves with frequencies higher than the plasma frequency can propagate in plasma, this broad maximum has a cutoff at $\omega < {{\omega }_{p}}$ and forms a spectral line at the plasma frequency, which is related to the presence of the factor $\operatorname{Re} \left( {\sqrt {\varepsilon \left( \omega \right)} } \right)$ in Eq. (4.2). In addition, the spectral line near ${{\omega }_{p}}$ is the most intense at $\Delta {{\omega }_{0}} = {{\omega }_{p}}$ (see Fig. 2), when the plasma fields in the region of interaction of the laser pulses are excited most efficiently.

Integration of Eq. (4.2) with respect to the frequency yields the angular dependence of THz radiation in the form

$$\frac{{d{{W}_{{{\text{rad}}}}}\left( \theta \right)}}{{dO}} = \frac{{\sqrt \pi \,{{\omega }_{p}}\omega _{0}^{4}{{\tau }^{5}}}}{{16}}{{\left( {\frac{{V_{E}^{2}}}{{4{{c}^{2}}}}} \right)}^{3}}{{W}_{L}}\,J\left( \theta \right),$$

((4.7))

where the function $J\left( \theta \right)$ is given by

$$\begin{gathered} J\left( \theta \right) = {{\left( {\frac{{{{k}_{p}}R}}{2}} \right)}^{2}}{{\sin }^{2}}\theta \int\limits_0^\infty {d\Omega } \operatorname{Re} \left( {\sqrt {\frac{{{{\Omega }^{2}} - 1}}{{{{\Omega }^{2}}}}} } \right) \\ \times \;\exp \left[ { - {{{\left( {\frac{{{{\omega }_{p}}\tau }}{2}} \right)}}^{2}}{{\Omega }^{2}}} \right] \\ \times \;\exp \left\{ { - \left( {{{\Omega }^{2}} - 1} \right)\left[ {{{{\left( {\frac{{{{k}_{p}}R}}{2}} \right)}}^{2}}{{{\sin }}^{2}}\theta } \right.} \right. \\ \left. {\left. { + \;{{{\left( {\frac{{{{\omega }_{p}}\tau }}{2}} \right)}}^{2}}{{{\cos }}^{2}}\theta } \right]} \right\}{{\left| {F\left( {\Omega ,\cos \theta } \right)} \right|}^{2}}. \\ \end{gathered} $$

((4.8))

Dependence (4.8) is illustrated in Fig. 4 in the form of a directional pattern. In the case of tight focusing of the laser pulses, the radiation is emitted perpendicular to the z axis (dashed line in Fig. 4), along which the laser pulses propagate. In the case of large-size focal spots, THz waves are emitted at small angles with respect to the z axis (dash-dotted line in Fig. 4). Such a directional pattern can be easily understood from analysis of expression (4.8). For tightly focused laser pulses, when ${{k}_{p}}R \ll 1$, energy (4.8) is mainly determined by the factor ${{\sin }^{2}}\theta$, reaching its maximum value at $\theta = {\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0em} 2}$, which corresponds to a dipole structure of the emitted radiation. In contrast, in the case of interaction of laser pulses with a large transverse size (${{k}_{p}}R \gg 1$), it follows from expression (4.8) that $\theta \to 0$, i.e., radiation is emitted nearly along the z axis.

Fig. 4.

Directional pattern $J\left( \theta \right)$ (4.8) of the THz radiation emitted from the region of interaction of counterpropagating laser pulses for $\delta = {{\Delta {{\omega }_{0}}} \mathord{\left/ {\vphantom {{\Delta {{\omega }_{0}}} {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}} = 1$, ${{{{\omega }_{p}}\tau } \mathord{\left/ {\vphantom {{{{\omega }_{p}}\tau } 2}} \right. \kern-0em} 2} = 1$, ${{{{\omega }_{p}}} \mathord{\left/ {\vphantom {{{{\omega }_{p}}} {{{\omega }_{0}}}}} \right. \kern-0em} {{{\omega }_{0}}}} = 0.01$, and $\gamma = {{{{\nu }_{{ei}}}} \mathord{\left/ {\vphantom {{{{\nu }_{{ei}}}} {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}} = 0.01$. The dashed, solid, and dash-dotted curves correspond to the dimensionless parameter ${{{{k}_{p}}R} \mathord{\left/ {\vphantom {{{{k}_{p}}R} 2}} \right. \kern-0em} 2}$ equal to 1, 2, and 4, respectively. The arrows show the propagation directions of the laser pulses ($\theta = 0$ and 180°).

The total energy of THz radiation can be obtained by integrating Eq. (4.3) with respect to frequency,

$${{W}_{{{\text{rad}}}}} = \frac{{{{\pi }^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}\omega _{0}^{2}{{\tau }^{2}}k_{0}^{2}{{R}^{2}}}}{4}{{\left( {\frac{{V_{E}^{2}}}{{4{{c}^{2}}}}} \right)}^{3}}{{W}_{L}}\,w,$$

((4.9))

where the function $w$ has the form

$$\begin{gathered} w = {{\left( {\frac{{{{\omega }_{p}}\tau }}{2}} \right)}^{3}}\int\limits_0^\infty {d\Omega } \operatorname{Re} \left( {\sqrt {\frac{{{{\Omega }^{2}} - 1}}{{{{\Omega }^{2}}}}} } \right) \\ \times \;\exp \left[ { - {{{\left( {\frac{{{{\omega }_{p}}\tau }}{2}} \right)}}^{2}}{{\Omega }^{2}}} \right]\int\limits_{ - 1}^1 {d\left( {\cos \theta } \right)} {{\sin }^{2}}\theta \\ \times \;\exp \left\{ { - \left( {{{\Omega }^{2}} - 1} \right)\left[ {{{{\left( {\frac{{{{k}_{p}}R}}{2}} \right)}}^{2}}{{{\sin }}^{2}}\theta } \right.} \right. \\ + \;\left. {\left. {{{{\left( {\frac{{{{\omega }_{p}}\tau }}{2}} \right)}}^{2}}{{{\cos }}^{2}}\theta } \right]} \right\}{{\left| {F\left( {\Omega ,\cos \theta } \right)} \right|}^{2}}. \\ \end{gathered} $$

((4.10))

The dependence of the total dimensionless energy (4.10) of THz radiation on the frequency difference of the laser pulses for several values of ${{{{\omega }_{p}}\tau } \mathord{\left/ {\vphantom {{{{\omega }_{p}}\tau } 2}} \right. \kern-0em} 2}$ is presented in Fig. 5. It can be seen that the energy reaches its peak values at ${{{{\omega }_{p}}\tau } \mathord{\left/ {\vphantom {{{{\omega }_{p}}\tau } 2}} \right. \kern-0em} 2} = 1$ and when the frequency difference is equal to the plasma frequency, $\Delta {{\omega }_{0}} = {{\omega }_{p}}$.

Fig. 5.

Total normalized energy of THz radiation (4.10) as a function of the frequency difference ${{\Delta {{\omega }_{0}}} \mathord{\left/ {\vphantom {{\Delta {{\omega }_{0}}} {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}}$ for ${{{{k}_{p}}R} \mathord{\left/ {\vphantom {{{{k}_{p}}R} 2}} \right. \kern-0em} 2} = 1$, ${{{{\omega }_{p}}} \mathord{\left/ {\vphantom {{{{\omega }_{p}}} {{{\omega }_{0}}}}} \right. \kern-0em} {{{\omega }_{0}}}} = 0.01$, and $\gamma = {{{{\nu }_{{ei}}}} \mathord{\left/ {\vphantom {{{{\nu }_{{ei}}}} {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}} = 0.01$. The dashed, solid, and dash-dotted curves correspond to the dimensionless pulse width ${{{{\omega }_{p}}\tau } \mathord{\left/ {\vphantom {{{{\omega }_{p}}\tau } 2}} \right. \kern-0em} 2}$ equal to 0.5, 1, and 1.5, respectively.

Let us compare the radiation energies at the plasma frequency and the doubled plasma frequency. Using formula (4.9) and expression (4.10) from [7], we find that the radiation energy at the plasma frequency exceeds that at the doubled plasma frequency if

$$\max \left( {\frac{{{{\nu }_{{ei}}}}}{{{{\omega }_{p}}}},\frac{{{{\gamma }_{L}}}}{{{{\omega }_{p}}}}} \right) > \frac{1}{{\omega _{0}^{2}{{\tau }^{2}}}},$$

((4.11))

where ${{\gamma }_{L}}$ is the collisionless damping rate of plasma waves.

Let us analyze the spatiotemporal distribution of the fields in the generated THz pulse. Applying inverse Fourier transformation with respect to time to Eq. (3.18), we obtain the following expressions for the electromagnetic fields:

$$\begin{gathered} B_{\varphi }^{{\left( 2 \right)}}\left( {{\mathbf{r}},t} \right) = \frac{{{{\omega }_{p}}}}{{{{\omega }_{0}}}}\frac{{\sqrt \pi \,\omega _{p}^{2}{{\tau }^{2}}k_{0}^{2}{{R}^{2}}{{k}_{0}}L}}{{\sqrt 2 {\kern 1pt} }}{{\left( {\frac{{{{V}_{E}}}}{{4c}}} \right)}^{3}} \\ \times \;{{E}_{{0L}}}\frac{{\sin \theta }}{{{{k}_{p}}r}}H\left( {{\mathbf{r}},t} \right), \\ \end{gathered} $$

((4.12))

$$\begin{gathered} E_{\theta }^{{\left( 2 \right)}}\left( {{\mathbf{r}},t} \right) = \frac{{{{\omega }_{p}}}}{{{{\omega }_{0}}}}\frac{{\sqrt \pi \omega _{p}^{2}{{\tau }^{2}}k_{0}^{2}{{R}^{2}}{{k}_{0}}L}}{{\sqrt 2 }}{{\left( {\frac{{{{V}_{E}}}}{{4c}}} \right)}^{3}} \\ \times \;{{E}_{{0L}}}\frac{{\sin \theta }}{{{{k}_{p}}r}}E\left( {{\mathbf{r}},t} \right), \\ \end{gathered} $$

((4.13))

where

$$\begin{gathered} H\left( {{\mathbf{r}},t} \right) = \operatorname{Re} \int\limits_0^\infty {d\Omega } \sqrt {\frac{{{{\Omega }^{2}} - 1}}{{{{\Omega }^{2}}}}} \\ \times \;\exp \left[ { - i{{\omega }_{p}}t{\kern 1pt} \Omega + i{{k}_{p}}r\sqrt {{{\Omega }^{2}} - 1} } \right] \\ \end{gathered} $$

$$\begin{gathered} \times \;\exp \left\{ { - \frac{{\omega _{p}^{2}{{\tau }^{2}}{{\Omega }^{2}}}}{8} - \left( {{{\Omega }^{2}} - 1} \right)} \right. \\ \times \;\left. {\frac{{k_{p}^{2}{{R}^{2}}{{{\sin }}^{2}}\theta + \omega _{p}^{2}{{\tau }^{2}}{{{\cos }}^{2}}\theta }}{8}} \right\} \\ \end{gathered} $$

((4.14))

$$\begin{gathered} \times \;\left\{ {{{W}_{1}}\left( \Omega \right) + \frac{{{{\omega }_{p}}}}{{2{{\omega }_{0}}}}\Omega \sqrt {{{\Omega }^{2}} - 1} } \right. \\ \times \;\left. {\cos \theta \left[ {\frac{{{{W}_{2}}\left( \Omega \right)}}{{\Omega - 2 + i\gamma }} - \frac{{{{W}_{{ - 2}}}\left( \Omega \right)}}{{\Omega + 2 + i\gamma }}} \right]} \right\}, \\ \end{gathered} $$

$$\begin{gathered} E\left( {{\mathbf{r}},t} \right) = \operatorname{Re} \int\limits_0^\infty {d\Omega } \\ \times \;\exp \left[ { - i{{\omega }_{p}}t{\kern 1pt} \Omega + i{{k}_{p}}r\sqrt {{{\Omega }^{2}} - 1} } \right] \\ \end{gathered} $$

$$\begin{gathered} \times \;\exp \left\{ { - \frac{{\omega _{p}^{2}{{\tau }^{2}}{{\Omega }^{2}}}}{8} - \left( {{{\Omega }^{2}} - 1} \right)} \right. \\ \times \;\left. {\frac{{k_{p}^{2}{{R}^{2}}{{{\sin }}^{2}}\theta + \omega _{p}^{2}{{\tau }^{2}}{{{\cos }}^{2}}\theta }}{8}} \right\} \\ \end{gathered} $$

((4.15))

$$\begin{gathered} \times \;\left\{ {{{W}_{1}}\left( \Omega \right) + \frac{{{{\omega }_{p}}}}{{2{{\omega }_{0}}}}\Omega \sqrt {{{\Omega }^{2}} - 1} } \right. \\ \left. { \times \;\cos \theta \left[ {\frac{{{{W}_{2}}\left( \Omega \right)}}{{\Omega - 2 + i\gamma }} - \frac{{{{W}_{{ - 2}}}\left( \Omega \right)}}{{\Omega + 2 + i\gamma }}} \right]} \right\}. \\ \end{gathered} $$

The time dependences of dimensionless fields (4.14) and (4.15) in the wave zone at a distance of ${{k}_{p}}r = 10$ from the coordinate origin are presented in Fig. 6. Oscillations of the electromagnetic field in the THz pulse occur at the plasma frequency and last for a relatively long time. In this case, the amplitude of the magnetic field is somewhat lower than that of the electric field, which is related to an addition factor $\sqrt {{{\left( {{{\Omega }^{2}} - 1} \right)} \mathord{\left/ {\vphantom {{\left( {{{\Omega }^{2}} - 1} \right)} {{{\Omega }^{2}}}}} \right. \kern-0em} {{{\Omega }^{2}}}}}$ under the integral in expression (4.14), which gives a small contribution when integrating in the vicinity of the plasma frequency, $\Omega \approx 1$.

Fig. 6.

Time dependences of dimensionless electric field (4.15) (solid line) and dimensionless magnetic field (4.14) (dashed line) in the wave zone (${{k}_{p}}r = 10$) of the THz pulse emitted at the angle $\theta = {\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0em} 2}$ for ${{k}_{p}}R = 2$, ${{\omega }_{p}}\tau = \,2$, ${{\Delta {{\omega }_{0}}} \mathord{\left/ {\vphantom {{\Delta {{\omega }_{0}}} {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}} = 1$, ${{{{\omega }_{p}}} \mathord{\left/ {\vphantom {{{{\omega }_{p}}} {{{\omega }_{0}}}}} \right. \kern-0em} {{{\omega }_{0}}}} = 0.01$, and $\gamma = {{{{\nu }_{{ei}}}} \mathord{\left/ {\vphantom {{{{\nu }_{{ei}}}} {{{\omega }_{p}}}}} \right. \kern-0em} {{{\omega }_{p}}}} = 0.01$.

5 CONCLUSIONS

Here, we have analyzed generation of THz waves under interaction of two counterpropagating laser pulses with different frequencies in underdense plasma. Spectral, angular, and energy characteristics of THz waves have been investigated as functions of the frequency difference of the laser pulses. It is shown that an additional broad peak near the plasma frequency appears in the spectrum of THz radiation when the frequencies of the laser pulses are different. The amplitude of this peak can exceed that of the spectral line at the second harmonic of the plasma frequency even in the case of a relatively small frequency difference. It is established that the peak at the plasma frequency is the highest when the frequency difference is equal to the plasma frequency, which is related to the excitation of strong plasma oscillations in the region of interaction of the counterpropagating laser pulses. It is demonstrated that the directional pattern of THz radiation critically depends on the transverse size of the laser pulses. In the case of tight focusing, the THz waves are emitted perpendicular to the axis of propagation of the laser pulses. For pulses of larger transverse dimensions, the radiation is emitted at small angles close to the propagation direction of the laser pulses. It is shown that the total energy of the THz pulse reaches its maximum value when the frequency difference of the laser pulses is equal to the plasma frequency. The time profile of the THz signal in the wave zone under the conditions in which radiation at the plasma frequency dominates is investigated. It is shown that field oscillations in the THz pulse occur at the plasma frequency and last for a relatively long time.

Let us discuss the applicability conditions of the proposed theory. The strongest restriction on the parameters of the laser pulses is imposed by the condition of electron density fluctuations being small ($\delta N_{e}^{{(1)}} < {{N}_{{0e}}}$), because the perturbation theory approach can be used only in this case. This inequality, together with Eq. (3.1), yields the condition

$$\frac{{V_{E}^{2}}}{{{{c}^{2}}}} < \frac{{{{\omega }_{p}}}}{{{{\omega }_{0}}}}\frac{1}{{{{\omega }_{0}}\tau }} \ll 1,$$

((5.1))

which limits the maximum intensity of the laser pulses. This restriction is related to the fact that density perturbations in the plasma oscillations become strong even at relatively low intensities of laser radiation.

In the present work, we have analyzed generation of THz waves under interaction of two counterpropagating laser pulses in underdense plasma. In this case, plasma was assumed to be homogeneous, preformed, and fully ionized. Under the conditions of modern experiments, such plasma is formed when laser radiation acts upon a neutral gas jet in a vacuum chamber. To implement the scheme of THz radiation generation discussed in the present work, the gas jet has to be preionized by an intense laser pulse and the counterpropagating laser pulses should be focused into thus preformed underdense plasma. The THz electromagnetic fields generated upon interaction of counterpropagating laser pulses in fully ionized plasma are proportional to the fourth power of the laser field amplitude (see Eqs. (4.12), (4.13)). This is because only longitudinal plasma fields (3.1)–(3.3) are driven in homogeneous plasma in the quadratic approximation with respect to the laser field amplitude. Electromagnetic waves are generated as a result of nonlinear interaction of the plasma fields. They are described by set of equations (3.5)–(3.8), and their amplitudes are proportional to the fourth power of the laser field.

In the case of interaction of laser pulses focused into a neutral gas, gas ionization by the pulse front creates a region with a sharp gradient of the electron density, which leads to the generation of transient radiation [8, 21]. Its maximum energy for ${{\omega }_{p}}\tau \approx 1$ and ${{k}_{p}}R \approx 1$ is given by (see expression (5.3) in [21])

$$W_{{{\text{rad}}}}^{{\left( 1 \right)}} \approx 4 \times {{10}^{{ - 3}}}\frac{{\omega _{p}^{2}}}{{\omega _{0}^{2}}}\frac{{V_{E}^{2}}}{{{{c}^{2}}}}{{W}_{L}}.$$

((5.2))

Note that the energy of THz radiation (5.2) generated at the sharp density gradient is proportional to $E_{{0L}}^{4}$, while the amplitude of THz radiation depends on the laser field amplitude squared. According to Eqs. (4.9) and (4.10), for the same parameters ${{\omega }_{p}}\tau \approx 1$ and ${{k}_{p}}R \approx 1$, the energy of THz radiation generated upon interaction of two laser pulses is proportional to $E_{{0L}}^{8}$ and, for $\Delta {{\omega }_{0}} = {{\omega }_{p}}$, is given by

$$W_{{{\text{rad}}}}^{{\left( 2 \right)}} \approx 1.5 \times {{10}^{{ - 3}}}\omega _{0}^{2}{{\tau }^{2}}k_{0}^{2}{{R}^{2}}\frac{{V_{E}^{6}}}{{{{c}^{6}}}}{{W}_{L}}.$$

((5.3))

Comparing expressions (5.2) and (5.3), we find that, when

$$0.4\,\omega _{0}^{2}{{\tau }^{2}}{{k}_{0}}R\frac{{V_{E}^{2}}}{{{{c}^{2}}}} > 1,$$

((5.4))

the energy emitted upon interaction of two laser pulses exceeds the energy of their transient radiation. For example, if pulses with the frequency ${{\omega }_{0}} =$ 1.5 × 10¹⁵ s^–1 and duration $\tau = 15{\kern 1pt}$ fs are focused into a spot of radius $R = 11{\kern 1pt}$ μm, condition (5.4) is satisfied for intensities ${{I}_{L}} > 6 \times {{10}^{{13}}}$ W/cm². For longer laser pulses, THz radiation emitted from the region of their interaction dominates the transient radiation at even lower intensities.

To conclude, let us estimate the energy of THz radiation for typical parameters of laser–plasma interaction. Let laser pulses with frequencies ${{\omega }_{0}} =$ 1.5 × 10¹⁵ s^–1, frequency difference $\Delta {{\omega }_{0}} = 1.3 \times {{10}^{{14}}}$ s^–1, pulse durations $\tau$ = 15 fs, cross-sectional radii $R =$ 11 μm, and intensities ${{I}_{L}} = 3.6 \times {{10}^{{15}}}$ W/cm² (the corresponding energies and powers of the pulses are ${{W}_{L}} \approx 0.4$ mJ and ${{P}_{L}} \approx 14$ GW, respectively) propagate in opposite directions in an underdense plasma characterized by the electron density ${{N}_{{0e}}} = 5.4 \times {{10}^{{18}}}$ cm^–3 and temperature ${{T}_{e}} = 50$ eV.

In this case, the electromagnetic waves are emitted at the frequency ${{\nu }_{{{\text{THz}}}}} \approx 21$ THz (${{\lambda }_{{{\text{THz}}}}} \approx 14$ μm) nearly perpendicular to the axis of propagation of the pulses. According to Eqs. (4.9) and (4.10), the total energy of THz radiation is ${{W}_{{{\text{rad}}}}} \approx {{10}^{{ - 4}}}{{W}_{L}} \approx$ 40 nJ. In the case under consideration, for the ion charge number $Z = 5$, the electron–ion collision frequency is ${{\nu }_{{ei}}} \approx {{10}^{{ - 2}}}{{\omega }_{p}} \approx 1.3 \times {{10}^{{12}}}$ s^–1, which exceeds the collisionless damping rate of plasma waves, ${{\gamma }_{L}} \approx 1.7 \times$${{10}^{{ - 3}}}{{\omega }_{p}} \approx 2.2 \times {{10}^{{11}}}$ s^–1. Taking into account inequality (4.11), we find that the energy of radiation at the plasma frequency exceeds that of radiation at the doubled plasma frequency. In this case, the applicability condition of the theory given by inequalities (5.1), which put a limit on the amplitude of density perturbations in plasma oscillations, is satisfied.