2. Boundary value problem for laser radiation

Let an s-polarized laser pulse with the carrier frequency ${\omega _0}$ and the time duration $\tau$ significantly exceeding the oscillation period $(\tau \gg 1/{\omega _0})$ be incident on the plasma, which occupies the region of space $z > 0$, from vacuum $(z < 0)$ at the angle $\alpha$ with respect to the normal to the plasma–vacuum interface. We assume that the size of the pulse in the direction of the $y$-axis significantly exceeds its size along the x and z axes. Then the electric field of the incident Gaussian laser radiation in vacuum $(z < 0)$ near the plasma boundary $(|z|\ll {k_0}R_x^2)$ can be represented in the following form:

(2.1)\begin{equation}\boldsymbol{E}_L^{\textrm{inc}}(\boldsymbol{r},t) = \frac{1}{2}{\boldsymbol{e}_y}{E_{0L}}\exp \left\{ { - \textrm{i}{\omega_0}\left( {t - \frac{{z^{\prime}}}{c}} \right) - \frac{1}{{2{\tau^2}}}{{\left( {t - \frac{{z^{\prime}}}{c}} \right)}^2} - \frac{{{{x^{\prime}}^2}}}{{2R_x^2}}} \right\} + \textrm{c}\textrm{.c}\textrm{.,}\end{equation}

where ${E_{0L}}$ is the amplitude of the laser field, $z^{\prime} = z\cos \alpha + x\sin \alpha$ is the axis characterizing the direction along which the incident pulse propagates, the axis $x^{\prime} = x\cos \alpha - z\sin \alpha$ is perpendicular to the $z^{\prime}$ axis, ${\boldsymbol{e}_y}$ is the basis vector of the y axis in the Cartesian coordinate system, ${R_x}$ is the transverse size of the laser pulse (along the $x^{\prime}$ axis), which we will consider to be much larger than the wavelength of laser radiation ${\lambda _0} = 2\mathrm{\pi }c/{\omega _0}$, ${k_0} = {\omega _0}/c$ is the wavenumber, $L = c\tau$ is the longitudinal size of the laser pulse, c is the speed of light and c.c. is the complex conjugate. This configuration of the focal spot in the form of a line at the plasma boundary takes place when laser radiation is focused by a cylindrical lens. We will assume that the frequency of laser radiation ${\omega _0}$ significantly exceeds the plasma frequency ${\omega _p} = \sqrt {4\mathrm{\pi }{e^2}{N_{0e}}/{m_e}}$, this condition corresponds to a rarefied plasma with the electron density ${N_{0e}}$ that is significantly less than the critical value ${N_{\textrm{cr}}} = {m_e}\omega _0^2/(4\mathrm{\pi }{e^2})$, that is, the inequalities ${\omega _0} \gg {\omega _p}$, ${N_{0e}} \ll {N_{\textrm{cr}}}$ are satisfied, where $e,\;{m_e}$ are the charge and mass of the electron. Using a Fourier transform with respect to time and x coordinate, we write the total field of the laser radiation in vacuum, which is the superposition of the incident and reflected pulses, in the following form (see Appendix A)

(2.2)\begin{align} {\boldsymbol{E}_L}(\boldsymbol{r},t) & = \dfrac{1}{2}{\boldsymbol{e}_y}{E_{0L}}\dfrac{{2\mathrm{\pi }{R_x}\tau }}{{\cos \alpha }}\int_{ - \infty }^{ + \infty } {\dfrac{{\textrm{d}\omega }}{{2\mathrm{\pi }}}} \exp \left[ { - \textrm{i}\omega t - \dfrac{{{{(\omega - {\omega_0})}^2}{\tau^2}}}{2}} \right]\nonumber\\ & \quad \times \int_{ - \infty }^{ + \infty } {\dfrac{{\textrm{d}{k_x}}}{{2\mathrm{\pi }}}} \exp \left\{ {\textrm{i}{k_x}x - \dfrac{{{{[{k_x} - (\omega /c)\sin \alpha ]}^2}R_x^2}}{{2{{\cos }^2}\alpha }}} \right\}\nonumber\\ & \quad \times \left\{ {\exp \left( {\textrm{i}z\sqrt {\dfrac{{{\omega^2}}}{{{c^2}}} - k_x^2} } \right) + R(\omega ,{k_x})\exp \left( { - \textrm{i}z\sqrt {\dfrac{{{\omega^2}}}{{{c^2}}} - k_x^2} } \right)} \right\} + \textrm{c}\textrm{.c}\textrm{.},\quad z \le 0, \end{align}

where $R(\omega ,{k_x})$ is the Fourier image of the reflection coefficient for s-polarized radiation, which is determined from the boundary conditions. The electric field of the laser pulse in plasma, by analogy with formula (2.2), can also be written in the form of Fourier integrals

(2.3)\begin{align} {\boldsymbol{E}_L}(\boldsymbol{r},t) & = \dfrac{1}{2}{\boldsymbol{e}_y}{E_{0L}}\dfrac{{2\mathrm{\pi }{R_x}\tau }}{{\cos \alpha }}\int_{ - \infty }^{ + \infty } {\dfrac{{\textrm{d}\omega }}{{2\mathrm{\pi }}}} \exp \left[ { - \textrm{i}\omega t - \dfrac{{{{(\omega - {\omega_0})}^2}{\tau^2}}}{2}} \right]\nonumber\\ & \quad \times \int_{ - \infty }^{ + \infty } {\dfrac{{\textrm{d}{k_x}}}{{2\mathrm{\pi }}}} T(\omega ,{k_x})\exp \left\{ {\textrm{i}{k_x}x + \textrm{i}z\sqrt {\dfrac{{{\omega^2}}}{{{c^2}}}\varepsilon (\omega ) - k_x^2} - \dfrac{{{{[{k_x} - (\omega /c)\sin \alpha ]}^2}R_x^2}}{{2{{\cos }^2}\alpha }}} \right\}\nonumber\\ & \quad + \textrm{c}\textrm{.c}\textrm{.},\quad z \ge 0, \end{align}

where $T(\omega ,{k_x})$ is the Fourier image transmission coefficient for the s-polarized wave, $\varepsilon (\omega ) = 1 - \omega _p^2/{\omega ^2}$ is the dielectric permittivity of the plasma at the frequency $\omega$. From the continuity conditions for the tangential components of the electric and magnetic fields of laser radiation at the plasma–vacuum interface, we find expressions for the Fourier components of the reflection and transmission coefficients

(2.4)\begin{equation}\left. {\begin{array}{*{20}{c}} {R(\omega ,{k_x}) = \dfrac{{\sqrt {({\omega^2}/{c^2}) - k_x^2} - \sqrt {({\omega^2}/{c^2})\varepsilon (\omega ) - k_x^2} }}{{\sqrt {({\omega^2}/{c^2}) - k_x^2} + \sqrt {({\omega^2}/{c^2})\varepsilon (\omega ) - k_x^2} }},}\\ {T(\omega ,{k_x}) = \dfrac{{2\sqrt {({\omega^2}/{c^2}) - k_x^2} }}{{\sqrt {({\omega^2}/{c^2}) - k_x^2} + \sqrt {({\omega^2}/{c^2})\varepsilon (\omega ) - k_x^2} }}.} \end{array}} \right\}\end{equation}

Let us consider the electric field of laser radiation in plasma near the plasma–vacuum interface, when $z \to + 0$. Taking into account the inequalities ${\omega _0}\tau \gg 1,\;{k_0}{R_x} \gg 1$, from (2.3) we find the following field distribution near the plasma boundary:

(2.5)\begin{align} {\boldsymbol{E}_L}(x,z ={+} 0,t) & = {\boldsymbol{e}_y}\dfrac{{2{E_{0L}}\cos \alpha }}{{\cos \alpha + \sqrt {{{\cos }^2}\alpha - N({\omega _0})} }}\nonumber\\ & \quad \times \exp \left[ { - \dfrac{1}{{2{\tau^2}}}{{\left( {t - \dfrac{x}{c}\sin \alpha } \right)}^2} - \dfrac{{{x^2}}}{{2R_x^2}}{{\cos }^2}\alpha } \right]\cos \left[ {{\omega_0}\left( {t - \dfrac{x}{c}\sin \alpha } \right)} \right], \end{align}

where $N({\omega _0}) = \omega _p^2/\omega _0^2 = {N_{0e}}/{N_{\textrm{cr}}}$ is the dimensionless electron density. As noted earlier (Frolov Reference Frolov2020), when a laser pulse is incident at the angle of total reflection and the condition

(2.6)\begin{equation}{\cos ^2}\alpha = N({\omega _0}),\end{equation}

or ${\rm sin}^{2}\alpha = \varepsilon ({\omega _0})$ is fulfilled, the amplitude of the laser radiation electric field in the plasma doubles compared with the amplitude of the incident field in vacuum, as follows from the comparison of (2.1) and (2.5). It should be noted that, when a laser pulse falls on the boundary of very rarefied plasma $({\cos ^2}\alpha \gg N({\omega _0}))$, it crosses the plasma–vacuum interface without reflection (see formula (2.5)) since the electric field of the incident (2.1) and transmitted (2.5) laser pulses coincide in magnitude. However, for an arbitrary finite value of the electron density, there is the certain angle of incidence for which the total reflection of the laser pulse takes place and this angle is determined from relation (2.6). This effect is similar to the effect of total internal reflection of light that falls from an optically denser medium onto the boundary of a medium with a lower refractive index, since vacuum is an optically denser medium (the refractive index is 1) than plasma whose refractive index is less than unity $(\sqrt {\varepsilon ({\omega _0})} < 1)$. In accordance with formula (2.6), we conclude that the total reflection of laser radiation from rarefied plasma occurs at grazing angles and the lower the electron density, the closer this angle is to $\pi /2$. Thus, the total reflection of a laser pulse from plasma with a sufficiently low electron density occurs when it propagates almost along the boundary. It follows from formula (2.5) that the electric field of laser radiation in plasma near the boundary under condition (2.6) has a value twice as high as the field of the incident laser pulse (2.1). This effect takes place due to the fact that, under the condition of total reflection (2.6), the electric field of laser radiation at the plasma boundary (2.6) is equal to the sum of the incident and reflected fields, which are equal in absolute value.

Using the expression for the electric field (2.5) and the definition of the potential of laser radiation ponderomotive forces $\varPhi (\boldsymbol{r},t)$ acting on plasma electrons

(2.7)\begin{equation}\varPhi (\boldsymbol{r},t) = \frac{e}{{2{m_e}}}\left\langle {{{\left( {\int_{ - \infty }^t {\textrm{d}t^{\prime}{\boldsymbol{E}_L}(\boldsymbol{r},t^{\prime})} } \right)}^2}} \right\rangle ,\end{equation}

where angle brackets $\langle \cdots \rangle$ mean averaging over the period of high-frequency oscillations $2\mathrm{\pi }/{\omega _0}$, we find the ponderomotive potential $\varPhi (\boldsymbol{r},t)$ near the plasma boundary

(2.8)\begin{align}\varPhi (x,z ={+} 0,t) & = \frac{{eE_{0L}^2}}{{4{m_e}\omega _0^2}}{\left|{\frac{{2\cos \alpha }}{{\cos \alpha + \sqrt {{{\cos }^2}\alpha - N({\omega_0})} }}} \right|^2}\nonumber\\ & \quad \times \exp \left\{ { - \frac{1}{{{\tau^2}}}{{\left[ {t - \frac{x}{c}\sin \alpha } \right]}^2} - \frac{{{x^2}}}{{R_x^2}}{{\cos }^2}\alpha } \right\},\end{align}

which differs from the corresponding expression (14) in (Frolov Reference Frolov2020) by taking into account the transverse size of the laser pulse. The ponderomotive action of laser radiation with potential (2.8) leads to the excitation of THz fields in the plasma and their emission into vacuum. It should be noted that the ponderomotive effect is significantly enhanced when the laser pulse is incident at the angle of total reflection (2.6), since in this case the potential (2.8) increases by a factor of 4 compared with the case of incidence at smaller angles, when the condition ${\cos ^2}\alpha \gg N({\omega _0})$ is satisfied.

3. Excitation of THz fields in vacuum and plasma

To describe the generation of THz radiation, we will use the time-averaged Maxwell equations for the electric $\boldsymbol{E}(\boldsymbol{r},t)$ and magnetic $\boldsymbol{B}(\boldsymbol{r},t)$ fields

(3.1a,b)\begin{equation}rot\boldsymbol{B}(\boldsymbol{r},t) = \frac{1}{c}\frac{\partial }{{\partial t}}\boldsymbol{E}(\boldsymbol{r},t) + \frac{{4\mathrm{\pi }}}{c}e{N_e}(z)\boldsymbol{V}(\boldsymbol{r},t),\quad rot\boldsymbol{E}(\boldsymbol{r},t) ={-} \frac{1}{c}\frac{\partial }{{\partial t}}\boldsymbol{B}(\boldsymbol{r},t),\end{equation}

as well as the equation for the electron velocity $\boldsymbol{V}(\boldsymbol{r},t)$, taking into account the ponderomotive effect of laser radiation (see, for example, Frolov Reference Frolov2020)

(3.2)\begin{equation}\left( {\frac{\partial }{{\partial t}} + {\nu_{ei}}} \right)\boldsymbol{V}(\boldsymbol{r},t) = \frac{e}{{{m_e}}}[\boldsymbol{E}(\boldsymbol{r},t) - \boldsymbol{\nabla }\varPhi (\boldsymbol{r},t)],\end{equation}

which is obtained in the non-relativistic approximation $|\boldsymbol{V}(\boldsymbol{r},t)|\ll c$ under the condition of rare electron collisions ${\nu _{ei}}\tau \ll 1$, where ${N_e}(z) = {N_{0e}}\theta (z)$ is the coordinate-dependent electron density, $\theta (z)$ is the Heaviside unit step function and ${\nu _{ei}}$ is the frequency of electron--ion collisions.

Using a Fourier transform with respect to time and x coordinate from the set of (3.1), (3.2) we find the following equation for the low-frequency magnetic field ${B_y}(\omega ,{k_x},z)$:

(3.3)\begin{align} & \dfrac{\textrm{d}}{{\textrm{d}z}}\left\{ {\dfrac{1}{{\varepsilon (\omega ,z)}}\dfrac{\textrm{d}}{{\textrm{d}z}}{B_y}(\omega ,{k_x},z)} \right\} + \left[ {\dfrac{{{\omega^2}}}{{{c^2}}} - \dfrac{{k_x^2}}{{\varepsilon (\omega ,z)}}} \right]{B_y}(\omega ,{k_x},z)\nonumber\\ & \quad = \dfrac{\omega }{c}{k_x}\left\{ {\dfrac{{\omega_p^2(z)}}{{\omega (\omega + i{\nu_{ei}})\varepsilon (\omega ,z)}}\dfrac{\textrm{d}}{{\textrm{d}z}}\varPhi (\omega ,{k_x},z) - \dfrac{\textrm{d}}{{\textrm{d}z}}\left[ {\dfrac{{\omega_p^2(z)\varPhi (\omega ,{k_x},z)}}{{\omega (\omega + i{\nu_{ei}})\varepsilon (\omega ,z)}}} \right]} \right\}, \end{align}

at the same time the components of the electric field ${E_x}(\omega ,{k_x},z)$, ${E_z}(\omega ,{k_x},z)$, are determined by the following relationships:

(3.4)\begin{equation}\left. {\begin{array}{*{20}{c}} {{E_x}(\omega ,{k_x},z) ={-} \dfrac{{i{\kern 1pt} c}}{{\omega \varepsilon (\omega ,z)}}\dfrac{\textrm{d}}{{\textrm{d}z}}{B_y}(\omega ,{k_x},z) - \dfrac{{i{\kern 1pt} {k_x}\omega_p^2(z)}}{{\omega (\omega + i{\nu_{ei}})\varepsilon (\omega ,z)}}\varPhi (\omega ,{k_x},z)}\\ {{E_z}(\omega ,{k_x},z) ={-} \dfrac{{c{k_x}}}{{\omega \varepsilon (\omega ,z)}}{B_y}(\omega ,{k_x},z) - \dfrac{{\omega_p^2(z)}}{{\omega (\omega + i{\nu_{ei}})\varepsilon (\omega ,z)}}\dfrac{\textrm{d}}{{\textrm{d}z}}\varPhi (\omega ,{k_x},z)} \end{array}} \right\},\end{equation}

where $\varepsilon (\omega ,z) = 1 - \omega _p^2(z)/{\omega ^2}$, $\omega _p^2(z) = 4\mathrm{\pi }{e^2}{N_e}(z)/{m_e}$ and $\varPhi (\omega ,{k_x},z)$ is the Fourier image of the ponderomotive potential. The solution of (3.3) for the low-frequency magnetic field in a plasma and in a vacuum, taking into account the boundary conditions of the continuity for ${E_x}(\omega ,{k_x},z)$ and ${B_y}(\omega ,{k_x},z)$, has the following form:

(3.5)\begin{align}{B_y}(\omega ,{k_x},z) & = \frac{{\textrm{i}\omega {k_x}}}{c}\frac{{\omega _p^2}}{{\omega (\omega + i{\nu _{ei}})}}\frac{{\varPhi (\omega ,{k_x},z = 0)}}{{D(\omega ,{k_x})}}\exp \left\{ { - \textrm{i}{\kern 1pt} z\sqrt {\frac{{{\omega^2}}}{{{c^2}}} - k_x^2} } \right\},\quad z \le 0,\end{align}

(3.6)\begin{align}{B_y}(\omega ,{k_x},z) & = \frac{{\textrm{i}\omega {k_x}}}{c}\frac{{\omega _p^2}}{{\omega (\omega + i{\nu _{ei}})}}\frac{{\varPhi (\omega ,{k_x},z = 0)}}{{D(\omega ,{k_x})}}\exp \left\{ {\textrm{i}{\kern 1pt} z\sqrt {\frac{{{\omega^2}}}{{{c^2}}}\varepsilon (\omega ) - k_x^2} } \right\},\quad z \ge 0,\end{align}

where the function $D(\omega ,{k_x})$ is given by the formula

(3.7)\begin{equation}D(\omega ,{k_x}) = \varepsilon (\omega )\sqrt {\frac{{{\omega ^2}}}{{{c^2}}} - k_x^2} + \sqrt {\frac{{{\omega ^2}}}{{{c^2}}}\varepsilon (\omega ) - k_x^2} .\end{equation}

It follows from formulas (3.5), (3.6) that the low-frequency magnetic field in plasma and in vacuum is determined by the Fourier transform of the ponderomotive potential at the plasma boundary $\varPhi (\omega ,{k_x},z = 0)$, which, in accordance with (2.8), has the form

(3.8)\begin{align} \varPhi (\omega ,{k_x},z = 0) & = \dfrac{{eE_{0L}^2}}{{4{m_e}\omega _0^2}}{\left|{\dfrac{{2\cos \alpha }}{{\cos \alpha + \sqrt {{{\cos }^2}\alpha - N({\omega_0})} }}} \right|^2}\nonumber\\ & \quad \times \dfrac{{\mathrm{\pi }R\tau }}{{\cos \alpha }}\exp \left\{ { - \dfrac{{{\omega^2}{\tau^2}}}{4} - {{\left( {{k_x} - \dfrac{\omega }{c}\sin \alpha } \right)}^2}\dfrac{{R_x^2}}{{4{{\cos }^2}\alpha }}} \right\}. \end{align}

The obtained expression for the magnetic field (3.5), taking into account formulas (3.7), (3.8), allows us to analyse the characteristics of the THz pulse, which is emitted from the plasma into vacuum.

4. Spectral, angular and energy characteristics of THz radiation in vacuum

Using the inverse Fourier transform with respect to the spatial coordinate from formula (3.5), we find the THz magnetic field in vacuum

(4.1)\begin{align}{B_y}(\omega ,\boldsymbol{r}) = \frac{{\omega _p^2}}{{c(\omega + i{\nu _{ei}})}}\frac{\partial }{{\partial x}}\int_{ - \infty }^{ + \infty } {\frac{{\textrm{d}{k_x}}}{{2\mathrm{\pi }}}} \exp \left\{ {\textrm{i}{k_x}x - \textrm{i}z\sqrt {\frac{{{\omega^2}}}{{{c^2}}} - k_x^2} } \right\}\frac{{\varPhi (\omega ,{k_x},z = 0)}}{{D(\omega ,{k_x})}},\quad z \le 0.\end{align}

We will consider the field (4.1), (3.8) at large distances from the plasma boundary in the wave zone, when the condition $r = \sqrt {{x^2} + {z^2}} \gg L,R,c/\omega$ is satisfied. In this case, the integral in formula (4.1) can be calculated using the stationary phase method (Olver Reference Olver1974). For large values of the exponential function argument, the main contribution to the integral is made by the small neighbourhood near the stationary point, the position of which is determined by the formula

(4.2)\begin{equation}{k_{x,s}} = \frac{{{\kern 1pt} \omega }}{c}\sin \theta ,\end{equation}

where $\theta$ is the angle between the direction of observation and the positive direction of the z axis. The position of the stationary point (4.2) is found from the condition that the derivative of the argument of the exponential function in formula (4.1) is equal to zero. Calculating the integral in formula (4.1) taking into account the contribution of the stationary point (4.2) for the Fourier transform of the THz magnetic field in vacuum, we obtain the following expression:

(4.3)\begin{align} {B_y}(\omega ,\boldsymbol{r}) & = \dfrac{{\omega _p^2}}{{{\omega _0}(\omega + i{\nu _{ei}})}}\dfrac{{{V_E}}}{{4c}}{E_{0L}}\dfrac{{\mathrm{\pi }{R_x}\tau }}{{\cos \alpha }}{\left|{\dfrac{{2\cos \alpha }}{{\sin \alpha + \sqrt {{{\cos }^2}\alpha - N({\omega_0})} }}} \right|^2}\sqrt {\dfrac{\omega }{{2\mathrm{\pi }rc}}} \nonumber\\ & \quad \times \dfrac{{\sin \theta |\cos \theta |}}{{\varepsilon (\omega )|\cos \theta |+ \sqrt {\varepsilon (\omega ) - {{\sin }^2}\theta } }}\nonumber\\ & \quad \times \exp \left\{ {\textrm{i}\dfrac{\mathrm{\pi }}{4} + \textrm{i}\dfrac{\omega }{c}r - \dfrac{{{\omega^2}{\tau^2}}}{4} - {{\left( {\dfrac{{\sin \theta - \sin \alpha }}{{\cos \alpha }}} \right)}^2}\dfrac{{{\omega^2}R_x^2}}{{4{c^2}}}} \right\},\quad z \le 0, \end{align}

where ${V_E} = e{E_{0L}}/({m_e}{\omega _0})$ is the amplitude of the electron oscillation velocity in the laser field (2.1). Using formula (4.3), we find the energy that is radiated into the unit frequency interval $\textrm{d}\omega$ and is carried through the unit cylindrical area $\textrm{d}\boldsymbol{f} = {\boldsymbol{e}_r}r\,\textrm{d}\theta \,\textrm{d}y$ along the normal to it

(4.4)\begin{equation}\textrm{d}W(\omega ,\theta ) = \frac{c}{{8{\mathrm{\pi }^2}}}\{ [\boldsymbol{E}(\omega ,\boldsymbol{r}) \times {\boldsymbol{B}^ \ast }(\omega ,\boldsymbol{r})] + \textrm{c}\textrm{.c}\textrm{.}\} \,\textrm{d}\omega \,\textrm{d}\boldsymbol{f} = \frac{c}{{4{\mathrm{\pi }^2}}}|{B_y}(\omega ,\boldsymbol{r}){|^2}r\,\textrm{d}\theta \,\textrm{d}y\,\textrm{d}\omega .\end{equation}

Taking into account formulas (4.3), (4.4), we find the energy radiated in the unit interval of frequencies $\textrm{d}\omega$ and angles $\textrm{d}\theta$

(4.5)\begin{align} \dfrac{{\textrm{d}W(\omega ,\theta )}}{{\textrm{d}\omega \,\textrm{d}\theta }} & = \dfrac{{\omega _p^2}}{{\omega _0^2}}\dfrac{{{\omega _p}\tau {k_p}{R_x}{\kern 1pt} \omega }}{{{\omega ^2} + \nu _{ei}^2}}\dfrac{{V_E^2}}{{{c^2}}}\dfrac{{{W_L}}}{\mathrm{\pi }}\dfrac{{{{\cos }^2}\alpha }}{{|\cos \alpha + \sqrt {{{\cos }^2}\alpha - N({\omega _0})} {|^4}}}\nonumber\\ & \quad \times \dfrac{{{{\sin }^2}\theta {{\cos }^2}\theta }}{{{{\left|{\varepsilon (\omega )|\cos \theta |+ \sqrt {\varepsilon (\omega ) - {{\sin }^2}\theta } } \right|}^2}}}\nonumber\\ & \quad \times \exp \left\{ { - \dfrac{{{\omega^2}{\tau^2}}}{2}\left[ {1 + \dfrac{{R_x^2}}{{{L^2}}}{{\left( {\dfrac{{\sin \theta - \sin \alpha }}{{\cos \alpha }}} \right)}^2}} \right]} \right\},\quad z \le 0, \end{align}

where ${W_L} = E_{0L}^2{R_x}{R_y}L/8$ is the energy of the laser pulse, ${R_y}$ is the size of the laser pulse along the y axis and ${k_p} = {\omega _p}/c$. If we integrate over frequencies in formula (4.5), then we find the THz radiation pattern

(4.6)\begin{align} \dfrac{{\textrm{d}W(\theta )}}{{\textrm{d}\theta }} & = \dfrac{{{R_x}}}{L}\dfrac{{V_E^2}}{{{c^2}}}\dfrac{{{W_L}}}{{\pi }}J(\theta ),\nonumber\\ J(\theta ) & = \dfrac{{\omega _0^2{\tau ^2}{N^2}({\omega _0}){{\sin }^2}\theta {{\cos }^2}\theta {{\cos }^2}\alpha }}{{|\cos \alpha + \sqrt {{{\cos }^2}\alpha - N({\omega _0})} {|^4}}}\nonumber\\ & \quad \times \int_0^\infty {\dfrac{{\textrm{d}\varOmega {\varOmega ^3}}}{{|({\varOmega ^2} - 1)|\cos \theta |+ \varOmega \sqrt {{\varOmega ^2}{{\cos }^2}\theta - 1} {|^2}}}} \nonumber\\ & \quad \times \exp \left\{ { - \dfrac{{N({\omega_0})\omega_0^2{\tau^2}{\varOmega^2}}}{2}\left[ {1 + \dfrac{{R_x^2}}{{{L^2}}}{{\left( {\dfrac{{\sin \theta - \sin \alpha }}{{\cos \alpha }}} \right)}^2}} \right]} \right\}. \end{align}

The spectrum of THz radiation is calculated by integrating in formula (4.5) over the angles

(4.7)\begin{align} \dfrac{{\textrm{d}W(\varOmega )}}{{\textrm{d}\Omega }} & = \dfrac{{{R_x}}}{L}\dfrac{{V_E^2}}{{{c^2}}}\dfrac{{{W_L}}}{\mathrm{\pi }}I(\varOmega ),\nonumber\\ I(\varOmega ) & = \dfrac{{\omega _0^2{\tau ^2}{N^2}({\omega _0}){{\cos }^2}\alpha }}{{|\cos \alpha + \sqrt {{{\cos }^2}\alpha - N({\omega _0})} {|^4}}}{\varOmega ^3}\nonumber\\ & \quad \times \int_{\mathrm{\pi }/2}^{3\mathrm{\pi }/2} {\dfrac{{\textrm{d}\theta {{\sin }^2}\theta {{\cos }^2}\theta }}{{|({\varOmega ^2} - 1)|\cos \theta |+ \varOmega \sqrt {{\varOmega ^2}{{\cos }^2}\theta - 1} {|^2}}}} \nonumber\\ & \quad \times \exp \left\{ { - \dfrac{{N({\omega_0})\omega_0^2{\tau^2}{\varOmega^2}}}{2}\left[ {1 + \dfrac{{R_x^2}}{{{L^2}}}{{\left( {\dfrac{{\sin \theta - \sin \alpha }}{{\cos \alpha }}} \right)}^2}} \right]} \right\}, \end{align}

where $\varOmega = \omega /{\omega _p}$ is the dimensionless frequency. The total energy of THz radiation can be calculated by integrating over angles in formula (4.6) or over frequencies in formula (4.7)

(4.8)\begin{align} W & = \dfrac{{{R_x}}}{L}\dfrac{{V_E^2}}{{{c^2}}}\dfrac{{{W_L}}}{\mathrm{\pi }}w,\quad w = \dfrac{{\omega _0^2{\tau ^2}{N^2}({\omega _0}){{\cos }^2}\alpha }}{{|\cos \alpha + \sqrt {{{\cos }^2}\alpha - N({\omega _0})} {|^4}}}\int_{\mathrm{\pi }/2}^{3\mathrm{\pi }/2} {\textrm{d}\theta } {\sin ^2}\theta {\cos ^2}\theta \nonumber\\ & \quad \times \int_0^\infty {\dfrac{{\textrm{d}\varOmega {\varOmega ^3}}}{{|({\varOmega ^2} - 1)|\cos \theta |+ \varOmega \sqrt {{\varOmega ^2}{{\cos }^2}\theta - 1} {|^2}}}} \nonumber\\ & \quad \times \exp \left\{ { - \dfrac{{N({\omega_0})\omega_0^2{\tau^2}{\varOmega^2}}}{2}\left[ {1 + \dfrac{{R_x^2}}{{{L^2}}}{{\left( {\dfrac{{\sin \theta - \sin \alpha }}{{\cos \alpha }}} \right)}^2}} \right]} \right\}. \end{align}

We will further assume that the laser pulse energy, its wavelength and duration are fixed values, whereas the size of the focal spot and radiation intensity (these parameters are determined by the degree of focusing), as well as the angle of incidence are variable. The electron density of the plasma can also be a variable, which is determined by the type of target irradiated by the laser.

The total energy of THz radiation (4.8) as a function of the incidence angle for different degrees of laser radiation focusing is shown in figure 1. It follows from figure 1 that the total energy of THz radiation is maximum when the laser pulse is grazingly incident at the angle of total reflection (2.6) onto a rarefied plasma whose density for the indicated parameters is almost three orders of magnitude less than the critical value. Also, the energy of the THz pulse increases with a decrease in the size of the laser radiation focal spot. If condition (2.6) is satisfied and the laser pulse is incident at the angle of total reflection, then the maximum value of the total energy of THz radiation (4.8) takes the form

(4.9)\begin{align} {W_{\max }} & = \dfrac{{{R_x}}}{L}\dfrac{{V_E^2}}{{{c^2}}}\dfrac{{{W_L}}}{\mathrm{\pi }}{w_{\max }},\quad {w_{\max }} = N({\omega _0})\omega _0^2{\tau ^2}\int_{\mathrm{\pi }/2}^{3\mathrm{\pi }/2} {\textrm{d}\theta } {\sin ^2}\theta {\cos ^2}\theta \nonumber\\ & \quad \times \int_0^\infty {\dfrac{{\textrm{d}\varOmega {\varOmega ^3}}}{{|({\varOmega ^2} - 1)|\cos \theta |+ \varOmega \sqrt {{\varOmega ^2}{{\cos }^2}\theta - 1} {|^2}}}} \nonumber\\ & \quad \times \exp \left\{ { - \dfrac{{N({\omega_0})\omega_0^2{\tau^2}{\varOmega^2}}}{2}\left[ {1 + \dfrac{{R_x^2}}{{{L^2}}}{{\left( {\dfrac{{\sin \theta - \sqrt {1 - N({\omega_0})} }}{{\sqrt {N({\omega_0})} }}} \right)}^2}} \right]} \right\}. \end{align}

It follows from formula (4.9) that the dimensionless energy has the maximum value

(4.10)\begin{align}{w_{\max }}({R_x} \to 0) = \omega _p^2{\tau ^2}\int_{\mathrm{\pi }/2}^{3\mathrm{\pi }/2} {\textrm{d}\theta } {\sin ^2}\theta {\cos ^2}\theta \int_0^\infty {\frac{{\textrm{d}\varOmega {\varOmega ^3}\exp \{ - \omega _p^2{\tau ^2}{\varOmega ^2}/2\} }}{{|({\varOmega ^2} - 1)|\cos \theta |+ \varOmega \sqrt {{\varOmega ^2}{{\cos }^2}\theta - 1} {|^2}}}} ,\end{align}

in the limit ${R_x} \to 0$, that is, for tight focusing of the laser pulse. In this case, it should be taken into account that the minimum value of the focal spot size is limited by the wavelength of laser radiation ${\lambda _0}$. The dimensionless energy of THz radiation for the laser pulse for extremely tight focusing (4.10) as a function of the plasma density is shown in figure 2.

Figure 1.

The THz radiation energy (4.8) as a function of the laser pulse incidence angle at ${\omega _0}\tau = 100$ for different electron densities and for ${({R_x}/L)^2} = 4$ (a), ${({R_x}/L)^2} = 1$ (b), ${({R_x}/L)^2 = 0.25}$ (c).

Figure 2.

The energy of THz radiation as the function of the electron density (4.10) when the tight focused $({R_x} \to 0)$ laser pulse is incident at the angle of total reflection (2.6) at ${\omega _0}\tau = 100$.

For the duration of the laser pulse $\tau = 100/{\omega _0}$, the energy is maximum for the density ${N_{0e}} \approx 1.6 \times {10^{ - 4}}{N_{\textrm{cr}}}$, which corresponds to the parameter ${\omega _p}\tau \approx 1.3$. The energy of THz radiation (4.9) for the indicated value of the electron density as a function of the focal spot size is shown in figure 3, from which it follows that the THz energy noticeably increases with a decrease in the focal spot size of the laser pulse. If at ${R_x} = L$ the dimensionless energy of THz radiation is $w \approx 0.12$, then with a decrease in the size of the laser pulse focal spot, a sharp increase in the energy of THz radiation takes place, and at the extremely tight focusing ${R_x} \to 0$, THz energy reaches the value $w \approx 33$, which is more than two orders of magnitude greater than at ${R_x} = L$.

Figure 3.

The energy of THz radiation (4.9) as the function of the degree of laser pulse focusing when it is incident at the angle of total reflection (2.6), at ${\omega _0}\tau = 100$, ${N_{0e}} = 1.6 \times {10^{ - 4}}{N_{\textrm{cr}}}$.

The THz radiation pattern (4.6) is shown in figure 4. At large sizes of the laser pulse focal spot, THz waves are emitted in the direction approximately coinciding with the direction of the reflected laser pulse propagation (see figure 4a).

Figure 4.

The THz radiation pattern for the incidence of the laser pulse with duration $\tau = 100/{\omega _0}$ at the angle of total reflection (2.6) on the plasma with the electron density ${N_{0e}} = {10^{ - 3}}{N_{\textrm{cr}}}$ at various degrees of focusing. The arrows show the direction of propagation of the incident and reflected laser pulses.

This result follows from formula (4.6) under the condition ${R_x} \gg L$ and was previously presented for an unfocused laser pulse $({R_x} \to \infty )$ in a number of publications (see, for example, Frolov Reference Frolov2020, Reference Frolov2021). A decrease in the size of the laser pulse focal spot leads to the fact that the radiation pattern of THz radiation becomes wider and shifts towards the normal to the plasma boundary. The case of extremely tight focusing of laser radiation is shown in figure 4(b). In this case, radiation with lower energy appears in the specular direction in the range of angles $\mathrm{\pi } \le \theta \le 3\mathrm{\pi }/2$.

The energy of THz radiation as a function of frequency (4.7) is shown in figure 5 for different degrees of the laser radiation focusing.

Figure 5.

The spectrum of THz radiation when a laser pulse with duration $\tau = 100/{\omega _0}$ is incident at the angle of total reflection (2.6) onto the plasma with the electron density ${N_{0e}} = {10^{ - 3}}{N_{\textrm{cr}}}$ at various degrees of focusing.

A decrease in the size of the focal spot leads to an increase in the height of the spectral line and its shift to the region of higher frequencies. If, for a large transverse dimension of the laser pulse, the maximum in the THz radiation spectrum corresponds to a frequency of the order of the reciprocal duration ${\omega _{\max }} \approx 1/\tau$, which is less than the plasma frequency, then, with a decrease in the transverse dimension, the frequency approaches the plasma frequency and can become equal to it.