Terahertz waves emission from plasma under action of p-polarized tightly focused laser pulse

Abstract

The theory of terahertz (THz) waves generation at the incidence of the focused p-polarized laser pulse on the plasma boundary is developed. The angular, spectral, and energy characteristics of the THz signal are studied as the function of the focusing degree and the incidence angle of laser radiation, as well as the plasma density. The frequency dependence of the THz signal energy is considered and it is established that the emission spectrum contains the broad line, and its position for tight focusing of laser radiation is determined by its reciprocal duration. The radiation pattern of the THz pulse is studied and it is shown that the decrease in the size of the laser pulse focal spot leads to the increase in the angle of radiation with respect to the normal to the plasma boundary, and for the tight focusing of the laser pulse, THz waves are emitted almost along the plasma-vacuum interface. It is established that the energy of the THz pulse noticeably increases when the p-polarized laser pulse is incident almost along the normal onto the near-critical plasma at the angle of total reflection, and the transverse size of the laser pulse decreases. It is shown that the emission of high-power THz pulses with the sufficiently high conversion rate occurs at the almost normal incidence of ultrashort tightly focused laser radiation on the boundary of near-critical plasma with rare electron collisions.

Appendix: Boundary value problem for p-polarized laser pulse

To solve the boundary value problem when the p-polarized laser pulse (1) is incident on a semi-bounded plasma, we will use the Maxwell equations

$$ \begin{aligned} rot{\mathbf{E}}_{L} \left( {{\mathbf{r}},t} \right) & = - \frac{1}{c}\frac{\partial }{\partial t}{\mathbf{B}}_{L} \left( {{\mathbf{r}},t} \right), \\ rot{\mathbf{B}}_{L} \left( {{\mathbf{r}},t} \right) & = \frac{1}{c}\frac{\partial }{\partial t}{\mathbf{E}}_{L} \left( {{\mathbf{r}},t} \right) + \frac{4\pi e}{c}N_{0e} {\mathbf{V}}_{L} \left( {{\mathbf{r}},t} \right), \\ \end{aligned} $$

(A1)

where ${\mathbf{E}}_{L} \left( {{\mathbf{r}},t} \right),\;{\mathbf{B}}_{L} \left( {{\mathbf{r}},t} \right)$ are the electric and magnetic fields of laser radiation, and the electron velocity ${\mathbf{V}}_{L} \left( {{\mathbf{r}},t} \right)$ is determined from the equation

$$ \left( {\frac{\partial }{\partial t} + i\nu_{ei} } \right){\mathbf{V}}_{L} \left( {{\mathbf{r}},t} \right) = \frac{e}{{m_{e} }}{\mathbf{E}}_{L} \left( {{\mathbf{r}},t} \right). $$

(A2)

In this case, from the set of equations (A1), (A2) follows the equation for the $y$-component of the laser pulse magnetic field $B_{L} \left( {{\mathbf{r}},t} \right) = {\mathbf{e}}_{y} \cdot {\mathbf{B}}_{L} \left( {{\mathbf{r}},t} \right)$

$$ \frac{{\partial^{2} }}{{\partial t^{2} }}B_{L} \left( {{\mathbf{r}},t} \right) + \omega_{p}^{2} B_{L} \left( {{\mathbf{r}},t} \right) - c^{2} \Delta B_{L} \left( {{\mathbf{r}},t} \right) = 0. $$

(A3)

where $\omega_{p}$ is the plasma frequency, $\Delta$ is the Laplace operator.

To solve Eq. (A3), we will use the Fourier transform in time and coordinate

$$ \begin{aligned} B_{L} \left( {{\mathbf{r}},t} \right) & = \int\limits_{ - \infty }^{ + \infty } {\frac{{{\text{d}}\omega }}{2\pi }} \int\limits_{ - \infty }^{ + \infty } {\frac{{{\text{d}}k_{x} }}{2\pi }} \exp \left( { - i\omega t + ik_{x} x} \right)\\ & \quad \times B_{L} \left( {\omega ,k_{x} ,z} \right), \\ B_{L} \left( {\omega ,k_{x} ,z} \right) & = \int\limits_{ - \infty }^{ + \infty } {{\text{d}}t} \int\limits_{ - \infty }^{ + \infty } {{\text{d}}x} \exp \left( {i\omega t - ik_{x} x} \right)B_{L} \left( {{\mathbf{r}},t} \right). \\ \end{aligned} $$

(A4)

Then from (A3) follows the ordinary differential equation for the Fourier transform of the magnetic field

$$ \frac{{{\text{d}}^{2} }}{{{\text{d}}z^{2} }}B_{L} \left( {\omega ,k_{x} ,z} \right) + \left[ {\frac{{\omega^{2} }}{{c^{2} }}\varepsilon \left( \omega \right) - k_{x}^{2} } \right]B_{L} \left( {\omega ,k_{x} ,z} \right) = 0, $$

(A5)

whose solution, taking into account the continuity of the tangential components of the field $E_{L,x} \left( {\omega ,k_{x} ,z} \right)$ [see Eq. (4)] and $B_{L} \left( {\omega ,k_{x} ,z} \right)$, has the form

$$ \begin{aligned} &B_{L} \left( {\omega ,k_{x} ,z} \right) \\ &\quad = B_{0} \left( {\omega ,k_{x} } \right)\Bigg\{ \exp \left( {iz\sqrt {\frac{{\omega^{2} }}{{c^{2} }} - k_{x}^{2} } } \right) \\ & \qquad + R\left( {\omega ,k_{x} } \right)\exp \left( { - iz\sqrt {\frac{{\omega^{2} }}{{c^{2} }} - k_{x}^{2} } } \right) \Bigg\},\;\;z \le 0, \\ &B_{L} \left( {\omega ,k_{x} ,z} \right) \\ & \quad = B_{0} \left( {\omega ,k_{x} } \right)T\left( {\omega ,k_{x} } \right)\exp \left( {iz\sqrt {\frac{{\omega^{2} }}{{c^{2} }}\varepsilon \left( \omega \right) - k_{x}^{2} } } \right),\;\;z \ge 0, \\ \end{aligned} $$

(A6)

where the reflection coefficient $R\left( {\omega ,k_{x} } \right)$ and transmission coefficient $T\left( {\omega ,k_{x} } \right)$ for p-polarized radiation determined by formula (6). Equating the expression $B_{0} \left( {\omega ,k_{x} } \right)\exp \left\{ {iz\sqrt {\left( {{{\omega^{2} } \mathord{\left/ {\vphantom {{\omega^{2} } {c^{2} }}} \right. \kern-0pt} {c^{2} }}} \right) - k_{x}^{2} } } \right\}$ from (A6) to the Fourier transform of the incident pulse (1) near the boundary $z = 0$, for $B_{0} \left( {\omega ,k_{x} } \right)$ we obtain the following formula

$$ \begin{aligned} B_{0} \left( {\omega ,k_{x} } \right) & = B_{L}^{inc} \left( {\omega ,k_{x} ,z = 0} \right) \\ & = \frac{{E_{0L} \pi R\tau }}{\cos \alpha }\Bigg\{ \exp \left[ { - \frac{{\left( {\omega - \omega_{0} } \right)^{2} \tau^{2} }}{2}} \right] \\ & \qquad + \exp \left[ { - \frac{{\left( {\omega + \omega_{0} } \right)^{2} \tau^{2} }}{2}} \right] \Bigg\}\\ & \qquad \times \exp \left\{ { - \frac{{\left( {k_{x} - \left( {{\omega \mathord{\left/ {\vphantom {\omega c}} \right. \kern-0pt} c}} \right)\sin \alpha } \right)^{2} R^{2} }}{{2\cos^{2} \alpha }}} \right\}. \\ \end{aligned} $$

(A7)

Substituting formula (A7) into (A6) and using the inverse Fourier transform (A4), we obtain Eqs. (2), (3), (5) for the magnetic field of p-polarized laser radiation in vacuum and plasma.