Relativistic self-focusing and self-compression of Gaussian laser pulses through the magnetized plasma with density ramp-up

The 3th Iranian conference on engineering and physics of plasma, Tabriz, Iran

This article has been extracted from the Physics of Plasma article to present in the domestic conference.

In this work, the Spatio-temporal evolution of Gaussian laser pulses propagated through the plasma is investigated in the presence of the axial external magnetic field. The coupled equations of self-focusing and self-compression are obtained by the paraxial approximation. The effect of axial magnetic field on simultaneously self-focusing and self-compression of a laser pulse is studied for homogeneous and ramp-up plasmas. Results show that using both axial magnetic field and ramp structure result in producing pulses with the less spot size and compression length.

Chirped Pulse Amplification (CPA) is a common method to amplify the laser pulse that uses dielectric gratings. These dielectrics cannot tolerate high intensities and perform below the 30 fs. Because of these limitations, a new method needs to be used to achieve higher intensities and shorter pulses. The plasma medium is proposed as an alternative to amplify laser pulses. With the advent of ultra-high intensity laser pulses, the relativistic self-compression (RSC) of laser pulses in the plasma is proposed as an amplification method. The high electric field associated with the propagation of high power laser pulses leads to a quiver motion of electrons on the order of the speed of light in a vacuum. This caused a significant increase in the mass of electrons and a consequent increase in the dielectric constant of the plasma, which leads to the dependence of the refractive index on the intensity. Therefore, the relativistic mass variation during the laser-plasma interaction is the primary source of nonlinearity. The transverse gradient of the nonlinear refractive index is responsible for relativistic self-focusing (RSF), while the longitudinal gradient of the refractive index leads to the RSC. In the relativistic self-phase modulation (RSPM), the intensity dependence of the refractive index in the axial dimension produces the frequency chirp. Therefore, the leading edge of the pulse shifts to lower frequencies while the trailing edge shifts to higher frequencies. Since the plasma has the anomalous dispersion property, the higher frequencies (back of the pulse) move faster than, the lower frequencies (front of the pulse), and consequently, the pulse would be compressed while it propagates through the plasma.

In this regard, we aimed to investigate both effects of the external axial magnetic field and plasma inhomogeneity on simultaneously self-focusing and self-compression of a laser pulse through the plasma.

Consider a collisionless magnetized plasma with the static magnetic field \(B_0z\) and the density ramp-up profile which is presented by the following equation:

\begin{equation}\label{equ1}
n(z)=n_0(1+d \times z)
\end{equation}

where n₀ is the plasma density in z=0, and d is an adjustable constant which represents the slope of the ramp.

The electrical field is in the form:

\begin{equation}\label{2}
E(r,z,t)=A(r,z,t)exp\left[-i\left(\omega t-\int_0^z
k(z)dz\right)\right],
\end{equation}

\begin{equation}\label{11}
A(r,0,t)=A_0exp\left(-\frac{r^2}{2r_0^2}\right)exp\left(-\frac{\tau^2}{2\tau_0^2}\right),
\end{equation}

Here \(r_0\), \(\tau_0\), \(\omega\), \(k=\sqrt{\varepsilon_0}\omega/c\) are spot size of pulse, initial temporal lenght of pulse, laser frequency, and wave number repectively. Dielecrice function of Plasma in the presence of the lonfitunal Magnetic field is relativistic:

\begin{equation}\label{3}
\varepsilon=1-\frac{\omega^2_p}{\gamma\omega\left(\omega-\frac{\omega_c}{\gamma}\right)},
\end{equation}

\(\omega_c=eB_0/(mc)\) is the cyclotronic frequency and \(\gamma\) is relativistic factor:

\begin{equation}\label{4}
\gamma=\left[1+A^2\left(1-\frac{\omega_c}{\gamma\omega}\right)\right]^{1/2}.
\end{equation} 

For \(\omega_c/\gamma\omega<1\), the equation \ref{4} can be solved with the Repetition method. Therefore, by applying \(\gamma=\sqrt{1+A^2}\) for \(\omega_c=0\) value, the right side of equation \ref{4} will be: