Faraday rotation and polarization-modulated intense femtosecond laser pulses in a field-ionizing gaseous medium

In this paper we investigate the propagation of an intense linearly polarized laser through an ionizing gaseous medium in the presence of an axial strong magnetic field, addressing the modulation of laser polarization. Our simulation indicates that the laser polarization can be dramatically modulated and shows complicated temporal patterns (Lissajous curves). This striking phenomenon can be attributed to the collective movement of ionized electrons, in contrast to the traditional Faraday rotation in which the rotation angle of the laser polarization derived from the linear response of the medium is time independent. We take the weighted average of the rotation angle over the whole pulse duration and find that it explicitly relies on strong magnetic strength as well as the incident laser intensity. Our finding has implications in strong magnetic diagnosis, laser intensity calibration, and the generation of polarization-modulated light sources.

Figure 1

Diagram of the laser field and the initial velocity of an ionized electron. The laser field $\mathbf{E}\left(\tau \right)$ lies in the $xy$ plane at an angle $\theta$ to the $\mathbf{x}$ axis and the initial velocity ${\mathbf{V}}_{\perp }$ lies in the plane perpendicular to the polarization of the electric field at a random angle ${\varphi }_{\xi }$ to the $\xi$ axis.

Figure 2

Profile of the output electric field at a distance of $\xi =1\text{cm}$: (a) the relative delay of the $x$ and $y$ components of the output electric field in the time domain and (b) the synthesized laser pulse (the zero plane is colored light green to intuitionally show the rotation of the laser polarization). Here the incident laser intensity is $I_0=2\times {10}^{15}{\mathrm{W}/\mathrm{cm}}^2$ and the external magnetic strength $B_0=1500$ T.

Figure 3

Lissajous diagram for the output laser pulse. The averaged rotation angle is calculated as $\langle \theta \rangle \approx 16.{89}^{\circ }$ and $\delta \theta \approx {12}^{\circ }$ for the parameters $I_0=2.0\times {10}^{15}{\mathrm{W}/\mathrm{cm}}^2$ and $B_0=1500\text{T}$. The slope of the green dashed line is given by $k=tan\langle \theta \rangle$.

Figure 4

Electric peaks in the first quadrant and the corresponding rotation angle at the distance $\xi =1$ cm for the parameters $I_0=2.0\times {10}^{15}{\mathrm{W}/\mathrm{cm}}^2$ and $B_0=1500\text{T}$. The slope of the green dashed line is given by $\overline{\omega }=6.1283$ deg/fs.

Figure 5

Evolution of the average rotation angle and the distribution of ionized electron density during the laser's propagation. It is clear that the densities of ionized electrons almost overlap for three different magnetic strengths. Here the incident laser intensity is $I_0=2.0\times {10}^{15}{\mathrm{W}/\mathrm{cm}}^2$.

Figure 6

Rotating speeds of the laser polarized plane as functions of the distance $\xi$. Here the parameters are chosen as the incident laser intensity $I_0=2.0\times {10}^{15}{\mathrm{W}/\mathrm{cm}}^2$ and the magnetic strength $B_0=1200$, 1500, and 1800 T.

Figure 7

Average rotation angles as functions of different external strong magnetic fields for several incident laser intensities at the distance $\xi =1\text{cm}$. The inset shows the dependence of slope $R$ on the incident laser intensity $I_0$, which also demonstrates good linearity.

Figure 8

Values of $\alpha$ as functions of propagation distance $\xi$. The blue closed circles represent the sampling values calculated from the numerical simulations of the laser pulse.

Figure 9

Dependence of the rotating speeds of the laser polarized plane on the incident laser intensity and magnetic strength at the distance $\xi =1$ cm. The inset shows the slopes $C_{\overline{\omega }}$ depending on the incident laser intensity $I_0$. The spline fitting is represented by the blue line.

Figure 10

Evolution of different partitions of the laser energy during its propagation. The red solid line represents the evolution of the energy density of the $x$ component of laser field and the blue dashed line represents that of the $y$ component of the laser field. The parameters are $I_0=2.0\times {10}^{15}{\mathrm{W}/\mathrm{cm}}^2$ and $B_0=1500\text{T}$.

Figure 11

Dependence of energy transfer and conversion on the incident laser intensity and external magnetic strength. The laser intensity ranges between $1.6\times {10}^{15}$ and $2.2\times {10}^{15}$ W/${\mathrm{cm}}^2$ by a spacing of ${10}^{14}$ W/${\mathrm{cm}}^2$ and the magnetic strength varies from 100 to 2000 T by a spacing of 100 T, with the laser intensity $I_0=2.0\times {10}^{15}{\mathrm{W}/\mathrm{cm}}^2$. (a) $P_x$, (b)$P_y$, and (c) $P_{\mathrm{loss}}$.