Project 3b: Wave propagation in magnetized plasmas:

The following several cells will plot relevant dispersion relations for magnetized plasmas near to what we will be simulating.

These are for waves in a plasma that has an external magnetic field such that  $\Omega_{ce} = 0.5 \omega_{pe}$ and  $\Omega_{ce} = 2.0 \omega_{pe}$.

The goal of this project is to study what happens when an electromagnetic wave is sent into a magnetized plasma in which the wave moves across the magnetic field.

You will simulate plasmas in which each plasma electron is initialized with positions (only in  $x$ or what we call  $x_1$) such that the plasma has a nonuniform density profile. The spacing between the electrons is proportional to the inverse of the density. Each electron is also initialized with velocities ( $v_1$,  $v_2$,  $v_3$) = ( $0.005c$,  $0.005c$,  $0.005c$) or momentum ( $mv_1$,  $mv_2$,  $mv_3$) from a Maxwellian in each direction. The particles then begin to move in the self-consistent fields that their current and charge density produce. The peak density of the plasma in each case corresponds to  $\omega_{p0}$=1.

For all cases, a constant magnetic field exists in the  $x_3$ direction. It will be given a value corresponding to either  $\omega_c/\omega_{p0}=0.5$ or  $\omega_c/\omega_{p0}=2.0$.

At  $t=0$, an electromagnetic (EM) wave is launched by an antenna from the left hand boundary along the  $\hat x_1$ direction. The pulse grows in amplitude over  $t=50\omega_{p0}^{-1}$ and then remains constant. In some cases the EM wave is polarized in the  $E_2$ direction while in other cases it is polarized in the  $E_2$ and  $E_3$ directions at an angle of  $\pi/2$.

• The length of the simulation window is  $500 c/\omega_p$.
• There is no plasma between  $x_1=0$ and  $x_1=40 c/\omega_0$.
• The plasma density rises linearly from  $x_1=40$ to  $x_1=400 c/\omega_0$ to a value  $n_0$ and the corresponding plasma frequency is  $\omega_{p0}$.
• The density then remains constant at  $n_0$ from  $x_1=400$ to  $x_1=500 c/\omega_0$.
• The simulation will run for a time 1100  $1/\omega_{p0}$.
• The simulation uses 50,000 particles.

You will launch a wave with its electric field pointing in the  $x_2$ direction with  $\omega_0=0.7 \omega_{p0}$. The peak value of the electric field will be  $E_0=0.007 mc\omega_0/e$.

Run the simulation below:

Plot the density:

You will next plot  $E_2(t, x_1)$. The color corresponds to the amplitude of the EM wave. In addition, lines corresonding to the value of  $x_1$ where the incoming EM wave's frequency  $\omega_0$ equals  $\omega_R$,  $\omega_H$,  $\omega_p$, and  $\omega_L$ are also shown.

• Did the wave start out with a frequency above or below  $\omega_R$?
• The wave is reflected. How can you tell?
• At which cutoff is it reflected?

Next, let's zoom in a bit to see the behavior near the turning point:

• What is the phase and group velocity when it is reflected?
• Is this what you expect from the dispersion relation?

This case is exactly the same as the first with the exception of the magnetic field amplitude.

• The length of the simulation window is  $500 c/\omega_p$.
• There is no plasma between  $x_1=0$ and  $x_1=40 c/\omega_0$.
• The plasma density rises linearly from  $x_1=40$ to  $x_1=400 c/\omega_0$ to a value  $n_0$ and the corresponding plasma frequency is  $\omega_{p0}$.
• The density then remains constant at  $n_0$ from  $x_1=400$ to  $x_1=500 c/\omega_0$.
• The simulation will run for a time 1100  $1/\omega_{p0}$.
• The simulation uses 50,000 particles.

You will launch a wave with its electric field pointing in the  $x_2$ direction with  $\omega_0=0.7 \omega_{p0}$. The peak value of the electric field will be  $E_0=0.007 mc\omega_0/e$.

Run the simulation below:

Plot the density:

Here we plot the wave pattern as it propagates up the density gradient:

• Did the wave start out with a frequency above or below  $\omega_R$ when the density is 0?
• The wave is now transmitted. How can you tell?
• Can you explain why the wave does not get reflected in this case? (Hint: What is  $\omega_L$ at the peak density?)

The third case is the same as the second with the exception of the laser frequency and amplitude (the amplitude does not change the results).

• The length of the simulation window is  $500 c/\omega_p$.
• There is no plasma between  $x_1=0$ and  $x_1=40 c/\omega_0$.
• The plasma density rises linearly from  $x_1=40$ to  $x_1=400 c/\omega_0$ to a value  $n_0$ and the corresponding plasma frequency is  $\omega_{p0}$.
• The density then remains constant at  $n_0$ from  $x_1=400$ to  $x_1=500 c/\omega_0$.
• The simulation will run for a time 1100  $1/\omega_{p0}$.
• The simulation uses 50,000 particles.

You will launch a wave with its electric field pointing in the  $x_2$ direction with  $\omega_0=0.3 \omega_{p0}$. The peak value of the electric field will be  $E_0=0.003 mc\omega_0/e$.

Run the simulation below:

Plot the density:

Next you will plot  $E_2(t, x_1)$.

• Why is the wave now reflected?
• Why is the wave now reflected at the density for which  $\omega_0$= $\omega_L$?
• What is the phase and group velocity when it is reflected?

In the last case, the density will rise sharply from 0, then stay constant at  $\omega_p$= $\omega_{p0}$, and then fall sharply back to zero:

• The length of the simulation window is  $100 c/\omega_p$.
• There is no plasma between  $x_1=0$ and  $x_1=10 c/\omega_0$.
• The plasma density rises sharply at  $x_1=100$ to a value  $n_0$ and the corresponding plasma frequency is  $\omega_{p0}$.
• The density then remains constant at  $n_0$ from  $x_1=10$ to  $x_1=90 c/\omega_0$.
• The density decreases sharply to 0 at  $x_1=90$ and remains 0 until to  $x_1=100 c/\omega_0$.
• The simulation will run for a time 450  $1/\omega_{p0}$.
• The simulation uses 50,000 particles.

In this case you will launch a wave with  $\omega_0 = 2.2 \omega_{p0}$ and its electric field will have equal values for Ey and Ez ( $E_2$ and  $E_3$). The peak value of the electric field will be  $E_0=0.022 mc\omega_0/e$.

Run the simulation below:

Plot the density:

Next you will plot  $E_2(t, x_1)$ and  $E_3(t,x_1)$. First let's look at the entire time window:

• What type of wave does the  $E_2$ correspond to?
• What type of wave does the  $E_3$ correspond to?

Although they look similar, we show next that these two "waves" have slightly different group and phase velocities. This can be seen by looking at the data in a small time window. The two different waves arrive at the wall ( $x_1$=100) at slightly different times.

Next we plot dots of the values of the electric field on an  $E_2$ vs.  $E_3$ plot. Each dot corresponds to a different time. The colors correspond to different values of  $x_1$. Blue is for an  $x_1$ in the vacuum before the EM wave enters the plasma, red is for an  $x_1$ in the plasma, and green is for an  $x_1$ in the vacuum after the wave exits the plasma.

• the blue line is the polarization as a function of time at  $x_1=2.0$ 
• the red line is the polarization as a function of time at  $x_1=48.0$ 
• the green line is the polarization as a function of time at  $x_1=96.0$ 

Can you explain this plot? The blue dots should form a straight line, but some of the signal is reflected which makes it more complicated. Understanding this result will be helpful for the final exam.