This post covers work from 11/01-01/09

Paper Updates

In november, my first paper got published in Contributions to Plasma Physics! Additionally, I sent in the second synthetic data paper to the New Journal of Physics (which got desk rejected) and then to APL Machine Learning. Nathaniel also submitted the Wright-Patt paper to APL Machine Learning for one of their special issues.

Additional Titan Sims

I have also updated the Titan Paper with my simulation efforts which I feel are pretty complete at this time. Some new plots can compare teh proton (left) and electron (right) energy spectra

and conversion efficiency

which shows clearly that the double pulse energies proton were enhanced the most, but also interestingly shows that the overall conversion efficiency of laser energy into proton energy was lower than the pre-plasma targets. We also see significantly enhanced electron energies and conversion efficiencies with the preplasma present which makes sense. Overall, the simulations seem to make sense.

I also added a table that includes simulations from half and quarter of the energy (without preplasma). The same qualitative trends are observed but the maximum energy and conversion eficiencies are lowered in a manner approximately proportional to the energy.

I also did simulations with aluminum which showed overall higher energies and conversion efficiencies than gold, but a smaller enhancement from single to double pulse.

Theory behind eTNSA

Since December, I have been trying to work on a theory behind the enhanced TNSA process. As a starting point, I wanted to see how the constructive interference of two pulses leads to higher fields when the incident angles of the two pulses are not necessarily the same. First, I wrote down the equations for the electric field of a gaussian beam. Then I did a coordinate rotation transformation to find the equations of a gaussian beam when incident at a particular angle. Then I considered the double pulse and single pulse cases only looking at the overall envelope function for the electric field (i.e. ignoring the oscillating part in time and space). My overall findings were the following:

If $\theta_1$ and $\theta_2$ are the incident angles from two gaussian pulses with each half the energy of a single pulse of one of the two angles, then we can expect the enhancement of the electric field in the direction normal to the target (at the location of the target front $x = 0$) to be

\begin{equation} \alpha = \frac{1}{\sqrt{2}} ( 1 + \frac{\sin \theta_2}{\sin \theta_1} ) \end{equation}

where $\theta_2$ is the smaller of the two angles. When $\theta_1 = \theta_2$ we get perfect constructive interference, but we will get less constructive interference when the angles are not equal. Implicit in this analysis is that only the electric fields in the direction normal to the target are important. Furthermore, the enhancement of the electric field will go up to a maximum of $\sqrt{2}$ at a distance $x^*$ in front of the target given by

\begin{equation} x^* = \frac{w_0 \ln \lvert \sin \theta_1 / \sin \theta_2 \rvert}{2 \sin(\theta_1 + \theta_2) \sin \lvert \theta_2 - \theta_1 \rvert} \end{equation}

This is typically on the order of the beam waist $w_0$. However, this shouldn’t be relevant because the preplasma expansion (i.e. location of critical density of expanded target) should be much smaller than the beam waist. Below is a graphic depicted these results for angles of 20 and 40

On the left, we see the electric field (only along the line of sight in the direction normal to the target and passing through the center) in the normal direction from the double pulse, single pulse at 20 deg, and single pulse at 40 deg. The x-axis is $r = \sqrt{x^2 + y^2} = \sqrt{2} x$ because we are on the diagonal line $y = x$. Due to the gaussian-like drop off of the electric field in radial direction of the beam itself, we see that the 20 degree pulse decays slower because its incidence angle is closer to the normal. The 20 degree reaches a higher electric field because a p-polarized laser pulse will have a higher electric field in the normal direction if it is at a higher incidence angle. On the right side, the enhancement (generalized for $r \neq 0$) is plotted which peaks at $x = x^*$. I have a orange dashed line called theory which completely ignores the effect of the evolving beam waist w(z) and temporal pulse profile which turn out to not matter.

By the way, below is a picture of the gaussian beam equations I have (on the right) in comparison to an actual simulation ran (on the left)

which look almost identical. The simulation seems to have slightly higher electric fields in the interference region, but this is due to reflections from the target which I have not accounted for in my equations. I include this as proof that the gaussian beam equations I am using are actually correct. I am not writing down the full equations here because that would take a lot of time, but I looked at the wikipedia article for the gaussian beam.

From here, I am not actually sure where to go. To me, the differences in the total interference are actually less than what Ferri had predicted. But somehow, this leads to enhancement in the TNSA fields and