I did some work to justify the etching velocity addition to our fuchs model with mixed results. I made some updates to the Fuchs Spectrum Data Generation Notebook and uploaded it to GitHub: namely changing the noise to be log-gaussian. I also started working on the Titan Sims that Nathaniel had been running for the upcoming LaserNet experiment in February.

Etching Velocity Simulations Continued

Last time, I ran some etching velocity simulations but didn’t get conclusive results. This time, I ran a simulation with 2.5x bigger grid and I made the boundary conditions periodic in the y direction. I also implemented the absorption energy bug fix (t_profile = sin(pi*time/(80*femto))*(time lt 80*femto)) that we found a while ago to make absorption of laser energy through the open boundary work as intended. Below, I plot a graph comparing field energies

In red, I’ve plotted the field energy divided by the total energy in the simulation over time as a percentage (and then took 100 minus that to get the percent reduction). By my calculations I’ve done previously, I’ve estimated that $50\%$ of the energy of the pulse should be diminished in 330fs. From this graph, it looks like the energy reduction doesn’t really start until around 60 fs, which is something I’ve accounted for in the dashed lines. The green dashed line assumes a constant energy loss rate to make 50\% of the pulse deplete in 330fs. The paper, to me, assumes a constant etching velocity of the front of the pulse, which would be plotted in blue. Since the pulse has a sine-squared profile, very little of the pulse is depleted first, and the amount of depletion increases over time (until half the pulse is depleted). The simulation results are not completely off from the blue dashed line.

Next, I wanted to compare the electric field profiles. First, I looked at the magnitude of the electric field $\vert E \vert$ evaluated on the laser axis.

Here, we can see that the electric fields with particles can get higher than the electric fields without particles due to the electrostatic fields. The electric fields have a lot of spikes, so I thought it would be more instructive to do a rolling average of the fields (over $\pm 1 \mu m$) to get the following plot

Here, it becomes more clear the the particles appear to slow down the pulse, but the etching nature is not that clear. I can even plot the difference in the electric field without and with particles which would be positive if the pulse is less intense than expected and negative if it is more intense than expected.

Most of the positive portion of the blue curve is on the right half of the pulse, but we also have a lot in the middle of the pulse as well. It seems that field energy is taken out fo the pulse, but not just on the front edge.

Adding Pre-plasma to LaserNet Sims

Nathaniel has been running some sims comparing aluminum/gold targets with single or double pulses to compare the TNSA protons. I have downloaded EPOCH 4.18 and at first tried to re-run his simulations, but second, going to try to run simulations with an exponential scale length pre-plasma to see what kind of effect that has.

Overview of Simulations

The target itself is either made of aluminum or gold and is 15 microns thick with a length of 36 microns. It is angled at 45 degrees (top left to bottom right) and additionally has a thin layer of protons of 0.5 microns on the rear side of the target to enhance the TNSA process. The grid itself is 28x28 microns. The laser has a wavelength of 1.053 microns with a peak energy of 2.474 Joules (per pulse) but across the whole 700fs shot there will be around 100J of energy delivered to the target. The target density is 6.02e22 and 5.9e22 ions per cm3 for Aluminum and Gold.

Modified Pre-plasma Density and Scale Length

We want to impose a pre-plasma that exponentially decays from the front of the target.

\begin{equation} n(d) = n_0 \exp(-d/L) \end{equation}

where $L \sim 1 \mu m$ is some pre-plasma scale length that we choose, $d$ is the distance away from the target (which itself has thickness $\Delta d$), and $n_0$ is the initial density of the target. After expansion, the density $n_0$ will decrease slightly to $n_0^* < n_0$ to account for the exponential tail of plasma in front of the target. If $n_\text{cut} $ is the number density that we’re cutting off at, we wish to find the distance $d^* $ that corresponds to this cutoff and the new density plateau $n_0^*$.

We start with the following two equations

\begin{align} n_\text{cut} &= n_0^* \exp(-d^*/L) \\ n_0 \Delta d &= n_0^* \Delta d + \int_0^{d^*} n_0^* \exp(-x/L) dx \end{align}

which define the density cutoff $n_\text{cut}$ and conservation of particle number pre and post expansion. We can solve this simultaneous set of equations to get

\begin{equation} n_0^* = \frac{n_0 \Delta d + n_\text{cut} L}{\Delta d + L} \end{equation}

and

\begin{equation} d^* = L \ln(\frac{n_0^*}{n_\text{cut}}) = L \ln \left( \frac{n_0 \Delta d + n_\text{cut} L}{n_\text{cut} (\Delta d + L)} \right) \end{equation}

Log-Normal Noise Distribution

The log-normal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed (see here for details). The advantage of the log-normal distribution over the normal didstribution is that since $\ln(x)$ is normally distributed, the variable itself will involve an exponential, which cannot be negative. There are other functions with this property (see functions supported on semi-infinite intervals), but the log-normal is the most similar to the normal distribution. When the noise level is low, the log-normal and normal even look similar. Here is a comparison of the two distributions being applied to varying levels of noise added to a test function

Wikipedia states that in order to produce a distribution with desired mean $\mu_X$ and variance $\sigma_X^2$, one uses

\[\begin{align} \mu &= \ln \left( \frac{\mu_X^2}{\sqrt{\mu_X^2 + \sigma_X^2}} \right) \\ \sigma^2 &= \ln(1 + \frac{\sigma_X^2}{\mu_X^2}) \end{align}\]

We can define our desired noise level as $\alpha = \sigma_X/\mu_X$. For example: $\alpha=0.1$ would correspond to $10\%$ noise. Doing this redefines our transformed $\mu$ and $\sigma$ as

\[\begin{align} \mu &= \ln \left( \frac{\mu_X}{\sqrt{1 + \alpha^2}} \right) \\ \sigma &= \sqrt{\ln(1 + \alpha^2)} \end{align}\]

which is what I feed into the log-normal distribution to generate the noisy data.

Things to Do

• Preprare Presentation for Annual Review in January
• Figure out pre-plasma expansion from Le Pape (OL 2009) and Simpson (PoP 2021) papers and see if they match up to around 1/2 micron scale length
• Look through code Nathaniel sent me to plot energy distribution of simulation
• Think of how noise level for proton spectrum could be modified: don’t think it makes sense to directly apply to bins
• Work on (or try to find) some sort of diagnostic that keeps track of the protons that leave the simulation and what their energies were
• Think of optimization routine and how we might pick a new region of parameter space to sample