In December, I worked on using a LWFA paper’s results on propagation of lasers in underdense plasmas to help explain the asymmetry in the maximum proton energy spectrum with respect to negative and positive focal distance. I also sent in an application for student support for the NIF conference in Jan/Feb 2024. Additionally, I began working on 2D sims to try to confirm if the math behind this asymmetry makes sense.

Explanation of Max Proton Energy Spectrum

Last Time, I worked on trying to add a way to account for the asymmetry seen in the maximum proton energy spectra seen in Loughran et al. and Morrison et al..

Both of these paper show that the highest electron energies are produced when the target is at the peak focus of the laser, but there is some suppression of the proton energy spectra at peak focus. Loughran et. al. gives the following explanation:

strong suppression of proton flux at the highest intensity may indicate that the acceleration process is further compromised at the highest laser intensities by the contrast levels of our laser, with the prepulses and amplified spontaneous emission (ASE) causing adverse pre-heating of the target.

Morrison et. al. also has an explanation

If the pre-pulse perturbs the back surface of the target, the TNSA process will be stifled. Higher pre-plasma conditions that shut down the TNSA process also would contribute to more energetic electron acceleration. … lower peak proton energy … can be attributed to the pre-pulse at least partially punching through the target.

I have been told that pulse contrast improvements have been made to the laser at WPAFB, which should mitigate this dip in the maximum proton energy at peak focus $z=0$.

Additionally, there appears to be some asymmetry between negative (target is in front of the peak focus) focal length and positive (target is behind the peak focus) focal length $z$. In Loughran et. al. the peaks are almost equal, with just a small increase in max proton energy at negative $z$. This is a less understood effect and at least is not explained in these two aformentioned papers.

Asymmetry explained through Decker et. al.

Last time, I started obtained an expression for what I called the “etching distance”

\begin{equation} L_\text{etch} = \int v_\text{etch} dt = \int \frac{n_\text{max} \exp(-x / (c_s t_0)) e^2 c}{m_e \epsilon_0 \omega^2} dt \end{equation}

The interpretation of $L_\text{etch}$ is that it is the physical distance of the laser-pulse that gets diminished. The laser pulse has a physical distance of $c \tau_\text{FWHM}$. As a result, $L_\text{etch}/c \tau_{FWHM}$ would be the fraction of the laser energy that gets depleted. To proceed, we need to do two things:

1. Set $dx = c dt$ so that we can integrate with respect to position and not time.
2. Set the bounds of the integral to $x_0 = c_s t_0 ( \ln(n_\text{max}) - \ln(n_\text{crit}) )$ and $x_f = 5.517 c_s t_0$.

Additionally, we need to account for the asymmetry between the situation where the target is at negative $z$ versus positive $z$. We choose to do this by saying that there is a minimum value of $a_0$ for which we start to see the laser depletion effects. We choose a cutoff of $a_0 = 0.7$ (which is around $10^{18}$ W/cm2). This may modify ($x_0$, $x_f$) to encompass a smaller range (a, b). If $z^+$ and $z^-$ are the focal distances where $a_0 = 0.7$, then the conditions on the bounds $a$ and $b$ are

if x_f' < z- < x_0': b' = z-

if x_f' < z+ < x_0': a' = z+

if z- < x_f' < x_0' < z+: no decay

if z+ < x_f' or x_0' < z+: no decay

if x_f < x_0: no decay

And if none of these criteria are satisfied, $a = x_0$ and $b = x_f$. After these considerations, the final etching distance would be given by

\begin{equation} L_\text{etch} = \frac{e^2 n_\text{max} c_s t_0}{\epsilon_0 m_e \omega^2} (\exp(-a/(c_s t_0)) - \exp(-b/(c_s t_0)) ) \end{equation}

and always equal to 0 in the no decay cases leading to a decay factor of $1 - L_\text{etch} / (c \tau_\text{FWHM})$ that will be multiplied by the main pulse intensity. As an example, here is what this modification looks like for $I = 10^{19}$ W/cm2, $d = 0.5 \mu m$ and $a_0 = 0.7$ to be the cutoff

2D Simulations

I ran a simulation from $0 \rightarrow 130$ fs and x: $-20 \mu m \rightarrow 20 \mu m$ y: $-5 \mu m \rightarrow 5 \mu m$ of a laser with peak intensity $I_0 = 1 \times 10^{19}$ W/cm2 ($E_0 = 8.68 \times 10^{12}$ V/m) interacting with a uniform plasma of protons and electrons of density $1.04 \times 10^{20}$ cm$^{-3}$. The laser had a beam waist of $w_0 = 1.8 \mu m$ and wavelength of $\lambda = 0.80 \mu m$ which results in a Rayleigh Length $z_R = 12.72 \mu m$. I used 25 particles per cell for both protons and electrons and set the initial temperature of both to $1eV$. I just shot one laser pulse of $\tau = 40$ fs FWHM.

Pulse Energy

First, I ran a simulation without particles to just make sure that the theoretical estimate of the pulse energy is accurate with the simulation’s output of energy in fields

and it is. I did this a while ago, but I don’t think I wrote it anywhere, so I developed an expression for the energy in a gaussian laser pulse that will be dependent on whether the laser is in 2D or 3D.

\[\begin{equation} E = \begin{cases} (\pi/2) I_0 w_0^2 \tau & \text{ if 3D} \\ \sqrt{\pi/2} I_0 w_0 \tau & \text{ if 2D} \end{cases} \end{equation}\]

Additionally, as the pulse moves into and out of the simulation boundaries, the energy will increase and decrease. We account for this by considering the following modifications for the energy $E$ (where $L$ is the length of the simulation box that the pulse travels)

\[\begin{equation} E_\text{pulse}(t) = \begin{cases} 0 & \text{ if $t < 0$} \\ E( t/(2 \tau) - \sin(\pi t / \tau) / (2 \pi) ) & \text { if $0 < t < 2 \tau$} \\ E & \text{ if $2 \tau < t < L/c$} \\ E (1 - (t - L/c)/(2 \tau) + \sin(\pi (t - L/c)/\tau)/(2 \pi) ) & \text{ if $L/c < t < L/c + 2 \tau$} \\ 0 & \text{ if $t > L/c + 2 \tau$} \end{cases} \end{equation}\]

Another thing to think about simulation wise is how the electric field amplitude gets modified at the boundary because it is different for 2D and 3D. $E(z) = E_0 \sqrt{w_0/w(z)}$ in 2D and $E(z) = E_0 (w_0 / w(z))$ in 3D where $w(z) = w_0 \sqrt{1 + (z/z_R)^2}$ is the beam-waist as a function of the on-axis distance $z$.

Etching Results

The simulation with particles ran until $L/c \approx 130$ fs and we mainly focus on the region $2 \tau \rightarrow L/c$ or $80 \rightarrow 130$ fs becuase that is when the full pulse is traveling through the simulation. First, we look at the field energy.

Here we see that the $\color{red}{\text{simulation field energy}}$ is slightly lower than the $\color{gray}{\text{theoretical energy}}$ in the laser pulse shown in the dashed line. This is to be expected as some of the energy should transfer to the protons and electrons. In the next plot, I plot the percent reduction in energy over time of the simulation field energy as compared with either the $\color{red}{\text{simulation field energy}}$ or $\color{gray}{\text{theoretical energy}}$.

From this plot, we can clearly see a reduction in the energy stored in the pulse, that roughly linearly increases in time (especially after 80fs which is when the full pulse is immersed in the plasma). From our etching velocity calculations, we can expect around a $50 \%$ reduction in $100 \mu m$ or $1.5 \%$ reduction in $3 \mu m$ (equivalent to $1.5 \%$ reduction in $10 fs$). From the plot, wee see a max slope of around $\color{red}{0.5 \%}$ or $\color{gray}{1 \%}$ per 10 fs, so it seems that our simulation underpredicts the amount of energy loss by a factor of 1.5-3.

Another thing we could look at is the amount of the pulse that gets “etched”. The etching velocity is $0.06 c$ which corresponds to $0.18 \mu m$ per $10 fs$. The pulse itself has length $c (2\tau) = 24 \mu m$ and in the $50$ fs between $80$ and $130$ fs, we would etch the pulse less than $1 \mu m$. This amount would be difficult to notice, but I plotted the sum of $E_x^2 + E_y^2$ at a particular $x$ over all $y$ as a function of $x$ and visually inspected the pulses shown below

I measured the left pulse to be $22.96 \mu m$ and the right pulse to be $22.59 \mu m$ which results in $0.37 \mu m$ over the course of $50$ fs which falls within that 1.5-3 times less than expected amount of etching. I can try to re-run a simulation with a larger distance the pulse travels, but that would require making the simulation box bigger and we would have to run the simulation for longer. The Debye length $\lambda_D = 0.729$ nm is already being under-resolved by $\Delta x = 4$ nm, so increasing the grid size may not be a good option.

Update on ML

Jack and I have updated the Fuchs model (v4.2) to include this asymmetry modification. The good thing about this model is that the dip combined with the asymmetry is enough to give the polynomial regression a somewhat harder time at making a good model fit on the data. However, increasing the degree of the polynomial from 5 to 7 or 9 makes the polynomial regression have a visually better fit and usually have a lower percentage error than the trained NN model. Since we have so few input parameters, the 7 or 9 degree polynomial can still fit relatively quickly on a matter of seconds running with CuML on GPU.

For example (selecting for minimum thickness), this is the NN prediction

Here is the degree 5 polynomial

degree 7

and degree 9

All of the models are able to make good predictions when the dip goes away (e.g. for larger target thickness). Below is the degree 5 polynomial plotting predictions for maximum thickness

where the x-axis is the focal distance $z$ and the y-axis is the peak proton energy. Moving forward, Jack and I think that focusing on the small data regime would be more instructive using these pre-pulse modifications and an appendix to the paper could cover the big data stuff with training a neural network on 1 million points. We could gear this paper as more of a how-to for when we actually have real experimental data. Another thing we should do is increase the noise and explore non-gaussian sources of noise that wouldn’t make the energy become negative if the noise becomes too large.

Additionally, I talked with Nathaniel and he said that WPAFB would probably have the laser set up by January, but the real-time control with EPIX would likely not be ready. As a result, we would really only be able to do ML with the dataset after data collection and not during the run.

Things to Do

• Re-download newer version of rapids/pytorch that Jack is using
• Experiment with adding noise on the order of $60 \%$ which is a more realistic estimate that Brady mentioned to us. See if models can still have any use with this.
• Think of things that we could do for the experiment in January.