This post covers work from 01/09-03/07.

Synthetic Data Paper

We received news of the synthetic data paper being rejected on Wednesday (03/05), we are planning on taking some of the reviewer feedback and submitting it to another journal soon within the next few weeks.

Some more titan simulation plots

I created two additional plots. One shows a stackplot of the energy in the simulation

where the dashed line shows the expected energy injected into the simulation from a simulation where the target is not there (and is approximately the expected energy from the gaussian beam). The top row is single pulse and left column is no preplasma. I converted the 2D energy to the 3D energy by multiplying by a factor of $w_0 \sqrt{2/\pi}$ which comes from the ideal gaussian beam formula. Then, I created a plot showing the maximum proton energy over time in the simulation:

This shows the proton energy peaking between 1600-1800fs into the simulation for all 4 cases.

Double Pulse Theory

I tried to come up with a theory behind the double pulse phenomenon by considering the resonance absorption model in Kruer’s ``Physics of Laser Plasma Interactions’’ (Chapter 4.2). I also created a jupyter notebook that is available on my github repository. Kruer’s model describes an obliquely incident p-polarized plane wave incident on a target located between x=0 and x=L. This target has a linear density gradient meaning that $n(x) = n_c (x/L)$ (where $n = 0$ when $x < 0$ and $n=n_c$ when $x > L$). He then considers what the electric field would be at $x = L$ (the critical density surface). It turns out that the true reflection point is given by $x_c = L \cos(\theta)^2$ which occurs before the critical density surface for oblique incidences. Thus, the value of the electric field can be determined from two factors

1. The “swelling” of the electric field while traveling through the under-dense region (from $x=0$ to $x=L \cos(\theta)^2$) which is mathematically described by an Airy function.
2. The exponential decay of the electric field between $x=L \cos(\theta)^2$ and $x=L$.

Taking into account both of these considerations, Kruer finds the electric field to be given by $E_d$ at the resonance region $x=L$

\begin{equation} E_d \equiv \frac{E_\text{FS}}{\sqrt{2 \pi \omega L / c}} \phi(\tau) \end{equation}

where $\omega$ is the laser frequency, $c$ is the speed of light, and $E_\text{FS}$ is the free space solution for the electric field. $\phi(\tau)$ is given by

\begin{equation} \phi(\tau) = 2.3 \tau \exp(-2 \tau^3 / 3) \end{equation}

where $\tau \equiv (\omega L / c)^{1/3} \sin(\theta)$. This is the same resonance absorption formula that is illustrated in my thesis where Kruer notes that $\phi(\tau)^2 / 2$ approximates the absorption fraction. The problem with this formula for the electric field is that there is a factor of $L^{1/6}$ in the denomenator (after combining the $\sqrt{L}$ with $\tau$’s L dependence) which inflates the electric field when $L=0$. This makes it so that we will always prefer $L$ to be as small as possible and prefer an angle $\theta$ as high as possible. Additionally, this leads to the condition where we want the angles to always match.

I tried some alternative ways of approaching this problem: I incorporated the gaussian dependence of the laser beam but this actually has no effect on the formulas (mainly because $w_0 \gg \lambda$). I only found one way to proceed and that is by dropping the $L$ dependence (outside of $\tau$). As a result, I am simply setting $E \propto \phi(\tau)$. Using this relationship, I can now get more a more interesting electric field prediction as a function of two incident angles $\theta_1$ and $\theta_2$. Below is an example plot that compares the electric field of a combined double pulse (in arbitrary units) with that of the max of either one of the single pulses (with same total energy).

There are some general features that I can point out. First, on the graph on the right, we have the (maximum possible) enhancement that the double pulse can introduce. Just like before, when the angles are equal, the enhancement is $\sqrt{2}$ for all scale lengths $L$, but when the angles are not equal, it is only greater than 1 in some region between $x=x_i$ and $x=x_f$. The locations of $x_i$ and $x_f$ are given by the following formula

\begin{equation} x = \frac{3c}{2 \omega (\sin^3(\theta_1) - \sin^3(\theta_2))} \left( \log(\sin(\theta_1)/\sin(\theta_2)) \pm \log(\sqrt{2} \alpha - 1) \right) \end{equation}

where $\alpha = 1$ indicates the critical points corresponding to an enhancement factor of one. The peak enhancement is found at the singular point where $\alpha=\sqrt{2}$ where both solutions reduce to the same number. So, if we are not allowed to pick $\theta_1 = \theta_2$ due to experimental challenges, we can now analyze which angles would optimize enhancement if we know that the scale length $L$ takes on a certain value. By defining $\theta_1 > \theta_2$, I can calculate the enhancement region (L where $\alpha > 1$) for a given $\theta_1$ and angle difference $\Delta \theta = \theta_1 - \theta_2$). This is shown below for three different angular separations:

which shows that larger scale lengths are allowed for smaller angular separations. Furthermore, I can make a plot of the electric field as a function of the two angles for two different scale lengths $L$

which shows that longer scale lengths prefer both angles to be smaller and shorter scale lengths prefer both angles to be larger.

I think that main conclusions this approach can support is that to get a larger enhancement for a given angular separation $\Delta \theta$, we want the scale length to be small. But a scale length can be beneficial which is on the order of the laser wavelength or a little bit smaller.

Thinking about Double Pulse Setup at Wright-Patt

This discussion on the EPOCH github seems to be a way to add chirp to a pulse, i.e. a time-dependent wavelength which is something that could be done to a femtosecond pulse to increase its pulse length to a picosecond. So, I would just need to figure out what the hypothetical pulse length would be and how much of the wavelength would change throughout the pulse.