In my last post, I used a couple of simple assumptions about the modified density distribution of the plasma to motivate an expression for $n_{max}$, the maximum density of the new target. Using this result, we can derive the new effective thickness of the target, $d_{eff}$, to pass into the Fuchs function.

The general procedure will be as follows: first, the electron temperature of the plasma, $T_e$, must be calculated. This can be done by assuming the ~picosecond-duration prepulse happens on too short a timescale for Adiabatic effects to significantly contribute to the energy; as such, we can write a conservation equation for power that allows us to relate the intensity of the pre-pulse to the temperature to which it heats the plasma. Next, we use that electron temperature to calculate $C_s$ (the speed of sound in that medium). Using $C_s$ and $t$, the time delay between the pre- and main-pulses’ arrivals, we can estimate whether the quantity $n_{max}$ will be lower than critical density using the previous result; if it is, the laser would penetrate the entire target with no interaction. If $n_{max}>n_{crit}$, however, we can solve the exponential decay function for the first point along the curve where $n_{max}=n_{crit}$, which we then can mirror along the y-axis to find the effective thickness. Easy.

To begin, we know that the overall power will be conserved over the duration of the pre-pulse:

\begin{equation} P_{laser} - P_{loss} - P_{adiabatic} = P_{heating} \end{equation}

Due to the short duration (~1 picosecond) of the pre-pulse, we can safely approximate the effects due to adiabatic power loss and heating to be zero. Thus:

\begin{equation} P_{laser} - P_{loss} = 0 \end{equation}

We know that the power of the laser is proportional to some constant times the Intensity and the power due to radiative losses is proportional to some constant times the temperature of the plasma raised to the fourth power: $P_{laser} = k_1 I$; $P_{loss} = k_2 T^4$. Thus:

\begin{equation} k_1 I - k_2 T^4 = 0 \end{equation}

\begin{equation} T = \left(\frac{k_1 I}{k_2}\right)^{1/4} \end{equation}

Trivially. Now, let’s assume there exists some known values $T_{pre,0}$ and $I_{pre,0}$ for which this equation is satisfied. Experimentally, we know that a laser intensity of $10^{12} W cm^{-2}$ will correspond to a plasma temperature of $100 eV$. As such, we can eliminate the constants and rewrite the equation purely in terms of this known pair:

\begin{equation} T = T_{pre,0} \left(\frac{I}{I_{pre,0}}\right)^{1/4} \end{equation}

Now, we simply note that the intensity of the pre-pulse will be some contrast factor (let’s call it $\kappa$) multiplied the main-pulse intensity; in our experimental setup, this contrast happens to be $10^{-7}$. Thus:

\begin{equation} T_e = T_{pre,0} \left(\frac{\kappa I_{main}}{I_{pre,0}}\right)^{1/4} \end{equation}

Giving us the electron temperature in eV. This value for temperature can now be used to calculate $C_s$, the speed of sound in the plasma. Gibbon’s $\textit{Short Pulse Laser Interactions with Matter}$ gives a simplified formula for $C_s$ in terms of the temperature in eV:

\begin{equation} C_s = 3.1\text{e}7 \sqrt{\frac{T_e}{keV}} \sqrt{\frac{Z^*}{A}} \frac{cm}{s} \end{equation}

Where $Z^*$ represents the effective nuclear charge and $A$ the atomic number. As this experiment is concerned with the acceleration of protons, we know that $Z^* = A = 1$; a simple unit conversion tells us $3.1\text{e}7 \frac{cm}{s} = 3.1\text{e}2 \frac{\mu m}{ns}$, so:

\begin{equation} C_s = 3.1\text{e}2 \sqrt{\frac{T_e}{keV}} \frac{\mu m}{ns} \end{equation}

Finally, we have a value for $C_s$ with which we can calculate $n_{max}$, per the previous article’s results. Gibbon provides an equation for $n_{crit}$ as:

\begin{equation} n_{crit} = 1.1\text{e}21 \left(\frac{\lambda}{\mu m}\right)^{-2} cm^{-3} \end{equation}

For our laser wavelength of 800 nm, this value is $1.72\text{e}21$. So, we use our equation for $n_{max}$:

\begin{equation} n_{max} = \frac{n_0 d}{d + 2C_s t} \end{equation}

And if this value is $\leq n_{crit}$, the main pulse will simply phase through the plasma with no interaction, representing a return condition of all 0’s in the code. However, if $n_{max} > n_{crit}$, we can solve the positive-valued side of the curve for $x_0 | n(x_0) = n_{crit}$ to get the maximum value of $x$ for which the plasma is still dense enough to interact with the laser. Doubling this value represents mirroring this solution along the y-axis (as the decay is symmetric), giving a horizontal line segment representing effective thickness, $d_{eff}$. So, using our equation for $n(x)$ as before (taking $d$, the initial thickness, as a parameter) and solving for $x_0$:

\begin{equation} n_{max} e^{-(\frac{x_0 - d/2}{C_st})} = n_{crit} \end{equation}

\begin{equation} e^{-(\frac{x_0 - d/2}{C_st})} = \frac{n_{crit}}{n_{max}} \end{equation}

\begin{equation} -\left(\frac{x_0 - d/2}{C_st}\right) = \ln{n_{crit}} - \ln{n_{max}} \end{equation}

\begin{equation} \frac{x_0 - d/2}{C_st} = \ln{n_{max}} - \ln{n_{crit}} \end{equation}

\begin{equation} x_0 - \frac{d}{2} = C_s t \left(\ln{n_{max}} - \ln{n_{crit}}\right) \end{equation}

\begin{equation} x_0 = \frac{d}{2} + C_s t \left(\ln{n_{max}} - \ln{n_{crit}}\right) \end{equation}

Finally, we can use this to find $d_{eff} = 2x_0$:

\begin{equation} d_{eff} = d + 2C_s t \left(\ln{n_{max}} - \ln{n_{crit}}\right) \end{equation}

Giving an equation for the effective thickness of the target to be passed into the Fuchs function.