Laser-Ion Acceleration

Exercise 11 – Bunch timing and time-resolved deposition in water

Week 11 · discussed in the tutorial session · submit solutions by e-mail as agreed in class

This sheet accompanies Lecture 11. Part A treats focused-proton bunch timing; Part B the time-resolved energy deposition and solvated-electron transmission in water.

Part A — Spatial and temporal characteristics of a focused proton bunch

Consider a CALA-like setup: a \(30\,\mathrm{fs}\) laser pulse generates a proton spray with energies up to \(20\,\mathrm{MeV}\). A quadrupole doublet captures the central spray and focuses a design energy of \(10\,\mathrm{MeV}\) onto a platform \(2\,\mathrm{m}\) away. About \(10^{6}\) protons reach a \(\sim 1\,\mathrm{mm}\) spot; the transported relative energy spread is \(\sim 5\%\).

  1. Estimate the mean penetration depth of the protons (e.g. NIST PSTAR, water or tissue-equivalent).
  2. Estimate the variance of the range due to (i) range straggling and (ii) the initial energy spread. Which contribution dominates?
  3. Estimate the duration of the proton bunch at impact (TOF difference across the energy band).
  4. Estimate the trajectory / slowing-down time of the mean-energy proton while it stops. Conclude whether the deposition can be treated as instantaneous for µs-scale acoustics.

Part B — Time-resolved deposition and transmission in water

LION / CALA pump–probe context (Prasselsperger+): protons from a thin foil / water leaf with a typical spectrum extending to \(20\,\mathrm{MeV}\) and amplitude of order \[ \frac{\mathrm{d}^2 N}{\mathrm{d}E_{\mathrm{kin}}\,\mathrm{d}\Omega} \approx 10^{6}\,\frac{\text{protons}}{1\%\,E_{\mathrm{kin}}\cdot\mathrm{msr}} \] (cf. ALPA proton database). A water cell sits \(L=1.7\,\mathrm{cm}\) behind the target; a square aperture of side \(d=500\,\mu\mathrm{m}\) selects a pencil beam (aperture ≪ beam size at that distance). Assume all protons are born at \(t=0\).

Schematic pencil beam into water for Part B.
Geometry reminder for Part B (Lecture 11, Fig. 11.3).
  1. Deposited power density \(\pi(z,t)\). Calculate the deposited power density as a function of depth \(z\) into the sample and time \(t\). Use Bethe–Bloch / range–energy relations from Exercise 10 and weight by the spectrum through the aperture. (Equivalently: construct the cumulative energy density \(\varepsilon(z,t)=\int_{-\infty}^{t}\pi(z,t')\,\mathrm{d}t'\).)
  2. Transmission image. Convert the two-dimensional energy-density map into a transmission image by assuming that the solvated electron dominates absorption (Beer–Lambert; \(T=\mathrm{e}^{-\alpha}\), \(\alpha=\beta\,c\,l\)). State clearly which lifetime / yield assumptions you make.

Tip: Part A shows that stopping is ≪ ns; Part B asks for the structured \(\varepsilon(z,t)\) that optical probes can see via \(e_{\mathrm{aq}}^-\) (Lecture 11 §4–5).