Laser-Ion Acceleration

Exercise 14 – Sudden deposition and the acoustic trace

Week 14 · discussed in the tutorial session · submit solutions by e-mail as agreed in class

Accompanies Lecture 14: iono-acoustic wave for sudden energy deposition (related timing estimates are in Exercise 11, Part A).

Task

  1. Assume that the energy deposition can be considered sudden, i.e. the deposited energy density may be written \[ \varepsilon(\mathbf{r},t)=\varepsilon(\mathbf{r})\,\delta(t), \] with \(\delta\) the Dirac delta. Starting from the thermoacoustic wave equation of Lecture 14, derive a formula for the pressure \(p(\mathbf{r},t)\) under this condition.
  2. Discuss the result for a large detector–source distance (far field): what is the relation between the temporal shape of the pressure pulse and the spatial distribution of the deposited energy?
  3. (Optional.) Specialise to a beam-like Bragg distribution \(\varepsilon(z)\) with small transverse extent and argue why I-BEAT’s on-axis trace is essentially \(\propto-\partial\varepsilon/\partial z\) (compression leads; arrival time fixed by the detector depth).

Useful routes (any one is fine): (i) spherical / Kirchhoff representation of the 3D wave solution; (ii) d’Alembert initial-value formula with \(p(\mathbf{r},0)\propto\varepsilon(\mathbf{r})\) and \(\partial_t p|_{t=0}=0\); (iii) 1D reduction along the beam axis. Identify near-field vs. far-field terms (\(\propto 1/R^2\) vs. \(\propto 1/R\)).