Laser-Ion Acceleration

Exercise 9 – Radiation pressure, light sail, and \(E_{\mathrm{kin}}\propto E_L\)

Week 9 · discussed in the tutorial session · submit solutions by e-mail as agreed in class

This sheet accompanies Lecture 9: radiation pressure, light-sail / Star-CHIP estimates, and a comparison of optimised TNSA vs. optimised RPA scaling with laser energy.

1. Charged carbon disc

  1. Consider a pure carbon disc with radius \(R=2\,\mu\mathrm{m}\) and thickness \(d=100\,\mathrm{nm}\) from which all electrons are removed.
    1. Derive the electrostatic potential along the symmetry axis.
    2. What maximum kinetic energy \(E_{i,\infty}\) can a proton gain?
    3. How long does it take to gain half of this energy, and how far has the ion moved by then? (Tip: conceptually related to the Crow test-particle problem in Exercise 8.)

2. Light-sail / Star-CHIP estimate

  1. Assume a continuous \(1\,\mathrm{GW}\) laser and a perfectly reflecting ultralight sail with total mass \(M=1\,\mathrm{g}\).
    1. For how long must you fire the laser to reach \(\beta=0.7\)?
    2. Estimate the distance travelled when that velocity is reached.
    3. What beam diameter is needed so that diffraction does not miss the sail over that distance?
    4. Briefly discuss why a “STAR-CHIP” is hard, and how higher power improves the numbers.
    Use the relativistic sail equation \(\mathrm{d}p/\mathrm{d}t=(2P/c)(1-\beta)/(1+\beta)\).

3. New: optimised TNSA vs. optimised RPA — \(E_{\mathrm{kin}}\propto E_L\)

Goal: show explicitly that both optimised models predict ion kinetic energy linear in laser energy \(E_L\), for different physical reasons.

  1. Optimised TNSA (short pulses). In the Schreiber / charged-disc picture of Exercise 8, \(E_\infty\propto\sqrt{\eta P_L}\) and acceleration lasts for \(\sim\tau_L\). At the optimal pulse duration \(\tau_{L,\mathrm{opt}}\) (Question 4 of Exercise 8), the ion energy is a fixed fraction of \(E_{i,\infty}=q_i E_\infty\). Show that, for fixed spot radius \(R\) and absorption \(\eta\), \[ E_{\mathrm{kin,TNSA}}^{\mathrm{(opt)}} \propto E_L \] (equivalently: shorter pulses at higher power for fixed energy, or the \(\tau_{L,\mathrm{opt}}\) scaling you derived). State clearly which quantities you hold fixed.
  2. Optimised RPA (matched thickness). For a light sail with perfect reflection, \(\beta\approx 2E_L/(Mc^2)\) (non-relativistic). The depletion-layer constraint requires a minimum thickness \(l_d\propto a_L\propto\sqrt{I_L}\), hence a minimum mass \(M_{\min}\propto\sqrt{E_L}\) at fixed spot size (or argue the analogous scaling with intensity). Show that the kinetic energy of the sail then satisfies \[ E_{\mathrm{kin,RPA}}^{\mathrm{(opt)}} \propto E_L. \]
  3. Comparison. In a short paragraph, contrast the two routes: what is being optimised in each case (pulse duration vs. foil thickness), what fails if you are not optimal (too long a pulse for TNSA; too thick/thin a foil for RPA), and why both can still yield \(E_{\mathrm{kin}}\propto E_L\) once optimised. Optional: sketch \(E_{\mathrm{kin}}(E_L)\) for a non-optimised TNSA scaling you know from Lecture 8 (e.g. fixed \(\tau_L\)) next to the optimised lines.