Laser-Ion Acceleration

Exercise 2 – Peak intensity and material limits

Week 2 · discussed in the tutorial session · submit solutions by e-mail as agreed in class

This sheet accompanies Lecture 2. Work through the focusing geometry first, then the ATLAS numbers.

Part A — Focusing sunlight

Solar constant at Earth: \(I_\odot \approx 1.36\,\mathrm{kW\,m^{-2}}\). Angular diameter of the solar disk: \(\theta_\odot \approx 0.5^\circ\) (Sunlight is incoherent — a lens images the Sun rather than focusing a point source).

  1. Estimate the maximum intensity that you can create by “focusing” sunlight with a lens of focal length \(f = 150\,\mathrm{cm}\) and diameter \(D = 30\,\mathrm{cm}\). (First find the size of the solar image in the focal plane, then the collected power.)
  2. How large must the lens be to collect \(1\,\mathrm{PW}\) of sunlight? If the \(f\)-number \(f/D\) remains the same as in (1), how intense is the light in the image?
  3. To what temperature would you heat matter in the image if we assume that all light energy is absorbed? (Order of magnitude / Stefan–Boltzmann is fine; comment on whether thermal equilibrium is meaningful.)

Part B — ATLAS-scale peak intensity

Use ATLAS 3000 order-of-magnitude parameters: peak power \(P \approx 1\,\mathrm{PW}\), wavelength \(\lambda \approx 800\,\mathrm{nm}\), focal spot diameter \(\approx 5\,\mu\mathrm{m}\) (FWHM). Treat the focus as a uniform disk unless stated otherwise.

  1. Compute the peak intensity \(I_0\) in the focus.
  2. Derive the peak electric-field amplitude \(E_0\) from \(I_0 = \tfrac{1}{2}\varepsilon_0 c E_0^2\).
  3. If the pulse energy were absorbed uniformly in a solid density target of thickness \(1\,\mu\mathrm{m}\), estimate an equivalent temperature (order of magnitude is sufficient). Is a thermal-equilibrium picture meaningful on the laser time scale?
  4. Using the non-relativistic estimate from Lecture 2, \(E_k = q^2 E_0^2 / (2 m_e \omega_0^2)\) with \(\omega_0 = 2\pi c/\lambda\), what is the maximum oscillatory kinetic energy of a free electron? Is the non-relativistic approximation justified?

Part C — No solid converter survives

  1. Compare your \(E_0\) from (6) with a typical atomic binding field (\(\sim 10\,\mathrm{GV\,m^{-1}}\), order of magnitude). What happens to a metal or dielectric foil before the peak of the pulse?
  2. Briefly explain why both the dielectric grating (Lecture 2, §3) and the rigid light-sail mirror (§4) must be replaced by a plasma at these intensities — and why that motivates Lecture 3.

Part D — Coherent acceleration: Coulomb explosion

Lecture 2 §4 introduced Veksler’s idea of coherent acceleration: the accelerated object participates in generating the field that pushes it. Here is a purely electrostatic cousin of that idea — no laser required in the thought experiment.

  1. Consider a single hydrogen atom. Suppose you remove its electron instantaneously (leave the proton where it is). Does the proton accelerate? Why / why not?
  2. Now take two hydrogen atoms a distance \(d\) apart and remove both electrons instantly. Describe what happens to the two protons. Estimate their asymptotic kinetic energy if they start at rest a distance \(d\) apart (order of magnitude is fine; use \(U = e^2/(4\pi\varepsilon_0 d)\)).
  3. Repeat the argument for \(N = 4,\;8,\;16,\;\ldots\) protons left behind in a compact cluster after all electrons are stripped at \(t=0\). Argue qualitatively how the energy per proton and the collective push scale with \(N\) (compare a single “lonely” proton with a coherent multi-proton explosion). In what sense is this also coherent acceleration, in the spirit of Veksler’s idea?