Laser-Ion Acceleration

Exercise 5 – Transparency, ponderomotive force, and collisional absorption

Week 5 · discussed in the tutorial session · submit solutions by e-mail as agreed in class

This sheet accompanies Lecture 5 (transparency, ponderomotive force, and collisional absorption). MATLAB helper (optional): e03_collisionalabsorption.m in the course Exercises/ folder.

Part A — Relativistic transparency example

Use \(\lambda_0 = 800\,\mathrm{nm}\) for Ti:sapphire. For a Gaussian focus, relate pulse energy and FWHM spot size to peak intensity (see Exercise 3). Fully ionized water: \(n_e = 10 N_A \rho / M\) with \(M \approx 18\,\mathrm{g/mol}\), \(\rho \approx 1\,\mathrm{g/cm^3}\). Liquid hydrogen: \(n_e = 2 N_A \rho / M\) with \(M \approx 2\,\mathrm{g/mol}\), \(\rho \approx 0.07\,\mathrm{g/cm^3}\).

  1. A 30 fs Ti:sapphire pulse with 10 J energy is focused to a spot of 5 µm FWHM diameter. Determine the peak intensity \(I_0\) on target and the peak dimensionless vector potential \(a_0\).
  2. Calculate the free-electron density \(n_e\) and the collisionless skin depth \(\delta = c/\sqrt{\omega_p^2 - \omega^2}\) (or state when the plasma is overdense and give \(\delta \approx c/\omega_p\)) for fully ionized water and fully ionized liquid hydrogen.
  3. Using the simple relativistic transparency criterion \(n_e < \gamma\, n_c\) with \(\gamma \approx \sqrt{1 + a_0^2/2}\) (cycle-averaged quiver motion), decide whether each target would be relativistically transparent — if you neglect all other dynamics.
  4. Explain why a target can become transparent anyway during the pulse, even when the static estimate says overdense. Connect your answer to the underdense/overdense discussion in Lecture 5, §5.

Part B — Ponderomotive force from perturbation theory

Non-relativistic limit, linearly polarized field \(\mathbf{E}(\mathbf{r},t) = \hat{\mathbf{e}}\,E_0(\mathbf{r})\cos(\omega_0 t)\), particle charge \(q = Ze\), mass \(m\). Assume the oscillation amplitude \(\Delta x \sim ZeE_0/(m\omega_0^2)\) is small compared to the scale of variation of \(E_0(\mathbf{r})\).

  1. Solve the first-order equation of motion for \(\mathbf{v}_1(t)\) and \(\mathbf{x}_1(t)\) about a fixed guiding center \(\mathbf{r}_0\), neglecting spatial variation of \(E_0\) and the magnetic force at this order.
  2. Expand \(E_0(\mathbf{r})\) to first order in \(\mathbf{x}_1\). Use \(\nabla\times\mathbf{E} = -\partial_t\mathbf{B}\) to obtain the magnetic field associated with the spatially varying amplitude (to the order needed).
  3. Insert \(\mathbf{x}_1\) and \(\mathbf{v}_1\) into the Lorentz force and cycle-average over one optical period. Show that the secular force is \[ \langle \mathbf{F}_p \rangle = -\,\frac{Z^2 e^2}{4 m \omega_0^2}\,\nabla E_0^2(\mathbf{r}_0) = -\nabla U_p, \] with \(U_p = Z^2 e^2 E_0^2/(4 m \omega_0^2)\).
  4. Briefly state how the result changes for circular polarization (factor of 2 in \(U_p\) at the same time-averaged intensity). Compare with the focused-pulse simulator in Lecture 5.

Part C — Complex dielectric function and collisional absorption

Drude model with collision frequency \(\nu\): \(\varepsilon(\omega) = 1 - \omega_p^2/[\omega(\omega + i\nu)]\). For a vacuum–plasma interface at normal incidence, the Fresnel formula gives reflectivity \(R = |(1 - \eta)/(1 + \eta)|^2\) with complex \(\eta = \sqrt{\varepsilon}\).

  1. Starting from the equation of motion \(m_e(\dot{\mathbf{v}} + \nu\mathbf{v}) = -e\mathbf{E}\) for a cold electron fluid, show that the dielectric function is \(\varepsilon(\omega) = 1 - \omega_p^2/[\omega(\omega + i\nu)]\).
  2. Derive the reflectivity \(R(\omega_p/\omega)\) at normal incidence from the vacuum–plasma boundary condition. Distinguish the cases \(\omega_p < \omega\) (propagating) and \(\omega_p > \omega\) (evanescent).
  3. Plot \(R\) versus \(x = \omega_p/\omega\) for at least two collisionalities, e.g. \(\nu/\omega = 10^{-3}\) and \(\nu/\omega = 0.1\) (see e03_collisionalabsorption.m).
  4. For \(\omega_p < \omega\), explain why a naive “absorption = \(1 - R\)” can mislead: transmitted light carries energy into the underdense plasma. What happens to \(R\) as \(\omega_p \to \omega\) from below and from above?