Laser-Ion Acceleration

Exercise 4 – Ionization thresholds and relativistic electron motion

Week 4 · discussed in the tutorial session · submit solutions by e-mail as agreed in class

Part A — Field ionization

Review Lecture 4, §2.2 (barrier suppression) before you start. Ionization potentials can be looked up at NIST ASD.

  1. Derive the electric field \(E_{\text{th}}\) and the corresponding peak intensity \(I_{\text{th}}\) for which the combined Coulomb + static-field potential \(U(x) = -e^2/(4\pi\varepsilon_0 x) - eE_0 x\) is suppressed to the level of the ionization potential \(I_p\) (barrier top at the bound-state energy). State clearly where you set \(|U_m| = I_p\).
  2. Skim B. Chang et al., Phys. Rev. A 47, 4193 (1993) and extract the threshold intensity that ionizes 99% of a given charge state to the next one during a pulse of duration \(\tau_L\). A convenient form (with \(I_p\) in Rydberg, i.e. units of 13.6 eV, and \(\tau_L\) in seconds) is \[ I_{99} = 1.53\times 10^{16}\, \left(\frac{I_p}{13.6\ \mathrm{eV}}\right)^3 \left[ \ln\!\left( 854.5 \times 4.1\times 10^{16}\, \tau_L \frac{I_p}{13.6\ \mathrm{eV}} \right) \right]^{-2} \ \mathrm{W\,cm^{-2}}. \] Explain briefly how the authors define their atomic units and why the argument of the logarithm contains \(\tau_L I_p\).
  3. For oxygen, the ionization potentials (eV) of successive charge states are 13.618, 35.121, 54.936, 77.414, 113.899, 138.119, 739.327, 871.410. Compute the threshold intensity for ionizing O5+→O6+ and O6+→O7+ using both your result from (1) and the Chang formula from (2) for \(\tau_L = 30\,\mathrm{fs}\) and \(\tau_L = 300\,\mathrm{fs}\).
  4. Generate a log–log plot of threshold intensity versus charge state \(Z\) for oxygen (at least \(Z = 1\ldots 8\)). Overlay
    • the barrier-suppression estimate from (1),
    • the Chang curves for both pulse durations from (2),
    • and, as a multiphoton reference, the intensity at which the Keldysh parameter equals unity, \[ \Gamma = \frac{\omega\sqrt{2 m_e I_p}}{e E_0}=1 \quad\Rightarrow\quad I_{\Gamma=1} = \frac{c\,\varepsilon_0\, m_e\,\omega^2 I_p}{e^2} = \frac{4\pi^2 c^3\,\varepsilon_0\, m_e\, I_p}{e^2\lambda^2} \] (use \(\lambda=800\,\mathrm{nm}\); here \(I_p\) is the ionization potential of that charge state). Recall: \(\Gamma\gg 1\) favours multiphoton ionization, \(\Gamma\ll 1\) tunnelling / field ionization.
    Discuss which model is appropriate in which intensity regime.

Part B — Relativistic equation of motion in a plane-wave Gaussian pulse

Use normalized units with \(c = 1\). A plane wave propagates along \(+z\). With envelope parameter \(\tau\) (related to the FWHM pulse duration) and polarization parameter \(\delta\) (\(\delta = 1\) linear, \(\delta = 1/\sqrt{2}\) circular), the fields are \[ \mathbf{E} = E_0\,e^{-(t-z)^2/\tau^2}\,\mathrm{Re} \left[ e^{i(t-z)} \left(\delta\,\hat{x} + \sqrt{1-\delta^2}\,e^{i\pi/2}\,\hat{y}\right) \right], \] \[ \mathbf{B} = \hat{z}\times \mathbf{E}. \] For an electron, \(\mathbf{F} = -e(\mathbf{E} + \mathbf{v}\times\mathbf{B})\).

  1. Write the relativistic equations of motion as a six-component ODE for \((p_x, p_y, p_z, x, y, z)\) with \(\mathbf{v} = \mathbf{p}/\gamma\) and \(\gamma = \sqrt{1 + |\mathbf{p}|^2}\) (electron mass and charge absorbed into normalized units). Implement a numerical solver (MATLAB ode23/ode45 or equivalent).
  2. For \(E_0 = 1\) (relativistic threshold in these units) and linear polarization, plot \(x(t)\), \(z(t)\), and the trajectory \(x(z)\). Start from rest at the origin. Comment on the figure-of-eight structure and how it differs from a monochromatic plane wave.
  3. Repeat for circular polarization (\(\delta = 1/\sqrt{2}\)) at the same \(E_0\). How do the transverse and longitudinal excursions change?
  4. Compare your numerical results with the analytic plane-wave discussion in Paul Gibbon, Short Pulse Laser Interactions with Matter (Imperial College Press, 2005), pp. 31–36: conserved transverse canonical momentum, the absence of net energy gain, and the scaling of the quiver amplitude with \(a_0\). Where does the finite Gaussian envelope break the strict plane-wave conservation laws?