This sheet accompanies Lecture 6 and uses the interactive 1D3V PIC code 1D3V PIC simulator. The laser propagates along \(z\). Before running any case, decide which diagnostics you monitor so comparisons become meaningful: fields \(E_x(z,t)\), \(E_z(z,t)\); densities \(n_e(z,t)\), \(n_i(z,t)\); electron and ion phase space \(p_z\) vs \(z\); and total field and particle kinetic energies.
- Critical density: \(\displaystyle n_c=\varepsilon_0 m_e \omega_0^2/e^2\).
- Plasma frequency: \(\displaystyle \omega_p=\sqrt{n_e e^2/(\varepsilon_0 m_e)}\).
- Plasma wavelength: \(\displaystyle \lambda_p = 2\pi c/\omega_p\) (underdense, weakly relativistic).
- Cold-plasma dispersion: \(\displaystyle \omega^2=\omega_p^2+c^2 k^2\).
- Group velocity: \(\displaystyle v_g = c\sqrt{1-\omega_p^2/\omega^2}\approx c\sqrt{1-n_e/n_c}\) (for \(\omega\approx\omega_0\)).
- Relativistic critical density: \(\displaystyle n_{c,\mathrm{rel}}\approx \gamma\,n_c\) with \(\gamma\sim \sqrt{1+a_0^2/2}\) (linear polarization).
Part A — Diagnostics & parameters
Open the PIC tool in a separate tab if the embedded version in Lecture 6 is too small: fullscreen PIC tool.
- List the core diagnostics you should always monitor when comparing runs (fields, densities, phase space, energies — be specific about which components).
- Which parameters should you vary most frequently to move between underdense propagation, near-critical slow-down, wakefield excitation, and overdense reflection? Name at least: \(n_e/n_c\), \(a_0\), pulse duration, and slab thickness \(L\).
Part B — Underdense / near-critical
1. Weak few-cycle pulse through very underdense plasma
Goal: observe propagation, weak dispersion, and identify \(\lambda_p\).
Suggested setup: \(a_0 \ll 1\); \(n_e/n_c \sim 10^{-3} \ldots 10^{-2}\); few-cycle pulse; long enough box to see the wake behind the pulse.
- From \(E_z(z)\) or \(n_e(z)\) behind the pulse, estimate \(\lambda_p\) and compare to \(\lambda_p = 2\pi c/\omega_p\).
- Track the pulse peak vs time to estimate \(v_g\) and compare with \(v_g/c \approx \sqrt{1 - n_e/n_c}\).
2. Weak few-cycle pulse through near-critical plasma
Goal: observe slow-down and stronger dispersion close to \(n_c\).
Suggested setup: \(a_0 \ll 1\); \(n_e/n_c \sim 0.3 \ldots 0.9\).
- Measure the reduction in group velocity and compare with \(v_g/c \approx \sqrt{1 - n_e/n_c}\).
- Describe pulse stretching and temporal reshaping due to dispersion. What changes in the field and density plots compared to case 1?
3. Strong few-cycle pulse through moderately dense plasma
Goal: plasma wave excitation (wakefield), wave breaking, and electron acceleration signatures.
Suggested setup: \(a_0 \gtrsim 1\); \(n_e/n_c \sim 10^{-2} \ldots 10^{-1}\).
- Identify large-amplitude \(E_z\) behind the pulse and electron density modulation. What is the signature of wave breaking in phase space \(p_z(z)\)?
- Look for electron trapping: a population gaining large \(p_z\) and co-moving with the wake. Briefly explain why 1D3V can show this more cleanly than 2D/3D, yet still builds useful intuition.
Part C — Overdense / thin foils
4. Weak pulse on an overdense plasma slab
Goal: reflection as a natural consequence of \(n_e > n_c\).
Suggested setup: \(a_0 \ll 1\); \(n_e/n_c \sim 5 \ldots 50\).
- Observe formation of a standing wave in front of the surface and skin-depth decay inside the slab. Estimate \(\delta\) from the field profile and compare with \(c/\omega_p\).
- Comment on the energy balance: where does most of the laser energy go (reflected field vs particle heating)?
5. Increase amplitude: ion acceleration and expansion
Goal: front-side vs rear-side fields, and the competition of expansions.
Suggested setup: moderate overdense slab (thick target); start with \(a_0 \sim 1\) and increase; include mobile ions.
- Describe front-side charge separation fields (radiation pressure / hole-boring tendencies) and rear-side sheath formation (TNSA-like acceleration). How does ion acceleration change when you vary slab thickness \(L\)?
6. Thin slab and relativistic transparency
Goal: observe transmission onset as you reduce \(L\) and/or increase \(a_0\).
Suggested knobs: keep density fixed and reduce \(L\); or keep \(L\) fixed and increase \(a_0\).
- Document partial transmission and pulse reshaping inside the slab. Relate reduced reflectivity to the effective \(n_{c,\mathrm{rel}} \approx \gamma\, n_c\). How do front and rear fields start to “communicate” as transparency sets in?
7. Surface high harmonics (optional)
Goal: identify harmonic generation in reflected fields when the surface oscillates relativistically (relativistically oscillating mirror, ROM).
Suggested setup: overdense slab, steep front surface; \(a_0 \gtrsim 1\); short (few-cycle) pulse.
- Record the reflected \(E_x(t)\) at a probe point in front of the target. Take a Fourier transform and look for harmonics of \(\omega_0\). Correlate harmonic strength with surface motion and density oscillations.
ROM picture (conceptual hint for task 13)
In the ROM picture, the reflecting surface moves with relativistic velocity. The reflected field experiences a time-dependent Doppler shift, generating a comb of high harmonics. Even in 1D3V you can see the essential ingredients: a sharp density interface and relativistic transverse motion.