0. Setup: what to look at
We use the interactive code 20251126_pic1d3v_with_CHATGPT.html.
The laser propagates along \(z\).
Before running any case, decide which diagnostics you should always monitor, so comparisons become meaningful.
Interactive tool (embedded). If it does not load in your browser, open it directly: PIC tool (new tab).
Core diagnostics (recommended)
- Fields \(E_x(z,t)\), \(E_z(z,t)\), and optionally \(B_y(z,t)\).
- Densities \(n_e(z,t)\), \(n_i(z,t)\) (watch steepening, cavitation, compression).
- Phase space \(p_z\) vs \(z\) (electrons and ions separately).
- Energy total field energy and particle kinetic energy (to see absorption vs reflection).
Parameters you should frequently vary
- Density \(n_e/n_c\) (underdense \(\ll 1\), near-critical \(\sim 1\), overdense \(\gg 1\)).
- Pulse amplitude \(a_0\) (weak: \(a_0\ll 1\); relativistic: \(a_0\gtrsim 1\)).
- Pulse duration (few-cycle vs multi-cycle).
- Slab thickness \(L\) (thick target vs thin foil).
Quick formulas you may need
- 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 (EM wave in underdense plasma): \(\displaystyle \omega^2=\omega_p^2+c^2 k^2\).
- 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 \(\gamma\sim \sqrt{1+a_0^2/2}\) for linear polarization.
A. Weak few-cycle pulse through very underdense plasma
Goal Observe propagation, weak dispersion, and identify \(\lambda_p\).
Suggested setup
- \(a_0\ll 1\) (linear polarization).
- \(n_e/n_c \sim 10^{-3} \ldots 10^{-2}\).
- Few-cycle pulse; long enough simulation box to see the wake behind the pulse.
What to measure
- 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 pulse peak vs time to estimate \(v_g\).
B. Weak few-cycle pulse through near-critical plasma
Goal Observe slow-down and stronger dispersion close to \(n_c\).
Suggested setup
- \(a_0\ll 1\).
- \(n_e/n_c \sim 0.3 \ldots 0.9\).
What to look for
- Group velocity reduction \(v_g/c \approx \sqrt{1-n_e/n_c}\).
- Pulse stretching and spectral/temporal reshaping due to dispersion.
C. Strong few-cycle pulse through moderately dense plasma
Goal Plasma wave excitation (wakefield), wave breaking, and electron acceleration signatures.
Suggested setup
- \(a_0 \gtrsim 1\) (enter relativistic regime).
- \(n_e/n_c \sim 10^{-2} \ldots 10^{-1}\) (wakefield-friendly).
What to look for
- Large-amplitude \(E_z\) behind the pulse and electron density modulation.
- Wave breaking: multivalued phase space \(p_z(z)\), density spikes, onset of trapping.
- Electron acceleration: a population gaining large \(p_z\) and co-moving with the wake.
Interpretation hints
In 1D, wave breaking and trapping can appear “too clean” compared to 2D/3D, but it is excellent for building intuition: you literally see the phase-space folding that enables trapping.
D. Weak pulse on an overdense plasma slab
Goal Reflection as a natural consequence of \(n_e>n_c\).
Suggested setup
- \(a_0\ll 1\).
- \(n_e/n_c \sim 5 \ldots 50\) (overdense).
What to look for
- Formation of a standing wave in front of the surface.
- Skin-depth decay inside the slab.
- Energy balance: most energy stays in fields + reflected wave; little particle heating (depending on numerical heating).
E. 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 (choose realistic mass ratio if possible; otherwise use a larger \(m_i/m_e\) but still mobile).
What to look for
- Front-side charge separation fields (radiation pressure effects, hole boring tendencies).
- Rear-side sheath formation (TNSA-like acceleration), emergence of fast ion population.
- How ion acceleration changes when you vary slab thickness \(L\).
F. 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 slab thickness \(L\).
- Keep \(L\) fixed and increase \(a_0\).
What to look for
- Partial transmission and pulse reshaping inside the slab.
- Reduced reflectivity as effective \(n_{c,\mathrm{rel}}\) increases.
- Changes in ion acceleration when front and rear fields start to “communicate”.
G. Surface high harmonics (relativistically oscillating mirror)
Goal Identify harmonic generation in reflected fields when the surface oscillates relativistically.
Suggested setup
- Overdense slab, steep front surface (as steep as your initial conditions allow).
- \(a_0 \gtrsim 1\) and short pulse (few-cycle can be very illustrative).
What to do
- 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.
Why this works (conceptual)
In the ROM picture, the reflecting surface moves with relativistic velocity. The reflected field experiences a time-dependent Doppler shift, which generates a comb of high harmonics. Even in 1D3V, you can see the essential ingredients: a sharp density interface and relativistic transverse motion.