thunder4D

Transverse-resolved Herriott-cell UPPE solver for Nonlinear Dynamics close to Experimental Reality, 4D: a 4D (x, y, z, t) UPPE solver for multipass cells. Python package: thunder4d.

Full documentation: https://victorkolt.github.io/thunder4D/.

thunder4D simulates nonlinear pulse propagation in a gas-filled Herriott-type multipass cell (MPC) with full spatial and temporal resolution: the field is A(x, y, t) and it is propagated along z. It is built to study spatial and spatio-spectral effects that 1D or radially symmetric codes cannot capture:

• spatial effects in non-Gaussian inputs (clipped, aberrated, high-order beams),
• spatio-spectral homogeneity, radial chirp,
• time-dependent nonlinear mode mismatch,
• influence of space-time couplings (spatial chirp, pulse-front tilt, …) on the output.

The code is pure Python. NumPy (≥ 2.0) is required. SciPy is optional but strongly recommended, because scipy.fft is about 8× faster than numpy.fft for the 2D transforms. Matplotlib is only needed for the examples. An NVIDIA GPU is optional: with CuPy installed (pip install cupy-cuda12x), suggest_grid(..., device="auto") (the default) runs the propagation on the GPU, and device="cpu" / "gpu" force one or the other. thunder4d.gpu_available() tells you whether CuPy works; thunder4d.to_host(array) converts a device array to NumPy. Everything returned by thunder4d.diagnostics is a NumPy array on either device.

Quick start

import thunder4d as m
from thunder4d import diagnostics as dg

# medium and cell
gas  = m.Material("Ar", pressure_bar=1.0)
cell = m.MPC.herriott(R=0.5, N=20, k=3, branch="planar", medium=gas)  # 20 passes, 54 deg Gouy/pass, re-entrant

# grid sized from the cell
grid = m.suggest_grid(cell, lambda0=1030e-9, lambda_min=900e-9, lambda_max=1200e-9, pulse_fwhm=200e-15, box_factor=12)
print(grid.describe())

# 1) spectrum (+ phase)
E = m.gaussian_spectrum(grid, fwhm_fs=200)
E = m.add_spectral_phase(grid, E, gdd_fs2=0)

# 2) nonlinear mode matching, computed from a probe pulse of the right energy
probe = m.Pulse.at_focus(grid, E, m.Beam(m.gaussian(1e-3)), 1.2e-3)
mm = cell.match(probe)

# 3) the actual input beam
pulse = m.Pulse.at_focus(grid, E, m.Beam(m.gaussian(mm["w0"])), 1.2e-3)

# 4) run
sim = m.MPCSimulation(pulse, cell, phi_max=0.02, mode_reference=mm["w0"], profile_wavelengths=[1010e-9, 1030e-9, 1050e-9], profile_bandwidth=2e-9)
sim.run()
sim.save("run.npz")

# 5) analyse the output
x, lam, S = dg.spectrogram_x_lambda(grid, sim.Aw)
c = dg.compress(grid, sim.Aw)
print(c["fwhm_fs"], c["gdd_fs2"])

Physical model

The field is a complex envelope A(x, y, t) with carrier exp[i(k₀z − ω₀t)], normalised so that |A|² is the intensity in W/m². It is propagated with the unidirectional pulse propagation equation (UPPE), written in the frame moving at the group velocity at λ₀:

∂A(kx, ky, Ω)/∂z = i [ kz(kx, ky, ω) − k₀ − k₁Ω ] A  +  i γ(Ω) FT_t[ |A|² A ]

kz   = sqrt( k(ω)² − kx² − ky² )        (non-paraxial, full gas dispersion)
k(ω) = n(ω) ω / c,   ω = ω₀ + Ω,   k₁ = dk/dω at ω₀
γ(Ω) = n₂ ω/c · n(ω₀)/n(ω)               (self-steepening included)

• Linear operator: it is exact. It includes diffraction without the paraxial approximation, gas dispersion to all orders, and evanescent components, which decay.
• Kerr term: it is the instantaneous electronic response. Its frequency dependence gives self-steepening. With self_steepening=False it becomes γ = n₂ω₀/c.
• Edge taper: the nonlinear source is tapered to zero at the edges of the spectral window to avoid aliasing.

Numerical scheme

• Home domain: the solver keeps the field in the mixed domain (x, y, Ω).
  • The linear step needs only 2D spatial FFTs, because Ω is already a coordinate.
  • The Kerr term does not depend on (kx, ky), so it needs only 1D temporal FFTs.
• Splitting: symmetric Strang splitting (linear half-step, nonlinear step, linear half-step). Consecutive linear half-steps are merged.
• Nonlinear step: fourth-order Runge–Kutta.
• Adaptive step size: h = φ_max / (k₀ n₂ I_peak), where φ_max is the peak nonlinear phase allowed per step. It is limited to dz_max (default d/8, a quarter of a half-pass) and quantised to dz_max · 2^(−n/4). The quantisation lets the linear propagators exp(iDh) be reused from a small cache.

Performance and memory

• Cost per step: 2 spatial 2D FFTs, 8 temporal 1D FFTs (RK4) and a few array operations. With SciPy on one core, a 150 × 150 × 135 grid takes about 0.1–0.4 s per step. A 20-pass run at B ≈ 16 rad needs about 900 steps.
• Steps per pass: they scale with the B-integral per pass divided by phi_max.
• Memory: about 8–10 field arrays: the field, RK4 buffers, the linear operator D, up to cache_size cached propagators and the cached mirror transfer. One complex64 array is Nx·Ny·Nt·8 bytes, so 256 × 256 × 256 is 134 MB.
• Speed-ups:
  • SciPy backend with all cores (fft_backend="scipy", workers=-1, the default with "auto").
  • record_every > 1, or near_field=False (near-field diagnostics, i.e. of the collimated beam, use per-colour matrix transforms).
  • Fewer profile_wavelengths.
  • A smaller cache_size if memory is tight.
• Long runs: use run(max_seconds=…) with checkpoint / resume.

Limitations

Not included

• Ionisation and plasma: fine for MPCs operated well below ionisation thresholds.
• Raman response: relevant for N₂, air and other molecular gases.
• Off-axis geometry: the real cell is folded, so the beam hits the mirrors at small angles. The resulting astigmatism and the actual spot pattern are not modelled; the unfolded on-axis picture is used.
• Non-uniform media: gas plates, windows or pressure gradients are not built in. The nonlinear mode matching formula also assumes a uniformly filled cell with constant power during a pass.
• Vectorial effects: the field is scalar, with a single polarisation.