Computer simulations of high-harmonic generation in a thin silicon slab indicate that the spectrum can change as light propagates through the material. When the modeled propagation depth increased to 5 µm, harmonic lines broadened, moved toward higher frequencies — a blue shift — and grew more intense, while the fundamental peak fell slightly.
The modeled signal also depended on where it was observed. Reflected harmonic lines were narrower and less blue-shifted than transmitted lines, while transmitted harmonics had higher intensity. Below the direct bandgap, reflected harmonics were overall independent of slab thickness, but transmitted harmonics changed in intensity and central frequency as thickness varied.
A calculation built to keep both scales in view
The work, posted as a version 1 arXiv preprint on 20 August 2026, asks whether a computational framework can capture propagation effects in strong-field high-harmonic generation in solids while keeping the underlying microscopic response interpretable. It combines semiconductor Bloch equations, used here to model the material’s microscopic electronic response, with a finite-difference time-domain solution of Maxwell’s equations; both are integrated with a fourth-order Runge–Kutta method.
The microscopic part uses length-gauge semiconductor Bloch equations restricted to two bands. In the demonstration, the silicon model had a direct minimum band gap of about 3.3 eV and was driven by a 2.0 µm laser pulse with an 18.3 THz bandwidth and a 24 fs full-width half-maximum Gaussian envelope.
Without propagation, the isolated microscopic model provided the comparison point: its harmonic-generation efficiency decreased after the seventh harmonic, with spectral distortion near band edges.
The shortcut did not always match
The authors also compared the full Maxwell calculation with UPPE, an alternative propagation model. In the thin-slab test, the third- and fifth-order harmonic intensities were similar under the two methods.
That agreement did not extend across the spectrum. UPPE produced a sharp cutoff after the ninth harmonic that was absent from the equivalent Maxwell calculation; the paper linked the discrepancy to a refractive-index filter that zeroed UPPE’s kernel at those frequencies. Changing the refractive-index model removed the sharp cutoff and suggested that UPPE slightly overestimated transmitted harmonic intensity in the short-distance benchmark.
What the model leaves open
One test of a low-frequency resonance below the driving frequency produced minimal differences in output harmonics over 5 µm. The analysis says stronger effects may appear over longer propagation distances or with stronger absorption, so the short-distance result does not settle how the resonance would behave under those conditions.
The study’s main comparisons are selected simulations rather than an experimental sample: the demonstration centers on silicon and a two-band microscopic model, and no direct experimental validation or quantitative uncertainty analysis is reported. The reported changes should therefore not be read as evidence that the same effects will have the same size in other materials, crystal axes, thicknesses or driving fields.
For the authors, the practical message is that propagation should be modeled when transmitted high-harmonic spectra are used to make claims about microscopic electronic structure. The preprint does not show that UPPE is generally equivalent to full Maxwell propagation: its agreement was limited to selected low-order harmonics in thin slabs, while the broader study tested a specific silicon setup.
Paper data and sources
Original title: Computational Methods of Wave Propagation for Semiclassical Models of High Harmonic Generation in Bulk Solids
Authors: Ava N. Hejazi, Nicholas Karpowicz, Gregory D. Scholes, Julia M. Mikhailova
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text