28 -- Delayed light: a plate ghost, a mirror-substrate echo and a coating pedestal on the tail grid¶

A 100 fs pulse passes a fused-silica window with a wavelength-dependent AR coating and reflects from an HR mirror on a 12.7 mm substrate. The light that is delayed -- the window's ghost, the echo from the mirror's back face, the coating's ripple pedestal -- is 10⁻⁶ of the pulse and 49 to 124 ps behind it. A cross-correlator reads it. The nonlinear grid stays 4.3 ps long: the delayed light lives on the tail grid (Delayed light).

The overview figure of this example's run (python make_figures.py redraws it; the file is hero.png).

cd docs/examples/Diagnostics/delayed-light
ultrafast run cell.ufs run.json -o results

"grid": { "tail": true }          // run.json: the main grid is sized for the pulse alone

Expected output (a fresh run):

  grid.tail: ghosts, echoes and other satellites are carried on the long tail grid; the main window is sized for the pulse only.
  Grid auto-sized: N = 512, window = 4.267 ps, dt = 8.333 fs.
  energy 9.874E-07 J (T = 98.74 %), FWHM 100.1 fs (TL 100 fs), peak 9.268E+06 W, B = 0 rad

summary.json -> tail: N = 131 072 (256 x the main grid), window 1.09 ns, 1.7e-10 J (0.017 % of the output) in the tail. The loss budget carries it as delayed light (tail).

What the cross-correlator reads (contrastAt_*)¶

delay	source	engine	by hand
+48.8 ps	window ghost	-58.2 dB	R(1030 nm)² with R interpolated from the AR table (0.0012): 1.5e-6 = -58.2 dB
+123.96 ps	mirror echo	-54.7 dB	T² R_b / R = 0.01² x 0.0337 / 0.99 = 3.4e-6 = -54.7 dB; delay 2 n_g d / c
+3 ps	coating ripple (phase 0.02 rad rms, within 5 ps)	-52.6 dB	first-order pedestal 4e-4 of the energy spread over 0 .. 5 ps

The delays are the group delays of the exact transfer functions (exp(i 2 n(ω) ω d cos θ / c)), not numbers typed in: 48.8 ps for the 5 mm window and 123.96 ps for the 12.7 mm substrate. The notes of each element say so. The cross-correlator makes its SHG gate from the main field only; the probe channel sees main + tail.

Things to try¶

• Replace "grid": {"tail": true} by "grid": {"auto": false, "n": 2048, "windowS": 20e-12}: the run warns, by name, that the window ghost (+48.8 ps) and the mirror echo (+123.96 ps) wrap around, with their levels and energies. Add "extendForSatellites": true to grow the window, or "strict": true to make it an error.
• Set "substrate": {"material": "FusedSilica", "thicknessM": 12.7e-3, "backReflectance": 1e-3}: an AR-coated back face lowers the echo by 34 (15 dB).
• Add "delayedScatter": {"fraction": 1e-4, "delay": {"kind": "uniform"}} to the mirror: scatter that returns as light, uniformly over 0 .. 124 ps.