31 -- A resonator and a partially coherent beam¶

Two short runs with one-frequency ("shape": "cw") sources.

A ring resonator (lens-guide.ufs): a lens of f = 0.5 m every 0.6 m, with a 1.1 mm aperture. The tracker gives the round-trip matrix and the Gaussian eigenmode; the transverse grid adds the Fox-Li iteration, a wave calculation that knows nothing of ABCD matrices (Resonators).

A Gaussian-Schell beam (beam.ufs, run-m2.json): a 0.5 mm waist with M² = 2.5, one Rayleigh range of free space. "coherence" turns the run into 11 coherent modes (those above 10⁻⁴ of the energy) (Partial coherence).

cd docs/examples/Branched/cavity-and-coherence
ultrafast run lens-guide.ufs run-cavity.json -o results-cavity
ultrafast run beam.ufs run-m2.json -o results-m2
python report.py

(A + D) / 2      0.4000   analytic 0.4000
mode radius (mm) 0.3876   analytic 0.3876
Fox-Li radius    0.3877   round-trip loss 0.003 %, 29 round trips
waist (mm)       0.4998   analytic 0.5000
at z_R (mm)      1.3456   analytic 1.3463

The Fox-Li mode has the radius of the ABCD eigenmode, and the aperture, 2.8 mode radii, costs 0.003 % per round trip. The Gaussian-Schell beam has the same waist as a coherent one but is 1.9 times larger after a Rayleigh range (1.346 mm against 0.707 mm), as w(z) = w₀ √(1 + (M² z / z_R)²) says.

Things to try¶

• Close the aperture to 0.6 mm (1.5 mode radii): the loss rises and the mode shrinks and departs from the Gaussian.
• Make the cavity unstable (lengthM of 1.2 m each, L > 4 f): a warning, and no eigenmode.
• "m2": 1: one mode, the coherent beam.