Axis path equals general path; index-matched sphere invisible; NA → 0 is the plane wave

← Verification · Spheres · Focused beam
Reference
axis path equals general path; index-matched sphere invisible; NA → 0 is the plane wave
Kind
analytic / method: a closed form, limit or conservation law: independent of any code
Calculated
detected power
Agreement
1e-12
The same coated sphere, λ = 3 µm: a focused beam as its NA closes, against the plane wave A focused beam's extinction, divided by its intensity at the focus and the plane-wave cross section, tends to 1 as the NA closes and the beam becomes a plane wave across the sphere; a tightly focused beam sees less of it.
0.10.20.30.40.50.60.80.850.90.951NAextinction / (|E(0)|² σ_ext)
focused extinction / (|E(0)|² σext), PyStratify plane wave (= 1)
Hover to read values.
  • difference from 1 at NA 0.01: 5.72×10⁻⁵
  • PyStratify the lines, the reference the circles
The coated sphere at λ = 3 µm, focus moved along the axis: the fast on-axis path against the general one On the axis the beam couples only to azimuthal orders m = ±1, and PyStratify takes a shortcut; the general path, for a focus anywhere, must give the same numbers. NA 0.5/0.23 in, 0.7/0.1 collected.
-3-2-101230.640.660.680.70.720.74focus position z [µm]power / incident
extinction, on-axis path (PyStratify) detected, on-axis path (PyStratify) extinction, general path detected, general path
Hover to read values.
  • largest relative difference: 9.99×10⁻¹⁶
  • PyStratify the lines, the reference the circles

Run it yourself: focused_checks.py · test_focused.py · figures.py · FOCUSED_CHECKS.md on GitHub, PyStratify 97e52a9