Köhler against a plane-wave cone average; energy; NA → 0 is the plane wave

← Verification · Cylinders · Focused beam

Reference: Köhler against a plane-wave cone average; energy; NA → 0 is the plane wave
Kind: method / analytic: an independent formulation inside this repo (a scattering matrix, a plane-wave superposition, a quadrature): proves the formulation
Calculated: detected power, absorbance
Agreement: 1e-8

The coated cylinder (0.7/1.2 µm) under Köhler illumination (NA 0.6/0.2, collection 0.7/0.3, field stop 30 µm): against a cone average of plane waves Köhler illumination is incoherent across the pupil: the cylinder sees each plane wave of the illumination cone on its own, over the chord the field stop lights. The reference averages the oblique plane-wave extinction widths and the scattered power that lands in the collection annulus over the cone (focused_checks.py); x is across the axis.

extinction, x, PyStratify detected, x, PyStratify extinction, y, PyStratify detected, y, PyStratify extinction, x, cone average detected, x, cone average extinction, y, cone average detected, y, cone average

Hover to read values.

largest relative difference: 3.95×10⁻¹⁵
PyStratify the lines, the reference the circles

The coated cylinder (0.7/1.2 µm) at λ = 3 µm: a focused beam as its NA closes, against the plane wave A focused beam's extinction, divided by its |E|² integrated along the cylinder's axis and the plane-wave extinction width, tends to 1 as the NA closes; the test asserts it is within 2e-3 at NA 0.02 and closer than at 0.04, and that a lossless cylinder absorbs nothing.

x polarized (across the axis), PyStratify y polarized (along the axis), PyStratify plane wave (= 1)

Hover to read values.

difference from 1 at NA 0.01: 4.04×10⁻⁵
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