Step-index fibre dispersion relations (TE₀₁, TM₀₁; Snyder & Love)

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Reference: step-index fibre dispersion relations (TE₀₁, TM₀₁; Snyder & Love)
Kind: analytic: a closed form, limit or conservation law: independent of any code
Calculated: guided poles behind the guided channel; rates smooth through a mode's cutoff
Agreement: 1e-10 in n_eff; curvature < 2e-3

A step-index fibre, n = 2, radius 150 nm, in air: guided modes against the dispersion relations Snyder & Love's TE₀₁ and TM₀₁ relations, J₁(u)/(u J₀(u)) + c K₁(w)/(w K₀(w)) = 0, solved by bisection, against the poles PyStratify finds behind a dipole's guided channel; HE₁₁ (no closed form of this kind, and no cutoff) is shown as found; the shorter wavelengths guide higher modes too, not drawn. TE₀₁ and TM₀₁ reach cutoff at 0.392 µm, where V = 2.405.

TE₀₁, PyStratify TM₀₁, PyStratify HE₁₁, PyStratify TE₀₁, dispersion relation TM₀₁, dispersion relation

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largest |n_eff − n_eff(relation)| for TE₀₁ and TM₀₁: 4.84×10⁻¹⁴
PyStratify the lines, the reference the circles

An electric dipole 100 nm outside a step-index fibre (n = 2, radius 150 nm) through the TE₀₁/TM₀₁ cutoff TE₀₁ and TM₀₁ are cut off near 0.678 µm: the guided power of the radial and azimuthal dipoles, which they carried, drops there and reappears as escaping power (HE₁₁ stays guided), while the total, from the Green's function, passes through smoothly (the test bounds its second difference over 0.66–0.70 µm by 2e-3). Every mode PyStratify finds is checked against the dispersion relations in the fibre-dispersion plot.

radial, total (PyStratify) azimuthal, total (PyStratify) axial, total (PyStratify) radial, guided (PyStratify) azimuthal, guided (PyStratify) axial, guided (PyStratify) radial, escape + guided azimuthal, escape + guided axial, escape + guided

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largest relative difference, total against escape + guided: 1.16×10⁻¹²
PyStratify the lines, the reference the circles

Run it yourself: unified_checks.py · test_unified.py · figures.py · UNIFIED_CHECKS.md on GitHub, PyStratify 97e52a9