Micro Bend Loss

f(z) = A sin(2πz/Λ), curvature C = A(2π/Λ)², δ = β − k n₂, Λc = 2π/δ

α = (π²/2) (k n₂)³ C² H′(k*)², k* = √(2 k n₂ s), s = 2π/Λ − δ > 0

Gaussian mode: α = (π/8) (k n₂)⁴ w⁶ C² s e^(−k n₂ w² s)

First-order coupled-mode theory: the local curvature tilts the index profile (n²(1 + 2xC)), coupling LP01 to radiation modes whose propagation constant differs by 2π/Λ. Following Petermann (1976), the radiation modes are plane waves in the cladding, so the loss depends on the slope H′ of the mode's Hankel transform at k*. H′ is computed in closed form for the exact step-index LP01 field (Lommel integrals); δ uses the exact LP01 β. Petermann's Gaussian mode is about 2× low near the default period and far too low at short periods, where the field's edge at the core boundary dominates. The Gaussian coupling reproduces Marcuse's tilt loss (π n w θ/λ)² for a sharp kink.

Textbook approximation for one weak, infinitely long sinusoidal deformation; real profiles are rounded, so the step-index field overstates the loss at very short periods. Real cables have random deformation spectra, and the discrete cladding modes give resonances near Λc that this continuum model smooths out. No elasto-optic correction.