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V-b Curve

Symmetric step-index slab waveguide, TE/TM modes. Single-mode condition: V < π/2.

Wavelength (μm) Core thickness d (μm) Δ (%) Polarization

n_core (from Δ)

—

n_clad (silica, Sellmeier)

—

V (operating)

—

V cutoff (TE_1)

π/2 ≈ 1.5708

b (fundamental TE_0)

—

n_eff (fundamental TE_0)

—

Single mode?

—

Equations

V = (k₀·a)·√(n_core² − n_clad²), a = d/2, k₀ = 2π/λ

b = (n_eff² − n_clad²) / (n_core² − n_clad²) ∈ [0, 1]

Eigenvalue equation (u = κa, w = γa, u² + w² = V²):
TE: even → u·tan(u) = w, odd → −u·cot(u) = w
TM: even → u·tan(u) = (n_core/n_clad)²·w, odd → −u·cot(u) = (n_core/n_clad)²·w

Cutoff for mode m (TE and TM): V_c = m·π/2. Single-mode region: V < π/2.

Reference implementations

vb_curve.py VBCurveCalculator.java