Graded index fibre ray transmission

Features of graded-index fibre

• The core refractive index is not uniform: it is highest at the axis and decreases gradually to the cladding value n2 at the core boundary.
• Usually multimode, with core diameter 50 or 62.5 µm and cladding 125 µm; NA about 0.2–0.3.
• Rays follow curved, sinusoidal paths instead of zig-zags.
• Much lower intermodal dispersion than step-index multimode fibre, so bandwidth is about 100 times higher (~0.5–2 GHz·km).
• Accepts LED or VCSEL sources and is easier to couple than single-mode fibre.
• More costly to make than step-index multimode fibre.

Refractive index profile

n(r) = n1 · √( 1 − 2Δ (r/a)^α )     for r < a  (core)
n(r) = n1 · √( 1 − 2Δ ) = n2        for r ≥ a  (cladding)

Δ = (n1² − n2²) / (2n1²) ≈ (n1 − n2)/n1

where a is the core radius and α is the profile parameter: α = 1 triangular, α = 2 parabolic (the usual choice, near optimum), α → ∞ step index.

        n
        ^
     n1 |      .--.
        |    /      \        parabolic core
        |   /        \
     n2 |--'          '---   cladding
        +-----+---+---+----> r
            -a    0   a

Ray transmission in multimode graded-index fibre

 cladding ________________________________
           .-.       .-.       .-.
 core    /    \    /    \    /    \   outer ray
 axis ---*----------------------------->  axial ray
           \  /     \  /     \  /
 cladding ________________________________

• A ray entering at an angle moves into regions of lower index. By Snell's law it bends continuously away from the normal, until it turns back towards the axis. It never reaches the cladding in the ideal case; the turning is gradual refraction rather than a single sharp reflection.
• So rays travel in sinusoidal (meridional) or helical (skew) paths that repeatedly cross the axis.
• The axial ray takes the shortest path but in the highest-index (slowest) glass. Oblique rays take longer paths but spend most of their time in lower-index (faster) glass, since velocity v = c/n(r).
• With a near-parabolic profile, the extra path length is almost exactly compensated by the higher speed, so all modes arrive at about the same time. Intermodal pulse spreading falls from about L·n1Δ/c (step index) to about L·n1Δ²/(8c) (parabolic), giving much larger bandwidth.
• The local numerical aperture NA(r) = √(n(r)² − n2²) is maximum at the axis and zero at the core edge, so the fibre collects less light than a step-index fibre of the same n1 and n2.