Nanowire Optical Waveguide: Mode Confinement

Nanowire & Light

Core material Silicon (n=3.5)

Wire radius 60 nm

Light wavelength (vacuum) 550 nm

Guided mode shown

Waveguide State

V-number

0.00

Guided modes

1 / 4

Selected mode n_eff

1.000

● GUIDED

How it works

A semiconductor nanowire surrounded by air acts as a cylindrical step-index optical waveguide — the same physics as a fiber-optic core, just a few hundred nanometres wide. Whether it confines light at all is set by the dimensionless V-number:

V = (2π a / λ) · √(n₁² − n₀²)
  a  = wire radius, λ = vacuum wavelength
  n₁ = core index (semiconductor), n₀ = 1 (air cladding)

Each guided mode (LPₗₘ in the linearly-polarised approximation) only exists once V exceeds its cutoff — the m-th zero of the Bessel function J_l−1. LP01 has zero cutoff and is always guided, which is why a nanowire never fully "turns off"; higher modes switch on one by one as the wire gets fatter or the light gets bluer:

LP01 cutoff V = 0        (always guided)
LP11 cutoff V = 2.405    (1st zero of J0)
LP21 cutoff V = 3.832    (1st zero of J1)
LP02 cutoff V = 3.832    (1st zero of J1)

For a guided mode, the transverse field is a Bessel function J_l inside the wire and a decaying modified Bessel function K_l outside it (the evanescent tail). Its transverse wavenumber u is interpolated between the exact cutoff value (u = V_c, tail fully delocalised) and the exact large-V asymptote (u → j_l,m, tail collapsed onto the core) — both endpoints are real Bessel-zero values; the interpolation shows the correct qualitative behaviour without a full transcendental root search:

w = √(V² − u²)
b = 1 − (u/V)²                (normalised propagation constant)
n_eff = √(n₀² + b·(n₁² − n₀²))

The scene shows several cross-sections along the wire: dot height and colour encode |E(r,φ)|² for the selected mode, and each ring's overall brightness is modulated by cos(β·z − ωt) to animate the wave travelling along the wire — this is exactly the confinement mechanism behind nanowire photodetectors, LEDs and lasers, and why "how tightly is light held inside the wire" is a real design question, not just an intro-textbook abstraction.

Field intensity |E|² Nanowire core

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