Optical Fibre: How Light Follows a Bent Thread

Aim the laser at the end of the fibre and watch the ray zigzag inside thanks to total internal reflection. Change the entry angle, the refractive indices of the core and cladding and the bend radius: the simulator shows the acceptance cone, the numerical aperture NA = √(n₁² − n₂²) and the place where light starts to leak out.

Fibre type (n₁ / n₂)

θ entry angle °

R bend radius mm

n₁ core

n₂ cladding

y₀ entry point from the axis mm

Acceptance cone (aperture)

5 rays at different angles

Test yourself: predict the result

The core is drawn 0.4 mm thick. A real telecom fibre has a core of about 9 µm, so it can be bent with a radius of centimetres: bend loss depends on the ratio of the bend radius to the core thickness. Losses at the end faces (Fresnel) and leakage wherever the angle of incidence on the wall is below critical are taken into account.

In the 1840s Jean-Daniel Colladon demonstrated a “light fountain” in Geneva: light sent into a jet of water travelled inside it, bending along with the jet (choose the “Water jet” type). Today almost the whole internet runs through glass fibres as thin as a human hair: light pulses cross oceans in submarine cables. Charles Kao received the 2009 Nobel Prize for his work on fibre-optic communication. Try bending the fibre more sharply — light starts leaking at the bend, and the power graph shows steps of loss.

KEY FORMULAS

Absolute refractive index

$$n = \frac{c}{v}, \qquad \lambda = \frac{\lambda_0}{n}$$

c = 3·10⁸ m/s is the speed of light in a vacuum, v the speed of light in the medium, λ₀ the wavelength in a vacuum, λ in the medium

▶ Detailed Explanation

The refractive index shows how many times slower light travels in a substance than in a vacuum: for air n ≈ 1.0003 (problems use 1), for water 1.33 (v ≈ 2.26·10⁸ m/s), for glass about 1.5 (v = 2·10⁸ m/s), for diamond 2.42 (v ≈ 1.24·10⁸ m/s). A medium with a larger n is called optically denser.

When light passes from one medium to another, its frequency ν does not change (as many crests leave the boundary as arrive), but the speed v = λν decreases — so the wavelength decreases too: λ = λ₀/n. Switch on the wavefronts in the simulator: in glass they are closer together and slower.

Critical angle of total internal reflection

$$\sin\alpha_0 = \frac{n_2}{n_1}, \qquad n_1 > n_2$$

Light goes from an optically denser medium (n₁) into a less dense one (n₂). For α > α₀ there is no refracted ray — all the light is reflected

▶ Detailed Explanation

From the law of refraction sin β = (n₁/n₂)·sin α. For n₁ > n₂ the right-hand side grows faster than sin α and reaches one at the angle α₀: β = 90°, the refracted ray grazes the boundary. For larger angles the equation has no solution — the light is totally reflected back. At a boundary with air: water — α₀ ≈ 48.8°, glass — 41.8°, diamond — 24.4°. For glass (1.5) against water (1.33): sin α₀ = 1.33/1.5 = 0.887, α₀ ≈ 62.5°.

The transition to TIR is smooth: as α approaches α₀ the refracted ray fades and the reflected one brightens (Fresnel equations). Total internal reflection underlies optical fibres, prism binoculars and periscopes, and reflectors; diamond’s small critical angle makes light bounce many times inside a cut stone, so it sparkles.

Numerical aperture of a fibre

$$\mathrm{NA} = \sin\theta_{max} = \sqrt{n_1^2 - n_2^2}$$

θ_max is the largest angle between a ray and the fibre axis at which light is still guided; n₁ is the core, n₂ the cladding, with air outside

▶ Detailed Explanation

A ray enters the end face at angle θ: sin θ = n₁·sin θ₁. Inside it hits the wall at φ = 90° − θ₁, and TIR requires sin φ = cos θ₁ ≥ n₂/n₁. The largest allowed θ₁ gives sin θ_max = n₁·sin θ₁ = n₁·√(1 − n₂²/n₁²) = √(n₁² − n₂²).

Plastic fibre (1.49/1.41): NA ≈ 0.48, θ_max ≈ 28.8°. Telecom fibre (1.468/1.462): NA ≈ 0.13, θ_max ≈ 7.6°. A glass rod in air (1.5/1.0): NA > 1 — it captures light at any angle. The larger the NA, the easier it is to couple light into the fibre, but the more the pulses spread out.

Travel time and modal dispersion

$$t = \frac{L\,n_1}{c}, \qquad \Delta t = \frac{L\,n_1}{c}\left(\frac{n_1}{n_2} - 1\right)$$

t is the time for the axial ray to traverse a fibre of length L; Δt is the delay of the steepest guided ray (φ = α₀)

▶ Detailed Explanation

The axial ray travels at c/n₁: light crosses 1000 km of glass fibre (n = 1.5) in 5 ms. An inclined ray zigzags and its path is 1/cos θ₁ times longer; for the steepest guided ray cos θ₁ = n₂/n₁, hence Δt.

For a 1 km plastic fibre (1.49/1.41) Δt ≈ 0.28 µs: pulses shorter than this merge, so the data rate is limited. Telecom fibre with close indices (1.468/1.462) gives Δt ≈ 20 ns per kilometre, and a single-mode fibre with a 9 µm core has only one path, so there is no modal dispersion at all.

How a fibre works. An optical fibre is a thin glass or plastic thread made of two layers: a core with refractive index n₁ and a cladding with a slightly smaller n₂. A ray inside the core hits the boundary with the cladding at a large angle φ. If φ exceeds the critical angle α₀ (sin α₀ = n₂/n₁), total internal reflection occurs — the light is reflected without loss and, bouncing again and again, travels along the fibre even along a curved path.

Acceptance cone and numerical aperture. Not every ray entering the end face stays inside. Entering at angle θ to the axis, the ray refracts: sin θ = n₁·sin θ₁ — and hits the wall at φ = 90° − θ₁. The condition φ ≥ α₀ gives the maximum entry angle sin θ_max = √(n₁² − n₂²) = NA — the numerical aperture. Rays within the cone of half-angle θ_max (the green cone) are guided; the others leak through the walls on the first reflections. Plastic fibre has NA ≈ 0.5, telecom fibre about 0.1–0.2.

Bending. At a bend the outer wall turns towards the ray, and the angle of incidence on it decreases. If the bend radius is small, the angle falls below critical: TIR breaks down and some light escapes into the cladding. That is why optical cables have a minimum bend radius. In the simulator you can see it as flashes at the bend and steps on the power graph P(s).

Speed and dispersion. In the core light travels at v = c/n₁ ≈ 2·10⁸ m/s. Rays entering at different angles travel different paths: in a straight fibre an inclined ray zigzags and lags behind the axial one, and at bends the rays also change their tilt. So a short pulse spreads out at the output — this is modal dispersion. It is reduced by making the difference between n₁ and n₂ small, or the core so thin (about 9 µm) that only one path remains. That is how the single-mode fibre of backbone links is made.

End-face losses. At the input and output some light is reflected by the end faces according to the Fresnel equations: about 4 % per glass surface at normal incidence. The simulator takes these losses into account as well.

Optical Fibre: How Light Follows a Bent Thread

KEY FORMULAS

Absolute refractive index

Critical angle of total internal reflection

Numerical aperture of a fibre

Travel time and modal dispersion

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Problems on this topic