Optical Fiber: Light Guided by Total Internal Reflection

An optical fiber is a hair-thin strand of glass that carries light over kilometers with almost no loss — and it does so using one of the simplest principles in optics: total internal reflection (TIR). Light enters one end of the fiber, bounces off the internal walls at angles greater than the critical angle, and zigzags all the way to the other end without escaping through the sides.

This technology underpins the modern internet. The backbone of global communications — undersea cables connecting continents, trunk lines between cities, and increasingly the "last mile" to homes — is made of optical fibers carrying data encoded as pulses of laser light.

The structure of an optical fiber

An optical fiber has two concentric layers:

1. Core: The central glass cylinder where light travels. Typical diameter: 8–62.5 μm (micrometers). Refractive index $n_{1} = 1.50$ (varies by design).

2. Cladding: A glass layer surrounding the core with a slightly lower refractive index $n_{2} = 1.48$. This small difference ($\Delta n = 0.02$) is enough to create TIR. Typical outer diameter: 125 μm.

The cladding is not just "insulation" — it's an essential optical component. Without it, the core would be surrounded by air ($n = 1$), which would work for TIR but would make the fiber extremely fragile and susceptible to surface contamination ruining the reflection.

Critical angle at the core-cladding interface

TIR occurs when light in the core hits the core-cladding boundary at an angle greater than the critical angle:

$\theta_{c} = \arcsin\!\left(\dfrac{n_{2}}{n_{1}}\right) = \arcsin\!\left(\dfrac{1.48}{1.50}\right) = \arcsin(0.9867)$

$\theta_{c} \approx 80.6°$

This critical angle is very large — 80.6° from the normal to the core-cladding surface. This means the light must hit the boundary at a very grazing angle (nearly parallel to the surface) to undergo TIR. Only rays that travel nearly straight down the fiber (at small angles to the axis) satisfy this condition.

The acceptance cone and numerical aperture

Not all light entering the fiber end will be guided. Only rays that enter within a certain cone — the acceptance cone — will hit the core-cladding boundary at angles exceeding $\theta_{c}$ and be guided by TIR. Rays outside this cone hit the boundary at too steep an angle, refract into the cladding, and are lost.

The half-angle of the acceptance cone, $\theta_{a}$, is found by tracing a ray that just barely undergoes TIR (hits the boundary at exactly $\theta_{c}$):

At the fiber end face, Snell's law gives (assuming air outside):

$n_{\text{air}} \sin\theta_{a} = n_{1} \sin\theta_{r}$

where $\theta_{r}$ is the refraction angle inside the core. This ray then hits the side wall at the complement of $\theta_{r}$, which must equal $\theta_{c}$:

$\theta_{r} = 90° - \theta_{c}$

Substituting:

$\sin\theta_{a} = n_{1}\sin(90° - \theta_{c}) = n_{1}\cos\theta_{c}$

Using $\cos\theta_{c} = \sqrt{1 - \sin^{2}\theta_{c}} = \sqrt{1 - (n_{2}/n_{1})^{2}}$:

$\sin\theta_{a} = n_{1}\sqrt{1 - \dfrac{n_{2}^{2}}{n_{1}^{2}}} = \sqrt{n_{1}^{2} - n_{2}^{2}}$

This quantity $\sqrt{n_{1}^{2} - n_{2}^{2}}$ is called the numerical aperture (NA):

$\text{NA} = \sqrt{n_{1}^{2} - n_{2}^{2}} = \sqrt{1.50^{2} - 1.48^{2}} = \sqrt{2.25 - 2.1904} = \sqrt{0.0596} \approx 0.244$

$\theta_{a} = \arcsin(0.244) \approx 14.1°$

So light must enter the fiber within a ±14.1° cone to be guided. This is a narrow cone, which is why aligning fibers requires precision equipment.

Types of optical fibers

Step-index multimode fiber (what we've been describing): Core diameter 50–62.5 μm, abrupt $n$ change at core-cladding boundary. Multiple ray paths (modes) exist. Rays at different angles travel different total distances, causing modal dispersion — different modes arrive at different times, blurring the signal. Good for short distances (LANs, within buildings) at lower data rates.

Graded-index multimode fiber: The refractive index decreases gradually from the center to the edge (parabolic profile). Rays farther from the axis travel through lower-$n$ material and speed up, partially compensating for their longer path. This reduces modal dispersion. Same 50–62.5 μm core.

Single-mode fiber: Extremely thin core (~8–9 μm), so narrow that only one mode (the fundamental mode) can propagate. No modal dispersion. This is the gold standard for long-distance telecommunications — undersea cables, city-to-city backbones. Requires laser sources (not LEDs) because the light must be coupled into the tiny core.

Signal loss in fibers

Fibers aren't perfect. Light is gradually lost through:

• Absorption: The glass itself absorbs a tiny amount of light. Modern ultra-pure silica fibers have minimum absorption at $\lambda = 1550$ nm (the "C-band"), where loss is only about 0.2 dB/km. This wavelength was chosen specifically because the glass is most transparent there.
• Scattering: Tiny density fluctuations in the glass scatter light (Rayleigh scattering, proportional to $1/\lambda^{4}$). This is the dominant loss mechanism at shorter wavelengths.
• Bending losses: If the fiber is bent too sharply, rays that would normally undergo TIR hit the boundary at angles below $\theta_{c}$ and escape. This is why fibers have a minimum bend radius.

With 0.2 dB/km loss, a signal can travel about 100 km before needing amplification (an erbium-doped fiber amplifier, EDFA). Undersea cables have amplifiers every 60–80 km.

Data capacity

A single fiber can carry enormous data rates. Modern dense wavelength-division multiplexing (DWDM) uses 80–160 different laser wavelengths (colors), each carrying 100–400 Gbps, for a total capacity of 10–80 Tbps per fiber. A single undersea cable bundle contains 8–24 fiber pairs and can carry hundreds of terabits per second — enough for millions of simultaneous HD video streams.

Common mistakes

• Thinking the critical angle is small. For optical fibers, $\theta_{c}$ is very large (close to 90°) because $n_{1}$ and $n_{2}$ are very close. This means the acceptance angle is small — only rays nearly parallel to the axis are guided.
• Confusing the acceptance angle with the critical angle. They're complementary-ish but not the same. $\theta_{c}$ is measured from the normal to the side wall. $\theta_{a}$ is measured from the fiber axis at the entry face. They're related but distinct angles.
• Thinking TIR is 100% efficient. At the interface, yes. But the fiber still has absorption and scattering losses. TIR just means no energy is lost at each bounce.

Try it in the visualization

Adjust the core and cladding refractive indices and watch the critical angle and acceptance cone change. Drag a ray's entry angle — if it's within the acceptance cone, it zigzags through the fiber; if it's outside, it leaks into the cladding. Switch between step-index and graded-index to see the ray paths curve in a graded fiber. Toggle the mode diagram to see how core diameter determines the number of guided modes.