Optical Fiber

How light gets trapped

An optical fiber has two concentric glass regions: a central core with refractive index n_core, and a surrounding cladding with a slightly lower index n_clad. The whole thing is drawn from a glass preform into a strand typically 125 µm in outer diameter — about the thickness of a human hair — and wrapped in a protective polymer coating.

The physics that confines the light is total internal reflection. When a ray inside the core hits the boundary with the cladding, Snell's law governs what happens. If the angle of incidence (measured from the normal) exceeds the critical angle

θc = arcsin(n_clad / n_core)

then no light refracts into the cladding at all — 100% of it reflects back into the core. Unlike a metal mirror, which absorbs a few percent at every bounce, total internal reflection is lossless in principle. A ray can bounce millions of times along a kilometer of fiber and lose almost nothing to the reflection itself.

For a typical fiber with n_core = 1.4675 and n_clad = 1.4622 (a relative index difference Δ ≈ 0.36%), the critical angle is θc = arcsin(1.4622/1.4675) = 85.1°. So only rays travelling within about 4.9° of the fiber axis stay guided — a narrow cone, but enough.

Acceptance cone and numerical aperture

Not every ray you shine at the fiber face gets guided. A ray entering at too steep an angle will hit the boundary below θc and leak out. The half-angle of the cone of rays that are accepted defines the numerical aperture:

NA = n_0 · sin(θ_max) = √(n_core² − n_clad²)

where n₀ ≈ 1 is the index of air. The numerical aperture is a single number that captures both how much light the fiber can collect and how steeply rays bounce inside it. For the Δ = 0.36% example, NA = √(1.4675² − 1.4622²) ≈ 0.125, giving an acceptance half-angle θ_max ≈ 7.2°. Multimode fibers with larger Δ reach NA ≈ 0.20–0.29, accepting a much wider cone — handy for coupling cheap LEDs, but it lets in many steep rays that cause dispersion.

Modes and the V-number

Ray optics is a useful cartoon, but light is a wave. Solving Maxwell's equations in the cylindrical waveguide shows the fiber supports a discrete set of modes — stable transverse field patterns that propagate without changing shape. How many modes a fiber carries is set by the dimensionless normalized frequency, or V-number:

V = (2π · a / λ) · NA

where a is the core radius and λ the free-space wavelength. When V < 2.405 (the first zero of the Bessel function J₀), only the fundamental LP₀₁ mode survives — the fiber is single-mode. Above that, the number of guided modes grows roughly as V²/2. This is exactly why single-mode fiber needs such a small core: to keep V below cutoff at 1550 nm you need a ≈ 4–5 µm, hence the ~9 µm core diameter.

Property	Single-mode (SMF)	Multimode (MMF)
Core diameter	8–9 µm	50 or 62.5 µm
Modes carried	1 (LP₀₁)	hundreds to thousands
Typical NA	0.12–0.14	0.20–0.29
Source	laser (1310 / 1550 nm)	VCSEL / LED (850 nm)
Dominant dispersion	chromatic	modal
Reach	tens to hundreds of km	~100–550 m
Use case	long-haul, metro, FTTH	data center, LAN

Attenuation — why glass is now so clear

Loss in fiber is measured logarithmically in decibels per kilometer:

α (dB/km) = (10 / L) · log₁₀(P_in / P_out)

Three mechanisms dominate. Rayleigh scattering off frozen-in density fluctuations in the glass scales as 1/λ⁴, so it falls steeply toward longer wavelengths. The infrared absorption tail of vibrating Si–O bonds rises sharply beyond ~1700 nm. And historically a peak of OH⁻ absorption sat near 1383 nm (the "water peak"), now eliminated in modern low-water fibers. The minimum of all these contributions lands at 1550 nm, where loss bottoms out around 0.2 dB/km — close to the fundamental Rayleigh limit for silica.

To appreciate how transparent that is: 0.2 dB/km means light keeps about 95.5% of its power per kilometer, and you could see clearly through several kilometers of this glass. Corning's 1970 breakthrough was the first fiber below 20 dB/km — already a thousand-fold better than the best optics of the day, and the threshold Charles Kao had argued in 1966 would make fiber telecom viable, work that won him the 2009 Nobel Prize.

Wavelength / window	Typical loss	Note
850 nm	~2.5 dB/km	multimode, cheap VCSELs
1310 nm (O-band)	~0.33 dB/km	zero dispersion in standard SMF
1550 nm (C-band)	~0.20 dB/km	minimum loss; EDFA amplification
1383 nm (water peak)	historically ~2 dB/km	removed in low-water fiber
Window glass (compare)	~10⁴ dB/km	opaque after a few meters

Dispersion — why pulses spread

Attenuation limits how far a signal reaches; dispersion limits how fast you can send bits, because it smears sharp pulses until adjacent ones overlap. In multimode fiber the main culprit is modal dispersion: a steep zig-zag ray travels a longer path than an axial ray, so the modes arrive at different times. Graded-index multimode fiber fights this by tapering the index profile so steep rays speed up in the lower-index outer region, roughly equalizing arrival times.

Single-mode fiber has no modal dispersion, but it still suffers chromatic dispersion: different wavelengths in a real (non-monochromatic) pulse travel at slightly different group velocities. Standard SMF has near-zero chromatic dispersion at 1310 nm and about +17 ps/(nm·km) at 1550 nm. Engineers manage it with dispersion-compensating fiber, dispersion-shifted designs, and coherent digital signal processing that undoes the spreading electronically.

Worked numbers

Quantity	Calculation	Result
Critical angle (Δ = 0.36%)	arcsin(1.4622 / 1.4675)	85.1°
Numerical aperture	√(1.4675² − 1.4622²)	0.125
Acceptance half-angle	arcsin(0.125)	7.2°
Power after 100 km at 0.2 dB/km	10^(−20/10)	1% remains
Single-mode cutoff core radius (1550 nm, NA 0.12)	2.405·λ / (2π·NA)	≈ 4.9 µm
Speed of light in core (n = 1.4675)	c / n	≈ 2.04 × 10⁸ m/s

JavaScript — fiber calculations

// Critical angle for total internal reflection (degrees)
function criticalAngle(nCore, nClad) {
  return Math.asin(nClad / nCore) * 180 / Math.PI;
}
console.log(criticalAngle(1.4675, 1.4622).toFixed(1)); // 85.1 deg

// Numerical aperture and acceptance half-angle
function numericalAperture(nCore, nClad) {
  return Math.sqrt(nCore * nCore - nClad * nClad);
}
const NA = numericalAperture(1.4675, 1.4622);
console.log(NA.toFixed(3));                                  // 0.125
console.log((Math.asin(NA) * 180 / Math.PI).toFixed(1));     // 7.2 deg (acceptance)

// V-number: V < 2.405 => single mode
function vNumber(coreRadius_um, wavelength_um, NA) {
  return (2 * Math.PI * coreRadius_um / wavelength_um) * NA;
}
console.log(vNumber(4.5, 1.55, 0.125).toFixed(2)); // 2.28 -> single mode

// Approximate number of guided modes (step-index, large V)
function numModes(V) {
  return Math.max(1, Math.round(V * V / 2));
}
console.log(numModes(vNumber(25, 0.85, 0.275))); // multimode: thousands

// Power surviving a length of fiber given loss in dB/km
function powerFraction(lengthKm, lossDbPerKm) {
  const dB = lengthKm * lossDbPerKm;
  return Math.pow(10, -dB / 10);
}
console.log((powerFraction(100, 0.2) * 100).toFixed(1) + '%'); // 1.0% after 100 km
console.log((powerFraction(1, 0.2) * 100).toFixed(1) + '%');   // 95.5% per km

// Group delay spread from chromatic dispersion (ns)
function chromaticSpread_ns(D_ps_nm_km, spectralWidth_nm, lengthKm) {
  return D_ps_nm_km * spectralWidth_nm * lengthKm / 1000; // ps -> ns
}
console.log(chromaticSpread_ns(17, 0.1, 80).toFixed(2)); // 0.14 ns over 80 km

Where optical fiber shows up

• Telecom backbone. Transoceanic submarine cables and continental long-haul links carry essentially all internet traffic on single-mode fiber, with EDFAs every ~80–100 km.
• Fiber to the home (FTTH). GPON and XGS-PON deliver gigabit broadband over a single fiber shared by a passive splitter.
• Data centers. Short multimode runs (OM3/OM4/OM5) connect servers and switches at 100–800 Gbps using 850 nm VCSEL transceivers.
• Medicine. Endoscopes and fiber bundles relay images and deliver laser light inside the body; specialty fibers guide surgical lasers.
• Sensing. Fiber Bragg gratings and distributed acoustic sensing turn a fiber into a kilometers-long strain, temperature, or vibration sensor.
• Fiber lasers and amplifiers. Rare-earth-doped cores (Er, Yb) make high-power lasers and the EDFAs that revolutionized long-haul transmission.

Common mistakes

• Measuring the angle from the wrong reference. The critical angle is measured from the normal to the boundary, not from the fiber axis. Rays must hit the wall above θc (grazing) to stay guided.
• Thinking the core must be hollow. The core is solid glass; light is guided through it, not down an empty tube. Hollow-core photonic-crystal fibers exist but are a specialized exception.
• Confusing attenuation with dispersion. Attenuation reduces power (limits reach); dispersion spreads pulses (limits bandwidth). A link can be loss-limited or dispersion-limited depending on length and bit rate.
• Assuming a bigger NA is always better. A large NA collects more light but admits steep rays that worsen modal dispersion — fine for short multimode links, bad for long-haul.
• Ignoring bend loss. Total internal reflection only holds if rays stay above θc. Tight bends push outer rays below the critical angle and they refract out, so standard fiber has a minimum bend radius.
• Forgetting the wavelength dependence of mode count. A fiber that is single-mode at 1550 nm may be multimode at 850 nm, because V scales as 1/λ.