WAVEGUIDES AND OPTICAL FIBERS

An undersea internet cable is a fistful of glass strands, each one finer than a human hair, carrying roughly a million simultaneous Zoom calls between continents. The light that does the carrying was launched from a semiconductor laser the size of a grain of rice in Virginia, and it will emerge twelve thousand kilometres later near Bude, Cornwall, having lost maybe four-fifths of its intensity along the way. Nothing in everyday experience suggests this is possible. A flashlight beam attenuates inside a metre of fog. How does anyone get a photon across the Atlantic?

The answer is §09.4 turned into an engineering discipline. A step-index optical fiber is a cylinder of glass with two regions: an inner core of refractive index n₁ surrounded by a cladding of slightly lower index n₂ < n₁. Any light ray inside the core that hits the core-cladding wall at an angle steeper than the critical angle θ_c = arcsin(n₂/n₁) is totally internally reflected — it cannot escape, by the same Snell-law geometry that made water surfaces into mirrors. Chain those reflections end to end and a ray can zig-zag down the fiber for kilometres without ever touching the plastic jacket on the outside. The whole internet runs on total internal reflection, exactly as Snell wrote it down in 1621.

Which rays from the outside world make it into the core? A ray striking the flat end-face refracts on entry by Snell, then must satisfy the wall-TIR condition once inside. Chain those two constraints and you get a single angle — the widest off-axis tilt in ambient air that the fiber can capture — whose sine has a clean closed form:

NA is the numerical aperture. It has no units; it is a sine. A typical single-mode telecom fiber has n₁ = 1.4682, n₂ = 1.4629, giving NA = √(1.4682² − 1.4629²) ≈ 0.132 — an acceptance cone of only about 7.6° from the axis. A multimode step-index plastic fiber with NA = 0.5 opens a 30° cone. The tradeoff shows up in every link budget in the industry: wide cones collect more light from an LED but admit more modes (many simultaneous ray paths, each taking a different zig-zag length) which smears short pulses into wide ones. Narrow cones are stricter — single-moded, long-distance — and need precise laser alignment.

Slide n_core up and the cone opens — the harder the index contrast, the more off-axis light you can funnel in. Slide the cladding index up toward the core and NA collapses to zero: the two glasses become identical and there is no boundary to TIR against. The fiber stops being a waveguide at all.

In a multimode step-index fiber a ray going straight down the axis takes length L to cross; a ray bouncing near θ_c zig-zags through a total path length L/cos(90° − θ_c) = L/sinθ_c. Both paths carry the same data pulse, but they arrive at different times. A short laser flash launched as one pulse splits on arrival into a smear of replicas — modal dispersion. Over kilometres of fiber the smear becomes longer than the pulse, and the optical link stops working.

Two fixes, both industry-standard. The first is to thin the core until only a single mode will propagate — below a cutoff diameter ~2.4 · λ / (π · NA), all higher-order ray paths are killed off by quantum-like boundary conditions at the core wall, and everything travels as the one remaining ground mode. This is single-mode fiber, the workhorse of long-haul telecom. The second is the graded-index fiber: instead of a hard step in n, the core has a smoothly tapered parabolic index, highest on axis and falling off toward the cladding. Rays spending time off-axis travel through lower-n regions where light moves faster, so the longer zig-zag paths go at higher speed and arrive at the same time as the axial ray. The arithmetic balances; modal dispersion is beaten without throwing away the multimode advantage of easy launch.

Fibers are not the only waveguide. At microwave wavelengths (centimetres), a hollow rectangular metal tube of internal dimensions a × b does the same job by boundary conditions instead of TIR — the transverse electric field must vanish at a perfect conductor, so only standing-wave patterns that fit the cross-section are allowed to propagate. Each allowed pattern is a mode, labelled TE_mn for transverse-electric with m half-wavelengths across a and n across b. Each mode has a sharp cutoff frequency below which it simply does not transmit:

The dominant TE₁₀ mode (lowest cutoff at c/(2a)) is the standard operating channel. WR-90, the X-band workhorse used in every airport surveillance radar, is a standardised 22.86 × 10.16 mm rectangle — which gives f_c(TE₁₀) = 3·10⁸ / (2 · 0.02286) ≈ 6.56 GHz. Run it from about 8.2 to 12.4 GHz and the next-higher mode is still below cutoff, so the guide transmits only TE₁₀ and acts exactly like a single-mode optical fiber at its own wavelength. Below 6.56 GHz the pipe is a brick wall.

This is where Heaviside shows up. In §06.7 we met the telegrapher's equations — Oliver Heaviside's 1880 model of an undersea cable as a lumped LC-ladder of tiny inductors and capacitors stretched along a wire. Take that ladder and shrink the spacing to zero; you get a distributed transmission line — the local wave equation on a pair of conductors. Now wrap the two conductors into a rectangular metal pipe: same distributed equations, same reflection coefficients at discontinuities, but with an extra boundary-condition constraint that selects a discrete mode spectrum and imposes a cutoff frequency. Every microwave waveguide, every coaxial cable, every optical fiber is a direct descendant of the transatlantic telegraph cables Heaviside was patching up in the 1870s. The engineering vocabulary — impedance, reflection, standing wave — transferred unchanged.

The last piece is why telecom fiber runs at the wavelengths it does. Glass is beautifully transparent in the visible, but not uniformly. Silica fiber loss across 800–1700 nm has two sculptors: short-wavelength Rayleigh scattering from density fluctuations frozen into the glass during drawing (α ∝ 1/λ⁴, the same law that makes daylight skies blue), and long-wavelength IR absorption from Si-O bond vibrations (exponentially rising past about 1600 nm). Between them lies a bowl — and inside the bowl, three operating windows.

850 nm was the first usable window — cheap GaAs lasers, short-reach multimode, roughly 2 dB/km loss (half the light gone every 1.5 km). 1310 nm is the second — standard single-mode fiber has zero material dispersion there, so pulses don't spread as they propagate, even though the loss floor is ~0.35 dB/km. 1550 nm is the promised land: 0.17 dB/km, the absolute minimum of the bowl. A pulse launched at 1550 loses half its intensity every 18 km, which means you can run 100 km of fiber between amplifiers, and the amplifier itself — an erbium-doped fiber amplifier (EDFA) — happens to gain cleanly in exactly that band because of an atomic accident of the erbium level diagram.

All three facts — the TIR rule that keeps light inside the core, the NA-sized acceptance cone that lets you couple in, the 1550-nm loss minimum that lets you go for a hundred kilometres between repeaters — combine into the cable that carries this paragraph to your screen. Every time you open a video call across an ocean, Snell's arcsin from 1621 and Heaviside's ladder from 1880 are cooperating across a thousand kilometres of hair-thin glass.

The first commercial transatlantic optical cable, TAT-8, went live in 1988 at 1310 nm and carried 40,000 simultaneous voice channels on two fiber pairs — a hundred-fold leap in capacity over its copper predecessor, at a fraction of the power budget. TAT-14, a generation later at 1550 nm with wavelength-division multiplexing, carried the same data rate on a single fiber. The present generation stuffs eighty independent wavelengths into each strand, each wavelength carrying roughly 100 Gb/s, and lights up hundreds of terabits per second between Virginia and Cornwall on a cable the diameter of a garden hose. There is no deep physics in that progression — only the persistent exploitation of a single arcsin and a single loss minimum. The internet is not an abstraction. It is a ray inside a pipe, pretending it is a wave, refusing to leave.