Illustrated Physics
physics, drawn
Concept

Optical fibre — where it appears

A thread of glass that guides light along itself, by total internal reflection or by a graded index. Silica's loss is least at 1,550 nm and its dispersion vanishes at 1,273, so the two properties a link wants are at different wavelengths and the design is a negotiation between them.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

The critical angle for n = 1.5 into n = 1. Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 41.8°, beyond which no refracted ray exists at all.

The angle past which light cannot leave

Snell's law asks for the sine of an angle greater than one, and no such angle exists. What happens instead is a perfect mirror made out of nothing but a change of speed.

optics · Total internal reflection The wavelength at which a fibre stops smearing a pulse. Group-delay dispersion against wavelength for fused silica, computed by differencing Malitson's Sellmeier fit twice rather than read off a table, in picoseconds of spread per nanometre of source width per kilometre of fibre. The material curve crosses zero at 1273 nm — a property of the glass, fixed by where the ultraviolet and infrared absorptions balance, and not adjustable. Below it a fibre is normally dispersive and above it anomalously so, and at 1550 nm, where silica is most transparent, it is 21.9. The second curve adds a waveguide term, which is negative because a mode confined by a core spreads into the cladding differently at different wavelengths, and which depends on the fibre's geometry rather than on the glass; it is drawn here as a constant offset chosen to put the total zero at 1310 nm, which is what standard single-mode fibre is built to do. The consequence in a system is the thing worth carrying away: a source one nanometre wide sent 100 km through fibre at 1550 nm arrives 1833 picoseconds broader than it left, against 0.0 at the zero. That is the whole reason a network runs at one wavelength rather than another, and the reason the two quantities an engineer wants — lowest loss and zero dispersion — sit at different wavelengths and have to be reconciled by design rather than chosen.

The wavelength a fibre does not smear

Fused silica's index has a second derivative that passes through zero at 1,273 nanometres, and that is not a design choice — it is where the ultraviolet and infrared absorptions balance. A pulse sent at that wavelength arrives the shape it left. A pulse at the wavelength of least loss arrives 1,800 picoseconds wider after a hundred kilometres.

optics · Dispersion A cone of 14.3°, from two indices and nothing else. On the left, the acceptance cone of a step-index fibre with a core index of 1.4677 and a cladding of 1.4624. A ray entering steeper than the cone reaches the wall inside the critical angle and is refracted out at the first bounce; one inside it is trapped. The sine of the half-angle is √(n₁² − n₂²) = 0.125, which is 7.2° in air. On the right, that number against the fractional index difference between core and cladding. Nothing about the core's diameter appears: a fibre a hundred times thicker accepts exactly the same cone, and takes a hundred times the area's worth of light through it.

The cone a fibre will accept

A fibre takes light from a cone whose half-angle depends on two refractive indices and nothing else — not on how thick it is, not on how long, not on what is shining into it. That single number, squared and multiplied by the core's area, is all the light it will ever carry.

optics · Etendue A bend of 5 mm, and where the field has to give up. The transverse profile of the guided mode, with the core shaded and the radius at which a bend of 5 millimetres would require the field to outrun the cladding marked. Inside the core the field is a cosine; outside it decays, and the decay is what keeps the mode together. Bending the guide imposes a rigid rotation, so the field a distance x from the axis must travel faster in proportion to x — and past 13.5 micrometres it would have to travel faster than the cladding permits. There the field can no longer be evanescent, and what is there radiates away. The amount there is 1.63e-2 of the peak, which is why the loss is negligible until the caustic moves in, and then is not.

The mode that will not turn a corner

Bend a waveguide and the field has to go round with it, which means the part furthest from the centre has to travel faster. Past a certain distance it would have to travel faster than the surrounding medium allows, and everything out there radiates away — which is why bend loss is exponential in the radius and arrives all at once.

waves · Guided waves A delay that grows as the square root of the length. The differential group delay between the fastest and slowest polarisation states of a fibre built from sections 100 m long, each a slightly birefringent waveplate with its axis at a random angle, against length up to 400 km. Three individual fibres are drawn faint, and the root-mean-square delay over 300 of them heavy. The ensemble grows as a power 0.50 of the length — the square root, because each section rotates the polarisation it receives before adding its own delay, so the delays add like the steps of a random walk in three dimensions rather than like lengths laid end to end. At 100 km the mean delay is 5.18 ps against 5.00 ps for a coefficient of 0.5 ps/√km. Had the axes all been aligned, the same sections would have added to 172 ps at that length, growing in proportion, and the drawn dashed line leaves the frame within a few kilometres. The randomness is what keeps the delay small, and it is also what makes it impossible to compensate with a fixed device.

The delay that is a random variable

A fibre's core is very slightly elliptical, so its two polarisations travel at very slightly different speeds — and the ellipse turns, at random, every hundred metres or so. The delays of the pieces do not add. They random-walk, so the total grows as the square root of the length, follows the same distribution as the speeds of gas molecules, differs from one wavelength to the next, and on any particular day may be three times its average.

optics · Dispersion Four rays that arrive together. Meridional rays through a fibre whose index falls as the square of the distance from the axis, launched at four angles, drawn against distance along the fibre in millimetres and radius in units of the core radius. Each path is integrated from the ray equation. A steep ray swings out to where the index is lower and travels faster; a shallow one stays near the axis where the index is highest and travels slowest, and in a parabolic profile the two effects cancel: the four rays cross the axis within 0.36 per cent of a pitch of each other, measured off the traced paths rather than assumed. The pitch is 1.11 millimetres and it contains no launch angle, which is the whole of the result. Nothing about what the fibre accepts has changed — the acceptance cone is what the invariant fixed and it is untouched — and everything about when the light arrives has.

The same cone, and a different arrival

The invariant fixes what a guide accepts and says nothing about when it arrives, and the two turn out to be nearly independent. Shaping the index so that the rays which travel furthest also travel fastest cuts the spread in arrival times by a factor of five hundred, at the cost of exactly half the light — and the acceptance cone the invariant governs is untouched throughout.

optics · Etendue Two guides, and the two solutions they have. The transverse field of the two modes a pair of identical slab guides supports, against position across them, for guides half a micrometre wide separated by a gap of 300 nanometres at a wavelength of 1550 nanometres. A single guide has one fundamental mode; two guides side by side have two, and neither of them lives in one guide. The symmetric one is a single hump spanning both, the antisymmetric one has a node exactly between them — checked here to be exactly zero rather than nearly so — and they have slightly different propagation constants because the symmetric one has more of its field in the high-index gap region. That difference, computed from the slab's own dispersion condition, is 4.60e-2 per micrometre. Everything else about a coupler follows from it: a wave launched into one guide alone is the sum of the two supermodes in equal parts, they run at different speeds, and the interference between them moves the power from one guide to the other and back. There is no leakage in the account anywhere — only two solutions beating.

Two tails that swap everything

Bring two guides close enough for their evanescent tails to overlap and they do not leak a little power into each other. They exchange all of it, and then exchange it back, over a length fixed by the splitting between two modes that belong to neither guide — so a coupler is cut to a length rather than tuned to a ratio, and the length depends exponentially on a gap of a few hundred nanometres.

waves · Guided waves

Named alongside it

The objects these essays reach for when they reach for this one.

Guided wavesRefractive indexEvanescent waveNumerical apertureTotal internal reflectionBoundary conditionsCritical angleDispersionNormal modesPulse broadeningSnell's lawAttenuation

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