Two glasses, and how differently they disagree with themselves

At its defaults it draws two glasses, and how differently they disagree with themselves. The refractive index of N-BK7 and F2 across the visible, each computed from its manufacturer's Sellmeier coefficients. Both curves fall from blue to red, which is why any single lens has a shorter focal length for blue light than for red. What separates the two glasses is not where they sit but how steeply they fall: over the same interval F2 changes index by 17.1 parts in a thousand and N-BK7 by only 8.1. The ratio of a glass's index-minus-one to that difference is its Abbe number, which is 64.2 for the crown and 36.4 for the flint — a single figure of merit saying how much bending is bought per unit of colour spread, and the only property of a glass the achromatic condition uses. The three vertical lines are the Fraunhofer wavelengths the definition is stated at.

chromatic-focus is one function in lib/figures/optics.js — rays, lenses, mirrors and what light does to a surface. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

The refractive index of N-BK7 and F2 across the visible, each computed from its manufacturer's Sellmeier coefficients. Both curves fall from blue to red, which is why any single lens has a shorter focal length for blue light than for red. What separates the two glasses is not where they sit but how steeply they fall: over the same interval F2 changes index by 17.1 parts in a thousand and N-BK7 by only 8.1. The ratio of a glass's index-minus-one to that difference is its Abbe number, which is 64.2 for the crown and 36.4 for the flint — a single figure of merit saying how much bending is bought per unit of colour spread, and the only property of a glass the achromatic condition uses. The three vertical lines are the Fraunhofer wavelengths the definition is stated at.

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 differential group delay of 3000 independent 100 km fibres, each built from 1000 randomly oriented sections, as a histogram, against the Maxwell distribution with the same mean. The mean is 5.01 ps and the ratio of mean to root-mean-square is 0.921, against 0.921 for a Maxwellian — the length of a vector whose three components are independent Gaussians, which is what a long random walk in three dimensions produces. The same distribution describes one fibre watched over time or across wavelength, because temperature and vibration keep reshuffling the sections' axes. So a fibre does not have a delay; it has a mean delay, and on any given day at any given wavelength it may be well above it.

The probability that the differential group delay exceeds a given multiple of its mean, on a logarithmic scale, from the Maxwell distribution, with the fraction of 3000 simulated fibres beyond each of four multiples marked and checked against it within its statistical error. The tail falls faster than exponentially, and it does not stop: beyond twice the mean the probability is 1.71e-2, beyond three times it 4.20e-5 — about 22 minutes a year for a link watched continuously. A system is therefore not designed to survive its fibre's delay but to survive it except for an agreed fraction of the time, and three times the mean is the multiple a designer usually picks.

The differential group delay of a single 100 km fibre, built from 1000 randomly oriented sections, across a band 4.4 THz wide — about the width of the erbium amplifier band — at 901 frequencies. It is not a constant of the fibre: it wanders between 0.6 and 9.8 ps, with a band average of 4.73 ps. The delay vector loses half its correlation over about 73 GHz, which is of the order of one over the mean delay: frequencies further apart than that see effectively independent fibres. A channel sitting on a peak is unlucky and its neighbour a few channels away may not be, and nothing about the fibre's construction says which will be which.

The longest fibre a signal can cross before its mean polarisation delay reaches a tenth of one bit period, against the bit rate, on logarithmic axes, for fibres with PMD coefficients of 0.05, 0.1, 0.5, 1 ps/√km. Because the delay grows as the square root of length, the tolerable length falls as the inverse square of the bit rate — quadruple the rate and a sixteenth of the distance remains. At 0.05 ps/√km: 40,000 km at 10 Gb/s, 2,500 km at 40 Gb/s, 400 km at 100 Gb/s; at 0.1 ps/√km: 10,000 km at 10 Gb/s, 630 km at 40 Gb/s, 100 km at 100 Gb/s; at 0.5 ps/√km: 400 km at 10 Gb/s, 25 km at 40 Gb/s, 4.0 km at 100 Gb/s; at 1 ps/√km: 100 km at 10 Gb/s, 6.3 km at 40 Gb/s, 1.0 km at 100 Gb/s. Fibre laid before the effect mattered carries coefficients near the top of that range, which is why links that were comfortable at ten gigabits a second became unusable at forty without a change to the glass.

physicscheck asserts something about chromatic-focus that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.