A packet on deep water, ω = √(gk), 1.6 s apart

At its defaults it draws a packet on deep water, ω = √(gk), 1.6 s apart. A wave packet built from a Gaussian spread of wavenumbers about 1.57 per metre, drawn at two times 1.6 seconds apart, with its computed envelope ghosted around it. Between the two frames the envelope's peak moves 2.01 metres and the marked crest moves 3.95 metres, so the packet travels at 1.26 metres per second and the crests at 2.47 — a ratio of 0.51.

wave-packet is one function in lib/figures/waves.js — travelling, standing, adding and shifting. 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.

A wave packet built from a Gaussian spread of wavenumbers about 1.57 per metre, drawn at two times 1.6 seconds apart, with its computed envelope ghosted around it. Between the two frames the envelope's peak moves 2.01 metres and the marked crest moves 3.95 metres, so the packet travels at 1.26 metres per second and the crests at 2.47 — a ratio of 0.51.

Inside a tapered coupler 800 µm long at 1550 nm. Above: the fraction of the light in the first guide along the coupler, from integrating the coupled-mode equations, and dashed, the share of the first guide in the local supermode the light was launched into. They agree to within 8.3 percentage points along the whole length: the light does not beat between the guides but follows the supermode as that supermode changes from being in the first guide to being in the second. Below: the two supermodes' propagation constants relative to their average, which approach each other and repel across a gap of twice the coupling, 0.084 per micrometre, at the point where the guides are equally wide; dashed, the two single guides' constants, which cross.

The fraction of light that crosses from one silicon guide to its neighbour, 0.5 µm wide and 250 nm apart, against wavelength from 1300 to 1800 nm. The directional coupler is two identical guides cut to 37.6 µm, the length that transfers everything at 1550 nm; at other wavelengths the evanescent tails reach further or less far and the transfer falls to 3 per cent at the edge of the range. The tapered couplers have their widths swept 200 nm in opposite directions along their length, so the two guides' propagation constants cross in the middle. directional, 37.6 µm: above 95 per cent over 94 nm; tapered, 300 µm: above 95 per cent over 313 nm; tapered, 800 µm: above 95 per cent over 438 nm. Each curve comes from integrating the coupled-mode equations along the coupler, with the guide modes and their coupling solved afresh at every wavelength.

At 1550 nm, the fraction of light transferred against the length of the coupler. The directional coupler of two identical guides transfers all of its light at 37.6 µm and then gives it back, oscillating for ever with length. The tapered coupler, with the same guides, gap and 200 nm width swing, transfers little when short, because the crossing is swept too fast for the light to follow, and approaches complete transfer as it lengthens: above 95 per cent from 195 µm, 5 times the directional coupler's length. The solid curve is the coupled-mode integration; the dots are the Landau–Zener formula for the chance of jumping the gap, from the coupling and the rate at which the taper sweeps the detuning, and the two agree.

At 1550 nm, the fraction transferred against length for two tapers with the same guides, gap and 200 nm width swing. The linear taper changes the widths at a steady rate along its length and stays above 95 per cent from 280 µm. The shaped taper changes them quickly near its ends, where the guides are far from matched and the pair modes far apart, and slowly near the middle, where they cross and the gap between the modes is smallest — keeping the rate at which the light's mode turns constant along the whole length. It stays above 95 per cent from 40 µm, 86 per cent shorter. Both come from integrating the coupled-mode equations along the taper.

At 1550 nm, the fraction transferred when the two guides are not made as drawn: one wider and the other narrower by the amount across, a fabrication error of a few nanometres being ordinary. The directional coupler cut to 37.6 µm transfers everything only when the guides match, and with a 10 nm mismatch it transfers 67 per cent, with 20 nm 15 per cent: a detuning between the guides caps how much can cross. The linear tapered coupler 800 µm long transfers 99.7 and 99.8 per cent at the same errors, because a mismatch only moves the point along the taper where the two guides' constants cross, and the slow taper is slow everywhere. The shaped taper 60 µm long, which reaches full transfer in a fraction of the length by being slow only in its middle, transfers 82 and 64 per cent: a mismatch moves the crossing out of the slow middle into a part of the taper that hurries, where the light cannot follow.

physicscheck asserts something about wave-packet 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.