Snell's law, found by searching

At its defaults it draws snell's law, found by searching. Paths from a point in a medium of index 1 to a point in one of index 1.5, and the optical path length of each against where it crosses the boundary. The curve is that length; the marked point is its minimum, located by golden-section search and not by any use of a law of optics. The angles there are 55.80° and 33.46°, which satisfy n₁sin θ₁ = n₂sin θ₂ to 1.0e-8. Every other drawn path is longer, and the flatness of the curve near the bottom is why light is not fussy: a path a tenth of the way off costs almost nothing.

fermat-path 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.

Paths from a point in a medium of index 1 to a point in one of index 1.5, and the optical path length of each against where it crosses the boundary. The curve is that length; the marked point is its minimum, located by golden-section search and not by any use of a law of optics. The angles there are 55.80° and 33.46°, which satisfy n₁sin θ₁ = n₂sin θ₂ to 1.0e-8. Every other drawn path is longer, and the flatness of the curve near the bottom is why light is not fussy: a path a tenth of the way off costs almost nothing.

The root-mean-square width of a collimated Gaussian beam in a Kerr medium, relative to its starting width, against distance in Rayleigh ranges, at powers of 0, 0.5, 1, 1.5 and 3 times the Gaussian threshold, the wavelength squared over 2π n0 n2. This curve is exact, not a model: the mean square width of any beam obeyed by the two-dimensional cubic wave equation is a parabola in distance, its coefficient set by one minus the power over the threshold, whatever happens to its shape. At low power the beam spreads by diffraction. At exactly the threshold its width never changes. Above it the parabola reaches zero at a finite distance — 1.41 Rayleigh ranges at 1.5 times, 0.71 at three times — which a width cannot do, so the beam must have collapsed to a point at or before it. The equation stops being true there; the glass usually breaks first.

Rays of a 1064 nm beam 2 mm across (1/e² radius 1 mm) entering glass with a nonlinear index of 2.7 × 10⁻²⁰ square metres per watt, at 4 times the critical power, traced by Fermat's condition through the index the beam's own intensity raises — 3.0 × 10⁻⁷ higher on the axis than far outside it — with the beam's profile held fixed. Rays near the axis cross it after 1.73 m, rays starting at the 1/e² radius after 3.63 m: the lens is stronger in the middle, so it is aberrated. The dashed lines are the same beam's ordinary diffraction spread at low power, its 1/e² radius growing over a Rayleigh range of 4.28 m. A real beam narrows as it focuses, which makes its own lens stronger, and the focusing runs away.

Two angles for a 1064 nm beam in the same glass, against the beam's radius on logarithmic axes: the angle at which diffraction spreads it (black), λ/πn0w, and the angle at which its own index, raised by its mean intensity, would bend its rays inward — the square root of twice the index rise over the base index (coloured, at 0.5, 1, 2, 4 times the critical power). Squeezing a beam of fixed power into half the width raises its intensity fourfold, which doubles the inward angle; it also doubles the diffraction angle. Every line has the same slope, so they never cross, and which one wins is set by the power alone: at the critical power the two angles agree to within 3 per cent at every size, a ray estimate that lands close to the exact answer. The critical power for this glass is 4.3 MW, whether the beam is a millimetre across or a hundred micrometres.

The distance at which a collimated Gaussian beam collapses, in Rayleigh ranges, against its power as a multiple of the critical power, on logarithmic axes: Marburger's fit to computed collapses (solid) and the bound the exact width law sets (dashed), beyond which the beam cannot survive. Just above the critical power the collapse distance runs to infinity; at ten times it is 0.159 Rayleigh ranges, and above that it falls only as the inverse square root of the power. For a laser pulse in air at 800 nm, 1 cm across, the critical power is about 3.0 GW and the Rayleigh range 98 m, so a 100 GW pulse collapses after about 7 m — which is how terawatt pulses come to leave thin luminous filaments running for hundreds of metres through the sky.

The critical power for self-focusing — 3.77 times the wavelength squared over 8π n0 n2 — for six media at the wavelengths they are usually used at, on a logarithmic axis, with typical published nonlinear indices. Air needs 3.0 GW, because its nonlinear index is a thousand times smaller than a solid's; water 1.8 MW; fused silica 4.3 MW at 1064 nm and 2.1 MW at 800 nm, since the power goes as the wavelength squared; carbon disulphide, whose molecules turn to line up with the light, 35 kW; silicon in the telecommunications band 23 kW. Continuous lasers almost never reach these powers; pulsed ones pass them routinely, which is why every high-power laser is designed around this number.

physicscheck asserts something about fermat-path 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.