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 fraction of light transmitted straight through a 5 mm layer of N-BK7 grains packed to half its volume in the matching liquid, against wavelength, for grains of radius 5, 10 and 20 μm, from van de Hulst's extinction for large, weakly refracting particles. At the crossing the grains vanish and the layer is clear. A few tens of nanometres away each grain shifts the light passing through it in phase by enough to scatter a little of it out of the beam, and a thousand grains in a row scatter all of it: the passband is 51.7 nm wide at half height for 5 μm grains, 36.3 nm for 10 μm, 25.5 nm for 20 μm. Larger grains make a narrower filter, because the phase a grain imposes grows with its size; the same mismatch of index that a fine powder lets through, a coarse one scatters.

The refractive index against wavelength of N-BK7 glass, from its Sellmeier fit, and of an immersion liquid chosen to match it at the sodium d line, 589.3 nm, where both are 1.5167; the liquid disperses more, with an Abbe number of 35 against the glass's 64.2. At 450 nm the liquid's index exceeds the glass's by 73.2 × 10⁻⁴; at 650 nm it falls short by 17.5 × 10⁻⁴. The difference changes by −34.1 × 10⁻⁶ per nanometre near the crossing. Only at the crossing are the two materials optically the same; everywhere else light passing from one to the other is bent and reflected, a little more the further from the crossing it is.

The fraction of light reflected at normal incidence by the surface of N-BK7 glass immersed in the matching liquid, against wavelength, on a logarithmic scale, from Fresnel's formula, the square of the difference of the two indices over their sum. In air the same glass reflects 4.2 per cent. In the liquid it reflects 5.7·10⁻⁶ of blue light at 450 nm and 3.4·10⁻⁷ of red at 650 nm, and nothing at all at 589 nm. A glass rod lowered into such a liquid disappears, except that its outline glows faintly: at the edges, where light grazes the surface and even a small index difference bends and reflects it, the wavelengths on either side of the match are turned aside and the match is not, so the outline is tinged blue on one side of the colour of the match and orange on the other.

The extinction efficiency of a large, weakly refracting sphere — the light it removes from a beam, as a multiple of its cross-section — against the phase difference ρ = 2ka|m − 1| between a ray through its middle and one past it, on logarithmic scales, from van de Hulst's anomalous-diffraction formula. For small ρ the efficiency goes as half the square of ρ: halve the index mismatch and the grain removes a quarter as much. Near ρ = 4 it peaks at 3.17 and at large ρ it settles at 2, twice the grain's shadow, because it diffracts as much light as it blocks. A single 10 μm grain at 550 nm reaches ρ = 1 only when the indices differ by 0.0066, far from the crossing. But a packed layer holds so many grains that it goes opaque while each is still deep in the weak regime where extinction goes as the square of ρ: the 5 mm layer drawn beside this halves its light at ρ = 0.09, an index difference of about 0.0006.

The same layer of 10 μm grains at 15, 20 and 25 °C. The liquid's index falls by 4.0 × 10⁻⁴ for each kelvin of warming while the glass's barely changes, so the crossing slides along the spectrum: 661 nm at 15 °C, 589 nm at 20 °C, 538 nm at 25 °C — −11.5 nm for every kelvin. A Christiansen filter is therefore a tunable filter whose knob is temperature, and an untuned one is useless unless held still to a fraction of a degree; the passband changes colour in the hand as the hand warms it.

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.