The Cornu spiral, and the chords that are amplitudes

At its defaults it draws the cornu spiral, and the chords that are amplitudes. Fresnel's two integrals plotted against each other, traced from v = -4.2 to 4.2. Arc length along the curve is v — the distance along the wavefront in units of √(λL/2) — and the amplitude arriving from any stretch of that front is the straight chord between the stretch's two endpoints, not the length of curve between them. The two eyes at ±(½, ½) are where the far parts of an unobstructed front pile up, so the whole open front is the chord between them, of length 1.4142. A straight edge blocks half of it and leaves a chord of 0.7071 — exactly half the amplitude, so a quarter of the intensity, at the geometrical shadow's edge. The chord drawn here runs to v = 1.217, is 1.6556 long, and is the longest chord from the eye that exists: its square over the open chord's square is 1.370, which is why the brightest fringe outside a shadow is brighter than no obstacle at all.

fresnel-edge 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.

Fresnel's two integrals plotted against each other, traced from v = -4.2 to 4.2. Arc length along the curve is v — the distance along the wavefront in units of √(λL/2) — and the amplitude arriving from any stretch of that front is the straight chord between the stretch's two endpoints, not the length of curve between them. The two eyes at ±(½, ½) are where the far parts of an unobstructed front pile up, so the whole open front is the chord between them, of length 1.4142. A straight edge blocks half of it and leaves a chord of 0.7071 — exactly half the amplitude, so a quarter of the intensity, at the geometrical shadow's edge. The chord drawn here runs to v = 1.217, is 1.6556 long, and is the longest chord from the eye that exists: its square over the open chord's square is 1.370, which is why the brightest fringe outside a shadow is brighter than no obstacle at all.

Fresnel's two integrals plotted against each other, traced from v = -4.2 to 4.2. Arc length along the curve is v — the distance along the wavefront in units of √(λL/2) — and the amplitude arriving from any stretch of that front is the straight chord between the stretch's two endpoints, not the length of curve between them. The two eyes at ±(½, ½) are where the far parts of an unobstructed front pile up, so the whole open front is the chord between them, of length 1.4142. A straight edge blocks half of it and leaves a chord of 0.7071 — exactly half the amplitude, so a quarter of the intensity, at the geometrical shadow's edge. The chord drawn here runs to v = 1.217, is 1.6556 long, and is the longest chord from the eye that exists: its square over the open chord's square is 1.370, which is why the brightest fringe outside a shadow is brighter than no obstacle at all.

Intensity across the edge of a shadow cast by a straight edge in 550 nm light, at 3 screen distances, in units of the unobstructed intensity. Geometrical optics predicts a step: full brightness on one side of zero and nothing on the other. What is there instead is a set of fringes outside the shadow, decaying outward, and a smooth fade to darkness inside it with no fringes at all. Exactly at the geometrical edge the intensity is a quarter, not a half, because it is the amplitude that halves. The first and brightest fringe is 1.37 times the unobstructed intensity — brighter than if the edge were not there — and it sits at 0.202 mm at 0.1 m, 0.451 mm at 0.5 m, 0.903 mm at 2 m, moving outward as the square root of the distance, which is the one thing in the pattern that is not scale-free.

The intensity across the shadow of a slit 0.70 mm wide in 550 nm light, at four screen distances, labelled by the Fresnel number — the number of half-period zones the aperture holds. The shaded band is the geometrical shadow of the opening. At a large Fresnel number the screen is close, the pattern fills the geometrical opening and is covered in ripples, and the edges are where the fringes are; as the screen moves away the ripples merge, the pattern spills out of the opening, and by N = 0.25 it is the single broad lobe of far-field diffraction with the aperture no longer recognisable in it. Nothing changes but the distance: the near field and the far field are one calculation at two values of one number.

Fresnel's two integrals plotted against each other, traced from v = -6.5 to 6.5. Arc length along the curve is v — the distance along the wavefront in units of √(λL/2) — and the amplitude arriving from any stretch of that front is the straight chord between the stretch's two endpoints, not the length of curve between them. The two eyes at ±(½, ½) are where the far parts of an unobstructed front pile up, so the whole open front is the chord between them, of length 1.4142. A straight edge blocks half of it and leaves a chord of 0.7071 — exactly half the amplitude, so a quarter of the intensity, at the geometrical shadow's edge. The chord drawn here runs to v = 2.345, is 1.5487 long, and is the longest chord from the eye that exists: its square over the open chord's square is 1.199, which is why the brightest fringe outside a shadow is brighter than no obstacle at all.

Intensity across the edge of a shadow cast by a straight edge in 30000 nm light, at 3 screen distances, in units of the unobstructed intensity. Geometrical optics predicts a step: full brightness on one side of zero and nothing on the other. What is there instead is a set of fringes outside the shadow, decaying outward, and a smooth fade to darkness inside it with no fringes at all. Exactly at the geometrical edge the intensity is a quarter, not a half, because it is the amplitude that halves. The first and brightest fringe is 1.37 times the unobstructed intensity — brighter than if the edge were not there — and it sits at 2.582 mm at 0.3 m, 4.713 mm at 1 m, 8.164 mm at 3 m, moving outward as the square root of the distance, which is the one thing in the pattern that is not scale-free.

physicscheck asserts something about fresnel-edge 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.