Illustrated Physics
physics, drawn
Concept

Rainbow angle — where it appears

The deviation at which rays leaving a spherical drop bunch together, being a stationary point of deviation against where the ray struck. Its value is about 42 degrees for one internal reflection, and because it varies slightly with wavelength the bunching happens at a different angle for each colour.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Rays through a raindrop. Parallel rays entering a spherical drop at different heights, refracting in, reflecting once from the back, and refracting out. The outgoing rays crowd together near one particular direction, and that crowding is the bow.

The angle the rainbow has to be, and why nobody chose it

A rainbow is at forty-two degrees because a function has a minimum there. Nothing about water, light or weather picks the number — it falls out of running Snell's law three times through a sphere.

optics · Dispersion 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.

The path that does not change

Reflection, refraction and the angle of the rainbow are not three laws. They are one condition — that the optical path length is stationary — and the word stationary rather than shortest is the whole of what makes an elliptical mirror and a rainbow the same statement.

optics · Fermat A caustic, by rays and by waves. The brightness across a fold caustic, computed two ways. Geometric optics gives the rising curve: on the illuminated side two rays arrive at every point and the intensity goes as the inverse square root of the distance from the caustic, so it becomes infinite exactly at it; on the other side no ray arrives at all and the intensity is zero. The wave answer is the squared Airy function, and it disagrees in three ways that are all observable. It is finite, peaking at 1.0188 in the scaled variable rather than at the caustic itself, so the brightest line is displaced onto the bright side. It oscillates, with maxima at -1.02, -3.25, -4.82, -6.16 — those are the supernumerary fringes, and they are not interference between two separate objects but between the two rays the caustic joins. And it leaks: on the dark side, where geometry forbids any light, the Airy function decays exponentially rather than stopping, which is the same mathematics as tunnelling and is why the edge of a shadow is soft before diffraction from any aperture is considered. The two curves agree far from the caustic, which is where the ray picture is a good approximation and where they have been matched here.

The fringes below the rainbow

Geometric optics puts the whole rainbow at one angle and predicts an infinite brightness there. What is seen instead is a peak displaced inside that angle, followed by a train of pink and green arcs — and their spacing is a measurement of the raindrops, because a caustic's structure is set by the wavelength to the two-thirds power over the drop radius to the two-thirds.

optics · Diffraction

Named alongside it

The objects these essays reach for when they reach for this one.

Refractive indexSnell's lawStationary pointAiry functionAntisolar pointCausticDiffractionDispersionFermat's principleGeometric opticsOptical path lengthReflection

All concepts