Illustrated Physics
physics, drawn
Series

Fermat — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · optics
  2. Rays that turn round without meeting anything. Rays leaving an eye 1.5 m above a road, at 0.18°, 0.28°, 0.36°, 0.42°, 0.52° below the horizontal, in air whose refractive index is reduced by 3.0e-5 at the hot surface and recovers over 5 cm. Each is traced by integrating the ray equation, with the conserved quantity n cos θ fixed by where and how the ray set out. The shallow ones come back up without touching anything — the lowest gets to 0.6 cm above the surface and turns — and a ray that returns to eye level from below is seen as sky lying on the road. The steep ones run out of gradient first and hit it. The dividing angle is 0.444°, which is what the whole effect is made of: an unremarkable temperature difference and an angle a fiftieth the width of the Moon. Heights are exaggerated 128× against distances; at true scale every ray here would be indistinguishable from the axis. The conserved n cos θ holds to 4.3e-14 over every trace, which is what says the turns are the physics and not the integrator.

    The ray that bends without a surface

    Snell's law is about a boundary, and light bends in air where there is no boundary anywhere. Let the index vary continuously and the law of angles becomes a differential equation — one that carries a conserved quantity, forbids the ray from reaching certain heights, and turns a hot road into a mirror a hundred metres long.

    part 2 · optics
  3. Stationary, and not always shortest. Three mirrors through the same point, with the same source and the same detector, and the length of the reflected path against where on the mirror the ray strikes. The actual ray is the one at the centre in all three, because the tangent there is horizontal and the two angles are equal whatever the curvature. Its path is a minimum at 0.6× the ellipse's curvature, every path equal at 1× the ellipse's curvature, a maximum at 1.6× the ellipse's curvature. On the most curved mirror every path the light did not take is shorter than the one it did, so the principle cannot be stated as least time and survive; it has to be stated as stationary time, and the difference is not a technicality.

    The path that takes the longest time

    Light is said to take the quickest route. Put a source and a detector in front of a mirror curved a little more than the ellipse through them, and the route it takes is slower than every route beside it — by construction, not by exception. The principle was never about least; it was about stationary, and the difference is where optics stops being geometry.

    part 3 · optics
  4. The surface that focuses 1 into 1.52 with no error at all. A Cartesian oval: the locus of points for which 1 times the distance from the object plus 1.52 times the distance to the image is a constant. Every ray drawn takes exactly the same optical path, so every one arrives at the image point — not nearly, and not for small angles, but exactly, for rays at any angle the surface reaches. There is no spherical aberration because there is no approximation: this is what Fermat's principle asks for, solved rather than expanded. The surface is not a sphere, not a conic in general, and not anything a grinding machine makes easily, which is most of why lenses are spherical and aberrated instead.

    The surface that images one point exactly

    Fermat's principle says every ray from an object to its image must take the same time. Written as an equation, that is a curve Descartes found — a surface with no aberration at all, at any angle. It exists, it is easy to compute, and it images exactly one pair of points and nothing else.

    part 4 · optics

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