Illustrated Physics
physics, drawn
Series

Fermat — the series

8 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
  5. Every ray from one point, meeting again at another. Rays from a point half a radius from the centre of Maxwell's fish-eye, n = n₀/(1 + r²/R²), traced in 11 directions round the full circle. Every one is an arc of a circle, and every one passes through the point on the other side of the centre at distance R²/r from it — 2.00 radii — which is the image. The worst miss is 8.8e-4 radii, the size of the integration's own error. Rays leaving in opposite directions, rays leaving at right angles, rays that go the long way round: all arrive. The faint circles are contours of the index, which falls from n₀ at the centre to n₀/2 at one radius and n₀/5 at two.

    The medium that images every point

    A single refracting surface can be shaped to image one point perfectly and no other. A medium whose index falls smoothly away from a centre can do better: in Maxwell's fish-eye every ray from any point, leaving in any direction, arrives at one image point, and every such path takes exactly the same time. It is a perfect instrument for all of space at once — and the image it forms is sharp everywhere and a faithful copy nowhere.

    part 5 · optics
  6. A parallel beam brought to a point on the far side. Parallel rays sent into Luneburg's sphere, whose index falls from the square root of two at the centre to one at the rim, from the left. Each beam is brought to a single point on the rim diametrically opposite the direction it came from; the worst of the 11 rays misses by 2.0e-3 radii. The sphere has no axis — every direction is the same to it — so a beam from any direction is focused as well as any other, and there is no such thing as an off-axis aberration.

    The lens with no axis

    Every ordinary lens has an axis, and every one of its hardest aberrations is a penalty for looking away from it. A sphere has no axis, so a sphere cannot have off-axis aberrations — but a glass ball has spherical aberration instead, because one index and one curvature are too few to focus every ray. Luneburg's sphere, whose index falls from √2 at the centre to 1 at the rim, keeps the symmetry and removes the aberration: it brings parallel light from every direction to a perfect point on its own surface.

    part 6 · optics
  7. Rays and the surfaces of equal optical path. Rays from a point source 20 mm in front of a curved glass surface (radius 4 mm, index 1.5), traced exactly, and the surfaces on which the optical path from the source — distance times index — takes the same value (curves, every 2.5 mm of path). In air the surfaces are spheres round the source. In the glass the rays from the edge of the surface cross the axis at 12.8 mm, well short of the paraxial focus at 20.0 mm, and the equal-path surfaces are no longer spheres. They are still at right angles to every ray: measured on a fan of 641 rays over 10 surfaces in the glass, the largest departure from 90° is 0.005°. Further in, the surfaces fold into cusps where neighbouring rays cross, and there a surface has no single normal.

    The surface every ray is normal to

    Light from a point, bent by any number of lenses and mirrors, comes out as a tangle of rays that no longer meet anywhere. Hidden in the tangle is a surface every ray crosses at a right angle: the set of points the light reaches at the same optical time. That a surface always exists is a theorem two centuries old, and it is the reason a modern wavefront sensor works — an array of tiny lenses measures only the direction each patch of light is travelling, and the measurements can be added up into a shape only because they are the slopes of one.

    part 7 · optics
  8. The width a self-focusing beam is forced to. 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.

    The beam that focuses itself

    In glass the refractive index rises very slightly with the intensity of the light, so a beam that is brightest in the middle writes a converging lens into the glass it is crossing. Diffraction spreads it at the same time. Both effects grow in the same proportion as the beam is narrowed, so neither can win by a change of size — only by a change of power. Above a critical power, 4.3 megawatts in fused silica, the beam is forced to collapse within a finite distance, and every high-power laser is built around not letting that happen where it is not wanted.

    part 8 · optics

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