Classical Optics

Descartes had the right law from the wrong premise: his derivation required light to move faster in glass than in air. Pierre de Fermat objected in 1662, and replaced the premise with a principle. Of all paths from a point in air to a point in water, light takes the one that takes the least time. Since it travels more slowly in water, the quickest route is not the straight one — it spends more of its length in the fast medium — and working out which route minimises the total time gives exactly the sine law.

The principle is more important than the problem. It says that the path is determined by making a quantity extremal, and that way of thinking propagated: Maupertuis and Euler to least action in mechanics, Lagrange and Hamilton to the whole structure of classical dynamics, Feynman to the sum over histories. The calculus of variations was built to handle problems of this shape, and the first of them was a ray of light entering water.

Isaac Newton let a thin beam of sunlight through a prism and got a band of colours. That much was known. His decisive addition was to take one colour out of the band and send it through a second prism: it emerged the same colour, bent by the same amount, unchanged. So the prism does not manufacture colour; it sorts a mixture that was in the white light all along. The experimentum crucis was published in 1672, attacked immediately by Hooke, and defended by Newton for the rest of his life.

Christiaan Huygens built the other theory. In the Traité de la Lumière of 1690, light is a disturbance spreading through a medium, and every point on a wavefront is treated as a source of a new spherical wavelet whose envelope gives the front an instant later. From this he derived reflection, derived refraction with the index as a ratio of speeds, and — the triumph — explained the double image formed by a calcite crystal, which no account made of particles could approach. His theory required light to go slower in glass, which was correct and, at the time, unmeasurable.

Two theories, both quantitative, both covering nearly all the evidence, separated by a measurement nobody could make for 160 years. Newton's prestige settled the matter socially rather than physically, and the particle view prevailed in Britain until Thomas Young put two slits in front of a light source.

Io circles Jupiter every 42.5 hours and is eclipsed by it each orbit, so the Jupiter system is a clock visible from Earth. Ole Rømer compiled eclipse times at the Paris Observatory through the 1670s and found they did not keep step with a uniform period. When the Earth was approaching Jupiter the eclipses came early; when receding, late. The discrepancy accumulated to something like twenty-two minutes between the two extremes.

His interpretation: the eclipses happen when they happen, and the news of them travels at a finite speed. When the Earth is on the far side of its orbit the news has an extra distance to cross, equal to the diameter of the Earth's orbit. So

$$c = \frac{\text{diameter of Earth's orbit}}{\text{accumulated delay}}.$$

Take the modern value for the diameter, $2 \text{ AU} = 2.99 \times 10^{11}$ m, and Rømer's twenty-two minutes, 1,320 seconds:

$$c = \frac{2.99 \times 10^{11}}{1320} = 2.3 \times 10^{8} \text{ m/s}.$$

That is 25% low, and the error is entirely in the delay. With the true value of $c$, the crossing takes

$$\frac{2.99 \times 10^{11}}{2.998 \times 10^{8}} = 997 \text{ s} = 16.6 \text{ minutes},$$

so Rømer's figure was about a third too large. Timing an eclipse of a moon by eye, through a seventeenth-century refractor, with Jupiter's shadow edge being soft, is good to a minute or two at best, and the orbital period itself had to be fitted from the same contaminated data. Note also what he did not need: the size of the Earth's orbit. Rømer published a time, eleven minutes one way, and a prediction that an eclipse in November 1676 would run late by that much. The prediction held. Huygens supplied the astronomical unit and turned the time into a speed.

Two things are worth taking from this. First, a 25% measurement that settles a qualitative question — light is not instantaneous — is worth more than a precise measurement of something nobody disputes. Second, the method is differential: Rømer never measured a one-way travel time, only how the delay changed as the geometry changed, which is why a crude clock sufficed. The terrestrial measurements of Fizeau and Foucault in the 1850s used the same trick with a spinning wheel instead of a planet, and they finally showed that light slows in water, as Fermat and Huygens had required and Newton's followers had denied.

Nothing in this chapter was overturned. Rays, refraction and least time survive as the short-wavelength limit of the wave theory, and they are still how a lens is designed: a modern optical design program traces rays through surfaces using Ibn Sahl's relation, several billion times, to balance aberrations across a field of view. A theory that is superseded in its account of what light is can remain exactly right about what light does, within its range.

What the period never addressed is emission and absorption. Both Newton's corpuscles and Huygens's wavefronts describe light in transit and say nothing about how a hot body comes to glow, why it glows in the particular distribution of colours it does, or what happens when light meets an atom. Those questions were not asked seriously until the spectroscopists found that each element emits its own sharp lines, and the answers broke classical physics altogether — the subject of old quantum theory.

One question from this era is still not closed, and it concerns momentum rather than energy. Light carries momentum, and inside a transparent medium there are two defensible expressions for how much, differing by the square of the refractive index — Minkowski's and Abraham's, proposed in 1908 and 1909. Experiments have been read as supporting each. The modern view is that the two answer different questions, one about the momentum carried by the field and one about the momentum transferred to matter, and that the dispute was about dividing a total that cannot be divided uniquely. It matters in practice wherever light is used to push something, which is to say in optical trapping and in every measurement where a mirror recoils.

The immediate successor to this chapter, though, is the question Newton and Huygens could not settle between them. It was settled by two slits and a card, and the answer was that light is a wave — which is where wave optics begins, and which turned out not to be the final word either.